Silicon anodization as a structuring technique : literature review, modeling and experiments 978-3-658-19238-9, 3658192380, 978-3-658-19237-2

Table of contents :
Front Matter ....Pages I-XXIX
Introduction (Alexey Ivanov)....Pages 1-5
State of the art (Alexey Ivanov)....Pages 7-91
Experimental, characterization and simulation methods (Alexey Ivanov)....Pages 93-123
Microscale study of anodization process (Alexey Ivanov)....Pages 125-143
Anodization process as a structuring technique (Alexey Ivanov)....Pages 145-225
General conclusions (Alexey Ivanov)....Pages 227-229
Back Matter ....Pages 231-316

Citation preview

Alexey Ivanov

Silicon Anodization as a Structuring Technique Literature Review, Modeling and Experiments

Silicon Anodization as a Structuring Technique

Alexey Ivanov

Silicon Anodization as a Structuring Technique Literature Review, Modeling and Experiments

Alexey Ivanov Freiburg im Breisgau, Germany Dissertation Albert-Ludwigs-Universität Freiburg, 2016

ISBN 978-3-658-19237-2 ISBN 978-3-658-19238-9  (eBook) https://doi.org/10.1007/978-3-658-19238-9 Library of Congress Control Number: 2017949671 Springer Vieweg © Springer Fachmedien Wiesbaden GmbH 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer Vieweg imprint is published by Springer Nature The registered company is Springer Fachmedien Wiesbaden GmbH The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Dedicated to my family.

Contents

Abstract

XI

Zusammenfassung Nomenclature

XV XIX

1 Introduction 1.1 Overview: silicon anodization . . . . . . . . . . . . . . . . . . 1.2 Objectives and outline of the work . . . . . . . . . . . . . . .

1 1 3

2 State of the art 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Structuring techniques for semiconductor based smart systems 2.2.1 Principles and characteristics of structuring processes 2.2.2 Fabrication strategies for smart systems . . . . . . . . 2.2.3 Structuring techniques . . . . . . . . . . . . . . . . . . 2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mass and charge transfer in silicon and electrolyte . . . . . . 2.3.1 Band diagram and transfer of charge and mass in bulk p-type silicon . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Band diagram and transfer of charge and mass in bulk electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Interface silicon-electrolyte . . . . . . . . . . . . . . . 2.4 Silicon anodization: current-voltage characteristics and dissolution mechanisms . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Cathodic polarization and open circuit . . . . . . . . . 2.4.2 Pore formation . . . . . . . . . . . . . . . . . . . . . . 2.4.3 First and second plateaus of electropolishing . . . . . 2.4.4 Position of the first and the second current peaks . . . 2.5 Silicon anodization: influence of process parameters . . . . .

7 7 7 9 10 12 18 19 19 21 25 35 35 37 44 46 48

VIII

2.6

2.7

2.8

Contents 2.5.1 Current density . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Etch time . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Electrolyte mixture and concentration . . . . . . . . . 2.5.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Substrate doping density . . . . . . . . . . . . . . . . 2.5.6 Crystal orientation . . . . . . . . . . . . . . . . . . . . 2.5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . Application of silicon anodization for structuring . . . . . . . 2.6.1 Fabrication aspects . . . . . . . . . . . . . . . . . . . . 2.6.2 Process flow for structuring with anodization process . 2.6.3 Shape control in anodization process . . . . . . . . . . 2.6.4 Structuring by pore formation (sacrificial porous silicon) 2.6.5 Structuring by electropolishing . . . . . . . . . . . . . Macroscale models of silicon anodization for structuring . . . 2.7.1 Charge flow models - primary and secondary current distribution . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Mass transport and tertiary current distribution models Conclusions on state of the art . . . . . . . . . . . . . . . . .

50 54 58 60 64 67 69 69 69 73 74 80 81 83 83 87 89

3 Experimental, characterization and simulation methods 93 3.1 Fabrication methods . . . . . . . . . . . . . . . . . . . . . . . 93 3.1.1 Electrolytes and other chemicals . . . . . . . . . . . . 93 3.1.2 Sample preparation . . . . . . . . . . . . . . . . . . . 94 3.1.3 Sample anodization and post-processing . . . . . . . . 105 3.2 Characterization of samples . . . . . . . . . . . . . . . . . . . 109 3.2.1 Mass measurement . . . . . . . . . . . . . . . . . . . . 109 3.2.2 Microscopic images . . . . . . . . . . . . . . . . . . . . 110 3.2.3 Surface quality characterization . . . . . . . . . . . . . 111 3.2.4 Etch shape measurements . . . . . . . . . . . . . . . . 111 3.3 Macroscale simulations of the anodization process . . . . . . . 114 3.3.1 Model geometry . . . . . . . . . . . . . . . . . . . . . 114 3.3.2 Mesh configuration . . . . . . . . . . . . . . . . . . . . 116 3.3.3 Time-dependent solver configuration and stability issues120 3.3.4 Further general model parameters . . . . . . . . . . . 120 3.3.5 Specific parameters for secondary current distribution models . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.3.6 Hardware configuration . . . . . . . . . . . . . . . . . 122 4 Microscale study of anodization process 125 4.1 Study of pore formation regime with full wafers . . . . . . . . 125

Contents

IX

4.1.1 4.1.2 4.2

4.3

Current density sweep . . . . . . . . . . . . . . . . . . 126 Substrate resistivity sweep in the range 0.001–20 Ohm cm for p-type silicon . . . . . . . . . . . . . . . . . . . . . 129 Extended study of pore formation and electropolishing regimes with small samples . . . . . . . . . . . . . . . . . . . 132 4.2.1 Extended current density sweep in 29.93 m% HF with ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.2.2 Variation of HF concentration . . . . . . . . . . . . . . 139 4.2.3 Influence of porous silicon removal process on surface quality . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5 Anodization process as a structuring technique 145 5.1 Process localization with frontside masking . . . . . . . . . . 145 5.1.1 Preliminary study . . . . . . . . . . . . . . . . . . . . 145 5.1.2 Influence of opening size and current density on time development of etch forms for low-doped p-type silicon 148 5.1.3 Influence of opening size and current density on etch forms for highly-doped p-type silicon . . . . . . . . . . 163 5.1.4 Proximity effects for neighboring openings for lowdoped p-type silicon . . . . . . . . . . . . . . . . . . . 170 5.1.5 Study of etch form development for pre-structured cavities . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 Process localization with backside local contacts . . . . . . . 179 5.2.1 Etch form development for single circular backside contact . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2.2 Influence of spacing between two backside rectangular openings on etch form . . . . . . . . . . . . . . . . . . 187 5.3 Combination of frontside masking and backside local contact 191 5.4 Macroscale simulation of anodization process . . . . . . . . . 194 5.4.1 Primary current distribution model for low-doped ptype silicon . . . . . . . . . . . . . . . . . . . . . . . . 195 5.4.2 Primary current distribution model for highly-doped p-type silicon . . . . . . . . . . . . . . . . . . . . . . . 207 5.4.3 Secondary current distribution model for low-doped p-type silicon . . . . . . . . . . . . . . . . . . . . . . . 216 5.4.4 Secondary current distribution model for highlydoped p-type silicon . . . . . . . . . . . . . . . . . . . 219 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

X

Contents

6 General conclusions

227

Acknowledgments

231

List of relevant publications

233

A Appendix 239 A.1 Conversion of concentration units for aqueous HF electrolytes 239 A.1.1 Concentration units in chemistry . . . . . . . . . . . . 239 A.1.2 Reference data . . . . . . . . . . . . . . . . . . . . . . 241 A.1.3 HF:water solutions . . . . . . . . . . . . . . . . . . . . 242 A.1.4 HF:water:ethanol solutions . . . . . . . . . . . . . . . 250 A.2 Conversions of other anodization parameters . . . . . . . . . 256 A.2.1 Dissolution valence calculated from porosity and thickness of porous layer . . . . . . . . . . . . . . . . . 256 A.2.2 Porosity and density of porous silicon . . . . . . . . . 258 A.3 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . 259 A.3.1 Experimental data for sec. 5.1.2 . . . . . . . . . . . . . 259 A.3.2 Comparison of the final current densities for sec. 5.1.3 264 A.4 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A.4.1 Primary current distribution model for low-doped ptype silicon in sec. 5.4.1 . . . . . . . . . . . . . . . . . 265 A.4.2 Primary current distribution model for highly-doped p-type silicon in sec. 5.4.2 . . . . . . . . . . . . . . . . 270 A.4.3 Secondary current distribution model for low-doped p-type silicon in sec. 5.4.3 . . . . . . . . . . . . . . . . 273 A.4.4 Secondary current distribution model for highlydoped p-type silicon in sec. 5.4.4 . . . . . . . . . . . . 276 Bibliography

279

Abstract This work investigates application of silicon anodization process as a threedimensional structuring technique, where silicon is transformed into porous silicon as a sacrificial layer or directly dissolved in electropolishing regime. In the presented work, first, a detailed state of the art with overview of micro-structuring techniques and theory on silicon-electrolyte interface and the silicon anodization process is given. A systematic analysis of the data from the available references showed influence of various process parameters on the properties of anodization process and revealed some inconsistencies in the data which had to be verified in the work. Systematic study of parameters of the anodization process for p-type silicon in 29.93 m% HF electrolyte with ethanol was performed. Dependence of porosity and growth rate of porous silicon, dissolution valence, interface surface roughness, and mean fractal dimension on applied current density and substrate resistivity during the process was obtained. The results of the study provided a solid base for the experiments and models. Improvement of surface quality by electropolishing of silicon samples in electrolytes with different HF concentrations was studied. It was shown that average roughness of about 1 nm can be obtained by electropolishing in aqueous 7 m% HF electrolyte at current densities in the range 100–300 mA/cm2 . Influence of etch solutions used for removal of porous silicon as a sacrificial layer was investigated. It was shown that with a treatment of blank not anodized silicon substrates in 1 m% KOH or 1:5 mixture by volume of resist developer AZ351B in water for less than 30 min, average surface roughness of the samples remains below 10 nm. The main focus of the work was the study of macroscale behavior of the anodization process, that is the etch form development under application of frontside insulating masks and/or local backside contacts. Here, the basic test structure was p-type silicon sample with a circular opening in the frontside insulating masking. For the first time, etch form development for this kind of structures for low-doped (10–16 Ω cm) and highly-doped

XII

Abstract

(0.01–0.1 Ω cm) p-type silicon samples was investigated and compared. For both doping levels, it was shown experimentally that during the process etch forms undergo a transformation from convex shape to concave. It was found that the critical depth at which the shapes transform from convex to concave (named threshold depth) did not depend on the resistivity of samples (the low- and highly-doped p-type silicon), but depended nearly linearly on the diameter of the frontside opening. Additionally, it was found that the anisotropy factor of the process for these structures remained at large values in the range 0.45–0.75 when the process was running at high current densities in electropolishing regime, and decreased strongly to the values ±0.2 for anodization performed at lower current density, i.e., when the process was running in a mixed regime of pore formation and electropolishing. Significant part of the work was devoted to development and verification of novel time-dependent macroscale models describing etch front movement in both porous silicon formation and electropolishing regimes for the case of silicon anodization through an opening in a frontside mask. The models described primary and secondary current distributions. Comparing to the experiments in this work, the models showed similar etch form development. However, only partial matching of the results in regard to threshold depth and anisotropy was observed, and ways to improve the models were proposed. Another important finding of the work is that the significantly larger positive values of anisotropy factor in the experiments in comparison to the developed electrical models revealed some anisotropic mechanism of anodization process in the chosen conditions, which could not be explained with the charge distribution in the anodization cell. Additional studies on cross-influence of neighboring openings in a frontside masking, anodization of pre-structured cavities, and etch form development for local backside contacts (with and without frontside masking) were performed. In the study of the process localization with local backside contacts, the “pixel concept” was proposed. In this “pixel concept”, a basic etch form for an elementary local backside contact (a “pixel”) was used to obtain a good estimation of an etch form resulting from anodization of a sample without frontside masking and with multiple equal local backside contacts. The results of the work provide an important contribution to the scientific community and industry for future applications of the anodization process as a structuring technique. The results on etch form development in the anodization process, the developed macroscale models of silicon anodization, and the proposed method of “pixel concept” will help to reduce the time needed to optimize the process for fabrication of specific structures in

Abstract

XIII

industrial scale. The types of etch shapes studied in the work are especially interesting for applications in fluidic, optical, and lab-on-chip micro-devices.

Zusammenfassung Diese Arbeit befasst sich mit Siliziumanodisierungsprozessen als elektrochemische dreidimensionale Strukturierungsverfahren, wobei Silizium in poröses Silizium als Opferschicht umgewandelt wird oder durch direkten Abtrag (Elektropolieren) entfernt wird. In der vorliegenden Arbeit wurde zuerst ein ausführlicher Stand der Technik zu Mikrostrukturierungsverfahren, zu theoretischen Grundlagen der Silizium-Elektrolyt-Grenzfläche und zum Anodisierungsprozess erarbeitet. Eine systematische Analyse der Daten aus der Literatur wurde durchgeführt und zeigte den Einfluss verschiedener Prozessparameter auf die Eigenschaften des Anodisierungsprozesses. Zudem wurde Bedarf für eine unabhängige Prüfung der Daten erkannt. Zunächst wurden systematische Untersuchungen der Prozessparameter für p-Typ-Silizium in 29,93 m% Flusssäurelösung mit Ethanol durchgeführt. Dabei wurde der Einfluss der eingesetzten Stromdichte und des Substratwiderstandes auf die Porosität und die Wachstumsrate des porösen Siliziums, die Reaktionsvalenz und die Güte der Grenzfläche zwischen dem porösen Silizium und dem Substrat ermittelt. Die Ergebnisse der Studie bildeten die Grundlage für die weiteren Experimente und die Entwicklung der Modelle. Die Untersuchung zur Verbesserung der Oberflächengüte von Siliziumproben in Elektrolyten mit unterschiedlichen HF-Konzentrationen zeigte, dass ein Mittenrauwert von ungefähr 1 nm durch das Elektropolieren in wässriger 7 m% HF Elektrolytlösung bei Stromdichten im Bereich 100–300 mA/cm2 erreicht werden kann. Für die Entfernung des porösen Siliziums als Opferschicht wurde der Einfluss der Ätzlösung auf die Oberflächengüte von blanken, nicht anodisierten Siliziumsubstraten untersucht. Es konnte gezeigt werden, dass bei der Behandlung mit 1 m% KOH oder 1:5-Gemisch von Fotolackentwickler AZ351B im Wasser für weniger als 30 min die Oberflächenrauheit unter 10 nm bleibt. Der Schwerpunkt der Arbeit lag auf der Untersuchung des makroskopischen Verhaltens des Anodisierungsprozesses, d.h. der Untersuchung der Ätzfor-

XVI

Zusammenfassung

mentwicklung unter Anwendung der unterschiedlich strukturierten nichtleitenden Vorderseitenmaskierungen und/oder der lokalen Rückseitenkontakten. Als bevorzugte Teststrukturen wurden p-Typ-Silizium-Proben mit einer kreisförmigen Öffnung in der nichtleitenden Vorderseitenmaskierung genommen. Im Rahmen dieser Arbeit wurde zum ersten Mal für solche Proben aus niedrigdotiertem (10–16 Ω cm) und hochdotiertem (0.01–0.1 Ω cm) p-Typ-Silizium der zeitabhängige Verlauf der Veränderung der Ätzformen untersucht. Es wurde für beide Dotierungsbereiche experimentell gezeigt, dass während des Prozesses die Ätzformen einen Übergang von konvexer zu konkaver Form durchlaufen. Es wurde festgestellt, dass die kritische Ätztiefe, bei der die Formen von konvexer zu konkaver Form transformieren (hier als Schwellentiefe benannt) vom spezifischen Widerstand der Proben (für die niedrig- und hochdotierten p-Typ-Siliziumproben) kaum abhängig ist, aber eine nahezu lineare Abhängigkeit vom Durchmesser der Vorderseitenöffnung aufweist. Darüber hinaus wurde festgestellt, dass der Anisotropiefaktor des Prozesses bei Elektropolieren hoch im Bereich 0.45–0.75 war. Mit Abnahme der Stromdichte (d.h. der Prozess wandelt sich von reinem Elektropolieren zu einem Mischmodus aus Porenbildung und Elektropolieren) verringerte sich der Anisotropiefaktor stark bis auf ±0.2. Des Weiteren wurde die Untersuchung des makroskopischen Verhaltens mit der Entwicklung von neuartigen zeitabhängigen makroskopischen Modellen des Anodisierungsprozesses ergänzt. Die Modelle beschreiben mit primären und sekundären Stromverteilungen die Bewegung der Ätzfront sowohl bei Porenbildung als auch bei Elektropolieren für den Fall der Anodisierung durch eine Öffnung in einer nichtleitenden Vorderseitenmaskierung. Der Vergleich mit den in der Arbeit durchgeführten Experimenten zeigte, dass die Modelle eine ähnliche Entwicklung der Ätzformen aufweisen. Die Ergebnisse stimmten in Bezug auf Schwellentiefe und Anisotropiefaktor jedoch nur zum Teil überein; Möglichkeiten und Vorschläge für die Optimierung der Modelle wurden diskutiert. Die deutlich größeren Werte des Anisotropiefaktors im Experiment im Vergleich zu den entwickelten elektrischen Modellen weisen auf einen anisotropen Mechanismus des Anodisierungsprozesses hin, das nicht alleine durch Stromverteilung erklärt werden konnte, was als weitere wichtige Erkenntnis der Arbeit festgehalten werden kann. Bei der Untersuchung für die Proben mit lokalen Rückseitenkontakten wurde ein sogenanntes ”Pixel-Konzept” vorgeschlagen, das eine gute Schätzung der Ätzformen, die durch Anodisierung der Proben ohne Vorder-

Zusammenfassung

XVII

seitenmaskierung mit mehreren Rückseitenkontakten erzeugt werden, ermöglicht. Die Ergebnisse der Arbeit leisten einen wertvollen wissenschaftlichen Beitrag und erweitern die Einsatzmöglichkeiten des Anodisierungsprozesses als dreidimensionale Strukturierungstechnik für zukünftige industrielle Entwicklungen. Die Ergebnisse zur Ätzformentwicklung, die entwickelten Makromodelle und die vorgeschlagene Methode des ”Pixel-Konzepts” tragen dazu bei, dass die Zeit, die für die Optimierung des Verfahrens zur Herstellung von spezifischen Strukturen im industriellen Maßstab benötigt wird, deutlich reduziert werden kann. Die in der Arbeit untersuchten Ätzformen sind besonders interessant für Anwendungen für fluidische, optische und Lab-on-Chip Mikrosysteme.

Nomenclature Abbreviations AFEM

ammonium fluoride etch mixture

AFM

atomic force microscope

APSM

Bosch Advanced Porous Silicon Membrane technology

BET

Brunauer-Emmett-Teller theory

BHF

buffered hydrofluoric acid

BJH

Barrett-Joyner-Halenda theory

CB

current burst

CBM

current burst model

CMOS

complementary metal-oxide-semiconductor

DMSO

dimethylsulfoxide

DRIE

deep reactive ion etching

EBM

electron beam machining

ECAM

electrochemical arc machining

ECDM

electrochemical discharge (micro)machining

ECM

electrochemical (micro)machining

EDL

electric double layer

EDM

electro discharge (micro)machining

FEM

finite element method

XX

Nomenclature

FIB

focused ion beam

HARMS

high aspect ratio microsystem

HF

hydrofluoric acid

HI-PS

method of porous silicon formation after hydrogen ion implantation

HNA

mixture of hydrofluoric acid, nitric acid and acetic acid in water

IHP

inner Helmholtz plane

IUPAC

International Union of Pure and Applied Chemistry

LBM

laser beam machining

LIGA

(German) Lithographie, Galvanik und Abformung, stands for deep-etch x-ray lithography, electroplating and molding – fabrication technology used to create high-aspect-ratio microstructures

LPCVD

low-pressure chemical vapor deposition

LSM

laser-scanning microscope

MEMS

micro electro mechanical system

µTAS

micro total analysis system

MIS

metal-insulator-semiconductor structure

𝑛D

𝑛-dimensional, where 𝑛 is the number of dimensions, e.g. 1, 2, 2.5, 3, etc.

NEMS

nano electro mechanical system

OHP

outer Helmholtz plane

PMMA

poly methyl methacrylate

PS

porous silicon

PTFE

polytetrafluoroethylene

XXI

Nomenclature PVDF

polyvinylidene fluoride

RAP

reactive atom plasma machining

RDS

rate-determining step

RF

radio frequency

RIE

reactive ion etching

RMS

root mean square

RT

room temperature

SACE

spark assisted chemical engraving

SCE

saturated calomel electrode

SCR

space charge region

SEM

scanning electron microscope

SI

international system of units, (French) Le Système International d’Unités

SLIM

spectroscopic liquid infiltration method

SOI

silicon-on-insulator

TMAH

tetramethyl ammonium hydroxide

ULSI

ultra large scale integration

USM

ultrasound (micro)machining

UV

ultra-violet

VLSI

very large scale integration

WECDM

wire electro chemical discharge (micro)machining

WECM

wire electro chemical (micro)machining

WEDG

wire electro discharge grinding

XXII

Nomenclature

Symbols 𝐴

electrode area, or anodization area

𝑎

value of polarization at the unit current (in the Tafel equation)

𝐴f

anisotropy factor

𝐴layout

total open area on a sample with insulating frontside masking of a given layout

𝛼

charge transfer coefficient (with value between zero and unity)

𝐴open

area of opening in frontside mask

𝐴sample

open area defined by sample/wafer holder

𝐴total final

total final anodization area for all structures on a sample

𝑎X

activity coefficient of species X

𝑏

Tafel slope (in the Tafel equation)

𝑏

molality (in this sense the symbol 𝑏 is used only in Appendix)

𝐶𝑖

compensation coefficient used for calculation of superpositioned etch profiles in description of pixel concept

0 𝑐mol

formal (total) amount concentration of a species

eq 𝑐mol

equilibrium amount concentration of a species

𝑐mol X

amount concentration (molar concentration) of a species X

𝑐Ox S

surface concentration of the oxidized species Ox

𝑐Ox V

volume (bulk) concentration of the oxidized species Ox

𝑐Red S

surface concentration of the reduced species Red

𝑐Red V

volume (bulk) concentration of the reduced species Red

𝑐vol 𝑖

volume concentration of a constituent 𝑖 in a mixture

XXIII

Nomenclature 𝐷

diffusion constant of electrons (n), holes (p), or another species specified in subscript

𝑑

thickness of porous silicon layer

𝛿

thickness of the Helmholtz layer (in the electric double layer)

Δ𝜙H

potential drop in the Helmholtz layer

Δ𝜙SCR

potential drop in the space charge region in silicon

𝑑etch

etch depth

𝐷mean

mean fractal dimension

𝐷open

diameter of opening in frontside masking layer

𝑑𝑟

displacement in polar direction of cylindrical coordinate system

𝑑th

threshold depth - structure depth at which structure shape at the bottom switches from convex to concave

𝑑𝑧

displacement in longitudinal direction of cylindrical coordinate system

𝐸

electric vector field

𝑒

elementary charge, 𝑒 ≈ 1.602 176 487×10−19 C

𝐸A

acceptor energy level in semiconductor band diagram

𝐸a

activation energy

𝐸C

bottom level of the conduction band

𝐸F

Fermy level

𝐸G

band gap

𝐸Ox

average energy level of oxidized species

𝐸Red

average energy level of reduced species

𝐸redox

redox potential

XXIV

Nomenclature

𝜂

overpotential

𝐸V

top level of the valence band

𝐹

Faraday constant, 𝐹 = 𝑒𝑁A

𝜙

electric potential

𝜙𝑖

volume fraction of a constituent 𝑖 in a mixture

𝜙𝑙

bulk potential of the liquid (in solid-liquid interface)

𝜙𝑙0

initial electrolyte potential in secondary current distribution models

𝜙𝑙 eq

equilibrium potential of the liquid (in solid-liquid interface)

𝜙s

bulk potential of the solid (in solid-liquid interface)

𝜙s0

initial electrode (silicon) potential in secondary current distribution models

𝜙s eq

equilibrium potential of the solid (in solid-liquid interface)

𝛾

reduction coefficient used in the models for the part of the etch front boundary near the symmetry axis (𝛾1 ), and for the part of the etch front boundary near the mask (𝛾2 )

𝛾X

activity coefficient of species X

𝐼

current

𝐼0

exchange current

𝐼0

total current flowing through an elementary etch form, used in description of pixel concept

𝐼a

anodic current

𝐼act

activation polarization current

𝐼c

cathodic current

𝐼leak

leakage current of an anodization setup

XXV

Nomenclature 𝐼lim

limiting current (in diffusional concentration polarization)

𝐼v

light intensity

𝐼work

working current of an anodization setup (current going through silicon sample and driving the anodization process)

𝑖X

flux of species X

𝑗

current density vector field

𝑗

current density

𝑗0

exchange current density

𝑗a

average current density as arithmetic mean value between the initial current density 𝑗init and the final current density 𝑗area

𝑗area

current density calculated for the anodization area which is increasing during the etching of a structure through an opening in frontside masking layer

𝑗d

diffusion current density

𝑗k

kinetic current density

𝑗ox

value of current density at second current peak in polarization curve of silicon in fluoride-based electrolyte

𝑗PSL

value of current density at first current peak in polarization curve of silicon in fluoride-based electrolyte

𝑗vol

final current density calculated based on the volume of etched structure

𝑘°

standard heterogeneous rate constant

𝜅

curvature

𝑘b

Boltzmann constant, 𝑘b ≈ 1.380 6504×10−23 J/K

𝐾eq

equilibrium constant of a reaction

𝐾HF

equilibrium constant for the reaction of dissociation of HF

XXVI

Nomenclature

𝐾(HF)2

equilibrium constant for the reaction of dissociation of (HF)2

𝐾HF2 −

equilibrium constant for the reaction of dissociation of HF2 −

𝑙

as subscript means liquid (electrolyte)

𝑚1

mass of a silicon sample before anodization, also 𝑚bulk

𝑚2

mass of a silicon sample after anodization, also 𝑚PS (assuming the sample is transformed to porous silicon completely)

𝑚3

mass of an anodized silicon sample after removal of porous silicon

𝑚𝑖

mass of a constituent 𝑖 in a mixture

𝑚solvent

mass of a solvent

𝑚total

mass of a mixture

𝜇

mobility of electrons (n), holes (p), or another species specified in subscript

𝑀X

molar mass of a species X

𝑛

concentration of electrons

𝑁A

Avogadro constant, 𝑁A ≈ 6.022 141 79×1023 mol−1

𝑛e

reaction valence

𝑛F

number of fluorine atoms consumed to dissolve one silicon atom

𝑁𝑖

amount of a constituent 𝑖 in a mixture

n

as subscript means electrons

n+

high n-type doping

𝑁solute

amount of a solute

𝜈

stoichiometric coefficient for the etch rate equation in simulation

XXVII

Nomenclature 𝑝

concentration of positive charges (holes)

p

as subscript means positive charges (holes)

p

low p-type doping

𝑃%

porosity

p+

high p-type doping

𝑞

charge

𝑅

radius of curvature

𝑅

universal gas constant, 𝑅 = 𝑁A 𝑘b

𝑟

polar axis coordinate in cylindrical coordinate system (in the simulation)

𝑅a

average roughness

𝑅etch

etch rate

𝑅etch L

lateral etch rate

𝑅etch V

vertical etch rate

𝑅F

consumption rate of fluorine at reaction site

𝜌el

electrical resistivity

𝜌X

density of a substance or quantity X

𝑅leak

leakage resistance of an anodization setup

𝑅model

radius of anodization cell in simulation

𝑅open

radius of opening in frontside masking

𝑅PS

porous silicon growth rate

𝑅q

root-mean-square roughness

𝑅Si

silicon dissolution rate

XXVIII

Nomenclature

𝑅work

working resistance of an anodization setup (resistance of silicon sample including interface resistances of siliconelectrolyte contacts)

s

as subscript means solid (electrode)

𝑆

selectivity of an etch process

𝑆𝐺

specific gravity

𝑠𝑖

spacing between two backside contacts for a sample 𝑖 used in description of pixel concept

𝑆𝑖 (𝑥)

measured etch profile (depth as a function of 𝑥-coordinate) for a sample 𝑖 used in description of pixel concept

𝜎

conductivity

𝑆𝑖∗ (𝑥)

etch profile (depth as a function of 𝑥-coordinate) for a sample 𝑖, obtained by superposition of elementary profiles used in description of pixel concept

𝑇

absolute temperature in Kelvin

𝑡etch

etch duration, etch time, anodization time, also as simply 𝑡

𝑈

potential drop

𝑈eq

equilibrium potential

0 𝑈eq

standard electrode potential

𝑈OC

open-circuit potential

𝑈PSL

potential at first current peak in polarization curve of silicon in fluoride-based electrolyte

𝑉

etched volume of a structure

𝑉

volume of a mixture or of its constituent

𝜐etch

etch front velocity

𝑤0

width of the backside contact for the elementary profile 𝑆0 (𝑥) used in description of pixel concept

XXIX

Nomenclature 𝑤0.5

structure width at half depth

𝑤etch

structure width

𝑤back

width of a backside local contact

𝑤front

width of a frontside opening

𝑤𝑖

mass fraction of a constituent 𝑖 in a mixture

𝑧

longitudinal axis coordinate in cylindrical coordinate system (in the simulation)

𝑧X

charge number of species X

1. Introduction 1.1. Overview: silicon anodization In the last decades, miniaturization of sensors and actuators opened new possibilities for applications of these devices in all fields of our lives. In order to fabricate such devices, different microstructuring techniques mostly coming from microelectronics technology are used. However, every technique is limited in choice of materials and achievable geometries, and often requires sophisticated and expensive equipment. Therefore, demand for new innovative microstructuring methods remains. One such innovative process is electrochemical etching of silicon (anodization). In this work, application of silicon anodization as a structuring techniques was studied. Silicon anodization is an electrochemical process of silicon etching in hydrofluoric acid based electrolytes. In this process, two regimes can be distinguished: pore formation resulting in porous silicon layer and electropolishing. Processes running during silicon anodization can be analyzed on both microand macroscale. Microscale here describes features in the order of nanometers to a few micrometers. In this scale, self-organization mechanisms of pore formation for porous silicon at low applied potential, and porous oxide at high potential are running. Additionally, between the porous silicon and porous oxide regimes, electropolishing takes place providing for pronounced etching of prominent points of a silicon surface by formation of nanometerthick anodic oxide and its subsequent etching. Scientific questions of significant importance do arise on the microscale such as pore formation mechanisms, influence of process parameters (e.g. temperature, substrate doping level and type, and electrolyte composition) on morphology and dimensions of pores, role of silicon-electrolyte interface layers (e.g. Helmholtz layer in electrolyte and space charge region in silicon), surface roughness and topography. On the macroscale, the point of investigation is the shape of the porous silicon volume in pore formation regime or of the cavity in electropolishing © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_1

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1 Introduction

regime, left after the process. Here, the scientific questions include study of the etch form development and influences of process conditions, such as electrolyte composition, current density and potential distribution in the electrochemical cell, masking materials, and other localization techniques, on the etch front propagation. Although the macroscale behavior of the process is naturally based on the microscale physics, the macroscale effects can be often well described with simplified macroscale mechanisms without detailed microscale models. Self-organization mechanisms on the microscale level during pore formation have inspired researches for many applications in the last three decades [1–48]. For example, porous silicon of various morphologies has found applications as filter and sensing layer for micro-fluidic/lab-on-chip devices [3, 47, 49–51], electrodes with high internal area for batteries [52, 53], proton-conducting membranes for fuel cells [54–58], and as etch technique with adjustable anisotropy for smart systems [3, 11, 17, 22, 32, 39, 45, 59–63]. Porous silicon technology is also used for fabrication of silicon-on-insulator (SOI) structures in microelectronics [64–67], and it has proved to be a very flexible technique for fabrication of porous silicon based photonic crystals [16, 18–20, 25, 36, 42, 43, 68, 69]. Application of silicon anodization process in electropolishing regime offers superior surface quality [47, 70]. Originally investigated for polishing of silicon substrates for silicon microelectronics industry [71, 72], electropolishing of silicon is used nowadays, for example, for release of free standing porous silicon structures [30, 31]. Although a lot of research on the topic of silicon anodization has been done, and a lot of applications of silicon anodization process have been demonstrated by researchers worldwide, only some of these applications (e.g. Bosch Advanced Porous Silicon Membrane technology, APSM [73, 74]) have been transferred from research to industry. The main reason for this is low reproducibility of results due to many influencing factors and process parameters (applied voltage, temperature, illumination conditions, electrolyte flow, wafer position in the anodization cell, type of electrical contact to wafer, wafer doping, surface quality, etc. [2, 22, 75–77]). These factors prevent so far the process of silicon anodization from becoming a standard industrial technique.

1.2 Objectives and outline of the work

3

1.2. Objectives and outline of the work The aim of this research is to investigate the process of silicon anodization as a three-dimensional structuring technique for fabrication of shaped cavities in silicon for micro devices. Both regimes of the process, namely porous silicon formation and electropolishing, are considered. The work especially emphasizes the macroscale behavior of the process, i.e., study of etch fronti movement and shapes of resulting etch forms, leaving other exciting topics of the process, such as microscale pore formation mechanisms, beyond the scope of the work. Significant part of the work is devoted to development and verification of time-dependent macroscale models describing etch front movement in both porous silicon formation and electropolishing regimes. As literature research showed, first attempts on building charge-flow models have been made. However, to the best of our knowledge, no systematic analysis with quantitative comparison to experiments were done. Since application of any structuring process for generation of precise etch forms requires full understanding of underlying mechanisms, development of such models is the inevitable step on the way to industrial applications. In this work, among other goals, primary and secondary current distribution models for silicon anodization through an opening in a frontside mask are developed and compared to experiments. In order to control the process, achieve good precision and reproducibility of the resulting etch forms, and develop macroscale models for the process, it is necessary to understand the role of the process parameters. There are various parameters which influence the silicon anodization process on micro- and macroscale. In this work, silicon doping level and electrolyte concentration are chosen for evaluation as the most significant parameters besides the applied current/voltage. Besides the shape of the etch forms produced in the process, the question of surface quality is also addressed in the work. Optimization of anodization parameters in regard to surface quality is performed in the work. Influence of porous silicon etchants on surface quality is studied as well. i Terms

etch front and etch rate in the work are used for both porous silicon formation and electropolishing. In case of electropolishing they are used in conventional way. In case of porous silicon formation, the term etch front refers to the interface between porous silicon and solid silicon moving into the depth of silicon substrate as the process proceeds, with the corresponding movement velocity known as etch rate or growth rate.

4

1 Introduction

The process is tested in the work on fabrication of various elementary cavities in silicon. The main test structure in the work consists of silicon sample with a circular opening in a frontside insulating layer. Additionally, the combination of the anodization process with other etch techniques and the application of backside local contacts are investigated. Although the work concentrates on the silicon anodization process, the results of the work can be applied to some extent to other electrochemical etching processes (e.g., anodization of other semiconductors or metals). The work has to give answers to the following scientific questions: • Convex and concave shapes of the cavities anodized through an opening in frontside insulating masking have been reported. Can these two distinct types of shapes represent the development of the same etch form at different moments of time of the process? • For the case of cavities anodized in high-concentrated electrolyte through an opening in an insulating masking, can the electrical charge flow be considered as a shape determining mechanism for both convex and concave shapes? In other words, can a primary or secondary current distribution model neglecting mass transport effects be used for this case to describe shape development? • Anodization of silicon substrates with local backside contacts of simple geometry (circular, rectangular) results in funnel-like structures. Can the shape of a cavity on a sample with a backside contact of complex geometry be represented as a superposition of etch forms obtained for elementary backside contacts (“pixels”)? • At which conditions electropolishing of silicon substrates can result in a surface roughness comparable to the roughness of polished blank silicon wafers with average roughness in order of few nanometers? • Electropolishing can be used to modify shape of cavities etched with another etch process. How does short electropolishing of prestructured cavities change the shape of the cavities? • To remove sacrificial porous silicon, 1 m% KOH or a solution with 1:5 volume ratio of AZ351B:water can be used. How do these solutions change surface roughness of polished silicon? The work is partially based on selected experimental results obtained by the author for several research projects at the Institute of Applied Research (Institut für Angewandte Forschung, IAF) of the Furtwangen University (Hochschule Furtwangen, HFU), with one of the projects done in

1.2 Objectives and outline of the work

5

cooperation with the Department of Microsystems Engineering (Institut für Mikrosystemtechnik, IMTEK) of the University of Freiburg (AlbertLudwigs-Universität Freiburg). The presented investigations are connected together by the general topic of microscale and macroscale structuring mechanisms in silicon anodization process. Some of the results were already published previously. In most cases, those publications are not cited in the text, but summarized in the list of relevant publications in the end of this report. The report is divided into the following chapters: In chapter 2, first, an overview of microstructuring methods is presented. Then, the state of the art on silicon anodization process, with emphasis on the micro- and macroscale processes, is described. An extensive systematic analysis of the data from the available references showing influence of various process parameters on the properties of anodization process is presented. The chapter also provides some background information on electrochemical phenomena in the anodization process and a short overview on applications of the process as an etch technique. In chapter 3, experimental methods and characterization techniques used in the work are explained. Experimental setups, silicon sample preparation steps, characterization aspects, and approaches for the macroscale models are described. In chapter 4, microscale aspects of the anodization process, namely, influence of current density, substrate doping level, and electrolyte composition on characteristics of the resulting layers and surfaces, are studied. In chapter 5, structuring of silicon with anodization process is studied on the macroscale. Experiments on etch front movement for various localization techniques are supported by macroscale models with primary and secondary current distributions. The report ends with concluding remarks, acknowledgments, list of own relevant publications, appendices, and bibliography. This book is based on the PhD thesis titled “Investigation of silicon anodization process as a three-dimensional structuring technique” prepared within the frame of the PhD project which was defended by the author on 30th of September 2016 at the University of Freiburg. The examination committee consisted of Prof. Dr. Jürgen Wilde (chair), Prof. Dr. Oliver Paul (co-chair), Prof. Dr. Peter Woias (advisor), Prof. Dr. Holger Reinecke (examiner), Prof. Dr. Ulrich Mescheder (examiner and co-advisor).

2. State of the art 2.1. Introduction Electrochemical etching of silicon in fluoride containing electrolytes, also named as anodization process, is a fascinating process which provides material for research for more than half a century. The first report on anodization of silicon (and germanium) was published by Uhlir [78]. Further investigations of silicon anodization were performed ten years later by Memming and Schwandt [79]. The process was used firstly only as a technique to polish wafers [71, 72]. In the 1990s the process of silicon anodization got great interest of the scientific community after the publication of Canham about his discovery of photoluminescence from one of the products of the process - porous silicon [1]i . At that time other remarkable papers on silicon anodization and applications of porous silicon as a new material for photoluminescent devices, as well as adsorption material, insulating material, composition material, and sacrificial layer material for microelectronic and micromechanical devices were published by Eddowes [81], Zhang et al. [76, 82], Lehmann et al. [32, 83–86], Lang et al. [5–7], and others [9, 12–14, 21, 27, 28, 33, 37, 38, 45, 66, 67, 87–103]. In this chapter, state of the art on theory and technology of the process is presented. An emphasis is given to the electrochemical backgrounds of the process and the role of process parameters. A short overview of other structuring techniques for semiconductor based smart systems is also presented. 2.2. Structuring techniques for semiconductor based smart systems The history of semiconductors started in the first half of the 19th century with the discovery of first semiconductor material – silver sulfide – by i The

importance of this discovery for the scientific community is confirmed by the fact that for his discovery Canham was named as probable Nobel Prize candidate in physics by Thomson Reuters in 2012 [80].

© Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_2

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2 State of the art

Michael Faraday in 1833 [104]. After the discovery of the phenomenon of unilateral conduction of metal-sulfide by Karl Ferdinand Braun [105] and its practical application in the first semiconductor device – Cat’s-whisker detector – for radio wave detection [104] in the beginning of the 20th century, the next milestone of the semiconductor device technology came with the invention of the first transistor in 1947 [106], which led to the beginning of the microelectronics era of – mostly silicon based – very and ultra large scale integration (VLSI and ULSI) devices. Since then, the need for miniaturization of semiconductor devices encouraged the invention of numerous technologies for production of high-quality crystalline silicon wafers and structuring (mostly planar) of silicon and the accompanying materials. Miniaturization technologies of microelectronics enabled also miniaturization of mechanical devices. The idea of miniaturization of mechanical systems was first pronounced by Richard Phillips Feynman in his famous lecture “There’s Plenty of Room at the Bottom” at an American Physical Society meeting in 1959. This talk is considered as a starting point of micromachining and nanotechnology [107] as well as the beginning of the era of micro- and nanoelectromechanical systems (MEMS and NEMS). Characteristic length scale of MEMS and NEMS is considered to be in order from micrometer to few millimeters and below micrometer, respectively. Advantages of micromachining, such as reduction of costs due to batch processing and integration with electronic components on a chip, led to a rapid increase of popularity of MEMS and NEMS and based on them smart systems starting from 1980s. At first, fabrication of MEMS devices was done with the technological processes of planar microelectronics. One of the first mass applications were accelerometers, pressure sensors, and ink-jet nozzles. Further advancements of the structuring techniques for fabrication of three-dimensional (3D) etch forms in 1990s allowed the fabrication of more complex systems. Today miniaturized mechanisms and devices can be found in many application fields. The choice of materials for smart systems is typically limited to the materials compatible to the complementary metal-oxide-semiconductor (CMOS) technology, because this way integration of smart systems into electronic chips can be guaranteed [108]. Silicon was and remains the most popular material for smart systems due to its good mechanical properties, properties of its oxide, well-studied technological processes, and the fact that it is a second abundant material on Earth. This chapter provides an overview

2.2 Structuring techniques for semiconductor based smart systems

9

of main fabrication strategies and structuring processes for silicon based MEMS and NEMS. 2.2.1. Principles and characteristics of structuring processes Micromechanical structures are usually formed by removing/etching material in some regions of a substrate. An etching process is characterized by etch rate 𝑅etch [length/time] (equal to etch front velocity 𝜐etch ) and etch time 𝑡etch [time], which define together etch depth 𝑑etch [length]. Localization of a structuring process can either be achieved by the process itself (e.g., laser beam machining) or by auxiliary masking layers working as a protection for areas which should remain after the process. In case of using protecting masking materials which provide good protection for the duration of the whole etching process, a masking material with better/higher chemical stability (low etch rate) compared to the underlying layer for the given etching process is required. Hence, the etching processes are classified according to their selectivity 𝑆 for different materials. Selectivity is defined as etch rate ratio for material to be structured and masking material (s. eq. (2.1)). Selectivity defines how deep (or how long) the etch process can be run for a given thickness of the protecting mask. 𝑆=

𝑅etch (material) 𝑅etch (mask)

(2.1)

Other important issues for an etching process with a protective masking film are the adhesion of the masking film to the substrate and mechanical properties (internal stress) of the film. In case of low adhesion, the masking material can detach completely from the substrate before the etching process is finished. For example, this problem occurs during etching of silicon nitride in buffered hydrofluoric acid solution (BHF) with photoresist as a protecting mask. In this case, special adhesion promoters are applied before deposition of photoresist, and intermediate baking steps during the etch process are used in order to prevent the photoresist from detaching. Additionally, internal stress in the masking film might result in bending or breaking of the film in case of large underetching. Etch shape during etching process is typically defined by vertical 𝑅etch V and lateral 𝑅etch L etch rates. Lateral etch rate defines underetching of a masking material. If 𝑅etch V = 𝑅etch L , etching process is called isotropic, otherwise

10

2 State of the art

the process is anisotropic. Anisotropy of an etching process can be related to crystallographic orientation of a crystalline material (e.g., silicon etching in KOH or TMAH), or induced by the etching process itself (e.g., reactive ion etching, where more anisotropy can be achieved by directional electric field). Anisotropy factor 𝐴f defines anisotropy of the process (eq. (2.2)) for simple shapes [7]. If 𝑅etch V ≫ 𝑅etch L (s. Fig. 2.1a), then 𝐴f → 1, meaning only vertical etching, as can be achieved, for example, with deep reactive ion etching (DRIE) [109]. For isotropic processes, 𝐴f = 0 (s. Fig. 2.1b). 𝐴f = 1 −

𝑅etch L 𝑅etch V

(2.2)

(a)

(b)

Af = 1

Af = 0

Figure 2.1.: Wafer cross-section showing (a) anisotropic and (b) isotropic etch structures. Although for planar microelectronic devices etch shape is often not of main concern, it plays very important role in MEMS, where real 3D structures are required in order to fabricate particular mechanisms or structural elements. Anisotropic etching of silicon (e.g., DRIE) can result in vertical sidewalls, whereas isotropic (e.g., wet) etching results in spherical concave shapes. Thus the resulting shape and depth are determined by the anisotropy factor of the etch process. For conventional techniques with a fixed anisotropy factor, it is not possible to produce structures with different 3D shapes and depths in one step. 2.2.2. Fabrication strategies for smart systems There are two main fabrication routes used for silicon-based smart systems: surface micromachining and bulk micromachining [108, 110]. Surface micromachining processes came from microelectronics technology. As the name suggests, the fabrication processes used in surface micromachining are performed on thin films on one side of a wafer. The starting point is normally a crystalline silicon wafer (substrate). Devices are

2.2 Structuring techniques for semiconductor based smart systems

11

formed by many steps of deposition and subsequent structuring by etching of films on the substrate. Structuring processes in planar microelectronic technology are typically relying on sophisticated methods for transfer of two-dimensional (2D) patterns into 3D structures by lithography and subsequent etching. Whereas the lateral definition and resolution are mainly determined by the lithography as primary structuring step, the vertical dimension and shape depend on the etch process. Examples of surface micromachined smart systems are RF capacitors [111], pressure sensors [112], angle sensors [113], and micromirror devices [114, 115]. In general, surface micromachined smart systems have limitations in depth of their structures. In the bulk micromachining of smart systems, microelements are formed in volume of a silicon substrate. This way larger and deeper elements than in surface micromachining can be fabricated. In contrast to the surface micromachining technology, in bulk micromachining typically both sides of a wafer are processed, which makes the process less compatible to the CMOS technology. In order to overcome limitations of planar microelectronics technology and generate deep 3D cavities, several sophisticated approaches have been investigated, e.g., gray-scale lithography [116, 117] (s. also sec. 2.2.3.2) and stereo lithography [118–120]. In order to obtain reproducible etch results, it is important to have precise control of etching process parameters and/or apply special etch stop techniques [7]. One such etch stop technique which is very popular nowadays is based on the use of SOI wafers [121]. Examples of bulk micromachined smart systems are pressure sensors [122], accelerometers [123,124], thermoelectric generators [125], tactile sensors [8], micro force sensor [126], focusing mirror device [127], and shear-stress sensor [128]. To decrease the production costs for MEMS, there is a trend in the last decade to replace silicon with cheaper materials, such as polymers. In this case, structures etched into silicon or other materials can be used as micro-molds to form plastic elements by some molding technique, such as injection molding or hot embossing [120]. A quite often used process here is the LIGA-technique (s. sec. 2.2.3.2) for high aspect ratio microsystems (HARMS), although silicon plays in this technology only a minor or no role [110, 129–136]. Another nanostructuring process based on replication is the nanoimprint process [137–139] which, however, relies on other structuring techniques for fabrication of the imprints.

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2.2.3. Structuring techniques 2.2.3.1. Conventional wet and dry etch techniques There are various conventional structuring techniques developed in microelectronics and microsystems technologies for structuring silicon. In general, these processes can be divided in dry and wet processes. In dry etch processes the material is removed by gaseous etchant in form of vapor or plasma. The principles of removing material are either chemical, physical or physical-chemical [7,140]. Since the free mean path of a particle in a gas is inversely proportional to the gas pressureii , in general, increasing the gas pressure provides a more isotropic process. An example of physical etching is sputter etching, where ions of chemically inert gases, such as argon, bombard the surface after being accelerated in high-frequency electric field. With this process, high anisotropy can be achieved due to directed attack of the surface by the accelerated ions. Selectivity in the process is provided by the fact that soft materials (e.g., resist) are etched slower than hard materials. The formed cavities, although showing high anisotropy, are subjected to the scattering effects, such as micro-trenching (high etch rate near the edges of cavities due to ion reflection from the cavity sidewalls), and redeposition of the removed material to the mask edges (Fig. 2.2a) [141, 142]. In chemical plasma etch processes, chemically active gases, such as the halogens F, Cl or Br [140], are used, and the processes run with chemical reaction of the radicals generated in the plasma, when they are transported to the etch site, adsorbed there, and then react with the surface atoms. Another examples of dry chemical etching are vapor-phase etch processesiii , where etchants in vapor/gas phase, e.g., HF for silicon dioxide etching or XeF2 for silicon etching, are used [132, 143, 146–149]. The process selectivity of dry chemical etch processes is defined by the chemical reactions and the phase of the products: volatile products can move away from the reaction site without disturbing the reaction; solid products, in contrast, passivate the ii In

an ideal gas when the particle speed is given as Maxwell-Boltzmann distribution, the mean free path 𝑙 for a given diameter of the particles 𝑑 and the gas pressure 𝑝 can 𝑘b 𝑇 be calculated with the formula: 𝑙 = √2𝜋𝑑 2 𝑝 , where 𝑘b is the Boltzmann constant and 𝑇 is the absolute temperature in Kelvin. iii There are also classifications of etch processes, where only plasma/discharge etch processes are meant as dry etch processes, and vapor etching is considered as a separate class [132].

2.2 Structuring techniques for semiconductor based smart systems (b)

(a) mask

(c)

13

(d)

redeposition ion

micro-trenching

Figure 2.2.: Schematic cross-section of etch forms for (a) sputter etching with micro-trenching and material redeposition [142], (b) isotropic etching without diffusion-limitation (e.g., with slow etch rate) [143], (c) diffusion-controlled isotropic wet etching without agitation [144,145], and (d) isotropic wet etching with agitation [143]. surface, and the process might stop. The process of chemical dry etching is load-dependent, i.e., for larger exposed wafer areas slower etch rates are observed. Typically, high pressure is used, and the process results in isotropic shapes. In physical-chemical processes (for example, reactive ion etching - RIE), the reaction is additionally activated with ion bombardment (physical action). The more physical component, the more anisotropic is the etch process. Therefore, the anisotropy of such a process can be to some extent controlled by the process parameters. During the process, at specific operation conditions the surface can get passivated by polymer layers formed from present halogens and carbons inside the reactor gases. The subsequent etching step results in vertical etching of the passivating layer and the material, leaving the vertical sidewalls passivated. This way, switching between etching and passivating steps can provide nearly anisotropic etching in the DRIE Bosch process. In wet etch techniques, etchants in liquid phase are used. Although in many fabrication processes wet etch processes have been replaced with dry etch processes, wet etch processes are still widely used as structuring techniques for smart systems because of their advantages such as setup simplicity and low cost. An example of isotropic silicon etchant is HNA, that is a mixture of HF, nitric acid (HNO3 ), and acetic acid (CH3 COOH) in water [150, 151].

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2 State of the art

In this solution, silicon is first oxidized in nitric acid and subsequently etched away by hydrofluoric acid [143]. For an isotropic wet etch process, where diffusion does not control the process (due to high diffusion constants of the reactants and reaction products, or due to slow etch rates), the isotropic shape develops from a shape with flat bottom to a concave shape, when the etch depth increases over the width of the opening in the mask (s. Fig. 2.2b). With diffusion limitation, the isotropic etch process might result in pronounced etching of the structures near the edges of the mask in the beginning of the process, giving convex etch form, with later transformation of the etch form to the concave form (s. Fig. 2.2c) [144,145]. When agitation of the etch solution, e.g., stirring, is applied to overcome diffusion limitation, concave forms from the beginning of the process can be obtained (s. Fig. 2.2d) [143]. Thus, certain etch shape control for the isotropic wet etch process can be done and has been used in the past. Anisotropy in a wet etch process can be also caused by anisotropic properties of the material to be etched. For monocrystalline silicon, such anisotropy can be provided by its crystal lattice. Typical anisotropic silicon etchants are KOH and TMAH. These solutions show very slow etching of (111)-planes in comparison to the etch rate for (100)-planesiv , resulting in well controlled V-groove shapes for (100)-oriented silicon wafers (s. Fig. 2.3a) and perpendicular (vertical) sidewalls for (110)-oriented silicon wafers (s. Fig. 2.3b) [143]. (a)

(100) (111) 54.74

(b)

(110) (111)

◦

Figure 2.3.: Schematic cross-section of etch forms in anisotropic silicon etching for (a) (100)-substrate resulting in V-groove and (b) (110)-substrate resulting in vertical walls [143]. Electrochemical etching processes, as a special group of wet chemical processes, provide additional degree of control through dependence of their etch iv The

etch-rate ratio 𝑅etch (111) /𝑅etch (100) is concentration and temperature dependent [152].

2.2 Structuring techniques for semiconductor based smart systems

15

rates on charge flow. This way, for example, in anisotropic etching of silicon in alkaline solutions, the etch rate of the process can be controlled by applying anodic potentials to the silicon [153]. Silicon etching in hydrofluoric acid as a specific example of electrochemical etching processes for silicon structuring, and as main topic of this work will be covered in detail in sec. 2.6. It is worth noting that although silicon was and remains the material for most of studies of the anodization process, there are also other semiconducting and conductive materials that can be anodized, such as GaAs and other compound semiconductors [154] or metals, e.g., aluminum [155]. 2.2.3.2. Micromachining methods The conventional wet and dry etch techniques are typically applied to the full area of a wafer, therefore they provide high throughput. However, they are limited in definition of real 3D shapes. Several advanced techniques for 3D structuring at wafer-level have been developed. For example, with grayscale lithography, 3D shapes formed in photoresist by exposure with locally modulated light intensity are transferred into the underlying material during RIE process (s. Fig. 2.4) [117]. Another technique for real 3D shapes is 3D micro-stereolithography, where polymer is cured with focused UV light at very small energy, at which hardening of the polymer only occurs at the focal point, and complex 3D structures as small as several micrometers are defined [156]. In the LIGA process utilizing X-ray lithography, quasi-3D or 2.5D structures (i.e., 2D layout in 𝑥, 𝑦-plane projected to a fixed depth in 𝑧-axis) with typical feature size of 1 µm are formed in thick photoresist and transferred to another material, for example, with electroplating [110, 129– 136, 138]. Besides these wafer-level processes, there is a number of micromachining techniques for local material removal, which can provide very high precision in nanoscale, however, they require processing of structures in a sequence one after anotherv . Whereas well established for macrostructuring, for microstructuring such processes are mostly in the research-and-development stage. These processes can be considered as a separate class of structuring techniques. Examples of such techniques are laser beam machining (LBM), v Batch

processing with wet microstructuring processes (e.g., micro-ECM) is also possible, to some extent, by using arrays of tool-electrodes.

16

2 State of the art Deposition of photoresist Grayscale lithography Deep reactive ion etching

Final structure

Figure 2.4.: Process flow for grayscale lithography with structure transfer process from photoresist to substrate. electrochemical micromachining (micro-ECM), electro discharge micromachining (micro-EDM), etc. Dry microstructuring processes are listed in Tab. 2.1. The overview of wet microstructuring processes (s. Fig. 2.5) is given in Tab. 2.2. Conventional mechanical material processing techniques, such as drilling and milling [157], or micro ultrasound machining (USM) [138, 158], which require mechanical contact between the workpiece and a tool or abrasive particles, as well as electrochemical cutting techniques (e.g., wire electrodischarge grinding - WEDG, wire-ECDM - WECDM [159–161], wire-ECM WECM [162]) are excluded from this overview. Excluded are also variations of the microstructuring processes for nanomachining, such as application of scanning tunneling microscope, where the tip plays the role of the tool for nano-ECM [138, 139, 163, 164].

17

2.2 Structuring techniques for semiconductor based smart systems (a) ECM

(b) EDM, ECAM

tool-electrode

tool-electrode

zy x

zy x

charge flow

workpiece (counter electrode)

(c) SACE tool-electrode

zy x

plasma plasma

workpiece (counter electrode)

counter electrode workpiece

Figure 2.5.: Wet micromachining methods shown in schematic crosssection (electrolyte not shown): (a) ECM, (b) EDM or ECAM, (c) SACE.

Table 2.1.: Dry micromachining methods Structuring method

Principle

Processed materials

Resolution

Typical applications

RAPa [165]

chemical

silicon, quartz, silicon nitride, silicon carbide, etc.

micrometers

optical components in glasses and ceramics

EBMb [138, 165, 166]

thermal vaporization via electron impact

any

5 nm

structures with very high aspect ratio, microholes, molds

FIBc [138,165,167–169]

mechanical (ion impact)

any

10 nm

MEMS, AFM tips, microtools, molds

Long-pulse-LBM and CW-LBMd [165, 166, 170]

photon absorption, thermal evaporation or melting

light absorbing materials

50 µm

cutting, scribing

Ultrashort-LBMe [138, 170–172]

evaporation through ionization

any

100 nm

microstructures, microtools

a RAP

- reactive atom plasma machining. - electron beam machining. c FIB - focused ion beam machining. d Long-pulse-LBM and CW-LBM - long-pulse and continuous wave laser beam machining with pulses of nanosecond and longer width. e Ultrashort-LBM - ultrashort-pulse laser beam machining with pulses of femtosecond and shorter width. b EBM

18

2 State of the art

Table 2.2.: Wet micromachining methods Structuring methoda

Principle

Processed materials / workpiece

Electrolyte

Working electrodes

Resolution

ECMb [138, 162, 163, 173–178]

electrochemical etching, anodic dissolution

(semi-)conducting materials

conductive, chemically active

tool and workpiece

0.5 µm

EDMc [138, 156, 163, 165, 181–194]

thermal vaporization via spark discharge plasma between the tool and workpiece

non-conducting materials with conductive carbon film formed after discharge or (semi-)conducting materials

non-conductive (hydrocarbon oils, e.g., kerosene, or deionized water)

tool and workpiece

0.5 µm

ECAMd [138, 195–199]

s. ECM + EDM

(semi-)conducting materials

low-resistivity deionized water

tool and workpiece

15 µm

SACEe [198–216]

thermal vaporization of workpiece in close vicinity to the tool, on which a plasma layer is formed by discharge between the tool and electrolyte in a surface gas layer (hydrogen and water vapor)

any material

conductive, e.g., aqueous solutions of salts

tool and electrolyte (with a counter big area electrode remotely located in the electrolyte)

10 µm

a The

word “micro” is omitted in the names of the micromachining techniques here, because the discussed processes can be performed on both macro- and microscale. b ECM - electrochemical machining; other variations: pulse electrochemical machining (PECM), jet electrochemical machining (JECM), laser assisted ECM, vibration/ultrasonic assisted ECM. c EDM - electrical discharge machining; typically (semi-)conducting materials can be structured with EDM. However, structuring of non-conductive materials have been also demonstrated [179,180]: in this case, a carbon film on a non-conducting workpiece is formed after discharge in electrolyte (oil), and works as counter electrode for a next discharge, after which again the carbon film is formed, and the process repeats. d ECAM - electrochemical arc machining (ECAM); other names: electrochemical discharge machining (ECDM), simultaneous ECM and EDM (SECDM). The name ECDM is also used to describe the microstructuring process SACE for non-conductive and conductive materials (s. below). In general, literature review in the field reveals lack of unified conventions on names of various wet micromachining processes. e SACE - spark assisted chemical engraving; other names: electrochemical spark machining (ECSM), electrochemical discharge micromachining (ECDM).

2.2.4. Conclusions In this section, an overview of microstructuring methods, with emphasis on structure shape, has been presented. Conventional wet and dry etch

2.3 Mass and charge transfer in silicon and electrolyte

19

techniques provide only specific etch shapes of fixed anisotropy. From the other point, the techniques giving real 3D shapes require expensive facilities, such as micro-stereolithography or FIB, and are typically suffering from very long processing times and associated costs of final product. Electrochemical etching of silicon in hydrofluoric acid based electrolytes can be applied as an alternative structuring method for fabrication of 3D forms in silicon, where etch profile (shape and surface quality) can be controlled with various localization techniques (electric potential, doping, light, and more) and a precision in the nanometer range. As common for wet chemical processes, silicon anodization does not require costly equipment, and can be performed at wafer-level, providing high throughput. In the next sections, this process will be described in detail. 2.3. Mass and charge transfer in silicon and electrolyte 2.3.1. Band diagram and transfer of charge and mass in bulk p-type silicon The schematic band diagram of p-doped silicon at room temperature is shown in Fig. 2.6, where 𝐸F is the Fermi level, 𝐸C is the energy at the bottom of the conduction band, 𝐸V is the energy at the top of the valence band, 𝐸G = 𝐸C −𝐸V is the band gap (𝐸G = 1.12 eV for silicon at 300 K [106]), and 𝐸A is the acceptor levelvi . The difference 𝐸A − 𝐸V defines the ionization energy required to excite holes from the acceptor energy level to the valence band. Transport of electrons and holes in silicon with concentrations 𝑛 and 𝑝, respectively, can be described by Nernst-Planck equation accounting for drift and diffusion [106, 221–223]vii : 𝐸 + 𝑒𝐷n ∇𝑛, 𝑗p = 𝑒𝜇p 𝑝𝐸 𝐸 − 𝑒𝐷p ∇𝑝 𝑗n = 𝑒𝜇n 𝑛𝐸

(2.3)

where 𝐸 is the electric field, 𝑗 is the current density field, 𝑒 is the elementary charge, 𝜇n/p are the carrier mobilities, and 𝐷n/p are the diffusion constants of electrons (subscript n) and holes (subscript p). vi In

the work an effort was made to write symbols according to the rules of the Le Système International d’Unités (SI) [217], International Union of Pure and Applied Chemistry (IUPAC) [218, 219] and the international standard ISO 80000-9 [220]. In some cases, conventional symbols are used, which are not SI-conform. vii These references are used through this section.

20

2 State of the art

E

EC EG EF EA EV

Figure 2.6.: Band diagram of p-type silicon. At 300 K, 𝜇n = 1450 cm2/(V⋅s), 𝜇p = 500 cm2/(V⋅s), 𝐷n = 37.5 cm2/s, and 𝐷p = 13 cm2/s [221]. The total current density is 𝑗 = 𝑗n + 𝑗 p

(2.4)

For non-degenerate semiconductors (i.e., for semiconductors with 𝐸C − 𝐸F ≫ 𝑘b 𝑇 ) the mobility of charge carriers is related to their diffusion constants according to the Einstein relation: 𝐷=(

𝑘b 𝑇 )𝜇 𝑒

(2.5)

The conductivity 𝜎 is defined by the mobilities and the concentrations of the carriers: 𝜎 = 𝑒𝜇n 𝑛 + 𝑒𝜇p 𝑝

(2.6)

In the models of silicon anodization, transport of holes in p-type silicon substrate is often described by current density distribution, assuming that holes as major carriers almost solely provide this current density [3, 13, 29, 30, 45, 62, 96, 110, 224]. In this case, simplified description of silicon substrate as a material with isotropic conductivity defined by doping level is sufficient. For the silicon substrates with resistivity in the ranges 10–20 Ω cm and 0.01–0.1 Ω cm used in the models in this work, the conductivity 𝜎Si is in the ranges 5–10 S/m and 103 –104 S/m, respectively.

2.3 Mass and charge transfer in silicon and electrolyte

21

2.3.2. Band diagram and transfer of charge and mass in bulk electrolyte 2.3.2.1. Band diagram An electrolyte contains reduced and oxidized species characterized with the corresponding energy levels 𝐸Red and 𝐸Ox reflecting the average energy levels of all individual redox species in reduced or oxidized form, respectively (Fig. 2.7). These energy levels 𝐸Red and 𝐸Ox are the most probable energy levels for the reduced and oxidized species, and together they determine the redox potential 𝐸redox = 0.5 ⋅ (𝐸Red + 𝐸Ox ), which can be considered as the effective Fermi level of electrolyte similarly to the Fermi level in semiconductors. The redox potential describes the tendency of the species to accept or give up electrons [4].

E

EOx Eredox ERed

Figure 2.7.: Electron energy levels of a redox couple in electrolyte [4].

2.3.2.2. Composition of an HF-based electrolyte Anodization of silicon is conducted in fluoride containing electrolytes, typically based on hydrofluoric acid (HF). According to Föll et al. [22], electrolytes for silicon anodization can be classified in aqueous electrolytes and organic electrolytes. The aqueous electrolytes, as the name suggests, include mixtures of HF and water in various proportions. Additionally, such electrolytes might contain ethanol, acetic acid or some other surfactants which are applied in order to reduce surface tension or to adjust pH-value or viscosity. In the organic electrolytes, mixtures of HF with organic substances

22

2 State of the art

such as acetonitrile (MeCN) or dimethylsulfoxide (DMSO) are applied. This classification might be not quite obvious at first sight, however is well justified by Föll and co-authors, and both classes of electrolytes provide peculiar features distinguishing them in anodization process [22]. Most commonly used electrolytes for silicon anodization are aqueous electrolytes based on HF and water with or without addition of ethanol, and such electrolytes are meant through this report, unless otherwise stated. Hydrofluoric acid based aqueous electrolyte is a complex system of ions and not dissociated molecules. Because HF is a weak acid, besides undissociated HF molecules, there are small amounts of HF2 − , F− , and (HF)2 species present in such electrolyte according to the following dissociation (ionization) and association reactions [2, 225–228]: HF

𝐾HF ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ H+ + F −

(2.7)

𝐾HF2 − ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ HF2 −

(2.8)

hydrogen fluoride

HF + F−

bifluoride

𝐾

2HF

(HF)2 ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ (HF)2

(2.9)

hydrogen fluoride dimer

with 𝐾HF =

𝑎H+ ⋅ 𝑎F− , 𝑎HF

𝐾HF2 − =

𝑎HF2 − 𝑎HF ⋅ 𝑎F−

and

𝐾(HF)2 =

𝑎(HF)

2

𝑎HF 2

(2.10)

where 𝐾HF , 𝐾HF2 − , and 𝐾(HF)2 are the equilibrium constants for the reactionsviii , and 𝑎 is the activity of species given in subscript. The smaller the 𝐾eq a reaction in equilibrium of the form 𝛼A + 𝛽B... ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ 𝜌R + 𝜎S..., the equilibrium ρ σ 𝑎 ⋅𝑎 ... constant is 𝐾eq = 𝑎 Rα ⋅𝑎S β ... , where the symbol of the form 𝑎X is the activity of a B A species X, which can be defined in relation to the concentration of the species X (concentration 𝑐mol X ) through the activity coefficient 𝛾X : 𝑎X = 𝛾X ⋅ 𝑐mol X . Activity defines how much of the available species actually take part in the reactions, i.e., how close the solution behaves to the ideal solution. Since for diluted solutions activity coefficients approach 1, concentrations are often used instead of activities to find equilibrium constant. In practice values of activity coefficients are mostly unknown, and concentrations are applied also for concentrated solution [229]. Therefore, in further equations for simplicity also concentrations will be used instead of activities in this work. Dissociation constant (also called as ionization constant) is a specific type of equilibrium constant for reactions of dissociation [230].

viii For

2.3 Mass and charge transfer in silicon and electrolyte

23

equilibrium constants, the less amount of the reaction products is available in a solution in equilibrium. The equilibrium constants in an electrolyte depend on the concentration of the electrolyteix , or, more exactly, on the ionic strength of the electrolyte [225, 231], however, according to Warren [226] this effect is small. Up-to-date publications on the topic provide the following values of the equilibrium constants corrected for zero ionic strength at 25 °C: 𝐾HF = 6.71×10−4 mol/dm3 , 𝐾HF2 − = 3.861 dm3/mol, and 𝐾(HF)2 = 2.703 dm3/mol [2, 227]. With these constants, equilibrium concentrations of all components in a HF:water solution can be calculated (s. Appendix, sec. A.1.3.4 and sec. A.1.3.5). The dimerization reaction (2.9) is not always taken into account. However, as can be seen from the calculations in sec. A.1.3.5, this reaction changes significantly the equilibrium concentrations in the solution. Besides the above mentioned species, a concentrated HF solution might contain other species, such as more complex polymers of HF (e.g., hexamer (HF)6 , tetramer (HF)4 ) [2,226] or the fluoronium ion H2 F+ [231–234]. It is often assumed that these species are present in very small concentrations, and can therefore be neglectedx . The role of other substances added to the electrolyte in chemical reactions is also often neglected. However, ethanol, for example, can reduce dissociation of HF and decrease the activity of F− [2, 236]. For example, 𝐾HF = 3.214×10−5 mol/dm3 and 𝐾HF2 − = 12.882 dm3/mol for the solution with 50 m% ethanol at 25 °C and zero ionic strength [237]. The effect of the ethanol mass fraction on the equilibrium constants and calculation of the equilibrium concentrations of the components in HF:water:ethanol mixture are shown in Appendix, sec. A.1.4.4. There is also water dissociation reaction in the electrolytexi : 𝐾H 2 O H2 O ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ H+ + OH− ix There

(2.11)

are various concentration units with multiple different notations introduced in chemistry through the history of its development and for ease of calculations in certain cases. An overview of different concentration units and conversions between them used in the work are explained in Appendix, sec. A.1. x Polymerized species of the form (HF) F− start playing an important role in solutions 𝑥 with the total F concentration above 15 mol/l [235]. xi Proton H+ after the ionization quickly connects to a molecule of water to form H O+ 3 [238], therefore the reaction of water dissociation is often written in the form 𝐾H O 2 2H O ͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍͍ H O+ + OH− [229]. 2

3

24

2 State of the art

where 𝐾H2 O = 1.8×10−16 mol/dm3 at 25 °C, which is very small, meaning almost no dissociation in pure water [238, 239]. Additionally, during electrochemical dissolution of silicon according to the reactions (2.29) and (2.34) which will be discussed in sec. 2.4, there is also silicon hexafluoride (hexafluorosilicate anion) SiF6 2− as final stable reaction product in the electrolyte [94]. 2.3.2.3. Mass and charge transfer phenomena in bulk HF-electrolyte The transport of species X in an electrolyte by drift and diffusion, and neglecting convection can be described with the Nernst-Planck equation [240]: 𝑖X = 𝑧X 𝜇X 𝑐X 𝐸 − 𝐷X ∇𝑐X

(2.12)

where 𝑖X is the flux, 𝜇X is the mobility, 𝑧X is the charge number, 𝐷X is the diffusion constant, and 𝑐X is the concentration of the species X [1/m3 ], and the current density is 𝑗X = 𝑒𝑧X𝑖X

(2.13)

Different values of the diffusion constants for the species in HF-based electrolytes have been reported. Monk reported 𝐷HF = 3.1×10−5 cm2/s for the experiments of diffusion-limited etching of sacrificial silicon dioxide layers in HF-water solution at 25 °C [241, 242]. J. van den Meerakker found experimentally with rotating disk electrodes 𝐷HF = 2.2×10−5 cm2/s in HF-water-ethanol solution at 20 °C [243]. Values for other species are: 𝐷F− = 1.473×10−5 cm2/s, 𝐷HF2 − = 2.35×10−5 cm2/s, 𝐷H+ = 9.315×10−5 cm2/s [2], 𝐷OH− = 5.3×10−5 cm2/s [228]. It is important to note that the diffusion constants depend on the concentration of the components in the electrolyte [228]. For silicon hexafluoride SiF6 2− , Suratwala et al. provide concentration independent diffusion constant of 5×10−8 cm2/s “based on the values described for oils with similar viscosity” [244], which is suspiciously much smaller than typical values of diffusion constants in the order of 10−5 cm2/s, and cannot be taken for granted. Lehmann [2] provides from [245] values of 𝐷SiF6 2− at room temperature (RT) to be from 1.2×10−5 cm2/s for 0.83 mol/dm3 to 0.45×10−5 cm2/s

2.3 Mass and charge transfer in silicon and electrolyte

25

for 2.5 mol/dm3 which are more trustworthy, considering that diffusion constants in a system are typically of similar magnitude. The Einstein relation (eq. (2.5)) connecting diffusion constant of species to its mobility in thermodynamic equilibrium is also applicable for electrolytes. Diffusion limitation of reactants to and products from the reaction site can be neglected for concentrated electrolytes (meaning high supply of the reactants) with high diffusion constants and low etch rates provided by low applied current density. Then, neglecting also convection, a simplified representation of such electrolyte as conductive media with isotropic conductivity 𝜎𝑙 defined by concentration of ions can be used. Hill and Sirkar provided the values of conductivity of water-based HF mixture at 18 °C [246]: the conductivity increases nearly linearly with increase of HF concentration from 5.93 S/m for 4.8 m% HF to 34.11 S/m for 29.83 m% HF. 2.3.3. Interface silicon-electrolyte 2.3.3.1. Band diagram and models of the silicon-electrolyte interface When silicon with native oxide is immersed into HF-based electrolyte, the native oxide is etched, and the surface is predominantly terminated with hydrogen forming hydride Si-H or trihydride Si-H3 for ideal (111) surface and dihydride Si-H2 for ideal (100) surface differentiated due to the difference in the number of dangling bonds [2,4,35,247]. Non-ideal silicon surface is rough on an atomic scale, therefore all three hydrides may be present, as well as other adsorbed impurities, such as metals [4]. After immersion in the electrolyte, thermodynamic equilibrium between the electrolyte and silicon is achieved when their Fermi levels become equal, i.e., 𝐸F = 𝐸redox (s. Fig. 2.8) [4, 13, 87]. In this situation for p-type silicon holes will move deeper into silicon away from the electrolyte, leaving negatively charged acceptor ions, as shown in Fig. 2.8b for generated electron-hole pair. Thus, excess charge (negative) is built in the silicon near the contact to the electrolyte. This region is called space charge region (SCR). Electric field resulting due to the charge redistribution in SCR is shown by bending of the bands. Typical thickness of the SCR is in order of micrometer for moderate/low doped silicon [4]. On the electrolyte side, an ionic layer is established in order to compensate the excess charge in the semiconductor, and an interfacial electric double layer (EDL) between the silicon surface and the electrolyte is formed.

26

2 State of the art

(a)

p-type silicon

electrolyte

EC

(b)

p-type silicon electrolyte

EC e−

Eredox EF EV

EF EA EV

Eredox h+

SCR

uncompensated ionized acceptor atoms

Figure 2.8.: Band diagram (a) before and (b) after the contact of p-type silicon with electrolyte. There are various models of the ionic electric double layer at the interface solid-electrolyte [248]. In the classical Helmholtz model, the EDL is formed by layer of ions packed at the interface, forming with the solid surface two oppositely charged planes representing a capacitor separated by distance 𝛿 which is defined by radius of ions (s. Fig. 2.9a). The Helmholtz model explains capacitance observed on the interface solidelectrolyte, but does not show any influence of the electrolyte concentration and other factors [248]. Further improvements were introduced by Frumkin, Gouy, Chapman, Stern, Grahame, Bockris, and others [248, 251]. The comprehensive model of the EDL consists of the following regions (s. Fig. 2.9b): • The Helmholtz layer of ions attracted to the solid electrode by the excess charge in the SCR and also by the polar water (or, in general, solvent) molecules. The outer border of the Helmholtz layer at the distance 𝛿 defines the outer Helmholtz plane (OHP). The region from the OHP till the solid surface is also called the compact layer of the EDL. • Further in the electrolyte after the compact layer lies the diffuse ionic layer (Gouy-Chapman layer) which is formed by both anions and cations under the effect of thermal motion. • Directly on the solid surface specifically adsorbed cations and anions are located. The locus of these ions is called the inner Helmholtz plane (IHP). The compact layer is typically few angstroms thick. Thickness of the diffuse ionic layer is strongly dependent on the concentration of the electrolyte, and

27

2.3 Mass and charge transfer in silicon and electrolyte

(a)

(b)

solid electrolyte

p-Si electrolyte SCR

IHP OHP

Legend: anion cation

0δ

ionized acceptor atoms water molecule

x

0

φl

x

δ

compact layer

diffuse ionic layer

φ

∆φH ∆φSCR φs

Figure 2.9.: Electric double layer models: (a) Helmholtz model for solidelectrolyte interface; (b) general electric double layer model with potential distribution 𝜙 for p-Si, where Δ𝜙SCR and Δ𝜙H are the potential drops in the SCR and the Helmholtz layer, 𝜙s and 𝜙𝑙 are the bulk potentials of the solid (silicon) and liquid (electrolyte), and IHP and OHP are the inner and outer Helmholtz planes, respectively; not in scale; increase of potential in silicon and electrolyte due to finite conductivity is not shown [88,248–250]; the legend is for the figure (b) only. is typically below 300 Å [4, 250]. For electrolytes with a high concentration (> 0.1 M), the potential drop in the diffuse ionic layer can be neglected [4], thus the main potential drop on the solution side of the EDL occurs in the Helmholtz layer (s. at the bottom in Fig. 2.9b). The potential difference between the potential in the bulk solid (silicon) 𝜙s and the potential in the bulk liquid (electrolyte) 𝜙𝑙 determines the total potential drop in the EDL 𝑈 = 𝜙s − 𝜙𝑙 which is also called the Galvani potential [252]. Due to the small thickness of the EDL, the capacitance at the interface silicon-electrolyte, although smaller than for metal-electrolyte interface, is rather high (few Faradays per square centimeter [4]) and must be taken into account when working in a time-dependent power supply mode.

28

2 State of the art

2.3.3.2. Polarization of silicon electrode The electrochemical reactions of the silicon anodization involve a charge transfer between silicon and electrolyte. The silicon-HF system forms an electrolytic cell, i.e., a cell in which a non-spontaneous reaction is driven by an external source of current, in contrast to galvanic cells producing electricity as a result of spontaneous reaction occurring in them. Therefore, in order to proceed, the silicon electrode anodization reactions require the driving force provided by the surface overpotentialxii , and the rate of the electrode reaction and, consequently, the current density are related to the surface overpotential. On the other hand, the reaction rate is also dependent on the surface concentrations of reactants from both solid (e.g., holes in forward biased p-Si) and liquid (e.g., HF2 − , F− , and HF in HF-based electrolyte). Two specific cases can be distinguished with limitation of charge transfer at the interface (charge transfer controlled) or in the electrolyte (mass transfer controlled) as described below. Control by the rate of the charge transfer - activation polarization Excess fluoride components induced by the concentrated electrolyte at relatively low applied current density (and with sufficiently high diffusion constants for the reactants and the product) provide the process where mass transport limitation phenomena in the electrolyte can be neglected and the reactions are controlled by the rate of charge transfer. In this case, the current-voltage curve of the silicon-electrolyte junction is very similar to the current-voltage curve of the silicon-metal junction (Schottky diode) (s. Fig. 2.10) [2, 79, 81, 98]. This exponential increase of current 𝐼 with increase of applied overpotential 𝜂 can be described with Tafel expression which is characteristic for activation-controlled reactions [4, 248, 253]: 𝜂 = 𝑎 − 𝑏 log 𝐼

(2.14)

where 𝑎 is the value of polarization at the unit current and 𝑏 is called the Tafel slope (s. Fig. 2.11b): 𝑏 = 2.303 ⋅ xii also

𝑅𝑇 𝛼𝑛e 𝐹

called overvoltage [253]

(2.15)

29

2.3 Mass and charge transfer in silicon and electrolyte j, mA/cm2

8 4

–1500 –1000 –500

500

1000 U, mV

-4 -8

Figure 2.10.: Current density vs. electrode potential for a p-type (1 Ω cm) silicon electrode in 10 M hydrofluoric acid electrolyte measured in the dark; reproduced from Memming and Schwandt [79] © 1966 with permission from Elsevier. where 𝑅 is the universal gas constant, 𝛼 is the dimensionless charge transfer coefficient (between zero and unity), 𝑛e is the reaction valence, and 𝐹 is the Faraday constant.

I

(a)

Ia U = Ueq

(b)

Ic



Tafel slope b

Inet

I0



log II0

η

η

Figure 2.11.: Polarization curve for the redox process Ox + 𝑛e e− ↔ Red in charge transfer limited mode, assuming equal concentrations of the oxidized Ox and reduced Red forms of the redox couple, and 𝛼 = 0.5; the dashed lines show the anodic 𝐼a and cathodic 𝐼c components: (a) in linear scale, (b) with the ordinate showing absolute values of normalized current in logarithmic scale [253].

30

2 State of the art

More complicated polarization curves might have several Tafel sections, each with a specific slope. Typical values of Tafel slope 𝑏 for p-Si in pore formation regime are in the range 40–110 mV/decade [4] corresponding to 𝛼𝑛e in the range 0.53–1.45. In general, for a redox half-reaction Ox + 𝑛e e− ↔ Red (s. Fig. 2.12) on an electrode with the oxidized Ox and reduced Red forms of species in the charge transfer limited mode both reactions, forward and backwardxiii , are running at the same time, meaning non-zero anodic and cathodic currents 𝐼a and 𝐼c even when the net current 𝐼 = 𝐼a + 𝐼c is equal to zero at the equilibrium potential 𝑈 = 𝑈eq (s. Fig. 2.11a) [253]. The equilibrium potential depends on the surface concentrations (i.e., the concentrations near the electrode) 𝑐Ox S and 𝑐Red S of the oxidized Ox and reduced Red species according to the Nernst equation [253]: 0 𝑈eq = 𝑈eq + 2.303 ⋅

𝑅𝑇 𝑐 ⋅ log Ox S 𝑛e 𝐹 𝑐Red S

(2.16)

0 where 𝑈eq is the standard electrode potential, which is the electrode potential at the unit concentrations.

Electrode

n e e−

Electrolyte

Red Ox

Red → Ox + nee−

Figure 2.12.: Oxidation half-reaction on an electrode. The potential difference of the solid-liquid half-cell is given by 𝑈 = 𝜙s − 𝜙𝑙 . In terms of the overpotential 𝜂 = 𝑈 − 𝑈eq = 𝜙s − 𝜙𝑙 − 𝑈eq xiii In

(2.17)

the text here forward for a reaction or current means the same as anodic, and backward means the same as cathodic according to Wang [253]. Another notation can be found in the literature, e.g., in Bagotsky’s book [248]: forward and backward denote there the partial reactions and the corresponding partial currents as in the Wang’s book [253]; anodic and cathodic describe, in contrast, net properties of the redox system (overpotential or net current) in the corresponding quadrants of the current-voltage plot.

2.3 Mass and charge transfer in silicon and electrolyte

31

and for the equilibrium case 𝑈 = 𝑈eq = 𝜙s eq − 𝜙𝑙 eq this means 𝜂 = 0. The absolute magnitude of 𝐼a and 𝐼c at 𝑈 = 𝑈eq defines the exchange current 𝐼0 = |𝐼a |𝑈 =𝑈 = |𝐼c |𝑈 =𝑈 for the electrode area 𝐴, and the exchange current eq

eq

density 𝑗0 = 𝐼0 /𝐴. The exchange current in the silicon anodization process is very sensitive to trace amounts of metals such as Cu or Au in the electrolyte, which can give change of 𝐼0 in order of 103 [4]. For low-doped p-type silicon with resistivity 1–10 Ω cm, 𝑗0 = 7×10−6 A/cm2 was reported [98].

In terms of the exchange current 𝐼0 , the constant 𝑎 in eq. (2.14) is given as: 𝑎 = 2.303 ⋅

𝑅𝑇 log 𝐼0 𝛼𝑛e 𝐹

(2.18)

The exchange current density 𝑗0 is proportional to the standard heterogeneous rate constant 𝑘° [cm/s] for the particular reaction of the reactants with the electrode material and the concentration of the species [253, 254]: 𝑗0 = 𝑛e 𝐹 𝑘°𝑐Ox V (1−α) 𝑐Red V α

(2.19)

where 𝑐Ox V and 𝑐Red V are the bulk concentration of the oxidized Ox and reduced Red forms of the redox couple. The formula is valid for first-order reactions. The charge transfer coefficient 𝛼 defines the symmetry of both forward and backward reactions, with typically assumed 𝛼 = 0.5 meaning symmetric curves as shown in the example in Fig. 2.11a. It is worth noting that in some literature, as for example in the Bagotsky’s book [248], the charge transfer coefficient is defined as the product of 𝛼 and the reaction valence, i.e., (𝛼𝑛e ), with the range from zero to 𝑛e . The Tafel equation can be also written in the following form [253]: 𝐼 = 𝐼0 ⋅ exp (±𝛼 ⋅

𝑛e 𝐹 𝜂 ) 𝑅𝑇

(2.20)

where the minus sign refers to the cathodic reaction (cathodic Tafel curve), and the plus sign to the anodic reaction (anodic Tafel curve). The Tafel equation is valid for relatively high overpotentials (𝜂 > 118 mV/𝑛e ), when the electrode reaction becomes irreversible, meaning that there is only forward or backward reaction on the electrode with the

32

2 State of the art

corresponding anodic or cathodic current depending on the sign of the applied overpotential (s. Fig. 2.11b) [253]. The Butler-Volmer equation is the generalized equation for polarization curve of a redox process in the charge transfer limited mode, that describes the current-voltage dependence also at low overpotentials and for both cathodic and anodic reactions [253]: 𝐼 = 𝐼0 [exp (𝛼 ⋅

𝑛e 𝐹 𝜂 𝑛 𝐹𝜂 ) − exp (−(1 − 𝛼) ⋅ e )] 𝑅𝑇 𝑅𝑇

(2.21)

In the region of low polarizations eq. (2.21) can be reduced to the linearized form [253]: 𝐼 = 𝐼0 ⋅

𝑛e 𝐹 𝜂 𝑅𝑇

(2.22)

The concentration dependent form of the Butler-Volmer equation takes into account concentration dependence of the exchange current density [254]: 𝐼 = 𝐼0 [

𝑐Red S 𝑛 𝐹𝜂 𝑐 𝑛 𝐹𝜂 ⋅ exp (𝛼 ⋅ e ) − Ox S ⋅ exp (−(1 − 𝛼) ⋅ e )] (2.23) 𝑐Red V 𝑅𝑇 𝑐Ox V 𝑅𝑇

Control by the mass transport of reactants - concentration polarization If the reactants or the products have small diffusion constants, or the concentration of the reactants in the electrolyte is low, or etch rate is high due to high applied current density, the electrochemical process can become controlled by the mass transportxiv of the products and reactants in the electrolyte. In case of pure diffusional concentration polarization, with increase of the applied potential, current approaches the limiting value 𝐼lim (s. Fig. 2.13, curve 2) [248]. This limiting diffusion current neither depends on the electrode potential nor on the rate of the electrode reaction, but on concentrations and diffusion constants of species, and geometry of the electrode. The polarization curve under mass transport control is strongly dependent on the scan rate at which it was measured. xiv As

described in sec. 2.3.2.3, only mass transport by diffusion and electromigration is considered in the whole work, neglecting convection.

33

2.3 Mass and charge transfer in silicon and electrolyte

I

1 2

Ilim 3

η

0

Figure 2.13.: Current-voltage curves for 1 - activation polarization, 2 diffusional concentration polarization, and 3 - combined polarization; figure adapted by author from Bagotsky [248]. Virtually every real electrochemical reaction of dissolution is the combination of at least two steps in series, which are (a) the chemical dissolution step defined by reaction kinetics and (b) the mass transport of the products and the reactants in the electrolyte [227]. This means mixed regime of activation polarization and concentration polarization, where the current-voltage curve cannot follow the exponential behavior of the Butler-Volmer and the Tafel equations and limits itself with 𝐼lim (s. Fig. 2.13, curve 3). The corresponding current densities of the single reaction steps of mass transport (diffusion) 𝑗d and reaction kinetics 𝑗k are contributing to the total current density 𝑗 according to the following equation [248]: 1 1 1 = + 𝑗 𝑗d 𝑗k

(2.24)

If the limiting diffusion current 𝐼 lim is known for electrochemical system with mixed polarization, the polarization curve for this system (s. Fig. 2.13, curve 3) can be described applying the activation polarization equations in the following form: 𝐼=

𝐼lim 𝐼act 𝐼lim = 𝐼lim + 𝐼act 1 + 𝐼𝐼lim

(2.25)

act

where 𝐼act is obtained according to the Tafel equation (2.14) or ButlerVolmer equation (2.21). In this case, explicit description of the concentration distribution according to the Nernst-Planck equation (2.12) is not necessary, which helps to reduce numerical calculations of electrochemical systems.

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2 State of the art

It is important to note that the described idealized polarization curves account only for the diffusional mass transport in electrolyte, neglecting movement of species due to migration and convection. For neglecting convection, this simplified case is achieved in analytical electrochemistry by ensuring steady conditions without steering. To neglect effect of migration of the reacting ions, a supporting electrolyte is added. Such a supporting electrolyte contains much more ions which participate in charge flow in the electrolyte but do not participate in the charge transfer at the interface electrode-electrolyte, i.e., do not participate in the electrode reactions. In real systems such simplification are not always possible, therefore more complex polarization curves are observed. Classification of electrochemical systems on current distribution Summarizing the theory on mass and charge transfer phenomena in the electrolyte, electrode, and their interface, for modeling of an electrochemical cell the following classification of current distributions in electrochemical systems is applied [219, 255, 256]: • Primary current distribution takes into account only current conduction according to the Ohm’s law in the electrolyte and in the electrode neglecting the mass transport phenomena and activation polarization at the interface electrode-electrolyte. Prerequisites are that the electrochemical reactions are so fast that the losses on reaction activation can be neglected (assuming activation overpotential 𝜂 = 0) and the electrolyte is homogenous meaning concentration gradients can be also neglected. Then: 𝑗𝑙 = −𝜎𝑙 ∇𝜙𝑙 ,

∇ ⋅ 𝑗𝑙 = −

𝜕𝜌q 𝜕𝑡

(2.26)

𝑗s = −𝜎s ∇𝜙s ,

∇ ⋅ 𝑗s = −

𝜕𝜌𝑞 𝜕𝑡

(2.27)

𝜂 = 0,

𝑈eq = 𝜙s − 𝜙𝑙

(2.28)

where subscript 𝑙 denotes the electrolyte (liquid) and s the electrode (solid), 𝜌𝑞 is the density of charge, and 𝑗 is the current density vector field. In other words, the distribution is primary, when the solution bulk resistance is much higher than the interfacial impedance.

2.4 Silicon anodization: current-voltage characteristics and …

35

• Secondary current distribution: in this case the mass transport is still neglected (homogenous electrolyte), but the interfacial overpotential is taken into account. Then, for the electrode and the electrolyte equations (2.26) and (2.27) are still valid, and the overpotential is described according to equations (2.17) and (2.23). Secondary distribution is observed when the interfacial impedance is determined by processes of charging or charge transfer, or when the overpotential of the interfacial processes and the ohmic voltage drop in the solution bulk are comparable. • Tertiary current distribution considers both interfacial overpotential at the interface solid-liquid as well as the mass transport phenomena in the bulk electrolyte according to the Nernst-Planck equation (2.12) and further equations discussed above in this section. 2.4. Silicon anodization: current-voltage characteristics and dissolution mechanisms In this section, the basic theory explained above will be applied to discussion of the current-voltage curve and dissolution mechanisms in the silicon anodization process. Silicon as a semiconductor shows rectifying behavior in any electrolyte [257]. However, the polarization curves for p-type and n-type silicon in aqueous HF electrolytes are more complex than the ones presented in Fig. 2.13. Typical polarization curves for p-type and n-type silicon in aqueous HF electrolytes are shown in Fig. 2.14. On these curves, distinct regions of various polarization types can be detected, as will be discussed in the subsections below. 2.4.1. Cathodic polarization and open circuit At negative potentials 𝑈 ≤ 𝑈OC the polarization curves for p-type silicon and for n-type silicon with sufficient illumination show to some extent basic similarities to normal Schottky diode behavior expected from a semiconductor/electrolyte interface [89]. When p-type silicon gets in contact with HF electrolyte, band bending occurs in silicon, with the space charge region of ionized acceptor atoms (negative) being formed in silicon near the

36

2 State of the art j, mA/cm2

n-Si 20 jPSL

4

j, mA/cm2

p-Si oscillations

jox 3

oscillations

20 jPSL

jox

2 1

forward bias

UOC

reverse bias

H2 -evolution

5

U, V

reverse bias

1 2

3

forward bias

UOC

UPSL

5 U, V

4 H2 -evolution

electropolishing porous silicon formation

electropolishing porous silicon formation

Figure 2.14.: Typical current-voltage relationships for n- and p-type silicon in contact with aqueous HF; 𝑈OC - open-circuit potential; 1 – no illumination; 2–4 – with illumination of intensity 𝐼v (𝐼v 4 > 𝐼v 3 > 𝐼v 2 ); SCE - saturated calomel electrode; reproduced from Föll et al. [22] © 2002 with permission from Elsevier, with minor changes. junction (depletion region), and the open-circuit potential 𝑈OC is formedxv . Under cathodic polarization (𝑈 < 𝑈OC ), the band bending increases. The depletion region increases in depth, and the Schottky diode remains reversebiased. Electrons generated in the conduction band (e.g., due to photoand thermogeneration) determine the reverse current. For n-type silicon, cathodic polarization provides forward bias for the Schottky diode, and exponential increase of current is observed. Both n- and p-type silicon remain mostly stable to the HF solution, however, there might be surface modifications (up to 0.1 µm in depth) due to the formation of silicon hydrides, especially at higher cathodic potentials, as shown in Fig. 2.15 [4,35]. At high cathodic overpotentials, the reduction of water at Si/HF interface with evolution of hydrogen gas also takes place [4, 77, 89]. When the potential is increased to the negative (vs. SCE) open-circuit value 𝑈OC , current decreases to zero, and chemical processes are in equilibrium.

xv In

the text, the term open-circuit potential 𝑈OC is used in the same sense as the equilibrium potential 𝑈eq introduced in sec. 2.3.3, which is true only for the electrodeelectrolyte systems where the open-circuit potential is at the equilibrium condition (reversible reactions), s. more in sec. 1.6 in [4], in sec. 8.3 in the Memming’s book [250], and in sec. 16.1 in Bagotsky’s book [248], where non-equilibrium open-circuit potential causes dissolution (corrosion) of metals.

2.4 Silicon anodization: current-voltage characteristics and … UOC H2 -evolution

hydride H H Si H Si H H Si H Si H H Si Si Si Si Si

UPSL SiO2 -formation

37

U

oxide

H Si

H H OH H OH O O Si O O Si Si Si Si Si Si Si O O SiO Si Si Si O Si Si

silicon

Figure 2.15.: Schematic representation of the surface condition of silicon in fluoride-containing aqueous electrolyte as a function of applied voltage; Si - silicon atom, H - hydrogen atom, O - oxygen atom; reproduced from Zhang [4] © 2001 with permission of Springer. 2.4.2. Pore formation 2.4.2.1. Current-voltage region Dissolution of silicon in aqueous HF electrolytes occurs only under anodic polarization [227].xvi For p-type silicon this means forward-biased Schottky contact. Illumination of the p-type silicon under anodic polarization does not influence much the I-V curve. For low- and moderately-doped n-type silicon illumination plays a significant role under anodic polarization. If the process runs without illumination, the I-V curve represents the reversedbiased Schottky contact, with breakdown taking place at a certain high potential. If illumination is applied to backside or frontside of the n-type silicon wafer, there is an additional photogenerated current, and for high illumination levels the curve follows the one for p-type silicon (s. Fig. 2.14). The fact that the dissolution of silicon is quite intense for p-type material where holes are majority carriers, whereas for n-type silicon the reaction is very slow without illumination or other hole generation mechanisms, is an indication that holes are required for silicon dissolution process. For low applied voltage, for both p-type and n-type silicon samples, and assuming high illumination intensity for n-Si, the polarization curve (s. Fig. 2.10 and Fig. 2.14) can be well described with the Tafel expression (2.15) or the Butler-Volmer equation (2.21) as explained in sec. 2.3.3.2 [79, 81, 89, 98]. xvi The

very slow process of chemical dissolution of silicon in HF electrolytes [75, 77, 258] is neglected here.

38

2 State of the art

This exponential increase of current ends with the first current peak, or porous silicon layer peak, at potential 𝑈PSL and current density 𝑗PSL (s. Fig. 2.14). The first current peak defines the end of the pure pore formation regime. 2.4.2.2. Reaction In the region 𝑈OC < 𝑈 < 𝑈PSL (which is also referred to as region 𝑗 < 𝑗PSL or Tafel region), direct dissolution of silicon occurs at the silicon surface representing the secondary anode, with consumption of one hole h+ and release of one electron e− per silicon atom, which gives the reaction valence (dissolution valence) 𝑛e = 2 [2]: Si + 4HF2 − + h+ ⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯⃯ SiF6 2− + 2HF + H2 + e− silicon hexafluoride

(2.29)

However, the actual measured effective dissolution valence in the pore formation mode is typically varying between 1.3 and 2.8 [4, 30, 71, 78, 259] (for example 𝑛e = 2.7 for n-macropores [77]), which means that other parallel reactions take place during the anodization. For example, Balagurov distinguishes in the anodization process between the electrochemical dissolution and pure chemical etching that can reduce the effective dissolution valence and is temperature-dependent [259]. Considering the reaction (2.29), one can see that supply of holes coming from the silicon substrate and F-ions in form of bifluoride anions HF2 − are required. Strong dependence of 𝑗PSL on HF concentration was shown by many researchers [12, 71, 76, 81, 85, 90, 94, 243]xvii . This indicated that the pore formation regime is favored when supply of holes from the silicon substrate is the limiting factor, and not the supply of F-ions from the electrolyte. Many mechanisms of silicon dissolution in the pore formation regime have been proposed. One of the most accepted theories is the Gerischer mechanism revised and extended by Kolasinski [247]. The silicon surface covered with hydrogen (Fig. 2.15) is very inert, and according to Gerischer/Kolasinski the substitution of hydrogen atom with fluorine occurs after a hole comes from the silicon bulk and is captured near the surface in Si-Si bond. After xvii More

on this is given in sec. 2.4.4.

2.4 Silicon anodization: current-voltage characteristics and …

39

some intermediate steps this leads to dissolution of a silicon atom leaving again hydrogen-terminated silicon surface. Dissolution of silicon atoms occurs selectively in this mode. This way, pores of various shapes are etched into silicon, and porous silicon is formed. The dark layer of porous silicon was at first considered to be amorphous redeposited silicon layer [71, 79]. Further studies with X-ray analysis showed that the skeleton of porous silicon remains mostly crystalline [260], although evidence of some amorphous material was also detected with Raman spectroscopy [261]. 2.4.2.3. Classification of pores Generally, pores are etch pits with their depth exceeding their width/diameter. There are different classifications of pores in porous silicon. The generally used classification is based on pore size (diameter) according to the IUPAC specifications for conventional porous materials. According to this classification, there are three types of pores: • micropores (pore diameter < 2 nm)xviii • mesopores (pore diameter from 2 nm to 50 nm) • macropores (pore diameter > 50 nm) Pore size, specific surface area, and pore size distribution can be evaluated using Brunauer-Emmett-Teller (BET) and the Barrett-Joyner-Halenda (BJH) theories from measured adsorption/desorption isotherms [262, 263]. The classification based on pore size takes into account only the size of pores, without considering their morphologies, which in case of porous silicon can vary significantly depending on process conditions, even within one sample. Two distinguished morphologies of porous silicon are straight cylindrical pores (mostly in macropore-size) with very large aspect ratios (s. Fig. 2.16a) and 3D sponge-like (mostly microporous or mesoporous) structures (s. Fig. 2.16b) [22, 77, 264]. Various intermediate cases possessing features of these two morphologies (e.g., macropores with extra branching) have been reported [2, 77]. Typically, micropores and mesopores of sponge-like morphology are obtained from p-Si, and straight macropores (with backside illumination) from n-Si. More exotic cases like macropores xviii Microporous

and mesoporous materials sometimes are also called nanoporous due to pore size in nanometer range.

40

2 State of the art

in p-Si (s. Fig. 2.16a) require special process conditions (e.g., organic electrolytes). (a)

(b)

Figure 2.16.: Two specific types of porous silicon morphologies: (a) p-Si macropores with straight cylindrical pores, reprinted from Föll et al. [22] © 2002 with permission from Elsevier; (b) p-Si micropores with sponge-like morphology, reprinted from Cullis et al. [264] © 1997 with permission from AIP Publishing LLC. There are also other classifications of pores in silicon based on process conditions (electrolyte, illumination, etc.), on crystallographic dependence, etc. [22, 77]. 2.4.2.4. Characterization of the pore formation process and the porous silicon The main parameter characterizing porous material is porosity 𝑃% in percent, which for porous silicon can be found with the gravimetric method according to the following equation: 𝑃% =

𝑚1 − 𝑚 2 ⋅ 100 𝑚1 − 𝑚 3

(2.30)

where 𝑚1 , 𝑚2 , and 𝑚3 are the mass of silicon sample before anodization, after anodization, and after removal of porous silicon, respectively. Porosity defines the effective refractive index of the porous silicon according to the effective-medium-approximation models, e.g., the Bruggeman’s model [68], which is applied in fabrication of porous silicon based photonic crystals consisting of multiple layers with modulated refractive index.

2.4 Silicon anodization: current-voltage characteristics and …

41

When the values 𝑚1 , 𝑚2 , and 𝑚3 are known, additionally, the growth rate 𝑅PS and the dissolution valence 𝑛e of the anodization process can be found: 𝑅PS =

𝑛e =

𝑚1 − 𝑚3 𝜌Si ⋅ 𝐴 ⋅ 𝑡etch

𝐼 ⋅ 𝑡etch 𝑀Si ⋅ 𝑒 (𝑚1 − 𝑚2 ) ⋅ 𝑁A

(2.31) (2.32)

where 𝐴 is the anodization area, 𝐼 is the current during the anodization of time 𝑡etch , 𝑒 is the elementary charge, 𝜌Si and 𝑀Si are the density and the molar mass of silicon, and 𝑁A is the Avogadro constant. The expression for the growth rate in this form is valid for the quasi-3D case, where the anodization process runs strictly in depth with flat etch front plane, meaning almost constant value of the anodization area during the process. The expression is applicable, therefore, for anodization of large area samples (with etch depth ≪ area sidelength), and not suitable for samples with the etch depth comparable to the area sidelength, where significant part of anodization process is running laterally, and thus the anodization area increases significantly during the process. The growth rate and the dissolution valence can be also found by etching porous silicon away and measuring the etch depth. This method, as well as the gravimetric method, is destructive, however, there are also non-destructive characterization methods. One such non-destructive optical technique – spectroscopic liquid infiltration method (SLIM) – was described by Segal et al. [265]. In this method, interferometric reflectance spectrum of the porous layer is measured for different filling liquids with known refractive indices. Then, the data are fitted to the Bruggeman’s effective-medium approximation. This way, porosity and thickness of the porous layer can be determined. 2.4.2.5. Pore formation mechanisms Two phases during pore formation can be distinguished: pore initiation (nucleation) and pore growth. Since the discovery of porous silicon formation in anodization process, different models for the two phases have been suggested. Although the proposed mechanisms could not succeed explaining all effects, they helped to understand influence of various process parameters and, to some extent, to predict the results.

42

2 State of the art

As was discussed in sec. 2.3.3, upon contact of electrolyte to silicon, space charge region depleted from free charge carriers forms in silicon. According to Theunissen [266] and, later, Beale et al. [87], the driving factor for pore initiation and growth is current focusing on surface inhomogeneities, where a breakdown voltage of the SCR is expected to be lower. Later, Searson et al. [267] have showed analytically that the depletion layer at the tip of an elliptical pore is indeed thinner. Thus, according to Beale et al., the dissolution process starts at surface defects. Once the first silicon atoms are removed, current focusing on the pore tips provides further growth of the pores in a positive feedback mechanism. SCR passivates already porosified material and defines distance between single pores, i.e., thickness of pore walls [83]. Zhang [82] showed that when the distance between two neighboring pores gets larger than the double thickness of SCR, then a new pore between the existing two should grow. Thus, porous silicon layer is fully depleted from free carriers, and the dissolution process continues only at the interface between bulk silicon and porous silicon, where holes coming from bulk silicon reach pore tips by tunneling or avalanche breakdown through the SCR [89, 268]. The effect of pore wall passivation by SCR matches well with experiments and explains very good growth of macropores in n-type silicon [59, 83, 269]. However, the mechanism of pore initiation by current focusing on surface defects was later questioned with the in-situ study by Allongue et al. of KOH etching with scanning tunneling microscope [154,270]. The authors have demonstrated that pore pits on silicon surface occur independently of atomic defects at the initial stages. On the other hand, the effect of current focusing is successfully applied for growth of regular arrays of macropores in n-type silicon, where controlled pore initiation is achieved with inverted four-sided pyramidal etch-pits formed with photolithography and anisotropic KOH etching [83, 271]. Smith et al. [89, 95] have proposed the diffusion-limited model of pore initiation and growth, where a hole in a process of stochastic random walk diffuses to the surface and initiates silicon atom dissolution. Next holes, with higher probability, will come to the formed pits and provide further pore growth. The Beale’s field effect model and the diffusion-limited model can explain both pore initiation and pore growth. Another proposed model for pore growth is based on quantum effects in nanometer-sized features of porous silicon [84,89,272,273]. It is known that small silicon crystallites have larger band gap than bulk silicon. This effect was used to explain strong photoluminescence in visible range in porous silicon, with nanocrystalline porous

2.4 Silicon anodization: current-voltage characteristics and …

43

material consequently called “nanowires” or “nanodots”. Due to difference in band gaps between porous silicon and bulk silicon, holes from bulk silicon are blocked from penetrating small crystallites of porous silicon, and no holes are generated in the pore wallsxix . This way, already formed nanoporous structures remain passivated during further anodization. The model originally was applied to explain formation of microporous silicon. However, Föll [272] applied the quantum model to explain growth of macropores in n-type silicon. According to Föll, in case of macropores of micrometer-size, where no quantum effects are expected, pore walls are covered with thin layer of micropores which passivate the macropore-walls. A model that attempts to explain all processes of the silicon anodization on the microscale (formation of all kinds of pores, and current oscillations in the electropolishing regime, s. sec. 2.4.3) was developed at the University of Kiel by Föll, Carstensen, Christophersen et al. [101,274–276]. The model named current burst model (CBM) assumes that all silicon dissolution or oxidation processes during anodization process occur inhomogeneously in time and space as current bursts (CB). Interaction between single CBs is described by surface passivation with hydrogen (in case of direct silicon dissolution - pore formation) or oxide (in case of electropolishing), which defines the probability of the next CB at a given location. Effect of SCR is also included in the model. Besides silicon, the model covers also anodization of III-V semiconductors. Martín-Palma et al. [277] have performed examination of porous silicon with high resolution electron microscope, and proposed another mechanism of pore growth. They observed that dissolution of silicon atoms during pore formation causes dislocations in the silicon matrix, which are more electronically active, thus the dissolution continues through these areas. The anodization process is not always a stable process. For example, for p-type silicon, micropores can occur first, and later develop into macropores [85, 278]. The question of process stability was studied by several research groups with help of etch front stability analysis. The stability analysis was initially applied for growth of a sphere precipitate particle in a uniformly saturated solution [279]. In 1990s the stability analysis approach was applied to silicon anodization process. Kang and Jorné [280] studied morphology of porous silicon with stability analysis according to the Mullins-Sekerka theory [279], taking into account diffusion of holes in silicon, diffusion of ions in electrolyte, and surface tension. They showed for xix The

effect is also called charge confinement or quantum-wire effect.

44

2 State of the art

n-type silicon, that porous silicon formation is favored when the process is controlled by the supply of holes from silicon bulk. Wehrspohn et al. [103] proposed an electrostatic model aimed on explanation of macropore formation in low-doped p-type silicon. There the role of difference between resistivity of silicon and of electrolyte on mechanism of pore formation was studied. It was shown that for low-resistive silicon (high and moderate ntype or p-type doping, with sufficient illumination for moderate n-type doping), micropores are formed substantially, and in opposite case (low-doped n-silicon and low-doped p-type silicon) macropores are observed. Additionally, in case silicon is more conductive than electrolyte, the growth front is more stable. In their further paper they extended the model and showed conditions for instability of the interface “bulk silicon - porous silicon” for p-type silicon of resistivities in the range 10−1 –104 Ω cm [278]. Stability analysis was also used to study how the process switches between pore formation (divalent etching) and electropolishing (tetravalent etching, s. sec. 2.4.3). Valance [94, 99] has further developed the model of Kang and Jorné [280], and showed that the electrolyte-silicon interface gets unstable when the dissolution current is below a critical value which depends linearly on the HF concentration. In their model, unstable electrolyte-silicon interface indicated pore formation and stable interface meant electropolishing. The stability criteria fits to some extent to the experimentally observed dependence of limiting current density 𝑗PSL on HF-concentration (s. Fig. 2.17). Rauscher and Spohn in their Laplacian growth model also showed that instability is maximal for small current densities (pore formation), and a transition to a stable interface at high current takes place [228, 281, 282]. 2.4.3. First and second plateaus of electropolishing At potentials larger than the first peak on the polarization curve (𝑈 > 𝑈PSL ), two regions can be distinguished - first and second plateaus of electropolishing, separated from each other by the second peak with current density 𝑗ox at potential of few volts vs. SCE (s. Fig. 2.14). The reactions in the first plateau of electropolishing are considered to be under mixed control conditions described with Koutecky-Levich equation (2.24) [257], and the dissolution reaction gradually changes from the divalent reaction of direct silicon dissolution (s. eq. (2.29)) to tetravalent reaction of silicon oxidation [2]: Si + 2H2 O + 4h+ → SiO2 + 4H+

(2.33)

2.4 Silicon anodization: current-voltage characteristics and …

45

The formed anodic oxide is etched in hydrofluoric acid [283]: SiO2 + 6HF → SiF6 2− + 2H2 O + 2H+

(2.34)

Thus the reaction of tetravalent silicon dissolution has a two-step mechanism. The oxide etching step (2.33) is considered to be the rate-determining step (RDS). The limitation of the oxide etching step comes not due to the reaction kinetics but due to diffusion-limited transport of fluoride species to the reaction site. This way, hillocks of the oxide are etched faster, and electropolishing of the surface occurs. The reaction of oxide dissolution shown above is only one possible reaction. Alternative reactions with other (minor) components of the aqueous HF solution, for example (HF)2 and HF2 − , were also proposed [2, 284]: SiO2 + 2HF2 − + 2HF → SiF6 2− + 2H2 O

(2.35)

SiO2 + 3HF2 − + H+ → SiF6 2− + 2H2 O

(2.36)

The reactions of oxide dissolution involving HF2 − are considered to be more likely because of higher reactivity of HF2 − in comparison to HF [247]. Good overview of various mechanisms of silicon oxide dissolution is given in the work of Mitra and Rimstidt [285]. The silicon surface at the first plateau of electropolishing is covered with porous “wet” hydroxylated oxide, and the surface remains rough on the nanometric scale [257, 286, 287]. In the second plateau of electropolishing, the reaction is nearly tetravalent, and the surface is covered with “dry” compact oxide which thickness increases with increase of potential [257,286]. At this conditions, electropolishing takes place in the diffusion-controlled regime, which leads to smoothing of the surface [71]. That electropolishing occurs in the plateau region of the anodization polarization curve, where current is assumed to be independent of voltage, is not unexpected, and is common for many other electropolishing processes [284]. The dissolution rate in this region may be expressed in terms of current according to the Faraday’s law of electrolysis, although the chemical step of oxide dissolution in HF is dependent on the kinetic constants. At higher potentials in the second plateau of electropolishing, additionally oscillations of current or voltage (depending on which

46

2 State of the art

parameter is fixed) are observed. Good explanation of these oscillations gives the current-burst model described above (s. sec. 2.4.2.5). At even higher potentials typically between 10 V and 20 V (not shown in Fig. 2.14), again strong increase of current is observed. In this region increased silicon dissolution occurs with formation of mesoporous silicon oxide (silica) layer [257, 288]. Further increase of potential leads to formation of macrostructures of silica and evolution of oxygen [289, 290]. 2.4.4. Position of the first and the second current peaks There is some controversy in the reviewed papers on the observations of the first and the second peaks on the polarization curves shown in Fig. 2.14. For example, Turner [71] observed only one peak dividing the process between pore formation and electropolishing. He found that the value of this limiting current density peak (named in the work as critical current density) is determined by the mass transport of HF to the reaction site. He showed that this critical current density for the horizontally arranged silicon electrodes was linearly dependent on HF concentration and temperature and inversely dependent on the fourth root of the electrolyte viscosity. Turner also reported that the role of electrolyte agitation due to natural convection and hydrogen bubbles has a significant influence on the value of the critical current density. For vertically arranged silicon electrodes he observed a less sharp peak indicating that the switching from pore formation to electropolishing at first took place only in some regions of the silicon electrode and then spread to its entire surface. Zhang et al. [76] also observed only one current peak on the current-voltage curves, after which pure electropolishing took place, meaning the second current peak 𝑗ox . However, they also showed a transition region below this peak, where the exponential Tafel part changes to a slower linear increase. Additionally, Zhang et al. reported linear dependence of the critical current density at this peak on HF bulk concentration, and showed that there is only a weak dependence of the critical current density on the doping type and level (s. Fig. 2.17). The linear dependence of the critical current densities on HF concentration was also observed by Eddowes [81], Searson and Macaulay [90], Barret et al. [12], and others. In contrast to the above references, non-linear dependencies of the critical current densities on HF concentration were also reported. The most cited

2.4 Silicon anodization: current-voltage characteristics and …

47

electropolishing

porous silicon formation

Figure 2.17.: Map of porous silicon formation and electropolishing modes for different values of current densities, HF concentrations in electrolyte, and doping level and type; the solid line shows 𝑗ox , the region between the solid and the dotted lines is the transition region between pore formation and electropolishing; reproduced from Zhang et al. [76] by permission of The Electrochemical Society. and applied is the dependence of order 1.5 found empirically for 𝑗PSL by Lehmann [85]: 𝑗PSL = 𝐶 𝑤HF 1.5 exp (−

𝐸a ) 𝑘b 𝑇

(2.37)

where 𝐶 = 3300 A/(cm2 m%1.5 ), activation energy 𝐸a = 0.345 eV, and 𝑤HF is the HF mass fraction in percent. The measured valence values indicated that switching of the process from pore formation to electropolishing occurred in a wide region close to 𝑗PSL (s. below in Fig. 2.18a, the switching appears to take place in a narrow region due to the logarithmic scale). For the value of the second current peak 𝑗ox , also different dependencies have been reported. For example, Eddowes [81] showed a second order dependence. Other orders of the dependence were summarized by Hassan et al. [291]. With addition of ethanol to the solution, remarkable change of the polarization curve with reduction almost to a half of current for both peaks and the

48

2 State of the art

first plateau was reported [292]. For the electrolyte containing 1:1 volume parts of 50 m% HF and ethanol, used in this work, Lehmann et al. [268] reported 𝑗PSL ≈ 600 mA/cm2. However, the fit function proposed by Lehmann in his book [2] (Fig. 4.6) shows only a valence of approximately 2.6 at 1000 mA/cm2. No values of 𝑗ox for the ethanoic electrolyte could be found in the literature. Hassan et al. [291], and van den Meerakker and Mellier [243] emphasized the mixed effect of diffusional and kinetic aspects of the anodization process. Hassan et al. showed that at low HF concentration of 0.03 M the process is kinetically controlled at all current densities up to the oscillations after the second electropolishing plateau. At higher concentrations, diffusion control starts playing an important role. To summarize, due to complexity of the anodization process, the peak current density values are not constant for a given electrolyte concentration, but depend on other parameters. The mass transport dependence means that geometrical factors (such as thickness of protective masking layer, anodization area dimensions, etc.) play an important role on the peak current density values, as known for diffusion-limited wet-etching processes [145]. Therefore, the values given in literature should be used with caution only as rough estimations. 2.5. Silicon anodization: influence of process parameters Any process or system can be characterized in terms of input and output signals, or as applied action to the system and resulting reaction of the system. For the process of silicon anodization, input signals are the process parameters, and the output is the resulting structure of the anodized silicon surface with corresponding characteristic properties. Main process parameters and characteristic properties of the resulting etch structures in the anodization process for pore formation and electropolishing regimes are summarized in Tab. 2.3.

2.5 Silicon anodization: influence of process parameters

49

Table 2.3.: Silicon anodization process parameters and resulting properties Process parameters (input)

Characteristic output properties

• current density 𝑗 (or – more difficult to control – potential difference 𝑈 )

• morphology (micro/meso/macro porous silicon, electropolished surface, or mixed)

• etch time 𝑡etch

• etch rate 𝑅etch

• illumination conditions

• dissolution valence 𝑛e

• electrolyte composition, HF concentration, viscosity, additional agitation

• porosity of porous silicon 𝑃%

• temperature 𝑇 • substrate doping type • substrate doping density 𝑁a or substrate resistivity 𝜌el

• quality of the electropolished surface or of the interface “porous silicon - bulk silicon”, e.g., in terms of average surface roughness 𝑅a or root-mean-square (RMS) roughness 𝑅q

• crystal orientation

Influences of the process parameters on the characteristic properties of the process are discussed in this section. The overview is emphasized on microporous and mesoporous sponge-like silicon, because macropores are less suitable for application of anodization as a structuring technique, and thus not studied in this work.xx The doping type in this overview is limited to p-type for the same reasons, since micropores and mesopores are generated more readily in p-type silicon. For p-type silicon, illumination conditions do not play significant role, and are therefore not discussed. xx Macroporous

silicon has a few specific applications as a structuring technique. Long vertical macropores can be used as is, for example, as channels with high aspect ratio in electrodes with high inner surface for capacitors or batteries [32]. Modulation of current during the process can result in quasi-3D structures used as photonic crystals [293]. Additionally, not only long channels, but also trenches can be formed with anodization process in macroporous regime [11, 39, 59]. This way, micromechanical structures for MEMS can be fabricated [294]. With increased current density, pore overlapping can be achieved, resulting in formation of nanowires [61]. Either way, the process with macropores is limited to these specific structures and is not applicable for fabrication of arbitrary 3D cavities, therefore it does not meet the goals of this work.

50

2 State of the art

For this section, data from different references were collected and compared to each other in frames of this work. If multiple papers from the same authors were available, cross-checking of the data was performed. Where necessary, conversion of units and derivation of additional parameters were performed. References to all datasets are provided. 2.5.1. Current density Current density is a very important parameter of the anodization process, because it defines the dissolution reaction pathway of the process (divalent pore formation vs. tetravalent electropolishing)xxi . Thus, current density defines the dissolution valence 𝑛e of the anodization process (s. Fig. 2.18a). Besides this fast change of the dissolution valence value at the critical current density, slow increase is observed in pore formation and electropolishing regimes. The effect of increasing dissolution valence with increasing current in pore formation regime is observed in both HF:water and HF:ethanol:water electrolytes (s. Fig. 2.18a,b). Data published in several papers suggest that the porosity of micro/mesoporous silicon in general increases with increase of current density [87, 260, 295–298], although a short reduction at low current densities is also detectable, with minimum at 10–30 mA/cm2 (s. datasets “0.1 Ω cm”, “1 Ω cm”, and “5 Ω cm” in Fig. 2.19a, and dataset “0.01 Ω cm” in Fig. 2.19b). According to Setzu et al. [299], the increasing part can be explained by considering electropolishing regime, where the porosity is 100 %. Then, by lowering the current, the process is moved away from the electropolishing regime, and the porosity decreases. On the microscale, assuming that the dissolution valence 𝑛e is known and the dissolution process is 100 % efficient, i.e., all charges are consumed for xxi To

be more exact, the reaction pathway is defined with the applied potential difference (s. Fig. 2.14). In galvanostatic power supply, when current density near the first or the second peaks on the polarization curve is set, there are two corresponding possible values of overpotential, one smaller and one larger than the value of overpotential at the peak. However, one can expect that the electrochemical system tends to the minimal energy, therefore, in this case the smaller value of the overpotential is established. In general, galvanostatic control is preferred over the potentiostatic control, because values of overpotential are often not well defined and documented in electrochemical experiments, e.g., because reference electrodes are not used, therefore it is more convenient to use current-control in the experiments and make comparison of process parameters vs. current density.

2.5 Silicon anodization: influence of process parameters (a) HF:water, 2.37m% HF

51

(b) HF:water:ethanol, 29.93 m% HF

j PSL

Figure 2.18.: (a) Dependence of the dissolution valence on applied anodic current density for a low-doped p-type and a strongly illuminated lowdoped n-type silicon samples anodized in water:HF solution (2.37 m% HF) at RT (reproduced from Lehmann [85] by permission of The Electrochemical Society); (b) p-type silicon of different resistivity anodized in water:HF:ethanol solution (29.93 m% HF, 1:1 parts by volume of 50 m% HF and ethanol) at RT in pore formation regime (based on data from Frohnhoff et al. [273] and Lehmann et al. [268]); here and further in the chapter, points of each dataset are connected with lines for clarity. the dissolution reactions, current density 𝑗 defines the silicon dissolution rate 𝑅Si according to the Faraday’s law of electrolysis: 𝑅Si =

𝑗 𝑀Si ⋅ 𝑛e 𝑒 𝜌Si 𝑁A

(2.38)

In the electropolishing regime, the dissolution of silicon runs through intermediate step of anodic oxidation. However, the reaction is considered to be fast enough, so that the main limitation is provided through the transport of the fluoride species to the oxide film. In turn, the transport of the ions in electrolyte is controlled by the applied electric field defining migration velocity. Therefore, the velocity of etch front in electropolishing regime 𝑅etch (neglecting extreme case of porous silica growth at high potentials) is assumed to be equal to this dissolution rate, and thus to the applied current density: 𝑅etch = 𝑅Si

(2.39)

52

2 State of the art

(a) HF:water, 30 m% HF 0.01 Ω cm 0.1 Ω cm 1 Ω cm 5 Ω cm 25 Ω cm

Porosity, %

60 55 50 45 40

70

Porosity, %

65

(b) HF:water:ethanol, 29.93 m% HF

60 50 0.01 Ω cm 0.2 Ω cm 8 Ω cm

40

35 30

30

0

50

100

Current density, mA/cm2

0

50

100

150

200

Current density, mA/cm2

Figure 2.19.: Porosity of microporous silicon as a function of current density for p-type silicon anodized in (a) water:HF solution of 30 m% HF (based on data from Beale et al. [87]) and (b) 1:1, HF(50 m%):ethanol electrolyte (based on data from Frohnhoff et al. [273]). In galvanostatic control, this means that for a defined charge flow, the electrochemical system is forced to provide enough ions to the reaction site by adjusting the electrical field. In case of pore formation, etch rate is defined with the movement velocity of the interface between bulk silicon and porous silicon. Due to selective etching of pores into silicon, the dissolution rate 𝑅Si defined with current density has to be corrected for the porosity 𝑃% of the formed porous layer: 𝑅etch =

𝑅Si ⋅ 100 𝑃%

(2.40)

i.e., a highly porous layer will grow slower than a less porous layer at the same dissolution rate. In eq. (2.40) it is assumed that the dissolution process runs always at the interface “porous silicon - bulk silicon”, that is confirmed with the fabrication of 1D photonic crystals, where porosity (and therefore the refractive index) at each moment of anodization, i.e., at each depth of the interface “porous silicon - bulk silicon”, is defined by the applied current density, and the already generated modulation of the porosity remains unchanged during further anodization.

53

2.5 Silicon anodization: influence of process parameters

Measured growth rates as a function of current density in HF:water and in HF:water:ethanol electrolytes reveal nearly linear dependence (s. Fig. 2.20).

(a) HF:water, 50 m% HF

(b) HF:water:ethanol, 29.93 m% HF 103

100 80 60 40

0.006 Ω cm 1.6 Ω cm

20 20

40

60

80

Current density, mA/cm2

100

Growth rate, nm/s

Growth rate, nm/s

120

102

101 0.005 Ω cm, meso-PS 1 Ω cm, micro-PS 100 0 10

101

102

103

Current density, mA/cm2

Figure 2.20.: Dependence of growth rate on current density for ptype silicon of different resistivity in (a) aqueous HF solution with 50 m% HF (based on data from Arita and Sunohara [295]) and (b) 1:1, HF(50 m%):ethanol electrolyte (based on data from Lehmann and Rönnebeck [86]). Surface quality after electropolishing or after removal of porous silicon (interface roughness) can be characterized in terms of surface roughness, which gives information about height variations, and fractal dimension, which characterizes also the surface morphologyxxii . Lérondel et al. [300] have found that the root-mean-square (RMS) roughness of the interface “porous silicon - bulk silicon” decreases with increase of current density (s. Fig. 2.21). Similar effect was observed by Ha et al. [23]. This increase can be explained from the point of electropolishing regime, where roughness decreases to atomic scale. The further the process is moved away from the electropolishing regime by decreasing current density, the rougher the etch front is achieved [299]. It is worth noting that the quality of the surface under porous silicon depends also on the process of porous silicon removal, as will be discussed in sec. 2.6.2. xxii More

information on surface characterization is given in sec. 3.2.3.

54

RMS roughness, nm

2 State of the art

10

PS thickness 2 µm PS thickness 10 µm

2

101

100

101

102

Current density, mA/cm2

Figure 2.21.: Influence of current density on interface surface roughness for 4–6 Ω cm p-type silicon with thickness of the removed porous silicon 2 µm and 10 µm, based on data from Lérondel et al. [300]. Electropolishing can provide surface of superior quality. Tjerkstra [47] has reported roughness of lower than ±5 nm (estimated from SEM micrographs) after electropolishing at potentials corresponding to the first and second plateaus in electrolytes with HF concentration in the range from 0.5 % to 10 % for lightly-doped (5–10 Ω cm) and heavily-doped (0.01–0.018 Ω cm) (100) p-type silicon. Comparable results have been reported by Rappich et al. for (111) n-type silicon substrates [301]. In contrast, Artmann and Frey have reported roughness of up to 200 nm for 15 Ω cm p-type silicon in 5 % HF electrolyte with ethanol and current density between 10 mA/cm2 and 75 mA/cm2 [10]. However, they used electropolishing to undercut a lateral oscillating structure of a gyroscope, thus additional roughness could be induced by the underetched elements. 2.5.2. Etch time In HF:water electrolytes, dissolution valence shows strong dependence on etch time (s. Fig. 2.22a), with almost no difference between the values of dissolution valence for a given etch time for various current densities in pore formation regime. As shown above, current density defines the growth rate (s. Fig. 2.20) and, consequently, the thickness of the porous layer. Since porous silicon layers of different thickness, but formed with the same etch time, reveal almost the same dissolution valence, there is no diffusion limitation induced by already formed porous layer in this case. Because almost

55

2.5 Silicon anodization: influence of process parameters

no dependence on current density is visible, dependence of the dissolution valence on etch time can be explained with chemical dissolution, running simultaneously with the electrochemical dissolution [75, 77, 258]. Then, less electrons are consumed per a dissolved silicon atom, and the dissolution valence decreases. Chemical dissolution was reported to be higher for low concentrated HF solutions [302–305], which is attributed to the increased OH− concentration (increased pH value) in the low concentrated HF solution [4]. In contrast, Hedrich et al. reported almost no effect of the etch time on the dissolution valence in HF:water:ethanol electrolyte (s. Fig. 2.22b) [24], thus it can be concluded, that there is no/little chemical etching running in the HF:water:ethanol electrolyte. However, this statement should be used with caution, because only one value of porosity, from which the dissolution valence is calculated, is provided for each current density in [24].xxiii

(a) HF:water, 30 m% HF 5 mA/cm2

2.20

10 mA/cm2

2.4

20 mA/cm2

2.3

50 mA/cm2

2.2 2.1 2.0 1.9 102

Dissolution valence

2.5

Dissolution valence

(b) HF:water:ethanol, 29.93 m% HF

2.15

2.10

2.05 103

Etch time, s

104

20 mA/cm2

35 mA/cm2

25 mA/cm

2

40 mA/cm2

30 mA/cm

2

1000

1500

Figure 2.22.: Dissolution valence as a function of etch doped p-type silicon in pore formation regime with 1.5 Ω cm in 30 m% HF electrolyte (HF:water) (based Unno et al. [302]); (b) resistivity 2–9 Ω cm in 29.93 m% (1:1 of HF(50 m%):ethanol) (based on data from Hedrich

xxiii There

2000

Etch time, s

time for low(a) resistivity on data from HF electrolyte et al. [24]).

are real data on thickness and etch time for each data point, though. The calculation of the dissolution valence was done according to eq. (A.60) in Appendix.

56

2 State of the art

Porosity was reported to depend strongly on etch time in HF:water electrolytes, as shown in Fig. 2.23a for 30 m% HF. The effect is more pronounced for higher current densities. The dependence versus porous layer thickness is similar for different current densities (s. Fig. 2.23b). Increase of porosity with duration of contact of porous silicon with HF electrolyte can be also explained with chemical dissolution. Halimaoui [303] showed that keeping an already formed 1 µm thick porous silicon layer in ethanoic solution of 5 % HF without electrical supply resulted in increase of porosity from 51 % to over 95 % after 30 minutes. Similarly, Hamm et al. [305] have studied dependence of porosity of porous silicon of p-type samples with resistivity 10–20 Ω cm on etch time till 200 minutes in different mixtures of HF(47 m%) and ethanol at 2 mA/cm2 . They found an increase of porosity with etch time for high concentrated electrolytes (28 m% HF and 35 m% HF), and a decrease of porosity with etch time for lower concentrated electrolytes (14 m% HF and 22 m% HF). In contrast, for 29.93 m% HF electrolyte (1:1 of HF(50 m%):ethanol), Hedrich et al. [24] reported that porosity is independent of the porous layer thickness in the range up to about 66 µm. (a)

(b)

66

66

Porosity, %

Porosity, %

64 62 60 58 56

5 mA/cm

54

10 mA/cm2

102

2

20 mA/cm

103

Etch time, s

64 62

5 mA/cm2 10 mA/cm2

60

20 mA/cm2

2

50 mA/cm2 104

50 mA/cm2

58 5

10

15

20

Thickness, µm

Figure 2.23.: Porosity vs. (a) etch time and (b) porous layer thickness for low-doped p-type silicon with resistivity 1.5 Ω cm in 30 m% HF electrolyte (HF:water) (based on data from Unno et al. [302]). Assuming constant porosity, the growth rate of porous silicon does not depend on the etch time for a given current density, in both HF:water and

57

2.5 Silicon anodization: influence of process parameters

HF:water:ethanol electrolytes, as demonstrated by Unno et al. [302] for current densities in the range 10–200 mA/cm2 for thicknesses up to about 16 µm (Fig. 2.24a) and by Hedrich et al. [24] for current densities in the range 20–40 mA/cm2 for thicknesses up to about 66 µm (s. Fig. 2.24b). However, mass transport limitation can be expected for higher current densities and thicker porous layers, which can change local HF concentration near the interface between bulk silicon and porous silicon. As a consequence, porosity and growth rate might change in depth. According to Korotcenkov and Cho [77], for very thick porous layers (150 µm), 20 % difference in HF concentration between that at the etch front and that in the bulk solution was reported.

(a) HF:water, 30 m% HF

(b) HF:water:ethanol, 29.93 m% HF 35

10 mA/cm2 20 mA/cm2 50 mA/cm2

100

100 mA/cm2 200 mA/cm2 50

0 101

102

103

Etch time, s

104

Growth rate, nm/s

Growth rate, nm/s

150

30 25 20 15

20 mA/cm2

35 mA/cm2

10

25 mA/cm2

40 mA/cm2

5

30 mA/cm2 1000

1500

2000

Etch time, s

Figure 2.24.: Growth rate vs. etch time for low-doped p-type silicon with (a) resistivity 1.5 Ω cm in 30 m% HF electrolyte (HF:water) (based on data from Unno et al. [302]); (b) resistivity 2–9 Ω cm in 29.93 m% HF electrolyte (1:1 HF(50 m%):ethanol) (based on data from Hedrich et al. [24]). Lérondel et al. [300] studied roughness of the interface between bulk silicon and porous silicon, and reported increase of RMS roughness with thickness for p-type silicon samples anodized in HF:water:ethanol electrolyte (s. Fig. 2.25). The increase is more pronounced for lower current densities (s. also Fig. 2.21): for 20 mA/cm2, an increase from about 1 nm up to about 52 nm was measured for the thickness of about 60 µm. For thicker porous layers, roughness comes to saturated values. These results match those reported by Setzu et al. [299].

58

2 State of the art

RMS roughness, nm

4−6 Ω cm, 20 mA/cm2 4−6 Ω cm, 111 mA/cm2 0.1 Ω cm, 40 mA/cm2 10

0.01 Ω cm, 80 mA/cm2

1

100 102

103

104

105

Thickness, nm

Figure 2.25.: Dependence of the root mean square roughness of the interface between bulk silicon and porous silicon on thickness of porous layer for p-type silicon samples anodized in HF:water:ethanol electrolyte (based on data from Lérondel et al. [300]). 2.5.3. Electrolyte mixture and concentration As shown in Fig. 2.17, HF concentration in electrolyte defines the critical current densities between pore formation and electropolishing regimes [89]. Therefore it influences the corresponding dependence of the dissolution valence on the current density (s. Fig. 2.18a). The effect of HF concentration on the dissolution valence for p-type silicon in pore formation regime is shown in Fig. 2.26. The effect is more pronounced for lower-doped silicon. Similar dependence was reported by Kordas et al. [236] and Unno et al. [302]. In electropolishing regime, the dissolution valence is reported to have no dependence on electrolyte concentration [2]. Porosity of the micro/meso porous layer decreases with increase of HF concentration for highly-doped p-type silicon anodized in HF:water and HF:water:ethanol electrolytes as shown in Fig. 2.27. According to Kordas et al. [236], with decrease of HF concentration, the porosity is raised until there is enough HF serving F− -ions in the electrolyte in the pores providing charge carriers. Therefore, at lower HF concentration higher porosity is observed. Similar dependence was also reported in HF:water electrolyte for low-doped silicon (s. Fig. 2.28, datasets from Beale et al. and Unno et al.). However, at low concentrations below 12 m% HF, Iraji zad et al. [306] have measured increase of porosity with increase of current density (s. Fig. 2.28).

59

2.5 Silicon anodization: influence of process parameters (a) 0.01 Ω cm

(b) 25 Ω cm 2.5

10 mA/cm2

Dissolution valence

Dissolution valence

2.60 2.55 2.50 2.45 10 mA/cm2 2.40

30 mA/cm2

2.35

100 mA/cm2 20

30

40

30 mA/cm2

2.4

100 mA/cm2

2.3 2.2 2.1

50

20

HF mass fraction, m%

30

40

50

HF mass fraction, m%

Figure 2.26.: Dissolution valence vs. HF mass fraction at various applied current densities for p-type silicon of resistivity (a) 0.01 Ω cm and (b) 25 Ω cm in pore formation regime, based on data from Beale et al. [87].

(a) HF:water, 0.01 Ω cm 80

10 mA/cm2

60

30 mA/cm

2

50 mA/cm

2

100 mA/cm

10 mA/cm2

80

2

50 40

Porosity, %

70

Porosity, %

(b) HF:water:ethanol, 0.01 Ω cm 80 mA/cm2

70

240 mA/cm2

60 50 40

30

30

20 10

20

30

40

HF mass fraction, m%

50

10

20

30

HF mass fraction, m%

Figure 2.27.: Porosity of microporous silicon as a function of HF concentration for p-type silicon of resistivity 0.01 Ω cm anodized in (a) HF:water electrolyte (datasets for 10 mA/cm2 , 30 mA/cm2 , and 100 mA/cm2 are from Beale et al., and the dataset for 50 mA/cm2 is from Kordas et al. [87, 236]) and in (b) HF:water:ethanol electrolyte (based on data from Herino et al. [296]).

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10 mA/cm2 , Beale et al.

70

10 mA/cm2 , Unno et al.

Porosity, %

60

30 mA/cm2 , Beale et al.

50

32 mA/cm2 , Iraji zad et al.

40

100 mA/cm2 , Beale et al.

30

100 mA/cm2 , Unno et al.

20

HF:water, 0.4–2 Ω cm

10 10

20

30

40

50

HF mass fraction, m%

Figure 2.28.: Porosity of microporous silicon as a function of HF concentration in HF:water electrolyte for p-type silicon of resistivity 0.4–2 Ω cm (dataset from Iraji zad et al. [306]) and 1–1.5 Ω cm (datasets from Beale et al. and Unno et al. [87, 302]). For a given HF concentration in the range 10 M < 𝑐mol HF < 29 M, no significant effect of ethanol concentration in the electrolyte on porosity has been found [236]. Decrease of porosity with increase of HF concentration can explain observed increase of growth rate with increase of HF concentration (s. Fig. 2.29). There is almost no difference between the growth rate for low- and highlydoped p-type silicon at low current densities. With increase of current density, the curves get steeper. This effect is more pronounced for highly-doped silicon. 2.5.4. Temperature For p-type silicon, Turner [71] reported linear dependence of the critical current density for electropolishing on temperature in 5 m% HF, with slope change at ca. 30 °C (s. Fig. 2.30a). According to Turner, electropolishing is favored when a viscous layer near the reaction site causes mass transport limitation. Therefore, the critical current density is dependent on both the diffusion coefficient and viscosity of the HF electrolyte, and both parameters are temperature-dependent, resulting in the temperature dependence of the critical current density. The change of the slope at ca. 30 °C is due to start of convection stirring caused by thermal effects. Temperature dependence is

Growth rate, nm/s

2.5 Silicon anodization: influence of process parameters

120

0.006 Ω cm, 5 mA/cm2

100

0.006 Ω cm, 30 mA/cm2

61

0.006 Ω cm, 100 mA/cm2

80

25 Ω cm, 10 mA/cm2

60

25 Ω cm, 30 mA/cm2

40

25 Ω cm, 100 mA/cm2

20 0 10

20

30

40

50

HF mass fraction, m%

Figure 2.29.: Dependence of growth rate of porous layer on HF concentration in HF:water electrolyte for p-type silicon of resistivities 0.006 Ω cm (based on data from Arita and Sunohara [295]) and 25 Ω cm (based on data from Beale et al. [87]). also included in the Lehmann’s empirical dependence of the critical current density on HF concentration (s. eq. (2.37)). Balagurov et al. [259] have shown that increase of temperature in the range from about 18 °C to 65 °C decreases the dissolution valence from 2.23 to 1.58 for p-type silicon of resistivity 10 Ω cm anodized in 5:1 parts per volume of 41 % HF and 95 % ethanol at 10 mA/cm2 (s. Fig. 2.30b). The etch time was 8.5 minutes for all data points. The authors demonstrated that the dependence was due to temperature-dependent chemical dissolution of silicon which is significant at elevated temperatures even in this rather highconcentrated electrolyte: chemical dissolution rate increases exponentially from 4.71×10−5 nm/s at 18 °C to 9.62×10−4 nm/s at 65 °C. For temperatures in the range from −35 °C to 37 °C, Setzu et al. [299] and Blackwood et al. [307] reported decrease of porosity with increase of temperature (s. Fig. 2.31). According to Setzu et al. [299], this temperature dependence of porosity can be explained by considering the temperature dependence of the critical current density as was shown by Turner (s. Fig. 2.30a): with increase of temperature the critical current density increases, therefore by increasing the temperature at a given applied current density the process is moved away from the electropolishing regime (where porosity is 100 %), so the porosity decreases (s. also Fig. 2.19).

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(a)

(b)

j= 5.3 Tsample – 83

j= 2 Tsample + 16

Figure 2.30.: (a) Critical current density for electropolishing vs. temperature for 3.5 Ω cm p-type silicon, where the silicon sample was heated and its temperature measured, and the temperature of the solution was near room temperature or near the temperature of the sample (reproduced from Turner [71] by permission of The Electrochemical Society); (b) dissolution valence as a function of temperature in pore formation regime for p-type silicon of resistivity 10 Ω cm anodized for 8.5 minutes in 5:1 parts per volume of 41 % HF and 95 % ethanol at 10 mA/cm2 (based on data from Balagurov et al. [259]). For higher temperatures in the range from 18 °C to 65 °C, Balagurov et al. [259] have shown increase of porosity with temperature (s. Fig. 2.31, dataset “10 mA/cm2, 32 m% HF, Balagurov et al.”). The effect was explained again with chemical dissolution, as for the dissolution dependence in Fig. 2.30b. Slow increase of the growth rate with temperature was reported by Arita and Sunohara [295] for highly-doped p-type silicon anodized in HF:water electrolyte at temperatures in the range from about 4 °C to 66 °C and 100 mA/cm2 (s. Fig. 2.32a). Setzu et al. [299] showed stronger increase of the growth rate for low-doped p-type silicon anodized in HF:water:ethanol electrolyte at temperatures in the range from −35 °C to 25 °C for higher current densities (s. Fig. 2.32a). Setzu et al. [299] measured surface roughness of the interface between the porous silicon and bulk silicon for different temperatures and current densities. They showed that for thicknesses of the porous layer in the satu-

2.5 Silicon anodization: influence of process parameters 100

10 mA/cm2 , 32 m% HF, Balagurov et al. 16.6 mA/cm2 , 39 m% HF, Setzu et al.

90

Porosity, %

63

20 mA/cm2 , 30 m% HF, Blackwood et al.

80

166 mA/cm2 , 39 m% HF, Setzu et al. 333 mA/cm2 , 39 m% HF, Setzu et al.

70 60 50 −40

−20

0

20

40

60

Temperature, ◦ C

Figure 2.31.: Porosity vs. temperature for low-doped p-type silicon in HF:water:ethanol electrolytes (5 Ω cm in 29.93 m% HF, 8–9 Ω cm in 39 m% HF and 10 Ω cm in 32 m% HF) for different current densities (based on data from Balagurov et al., Setzu et al., and Blackwood and Zhang [259, 299, 307]). ration range (more than few micrometers, s. Fig. 2.25 in sec. 2.5.2), roughness increases with increase of temperature from −35 °C to 25 °C, with the effect being more pronounced for smaller current densities (s. Fig. 2.32b). Setzu et al. explained this dependence in the same way as for porosity by considering electropolishing regime and the temperature dependence of the critical current density from Turner (s. Fig. 2.30a): in the electropolishing regime, surface roughness decreases to atomic scale; therefore, when the process is moved further away from electropolishing regime at constant current by increasing temperature, the roughness increases. Wehrspohn et al. [103] proposed that additional temperature dependent effect on morphology of porous silicon and growth front can be due to temperature dependent resistivities of silicon and electrolyte. In case resistivity of electrolyte is higher than that of p-type silicon substrate, there is stabilizing mechanism of nanopore formation. In the opposite case, unstable macropore formation process is observed.

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(a)

(b) 0.006 Ω cm, 100

120

mA/cm2

Average roughness, nm

8−9 Ω cm, 16.6 mA/cm2 8−9 Ω cm, 166 mA/cm2

Growth rate, nm/s

8−9 Ω cm, 333 mA/cm2

300 200 100

100 80

16.6 mA/cm2 166 mA/cm2 333 mA/cm2

60 40 20

0 −20

0

20

40

Temperature, ◦ C

60

−30 −20 −10

0

10

20

30

Temperature, ◦ C

Figure 2.32.: (a) Growth rate as a function of temperature for p-type silicon of resistivity 0.006 Ω cm in 50 m% HF (HF:water electrolyte) (based on data from Arita and Sunohara [295]) and p-type silicon of resistivity 8–9 Ω cm in 38.84 m% HF (HF:water:ethanol electrolyte) (based on data from Setzu et al. [299]); (b) average roughness of the interface between the porous silicon and bulk silicon vs. temperature for p-type silicon of resistivity 8–9 Ω cm in 38.84 m% HF (HF:water:ethanol electrolyte) (based on data from Setzu et al. [299]). 2.5.5. Substrate doping density Dissolution valence increases with increase of doping densityxxiv in pore formation regime in HF:water electrolyte (Fig. 2.33a). In HF:water:ethanol electrolyte, similar curves, although, slightly lower, have been reported (Fig. 2.33b). In electropolishing regime, dissolution valence does not depend on doping density [2, 91]. Growth rate decreases with decrease of doping density in pore formation regime in HF:water electrolyte in the range of high doping density (s. Fig. 2.34a). At low doping densities, this decrease slows down or even switches to increase. In contrast, in HF:water:ethanol electrolyte, increasing growth rates for low resistivities and decreasing growth rate for higher resistivities have been reported by Lehmann et al. [268], with maximum in the range 0.01–0.1 Ω cm (s. Fig. 2.34b). The data from Berger et al. in [258] also follow the decreasing trend in Fig. 2.34b. xxiv Conversions

between resistivity and doping density, considering doping density dependent mobility, is performed with data from Sze [106].

65

2.5 Silicon anodization: influence of process parameters

(a) HF:water, 48–50 m% HF

3.2

10

mA/cm2 ,

30

mA/cm2 ,

(b) HF:water:ethanol, 29.93 m% HF 48 m%

3.2

3 mA/cm2

48 m%

3.0

30 mA/cm2

30 mA/cm2 , 50 m% 100

3.0

mA/cm2 ,

48 m%

150 mA/cm2 , 50 m%

2.8 2.6 2.4

Dissolution valence

Dissolution valence

3.4

32 mA/cm2

2.8

97 mA/cm2

2.6

300 mA/cm2

2.4 2.2 2.0

10−2

10−1

100

Resistivity, Ω cm

101

10−3

10−2

10−1

100

101

Resistivity, Ω cm

Figure 2.33.: Dissolution valence as a function of substrate resistivity for p-type silicon anodized in (a) HF:water electrolyte (the datasets for 48 m% HF are from Beale et al. [87], the datasets for 50 m% HF are from Arita and Sunohara [295]) and (b) 1:1 volume ratio of HF(50 m%) and ethanol (the datasets for 3 mA/cm2, 30 mA/cm2, and 300 mA/cm2 are from Lehmann et al. [268], the datasets for 32 mA/cm2, and 97 mA/cm2 are from Frohnhoff et al. [273]). Dependence of growth rate on doping density can explain waviness (also called macroroughness) in form of concentric rings with a spacing in the order of millimeters, often observed after removal of porous layer: during the crystal growth process, striations (resistivity oscillations in silicon substrates) are introduced into silicon crystal [300]. According to Zhang [308], doping density of p-type silicon influences strongly the size and shape of pores. The porous silicon formed in moderately-doped p-type silicon (1015 –1018 cm−3 ) has very small pores from 1 nm to 10 nm. In heavily-doped p-type silicon (> 1019 cm−3 ) pores are typically 10 nm to 100 nm wide. In low-doped p-type silicon (< 1015 cm−3 ), depending on electrolyte, macropores of diameters in the order of micrometers or micropores with diameters in the order of nanometers are formed. Porosity for p-type silicon in HF:water electrolyte of 48–50 m% HF increases with increase of resistivity up to about 1 Ω cm, and then comes to saturation or starts to decrease (s. Fig. 2.35a). In contrast, in the 1:1 mixture of HF(50 m%):ethanol, porosity decreases with increasing resistivity in the

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(a) HF:water, 48–50 m% HF

(b) HF:water:ethanol, 29.93 m% HF 2

3 mA/cm2

30 mA/cm2 , 48 m%

102

30 mA/cm2 , 50 m% 100 mA/cm2 , 48 m% 150 mA/cm2 , 50 m%

101

10−2 10−1 100 101

Resistivity, Ω cm

Growth rate, nm/s

Growth rate, nm/s

10 mA/cm , 48 m%

9 mA/cm2

102

30 mA/cm2 31 mA/cm2 63 mA/cm2 300 mA/cm2

10

1

100 −3 −2 −1 0 10 10 10 10 101

Resistivity, Ω cm

Figure 2.34.: Growth rate as a function of substrate resistivity for ptype silicon anodized in (a) HF:water electrolyte (the datasets for 48 m% HF are from Beale et al. [87], the datasets for 50 m% HF are from Arita and Sunohara [295]) and (b) 1:1 volume ratio of HF(50 m%) and ethanol (the datasets for 3 mA/cm2, 30 mA/cm2, and 300 mA/cm2 are from Lehmann et al. [268], the datasets for 9 mA/cm2, 31 mA/cm2, and 63 mA/cm2 are from Berger et al. in [258]). range 10−3 –10−2 Ω cm, and then increases again (s. Fig. 2.35b), as also confirmed with the datasets for 31 mA/cm2 and 63 mA/cm2 from Berger et al. According to Lérondel et al. [2, 300], the roughness frequency shows no strong dependence on doping density. The amplitude of surface height and, therefore, roughness increase from few nanometers for degenerately-doped material to several tens of nanometers for low-doped silicon (s. Fig. 2.25). However, the current density for these data was changed with doping density (higher-doped samples anodized at higher current density), therefore the effect of doping density on roughness cannot be separated from the effect of current density (s. Fig. 2.21), where decrease of roughness is observed with increase of current density. Wehrspohn et al. [103] proposed with their stability analysis of pore formation mechanism, that any source of noise, such as poor stirring in electrolyte near the bottoms of the pores, ionized impurities in the space charge region, etc., is more effectively damped for highly conductive substrates, resulting in low interface roughness.

2.5 Silicon anodization: influence of process parameters

(a) HF:water, 48–50 m% HF 70

(b) HF:water:ethanol, 29.93 m% HF 100

2

10 mA/cm , 48 m% 30 mA/cm2 , 48 m%

3 mA/cm2

90

30 mA/cm2

30 mA/cm2 , 50 m% 100 mA/cm2 , 48 m%

50

150 mA/cm2 , 50 m%

40

Porosity, %

Porosity, %

60

67

31 mA/cm2

80

63 mA/cm2 300 mA/cm2

70 60 50

30

40

20 10

−2

10

−1

10

0

10

1

Resistivity, Ω cm

10−3 10−2 10−1 100 101

Resistivity, Ω cm

Figure 2.35.: Porosity as a function of substrate resistivity for p-type silicon anodized in (a) HF:water electrolyte (the datasets for 48 m% HF are from Beale et al. [87], the datasets for 50 m% HF are from Arita and Sunohara [295]) and (b) 1:1 volume ratio of HF(50 m%) and ethanol (the datasets for 3 mA/cm2, 30 mA/cm2, and 300 mA/cm2 are from Lehmann et al. [268], the datasets for 31 mA/cm2 are from Berger et al. in [258]). 2.5.6. Crystal orientation Lehmann [85] showed that the critical current density for both (110) and (111) oriented samples was up to 30 % less than for (100) oriented samples, depending on the HF concentration. Christophersen et al. [309] reported almost no dependence of the dissolution valence for p-Si samples on the studied orientations, namely, (100), (5 1 1), (5 5 12), and (111). Beale et al. [87] reported that growth of individual pores is determined by stochastic nature of the initiation of protrusions, i.e., it is of a random nature and, therefore, not crystal orientation dependent. The overall trend in the growth direction is determined by the general direction of current flow. According to Lehmann [2], effect of crystal orientation on the growth rate of porous silicon 𝑅PS is not observed for microporous silicon layers etched into

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a moderately-doped p-type substrate. Guendouz et al. [310,311] observed a certain dependence for highly-doped silicon, with the growth rate for ⟨100⟩ direction being higher than for ⟨111⟩ (s. Fig. 2.36a). The orientation dependence results in more flat walls of the shape of the porous volume for (100) oriented substrates anodized through a mask, and with sufficient anodization duration a sharp bottom is evolved (Fig. 2.36b). Guendouz et al. named this kind of shape a “pseudo V-shape”. Tjerkstra [47] also reported on similar kind of shapes in (100) p-type silicon substrates of resistivities 0.01–0.018 Ω cm and 5–10 Ω cm anodized in electropolishing regime through a mask in electrolytes with HF concentrations in the range from 0.5 % to 10 %. He showed that the effect is more pronounced for anodization potentials corresponding to the first plateau of electropolishing than for the second plateau. In contrast, Baranov et al. [70] reported on isotropic, i.e., independent of the crystallographic orientation, electropolishing of silicon. (a)

Growth rate, nm/s

30

(b) p (100) p (111)

4 µm

1 min

8 µm

4 min

25 20

10 min

15 10 5 20

40

60

Current density, mA/cm2

Figure 2.36.: (a) Orientation dependence of growth rate for p-type silicon of resistivity 0.007–0.008 Ω cm anodized in HF:water:ethanol mixture with HF concentration 14.78 m%; (b) resulting etch form evolution for n-Si, with a “pseudo V-shape” formed at longer etching time (for p-Si, the result is to some extent similar); reproduced from Guendouz et al. [310] © 2000 with permission from Elsevier.

2.6 Application of silicon anodization for structuring

69

2.5.7. Conclusions In this section, general dependencies of anodization parameters on process conditions have been reviewed. By comparing results from various studies, some differences could be observed. This means, the process of silicon anodization is so sensitive, that even when considering same process conditions, some deviations exist between the results reported from different research groups. Therefore, not only the process conditions, such as current density and electrolyte concentration, but also general process flow can influence results, as, for example, influence of porosity on how quick a sample is taken from anodization cell after anodization (increase of porosity due to chemical dissolution). Therefore, establishing a reliable process with expected results cannot be done by only relying on the literature, and requires preliminary experiments with sweeping the main parameters, in order to get the reference data for own experiments. This was also one of the reasons to perform systematic study of substrate resistivity and current density dependencies on porosity, growth rate, and dissolution valence of p-type silicon in pore formation regime in this work (s. chapter 4) in order to have reliable data for the model development in chapter 5. 2.6. Application of silicon anodization for structuring 2.6.1. Fabrication aspects Working with HF requires strict following the safety precautions, since even minor contact with high concentrated HF can be lethal for human [2]. Chemically resistant safety gloves, full face protection, and, preferably, overall coat represent a typical “dress-code” for working with concentrated HF (s. Fig. 2.37). Additionally, good ventilation in the work place is prerequisite for handling HF. Environmental aspects for disposal of used electrolytes have to be also strictly fulfilled. Additional safety concerns arise from using high currents during the process, since even for a relatively small current density of 100 mA/cm2, the total current for a full area of a 4-inch wafer is about 6 A. Application of ethanol requires additional precautions against electric spark formation in order to avoid fire. On industrial scale, to improve safety and process reproducibility, fully-automated systems are introduced [312].

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Figure 2.37.: “Dress-code” for working with concentrated HF (photograph B. Müller, model A. Ivanov). Anodization process is conducted in HF-stable electrochemical cell typically fabricated from PTFE [75]. An example of double tank cell with vertical placement of a silicon wafer is shown in Fig. 2.38, where the silicon wafer separates the two tanks (one with anode and another with cathode) of the cell. There are also setups with horizontal placement of the wafer. Optionally, circulation of electrolyte with cooling/heating might be integrated into the setup. At least two electrodes must be present in the cell. As was explained above, the anodization process runs on the side of silicon substrate that is facing the cathode. This side of the substrate – frontside – plays the role of secondary anode, therefore the name anodization is used for the process. Electrical supply from anode to the opposite side of the substrate (backside) can be either also through electrolyte (“wet” backside contact), as shown in Fig. 2.38, or directly to the wafer (“dry” backside contact). For low-doped wafers, a highly-doped surface layer formed by ion implantation of boron (for p-type) or phosphorus (for n-type) is necessary in order to provide good uniform electrical contact of the substrate backside to the electrolyte or metal electrode. Electrodes immersed in HF are typically made of platinum (stable in HF), or silicon (sacrificial, consumed) for metal-free electrochemical cells. In case of high resistive n-type silicon, illumination of back- or frontside of the substrate is required in order to generate holes for the process. For this purpose, PMMA for low concentrated HF-electrolytes

71

2.6 Application of silicon anodization for structuring

till 15 % HF or sintered clear Al2 O3 (sapphire) windows which are stable even for higher concentrated HF-electrolytes are integrated into the cell [77]. cathode

anode wafer holder

electrolyte

silicon wafer secondary cathode

secondary anode

Figure 2.38.: Double-tank anodization cell setup with two electrodes (schematic view). Anodization process is performed under current or voltage control, i.e., in galvanostatic or potentiostatic regime, respectively. Galvanostatic regime provides better control of the process, since etch rate is typically proportional to the applied current density according to the Faraday’s law of electrolysis, assuming that leakage current (i.e., current flowing not through the substrate, but through other parts of the electrochemical cell) is negligible. In galvanostatic regime, it is important to remember that the active area (anodized area) of the process increases during the process [30, 313]. For formation of all types of pores and especially for macropores, it might be not quite correct to talk about average current density calculated based on the area of silicon surface exposed to the electrolyte, because the actual current density on tips of the pores is always larger and not well defined [77]. Additionally, for etching of macropores, adjustment of current during the process is performed for compensation of mass transport limitation in the pores for reactants and products in order to provide formation of straight pores with constant pore diameters [22, 85]. In potentiostatic regime, one has to take into account the voltage drop in the electrolyte, cables and other parts of the electrochemical circuit. For analytical purposes, in order to compensate for such unwanted voltage

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drops and non-linear current-voltage dependence of Pt/HF, and to know the potential difference applied to the interface silicon-electrolyte, reference electrodes (e.g., Ag/AgCl) placed close to the reaction site are used. These reference electrodes must be HF-stable, therefore glass reference electrodes are only applicable for low concentrated HF solutions [226]. Same as for other electrochemical processes [314, 315], oscillating or pulsed voltage/current supply might be applied for enhancement of the process uniformity and etch rate by accelerating transport of reactants and products in the electrolyte [316], and can lead to more efficient photo-luminescence from the porous silicon fabricated this way [317]. In order to further enhance uniformity or etch rate of the process, additional physical impacts typically applied in electrochemical processes, such as magnetic field [318] or ultrasound [319–321], can be employed during the anodization process. Ultrasonic waves provide better transport of reactants and products from the pores and promote vertical etching [37, 38, 322]. Application of strong magnetic field can enhance uniformity or direction of pore growth and improve photoluminescence of porous silicon fabricated this way [2, 97]. Drying of porous silicon layers after anodization and rinsing in water is a crucial step. Since porous layer of silicon may be very fragile, drying with a nitrogen gun or in a spinner can break the structure of porous silicon. Natural drying in air by water evaporation is also risky, because large capillary stress associated with the evaporation process often induces cracks or total collapse in porous silicon [1, 2, 30, 77]. In order to reduce surface tension, water-replacing liquids with lower surface tension can be applied. For example, porous silicon samples can be immersed in alcoholic solution (typically methanol or 2-propanol) subsequently after rinsing in DI-water, and then naturally dried in air. Even lower surface tension is provided by pentane, but in this case intermediate step of rinsing in, e.g., methanol is still necessary, because pentane is not miscible in water [77,323]. Other drying techniques include supercritical drying or freeze drying [297, 324–326]. Stability of porous silicon depends also on its thickness, with thicker porous layers, in general, being more susceptible to damages during drying process or further processing, e.g., oxidation.

2.6 Application of silicon anodization for structuring

73

2.6.2. Process flow for structuring with anodization process The basic process flow of application of the silicon anodization for structuring is presented in Fig. 2.39. The process can proceed in three ways: • at low current density, silicon is transformed into porous silicon; this sacrificial porous silicon layer is then removed, e.g., in 1 m% KOH (s. Fig. 2.39a); • when higher current density is applied, silicon is dissolved in electropolishing regime (s. Fig. 2.39b); • porous silicon formation and electropolishing regimes can be combined: first, porous silicon volume is formed at low current density, then the process is switched to electropolishing by increasing current density to release the porous volume; after this, removal of remaining porous silicon in KOH still might be necessary. When the process shown in Fig. 2.39a is used, it is necessary to remove sacrificial porous silicon volume. One of the advantages of porous silicon as a sacrificial layer is that it is very reactive due to its high inner surface. Thus, weak silicon etch solutions at room temperature can be used to remove it very quickly. Typically, dilute (0.1–1 m%) anisotropic silicon etch solutions (hydroxide solutions) such as KOH, NaOH or CMOS-compatible TMAH, also in form of photoresist developers, are applied. By dissolving some silicon in TMAH solution, reduction of etch rate of aluminum can be achieved [152], which is especially useful for substrates with aluminum elements. Silicon stain etch solution of hydrofluoric acid and nitric acid in water [150] has been also used for removal of porous silicon [327]. Another approach for removal of porous silicon is to convert it to porous oxide by thermal oxidation, and then remove it in some oxide etch process [24]. After fabrication of porous silicon layer and during further processing, porous silicon layer gets natively oxidized. To remove such oxidized porous silicon with alkaline solutions, an intermediate HF-dip can be necessary. During the etching of porous silicon, hydrogen bubbles are formed and can hinder further reaction. Addition of ethanol helps to solve this problem [38]. When the process with sacrificial porous silicon (s. Fig. 2.39a) is used, the roughness of the surface remaining after the process is influenced, on the one hand, by the process parameters (e.g., current density, s. Fig. 2.21, or etch depth, s. Fig. 2.25). On the other hand, the process of porous silicon

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cathode wafer holder

1% KOH

electrolyte

silicon wafer

porous silicon

Anodization: from 2D opening in frontside mask to 3D porous silicon volume

(b)

anode

Removing PS in 1% KOH

3D profile in silicon

cathode wafer holder electrolyte

silicon wafer

Anodization: from 2D opening in frontside mask to 3D profile in silicon

3D profile in silicon

Figure 2.39.: Process flow of structuring silicon wafer by anodization: (a) with sacrificial porous silicon; (b) in electropolishing regime; backside doping of the substrate is not shown. removal in weak KOH or other etchant can induce additional roughness [4]. The effect is more pronounced for concentrated KOH with concentration over 30 %. Such concentrated etchants, although rarely used for removal of porous silicon, might be necessary for etching porous silicon with low porosity [328]. 2.6.3. Shape control in anodization process The principle of shape control during the anodization process is based on localization of electrical and chemical access to the silicon substrate (s. Fig. 2.40). Chemical and electrical localization of the anodization process is achieved with such techniques as surface masking films, surface and volume doping, etc. Typical localization techniques are reviewed in this section.

75

2.6 Application of silicon anodization for structuring Shape control during anodization process Sample frontside Chemical localization Masking films with openings

Sample backside

Electrical localization Doped regions

Electrical contacts

Frontside local substrate electrodes

Electrical localization Masking films with openings

Electrical contacts

Backside local substrate electrodes

Figure 2.40.: Shape control in anodization process by chemical and electrical localization techniques. 2.6.3.1. Localization by surface coating (masking) Localization by protective thin films on sample is the common technique that is used for etch processes in microelectronics technology. Openings in such films on the frontside of the substrate define a 2D layout that is transferred into – slightly bigger due to underetching – 3D form under influence of mass transport phenomena in electrolyte and electrical current distribution in the substrate. In case of electrochemical etch process, such as silicon anodization, electrical properties of a film must be taken into account as they define the current flow through the substrate, thus influencing local reaction rate of a surface electrochemical reaction that is proportional to the local current density according to the Faraday’s law of electrolysis. There are several materials which are stable in hydrofluoric acid and can be used as protective masking during silicon anodization. Silicon nitride can be used for relatively short processes (< 1 hour depending on the silicon nitride properties, electrolyte concentration and temperature) [45, 93, 329]. Photoresist has shown to have weak adhesion to silicon during anodization process and can be used only for very short processes (< 10 min) [45, 96]. Etch rate of silicon dioxide in hydrofluoric acid is very high, so that it is hardly applicable for the anodization process [93]. As very stable protective layers carbon films, amorphous silicon carbide and, more recently, fluoropolymer films with etch rates below few ångströms per hour have

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been reported [45,329–332]. Films of noble metals such as platinum or gold are also very stable in hydrofluoric acid. However, they require some intermediate layer of, for example, NiCr to improve adhesion to silicon [5]. There are also other material stacks reported, such as polycrystalline silicon on oxide, where the poly-silicon layer remains stable to hydrofluoric acid as it is electrically isolated from the silicon substrate by the oxide layer [27]. The drawback of this solution is that the oxide layer is attacked from side by hydrofluoric acid, so that underetching of the poly-silicon film occurs. Ammonium fluoride etch mixture (AFEM, pad etch), consisting of 13.5–14.5 m% of ammonium fluoride, 32.6–33.8 m% of acetic acid, and water, has lower aluminum etch rates compared to that of hydrofluoric acid and was proposed as alternative electrolyte for anodization of wafers with aluminum structures [325]. Depending on the material used as patterned protective layer, electrical structures such as metal-insulator-semiconductor (MIS) or metalsemiconductor Schottky diode are formed and define the shape of the porous silicon volume (Fig. 2.41a-c) [45]. Insulating layers typically result in pronounced etch rate near the mask edges due to current crowding resulting in convex shapesxxv , although concave shapes have been also reported [333, 334]. In some cases, conductive films have shown to solve the problem of current crowding near the mask edges and produce concave shapes as well [45]. Dimensions of the anodized regions also influence local reaction rates: smaller regions are anodized faster than the larger ones [24, 335]. This effect was applied by Mescheder and Kovacs to form 3D volumes with microfeatures by 2D complex patterns of openings in a protective masking layer (s. Fig. 2.42) [224, 334, 336]. 2.6.3.2. Localization by doping The process of silicon anodization in fluoride containing electrolytes requires positive charges coming from silicon due to applied potential difference, photo-generation, thermo-generation, and tunneling. Therefore, without illumination only p-type silicon and moderately- or highly-doped n-type silicon can be anodized [13, 337], and p-type silicon in wide range of doping xxv Here

the terms concave and convex refer to the shape of the silicon surface remaining after removal of porous silicon.

77

2.6 Application of silicon anodization for structuring

(a)

M I S +

Iinversion

+

metal masking layer Imetal

p, p+, n+

Ietch

M I S

+

n-type (b)

(c)

insulating masking layer

Ietch

(d) -

insulating masking layer

n-implant masking layer

Iundercut

+

p, p+, n+

p-type

Ietch

Ietch

Figure 2.41.: Porous silicon shape depending on the masking material and the substrate doping: (a) insulating masking for n-type silicon; (b) insulating masking for p, p+ , and n+ -type silicon; (c) metal masking for p, p+ , and n+ -type silicon; (d) n-implant masking for p-type silicon; reproduced from Steiner and Lang [45] © 1995 with permission from Elsevier.

(a)

pattern of openings in masking layer

(b)

t1 t2 t3

silicon t1 < t 2 < t 3

Figure 2.42.: 3D shape control by 2D complex patterns of openings in a protective masking layer: (a) principle of the technique, cross-section of silicon substrate, the lines within the silicon substrate show etch front movement during the process (reproduced from Mescheder and Kovacs [224]); (b) example of structure obtained with the technique, topographical scan (reprinted from Ivanov et al. [334] © 2011 with permission from John Wiley and Sons). level from low-doped to highly-doped is anodized faster than moderatelydoped n-type silicon [338]. Additionally, n-type and p-type regions formed in surface layers or in depth of silicon substrate create pn-junctions which can be used to block current flow (s. Fig. 2.41d) [45].

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2 State of the art

With this technique, doped regions which are anodized selectively to the substrate or doped regions working as protective layer or etch stop can be defined. Typical combinations are highly-doped p-type regions which are etched selectively to low-doped p-type regions, or p-type regions etched selectively to n-type regions. Surface n-type doped layers in p-type substrate are used the same way as protective masking layers discussed above (s. Fig. 2.41d) [12, 89, 339–343]. Doping regions can be also formed in depth by high energy ion implantation [344] or epitaxial growth of another silicon layer on top of a doped layer as shown in Fig. 2.43 [14, 63, 344]. n-type silicon substrate with mask boron ion implantation (p-type buried layer) mask removal and epitaxial growth of n-type silicon formation of openings for anodization

Figure 2.43.: Process flow to form a buried p-type silicon region in n-type substrate.

2.6.3.3. Other localization techniques Similarly to the localization by doping, resistivity of silicon can be increased a few orders of magnitude with hydrogen ion implantation (HI-PS) and subsequent annealing [92, 345–349]. Additional shape control can be achieved by defining local electrical contacts on the backside or frontside of the substrate without chemical access (electrical backside/frontside localization). The idea is that porous silicon formation front is following the current lines. Then, with the help of local backside contacts, a current focusing can be achieved [62]. This way high

79

2.6 Application of silicon anodization for structuring

lateral porous silicon formation rates are possible for underetching the functional structures (s. Fig. 2.44). In a case of wet backside contact, when electrical contact to the backside is provided through electrolyte, local backside contacts can be defined with formation of local highly-doped p-Si regions on the backside of a highly resistive p-Si substrate [75]. With these highlydoped regions, local electrical contacts between electrolyte and the backside of the substrate are achieved, whereas remaining low-doped regions of the substrate form reversed biased Schottky contact to the electrolyte [89]. (a)

layer to be underetched

p-type silicon

p+ -type implantation

(b)

layer to be underetched

a-SiC:H

n-type implantation

Figure 2.44.: Current focusing: (a) wafer cross-section with full area backside contact (no current focusing); (b) wafer cross-section with local backside contact for current focusing; thin lines show the equipotential lines (etch fronts); reproduced from Zeitschel et al. [62] © 1999 with permission from Elsevier.

2.6.3.4. Conclusions The above listed shape control techniques can be also combined. According to the diagram (s. Fig. 2.40), all these localization techniques include an electrical component. Localized regions on the substrate with electrical contact to the substrate, such as the local backside contact in Fig. 2.44b, will be denoted in the work as local substrate electrodes (further in text: local electrodes). To make an overview of shape control techniques by process localization in electrochemical etching of silicon complete, there are also local electrolyte electrodes used in ECM or EDM as tool-electrodes (s. sec. 2.2.3.2). As a rule, in this work electrodes are called local if they directly define the shape of the etch form, e.g., have dimensions comparable to the dimensions of the etch form. In contrast to local substrate electrodes, there might be also undefined electrical contacts of electrolyte and the substrate covering entire area which is

80

2 State of the art

much bigger than the resulting etch form. Such electrodes will be denoted as global substrate electrodes. An example of backside global substrate electrode is wet contact of substrate backside with electrolyte, when the full backside of the highly resistive substrate is highly doped or highly-conductive substrate is used, and there are no masking films on the backside (s. Fig. 2.44a). To complete the classification of global and local electrodes, platinum electrodes typically used to provide electrical supply in case of double-tank anodization cell for anodization of full wafers will be called global electrolyte electrodes (s. Fig. 2.38). 2.6.4. Structuring by pore formation (sacrificial porous silicon) Typically, microporous or mesoporous silicon with sponge-like morphology and pore size not exceeding few tens nanometers is used as sacrificial layer. Thick layers of such material can be quickly formed (in comparison to other typical sacrificial layers, such as silicon dioxide) and removed (due to very high reactivity provided with large inner surface). Additionally, the surface remaining after removal of porous silicon has a roughness in the order of pore size, thus allowing fabrication of optical and fluidic microcomponents with high surface quality. Main applications of porous silicon as sacrificial layer are described in this section. One of the basic applications of sacrificial porous silicon is for etching cavities of different shapes in silicon substrate. Porous silicon in this case is just a by-product of the etch process. Various applications of sacrificial porous silicon for structuring have been proposed. Lang et al. [5] used sacrificial porous silicon to form flow channels. Undercutting of the mask was reduced by using heavily-doped p-type or n-type substrates with resistivities of 0.2 Ω cm and 0.01 Ω cm, respectively, and a metal mask. Joubert et al. [311] applied sacrificial silicon to form 75 µm deep pseudo-V-shaped grooves used for positioning of optical fibers in silicon integrated optical circuits. Vitanov et al. [327] reported on application of sacrificial porous silicon to form channels in silicon substrates for buried-contact silicon solar cells. Another useful application of sacrificial porous silicon is to fabricate free standing mechanical elements. In this case cavity, remaining after removal of the sacrificial layer, provides mechanical isolation of free standing elements from the substrate. This way, suspended beams or masses for various MEMS devices, such as force sensors, accelerometers, and gyroscopes, are fabricated [8, 10, 13, 350–353].

2.6 Application of silicon anodization for structuring

81

Deep cavities are also necessary to isolate functional free-standing elements thermally or electrically. Thermal isolation is an important issue for temperature-sensitive sensors or sensors working at elevated temperatures, such as bolometers, gas sensors or mass flow sensors [21, 24, 45, 354–357]. Electrical isolation with sacrificial porous silicon layer is applied for fabrication of RF semiconductor devices [358–360]. Some applications of sacrificial porous silicon in microelectronics were reviewed by the author elsewhere [361] (to be published). 2.6.5. Structuring by electropolishing Advantage of using electropolishing regime for structuring of 3D forms in silicon substrate is that the resulting cavities have very good surface quality which is sufficient for optical and fluidic applications. Kleimann et al. [29] applied electropolishing to create quasi-3D microstructures in form of protrusions such as needles, columns or tubes (s. Fig. 2.45a). The process localization is based on the SCR pore formation mechanism: with pre-structuring of a silicon substrate surface, a space charge region depleted from holes is formed, thus blocking the anodization there (s. Fig. 2.45b). Another approach to use anodization for structuring is to switch between pore formation and electropolishing regimes. This way, a sacrificial porous volume can be removed from the cavity, or left as an intermediate layer. Tjerkstra et al. have used such combined process to create multi-walled microchannels with intermediate porous silicon wall for micro total analysis systems (µTAS) and gas chromotographs [47, 362]. Similar approach was used by Kaltsas et al. to form buried microchannels, where a top porous silicon layer formed at the beginning of the process was used to seal the channel [363]. Lammel et al. [30, 31, 364] used pore formation with subsequent electropolishing to fabricate a free-standing tunable interference filter for a microspectrometer. First, a 1D photonic crystal was formed in pore formation mode by modulating current density (s. Fig. 2.46a). Then, the formed photonic crystal was released from the substrate with electropolishing and subsequent removal of the masking material. For support arms of the structure, anodization was done through a perforated mask, which resulted in non-uniform porosity in depth in these regions. During oxidation of the

82

2 State of the art (a)

(b)

(1)

(2)

(3)

Figure 2.45.: Microstructures formed in electropolishing mode: (a) SEM micrographs; (b) schematic illustration of the principle, where (1) is the cross-sectional view showing etching current line bending due to the electrical field in the SCR and the conditions for stable microstructure formation (with 𝐽1 in the electropolishing range and 𝐽2 = 0), (2) is the starting structure (n-type silicon with preformed step), and (3) is the vertical structure formation by electropolishing flat parts; reprinted from Kleimann et al. [29] © 2001 with permission from AIP Publishing LLC. structure in the final steps, this non-uniformity caused desired lift-up of the whole structure (s. Fig. 2.46b). (a)

(b)

Figure 2.46.: Free-standing tunable interference filter for microspectrometer [364]: (a) schematic diagram of current density during the phases of porosification and electropolishing of silicon; (b) SEM micrograph of a resulting structure; reprinted from Lammel et al. [364] © 2001 with permission from Elsevier.

2.7 Macroscale models of silicon anodization for structuring

83

2.7. Macroscale models of silicon anodization for structuring In order to use a process for structuring, i.e., formation of cavities of a specific 3D shape, it is important to be able to predict behavior of the process. For this, knowledge about the process, such as design rules, description of definite cases or models describing etch front movement, etc., are necessary. As was shown above in this chapter, the process of electrochemical etching of silicon is a rather complicated process featuring: • two regimes of reactions, resulting in pore formation and electropolishing; • mass transport of reactants and reaction products in electrolyte; • mixed polarization dependence resulting in tertiary current distribution; • additional influence of porous layer on mass transport phenomena in pore formation regime; • complex dependence of results on the process parameters. There are multiple models describing the process of silicon anodization on the microscale, which to some extent manage to explain formation of pores of various size and shape, predict their influence on the process parameters and explain polarization curves and oscillation mechanisms (s. sec. 2.4.2.5). In contrast, little effort was done so far to describe the macroscopic behavior of the process needed for practical 3D structuring. 2.7.1. Charge flow models - primary and secondary current distribution One of the basic ideas to describe the etch front movement (i.e., etch rate) in this electrochemical process is based on consideration of charge flow through the anodization cell with electrolyte and silicon substrate. In this case, only resistivities of the electrolyte and substrate are taken into account for calculation of charge flow at the applied potential, and etch front velocity is considered to be proportional to the local current density according to the Faraday’s law. This way, the current crowding induced with insulating frontside mask [333, 365] and the current-focusing technique to enhance lateral etching were explained qualitatively [62] as reviewed briefly below. Kim et al. [333] studied shape of etch forms anodized with an insulating silicon nitride mask. In the resulting semi-spherical shapes, they observed

84

2 State of the art

“primary undercutting”, that is lateral widening of a structure under the mask edge, and “secondary undercutting”, that is etching of the substrate in the surface layers under the mask, causing the mask to pile-off. Both effects have been explained with the current crowding. The primary undercutting was due to general flow of holes from big-area backside to small-area opening in the mask. The secondary undercutting was explained with the field-effect causing charges to accumulate near the interface between the mask and the substrate on the frontside, resulting in the promoted etching there. The model of current density distribution of holes in the substrate was calculated in the device simulator SILVACO (modules Athena and Atlas). Based on both the modeling and experimental results, Kim et al. concluded that the current crowding is significantly less for low-resistive p-type substrates. The model described by the authors showed only potential distribution in silicon substrate for the beginning of the process, i.e., for a flat initial etch front at etching time 𝑡etch = 0, thus it could not be applied to simulate real 3D etch forms resulting from anodization. Mescheder et al. [335,365] have used a similar approach to simulate the current density distribution of holes with another semiconductor device simulator TMA Medici for 2D cases and TMA Da Vinci for 3D cases. In their model the resistivity of electrolyte was not taken into account, and the model consisted of a low-doped silicon substrate with two metal electrodes connected to its frontside and backside (s. Fig. 2.47b), where the frontside electrode represented the contact silicon-electrolyte (s. Fig. 2.47a). This frontside contact was defined as Schottky contact [89]. The simulations showed that the current crowding is responsible for the experimentally observed convex shapes (s. Fig. 2.47c). Effect of local backside contacts on etch form could be also shown qualitatively. Again, the models simulated only the current density distribution for the initial state of the process, thus no etch form development was shown with the model. Later, application of the Medici program described in [335] was further extended by the author of this PhD in his master thesis work [366]. There, qualitative results for conducting masks under consideration of various work function values for the masking material and the electrolyte have been obtained [367]. It was demonstrated that the ratio between the work function values of the electrolyte and the masking material, i.e., the electrical properties of the contacts mask-silicon and electrolyte-silicon, define the resulting shape (convex or concave) (s. Fig. 2.48a). Additionally, an attempt to simulate the process for 𝑡etch > 0 with another static model has been performed. It was shown that even if the etch form develops concave in the beginning

2.7 Macroscale models of silicon anodization for structuring

(a)

85

(c)

front-side opening in insulating mask (contact with electrolyte) Si

back-side ”wet” contact to electrolyte

(b) x=0 y=0

y

front-side electrode (contact with electrolyte) Si

x

back-side electrode with ohmic contact

Figure 2.47.: Secondary current distribution model of the etch form in the anodization process: (a) cross-sectional view of experimental setup substrate with insulating masking layer on its frontside with opening; (b) the 2D model; (c) typical resulting hole current density distribution as surface for the 2D model (Ivanov 2005, unpublished). of the anodization process, later it converts to a convex form (s. Fig. 2.48b). The disadvantage of the used approach was that the models for both cases (𝑡etch = 0 and 𝑡etch > 0) were still static models, and no real etch front movement could be simulated. Therefore, the results could only be taken with caution, although to the moment of that work, these qualitative results confirmed experimental results, where only convex shapes could be obtained for insulating and metal masks. Kröner et al. [368] have developed a dynamic electrical finite element model (FEM) in ANSYS. Development of anodization forms through simple openings with axial symmetry, such as circles and rings, was shown and fitted to experiments. Growth of convex etch forms, transformation of convex forms to concave, influence of neighboring openings on each other and development of etch forms for local backside contact configurations could be demonstrated. However, the chosen resistivity values for the substrate and electrolyte were rather controversial. Current density dependent porosity and valence in pore formation regime have been also not considered. Additionally, application of general ANSYS software required great effort of

86

2 State of the art

(a) x=0

mask

electrolyte

(b)

mask

x=0 x

y=0

back-side electrode with ohmic contact

electrolyte

mask x

y=0

Si y

mask

Si y

back-side electrode with ohmic contact

Figure 2.48.: Secondary current distribution model of the etch form in the anodization process with conductive mask for (a) 𝑡etch = 0 and (b) 𝑡etch > 0 (concave etch form with depth 50 µm), reproduced from Ivanov [366]. scripting to describe the etch front movement, which seems to be not practical. To conclude on primary and secondary current distribution models, although they neglect mass transport limitation mechanisms in the electrolyte and the porous media, they can show quite good (at least, qualitative) results for etch form development for the anodization process in pore formation regime, where the process is considered to be limited by the charge transfer at the interface silicon-electrolyte, and not by the transport of ions in electrolyte. Additionally, in secondary current distribution models, activation polarization provides a stabilizing effect for the current distribution, therefore more smooth shapes are achieved than with primary current distribution models [254].

2.7 Macroscale models of silicon anodization for structuring

87

2.7.2. Mass transport and tertiary current distribution models For the process of silicon anodization, especially in electropolishing regime, mass transport phenomena are proposed to play an important role (s. sec. 2.4). Therefore, it can be expected that in this case a tertiary current distribution model is necessary in order to describe etch front movement. To the best of our knowledge, there are no such macro-scale models, so far, which could show etch form evolution during the anodization process. However, some ideas can be taken from models available for other (electro)chemical processes and microscale models of the anodization process as described below. 2.7.2.1. Models of chemical etching Mass transport phenomena have been considered in macroscale models of chemical dissolution in diffusion-controlled regime [144, 145, 369–371]. The transport of species is described there according to the Fick’s law of diffusion (s. Nernst-Planck equation (2.12) in sec. 2.3.2.3 in the absence of electric field). This way, a typical etch form development for etching through a mask could be demonstrated (s. Fig. 2.2b,c): In a diffusion-limited process, in the beginning, there is a higher supply of reactants to the reaction site near the mask edges, that results in higher etch rate near the mask edges (convex shape). Later in the process, when the etch form gets deep enough, the shape transforms to a concave. Chemical polishing of silicon in a mixture of an oxidizing agent (nitric acid) and an oxide-etching acid (hydrofluoric acid) was analyzed and modeled by Kulkarni et al. [372, 373]. Two-phase model was proposed, where reagents diffuse through a liquid film in the interface of electrolyte-silicon. This layer is called “exhausted layer” and is also known as the concentration boundary layer of the mass transport film. Optimal conditions for polishing and influence of extrinsic bubbles (introduced to the solution during the process) and intrinsic bubbles (formed in the dissolution reactions) have been worked out. 2.7.2.2. Microscale models of silicon anodization As was partly covered in sec. 2.4.2.5, there are multiple microscale models of silicon anodization process aimed on simulation of surface roughness, pore

88

2 State of the art

morphology or polarization curves. Some aspects of these models can be as well applied for a macroscale model of interest. Cattarin et al. [284] have discussed the steady-state electrodissolution current density (the plateau current density) in electropolishing regime. In this regime, the anodic oxide dissolution rate is in equilibrium with the anodic oxide formation rate, which is directly proportional to the applied current density. Three paths of silicon dioxide dissolution were considered, involving either HF or HF2 − separately, or a combination of both, with a second order kinetics. It was assumed that fast equilibrium exists for the dissociation reactions (s. sec. 2.3.2.2) both in the diffusion layer near the reaction site and in the bulk solution. For simplification, the concentration of hydrogen ions H+ was assumed to be constant everywhere, because the model referred to buffered solution. Equal diffusion constants for F− , HF, and HF2 − were assumed. The total current was the sum of the currents of the three proposed dissolution reactions. Then, with experimentally found value of the dissolution valence of 3.6, a dependence of the steady-state current density in electropolishing regime on total fluoride concentration, pH, and mass transfer rate were derived and found to be in good agreement with experiments. In their later work, Cattarin and Musiani [292] have extended their study to HF:water:ethanol electrolytes. Addition of ethanol was shown to change anodization current in the whole anodic range (pore formation and electropolishing). It was proposed that this change could be due to the changed dissociation equilibria in the solution with ethanol (s. sec. 2.3.2.2) and (for the electropolishing region) due to change in the kinetic constants for the process of anodic oxide dissolution. Liu and Blackwood [227] have analyzed the electropolishing regime of silicon anodization in the first plateau for p-type silicon of various resistivities in HF electrolytes of concentration in the range 0.01–15 m%. As in the work of Cattarin et al. [284], direct dependence of current density on dissolution rate of anodic oxide was assumed. However, the anodic oxide was considered to be hydrated oxide (SiO (OH)2 ) which dissolution involved only the undissociated HF molecules. The dissolution valence of 3.5 was applied. As a result, an expression of anodic oxide dissolution rate as a function of amount concentration of undissociated HF molecules was derived. 2.7.2.3. Macroscale models of other electrochemical processes Electrochemical dissolution of metals was modeled for various applications, such as dissolution through a mask [374–377], replication with ECM [378],

2.8 Conclusions on state of the art

89

ECM for drilling [166,379], micro-machining with electrolytic jet [380], electrochemical formation of micropins [381], wire-ECM [382], ECM with ultrashort pulses [383], general considerations on electrochemical forming [384], etc. In the COMSOL MultiphysicsTM software applied in the work for calculation of FEM models of the anodization process, there is a module specifically designed to simulate electrodeposition problems (electrodeposition module), and recently introduced a module for electrochemical dissolution problems (corrosion module)xxvi . In the electrodeposition module, there is an example of a model of copper electrodeposition in a trench, which can be as well used to model the opposite process of copper dissolution. However, application of the model to silicon anodization requires complete revision of the parameters, equations, and reactions.xxvii To conclude, first attempts on building charge-flow models have been made. However, to the best of our knowledge, no systematic analysis and comparison to experiments were done. In this work, among other goals, primary and secondary current distribution models for silicon anodization through an opening in a frontside mask are developed and compared to experiments. 2.8. Conclusions on state of the art In this chapter, state of the art on silicon anodization process, with emphasis on its application as a structuring technique, has been given. Overview of other structuring techniques for silicon has been also done, and theory and technology of silicon anodization process were discussed in detail. Based on this literature research, the following conclusions could be drawn:

xxvi As

confirmed by the representatives of the software, the modules do not differ from each other in theoretical backgrounds. In the work the electrodeposition module was used as the one available to the author. xxvii It is important to note that in general, although such software, as COMSOL MultiphysicsTM , includes all typically used equations for mass and charge transfer phenomena, all reactions and parameters for specific process have to be defined by a user. Thus, the role of the software in this case is only to minimize low-level programming for calculations and coupling of various differential equations in the time dependent environment of deforming mesh. This way, more attention can be paid to underlying theory.

90

2 State of the art • There are multiple structuring techniques for silicon to form 3D shapes. However, every technique has its own limitations and drawbacks. Thus, with further advancement, silicon anodization can deserve its place among other structuring techniques. • Various applications of silicon anodization as a structuring technique have been published. However, only few of them could develop themselves from laboratory-scale to industrial implementation, presumably due to low reproducibility of the result. Development of a macroscale model of the anodization process could simplify application of the process for specific application-oriented structures. • The multitude of parameters significantly influencing the process on the microscale makes comparison of experimental results from different research groups difficult, revealing some inconsistencies between the results. Therefore, it seems to be unavoidable to verify the data with own experiments. • Although there were some attempts to simulate partially the process on the macroscale, no systematic and quantitative comparison to experiments was done so far.

These conclusions, together with the research goals dictated by the research projects, in which the author was employed, with their specific technical and economical limitations, defined the goals and set the limits of this work: • study of the influence of current density, electrolyte composition, and wafer doping level on the etch rate, porosity, dissolution valence, and interface roughness; this study is limited due to the following considerations: – only p-type silicon is studied, as the one readily providing etching of microporous/mesoporous silicon that was applied (out of the scope of this work) for fabrication of 1D photonic crystals, and is more suitable for application as a structuring technique; – in the pore formation regime, only one electrolyte (1:1 (50 m% HF):ethanol) was applied, because this electrolyte has already proved (with external and internal studies) to be very suitable for generation of uniform micro- and mesopores; – in general, the number of data points, i.e., single anodization experiments, studied was limited from the point of financial and time resources available to specific projects.

2.8 Conclusions on state of the art

91

• investigation of etch front movement in anodization process for various localization techniques; as the main test structure, silicon substrate with a frontside insulating masking with a circular opening was selected as rather simple, but not yet studied in detail for silicon anodization process; • simulation of the etch front movement in a macroscale time-dependent models with primary and secondary current distributions, using the experimentally obtained parameters for porosity and dissolution valence, and comparison of the results to experiments.

3. Experimental, characterization and simulation methods 3.1. Fabrication methods 3.1.1. Electrolytes and other chemicals Semiconductor grade chemicals were used in the work. The summary of main chemicals is given in Tab. 3.1.

Table 3.1.: Summary of main chemicals used in the work Substance name

Chemical formula

Producer

Package mass/volume

Further information

Hydrofluoric acid 50 %

HF

Honeywell Specialty Chemicals Seelze, GmbH, Seelze, Germany

2.5 l

Product family PURANALr

Ethanol absolute

C2 H5 OH

Honeywell Specialty Chemicals Seelze, GmbH, Seelze, Germany

2.5 l

Product family MOS PURANALr , particle class 0

2-Propanol

C3 H7 OH

Honeywell Specialty Chemicals Seelze, GmbH, Seelze, Germany

2.5 l

Product family PURANALr

Potassium hydroxide pellets

KOH

Honeywell Specialty Chemicals Seelze, GmbH, Seelze, Germany

5 kg

Product family PURANALr

MicropositTM S1818TM G2 Positive Photoresist

complex composition, s. datasheet

Rohm and Haas Europe Trading ApS / The Dow Chemical Company, Lyngby, Denmark

946 ml

AZ 351B Developer

NaOH, (BH3 O3 ).xNa, etc.

AZ Electronic Materials GmbH, Wiesbaden, Germany

5l

Preparation of electrolytes was done mostly on the day of experiments or one day before. Before starting experiments, electrolyte was stirred manually © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_3

94

3 Experimental, characterization and simulation methods

to provide good homogeneity. During anodization, no stirring/circulation was applied. All electrolytes used in the work are listed in Tab. 3.2. Table 3.2.: Summary of electrolytes used in the work Electrolyte

Composition

3 m% HF

190 ml HF(50 m%) + 3410 ml water *

7 m% HF

430 ml HF(50 m%) + 3170 ml water *

15 m% HF

960 ml HF(50 m%) + 2640 ml water *

29.93 m% HF with ethanol

1:1 volume ratio of 50 m% HF and 100 m% ethanol

* The volumes are given for the use in the setup 2.

All experiments have been conducted in electrolytes at room temperature, which varied in the range from 18 °C to 20 °C. Mixing of ethanol and water is exothermic, therefore their mixing resulted in increase of temperature to over 30 °C. The experiments in this case were conducted one day after mixing, so that the temperature of electrolyte adjusts itself to the room temperature, and the temperature was also additionally controlled before experiments to ensure the stabilized conditions. For few experiments aimed on study of surface quality after anodization of full 4 inch wafers, high total current (over 10 A) was used. In this case, increase of electrolyte temperature was observed. Namely, for the total current of 12.44 A, resulting in the voltage drop of about 60 V and the power of 746 W, an increase of temperature of electrolyte in the setup 2 of 3 °C/min was measured with an HF-stable glass thermometer. At such high currents, anodization duration was limited to few minutes to avoid high temperatures that could alter the results and bring safety concerns. 3.1.2. Sample preparation 3.1.2.1. Wafers Czochralski-grown 4-inch prime grade wafers with crystallographic orientation (100) from various suppliers have been used in the work. Summary

95

3.1 Fabrication methods

of the wafer specifications is given in Tab. 3.3. For the polished surfaces, average surface roughness below 5 nm was measured. Table 3.3.: Summary of wafer specifications Doping type

Resistivity, Ω cm

Supplier

Thickness, µm

Surface quality (frontside/backside)

p (boron)

0.001–0.005

Addison Engineering, Inc., San Jose, USA

525 ± 25

polished / etched

p (boron)

0.005–0.01

MicroChemicals GmbH, Ulm, Germany

525 ± 25

polished / etched

p (boron)

0.005–0.01

Addison Engineering, Inc., San Jose, USA

525 ± 25

polished / etched

p (boron)

0.01–0.02

Si-Mat, Landsberg/Lech, Germany

525 ± 25

polished / etched

p (boron)

0.01–0.1

Si-Mat, Landsberg/Lech, Germany

525 ± 25

polished/polished

p (boron)

10–16

Si-Mat, Landsberg/Lech, Germany

525 ± 25

polished/polished

p (boron)

10–20

Si-Mat, Landsberg/Lech, Germany

525 ± 25

polished/polished

p (boron)

10–20

UniversityWafer Inc., Boston, USA

500

polished/polished

p (boron)

10–20

Siltronix SAS, Archamps, France

525 ± 25

polished/polished

3.1.2.2. Cleaning procedure Huang cleaning procedure, similar to the RCA standard cleaning procedure [385], was performed before hot-temperature steps, such as silicon nitride deposition, oxidation and ion implant annealing, and before anodization process after sample cutting with laser or after contact with a KOH solution.

96

3 Experimental, characterization and simulation methods

Two solutions were prepared for the procedure: • Solution 1 - Huang A (organic clean and particle clean): – 500 ml of NH4 OH (ammonium hydroxide, 25 m% of ammonia NH3 ); – 500 ml of 30 m% H2 O2 (hydrogen peroxide); – 2300 ml of deionized water. The solution was prepared by filling the water and the ammonia to the bath and heating up to (65 ± 5) °C. The peroxide was given to the solution right before performing the cleaning procedure. • Solution 2 - Huang B (ionic clean): – 500 ml of 27 m% HCl (hydrochloric acid); – 500 ml of 30 m% H2 O2 (hydrogen peroxide); – 2300 ml of deionized water. The solution was prepared by filling the water and the hydrochloric acid to the bath, and heating up to (65 ± 5) °C. The peroxide was given to the solution right before performing the cleaning procedure. The procedure was performed by immersion of wafers to the first solution (step 1 - Huang A) for 5 minutes and then to the second solution (step 2 - Huang B) for 5 minutes. Rinsing in deionized water was done after each step. Both steps were performed with ultrasonic agitation. In the procedure, the two steps were done twice. Finally, the wafers were rinsed in deionized water to the water conductivity value of 0.1 µS/cm. 3.1.2.3. Photolithography To structure silicon nitride on wafers and to make local ion implantation, photolithography was applied. Photoresist S1818TM of thickness about 1.5–1.7 µm was used. For exposure, 5-inch chrome masks with minimum feature size of 1 µm were produced either in the laboratory with the pattern generator GCA3600 (David W. Mann, GCA Corp., Burlington, USA) or ordered from Delta Mask B.V. (Enschede, Netherlands). Wafer exposure was done with a top and bottom side contact printer Karl Süss MA6 with a mercury arc lamp (Süss MicroTec, Garching, Germany). The process consisted of the following steps:

3.1 Fabrication methods

97

• (only for structuring of silicon nitride) adhesion promoter TI Prime (MicroChemicals GmbH, Ulm, Germany): – spin on at 4250 rotations/min for 25 s; – baking in oven at 130 °C for 10 min; • photoresist S1818TM spin on at 4250 rotations/min for 25 s; • pre-bake in oven at 90 °C for 20 min; • exposure (25 mW/cm2 and duration about 10 s)i ; • development in a solution of 1 part of AZ 351B and 5 parts of deionized water at 21 °C for 60 s; • post-bake in oven at 130 °C for 30 min. 3.1.2.4. Frontside silicon nitride deposition and structuring LPCVD low-stress silicon-rich silicon nitride deposited in the laboratory with thickness from 200 nm to 250 nm was used for the experiments with frontside localization of the process. With the low-stress nitride, influence of the frontside masking film on the etch rate near the mask edges is eliminated [96, 386]. Measured etch rate of the silicon nitride in the concentrated electrolyte (29.93 m% HF with ethanol) was up to 6 nm/min. Thus the chosen thickness of the nitride film allowed anodization processes up to about 40 minutes. Etching of the silicon nitride through a photolithographically structured photoresist film was done in two steps. First, the nitride film was thinned down till thickness of about 40 nm with RIE (SF6 ) at an etch rate of 13 nm/min as fast etching process with low selectivity. The remaining nitride film was etched selectively in buffered HF solution 10:1 at 31 °C in the dark with good selectivity to silicon, at an etch rate of 0.8 nm/min. If the BHF etch process took longer than 30 minutes, the wafers were taken out every 30 minutes, rinsed in deionized water and baked in oven for 30 minutes at 130 °C to enhance adhesion of the photoresist to the nitride layer.

i The

values varied due to altering of the mercury arc lamp

98

3 Experimental, characterization and simulation methods

3.1.2.5. Backside wet contact In order to provide charge flow through the interface “electrolyte - silicon sample backside” by making the contact ohmic for low-doped p-type silicon, a highly-doped p-type layer with surface doping level in excess of 1017 cm−3 was needed to form a pp+ structure. This was done with boron ion implantation. Calculations in a semiconductor process simulator CSuprem (Crosslight, Vancouver, Canada) version 2.11.15.0 showed that with the ion acceleration energy of 30 keV and the dose of 5 × 1015 cm−2 , the surface doping profile of about 8.6 × 1019 cm−3 is achieved, that is sufficient to provide reliable ohmic contact. The process of backside boron ion implantation consisted of the following steps: • frontside protection photoresist: – spin-on of photoresist S1818TM on the frontside at 4250 rotations/min for 3 s; – bake in oven at 130 °C for 30 min; • boron ion implantation with the ion acceleration energy of 30 keV and the dose of 5 × 1015 cm−2 (in case of local backside contacts, photoresist mask was used); • removal of the frontside photoresist in O2 -plasma reactor; • Huang cleaning (s. sec. 3.1.2.2); • implant anneal and diffusion process: – 900 °C, 2 min, O2 ; – 900 °C, 28 min, N2 (anneal); – 1000 °C, 60 min, N2 (diffusion). In the laboratory, the surface doping level was controlled after the doping process by measuring the sheet resistance with the four-point probe method [106]. The simulated doping profile for the given implantation parameters provides a sheet resistance of approximately 30 Ω/2. In general, backside sheet resistance below 50 Ω/2 was sufficient to provide reliable electrical contact.

3.1 Fabrication methods

99

3.1.2.6. Mask layouts Several simple mask layouts have been used for frontside and backside localization techniques with variation of opening dimensions and spacing between them to study basic behavior of silicon anodization as a structuring technique. To avoid the issue of small anodization area resulting in small total current and not well defined process conditions due to leakage current (s. sec. 3.1.3.1), most of the layout for frontside insulating masking consisted of multiple openings. To reduce number of experiments, size of the openings or spacing between them were varied on one sample. For frontside mask layouts, additionally to the total open area 𝐴open , ratio of this area 𝐴open to the sample area 𝐴sample defined by the corresponding sample holder (s. Tab. 3.9) was evaluated for each layout. Layout 1 - multiple circular openings of diameters 200–1000 µm This layout contains 20 circular openings with diameters ranging from 200 µm to 1000 µm with step of 200 µm (Fig. 3.1). The layout was used to study effect of opening diameter on etch form development. The range of opening diameters was chosen such that both diameters less than half wafer thickness and equal to double wafer thickness are present. The centers of the openings are placed 2 mm from each other to avoid merging of neighboring etch forms due to mask underetching. The openings of various diameters are distributed in the layout to reduce effect of neighboring openings on each other. To address single opening, the notation with {line number}-{column number} was used. This gives for the top right opening the name 1-5, and for the bottom right opening 4-5. The mask layout for a 4-inch wafer consisted of 7 circular samples of diameter 30 mm, each with the described layout 1. Total open area of the layout is 0.06912 cm2, which is 2.2 % of the open area defined by the sample holder (circular area of radius 1 cm). Layout 2-1 - single concentric ring with central circular opening, varied radiiii The layout 2-1 is shown in Fig. 3.2a. In this layout, the values of radii 𝑅1 , 𝑅2 , and 𝑅3 were varied in order to check whether an outer ring opening can help to eliminate the current crowding in the central opening (s. Tab. 3.4). ii Layouts

2-x were developed in a cooperation with Dr. F. Goldschmidtböing and Dr. M. Kröner from University of Freiburg im Breisgau / IMTEK.

c ol umn5

c ol umn4

c ol umn3

c ol umn2

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Ryy

l i ne1 l i ne2 l i ne3 l i ne4

6B;m`2 jXRX, GvQmi R @ +B`+mH` QT2MBM;b Q7 p`B2/ /BK2i2` 7`QK kyy ƒK iQ Ryyy ƒK rBi? bi2T Q7 kyy ƒKc ?2`2 M/ BM i?2 7m`i?2` HvQmib- }HH2/ +QHQ` b?Qrb i`MbT`2Mi `2;BQMb UrBM/QrbV QM i?2 KbFX ( a )l a y o ut2 1

( d)l a y o ut2 4

( b)l a y o ut2 2 5mm

d

R2 R1

R1

R1

R1

R1

R1

R1

d

R3

( c )l a y o ut2 3 5mm R2

R1

6B;m`2 jXkX, GvQmi k- ;`v `2b `2T`2b2Mi QT2MBM;b BM i?2 bi`m+im`2/ Hv2`, UV HvQmi k@R Ĝ bBM;H2 +QM+2Mi`B+ `BM; rBi? +2Mi`H +B`+mH` QT2MBM; M/ p`B2/ `/BB ԇ - ԇ - M/ ԇ c U#V HvQmi k@k Ĝ irQ QT2MBM;b Q7 2[mH p`B2/ `/Bmb ԇ rBi? +QMbiMi /BbiM+2 Q7 8 KK #2ir22M i?2B` +2Mi2`bc U+V HvQmi k@j Ĝ irQ QT2MBM;b Q7 /Bz2`2Mi `/BB ԇ M/ ԇ rBi? +QMbiMi /BbiM+2 Q7 8 KK #2ir22M i?2B` +2Mi2`b M/ QM2 `/Bmb p`B2/c U/V Hv@ Qmi k@9 Ĝ 7Qm` QT2MBM;b Q7 +QMbiMi `/Bmb ԇ rBi? p`B2/ /BbiM+2 ԓ #2ir22M i?2B` +2Mi2`bX h#H2 jX9X, amKK`v Q7 bKTH2b rBi? HvQmi k@R- r?2`2 Ӷ Bb M `2 Q7  +B`+mH` QT2MBM; Q7 /BK2i2` R KK- Ӷ ஈ yXd38 KK iF2M b  `272`2M+2 aKTH2 MK2

ԇ - KK

ԇ - KK

ԇ - KK

ӶPQFO - KK

ӶPQFO Ӷ

ӶPQFO ӶTBNQMF - W

k@R@R k@R@k k@R@j

yX8 yXdyd yX8

RX8 RX83R kXy

kXy kXy kX8

ஈ eXk3 ஈ eXk3 ஈ dX38

3 3 Ry

k k kX8

101

3.1 Fabrication methods Layout 2-2 - two openings of equal varied radius, constant distance between their centers

With this layout, cross-influence of two equal circular openings with varied radius and constant distance between their centers was studied (s. Fig. 3.2b). The values of varied radius 𝑅1 and the total open area are summarized in Tab. 3.5. Table 3.5.: Summary of samples with layout 2-2, where 𝐴0 is the area of a circular opening of diameter 1 mm, 𝐴0 ≈ 0.785 mm2 Sample name

𝑅1 , mm

Total area, mm2

𝐴open /𝐴0

𝐴open /𝐴sample , %

2-2-1 2-2-2 2-2-3

0.5 0.707 1

≈ 1.5708 ≈ 3.1406 ≈ 6.2832

2 ≈4 8

0.5 1 2

Layout 2-3 - two openings of different radii, constant distance between their centers, one radius varied In this layout, uniformity of the anodization process in case of openings of different radii was studied. The layout is shown in Fig. 3.2c. The values of varied radii 𝑅1 and 𝑅2 , and the total open area are summarized in Tab. 3.6. Table 3.6.: Summary of samples with layout 2-3, where 𝐴0 is the area of a circular opening of diameter 1 mm, 𝐴0 ≈ 0.785 mm2 Sample name

𝑅1 , mm

𝑅2 , mm

Total area, mm2

𝐴open /𝐴0

𝐴open /𝐴sample , %

2-3-1 2-3-2 2-3-3 2-3-4

0.866 1.323 1.658 1.937

0.5 0.5 0.5 0.5

≈ 3.1415 ≈ 6.2842 ≈ 9.4215 ≈ 12.5726

≈4 ≈8 ≈ 12 ≈ 16

1 2 3 4

Layout 2-4 - four openings of constant radius, varied distance between their centers With this layout, effect of distance between the openings of equal constant radius was studied. The layout is shown in Fig. 3.2d. The values of varied radius 𝑅1 , the distance 𝑑, and the total open area are summarized in Tab. 3.7.

102

3 Experimental, characterization and simulation methods

Table 3.7.: Summary of samples with layout 2-4, where 𝐴0 is the area of a circular opening of diameter 1 mm, 𝐴0 ≈ 0.785 mm2 Sample name

𝑅1 , mm

𝑑, mm

Total area, mm2

𝐴open /𝐴0

𝐴open /𝐴sample , %

2-4-1 2-4-2 2-4-3

0.707 0.707 0.707

2 3 4

≈ 6.2813 ≈ 6.2813 ≈ 6.2813

≈8 ≈8 ≈8

2 2 2

Layout 3-1 - multiple square openings of dimension 100–400 µm This layout contains four blocks of square openings of varied dimension from 100 µm to 400 µm with step of 100 µm (Fig. 3.3a). The bottom right block is a 12 × 12 array of openings of width 100 µm with spacing between them of 200 µm. The top right block is a 6 × 6 array of openings of width 200 µm with spacing between them of 400 µm. The top left block is a 4 × 4 array of openings of width 300 µm with spacing between them of 600 µm. The bottom left block is a 3 × 3 array of openings of width 400 µm with spacing between them of 800 µm. To address single openings in each block, the following notation is used with line numbers: opening number is {line number}-{column number}, where the line numbers are counted from top to bottom, and the column numbers are counted from left to right. This gives for the top right opening in the top right block the name 1-6, and for the bottom right opening 6-6. This mask layout was designed to be used on circular samples of diameter 30 mm. Total open area of the layout is 0.0576 cm2, with all blocks having equal area. The ratio of the open area to the sample area of 1 cm for this layout is 1.8 %. Layout 3-2 - multiple rectangular openings of width 100–400 µm This layout contains four blocks of rectangular openings of varied dimension from 100 µm to 400 µm with step of 100 µm (Fig. 3.3b). The bottom right block is a 12 × 3 array of openings of dimensions 100 µm × 400 µm with spacing between them of 200 µm in horizontal direction and 800 µm in vertical direction. The top right block contains two lines of openings of dimensions 200 µm × 800 µm with spacing between them of 400 µm in horizontal direction and 1200 µm in vertical direction. The top left block contains one line of four openings of dimensions 300 µm × 1200 µm with spacing between them of 600 µm in horizontal direction. The bottom left block contains one line of three openings of dimensions 400 µm × 1200 µm with spacing between them of 800 µm in horizontal direction. To address single openings in each block, same notation as described for the layout 3-1 was used. This mask layout

103

3.1 Fabrication methods (a) layout 3-1

(b) layout 3-2

Figure 3.3.: (a) Layout 3-1 – multiple square openings of dimension 100–400 µm, (b) layout 3-2 – multiple rectangular openings with length of shorter side 100–400 µm. was designed to be used on circular samples of diameter 30 mm. Total open area of the layout is 0.0576 cm2, same as for the layout 3-1, with all blocks having equal area. The ratio of the open area to the sample area of radius 1 cm for this layout is 1.8 %. Layout 4 - backside rectangular contacts with varied spacing In this layout, spacing between two backside rectangular contacts is varied (Fig. 3.4). The layout 4-1 is a special case of one single 100 µm wide contact, which can be seen as the case of two contacts of width 100 µm placed one over another, i.e. with negative spacing of –100 µm. Layout 5 - frontside rectangular window with varied width and backside single rectangular contact In this layout, backside and frontside are structured. The frontside layout defines a single rectangular window of length 8 mm and varied width in the range from 130 µm to 4160 µm (Fig. 3.5). The backside layout defines a single rectangular backside contact located centered to the frontside opening, with width of 100 µm and length of 8 mm.

104

3 Experimental, characterization and simulation methods

8mm

s

w

Layout

w, µm

s , µm

4-1

50

0

4-2

100

0

4-3

100

260

4-4

100

520

4-5

100

780

4-6

100

1040

Figure 3.4.: Layout 4 – backside rectangular contacts with varied spacing; layout 4-1 is a special case of one single 100 µm wide contact, which can be seen as the case of two contacts of width 100 µm placed one over another, i.e., with negative spacing of –100 µm, or as two contacts, each of width 50 µm, placed with zero spacing; layout 4-2 with two contacts, each of width 100 µm, and spacing of zero means one single contact of width 200 µm. Frontal view

Cross-section AA

w

8mm

w

Layout w, µm

100 µm A

A 100 µm

5-1

130

5-2 5-3 5-4

260 520 1040

5-5 5-6

2080 4160

Figure 3.5.: Layout 5 – frontside rectangular window in silicon nitride layer of length 8 mm and width 𝑤, and backside single rectangular contact of length 8 mm and width 100 µm. 3.1.2.7. Sample cutting For many experiments, it was not economical to anodize full 4-inch wafers, therefore smaller samples cut out of full 4-inch wafers were used. Samples were typically cut with a neodymium-doped yttrium aluminium garnet (Nd:YAG) industrial laser system LS9000 (LS Systems, Munich,

3.1 Fabrication methods

105

Germany) working in a range of microsecond pulses with an automated positioning stage with two degrees of freedom. The laser cutting process provided precision of positioning up to 10 µm with typical cut width of 100 µm. The disadvantage of the process was that, due to melting of the material, laser particles were produced and redeposited on the wafer surface. To remove them, the Huang cleaning procedure was used as described in sec. 3.1.2.2. During the work with the system, the author has optimized the process parameters with respect to particle contamination and process efficiency, and has written programs to cut samples of needed dimensions. The parameters during the cutting process were: • current for the electric arc lamp: 20 Aiii ; • pulse frequency: 5 kHz; • pulse width: 1 µs; • number of repetitions to cut a ca. 500 µm thick silicon wafer through: 8; • stage movement speed during cutting: 10 mm/s. For some experiments, to prepare quarter-wafer samples and 20 mm × 20 mm square samples, cleaving along ⟨110⟩ directions with a diamond tip scribe was used. This method, although faster and cleaner than the laser process, lacked of precision in case of internal crystal lattice defects, and obviously could not be used for preparation of the circular samples. 3.1.3. Sample anodization and post-processing 3.1.3.1. Anodization setup Two setups for anodization were used during the work. Both setups had the following common features: • configuration as a double-tank (double-chamber) anodization cell where a to be anodized silicon wafer in a wafer holder is clamped to provide charge flow between anode and cathode only through the wafer (s. Fig. 2.38); iii The

output power of the electric arc lamp reduced with time due to aging processes, therefore the current was adjusted few times during the work to achieve the required energy.

106

3 Experimental, characterization and simulation methods

• the setups are made from PVDF or PTFE; • sealing for chemical and electrical isolation is provided with sealing rings made of HF stable materials (EPDM, PTFE); • platinum meshed electrodes of circular shape are used as anode and cathode, with diameter equal or more than the diameter of a full wafer (3.25 inch or 4 inch, s. below) to provide homogenous charge flow in electrolyte to the wafer; • “wet” backside contact, i.e. electrical contact to the backside of the substrate is provided through electrolyte; • wafer, anode, and cathode are placed vertically and parallel to each other on one axis (s. Fig. 2.38); • anodization setups were closed during anodization to increase safety and reduce influence of ambient illumination; • no special illumination was used; the effect of ambient light for p-Si was proved to be negligible in preliminary tests. Other specifications of the anodization setups and corresponding wafer/sample holders are summarized in Tab. 3.8 and Tab. 3.9. Table 3.8.: Specifications of the anodization setups used in the work Anodization setup

Manufacturer

Electrolyte volume, ml

Distance wafer-cathode, mm

Distance wafer-anode, mm

setup 1: 3.25 inch

custom made

≈ 5300

≈ 90

50

setup 2: 4 inch (AMMT PSB4)

AMMT GmbH, Frankenthal, Germany

≈ 3600

≈ 75

≈ 105

For all used sample/wafer holders, electrical isolation measurements (i.e., leakage resistance 𝑅leak and leakage current 𝐼leak ) between the chambers have been done. The measurements were done with a p-type silicon wafer/sample of resistivity 10–20 Ω cm, covered on both sides with silicon nitride of 200 nm or more. With this thickness of silicon nitride, it was

107

3.1 Fabrication methods

Table 3.9.: Specifications of the wafer holders for the anodization setups Anodization setup

Wafer/ sample holder

Manufacturer

Sample size

Anodization area* 𝐴sample

Leakage resistance 𝑅leak , kΩ

setup 1, 3.25 inch

full wafer

custom made

diameter 3.25 inch, ≈ 82.55 mm

43.65 cm2, circular area of diameter 74.5 mm

n/a (not used in the work)

circular sample

custom made by the author of the work

diameter 30 mm

3.14 cm2, circular area of diameter 20 mm

3.077

full wafer

AMMT GmbH, Frankenthal, Germany

diameter 4 inch ≈ 100 mm

(60.85 ± 2.76) cm2, circular area of diameter about 88 mm

6.66

quarter wafer

AMMT GmbH, Frankenthal, Germany

quarter of 4 inch wafer

(8.41 ± 0.57) cm2, square area of 29 mm × 29 mm

3.1

square sample

custom made by the author of the work

20 mm × 20 mm(1.033 ± 0.082) cm2, 1.54 square area of about 10.2 mm × 10.2 mm with rounded corners with radius 1 mm

setup 2, 4 inch (AMMT PSB4)

* Anodization area is not clearly defined with a sealing ring of a sample/wafer holder, therefore observed deviation is also shown where detected. guaranteed that the nitride layer has no pinholes in it. Voltage sweep in the range from 0 V to 10 V was performed, and the leakage current values have been recorded. Dependence of the leakage current on the state of the wafer surface (dry or wet) has been observed. Lower and more stable leakage current values were obtained by wetting the wafer and the sealing rings with deionized water before the process. Therefore, in the experiments and for the leakage current measurements samples/wafers and the holders were always wetted before loading. Leakage resistance 𝑅leak is in parallel with the resistance of the silicon sample including interface resistances of silicon-electrolyte contacts (working resistance 𝑅work ). The total current 𝐼 is the sum of the current going through the wafer 𝐼work and the leakage current 𝐼leak :

108

3 Experimental, characterization and simulation methods

𝐼work = 𝐼 ⋅

𝑅leak 𝑅leak + 𝑅work

𝐼leak = 𝐼 ⋅

𝑅work 𝑅leak + 𝑅work

𝐼 = 𝐼work + 𝐼leak

(3.1)

Therefore, if working resistance gets comparable to the leakage resistance, that can be the case for small anodization area, when only small total current 𝐼 of few milliamperes is necessary to get high current density, significant part of applied total current flows not through the silicon sample, but as leakage current. Then, the anodization is running at a significantly reduced current, leading to erroneous results. When the valence and porosity are known, the real working current can be verified through the sample mass loss during anodization. For the experiments with low working resistance (anodization area of 1 cm2 and larger, as confirmed by evaluation of etch volume and mass), the effect of leakage current was negligible. When anodization area was limited only by the sealing rings of a wafer/sample holder, it is not well defined (s. deviation ranges for anodization area in Tab. 3.9). These deviations caused a systematic error in derivation of current density from total current. Additionally, anodization area was used for calculations of the etch rate from the mass loss. This error was taken into account in the experiments. As power supply, typically one of the two following computer controlled power sources were used: • CGN-04 Current Generator (n/a, Hungary) – maximum output voltage: 12 V – maximum output current: 10 A • Computer Controlled Laboratory Power Supply PSP 1500-060-060 (GMC-I Messtechnik GmbH, Nuremberg): – maximum output voltage: 60 V – maximum output current: 60 A – maximum output power: 1500 W The current output error of the given power sources was below 1 % of the set values, which was considered to be negligible in comparison to other systematic errors (especially the error due to the anodization area).

3.2 Characterization of samples

109

3.1.3.2. Rinsing and drying process after anodization For the study of microscale aspects of anodization process, where porous silicon structure had to be preserved for further characterization, special drying process with 2-propanol was applied. First, samples with porous silicon after anodization were rinsed in deionized water at reduced flow up to the water conductivity value of 0.1 µS/cm and then placed in 2-propanol for 10 minutes in order to reduce risk of cracking (s. sec. 2.6.1). After that, the samples were left to dry naturally in air for another 10 minutes and then characterized (e.g., mass measurement). When macroscale structuring with anodization was used, no special precautions in respect to the porous layer were taken during rinsing and drying. In this case, the samples were rinsed in deionized water to the conductivity value of 0.1 µS/cm and then dried with strong nitrogen flow. 3.1.3.3. Porous silicon removal process To remove porous silicon from the samples selectively to the bulk silicon, solution with 3:1 volume ratio of (1 m% KOH):ethanol at room temperature was used. Ethanol was used here as a surfactant to provide faster etching of porous silicon. The solution was prepared fresh for each process from solid KOH in form of pellets and deionized water. An indication of the removal process was active evolution of hydrogen bubbles. Typically, porous silicon was removed in 10 minutes, depending on the thickness and morphology of the porous layer. For few experiments, porous silicon did not remove readily in the KOH solution due to native oxide. After dipping the samples in concentrated 50 m% HF or buffered HF solution at RT to remove the oxide, the porous silicon could be removed completely in the KOH solution selectively in respect to bulk silicon. 3.2. Characterization of samples 3.2.1. Mass measurement To evaluate the porosity, porous silicon formation rate, and dissolution valence for study of microscale aspects of anodization process in chapter 4,

110

3 Experimental, characterization and simulation methods

mass measurements of samples before anodization, after anodization, and after porous silicon removal were conducted. Mass of the samples was measured with calibrated precision scales Sartorius BP211D (Sartorius AG, Göttingen, Germany) of precision class I. The scales provided measurement with accuracy (absolute measurement error) of ±0.5 mg as half of the verification value for these scales of precision class I [387] and precision (relative error, reproducibility) of 0.05 mg according to the data sheet. Since anodization for the microscale studies was performed only up to the depth of 10 µm, mass measurements were applied only to full 4-inch wafers and quarters of 4-inch wafers, because for smaller samples the mass differences for gravimetric measurements would result in too big measurement errors. The calculations were performed according to equations (2.30), (2.31), and (2.32). In order to achieve stable results of mass measurements, the following procedure was applied: • ensure that the sample is dry and clean; • place the sample on the measurement plate; • wait 3 min for the mass value to stabilize; • read the mass value; • remove the sample from the scales; • wait 3 min, so that the scales stabilize and show zero again; • proceed with a next sample. 3.2.2. Microscopic images Optical micrographs were taken with microscope Zeiss Axio Imager.A1m with AxioCam ICc 1 (Carl Zeiss AG, Oberkochen, Germany). Additionally, a scanning electron microscope JSM-5400 (Jeol, Japan) was used in the work. The microscope provided a magnification up to 200.000× and a practically achievable minimum feature size of 50 nm. When needed, gold of thickness 15–20 nm was sputtered on samples to enhance the quality of the images for non-conductive materials.

3.2 Characterization of samples

111

3.2.3. Surface quality characterization Surface quality of anodized samples (electropolished or after removal of porous silicon) was evaluated from atomic-force-microscope (AFM) topographical scans taken with Autoprobe CP (Park Scientific Instruments, Sunnyvale, USA) in contact AFM mode with MicroleversTM cantilever (model MSCT-AUMT-BF) using sharpened pyramidal tip of length 3 µm and the tip angle of 36°. From the scans, the following parameters were evaluated in program SPIP v.3.2.13.0 (Image Metrology A/S, Hørsholm, Denmark): • RMS roughness 𝑅q [388] • mean fractal dimension value 𝐷mean [389] RMS roughness characterizes only the height variation neglecting lateral information (i.e., spatial frequency). Moreover, it is very scale-sensitive. To provide more information on the studied three-dimensional surfaces, the mean fractal dimension was also evaluated as a measure of the spatial complexity of roughness. The fractal dimension is a number in the range from 0 to 3, where 0 represents a point, 1 – a line, 2 – a surface, and 3 – a volume (complex rough surface filling a volume). It was calculated in the program as 2 minus the slope of the log-log curve of the amplitude as a function of spatial frequency obtained from the Fourier analysis of the surface topography. This way, the fractal dimension values for different angles (directions) were obtained, and then the mean fractal dimension was calculated. Scan region for evaluation of the parameters was 10 µm × 10 µm, thus possible scale effects were not studied. Typically, several scan regions of 10 µm × 10 µm in different locations were taken for analysis on a sample, and the average values of the parameters and the deviation ranges were obtained this way. 3.2.4. Etch shape measurements Two-dimensional and three-dimensional topographical scans of the etched cavities were mostly taken with Dektak 150 stylus profiler (Veeco Metrology, Tucson, USA) and evaluated in Wyko Vision v.4.10 (Veeco Instruments Inc., Plainview, USA). The stylus with tip radius of 12.5 µm (Veeco Metrology, Tucson, USA) was used. The vertical resolution of the profiler is 16 nm in the height range up to 1 mm, the lateral resolution is 0.5 µm. On the topographical scans measured with this profiler, the steep slopes on both

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sides of the profiles were not exactly resolved, and appeared as straight lines. However, this did not affect the evaluation of the etch forms. On the plots, scan distance is given as 𝑥 or 𝑟, and elevation as 𝑧. Few scans were taken also with a laser-scanning microscope (LSM) Axioplan 2 Imaging with LSM 5 PASCAL and Axiocam MRC5 (Carl Zeiss AG, Oberkochen, Germany)iv , however, this method was found to be not suitable for measuring deep cavities with steep walls. For analysis of the measured profiles, author’s custom scripts in Python programming language v.3.4.1 with NumPy library in the Spyder 2.3.1 programming environment were used. As described in sec. 2.6.3, localization of the anodization process with frontside insulating masking film results often in convex shapes due to current crowding near the mask edges. The effect is sometimes called the edge effect. Such convex shapes were characterized with the following parameters (Fig. 3.6a): • structure depth 𝑑etch as the maximum depth value in the profile; • etch width 𝑤etch as the width of the cavity measured at 5 % of maximum depth; • radius of curvature 𝑅 and curvature 𝜅 = 𝑅1 in the central part of the cavity: to evaluate these parameters, arc fit to the central part of structures was applied; obtained this way the radius of curvature and the curvature for convex profiles were taken as negative values (s. Fig. 3.6); in the work, only curvature was used, because radius of curvature is poorly applicable for characterization of nearly flat shapes (𝑅 → ±∞); • surface area of the structure: surface area of the structure was evaluated based on the left-hand and the right-hand halves of the measured profile by rotating the half-profiles 360° around the axis going through the center of the structure; from the two values of the area for the lefthand and the right-hand half-structures, the mean value was taken as the area of the structure; taking the average value was done to compensate for the minor asymmetry observed in some measured profiles; • volume of the structure: similarly to the evaluation of the surface area, mean volume of the structure was obtained from the volumes iv These

measurements were done in University of Freiburg im Breisgau / IMTEK with great support of Dr. F. Goldschmidtböing and Dr. M. Kröner.

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3.2 Characterization of samples

calculated by integration of the measured left-hand and right-hand half-profiles in cylindrical coordinates; • the anisotropy factor based on the etch depth and length of mask underetching according to eq. (2.2).

R>0 0.6 w0.5

w0.5 R

d etch

d etch

R0

(c) funnel profile wetch= 4 w0.5 0.05 d etch

wetch

d etch

(b) concave profile

wetch 0.05 d etch

(a) convex profile

Figure 3.6.: Characterization of etch cavities of (a) convex, (b) concave, and (c) funnel shape. For concave shapes, the same parameters as for convex shapes were applied (Fig. 3.6b). The radius of curvature 𝑅 and the curvature 𝜅 in the center of a concave profile were taken as positive values. There were also other types of shapes observed during the development of etch shapes from convex to concave for frontside mask localization (s. Fig. 5.4). The parameters described above could also be applied for their characterization. Characterization of convex shapes in terms of edge effect factor [334] was not used in the work because of the limitation of the parameter to only convex shapes. The chosen method of shape evaluation with curvature is much more universal in this sense. Anodization of samples with local backside contacts without frontside masking films resulted in funnel-like profiles with broad and not clearly defined borders at the top surface (Fig. 3.6c). Thus, the structure width could not be clearly determined at the top surface. However, in average, over 90 % of the structure volume was within a quadruple of the width 𝑤0.5 measured at the half depth, therefore the full structure width 𝑤etch was defined as 𝑤etch = 4 𝑤0.5 . The anisotropy factor based on the full structure width 𝑤etch and depth 𝑑etch was also calculated. Additionally, curvature of the bottom part of the structure was evaluated by performing an arc fit to the central

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3 Experimental, characterization and simulation methods

region of width 0.6 𝑤0.5 which provided a good fit with the average coefficient of determination of 0.95. Two-dimensional topographical scans have been also used to evaluate the etch depth and etch rate for small samples of dimensions 20 mm × 20 mm used for study of microscale parameters of the anodization process in chapter 4, because evaluation from mass, as done for full-area anodized 4-inch wafers, would result in significant measurement errors for these smallarea samples. 3.3. Macroscale simulations of the anodization process Macroscale time-dependent finite-element simulations of the anodization process were performed in COMSOL MultiphysicsTM (COMSOL Inc., Burlington, USA), versions from 4.1 to 5.1, with the Electrodeposition (edsec) physics interface which is applicable for etching/corrosion processes as well. Primary and secondary current distribution models were solved for two doping levels of silicon - low-doped silicon of conductivity 7.5 S/m (corresponding to resistivity of approximately 13 Ω cm) and highly-doped silicon of conductivity 104 S/m (corresponding to resistivity of 0.01 Ω cm). In the following, general model description and parameters are explained. Description of parameters is limited to those which were changed from their default values in the software. Specific terms of the COMSOL software user interface are italicized. 3.3.1. Model geometry The case of electrochemical etching of silicon through a single circular opening in a frontside insulating masking film was studied in the work. To reduce calculation time, 2D model geometry with axial symmetry was applied. The geometry consisted of the following domains (Fig. 3.7): • electrolyte domain of radius and height of 5 mm; chosen dimensions represent the case of a single opening in the frontside masking layer, with outer boundaries of the cell located far enough to have no direct influence on the etch form development; • silicon substrate of thickness 525 µm and radius 5 mm;

115

3.3 Macroscale simulations of the anodization process

• void region representing insulating frontside masking film (e.g., silicon nitride) of thickness 1 µm on the silicon substrate with an opening of radius 𝑅open ; the model was solved for 𝑅open from 200 µm to 1000 µm with step of 200 µm; thickness of the silicon nitride layer in the experiments was below 300 nm; a larger value of 1 µm was used here in order to reduce number of mesh elements by avoiding very small features in the model geometry; replacing the narrow mask region with void shape helped to avoid bad quality mesh in this region and reduce number of mesh nodes; • “predefined etch form” of radius 𝑅open + 5 µm and thickness 5 µm for enhanced mesh movement; this domain belongs to the electrolyte region.

5

z , mm

cathode

electrolyte

void region as frontside mask

1µm

Ropen 0

0

m 5µ

5 µm

0.525

initial etched form

silicon

anode

, mm 5 r

Figure 3.7.: Geometry of the 2D model of the electrochemical cell with electrolyte (top region), silicon substrate (bottom region), and axial symmetry at 𝑟 = 0; schematic view, not in scale. Simulations showed that over 92 % of charge flow comes from the central region of the substrate backside of radius 1.5 mm (s. sec. 5.4.1 and sec. 5.4.2). Therefore, the anodization cell radius (radius of the electrolyte and silicon substrate domains) 𝑅model of 5 mm was defined for all models to ensure that it has negligible influence on the shape development. Additional simulations with reduced radius 𝑅model of 1.5 mm and 1 mm showed that there is almost

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3 Experimental, characterization and simulation methods

no difference between the results for the radius 𝑅model of 1.5 mm and 5 mm (s. sec. 5.4.1). Only at the value of the radius 𝑅model of 1 mm some effect of the reduced cell radius on the etch form development was obtained, because at such radius, in the model with the diameter of opening of 1000 µm etch form came laterally close to the sidewall of the substrate domain (distance from the etch form to the sidewall of 146 µm for structure depth of 510 µm). The top boundary of the electrolyte domain was defined as cathode, and the bottom boundary of the silicon substrate was defined as anode. The geometry was built by a generation sequence of elementary shapes (rectangles and polygons), and boolean operations (difference and union). The geometry was finalized as a union, which means that adjacent domains shared same nodes on the boundaries. To improve stability of the model and reduce number of mesh nodes, right angles on the mask edge and the initial etched form were replaced by fillets as shown in Fig. 3.7. 3.3.2. Mesh configuration For numerical simulations, building a proper mesh is very important, because it determines whether the solver can find a solution, and whether the found solution is accurate. The mesh of the model used in the work is shown in Fig. 3.8. Free triangular mesh was applied for all domains. The most important part of the geometry was the region in the electrolyte and substrate near the etch front, and it had to be meshed accordingly fine. In order to reduce number of mesh nodes in other regions, the important parts of the silicon and electrolyte domains were separated with additional borders shown in Fig. 3.8a with dashed lines. Different regions of the model were meshed separately, starting from the most important regions near the moving etch front boundary 𝐚. Parameters of the mesh in different domains, in order of meshing, are summarized in Tab. 3.10. For the parameters not specified manually, default settings of normal mesh were used. Total number of nodes of the initial mesh was up to approximately 42000, dependent on the radius of the frontside opening 𝑅open .

117

3.3 Macroscale simulations of the anodization process

(a)

(b) l

V

m

n III I d II i e

j h IV f

k

z , µm

z , µm

III

I

c a

j i

b

II g

r , µm r , µm Figure 3.8.: Mesh of the model: (a) general view; (b) zoomed-in view on the edge of the frontside mask; domains are referenced with roman numerals from I to V, and boundaries with bold lower case letters from 𝐚 to 𝐧.

Table 3.10.: Parameters of the model mesh Entity

Min. element size, µm

Max. element size, µm

Growth rate

Resolution of narrow regions

domain I boundary 𝐛 boundary 𝐝 domain II boundary 𝐧 domain III domain IV domain V

5 5 -

20 20 20 50 50 -

1.05 1.01 1.01 1.05 1.1 1.1 1.1 1.1

0.5 0.5 0.5 0.2 -

In order to test that the mesh is sufficiently fine, a convergence test [390] (solving with an approximately twice finer mesh) was done. The results produced with the finer mesh matched the results produced with the rougher mesh described here with deviations below 5 %. Therefore, the rougher mesh was considered as sufficiently fine and was used for all simulations.

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3 Experimental, characterization and simulation methods

During the process, movement of the etch front at each solved time step resulted in deformation of the mesh. In order to avoid too large mesh distortion, automatic remeshing function was activated. As a condition for remeshing, square root of the maximum element distortion being above or equal to two was defined. Free movement of the mesh was defined for the domains I, II, and IV. The remaining domains were fixed. Additionally, in order to improve stability of the solving, additional boundary conditions were applied to the following boundaries: • boundaries 𝐜, 𝐞, 𝐟 , and 𝐠: 𝑑𝑟 = 0, 𝑑𝑧 = 0 • boundaries of the domains I and II on the symmetry axis 𝑟 = 0: 𝑑𝑟 = 0 • boundaries 𝐛, 𝐡, and 𝐢: 𝑑𝑧 = 0 where 𝑑𝑟 and 𝑑𝑧 are the prescribed mesh displacement values in 𝑟 and 𝑧 direction, respectively. In the model, the process of silicon anodization, in both pore formation and electropolishing regimes, was simulated. However, generation of the porous layer or intermediate anodic oxide was not included, and direct removal of the material with current density dependent dissolution valence and porosity was simulated instead. It was observed that the movement of the etch front in the primary current distribution model for the case of the low-doped silicon of conductivity 7.5 S/m was not stable, and the initially smooth etch front transformed to a rough wavy interface, resulting in failure of finding solution. This instability was likely due to the chosen model parameters, namely, that the conductivity of the electrolyte was more than four times higher than that of the low-doped silicon substrate (s. below in sec. 3.3.5). With such conditions, an etched point in low conductive silicon substrate filled with high conductive electrolyte provided increased current to this point, thus pronounced etching in this point persisted, as previously discussed in stability analysis of anodization process (s. sec. 2.4.2.5) [103,278]. In the work, however, only the macroscopic etch front movement was of interest. Therefore, to stabilize the etch front, additional built-in smoothing function which smoothed locally the etch front boundaries 𝐚 and 𝐛 was applied. Special care was taken to check that this smoothing does not alter the etch form development on the macroscale. Additionally, at the end points of the etch front (i.e., the points, lying on the bottom boundary of the mask domain and on the symmetry axis),

3.3 Macroscale simulations of the anodization process

119

especially for the point adjacent to the mask, high etch rate was observed for the low-doped silicon secondary models. This high etch rate resulted in the etch front boundary getting not normal to the corresponding adjacent regions. No physical explanation of such high local etch rate could be found, because: • in the models, a void region (i.e., not solved for electric fields) was used as a mask, therefore the interface silicon-mask did not include any special phenomena such as field effect which could be responsible for accumulation of the charge under the mask resulting in high surface charge flow; • at the symmetry axis, one expects that the etch front boundary is a smooth function which means that the etch front boundary should remain normal to the symmetry axis. Therefore, this behavior was considered as an artifact of numerical calculations and was suppressed with additional reduction coefficients 𝛾1 for the part of the etch front near the symmetry axis, and 𝛾2 for the part of the etch front near the mask. The values of 𝛾1 for each moment of a simulated process were calculated by evaluation of the difference between the 𝑧 coordinate of the point of the etch front on the symmetry axis and the 𝑧 coordinate of the point of the etch front at 𝑟 equal to 5 % of the structure width. The value of 𝛾2 was calculated in a similar way considering the difference between the 𝑟 coordinates of the points on the etch front at the mask and at 𝑧 equal to 15 % of structure depth. These reduction coefficients were entered in the stoichiometric coefficient field together with the porosity (s. below). The reduction coefficients were applied with full strength at the end points, and with decreasing effect up to the first 15 % of the structure depth (for the part of the etch front near the mask) or 5 % of the structure width (for the part of the etch front near the symmetry axis). With these reduction coefficients, etch front boundaries were stabilized to stay normal to the mask and the symmetry axis. In order to provide comparable results between the models, the same mesh configuration including smoothing and reduction coefficients was used in all models developed in this work. In the models, where without reduction coefficients etch front boundary remained normal to the mask and the symmetry axis, application of the reduction coefficients showed no significant influence on the etch front movement.

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3 Experimental, characterization and simulation methods

3.3.3. Time-dependent solver configuration and stability issues In the simulation program, one can define time steps for which the model is solved, and time steps for which the solution (values of variables) is saved. In order to help the solver to find a solution, initial time step of 10 µs (for initial mesh and every remeshed model) was applied. The maximum time step in seconds was defined as 70 [A/cm2 ]/𝑗init , where 𝑗init in A/cm2 is the initial current density defined for the area of the opening in the 2 frontside insulating layer 𝐴open = 𝜋𝑅open . The models were solved for 𝑗init from 1 A/cm2 to 3.5 A/cm2 with step of 0.5 A/cm2 , and additionally for 0.05 A/cm2 , 0.1 A/cm2 , 0.2 A/cm2 , and 0.5 A/cm2 for the models with the highly-doped silicon, according to the experiments. During the process, due to increase of the anodization area, current density decreases significantly, so does the etch rate. Therefore, saving the data for smaller time steps in the beginning and bigger time steps later in the process was set to reduce the size of the saved data. This was realized by defining the saved times as {range(0,𝑡step ,50)}^3 which means an array of time values in seconds calculated as 𝑡 = (𝑚𝑡step )3 with 𝑚𝑡step in the range 0–50 and 𝑡step varied to store solutions for 𝑚 (serial number of solved solution) from zero up to 40–150. Further settings in the program which were used to improve the solving performance: • non-linear method: constant (Newton) with Jacobian update on every iteration • consistent initialization: on 3.3.4. Further general model parameters The anode was defined as electric ground. On the cathode, electrolyte current density was defined with the value 𝑗init 𝐴open /𝐴cathode , where 𝐴cathode is the area of the cathode (circular area of radius 5 mm). Thus, the models were solved for constant current corresponding to the initial current density and the area of the frontside openings. This was done in order to have the models comparable to the experiments. Since the electrodeposition module was applied in the work for the study of the dissolution process, the option Solve for depositing species concentrations was deactivated.

3.3 Macroscale simulations of the anodization process

121

Layers at the interface silicon-electrolyte (s. sec. 2.3.3.1) responsible for interfacial capacitance were not described in the model, assuming that for this model with constant current supply and macro-scale features of size over micrometer no significant influence of them could be expected. For each simulated current density value and opening diameter, the etch profiles were exported for all saved time values. From these data, evaluation of curvature and anisotropy factor in dependence of structure depth was done. Curvature was evaluated by performing an arc fit to the central 40 % region of the etch forms, which proved to provide correct values of curvature for all obtained etch forms. Due to remeshing during the simulation, small steps on the curves of curvature vs. structure depth were observed in some cases. Conductivity of electrolyte was set to 34.11 S/m (corresponds to 29.93 m% aqueous HF electrolyte, s. sec. 2.3.2). For the electrode reaction, current density dependent dissolution valence and porosity were defined as will be shown in sec. 5.4.1 and sec. 5.4.2. The dissolution valence 𝑛e was set in the electrode reaction node as number of participating electrons and reciprocal of porosity was set as stoichiometric coefficient 𝜈 to get the etch rate equation according to eq. (2.40) and eq. (2.38): 𝑅etch =

𝑗 𝜈𝑀Si ⋅ 𝑛e 𝑒 𝜌Si 𝑁A

(3.2)

3.3.5. Specific parameters for secondary current distribution models In secondary current distribution models, additionally to the consideration of conductivities of the electrode (silicon) and electrolyte, as in the primary current distribution models, charge transfer control resulting in activation overpotential at the interface electrode-electrolyte is taken into account as was explained in sec. 2.3.3.2. The activation overpotential is defined with charge transfer equations, such as, for example, Tafel equation (2.14) or Butler-Volmer equation (2.21). Considering the irreversible process of silicon anodization process, the polarization curves for p-type silicon electrode in an HF-based electrolyte in pore formation regime can be described with asymmetric Butler-Volmer equation with the transfer coefficient 𝛼 = 1 and dissolution valence 𝑛e = 2. Various references show that the equilibrium potential 𝑈eq is in the range from –0.89 V to –0.46 V vs. SHE [391–394].

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3 Experimental, characterization and simulation methods

The exchange current density 𝑗0 for low-doped p-type silicon (resistivity 1–10 Ω cm) is 7 × 10−6 A/cm2 , and for highly-doped p-type silicon (resistivity 0.0114–0.0126 Ω cm) is 8 × 10−4 A/cm2 [98]. With these parameters, polarization curves of the interface silicon-electrolyte according to the ButlerVolmer equation (2.21) are described with eq. (3.3) as shown in Fig. 3.9.

Current density, A/cm2

𝑗 = 𝑗0 [exp (

2𝐹 𝜂 ) − 1] 𝑅𝑇

1.0

(3.3)

low doped Si highly doped Si

0.8 0.6 0.4 0.2 0.0

−1.0 −0.8 −0.6 −0.4 −0.2

0.0

0.2

Overpotential η, V

Figure 3.9.: Polarization curves for anodization of low-doped (1–10 Ω cm) and highly-doped (0.0114–0.0126 Ω cm) p-type silicon at 20 °C. Due to this activation polarization, with initial potentials in the electrolyte and silicon domains equal to zero (default values), the software could not find consistent initial conditions. To help the solver find a solution for the initial conditions, the initial potential in the electrolyte 𝜙𝑙0 was defined according to eq. (3.3) for initial current density 𝑗init and 𝜙s0 = 0 as following: 𝜙𝑙0 =

𝑅𝑇 𝑗 ⋅ ln ( init + 1) 2𝐹 𝑗0

3.3.6. Hardware configuration The following hardware configuration was used:

(3.4)

3.3 Macroscale simulations of the anodization process

123

• CPU: Intelr CoreTM i7-3770 @ 3.40 GHz, 4 cores • video card: NVIDIAr GeForceTM GT520, 2048 MB • RAM: 16 GB • 120 GB SSD + 3 TB HDD • Windows 7 Professional (©Microsoft), 64-bit With this configuration, depending on number of mesh nodes, time steps, etc., solving of a time-dependent axisymmetric 2D simulation for one combination of parameters (silicon resistivity, 𝐷open , 𝑗init , type of current distribution) typically took up to 30 min.

4. Microscale study of anodization process In order to apply the process of silicon anodization for structuring and develop a model, it was necessary to investigate the process on the microscale and check whether our data match the results from other researchers. In this chapter, dependence of microscale parameters of the anodization process (porosity, dissolution valence, etch rate, and interfacial surface quality) on current density and wafer resistivity for p-type silicon wafers was studied. Additionally, surface quality for p-type silicon samples anodized in low concentrated electrolytes, as well as influence of porous silicon removal process on surface quality, were investigated. As was described in chapter 3, some of the main properties of the anodization process in pore formation regime (valence and porosity) have been obtained gravimetrically. Gravimetric measurements, however, were restricted to big area samples (preferably full area 4-inch wafers, also quarters of 4-inch wafers), due to mass measurement errors of the used precision scales. On the other hand, anodization experiments were limited by the maximum applied current which the anodization setup can withstand (up to 15–20 A for very short processes of few seconds). Thus, only low current density could be used for anodization of full 4-inch wafers. For higher current densities, extended studies have been conducted with smaller samples. For this reason, this chapter is divided into two sections. The first section describes the anodization experiments on full 4-inch wafers for low current density range below 90 mA/cm2, and is therefore limited to pore formation regime. In the second section, study of the extended range of current densities up to 3 A/cm2 , where electropolishing regime in low concentrated electrolytes was achieved, is described. 4.1. Study of pore formation regime with full wafers Several studies have been performed to investigate influence of the process conditions, namely, wafer resistivity and current density on properties of the anodization process in the pore formation regime, as described below. The experiments on full 4-inch p-type silicon wafers for the resistivity © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_4

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4 Microscale study of anodization process

ranges 0.001–0.005 Ω cm and 0.005–0.01 Ω cm have been conducted within this work. The experiments for the resistivity ranges 0.01–0.02 Ω cm and 10–20 Ω cm have been conducted previously in the research groupi , with surface roughness characterization and complete analysis being done by the author of this work. Conducting experiments on full wafers, although not economically favorable, provides good precision of results due to large anodization area, resulting in relatively small leakage current and big mass changes for the gravimetric analysis. For the p-type silicon wafers with resistivity in the range 10–20 Ω cm, backside was highly doped as explained in sec. 3.1.2 to provide ohmic contact of the wafer backside to electrolyte. All wafers were anodized in 29.93 m% HF with ethanol (s. Tab. 3.2), because this electrolyte has proved to provide growth of uniform porous silicon layers. Other experimental conditions for all wafers were the following: • anodization in the setup 2 at room temperature (s. Tab. 3.8); • current density range up to 84 mA/cm2 ; • thickness of the porous layer for all the samples (estimated before the experiments) kept about 10 µm by correspondingly adjusting the anodization duration to compensate for etch depth effects; 4.1.1. Current density sweep Influence of current density on the dissolution valence is shown in the left plot in Fig. 4.1. For all the studied resistivity ranges but 0.01–0.02 Ω cm, slight increase of the dissolution valence with increase of current density was observed. This result is in good agreement with the data from Frohnhoff et al. (s. Fig. 2.18b).

i Those

experiments for the resistivity ranges 0.01–0.02 Ω cm and 10–20 Ω cm were planned and performed by M.Sc. Prasad Jonnalagadda and Dr. Andras Kovacs (unpublished).

4.1 Study of pore formation regime with full wafers

0.001−0.005 Ω cm 0.005−0.01 Ω cm 0.01−0.02 Ω cm 10−20 Ω cm

60

Growth rate, nm/s

Dissolution valence

3.0 2.8 2.6 2.4 2.2

127

50 40 30 20 10

2.0 0

20

40

60

80

Current density, mA/cm2

0

0

20

40

60

80

Current density, mA/cm2

Figure 4.1.: Dissolution valence (left) and growth rate (right) of porous silicon vs. current density for p-type silicon wafers of different resistivity ranges anodized in 29.93 m% HF with ethanol at RT.

Almost linear dependence of the growth rate on current density was obtained as shown in Fig. 4.1 (right). The result fits well to the data from Lehmann and Rönnebeck for similar experimental conditions (Fig. 2.20b). A detailed discussion on the dependence of anodization rate on current density in pore formation and electropolishing regimes is given below in sec. 4.2.1. Moderate increase of porosity with current density can be seen in Fig. 4.2. These data fit quite well to the data from Frohnhoff et al. (Fig. 2.19b), with the porosity values for 10–20 Ω cm being about 5 % higher here.

128

4 Microscale study of anodization process 80 0.001−0.005 Ω cm 0.005−0.01 Ω cm 0.01−0.02 Ω cm 10−20 Ω cm

Porosity, %

70 60 50 40 30

0

20

40

60

80

Current density, mA/cm2

Figure 4.2.: Porosity of porous silicon vs. current density for p-type silicon wafers of different resistivity ranges anodized in 29.93 m% HF with ethanol at RT.

Roughness of the interface between bulk silicon and porous silicon for the highly-doped p-type silicon wafers (0.001–0.005 Ω cm and 0.005–0.01 Ω cm) remained below 10 nm (s. left plot in Fig. 4.3). In contrast, the lower-doped p-type silicon wafers showed higher roughness of up to 108 nm. It is important to note that this largest value of roughness was obtained for the wafer of resistivity in the range 10–20 Ω cm. For this doping level, there is only one data point measured, because other wafers used for the analysis were not available for this studyii , therefore this data point was not considered as a reliable result. In the measured data, in contrast to the data from Lérondel et al. (s. Fig. 2.21), no clear decrease of roughness with increase of current density could be observed. Mean fractal dimension varies in the range between 2.45 and 3 also without clear tendencies (s. right plot in Fig. 4.3).

ii As

explained above, the full wafers of resistivity 10–20 Ω cm were anodized previously in the research group of the author.

4.1 Study of pore formation regime with full wafers

Mean fractal dimension

RMS roughness, nm

102

101

0

20

40

60

80

Current density, mA/cm2

129

0.001−0.005 Ω cm 0.005−0.01 Ω cm 0.01−0.02 Ω cm 10−20 Ω cm

2.9 2.8 2.7 2.6 2.5 0

20

40

60

80

Current density, mA/cm2

Figure 4.3.: Interface RMS roughness (left) and mean fractal dimension (right) vs. current density for p-type silicon wafers of different resistivity ranges anodized in 29.93 m% HF with ethanol at RT.

4.1.2. Substrate resistivity sweep in the range 0.001–20 Ohm cm for p-type silicon Study of the process for four ranges of substrate resistivity for p-type silicon was performed. The effect of the substrate resistivity on the dissolution valence is shown in Fig. 4.4a. Clear decrease of the dissolution valence from 2.7–2.9 to 2.1–2.3 with increase of the substrate resistivity from 10−3 Ω cm to 10–20 Ω cm is observed, similar to the results reported before (s. Fig. 2.33b).

130

4 Microscale study of anodization process

(a)

(b) 100

(10.13 ± 0.82) mA/cm2

3.0

(10.13 ± 0.82) mA/cm2

(30.38 ± 2.67) mA/cm2

(30.38 ± 2.67) mA/cm2

(50.63 ± 4.11) mA/cm2

(50.63 ± 4.11) mA/cm2

(70.88 ± 5.76) mA/cm2

2.6

2.4

2.2

(83.61 ± 3.98) mA/cm2

80

Growth rate, nm/s

Dissolution valence

2.8

2.0 10−3

(70.88 ± 5.76) mA/cm2

(83.61 ± 3.98) mA/cm2

60

40

20

10−2

10−1

100

Resistivity, Ω cm

101

0 −3 10

10−2

10−1

100

101

Resistivity, Ω cm

Figure 4.4.: (a) Dissolution valence and (b) growth rate of porous silicon vs. resistivity of p-type wafers anodized in 29.93 m% HF with ethanol at RT and different current densities; the resistivity range is shown as horizontal error bars; the data in (a) were fitted with reciprocal functions for resistivity; the data in (b) were fitted with a polynomials of second order applied to common logarithm of resistivity.

The influence of the substrate resistivity on the growth rate is more complex (s. Fig. 4.4b). First, the growth rate increases with increase of the substrate resistivity in the range from 1 × 10−3 Ω cm to 2 × 10−2 Ω cm. Then, a decrease to the initial values takes place. Because only one resistivity range (10–20 Ω cm) defines this decreasing tendency, with all other points showing the opposite, one can hardly rely on the chosen fit curves based only on these data. However, the data published by Lehmann et al. (s. Fig. 2.34b, datasets for 3 mA/cm2, 30 mA/cm2 , and 300 mA/cm2 ) show quite similar behavior, thus, confirming our results.

131

4.1 Study of pore formation regime with full wafers

Porosity changes in opposite direction to the growth rate (first decrease, and later increase with increase of the resistivity, s. Fig. 4.5a), and the data also fit to the data published by Lehmann et al. (s. Fig. 2.35b, datasets for 3 mA/cm2, 30 mA/cm2 , and 300 mA/cm2 ). (a)

(b) (10.13 ± 0.82) mA/cm2

90

(10.13 ± 0.82) mA/cm2

102

(30.38 ± 2.67) mA/cm2

(30.38 ± 2.67) mA/cm2

(50.63 ± 4.11) mA/cm2

(50.63 ± 4.11) mA/cm2

(70.88 ± 5.76) mA/cm2

(83.61 ± 3.98) mA/cm2

RMS roughness, nm

80

70

Porosity, %

(70.88 ± 5.76) mA/cm2

(83.61 ± 3.98) mA/cm2

60

50

40

101

30

20 −3 10

10−2

10−1

100

Resistivity, Ω cm

101

10−3

10−2

10−1

100

101

Resistivity, Ω cm

Figure 4.5.: (a) Porosity of porous silicon and (b) interface RMS roughness vs. resistivity of p-type silicon wafers anodized in 29.93 m% HF with ethanol at RT and different current densities; the resistivity range is shown as horizontal error bars; the data in (a) were fitted with a polynomials of second order applied to common logarithm of resistivity.

As was already mentioned in sec. 4.1.1, significant increase of the interface roughness with increase of substrate resistivity was observed (Fig. 4.5b). This result confirmed the observation of Lérondel et al. (s. Fig. 2.25). It is important to note that due to rather non-uniform distribution of the chosen resistivity ranges, there are no data obtained for moderately-doped

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4 Microscale study of anodization process

substrates (0.1–1 Ω cm), therefore the datafits on the plots presented above should be used with caution. 4.2. Extended study of pore formation and electropolishing regimes with small samples As was mentioned above, one of the reasons to use samples smaller than full 4-inch wafers was that the total applied current in the anodization setups is limited due to safety concerns to few amperes for long processes, and to 15–20 A for very short processes of few seconds. Therefore, in order to perform anodization at high current densities up to 3 A/cm2, smaller samples were used. Additional reason for using small samples was economical. Quarters of 4-inch wafers and 20 mm × 20 mm samples were anodized. All experiments in this section were performed in the anodization cell of setup 2 (s. sec. 3.1.3.1) at room temperature. 4.2.1. Extended current density sweep in 29.93 m% HF with ethanol The aim of the extended current density sweep was to see change of porosity, valence, and anodization rate before, during and after the process change from pore formation to electropolishing regime. As was discussed in sec. 2.4.4, the critical current density values may vary significantly depending on mass transport and kinetic conditions. As a reference for the 29.93 m% HF ethanoic electrolyte, 𝑗PSL ≈ 600 mA/cm2 given by Lehmann et al. [268] was used. Therefore, current density range 0.1–3 A/cm2 was defined. Since p-type silicon substrates of resistivity 10–20 Ω cm were considered as main material for studies of anodization process as a structuring technique (sec. 2.6.1), detailed information on values of the dissolution valence, porosity, and anodization rate at elevated current densities had to be evaluated as well for this resistivity. Therefore, for this material, the current density sweep in the range 0.1–1.5 A/cm2 (total current up to 12.6 A) was performed with quarter wafers in order to be able to perform gravimetric measurements with acceptable precision. Experiments with higher currents were not possible due to voltage limitation of the used power supply. To get information on anodization process for even higher current densities, 20 mm × 20 mm samples had to be used. Two resistivity ranges,

4.2 Extended study of pore formation and electropolishing regimes …

133

0.01–0.1 Ω cm and 10–20 Ω cm, were selected, and anodization was done for current densities up to 3 A/cm2. Since the gravimetric measurements would deliver too big errors for these samples with small anodization area, only anodization rate was measured with stylus profiler. After anodization, the samples anodized at current densities below 500 mA/cm2 had a porous layer of light brown color. On the samples anodized at current densities of 500 mA/cm2 and above, diffusing gray surface has formed. The porous silicon layers generated at 100 mA/cm2 and 250 mA/cm2 were found to be mechanically very unstable: the films of thickness over few micrometers after contact to air started to break out in form of curls during unloading from the sample holder. This was critical for the quarter wafer samples as the mass of the samples after anodization could not be measured correctly in this case. Therefore, another sample with reduced thickness of approximately 1.5 µm had to be anodized, which appeared to be more stable and could be dried without damages with intermediate isopropanol dip. Due to reduced thickness, the mass measurement error for this sample was much larger than for the other quarter wafer samples due to smaller mass loss during anodization and removal of porous silicon. When immersed to the 1 m% KOH with ethanol, fast reaction of porous silicon etching with intensive hydrogen bubble evolution was observed in the first few seconds, indicating that all samples had porous silicon layer. Reduction of mass of the quarter wafer samples during the KOH etch step confirmed this. This result was quite unexpected, because for the samples anodized over 600 mA/cm2 [268], pure electropolishing was expected according to Fig. 2.18a. Thus, the process above 600 mA/cm2 was still running in the first electropolishing plateau between the two current peaks 𝑗PSL and 𝑗ox on the polarization curve (s. Fig. 2.14), i.e., in the mixed regime of pore formation and electropolishing. Therefore, for the given process conditions one can conclude that 𝑗ox > 3 A/cm2. Dependence of dissolution valence and porosity of porous silicon for the quarter wafer samples of resistivity 10–20 Ω cm on current density is shown in Fig. 4.6. The values obtained in sec. 4.1.1 have been also added to the plot, and they lie in one trend with the new points. As was already noticed in optical observation of the samples after anodization and during removal of porous silicon in KOH, pure electropolishing was not achieved even at 3 A/cm2. The plots of porosity and dissolution valence confirm this: the value of dissolution valence of 4 and porosity of 100 % were not reached,

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4 Microscale study of anodization process

although some increase was observed, meaning that with increase of current density there was an increase of the area of the etch front where the dissolution of silicon ran according to the reactions corresponding to the electropolishing regime (s. sec. 2.4.3). As mentioned above, reduced thickness of the sample anodized at 𝑗 ≈ 250 mA/cm2 caused larger error in the mass measurements, resulting in higher error of the porosity value for this sample.

95

3.0 2.8 2.6 2.4 full wafer quarter wafer logarithmic fit

2.2 2.0 0.0

0.5

1.0

Current density, A/cm2

1.5

Porosity, %

Dissolution valence

3.2

90 85 80 full wafer quarter wafer logarithmic fit

75 70 0.0

0.5

1.0

1.5

Current density, A/cm2

Figure 4.6.: Dissolution valence and porosity vs. current density for p-type silicon of resistivity 10–20 Ω cm in 29.93 m% HF with ethanol.

Anodization rate evaluated with the gravimetric method for the quarter wafer samples and with stylus profiler for the 20 mm × 20 mm samples in dependence of the applied current density during anodization is shown in Fig. 4.7. The values obtained in the plot match very well the data for the full wafers from sec. 4.1.1 and the data from Lehmann and Rönnebeck [86] (s. Fig. 2.20b). As in Fig. 2.20b, only slight increase of the rate with increase of doping level could be observed.

4.2 Extended study of pore formation and electropolishing regimes …

135

Etch rate, µm/s

1.2 1.0 0.8 0.6 0.01−0.1 Ω cm, 20 mm imes 20 mm 10−20 Ω cm, 20 mm imes 20 mm 10−20 Ω cm, quarter wafer linear fit to all points

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Current density, A/cm2

Figure 4.7.: Etch rate vs. current density for p-type silicon of resistivity ranges 0.01–0.1 Ω cm and 10–20 Ω cm in 29.93 m% HF with ethanol.

Theoretical dependence of etch rate on current density can be calculated from the Faraday’s law of electrolysis (s. eq. (2.38) and eq. (2.40)) with the following considerations. If we assume 100 % efficiency of the anodization process, i.e., all charges are consumed to dissolve silicon atoms, at low current density, according to the data in sec. 4.1.1 for the resistivity 10–20 Ω cm, one can expect that the porous silicon growth rate as a function of current density can be calculated from the Faraday’s law of electrolysis with the effective dissolution valence of 2.1 and porosity of 70 % (s. Fig. 4.8a, dashed line - “pore formation”). At current density 𝑗 > 𝑗ox , the etch rate should follow the Faraday’s law of electrolysis with the valence of 4 and porosity of 100 % (s. Fig. 4.8a, dotted line - “electropolishing”). These are the boundary conditions. With increase of current density from 𝑗PSL to 𝑗ox , the etch rate curve should change from the pore formation curve to the electropolishing curve (s. Fig. 4.8a, solid line - “general anodization rate”). To understand how the etch rate 𝑅etch behaves during the process change, one has to consider that the change of the reactions takes place when the concentration of the reacting species in the electrolyte near the reaction site (surface concentration) decreases till some critical value. Here it is assumed in a simplified form that in the reactions there is only one type of reacting species containing one fluorine atom. This surface concentration, in turn, is defined by the consumption rate of fluorine at the reaction site 𝑅F per

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4 Microscale study of anodization process

area, which can be calculated from the known consumption rate of silicon atoms per area multiplied by the number of fluorine atoms 𝑛F consumed to dissolve one silicon atom according to eq. (4.1): 𝑅F =

𝑗 ⋅𝑛 𝑛e 𝑒 F

(4.1)

Based on the reactions of the anodization process, in both regimes, pore formation (eq. (2.29)) and electropolishing (eq. (2.33) and (2.34)), final stable product of the reactions is silicon hexafluoride SiF6 2− [94], therefore, six fluorine atoms are consumed to dissolve one silicon atomiii . If we neglect other possible changes of the surface concentration, e.g., due to the movement of the reaction site and mass transport limitations of porous silicon layer, then, in order to maintain the surface concentration below the critical value, the fluorine consumption rate should remain constant or increase during the transition. An interesting conclusion from this assumption: constant or increasing fluorine consumption rate in the transition region implies connection between the values of 𝑗PSL and 𝑗ox : with increase of 𝑗PSL , the width of the transition region 𝑗PSL < 𝑗 < 𝑗ox should increase as well, as reported by others (s. Fig. 2.17). More specifically, for the constant fluorine consumption rate 𝑗𝑗PSL ≤ 𝑛𝑛e PSL is expected, where 𝑛e PSL and 𝑛e ox are the ox e ox values of silicon dissolution valence at 𝑗PSL and 𝑗ox , respectively. Expected fluorine consumption rate for pore formation and electropolishing, together with the transition, is shown in Fig. 4.8b. With the known dependence for the flux, the etch rate in the transition region from pore formation to electropolishing can be calculated: in case of the constant fluorine consumption rate during the transition, the value of the etch rate is expected 𝑃 to decrease according to the ratio 𝑃 % ox , where 𝑃% ox and 𝑃% PSL are the % PSL porosity values at 𝑗PSL and 𝑗ox , respectively, and 𝑃% ox = 100 % (Fig. 4.8a). The calculations of the boundary conditions, for both 0.01–0.02 Ω cm and 10–20 Ω cm, where the pore formation regime parameters are based on the experiment for full wafers (s. sec. 4.1.1), are summarized in Tab. 4.1.

iii Actually,

in the reaction of divalent silicon dissolution (eq. (2.29)), eight fluorine atoms are consumed, however, two of them remain available for the process as HF molecules.

4.2 Extended study of pore formation and electropolishing regimes …

(a)

137

(b)

j PSL

j ox

j PSL

j ox

Figure 4.8.: Expected dependencies of (a) etch rate and (b) fluorine consumption rate on current density in linear scale based on the mechanism of anodization process; the boundary curves “pore formation” and “electropolishing” were calculated according to Tab. 4.1.

Table 4.1.: Summary of the boundary conditions for etch rate 𝑅etch and fluorine consumption rate 𝑅F in pore formation and electropolishing regimes for 0.01–0.1 Ω cm and 10–20 Ω cm Resistivity, Ω cm

Regime

𝑃% , %

𝑛e

𝑛F

𝑅etch 𝑗 , cm2 [ nm s ⋅ mA ]

𝑅F 𝑗 , 1 cm2 [ s cm 2 ⋅ mA ]

0.01–0.02 0.01–0.02 10–20 10–20

pore formation electropolishing pore formation electropolishing

30 100 70 100

2.5 4 2.1 4

6 6 6 6

≈ 1.666 ≈ 3.124×10−1 ≈ 8.502×10−1 ≈ 3.124×10−1

≈ 1.498×1016 ≈ 9.362×1015 ≈ 1.783×1016 ≈ 9.362×1015

Now, let us compare the theoretical etch rate dependencies to the experimental data (for both the full wafers and the smaller samples) (s. Fig. 4.9).iv The experimental data for the full wafers is consistent with the data for the smaller samples. The data points remain in the boundaries calculated for pore formation and electropolishing, and with increase of current density, the anodization rate turned from the pore formation curve to the “electropolishing” curve. However, the pure electropolishing regime even at iv For

the high doping level, the data for the resistivity range of 0.01–0.02 Ω cm (full wafers) are combined with the data for the wider resistivity range of 0.01–0.1 Ω cm for the 20 mm × 20 mm samples.

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4 Microscale study of anodization process

3 A/cm2 was not reached, as was already concluded from the optical observations and the evaluated valence and porosity values. Since the values for 10–20 Ω cm obtained in this study match those reported by Lehmann and Rönnebeck (s. Fig. 2.20b, anodization rate values up to current density of 1 A/cm2 ), one can conclude that in the work of Lehmann and Rönnebeck pure electropolishing was likely also not achieved. 10–20 Ω cm

pore formation electropolishing measured etch rate

103

Etch rate, nm/s

Etch rate, nm/s

0.01–0.1 Ω cm

102

101

101

102

103

Current density, mA/cm

2

pore formation electropolishing measured etch rate

103

102

101

101

102

103

Current density, mA/cm2

Figure 4.9.: Anodization rate as a function of current density – experimentally measured values for p-type silicon for resistivity ranges 0.01–0.1 Ω cm (left) and 10–20 Ω cm (right) in 29.93 m% HF with ethanol; the boundary curves “pore formation” and “electropolishing” were calculated according to Tab. 4.1.

The observed increase of the etch rate in the transition region is quite strong. If the process in the transition region runs as discussed above, this means that the fluorine consumption rate increased significantly during the transition. Another explanation of this rather wide transition region could be not well defined experimental conditions due to non-uniform distribution of current density across the samples. Then, in terms of the average current density calculated from the total current and anodization area, one would get the start of the transition region at 𝑗area < 𝑗PSL and the end of the transition at 𝑗area > 𝑗ox . However, the obtained porous layers looked homogenous and the etch depth was quite uniform, thus the effect of non-uniform distribution of current density seems to be negligible.

4.2 Extended study of pore formation and electropolishing regimes …

139

To conclude, in this study the anodization process in the 29.93 m% HF ethanoic electrolyte for p-type silicon for two resistivity ranges (0.01–0.1 Ω cm and 10–20 Ω cm) in a wide current density range was studied. Dissolution valence and porosity for the 10–20 Ω cm samples were evaluated up to 1.5 A/cm2. Anodization rate was evaluated for both resistivity ranges up to 3 A/cm2. The pure electropolishing regime could not be observed in the study, meaning that 𝑗ox > 3 A/cm2 for the chosen process conditions. Measured anodization rate matched the data for the full wafers and the theoretical expectations based on the Faraday’s law of electrolysis. Additionally, anodization rate for the 0.01–0.1 Ω cm samples appeared to be close to that for the 10–20 Ω cm samples. 4.2.2. Variation of HF concentration Anodization process applied to form cavities in silicon might result in structures with low surface quality. Then, the surface quality can be improved in the subsequent electropolishing process performed at different anodization conditions. In order to find optimal anodization parameters to achieve minimum surface roughness, anodizations in electropolishing regime for several electrolyte compositions and current densities were performed in this study. The study concentrated on the low-doped p-type silicon of the resistivity range 10–20 Ω cm, because this type of substrates was chosen as the main material for experiments aimed on application of the anodization process as a structuring technique (s. sec. 2.6.1). The setup 2 with the 20 mm × 20 mm sample holder (s. Tab. 3.9 in sec. 3.1.3.1) was used. An effort was made to keep the total etch depth for all the samples about 10 µm by correspondingly adjusting the anodization time for each current density, estimated using the Farady’s law of electrolysis (eq. (2.38)), independent of the electrolyte composition. Electrolytes of HF concentration of 3 m%, 7 m%, and 15 m% were used. For all electrolytes, two (rounded) values of current density have been selected (𝑗1 ≈ 𝑗ox + 0.2 A/cm2 and 𝑗2 ≈ 𝑗ox + 0.4 A/cm2, where the corresponding 𝑗ox values were taken from Föll et al. [22]). The best results have been obtained for 7 m% HF electrolyte, therefore, a more detailed current density sweep was performed for this electrolyte. The values of measured surface roughness and fractal dimension are summarized in Fig. 4.10.

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4 Microscale study of anodization process

Mean fractal dimension

RMS roughness, nm

3.0

102

10

1

100

10−1

initial (not anodized) 3 m% HF 7 m% HF 15 m% HF

2.8 2.6 2.4 2.2 2.0 1.8

0

200

400

600

Current density, mA/cm2

0

200

400

600

Current density, mA/cm2

Figure 4.10.: Measured RMS surface roughness and mean fractal dimension (AFM) for the low-doped p-type silicon samples electropolished in various electrolytes at different current densities; the dataset “7 m% HF” 2 was fitted with an exponential function of the form 𝑎 + 𝑏 ⋅ 10(𝑐𝑥 +𝑑) .

The minimum roughness of the electropolished surface was achieved in 7 m% HF electrolyte. That the roughness for the 3 m% HF and 15 m% HF electrolytes was higher can be explained with not optimally chosen current densities: since the roughness of a surface under porous silicon in pore formation regime (𝑗 < 𝑗PSL ), in transition regime (𝑗PSL < 𝑗 < 𝑗ox ) and in electropolishing regime for 𝑗 ≫ 𝑗ox (as can be seen for the 7 m% HF electrolyte) is higher than under optimal electropolishing conditions (𝑗 > 𝑗ox ), the current density values for the 3 m% HF electrolyte are likely to be too high and the current density values for the 15 m% HF electrolyte too low for the optimal electropolishing. For the 15 m% HF electrolyte this is also indicated by drastic decrease of the RMS roughness from 374 nm at 0.5 A/cm2 to 0.78 nm at 0.7 A/cm2. Slight decrease of the RMS roughness for the 3 m% HF electrolyte from 36.5 nm at 0.2 A/cm2 to 26.5 nm at 0.4 A/cm2 does not follow the trend observed for the 7 m% HF electrolyte (increase of the roughness with increase of current density), however, the mean fractal dimension for the 3 m% HF electrolyte increases slightly with increase of current density, indicating that the surface gets more fractal, i.e., more complex. To conclude, the goal of the study was to find optimal parameters, at which surface quality comparable to the quality of the initial polished silicon wafer surface can be achieved, and such (or even superior) conditions have been

4.2 Extended study of pore formation and electropolishing regimes …

141

found for p-type silicon in the 7 m% HF electrolyte for the current densities in the range 100–300 mA/cm2 with RMS surface roughness in the range from 0.64 nm to 1.25 nm. 4.2.3. Influence of porous silicon removal process on surface quality When pore formation regime is used to form cavities in silicon, for some applications it might be important to achieve and maintain low surface roughness. Post-processing steps, such as removal of porous silicon in diluted KOH or other anisotropic etchant, might increase surface roughness. To check this, influence of porous silicon removal process performed either in 1 m% KOH or diluted photoresist developer AZ351B in water in volume proportion 1:5 was studied in the work on 10 mm × 10 mm blank not anodized polished p-type silicon samples of resistivity 10–20 Ω cm. The results for the RMS surface roughness and the mean fractal dimension in dependence of etch time at RT are shown in Fig. 4.11. As expected, the longer the etch time, the rougher is the surface, with roughness over 100 nm obtained after 2 hours in the KOH solution. Thus, the main conclusion of this study is that for typical duration of the porous silicon removal process below 0.5 hour only slight increase of roughness can be expected (in the KOH solution, after 30 min etch time, 𝑅a ≈ 2 nm and 𝑅q ≈ 2.7 nm). The data also suggested that the KOH solution has more impact on the surface roughness, however, comparison of the effectiveness (i.e., porous silicon removal rate) of the solutions was not performed in the work.

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4 Microscale study of anodization process

101

100

10−1

KOH 1 m% 1 : 5 AZ351B : water 0

1

2

3

Etch time, h

4

5

Mean fractal dimension

RMS roughness, nm

3.0 102

2.8 2.6 2.4 2.2 2.0

KOH 1 m% 1 : 5 AZ351B : water

1.8 0

1

2

3

4

5

Etch time, h

Figure 4.11.: Measured RMS surface roughness and mean fractal dimension (AFM) for not anodized polished p-type silicon samples after etching in 1 m% KOH and in diluted photoresist developer AZ351B solution; the datasets were fitted with error functions.

4.3. Conclusions In this chapter, microscale parameters of anodization process were studied. Dependencies of porosity, etch rate, dissolution valence, and surface quality on current density and doping level of p-type silicon were obtained. The results confirmed the data from references, and showed that pure electropolishing regime could not be achieved at current densities up to 3 A/cm2 for large area samples (area of 1 cm2 and larger) used in the study. Surface quality of structures in silicon formed with anodization or other etch processes can be further improved by additional electropolishing step. The study of surface quality after anodization in electrolytes with different HF concentration showed that electropolishing in 7 m% HF electrolyte at current densities in the range 100–300 mA/cm2 results in surface roughness in order of one nanometer. It is not enough to achieve a good surface quality with anodization process, but also to preserve the quality during subsequent porous silicon removal. The study of porous silicon removal step demonstrated that keeping polished silicon wafers in 1 m% KOH solution or solution with 1:5 volume ratio of AZ351B and water more than 30 min results in increase of surface roughness of one order of magnitude or even more.

4.3 Conclusions

143

To conclude, the results presented in this chapter supported planing and evaluation of experiments, as well as development of the models, in the next chapter, where application of silicon anodization as a structuring technique was studied.

5. Anodization process as a structuring technique Literature research in sec. 2.6 showed that some separate studies aimed on fabrication of specific structures were performed by different research groups. However, these studies only showed some effects, such as current crowding for insulating frontside masking (for example, Fig. 2.41), without considering complete time evolution of the etch shapes during the process at various conditions. Therefore, in this chapter, for the first time, detailed studies of shape control obtained through localization with basic localization techniques, namely, frontside masking and backside contacts, are presented. Experimental results are supported with novel time-dependent finite-elements simulations. Main part of this study was performed for a project that aimed on fabrication of silicon molds for injection molding of plastic lenses. Therefore, a question of surface quality of the resulting cavities is also covered in this chapter, and the results are compared to the data obtained in chapter 4. Unless otherwise stated, anodization was performed in 29.93 m% HF with ethanol. 5.1. Process localization with frontside masking 5.1.1. Preliminary study Localization of the anodization process with frontside insulating films is typically reported to produce convex (edge effect) shapes due to current crowding (s. Fig. 2.41b). In some reports, concave shapes have been also shown [47]. However, in the research group of the author, no concave shapes were achieved till this work. In the preliminary study, few experiments in potentiostatic and galvanostatic modes have been conducted for low-doped p-type silicon samples of resistivity 10–16 Ω cm. Silicon nitride frontside layer with multiple square and rectangular openings of various width (shorter side) in the range from 100 µm to 400 µm and, for rectangles, with four times bigger length, was used (s. layouts 3-1 and 3-2 in sec. 3.1.2.6). © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_5

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5 Anodization process as a structuring technique

In the potentiostatic mode, a three-electrode setup with a calomel reference electrode and the power source CGN-04 was used. The applied voltage was varied from 1.5 V to 10 V, with the process duration for all but 1.5 V being 5 min (for 1.5 V, 10 min due to much smaller expected etch rate). Some increase of the measured current during the process was observed, presumably due to increase of the anodization area as the etch form was growing. The current density was assumed to remain constant as automatically adjusted process parameter for the given constant potential difference, neglecting possible changes of the voltage drop in the diffuse ionic layer (s. sec. 2.3.3.1). It was found that with increase of the applied voltage from 1.5 V to 10 V the surface changes from uniform porous silicon to porous silicon with cracks, then to partly electropolished rough surface at 7 V (approximately 1.4 A/cm2 ), and finally to shiny electropolished surface at 10 V (approximately 1.96 A/cm2 ) as shown in Fig. 5.1. The resulting etch forms (after porous silicon removal, where needed) were convex in the pore formation regime and concave in the electropolishing regime, with both convex and concave shapes observed at the transition voltage of 7 V. In the galvanostatic mode with a two-electrode setup, an attempt to repeat the electropolishing results achieved at 10 V was done. The current in the range 23–202 mA, corresponding to initial current densities in the range from about 2 A/cm2 to 3.5 A/cm2 calculated for the initial anodization area (area of the openings in the silicon nitride masking layer) was applied for 1 min. Additionally to the samples with layouts 3-1 and 3-2, samples with single circular openings of diameter 1 mm were used. As expected, electropolishing of the cavities was achieved. However, only the structures with rectangular openings of width 100 µm were concave (s. upper left picture in Fig. 5.2). The structures with the bigger openings (rectangular openings with width 200–400 µm and the circular openings with diameter 1 mm) were electropolished and convex (s. bottom left picture in Fig. 5.2). With longer etch time of 5 min, wider rectangular structures with width of the opening up to 400 µm have also transformed into concave (s. central column in Fig. 5.2). At even longer etch time of 10 min, cavities anodized through 100 µm wide openings, located initially with spacing between the openings of 200 µm, have grown together (s. upper right picture in Fig. 5.2). Also there was some non-uniformity in the 10 min process resulted in a wavy surface (s. lower right picture in Fig. 5.2). Several conclusions have been drawn from this preliminary study. The found value of limiting current density 𝑗ox between 1.4 A/cm2 and 1.96 A/cm2 for square and rectangular openings of dimensions below 400 µm is about twice

5.1 Process localization with frontside masking 1.5 V, ~25 mA/cm², 10 min

147

10 V, ~2 A/cm², 5 min

Figure 5.1.: Change of surface quality (from rough to polished) (upper pictures) and cross-section shape (from convex to concave) (bottom pictures) for structures anodized in potentiostatic mode; the upper pictures are microscopic photographs of the front view of the structures, the bottom left picture is a topographic 2D scan by a profiler, and the bottom right picture is a SEM micrograph of two neighboring structures; silicon nitride mask was not removed and still visible on the upper pictures, including underetched parts in the structures. lower than the current density used in the study of microscale aspect of the anodization process (s. chapter 4) for samples with anodization area of 1 cm2 or larger. However, for those large-area samples no pure electropolishing could be achieved. Since the diffusion limitation for the smaller anodization area due to a small opening in the silicon nitride mask can be expected to be higher than for the big open area of 1 cm2, the results confirmed the dependence of the current density peaks on mass transport in electrolyte reported elsewhere (s. sec. 2.4.4). Thus, in general, one can achieve a diffusion-controlled process for a given electrolyte by increasing consumption of reactants (through increase of current density) or by geometrical limitation of the reactants’ transport with frontside masking layers or/and large etch depth, as observed in this study. Therefore, the values of the anodization parameters obtained in chapter 4 at high current densities cannot be applied directly to this section.

148

5 Anodization process as a structuring technique 5 min

10 min

o pe ningwidt h 4 00 µm

o pe ningwidt h 100 µm

1 min

Figure 5.2.: Etch front movement during anodization process at 3.5 A/cm2 for rectangular openings with width of 100 µm and 400 µm after 1 min, 5 min, and 10 min. Regarding the etch form development, the results suggested that at least in the electropolishing regime, where the process is known to be diffusionlimited, concave shapes are indeed achievable, similarly to the known shape development for diffusion-limited wet-etch processes (s. sec. 2.2.3.1). To understand more exactly etch form development in the pore formation and electropolishing regimes for configuration with frontside insulating masking, more detailed studies have been conducted. They are described in the following subsections. 5.1.2. Influence of opening size and current density on time development of etch forms for low-doped p-type silicon In this section, development of etch forms during anodization process through a circular opening in a frontside insulating masking for low-doped p-type silicon samples of resistivity 10–16 Ω cm at different constant total current values corresponding to initial current densities 𝑗init in the range 1.0–3.5 A/cm2 with steps of 0.5 A/cm2 was studied. The current density range was chosen more in the electropolishing regime, because one of the goals was to form clear concave shapes, which could be obtained by electropolishing in the preliminary studies. For each current density, the process

5.1 Process localization with frontside masking

149

with etch duration 1 min, 5 min, 10 min, and 20 mini was conducted. The process was performed in the setup 1 with circular samples of diameter 30 mm with frontside silicon nitride film structured with the layout 1 (circular openings of varied diameter 𝐷open from 200 µm to 1000 µm with steps of 200 µm, s. sec. 3.1.2.6 and Fig. 3.1). Summary of optical observations of the structures after anodization is provided in Tab. 5.1. Current density values 𝑗area given in brackets in the table are the values adjusted for the increased area in the end of the process (s. below in “Evaluation of current density in the structures”). From the optical observations one can conclude that the samples anodized at current densitiesii below 0.3–0.6 A/cm2 were anodized in pore formation regime. At higher current densities, mostly electropolishing in the first electropolishing plateau (0.3–0.9 A/cm2, rough shiny surface) and electropolishing in the second electropolishing plateau (from 0.8 A/cm2, smooth shiny surface) was obtained, confirming the results of the preliminary study in sec. 5.1.1. Thus, the limiting current densities roughly estimated from this experiment are the following: 𝑗PSL is 0.3–0.6 A/cm2 and 𝑗ox is 0.8–0.9 A/cm2. The wide ranges of the limiting current densities are presumably due to variation of opening diameters which provide different diffusion limitation, and non-uniform distribution of current density between the structures with different opening diameters (more on this below). After removal of porous silicon (where needed) and silicon nitride, the etch shapes in the lines 2 and 3 have been characterized according to sec. 3.2.4. Evaluation of current density in the structures The given initial current density values 𝑗init were calculated from the total current based on the total open area of the layout 1 (𝐴layout ). During anodization, the anodization area increased significantly, therefore the current density decreased in the process. To evaluate the changed values of the current density for each sample, surface area of each structure in the lines 2 and 3 on each sample after removal of porous silicon was calculated as described in sec. 3.2.4. Assuming that the total area of all structures on a sample 𝐴total final is the double sum of the areas of the structures in the lines 2 and 3, and that the current density is equal in all structures, average i Etch

duration of 20 min was used only up to 2.5 A/cm2. For higher current densities, the cavities would be etched through the wafer after 20 min. ii Final current density values 𝑗 area given in brackets in the table are meant here.

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5 Anodization process as a structuring technique

Table 5.1.: Summary of optical observations of the structures after anodization for the given initial current density 𝑗init and anodization duration; in brackets, current density values 𝑗area adjusted for the increased area in the end of the process (final current density) are shown. 𝑗init , A/cm2

Anodization duration, min 1

5

10

20

1.0

(880 mA/cm2 ) from the small openings to the big openings, the surface changes from light brown to rougher darker brown

(560 mA/cm2 ) from the small openings to the big openings, the surface changes from light brown to rougher darker brown

(402 mA/cm2 ) from the small openings to the big openings, the surface changes from light brown to rougher darker brown

(270 mA/cm2 ) from the small openings to the big openings, the surface changes from light brown to rougher darker brown

1.5

(1281 mA/cm2 ) all openings with smooth shiny surface

(823 mA/cm2 ) similar to the sample for 1 min, but in big openings dark rough surface on the bottom

(533 mA/cm2 ) brown surface in the openings; in the 200 µm openings surface is shiny and rough

(338 mA/cm2 ) separate light brown flakes on rough shiny surface

2.0

(1598 mA/cm2 ) all openings with smooth shiny surface

(1027 mA/cm2 ) all openings with smooth shiny surface

(637 mA/cm2 ) separate light brown flakes on rough shiny surface

(385 mA/cm2 ) separate light brown flakes on rough shiny surface

2.5

(1993 mA/cm2 ) all openings with smooth shiny surface

(1119 mA/cm2 ) all openings with smooth shiny surface

(815 mA/cm2 ) all openings with smooth shiny surface

(435 mA/cm2 ) separate light brown flakes on rough shiny surface

3.0

(2333 mA/cm2 ) all openings with smooth shiny surface

(1242 mA/cm2 ) all openings with smooth shiny surface

(922 mA/cm2 ) all openings with smooth shiny surface

3.5

(2629 mA/cm2 ) all openings with smooth shiny surface

(1427 mA/cm2 ) all openings with smooth shiny surface

(960 mA/cm2 ) all openings with smooth shiny surface

values of the final current density 𝑗area adjusted for the increased total area 𝐴total final were calculated for each initial current density for the given array of etch times: 𝑗area = 𝑗init ⋅

𝐴layout 𝐴total final

(5.1)

5.1 Process localization with frontside masking

151

where the total final area of the structures on a sample is the double sum of the structure areas of the openings (opening number 𝑘) in the lines 2 and 3 (line number 𝑙) of the sample: 3

5

𝐴total final = 2 ∑ ∑ 𝐴𝑙,𝑘

(5.2)

𝑙=2 𝑘=1

The assumption of equal current densities in all structures on a sample is quite controversial considering that openings of various diameters, although uniformly distributed, are present on one sample. The effect of non-uniform etch rate for structures of different size on one sample was reported previously by Hedrich et al. [24] and Mescheder et al. [335]. In order to evaluate the non-uniformity of the current distribution on a sample, the final current density values have been also calculated based on the etched volumes of the structures. For each opening 𝑘 in the lines 2 and 3, for the given array of etch times 𝑡𝑖 = [0, 1 min, 5 min, 10 min, 20 min] with the corresponding arrays of the structure area 𝐴𝑘,𝑖 and volumes 𝑉𝑘,𝑖 , where 𝑖 = [0 .. 4], the total charge 𝑞𝑘,𝑖 consumed for a structure was calculated according to the Faraday’s law of electrolysis for 𝑖 = [1 .. 4], assuming that all the structures were etched in electropolishing regime with valence of 4: 𝑞𝑘,𝑖 = 4

𝜌Si 𝑉𝑘,𝑖 𝑁A 𝑒 𝑀Si

(5.3)

Then the final current density values 𝑗vol 𝑘,𝑖 were calculated for the average area between the structures (𝑖 − 1) and 𝑖: 𝑗vol 𝑘,𝑖 =

𝑞𝑘,𝑖 − 𝑞𝑘,𝑖−1 2 ⋅ 𝑡𝑖 − 𝑡𝑖−1 (𝐴𝑘,𝑖 + 𝐴𝑘,𝑖−1 )

(5.4)

In Fig. 5.3, comparison of the final current density values calculated from the volume and area increase, 𝑗vol and 𝑗area , respectively, is shown for the structures anodized at the initial current densities of 1 A/cm2 and 2.5 A/cm2. The plots for the other initial current densities are given in Appendix in sec. A.3.1.1. From the optical observations above, it was concluded that in all openings on the samples anodized at the initial current densities of 1 A/cm2 and 1.5 A/cm2, and in some of the openings on the samples anodized at the

152

5 Anodization process as a structuring technique 𝑗init = 2.5 A/cm2 Final current density, A/cm2

Final current density, A/cm2

𝑗init = 1 A/cm2 1.2 1.0 0.8 0.6 0.4 0.2 200

400

600

800 1000

Diameter of opening, µm

2.5

j vol after 60 s j area after 60 s j vol after 300 s j area after 300 s j vol after 600 s j area after 600 s j vol after 1200 s j area after 1200 s

2.0 1.5 1.0 0.5 200

400

600

800 1000

Diameter of opening, µm

Figure 5.3.: Comparison of the calculated current density values 𝑗vol and 𝑗area in dependence on opening diameter for low-doped p-type silicon samples anodized at initial current densities of 1 A/cm2 and 2.5 A/cm2 ; fit with polynomials of 2nd order is applied to 𝑗vol datasets. initial current densities of 2 A/cm2 and 2.5 A/cm2 porous silicon was formed. This means that the dissolution valence there was below 4, and fewer silicon atoms had to be removed in the same volume due to porosity below 100 %. Therefore, the calculated 𝑗vol values for these openings are much larger than the real values and the 𝑗area values (s. left plot in Fig. 5.3). For the openings which were definitely electropolished, one can assume that the calculated values 𝑗vol are close to the real ones. The values of 𝑗vol for the electropolished structures mostly decrease with decrease of the opening diameter (s. right plot in Fig. 5.3). The dependence of the values on the opening diameter indicates that there is indeed redistribution of current density between the openings. This redistribution can be due to increase of electrical resistance of the diffuse ionic layer in the electrolyte or due to decrease of electrical resistance in the substrate for deeper structures. Since some of the structures to be characterized have been anodized in the pore formation regime, and some in the electropolishing regime, in order to maintain a uniform analysis of the data, the calculated 𝑗area values are mostly used in the following text and named as the final current density calculated from the increased area in the end of the process. Where the current density values are given without additional specification, the initial current density values are meant.

153

5.1 Process localization with frontside masking Shape development

In general, four types of shapes have been observed (s. Fig. 5.4): convex with sharp minima, convex with smooth minima, triple-hollow, and concave shapes.

x , µm

x , µm

z , µm

(d)

z , µm

(c)

z , µm

(b)

z , µm

(a)

x , µm

x , µm

Figure 5.4.: Experimentally observed types of shapes (profiler 2D scans): (a) convex shape with sharp minima (2.5 A/cm2, 1 min, opening 3-5 of diameter 1000 µm), (b) convex shape with smooth minima (2.5 A/cm2, 5 min, opening 3-5 of diameter 1000 µm), (c) triple-hollow shape (2.5 A/cm2, 5 min, opening 3-4 of diameter 800 µm), (d) concave shape (1.5 A/cm2, 20 min, opening 3-2 of diameter 400 µm); dashed lines show arc fit performed to the central part of the structures to evaluate curvature of the bottom shape. The observed results suggest that the shape evolution in the process undergoes transformations according to Fig. 5.4 from (a) to (d). For example, in Fig. 5.5, such shape transformation for the opening with diameter 1000 µm is shown for the samples anodized at initial current densities of 1.5 A/cm2, 2.5 A/cm2, and 3.5 A/cm2. Due to the limited number of etch time values studied, not all mentioned types of shape could be observed for each size of opening for every current density. For example, the etch form for 1000 µm opening does not develop to the triple-hollow and concave forms at 1.5 A/cm2, which are observed for the deeper structures obtained at the higher current densities (s. Fig. 5.5). The observed transformation of shapes can be explained with two mechanisms: diffusion mechanism and electrical mechanism (s. Fig. 5.6). In the diffusion mechanism (s. Fig. 5.6a), the mass transport of species with diffusion limitation in electrolyte is considered, and the shape development is same as known for diffusion-controlled wet chemical etch processes (s. Fig. 2.2c). The electrical mechanism (s. Fig. 5.6b) takes into account the

154

5 Anodization process as a structuring technique 𝑗init = 1.5 A/cm2

𝑗init = 2.5 A/cm2 0.0

0.0

0.0 1 min 5 min

−0.2

−0.2

1 min

z, mm

−0.1

1 min

z, mm

z, mm

𝑗init = 3.5 A/cm2

5 min

10 min

−0.3

−0.4

20 min 0.0

0.5

1.0

1.5

x, mm

0.0

10 min 0.5

1.0

x, mm

1.5

−0.2 5 min −0.4 0.0

10 min 0.5

1.0

1.5

x, mm

Figure 5.5.: Observed etch front movement during anodization of lowdoped p-type silicon through openings with diameter 1000 µm at various initial current densities. charge flow in the substrate and electrolyte: development of convex shape in the beginning of the process is due to current crowding near the mask edges due to high lateral flow of charges to the etch form; as the structure gets deeper, the direct charge flow from the backside of the substrate to the bottom of the etch form provides development of a concave shape. The triple-hollow shape can be the result of both mechanisms working not synchronously with each other, i.e., when one of the mechanisms already provides etching of a concave shape, whereas the other mechanism still conserves the high etch rate near the sidewalls of the etch form. Curvature and threshold depth Based on the observed shape transformations, the following dependence of curvature on etch time was expected as shown in Fig. 5.7a. First, there is some increase of convex shape (current crowding) in the beginning of the process. Later, the shape changes to concave (increase of curvature from negative to positive values) and, after reaching some maximum positive value, the curvature starts to decrease as the concave form grows. Curvature for all measured profiles was evaluated as described in sec. 3.2.4 by arc fit. Examples of the performed arc fit are shown on the measured topographical profiles in Fig. 5.4. The value of zero curvature for the initial etch form with zero depth neglecting the thickness of the silicon nitride mask was added to the datasets. Curvature change as a function of the structure depth for the openings of diameter 400 µm is shown in Fig. 5.7b. The plots

5.1 Process localization with frontside masking (a)

155

(b)

silicon

silicon

silicon

(a)

(b)

Curv ature, a.u.

Curv ature, mm–1

Figure 5.6.: Schematics of the mechanisms of shape conversion convexconcave shown in cross-section: (a) diffusion mechanism – effect of diffusion-controlled etching process (arrows represent flow of reactants to reaction site), (b) electrical mechanism – effect of current redistribution (arrows represent charge flow); thicker arrows mean stronger flow. Top: at the beginning of the process, bottom: after etching deep into silicon substrate.

0

Dopen = 400 µm 0

Structure depth, a.u.

Structure depth, µµm

Figure 5.7.: Curvature vs. structure depth: (a) expected schematic dependence based on the observed shape evolution; (b) evaluated dependence for cavities anodized in low-doped p-type silicon samples through openings of diameter 400 µm for various applied initial current densities with a data fit to obtain threshold depth values.

156

5 Anodization process as a structuring technique

for all other opening diameters are given in Appendix in sec. A.3.1.2. Although not clearly visible for all the varied parameters due to limited number of the data points, the measured data in general follow the expected curve in Fig. 5.7a. Valuable characteristic of the observed etch shape development obtained in this study is the threshold value 𝑑th of the structure depth, at which switching from convex shape (negative curvature) to concave shape (positive curvature) takes place. To find out the values of this threshold depth from the plots, the data had to be properly fitted. Due to the limited number of data points, it was decided to avoid automatic fitting with some function, and instead the fit was done manually. The fit curves are shown on the plots of curvature vs. structure depth presented above in Fig. 5.7b and in Appendix in sec. A.3.1.2. Based on the performed fit, the threshold depth values for each opening diameter and current density were evaluated. Dependence of the threshold depth on the initial current density and the opening diameter is shown in Fig. 5.8. For the smallest openings (diameter 200 µm), there is almost no dependence of the threshold depth on the current density in the studied range. Etch forms anodized in the bigger openings show decrease of the threshold depth with increase of the initial current. The dependence of the threshold depth on the opening diameter is quite strong, with bigger openings requiring deeper etching to result in concave shapes. The obtained decrease of the threshold depth and dependence of the threshold depth on the opening diameter can be explained with the diffusion mechanism of shape conversion described above: diffusion-controlled conditions are more readily achieved at higher current densities and, for a given etch time, for smaller openings. Thus, the results can be interpreted as showing an important role of the diffusion in electrolyte for the shape development in the given conditions. However, charge flow could also be responsible for this shape conversion, and the electrical models with primary and secondary current distributions developed in this work showed that this is indeed the case (s. sec. 5.4). Another interesting conclusion can be drawn considering that at the initial current density of 1 A/cm2, the process for all but 1 min duration was running mostly in the mixed pore formation / electropolishing regime due to increased anodization area. Thus, from the dependence of the threshold depth on the initial current density one can conclude that the transformation convex-concave can be achieved even in this mixed regime if the anodization is sufficiently long.

157

200 µm 400 µm 600 µm 800 µm 1000 µm

400 350 300 250 200 150 100 50 1

2

3

Current density, A/cm2

Threshold depth dth , µm

Threshold depth dth , µm

5.1 Process localization with frontside masking

400

1.0 A/cm2

350

1.5 A/cm2

300

2.0 A/cm2

250

2.5 A/cm2 3.0 A/cm2

200

3.5 A/cm2

150 100 50 500

1000

Diameter of opening Dopen , µm

Figure 5.8.: Measured threshold depth as a function of initial current density (left) and opening diameter (right) for the cavities anodized in lowdoped p-type silicon samples; values in the legend for the left plot are opening diameter, the values for the right plot are initial current density. Anisotropy factor In contrast to the curvature which characterizes the bottom part of the structures, anisotropy factor characterizes etched cavities as a whole. Anisotropy factor of the etched cavities was evaluated using the structure depth and length of mask underetching according to eq. (2.2). In Fig. 5.9 dependence of the anisotropy factor on the structure depth for the initial current density values of 1 A/cm2 and 3.5 A/cm2 for different opening diameters is shown. The plots for all other initial current densities are available in Appendix in sec. A.3.1.3. One can see from the plots that for smaller current densities the anisotropy factor mostly decreases with increase of depth for all the openings from about 0.65 to zero. For the structures anodized at higher current densities, the range of anisotropy factor gets more narrow, with vertical etch rate dominating over the lateral etch rate, and the trend changes from “first decreasing, then increasing” to “first increasing, then decreasing”. The same values of anisotropy are reached for the structures with bigger opening diameters at more depth than for the structures with smaller opening diameters. It is expected that for a purely diffusion-controlled process, the anisotropy factor remains about zero at any etch duration, i.e., in the beginning of the process, for convex shapes, and later for concave shapes. Based on the

158

5 Anodization process as a structuring technique 𝑗init = 1 A/cm2

𝑗init = 3.5 A/cm2 0.70

Anisotropy factor Af

Anisotropy factor Af

0.6 0.5 0.4 0.3 0.2 0.1 0.0

200 µm 400 µm 600 µm 800 µm 1000 µm

0.65 0.60 0.55 0.50 0.45 0.40

50

100 150 200 250

Structure depth, µm

100

200

300

400

Structure depth, µm

Figure 5.9.: Anisotropy factor in dependence of structure depth for the structures anodized at initial current density of 1 A/cm2 and 3.5 A/cm2 in low-doped p-type silicon samples through openings of different diameters; values in the legend are opening diameter. literature review (s. sec. 2.5.6), let us for now also neglect crystallographic dependence of the anodization rate, because the reported pseudo V-shapes (s. Fig. 2.36b) have not been observed in our case. Then, neglecting also the effect of the convection in the electrolyteiii , the main source of anisotropy in the anodization process must be the potential / charge flow distribution in the substrate. The electric mechanism of shape transformation proposed above suggests that the lateral etching is promoted in the beginning of the process due to current crowding. The deeper the structure gets, the larger the vertical etch rate should become. Therefore, one could expect decrease of the anisotropy factor in the beginning of the process, and its increase (or compensation of a decreasing trend) for deep structures. In the experiment, such behavior was observed only for some structures with big openings. For the most of the structures anodized at smaller current densities, only the decreasing trend was clearly observed. Since even at the beginning of the process (when significant current crowding is expected), the anisotropy faciii The

samples are placed vertically in the anodization cell. For the upper parts of the structures, slightly higher etch rate was typically observed. The effect is likely due to increased flow of electrolyte to the upper part of a structure due to evolved heat or hydrogen bubbles. However, all structures in the work were characterized in horizontal direction through the middle of a structure, therefore the observed asymmetry was left out of discussion.

5.1 Process localization with frontside masking

159

tor values are clearly above zero, the experimental data suggest that the main effect of the electric supply from the substrate seems to be the pronounced vertical etch rate due to vertical charge flow from the backside of the substrate to the structure. However, the electrical models developed in this work showed that this conclusion is not valid, and some other anisotropic mechanism not determined by charge flow must be responsible for the high anisotropy (s. sec. 5.4). At larger depth, it was shown above that the real current density decreased significantly from the initial value due to increase of the anodization area. Therefore, the effect of decreasing 𝐴f with increasing depth observed for the initial current densities below or equal to 2.5 A/cm2 might be not the result of the increased depth, but of the reduced current density. To check this, dependence of the anisotropy factor on the current density was also plotted. Since the current density values decrease significantly during the process, and the anisotropy factor is the result of the whole process, average current density values 𝑗a were introduced. As the average current density, the arithmetic mean value between the initial current density 𝑗init and the final current density 𝑗area were taken. Since with increase of the depth the range of the current density between the initial and the final values increases significantly, the plots of 𝐴f vs. the average current density were plotted separately for different ranges of the structure depth. The most valuable in this sense are the plots for the smaller depth ranges 0–100 µm and 100–200 µm, where the range of the current density is not wide (s. Fig. 5.10; other plots are given in Appendix in sec. A.3.1.4). One can clearly observe that the anisotropy factor is in the range about 0.4–0.7 for the structures anodized at the average current densities above approximately 1 A/cm2 corresponding to the electropolishing regime. In contrast, with decrease of the current density below 1 A/cm2, i.e., when moving from the electropolishing regime to the mixed regime of pore formation and electropolishing, which in the process conditions for the study was only observed for deeper structures, the anisotropy factor for openings of all diameters decreases drastically to zero. Since the peculiarity of the electropolishing is the concentration polarization, which is unlikely to be responsible for the enhanced vertical etching, and the models developed in the work showed that the charge flow cannot be responsible for this high anisotropy as well (s. sec. 5.4), the effect could be only explained with some other anisotropic mechanism in the electropolishing process. When the data points for all structure depth values are plotted together, one can see that the dependence of the anisotropy factor on current density

160

5 Anodization process as a structuring technique Structure depth 0–100 µm

Structure depth 100–200 µm 200 µm 400 µm 600 µm 800 µm 1000 µm

0.6

0.60

Anisotropy factor Af

Anisotropy factor Af

0.65

0.55 0.50 0.45 0.40 0.35

0.5 0.4 0.3 0.2 0.1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Average current density, A/cm2

Average current density, A/cm2

Figure 5.10.: Anisotropy factor vs. average current density 𝑗a for two ranges of structure depth; low-doped p-type silicon samples; error bars show the ranges between initial current densities and final current densities calculated for increased area in the end of the process; values in the legend are opening diameters. is hardly dependent on the structure depth (s. Fig. 5.11). Therefore, the observed above decrease of the anisotropy factor with increase of the depth at the initial current densities below or equal to 2.5 A/cm2 (s. left plot in Fig. 5.9 and further plots in Appendix in sec. A.3.1.3) might be indeed due to decrease of the real current density and approaching the pore formation regime. Surface quality in the etched forms For application of the anodization process as a structuring technique, it is also important to know how good the surface is after anodization and after porous silicon removal. Surface roughness for the samples anodized in this study was measured with AFM. Since these measurements are only possible for shallow and relatively big structures, AFM topography scans have been taken only from the samples anodized for 1 min in the center of the openings with diameter 1000 µm in the line 2 (s. Fig. 3.1 in sec. 3.1.2.6). The obtained RMS roughness and fractal dimension values in dependence of the final current density calculated for the increased area (𝑗area ) are shown in Fig. 5.12. As confirmed with the optical observations, roughness of the surface decreases significantly from 288 nm to approximately 7 nm when the

161

5.1 Process localization with frontside masking

Anisotropy factor Af

0.7

Dopen Dopen Dopen Dopen Dopen

0.6 0.5 0.4

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

Figure 5.11.: Anisotropy factor vs. average current density 𝑗a for all structure depth values in the range 0–500 µm; low-doped p-type silicon samples; values in the legend are opening diameters. final current density increases from 880 mA/cm2 to 1280 mA/cm2, indicating change of the regime from pore formation (or mixed pore formation / electropolishing) to pure electropolishing. The fractal dimension decreases correspondingly to a values close to two, meaning a change of the surface topography to a more flat one. In sec. 4.1.1 for low-doped silicon in pore formation regime at 9.66 mA/cm2 the value of RMS roughness of 108 nm was obtained, which is much smaller than the here obtained 288 nm at 880 mA/cm2 . Increase of roughness with increase of current density could be the result of the anodization mechanism changed from pure formation to the mixed pore formation and electropolishing regime, where electropolishing of local spots results in formation of needle-like structures [26]. Conclusions To conclude, in this section for the first time etch shape development in anodization process with frontside localization at various current densities in (and close to) electropolishing regime for low-doped p-type silicon of resistivity 10–16 Ω cm was investigated. Shape development from convex to triple-hollow and, finally, to concave shape was observed. Mechanisms explaining the observed etch form development based on charge flow in the cell

5 Anodization process as a structuring technique

Fractal dimension

RMS roughness, nm

162

102

101

100

1.0

1.5

2.0

Final current density, A/cm2

2.6 2.4 2.2 2.0 1.0

1.5

2.0

Final current density, A/cm2

Figure 5.12.: Measured RMS surface roughness and mean fractal dimension in centers of cavities anodized in low-doped p-type silicon for 1 min through openings of 1000 µm in dependence of final current density 𝑗area calculated for the increased area in the end of the process. and transport of species in the electrolyte were proposed. Influence of the opening diameter and the applied current density on the etch form development has been shown, and the curvature, threshold depth, and anisotropy factor values for various current densities and opening diameters have been evaluated. Based on the data, it was concluded that the shape transformation convex-concave is possible in electropolishing and the mixed regime of electropolishing and pore formation. Etch forms anodized in electropolishing regime had high anisotropy factor up to 0.7. In the mixed regime of pore formation and electropolishing, an anisotropy factor about zero was obtained. The simulation results presented later in this report (s. sec. 5.4) showed that the observed in this experiment high anisotropy in electropolishing regime cannot be explained with the current density redistribution in the substrate, but reveals some other anisotropic mechanism in anodization process. Additionally, surface quality in the etched forms for various current densities was measured: RMS roughness decreased from 288 nm at 880 mA/cm2 to about 10 nm at higher current densities. As the evaluation of the final current densities showed, one should use the given final current density values calculated from the total area increase with caution, because the real values calculated from the increase of individual volumes showed a certain dependence of the current density on the opening diameter. One could get more clear results by using a single opening diameter per sample, however, this would require significantly more resources to

5.1 Process localization with frontside masking

163

perform the study for the same variation of the parameters, therefore it is proposed as a future work. 5.1.3. Influence of opening size and current density on etch forms for highly-doped p-type silicon In this section, etch forms during anodization process at different constant total current values corresponding to initial current densities 𝑗init in the range 0.05–3.5 A/cm2 for highly-doped p-type silicon samples with resistivity in the range 0.01–0.1 Ω cm were studied. For each current density, only the process with constant etch time of 5 min was performed. The study was conducted the same way as the one for the low-doped silicon samples (s. sec. 5.1.2) with frontside silicon nitride film structured with the layout 1 (circular openings of varied diameter from 200 µm to 1000 µm with step 200 µm, s. sec. 3.1.2.6). Summary of optical observations of the structures after anodization is given in Tab. 5.2. Table 5.2.: Summary of optical observations of the structures after anodization together with values of initial current density 𝑗init and final current density 𝑗area calculated for initial and final anodization area of the structures 𝑗init , mA/cm2

𝑗area , mA/cm2

Surface after anodization

50 100

46.7 90.2

dark gray homogenous flat porous layer dark gray homogenous flat porous layer

200

171.9

dark gray homogenous flat porous layer

500

367.9

gray shiny porous surface, wavy

1000

625

darker than above, wavy

1500

855.7

shiny smooth polished surface

2000

1040.1

shiny smooth polished surface

2500

1130

shiny smooth polished surface

3000

1287.4

shiny smooth polished surface

3500

1423.1

shiny smooth polished surface

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5 Anodization process as a structuring technique

The given initial current density values 𝑗init were calculated from the total current based on the total open area for this layout. Same as in sec. 5.1.2, the final current density values based on the increased total anodization area (𝑗area ) and increased individual structure volume (𝑗vol ) have been evaluated (s. in Appendix in sec. A.3.2). As explained in sec. 5.1.2, the obtained final current density values from the increase of individual structure volume are not close to the real values for the structures anodized in pore formation regime. From the optical observations, anodization at 𝑗area ≥ 0.856 A/cm2 was in electropolishing regime. Therefore, only the 𝑗vol values for the initial current densities 1.5–3.5 A/cm2 are close to the real valuesiv . The first current peak (𝑗PSL ) seems to be between 171.9 mA/cm2 and 367.9 mA/cm2. The obtained limiting current densities and redistribution of the current density between the openings of different diameters in the second plot in sec. A.3.2 are similar to those observed for the samples of resistivity 10–16 Ω cm with the same layout (s. sec. 5.1.2). Although the calculated values 𝑗vol in the pore formation regime cannot be used in absolute scale, one can note quite a uniform distribution of the values between the openings of different diameters (s. the first plot in sec. A.3.2). After removal of porous silicon (where needed) and silicon nitride, the etch shapes in the lines 2 and 3 have been characterized according to sec. 3.2.4. Same four types of shapes obtained for the samples of resistivity 10–16 Ω cm with the same layout (s. sec. 5.1.2) have been observed also here. Although anodization was performed here only for 5 min, the resulting topographical scans suggest that the shape development runs the same way as for the low-doped samples, as shown in Fig. 5.13. Curvature and the threshold depth values were evaluated the same way as for the low-doped samples in sec. 5.1.2. Dependence of curvature on the structure depth is shown in Fig. 5.14 (left). General shape of the curves is similar to that obtained for the low-doped samples (s. Fig. 5.7b). However, since the process in this study was conducted without variation of etch duration, variation of the structure depth is achieved here only with variation of the applied current. Therefore, to compare the results for the samples of both resistivities, the data for the low-doped samples for anodization duration of 5 min were extracted from sec. 5.1.2 as shown in Fig. 5.14 (right). There are no data for small depth values for the low-doped samples, beiv Same

as in the study for the low-doped samples, 𝑗area values have been used for data analysis, to maintain same procedure for the structures anodized in pore formation and electropolishing regimes. Where the current density values are given without additional specification, the initial current density values are meant.

165

5.1 Process localization with frontside masking 𝐷open = 400 µm

𝐷open = 1000 µm

0.05 A/cm2

0

−50

−100

z, µm

z, µm

−50

−150

−100 −150 −200 −250

−200 −250

0.05 A/cm2

0

3.5 A/cm2 −200

0

x, µm

200

−300

3.5 A/cm2

−350 −500

0

500

x, µm

Figure 5.13.: Observed etch shapes for opening diameters 400 µm (left) and 1000 µm (right) for initial current densities 0.05 A/cm2 , 0.2 A/cm2 , 0.5 A/cm2 , 1 A/cm2 , 2 A/cm2, and 3.5 A/cm2 ; highly-doped p-type silicon samples, anodization duration 5 min, RT. cause the initial current density range started from 1 A/cm2 in that study, in contrast to 50 mA/cm2 for the highly-doped samples. Apart from that, the dependencies are quite similar. Comparing the threshold depth values for both resistivities (s. Fig. 5.15), one can conclude that there are only small differences between the values for the low- and highly-doped samples. One would expect that the change of substrate resistivity more than 100 times had influence on electric potential distribution in the substrate, which, in turn, would play a role in etch front movement through change of local current density. In contrast, the comparison in Fig. 5.15 lets us assume that the electrical mechanism of the etch form development for both resistivities has nearly equal or no effect on the development of etch forms at the studied conditions. However, as the electrical models presented later in the report showed, the charge flow provides significant difference between the results of the models with such low-doped and highly-doped silicon. Therefore, it seems that the mass transport effects in the electrolyte suppress the differences caused by different substrate resistivities in the experiments. Again, as for the low-doped samples, an interesting question is in what regime (pore formation or electropolishing) the transformation convexconcave took place. The question is quite difficult to answer due to the observed dependence of the limiting current density 𝑗ox on the diameter of

166

5 Anodization process as a structuring technique 10–16 Ω cm

6

Curvature, mm−1

Curvature, mm−1

0.01–0.1 Ω cm

4 2 0 −2

0

100

200

4 2 0 −2

300

Structure depth, µm

200 µm 400 µm 600 µm 800 µm 1000 µm

6

0

100

200

300

Structure depth, µm

Figure 5.14.: Curvature vs. structure depth for highly-doped samples (left) and low-doped samples (right) anodized for 5 min at various current densities; least-squares fit with polynomials of 2nd and 3rd order going through the point (0, 0) was performed; fitted curves were used to obtain threshold depth values except the curve for 1000 µm for low-doped samples, where the fit was not good due to only one point with positive curvature.

Threshold depth dth , µm

300

0.01−0.1 Ω cm 10−16 Ω cm

250 200 150 100 50 0

200

400

600

800

1000

Diameter of opening Dopen , µm

Figure 5.15.: Measured threshold depth as a function of opening diameter for highly-doped and low-doped samples anodized for 5 min at various current densities.

5.1 Process localization with frontside masking

167

openings, i.e., anodization area, and possible redistribution of the current density on a sample. However, rough estimation is possible. For example, the threshold depth for the 200 µm openings is about 40 µm, and this depth was achieved by anodization for 5 min at the initial current densities between 0.2 A/cm2 (depth about 31 µm) and 0.5 A/cm2 (depth about 70 µm). Therefore, the transformation for the 200 µm opening likely took place in the pore formation regime. For the 400 µm openings the transformation took place between the initial current densities of 0.5 A/cm2 and 1 A/cm2 with the corresponding final current densities of 367.9 mA/cm2 and 625 mA/cm2, which means that the shape development also took place in the pore formation or mixed pore formation / electropolishing regime. For the larger openings, the threshold depth in 5 min anodization process was achieved mostly in the electropolishing regime. Thus, it seems that the shape transformation convex-concave for the highly-doped samples, similarly to the low-doped samples, may be possible not only in pure electropolishing regime, but also in the mixed pore formation / electropolishing regime, and even in pure pore formation regime. The anisotropy factor in dependence of the structure depth and the average current densityv was also evaluated for the structures (s. Fig. 5.16). Same as for the low-doped samples in sec. 5.1.2, the anisotropy factor curves for structures of all opening diameters run similar to each other. Due to the specific process conditions in this study (5 min etch time for all values of the applied currents), the applied current density defined the etch depth, and therefore the dependencies of the anisotropy factor on the structure depth and the average current density are similar to each other. There is fast increase from negative values up to about 0.4 at small structure depth (small current density), and slow increase in the range 0.4–0.75 for deeper structures anodized at the average current densities above 0.8 A/cm2. As was shown for the low-doped samples, the change of the slope is likely due to the change of the process from the mixed regime to pure electropolishing, achieved in the experiment by increase of the current density. The observed negative anisotropy factor at 48 mA/cm2 means that lateral etching provided by lateral charge flow in the substrate to the etch form dominated over vertical etching for these structures.vi v Same

as for the low-doped samples, the arithmetic mean values of the current density between the initial current density 𝑗init and the final current density 𝑗area calculated for the increased total area were taken as the average current density 𝑗a . vi Extra check showed that the negative anisotropy factor values were obtained for welldefined symmetrical convex structures and the measurement was correct, thus these negative values are real. For example, the value of approximately –0.19 was obtained

5 Anodization process as a structuring technique

0.6 0.4 Dopen Dopen Dopen Dopen Dopen

0.2 0.0 −0.2 0

50

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

100 150 200 250 300 350

Structure depth, µm

Anisotropy factor Af

Anisotropy factor Af

168

0.6 0.4 Dopen Dopen Dopen Dopen Dopen

0.2 0.0 −0.2 0.0

0.5

1.0

1.5

2.0

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

2.5

3.0

3.5

Average current density, A/cm2

Figure 5.16.: Anisotropy factor in dependence of structure depth (left) and average current density 𝑗a (right) for structures anodized in highlydoped p-type silicon samples for 5 min; error bars in the right plot show the ranges between initial current densities and final current densities calculated for the increased area in the end of the process. For direct comparison to the data for the low-doped samples, again, an extraction of the data from sec. 5.1.2 for the etch duration of 5 min was done. The results shown in Fig. 5.17 are quite similar. The region of strong dependence on 𝑗a is wider for the highly-doped samples and ends at lower current density (below 0.8 A/cm2 ) than for the low-doped samples where the fast increase stops at current densities close to 1 A/cm2. However, it is not possible to make clear conclusions due to few data points available for the low-doped samples in this region. Another interesting observation from the plot of the anisotropy factor vs. current density for the low-doped samples is that the anisotropy factor for the smallest openings of 200 µm at 0.78 A/cm2 is already at the level corresponding to the electropolishing regime, whereas with increase of the opening diameter the anisotropy factor decreases to the values corresponding to the pore formation range or the mixed regime of electropolishing and pore formation. On the one hand, as was already discussed in sec. 5.1.2, it was observed that the critical current density between the pore formation and electropolishing regimes is lower, when frontside masking is used. for a structure etched through a 1000 µm opening, with the width of 1026 µm and the depth of 10.9 µm, giving lateral etch length of 13 µm and vertical etch length of 10.9 µm.

169

5.1 Process localization with frontside masking 10–16 Ω cm

0.6 0.4 Dopen Dopen Dopen Dopen Dopen

0.2 0.0 −0.2 0.0

0.5

1.0

1.5

2.0

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

2.5

3.0

Average current density, A/cm2

3.5

Anisotropy factor Af

Anisotropy factor Af

0.01–0.1 Ω cm 0.6 0.4

Dopen Dopen Dopen Dopen Dopen

0.2 0.0 −0.2 0.0

0.5

1.0

1.5

2.0

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

2.5

3.0

3.5

Average current density, A/cm2

Figure 5.17.: Anisotropy factor in dependence of applied current density for highly-doped samples (left) and low-doped samples (right) anodized for 5 min; the plot for the highly-doped samples is shown here for the same data range as for the low-doped samples for better comparison; error bars show the ranges between initial current densities and final current densities calculated for the increased area in the end of the process. The here observed dependence of the anisotropy factor on the opening diameter at 0.78 A/cm2 for the low-doped samples can confirm this observation: for smaller openings, the critical current density is lower, therefore at 0.78 A/cm2 there is electropolishing in those structures (high anisotropy factor), whereas in the bigger openings the pure electropolishing regime has not established yet (low anisotropy factor). On the other hand, one cannot exclude possible current density redistribution between the openings of different diameters, which could also explain the change of the process regime for an area-independent critical current density. In general, the anisotropy factor values for the highly-doped samples are slightly higher than for the low-doped samples (maximum values 0.715 vs. 0.69). This result confirms the observations of Steiner et al. [395], where a decrease of mask undercut (meaning increase of the anisotropy factor) during anodization of highly-doped silicon samples in comparison to lowdoped silicon samples was reported. This effect can be explained with the following considerations: for lower resistivity of the material, there is less electric potential differences in the substrate, therefore one can expect that the current crowding due to lateral charge flow from sides of a sample to the opening gets weaker in comparison to the vertical charge flow from the

170

5 Anodization process as a structuring technique

sample backside, and this leads to the higher anisotropy. The electrical models developed in this work confirmed this effect to some extent: for the model with the low-doped silicon samples and primary current distribution, the maximum anisotropy factor was approximately 0.22, whereas in the model for the highly-doped silicon samples the maximum anisotropy factor of approximately 0.38 was obtained (s. sec. 5.4). To conclude, in this section etch form development for highly-doped p-type silicon samples of resistivity in the range 0.01–0.1 Ω cm was presented. Similar shape development, as observed in sec. 5.1.2 for low-doped p-type silicon samples of resistivity in the range 10–16 Ω cm, was also obtained here. Direct comparison of the curvature and threshold depth values to the data for the low-doped samples showed that there is no any significant difference between them. Comparing this result to the electrical models presented later in this report, it was concluded that the mass transport effects can be responsible for this similarity by suppressing the differences caused by different current density distributions in the low-doped and highly-doped silicon substrates. Dependence of the anisotropy factor on the applied current density is also similar to the one observed for the low-doped samples, although the anisotropy factor values in electropolishing regime are slightly higher for the highly-doped samples, confirming the data from references. Verification of this conclusion was done in the electrical models solved for the low-doped and highly-doped p-type silicon (s. sec. 5.4). Because of the found similarities to the low-doped silicon, further experiments were done solely for the low-doped silicon as the common material used in the laboratory. 5.1.4. Proximity effects for neighboring openings for low-doped p-type silicon In this section, effect of neighboring structures on each other was studied for low-doped p-type silicon of resistivity 10–20 Ω cm. Setup 1 with circular samples of diameter 30 mm was used. Layout 2 of the openings in the silicon nitride frontside layer was applied (s. sec. 3.1.2.6, Fig. 3.2). All samples were anodized at current of 9 mA for 600 s. Summary of the samples and the process conditions is given in Tab. 5.3.

171

5.1 Process localization with frontside masking

Table 5.3.: Summary of samples and process conditions, where values of initial current density 𝑗init and final current density 𝑗area were calculated for initial and increased anodization area of the structures; sample names correspond to those described for layout 2 in sec. 3.1.2.6. Sample name

𝑗init , mA/cm2

𝑗area , mA/cm2

Sample name

𝑗init , mA/cm2

𝑗area , mA/cm2

2-1-1 2-1-2 2-1-3 2-2-1 2-2-2 2-2-3 2-3-1

143.2 143.2 114.6 573.0 286.6 143.2 286.5

111.4 109.1 90.0 300.8 198.6 121.3 199.1

2-3-2 2-3-3 2-3-4 2-4-1 2-4-2 2-4-3

143.2 95.5 71.6 143.3 143.3 143.3

120.7 86.0 66.4 115.7 116.3 115.8

As in the previous sections, total area of the samples was evaluated from the measured topographical scans, and this way the final current density values 𝑗area were calculated (s. Tab. 5.3). From the calculated initial and final current densities one can see that all structures were anodized in definite pore formation regime at 𝑗 < 𝑗PSL . The measured topographical scans of the etched cavities together with the corresponding opening layouts are shown in Fig. 5.18, Fig. 5.19, and Fig. 5.20 (in the shown layouts, the solid filled regions represent openings in the frontside insulating masks).

z, µm

scan path

s.2−1−1 s.2−1−2 s.2−1−3

0

layout 2-1

−20

−40

−60

0

2

4

6

x, mm

Figure 5.18.: Measured topographical scans of the structures with openings in frontside silicon nitride of layout 2-1 (single concentric ring with central circular opening, varied radii, s. sec. 3.1.2.6).

172

5 Anodization process as a structuring technique layout 2-2

layout 2-3

scan path

scan path

0

−50

z, µm

z, µm

0

−100

−150

s.2−2−1 s.2−2−2 s.2−2−3

0

2

4

x, mm

−50 −100 s.2−2−1 s.2−3−1 s.2−3−2

−150 6

8

0

2

s.2−3−3 s.2−3−4

4

6

8

x, mm

Figure 5.19.: Measured topographical scans of the structures with openings in frontside silicon nitride of layouts 2-2 (two openings of equal varied radius, constant distance between their centers, s. sec. 3.1.2.6) (left) and layout 2-3 (two openings of different radii, constant distance between their centers, one radius varied, s. sec. 3.1.2.6) (right). The profile for the sample 2-2-1 of layout 2-2 was also added to the right plot. Despite some asymmetry of the etched structuresvii , some conclusions can be drawn from the measured topographical scans. First, one can see that an increase of the opening area led to a decrease of the current density, which resulted in a lower etch rate and lower structure depth (compare samples 2-2-1, 2-2-2, and 2-2-3 in the left plot in Fig. 5.19). Another observation is that the current crowding did not only increase the etch rate on sides of a single structure, but also on the outer sides of a block of multiple structures, as was previously reported by Krüger et al. [96]. For example, in the structures 2-1-1...2-1-3 with an opening in form of a single concentric ring and a central circular opening (Fig. 5.18), the current crowding resulted in a pronounced higher etching of the outer edge of the ring opening. However, the outer ring did not help eliminating the current crowding for the central etch form. vii For

example, structure 2-2-1 in Fig. 5.19 (left): one would expect exact mirror symmetry between the left and the right structures anodized through equal openings.

173

5.1 Process localization with frontside masking layout 2-4 scan path upper line

s.2−4−1, lower line s.2−4−2, lower line s.2−4−3, lower line

scan path lower line

0

0

−20

−20

z, µm

z, µm

s.2−4−1, upper line s.2−4−2, upper line s.2−4−3, upper line

−40

−40 −60

−60 0

2

4

x, mm

6

0

2

4

6

x, mm

Figure 5.20.: Measured topographical scans of the structures with openings in frontside silicon nitride of layout 2-4 (four openings of constant radius, varied distance between their centers, s. sec. 3.1.2.6); the two upper structures and two lower structures are shown on separate plots. Another example showing current crowding in outer structures is the layout 2-4 with four openings of constant radius (s. Fig. 5.20): the outer edges of the block of two openings are etched deeper than the inner edges of the same openings. From the latter examples one can also see that the larger the distance between the openings in a block, the weaker is the effect, which is reasonable, because two structures infinitely separated from each other are expected to behave as two single structures. In Fig. 5.21a the effect for the samples 2-4-1...2-4-3 was quantified: there, the ratio of average depth values for the inner and the outer minima of the cavities in dependence of the distance between the centers of the openings is shown. The averaging is done for the upper and the lower structures on one sample, and the deviations from the average values are given as error bars. With further increase of the distance, the ratio is expected to approach unity.

174

5 Anodization process as a structuring technique

One more interesting observation is the current redistribution between the openings of different size on the samples 2-3-1...2-3-4 (s. right plot in Fig. 5.19). There, the structure anodized through the bigger opening has smaller depth. As was already mentioned in sec. 5.1.2, a similar effect was reported by Hedrich et al. [24] and Mescheder et al. [335]. The effect can be good explained when thinking of uniform charge flow over the sample area coming from cathode through the substrate to anode. The localization with frontside insulating layer makes the charge flow to squeeze through the limited open regions. The smaller the open regions, the more concentration of the charge is achieved, resulting in higher current density and etch rate. Therefore, if we assume that both structures receive equal current, and the etch depth is in direct proportion to the current density, then there must be an inverse proportion of the ratio of the depth values for the left and right structures 𝑑left /𝑑right to the initial area ratio 𝐴left /𝐴right . This dependence 𝑏 with a fit of a form 𝑓(𝑥) = 𝑎 + 𝑥+𝑐 is shown in Fig. 5.21b. (a)

(b)

0.85

dleft dright

0.90

0.95

1.0

Depth ratio

dinner average douter average

Depth ratio

1.1 1.00

0.9 0.8 0.7

0.80 2.0

2.5

3.0

3.5

4.0

Distance between centers of openings, mm

2

4

6

8

10

Initial area ratio

12

14

Aleft Aright

Figure 5.21.: (a) Dependence of the ratio of average depth values for the inner and the outer minima of the cavities from the plots in Fig. 5.20 (layout 2-4) on the distance between the centers of the openings, fitted with a function of a form erf(𝑐𝑥 + 𝑑), because for the argument going to infinity the depth ratio should approach unity; (b) dependence of the depth ratio on the initial area ratio of the left and right structures for the samples shown in the right plot in Fig. 5.19 (layout 2-3), fitted with 𝑏 a function of a form 𝑓(𝑥) = 𝑎 + 𝑥+𝑐 .

5.1 Process localization with frontside masking

175

To conclude, in this study few experiments have been performed with lowdoped p-type silicon samples to see the proximity effects for neighboring structures. Current redistribution between openings of different size and current crowding effects for multiple structures were observed. The evidence of current redistribution between the neighboring structures on a sample brings additional concern to the results of the studies of etch form development presented above in sec. 5.1.2 and sec. 5.1.3. However, the distance between the centers of the openings in those experiments was only 2 mm, in contrast to 5 mm here in the layout 2-3, and many more openings were distributed uniformly on a sample in those studies (s. Fig. 3.1), therefore, much less current redistribution is expected for those samples than observed here. 5.1.5. Study of etch form development for pre-structured cavities In this section, the etch form development after short electropolishing for low-doped p-type silicon samples (10–20 Ω cm) with frontside silicon nitride layer and cavities pre-structured with conventional etch techniques, namely anisotropic KOH etching and RIE, is presented. The goal of the study was to test on simple shapes whether anodization of pre-structured cavities can result in shapes which are not achievable with single anodization process. Anodization of RIE pre-structured cavities In the first experiment, a wafer with several uniformly distributed circular openings of diameter 1000 µm located in a distance of approximately 7 mm from each other was pre-structured in RIE to get a cavity with depth of approximately 206 µm, width of about 1157 µm, and a resulting anisotropy factor of approximately 0.625 calculated according to eq. (2.2) (s. Fig. 5.22, dataset “before anodization (RIE etched)”). Then, anodization was performed at an initial current density of approximately 1824 mA/cm2 for four minutes. Final current density for the increased area 𝑗area of 1279 mA/cm2 in the end of the process was calculated, thus the process run in electropolishing regime only. The resulting profile of a structure located in the center of the waferviii is shown in Fig. 5.22 (dataset “after anodization”). viii Minor

inhomogeneity of the process near the edges of the wafer was observed, presumably due to current crowding. To diminish this effect, in general, structures away from wafer edges were used for characterization.

176

5 Anodization process as a structuring technique

For comparison, profiles of two etch forms anodized to a comparable depth without pre-structuring are shown in Fig. 5.22 as well: one structure with single circular 1000 µm opening anodized in electropolishing regime at 5 V for 900 s with the current density decreasing from 1230 mA/cm2 to 820 mA/cm2 (dataset “not pre-structured, electropolished”), and another structure from sec. 5.1.2 anodized in pore formation regime for 20 min at the current density decreasing from 1500 mA/cm2 to 338 mA/cm2 (dataset “not pre-structured, pore formation”). The curves “after anodization” and “not pre-structured, electropolished” are very close to each other. This is an expectable result, because the anisotropy of the anodization process in electropolishing mode was shown above to be approximately 0.5–0.6 (s. sec. 5.1.2), that is close to the anisotropy of the here used RIE pre-structuring process of 0.625. In contrast, the cavity anodized in pore formation regime without pre-structuring with anisotropy factor of about zero shows a clear difference to the electropolished cavities.

before anodization (RIE etched) after anodization not pre-structured, electropolished not pre-structured, pore formation

z, µm

0

−100 −200 −300 0.0

0.5

1.0

1.5

2.0

x, mm

Figure 5.22.: Etch form development for a cavity pre-structured in RIE and then electropolished at current density 1824–1279 mA/cm2 for 240 s, compared to a structure anodized in electropolishing regime at 5 V for 900 s with current density decreasing from 1230 mA/cm2 to 820 mA/cm2 (dataset “not pre-structured, electropolished”), and a structure from sec. 5.1.2 anodized in pore formation regime for 20 min at current density decreasing from 1500 mA/cm2 to 338 mA/cm2 (dataset “not prestructured, pore formation”).

5.1 Process localization with frontside masking

177

Anodization of KOH pre-structured cavities In this experiment, two samples with layouts 3-1 and 3-2 have been prestructured in KOH (40 m% at 60 °C) for seven hours to achieve an etch depth of about 145–150 µm. The layouts 3-1 and 3-2 have been aligned along ⟨110⟩ direction, however, due to minor misalignment of ±2∘, the resulting width of the pre-structured cavities appeared to be few micrometers larger than the width of the openings in the silicon nitride layer. After the KOH process, electropolishing in 29.93 m% HF with ethanol at total current of 202 mA for one minute was performed. The total current of 202 mA corresponds to the initial current density of about 3.5 A/cm2 calculated for the total open area in the silicon nitride layer of 0.0576 cm2. Measured topographical scans of selected structures after electropolishing are shown in Fig. 5.23 and Fig. 5.24. The measurement here was done with a stylus with tip radius of 2.5 µm to measure correctly the sharp bottom parts of the structures. For comparison, the initial profiles after the KOH pre-structuring process are added to the plots. In general, one can clearly see that the electropolishing resulted in smoothing of the V-shapes due to higher etching of prominent parts. The structures anodized through the square openings with side length of 200 µm and 400 µm can be compared to the structures from sec. 5.1.2 anodized through the circular openings at the same initial current density of 3.5 A/cm2 till approximately the same final structure depth (Fig. 5.25). The factor between the areas of the square and circular openings of 4/𝜋 is neglected here. The anisotropy factor for the selected structures on both samples in dependence of the width of the openings in the frontside silicon nitride is given in Fig. 5.26. Values of the anisotropy factor for the cavities anodized without KOH prestructuring are also added to the plots. The anisotropy factor for the KOH pre-structured forms is slightly below one due to the minor misalignment of the layout to the ⟨110⟩ direction mentioned above, which resulted in increase of the width in comparison to the initial width of the openings in the silicon nitride layer. After electropolishing of these KOH etched structures, their anisotropy factor has reduced, however, it still remains larger than the anisotropy factor for the structures which were only electropolished without the KOH step (s. Fig. 5.26a). Since it was shown in sec. 5.1.2 that the anisotropy factor for the electropolished structures does not depend on the etch depth, one can expect that further electropolishing of

178

5 Anodization process as a structuring technique 100 µm, 100 µm, 200 µm, 200 µm,

300 µm, 300 µm, 400 µm, 400 µm,

KOH KOH + electropolishing KOH KOH + electropolishing

0

−50

z, µm

z, µm

0

−100 −150

KOH KOH + electropolishing KOH KOH + electropolishing

−50 −100 −150

400

500

x, µm

600

700

−200

400

600

800

x, µm

Figure 5.23.: Measured topographical scans of selected structures for samples with square openings after electropolishing; initial KOH profiles are reconstructed based on the width of the structures and known theoretical etch form development for KOH anisotropic etch process (s. Fig. 2.3); straight regions on both sides of the measured profiles near the top level are measurement artifacts; values in the legend mean the width of openings in silicon nitride layer. the structures anodized in a combined process will reduce the anisotropy factor further to about 0.6 observed for electropolishing in sec. 5.1.2. Nevertheless, the results show that short electropolishing of cavities pre-structured with a highly anisotropic process, such as KOH etching, can provide structures with smooth high-quality surface and anisotropy factor larger than that achieved with electropolishing only. The curvature of the structures achieved in the combined KOH+electropolishing process for the sample with square openings was also evaluated as described in sec. 3.2.4 (s. Fig. 5.27). One can note a linear decrease of the curvature with increase of the opening width. The curvature for the structures etched through the circular openings in single electropolishing process is smaller than for the presented combined process. To conclude, in this short study shape modification during electropolishing of the pre-structured cavities was investigated for low-doped p-type silicon. The combined process with RIE pre-structuring showed almost no difference to the single electropolishing process due to similar anisotropy factors

179

5.2 Process localization with backside local contacts 100 µm, 100 µm, 200 µm, 200 µm,

300 µm, 300 µm, 400 µm, 400 µm,

KOH KOH + electropolishing KOH KOH + electropolishing

0

0 −50

z, µm

z, µm

KOH KOH + electropolishing KOH KOH + electropolishing

−100

−50 −100 −150

−150 500

600

x, µm

700

−200

400

600

800

x, µm

Figure 5.24.: Measured topographical scans of selected structures for samples with rectangular openings after electropolishing; initial KOH profiles are reconstructed based on the width of the structures and known theoretical etch form development for KOH anisotropic etch process (s. Fig. 2.3); straight regions on both sides of the measured profiles near the top level are measurement artifacts; values in the legend mean width of openings in silicon nitride layer. of the two processes. The KOH pre-structuring process provides more distinct difference, however, one can expect that with longer electropolishing of the KOH pre-structured cavities, the shapes will resemble those obtained in a single electropolishing process due to the shape defining mechanisms of silicon anodization (e.g. charge flow in the cell and mass transport in electrolyte). 5.2. Process localization with backside local contacts 5.2.1. Etch form development for single circular backside contact In this section, etch form development during anodization process for lowdoped p-type silicon samples (10–20 Ω cm) without any films on the frontside and with single local circular backside contact of diameter 200 µm, 1000 µm or 1800 µm, defined with boron ion implantation, was investigated. Setup 2 with 20 mm × 20 mm sample holder was used. The total current of 10 mA was applied for samples with backside contacts of diameter 200 µm, and

180

5 Anodization process as a structuring technique Opening width 200 µm

Opening width 400 µm KOH KOH + EP EP only

0 −50

−50

z, µm

z, µm

0

−100

−100 −150

−150

−200 400

500

600

700

400

600

x, µm

800

1000

x, µm

Figure 5.25.: Comparison of the combined KOH+electropolishing process done for square openings of width 200 µm and 400 µm to the single electropolishing process done for circular openings of diameters 200 µm and 400 µm from sec. 5.1.2; straight regions on both sides of the measured profiles near the top level are measurement artifacts; EP - electropolishing. (b) 1.0

1.0

0.9

0.9

0.8 0.7 0.6 KOH KOH + electropolishing electropolishing only

0.5 0.4 100

200

300

Opening width, µm

400

Anisotropy factor Af

Anisotropy factor Af

(a)

0.8 0.7 0.6 0.5

KOH KOH + electropolishing

0.4 100

200

300

400

Opening width, µm

Figure 5.26.: Anisotropy factor vs. width of opening (a) for square openings with comparison to structures anodized without KOH prestructuring through circular openings from sec. 5.1.2 (the structures anodized for 5 min at initial current density of 3.5 A/cm2 ) and (b) for rectangular openings.

181

5.2 Process localization with backside local contacts 20

Curvature, mm−1

KOH + electropolishing electropolishing only 15

10

5

100

200

300

400

Opening width, µm

Figure 5.27.: Curvature as a function of opening width for sample with square openings, with comparison to single electropolishing process done for cirular openings of diameters 200 µm and 400 µm from sec. 5.1.2. 15 mA for samples with backside contacts of diameter 1000 µm and 1800 µm. Anodization duration was varied in the range from 5 min to 40 min. Anodization was done in the 29.93 m% HF electrolyte with ethanol. Optical observations after anodization showed that anodization run on the whole frontside area of the samples in contact with the electrolyte (about 10 mm × 10 mm), i.e., more than 5 mm distance from the structure centerix (s. Fig. 5.28, images (a) and (b) at the top). This indicates that despite the local backside contact, the anodized area is quite large. Due to the local backside contact, the local current density varied strongly across the frontside of the samples, therefore it is not useful to talk about anodization area (which is not well defined) and average current density. This also means that both pore formation and electropolishing could run simultaneously in different locations of the same sample. On the samples, the anodization far from the structure center resulted in a thin porous silicon film with multiple concentric color rings due to interference, especially good visible on the samples after short anodization process (Fig. 5.28a top). The changing color indicated an increase of the film thickness from the anodization area edges to the structure center, where the current density was the highest. On the samples anodized longer than ix The

structure center means the point on the frontside above the center of the local backside contact.

182

5 Anodization process as a structuring technique

20 min, the effect was hardly visible because of the increased thickness of the film (s. top picture in Fig. 5.28b). The porous film on all samples was removed in diluted KOH solution (s. bottom pictures in Fig. 5.28a and b). One can note a rather shiny central region on the structures after removal of porous silicon. This area was presumably electropolished. That this area was covered with porous silicon could indicate that in the beginning the local current density in this central area was below the critical current density, and it increased over the critical current density after the structure reached certain depth. (a)

(b)

(c)

(d)

Figure 5.28.: (a) and (b): frontside view of samples with backside circular contact of diameter 200 µm, which were anodized for (a) 10 min and (b) 40 min, before (top) and after (bottom) removal of porous layer; (c) backside hole etched through the sample with 200 µm backside contact after anodization for 40 min; (d) view of the hole in the chromium layer on the mask used to define the local backside contact of diameter 200 µm. The structure on the sample with 200 µm backside contact, anodized for 40 min, was etched through the sample, presumably in the last minutes of the process. In Fig. 5.28c the backside hole on the sample is shown. The shape of the backside hole matches closely the shape of the backside local contact defined by the mask shown in Fig. 5.28d. After removal of porous silicon, etch forms have been measured. In order to get 2D topographical scans going exactly through the centers of the

183

5.2 Process localization with backside local contacts

structures, 3D topographical scans of the etch forms have been measured first, and then 2D scans through the centers of the structures have been extracted (s. Fig. 5.29). 1000 µm

1800 µm

0

0

0

−100

−100

−50

−300 −400

−300

5–40 min 1

2

x, mm

3

−500

−100 −150

−400

−500 −600

−200

z, µm

−200

z, µm

z, µm

200 µm

−200

20–35 min 3

4

5

6

x, mm

7

8

10–35 min 2

4

6

8

x, mm

Figure 5.29.: Measured topographical scans through the centers of structures anodized with circular backside contact of diameter 200 µm, 1000 µm, and 1800 µm. As shown in Fig. 3.6, width of the structures 𝑤etch obtained by anodization of samples with backside local contact was defined as four times structure width at the half structure depth. Dependence of the structure width on depth is shown in Fig. 5.30a. As can be seen from the plot, the width of the structure, i.e., the width of the actively etched area, decreased during the process. This means that the process was becoming more localized as the structure was getting deeper. Etch rate 𝑅etch for the given sample numbers 𝑖 = [1 .. 𝑛] with the corresponding structure depth 𝑑etch 𝑖 and anodization duration 𝑡etch 𝑖 was calculated in the following way: 𝑅etch =

𝑑etch 𝑖 𝑡etch 𝑖

(5.5)

The etch rate in dependence of the structure depth is plotted in Fig. 5.30b. With increase of the structure depth for the samples with 200 µm backside contact, the etch rate increased significantly. An explanation of this

184

5 Anodization process as a structuring technique (b) 200 µm 1000 µm 1800 µm

8 6 4 2

200

Etch rate, nm/s

Structure width, mm

(a)

150

100 200 µm 1000 µm 1800 µm

50 0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

500

Structure depth, µm

Figure 5.30.: (a) Dependence of structure width on structure depth with linear fit and (b) etch rate in dependence of structure depth with a second order polynomial fit for dataset “200 µm”; on both plots, datasets for various diameters of backside contact are shown; error bars on the etch rate plot refer to the difference between the values obtained for several samples anodized for the same duration. effect is the following: as the structure gets deeper, the direct distance from the backside local contact to the bottom of the etched structure decreases, and due to larger conductivity of the electrolyte the resistance of the path decreases as well. This results in higher current density in the center of the structure, and the process runs faster in a positive feedback loop. The effect is more pronounced for the samples with smaller backside contacts. The effect seems to be reduced by some other effect, because with increase of the depth the increasing of the etch rate slows down. This reduction could be caused by the porous silicon layer limiting the flow of reactants to the etch front. As with the decreasing structure width vs. structure depth, increasing etch rate means that the process gets more and more localized to the center of the structure.

5.2 Process localization with backside local contacts

185

This effect is also good visible by looking at the anisotropy factor. It was calculated from the structure width 𝑤etch and depth 𝑑etch x : 𝑤 𝐴f = 1 − etch (5.6) 2𝑑etch The dependence of the anisotropy factor on the structure depth is shown in Fig. 5.31a. Due to the increase of the process localization in the center of the structure, the anisotropy factor increases strongly from high negative values (𝑤etch ≫ 𝑑etch ) to about zero. For the samples with larger backside contacts, the same anisotropy factor is achieved at larger depth. From the plot one can also conclude that the backside localization could not provide anisotropy above zero (even when taking not the quadruple width at half depth, but only single width at half depth), i.e., the width is larger than depth for all obtained structures. Curvature of the central part of the structures was obtained as described in sec. 3.2.4 by performing an arc fit to the central 60 % part of the width at the half structure depth. Initial point (0, 0) was added to the data for each dataset. Curvature in dependence of the structure depth is shown in Fig. 5.31b. Increasing curvature during the process confirms the increasing localization as the structure gets deeper. With smaller backside contact, shapes with smaller radius of curvature are achieved. In the measured topographical scans in Fig. 5.29 one can note that the shapes are not smooth, but rather rough with significant depth variation. To smooth the shapes, an additional electropolishing step can be performed. In the study, some selected samples were electropolished in 7 m% HF electrolyte at total current of 80 mA/sample to demonstrate feasibility of such approach. In Fig. 5.32 topographical scans for two samples with backside contact of diameter 1000 µm before/after the electropolishing step are shown. One can note an improved smoothness of the shapes after electropolishing. To conclude, in this section etch form development in low-doped p-type silicon samples (10–20 Ω cm) with single circular backside contacts of diameter x Application

of the anisotropy factor for such funnel-like structures without clearly defined walls, as in this experiment, is arguable, because structure width is not clearly measurable. Here, the quadruple width at the half structure depth was defined as described above. Another definition of the structure width would result in different values of the anisotropy factor. Therefore, the absolute values of the anisotropy factor are quite subjective. However, relative change of the anisotropy factor helps to understand the etch form development during the process and is of interest for us.

186

5 Anodization process as a structuring technique

(a)

(b) 200 µm 1000 µm 1800 µm

−20 −40 −60 −80

200 µm 1000 µm 1800 µm

−100 −120 0

100

200

300

400

500

Structure depth, µm

Curvature, mm−1

Anisotropy factor Af

0 3 2 1 0 0

100

200

300

400

Structure depth, µm

Figure 5.31.: (a) Anisotropy factor vs. structure depth fitted with a logarithmic function and (b) curvature vs. structure depth with a parabolic fit going through (0, 0); anisotropy factor was calculated according to eq. 5.6.

0

sample A before electopolishing sample A after electopolishing sample B before electopolishing sample B after electopolishing

z, µm

−100 −200 −300 −400 5.0

5.5

6.0

x, mm

Figure 5.32.: Effect of electropolishing on cavities formed at low current in (mostly) pore formation regime with backside contact of diameter 1000 µm; sample A was electropolished for 20 min at total current of 80 mA, sample B was electropolished for about 12 min at total current of 80 mA. 200 µm, 1000 µm, and 1800 µm was investigated. It was shown that absence of frontside protective masking results in a nearly unlimited anodization

5.2 Process localization with backside local contacts

187

area, i.e. anodization runs at the area which is much larger than the area of the backside contact. However, as the structure gets deeper, the process becomes more and more localized due to redistribution of charge flow more to the center of the structure. This results in increasing anisotropy factor, etch rate, and curvature in the center of the structure. The effect is more pronounced for the samples with smaller backside contacts. Increase of curvature during the process can be used in application of the process for fabrication of microlenses/micromirrors: this way, geometry of the optical elements can be adjusted. Additionally, it was found that the width of the structures was always larger than depth, and positive anisotropy factor values could not be achieved with this backside localization technique. Finally, it was shown that the rough shapes after anodization in 29.93 m% HF with ethanol can be improved with electropolishing in 7 m% HF electrolyte at total current of 80 mA/structure. 5.2.2. Influence of spacing between two backside rectangular openings on etch form In this section, influence of spacing between two backside rectangular openings on etch form was studied. Setup 2 with 20 mm × 20 mm sample holder was applied. Silicon samples of p-type with resistivity 10–20 Ω cm were used. Layout 4 was applied for backside local contacts (s. sec. 3.1.2.6). Anodization was performed at constant applied voltage of 5 V between the anode and cathode for 20 min in 29.93 m% HF with ethanol. One sample of each layout from 4-1 to 4-6 was anodized. Total measured average current showed increase from the sample of layout 4-1 to the sample with layout 4-6 from 41 mA to 68.2 mA. This can be the result of increasing actively anodized area, as the total width of the backside contacts gets bigger, and resulting decrease of the resistance of the samples. After anodization, the anodized frontside surface appeared dark due to formed porous silicon layer. After removal of the porous layer, topographical scans have been done across the structures (along the shorter side of the backside contacts) in the middle of the structures’ long side (s. Fig. 5.33). By measuring the profiles in the middle of the longer side, a quasi 2D case can be considered, because the length of the backside contacts is much larger than the total width of the contacts including spacing between them.

188

5 Anodization process as a structuring technique

layout 4 scan path width 100 µm, spacing 520 µm width 100 µm, spacing 780 µm width 100 µm, spacing 1040 µm

0

0

−50

−50

z, µm

z, µm

width 50 µm, spacing 0 µm width 100 µm, spacing 0 µm width 100 µm, spacing 260 µm

−100 −150 −200

−100 −150

0

2

4

x, mm

6

−200

2.5

3.0

3.5

4.0

4.5

x, mm

Figure 5.33.: Full view (left) and detailed view of central region (right) of topographical scans of etched cavities after removal of porous silicon for samples with two backside contacts of rectangular shape of dimensions 0.1 mm × 8 mm with spacing between them increasing from 0 (layout 4-1 with single contact of width 100 µm, or as written in the legend, two side-by-side contacts each 50 µm wide; layout 4-2 with two 100 µm wide contacts side-by-side giving a single contact of width 200 µm) to 1040 µm (layout 4-6); width refers to the width of each of the two contacts. For the sample with the layout 4-3, the shape of the etch form is similar to that observed for single backside contacts of the layouts 4-1 and 4-2 (s. sec. 5.2.1). For these samples, spacing between the contacts was smaller than the thickness of the samples (about 520 µm). In contrast, for the samples with the layouts 4-4, 4-5, and 4-6 with spacing from 520 µm to 1040 µm, two minima on the etch forms are observed, thus, for these samples, effect of each separate backside contact is visible. With the observed etch forms as a result of two backside contacts, it was proposed that the resulting etch form defined by multiple backside contacts is the result of superposition of the elementary etch forms, each defined with

5.2 Process localization with backside local contacts

189

single elementary backside contact. This concept was named “pixel concept”, where each single elementary backside contact (“pixel”) contributes to the resulting etch form. In this study, the single backside contact of width 100 µm (sample with the layout 4-1) represents a one-dimensional “pixel” - elementary shape. To see if the “pixel concept” is valid for the structures obtained in this study, the etch forms of the layouts 4-2..4-6 corresponding to the sample number 𝑖 = [1 .. 5] were reconstructed based on the etch form of the layout 4-1 (sample 𝑖 = 0) by superposition in the following way: 𝑆𝑖∗ (𝑥) = 𝐶𝑖 [𝑆0 (𝑥) + 𝑆0 (𝑥 − 𝑤0 − 𝑠𝑖 )]

(5.7)

where 𝑆𝑖∗ (𝑥) is the reconstructed etch profile (depth as a function of 𝑥-coordinate) for a sample 𝑖, obtained by superposition of the elementary profile 𝑆0 (𝑥), 𝑤0 is the width of the backside contact for the elementary profile 𝑆0 (𝑥) (100 µm in our case), 𝑠𝑖 is the spacing between two backside contacts for the sample 𝑖, and 𝐶𝑖 is the compensation coefficient equal to the ratio of the total current 𝐼𝑖 of the sample 𝑖 and double total current 𝐼0 of the sample with the elementary structure: 𝐶𝑖 =

𝐼𝑖 2𝐼0

(5.8)

The compensation coefficient 𝐶𝑖 was introduced as the current observed for the samples with two backside contacts was lower than double value of the current of our elementary structure. Thus, if we assume equal valence, i.e., equal volume etched per charge, we need to reduce the depth values of the composed profile to correspond to the total etched volume of the structure to be reconstructed. Comparison of the reconstructed forms 𝑆𝑖∗ to the real anodized etch forms 𝑆𝑖 shown in Fig. 5.34 reveals a remarkable matching. This means that the “pixel concept” works and can probably be applied to model arbitrary shapes obtained by complex layouts of backside local contacts. Moreover, if the approach works for more complex layouts, then, with this approach one can determine whether a desired shape can be formed by anodization with backside local contacts: if the desired shape can be represented as superposition of an elementary “pixel” shapes, the answer is positive. It is important to note that in real case current for the desired complex shape is not known, therefore the absolute depth of the resulting shape

190

5 Anodization process as a structuring technique 𝑠1 = 0 µm

𝑠2 = 260 µm elementary profile elementary profile, shifted constructed complex profile real complex profile

−50

z, µm

z, µm

−50

−100

−100 −150

−150

−200 0

1

2

0

x, mm

2

3

x, mm

𝑠3 = 520 µm

𝑠4 = 780 µm

−100 −150

𝑠5 = 1040 µm

0

0

−50

−50

z, µm

z, µm

−50

z, µm

1

−100 −150

0

1

2

x, mm

3

−100

−150

0

2

x, mm

4

0

2

4

x, mm

Figure 5.34.: Comparison of the real measured profiles of the anodized structures with the profiles reconstructed according to the “pixel concept”; 𝑠 is the spacing between the two backside contacts of width 100 µm. will most likely deviate from the reconstructed profile, therefore, adjustment of the anodization duration might be necessary. Additionally, due to the dependence of shape development for backside contact localization on depth determined in sec. 5.2.1, it might be necessary to take an elementary structure with depth comparable to the depth of the desired structure. To conclude, in this study influence of spacing between two rectangular backside contacts was studied in a quasi 2D case for low-doped p-type silicon. It was determined that for the samples with the spacing between the backside contacts lower than the sample thickness the resulting etch form is similar to the one obtained with a single backside contact. A novel method

5.3 Combination of frontside masking and backside local contact

191

of “pixel concept” was introduced and demonstrated to give correct results for reconstruction of the etch forms obtained in this study. If this “pixel approach” holds for arbitrary complex layouts of backside local contacts, it can provide a really helpful general solution to determine what types of shapes can be obtained with backside contact localization. This way, also a layout generator with a quite straightforward algorithm can be built, which gets a structure as input and generates a layout of backside local contacts based on the input shape. 5.3. Combination of frontside masking and backside local contact In this section, influence of frontside opening width in silicon nitride in combination with a single backside contact on etch form was studied. Setup 2 with 20 mm × 20 mm sample holder was applied. Silicon samples of p-type with resistivity 10–20 Ω cm were used. Layout 5 was applied with single rectangular window in frontside silicon nitride of length 8 mm and varied width in the range 130–4160 µm (s. sec. 3.1.2.6). The single rectangular backside contact centered to the frontside opening had width of 100 µm and length of 8 mm. Anodization was performed at constant applied voltage of 5 V between the anode and cathode for 20 min in 29.93 m% HF with ethanol. One sample of each layout from 5-1 to 5-6 was anodized. Total measured average current showed increase from the sample of layout 5-1 to the sample with layout 5-6 from about 22 mA to 44 mA. Similar behavior was observed in sec. 5.2.2 where samples with local backside contacts and free frontside were anodized. Again, as was explained in that section, this can be the result of increasing actively anodized area as the width of the frontside opening gets bigger, which results in decrease of the resistance of the samples. Summary of the values of total current and averagexi initial current density values calculated based on the area of the frontside opening is given in Tab. 5.4. After anodization, the anodized frontside surface appeared electropolished on the samples 5-1 and 5-2, and with porous silicon layer on the rest samples. Looking to the calculated initial current density values (s. Tab. 5.4) and comparing the values to the critical current density of 0.8–0.9 A/cm2, obtained in sec. 5.1.2, this result corresponds to the previous findings. xi Average

current densities over the area are meant here. Local current density values are expected to vary significantly across the anodization area due to the local backside contact.

192

5 Anodization process as a structuring technique

Table 5.4.: Summary of samples and process conditions, where values of the initial current density 𝑗init are calculated for the initial anodization area defined with the frontside opening in silicon nitride; sample names correspond to the names of the sub-layouts of the layout 5 in sec. 3.1.2.6. Sample name

𝐼, mA

𝑗init , mA/cm2

Sample name

𝐼, mA

𝑗init , mA/cm2

5-1 5-2 5-3

21.9 29 38

2106 1394 913

5-4 5-5 5-6

41.3 43.6 44.4

496 262 133

After removal of the porous layer and silicon nitride, topographical scans have been measured across the structures (parallel to the shorter side of the backside contact) in the middle of the structures’ long side (s. Fig. 5.35). By measuring the profiles in the middle of the longer side, a quasi 2D case can be considered, because the length of the backside contacts is much larger than the total width of the contacts including spacing between them.

width 130 µm width 260 µm width 520 µm width 1040 µm width 2080 µm width 4160 µm no frontside mask

0

z, µm

−50 −100 −150

Layout 5

8mm

−200 −250 0

1

2

3

x, mm

4

w scan path

5 100 µm

Figure 5.35.: Topographical scans of the etched cavities on the samples with a single backside contact and a single frontside opening according to layout 5: in the legend, width refers to the width of the frontside opening 𝑤front ; “no frontside mask” - profile of the structure obtained by anodization of a sample with single backside contact of width 100 µm without frontside masking from sec. 5.2.2. Inset at the right bottom corner shows layout 5, where the solid filled region is the frontside mask and the hatched region is the backside contact.

5.3 Combination of frontside masking and backside local contact

193

From the measured profiles, one can see that the backside localization resulted in concave form development for the samples 5-1, 5-2, and 5-3, i.e., for the samples with width of the frontside opening equal or smaller than the thickness of the wafer which is approximately 525 µmxii . The effect of current crowding typical for frontside localization with insulating masking and enhanced etching in the center of the structure typical for the backside contact localization was observed on the samples with larger width of the opening only. For comparison, the profile of the structure obtained by anodization of the single backside contact of width 100 µm without frontside masking from sec. 5.2.2 is added to the plot of the measured profiles. One can see that the larger the opening, i.e., the weaker the frontside localization of the process, the closer the shape of the etched form to the shape of the structure anodized at about the same total charge without frontside masking. One interesting question is whether such a combination of the frontside and backside localization can provide etching with high anisotropy. Anisotropy factor for the structures was evaluated according to eq.(2.2). In Fig. 5.36 dependence of the anisotropy factor on the ratio of the width of the frontside opening 𝑤front to the width of the backside contact 𝑤back is shown. In contrast to expectations, no high positive anisotropy factor values could be obtained with small frontside opening width, and even for the sample with frontside opening width of 130 µm, the anisotropy factor ratio is not higher than the one achieved for the samples with all-area backside contact in sec. 5.1.2 and sec. 5.1.3. On the plot in Fig. 5.36, there is a minimum of anisotropy factor for the sample with the width of the frontside opening of 1040 µm, however, due to rather different shape of the cavities, and depth dependent anisotropy factor for samples with local backside contacts as was observed in sec. 5.2.1, direct comparison of the values of 𝐴f on this plot should be avoided. To conclude, in this short study localization of anodization process with frontside insulating masking and backside local contacts was demonstrated for low-doped p-type silicon. Even with local backside contacts, use of frontside mask resulted in current crowding. It was also shown that with the chosen process parameters and layout geometry, no higher anisotropy factor values than obtained alone with frontside localization could be achieved. xii This

could be also the result of smaller frontside opening promoting the diffusion controlled process.

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5 Anodization process as a structuring technique

Anisotropy factor Af

0.55 0.50 0.45 0.40 0.35 0.30 0.25 0

10

20

30

40

wfront /wback

Figure 5.36.: Dependence of anisotropy factor on the ratio of the width of frontside opening 𝑤front to the width of backside contact 𝑤back . 5.4. Macroscale simulation of anodization process Simulation of anodization process with real process parameters in macroscopic scale is important for better understanding of the shape controlling mechanisms. The literature research showed that no systematic work on development and verification of such models has been done yet. As was observed in chapter 4, etch rate shows almost linear dependence on applied current density. Therefore, an indispensable goal of the model for this electrochemical process is to describe the charge flow through the silicon substrate and electrolyte. Additionally, the effects of activation polarization have to be checked. For these goals, two types of models were developed in the work: with the primary current distribution and with the secondary current distribution. Simulation of etch form development during anodization through a single circular opening in a frontside insulating masking layer was performed. For comparison, the corresponding experiments with the low-doped and highlydoped p-type silicon samples (s. sec. 5.1.2 and sec. 5.1.3) were taken. Thus, four models have been developed and solved: • primary current distribution model for the low-doped silicon samples • primary current distribution model for the highly-doped silicon samples • secondary current distribution model for the low-doped silicon samples

5.4 Macroscale simulation of anodization process

195

• secondary current distribution model for the highly-doped silicon samples For each model, the diameter of the frontside opening 𝐷open was varied from 200 µm to 1000 µm with steps of 200 µm. The models for the lowdoped silicon samples were solved according to the experiment in sec. 5.1.2 for five initial current densities 𝑗init in the range 1.0-3.5 A/cm2 . The models for the highly-doped silicon samples were solved according to the experiment in sec. 5.1.3 for ten values of initial current densities 𝑗init in the range 0.05-3.5 A/cm2 . As a result, a set of 150 models with unique parameter combinations was solved in the work. In the following, the results of the macroscale models are presented and compared to the experiments. 5.4.1. Primary current distribution model for low-doped p-type silicon 5.4.1.1. Introduction In the model here, etch form development during anodization process of ptype low-doped silicon samples (conductivity 7.5 S/m, resistivity 13 Ω cm) with a singular circular opening in the frontside masking layer from sec. 5.1.2 was simulated. As was discussed in sec. 5.1.2, the etch rate at high current densities evaluated for large area samples in chapter 4 was found to be not applicable to the small area structures. Therefore, the limiting current density 𝑗ox = 0.8–0.9 A/cm2 evaluated from the optical observations in sec. 5.1.2 was used. Based on this value, and the values of the porosity and the dissolution valence at current densities below 100 mA/cm2 from chapter 4, porosity and dissolution valence dependencies on current density were constructed for the simulation as shown in Fig. 5.37. The resulting etch rate calculated from the Faraday’s law of electrolysis with these dependencies according to eq. (2.38) and eq. (2.40) is shown in Fig. 5.38. Only the experimental data for current densities below 100 mA/cm2 from sec. 4.1.1 were used here, because at such low current densities the data were considered to be not dependent on the anodization area, i.e., not dependent on mass transport limitations 2 in electrolyte. For electropolishing, 𝑗ox = 900 mA/cm with 𝑛e = 4 and 𝑃% = 100 %, as obtained from optical observations in sec. 5.1.2, was used.

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5 Anodization process as a structuring technique

4.0

90

80 experiment reconstructed

70 0.0

0.5

1.0

Dissolution valence

Porosity, %

100

1.5

3.5 3.0 2.5

experiment reconstructed 0.0

Current density, A/cm2

0.5

1.0

1.5

Current density, A/cm2

Figure 5.37.: Functions of porosity (left) and dissolution valence (right) in dependence of current density constructed for the model based on available experimental data for p-type silicon of resistivity 10–20 Ω cm.

pore formation electropolishing experiment reconstructed

Etch rate, µm/s

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

Current density, A/cm2

Figure 5.38.: Etch rate in dependence of current density for the model with p-type silicon of resistivity 10–20 Ω cm, together with the boundary curves for pore formation and electropolishing regimes as described in sec. 4.2.1. 5.4.1.2. Results and discussion The model was solved for five values of initial current density and five values of opening diameter (s. sec. 3.3.1), giving in total 25 models. Etch form development from convex to concave was observed for the models with the opening diameter 200 µm and 400 µm. For the bigger opening diameters, convex forms developed into less convex forms, however, due to limited

197

5.4 Macroscale simulation of anodization process

depth of the substrate they could not transform into concave forms. Thus, the results are similar to the experiment in sec. 5.1.2, although no triplehollow structures (s. Fig. 5.4c) were observed. As example, etch form development for the models with opening diameter of 200 µm and 1000 µm, and initial current density of 1 A/cm2 is shown in Fig. 5.39. 𝐷open = 200 μm

z, µm

−100 −200 −300 −400 −500 −600 −400 −200

0

0.0 s

1341.4 s 2765.5 s 4527.4 s 6381.3 s 9333.7 s

0

200

261.3 s 647.2 s

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1167.7 s

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1743.1 s

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−100

516.0 s

z, µm

0

𝐷open = 1000 μm

400

600

3171.9 s

−1000

−500

0

500

1000

r, µm

Figure 5.39.: Etch form development simulated for the primary current distribution model for low-doped p-type silicon samples with opening diameters of 200 µm (left) and 1000 µm (right) at initial current density of 1 A/cm2 . In order to make sure that the taken parameters for the model are close to the experimental conditions, structure volumes from the experiment and the simulation were compared, as shown in Fig. 5.40 for the model with opening diameter of 1000 µm. The plots for all other opening diameters are shown in Appendix (sec. A.4.1.1). In average, structure volume in the simulation was approximately 4 % larger than in the experiment, which shows that the chosen porosity and valence functions for the simulation produced results close to those observed in the experiment in regard to the structure volume. Comparing the simulation results to the experiment in sec. 5.1.2, one can see that the simulated forms mostly show much lower anisotropy (s. Fig. 5.41). To make more quantitative comparison, curvature and anisotropy factor in dependence of structure depth were evaluated for each model, similarly to the characterization of etch forms done for the experiments. Dependencies of curvature and anisotropy factor on structure depth for the model with diameter of the opening of 200 µm and varied current density

198

5 Anodization process as a structuring technique 100

exp. : 1.0 A/cm2 sim. : 1.0 A/cm2

Structure volume, mm3

exp. : 1.5 A/cm2 sim. : 1.5 A/cm2 exp. : 2.0 A/cm2 sim. : 2.0 A/cm2 10

exp. : 2.5 A/cm2

−1

sim. : 2.5 A/cm2 exp. : 3.0 A/cm2 sim. : 3.0 A/cm2 exp. : 3.5 A/cm2 sim. : 3.5 A/cm2

10−2

0

5

10

15

20

Etch time, min

Figure 5.40.: Comparison of structure volume between the experiment (datasets starting with “exp.”) and the primary current distribution model (datasets starting with “sim.”) for low-doped p-type silicon samples with opening diameter of 1000 µm. are shown in Fig. 5.42. In Fig. 5.43, curvature and anisotropy factor vs. structure depth for the model with varied diameter of the opening and initial current density of 1 A/cm2 are presented. Curves for all other combinations of opening diameter and initial current density are shown in Appendix (s. sec. A.4.1.2 and sec. A.4.1.3). The curves for curvature obtained from the model are in general similar to those obtained experimentally in sec. 5.1.2. Additionally, in the last 50–100 µm remaining distance from the etch front till the sample backside, some stronger increase of the curvature values (for both negative and positive values) was observed, likely due to increase of the direct charge flow from the sample backside to the etch front. There is only small dependence of the curvature curves on initial current density, thus the effect of current density dependent porosity and dissolution valence seems to play little role in etch form development. In contrast, there is very strong dependence on the opening diameter. The effect is even better to see on the plots for the threshold depth values as shown in Fig. 5.44. There, the values for the opening diameter of 200 µm, 400 µm, and 600 µm were obtained as described earlier. For the opening diameters of 800 µm and 1000 µm, the concave shape was not reached. The threshold

199

5.4 Macroscale simulation of anodization process 𝐷open = 200 μm, 𝑗init = 3 A/cm2

0

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−50

−50

z, µm

z, µm

𝐷open = 200 μm, 𝑗init = 1 A/cm2

−100 −150

−150 −200

−200

−250

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r, µm

𝐷open = 1000 μm, 𝑗init = 1 A/cm2 0

0

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300

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𝐷open = 1000 μm, 𝑗init = 3 A/cm2

z, µm

z, µm

−100

−200 −300 −400

−250 −500

0

r, µm

500

−500

−500

0

500

r, µm

Figure 5.41.: Comparison of simulated and experimental etch shapes for opening diameters 200 µm (top) and 1000 µm (bottom) for initial current densities 1 A/cm2 (left) and 3 A/cm2 (right); anodization duration 1 min, 5 min, 10 min, and (only for 1 A/cm2 ) 20 min; low-doped silicon samples; solid lines are the experimental etch forms, dotted lines are the simulated etch forms. depth values in this case were obtained by extrapolation of the curvature curves to the positive quadrant, neglecting the decrease of the curvature in the last 50–100 µm. This extrapolation resulted in a higher error of the evaluated threshold depth values, which can explain the increasing scatter of the data points for these values of the opening diameter. On the plots of the threshold depth one can clearly see that the threshold depth is only slightly dependent on the initial current density, but depends

200

5 Anodization process as a structuring technique

1.0 A/cm2

0.2

1.5 A/cm2

Anisotropy factor Af

Curvature, mm−1

2 1 0 −1 −2 −3

0.1

2.0 A/cm2

0.0

2.5 A/cm2 3.0 A/cm2

−0.1

3.5 A/cm2

−0.2 −0.3 −0.4

0

100 200 300 400 500

Structure depth, µm

0

100 200 300 400 500

Structure depth, µm

Figure 5.42.: Curvature (left) and anisotropy factor (right) vs. structure depth for the primary current distribution model for low-doped p-type silicon samples with opening diameter of 200 µm; values in the legend are 𝑗init . strongly (almost linearly) on the opening diameter. The result is quite interesting, because initially, when a similar dependence was observed in the experiment in sec. 5.1.2, it was assumed that it is mostly determined with diffusional mass transport in the electrolyte. In contrast, these model results suggest that the dependence can be also explained with the charge flow (electrical mechanism) neglecting the mass transport phenomena in the electrolyte. Now let us compare the threshold depth curves with the experiment from Fig. 5.8. In the experiment, similar dependencies on the current density and opening diameter were obtained: weak dependence on current density and strong dependence on the opening diameter. However, the effect of current density was more pronounced for the bigger openings there. Additionally, the values of the threshold depth obtained by simulation are approximately twice higher than those obtained experimentally. The differences from the experiment let us conclude that the primary current distribution model cannot explain the etch form development completely. One possible reason for the lower threshold depth value in the experiment could be the mass transport limitations in the electrolyte as considered in the diffusion mechanism of shape transformation in sec. 5.1.2: in case this diffusion mechanism in

5.4 Macroscale simulation of anodization process

200 µm 400 µm 600 µm 800 µm 1000 µm

0.2

Anisotropy factor Af

Curvature, mm−1

2

1

0

−1 −2

201

0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0

100 200 300 400 500

0

Structure depth, µm

100 200 300 400 500

Structure depth, µm

Figure 5.43.: Curvature (left) and anisotropy factor (right) vs. structure depth for the primary current distribution model for low-doped p-type silicon samples and initial current density of 1 A/cm2 ; values in the legend are 𝐷open .

2.5 A/cm2

900

2

3.0 A/cm2

800

2.0 A/cm2

3.5 A/cm2

Threshold depth dth , µm

1.5 A/cm

800 600 400 200

Threshold depth dth , µm

1.0 A/cm2

200 µm 400 µm 600 µm 800 µm 1000 µm

700 600 500 400 300 200 100

200

400

600

800

1000

Diameter of opening Dopen , µm

1.0 1.5 2.0 2.5 3.0 3.5

Current density, A/cm2

Figure 5.44.: Threshold depth evaluated for the primary current distribution model for low-doped p-type silicon samples; values in the legends are 𝑗init and 𝐷open .

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5 Anodization process as a structuring technique

electrolyte works in parallel to the electrical mechanism, lower values of threshold depth can be the result. As was already clear from the comparison of the etch forms, the anisotropy factor of the simulated etch forms differed strongly from the experiment. In the simulation, the anisotropy factor decreased in the beginning of the process from zero to below –0.4, and then gradually increased with increase of etch depth up to approximately 0.1–0.2, with curves for the smaller openings lying higher than for the bigger ones. Anisotropy factor showed also a strong dependence on the average current density 𝑗a xiii , with clear minimum reached during the process, as shown in Fig. 5.45 for the opening diameter of 200 µm (for other opening diameters s. Appendix, sec. A.4.1.4). However, the position of the minima seems to be not related to the value of 𝑗ox , in contrast to the experiment, where the anisotropy factor was decreasing with depth from approximately 0.65 to zero (s. Fig. 5.9), and it was shown that the regime of the process (pore formation or electropolishing) might be responsible for this change (s. Fig. 5.11).

Anisotropy factor Af

𝑗ox 0.2

j init = 1.0 A/cm2

0.1

j init = 1.5 A/cm2

0.0

j init = 2.0 A/cm2

−0.1

j init = 3.0 A/cm2

j init = 2.5 A/cm2

−0.2

j init = 3.5 A/cm2

−0.3 −0.4 0.5

1.0

1.5

2.0

2.5

3.0

Average current density j a , A/cm2

Figure 5.45.: Anisotropy factor vs. average current density 𝑗a for the primary current distribution model for low-doped silicon samples and opening diameter of 200 µm; datasets are for different initial current densities; the vertical line shows position of 𝑗ox , i.e., the border between pure electropolishing regime (to the right from the line) and mixed pore formation and electropolishing process (to the left from the line). xiii As

the average current density 𝑗a , same way as in sec. 5.1.2, the arithmetic mean between the initial current density 𝑗init and the final current density 𝑗area was taken, where 𝑗area was calculated from the total area of the structure and total current.

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5.4 Macroscale simulation of anodization process

z , µm

z , µm

Strong decrease of anisotropy factor in the beginning of the process for all the models (s. the plots on the right-hand side in Fig. 5.42 and Fig. 5.43) means that the process was running with strong lateral etching. Therefore the charge flow to the etch front from sides of the sample was higher than that from the bottom of the sample, as can be clearly seen from the arrow plot showing the charge flow (s. Fig. 5.46).

r , µm

r , µm

Figure 5.46.: Cross-section of the primary current distribution model for low-doped p-type silicon samples, with arrows showing simulated current density vectors in linear scale and equipotential lines for opening diameter of 1000 µm and initial current density of 1 A/cm2 : overview (left) and detailed view of the right edge of the opening in masking (right). Darker gray is the silicon substrate, and light gray is the electrolyte. Frontside mask region is not visible at this magnification. One could assume that a rather big radius of the geometry in the model (i.e. radius of the substrate and electrolyte domains) of 5 mm caused this high lateral charge flow. However, this is not the case, because most of the current (over 92 %) comes from the central region of radius 1.5 mm on the sample backside (s. Fig. 5.47). To prove that the rather big cell radius in the model cannot be responsible for the negative anisotropy obtained in the simulation, the models for the opening diameters of 200 µm and 1000 µm with initial current density of 3.5 A/cm2 were solved for cell radius 𝑅model of 1 mm and 1.5 mm, and the resulting anisotropy factor was compared to the results of the models with 𝑅model = 5 mm (s. Fig. 5.48). The comparison showed no significant change of the anisotropy factor with reduction of the cell radius to 1 mm for the opening diameter of 200 µm and to 1.5 mm for

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5 Anodization process as a structuring technique

the opening diameter of 1000 µm. The difference of the etch form for the cell radius of 1 mm and diameter of opening of 1000 µm is due to etch form approaching laterally the sidewall of the substrate domain (distance from the etch form to the sidewall was only 146 µm for structure depth of 510 µm). Thus, for the cell radius of 5 mm used in the models, one would expect no significant influence of the backside contact dimensions and geometry on the etch form development.

0 s, depth 5 µm 15 s, depth 14 µm 118 s, depth 46 µm 397 s, depth 104 µm 941 s, depth 194 µm 1838 s, depth 321 µm 3177 s, depth 514 µm

Current density, mA/cm2

500 400 300 200 100 0 0.0

0.5

1.0

1.5

r, mm

2.0

2.5

Current fraction in region r < 1.5 mm, %

Another observation from the plots in Fig. 5.47 is that during the process, as the structure grew, current density redistributed to a more wider region with lower maximum of current density, and when the etch front got close enough to the backside contact (at etch time of approximately 1200 s and structure depth of 232 µm), increase of current density in the central region of the substrate backside took place. Since the shape of etch form remained convex for these parameters (opening diameter of 1000 µm and initial current density of 1 A/cm2 ), there was also local increase of current density at 3177 s at the point with minimal distance between the etch form and the substrate backside (s. left plot in Fig. 5.47, where at 3177 s the depth of the structure was 514 µm, i.e., only 11 µm distance to the backside contact).

95.5 95.0 94.5 94.0 93.5 93.0 92.5 0

10

20

30

40

50

Etch time, min

Figure 5.47.: Current density as a function of 𝑟-coordinate (left) and fraction of total current in the region 𝑟 < 1.5 mm as a function of etch time (right), at 𝑧 = −525 µm (sample backside) for the primary current distribution model for low-doped p-type silicon samples, opening diameter of 1000 µm and initial current density of 1 A/cm2 .

205

5.4 Macroscale simulation of anodization process

0.2

0.0 −0.1 −0.2 Rmodel = 5 mm Rmodel = 1.5 mm Rmodel = 1 mm

−0.3 −0.4 0

100

200

300

𝐷open = 1000 μm

0.2

400

Structure depth, µm

500

Anisotropy factor Af

Anisotropy factor Af

𝐷open = 200 μm 0.1

0.0

−0.2 Rmodel = 5 mm Rmodel = 1.5 mm Rmodel = 1 mm

−0.4 0

100

200

300

400

500

Structure depth, µm

Figure 5.48.: Influence of cell radius in the model 𝑅model on anisotropy of the simulated process for the primary current distribution model for low-doped p-type silicon samples; initial current density of 3.5 A/cm2 ; opening diameters in frontside insulating masking of 200 µm (left) and 1000 µm (right).

5.4.1.3. Conclusions on the primary current distribution model for low-doped p-type silicon In this section, the primary current distribution model for anodization of low-doped p-type silicon samples through a circular opening in a frontside insulating masking layer was developed and evaluated. Similar etch form development, i.e., transformation from convex to concave shape, as observed in the experiment in sec. 5.1.2, was obtained with the model. However, the experimentally observed triple-hollow shapes, as an intermediate state between convex and concave shapes, could not be obtained in the simulation. As proposed in sec. 5.1.2, the triple-hollow shapes could be the result of the electrical mechanism of shape transformation (charge flow) and diffusion mechanism of shape transformation (transport of species in electrolyte) working not synchronously, i.e. providing different threshold depth values: this way, if the threshold depth for the electrical mechanism is smaller than the one for the diffusion mechanism, at a certain structure depth between these two threshold depth values, the electrical mechanism starts forming the central hollow, whereas the diffusion mechanism still conserves the side

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5 Anodization process as a structuring technique

hollows, or vice versa. In the electrical model shown here we obviously could not observe this effect. Average relative volume deviation between the experiment and the simulation of –4 % showed that the chosen porosity and dissolution valence functions provided quite good simulation results in regard to the structure volume, in both pore formation and electropolishing regimes. Comparison of the simulation results to the experiment revealed some limitations of the model. Namely, the anisotropy of the simulated process was much less than observed in the experiment. Moreover, the model showed high negative anisotropy. It was shown that the chosen rather big anodization cell radius in the model of 5 mm could not be responsible for this effect. As in the experiment, strong (almost linear) dependence of the threshold depth on the opening diameter was obtained with the model. However, the simulated values of the threshold depth are at least twice bigger than the ones obtained experimentally. Additionally, in the experiment some dependence of the threshold depth on the initial current, especially for bigger openings, was observed, but almost no dependence on the initial current density was obtained in the model. It was assumed that this difference could be the result of missing description of the diffusion mechanism of shape transformation, i.e., concentration polarization in the electrolyte. Some difference from the experiment could be the result of using the dependencies of the dissolution valence and porosity reconstructed based on few data points at low current densities below 100 mA/cm2 from the experiments described in sec. 4.1.1 and only one point (𝑗ox ) at higher current densities based on optical observations in the experiments described in sec. 5.1.2 and sec. 5.1.3. However, in the chosen range of the initial current densities, the process was simulated close to or in pure electropolishing regime. Thus, influence of the reconstructed dependencies at low current densities was presumably small. It was assumed that more important reason for the different anisotropy factor values is that no activation polarization is considered in primary current distribution models: the activation overpotential (meaning secondary and tertiary current distribution models) is expected to make etch front movement more uniform thus providing more isotropic etch form development [254]. The case of secondary current distribution is described in this report after the section about the primary current distribution model for highly-doped p-type silicon.

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5.4 Macroscale simulation of anodization process

5.4.2. Primary current distribution model for highly-doped p-type silicon 5.4.2.1. Introduction In sec. 5.1.3, it was observed that the etch form development through a circular opening in a frontside insulating masking layer for the low-doped and highly-doped p-type silicon samples (resistivity ranges 10–16 Ω cm and 0.01–0.1 Ω cm, respectively) is nearly equal, which was considered as a quite surprising result considering the change of the sample resistivity of more than hundred times. In order to check, whether this similarity can be explained by charge flow through the substrate, the primary current distribution model for all opening diameters from 200 µm up to 1000 µm was solved for p-type silicon of resistivity 0.01 Ω cm (conductivity 104 S/m). In this model, similarly to the previous section, porosity and dissolution valence dependencies on current density were constructed for the simulation based on the experimental data for the full wafers of resistivity 0.01–0.02 Ω cm from chapter 4, as shown in Fig. 5.49. The resulting etch rate calculated from the Faraday’s law of electrolysis with these dependencies according to eq. (2.38) and eq. (2.40) is shown in Fig. 5.50. For electropolishing, the critical current density 𝑗ox was assumed to be equal to 0.9 A/cm2 , as approximately obtained from optical observations in sec. 5.1.3, with 𝑛e = 4 and 𝑃% = 100 %. 4.0

80 60 experiment reconstructed

40 0.0

0.5

1.0

Current density, A/cm2

1.5

Dissolution valence

Porosity, %

100

3.5

3.0 experiment reconstructed

2.5 0.0

0.5

1.0

1.5

Current density, A/cm2

Figure 5.49.: Functions of porosity (left) and dissolution valence (right) in dependence of current density constructed for the model based on the available experimental data for highly-doped p-type silicon (0.01–0.02 Ω cm).

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5 Anodization process as a structuring technique

Etch rate, µm/s

2.5

pore formation electropolishing experiment reconstructed

2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

Current density, A/cm2

Figure 5.50.: Etch rate in dependence of current density for the model for highly-doped p-type silicon (0.01–0.02 Ω cm), together with the boundary curves for pore formation and electropolishing regimes as described in sec. 4.2.1. 5.4.2.2. Results and discussion For comparison to the experiment for the highly-doped p-type silicon samples (s. sec. 5.1.3), this model was solved for ten values of initial current density and five values of opening diameter (s. sec. 3.3.1). In contrast to the simulation for the low-doped samples, for the highly-doped samples shape development from convex to concave was observed for all opening diameters. As example, etch form development for the models with opening diameter of 200 µm and 1000 µm, and initial current density of 1 A/cm2 is shown in Fig. 5.51. Again, as in the model for the low-doped samples, transformation of convex shape to concave proceeded without intermediate triple-hollow shape, in contrast to the experiments. Now let us compare the simulated etch forms to the experiment in sec. 5.1.3. In Fig. 5.52, etch forms from the experiment (solid lines) and simulation (dashed lines) for diameter of opening of 400 µm and 1000 µm, and etch time of 5 min are shown. In general one can see that the simulated etch forms are more isotropic than the real anodized structures. Since the compared shapes are for the same current and etch time (i.e., same total charge), one can observe that due to larger width the simulated forms are not as deep as in the experiment. In Fig. 5.53, the volume of the structures was compared. For structures anodized at initial current densities below 1 A/cm2 , the volume of the struc-

209

5.4 Macroscale simulation of anodization process 𝐷open = 200 μm 0

0

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864.1 s

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1532.8 s

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−500

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345.7 s

z, µm

z, µm

𝐷open = 1000 μm

0

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500

r, µm

2550.6 s

−500

0

500

1000

r, µm

Figure 5.51.: Etch form development simulated for the primary current distribution model for highly-doped p-type silicon samples and opening diameter of 200 µm (left) and 1000 µm (right) at initial current density of 1 A/cm2 . 𝐷open = 400 μm

𝐷open = 1000 μm

0

0 −50

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Figure 5.52.: Comparison of simulated and experimental etch shapes for opening diameters 400 µm (left) and 1000 µm (right) for initial current densities 0.05 A/cm2 , 0.2 A/cm2 , 0.5 A/cm2 , 1 A/cm2 , 2 A/cm2 , and 3.5 A/cm2 ; anodization duration 5 min; highly-doped p-type silicon samples; solid lines are experimental etch forms, dashed lines are simulated etch forms. tures in the experiment is up to 100 % smaller than in the simulation. Possible reasons for this high deviation are the following:

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• The porosity and valence functions reconstructed for the simulation deviate from real parameters at low current densities: this is unlikely, because especially at low current densities these functions are based on real experimental data. • Real initial current density could have been much smaller in the experiment than assumed due to leakage current comparable to the total applied current: the total current for such small initial current densities was rather small (for example, 35 mA for current density of 0.5 A/cm2 ); for such small total current effect of leakage current could be quite significant, resulting in less charge flowing through the sample. For the structures anodized at initial current densities of 1 A/cm2 and higher, the relative volume deviation between the experiment and simulation is similar to that observed for the low-doped silicon samples, with average relative deviation of approximately –10 %. The deviation decreased with increase of the opening diameter, which could be the result of the current redistribution between the openings on the samples in the experiment. However, the effect is opposite to that observed in sec. 5.1.4, where on the samples with openings of various dimensions higher current density was achieved in the smaller openings. Dependencies of curvature and anisotropy factor on structure depth for the model with diameter of the opening of 200 µm and varied current density are shown in Fig. 5.54. In Fig. 5.55, curvature and anisotropy factor vs. structure depth for the model with varied diameter of the opening and initial current density of 0.5 A/cm2 are presented. Curves for all other combinations of opening diameter and initial current density are shown in Appendix (s. sec. A.4.2.1 and sec. A.4.2.2). On the curvature plots, in contrast to the model for the low-doped p-type silicon samples, no amplification of the curvature at structure depth approaching the sample thickness could be observed. This result is not unexpected, because for this model with conductivity of silicon much higher than conductivity of electrolyte, decrease of the distance from structure bottom to the sample backside does not cause increase of the charge flow from the sample backside, but even reduces it further, as can be seen on the plots in Fig. 5.56. As in the model for the low-doped p-type silicon samples, one can see only small dependence of the curvature curves on initial current density, thus the effect of current density dependent porosity and dissolution valence seems

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5.4 Macroscale simulation of anodization process

exp.: sim.: exp.: sim.: exp.: sim.: exp.: sim.: exp.: sim.:

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Figure 5.53.: Volume of etch forms in the experiment (datasets “exp.”, markers) and simulation (datasets “sim.”, lines) vs. initial current density for the primary current distribution model for highly-doped samples.

0.05 A/cm2

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Anisotropy factor Af

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Figure 5.54.: Curvature (left) and anisotropy factor (right) vs. structure depth for the primary current distribution model for highly-doped samples and opening diameter of 200 µm; parameter in the legend is initial current density.

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3.0

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2.5 2.0 1.5 1.0 0.5 0.0

0.25 0.20 0.15 0.10 0.05

−0.5

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100 200 300 400 500

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100 200 300 400 500

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Figure 5.55.: Curvature (left) and anisotropy factor (right) vs. structure depth for the primary current distribution model for highly-doped p-type silicon samples and initial current density of 0.5 A/cm2 ; parameter in the legend is diameter of frontside opening. to play a little role in etch form development in the chosen current density range. Again, there is very strong dependence on the opening diameter, as can be seen on the threshold depth plots. Comparing the obtained values with the experiment for the highly-doped samples in sec. 5.1.3, one can see a remarkable matching of the experimental and simulated threshold depth values for the highly-doped samples (s. Fig. 5.58). It is important to note that the data in the experiment were obtained for varied initial current density and constant etch time of 5 min. However, this primary current distribution model showed almost no dependence on initial current density, therefore comparison to the simulated data for a constant initial current density seems to be eligible and provides more clarity in the plot. The anisotropy factor dependencies on structure depth from this model for the highly-doped p-type silicon are quite different from the curves for the model with the low-doped p-type silicon. For the highly-doped samples, no negative values of anisotropy factor were obtained. In general, except the beginning of the process, anisotropy factor varies in the range 0.2–0.4. For the bigger openings it increases with increase of depth, and for the smaller openings it decreases. For comparison to the experiment, anisotropy factor was evaluated for all current densities at etch time of 5 min (s. Fig. 5.59).

213

Current density, mA/cm2

500

Current fraction in region r < 1.5 mm, %

5.4 Macroscale simulation of anodization process

0 s, depth 5 µm 12 s, depth 13 µm 97 s, depth 43 µm 326 s, depth 98 µm 774 s, depth 190 µm 1511 s, depth 333 µm 2611 s, depth 514 µm

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Figure 5.56.: Current density as a function of 𝑟-coordinate (left) and fraction of total current in the region 𝑟 < 1.5 mm as a function of etch time (right), at 𝑧 = −525 µm (sample backside) for the primary current distribution model for highly-doped p-type silicon samples; opening diameter of 1000 µm; initial current density of 1 A/cm2 .

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Figure 5.57.: Threshold depth evaluated for the primary current distribution model for highly-doped samples; values in the legends are 𝑗init and 𝐷open .

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5 Anodization process as a structuring technique

Threshold depth dth , µm

300

simulation experiment

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Diameter of opening Dopen , µm

Figure 5.58.: Threshold depth vs. diameter of opening for the highlydoped samples: comparison of the primary current distribution model to the experiment from sec. 5.1.3; simulated data are for initial current density of 3.5 A/cm2 and resistivity of silicon 0.01 Ω cm; experimental data are for resistivity 0.01–0.1 Ω cm and different current densities in the range 0.05–3.5 A/cm. This allows us to see some similarity of the simulated data to the experiment (compare with Fig. 5.16). Namely, there is only slow increase of anisotropy factor as a function of the average current density or structure depth. However, in the experiment the anisotropy factor varied in the range 0.5–0.7, but in the simulation the range is 0.2–0.4. Moreover, in the simulation no any strong decrease of anisotropy factor at small current densities below 𝑗ox was observed, but only a slight increase, so the thesis that the anisotropy factor is dependent on anodization regime (pore formation or electropolishing) could not be proved with the primary current distribution model also for the highly-doped samples. 5.4.2.3. Conclusions on the primary current distribution model for highly-doped p-type silicon In this section, the primary current distribution model for anodization of highly-doped p-type silicon samples through a circular opening in a frontside insulating masking layer was developed and evaluated. As for the model

215

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Anisotropy factor Af

5.4 Macroscale simulation of anodization process

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D open D open D open D open D open

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= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

3

Average current density j a , A/cm2

Figure 5.59.: Simulated anisotropy factor in dependence of structure depth (left) and average current density 𝑗a (right) for structures anodized for 5 min from the primary current distribution model for highly-doped samples; error bars in the right plot show the ranges between initial current densities and final current densities calculated for the increased area in the end of the process; similar scales are taken as for the experiment in Fig. 5.16 for better comparison. with the low-doped p-type silicon samples (s. sec. 5.4.1) and the experiments for the low- and highly-doped p-type silicon, similar etch form development was observed. Large deviation of the structure volume between the experiment and the simulation up to 100 % for initial current densities below 1 A/cm2 could be a sign of unreliable experimental conditions at low applied current, when a significant part of the charge flow run not through the sample but as a leakage current. However, at initial current densities above 1 A/cm2 (with final current density for most of the model results below 𝑗ox ), average value of relative volume deviation between the experiment and the simulation was approximately –10%, meaning that the chosen porosity and dissolution valence functions provided satisfactory simulation results in regard to the structure volume in and close to electropolishing regime. In contrast to the model for the low-doped p-type silicon, the threshold depth values obtained in this model matched the experiment. However, the anisotropy factor, although not as different from the experiment as for the model with the low-doped silicon, was more than twice lower than in the experiment. Thus, the primary current distribution models, for both highly-doped and low-doped p-type silicon, could not show the same

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high anisotropy as it was observed experimentally, revealing some other anisotropic mechanism in the anodization process which was not considered in the models. As a next step, secondary current distribution models for both low-doped and highly-doped p-type silicon, are presented. 5.4.3. Secondary current distribution model for low-doped p-type silicon As was explained in sec. 2.3.3, secondary current distribution means that not only conductivities of electrolyte and electrodes are considered, but also the activation overpotential at the interface electrolyte-electrode. The activation overpotential for the model was defined in sec. 3.3.5. It is known that additional potential difference arising at the interface silicon-electrolyte due to the limited rate of the charge transfer makes the current density distribution at the interface more uniform [254]. Thus, the secondary current distribution models are expected to bring more isotropic etch forms, as will be checked in this section for the low-doped p-type silicon and in the next section for the highly-doped p-type silicon. 5.4.3.1. Results and discussion The etch form development for this secondary model is similar to the primary model, meaning similar dependencies of curvature and anisotropy factor on structure depth (s. Fig. 5.60, Fig. 5.61, and in Appendix sec. A.4.3.1 and sec. A.4.3.2). However, as expected, the process in this secondary model was more uniform, resulting in slightly smaller absolute values of curvature and anisotropy factor. The effect is not that strong though, and the high negative anisotropy of the process remains as in the primary model. The threshold depth showed strong dependence on the opening diameter, as in the primary model (compare Fig. 5.62 and Fig. 5.44). Additionally, in contrast to the primary model, dependence on the initial current density is also significant and increasing with increase of the opening diameter. As a consequence, the values of the threshold depth for the initial current densities of 1 A/cm2 and 1.5 A/cm2 are close to the experiment (s. Fig. 5.8). However, with increase of the initial current density for the bigger openings, the threshold depth obtained in the model increased, and in the experiment

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5.4 Macroscale simulation of anodization process

1.0 A/cm2

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1.0 0.5 0.0 −0.5 −1.0

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−0.2 −0.3

−1.5 0

100 200 300 400 500

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Figure 5.60.: Curvature (left) and anisotropy factor (right) vs. structure depth for the secondary current distribution model for low-doped p-type silicon samples and opening diameter of 200 µm; datasets are for different initial current densities; to compare with the results for the primary current distribution model in Fig. 5.42. decreased. The reason for this difference could be the missing mass transport description in the model: at higher current density, stronger diffusion limitation of species in electrolyte should result in smaller threshold depth. 5.4.3.2. Conclusions on the secondary current distribution model for low-doped p-type silicon In this section, the secondary current distribution model for anodization of low-doped p-type silicon through a circular opening in a frontside insulating masking layer was developed and compared to the primary current distribution model in sec. 5.4.1 and the experimental results in sec. 5.1.2. As expected, consideration of the activation overpotential led to more uniform etch process, resulting in smaller absolute values of the curvature and anisotropy factor of the etch forms, however, big negative values of the anisotropy factor, not observed experimentally, remained. In comparison to the primary current distribution model for the low-doped p-type silicon, more pronounced dependence of the threshold depth on the

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5 Anodization process as a structuring technique

200 µm 400 µm 600 µm 800 µm 1000 µm

0.1

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0

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Structure depth, µm

Figure 5.61.: Curvature (left) and anisotropy factor (right) vs. structure depth for the secondary current distribution model for low-doped p-type silicon samples and initial current density of 1 A/cm2 ; datasets are for different diameters of frontside opening; to compare with the results for the primary current distribution model in Fig. 5.43. initial current density was obtained in this model. As a result, the values of the threshold depth for the initial current densities of 1 A/cm2 and 1.5 A/cm2 appeared to be close to the values obtained experimentally. Thus, the secondary current distribution model provided results closer to the experiment than the primary model. In a tertiary current distribution model with concentration polarization (proposed as a future work), especially for higher current densities, one can expect that the etch rate should become even more uniform, resulting in smaller values of the threshold depth also for initial current densities above 1.5 A/cm2 . Regarding anisotropy of the etch forms, there is still big difference from the experiment. It can be expected that this high negative anisotropy could be further increased to become closer to zero with consideration of concentration polarization in electrolyte and silicon in a tertiary current distribution model. Nevertheless, a tertiary model would not explain the high positive anisotropy achieved in the experiments, thus the comparison of the experiment to the developed models reveals that there must be other factors responsible for the anisotropic behavior. For example, the anisotropic behavior could be induced due to crystallographic dependence of the etch

5.4 Macroscale simulation of anodization process

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Figure 5.62.: Threshold depth evaluated for the secondary current distribution model for low-doped p-type silicon samples; values in the legends are 𝑗init and 𝐷open ; values of threshold depth above 510 µm were obtained by extrapolation. rate and critical current density (s. sec. 2.5.6). Such effects were initially neglected in the work, because no specific “pseudo V-shapes” were observed in the experiments for the used (100)-oriented wafers. To check this idea, further experiments on wafers of different orientation are necessary beyond the frames of this work. 5.4.4. Secondary current distribution model for highly-doped p-type silicon In this section, similarly to the secondary current distribution model for the low-doped samples, a secondary current distribution model for the highlydoped p-type silicon samples is described. The activation overpotential for the model was defined as explained in sec. 3.3.5. 5.4.4.1. Results and discussion The etch form development in this secondary model was also similar to the corresponding primary model, with similar dependencies of curvature and anisotropy factor on structure depth (s. Fig. 5.63, Fig. 5.64, and in Appendix in sec. A.4.4.1 and sec. A.4.4.2). As for the low-doped p-type silicon model

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with secondary current distribution, activation overpotential led to smaller absolute values of curvature and anisotropy factor. The effect is very strong for low initial current densities. With increase of the initial current density, the curves get closer to the corresponding curves for the primary model.

Anisotropy factor Af

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−1

0.00 0

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0

100 200 300 400 500

Structure depth, µm

Figure 5.63.: Curvature (left) and anisotropy factor (right) vs. structure depth for the secondary current distribution model for highly-doped ptype silicon samples and opening diameter of 200 µm; datasets are for different initial current densities. As in the model for the low-doped silicon samples with secondary current distribution, in contrast to the primary model, there is a clear dependence of the threshold depth on the initial current density (compare Fig. 5.65 and Fig. 5.57). In contrast to the model for the low-doped samples with secondary current distribution, this dependence on the initial current density is the strongest at values of 𝑗init below 2 A/cm2 . As a result, only the threshold depth values for the initial current densities of 1 A/cm2 and above are close to the experiment (s. Fig. 5.15). The reason for this deviation of the model at low current densities remains unclear. Comparing the anisotropy factor values to the experiment in Fig. 5.16, one can see that the values of 𝐴f in the experiment were approximately twice higher than those in the presented model (s. Fig. 5.66). As for the low-doped samples, this difference cannot be explained with the missing description of the mass transport effects in the electrolyte, and presumably reveals some

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5.4 Macroscale simulation of anodization process 2.5

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Figure 5.64.: Curvature (left) and anisotropy factor (right) vs. structure depth for the secondary current distribution model for highly-doped ptype silicon samples and initial current density of 0.5 A/cm2 ; datasets are for different diameters of frontside opening.

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Figure 5.65.: Threshold depth evaluated for the secondary current distribution model for highly-doped p-type silicon samples; values in the legends are 𝑗init and 𝐷open .

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0.6

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other anisotropic mechanism of anodization process in the chosen conditions, not connected to the charge flow in the substrate.

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Figure 5.66.: Simulated anisotropy factor in dependence of structure depth (left) and average current density 𝑗a (right) for structures anodized for 5 min from the secondary current distribution model for highly-doped silicon samples; error bars in the right plot show ranges between initial current densities and final current densities calculated for the increased area in the end of the process; similar scales are taken as for the experiment in Fig. 5.16 for better comparison.

5.4.4.2. Conclusions on the secondary current distribution model for highly-doped p-type silicon In this section, the secondary current distribution model for anodization of highly-doped p-type silicon through a circular opening in a frontside insulating masking layer was developed and compared to the primary current distribution model in sec. 5.4.2 and the experimental results in sec. 5.1.3. Introduction of the activation overpotential led to more uniform etch process also for the highly-doped silicon samples, resulting in smaller absolute values of the curvature and anisotropy factor of the etch forms, especially for the smallest initial current densities. In comparison to the primary current distribution model for the highlydoped p-type silicon samples, more pronounced dependence of the threshold depth on the initial current density was obtained in this secondary current

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223

distribution model. As a result, only the values of the threshold depth for the initial current densities of 1 A/cm2 and larger appeared to be close to the values obtained experimentally. This result means that the secondary current distribution model for the highly-doped silicon samples shows more deviation from the experiment than the primary current distribution model where the threshold depth values for all initial current densities were close to the experiment. The big difference between the anisotropy factor values of the model and the experiment remained also for this secondary current distribution model, meaning that another anisotropy-increasing mechanism, such as orientation-dependent etch rate, is also present for this highly-doped p-type silicon. 5.5. Conclusions In this chapter, application of silicon anodization as a structuring technique for p-type silicon was studied. Two localization techniques have been tested: frontside insulating masking and local backside contacts, and combination of both. The main focus in the work was put on study of anodization process through an opening in a frontside insulating mask. Anodization through small openings of dimension below 1 mm in the frontside insulating masking showed that pure electropolishing in this case can be achieved at much lower current density than was obtained for large-area samples in chapter 4, resulting in 𝑗ox of equal of below 0.9 A/cm2 for both low-doped p-type silicon (10–16 Ω cm) and highly-doped p-type silicon (0.01–0.1 Ω cm). It was shown for the first time that for the case of anodization through a circular opening in an insulating frontside masking layer, etch forms undergo transformation from convex to triple-hollow and finally to concave shape. The shape development in the experiments appeared to be similar for the low-doped and highly-doped p-type silicon. The structures etched through bigger openings transformed from convex shape to concave at larger depth than the structures etched through smaller openings. Mechanisms explaining the observed etch form development based on charge flow in the cell and transport of species in the electrolyte were proposed. The transformation of etch shapes from convex to concave and dependence of the threshold depth on the diameter of opening was demonstrated with macroscale models considering primary and secondary current distributions. It was proposed that the experimentally observed triple-hollow shapes were the result of the

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electrical mechanism of shape transformation (charge flow) and the diffusion mechanism of shape transformation (transport of species in electrolyte) working not synchronously, i.e. providing different threshold depth values as explained in sec. 5.4.1.3, therefore such shapes were not obtained with the developed electrical models. For the first time, macroscale models based on real process parameters with primary and secondary current distributions were developed for the case of anodization through a frontside opening in an insulating mask. The models showed much smaller anisotropy of the process than obtained experimentally. Since diffusion-controlled process is expected to get the process even more isotropic, i.e., with anisotropy factor closer to zero, the missing description of the concentration polarization in electrolyte in the developed models would not explain higher positive anisotropy observed in the experiments. Thus, there must be other factors playing a role in the etch form development, such as dependence of the process on the crystallographic orientation of silicon samples (s. sec. 2.5.6). In the experiments it was shown that anisotropy factor increased from zero to 0.4–0.65 when the process changed from pore formation to electropolishing. Therefore, it might be expected that this crystallographic orientation dependent behavior starts playing a role in electropolishing regime, for example, in the process of anodic oxidation. To see the effect of crystallographic orientation on etch form development, experiments with silicon substrates of other orientation than (100) would be necessary. However, this effect was not studied in the work, and all experiments were performed on (100)-oriented wafers. Additional study was aimed on investigation of influence of frontside openings on each other for low-doped p-type silicon. Besides current crowding for single structures resulting in convex shapes, current crowding on the outer sides of a block of multiple structures could be observed as reported previously. Additionally it was observed that in case of openings of different size on a sample, there is current redistribution, resulting in the structures with smaller openings being etched faster than the structures with bigger openings. Besides the experiments and simulations for the case of anodization through an opening in a frontside insulating masking layer, etch shape development for local backside contacts (with full open frontside area) and combination of frontside insulating mask and backside local contacts for low-doped p-type silicon was studied. For a single backside contact it was shown that the process gets more and more localized as the structure gets deeper. However,

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localization of the process only with backside contacts could not provide structures with positive anisotropy. Study of shape development for multiple backside contacts for low-doped p-type silicon showed that effect of each backside contact could be recognized only if the backside contacts were located from each other at least at a distance equal to the thickness of the sample. Additionally, it was shown that an etch form obtained on a sample with multiple elementary backside contacts of equal size (“pixels”) can be reconstructed by superposition of elementary etch forms expected for each elementary contact. The method called “pixel concept” can help predicting the etch forms obtained for samples with backside local contacts without time-consuming simulation. In the experiment with combination of the process localization by frontside opening in an insulating mask and backside local contacts for low-doped p-type silicon (10–20 Ω cm), no higher anisotropy than obtained with only frontside localization could be obtained.

6. General conclusions In this work, application of silicon anodization process as a threedimensional structuring technique was investigated. In the state of the art, overview of micro-structuring techniques, theory on silicon-electrolyte interface and anodization process was given. An important part of this chapter was systematic collection and analysis of the data from the available references showing influence of various process parameters on the properties of anodization process. This analysis also showed that some discrepancy of the data between different research groups exists. The data collected in this section were then used for comparison to the results of the microscale study in chapter 4. In the study of microscale parameters in 29.93 m% HF electrolyte with ethanol in chapter 4, dependencies of anodization parameters, namely, porosity and growth rate of porous silicon, dissolution valence, interface surface roughness, and mean fractal dimension, on applied current density during the process and resistivity of silicon were investigated. No clear tendencies on current density could be derived for the roughness and fractal dimension, however, it was shown that roughness for highly doped p-type silicon (0.001–0.005 Ω cm and 0.005–0.01 Ω cm) remained below 10 nm, and increased with increase of substrate resistivity, which confirms the references. The results for growth rate, porosity, and dissolution valence were in agreement with the references and, for growth rate, with performed theoretical estimations based on the known mechanism of the process. The study also confirmed that no pure electropolishing is achieved in the 29.93 m% HF electrolyte with ethanol even at 3 A/cm2 for silicon samples with large open area in order of 1 cm2 or bigger. In the additional study in chapter 4, improvement of surface quality by electropolishing of silicon samples in electrolytes with different HF concentrations was studied. The best quality of anodized surface (average roughness of about 1 nm) was obtained for samples electropolished in aqueous 7 m% HF electrolyte at current densities in the range 100–300 mA/cm2 . With the optimized parameters, preparation of silicon surfaces for optical applications can be done. © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9_6

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6 General conclusions

Influence of porous silicon etch solutions (namely, 1 m% KOH and 1:5 mixture of resist developer AZ351B in water) on surface roughness was also studied in chapter 4. It was shown for blank not anodized silicon substrates that treatment in these solutions for more than 30 min results in rapid increase of surface roughness to over 100 nm. If the treatment time does not exceed 30 min, the surface roughness remains below 10 nm. The main focus of the work was the study of macroscale behavior of the anodization process, that is the etch form development under application of frontside insulating masks and/or local backside contacts, in chapter 5. Here, the main studied test structure was silicon sample with a circular opening in the frontside insulating masking. For this kind of samples with small open area, electropolishing was observed already at relatively low current density of 0.9 A/cm2 . Comparing this result to the microscale study in chapter 4 it was concluded that the critical current densities 𝑗PSL and 𝑗ox indeed depend on mass transport limitations of reactants and products as some references suggest. However, more systematic study of this effect would be helpful as a future work. For the first time, etch form development for low-doped (10–16 Ω cm) and highly-doped (0.01–0.1 Ω cm) p-type silicon samples with a circular opening in a frontside insulating masking was investigated and compared. For both doping levels, it was shown experimentally for the first time that during the process etch forms undergo a transformation from convex shape to concave. Mechanisms explaining the observed shape transformation based on charge flow in the cell and mass transport effects in electrolyte were proposed. It was found that the critical depth at which the shapes transform from convex to concave (named threshold depth) did not depend on the resistivity of samples (low- and highly-doped p-type silicon) and applied current (shown only for low-doped p-type silicon), but depended nearly linearly on the diameter of the frontside opening. Another remarkable finding here was that the anisotropy of the process for this type of structures remained at large positive values in the range 0.45–0.75 at current densities above 𝑗ox , and decreased strongly to small values of ±0.2 (the negative value observed only for the highly-doped silicon samples) with decrease of current density below 𝑗ox , i.e., when the process changed from pure electropolishing to a mixed regime of pore formation and electropolishing. For the first time, systematic finite-element simulations with real process parameters were performed for the case of anodization through a circular opening in a frontside masking layer. The models described primary and secondary current distributions, i.e., considered conductivities of electrolyte

6 General conclusions

229

and silicon (both the primary and secondary current distributions) and also included description of charge transfer at the interface silicon-electrolyte (in case of the secondary current distribution model). Comparing to the experiments in this work, the models showed similar etch form development from convex to concave etch form. However, only partial matching of the results in regard to threshold depth and anisotropy factor was observed. To some extent, this could be explained with missing description of concentration polarization in the models, as could be verified in tertiary current distribution models which are not covered in this work. Additionally, the significantly larger positive values of anisotropy factor in the experiments in comparison to the developed electrical models revealed some anisotropic mechanisms of anodization process in the chosen conditions, which could not be explained with charge flow in the substrate, and presents another important finding of the work. Thus, the results of the electrical models developed in the work partially matched the experimental observations. Further improvements of the models were suggested and discussed. Additional studies on mutual influence of neighboring openings in a frontside masking, anodization of pre-structured cavities, and etch form development for local backside contacts (with and without frontside masking) were performed. In the study of the process localization with local backside contacts, the “pixel concept” was proposed. In this “pixel concept”, a basic etch form for an elementary local backside contact (a “pixel”) was used to obtain a good estimation of an etch form resulting from anodization of a sample with multiple equal local backside contacts. The results of the work provide an important contribution to the scientific community and industry for future applications of the anodization process as a structuring technique, and will help to reduce the time needed to optimize the process for fabrication of specific structures in industrial scale. The types of etch shapes studied in the work are especially interesting for applications in fluidic, optical and lab-on-chip micro-devices. To conclude, this work provides new insight into application of silicon anodization as a structuring technique. However, due to the complexity of this electrochemical process, there are still some open questions, some of them revealed first in this work. Therefore, further work is necessary, especially in the direction of a complete model of anodization process considering mass transport phenomena and other effects playing a role in this fascinating process.

Acknowledgments First of all, I would like to thank Prof. Dr. Ulrich Mescheder of Furtwangen University for the given opportunity to work on this exciting field of research. He allowed me a lot of freedom to complete my work and provided useful discussions of my findings. I am also very grateful to Prof. Mescheder for helpful comments prior to publication of this report as a book. Prof. Dr. Peter Woias of University of Freiburg is acknowledged for support and the given possibility to accomplish this PhD at the University of Freiburg. Additionally, I would like to express my gratitude to Prof. Dr. Holger Reinecke of University of Freiburg for agreeing to co-examine my thesis. Special thanks should go to Dr. Andras Kovacs of Furtwangen University who introduced me to the field of silicon anodization technology and who was always available for scientific discussions of the work and gave many helpful advices on this report. I am indebted to all my colleagues in the micro- and nanosystems technology lab of Furtwangen University, and especially Ms Xenia Seng and Mr. Bernhard Müller, for assisting me with the preparation of samples. I would also like to acknowledge Prof. Dr. Victor Lyubimov and Prof. Dr. Vladimir Volgin, both of Tula State University, Russia, for fruitful discussions on electrochemical backgrounds for the simulations of the anodization process. Many thanks go to Mr. Michailas Romanovas of Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR) and Ms Caroline Armbruster of Furtwangen University for their very useful advices on the text of the report. The graduate school of Furtwangen University is acknowledged for organization of interesting courses and partial financial support to participate in a conference. At this point, I acknowledge also the professors of the Novosibirsk State Technical University, Russia, and especially Prof. Dr. Vladimir Makukha, © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9

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Acknowledgments

for giving me fundamental knowledge on technology of electronic and microelectronic devices. It would be unfair not to acknowledge the open source community of developers of Python (and especially of the libraries Matplotlib, NumPy, and SciPy), Spyder, OpenOffice, LATEX, LYX, Docear, JabRef, and Syncthing, whose software helped a lot to produce this high-quality report. My final acknowledgment goes to my parents Sergei and Natalia, my sister Anya, my daughter Emilia, and especially to my beloved wife And𝑧ela ̌ for their patience and spiritual support during this work.

List of relevant publications Book chapter A. Ivanov and U. Mescheder, Surface micromachining (sacricial layer) and its applications in electronic devices, in Porous Silicon: From Formation to Application. Volume 3: Optoelectronics, Microelectronics, and Energy Technology Applications, G. Korotcenkov, Ed. CRC Press (Taylor & Francis Group), 2016, ch. 6, pp. 129–141. Available: http://dx.doi.org/10.1201/b19042-9 Journal papers 1. M. Kroener, A. Ivanov, F. Goldschmidtböing, U. Mescheder, and P. Woias, “Finite-elements simulation for true 3D structure generation of anisotropic electrochemical wet-etching processes,” ECS Transactions, vol. 19, no. 26, pp. 93–102, 2009. [Online]. Available: http://dx.doi.org/10.1149/1.3247995 2. A. Ivanov, A. Kovacs, and U. Mescheder, “High quality 3D shapes by silicon anodization,” physica status solidi (a), vol. 208, no. 6, pp. 1383–1388, 2011. [Online]. Available: http://dx.doi.org/10.1002/pssa.201000163 3. A. Ivanov and U. Mescheder, “Silicon electrochemical etching for 3D microforms with high quality surfaces,” Advanced Materials Research, vol. 325, pp. 666–671, 2011. [Online]. Available: http://dx.doi.org/10.4028/www.scientific.net/AMR.325.666 4. A. Ivanov, U. Mescheder, and P. Woias, “Finite-elements simulation of etch front propagation in silicon electropolishing process,” ECS Transactions, vol. 58, no. 46, pp. 15–24, Apr 2014. [Online]. Available: http://dx.doi.org/10.1149/05846.0015ecst 5. U. Mescheder, I. Khazi, A. Kovacs, and A. Ivanov, “Tunable optical filters with wide wavelength range based on porous multilayers,” Nanoscale Research Letters, vol. 9, no. 1, p. 6, 2014. [Online]. Available: http: //www.nanoscalereslett.com/content/9/1/427 © Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9

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Conference proceedings 1. A. Ivanov, U. Mescheder, A. Kovacs, and R. Huster, “Einfluss unterschiedlicher Passivierungsschichten auf die Bildungsrate von porösem Silizium an Strukturkanten bei der 3D Formherstellung,” in Proceedings Mikrosystemtechnik-Kongress 2007: 15.–17. Oktober 2007, Dresden (Germany). Berlin and Offenbach: VDEVerl., 2007, pp. 529–532. [Online]. Available: http://www.vdeverlag.de/proceedings-en/563061112.html 2. A. Ivanov, U. Mescheder, M. Kröner, and P. Woias, “Formation of arbitrarily shaped 3D-forms in silicon by electrochemical wet-etching,” in Technical Digest, 19th Micromechanics and Microsystems Europe Workshop (MME 2008), September 28–30, 2008, Aachen (Germany), 2008, pp. 315–318. 3. A. Ivanov, A. Kovacs, U. Mescheder, S. Kuhn, and A. Burr, “Optimisation of surface quality of 3D silicon master forms for injection molding of optical micro elements,” in Proceedings MikrosystemtechnikKongress 2009: 12.–14. Oktober 2009, Berlin. Berlin and Offenbach: VDE-Verl., 2009, pp. 726–729. [Online]. Available: http://www.vdeverlag.de/proceedings-en/453183183.html 4. U. Mescheder, A. Ivanov, A. Kovacs, V. Lubimov, and A. Abitov, “Novel structuring processes for micro- and nanotechnology,” in Wissenschaftsforum im Rahmen der Baden-Württemberg Tage im Moskau, May 2–6, 2009, Moscow (Russia), 2009. 5. M. Kroener, A. Ivanov, F. Goldschmidtböing, U. Mescheder, and P. Woias, “Finite-elements simulation for true 3D structure generation of anisotropic electrochemical wet-etching processes: Abstract,” in 215th ECS Meeting, May 24–29, 2009, San Francisco (USA), 2009, pp. 93–102. 6. A. Ivanov, A. Kovacs, and A. Mescheder, “Nasschemisches Ätzen von nahezu beliebigen 3D-Strukturen in Silizium,” in MikroNano-Integration: Beiträge des 2. GMM-Workshops 3.–4. März 2010 in Erfurt, ser. GMM-Fachbericht, vol. 63. Berlin: VDEVerlag, 2010, pp. 65–68. [Online]. Available: http://www.vdeverlag.de/proceedings-de/453216012.html 7. A. Ivanov, A. Kovacs, and U. Mescheder, “High quality 3D shapes by silicon anodization,” in Porous Semiconductors-Science and Technol-

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ogy: Extended abstracts of the 7th internacional conference : Valencia, Spain, March 14–19, 2010, L. T. Canham and E. Matveeva, Eds. Valencia: Universidad de Valencia, Servicio de Publicaciones, 2010, pp. 197–198. 8. A. Ivanov and U. Mescheder, “Silicon electrochemical etching for 3D microforms with high quality surfaces,” in Advances in Abrasive Technology XIV: Selected, peer reviewed papers from the 14th International Symposium of Advances in Abrasive Technology (ISAAT 2011), September 18–21, 2011, Stuttgart, Germany, T. Tawakoli, Ed. Trans Tech Publications, 2011, pp. 666–671. 9. A. Ivanov, “Simulation of electrochemical etching of silicon with COMSOL,” in Proceedings COMSOL Conference 2011, October 26–28, 2011, Ludwigsburg / Stuttgart (Germany), 2011, 5 pages. [Online]. Available: http://www.ch.comsol.com/paper/simulation-ofelectrochemical-etching-of-silicon-with-comsol-11800 10. U. Mescheder, A. Kovacs, and A. Ivanov, “Electrooptical biosensor based on nanostructured multilayers,” in Proceedings XXVII. International Kando Conference 2011, 17.–18. November 2011, Budapest (Hungary), 2011, 9 pages. 11. A. Ivanov and U. Mescheder, “Dynamic simulation of electrochemical etching of silicon with COMSOL,” in Proceedings COMSOL Conference 2012, October 10-12, 2012, Milan (Italy), 2012, 7 pages. 12. A. Kovacs, A. Ivanov, A. Malisauskaite, and U. Mescheder, “Oberflächenstabilisierung nanostrukturierter poröser Siliziumschichten für sensorische Anwendungen,” in Mikro-Nano-Integration: Beiträge des 4. GMM Workshops, November 12–13, 2012, Berlin, ser. GMM-Fachbericht, vol. 74. Berlin and Offenbach: VDE-Verl, 2012, 5 pages. 13. A. Ivanov, U. Mescheder, and P. Woias, “Finite-elements simulation of etch front propagation in silicon electropolishing process: Abstract,” in Proceedings 224th ECS Meeting, October 27 – November 1, 2013, San Francisco, California (USA), 2013, 1 page. [Online]. Available: http://ma.ecsdl.org/content/MA2013-02/42/2463.abstract 14. A. Ivanov, U. Mescheder, V. Liubimov, V. Volgin, and M. Salomatnikov, “Innovative electrodes and processes for micro ECM,” in Proceedings German-Russian Forum Nanotechnology, May, 21–24, 2013, Tomsk, Russia, p. 19, 2013.

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15. A. Kovacs, A. Malisauskaite, A. Ivanov, U. Mescheder, and R. Wittig, “Optical sensing and analysis system based on porous layers,” in The 17th International Confernece on Miniaturized Systems for Chemistry and Life Sciences (MicroTAS 2013), October 27–31, 2013, Freiburg (Germany). Chemical and Biological Microsystems Society, San Diego, CA, USA, 2013, pp. 275–277. 16. A. Kovacs, A. Malisauskaite, A. Ivanov, and U. Mescheder, “Steuerung der Peaklage und der Sensitivität von siliziumbasierten porösen optischen Multischichten,” in Von Bauelementen zu Systemen: Proceedings Mikrosystemtechnik-Kongress 2013, 14–16 October 2013, Aachen (Germany). Berlin and Offenbach: VDEVerlag, 2013, pp. 185–188. [Online]. Available: http://www.vdeverlag.de/proceedings-de/453555041.html 17. A. Kovacs, A. Malisauskaite, A. Ivanov, and U. Mescheder, “Portable optical sensor using tunable optical multilayers,” in 2013 IEEE Sensors, November 4–6, 2013, Baltimore, Maryland (USA), 2013, pp. 1691–1694. 18. A. Ivanov, “Simulation of silicon anodization process,” in Proceedings German-Russian Young Scientists Conference, May 20–23, 2014, Tomsk (Russia). Tomsk Polytechnic University, 2014. 19. U. Mescheder, I. Khazi, A. Kovacs, and A. Ivanov, “Tunable optical filters with wide wavelength range based on porous multilayers,” in Porous Semiconductors - Science and Technology: Extended abstracts of the 9th international conference : Benidorm-Alicante, Spain, 09–14.03.2014, L. T. Canham and E. Matveeva, Eds. Valencia: Universidad de Valencia, Servicio de Publicaciones, 2014, 2 pages. 20. U. Mescheder, I. Khazi, A. Kovacs, and A. Ivanov, “MOEMS based concept for miniaturized monochromators, spectrometers and tunable light sources”, in Proceedings 1. Baden-Württemberg Center of Applied Research (BW-CAR) Symposium on Information and Communication Systems (SInCom), December 12, 2014, ISBN 978-3-00-048182-6, 2014, 6 pages. 21. A. Kovacs, A. Ivanov, and U. Mescheder, “Tunable narrow band porous photonic crystals for MOEMS based scanning systems”, Procedia Engineering, vol. 120, pp. 811–815, 2015, Eurosensors 2015, Freiburg, September 7–9. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1877705815023322

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22. A. Ivanov and U. Mescheder, Primary current distribution model for electrochemical etching of silicon through a circular opening, in Proceedings COMSOL Conference 2015, October 14–16, 2015, Grenoble (France), 2015, 7 pages. Final reports for research projects 1. U. Mescheder, R. Huster, and A. Ivanov, Verfahrensentwicklung zur Mikrospritzgussabformung von Si-Formeinsätzen mit mikrotechnisch hergestellten 3D-Profilen (MiSS-3D), Teilvorhaben: Technologieentwicklung und Herstellung von 3D Si-Formeinsätzen: Abschlussbericht zum Forschungsprojekt; Förderprogramm InnoNet; Laufzeit: 01.01.2007 – 28.02.2010. Furtwangen: Hochschule Furtwangen, 2010. 2. U. Mescheder and A. Ivanov, Innovative Elektroden und Prozesse für micro ECM: Abschlussbericht zum Forschungsprojekt ; Förderprogramm BMBF ERA-Net.Rus; Förderkennzeichen 01DJ11007; Laufzeit: 01.01.2012 – 31.12.2013. Furtwangen: Hochschule Furtwangen, 2014. Patent U. Mescheder, A. Ivanov, A. Kovacs, W. Kronast, and B. Müller, “Adaptiver optischer Filter bzw. spektral einstellbare Lichtquelle”, German Patent DE 10 2013 013 239 A1, 2013.

A. Appendix A.1. Conversion of concentration units for aqueous HF electrolytes A.1.1. Concentration units in chemistry There are various ways to characterize composition of a mixture, introduced in chemistry through the history of its development and for ease of calculations in certain cases, such as: • amount concentration (molar concentration) 𝑐mol 𝑖 as amount 𝑁𝑖 of a constituent 𝑖 in moles per volume 𝑉 of the mixture with unit mol/m3 or mol/l (mol/dm3 ): 𝑐mol 𝑖 =

𝑁𝑖 𝑉

(A.1)

The amount of a constituent in moles per 1 liter mixture was in the past named molarity and denoted as M (1 M = 1 mol/l ) [396]. • volume concentration 𝑐vol 𝑖 as ratio of volume 𝑉𝑖 of a constituent 𝑖 to the volume 𝑉 of the mixture, expressed typically with a denominator of 100 as percentage: 𝑐vol 𝑖 =

𝑉𝑖 𝑉

(A.2)

It is important to note that the volume concentration 𝑐vol 𝑖 should not be confused with the volume fraction 𝜙𝑖 , where the ratio of volume 𝑉𝑖 of a constituent 𝑖 to the sum of volumes of all constituents 𝑗 before mixing is considered: 𝜙𝑖 =

𝑉𝑖 ∑𝑉𝑗 𝑗

© Springer Fachmedien Wiesbaden GmbH 2018 A. Ivanov, Silicon Anodization as a Structuring Technique, https://doi.org/10.1007/978-3-658-19238-9

(A.3)

240

Appendix The volume of a mixture is generally not equal to the sum of volumes of all constituents before mixing. For example, because of the phenomenon of volume contraction, for ethanol-water binary mixtures, the volume after mixing reduces up to 4 % [397]. Therefore, in general 𝑐vol 𝑖 ≠ 𝜙𝑖 .

• mass fraction 𝑤𝑖 , which is the ratio of mass 𝑚𝑖 of a constituent 𝑖 to the sum of the masses of all constituents before mixing [398], expressed typically with a denominator of 100 as percentage by mass [m%], also often called percentage by weight [wt%], and in some older papers denoted as “w/o” [295]: 𝑤𝑖 =

𝑚𝑖 ∑𝑚𝑗

(A.4)

𝑗

In case no gas or energy is evolved or consumed upon mixing the components, the mass of the mixture 𝑚total is equal to the sum of masses of all components according to the principle of mass conservation, and then: 𝑤𝑖 =

𝑚𝑖 𝑚total

(A.5)

In this work, this formula is assumed to be valid for HF:water and HF:water:ethanol mixtures. • molality 𝑏, which is the ratio of the amount of a solute 𝑁solute to the mass of a solvent 𝑚solvent , typically given in mol/kg: 𝑏=

𝑁solute 𝑚solvent

(A.6)

• volume ratio of all constituents in the mixture, e.g. 1:1, HF(50 m%):ethanol meaning the mixture of equal volume parts of 50 m% HF and ethanol. • another unconventional concentration unit for mixtures of 50 m% HF and ethanol, which was often used in older papers about silicon anodization, is concentration in percent, which is neither the percentage by mass nor the percentage by volume, but a combination of the two,

241

Appendix

as explained by Halimaoui in [258]: in this case, given concentration 𝑐funny HF in % of a mixture 𝜙HF ∶ 𝜙ethanol is calculated as: 𝑐funny HF =

𝜙HF ⋅ 50 m% 𝜙HF + 𝜙ethanol

(A.7)

which for 1:1, HF(50 m%):ethanol gives 25 %. In the work this type of notation was strictly avoided. It is worth noting that according to IUPAC, concentration is the “group of four quantities characterizing the composition of a mixture (mass, amount, volume and number concentration) with respect to the volume of the mixture” [219]. Thus, strictly speaking, the mass fraction is not a measure of concentration. However, the term is used frequently in the sense of concentration by the scientific community, therefore it is unavoidable to use it in the same sense in this work. In papers, where concentration values are given in percent, volume concentration is assumed based on the definition of IUPAC, unless other definition is given. Different papers provide concentrations in different units, therefore the question of unit conversions during literature research is raised. The problematic of the unit conversion was discussed by Lehmann [2]. Conversions between the units used in the work are derived in the next sections. A.1.2. Reference data In some formulas in this and the next sections, in subscript in square brackets for a variable, its unit is given. For example, 𝑚HF [g] means mass of HF in grams.i The density of water 𝜌H2 O at 20 °C is 0.998 g/ml and 1.000 g/ml at 4 °C [399]. Conversion of the specific gravity 𝑆𝐺X of a substance X to its density 𝜌X : 𝜌X = 𝑆𝐺X 𝜌H2 O (4°C)

(A.8)

Using the specific gravity values for HF from Kirk-Othmer Encyclopedia of Chemical Technology [399], Vol. 11 and applying eq. (A.8), the density of aqueous hydrofluoric acid solution 𝜌HF as a function of HF mass fraction 𝑤HF at 20 °C can be well described with the following equation (s. Fig. A.1): 𝜌HF [g/ml] = 0.0036 ⋅ 𝑤HF [m%] + 0.998

(A.9)

242

Appendix

Density, g/ml

1.15

1.10

1.05

1.00 0

10

20

30

40

50

wHF , m%

Figure A.1.: Density of HF:water system vs. mass fraction of HF 𝑤HF at 20 °C. This equation is valid for 𝑤HF in the range from 0 to 50 m%. From the equation, 𝜌HF(50 m%) = 1.178 g/ml. The density of pure liquid HF at 20 °C 𝜌HF pure is 0.955 g/ml [399]. The molar mass of HF 𝑀HF is 20.006 g/mol. The density of absolute ethanol 𝜌ethanol at 20 °C is 0.79 g/ml [400]. The molar mass of ethanol 𝑀ethanol is 46.07 g/mol. A.1.3. HF:water solutions A.1.3.1. Conversion of volume concentration to mass fraction in HF:water mixture The volume concentration 𝑐vol HF in percent can be calculated from the mass fraction of HF 𝑤HF in m% according to the following formula: 𝑐vol HF [%] =

i In

0.0036𝑤HF [m%] 2 + 0.998𝑤HF [m%] 0.955

some places the unit is omitted for clarity.

(A.10)

243

Appendix

The mass fraction of HF 𝑤HF in m% can be obtained from the volume concentration 𝑐vol HF in percent with the following formula: 𝑤HF [m%] =

−0.998 + √0.996 + 0.01375𝑐vol HF [%]

(A.11)

0.0072

The formulas are valid at 20 °C for 𝑤HF in the range from 0 to 50 m%, because they make use of the dependence (A.9). For 𝑤HF = 50 m%, 𝑐vol HF = 61.68 %. The plot of the formulas is shown in Fig. A.2a. Density of the system HF:water in terms of the volume concentration: 𝜌HF [g/ml] =

(a)

(A.12)

2

(b)

50

HF:water

50

HF:water

40

wHF , m%

40

wHF , m%

0.998 + √0.996 + 0.01375𝑐vol HF [%]

30 20

30 20 10

10

0

0 0

10

20

30

40

50

60

cvol HF , %

0

5

10

15

20

25

30

cmol HF , mol/l

Figure A.2.: Mass fraction of HF 𝑤HF vs. (a) volume concentration 𝑐vol HF and (b) amount concentration 𝑐mol HF in HF:water mixture at 20 °C.

Derivation of the formulas: The volume concentration of HF:

244

Appendix

𝑐vol HF [%] = =

𝑉HF pure [ml] ⋅ 100 = 𝑉total [ml] 𝑚HF pure [g] /𝜌HF pure [g/ml] ⋅ 100 = 𝑚total [g] /𝜌total [g/ml]

= 𝑤HF [m%] = 𝑤HF [m%] =

𝜌total [g/ml] 𝜌HF pure [g/ml]

=

0.0036[g/ml] 𝑤HF [m%] + 0.998[g/ml] = 𝜌HF pure [g/ml]

0.0036[g/ml] 𝑤HF [m%] 2 + 0.998[g/ml] 𝑤HF [m%] 𝜌HF pure [g/ml]

(A.13)

For the quadratic equation 0.0036[g/ml] 𝑤HF [m%] 2 +0.998[g/ml] 𝑤HF [m%] −𝜌HF pure [g/ml] 𝑐vol HF [%] = 0 (A.14) the solution is 𝑤HF [m%] =

=

−0.998 ± √0.996 + 0.0144𝜌HF pure 𝑐vol HF = 0.0072 −0.998[g/ml] ± √0.996[g2 /ml2 ] + 0.01375[g2 /ml2 ] 𝑐vol HF [%] 0.0072[g/ml]

(A.15)

where only the solution with the plus sign is acceptable for positive values of 𝑤HF . Using this expression, the density of the system HF:water can be formulated in terms of the HF volume concentration: 𝜌HF [g/ml] = 0.0036𝑤HF [m%] + 0.998 = =

0.998[g/ml] + √0.996[g2 /ml2 ] + 0.01375[g2 /ml2 ] 𝑐vol HF [%] 2[g/ml] (A.16)

245

Appendix A.1.3.2. Amount concentration of HF to mass fraction in HF:water mixture

The amount concentration 𝑐mol HF in mol/l can be calculated from the mass fraction 𝑤HF in m% with the formula: 𝑐mol HF [mol/l] =

0.0036𝑤HF [m%] 2 + 0.998𝑤HF [m%] 2.0006

(A.17)

Reverse conversion: 𝑤HF [m%] =

−0.998 + √0.996 + 0.02881𝑐mol HF[mol/l] 0.0072

(A.18)

The formulas are valid at 20 °C for 𝑤HF in the range from 0 to 50 m%, because they make use of the dependence (A.9). For 𝑤HF = 50 m%, 𝑐mol HF = 29.44 mol/l. The plot of the formulas is shown in Fig. A.2b. Derivation of the formulas: The amount of HF: 𝑁HF [mol] = 𝑐mol HF [mol/l] 𝑉total [ml] ⋅ 10−3

(A.19)

Then the mass of pure HF: 𝑚HF pure [g] = 𝑁HF [mol] 𝑀HF [g/mol] = = 𝑐mol HF [mol/l] 𝑉total [ml] 𝑀HF [g/mol] ⋅ 10−3

(A.20)

The mass of the mixture can be obtained from the density of the mixture 𝜌total according to the dependence (A.9): 𝑚total [g] = 𝜌total [g/ml] 𝑉total [ml] = = (0.0036[g/ml] 𝑤HF [m%] + 0.998[g/ml] )𝑉total [ml] Then the mass fraction of HF:

(A.21)

246

Appendix

𝑤HF [m%] = =

=

𝑚HF pure [g] ⋅ 100 = 𝑚total [g] 𝑐mol HF [mol/l] 𝑉total [ml] 𝑀HF [g/mol] ⋅ 10−3 (0.0036[g/ml] 𝑤HF [m%] + 0.998[g/ml] )𝑉total [ml]

⋅ 100 =

𝑐mol HF [mol/l] 𝑀HF [g/mol] ⋅ 0.1 0.0036[g/ml] 𝑤HF [m%] + 0.998[g/ml]

(A.22)

Then the amount concentration: 0.0036[g/ml] 𝑤HF [m%] 2 + 0.998[g/ml] 𝑤HF [m%] 0.1𝑀HF [g/mol]

𝑐mol HF [mol/l] =

(A.23)

For the quadratic equation 0.0036𝑤HF [m%] 2 + 0.998𝑤HF [m%] − 0.1𝑀HF [g/mol] 𝑐mol HF [mol/l] = 0 (A.24) the solution is 𝑤HF [m%] =

−0.998 ± √0.996 + 0.00144𝑀HF[g/mol] 𝑐mol HF[mol/l] (A.25) 0.0072

where only the solution with the plus sign is acceptable for positive values of 𝑤HF . A.1.3.3. Molality of HF to mass fraction of HF in HF:water mixture Mass fraction of HF 𝑤HF in m% can be obtained from the molality 𝑏HF in mol/kg with the following expression: 𝑤HF [m%] =

100 1+

1000 𝑏HF 𝑀HF

(A.26)

The reverse conversion: 𝑏HF =

1000 𝑀HF ( 𝑤100 − 1) HF

(A.27)

247

Appendix Derivation of the formulas: Molality of HF: 𝑏HF [mol/kg] =

𝑁HF [mol] 𝑚H2 O [kg]

(A.28)

where the amount of HF: 𝑁HF [mol] =

1000 ⋅ 𝑚HF [kg] 𝑀HF [g/mol]

(A.29)

Then, the mass of water: 𝑚H2 O [kg] =

𝑁HF [mol] 1000 ⋅ 𝑚HF [kg] = 𝑏HF [mol/kg] 𝑏HF [mol/kg] 𝑀HF [g/mol]

(A.30)

Then, the mass fraction of HF: 𝑚HF [kg] ⋅ 100 = 𝑚HF [kg] + 𝑚H2 O [kg] 𝑚HF [kg] = ⋅ 100 = 1000⋅𝑚HF [kg] 𝑚HF [kg] + 𝑏 𝑀

𝑤HF [m%] =

HF [mol/kg]

=

100 1+

1000 𝑏HF [mol/kg] 𝑀HF [g/mol]

HF [g/mol]

(A.31)

A.1.3.4. Equilibrium concentrations of components without (HF)2 in HF:water mixture According to the dissociation/association equations (2.7) and (2.8), if the reaction of HF dimerization (eq. (2.9)) is neglected, then the equilibrium eq eq eq eq concentrations 𝑐mol HF , 𝑐mol H+ , 𝑐mol F− , and 𝑐mol HF2 − of the components HF, H+, F−, and HF2 −, respectively, for a given total (formal) HF concentration 0 𝑐mol HF can be calculated from the system of non-linear equations. The system of non-linear equations for four unknown equilibrium concentrations consists of the two reactions for the equilibrium constants and the two mass (concentration) conservation equations for hydrogen and fluorine: eq eq eq 𝑐mol HF 𝐾HF = 𝑐mol H+ 𝑐mol F−

(A.32)

248

Appendix

eq eq eq 𝑐mol HF 𝑐mol F− 𝐾HF2 − = 𝑐mol HF2 −

(A.33)

eq eq eq 0 𝑐mol HF = 𝑐mol HF + 𝑐mol H+ + 𝑐mol HF2 −

(A.34)

eq eq eq 0 𝑐mol HF = 𝑐mol HF + 𝑐mol F− + 2𝑐mol HF2 −

(A.35)

The calculations have been performed with the tools of SciPy (Python) library in Spyder ii , with 𝐾HF = 6.71 × 10−4 mol/l, 𝐾HF2 − = 3.861 l/mol 0 [2,227], and 𝑐mol HF in the range from 0 to 5 mol/l. The resulting equilibrium concentrations are shown in Fig. A.3a. The data fit well to the data available in the works of Hamer and Wu [401], and Varhaverbeke et al. [232]. It is important to note that the calculations are done for the constant given dissociation constants. However, these constants depend on the formal HF concentration, therefore for higher concentrations the calculation might be not correct. Additionally, at concentrations above 15 mol/l more complex polymers of HF start playing an important role, and have to be taken into account [235]. A.1.3.5. Equilibrium concentrations of components with (HF)2 in HF:water mixture In case the reaction of HF dimerization (eq. (2.9)) is considered, then the eq eq eq eq eq equilibrium concentrations 𝑐mol HF , 𝑐mol H+ , 𝑐mol F− , 𝑐mol HF2 − , and 𝑐mol (HF)2 of the components HF, H+, F−, HF2 −, and (HF)2 , respectively, for a given 0 total (formal) HF concentration 𝑐mol HF can be calculated from the system of five equations. The system for five unknown equilibrium concentrations consists of the three reactions for the equilibrium constants and the two mass (concentration) conservation equations for hydrogen and fluorine: eq eq eq 𝑐mol HF 𝐾HF = 𝑐mol H+ 𝑐mol F−

(A.36)

eq eq eq 𝑐mol HF 𝑐mol F− 𝐾HF2 − = 𝑐mol HF2 −

(A.37)

ii Spyder

- cross-platform IDE for scientific programming in the Python language

249

(a) without dimerization reaction

(b) with dimerization reaction

Equilibrium amount concentr., mol/l

Equilibrium amount concentr., mol/l

Appendix

HF:water 100 10−1 10−2 10−3 10−4

HF2 − HF

H+ F− 0

1

2

3

4

5

Formal HF amount concentr., mol/l

HF:water 100 10−1 10−2 H+ F− HF2 −

10−3 10−4

0

1

2

HF (HF)2 3

4

5

Formal HF amount concent., mol/l

Figure A.3.: Equilibrium concentrations in HF:water solutions at 25 °C and zero ionic strength: (a) neglecting the dimerization reaction, (b) with the dimerization reaction.

eq 2 𝑐eq mol HF 𝐾(HF)2 = 𝑐mol (HF)

(A.38)

2

eq eq eq eq 0 𝑐mol HF = 𝑐mol HF + 𝑐mol H+ + 𝑐mol HF2 − + 2𝑐mol (HF)

(A.39)

2

eq eq eq eq 0 𝑐mol HF = 𝑐mol HF + 𝑐mol F− + 2𝑐mol HF2 − + 2𝑐mol (HF)

2

(A.40)

The calculations have been performed the same way as in sec. A.1.3.4, with 𝐾HF = 6.71 × 10−4 mol/l, 𝐾HF2 − = 3.861 l/mol, 𝐾(HF)2 = 2.703 l/mol [2, 0 227], and 𝑐mol HF in the range from 0 to 5 mol/l. The resulting equilibrium concentrations are shown in Fig. A.3b. The data fit well to the data available in the work of Varhaverbeke et al. [232] and Liu and Blackwood [227]. As can be seen from the plot, the equilibrium concentration of the dimer (HF)2 0 gets higher than that of HF for 𝑐mol HF more than about 1.2 mol/l.

250

Appendix

A.1.4. HF:water:ethanol solutions A.1.4.1. Mass fraction of HF in HF(50 m%):ethanol mixture Typical electrolytes for anodization are prepared as a mixture of 50 m% HF with ethanol in a given volume ratio, e.g. 1:1. The mass fraction of HF 𝑤HF in m% at 20 °C can be obtained with the following formula: 𝑤HF [m%] =

0.589𝜙HF(50 m%) ⋅ 100 0.388𝜙HF(50 m%) + 0.79

(A.41)

where 𝜙HF(50 m%) is the volume fraction of the 50 m% HF in the range from 0 to 1. For the typical ratio 1:1 (i.e, 𝜙HF(50 m%) = 0.5), 𝑤HF = 29.93 m%. The plot of the formula is shown in Fig. A.5. 50

wHF , m%

40 30 20 10 0 0.0

0.2

0.4

0.6

0.8

1.0

φHF(50 m%)

Figure A.4.: Mass fraction of HF 𝑤HF vs. HF volume fraction 𝜙HF(50 m%) in HF(50 m%):ethanol mixture at 20 °C.

Derivation of the formula: The mass of pure HF 𝑚HF pure in the mixture: 𝑚HF pure [g] = 0.5𝑚HF(50 m%) [g] = = 0.5𝑉HF(50 m%) [ml] 𝜌HF(50 m%) [g/ml] = = 0.5𝑉total [ml] 𝜙HF(50 m%) 𝜌HF(50 m%) [g/ml]

(A.42)

251

Appendix and the total mass of the solution 𝑚total : 𝑚total [g] = 𝑚HF(50 m%) [g] + 𝑚ethanol [g] = = 𝑉total [ml] 𝜙HF(50 m%) 𝜌HF(50 m%) [g/ml] + + 𝑉total [ml] 𝜙ethanol 𝜌ethanol [g/ml] = = 𝑉total [𝜙HF(50 m%) 𝜌HF(50 m%) + (1 − 𝜙HF(50 m%) )𝜌ethanol ]

(A.43)

where 𝜙ethanol is the volume fraction of ethanol in the range from 0 to 1, with the sum of the HF and ethanol volume fractions 𝜙sum = 𝜙HF(50 m%) + 𝜙ethanol = 1 (i.e., 100 %). Then: 𝑤HF [m%] = =

=

𝑚HF pure [g] ⋅ 100 = 𝑚total [g] 0.5𝜙HF(50 m%) 𝜌HF(50 m%) [g/ml] 𝜙HF(50 m%) 𝜌HF(50 m%) + (1 − 𝜙HF(50 m%) )𝜌ethanol 0.589𝜙HF(50 m%) ⋅ 100 0.388𝜙HF(50 m%) + 0.79

⋅ 100 =

(A.44)

A.1.4.2. Volume concentrations of HF and ethanol to mass fraction of HF in HF:water:ethanol mixture Above the conversion from volume concentration to mass fraction in HF:water mixture has been obtained (sec. A.1.3.1). Here a more general case of HF:water:ethanol is considered, where the volume concentrations of HF and ethanol 𝑐vol HF and 𝑐vol ethanol are known. Derivation of a single formula for this case results in a bulky arithmetic and is omitted. Instead, only the result of the conversion and the way of derivation are shown. The resulting dependence of mass fraction 𝑤HF on the volume concentrations of HF and ethanol 𝑐vol HF and 𝑐vol ethanol is shown in Fig. A.5a. For the case of 1:1 mixture of HF(50 m%):ethanol, 𝑐vol HF = 30.84 % (corresponds to 61.68 % for the half-volume, which is 50 m% according to

252

Appendix

sec. A.1.3.1) and 𝑐vol ethanol = 50 %, the mass fraction of HF in the HF:water:ethanol mixture 𝑤HF [m%] = 29.93 m%, which is the same value as calculated in sec. A.1.4.1. With 𝑐vol ethanol = 0, the conversion reduces to the case described in sec. A.1.3.1.

HF:water:ethanol 1:1 HF(50m%):ethanol

HF:water:ethanol 1:1 HF(50m%):ethanol

Figure A.5.: Mass fraction of HF 𝑤HF in HF:water:ethanol mixture at 20 °C vs. volume concentrations of HF and ethanol 𝑐vol HF and 𝑐vol ethanol (left) and amount concentrations of HF and ethanol 𝑐mol HF and 𝑐mol ethanol (right). The range of the shown results is limited to the HF mass fraction of 50 m% in the HF:water part of the mixture. The values in the legend for the left plot are for 𝑐vol ethanol , and in the right plot for 𝑐mol ethanol .

Derivation of the formulas: When mixing hydrofluoric acid with water or ethanol with water, the total volume of these binary mixtures is not equal to the sum of volumes of the components. These effects are described for HF:water mixture with concentration dependent density (s. eq. (A.9)), and a corresponding dependence can be obtained for ethanol:water mixture [397]. However, the effect of mixing the three components (HF, water, and ethanol) together could not be found in the literature. Since the effect of volume change for HF:water is more pronounced than for ethanol:water (volume contraction more than 20 % for HF:water vs. below 4 % for ethanol:water), for the calculations here

253

Appendix

it is assumed, that only the volume change during mixing of HF and water takes place, therefore: (A.45)

𝑉total [ml] = 𝑉HF∶H2 O [ml] + 𝑉ethanol [ml] where for the given volume concentration 𝑐vol ethanol 𝑉ethanol [ml] = 𝑉total [ml]

𝑐vol ethanol [%] 100

(A.46)

Then 𝑉HF∶H2 O [ml] = 𝑉total [ml] − 𝑉ethanol [ml] = 𝑉total [ml] (1 −

𝑐vol ethanol [%] ) (A.47) 100

The volume of HF for the given volume concentration 𝑐vol HF : 𝑉HF [ml] = 𝑉total [ml]

𝑐vol HF [%] 100

(A.48)

To calculate the mass fraction of HF in HF:water:ethanol mixture, the masses of HF, ethanol, and HF:water mixture, 𝑚HF , 𝑚ethanol , and 𝑚HF∶H2 O , respectively, have to be found: 𝑚HF pure [g] = 𝑉HF [ml] 𝜌HF pure [g/ml] = 𝑐vol HF [%] = 𝑉total [ml] 𝜌HF pure [g/ml] 100 𝑚ethanol [g] = 𝑉ethanol [ml] 𝜌ethanol [g/ml] = 𝑐vol ethanol [%] = 𝑉total [ml] 𝜌ethanol [g/ml] 100 𝑚HF∶H2 O [g] = 𝑉HF∶H2 O [ml] 𝜌HF∶H2 O [g/ml] = 𝑐vol ethanol [%] = 𝑉total [ml] (1 − ) 𝜌HF∶H2 O [g/ml] 100

(A.49)

(A.50)

(A.51)

where the density of the mixture HF:water can be obtained according to ∗ eq. (A.12) from the volume concentration of HF 𝑐vol HF in the system HF:water:

254

Appendix

∗ 𝑐vol HF [%] =

=

𝑉HF [ml] ⋅ 100 = 𝑉HF∶H2 O [ml] 𝑐vol HF [%] 100 𝑐vol ethanol [%] − 100

𝑉total [ml] 𝑉total [ml] (1

)

⋅ 100 =

𝑐vol HF [%] (1 −

𝑐vol ethanol [%] 100

)

(A.52)

Then the mass fraction of HF in the HF:water:ethanol system can be obtained: 𝑤HF [m%] =

𝑚HF pure [g] ⋅ 100 𝑚HF∶H2 O [g] + 𝑚ethanol [g]

(A.53)

A.1.4.3. Amount concentrations of HF and ethanol to mass fraction of HF in HF:water:ethanol mixture Above the conversion from amount concentration to mass fraction in HF:water mixture has been obtained (sec. A.1.3.2). Here a more general case of HF:water:ethanol is considered, where the amount concentrations of HF and ethanol 𝑐mol HF and 𝑐mol ethanol are known. Similarly to sec. A.1.4.2, derivation of the final formula is omitted. The results of the conversion are shown in Fig. A.5b. For the case of 1:1 mixture of HF(50 m%):ethanol, 𝑐mol HF = 14.72 mol/l (corresponds to 29.44 mol/l for the half-volume) and 𝑐mol ethanol = 8.57 mol/l (selected to have 𝑉HF∶H2 O = 𝑉ethanol ), the mass fraction of HF in the HF:water:ethanol mixture 𝑤HF [m%] = 29.93 m% and the mass fraction of HF in the ∗ HF:water half-volume 𝑤HF = 50 m%. With 𝑐mol ethanol = 0, the conversion reduces to the case described in sec. A.1.3.2. Derivation of the formulas: Here the same assumption of volume contraction for HF:water neglecting the effect of volume contraction for ethanol:water mixture as in sec. A.1.4.2 is applied (eq. (A.45)). From the amount concentrations of HF and ethanol, their masses: 𝑚ethanol [g] = 𝑐mol ethanol [mol/l] 𝑉total [ml] 𝑀ethanol [g/mol] ⋅ 10−3

(A.54)

255

Appendix

𝑚HF pure [g] = 𝑐mol HF [mol/l] 𝑉total [ml] 𝑀HF [g/mol] ⋅ 10−3

(A.55)

Then the volumes of HF and ethanol: 𝑉HF [ml] =

𝑚HF pure [g] 𝑐mol HF [mol/l] 𝑉total [ml] 𝑀HF [g/mol] ⋅ 10−3 = (A.56) 𝜌HF pure [g/ml] 𝜌HF pure [g/ml]

𝑉ethanol [ml] = =

𝑚ethanol [g] = 𝜌ethanol [g/ml] 𝑐mol ethanol [mol/l] 𝑉total [ml] 𝑀ethanol [g/mol] ⋅ 10−3 𝜌ethanol [g/ml]

(A.57)

Then the volume of the HF:water mixture 𝑉HF∶H2 O can be obtained with eq. (A.45). Knowing the volume of HF and HF:water, the volume concen∗ tration of HF 𝑐vol HF in the HF:water mixture and the density 𝜌HF∶H2 O of the HF:water mixture can be found. Then the mass of the HF:water can be calculated. With the obtained values of masses of HF, ethanol, and HF:water, the mass fraction of HF in the HF:water:ethanol mixture can be calculated according to eq. (A.53). A.1.4.4. Equilibrium concentrations of components without (HF)2 in HF:water:ethanol mixture Luxenberg and Kim [237] showed that there is a strong dependence of the equilibrium constants for the reactions (2.7) and (2.8) on ethanol concentration in HF:water:ethanol solutions (Fig. A.6). Therefore, the equilibrium concentrations in a solution with ethanol will differ from those for the HF:water (s. above in sec. A.1.3.4). As example, here the calculation of the equilibrium concentrations for 50 m% ethanol is performed, where 𝐾HF = 3.214 × 10−5 mol/dm3 and 𝐾HF2 − = 12.882 dm3/mol at zero ionic strength and 25 °C. The calculation is done the same way as described in sec. A.1.3.4. The results are shown in Fig. A.7. For the formal HF concentration above 1 mol/l, the equilibrium concentration of ionized fluorine is about ten times less than that in HF:water solution. The equilibrium concentrations of H+ and HF2 − are also significantly reduced.

256

Appendix

20 o C 25 o C

10

−4

10−5

20 o C 25 o C

16

KHF2 − , l/mol

KHF , mol/l

10−3

14 12 10 8 6

10−6

0

10

20

30

40

50

60

Ethanol mass fraction, m%

70

0

10

20

30

40

50

60

70

Ethanol mass fraction, m%

Figure A.6.: Equilibrium constants 𝐾HF and 𝐾HF2 − in dependence of ethanol mass fraction in HF:water:ethanol mixture at zero ionic strength, based on data from Luxenberg and Kim [237]. For the calculation of the equilibrium concentrations with consideration of the dimerization reaction (eq. (2.9)), data on the equilibrium constant 𝐾(HF)2 in dependence of ethanol are missing, and therefore this calculation is not performed here. A.2. Conversions of other anodization parameters A.2.1. Dissolution valence calculated from porosity and thickness of porous layer As was given in sec. 2.4.2.4, effective dissolution valence during anodization can be calculated as following: 𝑛e =

𝑗𝐴𝑡etch 𝑀Si ⋅ 𝑒 𝑚etched 𝑁A

(A.58)

where 𝐴 is the anodization area, 𝑗 is the current density during the anodization of time 𝑡, 𝑒 is the elementary charge, 𝑀Si is the molar mass of silicon (28.083 g/mol), and 𝑚etched is the mass of dissolved silicon. The mass of dissolved silicon is normally measured directly by weighing the sample to be anodized before and after anodization. In case the data for

257

(a) HF:water:ethanol(50 m%)

(b) HF:water

Equilibrium amount concentr., mol/l

Equilibrium amount concentr., mol/l

Appendix

100 10−1 10−2 10−3 10−4

HF2 − HF

H+ F− 0

1

2

3

4

5

Formal HF amount concentr., mol/l

100 10−1 10−2 10−3 10−4

HF2 − HF

H+ F− 0

1

2

3

4

5

Formal HF amount concentr., mol/l

Figure A.7.: Equilibrium concentrations at 25 °C and zero ionic strength neglecting the dimerization reaction: (a) in HF:water:ethanol solutions, with 50 m% ethanol; (b) in HF:water solutions (the plot was already shown in Fig. A.3a, here it is placed for ease of comparison). the mass loss are not provided, it can be found with given porosity 𝑃% and thickness 𝑑 of the porous layer: 𝑚etched = 𝑚bulk ⋅

𝑃% 𝐴𝑑𝜌Si 𝑃% = 100 100

(A.59)

where 𝑚bulk is the mass of the silicon volume before its porosification and 𝜌Si is the density of silicon (2.329 g/ml at 25 °C [399]). Then, the dissolution valence can be calculated as: 𝑛e = 100

𝑗𝑡etch 𝑀Si ⋅ 𝑒 𝜌Si 𝑁A 𝑃% 𝑑

(A.60)

With the same equation, of course, missing porosity or etch rate can be also found, if the other parameters are given.

258

Appendix

A.2.2. Porosity and density of porous silicon The mass of dissolved silicon is related to the mass of porous silicon according to the following expression (assuming silicon sample is completely transformed into porous silicon): 𝑚etched = 𝑚bulk − 𝑚PS

(A.61)

Then, porosity 𝑃% can be calculated from the density of porous silicon 𝜌PS : 𝑚etched 𝑚 − 𝑚PS 𝑚 ⋅ 100 = bulk ⋅ 100 = (1 − PS ) ⋅ 100 = 𝑚bulk 𝑚bulk 𝑚bulk 𝑉 𝜌PS 𝜌 = (1 − (A.62) ) ⋅ 100 = (1 − PS ) ⋅ 100 𝑉 𝜌Si 𝜌Si

𝑃% =

259

Appendix A.3. Experimental data A.3.1. Experimental data for sec. 5.1.2 A.3.1.1. Comparison of the final current densities

The plots below show the comparison of the final current density values calculated either based on the increase of the total area of all structures on a sample (𝑗area ) or based on the etched volume for each individual structure (𝑗vol ). Fit with polynomials of 2nd order is applied to the 𝑗vol datasets.

𝑗init = 1.5 A/cm2 Final current density, A/cm2

Final current density, A/cm2

𝑗init = 1 A/cm2 1.2 1.0 0.8 0.6 0.4 0.2 200

400

600

800

1.4 1.2 1.0 0.8 0.6 0.4

1000

200

Diameter of opening, µm

400

600

800

1000

Diameter of opening, µm

𝑗init = 2 A/cm2

𝑗init = 2.5 A/cm2 Final current density, A/cm2

Final current density, A/cm2

2.5

2.0

1.5

1.0

0.5 200

400

600

800

Diameter of opening, µm

1000

2.0 1.5 1.0 0.5 200

400

600

800

Diameter of opening, µm

1000

260

Appendix

𝑗init = 3 A/cm2

𝑗init = 3.5 A/cm2 Final current density, A/cm2

Final current density, A/cm2

3.5 3.0 2.5 2.0 1.5 1.0 0.5

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

200

400

600

800

Diameter of opening, µm

1000

200

400

600

800

Diameter of opening, µm

1000

261

Appendix A.3.1.2. Curvature vs. structure depth

Curv a ture, mm–1

Custom fit on the plots below for all negative and increasing positive curvature values was performed in order to evaluate the threshold depth values as shown below. The legend is the same for all five plots.

𝐷open = 200 μm Structure depth, µm

Curv a ture, mm–1

Curv a ture, mm–1

𝐷open = 600 μm

𝐷open = 400 μm Structure depth, µm

Structure depth, µm

𝐷open = 1000 μm Curv a ture, mm–1

Curv a ture, mm–1

𝐷open = 800 μm

Structure depth, µm

Structure depth, µm

262

Appendix

A.3.1.3. Anisotropy factor vs. structure depth The legend is the same for all five plots, therefore it is shown only once.

Anisotropy factor Af

𝐷open = 200 μm 0.65

j init = 1.0 A/cm2

0.60

j init = 1.5 A/cm2

0.55

j init = 2.0 A/cm2 j init = 2.5 A/cm2

0.50

j init = 3.0 A/cm2

0.45

j init = 3.5 A/cm2

0.40 0.35 0.30 50

100

150

200

250

Structure depth, µm

𝐷open = 400 μm

𝐷open = 600 μm 0.7

Anisotropy factor Af

Anisotropy factor Af

0.6 0.5 0.4 0.3 0.2

0.6 0.5 0.4 0.3 0.2 0.1

0.1 50

100 150 200 250 300 350

50 100 150 200 250 300 350 400

Structure depth, µm

Structure depth, µm

𝐷open = 800 μm

𝐷open = 1000 μm 0.6

0.6

Anisotropy factor Af

Anisotropy factor Af

0.7

0.5 0.4 0.3 0.2 0.1 0.0

0.5 0.4 0.3 0.2 0.1 0.0

100

200

300

400

Structure depth, µm

500

100

200

300

400

Structure depth, µm

500

263

Appendix A.3.1.4. Anisotropy factor vs. average current density

As the average current density, the mean value between the initial current density (calculated for the area of the openings in silicon nitride on the sample) and the final current density 𝑗area (calculated for the increased total anodization area of all the structures on the sample in the end of the process) is given. The error bars show the corresponding ranges between the initial and the final current densities. Structure depth 0–100 µm Anisotropy factor Af

0.65

Dopen Dopen Dopen Dopen Dopen

0.60 0.55 0.50 0.45

= 200 µm = 400 µm = 600 µm = 800 µm = 1000 µm

0.40 0.35 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

Structure depth 200–300 µm 0.7

Anisotropy factor Af

Anisotropy factor Af

Structure depth 100–200 µm 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

Average current density, A/cm2

Structure depth 300–400 µm

Structure depth 400–500 µm Anisotropy factor Af

Anisotropy factor Af

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.6 0.5 0.4 0.3 0.2 0.1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

264

Appendix

A.3.2. Comparison of the final current densities for sec. 5.1.3

Final current density, A/cm2

The plots below show the comparison of the final current density values calculated either based on the increase of the total area of all structures on a sample (𝑗area ) or based on the etched volume for each individual structure (𝑗vol ). Fit with polynomials of 2nd order is applied to the 𝑗vol datasets.

j vol for j init = 50 mA/cm2 j area for j init = 50 mA/cm2

0.8

j vol for j init = 100 mA/cm2 0.6

j area for j init = 100 mA/cm2 j vol for j init = 200 mA/cm2

0.4

j area for j init = 200 mA/cm2 j vol for j init = 500 mA/cm2

0.2

j area for j init = 500 mA/cm2 j vol for j init = 1000 mA/cm2

0.0

200

400

600

800

1000

j area for j init = 1000 mA/cm2

Final current density, A/cm2

Diameter of opening, µm

2.2

j vol for j init = 1500 mA/cm2

2.0

j area for j init = 1500 mA/cm2

1.8

j vol for j init = 2000 mA/cm2

1.6

j area for j init = 2000 mA/cm2 j vol for j init = 2500 mA/cm2

1.4

j area for j init = 2500 mA/cm2

1.2

j vol for j init = 3000 mA/cm2

1.0

j area for j init = 3000 mA/cm2

0.8

j vol for j init = 3500 mA/cm2 200

400

600

800

Diameter of opening, µm

1000

j area for j init = 3500 mA/cm2

265

Appendix A.4. Simulated data

In the sections below, the data obtained from the models are presented. In order to help the reader, in the names of the subsections the abbreviations refer to the model (“primary model” and “secondary model” mean model with primary or secondary current distribution, respectively; “low-doped Si” and “highly-doped Si” refer the resistivity of silicon in the models). A.4.1. Primary current distribution model for low-doped p-type silicon in sec. 5.4.1 In this section, the data obtained from the primary current distribution model with single circular opening in a frontside insulating masking layer and low-doped p-type silicon (s. sec. 5.4.1) are shown. A.4.1.1. Comparison of the structure volume between simulation and experiment (primary model, low-doped Si) The volume of the simulated structures (datasets starting with “sim.”) is compared to the experiment (datasets starting with “exp.”) as described in sec. 5.1.2. Error bars at the data points from the experiment show the deviation of the values between the two evaluated structures with the same opening diameter on a sample. Datasets in the plots below are for the initial current density values. The legend is the same for all five plots. exp. : j init = 1.0 A/cm2 sim. : j init = 1.0 A/cm2

Structure volume, mm3

exp. : j init = 1.5 A/cm2 sim. : j init = 1.5 A/cm2 exp. : j init = 2.0 A/cm2 10

sim. : j init = 2.0 A/cm2

−2

exp. : j init = 2.5 A/cm2 sim. : j init = 2.5 A/cm2 exp. : j init = 3.0 A/cm2 sim. : j init = 3.0 A/cm2 exp. : j init = 3.5 A/cm2

𝐷open = 200 μm 10

sim. : j init = 3.5 A/cm2

−3

0

5

10

Etch time, min

15

20

266

Appendix exp. : j init = 1.0 A/cm2

exp. : j init = 2.5 A/cm2

sim. : j init = 1.0 A/cm2

sim. : j init = 2.5 A/cm2

exp. : j init = 1.5 A/cm2

exp. : j init = 3.0 A/cm2

2

sim. : j init = 3.0 A/cm2

2

exp. : j init = 3.5 A/cm2

sim. : j init = 2.0 A/cm2

sim. : j init = 3.5 A/cm2

sim. : j init = 1.5 A/cm

Structure volume, mm3

Structure volume, mm3

exp. : j init = 2.0 A/cm

10−1

10−2

𝐷open = 400 μm

0

5

10

15

10−1

𝐷open = 600 μm

10−2

20

0

5

Etch time, min

10

15

20

Etch time, min

Structure volume, mm3

Structure volume, mm3

100

10−1

10−1

𝐷open = 800 μm

𝐷open = 1000 μm

10−2 0

5

10

Etch time, min

15

20

10−2

0

5

10

Etch time, min

15

20

267

Appendix A.4.1.2. Simulated curvature vs. structure depth (primary model, low-doped Si)

The legend is the same for all five plots, therefore it is shown only once. j init = 1.0 A/cm2

Curvature, mm−1

2

j init = 1.5 A/cm2 j init = 2.0 A/cm2

1

j init = 2.5 A/cm2 0

j init = 3.0 A/cm2

𝐷open = 200 μm

j init = 3.5 A/cm2

−1 −2 −3

0

100

200

300

400

500

Structure depth, µm 0.6 0.4

𝐷open = 400 μm

1.0

Curvature, mm−1

Curvature, mm−1

1.5

0.5 0.0 −0.5 −1.0 −1.5

𝐷open = 600 μm

0.2 0.0 −0.2 −0.4 −0.6 −0.8

0

100

200

300

400

−1.0

500

0

Structure depth, µm

200

300

400

500

Structure depth, µm 0.00

0.0

𝐷open = 800 μm

−0.1

Curvature, mm−1

Curvature, mm−1

100

−0.2 −0.3 −0.4 −0.5 −0.6

−0.05

𝐷open = 1000 μm

−0.10 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40

0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

Structure depth, µm

500

268

Appendix

A.4.1.3. Simulated anisotropy factor vs. structure depth (primary model, low-doped Si) The legend is the same for all five plots, therefore it is shown only once. j init = 1.0 A/cm2

Anisotropy factor Af

0.2

𝐷open = 200 μm

0.1

j init = 1.5 A/cm2 j init = 2.0 A/cm2

0.0

j init = 2.5 A/cm2

−0.1

j init = 3.0 A/cm2 j init = 3.5 A/cm2

−0.2 −0.3 −0.4 0

100

200

300

400

500

Structure depth, µm 0.2

𝐷open = 400 μm

0.1

Anisotropy factor Af

Anisotropy factor Af

0.2

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0.1

𝐷open = 600 μm

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0

100

200

300

400

500

0

Structure depth, µm

200

300

400

500

Structure depth, µm

0.1

0.1

𝐷open = 800 μm

0.0

Anisotropy factor Af

Anisotropy factor Af

100

−0.1 −0.2 −0.3 −0.4 −0.5

𝐷open = 1000 μm

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

Structure depth, µm

500

269

Appendix

A.4.1.4. Simulated anisotropy factor vs. average current density (primary model, low-doped Si) As the average current density, the mean value between the initial current density (calculated for the area of the openings in the frontside masking layer) and the final current density 𝑗area (calculated for the increased total anodization area of a structure in the end of the process) is given. The legend is the same for all five plots, therefore it is shown only once. Anisotropy factor Af

0.2

j init = 1.0 A/cm2

𝐷open = 200 μm

0.1

j init = 1.5 A/cm2

0.0

j init = 2.0 A/cm2

−0.1

j init = 3.0 A/cm2

j init = 2.5 A/cm2

−0.2

j init = 3.5 A/cm2

−0.3 −0.4 0.5

1.0

1.5

2.0

2.5

3.0

Average current density j a , A/cm2

Anisotropy factor Af

𝐷open = 400 μm

0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5

Anisotropy factor Af

0.2

0.2

𝐷open = 600 μm

0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.5

𝐷open = 800 μm

0.1

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

0.0 −0.1 −0.2 −0.3 −0.4

0.1

Anisotropy factor Af

Anisotropy factor Af

Average current density, A/cm2

𝐷open = 1000 μm

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

−0.5 0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

270

Appendix

A.4.2. Primary current distribution model for highly-doped p-type silicon in sec. 5.4.2 In this section, the data obtained from the primary current distribution model with single circular opening in a frontside insulating masking layer and highly-doped p-type silicon (s. sec. 5.4.2) are shown. A.4.2.1. Simulated curvature vs. structure depth (primary model, highly-doped Si)

Curvature, mm−1

3 2 1

𝐷open = 200 μm

0

j init = 0.05 A/cm2

j init = 1.5 A/cm2

j init = 0.1 A/cm2

j init = 2.0 A/cm2

j init = 0.2 A/cm

2

j init = 2.5 A/cm2

j init = 0.5 A/cm

2

j init = 3.0 A/cm2

j init = 1.0 A/cm2

j init = 3.5 A/cm2

−1 0

100

200

300

400

500

Structure depth, µm

1.5 1.0 0.5 0.0

𝐷open = 400 μm

−0.5 −1.0

0

100

200

300

400

Curvature, mm−1

Curvature, mm−1

2.0 1.0 0.5 0.0

𝐷open = 600 μm

−0.5

500

0

Structure depth, µm

100

200

300

400

500

Structure depth, µm

0.6 0.4 0.2 0.0 −0.2

𝐷open = 800 μm

−0.4 0

100

200

300

400

Structure depth, µm

500

Curvature, mm−1

Curvature, mm−1

0.8 0.4 0.2 0.0 −0.2 −0.4

𝐷open = 1000 μm 0

100

200

300

400

Structure depth, µm

500

271

Appendix

A.4.2.2. Simulated anisotropy factor vs. structure depth (primary model, highly-doped Si) The legend is the same for all five plots, therefore it is shown only once. j init = 0.05 A/cm2

Anisotropy factor Af

0.35

j init = 0.1 A/cm2

0.30

j init = 0.2 A/cm2

0.25

j init = 0.5 A/cm2

0.20

j init = 1.0 A/cm2

0.15

j init = 1.5 A/cm2

0.10

j init = 2.0 A/cm2

𝐷open = 200 μm

0.05

j init = 2.5 A/cm2

0.00

j init = 3.0 A/cm2 0

100

200

300

400

j init = 3.5 A/cm2

500

Structure depth, µm 0.40 0.35

Anisotropy factor Af

Anisotropy factor Af

0.35 0.30 0.25 0.20 0.15 0.10

𝐷open = 400 μm

0.05 0.00

0.30 0.25 0.20 0.15 0.10

𝐷open = 600 μm

0.05 0.00

0

100

200

300

400

500

0

Structure depth, µm

100

200

300

400

500

Structure depth, µm 0.40 0.35

Anisotropy factor Af

Anisotropy factor Af

0.35 0.30 0.25 0.20 0.15 0.10

𝐷open = 800 μm

0.05

0.30 0.25 0.20 0.15 0.10

𝐷open = 1000 μm

0.05 0.00

0.00 0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

Structure depth, µm

500

272

Appendix

A.4.2.3. Simulated anisotropy factor vs. average current density (primary model, highly-doped Si)

Anisotropy factor Af

As the average current density, the mean value between the initial current density (calculated for the area of the openings in the frontside masking layer) and the final current density 𝑗area (calculated for the increased total anodization area of a structure in the end of the process) is given. Each plot is for one value of opening diameter. j init = 0.05 A/cm2

200 μm

0.35 0.30

j init = 0.1 A/cm2

0.25

j init = 0.2 A/cm2

0.20

j init = 0.5 A/cm2

0.15

j init = 1.0 A/cm2

0.10

j init = 1.5 A/cm2

0.05

j init = 2.0 A/cm2

0.00

j init = 2.5 A/cm2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

j init = 3.0 A/cm2

2

j init = 3.5 A/cm2

Average current density, A/cm

400 μm

0.35

Anisotropy factor Af

Anisotropy factor Af

0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.00

600 μm

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0

Average current density, A/cm2

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average current density, A/cm2

Anisotropy factor Af

Anisotropy factor Af

0.40

800 μm

0.35 0.30 0.25 0.20 0.15 0.10 0.05

1000 μm

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

273

Appendix

A.4.3. Secondary current distribution model for low-doped p-type silicon in sec. 5.4.3 In this section, the data obtained from the secondary current distribution model with single circular opening in a frontside insulating masking layer and low-doped p-type silicon (s. sec. 5.4.3) are shown.

Curvature, mm−1

A.4.3.1. Simulated curvature vs. structure depth (secondary model, low-doped Si) j init = 1.0 A/cm2

𝐷open = 200 μm

1.0

j init = 1.5 A/cm2

0.5

j init = 2.0 A/cm2

0.0

j init = 2.5 A/cm2

−0.5

j init = 3.0 A/cm2

−1.0

j init = 3.5 A/cm2

−1.5 0

100

200

300

400

500

0.4

𝐷open = 400 μm

0.5

Curvature, mm−1

Curvature, mm−1

Structure depth, µm

0.0 −0.5 −1.0

𝐷open = 600 μm

0.2 0.0 −0.2 −0.4 −0.6

0

100

200

300

400

500

0

100

Structure depth, µm

𝐷open = 800 μm

0.0 −0.1 −0.2 −0.3 −0.4 0

100

200

300

400

Structure depth, µm

300

400

500

𝐷open = 1000 μm

0.00

Curvature, mm−1

Curvature, mm−1

0.1

200

Structure depth, µm

500

−0.05 −0.10 −0.15 −0.20 −0.25 −0.30

0

100

200

300

400

Structure depth, µm

500

274

Appendix

A.4.3.2. Simulated anisotropy factor vs. structure depth (secondary model, low-doped Si) The legend is the same for all five plots, therefore it is shown only once. j init = 1.0 A/cm2

Anisotropy factor Af

0.1

𝐷open = 200 μm

j init = 1.5 A/cm2

0.0

j init = 2.0 A/cm2 j init = 2.5 A/cm2

−0.1

j init = 3.0 A/cm2 j init = 3.5 A/cm2

−0.2 −0.3 0

100

200

300

400

500

Structure depth, µm

0.1

𝐷open = 400 μm

0.0

Anisotropy factor Af

Anisotropy factor Af

0.1

−0.1 −0.2 −0.3 −0.4 0

100

200

300

400

−0.1 −0.2 −0.3 −0.4 −0.5

500

𝐷open = 600 μm

0.0

0

Structure depth, µm

200

300

400

500

Structure depth, µm

0.1

0.1

𝐷open = 800 μm

0.0

Anisotropy factor Af

Anisotropy factor Af

100

−0.1 −0.2 −0.3 −0.4 −0.5

𝐷open = 1000 μm

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

Structure depth, µm

500

275

Appendix A.4.3.3. Simulated anisotropy factor vs. average current density (secondary model, low-doped Si)

As the average current density, the mean value between the initial current density (calculated for the area of the openings in the frontside masking layer) and the final current density 𝑗area (calculated for the increased total anodization area of a structure in the end of the process) is given. The legend is the same for all five plots, therefore it is shown only once. Anisotropy factor Af

0.1

j init = 1.0 A/cm2

𝐷open = 200 μm

j init = 1.5 A/cm2

0.0

j init = 2.0 A/cm2

−0.1

j init = 2.5 A/cm2

−0.2

j init = 3.5 A/cm2

j init = 3.0 A/cm2

−0.3 0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2 0.1

𝐷open = 400 μm

Anisotropy factor Af

Anisotropy factor Af

0.1 0.0 −0.1 −0.2 −0.3 −0.4 0.5

1.0

1.5

2.0

2.5

3.0

−0.1 −0.2 −0.3 −0.4 −0.5

3.5

𝐷open = 600 μm

0.0

0.5

Average current density, A/cm2

0.0 −0.1 −0.2 −0.3 −0.4 −0.5 0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

1.5

0.1

𝐷open = 800 μm

Anisotropy factor Af

Anisotropy factor Af

0.1

1.0

2.0

2.5

3.0

3.5

Average current density, A/cm2

3.5

𝐷open = 1000 μm

0.0 −0.1 −0.2 −0.3 −0.4 −0.5

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

276

Appendix

A.4.4. Secondary current distribution model for highly-doped p-type silicon in sec. 5.4.4 In this section, the data obtained from the secondary current distribution model with single circular opening in a frontside insulating masking layer and highly-doped p-type silicon (s. sec. 5.4.4) are summarized.

Curvature, mm−1

A.4.4.1. Simulated curvature vs. structure depth (secondary model, highly-doped Si) j init = 0.05 A/cm2

3

j init = 2.0 A/cm2

j init = 0.2 A/cm2

j init = 2.5 A/cm2

j init = 0.5 A/cm2

j init = 3.0 A/cm2

2

j init = 3.5 A/cm2

j init = 0.1 A/cm 2 1

j init = 1.0 A/cm

0 −1

j init = 1.5 A/cm2

2

𝐷open = 200 μm 0

100

200

300

400

500

1.5

Curvature, mm−1

Curvature, mm−1

Structure depth, µm

1.0 0.5 0.0

𝐷open = 400 μm

−0.5 0

100

200

300

400

500

1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6

𝐷open = 600 μm 0

Structure depth, µm

100

200

300

400

500

Structure depth, µm

0.6

Curvature, mm−1

Curvature, mm−1

0.8

𝐷open = 800 μm

0.4 0.2 0.0 −0.2 −0.4 0

100

200

300

400

Structure depth, µm

500

0.4

𝐷open = 1000 μm

0.2 0.0 −0.2 −0.4

0

100

200

300

400

Structure depth, µm

500

277

Appendix

A.4.4.2. Simulated anisotropy factor vs. structure depth (secondary model, highly-doped Si) The legend is the same for all five plots, therefore it is shown only once. j init = 0.05 A/cm2

Anisotropy factor Af

0.30

j init = 0.1 A/cm2

0.25

j init = 0.2 A/cm2

0.20

j init = 0.5 A/cm2

0.15

j init = 1.0 A/cm2 j init = 1.5 A/cm2

0.10

j init = 2.0 A/cm2

0.05

j init = 2.5 A/cm2

𝐷open = 200 μm

0.00 0

100

200

300

400

j init = 3.0 A/cm2 j init = 3.5 A/cm2

500

0.35

0.35

0.30

0.30

Anisotropy factor Af

Anisotropy factor Af

Structure depth, µm

0.25 0.20 0.15 0.10 0.05

𝐷open = 400 μm

0.00

0.25 0.20 0.15 0.10

𝐷open = 600 μm

0.05 0.00

0

100

200

300

400

500

0

Structure depth, µm

200

300

400

500

Structure depth, µm

0.35

0.35

Anisotropy factor Af

Anisotropy factor Af

100

0.30 0.25 0.20 0.15 0.10

𝐷open = 800 μm

0.05

0.30 0.25 0.20 0.15 0.10

𝐷open = 1000 μm

0.05 0.00

0.00 0

100

200

300

400

Structure depth, µm

500

0

100

200

300

400

Structure depth, µm

500

278

Appendix

A.4.4.3. Simulated anisotropy factor vs. average current density (secondary model, highly-doped Si)

Anisotropy factor Af

As the average current density, the mean value between the initial current density (calculated for the area of the openings in the frontside masking layer) and the final current density 𝑗area (calculated for the increased total anodization area of a structure in the end of the process) is given. 0.30

j init = 0.05 A/cm2

𝐷open = 200 μm

j init = 0.1 A/cm2

0.25

j init = 0.2 A/cm2

0.20

j init = 0.5 A/cm2

0.15

j init = 1.0 A/cm2

0.10

j init = 1.5 A/cm2

0.05

j init = 2.0 A/cm2

0.00

j init = 2.5 A/cm2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

j init = 3.0 A/cm2

2

j init = 3.5 A/cm2

Average current density, A/cm

𝐷open = 600 μm Anisotropy factor Af

Anisotropy factor Af

𝐷open = 400 μm 0.3

0.2

0.1

0.0

0.3 0.2 0.1 0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0

Average current density, A/cm2

𝐷open = 800 μm

1.0

1.5

2.0

2.5

3.0

3.5

0.35 𝐷open = 1000 μm

Anisotropy factor Af

Anisotropy factor Af

0.35

0.5

Average current density, A/cm2

0.30 0.25 0.20 0.15 0.10 0.05

0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Average current density, A/cm2

3.5

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