Analysis of Piezoelectric Semiconductor Structures [1st ed.] 9783030482053, 9783030482060

Table of contents :
Front Matter ....Pages i-viii
Macroscopic Theory (Jiashi Yang)....Pages 1-11
Exact Solutions (Jiashi Yang)....Pages 13-29
Extension of Rods (Jiashi Yang)....Pages 31-88
Bending of Beams (Jiashi Yang)....Pages 89-111
Extension and Bending of Plates (Jiashi Yang)....Pages 113-140
Composite Structures (Jiashi Yang)....Pages 141-176
Thermal Effects (Jiashi Yang)....Pages 177-211
Back Matter ....Pages 213-230

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Jiashi Yang

Analysis of Piezoelectric Semiconductor Structures

Analysis of Piezoelectric Semiconductor Structures

Jiashi Yang

Analysis of Piezoelectric Semiconductor Structures

Jiashi Yang Lincoln, NE, USA

ISBN 978-3-030-48205-3 ISBN 978-3-030-48206-0 https://doi.org/10.1007/978-3-030-48206-0

(eBook)

© Springer Nature Switzerland AG 2020 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Piezoelectric materials may be dielectrics or semiconductors. In piezoelectric semiconductors, the distribution or motion of charge carriers is affected by mechanical loads through the electric field produced by piezoelectric coupling. Since the 1960s, there have been efforts on using piezoelectric semiconductors to make acoustic wave devices where the interaction between mechanical waves and electric currents is called the acoustoelectric effect. Relatively recently, various piezoelectric semiconductor structures have been synthesized such as ZnO fibers, tubes, belts, spirals, and films. They have been used to make energy harvesters for converting mechanical energy into electrical energy, transistors, and various physical as well as chemical sensors. Piezoelectric semiconductors are also used in nanostructures such as quantum wells, dots, and wires. The study of piezoelectric semiconductor materials and devices is growing rapidly. The recent studies on the topic have formed new research areas called piezotronics and piezo-phototronics. This book is on theoretical analysis of piezoelectric semiconductor structures or devices using the phenomenological or macroscopic theory. Although most semiconductor books follow a combined microscopic and macroscopic approach, there exist purely macroscopic semiconductor models with various levels of sophistication. This book employs the simplest drift-diffusion model for semiconduction, which is electrically coupled to the macroscopic theory of piezoelectricity. This approach is relatively simple physically. Mathematically, it involves coupled differential equations, ordinary or partial. Of course the phenomenological theory employed has its limitations, but it can provide some basic understanding of certain behaviors of piezoelectric semiconductors. In addition to electromechanical couplings, thermal effects such as thermoelastic and pyroelectric couplings are also discussed. Some equations are repeated in certain sections so that they can be read independently. Chapter 1 is a brief summary of the general three-dimensional phenomenological theory of piezoelectric semiconductors. A few exact solutions of the threedimensional equations are presented in Chap. 2. Then the three-dimensional theory is reduced to one- and two-dimensional models for the extension of thin rods, v

vi

Preface

bending of thin beams, and extension and bending of thin plates in Chaps. 3, 4, and 5, respectively, with solutions of static and dynamic problems. Chapter 6 is devoted to composite structures of piezoelectric dielectrics and nonpiezoelectric semiconductors in which the acoustoelectric effect is a consequence of the polarization and conduction in component phases. Chapters 7 specializes in thermal effects. The literature on piezoelectric semiconductor materials and devices are abundant, with quite a few review articles available. This book is limited to theoretical results based on purely macroscopic approach. In most cases only those papers whose results are directly used in the book are listed as references. Because of the nonlinearity associated with the drift current which is the product of the electric field and carrier concentration, theoretical analysis of piezoelectric semiconductors presents considerable mathematical challenges. Most of the sections of this book are based on the linearized theory for small loads or signals. They can provide some basic understanding but are often insufficient for device design. For nonlinear analysis, numerical methods are more effective. A few numerical analyses using COMSOL are also presented in this book. As a book on piezoelectric semiconductors, the focus is on mechanically induced carrier redistribution or motion. However, some topics treated such as the effects of doping and PN junctions do not rely on electromechanical coupling and are also relevant in nonpiezoelectric semiconductors.

Lincoln, NE, USA December, 2019

Jiashi Yang

Contents

1

Macroscopic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 One-Dimensional Problems of Crystals of Class (6mm) . . . . . . . 1.4 Antiplane Problems of Crystals of Class (6mm) . . . . . . . . . . . . 1.5 Thermal Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Magnetic Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

1 1 3 4 7 9 11 11

2

Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Uniform Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Abrupt PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Circular PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Antiplane Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Antiplane Waves in a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

13 13 14 18 24 27 29

3

Extension of Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 One-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Linearly Graded PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Smoothly Graded PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Periodic Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Linear Extension by End Forces . . . . . . . . . . . . . . . . . . . . . . . 3.7 Electrically Nonlinear Extension by End Forces . . . . . . . . . . . . 3.8 Local Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Local Extension and Compression . . . . . . . . . . . . . . . . . . . . . . 3.10 Periodic Extension and Compression . . . . . . . . . . . . . . . . . . . . 3.11 Harmonic Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Transient Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

31 31 35 36 40 44 45 48 54 58 59 61 67 72 vii

viii

Contents

3.14 Extension of a PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Small Fields Superposed on a Finite Bias . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 85 87

4

Bending of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1 One-Dimensional Equations for Bending . . . . . . . . . . . . . . . . . . 89 4.2 Reduction to Bending Without Shear Deformation . . . . . . . . . . . 93 4.3 Static Bending of a Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Time-Harmonic Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5 Transient Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.6 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5

Extension and Bending of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Recapitulation of Three-Dimensional Equations . . . . . . . . . . . . 5.2 Hierarchy of Two-Dimensional Equations . . . . . . . . . . . . . . . . 5.3 Zero- and First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Equations for Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Equations for Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Reduction to Bending Without Shear Deformation . . . . . . . . . . 5.7 Equations for Crystals of Class (6mm) . . . . . . . . . . . . . . . . . . . 5.8 Thickness-Shear Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Thickness-Shear Approximation . . . . . . . . . . . . . . . . . . . . . . . 5.10 Propagation of Thickness-Shear Waves . . . . . . . . . . . . . . . . . . 5.11 Equations for Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

113 113 114 117 119 121 124 125 128 129 131 132 140

6

Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Extension of Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Bending of Beams with e33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Bending of Beams with e15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Bending of a PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

141 141 147 155 161 168 175

7

Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Extension of Homogeneous Rods . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Effects of a Local Temperature Change . . . . . . . . . . . . . . . . . . . 7.3 Temperature Effects on PN Junctions . . . . . . . . . . . . . . . . . . . . . 7.4 Extension of Composite Rods . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Extension and Bending of Composite Beams . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 182 188 196 203 211

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Chapter 1

Macroscopic Theory

This chapter presents a concise summary of the basic equations of the phenomenological theory of piezoelectric semiconductors. The Cartesian tensor notation is used, along with the summation convention for repeated tensor indices and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index. A superimposed dot represents a time derivative.

1.1

Basic Theory

Piezoelectric materials may be dielectrics or semiconductors. Mechanical fields and mobile charges in piezoelectric semiconductors can interact through electric fields, which is called the acoustoelectric effect. The basic behaviors of piezoelectric semiconductors can be described by coupling the linear theory of piezoelectricity [1–4] and the macroscopic theory of nonpiezoelectric semiconductors [5, 6]. The three-dimensional phenomenological theory consists of the equation of motion (Newton’s law), the charge equation of electrostatics (Gauss’s law), and the conservation of charge for holes and electrons (continuity equations): T ji,j þ f i ¼ ρ€ui ,
  Di,i ¼ q p  n þ N þ D  NA , 1 _ thermal R‐G þ pj _ other processes , p_ ¼  J pi,i þ pj q 1 _ thermal R‐G þ nj _ other processes , n_ ¼ J ni,i þ nj q

ð1:1Þ

where T is the stress tensor, ρ the mass density, f the body force which is usually zero, u the mechanical displacement vector, and D the electric displacement vector. © Springer Nature Switzerland AG 2020 J. Yang, Analysis of Piezoelectric Semiconductor Structures, https://doi.org/10.1007/978-3-030-48206-0_1

1

2

1 Macroscopic Theory

q ¼ 1.6  1019 coul is the elementary charge. p and n are the concentrations of þ holes and electrons. N  A and N D are the concentrations of ionized accepters and p n donors from doping. J i and J i are the hole and electron current densities. In the following, we will neglect those terms in Eq. (1.1)3,4 associated with thermal recombination and generation as well as other processes of electric, magnetic, mechanical, and thermal origins. Constitutive relations accompanying Eq. (1.1) describing material behaviors can be written in the following form: T ij ¼ cEijkl Skl  ekij E k , Di ¼ eikl Skl þ εSik Ek , J pi ¼ qpμpij E j  qDpij p,j ¼ qpμpij E j  qpDpij ð ln pÞ,j ,

ð1:2Þ

J ni ¼ qnμnij E j þ qDnij n,j ¼ qnμnij E j þ qnDnij ð ln nÞ,j , where S is the strain tensor, E the electric field vector, cEijkl the elastic stiffness, eijk the piezoelectric constants, and εSij the dielectric constants. μpij and μnij are the carrier mobility. Dpij and Dnij are the carrier diffusion constants. qpμpij E j and qnμnij E j are the drift currents. qDpij p,j and qDnij n,j are the diffusion currents. The piezoelectric constitutive relations in Eq. (1.2) can be written in other forms in terms of other material constants [1–4]. For example, Sij ¼ sEijkl T kl þ d kij E k , Di ¼ d ikl T kl þ εTik Ek ,

ð1:3Þ

where sEijkl are the elastic compliance, dijk the piezoelectric constants, and εTij the dielectric constants. The mobility and diffusion constants, e.g., μp33, μn33, Dp33, and Dn33, satisfy the Einstein relation: μp33 μn33 q ¼ ¼ , Dp33 Dn33 kB T

ð1:4Þ

where kB is the Boltzmann constant and T the absolute temperature. The strain S and the electric field E are related to the mechanical displacement u and the electric potential φ through
 Sij ¼ ui,j þ u j,i =2, E i ¼ φ,i :

ð1:5Þ

1.2 Linearization

3

With successive substitutions from Eqs. (1.2) and (1.5), we can write Eq. (1.1) as six equations for u, φ, p, and n. This book is based on the macroscopic theory in Eqs. (1.1), (1.2), and (1.5) without the use of any information from the microscopic theory of semiconductors except Eq. (1.4) which serves as a relationship between the mobility and the diffusion constants. The right-hand side of Eq. (1.1)2 is in terms of p-n. Therefore, working with p-n and p+n instead of p and n may have some þ þ  mathematical advantages. N  A N D appears as a load in Eq. (1.1)2. When N A N D and f are zero, Eq. (1.1) becomes homogeneous. Then the state with u, φ, p, and n all being zero satisfies Eq. (1.1), which may be used as the reference state. For a finite piezoelectric semiconductor body, the electric field in the surrounding free space is usually neglected as an approximation. Then, on the surface of a finite body, the mechanical displacement vector or the traction vector, the electric potential or the normal component of D, and the carrier concentrations or the normal components of the current density vectors may be prescribed as boundary conditions. Integrating Eq. (1.1)2 over a volume V with boundary surface S and unit outward normal n, we obtain Z

Z ni Di dS ¼ S

V


  q p  n þ Nþ D  N A dV,

ð1:6Þ

which is a relationship among the boundary normal value of D and doping as well as carrier concentrations. Similar relationships can also be derived from the stress equation of motion and the continuity equations, in particular their static form.

1.2

Linearization

Because of the nonlinearity in Eq. (1.2)3,4 associated with the drift currents of holes and electrons which are the products of the unknown carrier concentrations and the unknown electric field, theoretical analyses of piezoelectric semiconductor devices may present considerable mathematical challenges. For many purposes, a linearized version of Eq. (1.2)3,4 is sufficient. Consider the following change of unknown variables from p and n to Δp and Δn: p ¼ p0 þ Δp, n ¼ n0 þ Δn,

ð1:7Þ

where, for simplicity, we have denoted p0 ¼ N  A,

n0 ¼ N þ D:

ð1:8Þ

4

1 Macroscopic Theory

Equation (1.7) makes the right-hand side of Eq. (1.1)2 homogeneous, while at the same time makes Eq. (1.2)3,4 nonhomogeneous. Consider the case of small Δp and Δn. We linearize Eq. (1.2)3,4 as J pi ffi qp0 μpij E j  qDpij ðp0 þ ΔpÞ,j ,

ð1:9Þ

J ni ffi qn0 μnij E j þ qDnij ðn0 þ ΔnÞ,j :

When p0 and n0 are uniform, Eq. (1.9) becomes homogeneous in E, Δp, and Δn. With Δp and Δn, Eq. (1.1)2–4 become Di,i ¼ qðΔp  ΔnÞ, ∂ðΔpÞ ¼ J pi,i , q ∂t ∂ðΔnÞ q ¼ J ni,i : ∂t

ð1:10Þ

Equation (1.10)1 also becomes homogeneous. Phenomenological theories involving electromechanical couplings and mechanical nonlinearities due to large deformations in semiconductors can be found in [7, 8].

1.3

One-Dimensional Problems of Crystals of Class (6mm)

We are mainly interested in ZnO which belongs to the crystal class of (6mm). Under the compact matrix notation [1–4] with indices p and q ranging from 1 to 6, when the c-axis of the crystal is along x3, the material tensors for ZnO can be represented by the following matrices: 0

c11 Bc B 21 B   B c31 cpq ¼ B B 0 B B @ 0

c12 c11

c13 c13

0 0

0 0

c31 0

c33 0

0 c44

0 0

0

0

0

c44

0

0

0

0

0

0 0

0 0

0 e15

e15 0

e31

e33

0

0

0



eip



0 B ¼@ 0 e31

1 0 0 C C C 0 C C, 0 C C C 0 A c66 1 0 C 0 A, 0

1.3 One-Dimensional Problems of Crystals of Class (6mm)

0

ε11   B εij ¼ @ 0

0 ε11

0

0

1 0 C 0 A, ε33

5

ð1:11Þ

where c66¼(c11–c12)/2. μij and dij have the same structure as εij. The matrices in Eq. (1.11) are the same as those of transversely isotropic piezoelectric materials such as polarized ceramics when the poling axis is along x3. Equation (1.11) yields the following piezoelectric constitutive relations: T 11 ¼ T 1 ¼ c11 S1 þ c12 S2 þ c13 S3  e31 E 3 , T 22 ¼ T 2 ¼ c12 S1 þ c11 S2 þ c13 S3  e31 E 3 , T 33 ¼ T 3 ¼ c13 S1 þ c13 S2 þ c33 S3  e33 E 3 , T 23 ¼ T 4 ¼ c44 S4  e15 E 2 , T 31 ¼ T 5 ¼ c44 S5  e15 E 1 , T 12 ¼ T 6 ¼ c66 S6 , D1 ¼ e15 S5 þ ε11 E1 , D2 ¼ e15 S4 þ ε11 E2 , D3 ¼ e31 ðS1 þ S2 Þ þ e33 S3 þ ε33 E 3 :

ð1:12Þ

To see the basic acoustoelectric coupling in ZnO, consider one-dimensional longitudinal motions along the c-axis satisfying u1 ¼ u2 ¼ 0, ∂/∂x1 ¼ 0, and ∂/ ∂x2 ¼ 0. In this case, the nontrivial fields are u3 ¼ uðx3 , t Þ, φ ¼ φðx3 , t Þ, Δp ¼ Δpðx3 , t Þ, Δn ¼ Δnðx3 , t Þ:

ð1:13Þ

From Eq. (1.5), the relevant strain and electric field components are S33 ¼ u3,3 , E 3 ¼ φ,3 :

ð1:14Þ

The relevant components of Tij, Di, J pi , and J ni are T 33 ¼ c33 S3  e33 E 3 ¼ c33 u3,3 þ e33 φ,3 , D3 ¼ e33 S3 þ ε33 E3 ¼ e33 u3,3  ε33 φ,3 ,

ð1:15Þ

6

1 Macroscopic Theory

J p3 ¼ qp0 μp33 E3  qDp33 ðΔpÞ,3 ¼ qp0 μp33 φ,3  qDp33 ðΔpÞ,3 , J n3 ¼ qn0 μn33 E3 þ qDn33 ðΔnÞ,3

ð1:16Þ

¼ qn0 μn33 φ,3 þ qDn33 ðΔnÞ,3 : In Eq. (1.16), the linearized constitutive relations in Eq. (1.9) are used and uniform doping with constant p0 and n0 is assumed. With successive substitutions, the relevant ones of Eq. (1.1) take the following form: c33 u3,33 þ e33 φ,33 ¼ ρ€u3 , e33 u3,33  ε33 φ,33 ¼ qðΔp  ΔnÞ,

ð1:17Þ

∂ ðΔpÞ ¼ p0 μp33 φ,33 þ Dp33 ðΔpÞ,33 , ∂t ∂ ðΔnÞ ¼ n0 μn33 φ,33 þ Dn33 ðΔnÞ,33 : ∂t

ð1:18Þ

For static problems, Eqs. (1.17) and (1.18) reduce to c33 u3,33 þ e33 φ,33 ¼ 0, e33 u3,33  ε33 φ,33 ¼ qðΔp  ΔnÞ, 0 ¼ p0 μp33 φ,33 þ Dp33 ðΔpÞ,33 , 0 ¼ n0 μn33 φ,33 þ Dn33 ðΔnÞ,33 :

ð1:19Þ ð1:20Þ

Equation (1.19) can be written as e33 φ , c33 ,33 ¼ qðΔp  ΔnÞ,

u3,33 ¼  ε33 φ,33

ð1:21Þ

where ε33 ¼ ε33 þ

e233 : c33

ð1:22Þ

ε33 is a piezoelectrically modified dielectric constant. From Eq. (1.20), we can write p0 μp33 φ , Dp33 ,33 n0 μ n ¼  n33 φ,33 : D33

ðΔpÞ,33 ¼  ðΔnÞ,33

ð1:23Þ

1.4 Antiplane Problems of Crystals of Class (6mm)

7

Subtracting the two equations in Eq. (1.23) from each other, we have q ðΔp  ΔnÞ,33 , k2 ε33

ð1:24Þ

 p  p0 μ33 n0 μn33 q k ¼ þ n : D33 ε33 Dp33

ð1:25Þ

φ,33 ¼  where 2

Substituting Eq. (1.24) into Eq. (1.21)2, we obtain a single equation for Δp  Δn: ðΔp  ΔnÞ,33 ¼ k2 ðΔp  ΔnÞ:

ð1:26Þ

From Eq. (1.10)1, it can be seen that Δp  Δn is closely related to the charge from doping and carriers. We can solve Eq. (1.26) for Δp  Δn, then the potential from Eq. (1.24), and then the displacement and carrier concentrations from Eqs. (1.21)1 and (1.23).

1.4

Antiplane Problems of Crystals of Class (6mm)

Antiplane problems [4] are relatively simple mathematically. Consider a cross section of an infinite cylindrical body of ZnO with the c-axis along the x3 axis which is perpendicular to the cross section, and study motions with ∂/∂x3 ¼ 0. In this case the equations split into a plane-strain problem [4] for u1 and u2, which is not coupled to the electrical field and therefore is not of interest here, and an antiplane problem for the following fields: u3 ¼ uðx1 , x2 , t Þ, φ ¼ φðx1 , x2 , t Þ, Δp ¼ Δpðx1 , x2 , t Þ, Δn ¼ Δnðx1 , x2 , t Þ:

ð1:27Þ

For antiplane problems the relevant strain and electric field components are 

S5 S4



 ¼

2S31 2S32



 ¼ ∇u,

E1 E2

¼ ∇φ,

ð1:28Þ

8

1 Macroscopic Theory

where ∇ is the two-dimensional gradient operator. The relevant components of Tij, Di, J pi , and J ni are 

T 31 ¼ ¼ c∇u þ e∇φ, T 32  D1 ¼ e∇u  ε∇φ, D2 ( p) J1 ¼ qp0 μp ∇φ  qDp ∇ðΔpÞ, J p2 ( n) J1 ¼ qn0 μn ∇φ þ qDn ∇ðΔnÞ, J n2 T5 T4





ð1:29Þ ð1:30Þ

ð1:31Þ

where we have denoted c ¼ c44, e ¼ e15, ε ¼ ε11, μp ¼ μp11, μn ¼ μn11, Dp ¼ Dp11, and Dn ¼ Dn11 . The linearized constitutive relations in Eq. (1.9) are used and uniform doping with constant p0 and n0 is assumed. With successive substitutions, the relevant ones of Eq. (1.1) take the following form: c∇2 u þ e∇2 φ ¼ ρ€u, e∇2 u  ε∇2 φ ¼ qðΔp  ΔnÞ, ∂ ðΔpÞ ¼ p0 μp ∇2 φ þ Dp ∇2 ðΔpÞ, ∂t ∂ ðΔnÞ ¼ n0 μn ∇2 φ þ Dn ∇2 ðΔnÞ, ∂t 2

ð1:32Þ

ð1:33Þ

2

where ∇2 ¼ ∂ =∂x21 þ ∂ =∂x22 is the two-dimensional Laplacian. For static problems, Eqs. (1.32) and (1.33) reduce to c∇2 u þ e∇2 φ ¼ 0, e∇2 u  ε∇2 φ ¼ qðΔp  ΔnÞ, 0 ¼ p0 μp ∇2 φ þ Dp ∇2 ðΔpÞ, 0 ¼ n0 μn ∇2 φ þ Dn ∇2 ðΔnÞ:

ð1:34Þ ð1:35Þ

Equations (1.34) and (1.35) can be decoupled as follows. From Eq. (1.34)1, e ∇2 u ¼  ∇2 φ: c Substituting Eq. (1.36) into Eq. (1.34)2, we obtain

ð1:36Þ

1.5 Thermal Couplings

9

ε∇2 φ ¼ qðΔp  ΔnÞ,

ð1:37Þ

where ε¼εþ

e2 : c

ð1:38Þ

We rewrite Eq. (1.35) as p0 μ p 2 ∇ φ, Dp n nμ ∇2 ðΔnÞ ¼ 0 n ∇2 φ: D

∇2 ðΔpÞ ¼ 

ð1:39Þ

Subtracting the two equations in Eq. (1.39) from each other, we have ∇2 φ ¼ 

q 2 ∇ ðΔp  ΔnÞ, εk2

ð1:40Þ

where k2 ¼

 p  p0 μ n0 μ n q : þ Dp Dn ε

ð1:41Þ

The substitution of Eq. (1.40) into Eq. (1.37) yields a single equation for Δp  Δn: ∇2 ðΔp  ΔnÞ ¼ k 2 ðΔp  ΔnÞ:

ð1:42Þ

Once Δp  Δn is obtained from Eq. (1.42), φ, u, Δp, and Δn can be determined from Eqs. (1.40), (1.36), and (1.39).

1.5

Thermal Couplings

Treatments of thermal effects in nonpiezoelectric semiconductors can be found in [9]. For a thermopiezoelectric semiconductor, the three-dimensional phenomenological theory couples the linear theory of thermopiezoelectricity [4, 10, 11] and the macroscopic theory of semiconductors. It consists of the following equation of

10

1 Macroscopic Theory

motion, the charge equation of electrostatics, the heat equation, and the conservation of charge for holes and electrons: T ji,j ¼ ρ€ui ,
  Di,i ¼ q p  n þ N þ D  NA , Θ0 η_ ¼ hi,i þ γ, qp_ ¼ J pi,i ,

ð1:43Þ

qn_ ¼ J ni,i , where h is the heat flux vector, γ the body heat source, Θ0 a reference temperature, and η the entropy density. γ includes body heat from various origins and may have a nonlinear expression such as the Joule or Ohmic heating due to conduction. Constitutive relations accompanying Eq. (1.43) can be written in the following form: T ij ¼ cijkl Skl  ekij Ek  λij θ, Di ¼ eikl Skl þ εik E k þ pi θ, hi ¼ κ ij θ,j þ κEij E j , η ¼ λkl Skl þ pk E k þ αθ, J pi ¼ qpμpij E j  qDpij p,j  qpDTp ij θ ,j

ð1:44Þ

¼ qpμpij E j  qDpij p,j  σ Tp ij θ ,j , J ni ¼ qnμnij E j þ qDnij n,j þ qnDTn ij θ ,j ¼ qnμnij E j þ qDnij n,j þ σ Tn ij θ ,j , where |θ| < < Θ0 is a small temperature change from Θ0. λij are the thermoelastic constants, pj the pyroelectric constants, κij the heat conduction coefficients, DTp ij and Tp E Tn the thermal diffusion constants [9], κ , σ , and σ the thermoelectric constants DTn ij ij ij ij p Tn n and D may be related to D and D through [9]: [12]. DTp ij ij ij ij DTp ffi

Dp , 2T

DTn ffi

Dn : 2T

ð1:45Þ

The heat flux and the currents in Eq. (1.44) are further restricted by the ClausiusDuhem inequality [12]. Similar to Eq. (1.3), the constitutive relations for thermopiezoelectric materials can also be written in other forms. In addition, there are other nonlinear forms of thermoelectric constitutive relations which can be found in [13].

References

1.6

11

Magnetic Couplings

When piezomagnetic effects are present and need to be considered, the piezoelectric constitutive relations in Eq. (1.2)1,2 can be generalized to [12] T ij ¼ cijkl Skl  ekij E k  hkij H k , Di ¼ eikl Skl þ εik Ek þ αik H k , Bi ¼ hikl Skl þ αik E k þ μik H k ,

ð1:46Þ

where B is the magnetic flux or induction, H the magnetic field, hijk the piezomagnetic constants, αij the magnetoelectric constants, and μij the magnetic permeability. B satisfies ∇  B ¼ 0. Under the well-known quasistatic approximation, ∇  H ¼ 0. Then a magnetic potential ψ can be introduced through H ¼  ∇ ψ. With the presence of the electric field, magnetic field, temperature gradient, electric conduction, and heat conduction, there are many cross effects such the thermomagnetic, Hall, and Ettingshausen effects [12]. Most of them are beyond the scope of the present book.

References 1. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969) 2. A.H. Meitzler, D. Berlincourt, F.S. Welsh III, H.F. Tiersten, G.A. Coquin, A.W. Warner, IEEE Standard on Piezoelectricity (IEEE, New York, 1988) 3. B.A. Auld, Acoustic Fields and Waves in Solids, vol 1 (Wiley, New York, 1973) 4. J.S. Yang, An Introduction to the Theory of Piezoelectricity, 2nd edn. (Springer, New York, 2018) 5. R.F. Pierret, Semiconductor Device Fundamentals (Pearson, Uttar Pradesh, 1996) 6. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981) 7. H.G. de Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat conducting deformable semiconductors. J. Math. Phys. 16, 938–957 (1975) 8. G.A. Maugin, N. Daher, Phenomenological theory of elastic semiconductors. Int. J. Eng. Sci. 24, 703–731 (1986) 9. S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, New York, 1984) 10. R.D. Mindlin, On the equations of motion of piezoelectric crystals, in Problems of Continuum Mechanics, ed. by J.R.M. Radok, (Society for Industrial and Applied Mathematics, Philadelphia, 1961), pp. 282–290 11. R.D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int. J. Solids Struct. 10, 625–637 (1974) 12. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, vol I (Springer, New York, 1990) 13. D.M. Rowe, CRC Handbook of Thermoelectrics (CRC Press, Boca Raton, 2016)

Chapter 2

Exact Solutions

This chapter gathers a few exact solutions satisfying the three-dimensional equations of piezoelectric semiconductors. Solutions like these are possible only in a few relatively special cases. Most of them are based on the linearized theory. Section 2.1 is simple but nontrivial. Section 2.2 is a one-dimensional problem mathematically in the sense that there is only one spatial variable. Sections 2.3, 2.4, and 2.5 are antiplane problems. Section 2.3 is axisymmetric and one-dimensional. The other two are two-dimensional.

2.1

Uniform Doping

þ N
A and N D in Eq. (1.1)2 may depend on spatial variables representing nonuniform þ doping. In the special case of uniform doping, N
A and N D are constants. In this case, it can be verified that

p ¼ N
A,

n ¼ Nþ D,

E ¼ 0, D ¼ 0, Jp ¼ 0, Jn ¼ 0, S ¼ 0,

ð2:1Þ

T ¼ 0,

satisfy the static form of Eqs. (1.1) and (1.2) when there is no body force, recombination, and generation. Hence, for a finite body without surface traction, charge, and currents, Eq. (2.1) is the solution of the problem. From Eq. (1.5), the displacement field can still have an undetermined rigid-body displacement. Similarly, the electric potential can still have an arbitrary constant.

© Springer Nature Switzerland AG 2020 J. Yang, Analysis of Piezoelectric Semiconductor Structures, https://doi.org/10.1007/978-3-030-48206-0_2

13

14

2.2

2 Exact Solutions

Abrupt PN Junction

Consider the interface between two piezoelectric half spaces of ZnO in Fig. 2.1 [1]. The two half spaces are doped into a p region where p > n, and an n region where n > p. Then diffusion occurs at the interface to form a PN junction. Since the doping is uniform separately in the p and n regions, respectively, it has a jump discontinuity at the interface and the corresponding PN junction is called an abrupt junction [2]. Mathematically, we need to treat the two half spaces separately and then apply boundary conditions at infinity as well as continuity conditions at the interface. The problem is one-dimensional without dependence on x1 and x2. The governing equations are taken from Eqs. (1.26), (1.24), (1.23), and (1.21)1: ðΔp
ΔnÞ,33 ¼ k2 ðΔp
ΔnÞ, φ,33 ¼


ð2:2Þ

q ðΔp
ΔnÞ,33 , k ε33

ð2:3Þ

2

p0 μp33 φ , Dp33 ,33 n0 μ n ¼ n33 φ,33 , D33

ðΔpÞ,33 ¼
ðΔnÞ,33

u3,33 ¼


ð2:4Þ

e33 φ , c33 ,33

ð2:5Þ

where k2 ¼


p  p0 μ33 n0 μn33 q e2 þ , ε33 ¼ ε33 þ 33 : p n D33 ε33 c33 D33

ð2:6Þ

At infinity we have the following boundary conditions: T 33 ð1Þ ¼ 0, J p3 ð1Þ

Fig. 2.1 An interface between two piezoelectric semiconductor half spaces

¼ 0,

D3 ð1Þ ¼ 0,

ð2:7Þ

J n3 ð1Þ ¼ 0:

Interface

p-doped

n-doped x3

2.2 Abrupt PN Junction

15

At the interface between the p- and n-doped regions, we have the following continuity conditions: u3 ð0
Þ ¼ u3 ð0þ Þ, φð0
Þ ¼ φð0þ Þ,

T 33 ð0
Þ ¼ T 33 ð0þ Þ, D3 ð0
Þ ¼ D3 ð0þ Þ,

nð0
Þ ¼ nð0þ Þ,

pð0
Þ ¼ pð0þ Þ,

J n3 ð0
Þ ¼ J n3 ð0þ Þ,

J p3 ð0
Þ ¼ J p3 ð0þ Þ:

ð2:8Þ ð2:9Þ

The above equations and boundary/continuity conditions are invariant under a rigidbody translation in the x3 direction and a shift of the electric potential through a constant. To fix the rigid-body displacement and the arbitrary constant in the electric potential so that the displacement and potential fields are unique, we choose the interface as a reference for the displacement and potential and impose u3 ð0Þ ¼ 0,

φð0Þ ¼ 0:

ð2:10Þ

We have the following additional conditions representing the global conservation of holes and electrons: Z

0


1 Z 0
1

Z

1

Δpdx3 þ Z

Δpdx3 ¼ 0, ð2:11Þ

0 1

Δndx3 þ

Δndx3 ¼ 0:

0

We use a prime for the material parameters in the p-doped left half space and a double prime for those in the n-doped region on the right. Equations (2.2), (2.3), (2.4), and (2.5) are linear ordinary differential equations with constant coefficients. Their general solution can be obtained in a straightforward and systematic manner. For x3 < 0, Δp
Δn ¼ C 1 exp ðk0 x3 Þ, q φ ¼
0 2 C1 exp ðk 0 x3 Þ þ C 2 x3 þ C 3 , ðk Þ ε033 " # e033 q 0 u3 ¼
0
0 2 C 1 exp ðk x3 Þ þ C2 x3 þ C3 þ C4 x3 þ C5 , c33 ðk Þ ε033 " # n00 μ0 n33 q Δn ¼ 0 n
0 2 C1 exp ðk0 x3 Þ þ C2 x3 þ C 3 þ C6 x3 þ C 7 , D 33 ðk Þ ε033 where C1 through C7 are undetermined constants. Similarly, for x3 > 0,

ð2:12Þ ð2:13Þ ð2:14Þ ð2:15Þ

16

2 Exact Solutions

Fig. 2.2 Doping profile

Interface Electrons

p0

Holes

n0

n0

p0

x

3

Δp
Δn ¼ C8 exp ð
k00 x3 Þ, q φ ¼
00 2 C 8 exp ð
k 00 x3 Þ þ C9 x3 þ C10 , ðk Þ ε0033 " # e0033 q 00 u3 ¼
00
00 2 C 8 exp ð
k x3 Þ þ C 9 x3 þ C10 þ C 11 x3 þ C12 , c33 ðk Þ ε0033 " # n000 μ00 n33 q 00 Δn ¼ 00 n
00 2 C 8 exp ð
k x3 Þ þ C9 x3 þ C 10 þ C 13 x3 þ C 14 : D 33 ðk Þ ε0033

ð2:16Þ ð2:17Þ ð2:18Þ ð2:19Þ

C1 through C14 are determined by Eqs. (2.7), (2.8), (2.9), (2.10), and (2.11). Some of them can be determined in a simple manner. At the end eight constants are determined by solving eight linear algebraic equations on a computer using MATLAB. As a numerical example consider the doping profile shown in Fig. 2.2 described by step functions. Specifically, we limit ourselves to the following doping profile with some symmetry or antisymmetry: p00 ¼ 10  1020 m
3 , p000 ¼ 7  1020 m
3 ¼ n00 , n00 ¼ 7  1020 m
3 , n000 ¼ 10  1020 m
3 ¼ p00 :

ð2:20Þ

The jumps of the step functions are not large so that the assumption of the linear 00 theory is not violated. When Eq. (2.20) is true, k0 ¼ k ¼ k. It can be seen from the above expressions that k is an important combination of parameters in piezoelectric semiconductors. For the parameters in Eq. (2.20), let
k20 ¼

 p0 μp33 n0 μn33 q q q þ ¼ ð p0 þ n0 Þ , k B T ε33 Dn33 ε33 Dp33

ð2:21Þ

which will be used as a unit for k. Figure 2.3 shows the concentrations of holes and electrons which were initially determined by the step functions in Fig. 2.2 but are now continuous because of diffusion. They are smooth functions after diffusion although they still change rapidly near the interface. When k varies, effectively, p0 + n0 varies when other parameters are fixed. From the expressions of the fields in the above, we expect that

2.2 Abrupt PN Junction

17

Fig. 2.3 Carrier concentrations near the interface. (a) p(x3) for holes. (b) n(x3) for electrons

when k increases, the fields vary more rapidly near the interface. This can be seen clearly in the figure. Figure 2.4 shows effectively the distribution of the net doping and mobile charge concentration near the interface, i.e., the right-hand side of Eq. (1.10)1 divided by q. This does not include the effective polarization charge. The distribution in the figure localizes near the interface and has a sign change there, showing the formation of a PN junction. These charges are often approximated by locally piecewise uniform charge distributions under the depletion layer approximation in common

18

2 Exact Solutions

Fig. 2.4 Δp
Δn near the interface

semiconductor textbooks. They can also be approximated by two oppositely charged point charges at some distance apart, forming a dipole or double layer. Figure 2.5 shows the electric field and potential near the interface, i.e., the so-called built-in electric field and potential in a PN junction. These are familiar results in semiconductor textbooks. Since the material is piezoelectric, the electric field causes mechanical deformation. The strain field associated with the PN junction is shown in Fig. 2.6 which localizes near the interface. In the present analysis, T33 ¼ 0. A small T33 can be included to study its effects on the PN junction without mathematical difficulty. However, the above linear analysis cannot describe the typical nonlinear currentvoltage relations or I-V curve of a PN junction. They will be treated by numerical methods later. Interfaces between metal conductors and semiconductors (MS junctions) are also common and are fundamentally important in devices. The analysis of an MS junction is similar to the above and is simpler in the sense that on the metal side there is a concentrated point charge described by a delta function [2] instead of a distribution.

2.3

Circular PN Junction

Figure 2.7 shows the cross section of a circular piezoelectric semiconductor cylinder with radius a embedded in another unbounded piezoelectric semiconductor material. The two materials are doped oppositely and form a PN junction at their interface [3].

2.3 Circular PN Junction

19

Fig. 2.5 (a) Electric field. (b) Electric Potential

Consider ZnO with the c-axis along x3. For antiplane problems, the governing equations are from Eqs. (1.42), (1.37), (1.36), and (1.39): ∇2 ðΔp
ΔnÞ ¼ k2 ðΔp
ΔnÞ,

ð2:22Þ


ε∇ φ ¼ qðΔp
ΔnÞ

ð2:23Þ

2

20

2 Exact Solutions

Fig. 2.6 Axial strain S3(x3)

Fig. 2.7 Two piezoelectric semiconductors and their circular interface

x2

a

x1 p-doped n-doped

e ∇2 u ¼
∇2 φ, c p μp ∇2 ðΔpÞ ¼
0 p ∇2 φ, D n0 μ n 2 2 ∇ ðΔnÞ ¼ n ∇ φ, D

ð2:24Þ ð2:25Þ

where
k ¼ 2

 p0 μ p n0 μ n q , þ n Dp D ε

ð2:26Þ

2.3 Circular PN Junction

21

ε¼εþ

e2 : c

ð2:27Þ

In polar coordinates, 2

∇2 ¼

2

∂ 1 ∂ 1 ∂ þ þ : ∂r 2 r ∂r r 2 ∂θ2

ð2:28Þ

At the interface between the p- and n-doped regions, we have the following continuity conditions: uða
Þ ¼ uðaþ Þ,

T 3r ða
Þ ¼ T 3r ðaþ Þ,

φða
Þ ¼ φðaþ Þ,

Dr ða
Þ ¼ Dr ðaþ Þ,

nða
Þ ¼ nðaþ Þ, pða
Þ ¼ pðaþ Þ, J nr ða
Þ ¼ J nr ðaþ Þ, J pr ða
Þ ¼ J pr ðaþ Þ:

ð2:29Þ ð2:30Þ

The above equations and boundary/continuity conditions are invariant under a rigidbody displacement in the x3 direction and a shift of the electric potential through a constant. To fix the rigid-body displacement and the arbitrary constant in the electric potential so that the displacement and potential fields are unique, we choose the interface as a reference and impose uðaÞ ¼ 0,

φðaÞ ¼ 0:

ð2:31Þ

We also have the following additional conditions representing the global conservation of holes and electrons: Z

a

Z0 a 0

Z

1

Δpdr þ Δndr þ

Za 1

Δpdr ¼ 0, ð2:32Þ Δndr ¼ 0:

a

In addition, the fields should be finite at r ¼ 0. There are restrictions on the fields at infinity too. We use a prime for the material parameters in the p-doped central region and a double prime for the material parameters in the n-doped outer region. For the central region, the solution finite at the origin is given by Δp
Δn ¼ C 1 I 0 ðk 0 r Þ,

ð2:33Þ

22

2 Exact Solutions

q C 1 I 0 ðk 0 r Þ þ C2 , ðk Þ ε " # e0 q 0 u ¼
0
0 2 C 1 I 0 ðk r Þ þ C 2 þ C 3 , c ðk Þ ε0 " # n00 μ0 n q 0 Δn ¼ 0 n
0 2 C 1 I 0 ðk r Þ þ C2 þ C 4 , D ð k Þ ε0 φ¼


0 2 0

ð2:34Þ ð2:35Þ ð2:36Þ

where I0 is the zero-order modified Bessel function of the first kind. C1 through C4 are undetermined constants. Similarly, for r > a, we have Δp
Δn ¼ C 5 K 0 ðk00 r Þ, q φ ¼
 2 C 5 K 0 ðk00 r Þ þ C 6 ln r þ C7 , k 00R ε00 " # e00 q 00 u ¼
00
 2 C 5 K 0 ðk r Þ þ C 6 ln r þ C7 þ C 8 ln r þ C9 , c k 00R ε00 " # n000 μ00 n q 00 Δn ¼ 00 n
 2 C5 K 0 ðk r Þ þ C6 ln r þ C 7 þ C10 ln r þ C11 , D k00 ε00

ð2:37Þ ð2:38Þ

ð2:39Þ

ð2:40Þ

R

where K0 is the zero-order modified Bessel function of the second kind. It vanishes at infinity. The following properties of I0 and K0 are needed in the numerical calculation: dI 0 ðξÞ ¼ I 1 ðξÞ, dξ

dK 0 ðξÞ ¼
K 1 ðξÞ: dξ

ð2:41Þ

C1 through C11 are determined by Eqs. (2.29), (2.30), (2.31), and (2.32). Some of them can be determined in a simple manner. At the end eight constants are determined by solving eight linear algebraic equations on a computer using MATLAB. As a numerical example, consider the doping profile shown in Fig. 2.8 with p00 ¼ 10  1020 m
3 , p000 ¼ 7  1020 m
3 ¼ n00 ,

n00 ¼ 7  1020 m
3 , n000 ¼ 10  1020 m
3 ¼ p00 :

ð2:42Þ

00

Under Eq. (2.42), we have k0 ¼ k ¼ k. We introduce k20 ¼


p  p0 μ33 n0 μn33 q q q þ ¼ ðp0 þ n0 Þ : k B T ε33 Dn33 ε33 Dp33

ð2:43Þ

2.3 Circular PN Junction

23

Fig. 2.8 Doping profile

Interface Electrons

p0′

Holes

n0′′

n0′ p0′′ O

a

r

Fig. 2.9 Effects of p00
n00 . (a) Δp
Δn showing the formation of a PN junction. (b) Built-in electric field Er. (c) Built-in electric potential φ(r)

Figure 2.9 shows the effects of p00
n00 , while k ¼ k0 is kept constant. When
n00 increases, the fields in the figure become stronger and vary more rapidly as expected. p00

24

2.4

2 Exact Solutions

Antiplane Crack

Consider a semi-infinite crack as shown in Fig. 2.10 [4, 5]. References [4, 5] are the same. One is a minor revision of the other with some change of the title. They were both published because Professor M. Kachanov, Editor-in-Chief of International Journal of Fracture at the time, forwarded both versions to the publisher separately by mistake. He also said that he had made the same mistake before when contacted by the author. We limit ourselves to the case of antiplane problems of crystals of class (6mm). The material is doped into a p-type semiconductor. The crack surfaces are traction free and unelectroded. The electric field inside the crack is neglected. There are no free charges and normal current on the crack faces. The governing equations are from Eqs. (1.42), (1.37), and (1.36) with n0 ¼ 0 and Δn ¼ 0: ∇2 ðΔpÞ ¼ k2 ðΔpÞ,

ð2:44Þ


ε∇2 φ ¼ qðΔpÞ, e ∇2 u ¼
∇2 φ, c

ð2:45Þ ð2:46Þ

where k2 ¼

p0 μ p q e2 , ε¼εþ : p D ε c

ð2:47Þ

In polar coordinates, Eq. (2.44) takes the following form:


2 2  ∂ ∂ 1 ∂ þ þ ðΔpÞ
k2 ðΔpÞ ¼ 0: ∂r 2 r∂r r 2 ∂θ2

ð2:48Þ

We look for Δpðr, θÞ ¼ Δpðr Þ sin

θ : 2

ð2:49Þ

Substituting Eq. (2.49) into Eq. (2.48), we obtain

Fig. 2.10 A semi-infinite crack

x2

r



x1

2.4 Antiplane Crack

25

" # ð1=2Þ2 d dðΔpÞ 1 dðΔpÞ ðΔpÞ ¼ 0: þ
1þ dðkr Þ dðkr Þ kr dðkr Þ ðkr Þ2

ð2:50Þ

Δpðr Þ ¼ C 1 I 1=2 ðkr Þ þ C 2 K 1=2 ðkr Þ ¼ C2 K 1=2 ðkr Þ,

ð2:51Þ

Its solution is

where I1/2 and K1/2 are the first- and second-kind modified Bessel functions of order 1/2: rffiffiffiffiffi rffiffiffiffiffi 2 2
x I 1=2 ðxÞ ¼ sinh x, K 1=2 ðxÞ ¼ e : πx πx

ð2:52Þ

C1 and C2 are undetermined constants. Since I1/2 is divergent for large values of its argument, we choose C1 ¼ 0. To find u we need to solve Eq. (2.46) which takes the following form after the use of Eqs. (2.51) and (2.45):


2 2  ∂ ∂ 1 ∂ eq θ þ C K ðkr Þ sin : þ u¼ 2 ε c 2 1=2 ∂r 2 r∂r r 2 ∂θ2

ð2:53Þ

Let uðr, θÞ ¼ uðr Þ sin

θ : 2

ð2:54Þ

The substitution of Eq. (2.54) into Eq. (2.53) yields


 2 ∂ ∂ 1 eq
þ u ¼ C 2 K 1=2 ðk r Þ: εc ∂r 2 r∂r 4r 2

ð2:55Þ

The homogeneous solution of Eq. (2.55) can be obtained easily. It can be verified that its particular solution is proportional to K1/2. Hence the general solution to Eq. (2.55) is   pffiffi 1 1 eq θ uðr, θÞ ¼ C 3 r þ C 4 pffiffi þ 2 C2 K 1=2 ðk r Þ sin 2 r k εc " # rffiffiffiffiffiffiffi pffiffi 1 1 eq 2
kr θ e ¼ C 3 r þ C 4 pffiffi þ 2 C2 sin , πkr 2 r k εc

ð2:56Þ

where C3 and C4 are undetermined constants. If we want u to be bounded at the crack tip, we must have

26

2 Exact Solutions

1 eq C4 þ 2 C2 k εc

rffiffiffiffiffi 2 ¼ 0: πk

ð2:57Þ

Then    pffiffi 1  θ uðr, θÞ ¼ C 3 r þ C 4 pffiffi 1
e
kr sin : 2 r

ð2:58Þ

Similarly, from Eq. (2.45) we have


 2 ∂ ∂ 1 q
2 φ ¼
C 2 K 1=2 ðk r Þ: þ 2 ε ∂r r∂r 4r

ð2:59Þ

The relevant solution is    pffiffi 1  θ φ ¼ C5 r þ C6 pffiffi 1
e
kr sin , 2 r

ð2:60Þ

where C6 is related to C2 by 1 q C6
2 C2 k ε

rffiffiffiffiffi 2 ¼ 0, πk

ð2:61Þ

and C5 is an undetermined constant. The strain, electric field, stress, and electric displacement components are h i  1 1 θ 2Srz ¼ pffiffi C 3
C 4 1
e
kr þ 2C 4 ke
kr sin , r 2 2 r h i   1 1 θ 2Sθz ¼ pffiffi C 3 þ C 4 1
e
kr cos , r 2 2 r h i  1 1 θ Er ¼
pffiffi C5
C6 1
e
kr þ 2C6 ke
kr sin , r 2 2 r h i   1 1 θ Eθ ¼
pffiffi C 5 þ C6 1
e
kr cos , r 2 2 r

ð2:62Þ

ð2:63Þ

h i  1 1 θ T rz ¼ pffiffi cC 3 þ eC 5
ðcC4 þ eC6 Þ 1
e
kr þ 2ðcC 4 þ eC 6 Þke
kr sin , r 2 2 r h i   1 1 θ T θz ¼ pffiffi cC 3 þ eC 5 þ ðcC4 þ eC6 Þ 1
e
kr cos , r 2 2 r ð2:64Þ

2.5 Antiplane Waves in a Plate

27

h i  1 1 θ Dr ¼ pffiffi eC3
εC5
ðeC 4
εC 6 Þ 1
e
kr þ 2ðeC 4
εC6 Þke
kr sin , r 2 2 r h i  1 1 θ Dθ ¼ pffiffi eC 3
εC5 þ ðeC 4
εC 6 Þ 1
e
kr cos : r 2 2 r ð2:65Þ It can be verified that the above solution satisfies the boundary conditions at the crack faces. When C2, C4, and C6 are all equal to zero, the solution reduces to that of a piezoelectric dielectric.

2.5

Antiplane Waves in a Plate

Consider shear-horizontal (SH) waves in a plate of crystals of class (6mm) [6]. The c-axis is along x3. The plate is unelectroded. In this problem, the electric field in the free space is considered (Fig. 2.11). This is an antiplane problem. The governing equations for the fields in the plate are from Eqs. (1.32) and (1.33): c∇2 u þ e∇2 φ ¼ ρ€u,

ð2:66Þ

e∇2 u
ε∇2 φ ¼ qðΔp
ΔnÞ, ∂ ðΔpÞ ¼ p0 μp ∇2 φ þ Dp ∇2 ðΔpÞ, ∂t ∂ ðΔnÞ ¼
n0 μn ∇2 φ þ Dn ∇2 ðΔnÞ: ∂t

ð2:67Þ

In the free space above and below the plate, the electric potential is governed by ∇2 φ ¼ 0, jx2 j > h, φ ! 0, x2 ! 1:

ð2:68Þ

The free space electric displacement is given by

Fig. 2.11 A piezoelectric semiconductor plate and coordinate system

x2 Free space Crystal Free space

2h

x1

28

2 Exact Solutions

Di ¼
ε0 φ,i :

ð2:69Þ

At the plate surfaces, there are six boundary conditions and four continuity conditions: T 23 ðhÞ ¼ 0, φðhþ Þ ¼ φðh
Þ,

J p2 ðhÞ ¼ 0,

J n2 ðhÞ ¼ 0,

D2 ðhþ Þ ¼ D2 ðh
Þ,

φð
hþ Þ ¼ φð
h
Þ,

D2 ð
hþ Þ ¼ D2 ð
h
Þ:

ð2:70Þ ð2:71Þ

We look for waves propagating in the x1 direction in the following form: 9 8 9 8 u > >A> > > > > > > =

= > < φ > exp ðηx2 Þ exp ½iðξx1
ωt Þ, ¼ > > >C> > > > Δp > > > > > ; : ; : D Δn

ð2:72Þ

where A, B, C, and D are constants. ω is real and positive. ξ has a positive real part. The substitution of Eq. (2.72) into Eqs. (2.66) and (2.67) results in four linear homogeneous algebraic equations for A, B, C, and D. For nontrivial solutions the determinant of the coefficient matrix of the equations has to vanish, which leads to a polynomial equation of degree eight for η. Let the eight roots of this equation be η(m)(ξ, ω) and the corresponding nontrivial solution of A, B, C, and D be A(m), B(m), C(m), and D(m). Only the ratios among A(m), B(m), C(m), and D(m) can be determined. Then the general solution to Eqs. (2.66) and (2.67) can be written as 9 8 ðmÞ 9 8 u > A > > > > > > > > > 8 = = X < BðmÞ > < φ >

ðmÞ exp η ¼ F ðmÞ x exp ½iðξx1
ωt Þ, 2 > > Δp > C ðmÞ > > > > > > > > ; ; m¼1 : ðmÞ > : Δn D

ð2:73Þ

where F(m) are undetermined constants. The solution in the free space is  φ¼

G exp ½ξðh
x2 Þ exp ½iðξx1
ωt Þ, x2 > h, H exp ½ξðh þ x2 Þ exp ½iðξx1
ωt Þ, x2 <
h,

ð2:74Þ

where G and H are undetermined constants. Equation (2.74) satisfies Eq. (2.68). Substituting Eqs. (2.73) and (2.74) into Eqs. (2.70) and (2.71), we obtain ten linear algebraic equations for F(m), G and H. For nontrivial solutions the coefficient matrix of these equations has to vanish, which yields an equation that determines the dispersion relations of ω versus ξ. The numerical procedure is carried out on a computer using MATLAB.

References

29

Fig. 2.12 Dispersion curves of antiplane waves 20

(10 9 rad/s)

15

B C

10

A D E

5 F 0 0

0 1 2

1

2

3 3 Re( )(

4 -1

4 5

5

-1

Im( )( m )

m )

For numerical results, consider a plate of ZnO with 2h ¼ 2 μm, p0 ¼ 2  1024m
3 and n0 ¼ 1.2  1023m
3. Typical dispersion curves found numerically are shown in Fig. 2.12. The dispersion curves are complex because conduction is dissipative.

References 1. Y.X. Luo, C.L. Zhang, W.Q. Chen, J.S. Yang, An analysis of PN junctions in piezoelectric semiconductors. J. Appl. Phys. 122, 204502 (2017) 2. R.F. Pierret, Semiconductor Device Fundamentals (Pearson, Uttar Pradesh, 1996) 3. Y.X. Luo, R.R. Cheng, C.L. Zhang, W.Q. Chen, J.S. Yang, Electromechanical fields near a circular PN junction between two piezoelectric semiconductors. Acta Mech. Solida Sin. 31, 127–140 (2018) 4. J.S. Yang, A semi-infinite anti-plane crack in a piezoelectric semiconductor. Int. J. Fract. 130, L169–L174 (2004) 5. J.S. Yang, An anti-plane crack in a piezoelectric semiconductor. Int. J. Fract. 136, L27–L32 (2005) 6. F. Zhu, S.H. Ji, J.Q. Zhu, Z.H. Qian, J.S. Yang, Study on the influence of semiconductive property for the improvement of nanogenerator by wave mode approach. Nano Energy 52, 474–484 (2018)

Chapter 3

Extension of Rods

Piezoelectric semiconductor rods or fibers of ZnO in extensional deformation are widely used in devices [1]. This chapter begins with the establishment of a one-dimensional model for the extension of thin rods, followed by a series of static and dynamic problems.

3.1

One-Dimensional Equations

Consider the piezoelectric semiconductor rod shown in Fig. 3.1. The shape of the cross section A may be arbitrary. The rod is assumed to be long and thin, i.e., its length is much larger than the characteristic dimension of the cross section. It is made from a piezoelectric semiconductor crystal of class (6mm) such as ZnO. The c-axis of the crystal is along the axis of the rod. The lateral surface of the rod is traction-free and is unelectroded. The electric field in the surrounding free space is neglected. These apply to the rest of the chapter. We are interested in the extensional deformation of the rod which can be described approximately by a set of one-dimensional equations for the axial displacement u3 and the axial stress T3 ¼ T33. An important approximation for extension of thin rods is the relaxation of the lateral stress components, i.e., T1 ¼ T2 ffi 0. In this case, from Eq. (1.3), S3 ¼ sE33 T 3 þ d 33 E3 , D3 ¼ d33 T 3 þ εT33 E 3 :

© Springer Nature Switzerland AG 2020 J. Yang, Analysis of Piezoelectric Semiconductor Structures, https://doi.org/10.1007/978-3-030-48206-0_3

ð3:1Þ

31

32

3 Extension of Rods x1

Fig. 3.1 A piezoelectric semiconductor rod of crystals of class (6mm) c

x3 x2

We solve Eq. (3.1) for T3 and D3 and obtain the following one-dimensional piezoelectric constitutive relations for thin rods: T 3 ¼ c33 S3  e33 E3 , D3 ¼ e33 S3 þ ε33 E3 ,

ð3:2Þ

where c33 , e33 , and ε33 are the effective one-dimensional elastic, piezoelectric, and dielectric constants. They are related to the usual three-dimensional material constants sEpq , εTij , and dip through c33 ¼ 1=sE33 , e33 ¼ d 33 =sE33 , ε33 ¼ εT33  d233 =sE33 :

ð3:3Þ

It can be verified that ε33 þ e233 =c33 ¼ εT33 : We will write εT33 as ε33 in the rest of this chapter whenever there is no ambiguity. The nonlinear one-dimensional constitutive relations for currents are taken from Eq. (1.2)3,4: ∂p , ∂x3 ∂n : J n3 ¼ qnμn33 E 3 þ qDn33 ∂x3 J p3 ¼ qpμp33 E 3  qDp33

ð3:4Þ

The one-dimensional field equations are from Eq. (1.1): ∂T 3 þ f 3 ¼ ρ€u3 , ∂x3
 ∂D3  , ¼ q p  n þ Nþ D  NA ∂x3 p ∂J qp_ ¼  3 , ∂x3 ∂J n3 qn_ ¼ : ∂x3

ð3:5Þ

The relevant strain-displacement relation and the electric field-potential relation are from Eq. (1.5):

3.1 One-Dimensional Equations

33

∂u3 , ∂x3 ∂φ : E3 ¼  ∂x3 S3 ¼

ð3:6Þ

Consider the special case of a rod in static equilibrium without body force, i.e., f3 ¼ 0. In this case, from Eq. (3.5)1, T 3 ¼ C1 ,

ð3:7Þ

where C1 is an integration constant. From Eq. (3.2)1, S3 ¼

1 ðC þ e33 E 3 Þ: c33 1

ð3:8Þ

Substituting Eq. (3.8) and Eq. (3.6)2 into Eq. (3.2)2, we obtain   e233 e33 e C þ ε33 þ D3 ¼ E ¼ 33 C  ε33 φ,3 : c33 1 c33 3 c33 1

ð3:9Þ

For equilibrium, from Eq. (3.5)3,4, J p3 and J n3 are constants. We consider the case when these constants are zero. Then ∂φ ∂p  qDp33 ¼ 0, ∂x3 ∂x3 ∂φ ∂n þ qDn33 ¼ 0: J n3 ¼ qnμn33 ∂x3 ∂x3

J p3 ¼ qpμp33

ð3:10Þ

With the use of Eq. (1.4), we rewrite Eq. (3.10) as 1 ∂p q ∂φ ¼ , p ∂x3 k B T ∂x3 1 ∂n q ∂φ ¼ : n ∂x3 k B T ∂x3

ð3:11Þ

Equation (3.11) can be integrated to produce   q p ¼ p0 exp  φ , kB T   q φ , n ¼ n0 exp kB T

ð3:12Þ

where p0 and n0 are integration constants. Physically they are the values of p and n at φ ¼ 0. We note that the definitions of p0 and n0 here in Eq. (3.12) are different from

34

3 Extension of Rods

the p0 and n0 in Eq. (1.8). Equation (3.12) implies that pn ¼ p0n0 which is a constant. Within the macroscopic theory we are using, p0 and n0 are integration constants and can be determined from, e.g., boundary conditions. Therefore, in general, p0 and n0 depend on the electromechanical loads and structural parameters in addition to material properties. The substitution of Eqs. (3.9) and (3.12) into Eq. (3.5)2 gives the following equation governing the electric potential: ε33 φ,33       q q  φ  n0 exp φ þ Nþ ¼ q p0 exp  ð x Þ  N ð x Þ : 3 3 D A kB T kB T

ð3:13Þ

For small φ, we make the following approximation in Eq. (3.12):   q p ffi p0 1  φ , kB T   q φ , n ffi n0 1 þ kB T

ð3:14Þ

which can describe small carrier concentration perturbations and low electric potential. Substituting Eq. (3.14) into Eq. (3.13), we obtain a linear equation for the potential: φ,33

 
0  q q 0 0 0 þ  φ þ N D ð x3 Þ  N A ð x3 Þ , ¼ p n  p þn ε33 kB T

ð3:15Þ

which can be further written as φ,33  k 2 φ ¼ 

q  0 p  n0 þ N þ ð x3 Þ  N  ð x3 Þ , D A ε33

ð3:16Þ

where  q q
0 1 ¼ , p þ n0 ε33 k B T λ2D ε k T λ2D ¼ 0 33 B0 2 : ðp þ n Þq

k2 ¼

ð3:17Þ

The general solution to Eq. (3.16) can be written as φ ¼ C2 cosh kx3 þ C 3 sinh kx3 þ

 q
0 p  n0 þ φp ðx3 Þ, k ε33 2

ð3:18Þ

where C2 and C3 are integration constants. φ p is a particular solution of the following nonhomogeneous equation:

3.2 Linear Doping

35

φp,33  k2 φp ¼ 

q  þ N ðx Þ  N  A ð x3 Þ : ε33 D 3

ð3:19Þ

Once φ is known, p and n can be obtained from Eq. (3.14). For the mechanical displacement, from Eqs. (3.8) and (3.6)2, we have, after the integration with respect to x3 once, u3 ¼

1 ðC x  e33 φÞ þ C 4 , c33 1 3

ð3:20Þ

where C4 is an integration constant. The polarization and the effective polarization charge density can be calculated from the following well-known expressions: P 3 ¼ D 3  ε0 E 3 , ρP ¼ Pk,k ¼ P3,3 :

ð3:21Þ

In the studies of other systems of mobile charges, e.g., plasmas and electrolytes, Eq. (3.5)2 is called the Poisson’s equation, Eq. (3.12) the Boltzmann distribution, Eq. (3.13) the Poisson-Boltzmann equation, Eq. (3.16) the Debye-Hückel equation, and λD the Debye-Hückel length.

3.2

Linear Doping

Consider the simple case of linear doping described by N A ð x 3 Þ ¼ b1 x 3 þ c 1 ,

Nþ D ð x 3 Þ ¼ b2 x 3 þ c 2 :

ð3:22Þ

Hence  Nþ D ðx3 Þ  N A ðx3 Þ ¼ bx3 þ c,

b ¼ b2  b1 ,

c ¼ c2  c1 :

ð3:23Þ

In this case a particular solution of Eq. (3.19) is simply φp ¼

q ðbx3 þ cÞ: ε33 k2

ð3:24Þ

Therefore, the direct contributions of linear doping as a particular solution to the electric potential and hence p as well as n according to Eq. (3.14) are also linear. The corresponding drift and diffusion currents are constants. Since the electric potential in Eq. (3.14) has to be small, the above observations are true only when |x3| is small.

36

3.3

3 Extension of Rods

Linearly Graded PN Junction

The PN junctions between two piezoelectric semiconductor half spaces and between a circular cylinder and its surrounding material treated in Sects. 2.2 and 2.3 are abrupt junctions with piecewise constant doping profiles. In reality, the doping profiles are gradually changing or graded. In this section we study linearly graded doping and PN junctions. Consider a rod with the doping profile in Fig. 3.2 [2]. The left half is mainly p-doped and the right half mainly n-doped. Between the two halves, there is a finite transition zone of width 2w in which the doping varies linearly. The PN junctions in Sects. 2.2 and 2.3 correspond to the special case of w ¼ 0. Quantitatively, the doping profile in Fig. 3.2 is described by

N A

8 >
: a2 , w < x3 < L,

ð3:25Þ

and

Nþ D

8 >
2w : b2 , w < x3 < L:

ð3:26Þ

Then, 8 > < d 1 ,  L < x3 < w,  Nþ d 3 þ d4 x3 , j x3 j< w,  N ¼ D A > : d2 , w < x3 < L,

Fig. 3.2 Linearly graded doping

a1

N

ð3:27Þ

N D+

− A

b2

b1

-L

a2 -w

0

w

L

x

3

3.3 Linearly Graded PN Junction

37

where d 1 ¼ b1  a1 , d 2 ¼ b2  a2 , b  a1 b2  a2 b  b1 a2  a1 þ , d4 ¼ 2  : d3 ¼ 1 2 2 2w 2w

ð3:28Þ

We consider a free rod with the following boundary and continuity conditions: T 33 ðLÞ ¼ 0, J p3 ðLÞ

¼ 0,

D3 ðLÞ ¼ 0, n J 3 ðLÞ ¼ 0,

u3 ðw Þ ¼ u3 ðwþ Þ, φðw Þ ¼ φðwþ Þ, pðw Þ ¼ pðwþ Þ, nðw Þ ¼ nðwþ Þ, T 33 ðw Þ ¼ T 33 ðwþ Þ, J p3 ðw Þ ¼ J p3 ðwþ Þ,

D3 ðw Þ ¼ D3 ðwþ Þ, J n3 ðw Þ ¼ J n3 ðwþ Þ:

ð3:29Þ ð3:30Þ ð3:31Þ

To make the displacement and potential unique, we impose u3 ð0Þ ¼ 0,

φð0Þ ¼ 0:

ð3:32Þ

In addition, we also have the following global charge neutrality condition: Z

L


L

p  N A



dx3 ¼ 0:

ð3:33Þ

Since Eq. (3.5)2 and the relevant boundary as well as continuity conditions imply that Z

L


L

 p  n þ Nþ D  NA



dx3 ¼ 0,

ð3:34Þ

Eqs. (3.33) and (3.34) together further imply that Z

L


L

n þ N þ D



dx3 ¼ 0:

ð3:35Þ

Therefore Eq. (3.33) is the only independent charge neutrality condition. Consider the special and relatively simple case when L ¼ 1. In this case, we have, for x3 <  w,

38

3 Extension of Rods

φp ¼

q d1 , ε33 k 2

 q
0 p  n0 þ φp ðx3 Þ, φ ¼ C 1 exp kðx3 þ wÞ þ 2 k ε33     q q p ffi p0 1  φ , n ffi n0 1 þ φ , kB T kB T e33 u3 ¼  φ þ C2 , c33 D3 ¼ ε33 φ,3 , T 3 ¼ 0, J p3 ¼ 0, J n3 ¼ 0,

ð3:36Þ

and, similarly, for x3>w, φp ¼

q d2 , ε33 k 2

 q
0 p  n0 þ φp ðx3 Þ, φ ¼ C 3 exp kðw  x3 Þ þ 2 k ε33     q q p ffi p0 1  φ , n ffi n0 1 þ φ , kB T kB T e33 u3 ¼  φ þ C4 , c33 D3 ¼ ε33 φ,3 , T 3 ¼ 0, J p3 ¼ 0, J n3 ¼ 0:

ð3:37Þ

Inside the transition zone where jx3 j < w, we have q ðd3 þ d 4 x3 Þ, ε33 k2 φ ¼ C5 sinh kðx3 þ wÞ þ C 6 sinh kðx3  wÞ  q
0 þ 2 p  n0 þ φp ðx3 Þ, k ε33     q q 0 0 pffip 1 φ , nffin 1þ φ , kB T kB T e u3 ¼  33 φ þ C 7 , c33 D3 ¼ ε33 φ,3 , T 3 ¼ 0, J p3 ¼ 0, J n3 ¼ 0:

φp ¼

ð3:38Þ

C1–C7, p0, and n0 are determined by Eqs. (3.29), (3.30), (3.31), (3.32), and (3.33) using a computer. As an example, consider the following case with some symmetry and/or antisymmetry: a1 ¼ b2 ¼ 1:0  1021 m3 , a2 ¼ b1 ¼ 0:8a1 , w ¼ 0:1 μm:

ð3:39Þ

3.3 Linearly Graded PN Junction

39

In this case, it is found that p0 ¼ n0 ¼ 0:9  1021 m3 ¼ ða1 þ a2 Þ=2 ¼ ðb1 þ b2 Þ=2, 1 ¼ 1:005  107 ffi 0:1 μm: k

ð3:40Þ ð3:41Þ

We denote the right-hand side of Eq. (3.5)2, i.e., the sum of the doping and mobile charges, by
  ρe ¼ q p  n þ N þ D  NA :

ð3:42Þ

Figure 3.3 shows the effects of 2w, i.e., the width of the transition zone, on ρe, the electric field, and the electric potential. As 2w decreases, the transition zone is narrower, and all fields change more rapidly there. ρeand the electric field become stronger. The built-in voltage is not sensitive to w because, as w decreases, the stronger electric field is over a narrower region, and thus the voltage as the spatial integration of the electric field does not change much. Far away from the transition zone, the fields are insensitive to w.

Fig. 3.3 Effects of the width of the transition zone. (a) ρe/q. (b) E3(x3). (c) φ(x3)

40

3 Extension of Rods

3.4

Smoothly Graded PN Junction

Consider a rod of length 2L with a general doping profile that varies smoothly [3]. The rod is in equilibrium without body force and currents, i.e., f3 ¼ 0, J p3 ¼ 0, and J n3 ¼ 0. In this case, from Eqs. (3.16), (3.14), and (3.20), the governing equations are q  0  p  n0 þ N þ D ð x3 Þ  N A ð x3 Þ , ε33     q q p ffi p0 1  φ , n ffi n0 1 þ φ , kB T kB T

φ,33  k 2 φ ¼ 

u3 ¼

ð3:43Þ ð3:44Þ

1 ðC x  e33 φÞ þ C 4 , c33 1 3

ð3:45Þ

 q q
0 : p þ n0 ε33 kB T

ð3:46Þ

where k2 ¼

We consider a free rod with the following boundary conditions: T 33 ðLÞ ¼ 0, J p3 ðLÞ

¼ 0,

D3 ðLÞ ¼ 0, n J 3 ðLÞ ¼ 0:

ð3:47Þ

To make the displacement and the electric potential unique, we impose u3 ðLÞ ¼ 0,

φðLÞ ¼ 0:

ð3:48Þ

Since we have set φ ¼ 0 at x3 ¼ L, we have pðLÞ ¼ p0 ,

nðLÞ ¼ n0 :

ð3:49Þ

We are interested in a PN junction near x3 ¼ 0. Since a PN junction produces a local charge distribution and a local electric field, when L is sufficiently large, p0 and n0 as defined by Eq. (3.49) can be determined from doping approximately, i.e., p0 ffi N  A ðLÞ,

n0 ffi N þ D ðLÞ:

ð3:50Þ

The general solution to Eq. (3.43) can be written as φ ¼C2 sinh kðx3 þ LÞ þ C 3 sinh k ðx3  LÞ  q
0 þ 2 p  n0 þ φp ðx3 Þ, k ε33

ð3:51Þ

3.4 Smoothly Graded PN Junction

41

where φ p satisfies φp,33  k2 φp ¼ 

q  þ N D ðx3 Þ  N  ð x3 Þ : A ε33

ð3:52Þ

Consider the following doping described by a power series in general:  Nþ D ðx3 Þ  N A ðx3 Þ ¼

1 X

an xn3 :

ð3:53Þ

n¼0

Let φp ¼

1 X

cn xn3 :

ð3:54Þ

n¼0

Substituting Eqs. (3.53) and (3.54) into Eq. (3.52), we obtain ðn þ 2Þðn þ 1Þcnþ2  k 2 cn ¼  n ¼ 0, 1, 2,   :

q a , ε33 n

ð3:55Þ

The solution to Eq. (3.55) is not unique. We choose a specific solution of Eq. (3.55) with c0 ¼ 0 and c1 ¼ 0. Then the rest of cn can be calculated from Eq. (3.55) as a recurrence relation. As an example of a PN Junction, we consider the specific doping described by the error function below: N A ðx3 Þ ¼ α1 þ β1 erf ðλx3 Þ,

Nþ D ðx3 Þ ¼ α2 þ β2 erf ðλx3 Þ:

ð3:56Þ

Then  Nþ D ðx3 Þ  N A ðx3 Þ ¼ a þ berf ðλx3 Þ, a ¼ α2  α1 , b ¼ β2  β1 :

ð3:57Þ

1 2 X λ2nþ1 erf ðλx3 Þ ¼ pffiffiffi ð1Þn x2nþ1 , n!ð2n þ 1Þ 3 π n¼0

ð3:58Þ

Since

we have, from Eqs. (3.53) and (3.57)1,

42

3 Extension of Rods

Fig. 3.4 Basic behavior of error function

1 2 X λ2nþ1 an xn3 ¼ a þ b pffiffiffi ð1Þn : x2nþ1 3 n! ð 2n þ 1 Þ π n¼0 n¼0

1 X

ð3:59Þ

We plot erf(λx3) in Fig. 3.4 which converges everywhere. Its behavior is suitable for describing the doping profiles of certain PN junctions with symmetry or antisymmetry. tan1(λx3) and tanh(λx3) have similar behaviors but are with a finite domain of convergence. As a numerical example, consider a rod with L ¼ 6 μm,

λ ¼ 107 m1 ,

α1 ¼ 0:9  1021 m3 , α2 ¼ 0:9  1021 m3 ,

β1 ¼ 0:1  1021 m3 , β2 ¼ 0:1  1021 m3 :

ð3:60Þ

 In this case a ¼ 0 according to Eq. (3.57) and N þ D  N A ¼ berf ðλx3 Þ is an odd function. Instead of Eqs. (3.48), (3.49), and (3.50), to make use of the symmetry or antisymmetry of the doping profile implied by Eq. (3.60), we simply impose

u3 ð0Þ ¼ 0,

φð0Þ ¼ 0,

p0 ¼ N  A ð0Þ ¼ α1 , n ¼ 0

Nþ D ð 0Þ

ð3:61Þ

¼ α2 ¼ α1 :

In Fig. 3.5, (a)–(c) show the effects of λ on the normalized doping and mobile charge ρe/q whose definition is in Eq. (3.42), the effective polarization charge ρP/q given by Eq. (3.21), the total charge (ρe + ρP)/q, the electric field, and the electric potential. ρe and ρP are with opposite signs, and ρe dominates. In addition to the distributed ρP, for a finite rod, in general, there may also exist concentrated effective polarization charges at the two ends of the rod. These concentrated charges depend on the values

3.4 Smoothly Graded PN Junction

43

Fig. 3.5 Effects of λ. (a) Doping and mobile charge ρe/q. (b) Polarization charge ρP/q. (c) Total charge (ρe + ρP)/q. (d) Electric field E3(x3). (e) Electric potential φ(x3)

of P3 at x3 ¼  L which in this specific example are not considered because for a PN junction the electric field, electric displacement, and polarization vanish sufficiently far away from the junction [3]. (d) and (e) of Fig. 3.5 show clearly that a PN junction has formed with a typical built-in electric field and a built-in potential. As λ increases, all fields change more rapidly as expected.

44

3.5

3 Extension of Rods

Periodic Doping

In this section we consider a rod with a periodically varying doping profile described by [2]: N A ðx3 Þ ¼ a1 þ b1 sin λx3 ,

Nþ D ðx3 Þ ¼ a2 þ b2 sin λx3 :

ð3:62Þ

Hence  Nþ D ðx3 Þ  N A ðx3 Þ ¼ a þ b sin λx3 , a ¼ a2  a1 , b ¼ b2  b1 :

ð3:63Þ

Then, for the particular solution of Eq. (3.19), we have φp ¼

  q a b þ sin λx 3 : ε33 k 2 k2 þ λ2

ð3:64Þ

In this case, it can be shown [2] that p0 and n0, i.e., the values of p and n at zero potential, are determined by the following equations: p0  n0 þ a ¼ 0, n0 a p0 ¼ 0 1 : 2n  a2

ð3:65Þ

In the special case when a2 ¼ a1, Eq. (3.63) implies that a ¼ 0. Then, from Eq. (3.65), we obtain p0 ¼ n0 ¼ a1. For a numerical example, we consider the case when a1 ¼ 1021m3, a2 ¼ a1, b1 ¼ 0.2a1, b2 ¼  b1, and λ ¼ 0.2k. Then 2π/ λ ffi 3 μm which can be viewed as some wavelength of the doping. We choose 2L ¼ 30 μm which is about ten times the doping wavelength 2π/λ. The doping and mobile charge density ρe whose definition is in Eq. (3.42), the electric field E3(x3), and the axial strain S3(x3) in this case are shown in Fig. 3.6. They are essentially periodic except at x3 ¼  L where there are some edge effects. ρe has some concentration near the ends.

3.6 Linear Extension by End Forces

45

Fig. 3.6 Fields produced by periodic doping. (a) ρe/q. (b) E3(x3). (c) S3(x3)

3.6

Linear Extension by End Forces

Consider the static extension of the rod shown in Fig. 3.7 [4]. It is within |x3| a, respectively, and apply boundary conditions and continuity (or jump) conditions. The general solution to Eqs. (3.118), (3.119), (3.120), (3.121), and (3.122) is Δn ¼ C1 ekx þ C 2 ekx ,  k T
φ ¼ B C1 ekx þ C 2 ekx þ C3 x þ C4 , qn0 qe 1  u¼ C 1 ekx þ C 2 ekx þ C 5 x þ C6 , 2 2 cε þ e k

ð3:124Þ

where C1–C6 are undetermined constants. Equation (3.124) is valid for each of the three regions in Fig. 3.12 with corresponding undetermined constants for each region. All together there are 18 undetermined constants. The rod is mechanically free and electrically open at infinity. Denoting a ¼ (a) and a+ ¼ (a)+, we write the boundary and continuity conditions as T ð1Þ ¼ 0,

Dð1Þ ¼ 0,

J n ð1Þ ¼ 0,

ð3:125Þ

3.8 Local Extension

57

uða Þ ¼ uðaþ Þ,

T ðaþ Þ  T ða Þ ¼ F=A,

φða Þ ¼ φðaþ Þ,

Dða Þ ¼ Dðaþ Þ,



þ

nða Þ ¼ nða Þ, 



þ

n

uða Þ ¼ uða Þ,

T ðaþ Þ  T ða Þ ¼ F=A,

φða Þ ¼ φðaþ Þ,

Dða Þ ¼ Dðaþ Þ,



þ

nða Þ ¼ nða Þ, T ðþ1Þ ¼ 0,

ð3:126Þ

þ

J ða Þ ¼ J ða Þ, n



ð3:127Þ

þ

J ða Þ ¼ J ða Þ, n

Dðþ1Þ ¼ 0,

n

J n ðþ1Þ ¼ 0:

ð3:128Þ

Equations (3.125), (3.126), (3.127), and (3.128) are 18 conditions, but some of them are not independent. For example, Jn(1) ¼ 0, Jn is continuous at a, and ∂Jn/ ∂x ¼ 0 imply that Jn(1) ¼ 0. Hence, to determine the displacement and potential fields uniquely, we need to choose 1 as a reference and set uð1Þ ¼ 0,

φð1Þ ¼ 0:

ð3:129Þ

The substitution of Eq. (3.124) for different regions into Eqs. (3.125), (3.126), (3.127), (3.128), and (3.129) results in a system of linear equations for the undetermined constants. They are solved on a computer using MATLAB. Then various electromechanical fields can be calculated. (a)–(c) of Fig. 3.13 show the effective polarization charge density as defined by Eq. (3.21), the electron concentration perturbation, and the electric potential produced by two different values of F when n0 ¼ 1021m3 and a ¼ 600 nm. There also exist concentrated polarization charges at x ¼  a. For the purpose of mechanically manipulating the electrical behavior of the rod, the most basic effect of F is that it produces a potential well followed by a potential barrier as shown in (c). With the presence of the potential well and barrier, a mobile charge cannot travel through the rod unless it has sufficient velocity or kinetic energy. Therefore, a local pair of concentrated forces forbids the passing of low-energy mobile charges through the rod. We note the resemblance of the central part of (b) to Fig. 3.8 for the electron distribution in the extension of a rod by end forces. (d) shows the effect of F on the current-voltage relation from a nonlinear numerical analysis by COMSOL [7]. It can be seen that when the applied voltage V is low, no current can flow through the rod in either direction. When a positive V exceeds a stress-dependent critical value, currents begin to flow in one direction of the rod but not in the other direction if the sign of the voltage is changed. When V exceeds a second stress-dependent critical value, currents can flow in both directions when the sign of V is changed. Therefore the local stress acts like a switch that has different behaviors in opposite directions. This provides a basic means for mechanically manipulating currents in a piezoelectric semiconductor rod. If the lateral surface of the rod is not traction-free, then the lateral stress T11 and T22 can affect the axial distribution of the electric fields in a similar way through the piezoelectric constant e31.

58

3 Extension of Rods

Fig. 3.13 Fields for different F. n0 ¼ 1021m3. (a) Effective polarization charge density ρP. (b) Electron concentration perturbation Δn. (c) Electric potential φ. (d) IV curve. V ¼ φ(L )φ(L ). I ¼ JnA

3.9

Local Extension and Compression

The fields and charge distributions obtained in the previous section by one pair of local forces can be used as building blocks for constructing more complicated electric fields and charge distributions mechanically. As an example, consider an unbounded rod under two local pairs of forces, one compressive and the other tensile as shown in Fig. 3.14 [6]. The solution to this problem is similar to that in the previous section, with more algebra because of more regions with different stresses, more undetermined constants, and more boundary and continuity conditions. The solution is not presented here, and we show the results directly in Fig. 3.15 for two different values of n0. Our main interest is in the big potential well with two small potential barriers in (c).

3.10

Periodic Extension and Compression

Fig. 3.14 A piezoelectric semiconductor rod under two local pairs of concentrated forces

59

F

F a

F 2b

F

x

a

Fig. 3.15 Fields for different n0. F ¼ 8.5 nN. (a) Effective polarization charge density ρP. (b) Electron concentration perturbation Δn. (c) Electric potential φ

3.10

Periodic Extension and Compression

In this section we consider the effect of the distributed body force f in Eq. (3.109)1 [6]. As an example, consider the case of a rod under a sinusoidal distribution of f described by f ¼ B sin ξx where B and ξ are constants. For an n-type rod with p ffi 0, it is straightforward to find the corresponding fields from Eqs. (3.118), (3.119), (3.20), (3.21), and (3.122) as

60

3 Extension of Rods

Fig. 3.16 Fields for different n0. (a) Effective polarization charge density ρP. (b) Electron concentration perturbation Δn. (c) Screened polarization charge ρP  qΔn. (d) Electric potential φ

Δn ¼ φ¼

ek 2  B sin ξx, cq k2 þ ξ2


kB T ek 2
 B sin ξx, q2 n0 c k 2 þ ξ 2

u¼

ð3:130Þ

e2 k B T k2 B1
2  B sin ξx þ 2 sin ξx: 2 2 cξ c q n0 k þ ξ2

We plot the charges and potential determined by Eq. (3.130) in Fig. 3.16 when B ¼ 8.5536  1011 N/m3 and ξ ¼ 3.4907  106/m. The charge and potential distributions are also sinusoidal. A series of periodically alternating potential barriers, wells, and charged regions are produced. They are similar to the fields in Fig. (3.6) due to periodic doping but are produced mechanically here. In this case, for the effective polarization charge, there is a distributed ρP only without concentrated effective polarization charges at discontinuities because all fields are continuous everywhere. (a) to (c) show that ρP is significantly but not completely screened by electrons.

3.11

3.11

Harmonic Vibration

61

Harmonic Vibration

Consider the finite piezoelectric semiconductor rod shown in Fig. 3.17. The left end is fixed. The right end is under the action of a time-harmonic extensional force [8]. Both ends are electroded with ohmic contacts. There is no interface nonlinearity, and the interface resistance between the crystal and electrodes is negligible at the two ends. The two end electrodes are connected by an output circuit whose impedance is Z in time-harmonic motions. Such a rod can operate as an energy harvester for converting mechanical energy to electrical energy. As an energy harvester, the screening of the effective polarization charge by the mobile charge in a semiconductor is undesirable. It reduces the piezoelectrically generated output voltage by the mechanical load. The governing equations are from Eqs. (1.1)1 with i ¼ 3, (1.10), (3.2), (1.9) with i ¼ 3, and (3.6): ∂T 3 ¼ ρ €u3 , ∂x3 ∂D3 ¼ qðΔp  ΔnÞ , ∂x3

ð3:131Þ

∂J p ∂ ðΔpÞ ¼  3 , ∂x3 ∂t ∂J n ∂ q ðΔnÞ ¼ 3 : ∂x3 ∂t

ð3:132Þ

q

T 3 ¼ c33 S3  e33 E3 , D3 ¼ e33 S3 þ ε33 E3 , ∂ðΔpÞ , ∂x3 ∂ðΔnÞ J n3 ¼ qn0 μn33 E3 þ qDn33 , ∂x3 J p3 ¼ qp0 μp33 E3  qDp33

Fig. 3.17 A piezoelectric semiconductor rod under a time-harmonic force

ð3:133Þ

ð3:134Þ

62

3 Extension of Rods

S3 ¼

∂u3 , ∂x3

E3 ¼ 

∂φ , ∂x3

ð3:135Þ

where uniform doping has been assumed. Δp and Δn are given in Eq. (1.7). p0 and n0 are defined by Eq. (1.8). We consider an n-type semiconductor with p ffi 0. For timeharmonic vibrations described by a common time factor exp(iω t), with successive substitutions, we can write Eqs. (3.131) and (3.132)2 as c33 u3,33 þ e33 φ,33 ¼ ρω2 u3 , e33 u3,33  ε33 φ,33 ¼ qðΔnÞ, iωqðΔnÞ ¼ qn0 μn33 φ,33 þ qDn33

∂ðΔnÞ : ∂x23

ð3:136Þ

We look for a solution of Eq. (3.136) in the following form: 9 8 9 8 > = = >

< u3 > ¼ B exp ðζx3 Þ, φ > > ; ; > : > : C Δn

ð3:137Þ

where A, B, C, and ζ are undetermined constants. Substituting Eq. (3.137) into Eq. (3.136), we obtain a system of linear homogeneous equations for A, B, and C: 2

c33 ζ 2 þ ρω2 6 e33 ζ 2 4 0

e33 ζ 2 ε33 ζ 2 n0 μn33 ζ 2

38 9 > =

7 5 B ¼ 0: > ; : > C Dn33 ζ 2  iω 0 q

ð3:138Þ

For nontrivial solutions of A, B, or C, the determinant of the coefficient matrix of Eq. (3.138) has to vanish, which leads to the following cubic equation for ζ 2:
  c33 ε33 þ e233 Dn33 ζ 6
 þ iωc33 ε33 þ iωe233 þ c33 qn0 μn33  ρω2 Dn33 ε33 ζ 4
 þ iρω3 ε33 þ ρω2 qn0 μn33 ζ 2 ¼ 0:

ð3:139Þ

Equation (3.139) has six roots for ζ. ζ (1) ¼ ζ (2) ¼ 0 is a repeated root. It is easy to show that ζ (1) ¼ ζ (2) ¼ 0 leads to 9 8 9 8 9 8 > > > = = =

< u3 > ¼ Bð1Þ x3 þ Bð2Þ 1 , φ > > > > ; ; ; : > : > : 0 0 Δn

ð3:140Þ

3.11

Harmonic Vibration

63

where B(1) and B(2) are undetermined constants. B(2) represents an arbitrary constant in the potential which does not affect the electric field and is immaterial. Therefore it will be dropped in the following. Denoting the four other roots of Eq. (3.139) by ζ (3) through ζ (6), we write the general solution of Eq. (3.136) as 9 9 8 8 8 9 ðmÞ > > > = = = X

6 < u3 >

exp ζ ðmÞ x3 , ¼ Bð1Þ x3 þ φ BðmÞ 1 > > > > ; ; ; m¼3 : ðmÞ > : : > Δn 0 β

ð3:141Þ

where B(3) through B(6) are undetermined constants, and α β

ðmÞ

ðmÞ


2 e33 ζ ðmÞ ¼
, 2 c33 ζ ðmÞ þ ρω2
2 n0 μn33 ζ ðmÞ ¼ , m ¼ 3, 4, 5, 6:
2 Dn33 ζ ðmÞ  iω

ð3:142Þ

The boundary conditions are u3 ð0Þ ¼ 0,

T 3 ðLÞ ¼

D3 ð0Þ ¼ 0,

F exp ðiω t Þ, S

D3 ðLÞ ¼ 0:

ð3:143Þ ð3:144Þ

We also have the following circuit equation: V ¼ J n ðLÞS, Z

ð3:145Þ

where V ¼ φ(L )  φ(0) is the piezoelectrically generated output voltage on Z. The substitution of Eq. (3.141) into Eqs. (3.143), (3.144), and (3.145) results in five linear equations for B(1) and B(3) through B(6). They are solved on a computer using MATLAB. As a numerical example, consider a ZnO rod whose geometric parameters are that the length L ¼ 600 nm, radius a ¼ 25 nm, and cross-sectional area S ¼ πa2 ¼ 1.9625  1015 m2. The magnitude of the end force F ¼ 0.5 nN. The 22 ‐3 reference carrier concentration n0 ¼ N þ D ¼ 10 m . For the mechanically forced vibration we are considering, to introduce some damping into the
mechanical  structure, we use a complex elastic constant c33 1 þ iQ1 in our numerical calculation which can describe viscous damping in time-harmonic motions. Q is the material quality factor. For convenience we introduce

64

3 Extension of Rods

ðnÞ ω0

ð2n  1Þπ ¼ 2L

rffiffiffiffiffiffi c33 , ρ

n ¼ 1, 2, 3,   ,

ð3:146Þ

which are the extensional resonance frequencies of the corresponding elastic dielectric rod. We also introduce Z0 ¼

1 ð1Þ iω0 C 0

,

C0 ¼ ε33

S L

ð3:147Þ

as a unit for the impedance Z of the output circuit. We calculate the open-circuit vibration characteristics when Z ¼ 1 first. The effect of Z will be examined later. We plot the modulus (absolute value) of the complex displacement at the right end of the rod versus the normalized driving frequency in Fig. 3.18a for two values of Q ¼ 50 and 100. These values of Q are relatively small for the material damping of common crystals. We exaggerate the damping a little so that the resonances are not too sharp and are more visible in the figure. In the frequency range shown, there are three resonances with frequencies ð1Þ ω(n). It can be seen that the values of ωðnÞ =ω0 are close to 1, 3, and 5 where

Fig. 3.18 (a) ju3(L )j and (b) θ ¼ tan1[Im{u3(L )}/ Re {u3(L )}] versus the normalized driving frequency showing resonances. Z ¼ 1

3.11

Harmonic Vibration

65

ð1Þ

ω0 ¼ 1:3198  1010 rad/s. Below the first resonance, there is a relatively small but finite response which is the static response in the low-frequency limit. The top of the first peak of the curve with Q ¼ 100 is 4.79  1011m which has been cut off in the figure for a better global view. When Q ¼ 200, 500, and 5000, the corresponding peak values of the first resonance are 6.35, 7.72 and 8.74  1011m, respectively, increasing monotonically because of less damping. When Q is very large, the amplitude at the first resonance becomes less sensitive to Q because in this case the dissipation due to semiconduction becomes more influential than material damping. (b) shows the phase angle defined by θ ¼ tan1[Im{u3(L )}/ ð1Þ Re {u3(L )}]. At resonances there is a phase jump of π. The jumps at ω=ω0 close to 2 and 4 are caused by Re{u3(L )} ¼ 0 and do not represent resonances. For an ð1Þ elastic rod, the resonance frequencies are exactly odd integral multiples of ω0 . For a piezoelectric dielectric rod, the three frequencies are higher because of the piezoelectric stiffening effect. If the piezoelectric rod is also semiconducting, the mobile charges screen the polarization charges, reduce the piezoelectric stiffening effect, ðnÞ and lower the frequencies. Therefore ω(n) are somewhat higher than ω0 . Next we examine the displacement fields at resonances (modes). Strictly speaking, vibration modes are defined by the eigenvalue problem of a free vibration analysis. They can also be felt from our forced vibration analysis by examining the displacement fields at resonances as follows. Based on the complex notation used, the physical field of the mechanical displacement can be obtained from the real part of the complex displacement through Re fu3 exp ðiω t Þg ¼ Re uR3




  uR3 þ iuI3 ð cos ω t þ i sin ωt Þ

¼ uR3 cos ω t  uI3 sin ωt, ¼ Re fu3 g, uI3 ¼ Imfu3 g,

ð3:148Þ

which depends on both the real and imaginary parts of the complex displacement. When Q ¼ 100, in Fig. 3.19, we plot both the real and imaginary parts of the axial mechanical displacement at the three resonance frequencies. The displacement vanishes at the fixed left end as dictated by the boundary condition. At the first resonance, the displacement does not have nodal points (zeros) except at the fixed left end. At the second and the third resonances, the displacement has one and two nodal points, respectively, in addition to the left end. These are typical behaviors of modes at resonances. Corresponding to the relatively high peak at the first resonance shown in Fig. 3.18a, the displacement at the first resonance is relatively large. As a mechanical-to-electrical energy converter, the output power and efficiency are of fundamental interest. The mechanical input power P1 at the right end of the rod and the electrical output power P2 on Z can be calculated from

66

3 Extension of Rods

Fig. 3.19 Displacement fields at the first three resonances. Z ¼ 1. Q ¼ 100. (a) Real parts. (b) Imaginary parts

1 P1 ¼ fF ½u_ 3 ðLÞ þ F  ½u_ 3 ðLÞg, 4 1 P2 ¼ fIV  þ IV  g, 4

ð3:149Þ

where an asterisk represents complex conjugate. The efficiency is given by P2/P1. Figure 3.20 shows P1, P2, and P2/P1 versus the driving frequency of the applied force when Z is real and represents a resistor load. P1 and P2 assume maxima at resonance frequencies as expected. As the ratio between two functions of the driving frequency, the behavior of the efficiency is somewhat involved, not necessarily assuming maxima at resonance frequencies.

3.12

Transient Vibration

67

Fig. 3.20 (a) Input power verses frequency. (b) Output power versus frequency. (c) Efficiency versus frequency. n0 ¼ 1  1022/m3

3.12

Transient Vibration

Consider the piezoelectric semiconductor rod in Fig. 3.21. The left end is fixed. At t ¼ 0, the right end is under the action of a suddenly applied extensional force F which produces an extensional stress f ¼ F/A there [9]. The one-dimensional governing equations are the same as Eqs. (3.131), (3.132), (3.133), (3.134), and (3.135). We consider a ZnO rod doped into an n-type

68

3 Extension of Rods

x1

Fig. 3.21 A piezoelectric semiconductor rod under the sudden application of an end force

x3

c

F

L

semiconductor with p ffi 0. As an approximation, we neglect the charge on the righthand side of Eq. (3.131)2. This decouples the problem into two one-way coupled subproblems. One consists of the equations of piezoelectricity for the mechanical displacement u3 and the electric potential φ. The other is the continuity equation for the electron concentration perturbation Δn. The two subproblems are treated one at a time below. The piezoelectricity subproblem is governed by ∂T 3 ¼ ρ€u3 , ∂x3

∂D3 ¼ 0: ∂x3

ð3:150Þ

Using Eqs. (3.133) and (3.135), we can write Eq. (3.150) as c33 u3,33 þ e33 φ,33 ¼ ρ€u3 , e33 u3,33  ε33 φ,33 ¼ 0:

ð3:151Þ

The boundary conditions are u3 ¼ 0, T3 ¼ f ,

D3 ¼ 0, D3 ¼ 0,

x3 ¼ 0, x3 ¼ L:

ð3:152Þ

The initial conditions are u3 ¼ 0,

u_ 3 ¼ 0,

φ ¼ 0,

t ¼ 0:

ð3:153Þ

From Eq. (3.151)2 and the boundary conditions on the electric displacement in Eq. (3.152), we have D3 ¼ e33 u3,3  ε33 φ,3 0:

ð3:154Þ

Then T 3 ¼ c33 u3,3 þ e33 φ,3 ¼ c33 u3,3 þ e33

e33 u ¼ bc33 u3,3 , ε33 3,3

ð3:155Þ

where bc33 ¼ c33 þ

e233 ε33

ð3:156Þ

3.12

Transient Vibration

69

is a piezoelectrically stiffened elastic constant. Using Eq. (3.151)2 to eliminate the potential, we write Eq. (3.151)1 as bc33 u3,33 ¼ ρ€u3 :

ð3:157Þ

The problem for u3 consists of Eq. (3.157) and the following initial and boundary conditions: u3 ¼ 0, x3 ¼ 0, bc33 u3,3 ¼ f , x3 ¼ L, u3 ¼ 0,

u_ 3 ¼ 0,

ð3:158Þ

t ¼ 0:

ð3:159Þ

To homogenize the boundary conditions in Eq. (3.158), we let u3 ¼ u þ

f x : bc33 3

ð3:160Þ

The problem for u is bc33 u,33 ¼ ρ €u,

ð3:161Þ

u ¼ 0, x3 ¼ 0, bc33 u,3 ¼ 0, x3 ¼ L,

ð3:162Þ

u¼

f x, bc33 3

u_ ¼ 0,

t ¼ 0:

ð3:163Þ

Mathematically, Eqs. (3.161), (3.162), and (3.163) are a rather standard problem. Its solution can be obtained in a straightforward manner by the method of separation of variables as uð x 3 , t Þ ¼

1 X m¼0

Am cos

  π þ 2mπ π þ 2mπ ct sin x3 , 2L 2L

ð3:164Þ

where  Z   2 L f π þ 2mπ  x3 dx3 , x3 sin Am ¼ L 0 2L bc33 rffiffiffiffiffiffi bc33 c¼ : ρ

m ¼ 0, 1, 2,   , ð3:165Þ

Once u is obtained, u3 and φ are obtained from Eqs. (3.160) and (3.154), respectively. For uniqueness, we let φ ¼ 0 at x3 ¼ 0. As an example, consider a ZnO rod with L ¼ 600 nm and cross-sectional area A ¼ 2.598  1014m2. The end force

70

3 Extension of Rods

Fig. 3.22 Distribution of u3 along the rod at different time instants. m ¼ 0–15. (a) t ¼ 4  1011s. (b) t ¼ 10.7  1011s. (c) t ¼ 18  1011s. (d) t ¼ 22  1011s

F ¼ 1.7 nN which produces an end stress of f ¼ 0.065 MPa. n0 ¼ 1023/m3. The series in Eq. (3.164) converges rapidly. Sixteen terms are used. Figure 3.22 presents a series of snapshots of the distribution of u3 along the rod at different time instants as the initial disturbance at the right end propagates to the left along the rod, hits the left end and gets reflected there, and then propagates back to the right end. The wave described by Eq. (3.161) is nondispersive and has a finite propagation speed. In (a), the right part of the rod has felt the disturbance with an essentially linearly distributed displacement, but the left part has notpfelt the disturbance yet. The extensional ffiffiffiffiffiffiffiffiffiffiffi wave speed can be determined as c ¼ bc33 =ρ ¼ 5546 m=s . In our numerical example, we have L/c ¼ 10.8  1011s which is the time for a disturbance to propagate through the entire rod, and it agrees with (b) very well. The propagation of φ is found to be similar to u3 and is not shown. The equation for Δn is obtained from Eqs. (3.132)2 and (3.134)2 as 2

∂ ðΔnÞ ∂ , ðΔnÞ ¼ n0 μn33 φ,33 þ Dn33 ∂t ∂x23 with the following boundary and initial conditions:

ð3:166Þ

3.12

Transient Vibration

71

J 3 ¼ qn0 μn33 E 3 þ qDn33

∂ðΔnÞ ¼ 0, ∂x3

Δn ¼ 0,

x3 ¼ 0, L,

t ¼ 0:

ð3:167Þ ð3:168Þ

To make the boundary conditions in Eq. (3.167) homogeneous, we let Δn ¼ b nþ

n0 μn33 φ: Dn33

ð3:169Þ

The initial-boundary-value problem for b n is ∂b n ∂b n n0 μn ∂φ ¼ Dn33 2  n33 , D33 ∂t ∂t ∂x3 2

∂b n ¼ 0, ∂x3 b n ¼ 0,

x3 ¼ 0, L,

ð3:170Þ ð3:171Þ

t ¼ 0:

ð3:172Þ

Equations (3.170), (3.171), and (3.172) form a standard mathematical problem. Its solution by separation of variables and Laplace transform is b nð x 3 , t Þ ¼

1 X r¼0

 cos

rπ x L 3

Z 0

t

   ffi 2 rπ pffiffiffiffiffiffiffi Dn33 ðt  τÞ dτ, ar ðτÞ exp  L

ð3:173Þ

where  Z L n0 μn33 ∂φ  n dx3 , r ¼ 0, D33 ∂t 0  Z L  n0 μn ∂φ 2 rπ ar ð t Þ ¼ x3 dx3 ,  n33 cos L 0 L D33 ∂t

1 a0 ð t Þ ¼ L

ð3:174Þ r ¼ 1, 2, 3  :

From the series solution in Eqs. (3.173) and (3.169), we show in Fig. 3.23 the electron concentration perturbation at the same time instants as those in Fig. 3.22. Fundamentally different from the mechanical disturbance governed by the hyperbolic wave equation in Eq. (3.161) with a finite wave speed, the motion of the electrons is governed by the parabolic equation in Eq. (3.170) with effectively an infinite speed for signal propagation. Therefore, in Fig. 3.23a, while mechanically the left part of the rod is still unperturbed, the electrons along the entire rod have felt the disturbance and been disturbed. Since the rod is electrically isolated, the total number of electrons is conserved. Therefore the integration of Δn over [0,L] has to vanish, which can be qualitatively seen in the figure. The series for Δn converges very rapidly. The results corresponding to seven and eight terms are indistinguishable when plotted in the scale of Fig. 3.23.

72

3 Extension of Rods

Fig. 3.23 Distribution of Δn along the rod at different time instants. m ¼ 0–15, r ¼ 0–7. (a) t ¼ 4  1011s. (b) t ¼ 10.7  1011s. (c) t ¼ 18  1011s. (d) t ¼ 22  1011s

3.13

Nonlinear Waves

Consider the rod shown in Fig. 3.24. It is doped into an n-type semiconductor with p ffi 0. We study the propagation of extensional waves with consideration of electrical nonlinearity [10]. The analysis is somewhat crude with several approximations. It is hoped that the solution can still capture some of the physics involved. The governing equations are as follows: T 3,3 ¼ ρ €u3 ,
 D3,3 ¼ q n þ N þ D , _ J n3,3 ¼ qn, T 3 ¼ c33 S3  e33 E3 , D3 ¼ e33 S3 þ ε33 E 3 , J n3

ð3:175Þ

ð3:176Þ

¼ qnμ33 E 3 þ qD33 n,3 , S3 ¼ u3,3 , E 3 ¼ φ,3 ,

ð3:177Þ

3.13

Nonlinear Waves

73

x1

Fig. 3.24 A piezoelectric semiconductor rod of crystals of class (6mm)

c

x3 x2

where μ33 ¼ μn33 and D33 ¼ Dn33. With the use of Eq. (3.177), we write Eq. (3.176) as T 3 ¼ c33 u3,3 þ e33 φ,3 , D3 ¼ e33 u3,3  ε33 φ,3 , J 3 ¼ qnμ33 φ,3 þ qD33 n,3 :

ð3:178Þ

The substitution of Eq. (3.178) into Eq. (3.175) yields three equations for u3, φ, and n: c33 u3,33 þ e33 φ,33 ¼ ρ €u3 , e33 u3,33  ε33 φ,33 ¼ qðN þ  nÞ, _ μ33 φ,33 n  μ33 φ,3 n,3 þ D33 n,33 ¼ n:

ð3:179Þ

The first two terms in Eq. (3.179)3 are nonlinear. Assuming low doping and low carrier concentration, we make an approximation by dropping the charges q(N+  n) on the right-hand side of Eq. (3.179)2. This reduces Eq. (3.179) into two one-way coupled systems of c33 u3,33 þ e33 φ,33 ¼ ρ€u3 , e33 u3,33  ε33 φ,33 ¼ 0,

ð3:180Þ

and _ μ33 φ,33 n  μ33 φ,3 n,3 þ D33 n,33 ¼ n:

ð3:181Þ

Equation (3.180) is the classical theory of piezoelectricity within which harmonic extensional waves in the rod are described by u3 ¼ A cos ψ, where

φ ¼ B cos ψ,

ð3:182Þ

74

3 Extension of Rods

ψ ¼ ξx3  ωt, bc33 ¼ c33 þ

ω2 ¼

e233 , ε33

bc33 2 ξ , ρ

B¼

e33 A: ε33

ð3:183Þ

The electric field accompanying the wave in Eq. (3.182) is given by E ¼ B0 sin ψ,

B0 ¼

e33 Aξ: ε33

ð3:184Þ

With the electric potential in Eq. (3.182) considered as known, Eq. (3.181) becomes linear in n. However, it has variable coefficients depending on both x3 and t through φ and is still a mathematically challenging problem. We consider uniform doping and denote n ¼ N þ D . We write n as the sum of a uniform static part n which is a constant and a dynamic part b nðx3 , t Þ caused by the waves in Eq. (3.182), i.e., nð x 3 , t Þ ¼ n þ b nðx3 , t Þ:

ð3:185Þ

Substituting Eq. (3.185) into Eq. (3.181), we obtain the following equation for b n: b n þ μ33 E3b n,3 þ D33b n,33 , n_ ¼ μ33 E3,3 n þ μ33 E3,3b

ð3:186Þ

where the electric field E is given by Eq. (3.184) which, when substituted into Eq. (3.186), results in b n þ μ33 B0 sin ψ b n,3 þ D33b n,33 : n_ ¼ μ33 nB0 ξ cos ψ þ μ33 B0 ξ cos ψ b

ð3:187Þ

The first three terms on the right-hand side of Eq. (3.187) are from the nonlinear drift current, and the last term is from the diffusion current. The first term does not depend on b n and is effectively a driving term from the wave in Eq. (3.182). Consider a special case of Eq. (3.187) first. If b n is very small and only the first term on the right-hand side of Eq. (3.187) is kept, the equation reduces to b n_ ¼ μ33 nB0 ξ cos ψ,

ð3:188Þ

which admits the following solution b n ¼ μ33 nB0

ξ sin ψ: ω

ð3:189Þ

Equation (3.189) describes a simple sinusoidal wave motion of the carriers driven by the electromechanical wave in Eq. (3.182).

3.14

Extension of a PN Junction

75

In general, when all of the terms in Eq. (3.187) are considered, we write the solution using the following trigonometric series: b n¼

M X

am cos mψ þ bm sin mψ,

ð3:190Þ

m¼1

where am and bm are undetermined constants. We substitute Eq. (3.190) into Eq. (3.187), write both sides of the resulting equation in terms of cosmψ and sinmψ using trigonometric identities, and set their corresponding coefficients equal. This yields a hierarchy of linear equations for am and bm. These equations are truncated and solved on a computer. As a numerical example, consider a rod with a cross-sectional dimension characterized by some diameter d ¼ 1 mm. n ¼ 1013 =m3 . For a wave with wavelength λ ¼ 10 mm which is much larger than the cross-sectional dimension, the wave number ξ ¼ 2π/λ ¼ 628.3 1/m. Then, from Eq. (3.183), the wave frequency is ω ¼ 3.49  106 1/s. The wave amplitude A will be varied in the computation. In the case when A ¼ 3  108m, the corresponding amplitude of the axial strain S3 ¼ u3,3 is Aξ ¼ 18.85  106. We calculate b n for increasing values of A in the range from 109m to 3  108m with five terms in the series in Eq. (3.190), i.e., M ¼ 5 which has been verified numerically to have sufficient accuracy for examining the qualitative behaviors of the waves. The results are summarized in Fig. 3.25. In (a), when A is small, the wave is essentially sinusoidal. As A increases, deviation from a sinusoidal wave becomes visible. The height of the crests is larger than the depth of the troughs, and the crests are narrower than the troughs. As A increases further in (b), the wave deviates from a sinusoidal one severely. These are believed to be due to the electrical nonlinearity and are to be verified numerically by, e.g., COMSOL.

3.14

Extension of a PN Junction

In this section we study the extension of a piezoelectric semiconductor rod with a PN junction (see Fig. 3.26). The rod has a unit cross-sectional area and is under an axial force F. It is electrically open at both ends. Both homogeneous and heterogeneous junctions will be considered. For a homogeneous junction, the material is uniform along the entire rod, but the two halves are doped differently. The left half is dominated by holes and the right half by electrons. For a heterogeneous junction, the materials of the two halves are different. We consider the case of a relatively simple heterogeneous junction obtained by reversing the c-axis of the material of the left half. When the c-axis is reversed, the relevant piezoelectric constants change signs. The problem is static. Below is a combination of a linear theoretical analysis and a nonlinear numerical analysis by COMSOL [11]. A nonlinear analysis using both the perturbation method and COMSOL can be found in [12].

76

3 Extension of Rods

Fig. 3.25 Effect of the wave amplitude A

Fig. 3.26 A PN junction in a piezoelectric semiconductor rod in extension

F

x3

p-doped 2L

n-doped

F

3.14

Extension of a PN Junction

77

The one-dimensional governing equations are T 33,3 ¼ ρ€u3 , D3,3 ¼ qðΔp  ΔnÞ, _ J p3,3 ¼ qp, n _ J 3,3 ¼ qn, T 33,3 ¼ c33 S33  e33 E 3 , D3 ¼ e33 S33 þ ε33 E 3 ,

ð3:191Þ

ð3:192Þ

J p3 ¼ qpμp33 E 3  qDp3 p,3 J n3

ffi qp0 μp33 E3  qDp3 ðΔpÞ,3 , ¼ qpμn33 E 3 þ qDn3 n,3

ð3:193Þ

ffi qn0 μn33 E3 þ qDn3 ðΔnÞ,3 , S33 ¼ u3,3 , E 3 ¼ φ,3 :

ð3:194Þ

The substituon of Eqs. (3.192), (3.193) and (3.194) into Eq. (3.191) yields four second-order ordinary differential equations for u3, φ, Δp, and Δn. The equations may be linear or nonlinear depending on whether the linearized or nonlinear version of Eq. (3.193) is used. The axial electric polarization P3 and the effective polarization charge density ρP can be calculated from the following expressions: P 3 ¼ D 3  ε0 E 3 , ρP ¼ P3,3 :

ð3:195Þ

We follow the simplified notation in Eq. (3.80). The boundary and continuity conditions are T ðLÞ ¼ F, DðLÞ ¼ 0, J p ðLÞ ¼ 0, J n ðLÞ ¼ 0, uð0 Þ ¼ uð0þ Þ, T ð0 Þ ¼ T ð0þ Þ, φð0 Þ ¼ φð0þ Þ, Dð0 Þ ¼ Dð0þ Þ, pð0 Þ ¼ pð0þ Þ, J p ð0 Þ ¼ J p ð0þ Þ, nð0 Þ ¼ nð0þ Þ, J n ð0 Þ ¼ J n ð0þ Þ, T ðLÞ ¼ F, DðLÞ ¼ 0, J p ðLÞ ¼ 0, J n ðLÞ ¼ 0: Δp and Δn satisfy the following global charge neutrality conditions:

ð3:196Þ ð3:197Þ ð3:198Þ ð3:199Þ

78

3 Extension of Rods

Z

L

L

Z Δpdx ¼ 0,

L L

Δndx ¼ 0:

ð3:200Þ

Only one of Eq. (3.200) is independent. To determine the displacement and potential fields uniquely, we choose a reference point, e.g., x ¼ a, and set uðaÞ ¼ 0, φðaÞ ¼ 0:

ð3:201Þ

Mathematically, we need to find solutions in the two regions in –L a, respectively, and apply boundary and continuity conditions. For the linear solution, in each region, we have a system of ordinary differential equations with constant coefficients. The final solution can be found in a straightforward manner. For x < a, the results are e ðeλ þ pcÞθ sinh ka exp ðkxÞ, c kðe2 þ cεÞ e ðeλ þ pcÞθ sinh ka exp ðkxÞ, S¼ c e2 þ cε

ð7:42Þ

ðeλ þ pcÞθ sinh ka exp ðkxÞ, kðe2 þ cεÞ ðeλ þ pcÞθ E¼ 2 sinh ka exp ðkxÞ, ðe þ cεÞ   eλ þ p θ sinh ka exp ðkxÞ, D¼ c     eλ cε þ p θ sinh ka 1  2 0 P¼ exp ðkxÞ, c e þ cε     eλ cε þ p θ sinh ka 1  2 0 exp ðkxÞ, ρP ¼ k c e þ cε

ð7:43Þ

u¼

φ¼

7.2 Effects of a Local Temperature Change

185

  k eλ þ p θ sinh ka exp ðkxÞ, Δn ¼  q c   eλ cε P þ p θ sinh ka 2 0 exp ðkxÞ, ρ  qΔn ¼ k c e þ cε

ð7:44Þ

where k2 ¼

n 0 μn q : Dn εT33

ð7:45Þ

In the central region with |x| < a, we have e ðeλ þ pcÞθ λ exp ðkaÞ sinh kx þ θðx þ aÞ, c kðe2 þ cεÞ c e ðeλ þ pcÞθ λ exp ðkaÞ cosh kx þ θ, S¼ c e2 þ cε c u¼

ðeλ þ pcÞθ exp ðkaÞ sinh kx, kðe2 þ cεÞ ðeλ þ pcÞθ E¼ 2 exp ðkaÞ cosh kx, e þ cε     eλ eλ D¼ þ p θ exp ðkaÞ cosh kx þ þ p θ, c c       eλ cε0 eλ þp θ 1 2 þ p θ, P¼ exp ðkaÞ cosh kx þ c c e þ cε     eλ cε þp θ 1 2 0 ρP ¼ k exp ðkaÞ sinh kx, c e þ cε   k eλ þ p θ exp ðkaÞ sinh kx, Δn ¼ q c   eλ cε P þ p θ 2 0 exp ðkaÞ sinh kx: ρ  qΔn ¼ k c e þ cε

ð7:46Þ

φ¼

ð7:47Þ

ð7:48Þ

For x > a, the fields are e ðeλ þ pcÞθ λ sinh ka exp ðkxÞ þ 2 θL, c kðe2 þ cεÞ c e ðeλ þ pcÞθ S¼ sinh ka exp ðkxÞ, c e2 þ cε

u¼

ð7:49Þ

186

7 Thermal Effects

ðeλ þ pcÞθ sinh ka exp ðkxÞ, kðe2 þ cεÞ ðeλ þ pcÞθ E¼ 2 sinh ka exp ðkxÞ, e þ cε   eλ þ p θ sinh ka exp ðkxÞ, D¼ c     eλ cε0 þ p θ sinh ka 1  2 P¼ exp ðkxÞ, c e þ cε     eλ cε þ p θ sinh ka 1  2 0 exp ðkxÞ, ρP ¼ k c e þ cε   k eλ þ p θ sinh ka exp ðkxÞ, Δn ¼ q c   eλ cε þ p θ sinh ka 2 0 exp ðkxÞ: ρP  qΔn ¼ k c e þ cε φ¼

ð7:50Þ

ð7:51Þ

For a numerical example, consider a ZnO rod with 2a ¼ 1.2 μm, n0 ¼ 1021/m3, and θ ¼ 0.1 K. As indicated by the above equations, a local temperature change produces many electromechanical fields. The situation is similar to those produced by local stresses. Our main interest is the local potential barrier and well shown in Fig. 7.5a which is caused by the thermally induced local polarization in (b). The polarization is under the screening effect of the electrons. As a linear solution, the potential barrier and well exhibit an antisymmetry about the origin. The above potential barrier and well predicted by a linear analysis neglect the inherent nonlinearity in the drift currents. As a comparison, for the same rod with a finite length of 2L ¼ 8.4 μm, we perform a nonlinear analysis using COMSOL with larger values of θ. The results are shown in Fig. 7.6. In (a), when θ ¼ 1 K, the nonlinear result looks qualitatively similar to the linear result. However, when θ ¼ 5 K, fundamentally different from the linear result, the potential barrier height and well depth become clearly different, losing the antisymmetry of the linear solution about the origin. This is typical for a nonlinear solution. If θ is increased further, (b) shows that the well becomes very deep and the barrier is barely visible. Next we consider the case when the two ends of a finite rod with length 2L is under an applied voltage 2V with the following boundary conditions: φðLÞ ¼ V,

φðLÞ ¼ V:

ð7:52Þ

Our main interest is the current-voltage relation or I-V curve. We use the nonlinear version of the constitutive relations for currents in Eq. (7.32) and perform a numerical analysis using COMSOL. Specifically, consider a rod with 2L ¼ 8.4 μm, 2a ¼ 1.2 μm, and n0 ¼ 1021/m3. The I-V curves of interest are shown in Fig. 7.7a for different θ, where Jn ¼ I. For a specific curve corresponding to a given θ, when the applied voltage is low, no currents can flow through the rod in either direction. When the applied voltage is high and above a critical value that is temperature

7.2 Effects of a Local Temperature Change

187

Fig. 7.5 Fields produced by a local temperature change predicted by the linear theory. (a) Potential barrier and well. (b) Electric polarization

dependent, currents can flow in the rod in both directions. In the case of θ ¼ 50 K, it can be seen that the critical voltages for positive or negative V are different. Therefore, there exists a range of the voltage in which currents can flow in one direction but not the other. Therefore the local temperature change has the effect of a switch which is off for low voltages and on for high voltages. This provides a basic means for thermally manipulating currents in a piezoelectric semiconductor rod. When θ ¼ 35 K, for specific values of the voltage that are low, critical, or high, (b) shows the corresponding distributions of the electric potential along the rod. When V ¼ 0.3 volts, the potential well still exists, blocking possible currents. When V ¼ 1.5 volts, it has overcome the potential well, and the potential distribution becomes monotonic. Hence currents can flow freely.

188

7 Thermal Effects

Fig. 7.6 Nonlinear solution for potential barrier and well by COMSOL. (a) θ ¼ 1, 3 and 5 K. (b) θ ¼ 20, 35, and 50 K

7.3

Temperature Effects on PN Junctions

In this section we examine the effects of a uniform temperature change on PN junctions in ZnO rods [3]. Consider a ZnO rod of length 2L as shown in Fig. 7.8. Both homogeneous and heterogeneous junctions are considered. For a homogeneous junction, the material is uniform except that the two halves are doped differently. For a heterogeneous junction, instead of using different materials, we simply reverse the c-axis of the left half. When the c-axis is reversed, the relevant piezoelectric and pyroelectric constants change signs. The temperature change θ is uniform. The problem is static. The rod is stress free and electrically open at both ends.

7.3 Temperature Effects on PN Junctions

189

Fig. 7.7 Electrical behavior under a voltage. (a) I-V curves. (b) Potential distribution when θ ¼ 35 K

Fig. 7.8 A PN junction in a ZnO rod

x

p-doped

n-doped

2L

The governing equations in Eqs. (7.29), (7.30), (7.31), (7.32), and (7.33) and the simplified notations in Eqs. (7.34) and (7.35) are employed. For an isolated rod, the boundary and continuity conditions are

190

7 Thermal Effects

T ðLÞ ¼ 0, DðLÞ ¼ 0, J p ðLÞ ¼ 0, J n ðLÞ ¼ 0, uð0 Þ ¼ uð0þ Þ,

T ð0 Þ ¼ T ð0þ Þ,

φð0 Þ ¼ φð0þ Þ,

Dð0 Þ ¼ Dð0þ Þ,

pð0 Þ ¼ pð0þ Þ, nð0 Þ ¼ nð0þ Þ,

J p ð0 Þ ¼ J p ð0þ Þ, J n ð0 Þ ¼ J n ð0þ Þ,

T ðLÞ ¼ 0, DðLÞ ¼ 0, J p ðLÞ ¼ 0, J n ðLÞ ¼ 0:

ð7:53Þ ð7:54Þ ð7:55Þ ð7:56Þ

Δp and Δn satisfy the following global charge neutrality conditions: Z

L

L

Z Δpdx ¼ 0,

L L

Δndx ¼ 0:

ð7:57Þ

Only one of Eq. (7.57) is independent. To determine the displacement and potential fields uniquely, we choose a reference point, e.g., x ¼ a, and set uðaÞ ¼ 0,

φðaÞ ¼ 0:

ð7:58Þ

We need to find solutions in the two regions with –L < x < 0 and 0 < x < L separately and apply boundary and continuity conditions. For the usual built-in fields in the junction, we perform a linear analysis analytically using the linearized constitutive relations in Eq. (7.32). In each region we have a system of ordinary differential equations with constant coefficients. The procedure for finding a general solution is straightforward. The junction may be heterogeneous. We use a prime for the material parameters in the left half and a double prime for those in the right half. The general solution for –L < x < 0 is Δp  Δn ¼ A1 sinh k 0 ðx  LÞ þ B1 sinh k 0 ðx þ LÞ, q φ ¼  0 2 T ½A1 sinh k0 ðx  LÞ þ B1 sinh k0 ðx þ LÞ þ C1 x þ C2 , ðk Þ ε0 33 u¼

e0 q ½A sinh k 0 ðx  LÞ þ B1 sinh k 0 ðx þ LÞ þ C 3 x þ C 4 , c0 ð k 0 Þ 2 ε0 T 1 33 Δp ¼

p00 μ0 p q ½A1 sinh k0 ðx  LÞ þ B1 sinh k0 ðx þ LÞ D0 p ðk0 Þ2 ε0 T33

þC 5 x þ C 6 ,

where A1, B1, and C1-C6 are undetermined constants and

ð7:59Þ ð7:60Þ ð7:61Þ

ð7:62Þ

7.3 Temperature Effects on PN Junctions

0 2

ðk Þ ¼



 p00 μ0 p n00 μ0 n q þ 0n : D0 p D ε0 T33

191

ð7:63Þ

Similarly, for 0 < x < L, we have Δp  Δn ¼ A2 sinh k00 ðx  LÞ þ B2 sinh k 00 ðx þ LÞ, q φ ¼  00 2 T ½A2 sinh k00 ðx  LÞ þ B2 sinh k 00 ðx þ LÞ þ C7 x þ C 8 , ðk Þ ε00 33 u¼

e00 q ½A sinh k00 ðx  LÞ þ B2 sinh k00 ðx þ LÞ þ C9 x þ C10 , c00 ðk 00 Þ2 ε00 T 2 33 Δp ¼

p000 μ00 p q ½A2 sinh k00 ðx  LÞ þ B2 sinh k 00 ðx þ LÞ D00 p ðk 00 Þ2 ε00 T33

ð7:64Þ ð7:65Þ ð7:66Þ

ð7:67Þ

þC 11 x þ C12 ,

where A2, B2, and C7-C12 are undetermined constants and  00 00 p  p0 μ n000 μ00 n q ðk Þ ¼ : 00 p þ 00 n 00 D D ε T33 00 2

ð7:68Þ

Equations (7.59), (7.60), (7.61), and (7.62) and Eqs. (7.64), (7.65), (7.66), and (7.67) are substituted into Eqs. (7.53), (7.54), (7.55), (7.56), (7.57), and (7.58), resulting in a system of linear equations for the undetermined constants. These equations are solved on a computer. The axial electric polarization P¼P3 is calculated from Eq. (7.41). Then the related volume effective polarization charge density ρP, the sum of the doping and mobile charge densities ρe, and the total charge density ρt can be calculated from the following expressions: ρP ¼ Pk,k ¼ P3,3 , ρe ¼ qðΔp  ΔnÞ, ρt ¼ ρe þ ρP :

ð7:69Þ

For a numerical example, consider a rod with 2L ¼ 30 μm. The doping profile is given by p00 ¼ 1021 m3, n00 ¼ 7  1020 m3, n000 ¼ 1021 m3, and p000 ¼ 7  1020 m3. It is chosen to be antisymmetric about the origin for simplicity. Consider three cases of temperature change with θ ¼ 0 K, 0.3 K, and 0.5 K, among which the case of θ ¼ 0 K is used as a reference. Figure 7.9 shows the temperature effects on the basic built-in fields near the junction for a heterogeneous junction with opposite c-axes in the two halves. In the case of a homogeneous junction, the corresponding figures show no visible temperature effects and therefore are not presented. Only a portion of the rod within |x| < 1 μm is shown, which is sufficient for examining the fields because they are

192 Fig. 7.9 Temperature effects on built-in fields in a heterogeneous junction. (a) Potential φ. (b) Electric field E. (c) Total charge density ρt

7 Thermal Effects

7.3 Temperature Effects on PN Junctions

193

local and near x ¼ 1 μm they have already stabilized. Near the rod ends at far away with x ¼ L, there exist field concentrations which cannot be seen in the figure. When θ ¼ 0 K, the curves are either symmetric or antisymmetric about the origin. When θ is nonzero, the curves lose their symmetry or antisymmetry. Specifically, (a) and (b) show the usual built-in electric potential and field. When plotting the electric potential, x ¼ 1 μm is the reference where the potential vanishes, i.e., a ¼ 1 μm in Eq. (7.58). It can be seen that although the built-in voltage determined by the potential difference at x ¼ 1 μm in (a) is essentially unaffected by the temperature change, the slope of the potential field or the built-in electric field in (b) is. When θ ¼ 0.5 K, the potential in (a) is continuous but not smooth at the origin. This is related to that the corresponding electric field in (b) in fact has a jump discontinuity at the origin which has been connected by a vertical line in the figure. The total charge ρt in (c) is directly responsible for the behavior of the electric field in (b). The difference between the homogenous and heterogeneous junctions is caused by the reversal of the c-axis in the left half of the rod, which leads to the fundamentally different polarizations shown in Fig. 7.10a, b for the homogeneous and heterogeneous junctions, respectively. Away from the junction and on its both sides, the P in (a) is the same but that in (b) has opposite signs. The different behaviors of P in (a) and (b) have implications on the effective polarization charge density ρP. For the homogeneous junction in (a), the corresponding ρP is essentially antisymmetric without any net charge and shows no visible temperature effect. Therefore it is not shown. For the heterogeneous junction in (b), the corresponding ρP is shown in (c) which produces some net positive charge near the junction. This affects the total charge density in Fig. 7.9c. In addition to the volume effective polarization charge density described by ρP, there also exists surface effective polarization charge density σ P on the interface of the PN junction due to the discontinuity of P there. For our junction interface, σ P can be calculated from σ P ¼ P  Pþ , P ¼ Pð0 Þ,

Pþ ¼ Pð0þ Þ:

ð7:70Þ

We list σ P for the same three temperature changes in Table 7.1. For a homogeneous junction or when θ ¼ 0 K, we have σ P ¼ 0. For a heterogeneous junction, σ P increases with θ. σ P is positive and contributes further to the net positive charge at the junction produced by ρP in Fig. 7.10c. We now consider the case when the two ends of the rod at x ¼ L are under an applied voltage 2V with the following boundary conditions: φðLÞ ¼ V,

φðLÞ ¼ V:

ð7:71Þ

Our main interest is the current-voltage relation or I-V curve. We use the nonlinear version of the constitutive relations for currents in Eq. (7.32) and perform a

194 Fig. 7.10 Polarization and its effective charges. (a) Polarization P, homogeneous junction. (b) Polarization P, heterogeneous junction. (c) ρP, heterogeneous junction

7 Thermal Effects

7.3 Temperature Effects on PN Junctions

195

Table 7.1 Temperature effect on effective polarization charge density at junction interface Charge (C/m2) Temperature (K) Homogeneous Heterogeneous Temperature (K) Homogeneous Heterogeneous Temperature (K) Homogeneous Heterogeneous

P θ¼0 4.5638  106 4.5638  106 θ ¼ 0.3 6.6335  106 4.4002  106 θ ¼ 0.5 8.0134  106 4.2911  106

P+

σP

4.5638  106 4.5638  106

0 0

6.6335  106 4.7274  106

0 3.2724  107

8.0134  106 4.8365  106

0 5.4541  107

Fig. 7.11 I-V curves of the heterogeneous junction. Jn ¼ I

numerical analysis using COMSOL. The I-V curves of the homogeneous junction are insensitive to the temperature change. The results for the heterogeneous junction are shown in Fig. 7.11. From Sect. 7.1, it is known that the following combination of the effective onedimensional material parameters pþ

eλ c

ð7:72Þ

plays a decisive role in the thermal behavior of a ZnO rod. It includes the effects of the effective one-dimensional pyroelectric, thermoelastic, and piezoelectric constants. With the use of Eq. (7.11), it can be shown that the combination in Eq. (7.72) is equal to the three-dimensional material constant p3. For our ZnO rod, we have

196

7 Thermal Effects

Fig. 7.12 I-V curves of the heterogeneous junction when the sign of e is artificially changed. Jn ¼ I

  p ¼ 1:1981  105 C=m2 K ,   eλ ¼ 0:50808  105 C=m2 K : c

ð7:73Þ

Equation (7.73) shows that p and eλ/c have opposite signs, and some cancelation is present in their combined effect. Therefore it can be expected that when p and e have the same sign, the temperature effect may be stronger. To verify this, we artificially change the sign of e in both halves of the junction but not that of p and plot the corresponding I-V curves in Fig. 7.12. Indeed, the temperature sensitivity is much stronger than that in Fig. 7.11.

7.4

Extension of Composite Rods

In this section we study the effects of a uniform temperature change on the extensional deformation of the composite rod shown in Fig. 7.13 [4]. It consists of two piezoelectric dielectric layers and a nonpiezoelectric semiconductor layer. (x,y,z) correspond to (x1,x2,x3). When the piezoelectric materials are polarized ceramics or crystals of class (6mm), the poling direction or the c axis is along the z axis. Equations (7.1), (7.2), (7.3), (7.4), and (7.5) are still valid. They include piezoelectric dielectrics and nonpiezoelectric semiconductors as special cases. We need to derive a set of one-dimensional equations from Eqs. (7.1), (7.2), (7.3), (7.4), and (7.5) for the extensional deformation of the composite rod under a uniform temperature change. For a thin composite rod, similar to Eqs. (7.6) and (7.7), we still have, for the entire rod, approximately

7.4 Extension of Composite Rods Fig. 7.13 Side view and cross section of a composite rod of piezoelectric dielectrics and nonpiezoelectric semiconductors

197

(1) PZT

c

z

(2) Si (1) PZT

c 2L

y h

x

2c h

b u3 ffi u3 ðz, t Þ,

φ ffi φðz, t Þ:

ð7:74Þ

In the semiconductor layer only, we also assume that Δp ffi Δpðz, t Þ,

Δn ffi Δnðz, t Þ:

ð7:75Þ

Then the axial strain and electric field can be approximated by S3 ¼ S33 ¼ u3,3 ,

E3 ¼ φ,3 ,

ð7:76Þ

which are common to all layers. To simplify the notation of the axial fields, we write u ¼ u3 , S ¼ S3 , T ¼ T 33 ¼ T 3 , E ¼ E 3 , P ¼ P3 , D ¼ D3 , J ¼ p

J p3 ,

J ¼ n

ð7:77Þ

J n3 :

We also denote the relevant material constants by s ¼ sE3333 ¼ sE33 , d ¼ d333 ¼ d 33 , ε ¼ εT33 , μp ¼ μp33 , μn ¼ μn33 , Dp ¼ Dp33 , Dn ¼ Dn33 ,

ð7:78Þ

α ¼ α33 : The fields and material constants for the piezoelectric layers will carry a superscript 1 in parentheses and those of the semiconductor layer a superscript 2 in parentheses.

198

7 Thermal Effects

For the piezoelectric layers, in terms of the elastic compliance, the constitutive relation for the axial strain and electric displacement are S ¼ sð1Þ T þ dð1Þ E þ αð1Þ θ, ð1Þ D ¼ d ð1Þ T þ εð1Þ E þ p3 θ:

ð7:79Þ

From Eq. (7.79) we obtain T ¼ cð1Þ S  eð1Þ E  λð1Þ θ, D ¼ eð1Þ S þ εð1Þ E þ pð1Þ θ,

ð7:80Þ

where d ð1Þ αð1Þ eð1Þ ¼ ð1Þ , λð1Þ ¼ ð1Þ , s s  ð1Þ 2 d d ð1Þ αð1Þ ð1Þ ¼ εð1Þ  ð1Þ , pð1Þ ¼ p3  ð1Þ : s s

cð1Þ ¼ ε

ð1Þ

1

sð1Þ

,

ð7:81Þ

Similarly, for the semiconductor layer, it is assumed to be nonpiezoelectric and hence does not have pyroelectric coupling. The relevant constitutive relations are S ¼ sð2Þ T þ αð2Þ θ, D ¼ εð2Þ E:

ð7:82Þ

Equation (7.82) can be rewritten into T ¼ cð2Þ S  λð2Þ θ, D ¼ εð2Þ E,

ð7:83Þ

where cð2Þ ¼

1

s

, ð2Þ

λð2Þ ¼

αð2Þ , sð2Þ

εð2Þ ¼ εð2Þ :

ð7:84Þ

The currents exist in the semiconductor layer only and therefore the superscript 2 in parentheses is neglected: dðΔpÞ , dz dðΔnÞ J n ffi qn0 μn E þ qDn : dz

J p ffi qp0 μp E  qDp

ð7:85Þ

7.4 Extension of Composite Rods

199

The total axial force and electric displacement in the composite rod are defined by integrations of T and D over the entire cross section and are found to be b ¼ bcS  beE  b T λθ, b ¼ beS þ bεE þ b D pθ,

ð7:86Þ

where bc ¼ cð1Þ Að1Þ þ cð2Þ Að2Þ , b λ ¼ λð1Þ Að1Þ þ λð2Þ Að2Þ , ð1Þ ð1Þ

bε ¼ ε A

ð2Þ ð2Þ

þε A ,

e ¼ eð1Þ Að1Þ , ð7:87Þ ð1Þ ð1Þ

b p¼p A :

A(1) and A(2) are the cross-sectional areas of the piezoelectric and semiconductor layers, respectively. The one-dimensional equation of motion is obtained by applying Newton’s second law to a differential element of the rod with length dz as shown in Fig. 7.14:  b ∂T þ f ðz, t Þ ¼ 2b ρð1Þ h þ ρð2Þ c €u, ∂z

ð7:88Þ

where f(z,t) is the axial load per unit length of the rod. Similarly, the one-dimensional charge equation can be obtained by considering the differential element in Fig. 7.14 under electrical loads: b dD ¼ Að2Þ qðΔp  ΔnÞ: dz

ð7:89Þ

The one-dimensional continuity equations are simply

Fig. 7.14 A differential element of the rod

Tˆ

fdz

dz

Tˆ  dTˆ

z

200

7 Thermal Effects

∂ ∂J p , ðΔpÞ ¼  ∂t ∂z n ∂ ∂J q ðΔnÞ ¼ : ∂z ∂t

q

ð7:90Þ

We consider the static extensional deformation of an electrically isolated free rod. The boundary conditions are b ðLÞ ¼ 0, T

b ðLÞ ¼ 0, D

J ðLÞ ¼ 0,

J p ðLÞ ¼ 0:

n

ð7:91Þ

Δp and Δn satisfy the following global charge neutrality conditions: Z

L

L

Z Δpdz ¼ 0,

L

L

Δndz ¼ 0:

ð7:92Þ

Only one of Eq. (7.92) is independent. To determine the mechanical displacement and the electric potential uniquely, we set uð0Þ ¼ 0,

φð0Þ ¼ 0:

ð7:93Þ

Solving Eqs. (7.88), (7.89), and (7.90) under Eqs. (7.91), (7.92), and (7.93) does not present any mathematical challenge. Once the equations are solved, the polarization vector and effective polarization charge density can be calculated from their definitions: P ¼ D  ε0 E ¼ ρ ¼ Pk,k P

ð1Þ

A dP ¼ : dz

b D  ε0 E, þ Að2Þ

ð7:94Þ

The expressions of the fields are found to be 

eb λþb pbc θ sinh kz , φ¼ ðbεbc þ e2 Þk cosh kL  eb λþb pbc θ cosh kz E¼ , ðbεbc þ e2 Þ cosh kL  e eb λþb pbc θ sinh kz b λθ þ z, u¼ ðbεbc þ e2 Þbck cosh kL bc

ð7:95Þ

ð7:96Þ

ð7:97Þ

7.4 Extension of Composite Rods

201

 e eb λþb pbc θ cosh kz b λθ þ , S¼ ðbεbc þ e2 Þbc cosh kL bc  b b λ þ p c θ sinh kz p eb pμ , p ¼ p0  0 p 2 D ðbεbc þ e Þk cosh kL  b b λ þ p c θ sinh kz n eb nμ , n ¼ n0 þ 0 n D ðbεbc þ e2 Þk cosh kL     b θ cosh kz λe D¼ 1 , þb p cosh kL bc Að1Þ þ Að2Þ     b ε 1 1 cosh kz λe P¼ þb p θ 0 ð2Þ  ð1Þ eε A bc A þ Að2Þ cosh kL   b θ λe b þ , þp ð1Þ bc A þ Að2Þ    b 1 ε0 1 sinh κz λe P ,  ρ ¼ kθ þb p ð1Þ ð 2Þ ð2Þ cosh κL e ε bc A þA A

ð7:98Þ

ð7:99Þ

ð7:100Þ ð7:101Þ

ð7:102Þ

ð7:103Þ

where  k2 ¼ eεA

ð2Þ

 p0 μ p n0 μ n q þ , Dp Dn eε

e2 ¼ bε þ : bc

ð7:104Þ

As a numerical example, consider a composite rod of PZT-5A and silicon with L ¼ 0.6 μm, h ¼ c ¼ 0.05 μm, and b ¼ 0.2 μm. p0 ffi 0. n0 ¼ 1023/m3. Figure 7.15 shows the redistribution of electrons. It can be seen that the rod considered is sensitive to temperature and a larger temperature change induces larger changes of the carrier distribution as expected. Δn is an order of magnitude smaller than n0 so that the linearized theory employed is valid. Figure 7.16 shows the carrier concentration perturbation for different n0 when θ¼0.1 K while all other parameters are kept the same as those for Fig. 7.15. The fields are sensitive to n0. From Eq. (7.100), we obtain Δn θ sinh kz , ¼γ n0 Θ0 cosh kL where

ð7:105Þ

202

7 Thermal Effects ×1022

Fig. 7.15 Electron concentration perturbation in the rod for different temperature change. n0 ¼ 1023/m3

1.0

Δ n (m-3)

0.5

0.0

-0.5

θ=0.1K θ=0.5K θ=1.0K

-1.0 -0.6

-0.4

-0.2

0.0

z (µm)

0.2

0.4

0.6

0.4

0.6

×1021

Fig. 7.16 Electron concentration perturbation in the rod for different n0. θ¼0.1 K

1.0

Δn (m-3)

0.5

0.0

-0.5

n0=1021/m3 n0=1022/m3 n0=1023/m3

-1.0 -0.6

-0.4

-0.2

0.0

0.2

z (µm) γ¼

q e b λþb pbc Θ , k B T bεeck Að1Þ þ Að2Þ 0

bc þ e2 =bε ec ¼ ð1Þ : A þ Að2Þ

ð7:106Þ

ec is an averaged, piezoelectrically stiffened elastic constant. At the end of a long rod where kL is large, we have sinhkL ffi cosh kL. Therefore γ describes the relative change of carrier concentration near the ends of the rod per unit relative temperature change. It can be used as a measure of the strength of the coupling effect of the thermally induced carrier redistribution in the rod. We are interested in the change of number of carriers in the differential element in Fig. 7.14. From Eq. (7.105), for a

7.5 Extension and Bending of Composite Beams

|g |(2cbdz)n0

1.5

203

×104

1.0

0.5

Si and LiNbO3 Si and BaTiO3 Si and PZT-5A 0.0

0

1

2

3

4

5

h/c Fig. 7.17 jγ j (2cbdz)n0 at the end of the rod versus h/c for different combinations of materials. dz ¼ 0.1μm. Θ0¼298 K. p0 ffi 0. n0 ¼ 1023/m3

differential element of the rod near its ends, per unit θ/Θ0, we have Δn(2cbdz) ffiγ(2cbdz)n0 whose absolute value is plotted in Fig. 7.17 versus h/c when h + c is fixed. The curves all have a maximum when h and c are not very different.

7.5

Extension and Bending of Composite Beams

In this section we study the behavior of the composite beam in Fig. 7.18 under a uniform temperature change [5]. It consists of a piezoelectric dielectric layer (layer (1)) such as ceramics poled along the axial direction and a nonpiezoelectric semiconductor layer (layer (2)) such as silicon. The cross-sectional areas of the piezoelectric and semiconductor layers are denoted by A(1) and A(2). The beam is unelectroded. The x3 axis is at the interface which is in both extension and bending under a uniform temperature change because of the lack of structural symmetry about the interface. Equations (7.1), (7.2), (7.3), (7.4), and (7.5) are still valid, with piezoelectric dielectrics and nonpiezoelectric semiconductors as special cases. We derive a set of one-dimensional equations first. For extension and bending without shear deformation in the x3-x2 plane, the bending and axial displacements are approximated by u2 ffi vðx3 , t Þ,

u3 ffi wðx3 , t Þ  x2 v,3 :

The axial strain S3 is expressed in terms of w and v through

ð7:107Þ

204

7 Thermal Effects

Fig. 7.18 Side view and cross section of a composite beam of a piezoelectric dielectric and a nonpiezoelectric semiconductor

x2 (1) PZT

x3

P

(2) Si L

x2 x1

h1 h2

b S3 ¼ w,3  x2 v,33 :

ð7:108Þ

Similarly, the electric potential and electric field over a cross section are approximated by φ ffi φðx3 , t Þ,

E 3 ¼ φ,3 :

ð7:109Þ

For the carrier concentration perturbations in the semiconductor layer, Δp ffi Δpðx3 , t Þ,

Δn ffi Δnðx3 , t Þ:

ð7:110Þ

We follow the simplified notations for the relevant axial fields and material parameters in Eqs. (7.77) and (7.78). For the piezoelectric layer, the one-dimensional constitutive relations and effective material constants in Eqs. (7.80) and (7.81) are still applicable. The constitutive relations take the following form with the use of Eqs. (7.108) and (7.109): T ¼ cð1Þ S  eð1Þ E  λð1Þ θ ¼ cð1Þ ðw,3  x2 v,33 Þ þ eð1Þ φ,3  λð1Þ θ, D ¼ eð1Þ S þ εð1Þ E þ pð1Þ θ

ð7:111Þ

¼ eð1Þ ðw,3  x2 v,33 Þ  εð1Þ φ,3 þ pð1Þ θ: Similarly, for the semiconductor layer, Eqs. (7.83) and (84) are still applicable. The one-dimensional constitutive relations take the following form with the use of Eqs. (7.108) and (109):

7.5 Extension and Bending of Composite Beams

205

T ¼ cð2Þ S  λð2Þ θ ¼ cð2Þ ðw,3  x2 v,33 Þ  λð2Þ θ, ð2Þ

ð7:112Þ

ð2Þ

D ¼ ε E ¼ ε φ,3 : The currents exist in the semiconductor layer only and therefore the superscript 2 in parentheses is neglected. We still have Eq. (7.85): dðΔpÞ , dx3 dðΔnÞ : J n ffi qn0 μn E þ qDn dx3

J p ffi qp0 μp E  qDp

ð7:113Þ

b shear force Q, bending moment M, and electric displacement The total axial force T, b are defined by the following integrations over a cross section: D Z

Z 0 Tdx2 þ b Tdx2 h2 0  ¼ cð1Þ w,3 þ eð1Þ φ,3  λð1Þ θ h1 b  cð1Þ v,33 2 cð2Þ v,33 2  h1 b þ cð2Þ w,3  λð2Þ θ h2 b þ h2 b, 2 2 Z Q ¼ T 32 dx1 dx2 ,

b¼ T

Z

Tdx1 dx2 ¼ b

h1

Z M¼

Z Tx2 dx1 dx2 ¼ b

h1

0 ð1Þ

Z Tx2 dx2 þ b

ð7:115Þ

0

h2

Tx2 dx2

cð1Þ w,3 þ eð1Þ φ,3  λ θ 2 h1 b 2 cð1Þ v,33 3 λð2Þ θ  cð2Þ w,3 2 cð2Þ v,33 3  h1 b þ h2 b  h2 b, 3 2 3 Z Z h1 Z 0 b D ¼ Ddx1 dx2 ¼ b Ddx2 þ b Ddx2 h2 0  ¼ eð1Þ w,3  εð1Þ φ,3 þ pð1Þ θ h1 b  eð1Þ v,33 2  h1 b þ εð2Þ φ,3 h2 b: 2

¼

ð7:114Þ

ð7:116Þ

ð7:117Þ

The constitutive relation for the shear force Q will be provided by the moment-shear force relationship as to be seen in the following. From the equations of motion (Newton’s second law) of the differential element of the beam in Fig. 7.19 in the x2 and x3 directions as well as its moment equation, we have

206

7 Thermal Effects

Fig. 7.19 A differential element of the beam under axial forces, shear forces, bending moments, and distributed mechanical loads

 b ,3 þ f 3 ðx3 , t Þ ¼ b ρð1Þ h1 þ ρð2Þ h2 w €, T  Q,3 þ f 2 ðx3 , t Þ ¼ b ρð1Þ h1 þ ρð2Þ h2 €v,

ð7:118Þ

M ,3  Q  m1 ðx3 , t Þ ¼ 0, where f2(x3,t) and f3(x3,t) are the transverse and axial loads per unit length of the beam at x2 ¼ 0. m1(x3,t) is the distributed moment per unit length of the beam. Equation (7.118)3 is the moment-shear force relationship needed. Equations (7.118)3 and (7.116) imply that cð1Þ w,33 þ eð1Þ φ,33 2 cð1Þ v,333 3 h1 b  h1 b 2 3 cð2Þ w,33 2 cð2Þ v,333 3  h2 b  h2 b þ m1 , 2 3

Q¼

ð7:119Þ

which serves as the constitutive relation for the shear force. Similar to the derivation of Eq. (7.118), by considering the differential element in Fig. 7.19 under electrical loads, the charge equation of electrostatics can be written as b ,3 ¼ qðΔp  ΔnÞAð2Þ : D

ð7:120Þ

The one-dimensional conservation of holes and electrons are still given by Eq. (7.90): ∂ ðΔpÞ ¼ J p,3 , ∂t ∂ q ðΔnÞ ¼ J n,3 : ∂t q

ð7:121Þ

7.5 Extension and Bending of Composite Beams

207

Consider static deformation of a free beam. There are no mechanical loads. f2 ¼ 0, f3 ¼ 0, and m1 ¼ 0. Equations (7.118), (7.120), and (7.121) reduce to b ,3 ¼ 0, T Q,3 ¼ M ,33 ¼ 0, b 3,3 ¼ qðΔp  ΔnÞAð2Þ , D 0¼ 0¼

J p3,3 , J n3,3 :

ð7:122Þ ð7:123Þ ð7:124Þ

For the mechanically free and electrically isolated beam we are considering, the boundary conditions are b ð 0Þ ¼ T b ðLÞ ¼ 0, T

M ð0Þ ¼ M ðLÞ ¼ 0, b ð 0Þ ¼ D b ðLÞ ¼ 0, Qð0Þ ¼ QðLÞ ¼ 0, D p p n J 3 ð0Þ ¼ J 3 ðLÞ ¼ 0, J 3 ð0Þ ¼ J n3 ðLÞ ¼ 0:

ð7:125Þ

Δp and Δn must satisfy the following global charge neutrality conditions: Z

L

Z Δpdx3 ¼ 0,

0

L

Δndx3 ¼ 0:

ð7:126Þ

0

Only one of Eq. (7.126) is independent. To determine the mechanical displacements and the electric potential uniquely, we set vð0Þ ¼ 0,

vðLÞ ¼ 0,

wðL=2Þ ¼ 0,

φðL=2Þ ¼ 0:

ð7:127Þ ð7:128Þ

Equations (7.122), (7.123), and (7.124) can be written as a fourth-order equation mainly for v, a second-order equation mainly for w, and second-order equations mainly for φ, Δp, and Δn, respectively, with couplings among them. The four equations can be further manipulated into φ,33  k2 φ ¼  v,333 ¼ αφ,33 , w,33 ¼ βφ,33 ,

ðC 1 þ C2 ÞAð2Þ , eε

ð7:129Þ

208

7 Thermal Effects

qp0 μp33 φ þ C1 , Dp33 qn μn qΔn ¼  0n 33 φ þ C 2 , D33

qΔp ¼ 

ð7:130Þ

where C1 and C2 are undetermined constants and Að1Þ ¼ h1 b,

Að2Þ ¼ h2 b, 6cð2Þ eð1Þ h1 h2 ðh1 þ h2 Þ  , α ¼ ð1Þ ð1Þ 4 c c h1 þ cð2Þ cð2Þ h42 þ 2cð1Þ cð2Þ h1 h2 2h21 þ 3h1 h2 þ 2h22   eð1Þ h1 cð1Þ h31 þ cð2Þ h22 ð3h1 þ 4h2 Þ  , β ¼  ð1Þ ð1Þ 4 c c h1 þ cð2Þ cð2Þ h42 þ 2cð1Þ cð2Þ h1 h2 2h21 þ 3h1 h2 þ 2h22 eð1Þ Að1Þ h1 eε ¼ εð1Þ Að1Þ þ εð2Þ Að2Þ þ α  eð1Þ Að1Þ β, 2  p  q p0 μ33 n0 μn33 ð2Þ k2 ¼ þ A : Dn33 eε Dp33

ð7:131Þ

ð7:132Þ

The general solution for φ, v, and w are found to be 1 ðC1 þ C2 ÞAð2Þ , k2eε v ¼ αðC3 cosh kx3 þ C4 cosh k ðx3  LÞÞ þ C 5 x23 þ C6 x3 þ C 7 , φ ¼ kC3 sinh kx3 þ kC 4 sinh kðx3  LÞ þ

ð7:133Þ

w ¼ βðkC 3 sinh kx3 þ kC4 sinh kðx3  LÞÞ þ C 8 x3 þ C9 , where C3-C9 are undetermined constants. Then Δp and Δn are determined from Eq. (7.130). The undetermined constants are governed by Eqs. (7.125), (7.126), (7.127), and (7.128). As an example, consider PZT-5A for the piezoelectric layer and Si for the semiconductor layer with L ¼ 0.6 μm, h1 ¼ h2 ¼ 25 nm, and b ¼ 50 μm. n0 ¼ p0 ¼ 1023/m3. Θ0 ¼ 298 K. Some of these parameters may be varied later. We are particularly interested in the temperature-induced redistributions of holes and electrons. They are described by the Δp and Δn in Fig. 7.20. Figure 7.21 shows Δp and Δn for different n0 when θ ¼ 0.5 K while all other parameters are kept the same as those for Fig. 7.20. The fields are sensitive to p0 and n0. Figure 7.22 shows the change of number of electrons in a differential element with length dz at the left end of the beam, i.e., Δn(0)h2bdz which may be used as a measure of the strength of the effect under consideration. Δn(0)h2bdz is plotted versus h1/h2 when h1 + h2 is fixed. The curves all have a maximum when h1 and h2 are not very different, which is reasonable.

7.5 Extension and Bending of Composite Beams Fig. 7.20 (a) Hole concentration perturbation. (b) Electron concentration perturbation

209

210 Fig. 7.21 Effects of p0 ¼ n0. (a) Hole concentration perturbation. (b) Electron concentration perturbation

7 Thermal Effects

References

211

Fig. 7.22 Δn(0)h2bdz versus h1/h2 for different combinations of materials. p0 ¼ n0 ¼ 1023/m3, θ ¼ 0.5 K, dz ¼ 10 nm

References 1. R.R. Cheng, C.L. Zhang, J.S. Yang, Thermally induced carrier distribution in a piezoelectric semiconductor fiber. J. Electron. Mater. 48, 4939–4946 (2019) 2. R.R. Cheng, C.L. Zhang, W.Q. Chen, J.S. Yang, Electrical behaviors of a piezoelectric semiconductor fiber under a local temperature change. Nano Energy 66, 104081 (2019) 3. R.R. Cheng, C.L. Zhang, W.Q. Chen, J.S. Yang, Temperature effects on PN junctions in piezoelectric semiconductor fibers with thermoelastic and pyroelectric couplings. J. Electron. Mater. 49, 3140–3148 (2020) 4. R.R. Cheng, C.L. Zhang, W.Q. Chen, J.S. Yang, Temperature effects on mobile charges in extension of composite fibers of piezoelectric dielectrics and non-piezoelectric semiconductors. Int. J. Appl. Mech., 11, 1950088 (2019) 5. Y.X. Luo, C.L. Zhang, W.Q. Chen, J.S. Yang, Thermally induced electromechanical fields in unimorphs of piezoelectric dielectrics and nonpiezoelectric semiconductors. Under review

Appendices

Appendix 1: List of Symbols Because of the multiple fields involved and the use of one-, two-, and threedimensional theories, some of the symbols have different meanings in different chapters or sections: ε0 μ0 q kB i * i, j, k, l p, q xk δij ρ fj uk Tkl Skl φ ψ Ei Pi Di Bi Hi p, n þ N
A , ND

Vacuum permittivity Vacuum permeability Elementary charge Boltzmann constant Imaginary unit Complex conjugate Three-dimensional tensor indices. Range: 1, 2, 3 Three-dimensional matrix indices. Range: 1, 2, 3, 4, 5, 6 Cartesian coordinates Kronecker delta Mass density Distributed body force Mechanical displacement vector Stress tensor Strain tensor Electrostatic potential Magnetic potential Electric field Electric polarization per unit volume Electric displacement Magnetic flux or induction Magnetic field Concentrations of holes and electrons Concentrations of ionized accepters and donors

© Springer Nature Switzerland AG 2020 J. Yang, Analysis of Piezoelectric Semiconductor Structures, https://doi.org/10.1007/978-3-030-48206-0

213

214

Appendices

p 0, n 0 p 0, n 0 Δp, Δn ρe ρt Qe ρP σP J pi , J ni cEijkl , sEijkl eijk, dijk εSij , εTij μpij , μnij Dpij , Dnij T Θ0 θ η hi λi,j pj κij Tn DTp ij , Dij

þ N
A , ND p|φ ¼ 0, n|φ ¼ 0 Carrier concentration perturbations Sum of doping and mobile charge densities Total charge density Charge Volume effective polarization charge density Area effective polarization charge density Current densities of holes and electrons Elastic stiffness and compliance Piezoelectric constants Dielectric constants Mobility of holes and electrons Diffusion constants of holes and electrons Absolute temperature Reference temperature Temperature deviation from Θ0 Entropy density Heat flux Thermoelastic constants Pyroelectric constants Heat conduction coefficients Thermal diffusion constants

Tn κEij , σ Tp ij , σ ij hijk αij μij

Thermoelectric constants

2

Piezomagnetic constants Magnetoelectric constants Magnetic permeability Electromechanical coupling factor

ð0Þ

Extensional force in rods Bending moment in beams Shear force in beams; material quality factor Effective material constants of rods/beams (Chaps. 3 and 4) Effective material constants of plates (Chap. 5) Effective material constants of plates (Chap. 5)

k215 , k15 N M Q c33 , e33 , ε33 cpq , eip , εij cðpq0Þ , eip γ rs, ψ ks, ζ kj κ, κ1, κ2 a, b, c, d b1 ω1 , ω k λD

Effective material constants of plates (Chap. 5) Thickness-shear correction factors of plates (Chap. 5) Two-dimensional tensor indices of plates (Chaps. 5 and 6) Fundamental thickness-shear frequency (Chaps. 5 and 6) Wave number Debye-Hückel length

Appendices

215

Appendix 2: Material Constants Vacuum permittivity ε0 ¼ 8.854  10
12 F/m Vacuum permeability μ0 ¼ 12.57  10
7H/m Elementary charge q ¼ 1.602  10
19 C Boltzmann constant kB¼1.381  10
23 J/K kBT/q ¼ 0.0259V at room temperature 300 K [1] Aluminum nitride (AlN) [2] ρ ¼ 3260 kg=m3

0

345 125 120 B 125 345 120 B B   B 120 120 395 cpq ¼ B B 0 0 0 B B @ 0 0 0 0

0 

0

0 0

0 118

0 0

0

118

0

0

0

0 0

 B eip ¼ @

0 0

0 0

0 0

1 0 0 C C C 0 C C  109 N=m2 0 C C C 0 A 110

1 0 C 0 A C=m2


0:48 0

0
0:48


0:58


0:58 1:55 0 0 0 0 1 8:0 0 0   B C εij ¼ @ 0 8:0 0 A  10
11 F=m 0 0 9:5

Barium titanate (BaTiO3) [3] ρ ¼ 5800 kg=m3

2

166:0 6 77:0 6 6   6 78:0 cpq ¼ 6 6 0 6 6 4 0 0 

2

77:0 166:0

78:0 78:0

0 0

0 0

78:0

162:0

0

0

0 0

0 0

43:0 0

0 43:0

0

0

0

0

0

 6 ejq ¼ 4 0
4:4

0 0
4:4

0

0

0 11:6 18:6 0

3 0 0 7 7 7 0 7 7  109 N=m2 0 7 7 7 0 5 44:5

11:6 0 0 0

3

7 0 5 C=m2 0

216

Appendices

2

11:2   6 εij ¼ 4 0

3 0 7 0 5  10
9 F=m 12:6

0 11:2

0

0

Cadium selenide (SdSe) [4]

2

ρ ¼ 4820 kg=m3 90:7

6 58:1 6 6   6 51:0 cpq ¼ 6 6 0 6 6 4 0 0 2   6 eip ¼ 4

58:1

51:0

0

0

90:7

51:0

0

0

51:0 0

93:8 0

0 15:04

0 0

0 0

0 0

0 0

15:04 0

0 0

0 0

0 0

0

0
0:21

3

0 7 7 7 0 7 7  109 N=m2 0 7 7 7 0 5 16:3

3 0 7 0 5 C=m2


0:21 0


0:24


0:24 0:44 0 0 2 3 9:02 0 0   6 7 εij ¼ 4 0 9:02 0 5 ε0 0 0 9:53

0

Germanium (Ge) [4, 5]

2

ρ ¼ 5327 kg=m3 128:9

48:3

48:3

0

6 48:3 128:9 48:3 0 6 6   6 48:3 48:3 128:9 0 cpq ¼ 6 6 0 0 0 67:1 6 6 4 0 0 0 0 0 0 0 0 2 0:1398932 0   6 0 0:1398932 εij ¼ 4 0

0

0

0

0 7 7 7 0 7 7  109 N=m2 0 7 7 7 0 5

0 0 0 67:1 0 0 0

67:1 3

0:1398932

μn ¼ 3900 cm2 =V  s μp ¼ 1900 cm2 =V  s

3

7 5  10
9 F=m

Appendices

217

Lithium niobate (LiNbO3) [6, 7] ρ ¼ 4700 kg=m3 ,

0

2:03 B 0:53 B B   B 0:75 cpq ¼ B B 0:09 B B @ 0

0:75 0:09 0:75
0:09

0 0

0:75
0:09

2:45 0

0 0:60

0 0

0

0

0

0:60

1 0 0 C C C 0 C C  1011 N=m2 0 C C C 0:09 A

0

0

0

0:09

0:75

0:53 2:03

0 0

0   B eip ¼ @
2:50 0:20

0

0 2:50

0 0

0:20

1:30

0

0

0

38:9   B εij ¼ @ 0 0

0 3:70 3:70 0 0

1

1
2:50 C 0 A C=m2 0

C 0 A  10
11 F=m 25:7

38:9 0

Lithium tantalate (LiTaO3) [6, 7] ρ ¼ 7450 kg=m3 ,

0

2:33 B 0:47 B B   B 0:80 cpq ¼ B B
0:11 B B @ 0 0 

PZT-2 [4]

eip



0:47 2:33

0:80
0:11 0:80 0:11

0 0

0:80

2:75

0

0


0:11 0

0 0

0:94 0

0 0:94

0

0

0


0:11

0

0 0 B ¼ @
1:6 1:6 0 0 0 36:3   B εij ¼ @ 0 0

0

0

2:6

0 1:9

2:6 0

0 0

0 36:3 0

0

1

0 0

1

C C C 0 C C  1011 N=m2 0 C C C
0:11 A 0:93 1
1:6 C 0 A C=m2 0

C 0 A  10
11 F=m 38:2

218

Appendices

ρ ¼ 7600 kg=m3 ,

0

13:5 6:79 B 6:79 13:5 B B   B 6:81 6:81 cpq ¼ B B 0 0 B B @ 0 0 0

0

6:81 6:81

0 0

0 0

11:3

0

0

0 0

2:22 0

0 2:22

0

0

0

0

0

0

0

  B eip ¼ @ 0
1:9

1 0 0 C C C 0 C C  1010 N=m2 0 C C C 0 A 3:36

0

9:8 0

C 0 A C=m2 0 1

0 0 9:8
1:9 9:0 0 0 504ε0 0   B 504ε0 εij ¼ @ 0

0 0

0

260ε0

0

C A

0

0

1

PZT-4 [3]

2

ρ ¼ 7600 kg=m3 138:5

6 77:37 6 6   6 73:64 cpq ¼ 6 6 0 6 6 4 0 0 2 

0

77:37

73:64

0

0

0

138:5 73:64

73:64 114:8

0 0

0 0

0 0

0 0

0 0

25:6 0

0 25:6

0

0

0

0

0

0

0

3

7 7 7 7 7  109 N=m2 0 7 7 7 0 5

30:6 12:72 0

3

 6 7 0 0 12:72 0 0 5 C=m2 ejq ¼ 4 0
5:2
5:2 15:08 0 0 0 2 3 13:06 0 0   6 7 εij ¼ 4 0 13:06 0 5  10
9 F=m 0 0 11:15

PZT-5A [3] ρ ¼ 7750 kg=m3

Appendices

219

2

99:201 6 54:016 6 6   6 50:788 cpq ¼ 6 6 0 6 6 4 0

 6 ejq ¼ 4

50:778 50:778

0 0

0 0

50:788

86:856

0

0

0 0

0 0

21:1 0

0 21:1

0

0

0

0

0

2 

54:016 99:201

0

0

0

3 0 0 7 7 7 0 7 7  109 N=m2 0 7 7 7 0 5 22:6

0

12:322

0

3

7 0 0 0 12:322 0 0 5 C=m2
7:209
7:209 15:118 0 0 0 2 3 15:3 0 0   6 7 15:3 0 5  10
9 F=m εij ¼ 4 0 0 0 15:0

PZT-5H [4] ρ ¼ 7500 kg=m3

0

12:6 7:95 B 7:95 12:6 B B   B 8:41 8:41 cpq ¼ B B 0 0 B B @ 0 0 0 0

8:41 8:41

0 0

0 0

11:7

0

0

0 0

2:30 0

0 2:30

0

0

0

0 0

  B eip ¼ @ 0
6:5

0

0

0

1 0 0 C C C 0 C C  1010 N=m2 0 C C C 0 A 2:33 17:0 0

0 0 17:0
6:5 23:3 0 0 1700ε0 0   B 1700ε0 εij ¼ @ 0

0 0

0

1470ε0

0

Silicon (Si) [4, 5] ρ ¼ 2332 kg=m3

0 0

1

C 0 A C=m2 0 1 C A

220

Appendices

0

16:57 B 6:39 B B   B 6:39 cpq ¼ B B 0 B B @ 0

6:39 16:57

6:39 6:39

0 0

0 0

0 0

6:39

16:57

0

0

0

0 0

0 0

7:956 0

0 7:956

0 0

0

0

0

0

7:956

0

0

11:7

0

  B εij ¼B @ 0 2 6 ¼6 4

0

C 0 C Aε0 11:7

0

C C C C C  1010 N=m2 C C C A

1

0

11:7

1

3

0:1035918

0

0

0

0:1035918

0

0

0

0:1035918

7 7  10
9 F=m 5

μn ¼ 1500 cm2 =V  s μp ¼ 450 cm2 =V  s Zinc oxide (ZnO) [4]

0

ρ ¼ 5680 kg=m3 20:97

12:11

B 12:11 20:97 B B   B 10:51 10:51 cpq ¼ B B 0 0 B B @ 0 0 0 0 0 0   B eip ¼ @ 0

10:51

0

0

10:51

0

0

21:09 0 0 4:247 0 0

0 0

0 0

0 0


0:573 1:32 0 8:55 0   B εij ¼ @ 0 8:55 0

4:247 0 4:43

0

0

1

0 C C C 0 C C  1010 N=m2 0 C C C 0 A

0 0

0
0:48


0:573

0


0:48 0 1

0

0 C 0 A ε0 10:2

1 0 C 0 A C=m2 0

Appendices

221

References 1. R.F. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, Reading, 1996) 2. K. Tsubouchi, K. Sugai, N. Mikoshiba, AlN material constants evaluation and SAW properties on AlN/Al2O3 and AlN/Si, in Proceedings of the IEEE Ultrasonics Symposium, (1981), pp. 375–380 3. F. Ramirez, P.R. Heyliger, E. Pan, Free vibration response of two-dimensional magneto-electroelastic laminated plates. J. Sound Vib. 292, 626–644 (2006) 4. B.A. Auld, Acoustic Fields and Waves in Solids, vol 1 (Wiley, New York, 1973) 5. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981) 6. A.W. Warner, M. Onoe, G.A. Couqin, Determination of elastic and piezoelectric constants for crystals in class (3m). J. Acoust. Soc. Am. 42, 1223–1231 (1967) 7. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)

Index

A Abrupt junction, 14 Aluminum nitride (AlN), 215 Antiplane crack, 24–26 Antiplane motions, 141 Antiplane problems, 7 Antiplane waves, 27–29 Axial electric polarization, 184, 191 Axial forces, 206

B Barium titanate (BaTiO3), 215 Beam arrays, 111 Beams bending, e15 axial electric field, 167 boundary conditions, 166 cantilever beam, 165 carrier concentration perturbations, 164 constitutive relations, 163 differential element, 165 electric field components, 163 geometric parameters, 167 global charge neutrality conditions, 166 holes and electrons conservation, 165 integration constant, 167 linear ordinary differential equations, 166 material constants, 163 moment equation, 165 nonpiezoelectric semiconductor layer, 161, 162 normal stress, 161 piezoelectric dielectric layer, 161 rotatory inertia, 165 shear deformation, 162

shear force, 164 strains and substitute, 163 stress components, 163 transverse shear force F, 161 undetermined constant, 166, 167 Beams bending, e33 bending moment M, 156 boundary conditions, 159 cantilever static bending, 158 differential element, 157, 160, 162 electric displacement, 157 electric potential and field, 155 elementary/classical, 157 global charge neutrality conditions, 160 integration, 157 lower ceramic layer, 156 mechanically induced redistribution, 160, 161 moment equation, 157 nonpiezoelectric semiconductor layer, 155 one-dimensional conservation, 158 piezoelectric semiconductor layer, 155 PZTs and BaTiO3, 160 reference carrier concentrations, 160 rotatory inertia, 157 shear deformation, 155 static, 160 stress and electric displacement, 156 stress approximation, 156 undetermined constants, 159 Bending equations (see One-dimensional equations) shear deformation, 93 static bending (see Cantilever static bending)

© Springer Nature Switzerland AG 2020 J. Yang, Analysis of Piezoelectric Semiconductor Structures, https://doi.org/10.1007/978-3-030-48206-0

223

224 Bending (cont.) time harmonic, 98 transient bending, 102 Bending equations compact matrix notation, 122 energy density function, 123 extensional displacements, 124 first-order constitutive relations, 123 first-order strains, 121 higher-order plate displacements, 121 mechanical, 121 relaxed material constants, 122 shear deformation, 121 thickness expansion/contraction, 122 thickness-shear deformation, 122 thickness-shear vibration, 123 transverse shear forces, 122 zero-order constitutive relations, 121 zero-order plate shear strains, 122 Bending moment M, 205 Bending without shear deformation, reduction, 124 Biasing fields definition, 85 dynamic incremental fields, 86, 87 one-dimensional equations, 85 piezoelectric semiconductor rod, 85, 86 Bleustein-Gulyaev wave, 146 Boltzmann constant, 2, 215 Boltzmann distribution, 35 Boundary-value problem, 105, 151 Buckling across nodal points, 111 axial force, 108 carrier concentration perturbation, 110 electromechanical coupling, 109 linear homogeneous equations, 109 load, 109 modes, 109, 110 piezoelectric dielectric beam, 110 piezoelectric semiconductor beam, 111 shear deformation, 108 zero deflection and bending moment, 109 ZnO beam, 108 Built-in electric field, 18, 23 Built-in potential, 18

C Cadium selenide (SdSe), 216 Cantilever static bending arbitrary constants, 97

Index boundary conditions, 96 carrier concentration perturbation, 96 electron concentration perturbation, 97 end force, 97, 98 geometric parameters, 97 linear algebraic equations, 97 linearized theory, 97 n-type semiconductor, 95 ZnO beam, 95 Carrier concentration perturbation, 101 Classical theory, 73 Clausius-Duhem inequality, 10 Composite rod extensions average extensional stress, 152 axial force, 149 boundary-value problem, 151 constitutive relations, 148 differential element, 150, 154 electric displacement, 150 electron concentration perturbation, 152–154 electrostatics charge equations, 150 global charge neutrality conditions, 151 hyperbolic functions, 153 initial carrier concentration, 153 mechanical displacement, 151 n-type semiconductor, 152 piezoelectric dielectric layers, 147, 148 piezoelectric layers, 155 polarization charge density, 151 semiconductor, 149 static, 150 strain-displacement relation, 148 three-dimensional material constants, 148 COMSOL commercial software, 53 COMSOL numerical solution, 106 Continuity equation, 1 Crystals of class equation constitutive relations, 126 electrostatics, 126, 127 extension, 125, 127 flexure, 127 material matrices, 125 semiconduction, 127, 128 shear, 125 stress relaxation, 126 thickness-shear correction, 126 ZnO, 125 Current-voltage relation/I-V curve, 57, 83, 186, 195 Curvilinear coordinate system, 132

Index D Debye-Hückel equation, 35 Debye-Hückel length, 35, 214 Dielectric constants, 148 Diffusion, 14, 16, 35, 74, 170, 172 Dispersion curves, 29 Dispersion relations, 28, 146 Dispersive wave, 147 Dissipative wave, 29 Doping profiles, 36 Drift, 35 Driving frequency, 100

E Efficiency, 65–67 Einstein relation, 2 Elastic constants, 144, 152 Elastic stiffness, 147 Electric fields, 143 Electrically nonlinear extension boundary conditions, 50 charge neutrality condition, 50 electric potential fields, 50 first-order problem, 51 first-order solution, 52 mechanical displacement, 50 notation, 48 perturbation, 50, 51 piezoelectric coupling, 54 piezoelectric semiconductor rod, 48 polarization/polarization charge density, 53 second-order problem, 51, 52 second-order solution, 52, 53 static, 49 uniform doping, 50 Electromechanical coupling, 109 Electromechanical fields, 177, 186 Electromechanical wave, 74 Electron concentration distribution, 53 Electron concentration perturbation, 108 Electron distribution, 106 Electrostatics charge equation, 150, 165 Energy density function, 123 Energy harvester, 61 Error function, 42 Ettingshausen effects, 11 Extension equations extensional displacement, 121 index convention, 120 in-plane extensional displacements, 119 plate material constants, 120 Poisson’s effect, 119, 120

225 stress relaxation, 120 zero-order equations, 119 Extensional deformation, 196 Extensional waves, 106

F Face-shear (FS), 145 Fiber, 79 Films, 113 First-order constitutive relations, 123 First-order shear deformation theory, 134 Flexure, 127 Four second-order equations, 179, 183 Free vibration analysis, 65 Free vibration modes, 104 Fundamental thickness-shear frequency, 214

G Gauss’s law, 1 Germanium (Ge), 216 Global charge neutrality conditions, 37, 151, 160, 166, 190

H Harmonic extensional waves, 73 Harmonic vibration boundary conditions, 63 circuit equation, 63 coefficient matrix, 62 complex conjugate, 66 displacement fields, 65, 66 effective polarization charge, 61 finite/static response, 65 linear equation, 63 linear homogeneous equations, 62 material damping, 64 n-type semiconductor, 62 open-circuit vibration characteristics, 64 piezoelectric semiconductor rod, 61 piezoelectric stiffening effect, 65 reference carrier concentration, 63 resonance frequencies, 64–66 undetermined constants, 63 uniform doping, 62 Heterogeneous junctions, 75, 172, 175, 188, 191 Homogeneous junction, 172, 174, 193 Homogeneous piezoelectric semiconductor rod, 153

226 Homogeneous rods extension axial strain and electric field, 178 carrier concentrations, 177 differential equations, 179 electromechanical fields, 177 electron concentration perturbation, 181, 182 four second-order equations, 179 free piezoelectric semiconductor rod, 177, 178 global charge neutrality conditions, 180 one-dimensional material constants, 178 pyroelectric/piezoelectric coupling, 180 static, 179 temperature-induced carrier concentration perturbation, 181 thermoelastic and piezoelectric effects, 181 thermoelectric couplings, 178 Hyperbolic functions, 47, 153 Hyperbolic wave, 71

I Index convention, 116, 144 Initial-boundary-value problem, 71 In-plane extensional displacements, 119 Integration, 154 Isotropic piezoelectric materials, 5

J Junction interface, 171

K Kirchhoff classical/elementary theory, 124

L Lamè coefficients, 132 Laplace transform, 71 Linear algebraic equations, 97 Linear analytical analysis, 183 Linear doping, 35 Linear extension, end forces axial stress, 47 carrier distribution, 45 charge neutrality conditions, 46 concentrated free charges, 46 electromechanical fields, 47 hyperbolic functions, 47 mechanical displacement, 46 mobile charge redistribution, 48

Index numerical calculation, 47 static extension, 45, 46 symmetric/antisymmetric fields, 47 uniform doping, 46 Linear homogeneous equations, 146 Linearization, 3 Linearized constitutive relations, 6, 183 Linearized theory, 13 Linearly graded PN junction boundary and continuity conditions, 37 charge neutrality conditions, 37 doping profiles, 36 electric field and potential, 39 global charge neutrality condition, 37 piezoelectric semiconductor, 36 symmetry/antisymmetry, 38 transition zone, 38, 39 Linear theoretical analysis, 75 Lithium niobate (LiNbO3), 217 Lithium tantalate (LiTaO3), 217 Local extension boundary and continuity conditions, 56 and compression, 58 displacement and potential fields, 57 electromechanical fields, 57 electron distribution, 48, 57 governing equations, 54 local stress, 57 nonlinear numerical analysis, COMSOL, 57 piezoelectric constant, 57 piezoelectric semiconductor rod, 54, 55 polarization charge density, 57, 58 potential barrier and well, 57 static problems, 55 successive substitutions, 55 undetermined constants, 56 ZnO rod, 56 Local temperature change effects axial electric polarization P, 184 boundary conditions, 186 COMSOL nonlinear analysis, 186 numerical analysis, 186 critical voltages, 187 four second-order equations, 183 I-V curve, 186 linear analytical analysis, 183 linearized constitutive relations, 183 material constants, 183 nonlinear solution, 186, 188 one-dimensional governing equations, 182 ordinary differential equations, 184 piezoelectric semiconductor rod, 183, 187

Index potential barriers, 182 potential distribution, 187 thermally induced local polarization, 186

M Macroscopic theory, 34 holes and electrons, 2 mechanical displacement vector, 3 mobility, 3 piezoelectric constitutive, 2 Magnetic couplings constants, 11 quasistatic approximation, 11 Magnetoelectric constants, 11 Material constants, 183 MATLAB, 16, 22, 172 Mechanical-to-electrical energy converter, 65 Membrane theory, 140 Metal conductors, 18 Mobile charge redistribution, 167 Modes, 65 Moment equation, 205 Moment-shear force relationship, 157, 205

N Nonlinearity, 186 Nonlinear waves amplitude, 75, 76 approximations, 72 classical theory, piezoelectricity, 73 COMSOL, 75 cross-sectional dimension, 75 electric field, 74 electric potential, 74 low doping/low carrier concentration, 73 nonlinear drift current, 74 n-type semiconductor, 72, 73 sinusoidal, 75 trigonometric series, 75 Non-piezoelectric, electromechanical coupling, 147 Non-piezoelectric semiconductor plate, 144 n-type semiconductor, 95

O One-dimensional equations approximation, 34 axial displacement, 31 carrier concentration perturbations, 34, 90

227 constitutive relations, 91 cross section, 92 deformation and weak fields, 89 displacement, 90 equilibrium, 33 integration constants, 33, 34 lateral stress component relaxation, 31 material constants, 91 mechanical displacement, 35 motion, 92 nonlinear constitutive relations, 32 one-dimensional field equations, 32 piezoelectric semiconductor rod, 31, 32 polarization charge density, 35 rod extensional deformation, 31 rod lateral surface, 31 semiconductor beam, ZnO, 89, 90 shear deformation, 93 strain-displacement/electric field potential, 32 strains and substitute, 91 stress components, 91 three-dimensional material constants, 32 zero-/first-order moments, 92 One-dimensional field equations, 179 One-dimensional material constants, 178 One-dimensional problem, 13 Open-circuit vibration characteristics, 64 Optimal thickness ratio, 167 Ordinary differential equations, 190

P p-doped nonpiezoelectric semiconductor plate, 141, 142 Periodic doping, 44, 45 Periodic extension/compression charges and potentials, 60 distributed body force effect, 59 polarization charges, 60 Phenomenological theory, 1 Piezoelectrically stiffened elastic constant, 69, 202 Piezoelectric constitutive relations, 5 Piezoelectric coupling, 54, 104, 180 Piezoelectric dielectric layers, 147 Piezoelectric semiconductors, 1, 13, 14 Piezoelectric stiffening effect, 65 Piezomagnetic effects, 11 Plate constitutive equations, 116 Plate thickness-shear strains, 124 PN junction, 14–18, 21, 22

228 PN junction extension boundary and continuity conditions, 77 carrier concentration perturbations, 78 COMSOL nonlinear solution, 79 numerical analysis, 85 numerical solution, 82 current-voltage relation/I-V curve, 83 global charge neutrality conditions, 77 heterogeneous, 75, 78, 79, 82–85 jump dicontinuity, 80 linear analytical vs. nonlinear numerical solutions, 82, 83 linear theoretical analysis, 75 one-dimensional governing equations, 77 piezoelectric semiconductor rod, 75, 76 polarization charge density, 77, 79, 81 polarization fields, 79, 80 second-order ordinary differential equations, 77 symmetric/antisymmetric curves, 80 undetermined constants, 79 PN junctions bending axial electric polarization P3, 169 carrier concentration distributions, 173 charge neutrality conditions, 171 electromechanical fields, 170 field equations, 168 heterogeneous, 172 homogeneous, 168, 174 junction interface, 171 MATLAB, 172 one-dimensional constitutive relations, 169 p and n distributions, 172 piezoelectric semiconductor beams, 168 piezotronic devices, 172 undetermined constants, 170 Poisson-Boltzmann equation, 35 Poisson’s effect, 119, 120, 122 Poisson’s equation, 35 Polarization charges, 43, 54, 60, 191, 193, 195, 200 Polarization vector, 151 Polarized ceramics (PZT), 147 Potential barriers, 60, 186, 187 Potential wells, 60, 186, 187 Power series, 113, 134 Pyroelectric constants, 10 Pyroelectric coupling, 180, 198

R Relaxed material constants, 122 Resonance frequencies, 64 Resonances, 100

Index Rotatory inertia, 124, 165

S Second-order equation, 207 Semiconductors, 18 Semi-infinite crack, 24 Shear correction factors, 122, 123 Shear deformation beam shear strain, 94 constitutive relations, 94, 95 extension, 93 field equations, 94 inertia, 95 Shear electromechanical coupling factor, 130 Shear force Q, 164, 205 Shear-horizontal (SH) waves, 27–29, 146 Shell structures bending shear deformations, 139 and twisting moments, 139 carrier concentration perturbations, 134, 136 curvilinear coordinate systems, 132 dependence, 134 differential shell element, 132, 133 electric displacement/surface charge, 138 mechanical resultants/surface loads, 137 membrane theory, 140 scalar field gradient, 133 shear correction factors, 138 strain-displacement relations, 132 stress relaxations, 139 three-dimensional constitutive relations, 138 transverse shear forces, 136 two-dimensional charge equations, 136 two-dimensional continuity equations, 137 two-dimensional equations, 132, 136 vector field divergence, 134 Silicon (Si), 147, 219 Sinusoidal wave motion, 74 Smoothly graded PN junction boundary conditions, 40 convergence domain, 42 doping profiles, 40, 42 error function, 41 local charge distribution, 40 mobile/polarization charge effect, 42, 43 recurrence relation, 41 Static bending analysis, 100 Static extension, 179 Stress relaxation, 143, 163 Surface waves anti-plane motions, 141 boundary/continuity conditions, 143

Index coefficient matrix, 146 compact matrix notation, 144 constitutive relations, 144, 145 decaying behavior, 142 dispersion relations, 146 dispersive and dissipative, 147 elastic and dielectric constants, 145 elastic half space, 147 electric fields, 143 electric potential, 145 FS, 145 plate thickness, 144 possible solutions, 142 propagation, 141 semiconductor plate, 143 SH, 146 two-dimensional Laplacian, 142 zero-order equations, 143 Symmetric/antisymmetric fields, 47

T Temperature effects, composite beams carrier concentration perturbations, 204 constitutive relations, 204, 205 differential element, 208, 211 electric potential and field, 204 electrostatics charge equation, 206 free and isolated beams, 207 global charge neutrality conditions, 207 hole/electron concentration perturbation, 209, 210 integrations, 205 moment equation, 205 moment-shear force relationship, 206 nonpiezoelectric semiconductor layer, 203, 204 one-dimensional conservation, 206 piezoelectric dielectric layer, 203, 204 second-order equation, 207 static deformation, 207 temperature-induced redistributions, 208 undetermined constants, 208 Temperature sensitivity, 196 Thermal couplings electrostatics, 9 holes and electrons, 10 Thermal effects, composite rods extension axial strain and electric field, 197 carrier concentration, 201, 202 carrier distribution, 201 cross-sectional areas, 199 differential element, 199, 202 electric displacement, 199

229 electron concentration perturbation, 202 extensional deformation, 196 global charge neutrality conditions, 200 material constants, 197 mechanical displacement, 200 nonpiezoelectric semiconductor layer, 196, 197 one-dimensional continuity equations, 199 piezoelectric dielectric layers, 196, 197 polarization charge density/vector, 200 pyroelectric coupling, 198 Thermally induced local polarization, 186 Thermal/temperature effects, PN junctions boundary and continuity conditions, 189 built-in fields, 190, 192, 193 global charge neutrality conditions, 190 heterogeneous, 188, 191, 195 heterogeneous vs. homogeneous junction, 193, 194 homogeneous, 193 linear equations, 191 material parameters, 190 numerical analysis, COMSOL, 195 one-dimensional material parameters, 195 polarization charge density, 191, 193, 195 symmetric/antisymmetric curves, 193 temperature sensitivity, 196 undetermined constants, 190, 191 ZnO rod, 188, 189 Thermoelastic constants, 10 Thermoelectric constants, 10 Thermoelectric couplings, 178 Thickness-shear approximation, 162 differential relationship, 130 electromechanical coupling factor, 130 long-wave approximation, 130 propagating waves, 129 small flexural deformation, 129 waves, 130 Thickness-shear correction, 126 Thickness-shear vibrations correction factor, 129 first-order plate equations, 128 nodal planes, 128 three-dimensional equations, 129 time-harmonic motions, 129 unelectroded plate, 128 Thickness-shear waves propagation dispersion relations, 131 dispersive and dissipative, 132 frequency, 132 homogeneous equation, 131 piezoelectric dielectric plate, 132 straight-crested waves, 131

230 Three-dimensional equations recapitulation carrier concentration perturbation, 114 piezoelectric constitutive relations, 113 uniform doping, 114 Three-dimensional material constants, 148 Three-dimensional phenomenological theory, 1 Time-harmonic boundary conditions, 99 differential equations, 99 linear homogeneous algebraic equations, 100 n-type semiconductor, 98 polynomial equations, 100 Time-harmonic bending carrier concentration perturbation, 101, 102 linear algebraic equations, 100 linear theory, 100 resonances, 100, 101 static bending analysis, 100 ZnO beam, 98, 99 Total charge density, 192 Transient bending approximation, 105 boundary and initial conditions, 103 electron concentration perturbation, 107, 108 electron distribution, 106 extensional waves, 106 free vibration modes, 104 horizontal reference state, 106 initial-boundary-value problem, 105 linear combination, 105 piezoelectric coupling, 104 shear deformation, 102 shear force boundary condition, 104 standard mathematical problem, 105, 106 theoretical analysis, 104 variable separation, 104 ZnO beam, 102, 103 Transient vibration end force, 69 hyperbolic wave equation, 71 initial and boundary conditions, 68, 69 initial-boundary-value problem, 71 integration, 71 Laplace transform, 71 one-dimensional governing equations, 67

Index piezoelectrically stiffened elastic constant, 69 piezoelectric semiconductor rod, 67, 68 piezoelectricity subproblem, 68 propagation, 70 time instants, 70–72 variables separation method, 69 Transverse shear force, 92, 155 Two-dimensional equations/model, hierarchy electromechanical loads, 116 index convention, 116 integration, 115 plate constitutive equations, 116 plate thickness, 116 power series, 114 uniform thickness, 114 Two-dimensional gradient operator, 142 Two-dimensional Laplacian, 8, 142 Two-dimensional problem, 13

U Undetermined constants, 143, 167 Uniform doping, 13, 46, 149, 157

V Variables separation method, 69 Vibration modes, 65 Voigt’s anisotropic plate elastic constants, 121

Z Zero- and first-order equations carrier concentration gradients, 117 perturbations, 117 displacements, 117 motion, 118 plate constitutive relations, 118 plate extension/bending problems analysis, 117 shear forces, 119 surface loads, 119 Zero-order equations, 143 Zinc oxide (ZnO), 220 ZnO piezoelectric semiconductor rod, 181