Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds [1 ed.] 9781614705826, 9781608761203

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

CONDENSED MATTER RESEARCH AND TECHNOLOGY SERIES

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PHYSICAL PROPERTIES OF THE LOW-DIMENSIONAL A3B6 AND A3B3C 62 COMPOUNDS

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Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

CONDENSED MATTER RESEARCH AND TECHNOLOGY SERIES Electron Beam Modification of Solids: Mechanisms, Common Features and Promising Applications Sergey V. Mjakin, Maxim M. Sychov and Inna V. Vasiljeva (Editors) 2009. ISBN: 978-1-60741-780-4

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Physical Properties of the Low-Dimensional A3B6 and A3B3C 62 Compounds Alexander M. Panich and Rauf M. Sadarly 2010. ISBN: 978-1-60876-120-3

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

CONDENSED MATTER RESEARCH AND TECHNOLOGY SERIES

PHYSICAL PROPERTIES OF THE LOW-DIMENSIONAL A3B6 AND A3B3C 62 COMPOUNDS

ALEXANDER M. PANICH AND Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

RAUF M. SARDARLY

Nova Science Publishers, Inc. New York

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

Copyright © 2010 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Panich, Alexander M. Physical properties of the low-dimensional A3B6 and A3b3C62 compounds / Alexander M. Panich and Rauf M. Sardarly. p. cm. Includes index. ISBN:  (eBook)

1. Low-dimensional semiconductors. 2. Chalcogenides--Electric properties. 3. Lattice dynamics. 4. Ferroelectricity. I. Sardarly, Rauf M. II. Title. QC611.8.L68P36 2009 537.6'226--dc22 2009035691

Published by Nova Science Publishers, Inc.  New York

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CONTENTS Preface

vii

Acknowledgments

xi

Chapter 1

Crystal Structure at Ambient Conditions

1

Chapter 2

Transport Properties under Ambient Conditions

11

Chapter 3

Experimental Studies of the Electronic Structure

21

Chapter 4

Band Structure Calculations

37

Chapter 5

Transport Properties and Semiconductor-Metal Phase Transitions under High Pressure

59

Chapter 6

Phase Transitions in Chain-Type Crystals under Ambient Pressure

79

Chapter 7

Ferroelectric Phase Transitions and Incommensurate States in Layered Crystals

87

Chapter 8

Chapter 9

Relaxor Behavior and Nanodomain Structure of Doped and Irradiated TlInS2 Crystals 3

6

3

3

133

6

Lattice Dynamics in A B and A B C 2 Compounds with TlSeType Structure

Chapter 10

149 3

6

Vibration Spectra of Mixed Crystals (Solid Solutions) of A B and

A3 B 3C26 Compounds

179

Chapter 11

Elastic Properties of Low-Dimensional Compounds

229

Chapter 12

Related Low-Dimensional Compounds

251

Chapter 13

Eventual Applications of TlX and TlMX2 Compounds

271

Conclusion

277

Index

279

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PREFACE Over the past decades there has been considerable interest in the physics of lowdimensional materials that exhibit highly anisotropic properties. In these crystals, the atomic arrangement is such that the electrons are constrained to move preferentially in only one or two directions, and thus the systems are described as having reduced dimensionality, which has some unusual consequences that are responsible for the intense development in this field. Examples of low-dimensional systems include graphite and graphite intercalation compounds, graphene, boron nitride, organic conductors, molybdenum bronzes, etc. Among them, there are several families of semiconductor crystals such as layered transition metal dichalcogenides MX2, where M = Ta, Ti, W, Nb and X = S, Se, Te, binary layered A3B6 semiconductors GaSe, GaS, InSe and GaTe, etc. The electronic structure of these crystals has been studied for decades both theoretically and experimentally because of their interesting quasi-two-dimensional crystallographic structure. The aim of the present monograph is to provide readers with a comprehensive review of the physical properties of TlX and TlMX2 (M = Ga, In; X = Se, S, Te) compounds. These binary and ternary chalcogenides belong to A3B6 and A3B3C62 families of low-dimensional semiconductors possessing a chain or layered structure. In most of the layered crystals, strong covalent chemical bonds are present inside the layers while the interactions between the layers are very weak and usually are described as van der Waals-like. As a result, there is strong dispersion of the energy bands parallel to the layers while only small dispersion is usually observed perpendicular to the layers. The aforementioned TlX and TlMX2 compounds are of significant interest because of their highly anisotropic properties, semiconductor and photoconductor behavior, and potential applications for optoelectronic devices. They exhibit non-linear effects in the current-voltage characteristics (including a region of negative differential resistance), switching and memory effects. The electrical conductivity of several chain-type crystals exhibit time oscillations and intermittency. Second harmonic optical generation has been reported in TlInS2. Layered TlMX2 compounds were the first lowdimensional semiconductors in which a series of phase transitions with modulated structures was discovered. The low-temperature phase of the above crystals was found to be ferroelectric, while the intermediate phase reveals an incommensurate state. Furthermore, being doped with some impurity atoms or subjected to gamma-irradiation, TlInS2 compound

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viii

Alexander M. Panich and Rauf M. Sardarly

exhibits relaxor behavior and formation of the nano-sized polar domains. Thallium sulphide and thallium selenide nanorods, which show the quantum confinement effects, have recently been synthesized and studied. Next, the interlayer interactions across the so-called van der Waals gap are particularly important for interface formation. The aforementioned materials were shown to be prospective for applications in laser technique and in non-linear optics, in electro-luminescence and switching devices, as modulators and frequency converters in infra-red and visible regions, as strain (gage) sensors, etc. Owing to the significant scientific and technical interest, a lot of experimental techniques, such as x-ray and neutron diffraction, specific heat and dielectric measurements, nuclear magnetic resonance and electron paramagnetic resonance, dielectric sub-millimeter spectroscopy, IR spectroscopy, Raman and Mandelshtam-Brillouin scatterings, inelastic neutron scattering, Mössbauer spectroscopy, etc. have been used to study these compounds. These methods mutually complement each other and allow receiving various information on different properties of the compounds. The great advantage of these studies is the opportunity to grow the sizable single crystals, which expanded the experimental potentialities of the investigators and makes the above crystals ideal model systems for study of the electronic interactions of this class of compounds. These interactions not only control the electronic characteristics, but may also affect the growth properties. Growing of mixed crystals (solid solutions) allows the design of materials with tailored properties for eventual applications in electronics and optoelectronics. The mixed crystals are also of significant scientific interest, particular in studying the lattice dynamics in anisotropic systems that exhibit compositional disorder due to breaking of the translation symmetry of the crystal lattice. Also, investigation of the mixed crystals allows extension of our knowledge about the mechanisms of the lattice instability and phase transitions. The monograph covers the recent literature and authors’ data on the crystal structure of the compounds, on their transport properties under ambient conditions, on experimental studies of the electronic structure, on band structure calculations, on transport properties, semiconductor-metal phase transitions and band structure under high pressure, on ferroelectric/structural phase transitions at ambient pressure, soft modes and incommensurate states, on lattice dynamics, on elastic properties and on physical properties of mixed crystals (solid solutions). Neutron diffraction and dielectric sub-millimeter spectroscopy measurements reveal that the subsequent phase transitions in the layered crystals are improper and displacive-type ferroelectric phase transitions. The electronic origin of the ferroelectric phase transitions in the chain-type and layer-type crystals is discussed. We also discuss in detail recently discovered relaxor behavior of doped and irradiated crystals, which is accompanied by the occurrence of nanodomains and quantum dots. We establish the main regularities of vibration spectra of the above-mentioned compounds caused by resonance and anharmonic interactions and by the lattice anisotropy and disorder. The density of the phonon states and elastic constants of the crystals are experimentally determined. The physical properties of some related low-dimensional compounds are also discussed. New perspectives in studying and application of the aforementioned semiconductors are considered. This monograph is addressed to physicists, materials scientists, engineers and students who are interested in the physics of low-dimensional compounds, lattice dynamics, phase transitions and ferroelectricity.

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

Preface

ix

Chapters 1 through 7, 12 and 13 were written by A. M. Panich, and Chapters 8 through 10 by R. M. Sardarly. Chapter 11 was mainly written by R. M. Sardarly with the assistance of A. M. Panich.

—A. M. Panich Department of Physics Ben-Gurion University of the Negev Be'er Sheva, Israel

—R. M. Sardarly

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Institute of Radiation Problems National Academy of Sciences of Azerbaijan Baku, Azerbaijan

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ACKNOWLEDGMENTS

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One of the authors (A.M.P.) thanks Professor S. Kashida for helpful discussions on the structure and phase transitions in the TlX and TlMX2 compounds and for providing us with the band structure calculations of TlSe prior to publication. We are grateful to the American Physical Society, the American Institute of Physics, the Physical Society of Japan, Elsevier Publishing, IOP Publishing, Verlag der Zeitschrift für Naturforschung, Wiley-VCH Verlag GmbH, Taylor and Francis Publishers, Scientific and Technological Research Council of Turkey (Tübitak) Publishing and Springer Science Publishing who kindly granted us permission to reproduce the figures in this book. Several figures are cited by courtesy of S. Kashida and A.I. Nadzhafov.

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Chapter 1

CRYSTAL STRUCTURE AT AMBIENT CONDITIONS The family of the binary and ternary TlX and TlMX2 (M = Ga, In; X = Se, S, Te) compounds belongs to a group of low-dimensional semiconductors that possess a chain or layered structure [1-14]. Room temperature x-ray diffraction (XRD) and neutron diffraction measurements showed that TlGaSe2, TlGaS2 and TlInS2 are layered compounds [1-6]. For example, the structure of TlGaSe2 belongs to the monoclinic symmetry, the space group is

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C 2 / c C26h with the lattice parameters a=10.772 Å, b=10.771 Å and c=15.636 Å, =100.6o, Z=16 [2,5,6,14]. The structure exhibits two gallium, two thallium and five selenium sites. It crystallizes as a structure with two anion layers stacked along the [001] (c*) direction in the unit cell. The structural motive of the layers (Figure 1.1) comprises large corner-linked Ga4Se10 tetrahedra consisting of four corner-linked GaSe4 tetrahedra [2, 6]. The average Ga-Se distance, 2.39 Å, is close to the sum of the covalent radii of Ga (1.26 Å) and Se (1.17 Å), respectively, and the average Se-Ga-Se angle is 109.5° [2], supporting the occurrence of covalent sp3 Ga-Se bonds. Two adjacent layers are turned relative to each other by 90° and kept together by Tl1+ ions, which are located in trigonal prismatic voids between the layers, on straight lines along the [110] and [1 1 0] directions that are parallel to the edges of the Ga4Se10 groups. Each Tl atom is surrounded by six Se atoms, forming trigonalprismatic TlSe6 polyhedra. The average Tl-Se bond length is 3.45 Å, a little bit shorter than the sum of the ionic radii of Tl1+ (1.50 Å for coordination number CN=6 [15]) and Se2- (1.98 Å). The average Tl-Tl spacing in the chains is of 3.81 Å. The shortest Tl-Tl contact between the two channels is 4.38 Å [2]. TlGaS2 and TlInS2 are isostructural to TlGaSe2 [1, 3-6]. The structural parameters of the compounds are given in Tables 1.1 and 1.2.

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Alexander M. Panich and Rauf M. Sardarly

2

c

Tl Tl Tl Se Se

SeSe

Tl

SeSe SeSe

Tl

Tl

Se

Se

Tl

Se Se

Se Se

Se

Tl

Tl Se

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Tl

Se

Tl Tl

Se

Tl Tl

Tl Se

a

b

Figure 1.1. Top: sketch of the layered structure of monoclinic TlGaSe2. The layers are composed of corner-linked Ga4Se10 tetrahedra. Tl ions are shown by black circles. (From [6] with permission, © 1982 American Physical Society.) Bottom: the same structure in more detail. Ga atoms are shown by small black circles.

Table 1.1. Structures of layered compounds at ambient temperature Compound TlGaS2

Structure monoclinic

Space group 6 2h

a, Å 10.2990

b, Å 10.2840

c, Å 15.1750

99.603o

C2 / c

TlGaSe2

monoclinic

C26h

C2 / c

10.772 10.779 10.90

10.771 10.776 10.94

15.636 15.663 15.18

TlInS2

monoclinic

C26h

C2 / c

TlS

monoclinic

C2

11.018

11.039

P41212

7.803

7.803

60.16= 4 15.039 29.55

TlS

tetragonal

C

Z 16

Ref. 4, 6

100.6o 99.993o 100.21o

16 16

2, 6, 14 3, 6

100.69o

128

11, 13

32

12

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Crystal Structure at Ambient Conditions

3

Table 1.2. Atomic coordinates in TlGaSe2 crystal in the space group C2/c Atom Tl1

Site 8f

Tl2

8f

Ga1

8f

Ga2

8f

Se1

4e

Se2

4e

Se3

8f

Se4

8f

Se5

8f

x 0.4632 0.4647 0.2163 0.2844 0.3981 0.100 0.1461 0.145 0 0 0 0 0.2047 0.207 0.2588 0.262 0.4541 0.048

y 0.1885 0.3109 0.0613 0.0623 0.1880 0.191 0.0639 0.438 0.9295 0.054 0.4468 0.574 0.4370 0.062 0.1882 0.310 0.3124 0.312

z 0.1078 0.1140 0.6158 0.3864 0.8378 0.162 0.3391 0.339 0.25 0.25 0.25 0.25 0.0695 0.071 0.2508 0.252 0.5732 0.438

Ref. 6 14 6 14 6 14 6 14 6 14 6 14 6 14 6 14 6 14

In contrast, TlSe, TlGaTe2, TlInTe2 and TlInSe2 show a chain structure [7-10], often called in literature as the B37 TlSe type. The crystal structure of TlSe [7, 10] belongs to the

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18 tetragonal symmetry, the space group is D4h -I4/mcm, the lattice parameters are a=b=8.02 Å

and c=6.79 Å, Z=8. TlSe is a mixed valence compound. Its formula should be more accurately written as Tl1+Tl3+(Se2-)2. Thus, in principle, TlSe may be considered as a ternary compound. The trivalent and univalent thallium ions occupy two crystallographically inequivalent sites. The Tl3+ cations form covalent (sp3) Tl-Se bonds and are located at the centers of Tl3+Se42- tetrahedra, which are linked by common horizontal edges and form linear chains along the c-axis. The Tl3+-Se distance, 2.67 Å, is close to the sum of the covalent radii of Tl (1.49 Å) and Se (1.17 Å), respectively; the Se-Tl-Se angle is 115°. Each univalent Tl1+ cation is surrounded by eight chalcogen atoms, which form slightly deformed Thomson cubes, which are skewed by a small angle. Columns of Thomson cubes with common square faces are parallel to the c axis and alternate with the columns of the aforementioned Tl3+Se42tetrahedra. Tl1+-Se distances are 3.43 Å, a little bit shorter than the sum of the ionic radii of Tl1+ (1.59 Å for CN=8 [15]) and Se2- (1.98 Å). The Tl1+ - Tl1+ and Tl3+ - Tl3+ distances in the chains are 3.49 Å [7], while the distances between these atoms in the (001) plane are 5.67 Å. Each Tl1+ ion has four Tl3+ neighbors at 4.01 Å in the a,b-plane, and, in reverse, each Tl3+ ion has four Tl1+ neighbors at the same distance. Projections of the TlSe structure on the a,b- and a,c-planes are shown in Figure 1.2.

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Alexander M. Panich and Rauf M. Sardarly

1/4, 3/4 0

1/2 0

1/2

1/2 1/4, 3/4

0 1/2

1/2

0

0

Tl1+ Tl

3+

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Se

Figure 1.2. Top: Projection of the TlSe chain structure on the a,b-plane. The numbers 1/4, 1/2 and 3/4 are the z/c numbers (i.e., positions of the atoms above the a,b-plane, expressed as a fraction of the unit cell dimension c). Bottom: Projection of the TlSe chain structure on the a,c-plane.

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Crystal Structure at Ambient Conditions

5

Table 1.3. Structures of chain-type compounds at ambient temperature Compound TlSe

Structure tetragonal

Space group 18 4h

D

I 4 / mcm

TlS

tetragonal

D418h

I 4 / mcm

TlGaTe2

tetragonal

D418h

I 4 / mcm

TlInSe2

tetragonal

D418h

I 4 / mcm

TlInTe2

tetragonal

D418h

I 4 / mcm

TlTe

tetragonal

D418h

I 4 / mcm

InTe

tetragonal

D418h

I 4 / mcm

a, Å 8.02 8.020(2) 7.77 7.785 8.429

c, Å 7.00 6.791(2) 6.79 6.802 6.865

Z 8

8.075 8.02 8.494

6.847 6.826 7.181

4 4

9 6 9

12.961 12.953 8.444

6.18 6.173 7.136

16 16 8

18 19 16, 17

8 8 4

Ref. 7 10 8 12 9

TlGaTe2, TlInTe2 and TlInSe2 are isostructural to TlSe. The structural parameters of the chain TlMX2 compounds are given in Tables 1.3 and 1.4. For comparison, the data on InTe [16, 17], which is isostructural to TlSe and is described by the formula In1+In3+(Te2-)2, are also included. Table 1.4. Atomic coordinates in the tetragonal TlMX2 crystals

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Compound TlGaTe2

TlInSe2

TlInTe2

TlSe

TlS

InTe

Atom Tl Ga Te Tl In Se Tl In Te Tl(1) Tl(2) Se Tl(1) Tl(2) S In(1) In(2) Te

Notation 4a 4b 8h 4a 4b 8h 4a 4b 8h 4a 4b 8h 4a 4b 8h 4a 4b 8h

x 0 0 0.1700(3) 0 0 0.1714(5) 0 0 0.1813(6) 0 0 0.1784(3) 0 0 0.1721 0 0 0.1821

y 0 0.5 0.6700(3) 0 0.5 0.6714(5) 0 0.5 0.6813(6) 0 0.5 0.6784(3) 0 0.5 0.6721 0 0.5 0.6821

z 0.25 0.25 0 0.25 0.25 0 0.25 0.25 0 0.25 0.25 0 0.25 0.25 0 0.25 0.25 0

For thallium monosulfide, TlS, both chain (tetragonal) and layered (monoclinic) modifications have been obtained [8, 11-13]. The former one is a mixed valence compound Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Alexander M. Panich and Rauf M. Sardarly

that is isostructural to TlSe and belongs to the tetragonal symmetry [8, 12]. The Tl3+-S distance, 2.55 Å, is in agreement with the sum of the covalent radii of Tl (1.49 Å) and S (1.07 Å). The S-Tl(III)-S angles are 117o and 96o, deviating from the tetrahedral value of 109.5o; the tetrahedra are elongated along the c-axis [13]. The Tl3+ - S - Tl3+ angle is about 84o. The Tl1+-S distance is 3.35 Å, a little bit shorter than the sum of the effective ionic radii of Tl1+ (1.59 Å for CN= 8 [15]) and S2- (1.84 Å [15]). The Tl1+ - Tl1+ and Tl3+ - Tl3+ distances in the chains are 3.40 Å [11], while the distances between these atoms in the (001) plane are 5.50 Å. The Tl1+ - Tl3+ distances in the a,b-plane are 3.89 Å. The latter, layered modification of TlS [11-13] is isostructural to TlGaSe2, with Tl3+ ions instead of Ga3+ ions, and belongs to the monoclinic symmetry. (More explicitly, the room temperature phase of the layered TlS is shown to be isostructural to the low-temperature phase of TlGaSe2, while the hightemperature phase of the layered TlS is isostructural to the room-temperature phase of TlGaSe2; we will discuss this point in Chapter 7). Furthermore, a layered tetragonal modification of TlS has also been reported [12]. The structural parameters of the chain and layered TlS modifications are given in Tables 1.1, 1.2, 1.3, and 1.4. The structure of TlTe (Figure 1.3) also belongs to the tetragonal symmetry and space group D418h

I 4 / mcm [18, 19]. However, it is considerably different from the structure of

TlSe. Instead of crystallizing as mixed valence compound Tl1+Tl3+(Te2-)2, it forms a polyanionic structure fragments, Tl (Ten )1 / n , revealing univalent Tl+ cations and a polytelluric

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counterpart with linear equidistant Te chains in the [001] direction at distances 3.0863 Å. One half of these chains is unbranched; the other one consists of linear [Te3]n units stacked crossshaped one upon the other.

Figure 1.3. Crystal structure of the room temperature phase of TlTe along the [001] direction. (From [19] with permission, © 2000 Elsevier.)

The chemically distinct Tl1+ and Tl3+ ions in TlSe, as well as In1+ and In3+ InTe, occupy two different crystallographic positions, preventing free transfer of electrons from the univalent to the trivalent ion. Therefore all aforementioned compounds are semiconductors,

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Crystal Structure at Ambient Conditions

7

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except for thallium telluride, whose room temperature phase shows semimetallic behavior, in contrast to the other compounds. The electronic properties of the reviewed compounds therefore strongly depend on their crystal structure. This problem will be discussed in the following chapters.

Figure 1.4. Dependences of the lattice parameters on the composition of TlGaS2-2xSe2x mixed crystals. (From [20] with permission, © 1993 Wiley.)

We note that TlS, TlSe, TlGaSe2, TlGaS2, TlInS2, TlInSe2, TlGaTe2 and TlInTe2 crystals allow a substitution among (i) S and Se anions and (ii) among Ga and In cations, respectively, and therefore form a continuous series of mixed crystals (solid solutions) of TlSxSe1-x, TlInxGa1-xS2, TlInxGa1-xSe2, TlInS2(1-x)Se2x, etc. in the whole range of concentrations (0 x 1). These compositions show a variation of the lattice parameters depending on x. For example, the composition variations of the lattice parameters in the TlGaS2-2xSe2x mixed layered crystals, reported by Gasanly et al. [20], reveal a linear decrease in the lattice parameters with increasing x from 0 to 1 with a change in the slope at x ~ 0.7 (Figure 1.4). At the same time, the crystals are monoclinic in the whole concentration range under study, since both TlGaS2 and TlGaSe2 belong to the same space symmetry group. However, when the first and the last member of series belong to different symmetry and space group (say, monoclinic TlGaSe2 and tetragonal TlInSe2), a structural phase transformation from monoclinic to tetragonal phase is observed at some value of x. For example, the x-ray diffraction

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investigation of the variation of the lattice parameters in the TlGaSe2-TlInSe2 mixed crystals by Gasanly et al. [21] revealed structural phase transition (monoclinic to tetragonal) due to the atom substitution, when the sum of the tetrahedral covalent radii of Ga (or In) and Se atoms in the Ga(In)Se4 tetrahedra reaches a critical value of about 2.54 Å. Aliev et al. [22] reported on the structural phase transition in TlSxSe1-x at x = 0.55. Recent detailed investigation of the TlGaSe2-TlInSe2 system by Nadzhafov [23] revealed that the single-phase mixed crystals exist in the regions of TlInSe2 concentrations of 0 - 55 and 70 – 100%, while in the region of 55–70% of TlInSe2 a disintegration into two phases is observed (Figure 1.5). Here in the range of 0–55% of TlInSe2 the mixed crystals show monoclinic TlGaSe2 structure ( -phase in Figure 1.5) whose lattice parameters increase with increasing of the TlInSe2 content, while in the range of 70–100% of TlInSe2 the crystals exhibit tetragonal TlInSe2 structure ( -phase in Figure 1.5) whose lattice parameters decrease with increasing the TlGaSe2 content.

Figure 1.5. Equilibrium diagram of the TlGaSe2–TlInSe2 system (a) and composition dependence of its microhardness (b) and specific electrical resistance (c). (Courtesy of A. Nadzhafov).

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For more detailed information about the structure of such mixed compounds, we refer the reader to references [20-34]. Furthermore, composition variations of the lattice parameters in Tl2xIn2(1-x)Se2 mixed layered crystals have been reported by Hatzisymeon et al. [35].

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Müller, D.; Poltmann, F. E.; Hahn, H. Z. Naturforsch. 1974, 29b, 117-118. Müller, D.; Hahn, H. Z. Anorg. Allg. Chem. 1978, 438, 258-272. Kashida, S.; Kobayashi, Y. J. Phys. Condens. Matter 1999, 11, 1027-1035. Delgado, G. E.; Mora, A. J.; Perez, F. V.; Gonzalez, J. Physica B 2007, 391, 385–388. Gasanly, N. M.; Marvin, B. N.; Sterin, K. E.; Tagirov, V. I.; Khalafov, Z. D. Phys. Status Solidi b 1978, 86, K49-K53. Henkel, W.; Hochheimer, H. D.; Carlone, C.; Werner, A.; Yes, S.; v. Schnering, H. G. Phys. Rev. B 1982, 26, 3211-3221. Ketelaar, J. A. A.; Hart, W. H.; Moerel, M.; Polder, D. Z. Kristallogr. 1939, 101, 396405. Hahn, H.; Klingler W.. Z. Anorg. Chem. 1949, 260, 110-119. Müller, D.; Eulenberger, G.; Hahn, H. Z. Anorg. Allg. Chem. 1973, 398, 207-220. Bradtmöller, S.; Kremer, R. K.; Böttcher, P. Z. anorg. allg.Chem. 1994, 620, 10731080. Kashida, S.; Nakamura, K.; Katayama, S. Solid State Commun. 1992, 82, 127-130. Kashida, S.; Nakamura, K. J. Solid State Chem. 1994, 110, 264-269. Nakamura, K.; Kashida, S. J. Phys. Soc. Japan 1993, 62, 3135-3141. Delgado, G. E.; Mora, A. J.; Perez, F. V.; Gonzalez, J. Cryst. Res. Technol. 2007, 42, 663-667. Shannon, R. D. Acta Cryst. 1976, A 32, 751-767. Schubert, K.; Dore, E.; Kluge, M. Z. Metallkunde 1955, 46, 216-224. Chattopadhyay, T.; Santandrea, R. P.; von Schnering, H. G. J. Solid State Chem. 1985, 46, 351-356 Toure, A. A.; Kra, G.; Eholie, R.; Olivier-Fourcade, J.; Jumas J. C. J. Solid State Chem. 1990, 87, 229-236. Stöwe, K. J. Solid State Chem. 2000, 149, 123-132. Gasanly, N. M.; Ozkan, H.; Culfaz, A. Phys. Status Solidi a 1993, 140, K1-K4. Gasanly, N. M. J. Korean Phys. Soc. 2006, 48, 914-918. Aliev, A. M.; Natig, B. A.; Safuat, B. Yu.; Shirinov, K. G. Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy 1991, 27, 622-623. Aliev, A.M.; Natig, B.A.; Safuat, B.Yu.; Shirinov, K.G. Inorg. Mater. 1991, 27, 523-524 (Engl. Transl.). 6

[23] A.Nadzhafov, Polytypism and polymorphism in A3B3C 2 crystals and their analogs. Dc. Sci. Dissertation, Baku, 2009. [24] Gasanly, N. M.; Ozkan, H.; Culfaz, A. Cryst. Res. Technol. 1995, 30, 109-113. [25] Gasanly, N. M.; Ozkan, H.; Culfaz, A. Phys. Status Solidi A 1995, 151, K23-K26. [26] Gasanly, N. M.; Culfaz, A.; Ozkan, H.; Ellialtioglu, S. Cryst. Res. Technol. 1994, 29, K51-K55.

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[27] Vinogradov, E. A.; Gasanly, N. M.; Goncharov, A. F.; Dzhavadov, B. M.; Tagirov, V. I. Fizika Tverdogo Tela (Leningrad) 1980, 22, 899-901. Vinogradov, E. A.; Gasanly, N. M.; Goncharov, A. F.; Dzhavadov, B. M.; Tagirov, V. I. Sov. Phys.- Solid State 1980, 22, 526-529 (Engl. Transl.). [28] Gasanly, N. M.; Dzhavadov, B. M.; Tagirov, V. I.; Vinogradov, E. A. Phys. Status Solidi b 1979, 95, K27-K30. [29] Guseinov, G. D.; Abdinbekov, S. S.; Godzhaev, M. M.; Agamaliev, D. G. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1988, 24, 144-145. Guseinov, G. D.; Abdinbekov, S. S.; Godzhaev, M. M.; Agamaliev, D. G. Inorg. Mater. 1988, 24, 119-121 (Engl. Transl.). [30] Aliyev, R. A.; Guseinov, G. D.; Najafov, A. I.; Aliyeva, M. Kh. Bull. Soc. Chim. France 1985, 2, 142-146. [31] Bakhyshov, A. E.; Akhmedov, A. M. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1979, 15, 417-420. Bakhyshov, A. E.; Akhmedov, A. M. Inorg. Mater. 1979, 15, 330-332 (Engl. Transl.). [32] Babanly, M. B.; Kuliev, A. A. Azerbaidzhanskii Khimicheskii Zhurnal 1977, 4, 110112. [33] Bidizinova, S. M.; Guseinov, G. D.; Guseinov, G. G.; Zargarova, M. I. Azerbaidzhanskii Khimicheskii Zhurnal 1973, 2, 133-137. [34] Gasanly, N. M. Acta Physica Polonica A 2006, 110, 471-477. [35] Hatzisymeon, K. G.; Kokkou, S. C.; Anagnostopoulos, A. N.; Rentzeperis, P. I. Acta Cryst. B 1998, 54, 358-364.

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Chapter 2

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TRANSPORT PROPERTIES UNDER AMBIENT CONDITIONS Numerous electrical conductivity, photoconductivity and optical measurements [1-74] have shown that all TlX and TlMX2 compounds examined in this monograph, with the exception of TlTe, are semiconductors at ambient conditions. As such, layered crystals exhibit significant (and also temperature-dependent) anisotropy of electric conductivity [1-4]. Mustafaeva et al. [1,3] and Aliev et al. [2] reported that the difference in the and values in thallium indium sulfide is of the order of magnitude [2,3], in thallium gallium sulfide – almost three orders of magnitude [2], while in thallium gallium selenide the ratio of / varies from 108 to 106 in the temperature range 90–250 K [1] (here is the in-plane conductivity and is the conductivity along the c* axis, i.e., perpendicular to the a,b plane). Also, Mustafaeva et al. [1,3] suggested the occurrence of the hopping conductivity between the states localized near the Fermi level both along and across the layers in the layered crystals TlGaSe2, TlGaS2 and TlInS2. Hanias et al. [4], who also measured the temperature dependence of the conductivity along and perpendicular to the c axis (Figure 2.1), reported / ratios as ~104, ~105 and ~101 for TlInS2, TlGaSe2 and TlGaS2, respectively. Monoclinic, layer-type semiconductor TlS crystal also shows anisotropic conduction [5,6]; the conductivity within the layer is about two orders of magnitude higher than that normal to the layer. These findings correspond to a two-dimensional-like (2D) behavior. The disagreement in the / values in the same compounds obtained by different authors is surprising and is presumably owing to the presence of different kinds of uncontrolled structural defects and impurities. Besides this, the proper orientation of crystals for measurements along the crystallographic axes, and knowledge of the real amount of chalcogen atoms, are among the other factors that strongly influence the results of measurements. Possible existence of polytypes may also be a reason for some controversies concerning the electrical and optical properties of these crystals. The values of the energy band gaps [4-12, 14-16] of the layered compounds are given in Table 2.1. The band gap variation in the mixed crystals with layered TlGaSe2-type structure has been reported in [17-22,74]. For example, in the mixed (TlGaSe2)1-x(TlInS2)x single crystals, which are monoclinic in the whole concentration range under study (since both TlInS2 and TlGaSe2 belong to the same space symmetry group), the energy band gap varies linearly with x at 0 x 0.4 and deviates from linearity at x = 0.6 [21,22]. However, in the

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TlGa1-xInxSe2 mixed crystals, in which TlGaSe2 is monoclinic and TlInSe2 is tetragonal, a structural phase transformation from monoclinic to tetragonal phase is observed at some value of x, causing an anomaly in the concentration dependence of the band gap [17,74] (Figure 2.2). Since the band gap of TlInSe2 is twice smaller than that of TlGaSe2 (Table 2.1), increase in the TlInSe2 concentration in the mixed TlGa1-xInxSe2 crystals causes decrease in the band gap [74].

Figure 2.1. Temperature dependence of conductivity of TlInS2, TlGaSe2, and TlGaS2 crystals along the c axis ( ||, crosses) and in the a,b plane ( , circles). (From [4] with permission, © 1992 Elsevier.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Table 2.1. Band gaps of TlX and TlMX2 compounds

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Compound TlGaSe2 TlGaS2 TlInS2 TlS layered monoclinic TlInSe2 TlSe TlS chain tetragonal TlGaTe2 TlInTe2 InTe

Thermal band gap, eV 2.1 – 2.2 2.55 - 2.64 2.45 – 2.56 0.9 1.12 0.56 - 0.71 0.94 0.84 0.7 - 0.8 0.34 ?

Optical band gap, eV 1.83- 2.23 2.38 – 2.54 2.28 –2.55 1.1 1.07 – 1.44 0.72 – 1.03 1.16 – 1.57 0.97 – 1.1 1.16

Ref. 4, 7-9, 14 4, 9, 10, 14, 16 4, 9, 11, 12, 15 5, 6 12, 15, 37 27-30 5, 13, 27, 36 40 12, 15, 37, 39 49-51

Figure 2.2. Compositional dependence of the band gap of TlGa1-x InxSe2 mixed crystals at liquid nitrogen temperature. (Courtesy of A. Nadzhafov.)

The Hall effect measurements allow the calculation of the effective masses of holes and electrons, the Hall mobility, and the carrier concentration. For example, the Hall measurements of TlGaSe2 crystals [23] revealed the extrinsic p-type conduction. The increase in the Hall mobility with decreasing temperature was found to be limited by the thermal lattice scattering. The Hall effect measurements of TlGaS2 crystals in the temperature range 200–350 K [24] showed that the Hall mobility is temperature-dependent and decreases with

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temperature following a logarithmic slope of ~ 1.6. Furthermore, these measurements revealed a conductivity type conversion from p- to n-type at 315 K. An anomalous behavior of the Hall voltage, which changes sign below 315 K, was also observed in TlInS2 [25] and was interpreted through the existence of deep donor impurity levels that behave as acceptor levels when are empty. In all aforementioned studies, the carrier concentration has been determined. Let us now turn to the transport properties of the chain-type compounds, starting from thallium selenide. Electrical conductivity and optical measurements [26-30] have shown that TlSe is a semiconductor with the energy gap measured by different authors as 0.6 - 1.0 eV at 300 K. Qualitatively, such behavior may be explained by the existence of structural constraints upon electron transfer between the chemically distinct Tl1+ and Tl3+ ions that occupy two different crystallographic positions. Guseinov et al. [31] and Hussein et al. [32] observed anisotropy of the electrical conductivity in the [110] and [001] directions. The Hall mobilities derived from these measurements at room temperature were of the order of = 42.66 cm2 V–1 s-1 and || = 112.2 cm2 V–1 s-1. Abdullaev et al. [33], who studied electrical resistance and magnetoresistance of TlSe single crystals in the temperature range from 1.3 to 300 K, also observed a difference in resistivity in two directions, i.e., parallel ( ) and perpendicular ( ) to the c-axis, in particular at low temperatures, from 1.3 to 5 K. However, this difference was rather small and varied from sample to sample, showing both and cases. Moreover, the samples with both activated and ―metallic‖ conductivities have been found. A ―metallic-like‖ behavior, observed in some samples, was attributed to the impurity conductivity. Trying to explain the absence of any definite anisotropy of the electrical conductivity, the authors [33] suggested to consider TlSe as a percolation system, in which quasi-one-dimensional trajectories of the electric current have no definite direction in the crystal. According to the measurements of Allakhverdiev et al. [34], TlSe shows under ambient conditions, while Rabinal et al. [35] reported that =1.92, also under ambient conditions. Nevertheless, one is led to the conclusion that TlSe exhibits a threedimensional-like (3D) electronic nature rather than one-dimensional (1D) nature, in spite of its chain-like structure. The explanation of such a behavior will be done in the next chapters. Semiconductor properties of the chain-type thallium monosulfide have been established by Kashida et al. [5,6] and Nagat [36]. Rabinal et al. [37] showed that the chain-like TlInSe2 and TlInTe2 crystals exhibit = 0.004 and 6.0, respectively, at ambient conditions. While the former can be attributed to 1D behavior, the latter definitely cannot be. Abdullaev et al. [38] have measured the electrical resistivity of the TlInTe2 crystals in the directions parallel and perpendicular to the chains. They found that || and are close to each other in the temperature range from 120 to 300 K. However, at T < 120 K the resistance perpendicular to the chains substantially exceeds the resistance along the chains, i.e., conductivity in the direction of the strong bond is larger than that in the direction of the weak bond. The authors suggested that the electrical conduction in the high-temperature range is predominantly provided by the thermally excited impurity charge carriers in the allowed energy band, while in the low-temperature range the conduction occurs through the charge carrier hopping between the localized states lying in a narrow energy band near the Fermi level, i.e., without excitation to the conduction band.

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The energy band gap values for the chain-type crystals [5,12,13,15,27-30,36,37,39,40] are given in Table 2.1. Measurements of the band gap variation in the mixed crystals with TlSe-type structure have been reported by Allakhverdiev et al. [41].

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Figure 2.3. Current-voltage characteristics measured at different temperatures in TlGaTe2. The Ohmic and negative differential resistance regions are apparent in these curves. (From [42] with permission, © 1993 American Physical Society.)

Figure 2.4. (a) Typical voltage oscillations in TlInTe2 as monitored in the negative differential resistance region for a fixed current value I = 1.15 mA. (b) and (c): enlargements of the regions –a- and –b- of the signal shown in the part (a) of this figure. (From [44] with permission, © 1994 American Physical Society.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Chain TlInSe2, TlGaTe2, and TlInTe2 crystals exhibit unusual current-voltage (I-V) characteristics (Figure 2.3) that consist of two parts: a linear (Ohmic) regime at low current densities and a nonlinear (S-type) regime at higher current densities [12,42-45]. In the latter regime, a well-pronounced region of negative differential resistance appears. Additionally, voltage oscillations (Figure 2.4) were observed in TlInTe2 and TlGaTe2 [12,43-45], revealing two components in the signal, a quasi-periodic component and a chaotic component. Such oscillations were explained [45] suggesting that the conductivity signal is formed by two concurring effects: (i) jumping between different levels of conductivity and (ii) fluctuations on these levels. Recently, Seyidov et al. [46,47] observed low-frequency oscillations in the current-voltage (I-V) characteristics of the layered TlGaSe2 crystals, when current is directed perpendicular to the layers. The oscillations were observed in the paraelectric phase of TlGaSe2, in the temperature range 120–200 K. The authors suggested that the current oscillations are due to nonlinear I-V characteristics of the crystal. The origin of the nonlinearity of the I-V characteristics was presumably explained by the formation of electret states and electrical domains due to instability in the electronic subsystem of the crystal. We also note that the I-V characteristics show a pronounced hysteresis. Abdullaev and Aliev [48] reported on the switching and memory effect in thallium indium selenide, TlInSe2, while Nagat et al. observed such an effect in thallium monosulfide [73]. For comparison, InTe crystal is a semiconductor [49-51] with the optical band gap of 1.16 eV [51]. (The thermal band gap value in InTe determined by Hussein [49] from Hall coefficient studies, 0.34 eV, seems to be incorrect). Analogously to the compounds mentioned above, InTe exhibits noticeable anisotropy of the conductivity [50,51]. The ratio was shown to be temperature dependent. The carrier mobility is also anisotropic and temperature dependent; there, the mobility perpendicular to c-axis, , which increases with temperature exponentially above 140 K with an activation energy of 0.03 eV, was attributed to the hopping mechanism due to the barriers between the chains. In contrast to the semiconductors mentioned above, the room temperature phase of thallium telluride exhibits semimetallic behavior [52,53], which is realized through the conducting, equidistant one-dimensional chains in the structure of TlTe [54]. The reviewed semiconductor compounds reveal pronounced photoconductive properties. Figure 2.5 shows the spectral variations of the photocurrent in TlGaSe2 and TlInSe2 single crystals and their solid solutions [74]. In pure crystals, the photoconductivity occurs due to the light excitation of carriers from the valence band to the conduction band, while in impure crystals the excitation of carriers from the impurity levels is also obtained. As a rule, several donor and acceptor energy levels in the compounds under review are obtained from the temperature dependence of the Hall coefficient, conductivity and photocurrent measurements. At fixed illumination, intensity the photocurrent increases with increasing temperature, revealing thermally activated behavior. The analysis of the photocurrent data allows one to investigate the recombination mechanism. For example, the illumination dependence of the photoconductivity in TlGaSe2 was found to exhibit sub-linear, linear and supra-linear recombinations at high, moderate and low temperatures, respectively [23]. The change in recombination mechanism was attributed to the exchange in the behavior of sensitizing and recombination centers. For more detailed information about photoconductive characteristics of TlX and TlMX2 crystals and their solid solutions, the reader is referred to Refs. [18,19,5,6,15,20,21,55-74].

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1.0 1

4

2

9

5

7

3

Iph, rel. units

17

8 6

0.8

0.6

0.4

0.2

0.6

0.8

1.0

1.2

1.4

λ·10 -6, m

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Figure 2.5. Spectral variations of photocurrent Iph in TlGaSe2 and TlInSe2 single crystals and their solid solutions TlGa1-xInxSe2 at T = 77 K. (1) x = 0; (2) x = 0.1; (3) x = 0.2; (4) x = 0.3; (5) x = 0.4; (6) x = 0.7; (7) x = 0.8; (8) x = 0.9; (9) x = 1.0. (Courtesy of A. Nadzhafov.)

REFERENCES [1]

[2] [3]

[4] [5] [6] [7]

Mustafaeva, S.N.; Aliev, V.A.; Asadov, M.M. Fizika Tverdogo Tela (St. Petersburg) 1998, 40, 48-51. Mustafaeva, S.N.; Aliev, V.A.; Asadov, M.M. Phys. Solid State 1998, 40, 41-44 (Engl. Transl.). Aliev, V. A.; Bagirzade, E. F.; Gasanov, N. Z.; Guseinov G. D. Phys. Status Solidi a 1987, 102, K109-112. Mustafaeva, S.N.; Aliev, V.A.; Asadov, M.M. Fizika Tverd. Tela (St. Petersburg) 1998, 40, 612-615. Mustafaeva, S. N.; Aliev, V. A.; Asadov, M. M. Phys. Solid State 1998, 40 561-563 (Engl. Transl.). Hanias, M.; Anagnostopoulos, A. N.; Kambas, K.; Spyridelis, J. Mater. Res. Bull. 1992, 27 25-38. Kashida, S.; Saito, T.; Mori, M.; Tezuka, Y.; Shin S.J. Phys: Condens. Matter 1997, 9, 10271-10282. Katayama, S.; Kashida, S.; Hori T. Jpn. J. Appl. Phys. 1993, 32, Suppl. 32-3, 639-641. Bakhyshov A.D.; Musaeva, L.G.; Lebedev, A.A.; Jakobson, M.A. Fiz. Tekh. Poluprovodn. 1975, 9 1548-1551. Bakhyshov A.D.; Musaeva, L.G.; Lebedev, A.A.; Jakobson, M.A. Sov. Phys.- Semicond. 1975, 9, 1021-1024 (Engl. Transl.).

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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[20]

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Alexander M. Panich and Rauf M. Sardarly Allakhverdiev, K. R.; Sardarly, R. M.; Wondre, F.; Ryan J. F. Phys. Status Solidi b 1978, 88, K5-K9. Allakhverdiev, K R.; Mammadov, T. G.; Suleymanov, R. A.; Gasanov, N. Z. J. Phys.: Condens. Matter 2003, 15, 1291-1298. Abay, B.; Guder, H. S.; Efeoglu, H.; Yogurtcu, Y. K. Phys. Status Solid.b 2001, 227, 469–476. Abutalybov, G. I.; Aliev, A. A.; Larionkina, L. S.; Nelman-zade, I. K.; Salaev, E. Yu. Fizika Tverd. Tela 1984, 26, 846-848. Abutalybov, G. I.; Aliev, A. A.; Larionkina, L. S.; Nelman-zade, I. K.; Salaev, E. Yu. Soviet Phys.- Solid State 1984, 26, 511-512 (Engl. Transl.). Hanias, M.; Anagnostopoulos, A.N.; Kambas, K.; Spyridelis, J. Physica B 1989, 160 154-160. Nagat, A. T.; Gamal, G. A.; Gameel, Y. H.; Mohamed, N. M. Phys. Status Solidi a 1990, 119, K47-K51. Karpovich, I. A.; Chervova, A. A.; Demidova, L. I. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1974, 10, 2216-2218. Guseinov, G. D.; Mooser, E.; Kerimova, E. M.; Gamidov, R. S.; Alekseev, I.V.; Ismailov, M. Z. Phys. Status Solidi 1969, 34 33-44. Qasrawi, A. F.; Gasanly, N. M. Phys. Status Solidi a 2005, 202, 2501-2507. Gasanly, N. M. J. Korean Phys. Soc. 2006, 48, 914-918. Aliyev, R. A.; Guseinov, G. D.; Najafov, A. I.; Aliyeva, M. Kh. Bull. Soc. Chim. France 1985, 2, 142-146. Bakhyshov, A. E.; Akhmedov, A. M. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1979, 15, 417-420. Bakhyshov, A. E.; Akhmedov, A. M. Inorg. Mater. 1979, 15, 330-332 (Engl. Transl.). Bakhyshov, A.E.; Lebedev, A.A.; Khalafov, Z.D.; Yakobson M.A. Fizika Tekhn. Poluprovodn. 1978, 12, 580-583. Bakhyshov, A.E.; Lebedev, A.A.; Khalafov, Z.D.; Yakobson M.A. Sov. Phys.- Semocond. 1978, 12 320-323 (Engl. Transl.). Godzhaev, M. M.; Guseinov, G. D.; Abdinbekov, S. S.; Alieva, M. Kh.; Godzhaev, V. M. Mater. Chem. Phys. 1986, 14, 443-453. Godzhaev, M. M.; Guseinov, G. D.; Kerimova, E. M. Izv. Akad. Nauk SSSR, Neorg. Mater. 1987, 23, 2087-2089. Godzhaev, M. M.; Guseinov, G. D.; Kerimova, E. M. Inorg. Mater. 1987, 23, 1827-1829. (Engl. Transl.). Qasrawi, A. F.; Gasanly, N. M. Semicond. Sci. Technol. 2004, 19, 505-509. Qasrawi, A. F.; Gasanly, N. M. Cryst. Res. Technol. 2006, 41, 174-179. Qasrawi, A. F.; Gasanly, N. M. Cryst. Res. Technol. 2004, 39, 439-447. Mooser, E.; Pearson, W. B. Phys.Rev. 1956, 101, 492-493. Itoga, R. S.; Kannewurf, C. R. J. Phys. Chem. Solids 1971, 32, 1099-1110. Allakhverdiev, K. R.; Gasymov, Sh. G.; Mamedov, T. G.; Salaev, E. Yu.; Efendieva, I. K. Phys. Status Solidi b 1982, 113, K127-K129. Pickar, P. B.; Tiller, H. D. Phys. Status Solidi 1968, 29, 153-158. Nayar, P. S.; Verma, J. K. D.; Nag, B. D. J. Phys. Soc. Jpn. 1967, 23, 144-149. Guseinov, G. D.; Akhundov, G. A. Doklady Akad. Nauk Azerbaidzhan. SSR 1965, 21, 8-13. Hussein, S. A.; Nagat, A. T. Cryst. Res. Technol. 1989, 24, 283-9.

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Transport Properties under Ambient Conditions

19

[33] Abdullaev, N. A.; Nizametdinova, M. A.; Sardarly A. D.; Suleymanov, R. A. J. Phys. Condens. Matter 1992, 4, 10361-10366. [34] Allakhverdiev, K.R.; Gasymov, S. G.; Mamedov, T.; Nizametdinova, M. A.; Salaev, E. Yu. Fizika Tekhn. Poluprovod. 1983, 17, 203-207. Allakhverdiev, K.R.; Gasymov, S. G.; Mamedov, T.; Nizametdinova, M. A.; Salaev, E. Yu. Sov. Phys.- Semicond. 1983, 17, 131-135 (Engl. Transl.). [35] Rabinal, M. K.; Asokan, S.; Godazaev, M. O.; Mamedov, N. T.; Gopal E. S. R. Phys. Status Solidi b 1991, 167, K97-100 [36] Nagat, A. T. J. Phys.: Condens. Matter 1989, 1, 7921-7924. [37] Rabinal, M. K.; Titus, S. S. K.; Asokan, S.; Gopal, E. S. R.; Godzaev, M. O.; Mamedov, N. T. Phys. Status Solidi b 1993, 178, 403-408. [38] Abdullaev, F. N.; Kerimova, T. G.; Abdullaev, N. A. Fizika Tverdogo Tela 2005, 47, 1180-1183. Abdullaev, F.N.; Kerimova, T.G.; Abdullaev, N.A. Physics Solid State 2005, 47, 1221-1224 (Engl. Transl.). [39] Godzhaev, E. M.; Zarbaliev, M. M.; Aliev, S. A. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1983, 19, 374-375. Godzhaev, E. M.; Zarbaliev, M. M.; Aliev, S. A. Inorg. Mater. 1983, 19, 338-339 (Engl. Transl.). [40] Nagat, A. T.; Gamal, G. A.; Hussein, S. A. Cryst. Res. Technol. 1991, 26, 19-23. [41] Allakhverdiev, K. R.; Bakhyshov; N. A.; Guseinov, S.S.; Mamedov, T. G.; Nizametdinova, M. A.; Efendieva, I. K. Phys. Status Solidi b 1988, 147, K99-104. [42] Hanias M. P.; Anagnostopoulos, A. N. Phys. Rev. B 1993, 47, 4261-4267. [43] Hanias, M. P.; Anagnostopoulos, A. N.; Kambas, K.; Spyridelis, J. Phys. Rev. B 1991, 43, 4135-4140. [44] Hanias, M. P.; Kalomiros, J. A.; Karakotsou, Ch.; Anagnostopoulos, A. N.; Spyridelis, J. Phys. Rev. B 1994, 49, 16994-16998. [45] Watzke, O.; Schneider, T.; Martienssen, W. Chaos, Solitons and Fractals 2000, 11, 1163-1170. [46] Seyidov, MirHassan Yu; Sahin, Y.; Erbahar, D.; Suleymanov, R. A. Phys. Status Solidi A 2006, 203, 3781-3787. [47] Seyidov, M.-H. Yu.; Suleymanov, R.A. Fizika Tverdogo Tela 2008, 50, 1169-1176. Seyidov, M.-H. Yu.; Suleymanov, R. A. Physics Solid State 2008, 50, 1219-1226 (Engl. Transl.). [48] Abdullaev, A. G.; Aliev, V. K. Mater. Res. Bull. 1980, 15, 1361-1366. [49] Hussein, S. A. Cryst. Res. Technol. 1989, 24, 635-638. [50] Pal, S.; Bose, D. N. Solid State Commun. 1996, 97, 725-729. [51] Parlak, H.; Ercelebi, C.; Gunal, I.; Ozkan, H.; Gasanly, N. M. Cryst. Res. Technol. 1996, 31, 673-678. [52] Jensen, J. D.; Burke, J. R.; Ernst, D. W.; Allgaier, R. S. Phys. Rev. B 1972, 6, 319-327. [53] Ikari, T; Hashimoto, Phys. Status Solidi b 1978, 86, 239-48. [54] Stöwe, K. J. Solid State Chem. 2000, 149, 123-132. [55] Kalkan, N.; Hanias, M. P.; Anagnostopoulos, A. N. Mater. Res. Bull. 1992, 27, 13291337. [56] Karpovich, I. A.; Chervova, A. A.; Demidova, L. I.; Leonov, E. I.; Orlov, V. M. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1972, 8, 70-72. [57] Bakhyshev, A.E.; Aliev, R.A.; Samedov, S.R.; Efendiev, Sh. M.; Tagirov, V.I. Fiz. Tekh. Poluprovodn. 1980, 14, 1661-1664. Bakhyshev, A.E.; Aliev, R.A.; Samedov,

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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[58]

[59] [60] [61] [62] [63] [64] [65] [66] [67]

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[68]

[69]

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[71] [72] [73]

Alexander M. Panich and Rauf M. Sardarly S.R.; Efendiev, Sh. M.; Tagirov, V.I. Sov. Phys.- Semicond. 1980, 14, 989-992 (Engl. Transl.). Bakhyshov, A.E.; Natig, B.A.; Safuat, B.; Samedov, S.R.; Abbasov, Sh. M. Fiz. Tekh. Poluprovodn. 1990, 24, 1318-1320. Bakhyshov, A.E.; Natig, B.A.; Safuat, B.; Samedov, S.R.; Abbasov, Sh. M. Sov.Phys.-Semicond. 1990, 24, 828-830 (Engl. Transl.). Kalomiros, J. A.; Kalkan, N.; Hanias, M.; Anagnostopoulos, A. N.; Kambas, K. Solid State Commun. 1995, 96, 601-607. Kalkan, N.; Kalomiros, J. A.; Hanias, M.; Anagnostopoulos, A. N. Solid State Commun. 1996, 99, 375-379. Kerimova, E. M.; Mustafaeva, S. N.; Kerimov, R. N.; Gadzhieva, G. A. Inorg. Mater. 1999, 35, 1123-1124. Qasrawi, A. F.; Gasanly, N. M. Phys. Status Solidi a 2003, 199, 277-283. Samedov, S. R.; Baykan, O. Intern. J. Infrared Millimeter Waves 2003, 24, 231-237. Ashraf, I.M.; Abdel-Rahman, M.M.; Badr, A.M. J.Phys.D: Appl. Phys. 2003, 36, 109113. Ashraf, I. M. J. Phys. Chem. B 2004, 108, 10765-10769. Kerimova, E. M.; Mustafaeva, S. N.; Mekhtieva, S. I. Prikladnaya Fizika 2004, 4, 8184. Bakhyshov, A. E.; Khalafov, Z. D.; Akhmedov, A. M.; Salmanov, V. M.; Tagirov, V. I. Fiz. Tekh. Poluprovodn. 1976, 10, 1950-1952. Bakhyshov, A. E.; Khalafov, Z. D.; Akhmedov, A. M.; Salmanov, V. M.; Tagirov, V. I. Sov. Phys.- Semicond. 1976, 10, 1163-1164 (Engl. Transl.). Bakhyshev, A. D.; Gasanova, L.; Lebedev, A. A.; Yakobson, M. A. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1977, 13, 366-368.Bakhyshev, A. D.; Gasanova, L.; Lebedev, A. A.; Yakobson, M. A. Inorg. Mater. 1977, 13, 306-308. (Engl. Transl.). Bakhyshov, A. E.; Lebedev, A. A.; Khalafov, Z. D.; Yakobson, M. A. Fizika Tekh. Poluprovodn. 1978, 12, 555-557. Bakhyshov, A.E.; Lebedev, A.A.; Khalafov, Z.D.; Yakobson, M.A. Sov. Phys.- Semicond. 1978, 12, 320-321 (Engl. Transl.). Baltremejunas, R.; Veleckas, D.; Zeinalov, N.; Kapturauskas, I. Fiz. Tekh. Poluprovodn. 1982, 16, 1696-1697. Baltremejunas, R.; Veleckas, D.; Zeinalov, N.; Kapturauskas, I. Sov. Phys.- Semicond. 1982, 16, 1085-1086 (Engl. Transl.). Aliyev, R. A.; Guseinov, G. D.; Najafov, A. I.; Aliyeva, M. Kh. Bulletin Soc. Chim. France 1985, 2, 142-146. Nagat, A. T.; Hussein, S. A.; Gameel Y. H.; Gamal, G. A. Cryst. Res. Technol. 1990, 25, 1195-1202. A. M. Panich J. Phys.: Condens. Matter 2008, 20, 293202/1-42. 6

[74] A.Nadzhafov, Polytypism and polymorphism in A3B3C 2 crystals and their analogs. Dc. Sci. Dissertation, Baku, 2009.

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Chapter 3

EXPERIMENTAL STUDIES OF THE ELECTRONIC STRUCTURE The electronic structure of the TlX and TlMX2 (M = Ga, In; X = Se, S, Te) compounds has been studied experimentally and theoretically in a number of papers. Experimental data have mainly been obtained by means of photoemission spectroscopy and nuclear magnetic resonance (NMR) techniques. In this chapter we review the experimental results, starting (for the convenience of the readers) from a brief introduction to the fundamentals of the above techniques.

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3.1. PHOTOEMISSION MEASUREMENTS X-ray photoelectron spectroscopy (XPS) is a quantitative spectroscopic technique in which a sample is irradiated with a beam of monochromatic x-rays, and the energies and the numbers of the resulting photoelectrons, escaping from the surface (typically from the depth of ~1 to 10 nm), are measured under ultra-high vacuum conditions. The collected photoelectrons result in a spectrum of the electron intensity as a function of the measured kinetic energy. The kinetic energies of the emitted photoelectrons are converted into binding energy values, which are characteristic of the chemical bonding and molecular orbital structure of the material. Photoemission spectroscopy measurements require clean surfaces that are usually obtained by cleaving the samples in situ under ultrahigh vacuum conditions or by scraping the samples using a diamond file in a preparation chamber. The working pressure should be sufficient to keep the samples free of detectable contamination for the duration of the experiment. Besides x-ray, the typical sources of photoelectron excitation are helium discharge lamp emitting ultraviolet radiation, and synchrotron radiation. A noticeable advantage in studying the electronic structure may be reached by measuring the angleresolved photoemission spectra that yield important information about dispersion of the electronic states and anisotropy of the electronic distribution. A very useful approach of spectra interpretation is the comparison of the data on (i) isostructural compounds such as TlSe, InSe, TlInSe2 and on (ii) series of compounds with variation of ionic/covalent state of the same element [1, 2].

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The first XPS study of the thallium chalcogenides Tl2S, TlS, Tl2Se, TlSe, Tl5Te3, TlTe and Tl2Te3 was carried out by Porte and Tranquard [1], who reported the measurements from both the core and the valence levels. In the thallium telluride series, two intensive doublets coming from the Te 4d and Tl 5d states are accompanied by two weak valence bands. The relative variations of the core level binding energies and the evolution in the valence band structure are consistent with an increase of ionic contribution from Tl(III) to Tl(I) compounds. In thallium sulfides, the authors observed an intense doublet coming from Tl 5d3/2 and Tl 5d5/2 core states, accompanied by two weak valence bands in the range from 0 to 10 eV. Valence band structures of thallium sulfides and selenides were analyzed with regard to the crystal structures. Particular attention was devoted to the structure derived in major part from the Tl 6s level. An explanation of the variations observed for this structure in various compounds was advanced, taking into account the peculiarity of Tl 6s level participation in the chemical bond.

Figure 3.1. Top: Photoelectron EDCs of TlInSe2 taken with photon energy 18 eV, and x-ray photoemission EDCs of TlSe and InSe, taken with photon energy 32 eV. The letter C labels one of the features of the spectrum. The horizontal scale is referred to the top of the clean TlInSe2 valence band. Bottom: EDC's taken on clean and Ge-covered TlInSe2. The nominal thickness of the Ge overlayer is shown at the right side of each curve. (From [2] with permission, © 1987 American Physical Society.)

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Experimental Studies of the Electronic Structure

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Kilday et al. [2] measured photoelectron energy distribution curves (EDCs) of TlInSe2 and compared them with those of isostructural compounds TlSe and InSe (Figure 3.1). The main peaks of the TlSe and TlInSe2 spectra, at 1.4 eV below the top of the TlInSe2 valence band Ev that was assigned to zero energy, correspond primarily to Se 4p states. The second feature at larger binding energies, from –5 to -7 eV, is primarily due to Tl 6s states. Its position is different for TlSe and for TlInSe2 owing to two inequivalent Tl sites in TlSe. One can also see in Figure 3.1 a feature at -3.5 eV below Ev for TlInSe2, which is not present for TlSe. On the other hand, a similar feature is present in the InSe XPS spectrum (peak C in Figure 3.1). That feature is due to hybridized px, py states related to In-Se bonds. Thus, one can tentatively interpret the –3.5 eV feature of InSe as due to hybridized In and Se orbitals. Analogously, the positions of the Tl 5d and In 4d core-level peaks in TlInSe2 measured at photon energy 29 eV, were found to be –12.15, -14.4 eV for Tl 5d and -16.7, -17.5 eV for In 4d [2]. We note that the Tl 5d positions for TlSe are –12.9, -15.1 eV [1]. The above difference between TlInSe2 and TlSe is in qualitative agreement with the replacement of trivalent Tl with trivalent In. The 4d level positions in TlInSe2 are close to those in InSe, -16.9 and –17.7 eV. This is the further evidence that In atoms in TlInSe2 are involved in the formation of covalent bonds with Se, like in InSe. Kashida et al. [3] reported on the measurements of photoemission spectra of chain and layered modifications of TlS using a synchrotron photon source (Figure 3.2). The EDCs were taken with different incident photon energies, from 40 to 120 eV. For comparison, the EDCs of Tl metal were also measured. The binding energies were determined from the Fermi edge EF of the Tl metal sample. The most prominent peaks observed around -13 and -15 eV come from Tl 5d3/2 and 5d5/2 core states. In the chain and layer types of TlS, the positions of these 5d core levels coincide within the experimental uncertainty. As the average valence of Tl atoms increases, the levels shift to higher binding energies (Table 3.1). The shift measures the charge transfer from the cations to the anions. However, the splitting of the 5d doublet corresponding to the two inequivalent cation sites in TlS, for Tl1+ and Tl3+, is not observed, as already reported in reference [1]. A qualitative explanation for this finding is the rise in the Madelung energy; that is, around the Tl3+ ion the anions are in closer distances than those around the Tl1+ ion, which compensates the charge transfer effect. The more asymmetric line shapes of the 5d levels in the chain-type TlS than those in the layer-type TlS were attributed to a higher density of states near the Fermi level in the chain-type TlS. This is consistent with the fact that the electrical conductivity of the chain-type TlS is higher than that of the layertype TlS. Table 3.1. Binding energies (eV) of the Tl 5d core levels [3] Sample Tl metal Tl2S TlS chain TlS layered

Tl valence 0 1 1, 3 1, 3

Binding energy, Tl 5d5/2 state 14.81 15.55 15.72 15.81

Binding energy, Tl 5d3/2 state 12.54 13.36 13.50 13.59

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Figure 3.2. Photoelectron EDCs of layer-type and chain-type TlS taken with different incident photon energies. The spectra are normalized so that the Tl 5d levels have the same height [3]. (From [3] with permission, © 1997 IOP Publishing.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

Experimental Studies of the Electronic Structure

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Above the core 5d levels, broad valence band peaks are observed (Figure 3.2), which are composed of two sub-bands. The upper valence band edges are found around 0.95–1.05 eV below EF. The edges are seen at almost the same positions in both the chain and layer types of TlS. The valence band A is composed mainly of the S 3p states, while the valence band B is derived from the Tl 6p states as well as from the S 3p and Tl 6s states. As the photon energy increases, the intensity of these peaks increases relative to that of the Tl 5d peaks. This change is due to the rise in the photo-ionization cross sections of the Tl 6s and S 3p states, relative to that of the Tl 5d states.

Figure 3.3. Comparison of the AIPES spectrum of TlGaTe2 with the calculated DOS (bottom). The dotted line shows the integral background. (From [4] with permission, © 2001 American Physical Society.)

Let us now discuss the angle-integrated photoemission (AIPES) spectra of TlGaTe2 [4] taken with He radiation (Figure 3.3), which show five features labeled as A – E in the He I spectrum. Two pronounced features A and C are not seen in the He II spectrum, and hence these structures are mainly attributed to Te 5p states, because the Te 5p cross section is dramatically reduced in going from He I to He II. On the other hand, structures D and E at 5 – 8 eV are seen in the He II spectrum, and hence these are assigned to Tl 6s and Ga 4s states. This assignment is confirmed by the muffin-tin projected partial density of states (DOS) given at the bottom of Figure 3.3. The He II spectrum, for which the cross sections of the Te 5p, Tl 6s and Ga 4s states are not so different as those for the He I spectrum, is in rather good

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Figure 3.4. ARPES spectra of TlGaTe2 in wide region (left panels) and narrow region (right panels). Top: for k|| || c. Bottom: for k|| c. Inset: measurement geometry showing the definition of k|| and for k . (From [4] with permission, © 2001 American Physical Society.)

agreement with the broadened total DOS. Very important information was obtained from the angle-resolved photoemission (ARPES) spectra (Figure 3.4). In the following, k denotes the electron momentum in the solid and k|| and k denote the components parallel and perpendicular to the (110) surface, respectively. In spite of the chain-like structure running along the c axis, one can clearly see dispersive features for both k|| || c and k|| c arrangements, i.e., the band dispersions in TlGaTe2 depend on the momentum not only parallel but also perpendicular to the chain direction. This weighty finding, indicating 3D rather than 1D character of the electronic structure, will be discussed below along the NMR data and calculated electronic band structure. Recently, an angle-resolved photoemission

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27

study of quasi-one-dimensional chain-type crystal TlInSe2 [5] also showed noticeable band dispersion in the direction normal to the chains.

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3.2. NUCLEAR MAGNETIC RESONANCE MEASUREMENTS 3.2.1. NMR in Thallium Compounds Nuclear Magnetic Resonance (NMR) is the resonance absorption of electromagnetic wave by a nuclear spin system subjected to an external magnetic field B0. NMR is an element-selective, inherently quantitative tool for studying the electronic structure, local crystal structure, dynamics and phase transitions in solids at the atomic level [6,7]. The electron-nuclear interaction causes a deviation of the NMR frequency from the Larmor frequency, which is called chemical shift (or shielding). The chemical shielding tensor is sensitive to the nature of chemical bonds and yields information relating to the electronic structure, electron density distribution and wave function hybridization. Phase transitions may be observed through the temperature dependences of the chemical shift and spin-lattice relaxation time. For the present series of compounds, 203Tl and 205Tl are the most attractive nuclear probes for NMR measurements. These nuclei are particularly sensitive to the effects of chemical bonding because of the strong indirect exchange coupling between the nuclear spins, J12I1I2, which is realized across the overlapping electron clouds. For nuclear spins of atoms A and B, this coupling can be expressed as a product of their hyperfine interaction constants and squared overlap integral of the electron shells [6,8]. In the solid thallium compounds, this coupling dominates over the dipole-dipole one and determines the line shape in single crystals, as well as in powder samples measured in low magnetic field, when the chemical shielding anisotropy is negligible. In the early 1980s, the author discovered [9] that the indirect exchange between nuclei could arise from the electron shell of a bridging atom or atomic group, by analogy with the Kramers mechanism of electron-spin exchange via a nonmagnetic bridge ion [10]. Just such an effect is realized in the compounds examined in this monograph. For these systems, the scalar exchange term of the spin Hamiltonian is given as ^

H

J11

Here spins

I

i, j

I

and

I

I

i

j

I I

I

III

+ J33 i, j

III

III

i

j

I I

+ J13 i, j

I

III

i

j

I I

(3.1)

belong to the Tl1+ and M3+ (sites I and III) respectively, J11 and J33

are the Tl1+ - Tl1+ and M3+ - M3+ exchange coupling constants among the spins of univalent and trivalent ions, respectively, and J13 is the Tl1+ - M3+ exchange interaction. (Note that here M = Tl(III), Ga and In). Due to low natural abundance of the 33S (f = 0.76%), 77Se (f = 7.56%), 123Te (f = 0.87%) and 125Te (f = 6.99%) isotopes having nuclear spins, one can neglect the spin-spin coupling between thallium and chalcogen nuclei. Van Vleck has shown [11] that in the crystal that contains two different types of the exchange-coupled spins I and I’, the contribution to the second moment of the NMR line S2 comes from the exchange interaction with the unlike nuclei only and is proportional to the

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Alexander M. Panich and Rauf M. Sardarly

abundance of the unlike isotope. Therefore the ratio of the second moments of two different isotopes is inversely proportional to the ratio of their abundances. For thallium, the natural abundances are f=29.5% for 203Tl and (1-f)=70.5% for 205Tl with (1-f)/f=2.39, which makes the aforementioned effect readily observable [9]. The first experimental evidence of such an effect (in thallium oxide Tl2O3 and metallic thallium) was reported in the classic paper of Bloembergen and Rowland [12]; then it was observed by the other authors (e.g., [13-21]).

3.2.2. Indirect Nuclear Exchange in Chain-Type Compounds The most impressive manifestation of the exchange coupling is observed in the single crystal of the chain semiconductor TlSe [22]. The low-field thallium spectrum at B0 c is given in Figure 3.5. In this orientation the chemical shifts of Tl1+ and Tl3+ ions coincide, and all thallium atoms are equivalent. Both 203Tl and 205Tl isotopes show single Lorentzian-like resonances with the second moments S2 = 360 and 150 kHz2 for 203Tl and 205Tl, respectively. The values of S2 are more than two orders of magnitude larger than the contributions of the dipole-dipole interactions of nuclear spins, estimated from the structure of TlSe as ~1 kHz2. The ratio of the second moments of two thallium isotopes, S2(203Tl)/S2(205Tl) = 2.4, is inversely proportional to the ratio of their natural abundances that is characteristic for the exchange coupling among Tl nuclei. The effective exchange constant J0 = (J112/2+J332/2+J132)1/2, calculated from the S2 values, is 45.1 kHz.

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205

Tl 203

Tl

o

(B0, c) = 90

200

100

0

-100

-200

-300

-400

-500

Frequency, kHz 203

205

Figure 3.5. Room temperature Tl and Tl NMR spectra of the TlSe single crystal at the resonance frequency 30.3 MHz. Applied magnetic field B0 = 1.2 T is perpendicular to the c axis. (From [22] with permission, © 2001 American Physical Society.)

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Chemical shift, ppm 4000

3000

2000

1000

0

-1000

90

o

60

53

-2000

o

o

42

o

angle (B0, c)=0 3+

Tl

Tl 197.8

197.6

197.4

o

1+

197.2

197.0

196.8

196.6

Frequency, MHz

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Figure 3.6. Angular dependence of the room temperature 205Tl NMR spectra of the single crystal of TlSe in high magnetic field (B0 = 8.0196 T). (From [22] with permission, © 2001 American Physical Society.)

The interchain exchange is readily seen in the high field NMR measurements (Figure 3.6), when the difference in chemical shifts of each isotope, belonging to the Tl1+ and Tl3+ ions, exceeds the exchange coupling between them. At B0 c, the 205Tl spectrum shows two separate lines attributed to the Tl1+ and Tl3+ ions. When the applied magnetic field B0 is tilted from the c axis, the two lines move to each other due to the angular dependence of the Tl1+ and Tl3+ chemical shifts, broaden and finally collapse. Such a behavior is characteristic for the exchange interaction between the nuclei at inequivalent sites. It means that besides the common intrachain Tl-Tl exchange interaction, the exchange coupling between nuclei of structurally inequivalent Tl1+ and Tl3+ ions, which reside in neighboring chains, is realized in TlSe. Analysis of the spectra in terms of the theory of exchange processes in NMR yields the value of J13 = 39.4 kHz. Comparing this value with J0 = 45.1 kHz and assuming that J11 and J33 are equal, one can calculate the intrachain exchange constants J11 = J33 =21.9 kHz. Thus intra- and interchain wave function overlaps are comparable (J13 is even larger than J11), and that is why TlSe is not a highly anisotropic compound, as noticed in Chapter 2. An analogous effect of the indirect nuclear exchange interaction has been observed in the chain semiconductors TlS [23] and TlGaTe2 [24]. The scalar terms of the exchange coupling constants Jij were evaluated from the S2 values of the 205Tl and 203Tl resonances; the separation of intrachain (Tl-Tl) and interchain (Tl-Ga) contributions to S2 was shown to be possible [24] in TlGaTe2 with neighboring Tl(I) and Ga(III) chains.

3.2.3. Indirect Nuclear Exchange in Layered Compounds Low-field 203Tl and 205Tl measurements of the powdered samples of layered TlX and TlMX2 compounds [25-27] show broad singlet lines (Figure 3.7). Their second moments are

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TlGaSe2

205

203

100

80

60

40

20

0

-20

-40

Tl

Tl

-60

-80

-100

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Frequency, kHz

205

Tl

203

100

80

60

40

20

0

-20

-40

-60

Tl

-80

-100

Frequency, kHz 203

205

Figure 3.7. Room temperature Tl and Tl NMR spectra of the powder TlGaSe2 in a magnetic field B0 = 1.228 T [27] (top) and of the powder layer-type TlS in a magnetic field B0 = 0.87 T [26] (bottom). (From [27] with permission, © 2004 Springer; from [26] with permission, © 2004 IOP Publishing.)

much larger than those resulted from the contributions of dipole-dipole interactions of nuclear spins and chemical shielding anisotropy and are indeed characteristic for the indirect exchange coupling among nuclei [9, 11-27]. In this case, however, the experimental line widths and second moments of 203Tl and 205Tl isotopes are close to each other. For example, powder layered TlS [26] shows a ratio of the second moments of 203Tl and 205Tl isotopes, at the resonance frequency 21.4 MHz, of 1.23 instead of 2.39 (after subtracting the chemical shielding anisotropy contribution). It means that the indirect exchange between structurally

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Experimental Studies of the Electronic Structure

31

inequivalent Tl1+ and Tl3+ ions plays a significant role and possibly dominates over the Tl3+ Tl3+ and Tl1+ - Tl1+ exchange. In such a case, all nuclei are unlike ones, and line broadening is realized not only for 203Tl-205Tl but also for 205Tl1+-205Tl3+ and 203Tl1+-203Tl3+ exchange interactions. Therefore the S2(Tl203)/S2(Tl205) ratio differs from 2.39. In the case that the resonances of uni- and trivalent Tl ions are not well resolved (e.g., in powder samples), two approaches for evaluation the exchange couplings Jij are used. The interlayer Tl1+ - Tl3+/M3+ interaction is extracted (i) from the second moment values of the 205Tl and 203Tl resonances and (ii) from the field dependence of the line width at low resonance frequencies, when the case of the ―fast‖ exchange (J >> ) is realized (here is the NMR frequency separation between two sites). In the latter case, the additional line broadening caused by the exchange interaction among the spins of the Tl(I) and M(III) atoms is proportional to 2 [28]: =

0+

2 2

/(4 J13)

(3.2)

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Such a behavior is readily observed in the experiment [22, 26]. For the layered TlS, using both these approaches and assuming that J11 and J33 are equal, we found J11 = J33 =11 kHz, and J13 =12 kHz [26]. These results show that the interlayer exchange coupling is comparable with the intralayer one.

Figure 3.8. Arrangement of atoms around the Tl1+ ion showing one of the Tl1+ - S – Tl3+ bonds in the layer-type TlS structure (view along the Tl1+ channel). (From [26] with permission, © 2004 IOP Publishing.)

Similar results have been obtained in the layered semiconductors TlInS2, TlGaS2 and TlGaSe2 [25] and have indicated an overlap of the wave functions of univalent Tl1+ and trivalent Ga3+ and In3+ ions through the intervening S or Se atoms. It was shown that the (i) closeness of the S2 values of the 203Tl and 205Tl isotopes and (ii) the considerable difference in the second moments of the thallium spectra of the isostructural compounds TlInS2 and TlGaS2 (even though their cell parameters are not at all different) are the evidence that the predominant contribution to S2 comes from the indirect exchange coupling of Tl spins with

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In, 69Ga and 71Ga spins. Such interaction implies a formation of weak M3+ - X - Tl+ chemical bonds (here X is the chalcogen atom) by means of directed sp- and p - orbitals. One such bond in the TlS structure [26] is shown in Figure 3.8.

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115

3.2.4. Wave Function Overlap and Electronic Structure As it was mentioned above, the indirect nuclear exchange coupling is realized due to overlap of the electron clouds of atoms. In the aforementioned compounds, the Tl1+-Tl1+, Tl3+-Tl3+ and Tl1+- M3+ distances exceed the sum of the ionic radii of the corresponding Tl1+, Tl3+ and M3+ ions and are therefore rather long to guarantee a significant Tl - Tl and Tl - M overlap. Since the chalcogen atoms are the first neighbors of Tl, one can conclude that the interchain and interlayer exchange couplings of nuclei are mostly caused by the overlap of the Tl1+ and M3+ electron wave functions of the Tl1+ - X - M3+ type across the intervening chalcogen atom. (The Tl3+ - X - Tl3+ coupling within the chains in TlSe, TlS and TlGaTe2 is evidently realized by means of the Tl3+ - X covalent bonds). The obtained wave function overlap should be an important mechanism in the formation of the uppermost valence bands, lower conduction bands and the entire electronic structure of the aforementioned compounds. The band structure of the semiconductors under review is consistent with a long-range indirect nuclear exchange coupling via intervening chalcogen atoms, analogous to the Kramers mechanism [10] of electron spin exchange via a nonmagnetic bridge ion. Common wave functions of thallium and chalcogen guarantee, via the electron-nuclear hyperfine interaction, an effective correlation of Tl nuclear spins. As shown by Bloembergen and Rowland [12], the indirect exchange coupling of nuclei is put into effect via intermediate excited electronic states. Thus, to describe the indirect nuclear exchange interaction via a bridge atom, we should discuss the excited electronic states of Tl1+ and M3+ mixed with the states of the bridge chalcogen ion. For Tl1+ in TlSe, such an interaction may be realized by means of mixing of Tl 6s2 electron states with unoccupied 4pstates of Se. The Tl(I) orbitals are likely sp-hybridized Tl wave functions, which increase the orbital overlap, since Tl p orbitals span a large range. Such a mixing of the empty Tl1+ 6p orbital into the filled Tl 6s level was predicted by Orgel [29]. Though formally the Tl3+ ion has a configuration 5d10, covalent Tl3+ - Se2- bonds with sp3 hybridization are realized, since the Tl3+-Se distance in tetrahedra is close to the sum of the covalent radii of Tl and Se. On the other hand, exchange interaction between nuclear spins of atoms A and B is of the order of (8 /3)

2 nA eh |

(0)|2

(8 /3)

2 nB eh |

(0)|2 / E

(3.3)

Here E is a suitable average of the energy difference between the conduction and valence bands. The interaction is realized by means of the s-parts of wave functions having a nonzero value | (0)|2 at the nucleus site. Thus the assistance of Tl 6s states is necessary, and one is led to consider the role of the outer 6s2 electron pair in interatomic interactions. Such an interaction involves excited states with the electronic configuration 6s2 due to their mixing with the empty 6p (and perhaps 6d) states of Tl. The wave functions of these thallium 6s6p and 6s6d states overlap with the p-orbitals of chalcogen. This model is in accordance with recent calculations [30] demonstrating that lone pairs cannot be completely localized and exhibit a presence in the bonding regions to varying degrees, from 10-15% to 44% of the total

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covalent bond order. In the case of the chain-like TlSe and TlS, we suggest that the outer 6s2 lone pair electrons of Tl1+ are delocalized and actually shared between the uni- and trivalent thallium ions in order to guarantee the exchange coupling of Tl3+ ion. The presence of some portion of the s-electron at the Tl3+ atom causes the electron-nuclear hyperfine interaction and explains the indirect exchange coupling of its nuclear spin. An analogous effect is suggested for the Tl1+ and Ga3+/In3+ ions in the chain and layered TlX and TlMX2 compounds. It seems that such an overlap and 6s2 pair activity are common properties of thallium chalcogenides. The obtained interchain wave function overlap should reduce the anisotropy of the physical properties of TlSe in comparison to the layered semiconductors AIIIBVI. That is why TlSe crystals posses a nearly three-dimensional electronic nature in spite of its chain-like structure. This can be seen from the experimental values of conductivity that are not much different along and normal to the c axis [31-33]. The same is true for the chain-like TlInSe2 compound [34]. This finding will be discussed below along the band structure calculations. We note that the considerable interlayer and interchain overlap observed in the aforementioned compounds affects their physical properties, e.g., it reduces the anisotropy of the elastic coefficient in comparison to the layered semiconductors AIIIBVI [35]. As shown above, the two-dimensional square lattice of TlSe in the (001) plane, with alternating univalent and trivalent ions, is not metallic at ambient temperature. However, significant overlap of electron wave functions in this plane allows us to expect an electron hopping between Tl1+ and Tl3+ ions at higher temperature, possibly accompanied by a phase transition into a metallic state. Similar nuclear exchange effects were also observed in the layered Tl2Te3 [21], Tl2Se [36] and chain TlTaS3 [37] semiconductor compounds. In Table 3.2, all currently known data on the indirect exchange coupling in the thallium-contained semiconductors [27,38] are collected for the convenience of the readers. We stress that not discriminating between the scalar and pseudo-dipolar interactions does not affect our conclusions, since both these interactions are realized due to the overlap of the electron clouds of atoms and imply an occurrence of a weak Tl - X- M chemical bond. Table 3.2. Parameters of the indirect nuclear exchange in solid thallium semiconductors and Tl metal Compound TlSe TlS (chain) TlS (layered) TlGaTe2 TlGaSe2 Tl2Te3 Tl2Se nanorods TlTaS3 Tl (metal) Tl (metal)

T, K 295 295 200 295 295 295 295 295 77 4.2

J11, kHz 21.9 27 11 7.7 7.3 22.9 21 8.6 17.5 37.5

J33, kHz 21.9 27 11

J13, kHz 39.4 5 12 1.6

1.4

Ref. [22] [23] [26] [24] [25] [21] [36] [37] [12] [13]

We note that both univalent and trivalent Tl atoms in the reviewed compounds show essential chemical shielding anisotropy (Figures 3.6 and 3.9) despite their formal spherically Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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symmetric 5d106s2 and 5d10 electron configurations. Thus one is led to consider sp (or dsp) hybridization of the Tl wave functions, which interact with the p-orbilals of the chalcogen atom and yield a strong deviation of the Tl electron cloud and electronic charge distribution from the spherical form. Thus the chemical shielding data support the aforementioned conclusions based on the analysis of the indirect exchange coupling. (Taking into account the coordination polyhedron around Tl1+, one can speculate that d-orbitals, perhaps in the form of dsp or d4sp, are also included, yielding a weak interaction with the neighbors. A contribution of d-states of Tl into the valence band of thallium selenides, sulphides and tellurides will be considered in the next chapter.) We note that chemical shielding results from the electronnuclear interaction and is inversely proportional to the band gap [28]; the main contribution therefore comes from the states near the top of the valence band and the bottom of the conduction band. This is also true for the indirect spin-spin coupling according to equation 3.3. The above findings on spin-spin coupling, chemical shielding and wave function overlap yield new physical insights into the electronic structure and properties of the chain and layered semiconductor compounds. The aforementioned experimental data are in good qualitative agreement with the band structure calculation of the aforementioned compounds to be discussed in the next chapter.

Chemical shift, ppm

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2400

2000

1600

1200

Tl

197.3

400

0

3+

Tl

197.4

800

197.2

1+

197.1

197.0

Frequency, MHz Figure 3.9. Room temperature 205Tl NMR spectrum of the powder sample of chain-type TlS in the magnetic field B0 = 8.0196 T (solid line) and calculated spectra of the Tl+ and Tl3+ components (dashed lines). The line shape is mainly caused by the chemical shielding anisotropy [23]. (From [23] with permission, © 2002 Elsevier.)

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REFERENCES [1] [2] [3] [4] [5]

[6] [7] [8] [9]

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

[21] [22] [23] [24] [25]

[26] [27]

Porte, L.; Tranquard, A. J. Solid State Chem. 1980, 35, 59-68. Kilday, D. G.; Niles, D. W.; Margaritondo, G.; Levy, F. Phys. Rev. B 1987, 35, 660663. Kashida, S.; Saito, T.; Mori, M.; Tezuka, Y.; Shin S.J. Phys: Condens. Matter 1997, 9, 10271-10282. Okazaki, K.; Tanaka, K.; Matsuno, J.; Fujimori, A.; Mattheiss, L. F.; Iida, S.; Kerimova, E. Mamedov, N. Phys. Rev. B 2001, 64, 045210/1-5. Mimura, K.; Wakita, K.; Arita, M.; Mamedov, N.; Orudzhev, G.; Taguchi, Y.; Ichikawa, K.; Namatame, H.; Taniguchi, M. J. Electron Spectrosc. Rel. Phenom. 2007, 156-158, 379 -382. Abragam, A. The Principles of Nuclear Magnetism, Oxford, Clarendon Press, 1961. Slichter, C. P. Principles of Magnetic Resonance, Berlin-Heidelberg-New York, Springer 1992. Shimizu, T. J. Phys. Soc. Jpn. 1961, 16, 1264-1265. Kholopov, E. V.; Panich, A. M.; Kriger, Yu. G. Zh. Eksp. Teor. Fiz. 1983, 84 10911096. Kholopov, E. V.; Panich, A. M.; Kriger, Yu. G. Sov. Phys.- JETP 1983, 57, 632635 (Engl. Transl.). Kramers, H.A. Physica 1934, 1, 184-192. Van Vleck, J. H. Phys. Rev. 1948, 74, 1168-1183. Bloembergen, N.; Rowland, T. J. Phys. Rev. 1955, 97, 1679-1698. Karimov, Yu. S.; Schegolev, I. F. Zh. Eksp. Teor. Fiz. 1961, 41, 1082-1090. Karimov, Yu. S.; Schegolev, I. F. Soviet Phys. -JETP 1962, 14, 772-778 (Engl. Transl.). Saito, Y., J. Phys. Soc. Jpn., 1966, 21, 1072-1081. Clough, S.; Goldburg, W. I. J. Chem. Phys. 1966, 45, 4080-4087. Vaughan, R. W; Anderson D. H. J. Chem. Phys. 1970, 52, 5287-5290. Villa, M.; Avogadro A. Phys. Status Solidi b 1976, 75, 179-88. Furukawa, Y.; Kiriyama, H. Chem. Phys. Lett. (1982, 93, 617-620. Hinton, J. F.; Metz, K. R.; Briggs, R. W. in: Annual Reports of NMR Spectroscopy, Ed. G.A. Webb, Academic Press, London, 1982; Vol.13,. pp. 211-319. Panich, A. M.; Belitskii, I. A.; Gabuda, S. P.; Drebushchak V. A.; Seretkin, Yu. V. Zh. Strukturnoi Khimii 1990, 31, 69-73. Panich, A. M.; Belitskii, I. A.; Gabuda, S. P.; Drebushchak V. A.; Seretkin, Yu. V. J. Struct. Chem. 1990, 31, 56- 63 (Engl. Transl.). Panich, A. M.; Doert, Th. Solid State Commun. 2000, 114, 371-375. Panich, A. M.; Gasanly, N. M. Phys. Rev. B 2001, 63 195201/1-7. Panich, A. M.; Kashida, S. Physica B 2002, 318, 217-221. Panich, A. M. Fizika Tverdogo Tela 1989, 31, 279-281. Panich, A. M. Sov. Phys.- Solid State 1989, 31, 1814-1815 (Engl. Transl.). Panich, A.M.; Gabuda, S.P.; Mamedov, N.T.; Aliev, S.N. Fiz. Tverd. Tela (Leningrad) 1987, 29, 3694-3695. Panich, A.M.; Gabuda, S.P.; Mamedov, N.T.; Aliev, S.N. Sov. Phys. Solid State 1987, 29, 2114-2116 (Engl. Transl.). Panich, A. M.; Kashida, S. J. Phys. Condens. Matter 2004, 16, 3071-3080. Panich, A. M. Appl. Magn. Reson. 2004, 27, 29-39.

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[28] Carrington, A; McLachlan, A. D. Introduction to Magnetic Resonance. NY: HarperandRow 1967. [29] Orgel, L. E. J. Chem. Soc. 1959, 4, 3815-3819. [30] Chesnut, D. B. Chem. Phys. 2003, 291, 141-152. [31] Itoga, R. S.; Kannewurf, C. R. J. Phys. Chem. Solids 1971, 32, 1099-1110. [32] Abdullaev, N. A.; Nizametdinova, M. A.; Sardarly A. D.; Suleymanov, R. A. J. Phys. Condens. Matter 1992, 4, 10361-10366. [33] Rabinal, M. K.; Asokan, S.; Godazaev, M. O.; Mamedov, N. T.; Gopal E. S. R. Phys. Status Solidi b 1991, 167, K97-100 [34] Rabinal, M. K.; Titus, S. S. K.; Asokan, S.; Gopal, E. S. R.; Godzaev, M. O.; Mamedov, N. T. Phys. Status Solidi b 1993, 178, 403-408. [35] Gasanly, N. M.; Akinoglu, B. G.; Ellialtioglu, S.; Laiho, R.; Bakhyshov, A.E. Physica B 1993, 192, 371-377. [36] Panich, A.M.; Shao, M.; Teske, C.L.; Bensch, W. Phys. Rev. B 2006, 74, 233305/1-4. [37] Panich, A. M.; Teske, C. L.; Bensch, W.; Perlov, A.; Ebert, H. Solid State Commun. 2004, 131, 201-205. [38] A. M. Panich J. Phys.: Condens. Matter 2008, 20, 293202/1-42.

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Chapter 4

BAND STRUCTURE CALCULATIONS

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4.1. BAND STRUCTURE CALCULATIONS OF THE CHAIN-TYPE COMPOUNDS Let us start from the band structure calculations of the tetragonal compounds having a chain-type structure. The first calculations of the band structure in TlSe-type compounds— TlSe, TlGaTe2, TlInTe2, TlInSe2—have been carried out by Gashimzade et al. [1-3], using (i) the concept of band representation and continuity chord and (ii) pseudopotential method; there, some unknown parameters of the atomic pseudopotential were determined by a fit of the experimental energy gaps near the fundamental optical edge to the theoretically calculated values. The 5d states of Tl atoms were not taken into account. The top of valence band and the bottom of conduction band were shown to be localized at different points on the surface of the Brillouin zone of the body-centered tetragonal lattice, namely at the symmetry point T(0, /a, 0) and on symmetry line D( /2 , /2 , ), respectively. The direct band transitions were found to be forbidden according to the symmetry rules of selection. Therefore, the compounds were shown to be indirect gap semiconductors with the band gap ~ 1 eV; these findings are in agreement with the existing experimental data [4-12]. The lowest four valence bands originate from s-states of chalcogen atoms, i.e., S, Se and Te. The next two valence bands were attributed to s-states of the trivalent cation located in the tetrahedral environment of chalcogen atoms. Then, a large group of ten valence bands follows, presumably taking their origin from the hybridized pz-states of chalcogen atoms and the px-, py-, pz-states of trivalent cations. At last, the highest two valence bands were attributed to the s-state of univalent Tl atoms. The band’s assignment is given in Table 4.1. (Here and below, to assist the reader in comparing the results of different authors, we have collected the band structure data in a series of tables).

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Table 4.1. Band assignment in TlSe, TlGaTe2, TlInTe2 and TlInSe2 according to Gashimzade et al. [1-3] Energy, eV -3 to 0 - 5 to – 2.5

Number of bands 2 10

Bands valence bands valence bands

-6 to – 5 -15 to –14

2 4

valence bands valence bands

Electronic states s-state of univalent Tl atoms. hybridized pz-states of chalcogen atoms and the px-, py-, pz-states of trivalent cations. s-states of the trivalent cation s-states of chalcogen atoms (Se, S, Te).

Table 4.2. Band assignment in TlSe according to Orudzhev et al. [13]

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Energy, eV

Number of bands 2 4

Bands

Electronic states

first conduction bands uppermost valence bands valence bands

s-states of Tl(I) and Tl(III) s-states of Tl(I) and Tl(III) s-states of Se

Figure 4.1. Band Structure of TlInSe2. The top of the valence band is taken to be zero. (From [14] with permission, © 2004 Elsevier.)

Orudzhev calculated [13] the charge distribution in TlSe using pseudopotential wave functions determined from the band structure calculation. This computation (Table 4.2) confirmed that the four lowest valence bands are composed from the s-states of Se and showed that the uppermost valence bands and the first two conduction bands near the Tl(I)

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and Tl(III) atoms originate from s-states of the Tl(I) and Tl(III) atoms, respectively. The charge distribution showed a noticeable maximum on the line joining the Tl(III) and Se atoms, indicating covalent bonds between these atoms. However, neither interchain Tl(I)-SeTl(III) overlap nor wave function delocalization, observed in the NMR experiments (Chapter 3), were discussed in the aforementioned paper [13]. The band structure calculation of the ternary chain-type TlInSe2 by Orudzhev et al. [14] (Figure 4.1), using a pseudo-potential method with allowance for non-locality of ionic pseudo-potentials, showed that the top of the valence band and the bottom of the conduction band in this compound are located in the symmetry point T (0.5, -0.5, 0.5) on the surface of the Brillouin zone. Though the authors contend that the comparison with the previous band structure calculations shows that the symmetry of the electron states at the band gap is the same for all TlSe-type linear chain compounds, one can notice that, according to [14], TlInSe2 is a direct gap semiconductor with band gap of 0.60 eV, in contradiction with the previous findings [1-3] showing it to be the indirect gap semiconductor. The experimental data presented in Chapter 2 also favor indirect scenario with twice as large thermal and optical band gaps. The calculated valence bands of TlInSe2 may be sorted into three groups (Table 4.3). Brillouin zone of TlInSe2 [15] is shown in Figure 4.2. Table 4.3. Band assignment in TlInSe2 according to Orudzhev et al. [14] Number of bands 10 4 4

Bands valence bands valence bands valence bands

Electronic states mainly p-states of Se and In(III) s-states of Tl(I) and In(III) s-states of Se

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Energy, eV - 4 to 0 - 6 to - 4 -11.8 to -13

Figure 4.2. Brillouin zone of TlInSe2 [15]. (From [15] with permission, © 2005 Japan Society of Applied Physics.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Recently, detailed calculations of the electronic structure of TlSe and TlGaTe2 have been reported by Ellialtoglu et al. [16], Kashida [17] and Okazaki et al. [18]. The first of them has been made by means of ab-initio pseudopotential method using density functional theory within the local-density approximation. The results of the calculations: the energy bands of TlSe, the normalized total densities of states (DOS), as well as the local DOS curves (not normalized) for the individual Se2-, Tl3+, and Tl+ ions, respectively, are shown in Figure 4.3.

Figure 4.3. Energy bands for TlSe, total DOS, and local DOSs for Se2−, Tl3+, and Tl+, in panels from left to right, respectively. DOS scale for the lower valence states due to Tl 5d and Se 4s electrons is reduced by a factor of 10. The top of the valence band is taken to be zero. (From [16] with permission, © 2004 American Physical Society.)

Table 4.4. Band assignment in TlSe according to Ellialt glu et al. [16] Energy, eV

Bands

Electronic states

Above 1.5 0 to 2

Number of bands 12 2

conduction bands conduction bands

0

1

-4 to 0 -7 to -4

10 4

uppermost valence band (tops at T) valence bands valence bands

-12 to -10.6 -13.7 to -13

20 4

valence bands valence bands

p states of Se, Tl(I) and Tl(III) ions antibonding mixture of Se 4p and Tl(III) 6s states; intermix slightly (around H) with the 12 upper conduction bands Mainly non-bonding Se 4p states and 6s states of Tl(I) Se 4p states and 6p states of Tl(I) and Tl(III) mostly 6s states of Tl(I) and some Se 4p states mixed with the 6s states of Tl(III) 5d states of Tl 4s-states of Se

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The calculated valence and conduction bands of TlSe may be sorted into six groups (Table 4.4). The bottom of the conduction band is located almost at the midpoint D1 = ( /a, /a, /2c) along the line D joining the points P and N, and corresponds to the irreducible representation D1. Two additional minima are situated along the symmetry line A that connects the points G and H. The band at point T is a little higher in energy. The energy gap is underestimated relative to the experimental value (~ 0.7 eV) due to a well-known artifact of the local density approximation (LDA) calculations. As a result, the bottom of the conduction band crosses the top of the valence band, and the indirect gap appears to be negative, leading to a semimetal band structure with a hole pocket at T and electron pocket at D points, respectively.

Figure 4.4. (a) Oblique surface plots and (b) top view of the total valence charge density for TlSe cut through the (004) plane containing both Tl+ (center and corners) and Tl3+ (edges) ions. Se2- ions shown are not in plane, but either above (larger balls) or below (smaller balls) by c/4. (From [16] with permission, © 2004 American Physical Society.)

Ellialtoglu et al. [16] have also built several charge density plots of TlSe (Figures 4.4 and 4.5), in which the Tl+ ions, having lost their 6p electrons, show s-like character due to the outermost 6s2 electrons participating in the valence bands. Tl3+ ions, on the other hand, donate their 6p and 6s electrons to bond formation and show some charge density extending towards Se2- ions and a negligible amount of d influence. Note that Figure 4.4 displays the total valence charge density contours for TlSe cut through the (004) plane containing both Tl+ (center and corners) and Tl3+ (edges) ions, while Figure 4.5 shows the plane that contains all three ions, however, that time the charge density contours depict the contributions from the lowest two conduction bands only. One can find from Figures 4.4 and 4.5 that most of the charge seem to be accumulated on the Se2- ion rather than on the Tl3+-Se2- bond, which is therefore found to be more ionic than covalent. The monovalent cation seems to be not bound to the chalcogens. These findings are in contrast with the well-established covalent nature of the Tl3+-Se2- bond, as well as with the pronounced charge accumulation at the Tl3+-Se2-

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covalent bonds found in the empirical calculation [13] and NMR data [19]. Absence of the overlap of thallium and selenium wave functions contradicts to the experimentally observed overlap [19] and nearly 3D behavior of TlSe [20,21]. The reason of this contradiction is unclear. Also, the authors did not consider the role of the Tl 6s2 lone pair.

Figure 4.5. Top: Contour plots of the total valence charge density for TlSe cut through the (220) plane containing Tl3+ and Se2- ions. Tl+ ions shown are not in plane. Bottom: Contour plots of (a) the total valence (bonding) charge density and (b) the charge density due to the lowest two conduction bands (antibonding) for TlSe cut through an incommensurate plane containing all three ions. (From [16] with permission, © 2004 American Physical Society.)

Another band structure calculation of TlSe has recently been made by Kashida [17], using the full-potential linear-muffin-tin-orbital (LMTO) program LMTART and LDA and taking into account the spin-orbit interaction. As the base functions, s, p and d orbitals were taken for each atom (5d, 6s and 6p for Tl, and 4s, 4p and 4d for Se). The space was divided into muffin-tin spheres and the interstitial region. Within the muffin-tin spheres, the wave function and potential are expanded using spherical harmonics, and for the interstitial region, the wave function and potential are Fourier transformed. The calculated band structure and DOS of TlSe [17] are shown in Figures 4.6 and 4.7. The calculated valence and conduction bands of TlSe are sorted into five groups (Table 4.5). The bottom of the conduction bands is located along the P( /a, /a, /c)–N( /a, /a,0) line around W(( /a, /a, /2c), while the top of the valence bands is located at T(2 /a,0,0). Though the measurements show that TlSe is a semiconductor, the calculated electronic band structure (Figure 4.6), however, suggests that TlSe is a semimetal that has an electron pocket on the PN line and whose valence band touches the Fermi surface at the T point. This discrepancy results from the well-known underestimation of the band gap characteristic of LDA calculations.

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Energy (eV)

Band Structure Calculations

43

5

0

-5

- 10

P

N

T

R

P

Partial DOS (arb. unit)

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Figure 4.6. Electronic band structure of the chain-type semiconductor TlSe. Maximum of the valence band at T is set to zero [17]. (Courtesy of S. Kashida.)

10 8 6

Tl1+ 5d

5d

4 6s

2

6p 6s 6p

0 8

5d

Tl3+

5d

6

6s

4 2

6p

6s

6p

0 8

Se

6 4 2 0

4p

4s 4s

-10

4p

-5

Ef

4d 5 Energy (eV)

Figure 4.7. Calculated angular momentum resolved densities of states of the chain-type semiconductor TlSe. The thick lines represent Tl 6s states and Se 4p states [17]. (Courtesy of S. Kashida.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Table 4.5. Band assignment in TlSe according to Kashida et al. [17] Energy, eV 0 to 3

Bands conduction bands

- 5 to 0 –6 to –5

valence bands valence bands

Around -10 -13 to -11

valence bands valence bands

Electronic states mixture Se 4s, 4p and Tl3+, Tl1+ 6s, 6p anti-bonding states mainly Se 4p states mixed with Tl 6s, 6p, states Tl3+ 6s states, Se 4s, 4p bonding states, Tl1+ 6s nonbonding states Se 4s, Tl3+ and Tl1+ 5d states Se 4s, Tl3+ and Tl1+ 5d states

Table 4.6. Wave functions of TlSe at the top of the valence band and at the bottom of the conduction band and corresponding atomic orbital coefficients [17]. z direction is taken along the c axis. (Courtesy of S. Kashida.). (a) The top of the valence band around W( /a, /a, /2c), E = -0.80 eV. Tl(1+):

6s 6px 6py 6pz 5dyz 0.283 0.015 0.016 0.077 0.052 T1(3+) 0.017 0.035 0.035 0.013 0.004 Se: 4s 4px 4py 4pz 4dyz 0.013 0.291 0.291 0.161 0.019 (b) The top of the valence band at T(2 /a,0,0),

5dzx 5d3z2 0.003 0.012 0.082 0.013 4dzx 4d3z2 0.007 0.004 E = 0.00 eV.

5dxy 0.003 0.082 4dxy 0.007

5dx2-y2 0.015 0.003 4dx2-y2 0.024

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T1(1+)

6s 6px 6py 6pz 5dyz 5dzx 5d3z2 5dxy 5dx2-y2 0.375 0.007 0.003 0.050 0.000 0.004 0.044 0.003 0.001 Tl(3+): 0.000 0.015 0.015 0.006 0.005 0.003 0.001 0.002 0.083 Se: 4s 4px 4py 4pz 4dyz 4dzx 4d3z2 4dxy 4dx2-y2 0.007 0.294 0.294 0.014 0.005 0.004 0.004 0.004 0.018 (c) The bottom of the conduction band around W( /a, /a, /2c), E=-0.16 eV. Tl(1+):

6s 6px 6py 6pz 5dyz 5dzx 5d3z2 5dxy 0.015 0.215 0.215 0.067 0.006 0.012 0.009 0.012 Tl(3+): 0.397 0.016 0.016 0.065 0.039 0.010 0.052 0.010 Se: 4s 4px 4py 4pz 4dyz 4dzx 4d3z2 4dxy 0.106 0.135 0.135 0.247 0.023 0.064 0.025 0.064 (d) The bottom of the conduction band at T(2 /a,0,0), E = 0.20 eV.

5dx2-y2 0.004 0.007 4dx2-y2 0.004

Tl(1+):

5dx2-y2 0.000 0.010 4dx2-y2 0.002

Tl(3+): Se:

6s 0.027 0.003 4s 0.139

6px 0.154 0.023 4px 0.059

6py 0.156 0.024 4py 0.056

6pz 0.607 0.120 4pz 0.048

5dyz 0.001 0.038 4dyz 0.025

5dzx 0.004 0.009 4dzx 0.019

5d3z2 0.003 0.000 4d3z2 0.057

5dxy 0.006 0.008 4dxy 0.019

The calculated band structure shows that the bands near the Fermi energy have a pronounced dispersion not only along the chains but also normal to the chain axis; the latter reflects the interchain interaction. Occurrence of such dispersion is in good agreement with the experimental NMR data [19] on the wave function hybridization and intra- and interchain

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overlap. Indeed, as noticed above, the main contribution to indirect exchange and chemical shielding comes from the states on the top of the valence band and at the bottom of the conduction band. The calculated angular momentum resolved wave functions at P–N line (W) and T point [17] (Table 4.6) show that near the top of the valence band, the density of states results mainly from the contributions of Tl1+ s, Se p and some amount of Tl3+ s and p states. Near the bottom of the conduction band, the density of states results mainly from the contributions of Tl1+ p, Se s, p, and Tl3+ s and p states; some amount of Tl1+ s-states also presents. This result reflects the occurrence of the Tl3+ 6s6p – Se 4s4p - Tl3+ 6s6p and Tl3+ 6s6p – Se 4s4p - Tl1+ 6s6p mixed states, yielding the intra- and interchain overlap of the Tl3+ - Se - Tl3+ and Tl3+ - Se - Tl1+ types. These electronic states give rise to the effective coupling of Tl nuclear spins observed in the NMR experiment and implies the formation of weak Tl3+ - S - Tl1+ chemical bonds by means of directed p, sp (and perhaps spd) orbitals. This calculation also demonstrates the presence of some portion of the s-electron at the Tl3+ atom that causes the electron-nuclear hyperfine interaction and explains the indirect exchange coupling of its spin. The calculation also agrees with the experimental fact that TlSe possesses a three-dimensional electronic nature (rather than a one-dimensional one) in spite of its chain-like structure. Let us now turn to the band structure and DOS calculations of the chain-type TlGaTe2 [18] by Okazaki et al., which has been carried out in the local-density approximation using a full-potential, scalar-relativistic implementation of the linear augmented plane-wave (LAPW) method. In this work, the band structure was also studied experimentally by means of photoemission spectroscopy (Chapter 3, section 3.1), focusing on the anisotropy of the electronic structure. The calculated DOS and band dispersion in TlGaTe2 [18] are shown in Figures 4.8 and 4.9 and Table 4.7. Although the experimental results show that TlGaTe2 is a semiconductor, the present calculations suggest that TlGaTe2 is a semimetal, which has a hole pocket at the M point and an electron pocket on the W line. This is due to the inherent deficiency of LDA. Really, because of the short Ga-Te bond length (~2.70 Å) within the GaTe4 tetrahedra, the strong Ga 4s–Te 5p interaction raises two of the 12 Te 5p bands above EF, thereby opening a band gap at the TlGaTe2 Fermi level. At the M point, for example, these antibonding Ga 4s–Te 5p bands occur at energies 0.9 and 2.3 eV, respectively. These unoccupied antibonding bands are overlapped by the Tl 6p and Ga 4p type states, producing a complicated conduction-band complex shown in Figure 4.9. Note that the lowest groups of the valence bands, between –15 to –8 eV, were not shown (maybe not calculated) by the authors. Analysis of the wave function for the valence-band maximum at the M point reveals that ~50% of the weight consists of Te 5px,y and ~20% Tl 6s. The interaction between the Tl 6s and Te 5p orbitals is analogous to that of the Ga-Te interaction, but is reduced by the fact that the Tl-Te bond length (~ 3.55 Å) is significantly larger than the corresponding Ga-Te value (~ 2.70 Å). As a result, this Tl-Te antibonding band falls at a lower energy and forms the valence-band maximum. The electronic properties near EF of p-type TlGaTe2 are determined by a combination of the interchain Tl-Te interactions as well as the intrachain TeTe hopping. Although the Tl-Te bond length is rather large, the interchain interactions are enhanced by the fact that each Tl has eight Te nearest neighbors. As a result, the expected one-dimensional features in the TlGaTe2 band structure are masked by interchain interactions. From the strong band dispersion perpendicular to the c axis around the M point, it is concluded that TlGaTe2 has a three-dimensional-like electronic structure in spite of the chain structure at least for the transport properties associated with the p-type carriers. This

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conclusion is in good agreement with the NMR data on TlGaTe2 [22]. The calculated electronic structure of TlGaTe2 also agrees well with the photoemission spectroscopy data mentioned above.

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Figure 4.8. Total and muffin-tin projected DOS of TlGaTe2. The top of the valence band is taken to be zero. (From [18] with permission, © 2001 American Physical Society.)

Figure 4.9. Calculated band structure of TlGaTe2. The top of the valence band is taken to be zero. (From [18] with permission, © 2001 American Physical Society.)

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Table 4.7. Band assignment in TlGaTe2 according to Okazaki et al. [18] Energy, eV 0 to 4

Number of bands -

-4 to 0 -7 to -4

10 4

Bands conduction bands valence bands valence bands

Electronic states two Te 5p bands overlapping with Tl 6p and Ga 4p states Te 5p states mainly Tl 6s and Ga 4s

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Figure 4.10. Band structure of TlInTe2. (From [23] with permission, © 2006 Wiley.)

The band structure of the ternary chain-like TlInTe2 compound was calculated [23] with the allowance for non-locality of ionic pseudo-potentials. The results of calculations of the electronic band structure of TlInTe2 are given in Figure 4.10. The top of the valence band is located in the high-symmetry point T (0, 2 /a, 0) on the surface of the Brillouin zone and belongs to the irreducible representation T3. The conduction band bottom, D1 on the line D (π/a, π/a, k), also corresponds to a surface point of the Brillouin zone. The energy gap is found to be 0.66 eV, which is close to the experimentally measured values [4,5,10,11]. The bottom of the conduction band in the T-point (T4) is a saddle point. The energy gap for direct transitions is approximately the same as the gap between the T3 and T4 band states. According to the calculated band structure shown in Figure 4.10, the valence band states can be separated into three groups (Table 4.8). The first group consists of 4 bands around –11 eV, which are originated from the 5s-states of Te. The second group between energies –3.5 and – 6 eV also includes 4 bands that are originated for the most part from 6s-states of Tl and 5sstates of In. The third group extends over the energies between 0 and –3.5 eV. This group is the most complex one and consists of 10 bands that overlap one another and are built from the 5p-states of Te, 6p-states of Tl, and 5p-states of In. It is important to mention a contribution of the monovalent Tl+ ions to the top of the valence band, which is the largest in the point T. Two lowest conduction bands are mainly due to contribution coming from In3+ ions, which are in ionic-covalent chemical bonding with Te2–.

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Table 4.8. Band assignment in TlInTe2 according to Wakita et al. [23] Energy, eV 0 to 4

Number of bands -

–3.5 to 0 –3.5 to –6 -10 to -11

10 4 4

Bands conduction bands valence bands valence bands valence bands

Electronic states mainly In states Te 5p, Tl 6p and In 5p-states Tl 6s and In 5s states Te 5s states

The electronic structures of the tetragonal (chain-type) thallium monosulfide TlS have been studied [24,25] using linear-muffin-tin-orbital (LMTO) calculations. The spin–orbit interaction was taken into account. The calculated Brillouin zones, band structures and DOS of this crystal are shown in Figures 4.11-4.13, respectively; the corresponding band assignments are given in Table 4.9. In the chain TlS, the bottom of the conduction band is located along the P–N line around W ( /a, /a, /c), while the top of the valence bands is located at T(2 /a, 0, 0). Thus this compound is an indirect gap semiconductor with the calculated indirect gap Eg = 0.07 eV from T (2 /a, 0, 0) to W ( /a, /a, /c), while the direct gap at T is 0.76 eV. The experimental gap estimated from the electrical conductivity is 0.94 eV [26]. The discrepancy may be attributed to drawback of the LDA, which tends to underestimate the energy gap. An inspection of the LMTO wave functions shows that the valence band top at T is mostly composed of S 3px,y and Tl1+ s with weights 0.81 and 0.17, while the conduction band bottom at W is mainly composed of S 3pz, Tl3+ s and Tl1+ px,y with weights 0.46, 0.36 and 0.26. This fact suggests that the top of the valence bands has some character of the Tl1+ s – S 3p antibonding state, and that the bottom of the conduction bands has some character of the Tl1+ p - S 3p - Tl3+ s antibonding states. The present band structure is very similar to that of tetragonal TlGaTe2 calculated by the LAPW method [18] where the valence band top is located at T and the conduction band bottom is located along the P–N line. Analogously to ref. [18], it could be argued that the band dispersion is fairly strong not only along the chain but also perpendicular to the chains; the latter is ascribed to the interchain interaction. These findings correlate with the NMR data [27] on the wave function overlap. We note that Shimosaka and Kashida [24] also calculated the charge densities (under the assumption of the non-overlapping muffin-tin spheres); the estimated charge outflows were found to be 0.18, 0.94 and 0.47 for Tl1+, Tl3+ and S2- atoms, confirming the covalent character of the bonding. Table 4.9. Band assignment in the chain-type TlS according to Shimosaka & Kashida [24] Energy, eV 0 to 3 - 4 to 0

Bands conduction bands valence bands

-7 to – 4.5 around - 11 -14 to -12

valence bands valence bands valence bands

Electronic states S 3p, Tl(III) s and Tl(I) p states mainly S 3p states mixed somewhat with the Tl(I) and Tl(III) p states Tl(I) 6s and the Tl(III) s – S 3s, 3p states S 3s states mixed with Tl(III) 5d states Tl 5d

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Figure 4.11. Brillouin zones of tetragonal chain-type TlS [24]. The symmetry points and lines used to calculate the dispersion relation are labeled. (From [24] with permission, © 2004 Physical Society of Japan.)

Figure 4.12. Band structures of tetragonal TlS. Maximum of the valence band is set to zero. (From [24] with permission, © 2004 Physical Society of Japan.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Figure 4.13. Calculated partial DOS for tetragonal TlS. Thick lines represent in (a) and (b) the Tl 6s states, and in (c) and (d) the S 3p states, respectively. (From [24] with permission, © 2004 Physical Society of Japan.)

4.2. BAND STRUCTURE CALCULATIONS OF THE LAYER-TYPE COMPOUNDS The room temperature structure of the monoclinic layer-type TlS is very complicated containing 288 atoms in the unit cell. Therefore in their calculations Shimosaka and Kashida [24] have used a prototypic structure and took into account 32 atoms (16 Tl and 16 S) and 10 empty cells in the asymmetry unit. Band structure of the compound was calculated neglecting the spin-orbital interaction. The results (Figure 4.14-4.16, Table 4.10) show that in the layered TlS, the top of the valence band and the bottom of the conduction bands are located at the Γ point. Thus monoclinic TlS should be a direct (at Γ point) gap semiconductor with Eg= 0.06 eV. This value is much smaller than the experimental value of 0.9 eV [26], which was determined from the temperature dependence of the electrical conductivity. The obtained result is an expected manifestation of the well-known LDA underestimate of the band gap. We note, however, that LDA gives relatively correct eigenstates and wave functions. The calculated DOS seems to qualitatively reproduce the experimental photoemission data [26]. Monoclinic TlS crystals show anisotropic conduction, the conductivity within the layer is about two orders of magnitude higher than that normal to the layer direction [28]. The above band calculation yields relatively large dispersion along the Γ–Z line, which does not explain the above anisotropy though does not disclaim it either. An inspection of the LMTO wave

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functions reveals that the valence band top at the Γ point is mainly composed of S 3s, 3p, Tl1+ s, 6p, and Tl3+ s, 6p, 5dx2-y2 states, while the conduction band bottom - of S 3s, 3p, Tl1+ s, p and Tl3+ s, p states, in which the aforementioned wave function are likely mixed into each other. The corresponding atomic orbital coefficients are given in Table 4.11. One can find that on the top of the valence band the contribution of the Tl1+ 6s states exceeds that of the Tl3+ 6s ones, while at the bottom of the conduction band the Tl3+ 6s wave function dominates here over the Tl1+ 6s one. The aforementioned wave function’s structure correlates well with the NMR data [25], i.e., with the experimentally observed indirect exchange coupling among thallium nuclei due to the overlap of the Tl1+ and Tl3+ electron wave functions across the intervening chalcogen atom discussed in the previous section and in Chapter 3. Shimosaka and Kashida [24] have also calculated the charge densities (under the assumption of the non-overlapping muffin-tin spheres); the estimated charge flows were found to be 0.83, 1.13 and 0.5 for Tl1+, Tl3+ and S2- atoms, confirming the covalent character of the bonding. Note that Figure 4.16 shows a clear difference between the DOSs of the equatorial and apical sulfur ions. The apical sulfur ions make larger contribution to the states just below the Fermi level than the equatorial sulfur ions.

Figure 4.14. Brillouin zone of monoclinic layer-type TlS. The symmetry points and lines used to calculate the dispersion relation are labeled. (From [24] with permission, © 2004 Physical Society of Japan.)

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Figure 4.15. Band structure of monoclinic TlS. The maximum of the valence band is set to zero. (From [24] with permission, © 2004 Physical Society of Japan.)

Table 4.10. Band assignment in the layer-type TlS according to Shimosaka and Kashida [24] Energy, eV 0 to 3

Bands conduction bands

-3.5 to 0

valence bands

- 8 to -5 -12 to –10.5 - 14 to -13

valence bands valence bands valence bands

Around -14.7

valence bands

Electronic states S 3p and Tl(I) p states, with some contribution of Tl(III) s, 6p states mainly S 3p and some Tl(III), Tl(I) 6s, p - S 3p states Tl(III) and Tl(III) 6s and S 3s, 3p states mainly S 3s states mixed with Tl(III) 5d states Tl(III) 5d states mixed with S 3s states, and Tl(III) 5d nonbonding states Tl(I) 5d

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Figure 4.16. Calculated partial DOS for monoclinic TlS. Thick lines represent in (a) and (b) the Tl 6s states, and in (c) and (d) the S 3p states, respectively. Note in (a) small Tl1+ 6s DOS peak just below EF, and in (d) relatively large 3p peak just below EF. (From [24] with permission, © 2004 Physical Society of Japan.)

Let us now discuss the band structure of the other layered compounds. Owing to the complexity of crystal structure, the published data show noticeable discrepancies in the calculated band structure. The first attempt to calculate the band structure of TlGaSe2 was done by Abdullaeva et al. [29] using empirical pseudopotential method. The authors considered the monoclinic modification of TlGaSe2 as a deformed tetragonal structure. It was found that the bottom of the conduction band is located at the point Т (0, /a, 0) of the Brillouin zone, while the maxima of the valence band are located at three points, N ( /2a, /2a, 0), Т (0, /a, 0) and A (0, 0, /2c), approximately at equal energies. Next pseudopotential calculation of the band structure of TlGaSe2 by Abdullaeva et al. [30] showed that the top of the valence band is located at the point. The bottom of the conduction band is located at the -Y line. The direct band gap was about 2.1 eV. No assignments of the electronic states have been done in those two papers. Recently, the electronic structure of the ternary thallium chalcogenides TlGaSe2 and TlGaS2 has been studied in more detail by Kashida et al. [31], who used LMTO method without taking into

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account the spin–orbit interaction. Results of these band structure calculations are shown in Figures 4.17 – 4.19 and Table 4.12. The calculated band dispersion shows that both compounds are indirect gap semiconductors. For both TlGaSe2 and TlGaS2, the top of the valence bands is situated at point Γ, where the wave function is mostly composed of mixed Tl 6s - Se 4p (for TlGaSe2) or Tl 6s - S 3p (for TlGaS2) states. For TlGaSe2, the bottom of the conduction band is located along the Z(0, 0, –0.5)–L(0.5, 0.5, –0.5) line, while for TlGaS2 the bottom of the conduction band is situated along the Γ–Y(0, 1, 0) line. The calculated orbital decomposition of the states near the Fermi level [31] shows that the wave function at the bottom of the conduction bands at Γ is composed of the mixed Tl 6p, Ga 4s and Se 4s (for TlGaSe2) or S 3s (for TlGaS2) states. The bottom of the conduction bands has the character of Tl 6p–Se 4p or S 3p antibonding state. These states have a relatively stronger character of Tl 6p states, than the corresponding states at Γ. Table 4.11. Wave functions and corresponding atomic orbital coefficients (a) on the top of the valence band and (b) at the bottom of the conduction band for the layer-type TlS. Indexes x, y, and z correspond to the a, b. and c-axes [25].

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(a) valence band top: Tl1+ 6s 6px 6py 0.153 0.002 0.002 Tl3+ 6s 6px 6py 0.029 0.001 0.001 S23s 0.022

3px 0.0168

3py 0.0168

6pz 0.031

5dyz 0.000

5dzx 0.000

5dz2 0.000

5dxy 0.002

5dx2-y2 0.015

6pz 0.040

5dyz 0.001

5dzx 0.002

5dz2 0.0142

5dxy 0.001

5dx2-y2 0.060

5dyz 0.000

5dzx 0.003

5dz2 0.009

5dxy 0.003

5dx2-y2 0.011

5dyz 0.002

5dzx 0.004

5dz2 0.018

5dxy 0.004

5dx2-y2 0.007

3pz 0.173

(b) conduction band bottom: Tl1+ 6s 6px 6py 6pz 0.018 0.018 0.018 0.189 3+ Tl 6s 6px 6py 6pz 0.151 0.006 0.006 0.018 2S 3s 3px 3py 3pz 0.093 0.020 0.020 0.097

TlGaSe2 and TlGaS2 are known as highly anisotropic semiconductors, where the conductivity within the layer is several orders of magnitude higher than that normal to the layer. The present band calculation shows that, due to the relatively strong hybridization between the Tl 6s and Se 4p or S 3p states, the dispersion of the valence band top state along the Γ–Z line is rather steep and probably does not explain well the observed anisotropy, but does explain the interlayer overlap observed in NMR [32,33]. Figure 4.18 shows that TlGaSe2

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is an indirect gap semiconductor with the indirect gap from Γ to Z–L of 1.24 eV, while the direct gap at the Γ-point is 1.25 eV.

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Figure 4.17. Brillouin zone of TlGaSe2 and TlGaS2. (From [31] with permission, © 2006 Wiley.)

Figure 4.18. Electronic band structures of TlGaSe2 and TlGaS2. EF values are given in the figure. (From [31] with permission, © 2006 Wiley.)

Figure 4.19. Partial densities of states of TlGaSe2 and TlGaS2. (From [31] with permission, © 2006 Wiley.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Table 4.12. Band assignment in TlGaSe2 and TlGaS2 according to Kashida et al. [31]. EF = 9.68 and 9.49 eV for TlGaSe2 and TlGaS2, respectively. Energy, eV 11 to 15

Bands conduction bands

5 to 10

Bottom of the conduction band at Γ Top of the valence band at Γ valence bands

3 to 5

valence bands

around – 1.5 - 4 to – 2.5 around – 5

valence bands

Electronic states Tl p states, mixed somewhat with Se 4p and Ga 4s states (for TlGaSe2) or with S 3p and Ga 4s states (for TlGaS2) Tl 6p, Ga 4s and Se 4s (for TlGaSe2) or S 3s (for TlGaS2) Se 4p (for TlGaSe2) or S 3p (for TlGaS2) states mixed with the Tl 6s states Se 4p (for TlGaSe2) or S 3p (for TlGaS2) nonbonding states mixed somewhat with Tl 6p and Ga 4p states Ga 4s and Tl 6s – Se 4p bonding states for TlGaSe2, Ga 4s and Tl 6s – S 3s bonding states for TlGaS2 Ga 3d and Tl 5d states

valence bands valence bands

Se 4s for TlGaSe2 and S 3s states for TlGaS2 Ga 3d and Tl 5d states

Next, it shows that TlGaS2 is also an indirect gap semiconductor, the direct gap at the Γ-point is 1.70 eV and the indirect gap from Γ to Γ–Y is 1.58 eV. These results are qualitatively in accord with the previous empirical pseudopotential band calculation [29,30] and those of optical absorption studies [34], where TlGaSe2 and TlGaS2 are assigned as indirect gap d

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semiconductors. The experimentally measured (by Hanias et al. [34]) energy gaps are E g = i

d

i

d

2.11 eV, E g = 1.83 eV for TlGaSe2 and E g = 2.53 eV, E g = 2.38 eV for TlGaS2 where E g i

and E g denote the direct and indirect gap, respectively. These values are larger than those calculated by Kashida et al. [31]. Preliminary calculations [31] showed that in TlInS2 both the top of the valance band and the bottom of the conduction band are located at Γ, the Fermi level is around 9.55 eV and the energy gap is 1.58 eV. This fact suggests that, in contrast to TlGaSe2 and TlGaS2, TlInS2 is a direct gap semiconductor. It is worth mentioning the paper by Yee and Albright [35] who investigated the bonding and structure of TlGaSe2 by tight binding calculations with an extended Hückel Hamiltonian. This calculation draws attention to the sp-hybridization of the Se and Tl wave functions and the role of lone electron pairs on the Se and Tl atoms. The bonding between Tl and Se was found to be reasonably covalent; the region around the Fermi level consists primarily of Tl 6s states antibonding to Se lone pairs. The authors suggested that the large dispersion of the atomic orbitals signals some Tl-Tl communication. They found that the empty Tl p orbitals mix with the Se lone pairs and create a net bonding situation, and that the Tl p orbitals play a decisive role in the Tl-Tl interactions. Based on the paper by Janiak and Hoffmann [36], who has shown that Tl1+ - Tl1+ interactions in molecular and solid-state systems can be turned into a net bonding situation by Tl p mixing into filled Tl s orbitals, Yee and Albright [35] suggested the same situation in TlGaSe2. The shortest Tl-Tl contact between the two channels

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is 4.38 A, and certainly there is no direct communication. However, this might occur though bond coupling in the Ga-Se framework. Furthermore, the authors found that a soft, double well potential exists for the Tl atoms to slide away from a trigonal prismatic to a (3+3) environment, and discussed the electronic factors which create this distortion and lead to ferroelectric phase transition. This mechanism will be considered in more detail in Chapter 7. Wagner and Stöwe [37] reported on self-consistent ab initio LMTO-ASA calculations of the electronic band structure and the crystal orbital Hamiltonian population function in the semimetallic TlTe. The calculations support a view of TlTe as a univalent Tl compound with two polyanionic partial structures, i.e., linear branched and unbranched chains. The branched Te2 chains show weaker Tl – Te interactions compared to Te3 chains. It was shown that in the energy range of 1 eV below EF, Tl mixing with Te-centered bands, which are not involved in strong homoatomic bonding interactions, is quite strong. The role of Te - Tl orbital interactions in formation of the electronic structure and in the electronic nature of phase transition was discussed. Summarizing, we conclude that although some results of different calculations are, in general, similar, the other ones exhibit noticeable differences. The results depend on the method of calculation, number of the wave functions, taking into account the spin–orbit interaction, etc. The more complicate the crystal structure, the more approximate the calculation. At least, several results of calculations correlate well with the experimental NMR data [38] on the spin-spin coupling, wave function overlap and chemical shielding effects and thus reflect real features of the electronic structure of the reviewed compounds. However, most of the band structure calculations did not place high emphasis on the stereochemical activity of the s2 lone pair. At the same time, the calculations often show qualitative agreement with the XPS curves.

REFERENCES [1]

Gashimzade, F. M.; Orudzhev, G. S. Dokl. Akad. Nauk Azerbaijan. SSR 1980, 36, 1823. [2] Gashimzade, F. M.; Orudzhev, G. S. Fiz. Tekh. Poluprovodn. 1981, 15, 1311-1315. Gashimzade, F. M.; Orudzhev, G. S. Sov. Phys.- Semicond. 1981, 15, 757-759 (Engl. Transl.). [3] Gashimzade, F. M.; Guliev, D. G. Phys. Status Solidi b 1985, 131, 201-206. [4] Hanias, M.; Anagnostopoulos, A. N.; Kambas, K.; Spyridelis, J. Physica B 1989, 160 154-160. [5] Guseinov, G. D.; Mooser, E.; Kerimova, E. M.; Gamidov, R. S.; Alekseev, I.V.; Ismailov, M. Z. Phys. Status Solidi 1969, 34 33-44. [6] Itoga, R. S.; Kannewurf, C. R. J. Phys. Chem. Solids 1971, 32, 1099-1110. [7] Allakhverdiev, K. R.; Gasymov, Sh. G.; Mamedov, T. G.; Salaev, E. Yu.; Efendieva, I. K. Phys. Status Solidi b 1982, 113, K127-K129. [8] Pickar, P. B.; Tiller, H. D. Phys. Status Solidi 1968, 29, 153-158. [9] Nayar, P. S.; Verma, J. K. D.; Nag, B. D. J. Phys. Soc. Japan 1967, 23, 144-149. [10] Rabinal, M. K.; Titus, S. S. K.; Asokan, S.; Gopal, E. S. R.; Godzaev, M. O.; Mamedov, N. T. Phys. Status Solidi b 1993, 178, 403-408.

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[11] Godzhaev, E. M.; Zarbaliev, M. M.; Aliev, S. A. Izvestiya Akad. Nauk SSSR, Neorg. Mater. 1983, 19, 374-375. Godzhaev, E. M.; Zarbaliev, M. M.; Aliev, S. A. Inorg. Mater. 1983, 19, 338-339 (Engl. Transl.). [12] Nagat, A. T.; Gamal, G. A.; Hussein, S. A. Cryst. Res. Technol. 1991, 26, 19-23. [13] Orudzhev, G. S.; Efendiev, Sh. M.; Dzhakhangirov, Z. A. Fizika Tverdogo Tela 1995, 37, 284-287. Orudzhev, G.S.; Efendiev, Sh.M.; Dzhakhangirov, Z.A. Sov. Phys.- Solid State 1995, 37, 152-153. (Engl. Transl.). [14] Orudzhev, G.; Mamedov, N.; Uchiki, H.; Yamamoto, N.; Iida, S.; Toyota, H.; Gojaev, E.; Hashimzade, F. J. Phys. Chem. Solids 2003, 64, 1703-1706. [15] Mamedov, N.; Wakita, K.; Akita, S.; Nakayama, Y. Japanese J. Appl. Phys. 2005, 44, 709-714. [16] Ellialtoglu, S.; Mete, E.; Shaltaf, R.; Allakhverdiev, K.; Gashimzade, F.; Nizametdinova, M.; Orudzhev, G. Phys. Rev. B 2004, 70, 195118/1-6. [17] Kashida, S, Electronic band structure of TlSe. Unpublished results. [18] Okazaki, K.; Tanaka, K.; Matsuno, J.; Fujimori, A.; Mattheiss, L. F.; Iida, S.; Kerimova, E. Mamedov, N. Phys. Rev. B 2001, 64, 045210/1-5. [19] Panich, A. M.; Gasanly, N. M. Phys. Rev. B 2001, 63 195201/1-7. [20] Abdullaev, N. A.; Nizametdinova, M. A.; Sardarly A. D.; Suleymanov, R. A. J. Phys. Condens. Matter 1992, 4, 10361-10366. [21] Rabinal, M. K.; Asokan, S.; Godazaev, M. O.; Mamedov, N. T.; Gopal E. S. R. Phys. Status Solidi b 1991, 167, K97-100. [22] Panich, A. M. Fizika Tverdogo Tela 1989, 31, 279-281.Panich, A. M. Sov. Phys.- Solid State 1989, 31, 1814-1815 (Engl. Transl). [23] Wakita, K.; Shim, Y.; Orudzhev, G.; Mamedov, N.; Hashimzade, F. Phys. Status Solidi A 2006, 203, 2841-2844. [24] Shimosaka, W.; Kashida, S. J. Phys. Soc. Japan 2004, 73, 1532–1538. [25] Panich, A. M.; Kashida, S. J. Phys. Condens. Matter 2004, 16, 3071-3080. [26] Kashida, S.; Saito, T.; Mori, M.; Tezuka, Y.; Shin S.J. Phys: Condens. Matter 1997, 9, 10271-10282. [27] Panich, A. M.; Kashida, S. Physica B 2002, 318, 217-221. [28] Katayama, S.; Kashida, S.; Hori T. Jpn. J. Appl. Phys. 1993, 32, Suppl. 32-3, 639-641. [29] Abdullaeva, S. G. Mamedov N. T. and Orudzhev, G. S. Phys. Status Solidi b 1983, 119, 41-48. [30] Abdullaeva, S. G.; Mamedov, N.T. Phys. Status Solidi b 1986, 133, 171-9. [31] Kashida, S.; Yanadori, Y.; Otaki, Y.; Seki Y.; Panich, A. M. Phys. Status Solidi a 2006, 203, 2666– 2669. [32] Panich, A.M.; Gabuda, S.P.; Mamedov, N.T.; Aliev, S.N. Fiz. Tverd. Tela (Leningrad) 1987, 29, 3694-3696. Panich, A.M.; Gabuda, S.P.; Mamedov, N.T.; Aliev, S.N. Sov. Phys.- Solid State 1987, 29, 2114-2116 (Engl. Transl.). [33] Panich, A. M. Appl. Magn. Reson. 2004, 27, 29-39. [34] Hanias, M.; Anagnostopoulos, A. N.; Kambas, K.; Spyridelis, J. Mater. Res. Bull. 1992, 27 25-38. [35] Yee, K. A.; Albright, T. J. Am. Chem. Soc. 1991, 113, 6474-6478. [36] Janiak, C.; Hoffmann, R. J. Amer. Chem. Soc. 1990, 112, 5924-5946. [37] Wagner, F. R.; Stöwe, K. J. Solid State Chem. 2001, 157, 193-205. [38] A. M. Panich J. Phys.: Condens. Matter 2008, 20, 293202/1-42.

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Chapter 5

TRANSPORT PROPERTIES AND SEMICONDUCTORMETAL PHASE TRANSITIONS UNDER HIGH PRESSURE

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5.1. TETRAGONAL (CHAIN-TYPE) CRYSTALS UNDER HIGH PRESSURE In this section, we review the influence of high pressure on the electronic properties of the chain-type TlX and TlMX2 crystals. The effect of uniaxial pressure applied along and perpendicular to the tetragonal c axis, as well as isotropic hydrostatic pressure on the electronic band structure of TlSe, was first reported by Gashimzade and Orudzhev [1]. The authors calculated the variation of the energies of the main extremums of the conduction and valence bands depending on the lattice constants a and c. The calculations predict a decrease of the band gap under pressure and, finally, a semiconductor-metal phase transition in TlSe between 2.2 and 2.7 GPa under uniaxial pressure and at ~5 GPa under isotropic hydrostatic pressure. Then Valyukonis et al. [2] and Allakhverdiev et al. [3-6] measured the pressure d

i

dependence of the direct and indirect energy gaps E g and E g in TlSe, TlInSe2, TlInS2, and TlInSe2(1-x)S2x (0 x 0.25) by analyzing the shifts of the fundamental absorption edges under applied pressure up to 5.5 GPa. (Here indices d and i denote direct and indirect gaps, respectively). The absorption coefficient was calculated from the transmission spectra using the value of the refractive index. Both direct and indirect gaps were found to linearly decrease with increasing pressure (Figure 5.1), i.e., showing negative pressure coefficients dEg/dP. d

These coefficients for TlSe were determined [2] as dE g / dP =-0.15 eV/GPa for Ē

dE gd / dP = -0.17 eV/GPa for Ē c, dE gi / dP = -0.09 eV/GPa for Ē

c,

i

c, dE g / dP = -0.11

eV/GPa for Ē c, respectively. (Here Ē is the electric field vector of the electromagnetic wave). These data are in satisfactory agreement with the calculated ones [5]. Some d

differences in dE g / dP measured by different authors (e.g., dE g / dP = -0.125 eV/GPa, i

and dE g / dP = 0.2 eV/GPa in TlSe [3,4]) may be caused by different quality of the investigated crystals. The pressure dependences of the band gap in the TlInSe2 crystal were d

i

determined as dE g / dP = -0.11 eV/GPa, and dE g / dP = 0.15 eV/GPa [4,7]. Since TlInSe2 Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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and TlInS2 form a continuous series of mixed crystals in the whole range of concentrations (0 x 1), it was interesting to study the properties of these crystals. In the mixed crystals i

TlInSe2(1-x)S2x, dE g / dP was shown to decrease slightly with increasing x, from –0.145 to – 0.13 eV/GPa for x = 0.05 to 0.25, respectively [4,5]. No phase transitions were observed up to pressures ~ 0.8 GPa in all the aforementioned crystals [3-5]. However, a structural phase i

transition, accompanied by a reversal of the sign of dE g / dP , was reported in TlInSe0.2S1.8

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under pressure P~ 0.6 GPa [5].

Figure 5.1. Pressure dependence of the direct (triangles) and indirect (circles) band gaps in TlSe at room temperature measured for Ē c (open symbols) and for Ē c (bold symbols). (From [2] with permission, © 1982 American Institute of Physics.)

The transition was shown to be of the first order and reversible with pressure. Valyukonis et al. [2] ascribed the aforementioned pressure dependence of the band gap to changes in the interchain interactions with pressure, which, in turn, may influence the positions of energy minima and maxima at the bottom of the conduction band and on the top of the valence band. i

Next, an abrupt change of dE g / dP at x~0.3 [7] indicates a structural transformation, most probably from the tetragonal TlInSe2-like crystal to the monoclinic TlInS2-like crystal. Furthermore, TlInSe1.4S0.6 shows a linear reduction of the band gap with increased pressure up to ~0.72 GPa; at this pressure the band gap changes abruptly. The authors [7] assigned this change to the structural transformation under pressure. It was found that increasing the amount of Se in the TlInSe2-TlInS2 system shifts the pressure-induced phase transition to higher pressure. We note that Allakhverdiev et al. [3-5] measured also the temperature dependence of the direct and indirect energy gaps.

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Transport Properties and Semiconductor-Metal Phase Transitions under High Pressure 61 Ves [8] has measured the Raman spectra of the chain thallium indium telluride (TlInTe2) crystal up to high pressure of 17 GPa at room temperature, using a diamond anvil cell. Three Raman peaks were observed in the low-pressure tetragonal phase. The pressure coefficients and the corresponding mode Grueneisen parameters of their frequencies were obtained. The modes were tentatively assigned by comparison to Raman spectra of related TlSe-type compounds. The frequency dependence of indicated that a hierarchy of bonding forces is present. The appearance and disappearance of a new Raman peak at about 1.75 GPa was presumably attributed to a gradual, pressure induced mutual replacement of In atoms by Tl atoms but not to an indication of the occurrence of a structural phase transition. However, a noticeable change in the slope of the pressure dependence of the Raman shift near 7.0 GPa was definitely assigned to a phase transition; furthermore, at even higher pressures indications for a second phase transition were found. We note that, to our knowledge, the aforementioned explanation of the pressure evolution of the Raman modes, based on a pressure-induced mutual interchange of In and Tl atoms [8], was not yet supported by the x-ray measurements. Investigation of the crystal structures of thallium sulfide and thallium selenide under very high pressure, up to 37 GPa, was carried out by Demishev et al. [9] using the XRD technique and a diamond anvil cell (Figure 5.2). Three first-order phase transitions were found in TlS: TlS I (TlSe-type)  TlS II (-NaFeO2-type) TlS III (distorted -NaFe2O-type) TlS IV (CsCl-type) at 5, 10, and 25 GPa, respectively. The transition sequence is reversible. The 18 space group of the first phase is D4h -I4/mcm, and the lattice parameters were determined as

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5 a = 7.77 Å, c = 6.79 Å. For the second phase, the space group is D3d -R3m, and the lattice

parameters are a = 3.945 Å, c = 21.788 Å, Z = 6. This phase II of TlS is metastable under normal conditions and reveals semiconductor properties. Here, the phase II ↔ phase III transition exhibits a hysteresis: the pressure of the direct transition II III is 10 GPa, while that of III II transition is 5.5 GPa. Phase IV of TlS appears near P = 25 GPa, being mixed with the phases I and III in the pressure range from 25 to 30 GPa. At P = 35.5 GPa, its x-ray pattern corresponds to the pure phase IV structure of CsCl type with a = 3.202 Å.

Figure 5.2. Pressure dependence of cell parameters and c/a ratio for tetragonal TlSe I and cubic TlSe II phases: (1) at/2, (2) ct/2, (3) acub, (4) ct/at (5) ccub/acub. The two-phase area is shown by dashed lines. From [9] with permission, © 1988 Wiley.) Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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62

The compression of thallium selenide up to P = 21 GPa [9] results in the first-order structural transition from the tetragonal TlSe I phase to the cubic TlSe II phase. The latter is of the CsCl-type. The transformation is realized by a shift of the selenium atoms from the position with x = 0.18, y = 0.68, z = 0 to that with x = 0.25, y = 0.75, z = 0. The authors suggest destroying of the covalent chains under phase transitions [9]. The pressure-induced transition from TlSe I into the TlSe II phase is accompanied by a reduction of the a/c ratio from 1.16 to 1 and a reduction of the relative volume of the unit cell V/V0 of 40%. The TlSe II phase shows a = c ~3.78 Å. The Se - Tl3+ and Se - Tl+ distances become equal to 2.9 Å, Tl+ Tl+ distance shortens as well, and hence the bond character is changed. Note that Se – Tl distance of 2.9 Å is longer than the sum of the covalent radii of Tl (1.49 Å) and Se (1.17 Å), 2.66 Å, but much shorter than the sum of the ionic radii of Tl1+ (1.59 Å for CN=8 [10] in the CsCl lattice) and Se2- (1.98 Å), i.e., 3.57 Å, and is close to the sum of the ionic radii of Tl3+ (0.98 Å for CN=8 [10]) and Se2-, i.e., 2.96 Å, respectively. Such a bond is intermediate between the ionic and covalent ones. Since Tl+ and Tl3+ ions in the phases I and II of TlS and phase I of TlSe occupy different crystallographic positions, the charge transfer between them seems to be unlikely, and these phase are suggested to show semiconductor properties. One might assume metallic properties for the TlS IV and TlSe II phases since in the CsCl-type lattice all thallium ions are structurally equivalent, and free charge transfer between them may be structurally allowed. This hypothesis has to be verified by electric conductivity measurements of TlS and TlSe under high pressure. We note that Demishev et al. [9] also determined the equations of state for TlS and TlSe. A pressure dependence of the electrical conductivity (as a rule, at room temperature) was reported by several authors. Kerimova et al. [7] obtained a gradual increase in the conductivity along the c axis in the undoped TlInSe2 crystal, from 1.4×10-6 Ohm -1cm -1 at ambient pressure to 7.5×10-5 Ω -1cm -1 at P=1.4 GPa. The authors realized that the pressure behavior of the conductivity may be well fitted by the equation ln

(P) = ln

(0) + AP ,

(5.1)

where A = d ln (P)/dP = 0.34 ×10-2 Ω-1cm -1GPa -1. Assuming an exponential variation of the electrical conductivity with pressure (P) =

(0) exp(-G P /2kT) ,

(5.2)

i

where G = dE g / dP , k is the Boltzmann constant and T is the temperature, the authors found that G = 2kTA [7]. To satisfy equation 5, parameter G should be negative, since A>0, and thus the band gap should decrease with increasing pressure, in accordance with that observed in the experiment. Anisotropy of the crystal structure of the TlMX2 compounds causes anisotropy of their electronic properties. The pressure dependence (up to ~0.8 GPa) of the anisotropy of the electrical conductivity and the Hall coefficient R in TlSe single crystal was measured by Allakhverdiev et al. [6,11,12]. This study showed a gradual (nearly linear) increase in conductivity and decrease in Hall coefficient with increased pressure (Figure 5.3). While the slopes of the R||(P) and R (P) curves are practically the same, the slopes of the ||(P) and

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(P) dependences are different, so that the anisotropy of the conductivity decreased with increasing P, and the ||(P) and (P) curves cross each other at P ~ 0.7 GPa. An analogous decrease in ( || - ) with increased temperature has also been observed in [12].

Figure 5.3. Pressure dependence of the parallel ( ||) and perpendicular (  ) components of the electrical conductivity of TlSe (subscripts || and denote current direction relative to the c axis) and of Hall coefficient R|| and R (subscripts || and denote magnetic field direction relative to the c axis). (From [11] with permission, © 1983 American Institute of Physics.)

Rabinal et al. [13] measured the electrical resistivity of TlSe up to 8 GPa and reported that TlSe undergoes a pressure-induced semiconductor-metal transition. Figure 5.4 shows the variation of the electrical resistivities || and parallel and perpendicular to the c axis, respectively, as a function of pressure at room temperature. At ambient conditions, || and are 400 and 208 Ω×cm, respectively. Both || and components decrease continuously with pressure and reach metallic values at about 2.7 GPa. Under ambient conditions, the || / ratio is 1.92. Under high pressure, || becomes unity at about 0.6 GPa, decreases continuously, and becomes as low as 0.016 at 5.0 GPa (since || drops more rapidly than ). We note a correlation of this finding with the data of ref. [6] that shows || < at ambient conditions. Furthermore, according to ref. [6], || increases faster than with increasing pressure up to 0.8 GPa, and the components become equal around 0.76 GPa. The measurements of Rabinal et al. [13] show a decrease in the band gap with increased pressure. The authors affirm that TlSe reveals positive temperature coefficient of the resistivity above 2.7 GPa, indicating the metallization of the samples [13].

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Figure 5.4. Pressure dependence of the electrical resistivities of TlSe parallel (triangles) and perpendicular (cycles) to the c axis at ambient temperature. (From [13] with permission, © 1991 Wiley.)

These findings are in good agreement with the calculations of Gashimzade et al. [1] who predicted a semiconductor-metal transition in TlSe at pressures of 2.2 – 5 GPa, but, however, conflict with the findings of Demishev et al. [9] who obtained a pressure-induced phase transition in TlSe at P = 21 GPa, suggesting it to be the semiconductor-metal transition. Furthermore, Rabinal et al. [13] found the anisotropy of the resistivity in TlSe at ambient conditions ( || >1) to be opposite to that expected from the chain structure ( || 100 cm-1) is characteristic of two-mode behavior since the Raman bands broaden on approaching medium concentration (0.1 < х < 0.4 and 0.6 < х < 0.9) and reveal new lines that are not characteristic of the extreme components. The bands 279, 233, 196 and 137 cm-1 for TlGaSe2 move to the low frequency region. Shift of 279 cm-1 line with variation of composition may be described assuming In atoms as heavier substitution impurity in TlGaSe2 crystal. Behavior of the other Raman lines of solid solutions TlGaxIn1-xSe2xS2(1-x) can not be explained using known criteria. Behavior of phonons with frequencies 233, 196 and 137 cm-1 is strongly influenced by new lines that are not observed in the extreme components. The frequency of TlInS2 vibration at 140 cm-1 decreases with increase of x. The frequencies of the lines at 284, 293 and 304 cm-1 are practically not changed, but these lines broaden and decrease in intensity and, moving over the solid solution region, collapse into a single line. The frequency of the band 346 cm-1 increases with increase in concentration up to x = 0.4, passes the region of no solubility and then decreases; at that, the intensity decreases as well. Behavior of new lines does not display common regularities. New bands in Raman spectra of solid solutions (TlGaSe2)x(TlInS2)1-x are probably caused by independently formatting solid solutions based on TlGaS2-TlInSe2 compounds, the Raman spectra of extreme components of which were studied in refs. [58,59]. One should also take into account that violation of translation symmetry in solid solution would cause violation of the wave vector selection rules and, as a consequence of this fact, phonons with non-zero vectors can participate in the first order scattering spectra. Some effects caused by the resonance interaction of vibrations are also possible. Thus analysis of Raman spectra of (TlGaSe2)x(TlInS2)1-x mixed crystal showed that the system is characterized by a multi-mode reorganization of the Raman-active lines in the solubility region. Simultaneous substitution in both cation and anion sublattices significantly complicates the interpretation of the spectra of solid solutions. Character and nature of the new Raman lines in the above-mentioned solid solution may probably be elucidated on examination of solid solutions (TlGaS2)x(TlInSe2)1-x, which, however, have a limited solubility region.

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Figure 10.23. Room temperature Raman spectra of (TlGaSe2)x(TlInS2)1-x crystals. (1) х = 0; (2) х = 0.1; (3) х = 0.2; (4) х = 0.3; (5) х = 0.4; (6) х = 0.6; (7) х = 0.7; (8) х = 0.8; (9) х = 0.9; (10) х = 1. (From [56] with permission, © 1980 Wiley.)

Figure 10.24. Dependence of frequencies of Raman-active phonons on composition of solid solution (TlGaSe2)x(TlInS2)1-x. (From [56] with permission, © 1980 Wiley.)

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3

6

Vibration Spectra of Mixed Crystals (Solid Solutions) of A B ...

215

10.5.2. IR Reflection Spectra Figure 10.25 shows IR reflection spectra of several compositions of solid solutions (TlGaSe2)x(TlInS2)1-x, while Figure 10.26 shows composition dependence of phonon frequencies for this system. Discrepancy in the number of the observed reflection bands with that predicted by means of symmetrical analysis (6 instead of 30) for the extreme components of the solid solution (x = 0 and x = 1 ) may be caused by weak oscillation forces of the unobserved IR-active phonons. In the band of the residual rays in crystals with x = 0.3 and 0.4, in the high frequency spectral region, 300-350 cm-1, there also appears a band (as a curve inflection in the spectra of Figure 10.25) that can hardly be assigned to a certain symmetry type owing to its very weak intensity in the polarization E ⊥ C . This band is probably

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caused by a contribution of Аu vibrations forbidden in the geometry E ⊥ C , which may be excited owing to oblique light incidence on crystal. Because of this fact the frequencies of that band are not shown in Figure 10.26.

Figure 10.25. Room temperature reflection spectra of solid solutions (TlGaSe2)x(TlInS2)1-x for polarization E ⊥ C for (1) x = 0; (2) х = 0.2; (3) х = 0.3; (4) х = 0.4; (5) х = 0.6; (6) х = 0.8; (7) х = 1. (From [56] with permission, © 1980 Wiley.)

As seen from Figures 10.25 and 10.26, composition dependence of frequencies is rather complicate. First, one can notice a pronounced single-mode behavior of the low-frequency phonons (ν < 100 cm-1) in the solubility region. Second, an additional band appears in the region 0.3 ≤ х ≤ 0.7, which is absent in the extreme components of the solid solution. Third, all high-frequency vibrations show multi-mode character in the solubility region, and additional diffuse bands appear. Appearance of the additional bands in the reflection spectra may be caused by substitution in both cation and anion sub-lattices. Therefore additional dipolar vibrations, which are characteristic of Ga-S and In-S atomic vibrations in TlGaS2-TlInSe2 sublattices,

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would independently occur in the medium concentration range of solid solutions (TlGaSe2)x(TlInS2)1-x. Indeed, LO-TO vibration with splitting of 355-325 cm-1 for x = 0.6 drops sharply to zero value with reduction of TlInS2 content [59]. The highest frequency reflection band of TlInS2 crystal with LO-TO splitting of 340 - 295 cm-1 seems to correspond to In-S vibration.

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Figure 10.26. Concentration dependence of IR-reflection-active phonon frequencies in solid solutions (TlGaSe2)x(TlInS2)1-x. ● —ТО phonon frequencies, ○ —LO phonon frequencies. (From [56] with permission, © 1980 Wiley.)

Increase in frequency of this vibration with increase of TlGaSe2 content can be explained by substitution of heavy In atom for lighter Ga. At that, for x = 0.6 this vibration transforms to the vibration of Ga – S atoms with LO-TO splitting of 355-325 cm -1, regular frequency reduction of which with increase of x (for x > 0.6) is caused by insertion of heavier Se atoms into the lattice. The vibration with LO-TO splitting of 216-199 cm -1 for x = 0.1 is observed up to x = 0.9 (Figures 10.25 and 10.26) and is absent for the extreme components of the solid solution. Its frequency is close to that of Eu vibration of TlInSe2 [58] but is shifted to the high-frequency region, which is caused by vibrations of lighter S atoms. We also notice that, as in the case of Raman spectra, crystal lattice disorder leads to appearance of phonons from whole Brillouin zone in the first order spectra. Resonance interaction of vibrations, which can hardly be taken into account in such complicate spectrum, is also possible. Thus the conducted analysis shows that the vibration spectra of solid solutions (TlGaSe2)x(TlInS2)1-x exhibit multi-mode behavior of Raman and IR-active optical phonons depending on composition. Moreover, there appear additional vibrations that are absent in the extreme components of the solid solution. Frequencies of these new vibrations are close to that of phonons of TlGaS2 (monoclinic space group) and TlInSe2 (tetragonal space group D418h ). Figures 10.27 and 10.28 show dispersion dependences of real (ε1(ω)) and imaginary (ε 2(ω)) parts of dielectric permittivity and Im[-ε-1(ω)] function. Figure 10.29 shows composition dependence of high-frequency (ε∞) and low-frequency (ε0) dielectric permittivity in solid solution. The former was calculated from the reflection spectra in the range 50002 7000 cm-1, while the latter was calculated from the Leiden-Saaks-Tailor ratio ε 0 = ε ∞ ∏ ω τ i . 2 i ωτ i

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Figure 10.29 shows decrease in high-frequency permittivity and increase in low-frequency permittivity on approaching medium concentration. Increase in ε0 is caused by contribution of new vibrations that were absent in the extreme components of solid solution, while decrease in ε∞ in the medium concentration range is probably caused by increase in conductivity due to larger contribution coming from TlInSe2 whose conductivity is higher than that of TlGaSe2 and TlInS2 [60-64].

10.6. ANGULAR DEPENDENCE OF OPTICAL PHONON (QUASIPHONON) SPECTRA IN А3В6 AND А3В3С 62 COMPOUNDS 10.6.1. Angular Dependence of Polar Optical Phonon Frequencies in Chain-Like TlSe, TlS and TlInSe2 Crystals Polar optical phonons that propagate in an arbitrary direction with respect to the principle axes of anisotropic crystal show angular dependence of the phonon frequencies on direction of wave vector q . Let us discuss phonon dispersion in TlSe, TlS and TlInSe2 compounds

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[65,66]. The frequencies of "pure" longitudinal and transversal phonons for these compounds are collected in Tables 9.4 and 9.9. Lang et al. [67] generalized the Born and Huan-Kun's theory [68] of the long-wave optical vibrations for the case of anisotropic crystals, defining arrangement of opacity bands as a function of angle θ between reflecting surface and optical axis of a crystal C . It was also shown that single-axis crystals exhibit two absorption bands that are divided by a transparency interval. Analogous expressions for angular dependence of phonon frequencies were obtained in [69]. Analysis of the experimentally observed angular dependence of polar phonon frequencies in layered crystals GaSe, GaS and InSe was reported by Abdullaev et al. and Allakhverdiev et al. [70-72] based on theory outlined by Onstott [73].Experimental data of angular behavior of IR-active phonons in α-quartz are discussed in ref. [73]. The authors of the above-mentioned experimental and theoretical works analyzed the cases of θ = 0 and 90o, i.e., E // C ( K ⊥ C ) (here K is the wave vector of the incident light) (unusual beam) and E ⊥ C ( K // C ) (usual beam) when one phonon appears in the band of residual beams. Besides, they neglected phonon damping. In this paragraph we generalize the Lang's theory for the multi-oscillator case, taking into account the phonon damping. The results are compared with experimental data on angular dependence of polar optical phonon frequencies in thallium selenide. We calculated dependences of the IR-active phonon frequencies in TlS, TlInS2 and solid solutions TlSxSe1-x as a function of angle θ between the reflection plane and optical axis of crystal. Figure 10.30 shows reflection spectra of thallium selenide for different θ. There are several features of angular behavior of IRactive phonon frequencies of TlSe: 1. Reflection spectra in s-polarized light (when electrical field vector of the incident light wave is perpendicular to the incident plane) coincide with one another at different orientations regarding to the C axis and look like curve 7 in Figure 10.30. Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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2. Reflection spectra in p-polarized light (when electrical field vector of the incident light wave is in the incident plane) vary significantly with variation of angle θ. Increase in angle θ yields А2u and Еu phonons appearance in reflection spectra. At that, according to the prediction of the theory of symmetry, two А2u phonons appear at θ = 0° and three Еu phonons – at θ = 90°, while for intermediate orientations (θ = 15, 30, 45, 60 and 75°) four reflection bands are observed. Only transversal phonons show angular dependence of frequencies, while longitudinal do not. Angular dependence of the reflection coefficient R is unchanged when the spectra are measured in the s-polarized light [67].

Figure 10.27. Dispersion of real (ε1) and imaginary (ε2) parts of dielectric permittivity and Im( −ε function in solid solution (TlGaSe2)x (TlInS2)1-x, where х = 1; 0.9; 0.8 and 0.7 for curves 1 - 4, respectively [80].

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−1

)

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Figure 10.28. Dispersion of real (ε1) and imaginary (ε2) parts of dielectric permittivity and Im(-ε-1) function in solid solution (TlGaSe2)x(TlInS2)1-x, where х = 0.4; 0.3; 0.2 and 0 for curves 5 - 8, respectively [80].

In the case that electrical vector of the light wave is in the incident plane, the expression for R is more complicate, and ε(ω)= εi + iε 2 will be angular dependent and given by

ε (ω , θ ) = [ε //−1 (ω ) sin 2 θ + ε ⊥−1 (ω ) cos 2 θ ] ~

n

ε // (ω ) = ε ∞// ∏ i =1

ω //2 li − ω 2 − iωγ // li ω //2 ti − ω 2 − iωγ // ti

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(10.24)

(10.25)

Alexander M. Panich and Rauf M. Sardarly

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ε ⊥ω ) = ε ∞⊥ ∏ i =1

ω ⊥2 li − ω 2 − iωγ ⊥li ω ⊥2 ti − ω 2 − iωγ ⊥ti

(10.26)

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Figure 10.29. Composition dependence of high-frequency (ε∞) and low-frequency (ε0) dielectric permittivity in solid solution (TlGaSe2)x(TlInS2)1-x, where, ○ — ε∞ and ● — ε0, respectively [80].

Figure 10.30. Room temperature reflection spectra of TlSe in p-polarized light for different angles θ between incident plane and optical axis of crystal. (1) 00; (2) 150; (3) 300; (4) 450; (5) 600; (6) 700; (7) 900 [80]. Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Here εII and ε ⊥ are the complex dielectric permittivities at θ=0° and θ=90°, respectively, II ε∞

⊥

and ε ∞ are the high-frequency dielectric permittivity at θ=0° and θ=90°, respectively, ω

and γ are the frequencies of longitudinal and transversal phonons and corresponding damping coefficients at θ=90°, and i is the number of oscillator. Points in Figure 10.31 show frequencies of optical phonons obtained from the reflection spectra (Figure 10.30) using Kramers-Kronig relation, while solid lines in Figure 10.31 confining the regions of total reflection (hatched regions) are received by means of computer calculations using Eqs. (10.24 – 10.26) for damping coefficient values γti = γli = 0.01 cm-1. The calculations made using real damping coefficients ( γ ⊥t = 12 cm-1; γ // t = 11 cm -1; γ ⊥l = 15 cm-1; γ // l = 2 cm -1), obtained from the experimental reflection spectra of TlSe for θ = 0 and 90°, accurately describe the experimental curves; furthermore, angular dispersion of the phonon frequencies does not reveal a transparency region that is separated by an opacity region between Еu (LO) and А2u (LO) high-frequency modes (Figure 10.31 c). The fact that the experimental angular dependence of lattice reflection in TlSe in the high frequency region does not reveal a transparency band separated by the opacity band is caused by large damping coefficients of longitudinal optical phonons in TlSe. Figure 10.31 (a,b) shows angular dependences of reflection bands for TlS and TlInSe2 crystals calculated by means of Eqs. 10.24–10.26; at that, damping coefficients were taken as 0.01 cm-1. Points correspond to longitudinal and transversal optical phonons. Figure 10.32 shows bands of total reflection in solid solution TlSxSe1-x calculated by means of Eqs. 10.24 10.26. Thus a variation of solid solution composition and angle θ between crystal plane and

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C axis allows selecting of any spectral region within 20 - 400 cm-1 in for reflection filters. It should be noticed that "pure" TO and LO phonons exist only in the extreme cases θ = 0 and 90°. For all intermediate values 0° < θ < 90° experiment exhibits tilted phonons (quasiphonons), which are transformed to the “pure” TO and LO phonons only for wave propagation along the principal axes of crystal.

10.6.2. Angular Dependence of Polar Optical Phonons in TlGaSe2, TlGaS2 and TlInS2 Compounds Long-wave optical phonons in layered crystals TlGaSe2, TlGaS2 and TlInS2 were studied by many authors [74-77, 2, 37-39, 78]. Angular behavior of the IR-active optical phonons in TlGaS2 was studied by Gasanly et al. [79]. The results of our calculations qualitatively coincide with experiment and calculations reported in ref. [79]. In this paragraph we report on calculation of angular behavior of phonon frequencies of Аu and Вu symmetry in single crystals TlGaSe2, TlGaS2 and TlInS2. In this calculation, we used phonon frequencies (Table 10.7) of symmetry Вu, K // C and Аu, K ⊥ C , and also Eqs. 10.24 - 10.26. We note that monoclinic structure of TlGaSe2, TlInS2 and TlGaS2 crystals should make them to be biaxial.

Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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Figure 10.32. Angular dependence of total reflection bands for TlSxSe1-x, where где х = 0.2; 0.4; 0.6 and 0.8 (regions 1–4, respectively) [80].

In such a case, usage of Eqs. 10.24–10.26 is not correct enough since they were received for uniaxial crystals. However, our IR reflection spectra measurements show that these compounds behave as uniaxial ones, which is caused by their weak biaxiality and also by their one-dimensional disordering of along the optical axis.

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Figure 10.33 shows angular dependence of phonon frequencies in TlGaSe2, TlInS2 and TlGaS2; experimental values of phonon frequencies determined from the reflection spectra (Table 10.7) are shown by circles. Hatched regions correspond to the bands of total reflection. Phonon frequencies on the lines bounding the hatched regions correspond to quasiphonons, i.e., the optical phonon frequencies are neither longitudinal nor transversal. The geometry of the experiments is shown in Figure 10.33a. Table 10.7. High frequency dielectric permittivity and optical phonon frequencies determined from reflection spectra TlGaSe2

TlInS2

E ⊥C

E ∥C

TlGaS2

E ⊥C

E ∥C

E ⊥C

E ∥C

ТО

LO

Е∞

ТО

LO

Е∞

ТО

LO

Е∞

ТО

LO

Е∞

ТО

LO

Е∞

ТО

LO

13 ,2

254 228 88 46 32

268 245 94 56 36

12

261 131 100 61

281 136 107 76

8.1

298 266 126 84 28

332 281 144 96 40

7.9

330 284 160 110 63

349 300 176 116 77

7.6

364 340 320 303 172 122 39

373 356 338 310 182 132 60

7.4

370 312 202 147 80

390 331 210 152 90

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Е∞

Figure 10.33. (a) orientation of the crystal plane regarding to optical axis C.(b,c,d) –dependence of bands of total reflection on angle θ between the plane and optical axes of TlGaSe2, TlInS2 and TlGaS2 crystals, respectively [80]. Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

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[55] Allakhverdiev, K.R.; Godjaev, M.M.; Nadjafov, A.I.; Sardarly, R.M.; Sov. Phys. - Solid State 1982, 24, 2533-2535. [56] Allakhverdiev, K.R.; Godzhaev, M.M.; Nadjafov, A.J.; Sardarly, R.M. Phys. Status Solidi b 1982, 112, K93-K96. [57] Guseynov, G.D.; Godzhaev, M.M.; Sardarly, R.M. Preprint 89, Baku, Az. SSR, 1984, 29. [58] Allakhverdiev, K.R.; Nizametdinova, M.A.; Salaev, E.Yu.; Sardarly, R.M.; Safarov, N.Yu.; Vinogradov, E.A.; Zhizhin, G.N. Preprint 8, Baku, Az. SSR, 1979, 32. [59] Khalafov, Z.D.; Gasanly, N.M.; Sardarly, R.M. Izv. Vissh. Uch. Zaved. Physica 1977, 2, 151-154. [60] Guseinov, G.D.; Ramazanzade, A.N.; Kerimova, E.M.; Ismailov, M.Z. Phys. Status Solidi b 1967, 22, K117-K122. [61] Guseinov, G.D.; Abdullaev, G.B.; Bidzinova, S.M.; Seidov, F.M.; Ismailov, M.Z.; Rashaev, A.M. Phys. Letters 1970, 33A, 421-422. [62] Bakhyshov, A.E.; Musaeva, L.G.; Lebedev, A.A.; Yakobson, M.A. Sov. Phys.Semicond. 1975, 9, 1548-1551. [63] Karpovich, I.A.; Chernova, A.A.; Demidova, L.I.; Leonov, E.I., Orlov, V.M. Inorg. Mater. 1972, 8, 70-72. [64] Karpovich, I.A.; Chernova, A.A.; Demidova, L.I. Inorg. Mater. 1974, 10, 2216-2218. [65] Babaev, S.S.; Tagiev, M.M.; Sardarly, R.M. Safarov, N.Yu. Proc. National (USSR) Conf. Phys. Semicond., Baku, 1982, 102-103. [66] Aliev, R.A.; Allakhverdiev, K.R.; Salaev, E.Yu.; Safarov, N.Yu.; Sardarly, R.M.; Steynshreiber, V.Ya. Reports AN Azerb. SSR, 1983, 39, 37-40. [67] Lang, I.G.; Pashabekova, U.S. Sov. Phys. - Solid State 1964, 6, 3640-3645. [68] Born, M.; Huang K. Dynamic Theory of Crystal Lattices; Clarendon Press: Oxford, 1988, 487 p. [69] Mertin, L.; Lamprecht, G. Phys. Status Solidi b 1970, 39, 573-580. [70] Abdullaev, G.B.; Allakhverdiev, K.R.; Babaev, S.S.; Mityagin, Yu.A. Nani, R.G.; Salaev, E.Yu.; Tagiev, M.M. Sov. Phys. - Solid State 1979, 21, 910-912. [71] Allakhverdiev, K.R.; Babaev, S.S.; Salaev, E.Yu.; Tagyev M.M. Phys. Status Solidi b 1979, 96, 177-182. [72] Allakhverdiev, K.R.; Babaev, S.S.; Nani, R.G.; Salaev, E.Yu.; Tagiev, M.M. Preprint 7, Baku, Az. SSR, 1979, 18. [73] Onstott, J.; Lucovsky, G. J. Phys. Chem. Solids 1970, 31, 2171-2184. [74] Allakhverdiev, K.R.; Sardarly, R.M.; Wondre, F.; Ryan, I.F. Phys. Status Solidi b 1978, 88, K5-K9. [75] Sardarly, R.M. Investigation of vibration spectrum TlSe and its threefold analogues methods of optical spectroscopy; Dissertation, Baku, 1978, 146. [76] Allakhverdiev, K.R Optical properties and vibration spectra of layered and chained crystals of groups A3B6, A3B3C62 and solid solutions on their basis; Dissertation, Baku, 1980, 288. [77] Allakhverdiev, K.R.; Nizametdinova, M.A.; Vinogradov, E.A.; Zhizhin, G.N.; Sardarly, R.M. Proc. Int. Conf. Lattice Dynamics, Flamarion, Paris, 1978, 95-98. [78] Aliev, R.A.; Allakhverdiev, K.R.; Sardarly, R.M.; Steynshreiber, V.Ya. Phys. Status Solidi b 1982, 112, K153-K156. [79] Gasanly, N.M.; Goncharov, A.F.; Melnik, N.N.; Ragimov, A.S.; Tagirov V.J. Phys. Status Solidi b 1983, 116, 427-443.

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[80] Sardarly R.M. Lattice dynamics and phase transition in A3B6 and A3B3C62 compounds and their solid solutions; Dissertation, Baku, 1985, 320p.

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Chapter 11

ELASTIC PROPERTIES OF LOW-DIMENSIONAL COMPOUNDS

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11.1. INTRODUCTION The structural anisotropy of the chain and layered crystals under consideration should cause anisotropy of their mechanical properties such as Young’s modulus and elastic constants. In the layered TlMX2 crystals, strong covalent chemical bonds (with some ionic contribution) are present inside the layers, while the interactions between the layers are very weak and usually are described as van der Waals-like ones. Some chain crystals also show strong intra-chain and weak inter-chain bonds, while in the others inter- and intra-chain interactions may be comparable. In layered crystals, interatomic distances within the layer are much shorter than the interplane spacing, which causes anisotropic behavior of the compounds. The most typical representatives of the layered crystals are graphite and boron nitride, whose structures consist of monoatomic layers. MoS2 and PbI2 reveal triple Mo-S-Mo and I-Pb-I layers, while GaS and InSe show quadruple S-Ga-Ga-S and Se-In-In-Se layers, respectively. The structure of Bi2Te3 consists of quinary Te-Bi-Te-Bi-Te layers. The layered TlMX2 crystals under consideration reveal more complicate structure that consists of heptatomic, "seven-storied" layers. The aforementioned anisotropy causes specific features of the phonon spectra of layered crystals, such as low-frequency modes corresponding to displacements of the layer relative to each other, small velocity of the acoustic mode propagation along the weak bond direction, quadratic dispersion law that is characteristic for vibrations propagating in the plane with the displacement vector perpendicular to the layer (so called TA⊥ -mode), etc. These peculiarities result in specific dynamics of the crystal lattice, which is reflected in such phonon-driven physical phenomena as heat capacity [1], thermal expansion [2] and thermal conductivity [3].

11.2. ELASTIC CONSTANTS IN LAYERED CRYSTALS Theory of elasticity [4] describes a small deformation of solid by means of deformation (strain) tensor

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1 ⎛ ∂U ∂U k U ik = ⎜⎜ i + ∂xi 2 ⎝ ∂xk

⎞ ⎟⎟ ⎠

(11.1)

where U i is the component of the displacement vector. Internal stress resulting from deformation is described by stress tensor σ ik . The force component per volume area is

F1 =

∂σ ik ∂σ k

(11.2)

According to the Hooke's law

σ ik = CiklmU lm

(11.3)

where Ciklm is the fourth-rank elasticity tensor, and its components are called elastic

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constants. Since elasticity deformation and stress tensors are symmetric, the number of independent constants is reduced to 21. Next, symmetry of a specific crystal lattice results in further reduction of the independent elastic constants. For example, graphite, boron nitride, gallium sulfide and selenide and indium selenides, having hexagonal symmetry that is characteristic of layered crystals, are described by five independent elastic constants:

C xxxx = C yyyy = C11

C zzzz = C33

C xxzz = C yyzz = C13

C xxyy = C12

C xzxz = C yzyz = C44 In-plane elastic properties of a hexagonal crystal are isotropic and are described by elastic constants C11 and C12. Elastic constant C33 determines Young's modulus in the direction perpendicular to the plane, while C13 defines the corresponding Poisson's ratio. Elastic constant C44 describes a stress arising from a displacement of the layers relative to each other. One of the basic techniques in measuring the elastic constants is a measurement of the velocity of sound wave propagation in crystal. The wave propagation velocity of three normal modes can be found by solving the Green-Christoffel equation:

Cijkl n j nl − δ ik ρυ 2 = 0

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(11.4)

Elastic Properties of Low-Dimensional Compounds

231

where ρ is the density and n j , nl are the directional cosines regarding to propagation direction. Choosing a suitable sound wave propagation direction, one can determine the elastic constants C11, C12, C33, C13 and C44. Usually so called Cauchy's relations [5] among elastic constants should be fulfilled. They arise from the condition of equilibrium of the crystal lattice that requires minimum of energy. We note that these relations are valid under assumption of taking into account only the central-force interaction among atoms. However, in layered crystals the intralayer interactions noticeably exceed the interlayer ones, thus the interaction between the distant in-plane neighbors may be of the same order of magnitude that the interaction between the next neighbors bound by a weak interlayer bond. Owing to the weakness of the interlaminar central-force interactions, non-central in-plane interactions should be taking into account, which makes the Cauchy's relations inapplicable for the layered crystals. However, the relations between the elastic constants exist and originate from the criterion of stability of the crystal lattice [5]. The crystal lattice is stable when the density of energy is a positively defined quadratic form, such as any small deformation results in the energy increase. The coefficients of quadratic form are described by a matrix

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⎡C11C12 .C13 C14 .C15 C16 ⎤ ⎢C C C C C C ⎥ ⎢ 21 22 23 24 25 26 ⎥ ⎢C 31C 32 C33 C 34 C 35 C 36 ⎥ ⎢ ⎥ ⎢C 41C 42 C 43 C 44 C 45 C 46 ⎥ ⎢C C C C C C ⎥ ⎢ 51 52 53 54 55 56 ⎥ ⎢⎣C 61C 62 C 63 C 64 C 65 C 66 ⎥⎦

(11.5)

thus this quadratic form is positively defined if determinants of all matrices of serial ranks (i.e., principal minors) are positive. Let us notice that physical properties of the layered TlMX2 crystals do not show in-plane anisotropy [6,7]. Therefore, to a good approximation, these crystals may be considered as quasi-hexagonal ones. For crystals with hexagonal symmetry, the matrix Eq. (11.5) is given by

⎡C 11 C 12 C 13 0 ... 0 ... 0 ⎤ ⎢C C C 0 ... 0 ... 0 ⎥ ⎢ 12 11 13 ⎥ ⎢C 13 C 11 C 33 0 ... 0 ... 0 ⎥ ⎢ ⎥ ⎢ 0 .... 0 ... 0 ...C 44 0 ... 0 ⎥ ⎢ 0 .... 0 ... 0 ... 0 ...C 0 ⎥ 44 ⎢ ⎥ C 66 ⎦⎥ 0 .... 0 ... 0 ... 0 ... 0 ... ⎣⎢

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(11.6)

Alexander M. Panich and Rauf M. Sardarly

232 and

(C

its

principal

minors

C33 − C132 )C442 C66 ,

11

are

equal

to

C66 ,

C44C66 ,

(C11 − C12 )[C33 (C11 + C12 ) − 2C132 ]C442 C66 .

2 C44 C66 ,

C33C442 C66 ,

Taking into account

that C66 = (C11 − C12 ) / 2 , the principal minors are positive under conditions

C 44 > 0

(C11 − C12 ) > 0

C11C 33 − C132 > 0

(11.7)

C 33 (C11 + C12 ) − 2C132 > 0 One more important relation between elastic constants may be found from the following considerations. Let us apply a stretching force p to a layer hexagonal crystal with x,y axes in plane and z axis perpendicular to the plane. In such a case the diagonal components of deformation tensor are given [4] by

⎤ ⎡ C33 / 2 1 U xx = p ⎢ + 2 ⎥ ⎣ 2(C11 − C12 ) (C11 + C12 )C 33 − 2C13 ⎦ ⎤ ⎡ C33 / 2 1 U yy = p ⎢− + 2 ⎥ ⎣ 2(C11 − C12 ) (C11 + C12 )C 33 − 2C13 ⎦

(11.8)

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⎤ ⎡ C13 U zz = p ⎢− 2 ⎥ ⎣ (C11 + C12 )C33 − 2C13 ⎦ The energy conservation law requires that stretching force may result in volume increase only, i.e.,

ΔV = U xx + U yy + U zz > 0 V

(11.9)

Taking into account that C11 , C12 > C13 , one can get from Eqs. 11.8 and 11.9 that

C33 > C13

(11.10)

Elastic constants of some layered crystals, obtained in the room temperature ultrasound propagation measurements, are listed in Table 11.1. One can find that graphite shows the largest anisotropy of the elastic properties, C11 / C33 ~ 30 , while the most of layered crystals reveal C11 / C33 ~ 2 − 4 . C33 does not vary a lot from crystal to crystal, while C11 does and stipulates the variation of the anisotropy. Such behavior is caused by the crystal structure, in

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Elastic Properties of Low-Dimensional Compounds

233

which the difference in the interlayer spacings (3.35 Å in graphite, 3.33 Å in BN, 3.81 Å in GaS, 3.84 Å in GaSe, 4.19 Å in InSe) is smaller than the difference in distances between atoms within the layer (r(C-C) = 1.421 Å in graphite, r(B-N) = 1.446 Å in BN, r(Ga-S) = 2.32 Å in GaS, r(Ga-Se) = 2.48 Å in GaSe, r(In-Se) = 2.53 Å) in InSe.

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Table 11.1. Elastic constants of layered crystals Crystals

Elastic constants, 1010 Pа

С (graphite)

С11 106

C12 18

C13 1,5

C33 3,7

С44 0,018-0,035

GaS GaSe InSe TiSe2 TaSe2 NbSe2 TlInS2 TlGaSe2

15,7 10,3 7,3 12 22,9 19,4 4,49 6,42

3,3 2,9 2,7 4,2 10,7 9,1 3,05 3,88

1,5 1,2 3,0

3,6 3,4 3,6 3,9 5,4 4,2 3,99 4,37

0,8 0,9 1,2 1,4 1,9 1,8 0,3 0,5

— — — —

Acousto-optic measurements of nonlinear elastic constants in KY(MoO4)2 [8] showed that noticeable anisotropy of mechanical strength in layered crystals is caused by extraordinary high anharmonicity of the inter-layer interactions. The value of C44 in graphite depends on the sample quality, such as layer stacking, defects, etc. and thus varies within the broad limits. Seldin and Nezbeda [9] showed noticeable reduction of the number of dislocations and stacking defects under -irradiation, which gives rise to increase in elastic constant C44 determined by inter-layer interactions. Measurement of elastic constant C13 in crystals, in particular by ultrasonic techniques, is usually difficult and is made with insufficient accuracy (around 30%). One of the obstacles is a necessity to prepare samples with crystal faces on the certain angles with the layers. Some information about elastic constant C13 may be obtained using relationships of the theory of elasticity [4]. Resonance frequency of bending vibrations of a thin rectangular plate with fixed end point is given by expression

fn

k n2 d 2 l2

E eff (11.11)

12

where d is the plate thickness, l is the plate length, is the density of material, and kn = 1.875; 4.694; …, is the constant corresponding to different the nth harmonics. In the chosen experimental geometry and under assumption that the crystal is quasi-hexagonal, an effective Young's modulus is Eeff coefficient

describing

E /(1 in-plane

2

)

C11 C132 / C33 , where

deformation.

Eq.(11.11)

is the Poisson's yields

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a

condition

Alexander M. Panich and Rauf M. Sardarly

234

C11 C132 / C33

Eeff

0 obtained above in Eq.(11.7). We note that reliable values of C13

are required for interpretation of the thermal expansion data. In hexagonal crystals, in-plane thermal expansion coefficient // is given by expression [10]

C C33 V (C11 C12 )C33 2C132

//

//

(C11

C33 C12 )C33 2C132

(11.12)

where V is the volume, C is the heat capacity, Cik are the elastic constants, // and  the average weighted Grüneisen parameters [11]. Since Eq.(11.7) shows that

(C11 C12 )C33 2C132

0 , Eq.(11.12) reveals that in-plane thermal expansion coefficient

may be negative under two conditions: (i) if the Grüneisen parameter // is negative and (ii) 0 , such as the second term in Eq.(11.12) is if C13 is large enough, i.e., C33 // C13 //

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larger than the first term. As a rule, layered crystals show significant anisotropy of thermal expansion. While thermal expansion coefficient (perpendicular to the layers) is large and positive, the in-plane coefficient // is small. Furthermore, in some crystals such as graphite and boron nitride // is negative in the wide temperature range, from 0 to 600 K [13,14]. Some authors [12-15] explain the negative in-plane thermal expansion in the abovementioned crystals assuming that large thermal expansion perpendicular to the layers is accompanied by a side compression (so called Poisson's compression), which is larger than the in-plane expansion. However, in such a case C13 C33 that contradicts Eq.(11.10). Belenkii et al. [2] showed that phonon spectra of layered crystals reveal bending vibrations that are characterized by negative Grüneisen parameters. At low temperature such vibrations contribute significantly to the density of states, which may lead to negative average weighted Grüneisen parameters [11]. An interesting discussion about temperature and pressure dependences of elastic constants С11, С12, С33, C13, С44 can be found in Ref. [16]. Such dependences provide us with the information about anharmonic interactions in crystals and are useful for calculations of thermodynamic parameters, e.g., of linear thermal expansion coefficient [2]. Relative variations of elastic constants in some layered crystals with uniform pressure P in the region of linear Cik(P) dependence are collected in Table 11.2. The Table shows that pressure dependence of the interlayer elastic constants is stronger than that of intralayer constants. Quasi-harmonic theory of crystals [17] predicts that temperature dependence of elastic constants is given by

Cij0 (1 Dij )

Cij 0

Here Cij is the elastic constant of static lattice,

(11.13)

is the average energy of oscillations,

and Dij is the constant that depends on the kind of crystal.

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Table 11.2. Pressure dependence of elastic constants in layered crystals at room temperature Crystals

1 Cik

Cik

11

a

C11

C12

C13

C33

С44

С (graphite)

4

6

21

26

8

GaS

8

14

—

63

GaSe

8

16

—

56

—

InSe

11

16

—

50

—

For kT

h

0

(here

is the average oscillation frequency)

0

0

yields Cij ~ Cij (1 Dij kT ) , while for kT

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, 10

h

0

h

0,

kT , and Eq.(11.13)

and Eq.(11.13) yields that the

temperature dependence of elastic constants approaches zero temperature with zero slope. This conclusion is supported by measurements of graphite that exhibits nearly linear increase in elastic constants on cooling at relatively high temperature, while the slope of Cij(T) becomes nearly zero on approaching T = 0. Graphite also shows stronger temperature dependence of the interlayer elastic constants in comparison with the intralayer constants. Pressure and temperature dependence of the elastic constants is an anharmonic phenomenon and occurs due to phonon-phonon interaction and lattice deformation caused by thermal expansion:

dCik dT

Cik T

V

Cik P

(11.14) T

where P, V and T are pressure, volume and temperature, respectively, is the volume thermal expansion coefficient, and is the volume compressibility coefficient. Here first and second terms come from phonon-phonon interaction and lattice deformation due to thermal expansion, respectively. Since

dCik dCik and are usually known from the experiments, dP dT

and using the equations

2

//

,

2

//

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(11.15)

236

Alexander M. Panich and Rauf M. Sardarly

C11 C12 2C13 , (C11 C12 )C33 2C132

//

C11 C12 2C13 (C11 C12 )C33 2C132

one can distinguish between two above-mentioned contributions. In graphite, relative temperature variation of elastic constant C33 that is attributed to the interlayer interaction is twice larger than that of "intralayer" constant C11 (0.115 and 0.062, respectively). At that, the contribution of thermal expansion to this variation is much larger for the "interlayer" elastic constant C33 than for the "intralayer" constant C11 (57 and 15 %, respectively). Analogous effect was observed in temperature dependences of vibration frequencies in the layered crystal GaS [19], which shows two Raman-active modes attributed to interlayer and intralayer vibrations. It was shown that relative temperature variation of the frequency of interlayer mode noticeably (four times) exceeds that of intralayer mode. At that, the contribution of the thermal expansion to this variation is much larger for the interlayer mode than for the intralayer mode (75 and 40 %, respectively). Analysis of the literature data on temperature behavior of the optical phonon frequencies in layered crystals such as GaSe and in molecular crystals such as As4S4 [20] leads to a conclusion that the contribution of the thermal expansion to the total variation of interlayer or intermolecular vibration is from 60 to 80 %, while such a contribution to the variation of frequency of the intralayer (or intramolecular) vibrations is only 20 - 40 %.

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11.3. ELASTIC CONSTANTS IN LAYERED CRYSTALS NEAR PHASE TRANSITIONS Presence of weak interlayer bonds in layered crystals increases the opportunity of temperature-induced structural phase transitions discussed in detail in Chapter 7. In this paragraph we show that the structural phase transitions cause a sharp ―softening‖ of the elastic constants in crystals. Phenomenological calculations of thermodynamic potential coefficients and elastic constant anomalies in the region of phase transitions were made by Sannikov et al. [21]. Ilisavskii et al. [22] have carried out careful measurements of the velocity of propagation of longitudinal and transverse ultrasonic waves along the layers and perpendicular to the layers in the temperature range 80–300 K. The incommensurate phases of TlInS2 and TlGaSe2 crystals, existing in the temperature ranges 200–214 K and 107–120 K, respectively, were observed as a sharp reduction (―softening‖) of the elastic constants as seen in Figures 11.1 and 11.2 [24]. Figure 11.1 shows that relative variations of the "interlayer" (C33) and "intralayer" (C11) elastic constants in TlGaSe2 are similar

( C11 / C11 ~ 5% , C33 / C33 ~ 4.5% ). However, in TlInS2 crystals the change of the "interlayer" elastic constant C33 under phase transition is twice larger than that of "intralayer" constant C11 ( C11 / C11 ~ 1,5% , C33 / C33 ~ 9% ). Such a behavior of the elastic constants in the region of phase transitions in TlInS2 is in agreement with the measurements of temperature dependences of the linear thermal expansion coefficient ij [23]. In the phase transitions region , which reflects crystal expansion in the direction of a weak bond (i.e.,

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Elastic Properties of Low-Dimensional Compounds

237

perpendicular to the layers), exhibits a sharp jump, while no noticeable jumps of in the inplane direction are observed. Using the data of Ilisavskii et al. [22] and taking into account Eq.(11.11), one can build the temperature dependences of the elastic constant C13 in TlInS2 and TlGaSe2 crystals

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Figure 11.1. Temperature dependence of elastic constants C11 and C33 in single crystal TlGaSe2. (From [24] with permission, © 2006 Springer.)

Figure 11.2. Temperature dependence of elastic constants C11 and C33 in single crystal TlInS2. (From [24] with permission, © 2006 Springer.)

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238

Figure 11.3. Temperature dependence of elastic constants C13 in single crystals TlInS2 and TIGaSe2. (From [24] with permission, © 2006 Springer.)

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(Figure 11.3). As seen from this Figure, elastic constant C13 in TlGaSe2 noticeably increases on cooling from ~ 220 K down to phase transition temperature showing C13 / C13 ~ 80% , while variation of this constant in TlGaSe2 is very small. This finding allows explaining a difference in temperature behavior of the linear thermal expansion coefficient ij in isostructural crystals TlGaSe2 and TlInS2. TlGaSe2 reveals essential increase in in the direction perpendicular to the layers and sharp decrease in the in-plane value of on cooling [25,26]. In our opinion, such a behavior is caused by a noticeable increase of the elastic constant C13 on cooling down to phase transition and by determinative contribution of the second term in Eq.(11.12). Another good tool in studying elastic constants and their behavior near phase transition is Mandelshtam-Brillouin light scattering. Brillouin frequency shift for inelastic light scattering is given by

c

( n i2

n s2 2n i n s cos )1 / 2 ,

(11.16)

where is the velocity of acoustic phonon, c is the light velocity in vacuum, ni and ns are the coefficients of light refraction in the directions of the incident ray and scattered light, respectively. In the case of ni = ns = n we have

2 n/ c

(11.17)

The dependence of the acoustic phonon velocity on elastic constants Cii is given by

(c ii / )1 / 2

(i = 1, 2, 3)

where ρ is the crystal density. One can receive from Eqs.(11.17) and (11.18) that Physical Properties of the Low-Dimensional A3B6 and A3B3C62 Compounds, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook

(11.18)

Elastic Properties of Low-Dimensional Compounds

cii

c2 (

)2 4n2

239

2

(11.19)

Laiho et al. [6, 27] have measured Mandelshtam-Brillouin scattering in ternary layered chalcogenides TlInS2 and TlGaSe2 in order to get information about the dynamics of acoustic phonon modes propagating either perpendicular to the layers (elastic mode C33) or in the layers (elastic mode C11). The Mandelshtam-Brillouin scattering in single crystals was measured using a Kr laser operating at λ = 568.2 nm and a triple pass Fabry-Perot interferometer to resolve the spectra of scattered light. Elastic constants Eq.(11.19) were calculated using values of mass density ρ = 5.71 g×cm–3 [28] and light refraction coefficients ni = 2.54 and ns = 2.37 [29]. In addition to the temperature dependence of Cii, the sound absorption coefficient, which is related to the full width of Brillouin peaks at half height ГВ as

(11.20),

was measured near the phase transition points. In TlInS2 the macroscopic physical quantities like the dielectric permittivity and the spontaneous polarization reveal no anisotropy in the (001) plane, which can be attributed to the similarity between the monoclinic a and b crystal axes, as well as probably to the formation of a disordered twin structure in the a,b plane. Temperature dependence of the

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elastic coefficient corresponding to the propagation of the longitudinal elastic mode with q // (001) is shown in Figure 11.4. Due to the properties of the crystal these data are related to an average elastic constant C1 in this plane. Noticeable anomaly is observed at T = 189 ± 2 К, which is the temperature of phase transition to an improper ferroelectric phase as described in Chapter 7.

Figure 11.4. Temperature dependence of the elastic coefficient corresponding to the propagation of the longitudinal elastic mode with q // (001) in TlInS2. (From [27] with permission, © 1988 Taylor & Francis.)

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To elucidate the meaning of the linear temperature dependence of C1(T) below 189 K, we use the Landau free energy expansion that involves coupling between the order parameter q and the components of the strain ei. Making the simplifying approximation that the order parameter has only one component, we get

F

q2 / 2

q 4 / 4 cii ei2 / 2

i

q 2 ei

i

q 2 ei2

(11.21)

wherefrom, using the standard method [30], one can get that

C0 ;

C (T ) C (T ) Here q

s

T>189K

C0

qs

2

2

2(

2

directly proportional to

i

e

;