Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory 9811522162, 9789811522161

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Rui Lan

Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory

Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory

Rui Lan

Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory

123

Rui Lan Jiangsu University of Science and Technology Zhenjiang, Jiangsu, China

ISBN 978-981-15-2216-1 ISBN 978-981-15-2217-8 https://doi.org/10.1007/978-981-15-2217-8

(eBook)

Jointly published with Xi’an Jiaotong University Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Xi’an Jiaotong University Press. ISBN of the Co-Publisher’s edition: 978-7-5693-1390-1 © Xi’an Jiaotong University Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

In recent years, nonvolatile storage technology has made some significant progress in many aspects, which has brought new opportunities for improving the storage efficiency of computer systems. Phase change random access memory (PCRAM) as one of new NVM technologies has attracted wide attention due to its high size scalability, fast read–write speed and high-density storage capability. In particular, because of its nonvolatility, byte addressability and other characteristics, PCRAM has the potential as both main memory and external memory, which may bring significant changes to the future storage architecture. Therefore, it is considered to be one of the most promising new NVM technologies that can completely replace DRAM. In PCRAM devices, phase change materials (PCMs) transform between amorphous and crystalline states reversibly, which transformation provides a dramatic change in electrical resistivity, and this resistivity change is the basis for data storage. This phase transformation process is controlled by Joule heating and cooling process. Although the volume of PCRAM is theoretically infinitely shrinkable, however, the heat generated by high programming current density largely limits the further reduction of the volume of PCRAM. Only by effectively restricting the diffusion of heat in the material, fast heating in the local minimum area and avoiding the cross-influence of heat on adjacent storage units can PCRAM devices with lower energy consumption and smaller volume be obtained. Therefore, thermophysical properties of PCMs, such as thermal conductivity, are essential for the material selection and structural optimization of phase change storage devices. This book focuses on the thermophysical properties of Ge-Sb-Te alloys which are most widely used phase change materials and the measuring technique. By introducing the details of the measuring procedure and parameter calibration, the book provides readers the accurate method to obtain the thermophysical properties of phase change materials and other related materials. The thermal and electrical conductivity data are used together to analyze the conduction mechanism. These properties data and the conduction mechanism will be beneficial for understanding the phase change materials and PCRAM industry simulation.

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I would like to express my sincere appreciation and thankfulness to my doctoral supervisor, Prof. Masahiro Susa. This book cannot be completed without his help. My sincere gratitude goes also to Prof. Rie Endo, Yoshinao Kobayashi, Ji Shi and all who give me kind help. At last, my deepest love is given to my family. Thank you so much for giving me trust and supporting to the fullest extent. All my achievements attribute to you. Zhenjiang, China

Rui Lan

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thermal Conductivity Theory . . . . . . . . . . . . 1.2.1 Electronic Thermal Conductivity (ke) . 1.2.2 Phonon Thermal Conductivity . . . . . . 1.2.3 Other Mechanisms . . . . . . . . . . . . . . . 1.3 Previous Studies on Ge–Sb–Te Alloy System 1.3.1 Phase Diagram . . . . . . . . . . . . . . . . . 1.3.2 Crystallographic Structure . . . . . . . . . 1.3.3 Electrical Resistivity . . . . . . . . . . . . . 1.3.4 Thermal Conductivity . . . . . . . . . . . . 1.3.5 Optical Properties . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Establishment of the Hot-Strip Method for Thermal Conductivity Measurements of Ge–Sb–Te Alloys . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Establishment of Hot-Strip Method . . . . . . . . . . . . . . . . . 2.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Measurement Setup Modifications . . . . . . . . . . . . 2.2.3 Parameters a and R0 Optimization . . . . . . . . . . . . 2.3 Sample and Experiment Procedure . . . . . . . . . . . . . . . . . . 2.4 Reliability Verification of the Hot-Strip Method . . . . . . . . 2.5 Application of the Hot-Strip Method to Ge–Sb–Te Alloys 2.5.1 Characterization of Sb2Te3 Alloys Before and After Measurements . . . . . . . . . . . . . . . . . . . 2.5.2 Thermal Conductivity Results of Sb2Te3 Alloy . . . 2.6 Uncertainty in Thermal Conductivity Measurement of Sb2Te3 Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.6.1 Uncertainty in Current (I ) . . . . . . . . . . . . 2.6.2 Uncertainty in DR, XT and dDV d ln t . . . . . . . . Dh 2.6.3 Uncertainty in Distance Between Potential Terminals (l) . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Sensitivity Coefficients . . . . . . . . . . . . . . . 2.6.5 Combined Standard Uncertainty . . . . . . . . 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Thermal Conductivities of Ge–Sb–Te Alloys . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sample and Experimental Procedure . . . . . . . . . . . . . . . . . . . 3.3 Characterization of Sb–Te and GeTe–Sb2Te3 Alloys . . . . . . . . 3.3.1 Characterization of Sb–Te Alloys . . . . . . . . . . . . . . . . 3.3.2 Characterization of GeTe–Sb2Te3 Alloys . . . . . . . . . . . 3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2Te3 Alloys . . 3.4.1 Thermal Conductivities of Sb–Te Alloys . . . . . . . . . . . 3.4.2 Thermal Conductivities of GeTe–Sb2Te3 Pseudobinary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Uncertainty in Thermal Conductivity Measurements of Ge–Sb–Te Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Uncertainty in Current (I ) . . . . . . . . . . . . . . . . . . . . . 3.5.2 Uncertainty in DR, XT and dDV d ln t . . . . . . . . . . . . . . . . . Dh 3.5.3 Uncertainty in Distance Between Potential Terminals (l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Sensitivity Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Combined Standard Uncertainty . . . . . . . . . . . . . . . . . 3.6 Comparison with Reported Data . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Electrical Resistivities of Ge–Sb–Te Alloys . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Four-Terminal Method . . . . . . . . . . . . . . . . 4.2.3 Measurement Setup and Procedure . . . . . . . 4.3 Characterization of GeTe–Sb2Te3 Alloys . . . . . . . . 4.4 Electrical Resistivities of GeTe–Sb2Te3 Alloys . . . . 4.5 Uncertainty in Electrical Resistivity Measurements of GeTe–Sb2Te3 Pseudobinary Alloys . . . . . . . . . . 4.5.1 Uncertainty in Resistance (R) . . . . . . . . . . . 4.5.2 Uncertainty in Diameter of Sample (d) . . . .

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4.5.3 Uncertainty in Distance Between Inner 4.5.4 Sensitivity Coefficients . . . . . . . . . . . . 4.5.5 Combined Standard Uncertainty . . . . . 4.6 Comparison with Reported Data . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Thermal Conduction Mechanisms and Prediction Equations of Thermal Conductivity for Ge–Sb–Te Alloys . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Thermal Conduction Mechanisms . . . . . . . . . . . . . . . . . . 5.2.1 Structural Similarity . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Sb–Te Binary Alloys . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Ternary GeTe–Sb2Te3 Pseudobinary Alloys . . . . . 5.2.4 GeTe Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Prediction Equations of Thermal Conductivity . . . . . . . . . 5.3.1 Sb–Te Binary Alloys . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Sb2Te3–GeTe Pseudobinary Alloys . . . . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Densities of Ge–Sb–Te Alloys . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Sessile Drop Method . . . . . . . . . . . . . . 6.2.1 Sample . . . . . . . . . . . . . . . . . . . 6.2.2 Sessile Drop Method . . . . . . . . . 6.2.3 Substrate Selection . . . . . . . . . . . 6.2.4 Parameter Calibration . . . . . . . . . 6.2.5 Archimedean Method . . . . . . . . . 6.3 Density Results . . . . . . . . . . . . . . . . . . 6.3.1 Substrate Selection . . . . . . . . . . . 6.3.2 Parameter Calibration . . . . . . . . . 6.3.3 Density Results . . . . . . . . . . . . . 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Comparison with Reported Data . 6.4.2 Application Discussion . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

About the Author

Ms. Rui Lan received the B.S. degree from Department of Material Science and Technology, Dalian University of Technology, Dalian, China in 2005, the M.S. and Ph.D. degrees from Department of Metallurgy and Ceramics Sciences, Tokyo Institute of Technology (TITECH), Japan in 2009 and 2012, respectively. She has worked in the lab of Prof. Matthias Wuttig in RWTH Aachen University for 1 year. Since 2013, she has been an Associate Professor at Department of Material Science and Technology, Jiangsu University of Science and Technology, Zhenjiang, China. Her research interests include Phase change storage materials, thermoelectric thin film materials and semiconductor materials.

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Chapter 1

Introduction

1.1 Background Since the first general-purpose electronic computer ENIAC came to the world in 1946, information and communication technology (ICT) is progressing with every passing day. The influence of information on the political, economic, social and cultural activities has been elevated to an absolutely important position. Human society has entered the information age. The amount of information, information dissemination speed and information progressing speed are growing at an exponential rate. With the information explosion and the popularity of digital electrical products, memory devices and the information carriers are required to fulfill higher performance and size miniaturization with higher storage density. Confronting such challenges, conventional memories such as flash memory have been facing the limitation of the operating speed and scalability. Phase change random access memory (PCRAM) is a promising non-volatile data storage technology for next-generation memory [1]. As shown in Fig. 1.1, in PCRAM devices, phase change materials (PCMs) transform between amorphous and crystalline states reversibly, where transformation provides a dramatic change in electrical resistivity, and this resistivity change is the basis for data storage [2]. Unlike flash memory, PCRAM stores information in bit-alterable memory and can be switched between one and zero more quickly without a separate erasing step, which makes the performance of PCRAM thousands times quicker than conventional hard drives. The volume of memory element can be further shrunk since it does not need enough space to store electrons. By introducing multi-level array technology, PCRAM can store multiple logical bits in each physical cell, making ultrahigh density PCRAM possible. Nowadays, the environment problem such as global warming has become more serious. Science and technology is a two-edged sword. Industrial civilization has brought about not only advanced productivity and ample material and spiritual

© Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_1

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1 Introduction

Fig. 1.1 Schematic diagram of phase change memory operation principle

life, but also over-exploitation of natural resources, excessive consumption of fossil energy and excessive alteration and destruction of natural ecosystems. The science and technology in the future should focus on the sustainable development of the human society. PCRAM is such a technology that can reduce the burden on the environment. Ultrahigh density will lead to less resource occupation, less product processing and lower energy consumption, and eventually, environment sustainability. The phase transformation process in PCRAM is controlled by Joule heating and cooling process. To switch the memory cell to the set state, the PCMs are heated above glass transition temperature and experience amorphous to crystalline transformation. Conversely, to reset it, PCMs are heated above their melting temperature and cooled down rapidly and thereby transform from crystalline to amorphous states. Fast and localized heating over a minimum area allows lower power consumption and higher scalability of PCRAM application [3]. Therefore, thermophysical properties such as thermal conductivity of PCMs are indispensible for the design and optimization of PCRAM device structures. However, in practice, there are many kinds of PCMs compositions used for PCRAM. It is impossible to measure the thermal conductivities of all these compositions. In that case, the accurate prediction equations of thermal conductivity directly from temperature and chemical composition are quite beneficial for industrial applications. Figure 1.2 shows the ternary phase diagram depicting different phase change alloys, their years of discovery as phase change alloys, and their uses in different optical storage products [4]. The shadow parts give suitable materials for PCMs that have been identified in the past few decades. Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys are important candidates for PCMs [4–8]. These alloys own unique property combination suitable for PCMs: their remarkable crystallization kinetics and the contrast between the amorphous and crystalline phases [8]. The amorphous state of these alloys can be stable for 10 years (3 × 108 s) at room temperature to keep good data retention but recrystallize within 10−8 s after temperature is increased for information writing. The 5–6 orders of magnitude difference in electrical resistivity between amorphous and crystalline states [9] is beneficial for binary codes.

1.1 Background

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Fig. 1.2 Ternary phase diagram depicting different phase-change alloys, their years of discovery as phase-change alloys and their uses in different optical storage products, reprint from Ref. [4], with permission from Nature Materials

Conventionally, the thermophysical properties of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have also drawn much attention in the respect that these alloys are chalcogenide. Onderka and Fitzner [10], Glazov et al. [11] and Endo et al. [12] have measured the electrical conductivities of Sb2 Te3 , and Konstantinov et al. [13] and Endo et al. [12] have measured those of Sb2 Te3 –GeTe pseudobinary alloys in the solid and liquid states as functions of temperature. The results have shown that, with increasing temperature, the resistivity increases in the solid state but decreases in the liquid state. On the basis of this temperature dependence of the resistivity, it was proposed that these alloys appear to exhibit metallic conduction in the solid state and semiconductor characteristics in the liquid state: such materials are usually called liquid semiconductors. It is known for metals that free electrons dominate both electrical and heat conduction, and thus, the relation between electrical and thermal conductivities can be given by the Wiedemann–Franz (WF) law [14, 15]. λ = Lσ T = L T /ρ

(1.1)

where λ is the thermal conductivity, σ is the electric conductivity, ρ is the electric resistivity, T is temperature and L is a constant called the Lorenz number, 2.45 × 10−8 W/K2 . Thermal conductivity data for actual metals indicate that the Lorenz number changes depending upon the kind of metal and temperature but the law can predict approximate values of thermal conductivity [16]. However, for crystalline silicon which is a typical semiconductor, the WF law cannot be used to predict the thermal conductivity. This difference between thermal conductivities for metals

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1 Introduction

and silicon has given the general understanding that free electrons dominate heat conduction in metals but phonons do in silicon. The difference in heat conduction dominates between metals and semiconductors, resulting in the different temperature dependence of thermal conductivity [17]. Hence, it is very interesting to measure the thermal conductivity of solid Sb–Te and Sb2 Te3 –GeTe alloys as a function of temperature and to discuss what entity dominates heat conduction and whether the WF law can predict the temperature dependence of the thermal conductivities, and this discussion is relevant to understanding the nature of Sb–Te binary and Sb2 Te3 – GeTe pseudobinary alloys. To discuss the WF law, the electrical resistivities of Sb–Te and Sb2 Te3 –GeTe pseudobinary alloys are required. For Sb–Te binary alloys, there are many reported data, which will be discussed in detail in Sect. 1.3.3. However, for Sb2 Te3 –GeTe pseudobinary alloys, there are few data available and not enough to ensure the accuracy. Therefore, it is necessary to measure the electrical resistivity of Sb2 Te3 –GeTe pseudobinary alloys as well as to use the reported electrical resistivities of Sb–Te alloys, to discuss the thermal conduction mechanism. There have been some investigations about thermal conductivities of Sb–Te and Sb2 Te3 –GeTe alloys; however, most of them focused on the properties of thin films [18–20]. It has been proved that the thermal conductivity of thin film is sensitive to the microstructure characterized by grain size, grain boundary, defect and structural anisotropy, and the heat conduction mechanisms should be discussed based upon the thermal conductivity of bulk materials [19, 20]. However, there are not enough thermal conductivity data for bulk materials of Sb–Te and Sb2 Te3 –GeTe alloys available, especially for high temperature. Density is another important thermophysical property, which is a basic property indispensible for discussing the nature and behavior of metals and alloys, and a key input parameter for the simulations of many other properties. In the PCRAM devices, Ge–Sb–Te alloys transform between amorphous and crystalline states to record data. As shown in Fig. 1.3, in the set process, Ge–Sb–Te alloys are heated above glass transformation temperature and changes to the crystalline state; in the reset process, Ge–Sb–Te alloys are first heated above the melting point and changes to the liquid state, and then are cooled down rapidly to change back to the amorphous state. In the practical memory element of PCRAM, only the programming volume transforms between these three phases in one bit. If the densities of Ge–Sb–Te alloys change Fig. 1.3 Schematic diagram of phase change during set and reset processes for Ge–Sb–Te alloys

Reset Liquid

Joule-heating Crystalline

Amorphous Set

1.1 Background

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considerably during the phase change process, the phase change will cause serious stress in the whole system and reduce the durability of PCRAM devices. Therefore, the densities of Ge–Sb–Te alloys in solid and liquid states are required for PCRAM device optimization. On the basis of the above background, consequently, this book is composed of six chapters. The content of each chapter is introduced briefly as following. Chapter 1 “Introduction” The importance of thermophysical properties of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys for PCRAM application and scientific signification has been explained. The thermal conductivity theory and the previous studies of the thermophysical properties have been introduced. Chapter 2 “Establishment of the Hot-Strip Method for Thermal Conductivity Measurements of Ge–Sb–Te Alloys” The hot-strip method has been established for the thermal conductivity measurements. The reliability of the hot-strip method has been verified by measuring the thermal conductivities of titanium and fused silica. The thermal conductivities of Sb2 Te3 alloy have been measured to confirm the applicability of the hot-strip method for the measurements of Ge–Sb–Te alloys. Chapter 3 “Thermal Conductivities of Ge–Sb–Te Alloys” The thermal conductivity results of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been given and compared with the reported data. The uncertainty of the thermal conductivity measurements has been estimated. Chapter 4 “Electrical Resistivities of Ge–Sb–Te Alloys” The four-terminal method has been introduced for electrical resistivity measurements. The electrical resistivity results of Sb2 Te3 –GeTe pseudobinary alloys have been given and compared with the reported data. The uncertainty of the electrical resistivity measurements has been estimated. Chapter 5 “Thermal Conduction Mechanisms and Prediction Equations of Thermal Conductivity for Ge–Sb–Te Alloys” The thermal conduction mechanisms of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been discussed based on the thermal conductivity and electrical resistivity results. The prediction equations have been proposed for the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys. Chapter 6 “Densities of Ge–Sb–Te Alloys” The sessile drop method has been introduced for the density measurements of Ge– Sb–Te alloys. The density results of Sb2 Te3 and Ge2 Sb2 Te5 alloys have been given and compared with the reported data. The Archimedean method has also been used to measure the densities of these alloys at room temperature to give a comparison to

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1 Introduction

the results by the sessile drop method. The significance of the results for application has been discussed. Chapter 7 “Summary and Conclusions” The general summary and conclusions of this thesis have been given.

1.2 Thermal Conductivity Theory Thermal conductivity λ is the property of a material’s ability to conduct heat. Thermal conductivity appears primarily in Fourier’s law for heat conduction and is defined as heat flow conducted across unit area per unit time per unit temperature gradient and expressed as follows [21]. J = −λ

dT dx

(1.2)

where J is the heat flow across unit area per unit time, T is temperature and x is the distance. Units of measurements for λ in the SI system are watt meter−1 kelvin−1 (W m−1 K−1 ). In the kinetic theory of gases, the thermal conductivity of a gas is given approximately by λ=

1 Cv υl 3

(1.3)

where C v is the heat capacity at constant volume per unit volume, υ is the average velocity of the molecules and l is the mean free path. A similar picture provides a model for heat conduction in solid. Heat in solids is conducted by various carriers: electrons, lattice vibrations (phonons), photons and in some cases magnons. The total thermal conductivity is additively composed of contributions from various types of carrier and can be expressed by λ=

1 Ci υi li 3 i

(1.4)

where the subscript i denotes the type of carrier. The different thermal conduction mechanisms will be discussed in the following.

1.2.1 Electronic Thermal Conductivity (λe ) In metals, where the density of electrons is high, the electronic component of the heat capacity (C e ) is in linear proportion to T, C e ∝ T, and the speed of free electrons

1.2 Thermal Conductivity Theory

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(υ e ) is typically 106 m s−1 and independent of T. Thus λe =

1 Ce υe le ∝ T le 3

(1.5)

Thus the temperature dependence of λe is governed by the temperature dependence of l e . Since the electrical conductivity is also determined by the electron mean free path l e , the relation between thermal conductivity and electrical conductivity can be given by the Wiedemann–Franz law (WF law) [14, 15]. The WF law is the most important law for electronic thermal conductivity. In 1853, Wiedemann and Franz observed that a good thermal conductor is also a good electrical conductor, and the consistency of (λe /σ ) among metals at a given temperature is still referred to as the WF law. By 1900 Drude had already produced an electron theory of electrical and thermal conduction in metals, and successfully predicted the Lorenz number which was found experimentally by Lorenz in 1872. In 1928, Sommerfeld corrected the Drude theory using quantum theory and obtained a more accurate value for Lorenz number [22]. In the Sommerfeld free-electron model, the transport properties of a metal are governed entirely by electrons near the Fermi surface (FS). The states well below the FS are completely full, and those well above the FS are completely empty, so that it is only electrons in states close to the FS that are able to respond to applied field. According to the quantum mechanical distribution function, the electronic heat capacity C e is given by Ce =

π 2 Ne k 2 T 2ε F

(1.6)

where N e is the density of free electrons, k is Boltzmann’s constant and εF is the Fermi energy. εF =

1 2 mev F 2

(1.7)



where me is the rest mass of electron, v F is the velocity of electron at the FS. Substituting Eqs. (1.6) and (1.7) to (1.5), the electrical thermal conductivity can be expressed as λe =

π 2 Ne k 2 T l e

(1.8)



3m e v F

According to Ohm’s law, on the other hand, the current density j produced by an

electrical field E is related to electrical conductivity σ by



j = σE

(1.9)

8

1 Introduction

Macroscopically,



j = −Ne ev F



(1.10)



In a field E, each electron experiences a force −e E, and an acceleration −e E/m e . Therefore,  e /m e v F = −e Eτ

(1.11)



where τe = le /v F is the relaxation time of the electrons. Using Eqs. (1.9), (1.10) and (1.11) together, the electrical conductivity σ can be expressed as σ =

Ne e2 τe me

(1.12)

Using Eqs. (1.8) and (1.12), the Lorenz number is thus given by L=

π 2k2 λ = = 2.45 × 10−8 (W K−2 ) σT 3e2

(1.13)

Table 1.1 gives some Lorenz numbers of pure metals obtained from experiments at 273 and 373 K [23]. The theoretical value is in good agreement with experimental values, which proves the success of the WF law in explanation of electronic thermal conductivity. Table 1.1 Lorenz numbers of metals obtained from experiments at 273 and 373 K [23]

Lorenz number in 10−8 W/K−2 Metal

273 K

373 K

Ag

2.31

2.37

Au

2.35

2.40

Cd

2.42

2.43

Cu

2.23

2.33

Ir

2.49

2.49

Mo

2.61

2.79

Pb

2.47

2.56

Pt

2.51

2.60

Sn

2.52

2.49

W

3.04

3.20

Zn

2.31

2.33

1.2 Thermal Conductivity Theory

9

1.2.2 Phonon Thermal Conductivity The thermal vibrations of atoms in a crystal which are described in terms of lattice waves contribute to the thermal conductivity. According to quantum theory, the energy of each wave varies by an integral number of quanta. Each quantum of energy is called a phonon, and each lattice wave contains an integral number of such phonons. The thermal vibrations of atoms can be regarded as a gas of phonons and treated by the kinetic theory of gases. From Eq. (1.4), the phonon thermal conductivity becomes λl =

1 Cl υl ll 3

(1.14)

The Debye theory [24, 25] gives the phonon component of the heat capacity C l : 

T Cl = 9N0 k ΘD

3  0

θ/T

x 4 ex dx (T < Θ D ) (e x − 1)2

(1.15)

where N 0 is the number of atoms per unit volume and D is the Debye temperature. C l varies according to T 3 at temperatures below the Debye temperature. The mean free path of the phonons increases very rapidly at sufficiently low temperatures, and the order of l is centimeters. This magnitude is dominated, in practice, by the finite size of the specimen. Usually, l is taken as a constant, equal to the diameter of the experimental rod. Thus, λl ∝ T 3 (T < Θ D )

(1.16)

At the temperature well above D , C l is independent of temperature and almost constant. vl may be taken as some average velocity of sound. The phonon thermal conductivity is therefore governed by the mean free path of phonons. In a perfect crystal, phonons are entirely unaffected by each other and the mean free path is infinite. In real crystals, however, phonons continuously interchange energy with each other due to the structural imperfections. Peierls [26] showed that the most important processes involve the exchange of energy between three phonons. Either one phonon is annihilated to yield two others or two phonons disappear to form a third. These three-phonon processes can be classified into two classes. First, there are the normal processes (N-processes) in which momentum is conserved: these processes do not contribute directly to the thermal resistance. Second, there are umklapp processes (U-processes) in which momentum is not conserved. It is the umklapp processes that limit the thermal conductivity [27]. According to the Peierls’ theory, phonons thermal conductivity is inversely proportional to the absolute temperature. Thus, λl ∝ 1/T (T < Θ D )

(1.17)

10

1 Introduction

Fig. 1.4 Thermal conductivities of high purity KCl from 1 to 200 K, reprint from Ref. [28], with permission from Physical Review

The simplest example for phonon thermal conductivity can be seen in insulative single crystals. Figure 1.4 gives thermal conductivities of pure KCl from 1 to 200 K [28]. It can be seen that above ~50 K, the slope of the curve is close to −1, showing that λl ∝ 1/T . Below the temperature of the thermal conductivity maximum, the slope tends to increase toward a value of three characteristic of many materials at low enough temperatures; here λl ∝ T 3 .

1.2.3 Other Mechanisms Electronic thermal conductivity and phonon thermal conductivity constitute the most important mechanisms in nearly all substances at nearly all temperatures. There are still other possibilities: for example, heat conduction by ambipolar diffusion, photons or magnons. When both electrons and holes are present in a semiconductor, the thermal conduction by bipolar diffusion should be considered. Bipolar diffusion means simultaneous diffusion of both electrons and holes that are thermally activated in a sample along a temperature gradient [21, 26, 29]. In an intrinsic semiconductor, there exist equal numbers of electrons and holes produced by the thermal excitation of electrons from the valence to the conduction band. The number of electrons, N, and the number of holes, P, are each given by   P = N = (NC N V )1/2 exp −E g /2k B T

(1.18)

where N C and N V are the effective densities of states in the conduction and valence bands, and E g is the energy gap. If there is a temperature gradient, the values of P or N will be higher in the hotter region and lower in the cooler region. Both electrons and holes will transport along the temperature gradient and recombine at the cold

1.2 Thermal Conductivity Theory

11

region for the local thermal equilibrium. Bipolar diffusion will not contribute to the electrical conductivity because free electrons and holes are transported in the same direction. However, bipolar diffusion does contribute to the thermal conductivity by the heat absorption for the ionization at the hot region and the heat liberation for the recombination of electrons and holes at the cold region. The presence of the radiative heat conduction by photons has been established in a number of semiconductors where the phonon thermal conductivity is rather low and the energy gap is fairly high. The radiation will pass through the transparent material undisturbedly, but if the radiation is absorbed then the photons diffuse through the material in a manner analogous to that exhibited by the lattice vibrational phonons. Replacing the mean free path in Eq. (1.4) by the reciprocal of the optical absorption coefficient α(ω) gives λr =

1 3



∞ 0

Cr (ω)νg dω α(ω)

(1.19)

where C r (ω) is the heat capacity of blackbody radiation in the frequency range ω to ω + dω, and vg is the group velocity of the radiation, which in a solid is not the same as the free space velocity of light. The evaluation of Eq. (1.19) leads to λr =

16 σ R n 2 T 3 α −1 3

(1.20)

where σ R is the Stefan-Boltzmann constant (5.670 × 10−8 W m2 K−4 ), and n is the refractive index of the material [21]. From this equation, it can be known that λr depends on two factors, temperature T and absorption coefficient α. With increasing temperature, λr increases. On the other hand, for the materials with low α such as glasses, the contribution of λr is great; reversely, for metals which have very high α, there is almost no contribution from λr . For some semiconductors, since α decreases with increasing temperature, λr must be treated carefully and two factors should be considered together. The thermal conductivity by magnons, or quantized spin waves has been found for some time in ferromagnetic insulators. It will not be discussed in details here since this mechanism is for some special kinds of materials.

1.3 Previous Studies on Ge–Sb–Te Alloy System 1.3.1 Phase Diagram The binary phase diagram from Sb–Te system, as shown in Fig. 1.5, has been established based on the thermodynamic assessment [30, 31]. The equilibrium phases in

12

1 Introduction

Fig. 1.5 Phase diagram of the binary system Sb–Te, reprint from Ref. [31], with permission from Journal of Phase Equilibria

the system are: the liquid, L, the terminal solid solution (Sb), the terminal solid solution (Te), and the intermediate phases δ, γ and Sb2 Te3 . Sb2 Te3 melts congruently at 617.7 °C and has a narrow non-stoichiometric range, whereas the γ and δ phases melt incongruently and have broad non-stoichiometric ranges [31–33]. The phase diagram in the pseudobinary Sb2 Te3 –GeTe system was first constructed by Abrikosov and Danilova-Dobryakova [34, 35] and later confirmed by Legendre et al. [33]. Figure 1.6 shows the phase diagram of the pseudobinary Sb2 Te3 –GeTe system. It can be seen that the system contains three ternary compounds (Ge2 Sb2 Te5 , GeSb2 Te4 and GeSb4 Te7 ), which melt peritectically at 903 K (630 °C), 889 K (616 °C) and 879 K (606 °C), respectively [34, 35]. However, Shelimova et al. [36] have found that there might exist a series compounds with the similar structures of stacked layers except those three ternary compounds mentioned. The binary phase diagram of Ge–Te system has also been shown in Fig. 1.7. The Ge–Te system contains one compound GeTe, and two eutectics, with 49.85 and 85 at.%Te, at 993 K (720 °C) and 658 K (385 °C) (e2 ), respectively [33, 37, 38] GeTe phase has a fairly broad non-stoichiometry range. The high temperature form (β) decomposes into two low-temperature forms (α and γ ) at 678 K (405 °C), depending on the deviation from stoichiometry.

1.3 Previous Studies on Ge–Sb–Te Alloy System

13

Fig. 1.6 Phase diagram of the ternary system Ge–Sb–Te, reprint from Ref. [33], with permission from Thermochimica Acta Fig. 1.7 Phase diagram of the binary system Ge–Te, reprint from Ref. [33], with permission from Thermochimica Acta

14

1 Introduction

1.3.2 Crystallographic Structure The phase diagram of Sb–Te system shows there are three intermediate phases δ, γ and Sb2 Te3 , in which the crystallographic structure of Sb2 Te3 has been determined to be rhombohedra [39, 40]. When turning to other phases, the crystallographic structures show similarity. Agafonov et al. [41] and Stasova [42] investigated structures of Sb2 Te and Sb2 Te2 , respectively, and found that the structure consisted of multiple layers stacked along the c axis and presented the combination of five-layer of Sb2 Te3 and two-layer stacks of Sb2 . Sun et al. [43] investigated the structure of δ-phase which contains Te from 16 to 37 at.% and also suggested the atomic arrangement model composed of Sb2 and Sb2 Te3 layers. Kifune et al. [44] presented the structural change mechanism of Sb–Te alloy in the Te composition range from 0 to 60 at.% by analyzing X-ray powder diffraction pattern. They concluded that the crystal structure of Sb–Te binary system changes continuously with Te concentration, as shown in Fig. 1.8, where the structural units of A7-type Sb, also seen in the structure of Sb2 Te [41] and Sb2 Te2 [42], are shown by yellow solid circles, and those of the NaCl-type atomic configuration which is characteristic of Sb2 Te3 structure are by blue parallelograms. These two units are located alternately along the direction of the c axis. These results show the existence of a series of successive intermetallic compounds which are formed by intergrowth of these two structural units on the atomic level on the left of Sb2 Te3 on the phase diagram. Sb2 Te3 –GeTe alloys have unique crystallization kinetics and amorphous phase can crystallize within 10−8 s after heated, which has drawn numerous studies investigating the phase change process as well as structural analysis of crystal and amorphous phases. Figure 1.9 [4] shows the crystal structures of a typical Ge–Sb–Te alloy. Te atoms occupy one lattice site, and Ge and Sb atoms randomly occupy the second lattice

Fig. 1.8 Crystal structure models of Sb–Te binary system. The stacking period M, chemical formula and Te concentration are shown under each model, reprint from Ref. [44], with permission from Acta crystallographica

1.3 Previous Studies on Ge–Sb–Te Alloy System

15

Fig. 1.9 Crystal structure of typical Ge–Sb–Te alloy, reprint from Ref. [4], with permission from Nature Materials

site of an atomic arrangement that is close to the rocksalt structure. However, this structure is not an ideal rocksalt structure and possesses a considerable amount of vacancies and local distortions [45–47]. Many studies have been made to give a model of local structure of amorphous phase. Kolobov et al. [48] have termed this transition between the amorphous and crystalline states an “umbrella flip”: this transition is not accompanied with rupture of strong covalent bonds and attributes the fast switching speed to the similarity in the atomic arrangement. From X-ray diffraction, Kohara et al. [49] have concluded that amorphous Ge2 Sb2 Te5 is characterized by the even-folded ring structure, which is schematically depicted in Fig. 1.10. The corresponding calculations have revealed that microscopically the amorphous structure takes over many relics of the crystal structure such as even-numbered rings and bond angles that centre around 90°, although the total pair correlation functions for the amorphous and crystalline states are rather different. The mechanisms of rapid phase changes in Ge2 Sb2 Te5 were proposed, as shown by the schematic representation in Fig. 1.11 [4]. In the crystal–liquid phase-change process (stage I), the atomic configuration in the crystal phase is disarranged by heating and melted into liquid. Meanwhile in the liquid–amorphous phase-change process (stage II), only even-numbered rings are constructed in a-Ge2 Sb2 Te5 . In the amorphous–crystal phase-change process (stage III), a-Ge2 Sb2 Te5 can transform to the crystal phase with only the transformation of the large-size even-numbered (8-, 10-, 12-fold) rings to NaCl-type structure (4- and 6-fold rings) with forming Ge(Sb)–Te bonds and without breaking bonds.

16

1 Introduction

Fig. 1.10 Amorphous structure of Ge2 Sb2 Te5 alloy, reprint from Ref. [49], with permission from Applied Physics Letters

Fig. 1.11 Mechanisms of rapid phase changes in Ge2 Sb2 Te5 , reprint from Ref. [4], with permission from Nature Materials

1.3.3 Electrical Resistivity The remarkable difference in the electrical resistivity between crystalline and amorphous states of PCMs is the basis of PCRAM applications. Therefore, many studies have recently been done to investigate the resistivities of thin films Sb–Te and Ge– Sb–Te in crystal and amorphous states [50–53]. Figure 1.12 shows an example of the temperature dependence of the resistivities for sputter-deposited films of Ge2 Sb2 Te5

1.3 Previous Studies on Ge–Sb–Te Alloy System

17

Fig. 1.12 Temperature dependence of resistivities for sputter-deposited phase-change films of compositions Ge2 Sb2 Te5 and doped SbTe, reprint from Ref. [53], with permission from Nature Materials

and doped SbTe [53]. It can be seen that both materials show several orders of magnitude difference between the amorphous and crystalline states. Ge2 Sb2 Te5 shows two transitions, one from the amorphous to fcc phase, and one less pronounced transition from the fcc to hexagonal phase. In this book, thermal conductivity of bulk materials has been measured and the thermal conduction mechanism has be clarified. The WF law is a useful tool to understand the thermal conduction mechanism, while the WF law needs the electrical resistivity of bulk materials to predict the thermal conductivity. The electrical resistivity of bulk Sb2 Te3 , the intermetallic compound in Sb–Te system, has been investigated by several researchers. Onderka et al. [10], Glazov et al. [11] and Gubskaya et al. [54] have measured the electrical resistivity of polycrystalline Sb2 Te3 from room temperature to the liquid state. All the data show a positive temperature coefficient in the solid state but negative in the liquid state. Eichler et al. [55] and Sherov [56] have measured the single crystal of Sb2 Te3 along the a and c axes, respectively. The results show that large difference exists between electrical resistivities of Sb2 Te3 along the a and c axes, which means that the electrical resistivity is anisotropic in the single crystal. The results of liquid Sb2 Te3 reported by different researchers [12, 57–59] have shown to some extent a small dispersion, while for the solid state the dispersion is large due to the difference in the microstructure and crystal orientation of the samples and the measurement methods. For other compositions of Sb–Te alloys, Gubskaya et al. [54] have given the electrical resistivity results in the solid state and Blakeway [59] and Glazov et al. [60] have given the results in the liquid state. For the Sb2 Te3 –GeTe system, few electrical resistivity data are available for bulk materials. The only electrical resistivity data as functions of composition and temperature have been given by Konstantinov et al. [13]. Abrikosov et al. [34] have just given the electrical resistivities of Ge–Sb–Te alloys at room temperature. Yanez-Limon et al. measured the hole concentrations of Ge–Sb–Te alloys and found that most Ge–Sb– Te alloys are p-type semiconductors with hole concentrations of approximately 102 to 1027 m−3 , depending on the material composition [61]. Recently, Endo et al. [12] have measured the electrical resistivity of Sb2 Te3 –GeTe alloys in the liquid state.

18

1 Introduction

1.3.4 Thermal Conductivity Similar to the electrical resistivity, the investigations of thermal conductivity also mainly focus on those of thin films. Kuwahara et al. [62] have measured the thermal conductivities of nanometer-scale Sb–Te alloys and found that the thermal conductivities decrease with increasing Te concentration and that a large change is observed in the vicinity of the composition ratio of Sb2 Te3 . Lyeo et al. [18] have measured on thin films of Ge2 Sb2 Te5 from 300 K (27 °C) to 673 K (400 °C) and found that from the low temperature conductivity of amorphous phase, the conductivity increases irreversibly with increasing temperature and undergoes large changes with phase transformations. Reifenberg et al. [19] and Yang et al. [63] have measured on the thermal conductivities of Ge2 Sb2 Te5 and Ge4 Sb1 Te5 films with different thickness, respectively, and found a strong thickness dependence of the thermal conductivity and large difference between thin films and bulk materials. The thickness dependence is thought to be attributed to a combination of thermal boundary resistance and microstructure imperfections. There have been some thermal conductivity data of bulk Sb2 Te3 [64–69] reported, most of which are for around room temperature and show a large dispersion. Yokota et al. [68] have measured the thermal conductivity of Sb2 Te3 from 90 to 300 K, which decreases with increasing temperature, while Fedorov et al. [69] have measured the thermal conductivity of Sb2 Te3 from 400 K to the liquid state and found that the thermal conductivity increases with increasing temperature even in the liquid state. Both of these two reports mentioned that ambipolar diffusion plays an important role in the thermal conductivity of Sb2 Te3 . Vukalovich et al. [57] have also measured the thermal conductivity of Sb2 Te3 in the liquid state and found that the thermal conductivity increases linearly with increasing temperature. To the best of the author’s knowledge, there are no results reported for other compositions of Sb–Te alloys, and the thermal conductivities of bulk Ge–Sb–Te alloys as a function of Te concentration has been given by Yanez et al. [61] only.

1.3.5 Optical Properties Sb–Te and Sb2 Te3 –GeTe alloys have also been used for optical storage which has already been a mature technology with rewritable compact discs (CDs) as the first generation, rewritable digital versatile discs (DVDs) as the second generation, and Blu-ray discs as the third generation [4, 70, 71]. Optical storage applications utilize small differences (approximately 20%) in the reflectivity [72] and thus, the optical properties are essential for the applications. For optical storage, the index of refraction, the extinction coefficient and the related absorption coefficient are of utmost importance because they determine the efficiency of the laser light absorption as well as the optical contrast [73]. However, in this work, the main topic is the electronic

1.3 Previous Studies on Ge–Sb–Te Alloy System

19

applications of PCMs. The optical properties mentioned above will not be discussed in detail. Optical band gap is another important optical parameter, which is beneficial to understand the electronic structure, and hence, the nature of these alloys. In addition, bipolar diffusion which is one of the important thermal conduction mechanisms introduced in Sect. 1.2.3 is closely related to the optical band gap. A great effort has been given to investigate the optical band gap of Sb2 Te3 films [74–77]. From these reports, the crystal band gap energies are in the range 0.15– 0.2 eV and amorphous band gap energies are in the range 0.55–0.8 eV, where 1 eV = 1.60218×10−19 J. The optical band gap of Ge2 Sb2 Te5 , the most commonly employed composition in Sb2 Te3 –GeTe pseudobinary system, also draws much interest. The reported optical band gap energies for amorphous phase range from 0.5 to 2 eV [9, 78–81], and those for the fcc phase range from 0.5 to 1.5 eV [9, 79–81]. For the hexagonal phase, one band-structure calculation [82] yields a zero band gap, but another group has suggested that this phase be a degenerate semiconductor [13]. Lee et al. [81] and Kato et al. [9] have analyzed the optical band gap of hexagonal phase to be 0.5 eV. Park et al. [83] have systemically measured the optical band gap of Sb2 Te3 –GeTe pseudobinary alloys and given the value for the crystal phase as 0.56 eV (GeTe), 0.42 eV (Ge2 Sb2 Te5 ), 0.44 eV (Ge1 Sb2 Te4 ), 0.32 eV (Ge1 Sb4 Te7 ) and 0.14 eV (Sb2 Te3 ). Bahl et al. [84] have also given the optical band gap of GeTe films as 0.1–0.2 eV for the crystal phase and 0.8 eV for the amorphous phase.

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21

61. J.M. Yanez-Limon, J. González-Hernández, J.J. A-Gil, I. Delgadillo, H. Vargas, Phys. Rev. B 52, 16321 (1995) 62. M. Kuwahara, O. Suzuki, N. Taketoshi, T. Yagi, P. Fons, J. Tominaga, T. Baba, Jpn. J. Appl. Phys. 46, 6863 (2007) 63. Y. Yang, C.-T. Li, S.M. Sadeghipour, H. Dieker, M. Wuttig, M. Asheghi, J. Appl. Phys. 100, 024102 (2006) 64. Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens, Thermal Conductivity of Metallic Elements and Alloys (Plenum Press, New York, 1970) 65. T.C. Harman, B. Paris, S.E. Miller, H.L. Georing, J. Phys. Chem. Solids 2, 181 (1957) 66. C.H. Champness, P.T. Chiang, P. Parekh, Can. J. Phys. 43, 653 (1965) 67. Sh. Movlanov, G.B. Abdullaev, A. Bashshaliev, A. Kuliev, I. Kerimov, Dokl. Akad. Nauk Azerbaidzhan 17, 375 (1961) [in Russian] 68. K. Yokota, S. Katayama, Jpn. J. Appl. Phys. 12, 1205 (1973) 69. V.I. Fedorov, L.S. Stil’bans, Teplofiz. Vys. Temp. 5, 232 (1967) 70. E.R. Meinder, A.V. Mijritskii, L. van Pieterson, M. Wuttig, Optical Data Storage: Phase Change Media and Recording (Springer, Berlin, 2006) 71. L.P. Shi, T.C. Chong, J. Nanosci. Nanotechnol. 7, 65 (2007) 72. S. Hudgens, B. Johnson, MRS Bull. 29, 829 (2004) 73. S. Raoux, Annu. Rev. Mater. Res. 39, 25 (2009) 74. S.D. Shutov, V.V. Sobolev, Y.V. Popov, S.N. Shestatskii, Phys. Status Solidi 31, K23 (1969) 75. A.L. Jung, H. Situ, Y.W. Lu, Z.T. Wang, Z. G. He, J. Non-Cryst. Solids 114, 55 (1989) 76. W. Procarione, C. Wood, Phys. Status Solidi 42, 871 (1970) 77. A. Von Middendorf, K. Dietrich, and G. Lanwehr, Solid State Commun. 13, 443 (1973) 78. E.G. García, A.M. Galván, Y. Vorobiev, E.M. Sánchez, J.G. Hernández, G. Martínez, B.S. Chao, J. Vac. Sci. Technol. A 17, 1805 (1999) 79. H.B. Yao, L.P. Shi, T.C. Chong, P.K. Tan, X.S. Miao, Jpn. J. Appl. Phys. Part 1 42, 828 (2003) 80. A. Pirovano, A.L. Lacaita, A. Benvenuti, F. Pellizzer, R. Bez, I.E.E.E. Trans, Electron Devices 51, 452 (2004) 81. B.S. Lee, J.R. Abelson, S.G. Bishop, D.H. Kang, By. Cheong, J. Appl. Phys. 97, 093509 (2005) 82. S. Yamanaka, S. Ogawa, I. Morimoto, Y. Ueshima, Jpn. J. Appl. Phys. Part 1 37, 3327 (1998) 83. J. Park, S.H. Baek, T.D. Kang, H. Lee, Y. Kang, T. Lee, D. Suh, K.J. Kim, C.K. Kim, Y.H. Khang, J.L.F.D. Silva, S. Wei, Appl. Phys. Lett. 93, 21914 (2008) 84. S.K. Bahl, K.L. Chopra, J. Appl. Phys. 41, 2196 (1970)

Chapter 2

Establishment of the Hot-Strip Method for Thermal Conductivity Measurements of Ge–Sb–Te Alloys

Abstract In this chapter, the possible problems for applying the hot-strip method to the thermal conductivity measurements of Ge–Sb–Te alloys have been discussed based on the practical measurement conditions, and the measurement setup and parameters have been modified. After the modification, the thermal conductivities of titanium and fused silica have been measured from 298 K up to about 800 K which is the temperature just below the melting point of Sb2 Te3 alloy using the hot-strip method since they have many reported data. The results of titanium and fused silica have verified that the hot-strip method is able to give a reliable thermal conductivity data. Consequently, the thermal conductivities of Sb2 Te3 alloy have been measured from 298 K up to about 800 K to confirm the applicability of the hot-strip method for the measurements of Ge–Sb–Te alloys since these alloys are easily oxidized and have high evaporation. By analyzing the characteristics of Sb2 Te3 sample before and after the measurements and comparing the thermal conductivity results with the reported data, it is approved that the hot-strip method can be applied to the thermal conductivity measurements of Ge–Sb–Te alloys. Keywords Hot-strip method · Measurement setup · Parameters modification · Thermal conductivity · Titanium · Fused silica · Sb2 Te3

2.1 Introduction As introduced in Chap. 1, there are few data available for Ge–Sb–Te bulk alloys and the accuracy of the data reported cannot be confirmed. Two major reasons are thought to be responsible for this insufficiency. The first is that the methods for thermal conductivity measurements are still not satisfactory enough to give reliable results. The second is that Ge–Sb–Te alloys themselves are easy to degenerate during the measurements, in particular at high temperatures. If these two problems are not dealt with appropriately, it is impossible to obtain the reliable thermal conductivities of Ge–Sb–Te alloys. There are several methods to measure the thermal conductivity of solids. These methods can be mainly classified into two categories, steady-state methods and dynamic methods [1]. In steady-state methods, such as the guarded hot-plate method © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_2

23

24

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

[2], the thermal conductivity is obtained from measurements of a temperature gradient together with the heat flux into or out of the sample. However, a serious drawback in the steady-state method is that a long time is required to obtain steady state and hence, the heat loss is inevitable. Moreover, the problems of maintaining uniform temperature gradient are enhanced as the measurement temperature is increased. In contrast, in dynamic methods, the temperature distribution throughout the sample varies with time; consequently, maintaining the steady state is not needed and the measurement time is much shorter, which leads to predominance in measurements at high temperature. The commonly applied dynamic methods include the laser flash method [3, 4] and the non-stationary hot-wire method [5–8]. The laser flash method is a reliable technique for thermal diffusivity measurement but inevitably requires values of other physical properties such as density and heat capacity of the sample to convert the results into thermal conductivity values. The hot-wire method can produce thermal conductivity values directly from the measurements without using values of other physical properties of the sample, which reduces the uncertainty of the final thermal conductivity results. The hot-wire method has been widely used to measure the thermal conductivities of liquids [9–13] and proved to be an excellent technique. To apply the hot-wire method for solid measurements, Susa et al. [14, 15] have improved the hot-wire method and developed a new method named “hot-strip method”. Using the hot-strip method, Yamasue et al. [15] have successfully measured the thermal conductivity of solid silicon. However, there is no evidence that the method they used can be directly used for thermal conductivity measurements of Ge–Sb–Te alloys since these alloys have high inclination to be oxidized and to evaporate during the measurements. The oxidation is mostly apt to form on the surface, which will strongly affect the thermal conductivity measurements. In addition, the Te element easily evaporates due to the high vapor pressure. The evaporation of Te will result in the degeneration of the samples and the formation of vacancies, both of which will affect the accuracy of the measurements. Furthermore, to apply the hot-strip method for Ge–Sb–Te alloys, some parameters and setups of this method have to be modified to accommodate the new materials, as mentioned in Sect. 2.2 in detail. On the basis of the above background, consequently, the present work aims: • to establish the hot-strip method for thermal conductivity measurements of Ge–Sb– Te alloys by modifying the measurement setup and optimizing the experimental parameters. • to verify the reliability of the hot-strip method by measuring the thermal conductivities of titanium and fused silica. • to confirm the feasibility of applying the hot-strip method for thermal conductivity measurements of Ge–Sb–Te alloys by measuring the thermal conductivity of Sb2 Te3 .

2.2 Establishment of Hot-Strip Method

25

2.2 Establishment of Hot-Strip Method 2.2.1 Principle The principle of this method is basically the same as that of the hot wire method [5–8, 12, 14, 15]. Figure 2.1 shows a conceptual diagram of the hot-strip method. A constant current (I) is supplied to the hot strip placed in the sample, which serves as both a heating element and a temperature sensor, and the temperature rise (T ) of the hot strip is recorded continuously. The solution of Fourier’s equation for a continuous line source of heat says the temperature increase in a line source at time t is given by the following Eq. (2.1) [5]. T =

Q (ln t + A) 4π λ

(2.1)

where Q is the heat generation rate per unit length of the wire, λ is the thermal conductivity, t is time and A is a constant. Differentiating Eq. (2.1) with respect to natural logarithm of time, ln t, the thermal conductivity of the sample is derived as Eq. (2.2). Q λ= 4π

Fig. 2.1 Conceptual diagram of hot-strip method



dT d ln t

(2.2)

26

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Usually, the temperature rise can be recorded using a thermocouple. However, the temperature rise in a sample having high thermal conductivity such as metal is too small to be recorded accurately by a thermocouple, and the heat loss through the thermocouple will also affect the accuracy of the results. On the other hand, the temperature rise of the hot strip will lead to the resistance change (R) of the strip itself. The term R has a simple relation with voltage change (V ) by Ohm’s law expressed by Eq. (2.3). V = I · R

(2.3)

In practice, the temperature rise is recorded as the voltage change of the hot strip based on the principle of the four-terminal method. The temperature coefficient of resistance of the hot strip α can be expressed using R as Eq. (2.4). α=

1 R · R0 θ

(2.4)

where R0 is the resistance of the hot strip at 0 °C and θ is the temperature change (θ is temperature in °C). Substituting R derived from Eq. (2.4) into Eq. (2.3) leads to Eq. (2.5). V = I · α · R0 · θ

(2.5)

When a constant current is applied to the hot strip, the resistance increase is within 1% of the resistance of the hot strip at temperature T. Therefore, the heat generation rate Q per unit length of the strip can be given using the resistance (X T ) of the hot strip per unit length at temperature T by Eq. (2.6). Q = I 2 XT

(2.6)

Combining Eqs. (2.5) and (2.6) with Eq. (2.2) gives the thermal conductivity (λ) of the sample as Eq. (2.7). I 3 · α · X T · R0 λ= 4π



dV d ln t

(2.7)

2.2.2 Measurement Setup Modifications The measurement setup was first established according to the previous work of Yamasue et al. [15]. Schematic diagrams of a sample and measurement setups are shown in Figs. 2.2 [16] and 2.3, respectively. A hot strip of Pt-13%Rh (40 mm length, 1 mm width, and 20 μm thickness) was sandwiched by two parts of the sample, where

2.2 Establishment of Hot-Strip Method

27

Fig. 2.2 Schematic diagram of sample setup, reprint from Ref. [16], with permission from Journal of Applied Physics

mica films (20–30 μm thickness) were used to keep insulation between the hot strip and the sample. Pt-13%Rh wires (50 μm diameter) were connected with the hot strip as potential terminals which measured the voltage change based on the fourterminal method. The distance between these two terminals was about 20 mm. Pt wires (0.5 mm diameter) were used as lead wires. As shown in Fig. 2.3, two pieces of sample were fixed together by a stainless-steel binder and put on a plate in a furnace in Ar atmosphere. The position of the hot strip and wires was fixed by alumina cement. A standard resistance was used to provide a constant current, and a digital multimeter was used to record the voltage. However, there are two drawbacks of this kind of setup. One is the stability of the hot-strip probe on the measurement surface. Figure 2.4a shows the schematic diagram of the hot-strip probe. Figure 2.4b and c show a side view and a cross-section view of the sample setup, respectively. The thickness of the hot strip was 20 μm, while the diameter of the potential terminal was 50 μm. From Fig. 2.4c, it can be seen that the whole probe was under an unstable state on the measurement surface. To increase the stability of the probe setup, two potential wires were modified to be set on both sides of the hot strip, shown in Fig. 2.5a. Now the cross-section view changes to be like Fig. 2.5b, and the stability of the hot-strip probe is greatly improved. A picture of the practical hot-strip probe is shown in Fig. 2.5c. The other drawback of the previous setup was in the orientation of the hot strip. As shown in Fig. 2.3, the sample was put in the furnace in the vertical direction and thus, the hot strip was also in the vertical direction. The length of the hot strip was about

28

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Fig. 2.3 Schematic diagram of sample setup

4 cm, which was close to the length of the uniform temperature zone of the furnace. To set the hot strip exactly within the uniform temperature zone, the position of the sample must be controlled strictly. Even a slight deviation in the position might bring about temperature difference between the top and bottom of the hot strip and result in large error in the thermal conductivity results. Therefore, in the present sample setup, the vertical direction of the sample was changed to be horizontal, which could make sure that the temperature of the hot strip everywhere is uniform. During the measurements, another problem was found: The normal stainless-steel binder reacted with Ge–Sb–Te alloys and could not be used to fix the samples due to the high reactivity of Ge–Sb–Te alloys. Figure 2.6 shows a picture of the sample and stainless-steel binder after a measurement. Therefore, two brass plates were used in place of the stainless-steel binder and were connected together by screws to fix the samples. Two alumina plates were put between the brass plate and the sample to prevent the reaction. A schematic diagram and a picture of the present sample setups are shown in Fig. 2.7.

2.2 Establishment of Hot-Strip Method

29

Fig. 2.4 Schematic diagrams of a hot-strip probe, b side view of sample setup and c cross-section view of sample setup

2.2.3 Parameters α and R0 Optimization To determine the thermal conductivity λ according to Eq. (2.7), all the parameters were obtained from the experiment should be known. The parameters I, X T and dV d ln t and R0 was determined by measuring the resistance of Pt-13%Rh per unit length at 0 °C. According to JIS, the ratio of the resistance (R) of Pt-13%Rh at temperature θ (0–1600 °C) to that at 0 °C is given by Eq. (2.8).  R R0 = −1.441 × 10−7 θ 2 + 1.557 × 10−3 θ + 1

(2.8)

According to Eq. (2.4), differentiating Eq. (2.8) with respect to temperature θ leads to α as Eq. (2.9). α = −2.882 × 10−7 θ + 1.557 × 10−3

(2.9)

The equation for α derived from the resistance given by JIS is used for industrial activities but the reliability for this measurement has not been validated. Furthermore,

30

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Fig. 2.5 Schematic diagrams of a hot-strip probe, b cross-section view of sample setup and c picture of practical hot-strip probe

Fig. 2.6 Picture of sample and stainless-steel binder after measurement

R0 was measured to be 11.1165 , which corresponds to the hot strip to be 0.88 mm wide. In the practical measurements, the hot strip was cut from a sheet of Pt-13%Rh by hand every time, and it was very difficult to keep the width of the hot strip exactly 0.88 mm. It is improper to use the R0 value obtained by one measurement every time. Accordingly, it is strongly required to determine values for α and R0 more accurately to obtain accurate thermal conductivity values. In Eq. (2.4), α has been defined as follows:

2.2 Establishment of Hot-Strip Method

31

Fig. 2.7 Schematic diagram and picture of present sample setup

α=

1 R · R0 θ

(2.4)

Here αT is newly defined as Eq. (2.10): αT =

1 R · RT θ

(2.10)

where RT is the resistance of the hot strip at a certain temperature T. Thus, αT is the temperature coefficient of resistance of the hot strip corresponding to temperature T. Comparison between Eqs. (2.4) and (2.10) enables the terms α · R0 to be replaced by αT · RT . Consequently, Eq. (2.7) can be modified to Eq. (2.11). λ=

I 3 · α T · X T · RT 4π



dV d ln t

(2.11)

The value of αT can be derived from the resistance RT of the hot strip measured as a function of temperature. This new way to determine the parameters is expected to increase the reliability and accuracy of thermal conductivity values; in fact, satisfactory results have been obtained using the parameters in this way, as described later.

32

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

2.3 Sample and Experiment Procedure Samples used were fused silica, titanium (99.9 mass%) and Sb2 Te3 (99.9 mass%), which were produced by Kojundo Chemical Laboratory. The cylinder sample of titanium (φ20 mm × 40 mm) was cut into two pieces along the longitudinal axis and two pieces of fused silica plate (40 mm × 20 mm × 10 mm) were prepared for measurements. Cylindrical samples of Sb2 Te3 (20 mm diameter and 40–50 mm length) were prepared by melting Sb2 Te3 powders (100 g) in a quartz crucible (20 mm inner diameter) at 973 K for 4 h in vacuum, followed by cooling at a rate of 50 K/h. A schematic diagram of the furnace setup and the procedure of the melting process are shown in Fig. 2.8. The top of the molten sample was placed below the position showing the highest temperature in the furnace so that solidification proceeded from the bottom to the top of the sample and the gas absorbed in the samples was able to escape without leaving pores in the samples. Depending on the compositions of alloys, the melting temperature was changed slightly. After cooling, the sample was taken out from the crucible and cut into two parts along the longitudinal axis, and the cross-sections were mechanically polished using emery papers up to #2000. The crystallographic structures of the samples were analyzed by an X-ray diffractometer Fig. 2.8 Schematic diagram of furnace setup and procedure of melting process

2.3 Sample and Experiment Procedure

33

(XRD), and the element distributions were analyzed by a scanning electron microscope with an energy-dispersive X-ray spectrometer (SEM-EDS). In addition, the chemical compositions of the samples were measured by X-ray fluorescence (XRF) analysis. The thermal conductivities were measured by the hot-strip method during the heating and cooling cycles from 298 K up to about 800 K, which are temperatures just below the melting point of Sb2 Te3 alloy. The sample was placed in a furnace in argon atmosphere and the measurements were not conducted until the temperature became stable. During the measurements, a constant current of 2 A was applied, the accurate value of which was measured using the standard resistance. The voltage between the potential wires was monitored continuously using a digital multimeter. The temperature of the sample was measured with a K-type thermocouple, and the reading was monitored continuously using a chart recorder to know the establishment of thermal equilibrium of the sample.

2.4 Reliability Verification of the Hot-Strip Method The thermal conductivities of fused silica and titanium were measured to justify the reliability of the hot-strip method since – fused silica is often used for thermal conductivity measurements and there are many reported data for comparison. In addition, the thermal conductivity values are close to those of Sb–Te alloys. – titanium is electrically conductive and also has many reported data of thermal conductivity. Figure 2.9 shows the thermal conductivities of fused silica as functions of temperature, together with the results obtained using the previous parameters of α and R0 and the reference data [17–21]. It can be seen that the thermal conductivities of fused silica increase monotonically with increasing temperature. The present data are smaller than the results obtained before the parameters optimization, and are in good agreement with the reported data at temperatures below 700 K, which testified that the parameter optimization improved the accuracy of the hot-strip method. Also it can be seen that above 700 K the present data are smaller than the data measured by Sugawara [19] and Stuckes [21] using the longitudinal steady-state heat flow methods but are larger than the data measured by Mendelssohn [20] using the radial heat flow method. This difference in thermal conductivities would arise from the difference in the measuring methods; in particular, the effect of radiation strongly depends on the method and is prone to larger thermal conductivity values for transparent materials such as fused silica at high temperature. Furthermore, fused silica is a non-equilibrium phase so that the structure is strongly related to the thermal process undergone. This would also result in the difference of thermal conductivities at high temperature.

34

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Fig. 2.9 Thermal conductivities of fused silica as functions of temperature, together with results obtained using previous parameters of α and R0 and reference data, reprint from Ref. [16], with permission from Journal of Applied Physics

Figure 2.10 shows the thermal conductivities of titanium as functions of temperature, together with the reported data [22–25]. The thermal conductivities of titanium were measured by the hot-strip method after parameters optimization only. It can be seen that the thermal conductivity of titanium decreases with increasing temperature from 200 to 500 K and then almost settles down to a constant value up to 1000 K. The present data are in good agreement with the reported data which show quite small scatter. The present measurements by the hot-strip method reproduce the thermal conductivities for both fused silica and titanium reasonably and can be applied to thermal conductivity measurements of solids. Fig. 2.10 Thermal conductivities of titanium as functions of temperature, together reported data, reprint from Ref. [16], with permission from Journal of Applied Physics

2.5 Application of the Hot Strip Method to Ge–Sb–Te Alloys

35

2.5 Application of the Hot-Strip Method to Ge–Sb–Te Alloys 2.5.1 Characterization of Sb2 Te3 Alloys Before and After Measurements Figure 2.11 shows pictures of the measurement surfaces for Sb2 Te3 sample (a) before and (b) after thermal conductivity measurements, respectively. It can be seen that there is no visible difference between two pictures. It seems that the measurement surface did not change during the thermal conductivity measurements. Figure 2.12 shows XRD profiles for the Sb2 Te3 sample at room temperature (a) before and (b) after thermal conductivity measurements, respectively. It can be seen from Fig. 2.12a that the XRD profile shows only diffraction peaks assigned to Sb2 Te3 corresponding to a rhombohedral crystallographic structure and there is no other phase existing in the sample. Comparison between Fig. 2.12a and b indicates that there is no difference between the two profiles, which proves that the crystallographic structure of the Sb2 Te3 sample did not change during the measurements. Figure 2.13 shows EDS mapping results for the Sb2 Te3 samples at room temperature before the thermal conductivity measurements. The figure indicates uniform Fig. 2.11 Pictures of measurement surfaces for Sb2 Te3 sample a before and b after thermal conductivity measurements

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Fig. 2.12 XRD profiles for Sb2 Te3 sample at room temperature a before and b after thermal conductivity measurements

1000 Intensity (arb unit)

36

(a)

600 400 200 0 20

1000 Intensity (arb unit)

Sb 2Te3

800

30

40 2θ / deg

50

(b)

60

Sb 2Te3

800 600 400 200 0 20

30

40 2θ / deg

50

60

distributions of Sb and Te, which further confirms that there is no other phase than Sb2 Te3 . The mapping result for oxygen element shows that the Sb2 Te3 sample was not oxidized before the thermal conductivity measurements. Figure 2.14 shows the EDS mapping results for the Sb2 Te3 samples at room temperature after the thermal conductivity measurements. The distributions of Sb and Te are still uniform, and no oxygen is detected. All the above results conclude that the composition and crystallographic structure of the Sb2 Te3 sample was not affected by the Te evaporation and that the sample was not oxidized during the thermal conductivity measurements.

2.5.2 Thermal Conductivity Results of Sb2 Te3 Alloy Figure 2.15 shows typical temperature rises of the hot strip as a function of logarithm of time obtained during measurements by the hot-strip method at (a) 298 K and (b) 789 K for Sb2 Te3 . It can be seen that there is a good linearity between T and ln t at each temperature, which supports the reliability of the present measurements. The thermal conductivities of Sb2 Te3 for both temperatures have been determined to be 2.5 and 2.6 W m−1 K−1 from Eq. (2.11) using the slope of the linear portion observed between 0.7 and 2 s.

2.5 Application of the Hot Strip Method to Ge–Sb–Te Alloys

37

O

30μ μm

Sb

Te

Fig. 2.13 EDS mapping results for Sb2 Te3 samples at room temperature before thermal conductivity measurements

O

30μ μm

Sb

Te

Fig. 2.14 EDS mapping results for Sb2 Te3 samples at room temperature after thermal conductivity measurements

38

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

Figure 2.16 shows the thermal conductivity for Sb2 Te3 as a function of temperature, in comparison with values measured by other investigators [26–31] and calculated from the Wiedemann–Franz (WF) law [32] using the resistivity data reported by Onderka [33] and the theoretical Lorenz number, 2.45 × 10−8 W/K2 [34]. Thermal conductivity data obtained during the heating cycle are in very good agreement with those obtained during the cooling cycle. The data obtained in the present work show interesting temperature dependence: the thermal conductivity decreases with increasing temperature up to approximately 600 K and then increases. It can also be seen from the figure that there is large discrepancy in the data recorded at room temperature. This discrepancy may result from the samples; for example, Harman et al. [27] used Sb2 Te3 sample with excess Sb, whereas Champness et al. [29] used Sb2 Te3 sample with excess Te. In addition, different thermal conductivity measurement methods may also account for this discrepancy, that is, Harman et al. [27] and Yokota and Yokota [31] used the comparative method, Champness et al. [29] the thermoelectronic method, and Movlanov et al. [30] the longitudinal heat flow method—they all made steady-state measurements. The data obtained at room temperature in the present work are close to those reported by Yokota and Katayama [31], who did not report any data for Sb2 Te3 at higher temperatures, and the difference is within 0.5 W m−1 K−1 . The present data are also in fairly good agreement with values calculated by the WF law at temperatures lower than about 600 K. By taking into account of the difference in samples and methods, it is considered that the present data are reliable and the hot-strip method can be applied for measuring the thermal conductivities of Ge–Sb–Te alloys. Figure 2.16 also shows thermal conductivity components of Sb2 Te3 due to free electron (λe ), phonon (λph ) and bipolar diffusion (λb ) estimated by Yokota and Katayama [31]. It can be seen that contribution from free electrons is as small as about 1 W m−1 K−1 and thus phonon diffusion dominates thermal conduction at temperatures lower than 300 K. With increasing temperature, however, both contributions to the thermal conductivity become smaller, whereas contribution from bipolar diffusion increases. This contribution is not noticeable at room temperature but increases remarkably at higher temperature to become comparable to contribution from phonon diffusion. As a consequence, the total thermal conductivity of solid Sb2 Te3 would begin to increase with increasing temperature.

2.5 Application of the Hot Strip Method to Ge–Sb–Te Alloys Fig. 2.15 Temperature rises of hot strip with time for thermal conductivity measurements of Sb2 Te3 at a 298 K and b 789 K

39

(a) 298 K

Δ T/K

6 4 2 0 10-1

100 t/s

101

1 t/s

10

8

(b) 789K

Δ T/K

6 4 2 0 10-1

Johnson [26] Harman [27] Hugh [28] Champness [29] Movlanov[30] Yokota[31] λph by Yokota [31] λe by Yokota [31] λab by Yokota[31] W-F law[32] This study-Heating This study-Cooling

8 6

λ /Wm-1K-1

Fig. 2.16 Temperature dependence of thermal conductivity for solid Sb2 Te3 by hot-strip method along with data measured and calculated by WF law, reprint from Ref. [34], with permission from Japanese Journal of Applied Physics

4 2 0 0

200

400

600

Temperature/K

800

40

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

2.6 Uncertainty in Thermal Conductivity Measurement of Sb2 Te3 Alloy Uncertainties of the measurement system and data analysis are determined according to the guide to the expression of uncertainty in measurement (GUM) [35]. As mentioned before, the thermal conductivity is given by Eq. 2.11, λ=

I 3 · α T · X T · RT 4π



dV d ln t

(2.11)

Here, αT =

1 R · RT θ

(2.10)

According to Eqs. (2.10) and (2.11), the major sources of uncertainty in thermal conductivity measurements are as follows: (i) the current applied during the measurement I, (ii) the resistance change with temperature R , (iii) the resistance X T of θ the hot strip per unit length at temperature T, which also leads to the fourth uncertainty source,  is, (iv) the distance between the potential terminals and (v) the  that of voltage change with respect to the logarithm of time. Table 2.1 derivative dV d ln t shows an example of the uncertainty budget on the thermal conductivity measurement of Sb2 Te3 at 789 K, which is the highest temperature for Sb2 Te3 measurement. Uncertainties are classified into Type A and Type B.

2.6.1 Uncertainty in Current (I) The current applied during the measurements was determined by the standard resistance and read by the digital galvanometer. The measurement limits of these devices cause the uncertainty in the current. The contributions have been estimated and the derived uncertainty is listed as “accuracy of devices” in Table 2.1.

2.6.2 Uncertainty in

R θ ,

XT and

dV d ln t

These three factors were all calculated from the slope of the profiles recorded directly from the measurements. The standard uncertainties of these factors are calculated from the standard deviation of the slope by the method of least square. The accuracy of devices mentioned in Sect. 2.6.1 also affects the respective uncertainty.

B

Accuracy from equipments (W/K)

B

Accuracy from equipments (W)

B

B

Gaging of caliper (m)

Linear coefficient of thermal expansion (K−1 )

A

B

Least square method (V /ln(s))

Gaging of galvanometer (V)

dV /dln t (V /ln(s))

A

Repeatability (m)

Length between terminals l (m)

A

Least square method (W)

X T (W)

A

Least square method (W/K)

dR/dT (W/K)

0.00078

0.02375

0.4716

0.00039

2.00251

Accuracy from equipments (A)

2.67283

B

Value

Current (I/A)

Type

Thermal conductivity λ (W K−1 m−1 )

Factor of uncertainty

7.07E−06

6.51E−07

5.77E−09

7.07E−05

0.000121106

4.80E−06

1.95E−05

4.80E−06

5.62E−06

4.80E−06

Value of uncertainty

7.10E−06

0.000140238

2.01E−05

7.39E−06

4.80E−06

Standard uncertainty, u(xi)

Table 2.1 Uncertainty budget of thermal conductivity determination for Sb2 Te3 at 789 K

−28.25587627

−20.53558439

13.42131509

16265.33432

9.482337963

Sensitivity coefficient ∂λ/∂ x i 0.120194496

Combined standard uncertainty, uc(λ)

2.6 Uncertainty in Thermal Conductivity Measurement of Sb2 Te3 … 41

42

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

2.6.3 Uncertainty in Distance Between Potential Terminals (l) The distance between potential terminals was measured at room temperature using a caliper. For the measurement at high temperature, thermal expansion should be considered, as listed in Table 2.1. The value of l at T K is calculated using the following equation: l = l293 [1 + αl (T − 293)]

(2.12)

where l293 represents the distance measured at 293 K and αl represents the linear coefficient of the thermal expansion of the hot strip. The uncertainty for l results from the combination of uncertainty for (i) measurement of the distance between potential terminals at 293 K [u(l293 )] and (ii) determination of the linear coefficient of thermal expansion [u(αl )]. The linear coefficient of thermal expansion for Pt-13%Rh has not been reported in the literature; therefore, the value of Pt, 8.99 × 10−6 K−1 , is used instead [36]. The accuracy of the αl is assumed to be 0.1 × 10−7 K−1 . The uncertainty of αl has been calculated as u(αl ) =

0.1 × 10−7 = 5.77 × 10−9 √ 3

(2.13)

2.6.4 Sensitivity Coefficients The sensitivity coefficients for thermal conductivity from the above factors are calculated on the basis of Eq. (2.11) using the estimated values listed in Table 2.1.

2.6.5 Combined Standard Uncertainty The combined standard uncertainty of thermal conductivity measurements is calculated from the standard uncertainties mentioned above and the sensitivity coefficients. The expanded uncertainty is estimated to be about 0.24 × 10−6 W m−1 K−1 , that is, 9%, with the coverage factor k = 2, providing a level of confidence of approximately 95%. Conventionally, the thermal conductivity measurement is thought to have large error, and the error within 10% is acceptable. Therefore, the hot-strip method can be used to measure the thermal conductivities of Ge–Sb–Te alloys even at high temperature, and the data obtained in this work are reliable with experimental uncertainty of about 9%.

2.7 Conclusions

43

2.7 Conclusions 1. The hot-strip method has been established for measurements of the thermal conductivity of Ge–Sb–Te alloys. 2. The reliability of the hot-strip method has been verified by the thermal conductivity measurements of titanium and fused silica. 3. The feasibility of applying the hot-strip method for Ge–Sb–Te alloys has been confirmed by the thermal conductivity measurements of Sb2 Te3 alloy. 4. The uncertainty of thermal conductivity measurements of Sb2 Te3 at high temperature has been calculated, indicating an uncertainty of 9%, which is within the acceptable error range for thermal conductivity measurement. The hot-strip method established in this chapter can be reasonably applied to measurements of thermal conductivities of Ge–Sb–Te alloys.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

J.E. Parrott, A.D. Stuckes, Thermal Conductivity of Solids (Academic, London, 1975) D. Salmon, Meas. Sci. Technol. 12, R89 (2001) W.J. Parker, R.J. Jenkins, C.P. Butler, G.L. Abbott, J. Appl. Phys. 32, 1679 (1961) F. Righini, A. Cezairliyan, High Temp. High Press. 5, 481 (1973) S.E. Gustafsson, E. Karawacki, M.N. Khan, J. Phys. D 12, 1411 (1979) S.E. Gustafsson, E. Karawacki, N.M. Khan, J. Appl. Phys. 52, 2596 (1981) S.E. Gustafsson, J. Appl. Phys. 53, 6064 (1982) S.E. Gustafsson, E. Karawacki, M.A. Chohan, J. Phys. D 19, 727 (1986) H. Fukuyama, T. Yoshimura, H. Yasuda, H. Ohta, Int. J. Thermophys. 27, 1760 (2006) M.V. Peralta-Martinez, M.J. Assael, M.J. Dix, L. Karagiannidis, W.A. Wakeham, Int. J. Thermophys. 27, 681 (2006) K. Yamamoto, T. Abe, S. Takasu, Jpn. J. Appl. Phys. 30, 2423 (1991) E. Yamasue, M. Susa H. Fukuyama, K. Nagata, Metall. Mater. Trans. A 30A, 1971 (1999) A. Miyamura, M. Susa, High Temp. High Press. 34, 607 (2002) M. Susa, K. Nagata, K.S. Goto, Trans. Jpn. Inst. Metals 29, 133 (1988) E. Yamasue, M. Susa H. Fukuyama, K. Nagata, J. Cryst. Growth 234, 121 (2002) R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 110, 023701 (2011) I.M. Abdulagatov, S.N. Emirov, T.A. Tsomaeva, KhA Gairbekov, S.Ya. Askerov, N.A. Magomedova, J. Phys. Chem. Solids 61, 779 (2000) E.D. Devyatkova, A.R. Petrova, I.A. Smirnov, Solid State Phys. B 11, 740 (1960). (in Russian) A. Sugawara, Physica 41, 515 (1969) K. Mendelssohn, J.L. Olsen, Phys. Rev. 80, 859 (1950) A.D. Stuckes, Phys. Rev. 107, 427 (1957) A. Miyamura, M. Shima, K. Kohara, M. Susa, J. Jpn. Inst. Met. 69, 332 (2005) L. Binkele, High Temp. High Press. 18, 599 (1986) V.E. Peletskii, High Temp. High Press. 17, 111 (1985) R.W. Powell, R.P. Tye, M.J. Hickman, Int. J. Heat Mass Transf. 8, 679 (1965) Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens, Thermal Conductivity of Metallic Elements and Alloys (Plenum Press, New York, 1970) T.C. Harman, B. Paris, S.E. Miller, H.L. Georing, J. Phys. Chem. Solids 2, 181 (1957)

44

2 Establishment of the Hot-Strip Method for Thermal Conductivity …

28. J.P. Mc Hugh: AD 410 434 (1962) 1–11. [cited in Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens, Thermal Conductivity of Metallic Elements and Alloys (Plenum Press, New York, 1970)] 29. C.H. Champness, P.T. Chiang, P. Parekh, Can. J. Phys. 43, 653 (1965) 30. Sh Movlanov, G.B. Abdullaev, A. Bashshaliev, A. Kuliev, I. Kerimov, Dokl. Akad. Nauk Azerbaidzhan 17, 375 (1961). [in Russian] 31. K. Yokota, S. Katayama, Jpn. J. Appl. Phys. 12, 1205 (1973) 32. R. Franz, G. Wiedemann, Ann. Phys. (Leipzig) 165, 497 (1853). [in German] 33. B. Onderka, K. Fitzner, Phys. Chem. Liq. 36, 215 (1998) 34. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, Jpn. J. Appl. Phys. 49, 078003 (2010) 35. ISO/IEC Guide 98 (1995) 36. Japan Society of Thermophysical Properties, Thermophysical Properties Hanbook (Yokendo, Japan, 2008)

Chapter 3

Thermal Conductivities of Ge–Sb–Te Alloys

Abstract In this chapter, the thermal conductivities of Sb–Te binary and Sb2 Te3 – GeTe pseudobinary alloys have been measured as functions of temperature and composition from room temperature up to below the respective melting temperature using the hot-strip method. The structures of Sb-rich Sb–Te single-phase alloys have shown similarity by the structure analysis as well as the thermal conductivities. The Te-rich Sb–Te alloys are two-phase alloys and the values of the thermal conductivity are close to those of Sb2 Te3 alloy. The Sb2 Te3 –GeTe pseudobinary alloys also show similarity on the structure and thermal conductivity. All the single-phase alloys in Ge–Sb–Te alloy system have an increase of the thermal conductivity above 600 K except GeTe alloy, the thermal conductivity of which decreases monotonically with increasing temperature. The thermal conductivities of Sb2 Te3 –GeTe pseudobinary alloys have also been measured as a function of time at 773 K to confirm whether the phase transformation occurs. The results show there is no change on the thermal conductivity with time. By analyzing the uncertainty of the measurements and comparing with the reported data, the thermal conductivity results are thought reliable and able to be used to explain the thermal conduction mechanisms. Keywords Thermal conductivity · Sb–Te binary alloys · Sb2 Te3 –GeTe pseudobinary alloys · Hot-strip method · Uncertainty

3.1 Introduction As introduced in Chap. 1, Sb–Te binary and GeTe–Sb2 Te3 pseudobinary alloys are important candidates for phase change materials (PCMs) in PCRAM devices [1– 5]. Ge–Sb–Te alloys exhibit a rapid and reversible amorphous-to-crystalline phase transformation that provides a pronounced difference in electrical resistivity, which provides the basis for the development of PCRAM [6, 7]. The phase transformation process in PCRAM is controlled by Joule heating and cooling process and thus, the thermal conductivities of Sb–Te binary and GeTe–Sb2 Te3 pseudobinary alloys are required for PCRAM operation optimization. Furthermore, the thermal conductivities of Ge–Sb–Te alloys also draw much interest in the respect that these alloys are chalcogenides which have special properties. © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_3

45

46

3 Thermal Conductivities of Ge–Sb–Te Alloys

The temperature dependences of the electrical resistivities of Ge–Sb–Te alloys have indicated that these alloys show metallic properties in solid states but semi-conductive in liquid states [8–11]. To understand the nature of Ge–Sb–Te alloys, it is essential to discuss the thermal conduction mechanisms by comparing the thermal conductivities with the electrical resistivities and also the temperature dependences of the properties. Therefore, the thermal conductivities of Sb–Te binary and GeTe–Sb2 Te3 pseudobinary alloys as functions of temperature and composition are indispensible both for industrial applications and for scientific understanding. There have been a few investigations about thermal conductivities of thin films of Sb–Te alloys [12, 13] and GeTe–Sb2 Te3 pseudobinary alloys [14–16]. However, it has been proved that the thermal conductivities of the thin films have thickness dependence and should be discussed based on those of bulk materials [17, 18]. To the best of authors’ knowledge, there have been no results reported for thermal conductivities of bulk Sb–Te alloys except Sb2 Te3 , the intermetallic compound in the binary system. Even for Sb2 Te3 , most of the thermal conductivity data are for around room temperature and show a large discrepancy [19–23]. The only thermal conductivity of bulk GeTe–Sb2 Te3 alloys as a function of Te concentration was given by Yanez-Limon et al. [24] at room temperature. At the present time, there are insufficient thermal conductivity data in the literature to understand the thermal conduction mechanisms and the nature of Ge–Sb–Te alloys. The major reason for the insufficiency is that there is an absence of reliable methods for measuring the thermal conductivities of solid Ge–Sb–Te alloys due to the high oxidization and evaporation proneness of these alloys. In Chap. 2, the hotstrip method has been established for thermal conductivity measurements of Ge–Sb– Te alloys and the reliability has been verified. Therefore, the thermal conductivities of Sb–Te binary and GeTe–Sb2 Te3 pseudobinary alloys can be determined by the hot-strip method. On the other hand, the thermal conductivities of bulk materials are strongly affected by the crystallographic structures. With increasing temperature, the phase transformation will lead to a change in the thermal conductivity. The phase diagram of GeTe–Sb2 Te3 alloys shows no phase transformations occurring in solid states for Ge2 Sb2 Te5 , GeSb2 Te4 and GeSb4 Te7 [25, 26]. However, Skoropanov et al. [27] have claimed that there might exist phase transformations for GeTe–Sb2 Te3 pseudobinary alloys at about 670 K by analyzing the linear expansion coefficients of these alloys. For GeTe alloy, it has been confirmed that there is a phase change from hightemperature form (β) to low-temperature form (α and γ) at 678 K (405 °C) [28, 29]. If the phase transformations occur, they will be reflected in the thermal conductivities. Meanwhile, several literatures have suggested that such phase transformation process will not complete unless the alloys are annealed for 300–500 h [30, 31]. In the conventional applications, Ge–Sb–Te alloys are usually not annealed for such a long time, and thus, there is a high possibility that the alloys do not complete the phase transformation and are the mixtures of high-temperature and low-temperature phases at room temperature. The thermal conductivities of such an alloy may change during the service for PCRAM applications. Therefore, the thermal conductivity

3.1 Introduction

47

change with time at one certain high temperature for Ge–Sb–Te alloys should be investigated to clarify the influence of such a phenomenon. Against the above background, consequently, in this chapter, the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys as functions of temperature and composition have been determined using the hot-strip method, and the thermal conductivity changes of Sb2 Te3 –GeTe pseudobinary alloys have been investigated at one certain high temperature withholding for a long time.

3.2 Sample and Experimental Procedure Samples used were Sb–x mol% Te (x = 14, 25, 44, 70 and 90) and nGeTe·mSb2 Te3 (n:m = 1:0, 2:1, 1:1, 1:2). Figures 3.1 and 3.2 show the phase diagrams of Sb– Te [32] and GeTe–Sb2 Te3 alloys [25, 26], where the circle represents the chemical compositions of the samples and the arrow gives the respective temperature ranges for conductivity measurements. To compare the results of the whole Sb–Te system, the thermal conductivities of Sb2 Te3 investigated in Chap. 2 are also shown in this chapter. Cylindrical samples of Sb–Te (20 mm diameter and 40–50 mm length) were prepared from Sb (99.9 mass%) and Te (99.9 mass%) powders and cylindrical samples of GeTe–Sb2 Te3 (20 mm diameter and 40–50 mm length) were prepared from Sb2 Te3 (99.9 mass%) and GeTe (99.9 mass%) powders. All the powder samples were produced by Kojundo Chemical Laboratory. Mixture (100 g) of these powders

Fig. 3.1 Phase diagram of Sb–Te alloys, reprint from Ref. [32], with permission from Journal of Phase Equilibria

48

3 Thermal Conductivities of Ge–Sb–Te Alloys

Fig. 3.2 Phase diagram of GeTe–Sb2 Te3 alloys, reprint from Ref. [25], with permission from Thermochimica Acta

precisely weighed to a desired composition was melted in a quartz crucible (20 mm inner diameter) at 973 K for 4 h in vacuum, followed by cooling at a rate of 50 K/h. The surface of the melting samples was set below the maximal temperature of the furnace so that solidification proceeded from the bottom to the top of the sample and the gas absorbed in the samples could escape without leaving pores in samples. Depending on the compositions of alloys, the melting temperature was changed slightly. A schematic diagram of furnace setup and the procedure of the melting process is shown in Fig. 3.3. Subsequently, the sample was taken out from the crucible and cut into two parts along the longitudinal axis, and the cross-sections were mechanically polished using emery papers up to #2000. Figures 3.4 and 3.5 show the Sb–Te and GeTe–Sb2 Te3 samples with different compositions after polishing. The crystallographic structures of the samples were analyzed by an X-ray diffractometer (XRD), and the element distributions were analyzed by a scanning electron microscope with an energy dispersive X-ray spectrometer (SEM-EDS). In addition, the chemical compositions of the samples were measured by X-ray fluorescence (XRF) analysis. The thermal conductivities were measured by the hot-strip method during the heating and cooling cycles. The measurements withholding for a long time were carried out for Ge2 Sb2 Te5 and GeTe alloys at 773 K for 168 h, respectively. The sample was placed in a furnace in argon atmosphere and the measurements were not to be conducted until the temperature became stable. During the measurements, a constant current of 2 A was applied, which was measured using the standard resistance. The voltage between the potential wires corresponding to temperature rise of the hot strip

3.2 Sample and Experimental Procedure Fig. 3.3 a Schematic diagram of furnace setup and b procedure of melting process

49

(a) Vacuum

Tmax

Mixture of powders

(b)

973 K 673 K

2h

4h

6h

Furnace cooling

was monitored continuously using a digital voltmeter. The temperature of the sample was measured with a K-type thermocouple, and the reading was monitored continuously using a chart recorder to know the establishment of thermal equilibrium of the sample.

3.3 Characterization of Sb–Te and GeTe–Sb2 Te3 Alloys 3.3.1 Characterization of Sb–Te Alloys Table 3.1 shows the chemical compositions of Sb–x mol% Te alloys analyzed by XRF. It can be seen that the analyzed compositions of samples are slightly different from the corresponding nominal compositions since the components might more or less have evaporated during the melting process. The error coming from XRF equipment might also account for the deviation. However, it can be understood that the final compositions of these alloys lie within the target phase regions. Figure 3.6 shows an example of the XRF results of Te concentration for different positions of the Sb56 Te44 sample. The red points in the picture show the positions analyzed. From the figure it can be seen that the composition fluctuates slightly depending on

50

3 Thermal Conductivities of Ge–Sb–Te Alloys

4cm

4cm

(a) Sb86Te14

(b) Sb75Te25

4cm

4cm

(c) Sb56Te44

(d) Sb2Te3

4cm

4cm

(e) Sb30Te70

(f) Sb20Te80

Fig. 3.4 Sb–Te samples with different compositions after polishing: a Sb86 Te14 ; b Sb75 Te25 ; c Sb56 Te44 ; d Sb2 Te3 ; e Sb30 Te70 ; f Sb20 Te80

the positions but there is no clear position dependence. The compositions shown in Table 3.1 are the averages of compositions at different positions. Figures 3.7 and 3.8[33] show the XRD profiles and the EDS mapping results for the Sb–Te samples at room temperature, respectively, which can be the clues to find the crystallographic structures of Sb–Te alloys. According to the phase diagram, the alloys fall into three groups: Sb2 Te3 , alloys having x < 60 and alloys having x > 60. The second and last groups are named Sb-rich and Te-rich alloys for convenience, respectively.

3.3 Characterization of Sb–Te and GeTe–Sb2 Te3 Alloys

51

4cm

4cm

(b) Ge2Sb2Te5

(a) GeTe

4cm

4cm

(c) GeSb2Te4

(d) GeSb4Te7

Fig. 3.5 GeTe–Sb2 Te3 samples with different compositions after polishing: a GeTe; b Ge2 Sb2 Te5 ; c GeSb2 Te4 ; d GeSb4 Te7

Table 3.1 Chemical compositions of Sb–Te samples analyzed by XRF

Nominal

Analyzed

mol% Te

mol% Sb

mol% Te

14

85.61

14.39

25

78.27

21.73

44

58.05

41.95

60

42.92

57.08

70

31.75

68.25

90

10.71

89.29

(1) The first group Sb2 Te3 has a rhombohedral crystallographic structure according to the XRD profile. The EDS mapping of this sample indicates uniform distributions of Sb and Te, which further confirms that there is no other phase than Sb2 Te3 . (2) All the three Sb-rich alloys (x = 14, 25 and 44) in the present work are identified as single-phase alloys from the XRD profiles. According to the phase diagram, the samples corresponding to x = 25 and 44 should take the δ and γ phases, respectively, which is consistent with the present results. However, for x = 14,

52

3 Thermal Conductivities of Ge–Sb–Te Alloys

Fig. 3.6 XRF results of Te concentration for different positions of Sb56 Te44 sample

Sb56Te44

y 4cm x the XRD profile shows only diffraction peaks assigned to Sb7 Te corresponding to X phase found by Kim et al. [34], which is in conflict with the two-phase region in the phase diagram. The diffraction peaks indicate that all the three Sb-rich alloys have a hexagonal structure although showing small shifts in the diffraction peaks. The EDS mapping results have also confirmed that all the Sb-rich alloys are single phases. (3) The Te-rich alloys are identified as alloys consisting of two phases Te and Sb2 Te3 from the XRD profiles, which is consistent with the phase diagram. The EDS mapping results for Te-rich alloys have also shown the existence of the two phases with lamellar structure clearly. The part with lighter color in Sb mapping

3.3 Characterization of Sb–Te and GeTe–Sb2 Te3 Alloys

Intensity (arb unit)

3000

53 Sb-14 % Te

Sb 7Te (X) Sb 2Te (δ)

Sb-25 % Te

2000

Sb 2Te2(γ)

Sb-44 % Te Sb-60 % Te(Sb2Te3)

1000

Sb 2Te3 Te

Sb-70 % Te Sb-90 % Te

0 20

30

40

50

60

2θ / deg Fig. 3.7 XRD profiles for Sb–Te alloys, reprint from Ref. [33], with permission from Journal of Applied Physics

is identified to be Sb2 Te3 , and the part with darker color to be Te. The alloy with x = 70 has more uniform distribution of the two phases than the alloy with x = 90.

3.3.2 Characterization of GeTe–Sb2 Te3 Alloys Table 3.2 shows the chemical compositions of mSb2 Te3 ·nGeTe (m:n = 0:1, 1:2, 1:1, 2:1) alloys analyzed by XRF. It can be seen that the analyzed compositions of samples are slightly different from the corresponding nominal compositions. The ratios of Ge to Sb are in fairly good agreement with nominal compositions but the concentrations of Te are all smaller than the nominal ones in the four kinds of GeTe–Sb2 Te3 alloys since Te element might have evaporated during the melting process due to its high vapor pressure. The error coming from XRF equipment might also account for the deviation. The compositions shown in Table 3.2 are the averages of compositions at different positions. Figures 3.9 and 3.10 show the XRD profiles and the EDS mapping results for the GeTe–Sb2 Te3 samples at room temperature, respectively. All the four kinds of GeTe–Sb2 Te3 alloys in the present work are identified as single-phase alloys from the XRD profiles. However, the peaks of XRD profiles for GeSb4 Te7 , GeSb2 Te4 and Ge2 Sb2 Te5 are similar to each other and assign to a rhombohedral structure which is the structure of Ge0.95 Sb2.01 Te4 alloys [35] although showing small shifts in the diffraction peaks. Matsunaga et al. [36] have suggested that GeTe–Sb2 Te3 pseudobinary alloys are crystallized into metastable single phases with NaCl-type structure independent of the compositions without sufficient heat treatments. The present results show consistency with their finding except that the structure is dif-

54

3 Thermal Conductivities of Ge–Sb–Te Alloys

Sb

Te

Sb-14%Te 30 m

Sb

Te

Sb

Te

Sb

Te

Sb

Te

Sb

Te

Sb-25%Te

Sb-44%Te

Sb-60%Te (Sb2Te3)

Sb-70%Te

Sb-90%Te

Fig. 3.8 EDS mapping results for Sb–Te alloys, reprint from Ref. [33], with permission from Journal of Applied Physics

ferent. GeTe has a rhombohedral crystallographic structure according to the XRD profile, which is consistent with the phase diagram. The EDS mapping results shown in Fig. 3.10 have also confirmed that all the GeTe–Sb2 Te3 alloys are single phases.

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys Table 3.2 Chemical compositions of GeTe–Sb2 Te3 samples analyzed by XRF

55

Nominal

Analyzed (nominal)

mSb2 Te3 ·nGeTe (m:n)

mol% Ge

mol% Sb

mol% Te

GeSb4 Te7 (2:1)

10.07 (8.33)

37.98 (33.3)

51.95 (58.3)

GeSb2 Te4 (1:1)

14.67 (14.3)

32.82 (28.6)

52.51 (57.1)

Ge2 Sb2 Te5 (1:2)

25.25 (22.2)

25.82 (22.2)

48.93 (55.6)

GeTe (0:1)

55.13 (50)

Fig. 3.9 XRD profiles for GeTe–Sb2 Te3 alloys

44.87 (50)

Ge 0.95 Sb 2.01 Te 4

Intensity (arb unit)

2000

GeTe GeSb 4Te 7

GeSb 2Te 4

1000

Ge 2Sb 2Te 5

GeTe

0 20

30

40

50

60

2θ / deg

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys 3.4.1 Thermal Conductivities of Sb–Te Alloys Figure 3.11 [33] shows the typical temperature rises of the hot strip as a function of logarithm of time obtained during measurements at 293 and 773 K for Sb75 Te25 by the hot-strip method. It can be seen that there is good linearity between T and ln t at each temperature, which supports the reliability of the present measurements. The thermal conductivities have been calculated to be 5.2 W m−1 K−1 at 293 K and 6.7 W m−1 K−1 at 773 K from Eq. 2.11 using the slopes of the linear portions in the time period 0.7–2 s. Figure 3.12 [33] shows the temperature dependence of Sb-rich alloys together with the data for Sb obtained by Konno et al. [37]. It can be seen that the thermal conductivities of all Sb-rich alloys keep roughly constant below approximately

56

3 Thermal Conductivities of Ge–Sb–Te Alloys Ge

Sb

Te

Ge

Sb

Te

Ge

Sb

Te

Ge

Te

GeSb4Te7 100 μm

GeSb2Te4

Ge2Sb2Te5

GeTe

Fig. 3.10 EDS mapping results for GeTe–Sb2 Te3 alloys

600 K and, above 600 K, and increase with increasing temperature. The thermal conductivities of the Sb-rich alloys decrease with increasing Te concentration. In contrast, the thermal conductivity data for Sb decrease with increasing temperature up to approximately 500 K but increase above approximately 600 K. Figure 3.13 [33] shows the temperature dependence of thermal conductivities for the Te-rich alloys together with the data measured for Sb2 Te3 in Chap. 2. It can be seen that the thermal conductivity of Sb2 Te3 has interesting temperature dependence: it decreases with increasing temperature up to approximately 600 K and then increases. The thermal conductivities of the Te-rich alloys decrease with temperature increase until their melting points are attained, and this behavior is similar to that of Sb2 Te3 below 600 K. The temperature dependencies of the Te-rich alloys are quite close to each other although the magnitude of thermal conductivity decreases with the Te concentration increase. The composition dependence of the thermal conductivities is related to the structure and will be discussed in Chap. 5. Figure 3.14 [33] shows the thermal conductivities of Sb–x mol% Te alloys as a function of the Te concentration at room temperature together with the data for Sb and Te recommended by Touloukian et al. [19]. The thermal conductivities of the Te-rich alloys at room temperature are close to those of Sb2 Te3 and Te. The thermal conductivities of the Sb-rich alloys at room temperature increase remarkably with decreasing Te concentration. The shadows show two single-phase ranges in the

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys

57

(a) 8

293K

T/K

6

4

2

0 10-1

100

t/s

(b) 10

T/K

101

773K

5

0 10-1

100

101

t/s Fig. 3.11 Temperature rise of heater with time for thermal conductivity measurements of Sb75 Te25 at a 298 K and b 773 K, reprint from Ref. [33], with permission from Journal of Applied Physics

phase diagram. It is obvious from Fig. 3.14 that the composition dependence is able to be separated to several segments by the single phases, although being drawn empirically. Except in the composition range inside of the δ phase which is slightly wider than other single-phase ranges, the thermal conductivity seems to decrease linearly with the Te concentration increase. The composition dependence of the thermal conductivity inside of the δ phase shows a certain extent complication.

58

3 Thermal Conductivities of Ge–Sb–Te Alloys [37]

Fig. 3.12 Thermal conductivities of Sb-rich samples as function of temperature, reprint from Ref. [33], with permission from Journal of Applied Physics

Fig. 3.13 Thermal conductivities of Te-rich alloys as function of temperature, reprint from Ref. [33], with permission from Journal of Applied Physics Fig. 3.14 Thermal conductivities of Sb–Te alloys at room temperature as function of Te concentration, reprint from Ref. [33], with permission from Journal of Applied Physics

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys

59

3.4.2 Thermal Conductivities of GeTe–Sb2 Te3 Pseudobinary Alloys Figures 3.15 [38] and 3.16 [39] show typical temperature rises of the hot strip as a function of logarithm of time obtained during measurements for Ge2 Sb2 Te5 and GeTe alloys at room temperature, and at 823 K and 773 K, respectively, by the hot-strip method. It can be seen that there is good linearity between T and ln t in each measurement, which supports the reliability of the present measurements. The thermal conductivities of Ge2 Sb2 Te5 have been calculated to be 2.4 W m−1 K−1 at 293 K and 2.4 W m−1 K−1 at 823 K from Eq. (2.11) using the slopes of the linear portions in the time period 0.7–2 s. The thermal conductivities of GeTe have been calculated to be 5.9 W m−1 K−1 at 293 K and 3.4 W m−1 K−1 at 773 K. Figure 3.17 shows the temperature dependence of thermal conductivities for the three GeTe–Sb2 Te3 alloys. It can be seen that the thermal conductivities of these

(a) 298 K 6

T/K

4

2

0 10-1

(b)

100

101

t/s 823 K

8 6

T/K

Fig. 3.15 Temperature rise of heater with time for thermal conductivity measurements of Ge2 Sb2 Te5 at a 298 K and b 823 K, reprint from Ref. [38], with permission from Journal of Electronic Materials

4 2 0 10-1

100

t/s

101

60

(a) 8

298K 6

T/K

Fig. 3.16 Temperature rise of heater with time for thermal conductivity measurements of GeTe at a 298 K and b 773 K, reprint from Ref. [39], with permission from Journal of Applied Physics

3 Thermal Conductivities of Ge–Sb–Te Alloys

4

2

0 -1 10

(b)

100

101

t/s 20

T/K

773K

10

0 -1 10

100

101

t/s Fig. 3.17 Thermal conductivities of GeTe–Sb2 Te3 alloys as function of temperature

Ge2Sb 2Te5 GeSb 2Te4 GeSb 4Te7

-1

/ Wm K

-1

3

2

1

400

600

T/K

800

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys

8

GeTe-heating GeTe-cooling

6

/ Wm-1K-1

Fig. 3.18 Thermal conductivities of GeTe alloy as function of temperature, reprint from Ref. [39], with permission from Journal of Applied Physics

61

4

2

0

400

600

800

1000

T/K

alloys have interesting temperature dependence similar to Sb2 Te3 : they decrease almost linearly with increasing temperature up to approximately 600 K and then increase. The temperature dependencies of these alloys are quite close to each other although the magnitude of thermal conductivities decreases with the GeTe concentration decrease. The composition dependence of the thermal conductivities is closely related to the structure and will be discussed in Chap. 5 in details. Figure 3.18 [39] shows the temperature dependence of thermal conductivities for GeTe alloy. The thermal conductivity of GeTe alloy shows different temperature dependence from those of other alloys: it decreases linearly with increasing temperature. The results during the cooling cycle are slightly smaller than those during the heating cycle, which difference might come from the looseness of the screws of fixing the samples due to the thermal expansion. Another possible source of the difference is the β to α phase transformation of GeTe alloys. However, the thermal conductivity results of GeTe alloy as a function of time at high temperature shown later indicate that the difference is not due to the phase transformation. Figure 3.19 shows the thermal conductivities of GeTe–Sb2 Te3 pseudobinary alloys as a function of the Ge concentration at room temperature. It can be seen that there is a minimum value of the thermal conductivity near the composition of 10 mol% Ge. The thermal conductivities of GeTe–Sb2 Te3 alloys are close to each other except that of GeTe alloy, whose thermal conductivity is about three times larger than those of other alloys. The thermal conductivities of GeSb4 Te7 , GeSb2 Te4 and Ge2 Sb2 Te5 alloys increase with an increase in GeTe concentration. The higher thermal conductivity of GeTe alloy than Sb2 Te3 alloy may account for this increase in the thermal conductivity. The possible composition dependence of the thermal conductivities has been drawn empirically and shown by the line. Figures 3.20 and 3.21[39] show the thermal conductivities for Ge2 Sb2 Te5 and GeTe alloys at 773 K as a function of time, respectively. It can be seen that in case of holding the samples at high temperature for 168 h, the thermal conductivities of both alloys are almost kept constant and do not change with time. From these results, it

62

8

298 K

Empirical drawing

-1

6 -1

/ Wm K

Fig. 3.19 Thermal conductivities of GeTe–Sb2 Te3 alloys at room temperature as function of Ge concentration

3 Thermal Conductivities of Ge–Sb–Te Alloys

4

2

0

0

10

20

Sb 2Te3 Fig. 3.20 Thermal conductivities of Ge2 Sb2 Te5 alloys at 773 K as function of time

30

40

50

GeTe

xGe / at%

4

/ Wm K

-1 -1

773 K

Ge2Sb 2Te5

3

2

1

0

0

50

100

150

t /h 6

773 K

5

/ Wm-1K-1

Fig. 3.21 Thermal conductivities of GeTe alloy at 773 K as function of time, reprint from Ref. [39], with permission from Journal of Applied Physics

GeTe

4 3 2 1 0

0

50

100

t/h

150

3.4 Thermal Conductivities of Sb–Te and GeTe–Sb2 Te3 Alloys

63

can be concluded that the thermal conductivity will not change with time even at high temperature. However, it is still unable to determine whether phase transformation occurs or not, which should be justified together with the electrical resistivity results.

3.5 Uncertainty in Thermal Conductivity Measurements of Ge–Sb–Te Alloys The determination of the uncertainties in the measurement system and data analysis has been introduced in Chap. 2. In the same way, the uncertainties in the thermal conductivity measurements of Ge–Sb–Te alloys are also discussed according to the guide to the expression of uncertainty in measurement (GUM) [40]. As mentioned before, the thermal conductivity is given by Eq. 2.11, λ=

I 3 · α T · X T · RT 4π



dV d ln t

(2.11)

Here, αT =

1 R · RT θ

(2.10)

According to Eqs. (2.10) and (2.11), the major sources of uncertainty in thermal conductivity measurements are (i) the current applied during the measurement I, (ii) the resistance change with temperature R , (iii) the resistance X T of the heater per θ unit length at temperature T, which also leads to the fourth uncertainty source, that ) of is, (iv) the distance between the potential terminals and (v) the derivative ( dV d ln t voltage change with respect to the logarithm of time. Table 3.3 shows an example of the uncertainty budget on the thermal conductivity measurement of Ge2 Sb2 Te5 at 823 K, which is the highest temperature for Ge2 Sb2 Te5 measurement. Uncertainties are classified into Type A and Type B.

3.5.1 Uncertainty in Current (I) The current applied during the measurements was determined by the standard resistance and read by the digital galvanometer. The measurement limits of these devices cause the uncertainty in the current. The contributions have been estimated and the derived uncertainty is listed as “accuracy of devices” in Table 3.3.

Accuracy from equipments (W/K)

B

Accuracy from equipments (W)

B

B

Gaging of caliper (m)

Linear coefficient of thermal expansion (K−1 )

A

B

Least square method (V/ln(s))

Gaging of galvanometer (V)

dV /dln t (V/ln(s))

A

Repeatability (m)

Length between terminals (m)

A

Least square method (W)

X T (W)

A

B

Least square method (W/K)

dR/dT (W/K)

B

0.00078

0.0247

0.38

0.00031

2.00323

Accuracy from equipments (A)

Current (I/A)

Value

2.40147

Type

Thermal conductivity λ (W K−1 m−1 )

Factor of uncertainty

7.07107E−06

1.09923E−06

5.77E−09

7.07107E−05

0.000121106

4.80E−06

4.20669E−05

4.80E−06

3.76E−06

4.80E−06

Value of uncertainty

7.156E−06

0.000140238

4.23399E−05

6.09702E−06

4.80E−06

Standard uncertainty, u(xi)

Table 3.3 Uncertainty budget of thermal conductivity determination for Ge2 Sb2 Te5 at 823 K

−17.32786985

−12.35773955

10.22195379

12615.59741

5.817119014

Sensitivity coefficient, ∂λ/∂xi 0.076938

Combined standard uncertainty, uc(λ)

64 3 Thermal Conductivities of Ge–Sb–Te Alloys

3.5 Uncertainty in Thermal Conductivity Measurements …

3.5.2 Uncertainty in

R θ ,

XT and

65

dV d ln t

These three factors have all been calculated from the slope of the profiles recorded directly from the measurements. The standard uncertainties of these factors are calculated from the standard deviation of the slope by the method of least square. The accuracy of devices mentioned in Sect. 3.5.1 also affects the respective uncertainty.

3.5.3 Uncertainty in Distance Between Potential Terminals (l) The distance between potential terminals was measured at room temperature using a caliper. For the measurement at high temperature, thermal expansion should be considered, as listed in Table 3.3. The value of l at T K is calculated using the following equation: l = l293 [1 + αl (T − 293)]

(3.1)

where l293 represents the distance measured at 293 K and αl represents the linear coefficient of the thermal expansion of the hot strip. The uncertainty for l results from the combination of uncertainty for (i) measurement of the distance between potential terminals at 293 K [u(l293 )] and (ii) determination of the linear coefficient of thermal expansion [u(αl )]. The linear coefficient of thermal expansion for Pt-13%Rh has not been reported in the literature; therefore, the value of Pt, 8.99 × 10−6 K−1 , is used instead [39]. The accuracy of the αl is assumed to be 0.1 × 10−7 K−1 . The uncertainty of αl has been calculated as u(αl ) =

0.1 × 10−7 = 5.77 × 10−9 √ 3

(3.2)

3.5.4 Sensitivity Coefficients The sensitivity coefficients for thermal conductivity from the above factors are calculated on the basis of Eq. (2.11) using the estimated values listed in Table 3.3.

66

3 Thermal Conductivities of Ge–Sb–Te Alloys

3.5.5 Combined Standard Uncertainty The combined standard uncertainty of thermal conductivity measurements is calculated from the standard uncertainties mentioned above and the sensitivity coefficients. The expanded uncertainty is estimated to be about 0.154 W m−1 K−1 , that is, 6%, with the coverage factor k = 2, providing a level of confidence of approximately 95%. Conventionally, the thermal conductivity measurement is thought to have large error, and the error within 10% is acceptable. Therefore, the data obtained for Ge–Sb–Te alloys in this work are reliable with experimental uncertainty of about 6%.

3.6 Comparison with Reported Data Figure 3.22 [33] shows the thermal conductivities of Sb–x mol% Te alloys at room temperature as a function of the Te concentration, together with the data for thin films reported by Kuwahara et al. [12] and the data for Sb and Te recommended by Touloukian et al. [19]. The present data seem close to those obtained by Kuwahara et al. for thin films [12] except the composition of Sb–33 mol% Te. Usually, the thermal conductivities of thin films are thought to be smaller than those of bulk materials [16] since there are many interfaces existing in thin films to interfere the thermal conduction. However, the present samples are polycrystalline, and the relation between grain orientation and heat flow direction during measurements will affect the magnitude of thermal conductivity. Figure 3.23 shows the thermal conductivities of GeTe–Sb2 Te3 alloys at room temperature as a function of the Ge concentration, together with the data reported by Yanez-Limon et al. [24], which are also for the bulk materials. The present thermal conductivity data for GeSb4 Te7 and GeSb2 Te4 are quite close to those obtained by Yanez-Limon et al. [24]. The present results for Sb2 Te3 and GeTe alloys are both Fig. 3.22 Thermal conductivities of Sb–Te alloys at room temperature as function of Te concentration, together with reported data, reprint from Ref. [33], with permission from Journal of Applied Physics

3.6 Comparison with Reported Data

67

Fig. 3.23 Thermal conductivities of GeTe–Sb2 Te3 alloys at room temperature as function of Ge concentration, together with reported data

greater than those reported data. The difference in the microstructure such as the grain orientation is thought to be the cause of the small difference for Sb2 Te3 alloy. The difference in the measurement methods may also contribute to the difference. However, the thermal conductivity result of GeTe alloy in this work is much larger than that reported. The difference in the results for GeTe alloy mainly comes from the composition difference of the sample. In the work of Yanez-Limon et al. [24], the composition of GeTe alloy is detected to be 52 mol% Ge–48 mol% Te, while in the present work the composition is 55.13 mol% Ge–44.87 mol% Te. From the phase diagram of Ge–Te alloys shown in Fig. 1.6, it can be seen that the compositions of the samples in both studies are located in the two-phase range and the samples consist of GeTe and Ge phases. The present sample has more Ge by 3 mol% than the sample of Yanez-Limon et al. [24]. The excess Ge phase enhances the thermal conduction of GeTe alloy since Ge has a much higher thermal conductivity than GeTe alloy as 60.2 W m−1 K−1 at room temperature [41]. Therefore, the present result for GeTe alloy is much larger than that of Yanez-Limon et al. [24]. The result of Ge40 Sb10 Te50 alloy is much smaller than the expected value shown by the empirical line in the figure. Yanez-Limon et al. [24] suggested that the exceptional smallness was because Ge40 Sb10 Te50 alloy was in cubic rock-salt structure different from other alloys with hexagonal phase and had a smaller free-charge density.

3.7 Conclusions 1. The thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been determined as functions of temperature and composition using the hot-strip method. 2. The thermal conductivity changes of Sb2 Te3 –GeTe pseudobinary alloys have been measured at 773 K for 168 h. The results show no changes in the thermal

68

3 Thermal Conductivities of Ge–Sb–Te Alloys

conductivities occurring with time even if the alloys are held at high temperature for a long time. 3. The uncertainty of thermal conductivity measurements of Ge2 Sb2 Te5 at high temperature has been calculated, indicating an uncertainty of 6%, which is within the acceptable error range for thermal conductivity measurement. The results of this work are thought to be reliable. 4. The present results have been compared with the reported data. The results at room temperature are in good agreement with the reported data.

References 1. H.J. Borg, M. van Schijindel, J.C.N. Rijpers, Jpn. J. Appl. Phys. 40, 1592 (2001) 2. N. Oomachi, S. Ashida, N. Nakamura, Jpn. J. Appl. Phys. 41, 1695 (2002) 3. M.H.R. Lankhorst, L. van Pieterson, M. van Schijndel, B.A.J. Jacobs, J.C.N. Rijpers, J. Appl. Phys. 42, 863 (2003) 4. A.V. Kolobov, P. Fons, A.I. Frenkel, A.L. Ankudinov, J. Tominaga, T. Uruga, Nat. Mater. 3, 703 (2004) 5. D.H. Kang, D.-H. Ahn, K.-B. Kim, J.F. Webb, K.-W. Yi, J. Appl. Phys. 94, 3536 (2003) 6. M. Wuttig, N. Yamada, Nat. Mater. 6, 824 (2007) 7. N. Yamada, E. Ohno, K. Nishiuchi, N. Akahira, J. Appl. Phys. 69, 2849 (1991) 8. B. Onderka, K. Fitzner, Phys. Chem. Liq. 36, 215 (1998) 9. V.M. Glazov, A.N. Krestovnikov, N.N. Glagoleva, S.B. Evgen’ev, Neorg. Mater. 2, 1477 (1966) 10. R. Endo, S. Maeda, Y. Jinnai, R. Lan, M. Kuwahara1, Y. Kobayashi, M. Susa, Jpn. J. Appl. Phys. 49, 065802 (2010) 11. P.P. Konstantinov, L.E. Shelimova, E.S. Avilov, M.A. Kretova, V.S. Zemskov, Inorg. Mater. 37, 662 (2001) 12. M. Kuwahara, O. Suzuki, N. Taketoshi, T. Yagi, P. Fons, J. Tominaga, T. Baba, Jpn. J. Appl. Phys. 46, 6863 (2007) 13. C. Wang, J. Zhai, S. Bai, X. Yao, Mater. Lett. 64, 2314 (2010) 14. H.-K. Lyeo, D.G. Cahill, B.-S. Lee, J.R. Abelson, Appl. Phys. Lett. 89, 151904 (2006) 15. J.P. Reifenberg, M.A. Panzer, Appl. Phys. Lett. 91, 111904 (2007) 16. Y. Yang, C.-T. Li, S.M. Sadeghipour, H. Dieker, M. Wuttig, M. Asheghi, J. Appl. Phys. 100, 024102 (2006) 17. C. Peng, M. Mansuripur, Appl. Opt. 39, 2347 (2000) 18. N. Tsutsumi, T. Kiyotsukuri, Appl. Phys. Lett. 52, 442 (1988) 19. Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens, Thermal Conductivity of Metallic Elements and Alloys (Plenum Press, New York, 1970) 20. T.C. Harman, B. Paris, S.E. Miller, H.L. Georing, J. Phys. Chem. Solids 2, 181 (1957) 21. C.H. Champness, P.T. Chiang, P. Parekh, Can. J. Phys. 43, 653 (1965) 22. Sh. Movlanov, G.B. Abdullaev, A. Bashshaliev, A. Kuliev, I. Kerimov, Dokl. Akad. Nauk Azerbaidzhan 17, 375 (1961). [in Russian] 23. K. Yokota, S. Katayama, Jpn. J. Appl. Phys. 12, 1205 (1973) 24. J.M. Yanez-Limon, J. G-Hernandez, J.J. A-Gil, I. Delgadillo, H. Vargas, Phys. Rev. B 52, 16321 (1995) 25. B. Legendre, Ch. Hancheng, S. Bordas, M.T. Clavaguera-Mora, Thermochim. Acta 78, 141 (1984) 26. N.Kh. Abrikosov, G.T. Danilova-Dobryakova, Izv. Akad. Nauk SSSR, Neorg. Mater. 1, 204 (1965)

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27. S.G. Karbanov, V.P. Zlomanov, A.V. Novoselova, Izv. Akad. Nauk SSSR, Neorg. Mater. 5, 1171 (1969) 28. L.E. Shelimova, O.G. Karpinskii, V.S. Zemskov, P.P. Konstantinov, Inorg. Mater. 36, 235 (2000) 29. N.Kh. Abrikosov, Semiconducting II–IV, IV–VI, V–VI Compounds (Plenum Press, New York, 1969), p. 66 30. S. Kuypers, G. van Tendeloo, J. van Landuyt, S. Amelinckx, J. Solid State Chem. 76, 102 (1988) 31. N. Frangis, S. Kuypers, C. Manolikas, J. Landuyt, S. Amelinckx, Solid State Commun. 69, 817 (1989) 32. G. Ghosh, J. Phase Equilib. 15, 349 (1994) 33. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 110, 023701 (2011) 34. W. Kim, G.Y. Chao, Can. Mineral. 28, 675 (1990) 35. O.G. Karpinsky, L.E. Shelimova, M.A. Kretova, J.-P. Fleurial, J. Alloys Compd. 268, 112 (1998) 36. T. Matsunaga, N. Yamada, K. Kifune, Y. Kubota, J. Crystallogr. Soc. Jpn. 51, 292 (2009) 37. S. Konno, Sci. Rep. Tohoku Imp. Univ. 8, 169 (1919) 38. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Electron. Mater. 47, 3184 (2018) 39. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 112, 053712 (2012) 40. ISO/IEC Guide 98 (1995) 41. Japan Society of Thermophysical Properties, Thermophysical Properties Hanbook (Yokendo, Japan, 2008)

Chapter 4

Electrical Resistivities of Ge–Sb–Te Alloys

Abstract In this chapter, the four-terminal method has been introduced for electrical resistivity measurements. The electrical resistivities of Sb2 Te3 –GeTe pseudobinary alloys have been measured as functions of temperature and composition from room temperature up to below the respective melting temperature. The electrical resistivities of all alloys increase with increasing temperature in the temperature range investigated. The electrical resistivity results of three ternary alloys show similarity as well as the structure. The measurements have also been carried out as a function of time at 773 K to confirm whether the phase transformation occurs. The results show the electrical resistivities of ternary Sb2 Te3 –GeTe pseudobinary alloys do not change while that of GeTe alloy increases slightly with time. This finding indicates that there are no phase transformation in the ternary alloys and the phase transformation of GeTe alloy progresses slowly even at high temperature. By analyzing the uncertainty of the measurements and comparing with the reported data, the electrical resistivity results are thought reliable and able to be used to explain the thermal conduction mechanisms. Keywords Electrical resistivity · Four-terminal method · Sb2 Te3 –GeTe pseudobinary alloys · Temperature · Composition · Time · Uncertainty

4.1 Introduction The remarkable difference in the electrical resistivity between crystalline and amorphous states of phase change materials is the basis of PCRAM applications. Many studies have been done to investigate the resistivities of thin films of Ge–Sb–Te alloys in crystal line and amorphous states [1–4]. On the other hand, the electrical resistivity is highly valuable to characterize solids, because the room-temperature resistivity for different materials spans more than 32 orders of magnitude [5]. Few properties have provided such a wealth of information on solids as that of charge-carrier transport. Two different types of solid can be distinguished on the basis of temperature dependencies of their resistivity: experimentally, the temperature coefficient of the resistivity is used to separate metallic (dρ/dT > 0) from insulating behavior (dρ/dT < 0). In addition, the thermal conductivity, the main topic in this work, has also a © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_4

71

72

4 Electrical Resistivities of Ge–Sb–Te Alloys

direct relationship with the electrical resistivity by the Wiedemann–Franz (WF) law [6, 7]. It is known for metals that free electrons dominate both electrical and heat conduction and, thus, the relation between electrical and thermal conductivities can be given by the WF law. λ = Lσ T = L T /ρ

(4.1)

where λ is the thermal conductivity, σ is the electrical conductivity, ρ is the electrical resistivity, T is temperature and L is a constant called the Lorenz number, 2.45 × 10−8 W/K2 . Therefore, the electrical resistivities as functions of temperature and composition are required to understand the temperature dependence of thermal conductivities of Ge–Sb–Te alloys and their thermal conduction mechanisms. For Sb–Te binary alloys, there are many reported data of the electrical resistivities for bulk materials. The electrical resistivity of bulk Sb2 Te3 alloy, the intermetallic compound in Sb–Te system, has been investigated by several researchers [8–17]. All the data show a positive temperature coefficient in the solid state but negative in the liquid state. The data of liquid Sb2 Te3 reported by different researchers [10, 14–16] have shown to some extent a small dispersion, while for the solid state the dispersion is quite large due to the difference of microstructure and the crystal orientation of the samples. For other compositions of Sb–Te alloys, Gubskaya and Evfimovskii [11] have given the electrical resistivity data in the solid state, and Blakeway [16] and Glazov et al. [17] have given the data in the liquid state. However, for Sb2 Te3 – GeTe pseudobinary alloys, there are few data available and not enough to ensure the accuracy. The only electrical resistivity data as functions of composition and temperature have been given by Konstantinov et al. [18]. Therefore, it is necessary to measure the electrical resistivity of Sb2 Te3 –GeTe pseudobinary alloys as well as to use the reported electrical resistivities of Sb–Te alloys, to discuss the thermal conduction mechanism of Ge–Sb–Te alloys. On the other hand, the electrical resistivities of bulk materials are strongly affected by the crystallographic structures. As introduced in Chap. 3, the GeTe–Sb2 Te3 alloys might have phase transformations in the solid state at high temperature for Ge2 Sb2 Te5 , GeSb2 Te4 and GeSb4 Te7 alloys [19], and for GeTe alloy, it has been confirmed that there is a phase change from high-temperature form (β) to lowtemperature form (α and γ ) at 678 K (405 °C) [20, 21]. With increasing temperature, the phase transformation will lead to an abrupt change in the electrical resistivity and thus, the electrical resistivity is able to give the evidence whether the phase transformation occurs or not. Meanwhile, several literatures suggested that such phase transformation process would not complete unless the alloys were annealed for 300– 500 h [22, 23]. In the conventional applications, Ge–Sb–Te alloys are usually not annealed for such a long time, and thus, there is a high possibility that the alloys do not complete the phase transformation and are the mixtures of high-temperature and low-temperature phases at room temperature. In that case, the abrupt change in the electrical resistivity will not be found; however, the electrical resistivity of such an alloy may change during the service for PCRAM applications. Therefore, the

4.1 Introduction

73

electrical resistivity change with time at one certain high temperature for Ge–Sb–Te alloys should be investigated to clarify the influence of such a phenomenon. Against the above background, consequently, in this chapter, the electrical resistivities of Sb2 Te3 –GeTe pseudobinary alloys have been determined as functions of temperature and composition, and the electrical resistivity changes of Sb2 Te3 – GeTe pseudobinary alloys have been investigated at one certain high temperature withholding for a long time.

4.2 Experimental 4.2.1 Sample Samples used were nGeTe·mSb2 Te3 (n:m = 1:0, 2:1, 1:1, 1:2), which are the same compositions as those of the samples for the thermal conductivity measurements. Figure 4.1 shows a phase diagram of GeTe–Sb2 Te3 alloys [24], where the circle represents the chemical compositions of the samples and the arrow gives the respective temperature range of each sample for electrical resistivity measurements. Cylindrical samples of GeTe–Sb2 Te3 (8 mm diameter and 70–80 mm length) were prepared from Sb2 Te3 (99.9 mass%) and GeTe (99.9 mass%) powders. All the powder samples were produced by Kojundo Chemical Laboratory. Mixture (25 g) of these powders

Fig. 4.1 Phase diagram of GeTe–Sb2 Te3 alloys, reprint from Ref. [24], with permission from Thermochimica Acta

74

4 Electrical Resistivities of Ge–Sb–Te Alloys

precisely weighed to a desired composition was melted in a quartz crucible (8 mm inner diameter) at 973 K for 4 h in vacuum, followed by cooling at a rate of 50 K/h. The surface of the molten samples was set below the maximal temperature of the furnace so that solidification proceeded from the bottom to the top of the sample and the gas absorbed in the samples could escape without leaving pores in the samples. Depending on the compositions of alloys, the melting temperature was changed slightly. The schematic diagram of furnace setup and the procedure of the melting process have been shown in Fig. 3.3. Subsequently, the sample was taken out from the crucible, and the surface was mechanically polished using emery papers up to #2000 for removing the potentially existing oxidation layer. Figure 4.2 shows GeTe– Sb2 Te3 samples with different compositions after polishing. The crystallographic structures of the samples were analyzed by an X-ray diffractometer (XRD), and the element distributions and chemical compositions were analyzed by a scanning electron microscope with an energy dispersive X-ray spectrometer (SEM-EDS).

8cm

(a) GeTe

8cm

(c) GeSb2Te4

8cm

(b) Ge2Sb2Te5

8cm

(d) GeSb4Te7

Fig. 4.2 GeTe–Sb2 Te3 samples with different compositions after polishing: a GeTe; b Ge2 Sb2 Te5 ; c GeSb2 Te4 ; d GeSb4 Te7

4.2 Experimental

75

Fig. 4.3 Conceptual diagram for four-terminal method, reprint from Ref. [25], with permission from Journal of Applied Physics

4.2.2 Four-Terminal Method The four-terminal method was used to measure the electrical resistivity of Sb2 Te3 – GeTe pseudobinary alloys. Figure 4.3 shows a conceptual diagram for the fourterminal method [25]. The current (I) was supplied between the outer electrodes, and the potential difference ( V ) was measured between the inner electrodes. The electrical resistance (R) of the sample between the inner electrodes can be derived on the basis of Ohm’s law as follows: R=

V I

(4.2)

In the experiment, the resistance was determined from the slope of a linear portion in the plot between I and ΔV. The electrical resistivity (ρ) of the sample can be derived using the resistance from the equation: ρ=

R·A l

(4.3)

where A and l represent the cross-sectional area and the distance between the inner electrodes, respectively. The value of A can be calculated from the following equation: A=

π d2 4

(4.4)

where d is the diameter of the sample.

4.2.3 Measurement Setup and Procedure W wires of 0.3 mm diameter were used as the electrodes and were round and fixed to the sample with alumina cement. However, alumina cement reacted with Ge–Sb–Te alloys and the contact condition between the sample and electrode got worse due to the oxidization. The oxidized surface of the sample by alumina cement is shown in Fig. 4.4a. Therefore, the way of fixing the electrodes was changed from smearing

76 Fig. 4.4 a Oxidized surface of sample by alumina cement, and b fixing way of electrodes

4 Electrical Resistivities of Ge–Sb–Te Alloys

(a)

8cm

(b)

8cm

alumina cement to the whole contact surface to smearing only at one point, shown as Fig. 4.4b. However, the long W wire was too tough to be fixed by the point-fixing. Pt wire is flexible to be fixed by the point-fixing but reacts with Ge–Sb–Te alloys at high temperature [26]. Therefore, the electrodes were arranged by connecting these two kinds of wires, as shown in Fig. 4.5a. Short W wires were round around the sample and fixed by alumina cement at one point, and long Pt wires (0.5 mm diameter) were connected with the W wire at one end and served as lead wires to connect with the power supply or multimeter at the other end. The picture of such arrangement is shown in Fig. 4.5b. The distance between the inner electrodes was about 40 mm, and the distance between the inner and outer electrodes was about 10 mm. The sample was sealed in a silica tube in vacuum atmosphere as shown in Fig. 4.6 and placed in the furnace. Electrical resistivity measurements were conducted in vacuum atmosphere from 298 K up to temperatures just below the melting points in both cooling and heating cycles. The waiting time at each temperature for the temperature to become stable is about 2 h. The supplied electric currents were − 0.3 to 0.3 A. The voltage between the potential wires was monitored continuously using a digital voltmeter. The temperature of the sample was measured with a K-type thermocouple, and the reading was monitored continuously using a chart recorder to know the establishment of thermal equilibrium of the sample. The measurements at 773 K were carried out for 168 h.

4.2 Experimental

77

Fig. 4.5 a Schematic of electrode arrangement, and b picture of electrode arrangement

Fig. 4.6 Schematic of measurement setup

Vacuum+Ar

78

4 Electrical Resistivities of Ge–Sb–Te Alloys

4.3 Characterization of GeTe–Sb2 Te3 Alloys Table 4.1 shows the chemical compositions of mSb2 Te3 ·nGeTe (m:n = 0:1, 1:2, 1:1, 2:1) alloys analyzed by SEM-EDS. The compositions shown in Table 4.1 are the averages of compositions at different positions. It can be seen that the analyzed compositions of samples are slightly different from the corresponding nominal compositions. The ratios of Ge to Sb are in fairly good agreement with nominal compositions but the concentrations of Te are all smaller than the nominal ones in the four kinds of GeTe–Sb2 Te3 alloys since Te element might have evaporated during the melting process due to its high vapor pressure. The present compositions of the samples for electrical resistivity measurements are quite close to those for thermal conductivity measurements. The samples will be called according to the nominal compositions in the below discussion for convenience. Figure 4.7 shows the XRD profiles for the GeTe–Sb2 Te3 samples at room temperature. All the four kinds of GeTe–Sb2 Te3 alloys in the present work are identified Table 4.1 Chemical compositions of GeTe–Sb2 Te3 samples analyzed by SEM-EDS

Nominal

Analyzed

mSb2Te3·nGeTe (m:n)

mol% Ge

mol% Sb

mol% Te

GeSb4 Te7 (2:1)

10.06 (8.33)

35.23 (33.3)

54.71 (58.3)

GeSb2 Te4 (1:1)

16.16 (14.3)

30.55 (28.6)

53.19 (57.1)

Ge2 Sb2 Te5 (1:2)

25.66 (22.2)

23.82 (22.2)

50.52 (55.6)

GeTe (0:1)

55.94 (50)

44.06 (50)

Ge 0.95 Sb 2.01 Te 4

Fig. 4.7 XRD profiles for GeTe–Sb2 Te3 alloys

Intensity (arb unit)

2000

GeTe GeSb 4Te 7

GeSb 2Te 4

1000

Ge 2Sb 2Te 5

GeTe

0 20

30

40

2θ / deg

50

60

4.3 Characterization of GeTe–Sb2 Te3 Alloys

79

as single-phase alloys from the XRD profiles. However, the peaks of XRD profiles for GeSb4 Te7 , GeSb2 Te4 and Ge2 Sb2 Te5 are similar to each other and assigned to a rhombohedral structure which is the structure of Ge0.95 Sb2.01 Te4 alloys [27] although showing small shifts in the diffraction peaks. Matsunaga et al. [28] have suggested that GeTe–Sb2 Te3 pseudobinary alloys are crystallized into metastable single phases with NaCl-type structure independent of the compositions without sufficient heat treatments. The present results are in consistency with their suggestions except that the structure is different. GeTe has a rhombohedral crystallographic structure according to the XRD profile, which is consistent with the phase diagram. Figure 4.8 shows the EDS mapping results for the GeTe–Sb2 Te3 samples at room temperature. The mapping results have also confirmed that all the GeTe–Sb2 Te3 alloys are single phases.

Ge

Sb

Te

Ge

Sb

Te

Ge

Sb

Te

Ge

Te

GeSb4Te7 30 μm

GeSb2Te4

Ge2Sb2Te5

GeTe

Fig. 4.8 EDS mapping results for GeTe–Sb2 Te3 alloys

80

4 Electrical Resistivities of Ge–Sb–Te Alloys

4.4 Electrical Resistivities of GeTe–Sb2 Te3 Alloys Figures 4.9 [29] and 4.10 [25] show typical voltages as a function of current obtained during measurements for Ge2 Sb2 Te5 alloy at (a) 308 K and (b) 823 K and for GeTe alloy at (a) 298 K and (b) 773 K by the four-terminal method, respectively. There is good linearity between the current and the voltage for all the measurements, which supports the reliability of the present measurements. The slopes of these straight lines represent the resistances of the samples. The electrical resistivities have been calculated using the derived resistances on the basis of Eq. (4.2) for Ge2 Sb2 Te5 alloy to be 2.31 × 10−6 m at 308 K and 1.03 × 10−5 m at 823 K, and for GeTe alloy to be 1.42 × 10−6 m at 298 K and 5.68 × 10−6 m at 773 K. Figure 4.11 shows the temperature dependence of electrical resistivities for the three ternary GeTe–Sb2 Te3 alloys obtained during the heating and cooling cycles. It can be seen that the electrical resistivities of these alloys are almost the same as each other within the measurement error. The electrical resistivity seems to be independent of the composition. They all increase with increasing temperature. This

(a)

[×10-4 ] 4

308 K

ΔV / V

2 0 -2 -4 -0.2

0

0.2

I /A

(b) [×10-3 ] 3 2

823 K

1

ΔV / V

Fig. 4.9 Typical voltages as function of current obtained during measurements for Ge2 Sb2 Te5 alloy at a 308 K and b 823 K, reprint from Ref. [29], with permission from Journal of Electronic Materials

0 -1 -2 -3

-0.2

0

I /A

0.2

4.4 Electrical Resistivities of GeTe–Sb2 Te3 Alloys

(a) -4 [×10 ] 4

298 K

2

ΔV / V

Fig. 4.10 Typical voltages as function of current obtained during measurements for GeTe alloy at a 298 K and b 773 K, reprint from Ref. [25], with permission from Journal of Applied Physics

81

0 -2 -4 -0.2

[×10-3]

1

0

0.2

0

0.2

I /A

(b) 773 K

ΔV / V

0

-1

-0.2

I /A -5 [×10 ] 1.4

1.2 1

ρ/Ω m

Fig. 4.11 Electrical resistivities of GeTe–Sb2 Te3 alloys as function of temperature

0.8

Ge 2Sb 2Te 5 heating Ge 2Sb 2Te 5 cooling GeSb 2Te 4 heating GeSb 2Te 4 cooling GeSb 4Te 7 heating GeSb 4Te 7 cooling

0.6 0.4 0.2 0

400

600

T/K

800

82 -5

[× 10 ] 1

First heating First cooling Second heating Second cooling

0.8

ρ/Ω m

Fig. 4.12 Electrical resistivities of GeTe alloy as function of temperature, reprint from Ref. [25], with permission from Journal of Applied Physics

4 Electrical Resistivities of Ge–Sb–Te Alloys

GeTe

0.6 0.4 m.p.

0.2 0

400

600

800

1000

T /K

result may come from the similar crystalline structures from the XRD profiles and the close compositions from SEM-EDS. Figure 4.12 [25] shows the temperature dependence of electrical resistivities for GeTe alloy obtained during the heating and cooling cycles of two measurements. The electrical resistivity of GeTe alloy increases linearly with increasing temperature; however, at the temperatures near the melting temperature, the increasing rate decreases appreciably, shown by the dash line in the diagram. This is undoubtedly due to the reflection of intrinsic conductivity, but the intrinsic conductivity branch could not be revealed, probably due to the insufficient purity of the GeTe sample. The data obtained during the heating and cooling cycles of the two measurements are in good agreement with each other. Figure 4.13 shows the electrical resistivities of GeTe–Sb2 Te3 pseudobinary alloys as a function of the Ge concentration at room temperature. It can be seen that the Sb2 Te3 alloy has the largest electrical resistivity and that GeTe alloy has the lowest. -6 [×10 ] 6

5

This work

298 K

Onderka

4

ρ/Ω m

Fig. 4.13 Electrical resistivities of GeTe–Sb2 Te3 alloys at room temperature as function of Ge concentration

3 2 1 0

0 Sb 2Te3

10

20

30

xGe / at%

40

50

GeTe

4.4 Electrical Resistivities of GeTe–Sb2 Te3 Alloys

83

The electrical resistivities of three ternary GeTe–Sb2 Te3 alloys are considered to be close to each other according to the above discussion about similar structures. Figures 4.14 and 4.15 [25] show the electrical resistivities for Ge2 Sb2 Te5 and GeTe alloys at 773 K as a function of time, respectively. It can be seen that there are different time dependences for these two alloys. The electrical resistivity of Ge2 Sb2 Te5 alloy decreases by about 3% for the first 20 h, which is within the measurement uncertainty, and then keeps almost constant. Therefore, the electrical resistivity of Ge2 Sb2 Te5 alloy cannot be thought to change with time at high temperature. On the other hand, the electrical resistivity of GeTe alloy keeps increasing with time and finally increases by about 9% after 168 h. The increase for GeTe alloy will be discussed in Sect. 4.6. Fig. 4.14 Electrical resistivities of Ge2 Sb2 Te5 alloys at 773 K as function of time

Fig. 4.15 Electrical resistivities of GeTe alloys at 773 K as function of time, reprint from Ref. [25], with permission from Journal of Applied Physics

84

4 Electrical Resistivities of Ge–Sb–Te Alloys

4.5 Uncertainty in Electrical Resistivity Measurements of GeTe–Sb2 Te3 Pseudobinary Alloys The uncertainties in the electrical resistivity measurement system and data analysis of GeTe–Sb2 Te3 alloys are discussed according to the guide to the expression of uncertainty in measurement (GUM) [30]. As mentioned before, the electrical resistivity is given by Eqs. (4.2) and (4.3). The major sources of uncertainty in thermal conductivity measurements are (i) the resistance of the sample, (ii) the diameter of the sample, and (iii) the distance between the electrodes. Table 4.2 shows an example of the uncertainty budget on the electrical resistivity measurement of Ge2 Sb2 Te5 alloy at 823 K, which is the highest temperature for Ge2 Sb2 Te5 measurement. Uncertainties are classified into Type A and Type B.

4.5.1 Uncertainty in Resistance (R) The resistance of the sample is calculated from the slope of the plot of voltage against current as shown in Fig. 4.9b. The standard uncertainty is calculated as the standard deviation of the slope by the method of least square. In addition, the measurement limits of the devices, that is, the digital multimeters and the standard resistance, also cause uncertainty in the resistance of the sample. The contributions have been estimated and the derived uncertainty is listed as “accuracy of devices” in Table 4.2.

4.5.2 Uncertainty in Diameter of Sample (d) The cross-sectional area of the sample (A) is determined by Eq. (4.4) using the diameter of the sample. The diameter was measured at room temperature using a caliper. For the measurement at high temperature, thermal expansions are considered, as listed in Table 4.2; for example, the value of d at T K is calculated using the following equation: d = d293 [1 + αd (T − 293)]

(4.5)

where d 293 and α d represent the diameter of the sample at 293 K, and the linear coefficient of thermal expansion of Ge2 Sb2 Te5 alloy, respectively. The uncertainty for d results from the combination of uncertainties for (i) measurement of the diameter of the sample at 293 K [u(d 293 )], (ii) determination of the linear coefficient of thermal expansion [u(α d )], and (iii) measurement of temperature [u(T )]. Therefore, the standard uncertainty of the value for d is written as

Accuracy from equipments (/K)

B

B

B

Gaging of caliper (m)

Linear coefficient of thermal expansion (K−1 )

Temperature (K)

A

B

B

B

Repeatability (m)

Gaging of caliper (m)

Linear coefficient of thermal expansion (K−1 )

Temperature (K)

Length between inner electrodes (m)

A

Repeatability (m)

Diameter of sample (m)

A

A

Least square method ()

1.00E−05

1.50E−05

3.61E−02

1.00E−05

1.50E−05

7.93E−03

7.77E−03

R ()

Value 1.06E−05

Type

Electrical resistivity ρ (m)

Factor of uncertainty

5.77E−01

5.77E−09

5.77E−05

1.21E−04

1.34E−04

5.77E−01

5.77E−09

5.77E−05

1.21E−04

1.34E−04

4.80E−06

1.47E−05

Value of uncertainty

1.34E−04

1.34E−04

1.55E−05

Standard uncertainty, u(xi)

Table 4.2 Uncertainty budget of electrical resistivity determination for Ge2 Sb2 Te5 at 823 K

−2.94E−04

2.68E−03

1.37E−03

Sensitivity coefficient, ∂ρ/∂xi

3.62E−07

Combined standard uncertainty, uc(ρ)

4.5 Uncertainty in Electrical Resistivity Measurements … 85

86

4 Electrical Resistivities of Ge–Sb–Te Alloys

u 2 (d) = [1 + αd (T − 293)]2 u 2 (d293 ) + [d293 (T − 293)]2 u 2 (αd ) + (d293 αd )2 u 2 (T ) (4.6) The value of u(d 293 ) in Eq. (4.6) is determined by the following equation using the uncertainty of the used caliper [ug (d 293 )] listed as Type B and the standard deviation of ten measurements [ur (d 293 )] listed as Type A. u 2 (d293 ) = u 2g (d293 ) + u r2 (d293 )

(4.7)

The linear coefficient of thermal expansion for Ge2 Sb2 Te5 alloy has been given a calculated equation between 290 and 670 K [31]. However, the coefficient calculated from this equation at 670 K is 1.24 × 10−4 , much larger than those for Ge, Sb and Te, which have been given to be 7.7 × 10−6 for Ge, 17.2 × 10−6 (//), 8 × 10−6 ( ) for Sb, and 16 × 10−6 (//), 27.2 × 10−6 ( ) K−1 for Te [32]. Furthermore, the linear coefficient of thermal expansion for chalcogenide Ge30 As10 Se30 Te(30−x) alloy has been reported to be 13.8–14.6 × 10−6 K−1 [33]. Therefore, the result calculated from the reported equation is thought incorrect and the value of 15 × 10−6 K−1 is used for the linear coefficient of thermal expansion for Ge2 Sb2 Te5 alloy instead. The accuracy of the α d is assumed to be 0.1 × 10−7 K−1 . The uncertainty of α d has been calculated as u(αd ) =

0.1 × 10−7 = 5.77 × 10−9 √ 3

(4.8)

4.5.3 Uncertainty in Distance Between Inner Electrodes (l) The distance between the inner electrodes was measured at room temperature by a caliper, and the thermal expansions were considered for measurements at high temperatures. The standard uncertainty has been derived in the same manner as for u(d), where the standard deviation of measurement has been derived from five measurements at room temperature, the accuracy of the caliper (0.05 mm) and the accuracy of the linear coefficient of thermal expansion for W (assumed to be 0.1 × 10−7 K−1 ).

4.5.4 Sensitivity Coefficients The sensitivity coefficients for thermal conductivity from the above factors are calculated on the basis of Eq. (4.3) using the estimated values listed in Table 4.2.

4.5 Uncertainty in Electrical Resistivity Measurements …

87

4.5.5 Combined Standard Uncertainty The combined standard uncertainty of electrical resistivity measurements is calculated from the standard uncertainties mentioned above and the sensitivity coefficients. The expanded uncertainty is estimated to be about 3.62 × 10−7 m, that is, 3.4%, with the coverage factor k = 2, providing a level of confidence of approximately 95%.

4.6 Comparison with Reported Data Figure 4.16 shows the electrical resistivities of three ternary GeTe–Sb2 Te3 alloys as a function of temperature, together with the data reported by Konstantinov et al. [18]. The different colors of the lines represent the alloys with different compositions corresponding to the colors of the symbols. The present data are smaller than the reported data and show no clear composition dependence. For the data obtained by Konstantinov et al. [18], the electrical resistivity decreases with increasing Ge concentration. At the temperature near the melting points, the electrical resistivities of all alloys in the reported work begin to decrease with increasing temperature, which behavior is not found in the present work. The difference between the present data and the reported data is thought to come from the concentration differences in the samples, which will be discussed in Chap. 5 in detail. There is no reason to explain the decrease of the electrical resistivity near the melting temperature in the reported data, and the data are old and thus, the accuracy of the measurements by Konstantinov et al. [18] should also be taken into account. Fig. 4.16 Electrical resistivities of three ternary GeTe-Sb2 Te3 alloys as function of temperature, together with reported data [18]

88

4 Electrical Resistivities of Ge–Sb–Te Alloys

Fig. 4.17 Electrical resistivities of GeTe alloys as function of temperature, together with data withholding for long time at 773 K and reported by Konstantinov et al. [18], reprint from Ref. [25], with permission from Journal of Applied Physics

Figure 4.17 [25] shows the electrical resistivity of GeTe alloy as a function of temperature, together with the data from long time measurements at 773 K and the data reported by Konstantinov et al. [18]. The present data are in good agreement with those obtained by Konstantinov et al. [18] near room temperature; therefore, it is thought that the concentration of the present GeTe sample is close to that of the reported work, which is due to the concentration of GeTe alloy much easier to control than other ternary GeTe–Sb2 Te3 alloys. With increasing temperature, the reported data increase rapidly and become larger than the present results. The reported data has an abrupt decrease near 700 K, which might correspond to the phase change between high-temperature form (β) and low-temperature form (α and γ ) [20, 21]. However, the electrical resistivities of three other ternary GeTe–Sb2 Te3 alloys also begin to decrease near 700 K. From this view point, the decrease might also come from the measurement method. GeTe alloy is conventionally considered to have metallic temperature dependence of electrical resistivity in the solid state since the nonstoichiometric Ge vacancies contribute to a high carrier density (~1026 to 1027 m−3 ) [34–36]. There is no reason to explain why the reported electrical resistivity of GeTe alloy continues to decrease with increasing temperature after the abrupt decrease. The present data increase monotonically with increasing temperature without an unexpected change. Withholding for a long time, the blue symbols in the figure show that the electrical resistivity of GeTe alloy gets close to the reported data at 773 K. These results indicate that the phase transformation of GeTe alloy progresses slowly even at high temperature and in the present GeTe sample, there might exist a small amount of high-temperature phase (β) at room temperature, which cannot be detected by XRD diffraction.

4.7 Conclusions 1. The electrical resistivities of Sb2 Te3 –GeTe pseudobinary alloys have been determined as functions of temperature and composition using the four-terminal method.

4.7 Conclusions

89

2. The electrical resistivity changes of Sb2 Te3 –GeTe pseudobinary alloys at 773 K withholding for a long time have been investigated. The results show that there are no changes in the electrical resistivity of ternary Sb2 Te3 –GeTe alloys occurring with time even if the alloys are held at high temperature for a long time. The electrical resistivity of GeTe alloy increases with time, which indicates that the phase transformation occurs in GeTe alloy at high temperature. 3. The uncertainty of electrical resistivity measurements of Ge2 Sb2 Te5 alloy at high temperature has been calculated, indicating an uncertainty of 3%, which is within the acceptable error range for electrical resistivity measurements. The data of this work are thought to be reliable and suitable for discussion about the thermal conductivities of Ge–Sb–Te alloys. 4. The present results have been compared with the reported data. The present results for ternary Sb2 Te3 –GeTe alloys are smaller than the reported data. The results for GeTe alloy are in good agreement with the reported data at room temperature but deviate at high temperature. The accuracy of the reported data should be checked.

References 1. E. Morales-Sanchez, E.F. Prokhorov, J. Gonzalez-Hernandez, A. Mendoza-Galvan, Thin Solid Films 471, 243 (2005) 2. S. Yoon, K. Choi, N. Lee, S. Lee, Y. Park, B. Yu, Jpn. J. Appl. Phys. 46, 7225 (2007) 3. S. Raoux, C.T. Rettner, J. Appl. Phys. 102, 094305 (2007) 4. M.H.R. Lankhorst, B.W.S.M.M. Ketelaars, R.A.M. Wolters, Nat. Mater. 4, 347 (2005) 5. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986) 6. G. Wiedemann, R. Franz, Annalen der Physik und Chemie 89, 497 (1853) 7. L. Lorenz, Annalen der Physik und Chemie 13, 422 (1881) 8. B. Onderka, K. Fitzner, Phys. Chem. Liq. 36, 215 (1998) 9. V.M. Glazov, A.N. Krestovnikov, N.N. Glagoleva, S.B. Evgen’ev, Neorg. Mater. 2, 1477 (1966) 10. R. Endo, S. Maeda, Y. Jinnai, R. Lan, M. Kuwahara, Y. Kobayashi, M. Susa, Jpn. J. Appl. Phys. 49, 065802 (2010) 11. G.F. Gubskaya, I.V. Evfimovskii, Russ. J. Inorg. Chem. 7, 834 (1962) 12. W. Eichler, G. Simon, Phys. Status Solidi (b) 86, K85 (1978) 13. P.N. Sherov, Neorg. Mater. 23, 1291 (1987) 14. M.P. Vukalovich, V.I. Fedorov, A.S. Okhotin, V.M. Glazov, Izvestiya Akademii Nauk SSSr, Neorg. Mater. 2, 844 (1966) 15. J.E. Enderby, L. Walsh, Philos. Mag. 991–1002 (1966) 16. R. Blakeway, Philos. Mag. 20, 965 (1969) 17. V.M. Glazov, A.N. Krestovnikov, N.N. Glagoleva, Inorg. Mater. 2, 392 (1966) 18. P.P. Konstantinov, L.E. Shelimova, E.S. Avilov, M.A. Kretova, V.S. Zemskov, Inorg. Mater. 37, 662 (2001) 19. S.G. Karbanov, V.P. Zlomanov, A.V. Novoselova, Izv. Akad. Nauk SSSR, Neorg. Mater. 5, 1171 (1969) 20. B. Legendre, Ch. Hancheng, S. Bordas, M.T. Clavaguera-Mora, Thermochim. Acta 78, 141 (1984) 21. N.Kh. Abrikosov, G.T. Danilova-Dobryakova, Izv. Akad. Nauk SSSR, Neorg. Mater. 1, 204 (1965)

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22. S. Kuypers, G. van Tendeloo, J. van Landuyt, S. Amelinckx, J. Solid State Chem. 76, 102 (1988) 23. N. Frangis, S. Kuypers, C. Manolikas, J. Landuyt, S. Amelinckx, Solid State Commun. 69, 817 (1989) 24. S. Bordas, M.T. Clavaguera-Mora, B. Legendre, Ch. Hancheng, Thermochim. Acta 107, 239 (1986) 25. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 112, 053712 (2012) 26. W.-S. Kim, J. Alloy. Compd. 252, 166 (1997) 27. O.G. Karpinsky, L.E. Shelimova, M.A. Kretova, J.-P. Fleurial, J. Alloys Compd. 268, 112 (1998) 28. T. Matsunaga, N. Yamada, K. Kifune, Y. Kubota, J. Crystallogr. Soc. Jpn. 51, 292 (2009) 29. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Electron. Mater. 47, 3184 (2018) 30. ISO/IEC Guide 98 (1995) 31. A.S. Skoropanov, B.L. Valevsky, V.F. Skums, G.I. Samal, A.A. Vecher, Thermochim. Acta 90, 331 (1985) 32. Japan Society of Thermophysical Properties, Thermophysical Properties Hanbook (Yokendo, Japan, 2008) 33. V.Q. Nguyen, J.S. Sanghera, I.D. Aggarwal, I.K. Lloyd, J. Am. Ceram. Soc. 83, 855 (2000) 34. N.Kh. Abrikosov, M.A. Korzhuev, L.E. Shelimova, Izvestiya Akademii Nauk SSSr, Neorg. Mater. 13, 1757 (1977) 35. B.F. Gruzinov, P.P. Konstantinov, B.Ya. Moizhes, Yu.I. Ravich, L.M. Sysoeva, Fiz. Tekh. Poluprovodn. 10, 497 (1976) [Sov. Phys. Semicond. 10, 296 (1976)] 36. M.A. Korzhuev, L.E. Shelimova, N.Kh. Abrikosov, Fiz. Tekh. Poluprovodn. 11, 296 (1977) [Sov. Phys. Semicond. 11, 171 (1977)]

Chapter 5

Thermal Conduction Mechanisms and Prediction Equations of Thermal Conductivity for Ge–Sb–Te Alloys

Abstract In this chapter, based on the thermal conductivity and electrical resistivity data obtained in Chaps. 3 and 4, the thermal conduction mechanisms of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been discussed. Both the thermal conductivity and electrical resistivity of Sb2 Te3 –GeTe pseudobinary alloys show similarity, which is directly related to the structure similarity. According to the structure analysis in this work and research in the literature, the Sb2 Te3 –GeTe pseudobinary alloys can be considered as solid solution. Therefore, the thermal conduction mechanisms of Sb2 Te3 –GeTe pseudobinary alloys should be the same. The same case is for Sb-rich Sb–Te alloys. The Wiedemann–Franz (WF) law has been used to predict the electrical thermal conductivity part using the electrical resistivity data. The results show that the free electrons contribute to the most part of the thermal conductivity and the phonon contribution can be neglected. Bipolar diffusion plays an important role at high temperature and accounts for the increase of the thermal conductivity. The exception can be seen in GeTe alloy, which shows metallic properties in the temperature range investigated. The prediction equations have been proposed for the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys. The comparison between the predicted data and experimental data proves that the prediction equations can be used for the industrial applications. Keywords Thermal conductivity · Electrical resistivity · Thermal conduction mechanism · Sb2 Te3 –GeTe pseudobinary alloys · Structure similarity · Wiedemann–Franz (WF) law · Phonon · Bipolar · Prediction equation

5.1 Introduction As introduced in Chap. 1, heat in solids is conducted by various carriers: electrons, lattice vibrations (phonons), photons and in some cases magnons. For metals, free electrons dominate both electrical and heat conduction and, thus, the relation between electrical and thermal conductivities can be given by the Wiedemann–Franz (WF) law [1, 2]. λ = Lσ T = L T /ρ © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_5

(5.1) 91

92

5 Thermal Conduction Mechanisms and Prediction Equations …

where λ is the thermal conductivity, σ is the electric conductivity, ρ is the electric resistivity, T is temperature and L is a constant called the Lorenz number, 2.45 × 10−8 W/K2 . For insulators and semiconductors, the phonon thermal conductivity makes the main contribution to the thermal conductivity. Electronic thermal conduction and phonon thermal conduction constitute the most important mechanisms in nearly all substances at nearly all temperatures. Except these two kinds of mechanisms, there are still other possibilities, for example, heat conduction by bipolar diffusion, photons or magnons; in particular, for the narrow-band semiconductors, bipolar diffusion plays an important role in the thermal conduction [3–5]. In Chaps. 3 and 4, the thermal conductivity and electrical resistivity have been measured for Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys. Using these data, in this chapter, the thermal conduction mechanisms are discussed based on the structure characteristics of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloy systems and the relationship between the thermal conductivities and electrical resistivities. The phase transformation process of Ge–Sb–Te alloys in PCRAM is controlled by Joule heating and cooling process. Therefore, the thermal conductivities of Ge– Sb–Te alloys are indispensible for the design and optimization of PCRAM device structures. However, in practice, there are many kinds of Ge–Sb–Te composition used for PCRAM. It is impossible to measure the thermal conductivities of all these compositions. In that case, the accurate prediction equations of thermal conductivity directly from temperature and chemical composition are quite beneficial for industrial applications. Therefore, the prediction equations of thermal conductivity for Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys are also established in this chapter.

5.2 Thermal Conduction Mechanisms 5.2.1 Structural Similarity Figure 5.1 has shown all the thermal conductivity data for Sb2 Te3 –GeTe pseudobinary alloys as a function of temperature obtained in this work. It can be obviously seen that there are two distinguishing kinds of temperature dependence. One is for the thermal conductivities of GeTe alloy, which decrease almost linearly with increasing temperature. The other is for the thermal conductivities of all other alloys including three ternary alloys and Sb2 Te3 alloy: the thermal conductivities decrease with increasing temperature up to about 600 K and then increase. The magnitudes of the thermal conductivities are all quite close to each other, but the temperature dependence for Sb2 Te3 alloy is slightly more obvious than those of ternary alloys. Figure 5.2 shows the electrical resistivity data for Sb2 Te3 –GeTe pseudobinary alloys as a function of temperature obtained in this work, together with the data for Sb2 Te3 alloy obtained by Onderka et al. [6]. The electrical resistivities seem to fall into three groups, although the difference is not so obvious. The electrical resistivity of GeTe alloy increases almost linearly with increasing temperature, while

5.2 Thermal Conduction Mechanisms

6

λ / Wm-1K-1

Fig. 5.1 Thermal conductivity data for Sb2 Te3 –GeTe pseudobinary alloys as function of temperature obtained in this work

93

4

2

0

400

600

800

1000

T/K GeTe-heat GeTe-cool Ge2Sb 2Te5-heat Ge2Sb 2Te5-cool GeSb 2Te4-heat

] ×10-5] 1

ρ/Ω m

Fig. 5.2 Electrical resistivity data for Sb2 Te3 –GeTe pseudobinary alloys as function of temperature obtained in this work, together with data for Sb2 Te3 alloy obtained by Onderka et al.

GeSb 2Te4-cool GeSb 4Te7-heat GeSb 4Te7-cool Sb 2Te3

GeTe Ge2Sb2Te5 GeSb2Te4 GeSb4Te7 Sb2Te3 by Onderka et al

0.5

0

400

600

800

T/ K

the electrical resistivities of other alloys increase more quickly than GeTe alloy and show the similarity in the tendency. The magnitude of the electrical resistivities of Sb2 Te3 alloy is larger than those of three ternary alloys which have the similar magnitude. From Figs. 5.1 and 5.2, it can be concluded that in the Sb2 Te3 –GeTe pseudobinary alloy system, the thermal and electrical properties of the three ternary alloys and Sb2 Te3 alloy are similar to each other, but different from that of GeTe alloy. This conclusion has been supported by the structural research of Matsunaga et al. [7]. Matsunaga et al. [7] have investigated the crystallographic structure of Sb2 Te3 – GeTe pseudobinary alloy systems and suggested that Sb2 Te3 –GeTe pseudobinary alloys exist as the semi-stable phases without annealing at high temperature for a

94

5 Thermal Conduction Mechanisms and Prediction Equations …

long time, and these phases are used in the practical optical discs. These semi-stable phases show the same NaCl structure independent of the compositions. Only lowtemperature GeTe phase shows a rhombohedral structure. The same authors [8] have also suggested that the crystal structures of Sb–Te binary system change continuously with Te concentration in the range from 15 to 60 mol%. The XRD results in Chaps. 3 and 4 have also confirmed the structural similarity for the single-phase alloys in the Sb–Te binary system and for the ternary alloys in the Sb2 Te3 –GeTe pseudobinary system. According to the discussion above, it can be considered that the single-phase Sb– Te alloys and ternary Sb2 Te3 –GeTe pseudobinary alloys act as solid solutions. The similarity in the crystallographic structures determines the thermophysical properties; therefore, the thermal conduction mechanisms are discussed separately for Sb–Te binary alloys, ternary Sb2 Te3 –GeTe pseudobinary alloys and GeTe alloy in the following sections.

5.2.2 Sb–Te Binary Alloys Figure 5.3a shows the compositional dependence of thermal conductivities of the Sb–Te single-phase alloys at 293 and 673 K in this work, together with the reported data [9, 10]. There are few reliable thermal conductivity data reported for Sb and Te at high temperature; thus, only the thermal conductivity of Sb at 673 K by Konno [9] is plotted in Fig. 5.3a. Figure 5.3b shows the electrical conductivities of Sb–Te single-phase alloys at 293 and 673 K obtained by Koyike [11, 12]. The electrical conductivity decreases with an increase in temperature and this dependence is common in metals. The dependence of electrical conductivity on chemical composition and temperature would be reasonable since the conditions of the samples used for the electrical resistivity measurements by Koyike [11] are the same with those used in this work. On the other hand, Te shows a very small electrical conductivity at room temperature because of its semiconductor characteristics, and it is known that phonons dominate the heat conduction in Te. Thus, Te is also excluded from the following discussion. Comparison between Fig. 5.3a and b indicates that the thermal and electrical conductivities of the Sb–Te single-phase alloys at room temperature have almost the same dependence on the Te concentration. This same dependence suggests that free electrons are predominant carriers for both the heat and electrical transports in the Sb–Te single-phase alloys although small contribution from phonons is also possible. Figure 5.3a, b also shows that the thermal conductivities of the Sb-rich alloys increase but the electrical conductivities decrease with increasing temperature, which suggests that there is another heat conduction mechanism operating at high temperature in addition to that by free electrons. Evidence can be seen in Figs. 5.4 and 5.5 [12], which show the temperature dependences of the thermal conductivities for Sb-rich alloys and Te-rich alloys, respectively. From these two diagrams, it can be seen that the thermal conductivities have quite different temperature dependences

5.2 Thermal Conduction Mechanisms

95

Fig. 5.3 a Thermal conductivities in this work and b electrical conductivities by Koyike [11] of Sb–Te single-phase alloys at 293 and 673 K as a function of Te concentration, reprint from Ref. [12], with permission from Journal of Applied Physics

λ / Wm-1K-1

15 Sb-14% Te Sb-25% Te Sb-44% Te

10

5

0

300

400

500

600

700

800

T/K Fig. 5.4 Temperature dependences of thermal conductivities for Sb-rich alloys, reprint from Ref. [12], with permission from Journal of Applied Physics

5 Thermal Conduction Mechanisms and Prediction Equations …

Fig. 5.5 Temperature dependences of thermal conductivities for Te-rich alloys, reprint from Ref. [12], with permission from Journal of Applied Physics

3

λ / Wm-1K -1

96

2

Sb 2Te3-heat Sb 2Te3-cool Sb-70% Te Sb-90% Te-heat Sb-90% Te-cool

1

0

400

600

800

T/K

below and above 600 K. Thus, the heat conduction mechanism below and above 600 K is discussed separately.

5.2.2.1

Below 600 K

Below 600 K, free electrons are supposed to dominate the heat conduction and thus the WF law [1, 2] would apply. Thermal conductivity data for actual metals indicate that absolute values of the Lorenz number change depending upon the kind of metal and temperature; however, the magnitude is in the order of 10−8 W/K2 [13]. Values of the Lorenz number of the alloys used in the present work have been calculated from the WF law using thermal and electrical conductivity data at room temperature in Fig. 5.3a, b, resulting in Fig. 5.6 [12]. It can be seen that the Lorenz numbers obtained are in the range of 1.8–2.8 × 10−8 W/K2 and fall down around the theoretical value. According to Eq. 5.1, the value of λ has been plotted against 3

L / 10-8WΩK-2

Fig. 5.6 Lorenz numbers for Sb–Te single-phase alloys as function of Te concentration at room temperature, reprint from Ref. [12], with permission from Journal of Applied Physics

2

1

0

0

20

40

60

x Te / mol%

80

100

5.2 Thermal Conduction Mechanisms

293K 473K 573K

20

λ / Wm-1K-1

Fig. 5.7 Plot of thermal conductivity as a function of σ T for Sb–Te single-phase alloys, reprint from Ref. [12], with permission from Journal of Applied Physics

97

15

10 5

0

0

200

400

600

800

σ T /10 6Ω –1m-1K

the product σ T for three temperatures below 600 K, resulting in Fig. 5.7 [12]. The thermal conductivity data can be approximated to a linear function of σ T having a slope of 2.2 × 10−8 W/K2 . Using this value, the WF law can be applied to roughly estimate the thermal conductivities of Sb–Te single-phase alloys. Figure 5.8 shows the electrical resistivity data of solid Sb2 Te3 reported by Onderka [6], Glazov [14] and Drasar [15], which can be regressed to the following equation: ρ = aT 2 + bT

(5.2)

where a and b are constants. From the regression line, the constants have been determined to be a = 1.525 × 10−5 and b = 2.698 × 10−3 . Substituting Eq. (5.2) into (5.1) gives the relationship between the thermal conductivity of Sb2 Te3 and temperature as follows: 15

ρ / 10-6Ω m

Fig. 5.8 Electrical resistivity data of solid Sb2 Te3 reported by Onderka [6], Glazov [12] and Drasar [14]

Glazov Onderka et al Drasar

10

5

0

0

200

400

T/K

600

800

5 Thermal Conduction Mechanisms and Prediction Equations … 0.6

0.5

-1

λ

–1

Fig. 5.9 Reciprocals of thermal conductivities for Sb2 Te3 as a function of temperature, reprint from Ref. [12], with permission from Journal of Applied Physics

/ W mK

98

0.4

0.3

300

400

500

600

700

800

T/K

1 a b = cT + d = T + λ L L

(5.3)

where c and d are constants. According to this equation, the reciprocals of thermal conductivities have been plotted against temperature, resulting in Fig. 5.9 [12]. It can be seen that below 600 K there is a linear relationship between temperature and the reciprocals of thermal conductivities. From the slope and intercept of the linear part, the constants are determined to be c = 6.55 × 10−4 and d = 0.206. Using the constants a, b, c, d obtained, the Lorenz numbers calculated from the slope and intercept of the linear part are 2.3 × 10−8 and 1.3 × 10−8 W/K2 , respectively. Considering the error of curving fitting, the results have been regarded to agree with the theoretical value of Lorenz number. On the other hand, the band gap energy of Sb2 Te3 has been reported to be 0.15–0.2 eV [16–19] where 1 eV = 1.60218 × 10−19 J, which is much smaller than that of typical semiconductors. Accordingly, it can be understood that Sb2 Te3 has a certain number of free electrons at room temperature and that these free electrons dominate the heat conduction below 600 K. The success of the WF law in explaining the thermal conductivity of Sb2 Te3 gives another evidence to say that Sb–Te single-phase alloys show metallic characteristics below 600 K and that free electrons dominate the heat conduction.

5.2.2.2

Above 600 K

Figure 5.10 shows the thermal conductivities of Sb2 Te3 , together with values calculated from the WF law using the Lorenz number 1.95 × 10−8 W/K2 derived for Sb2 Te3 in Fig. 5.6. It can be seen that the WF law can describe the thermal conductivities of Sb–Te single-phase alloys below 600 K but not above 600 K. The electrical conductivity does not increase with increasing temperature, and thus it cannot be expected that the number of free electrons contributing to electrical conduction increases at higher temperatures. Thus, another mechanism should be considered to

5.2 Thermal Conduction Mechanisms 5

This work-Heating This work-Cooling Calculated by WF law

4

λ / Wm-1K-1

Fig. 5.10 Thermal conductivities of Sb2 Te3 , together with values calculated by WF law using the Lorenz number 1.95 × 10−8 W/K2 derived for Sb2 Te3 , reprint from Ref. [12], with permission from Journal of Applied Physics

99

3 2 1 0

300

400

500

600

700

800

T/K

explain the thermal conductivities above 600 K in addition to free electrons. As temperature rises, phonon contribution is becoming smaller and can be neglected after all. One possible mechanism would be via bipolar diffusion. Bipolar diffusion means simultaneous diffusion of both electrons and holes which are thermally activated in a sample along the temperature gradient [3, 5, 20]. During the measurements of thermal conductivity by the hot-strip method, a temperature gradient is applied to the sample. The electrons and holes are created simultaneously by the thermal excitation of electrons from the valence band to the conduction band. The temperature gradient results in the number gradient of electrons and holes in a radial direction in the sample, and both electrons and holes are transported along the temperature gradient and recombine at the cold end, that is, sample surface. Bipolar diffusion does not contribute to the electrical conductivity since free electrons and holes are transported in the same direction. However, bipolar diffusion does contribute to the thermal conductivity by the heat absorption for the ionization at the hot end and the heat generation for the recombination of electrons and holes at the cold end. The bipolar diffusion is more effective at higher temperature and contributes to the increase of thermal conductivity. The total thermal conductivity can be written as follows: λ = λel + λb

(5.4)

where λel and λb are thermal conductivities by free electron and bipolar effect. For temperatures above 600 K, the value of λb has been estimated as the difference between the thermal conductivities of Sb2 Te3 measured and calculated from the WF law. The value of λb would be affected by two factors. One is the carrier density (n), which is related to the band gap energy for bipolar diffusion (ΔE a ) as following equation:   E a n ∝ exp − 2kT

(5.5)

100

5 Thermal Conduction Mechanisms and Prediction Equations …

where k is Boltzmann’s constant. The other factor is the mobility of the bipolar diffusion (μ), which is significantly affected by the presence of acoustic phonons and ionized impurities results in carrier scattering. For polar semiconductor such as GaAs optical-phonon scattering is significant and the mobility can be approximated in theory by [21]: μ ∝ T 1/ 2

(5.6)

In addition to the scattering mechanisms discussed above, however, other mechanisms also affect the actual mobility. The mobility varies as T and T 2.1 for n- and p-type GaAs instead of T 1/2 for the actual measurements, respectively [22]. Since Ge–Sb–Te alloys are p-type, the mobility is expressed in practice as follows: μ ∝ T 2.1

(5.7)

Therefore, the thermal conductivity contributed by bipolar diffusion can be written in theory and for practice, respectively, as follows:   E a λab = AT 1/ 2 exp − 2kT   E a λab = AT 2.1 exp − 2kT

(5.8) (5.9)

Plotting the logarithm of λab /T 1/2 and λab /T 2.1 against T −1 , respectively, has resulted in Figs. 5.11 and 5.12, which both show the linear relationship. The value of ΔE a can be derived from the slope of the linearity as 0.88 eV in theory and 0.70 eV for practice. Sb2 Te3 has a band gap as small as 0.15–0.2 eV; because of this, free electrons are generated below 600 K, leading to its metallic characteristics. Further Fig. 5.11 Plot of lnλb /T 1/2 as function of T −1 for Sb2 Te3 single-phase alloys

-2

-3

-1

-1

/ Wm K )

Sb 2Te3

ln(λ abT

-1/2

-4

-5

-6 1.2

1.3

1.4

1.5

T -1 /10 -3K-1

1.6

1.7

5.2 Thermal Conduction Mechanisms -13

Sb 2Te3

ln(λ ab T-2.1 / Wm -1K-1)

Fig. 5.12 Plot of lnλb /T 2.1 as function of T −1 for Sb2 Te3 single-phase alloys

101

-14

-15

-16 1.2

1.3

1.4 -1

1.5 -3

T / 10 K

1.6

1.7

-1

temperature increase gives electrons in valence bands energy enough to overleap the band gap for bipolar diffusion. It can be seen that band gap values obtained from Eqs. (5.8) and (5.9) are not quite different; in addition, it is not confirmed whether these equations are accurate for Ge–Sb–Te alloys. As a reference, Eq. (5.9) is temporarily chosen for the analyses of other alloys for convenience. The band gap for bipolar diffusion has also been derived as 0.78 eV for Sb–25%Te and 0.94 eV for Sb–44%Te, where the value of λb has been derived as λ − λ600K since the Lorenz number strongly depends on temperature.

5.2.3 Ternary GeTe–Sb2 Te3 Pseudobinary Alloys It has been demonstrated that Sb2 Te3 alloy and ternary Sb2 Te3 –GeTe pseudobinary alloys have similarity in the structure, which determines the similar behavior in the thermal conductivities and electrical resistivities. Therefore, in the same manner as for Sb–Te alloys, the WF law is also used as the important tool to discuss the thermal conduction mechanisms of ternary Sb2 Te3 –GeTe pseudobinary alloys. The compositional dependences of thermal conductivities in this work and the electrical conductivities obtained in this work and by Konstantinov et al. [20] and Onderka et al. [6] for the GeTe–Sb2 Te3 pseudobinary alloys at 298 K and 773 K are shown in Fig. 5.13a and b, respectively. The dependence of electrical conductivity by Konstantinov et al. [20] and Onderka et al. [6] on chemical composition and temperature would be reasonable although it is difficult to say that the magnitude itself is very accurate, since the present electrical conductivity data seem larger than those reported data. The data of GeTe alloy are also shown here just to give a comparison, and those data are not discussed in this section. Comparison between Fig. 5.13a and b indicates that the thermal and electrical conductivities of the ternary Sb2 Te3 –GeTe alloys at room temperature have almost

102

5 Thermal Conduction Mechanisms and Prediction Equations …

(a)

6

298 K 773 K

λ / Wm-1K-1

5 4 3 2 1

0 10 Sb 2Te3

(b)

20

30 40 xGe / mol%

50 60 GeTe

5 [×10 ] 8

σ / Ω -1m -1

6

This work 298K Konstantinov 298K Onderka 298K This work 773K Konstantinov 773K Onderka 773K

4 2 0

0

10

20

30

40

50

xGe / mol% Fig. 5.13 a Thermal conductivities in this work and b electrical conductivities by Konstantinov [20] of Sb2 Te3 –GeTe pseudobinary alloys at 293 and 773 K as function of Te concentration

the same dependence on the Te concentration. This same dependence suggests that free electrons are predominant carriers for both the heat and electrical transport in the ternary Sb2 Te3 –GeTe alloys although small contribution from phonons is also possible. Figure 5.13a, b also show that the thermal conductivities of the ternary Sb2 Te3 – GeTe alloys at 773 K keep almost the same values as those at room temperature, which comes from the decrease and then the increase of the thermal conductivities with increasing temperature. However, the electrical conductivities decrease with increasing temperature, which suggests that there is another heat conduction mechanism operating at high temperature in addition to that by free electrons. Evidence can be seen in Fig. 5.1. It can be seen that the thermal conductivities have quite different temperature dependences below and above 600 K. Thus, the heat conduction mechanism above 600 K is discussed separately in the next part.

5.2.3.1

Below 600 K

Below 600 K, free electrons are supposed to dominate the heat conduction and thus the WF law [1, 2] would be applied to predict the thermal conductivity using

5.2 Thermal Conduction Mechanisms

103

the electrical resistivity. However, there are few electrical resistivity data available for bulk Sb2 Te3 –GeTe pseudobinary alloys. The only electrical resistivity data as functions of composition and temperature were given by Konstantinov et al. [20]. There is no way to confirm the accuracy of the results by Konstantinov et al. [20]. Furthermore, from the investigations of Konstantinov et al. [20], it is known that any small deviation from the stoichiometric composition will result in a large difference in the electrical resistivity. Figure 5.14 shows an example of the electrical resistivity data for Ge2 Sb2 Te5 alloy and some alloys with small composition deviations from Ge2 Sb2 Te5 by Konstantinov et al. [20]. It can be seen that there are more than 50% difference at each temperature, which will bring about large error for the thermal conductivity prediction. Therefore, the possible range of the thermal conductivities is given using the largest and lowest resistivity data when applying the WF law to the data by Konstantinov et al. Figures 5.15, 5.16 and 5.17 show the thermal conductivities of the ternary Sb2 Te3 –GeTe pseudobinary alloys, together with values calculated from the WF law using the electrical resistivities obtained in this work and by Konstantinov et al. and the Lorenz number 2.45 × 10−8 W/K2 . It can be seen that the present thermal conductivity data for these three alloys are all within the range calculated from the WF law by the data of Konstantinov et al. [20]. Therefore, the supposition is reasonable that the free electrons dominate the thermal conduction below 600 K. The thermal conductivity data calculated using the electrical resistivity data in this work by WF law seem larger than the experimental results, which might come from the difference in the value of Lorenz number. Miyamura [23] has investigated the Lorenz numbers of several metals and mentioned that the Lorenz number changes largely depending on metals: some metals show larger and some smaller value than the theoretical value 2.45 × 10−8 W/K2 . The temperature dependence of Lorenz number is also varied from metal to metal. However, the magnitude is in the order Fig. 5.14 Resistivity obtained by Konstantinov as functions of temperature for (1) Ge1.94 Sb2 Te5 , (2) Ge1.96 Sb2 Te5 , (3) Ge2 Sb2 Te5 , (4) Ge2.02 Sb2 Te5 , (5) Ge2 Sb2 Te5.10 , reprint from Ref. [20], with permission from Inorganic Materials

104

5 Thermal Conduction Mechanisms and Prediction Equations … 4

Experimental results WF law by Konstantinov WF law by this work

λ / Wm-1K-1

3

2

1

0

300

400

500

600

700

800

T/K Fig. 5.15 Thermal conductivities of Ge2 Sb2 Te5 alloy, together with values calculated from WF law using electrical resistivities obtained in this work and by Konstantinov et al. 4

Experimental results WF law by Konstantinov WF law in this work

λ / Wm-1K-1

3 2 1 0

300

400

500

600

700

800

T/K Fig. 5.16 Thermal conductivities of GeSb2 Te4 alloy, together with values calculated from WF law using electrical resistivities obtained in this work and by Konstantinov et al. GeSb 4Te7-heat GeSb 4Te7-cool WF law by Konstantinov WF law in this work

5 4

λ / Wm-1K-1

Fig. 5.17 Thermal conductivities of GeSb4 Te7 alloy, together with values calculated from WF law using electrical resistivities obtained in this work and by Konstantinov et al.

3 2 1 0

300

400

500

600

T/K

700

800

5.2 Thermal Conduction Mechanisms

-2

3

2

-8

L / 10 WΩK

Fig. 5.18 Lorenz numbers for Sb2 Te3 –GeTe pseudobinary alloys as function of Te concentration at room temperature

105

1

0 0

20

40 60 xTe / mol%

80

100

of 10−8 W/K2 [13]. Because of the evaporation of the elements during the samplemaking process, it is impossible to control the exact stoichiometric composition of the samples in this work. The resistivity data by Konstantinov et al. can just give the possible range for the thermal conductivity. It is more reliable to apply the WF law using the electrical resistivity results obtained in the present work since the compositions and structures of the samples used for thermal conductivity and electrical resistivity measurements are much closer to each other. Values of the Lorenz number of alloys used in the present work have been calculated from the WF law using the thermal and electrical conductivity data at room temperature obtained in this work, resulting in Fig. 5.18. The results for Sb2 Te3 and GeTe alloys are also shown here to give a reference. The dash line shows the theoretical value for Lorenz number, 2.45 × 10−8 W/K2 . It can be seen that the Lorenz numbers obtained for the ternary alloys are in the range of 1.3–1.9 × 10−8 W/K2 and fall down around the theoretical value. According to Eq. 5.1, the value of λ has been plotted against the product T for four temperatures below 600 K, resulting in Fig. 5.19. The thermal conductivity data can be approximated to a linear function of σ T having a slope to be 1.56 × 10−8 W/K2 . Using this value, the WF law can be applied to roughly estimate the thermal conductivities of ternary Sb2 Te3 –GeTe pseudobinary alloys. The success of the WF law in explaining the thermal conductivity gives the evidence to say that ternary Sb2 Te3 –GeTe pseudobinary alloys show metallic characteristics below 600 K, similar to Sb2 Te3 alloy and that free electrons dominate the heat conduction, which is consistent with the structure similarity introduced before.

5.2.3.2

Above 600 K

Figure 5.20 shows the thermal conductivities of ternary Sb2 Te3 –GeTe pseudobinary alloys. It can be seen that the temperature dependences of the thermal conductivities for three ternary Sb2 Te3 –GeTe pseudobinary alloys are almost the same as each other

106

5 Thermal Conduction Mechanisms and Prediction Equations … 3

λ / Wm-1K-1

Fig. 5.19 Plot of thermal conductivity as a function of σ T for Sb2 Te3 –GeTe pseudobinary alloys

298K 378K 473K 573K

2

1

0

0

0.5

1 6

–1

-1

σ T /10 Ω m K Fig. 5.20 Thermal conductivities of ternary GeTe–Sb2 Te3 alloys as function of temperature

1.5 8

[×10 ]

λ / Wm-1K-1

3

2

1

400

600

800

T/ K Ge2Sb 2Te5-heat Ge2Sb 2Te5-cool GeSb 2Te4-heat

GeSb 2Te4-cool GeSb 4Te7-heat GeSb 4Te7-cool

because of the structure similarity, but distinguishingly different below and above 600 K for every alloy itself. From Fig. 5.13 it is known that the electrical conductivity does not increase with increasing temperature, and thus it cannot be expected that the number of free electrons increases at higher temperatures. Thus, another mechanism should be considered to explain the thermal conductivities above 600 K in addition to free electrons. As temperature rises, phonon contribution is becoming smaller and can be neglected after all. One possible mechanism would be via bipolar diffusion, which also contributes to the thermal conductivity increase of Sb2 Te3 alloys above 600 K and has been introduced in Sect. 5.2.2.2. The total thermal conductivity can be written the same as Eq. (5.4)

5.2 Thermal Conduction Mechanisms -13

ln(λ abT-2.1 / Wm-1K-1)

Fig. 5.21 Plot of lnλab as function of T −2.1 for Sb2 Te3 –GeTe pseudobinary alloys

107

-14

-15

-16

-17 0.0012

0.0013

0.0014

T

λ = λel + λb

-1

0.0015

-1

/K

(5.4)

where λel and λb are thermal conductivities by free electron and bipolar effect. Below 600 K, the thermal conductivities of all three ternary Sb2 Te3 –GeTe pseudobinary alloys have good linearity. By extrapolating the lines of the thermal conductivities below 600 K up to temperatures above 600 K, the electronic thermal conductivity parts can be obtained as shown by the dash lines in Fig. 5.20. The value of λb has been estimated as the difference between the thermal conductivities of ternary Sb2 Te3 – GeTe pseudobinary alloys measured and those estimated by the extrapolated lines. The value of λab would be represented by Eq. (5.9).   E a λab = AT 2.1 exp − 2kT

(5.9)

where A is a constant, ΔE a is the band gap energy for bipolar diffusion and k is Boltzmann’s constant. Plotting the logarithm of λb against T −2.1 for Ge2 Sb2 Te5 alloy has resulted in Fig. 5.21 as an example, which shows that there is a linear relationship between them. The value of ΔE a can be derived from the slope of the linearity as 0.87 eV. The band gap for bipolar diffusion has also been derived as 0.77 eV for GeSb2 Te4 alloy and 1.22 eV for GeSb4 Te7 alloy which has a certain extent deviation in thermal conductivities at high temperature and results in a slightly larger band gap.

5.2.4 GeTe Alloy Figure 5.22 shows the thermal conductivity for GeTe as a function of temperature, in comparison with values calculated from the WF law using the resistivity data reported by Konstantinov et al. [20] and in the present measurements [24]. The theoretical

Fig. 5.22 Thermal conductivity results for GeTe alloy as function of temperature, in comparison with values calculated from WF law using resistivity data reported by Konstantinov et al. [20] and in present measurements, reprint from Ref. [24], with permission from Journal of Applied Physics

5 Thermal Conduction Mechanisms and Prediction Equations … 8 GeTe-heating GeTe-cooling

WF law by Konstantinov

6

λ / Wm-1K-1

108

WF law in this work

4

2

0

400

600

800

1000

T/K

Lorenz number used in this calculation is 2.45 × 10−8 W/K2 . It can be seen that the thermal conductivity of GeTe alloy decreases monotonically with increasing temperature and does not increase at high temperature unlike other alloys. The thermal conductivity calculated from the WF law using the electric resistivity by Konstantinov et al. [20] shows an abrupt change at about 700 K since the electrical resistivity changes due to the phase change of GeTe alloy which has been discussed in Chap. 4. The thermal conductivities calculated from the WF law using the electrical resistivity data obtained in this work are in good agreement with the experimental thermal conductivity data. GeTe alloy is well known as a narrow band gap degenerate p-type semiconductor [25, 26]. Conventionally, it is considered that the non-stoichiometric Ge vacancies contribute to a high carrier density (~1026 to 1027 m−3 ) and to ensure the metallic temperature dependence of electrical resistivity of GeTe alloy in solid state [27–29]. Therefore, it can be concluded that free electrons dominate the thermal conduction of GeTe alloy at all temperature range in the solid state.

5.3 Prediction Equations of Thermal Conductivity The thermal conductivity prediction of Ge–Sb–Te alloys on the basis of the WF law requires accurate data of electrical conductivity. However, there are few accurate data of electrical conductivity available for Ge–Sb–Te alloys as introduced before. In addition, there are many kinds of compositions used in practice, so it is impossible to measure the thermal conductivities or electrical conductivities of all compositions. A more convenient method of prediction is required so as to estimate thermal conductivities of Ge–Sb–Te alloys directly from temperature and chemical composition. For example, Endo et al. [30] have investigated the thermal conductivity of Ni–Cr solution alloys and suggested an equation for thermal conductivity prediction as follows:

5.3 Prediction Equations of Thermal Conductivity

λ=

109

f e + T g + hY

(5.10)

where e, f, g and h are constants and Y = X (1 − X) where X is the mole fraction of solute component. The structure analysis has shown that the Sb2 Te3 –GeTe pseudobinary alloys except GeTe can also be considered as solid solution, and thus this equation can be applied to predict the thermal conductivities of Sb2 Te3 –GeTe pseudobinary alloys except GeTe alloy. Similar to the discussion on thermal conduction mechanism, the prediction equation establishments are also discussed separately for Sb–Te binary alloys and Sb2 Te3 –GeTe pseudobinary alloys.

5.3.1 Sb–Te Binary Alloys 5.3.1.1

Sb-Rich Single-Phase Alloys Below 600 K

As shown in Fig. 5.4, the thermal conductivities of the Sb-rich single-phase alloys have no temperature dependence below 600 K, and therefore the constant e equals zero. Thus, the temperature-dependent term can be eliminated from Eq. (5.10), leading to λ=

f g + hY

(5.11)

The intercept (λintercept ) for the Sb-rich single-phase alloys in Fig. 5.4 is equal to f/ (g + hY ), and its reciprocal has been plotted against Y, that is, X Te (1 − X Te ) where X Te is the mole fraction of Te, resulting in Fig. 5.23 [12]. The value of 1/λintercept is in linear proportion to X Te (1 − X Te ), and can be regressed to the following equation:   1/λintercept = 2.046X Te (1 − X Te ) − 0.1531 W m−1 K−1

-1 intercept

-1

/ W mK

0.4

λ

Fig. 5.23 Plot of λ−1 intercept as function of X Te (1 − X Te ) for Sb–Te single-phase alloys, reprint from Ref. [12], with permission from Journal of Applied Physics

(5.12)

0.3 0.2 0.1 0

0.15

0.2

XTe (1- XTe )

0.25

110

5 Thermal Conduction Mechanisms and Prediction Equations …

Combination of Eq. (5.11) with Eq. (5.12) produces a prediction equation for the thermal conductivity of Sb-rich alloys as follows: λ=

5.3.1.2

1 2.046X Te (1 − X Te ) − 0.1531

(5.13)

Single-Phase Alloys Above 600 K

On the basis of Eqs. (5.4) and (5.5), the thermal conductivities of single-phase alloys above 600 K can be expressed by the following equation: λ = λ600K

  E ab + A exp − 2kT

(5.14)

where λ600K is the thermal conductivities of single-phase alloys at 600 K, that is, Equation (5.14). Since the temperature dependence of the mobility is not confirmed and the simplification requirement of the prediction equation, the item related to the mobility is omitted, which does not affect the prediction results. Figure 5.24 shows the relationship between ln(λ − λ600K ), that is, ln λab and T −1 for all the singlephase alloys above 600 K [12]. The slopes of the straight lines are almost the same, which suggests that the band gap for bipolar diffusion is not dependent on the Te concentration. Using the slopes and intercepts obtained from this figure, a prediction equation has been proposed for single-phase alloys above 600 K as follows: 1 + λ= 2.046X T e (1 − X Te ) − 0.1531



   1.887 × 104 0.83 − 59.26 exp − X Te 2kT (5.15)

-1 -1

ln(λ–λ 600K / Wm K )

2

Sb-14% Te Sb-25% Te Sb-44% Te Sb-60% Te

1 0 -1 -2 -3 0.0012

0.0013

0.0014 -1

0.0015

0.0016

-1

T /K

Fig. 5.24 Plot of 1n(λ – λ600K ) as a function of T −1 for all Sb–Te single-phase alloys above 600 K, reprint from Ref. [12], with permission from Journal of Applied Physics

5.3 Prediction Equations of Thermal Conductivity

111

Actually, the second item on the right side of Eq. (5.15) is quite small and since below 600 K it can be neglected, so that Eq. (5.15) can be applied for Sb–Te singlephase alloys for all the temperature range investigated.

5.3.1.3

Two-Phase Alloys

In general, the thermal conductivity of two-phase alloys depends upon the volume of each component and its distribution [31]. It is frequently desirable to estimate a property of two-phase alloys from knowledge of the corresponding properties of the individual components. In the present work, the thermal conductivities of Sb–Te two-phase alloys is predicted by the lever rule using thermal conductivities of Sb2 Te3 and Te alloys. The simplest possible configuration model of Sb–Te two-phase alloys is the arrangement of two different single phases in parallel slabs since the present samples of Sb–Te two-phase alloys show lamellar structures, as shown in Fig. 5.25. The maximum conductivity is derived when the heat flow is parallel to the plane of the slabs and is given by λmix(max) = φ1 λ1 + φ2 λ2

(5.16)

On the other hand, the minimum conductivity is derived when the heat flow is perpendicular to the plane of the slabs and is given by λmix(min) =

λ1 λ2 ϕ1 λ2 + ϕ2 λ

(5.17)

where φ is the volume fraction, and the suffixes 1 and 2 represent constituting single phases. Figure 5.26 shows thermal conductivities as a function of the Te concentration calculated for the Te-rich alloys at room temperature from Eqs. (5.16) and (5.17) using the thermal conductivities of Sb2 Te3 (present data) and Te (reported data) [9], together with the present experimental data [24]. Both calculated values are in reasonable agreement with the present experimental data within the scatter of Fig. 5.25 Configuration model of Sb–Te two-phase alloys

Heat A Min

Max

B

A

B

3 Lever rule This work -1 -1

Fig. 5.26 Thermal conductivities calculated by lever rule for Te-rich two-phase alloys, together with experimental data in this work, reprint from Ref. [12], with permission from Journal of Applied Physics

5 Thermal Conduction Mechanisms and Prediction Equations …

λ / Wm K

112

2

1 60

Sb 2Te3

70

80

x Te / mol%

90

100

Te

thermal conductivities for Sb2 Te3 and Te. Furthermore, Sb2 Te3 and Te have thermal conductivities close to each other, which would make it difficult to conclude which equation is more reasonable. Figure 5.27 shows thermal conductivities for Sb–Te alloys measured and predicted for temperatures of 293 and 673 K. The circle and inverted triangle represent the present experimental data and the reported data [9], respectively, and the solid and dashed lines represent values calculated for single-phase alloys from Eq. (5.13) for 293 K and from Eq. (5.15) for 673 K. The dashed and single-dotted line represents values calculated for two-phase alloys from Eq. (5.16). The prediction equation for single-phase alloys has been applied to the concentration range of Te between 14 and 60 mol%, although the phase diagram shows two-phase alloys also existing in this range. The dashed-line frame shows the two single-phase ranges in the phase diagram and the blue frame shows the range to which the prediction equation has been applied. On the low Te mol% side, the values predicted by the prediction equation increase too fast and the prediction equation cannot be applied, and thus the prediction equation for two-phase alloys is applied. The calculated values are in Fig. 5.27 Thermal conductivities measured and predicted for Sb–Te alloys at 293 and 673 K, reprint from Ref. [12], with permission from Journal of Applied Physics

5.3 Prediction Equations of Thermal Conductivity

113

fairly good agreement with the experimental and reported data, which indicates that these prediction equations are useful in practice.

5.3.2 Sb2 Te3 –GeTe Pseudobinary Alloys 5.3.2.1

Below 600 K

As shown in Fig. 5.20, the temperature dependence of the thermal conductivities for ternary Ge–Sb–Te alloys is similar and these alloys can be considered as the solid solutions due to the same crystallographic structure as introduced before. Therefore, Eq. (5.10) is still used to establish the prediction equation for Sb2 Te3 –GeTe pseudobinary alloys. λ=

f e + T g + hY

(5.10)

According to Eq. (5.10), λ is plotted against T −1 , resulting in Fig. 5.28. Using the slope of the thermal conductivities below 600 K in Fig. 5.28, the constant e is roughly determined to be 162.76. The intercept (λintercept ) for the Sb2 Te3 –GeTe pseudobinary alloys in Fig. 5.28 is equal to f/ (g + hY ), and its reciprocal has been plotted against Y, that is, X Ge (1 – X Ge ) where X Ge is the mole fraction of Ge, resulting in Fig. 5.29. The value of 1/λintercept is in linear proportion to X Ge (1 – X Ge ), and can be regressed to the following equation: Fig. 5.28 Plot of λ as a function of T −1 for all ternary Sb2 Te3 –GeTe pseudobinary alloys

λ / Wm-1K-1

3

2

1 0.001

0.002 -1

0.003 -1

T /K

Ge2Sb 2Te5-heat Ge2Sb 2Te5-cool GeSb 2Te4-heat

GeSb 2Te4-cool GeSb 4Te7-heat GeSb 4Te7-cool

5 Thermal Conduction Mechanisms and Prediction Equations …

Fig. 5.29 Plot of λ−1 intercept as function of X Ge (1 − X Ge ) for Sb2 Te3 –GeTe pseudobinary alloys

1

λ-1intercept / W-1mK

114

0.8

0.6

0.4 0

0.1

0.2

XGe (1- XGe)

  1/λintercept = −2.6665X Ge (1−X Ge ) + 0.9892 W m−1 K−1

(5.18)

Combination of Eq. (5.6) with Eq. (5.18) produces a prediction equation for the thermal conductivity of Sb2 Te3 –GeTe pseudobinary alloys as follows: λ=

1 162.76 + T −2.6665X Ge (1 − X Ge )+0.9892

(5.19)

This equation is able to give not only the thermal conductivity of Sb2 Te3 –GeTe pseudobinary alloys below 600 K but also the electronic part of the thermal conductivity above 600 K since the electronic thermal conductivity above 600 K is thought to be along the extrapolation of the thermal conductivity below 600 K.

5.3.2.2

Above 600 K

The thermal conductivities of Sb2 Te3 –GeTe pseudobinary alloys above 600 K can be expressed by the following equation:   E ab λ = λel + A exp − 2kT

(5.20)

where λel is the thermal conductivities contributed by free electrons, that is, Eq. (5.19). Figure 5.30 shows the relationship between ln(λ − λel ), that is, lnλab and T −1 for Sb2 Te3 –GeTe pseudobinary alloys above 600 K. The slopes of the values for different alloys are almost the same, which suggests that the band gap for bipolar diffusion is not dependent on the Ge concentration. Using the slope and intercept obtained from this figure, a prediction equation has been proposed for Sb2 Te3 –GeTe pseudobinary alloys above 600 K as follows:

5.3 Prediction Equations of Thermal Conductivity 1

Ge 2Sb2 Te 5 GeSb 2 Te 4 GeSb 4 Te 7

0 -1

-1

ln(λ ab / Wm K )

Fig. 5.30 Plot of lnλab as function of T −1 for Sb2 Te3 –GeTe pseudobinary alloys

115

-1 -2 -3 1.2

1.3

1.4 -1

T /K

λ=

1.5 -3

-1

[×10 ]

  1 1.28 162.76 + + 6431.7 exp − (5.21) T −2.6665X Ge (1 − X Ge )+0.9892 2kT

In the same way with the prediction equations for Sb–Te single-phase alloys, Eq. (5.21) can also be used for the whole temperature range investigated. Figure 5.31 shows thermal conductivities for Sb2 Te3 –GeTe pseudobinary alloys measured and predicted for temperatures of 298, 673 and 773 K. The points and the lines represent the present experimental data for Sb2 Te3 –GeTe pseudobinary alloys and the values calculated from Eq. (5.19) for 293 K and from Eq. (5.21) for 673 K and 773 K, respectively. It has been known that the ternary Sb2 Te3 –GeTe pseudobinary alloys can be considered as solid solutions; however, there is no way to find the exact composition range within which the alloys are solid solutions. Therefore, it is still doubtful whether the prediction equation can be applied for all the composition range. From the figure it can be seen that the equation cannot give the correct values for the thermal conductivities of Sb2 Te3 and GeTe alloys. 6

298K 673K 773K 298K 673K 773K

5

λ / Wm-1K-1

Fig. 5.31 Thermal conductivities measured and predicted for Sb2 Te3 –GeTe pseudobinary alloys at 293, 673 and 773 K

4 3 2 1

0

10

20

30

x Ge / at%

40

50

116

5 Thermal Conduction Mechanisms and Prediction Equations …

The calculated values for the ternary alloys are in fairly good agreement with the experimental data, which indicates that these prediction equations can be used to predict the thermal conductivities of ternary Sb2 Te3 –GeTe pseudobinary alloys in practice.

5.4 Conclusions 1. The thermal and electrical behavior has been discussed based on the structural similarity of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys. It is proposed that single-phase Sb–Te alloys and ternary Sb2 Te3 –GeTe pseudobinary alloys can be considered as solid solutions. 2. The heat conduction mechanisms of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been clarified based on the thermal conductivity and electrical resistivity data. It is proposed that free electrons dominate the thermal conduction below 600 K and bipolar diffusion contributes to the increase in the thermal conductivity at higher temperatures, except for GeTe alloy, in which free electrons dominate the thermal conduction at the temperatures investigated. 3. The prediction equations have been proposed for the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys.

References 1. G. Wiedemann, R. Franz, Ann. Phys. Chem. 89, 497 (1853) 2. L. Lorenz, Ann. Phys. Chem. 13, 422 (1881) 3. J.E. Parrott, A.D. Stuckes, Thermal Conductivity of Solids, 1st edn. (Pion Limited, London, 1975) 4. R.G. Chambers, Electrons in Metals and Semiconductors (Chapman and Hall, London, New York, Tokyo, Melbourne, Madras, 1990) 5. J.R. Drabble, H.J. Goldsmid, Thermal Conduction in Semiconductors (Pergamon Press, Oxford, 1961) 6. B. Onderka, K. Fitzner, Phys. Chem. Liq. 36, 215 (1998) 7. T. Matunaga, N. Yamada, K. Kifune, Y. Kubota, J. Crystallogr. Soc. Jpn. 51, 292 (2009) 8. K. Kifune, Y. Kubota, T. Matsunaga, N. Yamada, Acta Crys. B61, 492 (2005) 9. S. Konno, Sci. Rep. Tohoku Imp. Univ. 8, 169 (1919) 10. Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens, Thermal Conductivity of Metallic Elements and Alloys (Plenum, New York, 1970), pp. 10–14, 366–371 11. M. Koyike, Master Thesis (Tokyo Institute of Technology, 2011) 12. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 110, 023701 (2011) 13. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986), p. 153 14. V.M. Glazov, A.N. Krestovnikov, N.N. Glagoleva, S.B. Evgen’ev, Neorg. Mater. 2, 1477 (1966) 15. C. Drasar, M. Steinharta, P. Lost’ak, H.-K. Shin, J.S. Dyck, C. Uher, J. Solid State. Chem. 178, 1301 (2005) 16. S.D. Shutov, V.V. Sobolev, Y.V. Popov, S.N. Shestatskii, Phys. Status Solidi, 31, K23 (1969) 17. W. Procarione, C. Wood, Phys. Status Solidi 42, 871 (1970)

References

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18. A. Von Middendorf, K. Dietrich, G. Lanwehr, Solid State Commun. 13, 443 (1973) 19. P.J. Price, Philos. Mag. 46, 1252 (1955) 20. P.P. Konstantinov, L.E. Shelimova, E.S. Avilov, M.A. Kretova, V.S. Zemskov, Inorg. Mater. 37, 662 (2001) 21. H. Ehrenreich, Phys. Rev. 120, 1951 (1960) 22. C. Jacoboni, C. Canali, G. Ottaviani, A.A. Quaranta, Solid State Electron. 20, 77 (1977) 23. A. Miyamura, Doctor Thesis (Tokyo Institute of Technology, 2004) 24. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, J. Appl. Phys. 112, 053712 (2012) 25. R.C. Miller, Thermoelectricity: Science and Engineering (Interscience, New York, 1961), p. 434 26. N.Kh. Abrikosov, O.G. Karpinskii, L.E. Shelimova, M.A. Korzhuev, Izvestiya Akademii Nauk SSSr. Neorg. Mater. 13, 2160 (1977) 27. N.Kh. Abrikosov, M.A. Korzhuev, L.E. Shelimova, Izvestiya Akademii Nauk SSSr. Neorg. Mater. 13, 1757 (1977) 28. B.F. Gruzinov, P.P. Konstantinov, B.Ya. Moizhes, Yu.I. Ravich, L.M. Sysoeva, Fiz. Tekh. Poluprovodn. 10, 497 (1976) [Sov. Phys. Semicond. 10, 296 (1976)] 29. M.A. Korzhuev, L.E. Shelimova, N.Kh. Abrikosov, Fiz. Tekh. Poluprovodn. 11, 296 (1977) [Sov. Phys. Semicond. 11, 171 (1977)] 30. R. Endo, M. Shima, M. Susa, Int. J. Thermophys. 31, 1991 (2010) 31. J.E. Parrott, A.D. Stuckes, Thermal Conductivity of Solids (Academic Press, London, 1975), pp. 79, 129–133

Chapter 6

Densities of Ge–Sb–Te Alloys

Abstract Ge–Sb–Te chalcogenide alloy has been widely investigated due to its applications in phase change random access memory (PCRAM). The density of Ge– Sb–Te alloy is a key factor for PCRAM applications. In this chapter, the densities of Sb2 Te3 and Ge2 Sb2 Te5 chalcogenide alloys in solid and molten states have been determined as a function of temperature by the sessile drop method. The density at room temperature was also determined by the Archimedean method to verify the reliability of data obtained by the sessile drop method. The density of solid alloys decreases linearly with increasing temperature and there is a discontinuous density decrease at the melting temperature due to the phase change. In the molten state, the density continues to decrease with increasing temperature. The molar volume of both chalcogenide alloys increases by about 4% from room temperature to just below the melting temperature and by 6% at the melting temperature from the solid to molten state, which may cause large stress for PCRAM devices. Keywords Density · Sessile drop method · Solid and molten states · Sb2 Te3 Ge2 Sb2 Te2 · Archimedean method · Molar volume

·

6.1 Introduction The chalcogenide Ge–Sb–Te alloys are candidate materials for PCRAM [1–5]. In the PCRAM devices, Ge–Sb–Te alloys transform repeatedly among liquid, amorphous and crystalline states to record data [6, 7]. During the data recording process, only a small volume in a micro-size-area memory element the phase transformation occurs. A large density change will cause serious stress to other part of the memory element and reduce the durability of PCRAM devices. Therefore, the densities of Ge–Sb–Te alloys in solid and liquid states are required for PCRAM device optimization and are key input parameters for the three-dimensional simulation of the time-transient behavior in a PCRAM memory element. From the discussion before, it is known that the GeTe–Sb2 Te3 pseudobinary alloys have crystallographic structures similar to each other, as well as the single-phase Sb– Te binary alloys. Kifune et al. [8] have also suggested that the crystal structures of Sb– Te binary system change continuously with Te concentration in the Te composition © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_6

119

120

6 Densities of Ge–Sb–Te Alloys

range from 0 to 60 at.%. The density is determined by the microstructure, and the densities of alloys with similar structures should also show similarity. Therefore, in this work, only two typical compounds Sb2 Te3 and Ge2 Sb2 Te5 alloys are chosen to be investigated, in that their densities can reflect the characteristics of Sb–Te binary and GeTe–Sb2 Te3 pseudobinary alloys. There are various methods for measuring the density of liquid metals and alloys, such as Archimedean method, pycnometric method, dilatometric method and liquid drop method [9]. The coefficients of thermal expansion are usually measured for the solids instead of measuring the density. The methods for the thermal expansion measurements of solids include interferometric method, X-ray diffraction method, push-rod dilatometer method and so on [10]. The sessile drop method is one kind of the liquid drop methods, which involves photographic profiles of a liquid drop. Following solidification, the specimen is then weighed and the volume of the liquid drop is calculated through geometrical analysis of the photographs [9]. A truly symmetrical liquid drop is required to obtain the accurate density data. To avoid a sessile drop of asymmetrical shape, a modified “large drop method” has been often used [11–13]. Another kind of the liquid drop method uses the levitation techniques [14–17], which is a containerless method beneficial for the highly reactive metallic elements. The large drop method and levitation method have their respective advantages for measuring the density of liquids, but it is difficult to adapt them to measure the density of solids. The relative simplicity of the device makes the sessile drop method easily applied for the density measurements of solids. Consequently, the densities of metals and alloys can be measured from solid to liquid states continuously by the sessile drop method. It is noteworthy that more efforts should be made to reduce the error during the measuring process. Therefore, the densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys are measured by the sessile drop method from solid to liquid states in this chapter. To confirm the accuracy of the results by the sessile drop method, the densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys are also measured by the Archimedean method at room temperature. Based on the results, discussion is made about the influence of density change during the phase transformation process for PCRAM application.

6.2 Sessile Drop Method 6.2.1 Sample Samples used were Sb2 Te3 and Ge2 Sb2 Te5 alloys. Cylindrical samples of Sb2 Te3 (6.5 mm diameter and 10 mm length) were prepared from Sb2 Te3 (99.9 mass%) powders, and cylindrical samples of Ge2 Sb2 Te5 (6.5 mm diameter and 40–50 mm length) were prepared from Sb2 Te3 (99.9 mass%) and GeTe (99.9 mass%) powders. All the powder samples were produced by Kojundo Chemical Laboratory. A mixture (10 g) of these powders with a desired composition was melted in a quartz crucible

6.2 Sessile Drop Method

121

(6.5 mm inner diameter) at 973 K for 15 min in Ar atmosphere, followed by cooling in the furnace. Subsequently, the sample was taken out from the crucible and cut into specimens with the height of about 10 mm along the transverse axis, and the cross-sections were mechanically polished using emery papers up to #2000, and the upside and underside surfaces were kept parallel as much as possible.

6.2.2 Sessile Drop Method The sessile drop method was used to measure the densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys. Figure 6.1 shows the experimental setup for density measurements by the sessile drop method [18]. The apparatus mainly consists of four parts: the furnace, light source (laser lamp), CCD camera and computer for image processing. In the center of the furnace, there is a horizontal sample stage. The furnace was equipped with a gas inlet and outlet and optical windows on both sides. The optical windows were water-cooled by cooling coils. The temperature of the sample was measured using an R-type thermocouple just near the sample surface. The atmosphere inside the furnace was controlled by Ar. A halogen lamp was placed at one side of the furnace to project the sample image. The CCD camera was placed at the opposite side of the halogen lamp to monitor the image continuously. The whole system was set on an

Fig. 6.1 Experimental setup for density measurements by sessile drop method, reprint from Ref. [18], with permission from High Temperatures-High Pressures

122

6 Densities of Ge–Sb–Te Alloys

Fig. 6.2 Image processing and density calculation process, reprint from Ref. [18], with permission from High Temperatures-High Pressures

aluminum rail system to make the optical alignment easier. The images of the sample obtained during the measurement were stored in a computer and analyzed with an image processing software. As shown in Fig. 6.2 [18], the image of the sample (a) was first separated from the substrate to become (b), followed by editing the edge (c). To determine the volume of the image, the drop was divided into discs of one pixel height, shown as (c). The volume of each disc (V i ) was calculated by the equation: Vi = πri2 h

(6.1)

where r i is the radius of each disc and h is the height. A reference sample gives the correspondence between the length and the pixel. The total volume was then derived by summing the volumes of the discs. V =

i=i 

πri2 h

(6.2)

i=1

Density (ρ d ) was then derived using the mass of the sample (m) and the determined volume (V ), ρd = m/V

(6.3)

6.2 Sessile Drop Method

123

The densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys were measured from room temperature to 973 K at intervals of about 100 K in Ar atmosphere. For every alloy, the density was measured at least three times at each temperature.

6.2.3 Substrate Selection To carry out the sessile drop method, a suitable substrate should be selected as the first step of the density measurements. As introduced before, Ge–Sb–Te alloys are highly reactive at high temperature. Therefore, a substrate not reacting with Ge– Sb–Te alloys is prerequisite for obtaining the reliable density data. Three kinds of oxides, SiO2 , Al2 O3 and MgO, were chosen as the candidates because they are all chemically inert with most metals and alloys. The contact angles of Sb2 Te3 alloy with these three kinds of substrates were measured at 923 K after holding for 1 h. After the measurements, the contacting parts of Sb2 Te3 alloy and the substrates were analyzed by SEM-EDS to check whether reactions occurred or not.

6.2.4 Parameter Calibration From Eq. (6.3), it can be known that in the sessile drop method the density values are finally calculated from the mass and volume of the sample and thus, the accuracy of the mass and volume determines the accuracy of the density results. For Ge–Sb–Te alloys, two problems restrain the accuracy of these two values. The first problem is that Ge–Sb–Te alloys are brittle and easily broken, which means that it is difficult to prepare samples with perfectly cylindrical shape. The upside and underside surfaces of the sample are also impossible to be kept absolutely parallel. Both the causes increase the error of the density. To solve this problem, the images of the sample at room temperature were taken from six different directions. The average volume of these six images was taken as the volume of the sample at room temperature. The volumes at higher temperature were calibrated using the average volume at room temperature. The second problem is that the Ge–Sb–Te alloys have high vapor pressure, especially in the liquid state. If evaporation is too strong, the mass will decrease and the density results will be affected. To calibrate the evaporation influence, the measurement procedure was adapted as shown in Fig. 6.3, where W means mass measurement. The sample was first heated up to one certain temperature and the picture was taken, and then the mass of the sample was measured after it was cooled down. The mass was measured every two temperatures in the solid state and at every temperature in the liquid state. The masses from the practical measurement at different temperatures were used as the masses for the density calculation.

124

6 Densities of Ge–Sb–Te Alloys

Solid

630

600 400

500

300 RT

W

W

Liquid

RT

700

680

660

640

W

9 W

W

W

W

Weight measurement

W

Fig. 6.3 Measurement procedures for solid and liquid states

6.2.5 Archimedean Method This method is based on the well-known principle of Archimedes. The outline of the Archimedean method is shown in Fig. 6.4. A solid sample of known weight w is suspended by a wire attached to the arm of a balance, and a weight w1 is observed. When the sample is immersed in distilled water, a new weight w2 is observed. The To balance

To balance

Distilled water

w

Weight=w1 Fig. 6.4 Outline of the Archimedean method

w Weight=w2

Sb2Te3

6.2 Sessile Drop Method

125

difference in the two weights, that is, the apparent loss of weight w (=w1 – w2 ) of the immersed sample, originates mainly from the buoyant force exerted by distilled water. The density of the sample, ρ, is given by ρd =

w ρwater w

(6.4)

The error of the Archimedean method is mainly from two aspects: the volume of the suspension wire and the surface tension, that is, the force acting against the suspension wire. Therefore, the densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys with different masses were measured by the Archimedean method at room temperature to reduce the error influence, and the results were compared with those obtained by the sessile drop method.

6.3 Density Results 6.3.1 Substrate Selection Figure 6.5 shows the contact angle results of Sb2 Te3 alloy on SiO2 , Al2 O3 and MgO substrates [18]. It can be seen that Sb2 Te3 alloy has the largest contact angle Fig. 6.5 Contact angle results of Sb2 Te3 on a SiO2 , b Al2 O3 and c MgO substrates, reprint from Ref. [18], with permission from High Temperatures-High Pressures

(a) SiO2 93°

(b) Al2O3 128°

(c) MgO 145°

126

6 Densities of Ge–Sb–Te Alloys

on the MgO substrate being 145° and the smallest on the SiO2 substrate being 93°. Figure 6.6 shows the EDS mapping results of the areas around the interfaces between Sb2 Te3 alloy and SiO2 , Al2 O3 and MgO substrates. For the SiO2 substrate, the Sb2 Te3 alloy separated from the substrate and resin got into the gap between the alloy and substrate when they were sealed by resin. The thin layers with different color in the mapping results of SiO2 substrate were not thought to come from the composition difference but from polishing. Sb or Te was also not found in the substrate sides for other two kinds of substrates. Therefore, it can be said that there were no reactions between Sb2 Te3 alloy and SiO2 , Al2 O3 and MgO substrates during holding for 1 h. However, the small contact angle between Sb2 Te3 alloy and SiO2 substrate suggests the reaction tendency between Sb2 Te3 alloy and SiO2 , and actually the reaction layer was observed in SiO2 for a longer holding time. Larger contact angles are known to be better for the sessile drop method. Therefore, MgO was selected as the substrate for density measurement of Ge–Sb–Te alloys.

Sb

Si

Te

O

Resin 2mm (a) SiO2 S

Te

S

Al

O

Mg

Te

O

2mm

2mm (b) Al2O3

(c) MgO

Fig. 6.6 EDS mapping results of areas around interfaces between Sb2 Te3 alloy and a SiO2 , b Al2 O3 and c MgO substrates

6.3 Density Results

127

Room temperature

0

180

60

120

240

360

Fig. 6.7 Images of the sample taken from six different directions at room temperature, reprint from Ref. [18], with permission from High Temperatures-High Pressures

6.3.2 Parameter Calibration Figure 6.7 shows the images of the sample taken from six different directions at room temperature [18]. From the pictures it can be clearly seen that the upside and underside surfaces are not parallel and the volume calibration is necessary. The deviation range of the six volumes from the average volume is between 1.7 and 1.8%. In this measurement, the calibration coefficient of the volume at high temperatures is calculated to be 0.24%. The mass of the sample in the solid state did not change and the evaporation can be ignored, while in the liquid state, the results showed 1–2% mass loss for every temperature, and the mass loss increased with holding time independent of the temperature.

6.3.3 Density Results Figures 6.8 [18] and 6.9 show the original images taken by the sessile drop method at different temperatures for Sb2 Te3 and Ge2 Sb2 Te5 alloys, respectively. It can be seen that the volume of the samples increases with increasing temperature, although it is not so obvious at lower temperatures. In the liquid state, the melt sample shows a satisfactory ellipse surface, which indicates that the surface of the sample was not contaminated by oxygen. Figure 6.10 shows the density results of Sb2 Te3 alloy as a function of temperature by the sessile drop method. The different marks show the density results from the measurements of different samples. The density decreases linearly with increasing temperature from room temperature up to the melting temperature. At the melting temperature, the density has a discontinuous decrease which comes from the phase change. In the liquid state, the decrease rate of the density is slightly larger than that in the solid state. The average density of Sb2 Te3 alloy at room temperature by four times measurements is 6.24 g cm−3 , with the uncertainty from −1.30 to 0.72%.

128

6 Densities of Ge–Sb–Te Alloys

RT

400

610

680

200

500

640

300

600

660

700

Fig. 6.8 Original images taken by sessile drop method at different temperatures for Sb2 Te3 alloys, reprint from Ref. [18], with permission from High Temperatures-High Pressures

RT

500

700

300

550

400

625

750

Fig. 6.9 Original images taken by sessile drop method at different temperatures for Ge2 Sb2 Te5 alloys

6.3 Density Results

129

Fig. 6.10 Density results of Sb2 Te3 alloys as function of temperature by sessile drop method, reprint from Ref. [18], with permission from High Temperatures-High Pressures

Figure 6.11 shows the density results of Ge2 Sb2 Te5 alloy as a function of temperature by the sessile drop method. The density shows the linear temperature dependence similar with that of Sb2 Te3 alloy and also has a discontinuous decrease at the melting temperature. The average density of Ge2 Sb2 Te5 alloy at room temperature by four times measurements is 6.25 g cm−3 , with the uncertainty from −1.48 to 0.98%. Figures 6.12 and 6.13 show the density results of Sb2 Te3 and Ge2 Sb2 Te5 alloys at room temperature by the Archimedean method for the samples with different masses. The masses of the samples are shown in the tables just below the figures. The average density of three Sb2 Te3 alloys is 6.39 g cm−3 , and the average density of two Ge2 Sb2 Te5 alloys is 6.34 g cm−3 . From the results of different samples, it can be seen that the mass of the sample does not affect the measurement of the Archimedean method. Fig. 6.11 Density results of Ge2 Sb2 Te3 alloys as function of temperature by sessile drop method

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Fig. 6.12 Density results of Sb2 Te3 alloy at room temperature by Archimedean method for samples with different masses

Fig. 6.13 Density results of Ge2 Sb2 Te5 alloy at room temperature by Archimedean method for samples with different masses

6.4 Discussion 6.4.1 Comparison with Reported Data Figure 6.14 shows the density results by the sessile drop method and the Archimedean method for Sb2 Te3 alloy at room temperature, together with the reported data by Anderson et al. [19]. The results by the sessile drop method are about 2% smaller

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Fig. 6.14 Density results by sessile drop method and Archimedean method for Sb2 Te3 alloy at room temperature, together with reported data, reprint from Ref. [18], with permission from High Temperatures-High Pressures

than those by the Archimedean method, which difference is thought to come from the error of the sample shapes. The 2% error is acceptable within the uncertainty range of the measurements. The results by the Archimedean method are close to the reported data by Anderson et al. [19], which were obtained by the XRD measurement and the densities were calculated from the lattice constant. From the comparison between the densities at room temperature, it can be said that the density measurements by the sessile drop method are reliable with about 2% uncertainty. Figure 6.15 shows the density results for Sb2 Te3 alloy as a function of temperature by the sessile drop method together with the reported data [18–21]. The reported data between 300 and 400 K were measured by Krost et al. [20], and the dotted

Fig. 6.15 Density results for Sb2 Te3 alloy as function of temperature by sessile drop method together with reported data, reprint from Ref. [18], with permission from High Temperatures-High Pressures

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line is extrapolated from the results by Krost et al. The present data are smaller than the reported and extrapolated data in the solid state, but the temperature tendency is the same as that of the extrapolated line. The density decrease with increasing temperature is not large in the solid state. It can be seen from the reported data that the temperature dependence of the density is almost the same in the solid and liquid states. The density data in the liquid state in this work show a slightly larger decrease rate than the reported data, which might result from the influence of the evaporation. Figure 6.16 shows the density results by the sessile drop method and the Archimedean method for Ge2 Sb2 Te5 alloy at room temperature, together with the reported data by Walter et al. [22] for thin films. There are no reported data found for bulk Ge2 Sb2 Te5 alloy above the room temperature. The present results by the sessile drop method and the Archimedean method are close to each other, and both of them are close to the result obtained by Walter et al. [22] by the XRR method. The results by the XRR method are smaller than those by the XRD method, which were calculated from the lattice parameters in the work of Walter et al. [22]. Figure 6.17 7

XRD by Walter et al [22] 6.5

ρd / gcm-3

Fig. 6.16 Density results by sessile drop method and Archimedean method for Ge2 Sb2 Te5 alloy at room temperature, together with reported data

6

XRR by Walter et al [22]

5.5

5

Sessile drop method

7 6.5

ρd / gcm-3

Fig. 6.17 Density results of liquid Ge2 Sb2 Te5 alloy at near melting temperature by sessile drop method in this work, together with reported data for thin films in amorphous state by Walter et al. [22]

Archimedean method

6

Amorphous

Liquid

5.5 5

4 6%

4.5 4

Sessile drop method XRR by Walter et al [22]

6.4 Discussion

120

Vm / cm3mol-1

Fig. 6.18 Molar volume of Sb2 Te3 alloy calculated from density data as function of temperature by sessile drop method, reprint from Ref. [18], with permission from High Temperatures-High Pressures

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110

100

Tmelting

90 400

600

800

1000

T/K

shows the density results of liquid Ge2 Sb2 Te5 alloy near melting temperature by the sessile drop method in this work, together with the results for thin films in the amorphous state by Walter et al. [22]. Usually, the structures of the materials in the liquid and amorphous states are similar to each other and thus, the densities which are directly determined by the structures in these two states should also have a certain extent similarity. Therefore, the density results of the amorphous Ge2 Sb2 Te5 alloy by Walter et al. [22] can be used as the reference data to compare with the results in the liquid state in this work. The present data in the liquid state are slightly smaller than the reported data for amorphous thin films. The average difference between the results in this work and by Walter et al. [22] is about 4.6%, which is thought reasonable.

6.4.2 Application Discussion Figures 6.18 [18] and 6.19 show the molar volume of Sb2 Te3 and Ge2 Sb2 Te5 alloys calculated from the density data as a function of temperature by the sessile drop method, respectively. The volume changes from room temperature up to just below the melting temperature for both alloys are about 4%. At the melting temperature, the volume changes from the solid to liquid state are both about 6%. The total 10% volume increase from room temperature up to melting temperature may cause large stress in the PCRAM element and affect the durability of the devices. Therefore, some composition adjustments and structure optimization should be done to reduce the influence of volume change during the phase transformation process.

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6 Densities of Ge–Sb–Te Alloys 200

Vm / cm3mol-1

Fig. 6.19 Molar volume of Ge2 Sb2 Te5 alloy calculated from density data as function of temperature by sessile drop method

180

160

Tmelting 140 400

600

800

1000

1200

T/K

6.5 Conclusions 1. The densities of Sb2 Te3 and Ge2 Sb2 Te5 alloys have been determined as a function of temperature by the sessile drop method and determined at room temperature by the Archimedean method. By comparing the results by the sessile drop method with those obtained by the Archimedean method and the reported data, it is thought that the present results are reliable with about 2% uncertainty. 2. The densities of solid Sb2 Te3 and Ge2 Sb2 Te5 alloys decrease linearly with increasing temperature and the tendencies of the density change are similar. There is a discontinuous density decrease at the melting temperature for both alloys due to the phase change. In the liquid states, the densities continue to decrease with increasing temperature. 3. The volume increases by about 4% from the room temperature to just below the melting temperature and by 6% at the melting temperature from the solid to liquid state, which may cause large stress for PCRAM devices.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

M. Wuttig, N. Yamada, Nat. Mater. 6, 824 (2007) H.J. Borg, M. van Schijindel, J.C.N. Rijpers, Jpn. J. Appl. Phys. 40, 1592 (2001) N. Oomachi, S. Ashida, N. Nakamura, Jpn. J. Appl. Phys. 41, 1695 (2002) M.H.R. Lankhorst, L. van Pieterson, M. van Schijndel, B.A.J. Jacobs, J.C.N. Rijpers, J. Appl. Phys. 42, 863 (2003) N. Yamada, E. Ohno, K. Nishiuchi, N. Akahira, J. Appl. Phys. 69, 2849–2856 (1991) S. Lai, IEDM Tech. Dig. (2003), p. 10.1.1 M. Wuttig, Nat. Mater. 4, 265 (2005) K. Kifune, Y. Kubota, T. Matsunaga, N. Yamada, Acta Crystallogr. B61, 492 (2005) T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals (Clarendon Press, Oxford, 1993)

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10. Japan Society of Thermophysical Properties, Thermophysical Properties Hanbook (Yokendo, Japan, 2008) 11. K. Mukai, F. Xiao, Mater. Trans. 43, 1153 (2002) 12. J.M. Molina, R. Voytovych, E. Louis, N. Eustathopoulos, Inter. J. Adhes. Adhes. 27, 394 (2007) 13. L. Fang, Y.F. Wang, F. Xiao, Z.N. Tao, K. MuKai, Mater. Sci. Eng. B 132, 164 (2006) 14. H. Fujii, T. Matsumoto, S. Izutani, S. Kiguchi, K. Nogi, Acta Mater. 54, 1221 (2006) 15. M. Adachi, M. Schick, J. Brillo, I. Egry, M. Watanabe, J. Mater. Sci. 45, 2002 (2010) 16. J. Brillo, I. Egry, T. Matsushita, Inter. J. Thermophys. 27, 1778 (2006) 17. W.-K. Rhim, K. Ohsaka, J. Cryst. Growth 208, 313 (2000) 18. R. Lan, R. Endo, M. Kuwahara, Y. Kobayashi, M. Susa, High Temp.-High Press. 46, 219 (2017) 19. T.L. Anderson, H.B. Krause, Acta Crystallogr. B 30, 1307 (1974) 20. A. Krost, U. Nowak, W. Richter, E. Anastassakis, Verhandl. DPG (VI) 15, 171 (1980) 21. K.J. Singh, R. Satoh, Y. Tsuchiya, J. Phys. Soc. Jpn. 72, 2546 (2003) 22. W.K. Njoroge, H.W. Woltgens, M. Wuttig, J. Vac. Sci. Technol. 20, 230 (2002)

Chapter 7

Summary and Conclusions

The thermophysical properties including thermal and electrical conductivity and density of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary chalcogenide alloys have great significance for the phase change random access memory (PCRAM) applications and scientific comprehension. The objectives of this book have been to provide the measuring technique and values of these properties for these alloys and to clarify the thermal conduction mechanisms. For the industrial applications, the prediction equations for thermal conductivities of these alloy systems have been proposed based on the thermal conductivity data and conduction mechanisms. Chapter 1 “Introduction” The importance of the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary chalcogenide alloys for PCRAM application and scientific signification has been explained. The thermal conductivity theory and the previous studies of the thermophysical properties have been introduced. Against the background, the objectives of this book have been represented. Chapter 2 “Establishment of the Hot-Strip Method for Thermal Conductivity Measurements of Ge–Sb–Te Alloys” The possible problems for applying the hot-strip method to the thermal conductivity measurements of Ge–Sb–Te alloys have been discussed based on the practical measurement conditions and the measurement setup and parameters have been modified. After the modification, the thermal conductivities of titanium and fused silica have been measured from 298 K up to about 800 K which is the temperature just below the melting point of Sb2 Te3 alloy using the hot-strip method since they have many reported data. The results for titanium and fused silica have verified that the hotstrip method is able to give the reliable thermal conductivity data. Consequently, the thermal conductivities of Sb2 Te3 alloy have been measured from 298 K up to about 800 K to confirm the applicability of the hot-strip method for the measurements of Ge–Sb–Te alloys since these alloys are easily oxidized and have high evaporation. By analyzing the characteristics of the Sb2 Te3 sample before and after measurements © Xi’an Jiaotong University Press 2020 R. Lan, Thermophysical Properties and Measuring Technique of Ge-Sb-Te Alloys for Phase Change Memory, https://doi.org/10.1007/978-981-15-2217-8_7

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and comparing the thermal conductivity results with the reported data, it is approved that the hot-strip method can be applied to the thermal conductivity measurements of Ge–Sb–Te alloys. Chapter 3 “Thermal Conductivities of Ge–Sb–Te Alloys” Using the hot-strip method established in Chap. 2, the thermal conductivities of Sb– Te binary and Sb2 Te3 –GeTe pseudobinary alloys have been measured as functions of temperature and composition from room temperature up to below the respective melting temperature. The structures of Sb-rich Sb–Te single-phase alloys have shown similarity by the structure analysis as well as the thermal conductivities. The Te-rich Sb–Te alloys are two-phase alloys and the values of the thermal conductivity are close to those of Sb2 Te3 alloy. The Sb2 Te3 –GeTe pseudobinary alloys also show similarity on the structure and thermal conductivity. All the single-phase alloys in Ge–Sb–Te alloy system have an increase of the thermal conductivity above 600 K except GeTe alloy, the thermal conductivity of which decreases monotonically with increasing temperature. The thermal conductivities of Sb2 Te3 –GeTe pseudobinary alloys have also been measured as a function of time at 773 K to confirm whether the phase transformation occurs. The results show there is no change on the thermal conductivity with time. By analyzing the uncertainty of the measurements and comparing with the reported data, the thermal conductivity results are thought reliable and able to be used to explain the thermal conduction mechanisms. Chapter 4 “Electrical Resistivities of Ge–Sb–Te Alloys” The four-terminal method has been introduced for electrical resistivity measurements. The electrical resistivities of Sb2 Te3 –GeTe pseudobinary alloys have been measured as functions of temperature and composition from room temperature up to below the respective melting temperature. The electrical resistivities of all alloys increase with increasing temperature in the temperature range investigated. The electrical resistivity results of three ternary alloys show similarity as well as the structure. The measurements have also been carried out as a function of time at 773 K to confirm whether the phase transformation occurs. The results show the electrical resistivities of ternary Sb2 Te3 –GeTe pseudobinary alloys do not change while that of GeTe alloy increases slightly with time. This finding indicates that there are no phase transformation in the ternary alloys and the phase transformation of GeTe alloy progresses slowly even at high temperature. By analyzing the uncertainty of the measurements and comparing with the reported data, the electrical resistivity results are thought reliable and able to be used to explain the thermal conduction mechanisms. Chapter 5 “Thermal Conduction Mechanisms and Prediction Equations of Thermal Conductivity for Ge–Sb–Te Alloys” Based on the thermal conductivity and electrical resistivity data obtained in Chaps. 3 and 4, the thermal conduction mechanisms of Sb–Te binary and Sb2 Te3 – GeTe pseudobinary alloys have been discussed. Both the thermal conductivity and electrical resistivity of Sb2 Te3 –GeTe pseudobinary alloys show similarity, which is directly related to the structure similarity. According to the structure analysis in

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this work and the research in literature, the Sb2 Te3 –GeTe pseudobinary alloys can be considered as solid solution. Therefore, the thermal conduction mechanisms of Sb2 Te3 –GeTe pseudobinary alloys should be the same. The same case is for Sbrich Sb–Te alloys. The Wiedemann–Franz (WF) law has been used to predict the electrical thermal conductivity part using the electrical resistivity data. The results show that the free electrons contribute to the most part of the thermal conductivity and the phonon contribution can be neglected. Bipolar diffusion plays an important role at high temperature and accounts for the increase of the thermal conductivity. The exception can be seen in GeTe alloy, which shows metallic properties in the temperature range investigated. The prediction equations have been proposed for the thermal conductivities of Sb–Te binary and Sb2 Te3 –GeTe pseudobinary alloys. The comparison between the predicted data and experimental data proves that the prediction equations can be used for the industrial applications. Chapter 6 “Densities of Ge–Sb–Te Alloys” The sessile drop method has been introduced for density measurements. The density of Sb2 Te3 and Ge2 Sb2 Te5 alloys has been determined as functions of temperature from solid to liquid state. The densities have also been measured at room temperature by the Archimedean method. By comparing the results using these two methods and reported data, it is thought that the present results are reliable with about 2% uncertainty. The densities of solid Sb2 Te3 and Ge2 Sb2 Te5 alloys decrease linearly with increasing temperature. There is a discontinuous density decrease at the melting temperature due to the phase change. In the molten state, the density continues decreasing with increasing temperature. The volume increases by about 4% from the room temperature to just below the melting temperature and by 6% at the melting temperature from the solid to molten state for both alloys, which may cause large stress for PCRAM devices. Chapter 7 “Summary and Conclusions” The book has been summarized and concluded.