Enhancement of the Figure of Merit of Silicon Germanium Thin Films for Thermoelectric Applications
by Lo¨ısd’Abbadie
Thesis submitted as a requirement for the Degree of Doctor of Philosophy
School of Materials Science and Engineering
Submitted: March 31st,2013 Supervisor: Prof. Sean Li PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES TheslsiD~ s""'
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Silicon Germanium thin films are !he most stable lbermoelcctric materials at high temperatures. Nonetheless, low efficiency and limited knowledge of such structures are still a challenge to research. Understanding !be various mechanisms taking place in !be maner and !heir relationships is !he next step to boost research and to enhance !be TE efficiencies. widening the range of applications. in this thesis, we develop a malbematical approach based on solid Sl3lC theory to calculate !he figure of merit from an electronic band structure. Added to DFr, this near ab initio melbod rapidly assesses virtual structures as possible TE materials. l.n this report, !be melbod is applied to bulk siliron germanium ~>ith good agreement with experimental results. Throughout tbe development of this method, we also conclude !bat TE semiconductors band-gaps are related to the range of temperature where !be material show higher val ues of zr. We also show that doping with donors and acceptors. which is a common enhancement, need optimi zation for high temperatures applications due to its contribution to the lbermal conductivity. Moreover !he carriers' mobility is a prevalent parameter but its calculation remains complex. So, we implement deformation potentiallbeory with DFr to calculate the electron-phonon interactions in silicon germanium alloys. We find no change of interactions wilb the alloy romposition. With enough computing power, these methods are applicable to low dimensions structures.ln addition to our lbeoretical study, we report a sputter deposition melhod of silicon germa:nium alloy thin ftlms wilb rontroUed composition and lhickoess, grown on a sputtered layer of silicon dioxide. XRD study shows the appearance of a crystal phase beginning at deposition temperatures of 650 C . Reflectivity and TEM provide a consistent measurement of deposition thickness in !he range of 20 to 100 om wilb average interfaces roughmess around 2 nm.
Dec:lonl1lon _.,... to disposition of project tl>esisldlssertalion
I heteb)f SratJ1 to lhe University of New Soo1h Wales or its agents the right to 0/ehlve and to make available my thesl$ or diosertatlon in whole or In part in lbe UniYe Ialso au1florise UniYe FOR OFFICE USE ONLY Date of eornpleti0<1 of requirements fe< Award: THIS SHEET IS TO BE GLUED TO THE INSIDE FRONT COVER OF THE THESIS Originality Statement ’I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or sub- stantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and con- ception or in style, presentation and linguistic expression is acknowledged.’ Date Signature 31/03/2013 -i- Copyright Statement ’I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I have either used in substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.’ Date Signature 31/03/2013 Authenticity Statement ’I certify that the Library deposit digital copy is a direct equivalent of the final of- ficially approved version of my thesis. No emendation of content has occured and if there are any minor variations in formatting, they are the result of the conversion to digital format.’ Date Signature 31/03/2013 -ii- Acknowledgement First and foremost I offer my gratitude to my supervisor, Prof Sean Li, who has sup- ported me throughout my thesis with his knowledge whilst encouraging me during difficult parts of the work. His optimistic nature always sees the bright side of any situation, setting you to new interesting directions. Without his support, this thesis would not have been completed. One could not wish for a friendlier supervisor. I would like to thank my co-supervisor Dr Chucheng Yang for his meticulous ob- servation. I hope him well with his new research team. In the various laboratories, I have been aided by Dr Tan Thiam Teck (alias TT ) for his help and formation on most of the equipment, his insight in the management of my samples, his available knowledge and his patience. I also want to thank Dr Mohammad Hussein Naseef AL Assadi for his help and his corrections with DFT simulations. I also thank Dr Yu Wang from the solid state and elemental analysis unit and Dr Charlie Kong from the electron microscope unit for their help on the equipment and their advices on how to perfect measurements. I am grateful to Prof Charles Christopher Sorrell for his patience and his sometimes needed pushes. To Lana Strizhevsky who orientated me with patience through admin- istration procedures. To my friends from very diverse horizons whom with I shared some refreshing discussions during lunch breaks. Finally, I thank my parents, Dr G´erardd’Abbadie and Marie-Andr´eeXiste-d’Abbadie, for supporting me throughout all my studies and who always encouraged me to travel and see the world myself. I think of my two brothers and sister, Micha¨el,Aude and Luc whom I lived away from for too long. My uncle, Gilles Espitalier-No¨el, and aunty Marie-Paul Espitalier-No¨elfor their regards. I also thank my partner, Amandine Re- nard, for her patience and her support during all this time. -iii- Abstract Silicon Germanium thin films are the most stable thermoelectric materials at high temperatures. Nonetheless, low efficiency and limited knowledge of such structures are still a challenge to research. Understanding the various mechanisms taking place in the matter and their relationships is the next step to boost research and to enhance the TE efficiencies, widening the range of applications. In this thesis, we develop a mathematical approach based on solid state theory to calculate the figure of merit from an electronic band structure. Added to DFT, this near ab initio method rapidly assesses virtual structures as possible TE materials. In this report, the method is applied to bulk silicon germanium with good agreement with experimental results. Throughout the development of this method, we also conclude that TE semiconductors band-gaps are related to the range of temperature where the material show higher values of ZT . We also show that doping with donors and acceptors, which is a common enhancement, need optimization for high temperatures applications due to its contribution to the thermal conductivity. Moreover the carriers’ mobility is a prevalent parameter but its calculation remains complex. So, we implement deformation potential theory with DFT to calculate the electron-phonon interactions in silicon germanium alloys. We find no change of interactions with the alloy composition. With enough computing power, these methods are applicable to low dimensions structures. In addition to our theoretical study, we report a sputter deposition method of silicon germanium alloy thin films with controlled composition and thickness, grown on a sputtered layer of silicon dioxide. XRD study shows the appearance of a crystal phase beginning at deposition temperatures of 650 ◦C . Reflectivity and TEM provide a consistent measurement of deposition thickness in the range of 20 to 100 nm with average interfaces roughness around 2 nm. -iv- CONTENTS Originality Statement ...... i Copyright Statement ...... ii Authenticity Statement ...... ii Acknowledgement ...... iii Abstract ...... iv List of Figures ...... viii List of Tables ...... xii 1 Introduction 1 2 State of the art thermoelectric materials 6 2.1 Development of thermoelectrics ...... 6 2.2 Challenges and drawbacks of modern thermoelectric materials . . . . . 9 2.3 Improvement of thermoelectric efficiency in superlattices ...... 12 2.4 Fabrication techniques of superlattices ...... 16 2.5 Summary ...... 21 3 Physics of thermoelectric materials 22 3.1 Mathematical description of the thermoelectric phenomenon ...... 22 3.1.1 Seebeck, Peltier and Thomson coefficients ...... 22 3.1.2 Figure-of-Merit ZT ...... 24 3.1.3 Peltier Thermoelectric Couple ...... 26 v CONTENTS 3.2 Understanding the transport of heat in solids ...... 28 3.2.1 Debye theory of the lattice vibration ...... 28 3.2.2 Heat transport by phonons ...... 30 3.2.3 Heat conduction by electrons ...... 32 3.3 Summary ...... 34 4 Development of a Mathematical Approach for the enhancement of semiconductor based thermoelectric materials 35 4.1 Review of Solid-state theory and Boltzmann transport of Semiconductors 36 4.1.1 Description of the electronic carriers density in an unperturbed system ...... 36 4.1.2 Carriers density in a perturbed system: Boltzmann Transport Equation ...... 41 4.2 Thermopower and electronic thermal conductivity calculation for a per- turbed system and identification of limiting/enhancing parameters . . . 44 4.2.1 Electrical conductivity ...... 44 4.2.2 Seebeck coefficient ...... 46 4.2.3 Electronic thermal conductivity ...... 48 4.2.4 Discussion ...... 49 4.3 Calculation results and variation of the thermoelectric efficiency with temperature ...... 52 4.3.1 Calculated thermopower for bulk germanium and silicon . . . . 52 4.3.2 Calculated electronic thermal conductivity ...... 56 4.3.3 Resulting thermoelectric figure of merit for intrinsic germanium and silicon ...... 58 4.4 Summary ...... 61 5 Density Functional Theory applied to silicon/germanium structures 62 5.1 Density Functional Theory ...... 62 5.1.1 The quantum problem: the Schr¨odingerequation ...... 62 5.1.2 implementation of DFT within CASTEP ...... 68 5.2 Electronic description of silicon/germanium systems ...... 71 5.2.1 Method for converging calculations ...... 71 5.2.2 Pseudopotentials and schemes ...... 72 -vi- CONTENTS 5.2.3 Calculated Electronic structures ...... 73 5.3 Summary ...... 80 6 Phonon-electron interactions: Linear Deformation Potential Theory implemented in the DFT 81 6.1 Theory ...... 82 6.1.1 principle of the deformation potential theory applied to study electron-phonon scattering ...... 82 6.1.2 Form of the scattering probability ...... 83 6.2 Stress and strain relationships for deformation potentials calculations . 85 6.3 Calculated deformation potentials for silicon, germanium and silicon germanium alloys ...... 87 6.4 Summary ...... 91 7 Fabrication of Si/Ge heterostructures for Thermoelectric applications 92 7.1 Silicon germanium alloy on silicon Thin films sputtering ...... 93 7.1.1 RF Sputtering ...... 93 7.1.2 Deposition process ...... 94 7.2 Characterization of the heterostructure ...... 96 7.2.1 Study of the phase by X-ray diffraction ...... 96 7.2.2 Determination of thin films’ thickness by reflectivity ...... 100 7.2.3 Imaging of heterostructures with transmission electronic microscopy105 7.2.4 Thin film composition by energy dispersive X-ray spectrometry 109 7.2.5 Band gap engineering with heterostructures ...... 114 7.2.6 Atomic Forces Microscopy measurement of the surface roughness 118 7.3 Summary ...... 122 8 Conclusion 123 Appendices 140 A Fermi’s Golden Rule 141 -vii- LIST OF FIGURES 2.1 Picture of a Peltier device that can be bought online [1] ...... 7 2.2 Structure of a thermoelectric device found on the market [2] ...... 8 2.3 Figure of merit of typical thermoelectric materials according to their year of discovery [3] ...... 9 2.4 Number of published papers dealing with thermoelectric materials from 1955 to 2008 [3] ...... 10 2.5 Crystal structure of non-filled and filled skutterudites [4] ...... 11 2.6 TEM image of a MBE grown silicon germanium superlattice [5] and a schematic description of a superlattice ...... 12 2.7 Effect of quantisation upon the electronic density of state according to number directions of size reduction [6] ...... 13 2.8 Critical thickness (in nm) for a molecular beam epitaxial grown silicon germanium alloy layer on a silicon wafer. pseudomorphic layers are indicated by open circles and relaxed layers by full circles. squares are results from People et al [7] and Fukuda et al [8]. (source S.C. Jain et al [9]) ...... 15 2.9 Chemical Vapour Deposition process for the deposition of thin films [10] 16 2.10 Schematic figure of the Pulsed Laser Deposition method to grow thin films ...... 17 2.11 Molecular Beam Epitaxy reactor chamber ...... 18 viii List of Figures 2.12 Schematic representation of a RF sputtering chamber with the plasma on the surface of the targets and the substrate directly below ...... 19 3.1 figures of merit for the most studied bulk materials according to the temperature [11] ...... 25 3.2 Semiconductor based Peltier device in the cooling (left) and generator (right) configurations ...... 27 4.1 Electronic band structure of Silicon at 300K (source:www.ioffe.ru) . . . 38 4.2 Electronic band structure of Germanium at 300K (source:www.ioffe.ru) 38 4.3 Calculation of n and p in the a) intrinsic and b) extrinsic (n-type) case of semiconductors ...... 42 4.4 Calculated of the self-consistent fermi level of germanium for phospho- rous and boron doping for different concentrations ...... 50 4.5 Calculated of the self-consistent fermi level of silicon for phosphorous and boron doping for different concentrations ...... 51 4.6 Calculated electrical resistivity of intrinsic germanium and experimental values [12,13] ...... 53 4.7 Calculated electrical resistivity of intrinsic silicon ...... 54 4.8 Calculated Seebeck coefficient of intrinsic germanium, experimental data [14] and previous simulation [12] ...... 55 4.9 Calculated Seebeck coefficient of intrinsic silicon and experimental data [15] ...... 56 4.10 Thermal conductivity by the lattice and by the electronic carriers for germanium ...... 57 4.11 Thermal conductivity by the lattice and by the electronic carriers for silicon ...... 58 4.12 Thermoelectric figure of merit of intrinsic germanium ...... 59 4.13 Thermoelectric figure of merit of intrinsic silicon ...... 60 5.1 Flow chart of a Self-Consistent Functional (SCF) to calculate the density n0 related to the minimum energy E0 of the system ...... 67 5.2 a) conventional cell of a fcc structure b) first Brillouin zone for fcc . . . 70 5.3 5 5 5 k-point mesh of the first brillouin’s zone of an fcc structure . 71 × × -ix- List of Figures 5.4 Simulated electronic band structure of Silicon and its partial density of state using LDA with an Ultra-soft pseudopotential and Ecut−off = 300eV 75 5.5 Simulated electronic band structure of Germanium and its partial den- sity of state using LDA with an Ultra-soft pseudopotential and Ecut−off = 500eV ...... 76 5.6 Densities of State for a) Silicon b) Germanium ...... 78 5.7 Densities of State for c) Si0.8Ge0.2 and d) Si0.5Ge0.5 ...... 79 6.1 The two electrons-phonon scattering processes ...... 82 6.2 Calculated uniaxial shear and dilatation deformation potential of the ∆ valley of silicon germanium alloys according to its composition . . . . . 87 6.3 Calculated uniaxial shear and dilatation deformation deformation poten- tial of the L valley of silicon germanium alloys according to its composition 88 6.4 Calculated dilatation deformation potential of the L valley of silicon germanium alloys according to its composition ...... 89 6.5 Contribution of intervalley scattering for + parallel and perpendicular ◦ in L valleys, and parallel and perpendicular in ∆ valley according 4 ∗ to Herring’s calculations [16] ...... 90 7.1 X-ray diffraction between two planes of the same family ...... 96 7.2 XRD equipment settings for the 2θ measurement ...... 97 7.3 Grazing angles diffraction of samples 1, 2, 3 and 4 from table 7.2 from bottom to top respectively...... 98 7.4 Illustration of Snell’s law ...... 100 7.5 Reflected and transmitted X-rays in a layered structure ...... 101 7.6 Typical reflectivity measurement of a simple surface (left) and of a layer on a substrate (right) with a thickness of 2π/∆q ...... 101 7.7 Reflectivity for a SiO2 single layer on a silicon wafer ...... 102 7.8 Reflectivity for a Si on SiO2 layers on a silicon wafer ...... 103 7.9 Reflectivity measurement of Sample 3...... 104 7.10 Schematic figure of a TEM ...... 105 7.11 ”Lift-out” method to mill a TEM sample using the FIB ...... 106 7.12 Grain structure by TEM imaging of Sample 3 ...... 107 7.13 thickness of SiO2 and SixGe1−x thin films by TEM imaging of Sample 3 108 -x- List of Figures 7.14 Interaction volume of the various electron-sample interactions. The X- ray resolution is around 1 micron ...... 109 7.15 Location of the 5 points where the EDX spectra was measured . . . . . 110 7.16 EDX spectra of point ’lois 1’ which is the silicon wafer ...... 110 7.17 Composition of the SiO2 layer at point ’lois2’ ...... 111 7.18 Composition of the SixGe1−x layers at point ’lois 3’ ...... 111 7.19 Composition analysis n◦ 2 of Sample 3 along the arrow ...... 112 7.20 schematic settings of a spectrophotometer ...... 114 7.21 Tauc diagram for of Ge on SiO2 (blue), Ge on Si wafer (green) and Si on SiO2 (red). The measured band gaps are, respectively, 0.74eV , 1.02eV and 1.04eV ...... 115 7.22 Tauc diagram for of Sample 2 (blue), Sample 3 (red) and Sample 4 (green). The measured band gap are, respectively, 1.04eV , 1.044eV and 1.014eV ...... 116 7.23 Cantilever tip for atomic force microscopy applications ...... 118 7.24 Measured surface roughness with atomic force microscopy of a bare sili- con wafer, of low and high temperature sputtered germanium layer and sample 3 and 4 described in table 7.2 ...... 119 7.25 AFM scan of a 2 by 2 µm area of sputtered germanium layer deposited at 650 ◦C ...... 119 7.26 AFM scan of a silicon dioxide layer deposited on a silicon wafer . . . . 120 7.27 AFM 3D image of a scan of a silicon layer deposited on a silicon dioxide layer ...... 121 7.28 AFM scan of a silicon layer deposited on a silicon dioxide layer . . . . . 122 -xi- LIST OF TABLES 5.1 Calculated parameters for Silicon with DFT ...... 73 5.2 Calculated parameters for Germanium using DFT. First norm-conserving pseudopotential includes 3d in core whereas the second norm-conserving pseudopotential take 3d electrons into account...... 74 6.1 Relationships between the shift of the band edge energy and the defor- mation potentials for a given valley, depending on the stress direction applied on the structure [17] ...... 86 7.1 Measured deposition rates obtained with sputtering ...... 93 7.2 Sputtering parameters for the deposition of SixGe1−x layers on SiO2 with power fixed at 100 W and vacuum at 3 mT orr ...... 95 xii CHAPTER 1 INTRODUCTION Thermoelectric materials convert a difference of temperature into electricity, and re- ciprocally generate cold from an electric current. Applications of such a device are numerous. In a time when global warming leads to the rise in global temperatures, the rise of sea levels and the gradual disappearance of the snow cover, menacing food supplies, agriculture and human health, modern societies must head toward the use of greener energies. Amongst the panel of available renewable energies, thermoelectric materials propose an interesting combination of sustainable energy and promising tool for new technologies. For example, thermoelectric devices implemented in a car can use the engine waste heat to power the car’s electronics. Or coupled with a solar cell, thermo- electric materials would complete the photocells by generating energy from the infrared spectra [18]. But used as cooling devices in a computer, thermoelectric materials can improve the computing power by 200%. Another application of thermoelectric mate- rials in new technologies is illustrated by the thermoelectric generators for probes in airspace programs. All in all, thermoelectric materials is a green new technology and its challenge emulates research in new directions. The challenge that faces thermoelectric research is to increase the efficiency of these devices. Since the discovery of the thermoelectric effect by Thomas Seebeck in 1821, efficiencies remained low for a long time, and thermoelectric applications remained 1 Chapter 1. Introduction marginals. Until 1995, these efficiencies, described by the more suited dimensionless figure of merit ZT , did not exceed the limit value of ZT = 1. But with technology progress, and the increasing interest for nanotechnologies, these efficiencies reach ZT = 3.5 in P bSeT e/P bT e quantum dots. Today, in a commercial thermoelectric device, state-of-the-art materials are bulk bi- nary semiconductors alloys, Bi2T e3, P bT e and SiGe which exhibit ZT values around 1. With the interest in nanotechnology, research found ZT values up to 2.4 in su- perlattices, quantum dots or nanocrystalline inclusion. New materials such as filled skutterudites have ZT equal to 1.4. Although Silicon germanium based thermoelectric materials do not have the highest values of ZT , owing to the stability of their structure, they are a serious contenders for high temperatures applications. The aim of this thesis is the study of efficiencies in low dimension silicon germanium thermoelectric materials, and the ability to predict any enhancement of the figure of merit. The underlying goal was to predict and theoretically explain, from a virtual structure made of semiconductors the possible improvement of thermoelectric devices. The increase of interest in new technology is following by a drastic improvement of computer simulations of new structures. These ab initio simulation are a gain in time and money for research teams as they can quickly verify the relevance of a new design. Furthermore, it is even more significant in the case of thermoelectric materials as the parameters and phenomena behind this technology are numerous and complex. Indeed, the thermoelectric research field deals with both electronic properties and thermal properties, know that there are some relationships between those. For this reason, we went in details in the semiconductor theory to link the density functionals calculations (DFT) directly to the figure of merit. And we hoped to compare the calculated results to the case of thin films. With the possibilities offered by DFT simulation for nanoscale structure, being able to straightforward calculate the figure of merit could boost the research in the field of thermoelectric devices In this thesis, we particularly focus on identifying the parameters that control the thermoelectric effect bearing in mind the enhancement of the figure of merit in silicon germanium structures. Especially in silicon germanium alloy thin films. These identified parameters are obviously the band gap, the carriers density, the fermi level related to doping and, above all, the mobility of electronic carriers. We start with the -2- Chapter 1. Introduction understanding of the electronic transport in bulk materials and take it to the size of thin films. In addition, with the prospect of predicting physical properties of a virtual structure, we used density functional theory (DFT) to simulate silicon germanium structures and calculate their figure of merit. Combined with DFT, we also apply the deformation potential theory to virtual structures in order to have a hint about possible electronic carriers’ mobility enhancement. Furthermore, we epitaxially grow some silicon germanium alloys thin film using sputtering. With a target alternating method for very short period of time, and with controlled parameters, we are able to grow different compositions of alloy with thick- nesses around 60 nm. -3- Chapter 1. Introduction Overview In chapter 2, we review the literature to explain and pinpoint the state-of-the-art re- search in the field of thermoelectric materials. We introduce the history of the thermo- electric effect, highlight the drawbacks and the challenges presented by today’s devices. Finally, we introduce the technology of thin films superlattices and their advantages for thermoelectric applications. We explain the different methods of fabrication of silicon germanium thin films. In chapter 3, we sum up the relevant theory about thermoelectric materials to understand and identify the possible enhancement of silicon germanium based ther- moelectric structures. We start with the three laws of thermoelectricity discovered by Seebeck, Peltier and Thomson. Then we explicit the notion of figure of merit which represents the thermoelectric efficiency of a material. We describe the most basic TE device: the Peltier device and its cooling and generator settings. And we review the theory of heat transport in solids which must be limited for TE improvements. In chapter 4, we develop a mathematical tool based on solid state theory to calcu- late the figure of merit of a semiconductor based thermoelectric material. We review the solid-state theory concerning semiconductors and calculate the electronic carriers’ density in a perturbed semiconductor using Boltzmann transport equation. Based on the DFT results from the following chapter, we calculate the figures of merit of silicon and germanium between 100 and 1000 K. In chapter 5, we simulate the electronic properties of silicon and germanium bulk as well as silicon germanium alloys. The results from these simulation are then used to calculate the figure of merit. This calculation is done in the previous chapter. In chapter 6, we apply the deformation potential theory to virtual crystal structures’ calculation using DFT. This aim is to calculate the electron-phonon interactions in alloys. We start to explain the theory developed by Herring and Vogt in 1956 [16] and we implement these results to DFT in order to obtain an ab initio calculation of the electronic mobilities. -4- Chapter 1. Introduction In chapter 7, we describe the silicon germanium alloy thin film deposition by sput- tering. We relate each deposition parameter to the characterization of the structure for a controlled deposition. The characterization deals with the composition, the film’s thickness, the crystal structure, the band gap and the interfaces roughness. This is done with X-ray diffraction, reflectivity, TEM imaging, EDX spectrum, UV/vis/NIR spectrophotometry and AFM scanning. -5- CHAPTER 2 STATE OF THE ART THERMOELECTRIC MATERIALS 2.1 Development of thermoelectrics In 1821, Thomas Seebeck discovered that a junction of different metals at different tem- peratures induces an electric current able to deflect a compass magnet [19]. Known as the Seebeck effect, the temperature difference at the junction of the two metals cre- ates an electric potential which conducts a current through a circuit. This potential is proportional to the temperature difference and this constant is called the seebeck coef- ficient. Thomas Seebeck’s discovery is the first hint of the thermoelectric phenomenon. In 1910, by characterizing the electrical conductivities and the seebeck coefficients of tellurium, antimony and bismuth alloys and copper-nickel alloys, Werner Haken de- fined the composition of good thermoelectric alloys in the quest of the thermoelectric material research [20]. In 1834, the French physicist Jean Charles Peltier observed the heating and the cooling at a junction of two dissimilar metals when an electric current was applied. Either cooling or heating was produced depending on the direction of the current flow. The heat absorbed or produced at the junction was proportional to the electric current and the constant is known as the Peltier coefficient. 6 Chapter 2. State of the art thermoelectric materials In 1851, William Thomson explained the Seebeck and the Peltier coefficient and their relationship as these two coefficients are linked to the same phenomenon [21]. The Seebeck effect and Peltier effect being thermodynamically linked, W. Thomson’s work on the thermodynamic theory enabled him to develop the third law of thermoelectrics. His theory states that the heat produced or absorbed when a current flows through a material with a temperature gradient is related to the Seebeck coefficient. Further work on these constant led to the definition of the efficiency called ther- moelectric figure of merit ZT . It shows that a good thermoelectric material must possess a high seebeck coefficient and a high electrical conductivity and a low thermal conductivity. Although the thermoelectric field of research was very active between the two world wars with applications for both military and civilian uses, the low efficiency and the argued upper limit of the figure of merit ZT = 1 made the research slow down after the 50’s. Figure 2.1: Picture of a Peltier device that can be bought online [1] In 1949, Abram Fedorovich Ioffe developed the modern theory of thermoelectric based of the figure of merit [22]. With the increase of interest for semiconductors for a century, Ioffe focused his study of thermoelectrics on semiconductors, concluding -7- Chapter 2. State of the art thermoelectric materials that high figures of merit were achieved with heavily doped semiconductors. He also worked with the reduction of the lattice thermal conductivity by point defects in alloys. Julian Goldsmith in 1954 stressed the significance of high mobility and low effective mass combined with low thermal conductivity in doped semiconductors. Later, Glen Slack introduced the ”phonon-glass, electron-crystal” approach for researching new high figure of merit thermoelectric materials [23]. Nowadays, many applications of thermoelectrics relies on doped semiconductor alloys and are mostly found in small cooling and power generation devices (cf figure 2.2 and 2.1). Figure 2.2: Structure of a thermoelectric device found on the market [2] In the past 15 years, with the rise of nanotechnologies and the need for efficient de- vices for electronic cooling and power generation, the field of thermoelectric materials is again pushed forward. New structures are investigated, and many complex ther- moelectric materials show promising efficiencies. From bulk alloys, research focuses -8- Chapter 2. State of the art thermoelectric materials on smaller structures with enhanced physical properties such as thin-film superlat- tices, nanocomposites, complex inorganic structures, crystal structures with rattlers and oxide thermoelectrics. 2.2 Challenges and drawbacks of modern thermo- electric materials Figure 2.3: Figure of merit of typical thermoelectric materials according to their year of discovery [3] With the burning issue of global warming and international tensions due to fos- sil fuel, today’s energy challenge relies on emerging renewable energies such as solar, hydroelectric and wind energies. Although thermoelectric efficiencies are still low, ther- moelectric devices nonetheless participate to the race toward sustainable green energies. For instance, thermoelectrics coupled to solar cell panels generate energy from infrared radiations which represents 48% of the solar energy which is lost by conventional solar panel [18]. New applications of thermoelectrics emerge such as biothermal batteries that would power pacemakers. For computers, optoelectronics, infrared detectors and electronics, localised thermoelectric cooling greatly enhances efficiencies. For example, cold computing gains speed up to 200% with some CMOS processors when cooled with a Peltier device. In the automobile industry, thermoelectric devices can capture en- -9- Chapter 2. State of the art thermoelectric materials gine’s waste heat to power internal electric circuits. In the airspace field, radioactive thermoelectric generators are used in deep-space missions like Voyager and Cassini. Despite the fact that these examples are very specifics, some TE devices are commonly found on the market in cooling devices such as seats coolers, cooling ice box, computer coolers. Peltier devices can be bought in electronic shops for any application the buyer wants [1]. Figure 2.4: Number of published papers dealing with thermoelectric materials from 1955 to 2008 [3] To reach a wider range of applications, research focuses on the main challenge of TE: the efficiency. For example, if TE devices could reach a zT value of 4, they would replace the classical compressor cooling systems in refrigerators by thermoelectric. Which means make a silent refrigerator with no use of the hazardous fluorocarbon refrigerants. This search for higher efficiencies is well illustrated by figure 2.3 which shows a slow increase of the discovered efficiencies up to the seventies due to the work on binary alloys, followed by a stagnation period until the mid-nineties and a sudden increase of the TE figure of merit with the use of more complex alloys and more complex structures, facilitated by the emerging nanotechnology. The trend of the TE efficiency can be explained by the interest gained by the thermoelectric field (cf figure 2.4). Between 50’s and the 70’s the interest of researchers increased but remained constant from the 70’s to the mid-90’s. Since 1995 to today, motivated by the new technologies, the interest grew drastically and the efficiencies increase. The main challenge of modern thermoelectric materials is still the efficiency. To increase the figure of merit, researcher often focus on the ”electron-crystal, phonon- -10- Chapter 2. State of the art thermoelectric materials Figure 2.5: Crystal structure of non-filled and filled skutterudites [4] glass” approach which states that efficiency enhancement is achieved by improving the electronic conductivity and by limiting the thermal conductivity [2, 11, 23, 24]. Semiconductors hence became promising materials thanks to the possible doping to increase the electronic properties at low to medium temperatures (higher temperatures with heavy doping), and alloying which in the case of silicon-germanium reduces the −1 −1 lattice thermal conductivity by a factor 10: κl = 0.0628W cm K for Si0.8Ge0.2 −1 −1 −1 −1 whereas κl = 1.3W cm K for Si and κl = 0.58W cm K for Ge at 300 K [25]. For thermoelectric applications, the optimized composition of silicon germanium alloy is 80% silicon and 20% germanium [6, 25]. Another possible reduction of the lattice thermal conductivity is achieved by using rattlers and point defects to scatter phonons [26]. Interfaces in thin-film superlattices are also used to scatter phonons without impeding too much the motion of electronic carriers [6, 27–31]. The induced strain at the interfaces of thin film was found to enhance the electronic carriers’ mobility [32–34]. Skutterudite materials are composed of a rare earth metal, a transition metal and a metalloid and they are cubic with space group Im3 which contains voids (cf. figure 2.5). These voids can be filled with low coordination atoms which scatter phonons, hence reducing the electronic conductivity, without reducing the thermal conductivity. Sales et al. found a zT value of 1.4 for filled skutterudites [35] while reduced thermal conductivities were reported in superlattices of skutterudite [36, 37]. The challenge in increasing the efficiency by reducing the lattice thermal conductivity is to limit the effect on the electronic conduction. -11- Chapter 2. State of the art thermoelectric materials 2.3 Improvement of thermoelectric efficiency in su- perlattices Benefits of superlattices A superlattice is a periodic structure of either thin films, nano wires or quantum dots of two or more materials (cf figure 2.6). Their thicknesses are typically around several nanometres and show effects of lower dimensional systems. There are three types of enhancement in superlattices. electronic quantisation due to size reduction • phonon scattering at interfaces • interfaces’ quality shows either strain or stacking defaults and dislocations • Figure 2.6: TEM image of a MBE grown silicon germanium superlattice [5] and a schematic description of a superlattice In quantum physics, the reduction of the dimensions of a conducting system leads to the quantisation of the electronic states. The quantisation of the electronic state changes the form of the electronic density state (cf. figure 2.7). Depending on the level of quantisation, electrons become more localised in the conduction band and so do holes in the valence band. This effect is very interesting particularly in tuneable semiconductors as it greatly enhance the electronic properties. -12- Chapter 2. State of the art thermoelectric materials Moreover, interfaces in a superlattice also bring a barrier against phonon propa- gation in the cross plane direction, hence lowering the thermal conductivity in this direction. Interfaces, depending on the type of growth can either induce strain or, if the strain is relieved, show dislocations and stacking default. In the first case, strain is known to enhance carriers mobilities while in the second case; defaults of the structure would scatter phonons, decreasing the thermal conductivity. Quantisation is a good method of increasing the electronic efficiency of materials. Indeed, reducing the size of a device to the characteristics electronic length (below hundreds of nanometre) leads to a quantum confinement of the carrier and hence a dif- ferent density of state than the one of the bulk. Figure 2.7 shows the different electronic density of states obtained according to the number of direction of size quantisation, form none (bulk material) to all 3 directions (quantum dot) [6]. Figure 2.7: Effect of quantisation upon the electronic density of state according to number directions of size reduction [6] Quantum wells structures have been widely studied and interesting devices based on semiconductor superlattices have been developed [6, 38–46]. The main aim of the use of thermoelectric superlattices is to enhance the electronic properties thanks to the specific form of the density of state in the plane of the superlattices. The problem comes from the heat conduction along the interfaces between successive layers. This increased thermal conduction limits the increase of the figure of merit. Another idea is to consider the conduction through the superlattices in the cross plane direction. In this direction, the interface would reflect the phonons while letting the electrons goes through. This method consists of decreasing the thermal conduction while trying to keep a good thermopower. To explain these two methods, scientists tend to focus either on the particle properties of the heat conduction or the wave -13- Chapter 2. State of the art thermoelectric materials aspect of the lattice vibration. The wave aspect of the heat conduction assumes a totally coherent regime and considers only the harmonic force interaction. According to this model, the reduction of the thermal conductivity of the lattice comes from the spectrum change in the superlattice that lowers the phonon group velocity. It predicts to lower the thermal conductivity by a factor 10 in Si-Ge superlattices. It also shows that, due to the tunnelling effect, when the different layers are less than 3-atomic layer thick, the thermal conductivity is recovered [47]. This theory based on the harmonic nature of the interactions in the crystal does not satisfy experimental results. The incoherent model based on Boltzmann equation uses the relaxation time in bulk material to understand the phonon reflection and transmission at interfaces. Short wavelength phonons increase interface scattering, drastically reducing the group velocity. This modelling satisfies the experimental results. It then can be concluded that in the in-plane conduction, diffuse interface scattering is responsible of the thermal conductivity reduction, and in the cross plane conduction the phonon reflection, the confinement and the diffuse scattering lowers the thermal conductivity [47]. G.Chen et al. conclude that diffuse interface scattering reduces the mean free path of phonons for the in plane conduction and reduces the phase coherence for the cross-plane conduction [45, 48, 49]. Consequently, silicon-germanium superlattices were reported to show a larger reduction of the thermal conductivity compared with that of an alloy in the cross plane direction [38, 50, 51]. Calculated figure of merit of 0.96 for a silicon-germanium superlattice was reported in 1999 [34], while Yang et al. showed a silicon germanium superlattice with a zT 8 times higher than the one of bulk material [52]. In Bismuth chalcogenides, p-type Bi2T e3/Sb2T e3 superlattices exhibits a zT of 2.4 at 300K while bulk materials only reach zT = 1 [53] and molecular beam epitaxial (MBE) grown non-optimized intrinsic P bSeT e-based quantum dot superlattice show a zT of 2 At room temperature (cf. figure 2.3) [54]. Silicon germanium based superlattices Silicon germanium thin film superlattices have been studied since early 70’s but suc- cessful growth were made in the mid 80’s [7, 55, 56]. The challenge to grow Si-Ge thin film superlattices comes from the lattice mismatch of 4% which induce disloca- tions, stacking faults and cracks in the layers [57]. Bean et al proved that high quality SixGe1−x layers can be grown provided that the growth deposition remains in the range -14- Chapter 2. State of the art thermoelectric materials 550-750 ◦C and that the layer’s thickness remains smaller than a critical thickness. Figure 2.8: Critical thickness (in nm) for a molecular beam epitaxial grown silicon germanium alloy layer on a silicon wafer. pseudomorphic layers are indicated by open circles and relaxed layers by full circles. squares are results from People et al [7] and Fukuda et al [8]. (source S.C. Jain et al [9]) Figure 2.8 shows the experimental critical thickness of a SixGe1−x layer grown by molecular beam epitaxial (MBE) on a silicon wafer according to the content of germa- nium [7, 8]. These results show a critical thickness of 50nm for pseudomorphic layers and 100 nm for relaxed layers for a Si0.8Ge0.2. These values decrease for a higher con- tent of germanium. Below this critical thickness, the interface counts few dislocations whereas above the critical value, dislocations and stacking defaults appear. -15- Chapter 2. State of the art thermoelectric materials 2.4 Fabrication techniques of superlattices With emerging new technologies focusing on fabrication and characterization of nano- sized materials, different possibilities for thin film superlattice deposition are available. Chemical Vapour Deposition Chemical Vapour Deposition (CVD) is amongst the simplest fabrication process. It consists in exposing a substrate to volatile precursors, which react on the surface of the substrate to create the desired structures. For silicon and germanium deposition, hydrides precursors are used: SiH4 and GeH4. For the deposition of an alloy layer, both precursors are injected in the reaction chamber. The gas flow in the chamber pushes the precursors on the deposition area (usually a substrate) which is held at a specific temperature. The particles deposit on the substrate, then desorption of the precursor takes place: silicon and germanium remain on the substrate to participate to the film growth whereas H atoms, which become volatile by-products are removed into the gas flow (see figure 2.9). Figure 2.9: Chemical Vapour Deposition process for the deposition of thin films [10] There are many types of CVD. In the case of silicon-germanium deposition, Low- Pressure CVD (LPCVD) can be used to reduce gas phase reactions and improve film -16- Chapter 2. State of the art thermoelectric materials uniformity . For silicon, germanium and silicon germanium alloys, pressure in the range of 10-100 Torr are used while the substrate is held at temperatures between 550 ◦C to 750 ◦C [58, 59]. Another type of CVD is the low temperature Metal Or- ganic CVD (MOCVD) which uses metalorganic precursors. Metal organic compounds contain metal with organic ligands. Venkatasubramanian et al reported superlattices depositions of Bi2T e3/Sb2T e3 with thicknesses 10 A˚ and 50A˚ respectively [53, 60]. These depositions were made at temperatures not exceeding 300 ◦C. Pulsed Laser Deposition Pulsed Laser Deposition (PLD) uses high power laser pulses to evaporate atoms or particles from a target. This evaporation of particles produces a plasma plume which expands rapidly away from the target. A substrate is place on the way of this plasma plume to collect the evaporated particles. These particles condense on the substrate surface, participating to the nucleation and film growth. Figure 2.10: Schematic figure of the Pulsed Laser Deposition method to grow thin films In this method of deposition, many variables affect the deposition of the thin film: the substrate temperature, the laser power, the vacuum and the properties of the thin film. Compared to other methods, PLD is known for its ability to respect the stoichiometric composition and for its high depositions rates. In the quest of silicon and germanium superlattice deposition Kramer et al used a XeCl excimer laser to grow a SiGe/Si superlattice under a 70 mTorr vacuum in 1992 [61]. For the deposition of -17- Chapter 2. State of the art thermoelectric materials silicon germanium alloy layers, silicon and germanium targets are alternated [62, 63]. This kind of deposition is done at pressure below 1 Torr [64]. Molecular Beam Epitaxy Molecular Beam Epitaxy (MBE) consists in thermally evaporating target material and project these evaporated particles in the form of molecular beam toward a heated crystalline substrate to grow epitaxial films. The high purity of deposited films strongly depends on the purity of the target material and on the ultra-high vacuum of the reactor chamber (cf figure 2.11). The advantage of the method comes from the low growth rate of deposition and the use of shutters on the effusion cells which enable a very accurate control of the deposition, hence of the thickness and the composition of the deposited thin films. In the deposition chamber, a reflection high-energy electron diffraction (RHEED) gun helps monitoring the deposition at a monolayer scale. Figure 2.11: Molecular Beam Epitaxy reactor chamber With the accuracy provided by the MBE, under the condition that deposited layers’ thickness remain under a critical value, deposition of pure silicon on pure germanium superlattices are achieved with a period no bigger than 10 A˚ [52]. Thanks to the iterative possibility of the process, 150 periods of 33 A˚ of silicon and germanium were -18- Chapter 2. State of the art thermoelectric materials reported in 1999 [51] and 100 periods of 20 A˚ silicon and germanium layers [65]. The Molecular Beam Epitaxy is the most accurate method for superlattice deposition but it is also the most complex and expensive equipment to run. sputtering deposition Figure 2.12: Schematic representation of a RF sputtering chamber with the plasma on the surface of the targets and the substrate directly below Radio Frequency (RF) magnetron sputtering is a common tool when depositing metal or ceramic material layers. The deposition takes place under vacuum with the help of a plasma (figure 2.12). Prior to any deposition, the reaction chamber is filled with either O2 or Ar gas or both. Then, using a negative voltage between the reactor chamber and the target holder, a cold plasma made of ions from the gas, electrons, photons and neutrons is formed on the surface of the targets. These ions and electrons in the plasma are accelerated toward the target. They hit the target which tear particles off the target. Then these particles deposit on the substrate. The substrate holder is -19- Chapter 2. State of the art thermoelectric materials rotating to obtain a uniform film. Shutters can be opened and closed in front of each target while the plasmas remain turned on. This enable alternated deposition of two or more materials. These types of depositions depends on the pressure of the chamber, on the composition and partial pressure of the gas reactants, on the RF power applied for the creation of the plasma, the time of deposition and the temperature of the substrate. For silicon and germanium deposition, this type of procedure produces amorphous thin films for substrate temperatures lower than 500 ◦C [66–68]. To obtain crystal layers, the deposited structures need annealing at temperatures above 550 ◦C . Jelenkovic et al anneal their silicon germanium alloy layer between 550 and 570 ◦C for 5 to 120 hours after deposition [69]. For silicon, germanium or silicon germanium alloy deposition, only argon gas is injected in the chamber to create a non-interacting plasma [68–72]. The partial pressure of argon in this case is usually set between 0.2 to 0.9 Pa [67]. For the deposition of an insulating SiO2 layer before any other deposition, O2 gas is injected in the chamber so the oxygen ions from the plasma can react with particles torn away from the silicon target to deposit SiO2 particles on the substrate [69,72,73]. In this case, the partial pressure ratio of O2 by Ar is 1/4. The inside pressure of the chamber is of the order of 5 mTorr. The deposition rates obtained with this technique go from 1 to 10 nm.min−1 [68] for a target power density in the 10 to 30 W.cm−2 range [74]. -20- Chapter 2. State of the art thermoelectric materials 2.5 Summary Thermoelectric materials is the only solid state device that exhibit heat and electricity reversible conversion. Although the phenomenon has been discovered nearly two cen- turies ago, the efficiencies of such technology remained limited until 1995. Of course, with low efficiencies, thermoelectric devices were applied in only specific cases such as airspace powering or small cooling devices. Only specific applications and niche market are concerned by these devices. However, once the challenge of low efficiency overcome, thermoelectric devices potential applications would be very significant on very diverse big market, such as car industry or green energies. Adding this to the emerging nanotechnology, this is the reason that motivates research to push dimension boundaries in devices or try completely new materials. And the motivation of research show some results since 1995 with values of the figure of merit up to 3.5 while before 1995, it was though that the upper limit was 1. In this thesis, we focus on the possibility of increasing the TE efficiency in silicon germanium thin film superlattices. They are not completely new materials, their first successful deposition was thirty years ago. But the underlying TE phenomena are still studied and research would benefit from understanding the TE mechanisms in these type of materials. -21- CHAPTER 3 PHYSICS OF THERMOELECTRIC MATERIALS 3.1 Mathematical description of the thermoelectric phenomenon 3.1.1 Seebeck, Peltier and Thomson coefficients The first sign of thermoelectricity discovered is the Seebeck effect. The Seebeck effect is the creation of a voltage ∆V in a junction of two materials a and b have a difference of temperature ∆T . This effect is measured by the seebeck coefficient αab and is expressed as follow: ∆V α = (3.1) ab ∆T The Seebeck coefficient is related to the intrinsic properties of the materials. It is very low for metals, around few µV/K, and is much larger for semiconductors, around several hundred µV/K [18]. Related to the Seebeck effect, the Peltier coefficient Πab is linked to the rate of 22 Chapter 3. Physics of thermoelectric materials heat Q exchanged at the junction of two dissimilar materials. If a current I is set through this junction, heat is absorbed or rejected at the junction depending in which direction the current travel. This phenomenon is mostly due to the difference in the Fermi energies level between the two materials that form the junction. Q = Πab.I (3.2) The relationship between the Seebeck and the Peltier coefficient is: Πab = αab.T (3.3) The Thomson effect is the heat absorption or generation of a conductor that un- dergoes a gradient of temperature, except for superconductors, when a current flows through it. If we consider the spatial coordinate x in the direction of the current flow, the Thomson coefficient is related to the gradient of the heat flux Q as following: dQ dT = τ.I (3.4) dx dx While the thermoelectric power generation α and the thermoelectric cooling Π require joining two different materials, the Thomson coefficient is relative to one con- ductor. In order to define absolute α and Π for a single material, measurement of the seebeck coefficient of a junction of P b and a superconductor set P b as a reference material. Consequently, the following relationship between the Seebeck, the Peltier and the Thomson coefficients are true for a single material: Π = αT (3.5) dα τ = T (3.6) dT The thermoelectric efficiency is limited by two main irreversible processes; the ther- mal conduction and the joule heating. In any material, a temperature gradient leads to -23- Chapter 3. Physics of thermoelectric materials an irreversible flow of heat in the opposite direction of the gradient. The joule heating is proportional to the square of the electric current while the Peltier effect is linear in current. Consequently, interesting thermoelectric materials must have a large Seebeck coefficient α, a low electrical resistivity ρ and low thermal conductivity κ. 3.1.2 Figure-of-Merit ZT The figure of merit ZT determines the efficiency of a material for thermoelectric ap- plications. ZT is a dimensionless variable. For a single material, the figure of merit ZT is defined by: α2σT α2T ZT = = (3.7) κ ρκ Where α: Seebeck coefficient ρ: Electrical resistivity σ: Electrical conductivity κ: Total thermal conductivity The thermoelectric power factor is defined as α2σT . Hence a TE material opti- mization relies on the increase of the power factor. Consequently, materials that fit for thermoelectric use must have a large electrical conductivity and a small thermal conductivity. For instance, narrow band gap semiconductors, doped to control the carrier concentration, show high figures of merit. Thermal conductivity is caused by two means: the lattice vibration (phonons) and the flow of electrons (which carries entropy). The most significant contribution comes from the lattice vibration. A high carrier concentration increases the electric conductivity but it also increases the elec- tronic thermal conductivity. So doping needs optimization. A solution would be to increase the mobility of the carrier to improve the electric conductivity with limiting the increase the thermal conductivity. A thermocouple is the association of two materials with opposed nature of carrier. There is an n-type material and a p-type material. The resulting figure of merit for such a couple is: 2 (αp αn) T ZT = 1/2− 1/2 2 (3.8) [(ρnκn) + (ρpκp) ] -24- Chapter 3. Physics of thermoelectric materials Figure 3.1: figures of merit for the most studied bulk materials according to the tem- perature [11] The efficiency ν of the thermocouple is given by the power output W over the input net heat flow Q (in the case of the power generation mode of the thermocouple). The efficiency is related to the figure of merit ZT : 1/2 W TH TC (1 + ZTM ) 1 ν = = − 1/2 − (3.9) Q TH (1 + ZTM ) + (TC /TH ) Where -25- Chapter 3. Physics of thermoelectric materials TH : hot side temperature TM : average temperature TC : cold side temperature In the equation of the efficiency above, if ZT tends to infinity, we get: TH TC lim = − (3.10) ZT →∞ TH This limit is the Carnot efficiency related to the Carnot cycle. The Carnot cycle is the hypothetic cycle which is reversible. This cycle shows the highest efficiency because of its lack of losses. Consequently, high values of ZT means that the efficiency of the device gets closer to the upper limit that is the Carnot efficiency. The Carnot efficiency is a reference for all the heating or cooling systems. A thermoelectric with a related ZT equal to 1 has an efficiency of 10% of the Carnot efficiency. To compare thermoelectrics with other commonly used systems, a home refrigeration is 30% of Carnot efficiency and the largest air conditioners for big building can reach 90%. Even if thermoelectrics devices have other applications than these examples, research is still working on increasing the figure of merit. An important aim would be to reach the efficiency of a home refrigeration, which means to obtain a ZT around 4 [75]. 3.1.3 Peltier Thermoelectric Couple To understand the notions introduced above, the Peltier thermoelectric couple is a device that explains the different phenomenon clearly. This thermoelectric couple is based the conducting connection of an n-type material and a p-type material. The couple can be set in two modes; the refrigeration mode and the power generation mode. In the first mode, a current flowing through the device induces a heat flow and the creation of a gradient of temperature. In the second mode, the power generation mode, a difference of temperature set a current in the device. In the refrigeration mode (figure 3.2(a)), the current move the carriers accordingly to their nature. Electrons move against the current and hole with the current. Both types of particles carry an electric charge and some heat. Consequently, both types of carriers are heading to the same side of the device bringing the heat on the same side -26- Chapter 3. Physics of thermoelectric materials Figure 3.2: Semiconductor based Peltier device in the cooling (left) and generator (right) configurations of the device. This heat flow removes heat from the metal connection: it is the active cooling. In the power generation mode (figure 3.2(b)), the metal connection between the two branches is set close to a heat source that induces the motion of the carrier toward a heat sink. This carrier flow creates a current in the device. In this Peltier cooling system, it is clearly seen that the cooling phenomenon and the joule heating are antagonists in the search of efficiency. The Peltier cooling is proportional to the Current whereas the joule heating is proportional to the current squared. Consequently a thermoelectric couple works at a specific optimal current which depends on the intrinsic properties of the thermoelectric materials. -27- Chapter 3. Physics of thermoelectric materials 3.2 Understanding the transport of heat in solids 3.2.1 Debye theory of the lattice vibration Debye interpolation at low temperature In the definition of the figure of merit, the total thermal conductivity is the sum of two components. First, the electronic thermal conductivity relies on the transport of the heat by the electronic carriers in a material. The second component is the transport of the heat by the vibration of the lattice in a crystal structure. Improving the efficiency of thermoelectrics implies either increasing the power factor or reducing the total thermal conductivity and improving the electronic conduction. Consequently, a good thermo- electric must have a low thermal conductivity. But the electronic contribution to the thermal conductivity is complex to reduce without reducing the electronic efficiency: one solution would be to decrease the density of electronic carriers and increase their mobility. Thus the main reduction of heat conduction is made on the contribution of the lattice vibration. In a crystal, the thermal conductivity of the lattice is defined by the vibration of the lattice that conducts elastic waves. Debye determined the specific heat capacity of a solid at low temperature by neglecting the dispersion of acoustic waves. Debye considered the crystal as an elastic continuum where the boundary conditions results in a finite number of possible vibration mode. Only the 3N modes of the lowest frequency are take in account, where N is the number of atom per unit of volume and 3N corresponds to the vibration of the N atoms in the 3-dimensional space. The number of mode dω per unit of volume and between the frequencies ν and ν + dν is: 2πν2dν dω = (3.11) V 3 V is the average of the speed of sound between the longitudinal and the transverse waves. The limit in the number of wave induces an upper limit of the frequency. This limit is the Debye frequency. Integrating the above equation on the 3N finite number of mode give: -28- Chapter 3. Physics of thermoelectric materials 4πν3 3N = D (3.12) V 3 Debye shows that the average of energy E in a mode of frequency ν is: hν E = (3.13) exp hν 1 kB T − Where h: Plancks constant kB : Boltzmans constant The three equations above enable to determine the internal energy of the crystal. Differentiating this internal energy with respect to the temperature, the specific heat at constant volume is given: 3 3 T Z θD/T x4ex 9π4 T C = 9Nk dx = Nk (3.14) v B θ (ex 1)2 15 B θ D 0 − D Where θD is the Debye temperature: hνD θD = (3.15) kB This value for the heat capacity is relevant for temperatures below the Debye tem- perature. Above Debye temperature the specific heat tends to evolve toward its upper limit 3N. Even if the continuum approximation made by Debye, it cannot describe a real vibration spectra. But the value obtain by this theory are very close to the experimental data. Consequently, the main behaviour of the specific heat is the conti- nuity between Debye approximation at low temperatures and the upper limit for high temperatures. The upper limit for high temperatures Einsteins quantum theory of the heat capacity of solids is based on the simplifying assumption that all 3N vibration mode of a 3 dimensional crystal structure had the -29- Chapter 3. Physics of thermoelectric materials same frequency. This model does not fit with the experimental values at low tempera- tures but is correct at high temperatures. It confirms the empirical equation found by Dulong and Petit saying that the heat capacity , for high temperatures, is equal to 3R for a mole of atom. In our case, C = 3N at high temperatures. 3.2.2 Heat transport by phonons Thermal conduction by phonons The Debye theory and the Einstein theory are both based on the harmonic oscillations of the atoms in the crystal structure. These methods gave pertinent model which are close to the experimental data. The assumption that this conduction relies only on the harmonic vibrations leads to the conclusion that the conduction is infinite. Conse- quently, the anharmonic oscillation must be taken into account. Peierls introduced the concept of phonon wave packet, considering the quantization of the vibration wave, to consider the anharmonic lattice vibration. The phonons are considered as heat carrier responsible of the heat conduction of the lattice vibration. The limit of the thermal conduction is the consequence of the collision of phonons that do not conserve momentum. The kinetic theory of gases applied to the phonon gas gives the lattice conductivity κL as following: 1 κ = C V l (3.16) L 3 v t Where Cv is the specific heat discussed above and deduced from the vibration of the lattice, V is the average speed of phonons which can be approximate by the constant speed of sound and lt the mean free path of the phonons. The above expression is deduced from the theory of gas applied to phonons, but is hardly true in the case of real gas: the correction factor 1/3 does not fit the results for real gas, but gives a good approximation in the case of phonons. In pure crystals, phonons are created in the hot region and they travel toward a cooler region transporting heat. At high temperatures, the scattering comes mainly from phonon-phonon scattering. The main process involves three phonons and includes two types of scattering. The normal process (N-process) conserves both energy and mo- -30- Chapter 3. Physics of thermoelectric materials mentum, whereas the Umklapp process (U-process) does not conserve the momentum. The U- process controls the thermal conductivity at high temperatures. Considering the first Brillouin zone, the N-process of a collision between two phonons with wave vec- tor q1 and q2 result in the creation of a third phonon with a wave vector q3 = q1 +q2. In this process, the wave vector remains in the first brillouin zone and the momentum of the particles is conserved. In the U-process, q1 + q2 get out of the first brillouin zone and the reciprocal vector G is introduce in the reaction in order to consider the resultant wave vector q3 in the first brillouin zone. The reaction is:q1 + q2 = q3 + G, so the momentum of the particles is diminished during the collision At low temperature, T << θD, because of the phonons small wave vector, the resulting wave vector from a collision remains in the first Brillouin zone. Consequently the U-process is very limited for low temperatures. The number of phonons that undergo the Umklapp process is proportional to exp( θ /2T ) which makes the mean − D free path proportional to exp(+θD/2T ). This exponential variation of the mean free path is responsible of the dependence of the thermal on the temperature in a pure crystal with no defects. When the temperature reaches θD, the Umklapp process of phonon-phonon collisions becomes significant as the phonon energy is comparable to the Debye energy kBθD. Scattering of phonons by defects In a dielectric, where the heat conduction by the phonon is predominant, the thermal conductivity reaches a maximum when the temperature decreases. Then, the thermal conductivity is proportional to T 3 when the temperatures reaches 0K. Knowing that the mean free path of phonons increases when the temperature decreases, the decrease of the thermal conductivity implies that this mean free path of phonons reaches a critical value. Casimir related the drop of the thermal conductivity to the phonon scattering at crystal boundaries which is made possible by long mean free path reaching of the order of the dimension of the crystal structure . Defects are found in most real crystals. They can be impurities, vacancies or even local variations of elasticity and density in solid solution (e.g. SiGe alloys). As the relaxing time for scattering is proportional to ν−4 , the scattering is negligible for low frequencies. Adding to that, the umklapp process is also neglected at low temperatures. The normal process collision being the dominant process, the thermal -31- Chapter 3. Physics of thermoelectric materials conductivity would be important at low temperatures. This result does not stick with the experiments. Consequently, the normal process is assumed to redistribute phonons into mode for which the scattering is stronger. Callaways treatment of the problem using the relaxation time instead of the mean free path gives a method to include the N-process . His method is based on non momentum-conserving processes that cause phonon to relax in no equilibrium distribution. Then two relaxation times are considered: the relaxation time τR for the momentum changing process and τN for the N-process. Callaways method gives the expression an effective relaxation time τeff according to the relaxation times cited above and a multiplying factor β that corrects the conservation of the momentum by N-process. 1 1 1 β −1 1 β −1 = + 1 + = 1 + (3.17) τeff τN τR τN τC τN 1 1 −1 with: τC = + (3.18) τN τR The complex expression of β can be expressed as followed: !−1 Z θD/T x Z θD/T x τC 4 e 1 τC 4 e β = x x 2 dx 1 x x 2 dx (3.19) τN (e 1) × τN − τN (e 1) 0 − 0 − Assumption can be made to simplify this expression. Callaway showed it is possible to obtain analytical solutions for the integrals expressed above. 3.2.3 Heat conduction by electrons Like phonons, electronic carriers conduct heat through thermoelectric materials. In- deed electrons coming from a hot region carry more energy than electrons coming from a cold region. Consequently, the flow of carriers in a material induces a heat flow. The kinetic theory gives the electronic thermal conductivity: 2 2 1 π nkBT τ κe = CvvFl = (3.20) 3 3me -32- Chapter 3. Physics of thermoelectric materials with: Cv=specific heat per unit of volume l= mean free path of carriers vF=mean speed of carriers responsible of the thermal conductivity (only carriers state within kBT around the Fermi energy level) τ= collision time The electronic conductivity depends on the density of carrier, the temperature, the collision time and the mass of the carriers. This thermal conductivity depends on most parameters that characterize the current. Consequently, the electronic contribution of the thermal conductivity cannot be reduced without affecting the electronic conductiv- 2 ity as σ = ne τ/me. For metals, the ratio of electrical and thermal conductivities was found to depend only on the temperature. This ratio is independent of the electron gas parameter and is known as the Wiedemann-Franz law: κ e = LT (3.21) σ Where L is the Lorentz number and equal to 2.45 10−8 W.Ω.K−2. This value is × the absolute value of the Lorentz number. This value is approximately the same for all metals but is not applicable to semiconductors. -33- Chapter 3. Physics of thermoelectric materials 3.3 Summary In this chapter, we review all the basic theoretical work done on TE materials. The three thermodynamic laws of TE materials are explained by the Seebeck, Peltier and Thomson coefficients. They are related. We focus more on the Seebeck coefficient because of the clarity of its definition and the simple expression of the figure of merit. Through the Peltier device, we prove how an increase of the figure of merit would significantly improve the status of thermoelectric in our society. But, we must over- come the issue of efficiency. The figure of merit shows that the Seebeck coefficient must be increased as well as the electronic conductivity whereas lattice and electronic contribution to the thermal conductivity must be attenuated. So we find important to explain the mechanisms of lattice vibration and electronic transport of entropy that participate to the thermal conductivity. -34- CHAPTER 4 DEVELOPMENT OF A MATHEMATICAL APPROACH FOR THE ENHANCEMENT OF SEMICONDUCTOR BASED THERMOELECTRIC MATERIALS The enhancement of thermoelectric materials relies on the increase of the Seebeck effect and the electrical conductivity and the decrease of the thermal conductivity. This quest of improvement corresponds to the ”electron-crystal, phonon-glass” approach introduced by Slack in 1995 [23]. This chapter focus on the mathematical description of the electronic behaviour of the semiconductor structure of silicon and germanium. From the electronic density of state (DOS) calculated by density functional theory (DFT) for both silicon and germanium and using the solid state theory and the Boltzmann transport equation, we develop the expression of the electrical variables that appears in the definition of the thermopower in the expression of the figure of merit. These variables are the electrical conductivity, the seebeck coefficient and the electronic thermal conductivity. 35 Chapter 4. Development of a Mathematical Approach for the ... This mathematical approach help identifying the parameters the influences the overall efficiency. In the first part of this chapter, we quickly explain the solid state theory of semicon- ductors and introduce the Boltzmann transport equation that leads to the description of an unperturbed system. Then we develop the expression of the conductivity, seebeck coefficient and electronic thermal conductivity. We finish with tracing the calculated figure of merit of silicon and germanium. 4.1 Review of Solid-state theory and Boltzmann transport of Semiconductors The solid state physics focus on the understanding of the physic involved in the matter. Relying on mathematical tools such as group theory, solid state theory drastically improved its understanding of matter mainly through the 19th century. For instance, the tight binding model developed in 1928 by Felix Bloch, expresses the differences between insulators, semiconductors and metals. It defined semiconductors as insulator with a narrow band gap that electrons can overcome with a significant excitation. Even though this model is nearly a hundred years old, it explain correctly the behaviour of materials according to their electronic band structures and band gaps in the static regime. Even if some results are simple, they still hold a correct decription of the matter. And this is valid for semiconductors. In this part we will review the solid state physics describing semiconductors to express the physical parameters that appear in the figure of merit ZT. This will help identify the different possibilities to improve the thermoelectric efficiency of silicon- germanium based devices. This approach bears upon the band theory and the semi classical statistical physics. 4.1.1 Description of the electronic carriers density in an un- perturbed system Felix Bloch based his work on the description of the wave function of a particle in a periodic potential. This description relies on Bloch waves which are expressed as follow: -36- Chapter 4. Development of a Mathematical Approach for the ... ik.r ψnk(r) = e unk(r) (4.1) With unk(r) being the periodic function of the crystal. Hence, the Bloch wave function is defined in the real space with r and in the space of the wave vector k, the k-space. The k-space can be limited to the First Brillouin Zone (BZ) of the reciprocal lattice because the crystal is invariant under translation out of the first Brillouin zone. The index n is the band index and for each n, ψnk(r) evolves continuously with k. The Bloch wave vector family (ψnk(r))n forms an orthonormal basis of the wave space. Which means that any particle in a crystal has a wave vector that can be described by a linear combination of the Bloch wave vectors: ∞ X ψ(k, r) = anψnk(r) (4.2) 0 This result shows that any particle in a crystal can be described if the Bloch vectors are known in the k-space. Hence Bloch’s theorem states that, for each k, basis vectors are eigenvectors of the Hamiltonian. For every eigenvector, there is a corresponding eigenvalue Enk. Hψnk = Enkψnk (4.3) Consequently, the energy levels of particles in a periodic potential can be expressed with the eigenvalues in the k-space. It defines the band structure of the solid (cf. figures 4.1 and 4.2). Note that the band structure depends on the choice of the basis. The electronic band structure diagram describes the possible energy values for an electron or a hole along the main crystal directions. For semiconductors, this band diagram is divided in two parts by the band gap. The lower part is the valence band and the upper part is the conduction band. When an electron is in the valence band, it cannot move freely in the material, it is fixed in the covalent bonding. If an electron is in the conduction band, it can move freely and hence participate to the electric conduction in a material. Similarly, a hole in the conduction band does not contribute to the conduction but one in the valence band does. Consequently, the study of the electronic properties of a material relies on the study of electrons in the conduction -37- Chapter 4. Development of a Mathematical Approach for the ... Figure 4.1: Electronic band structure of Silicon at 300K (source:www.ioffe.ru) Figure 4.2: Electronic band structure of Germanium at 300K (source:www.ioffe.ru) -38- Chapter 4. Development of a Mathematical Approach for the ... band and holes in the valence band. Figures 4.1 and 4.2 show the band structures of Silicon and Germanium. The level energy corresponding to zero is set at the top of the valence band and the band gap Eg is the difference between the lowest point of the conduction band and the highest energy of the valence band. For both Silicon and Germanium, the band gap is indirect which means that the bottom of the conduction band is not at the Γ-point. To complete the knowledge of the repartition of the electronic carriers in the band structure, one must consider the electronic Density Of State g(E) (DOS). It is the distribution of the possible states of the electronic carriers over the energies. In the solid state theory, the form of the extremums of the energy band is usually approximated 2 2 ∗ by the square function of k, E = ~ (k k0) /2m . This leads to an approximation of − the DOS g(E) as the square root of the energy. To know the probability to have an electronic carrier in a given state, the solid state theory relies on the Fermi-Dirac distribution. This distribution incorporates the dependence of the electronic properties of SC with the temperature and the Fermi level. 1 f(E) = (4.4) 1 + exp[(E Ef )/kBT ] − Ef is the Fermi energy, T is the temperature in Kelvin and kB the Boltzamann’s constant. By definition, the Fermi level is the energy of highest occupied state for an electron at absolute zero temperature. The density of state and the Fermi-Dirac distribution enables the calculation of the electronic carriers’ density, free electrons in the conduction band and free holes in the valence band. The number of electrons with energies between E and E + dE is: n(E)dE = g(E) f(E) (4.5) × So the total free electron density is given by integrating from the bottom of the conduction band: Z ∞ n = gc(E)fe(E)dE (4.6) Ec -39- Chapter 4. Development of a Mathematical Approach for the ... Similarly, the total hole density in the valence band is: Z Ev p = gv(E)fh(E)dE (4.7) ∞ with: gc/v: DOS for conduction/valence band fe/h: Fermi-Dirac distribution function for electrons/holes To sum it up, the expression of the electronic carriers’ contributing to the con- duction depends on the band structure, the electronic DOS, the Fermi level and the temperature. A simple approximation can be done on the form of the band structure edges, which leads to an approximated expression of the electronic DOS. The temper- ature is an easy variable that the user can set. The Fermi level depends on the charge neutrality equation: n + NA = p + ND (4.8) Where NA and ND are the concentration of acceptors and donors dopants in the material respectively. n and p are the concentrations of the electrons in the conduction band and the holes in the valence band. A SC can be in two states: the intrinsic state (no dopant) and the doped state. In the intrinsic state, the Fermi level is approximately in the middle of the band gap and the above equation becomes: n = p (4.9) In the extrinsic state, a semiconductor is usually doped with only one type of dopant, either acceptors or donors. In this case, the charge neutrality equation is used to calculate the Fermi level. Consequently, n-type doping rises the Fermi level close to the conduction band edge easily promoting electrons in the conduction band. In the case of p-type doping, the Fermi level is lowered close to the maximum of the valence band promoting holes in the valence band as electronic carriers. When approximations -40- Chapter 4. Development of a Mathematical Approach for the ... are made concerning the band structure, the Fermi level can be expressed by different concentration of dopants. In our study, we obtain the electronic band structure and the density of state using Density Functional Theory (DFT). We use these parameters to express the electronic carriers concentration and solve the charge neutrality equilibrium equation self-consistently. In the previous part, we considered the basic results of solid state theory to describe the electronic state of the material. These results are applicable under equilibrium. Thermoelectric materials are designed to work with external forces and continuous flux of electronic carriers which implies a non equilibrium state of the matter. The Boltzmann Transport Equation (BTE) describes the equilibrium state in the case of external forces and continuous flux. This equation is simplified to express the figure- of-merit. 4.1.2 Carriers density in a perturbed system: Boltzmann Trans- port Equation The Boltzmann transport equation expresses the perturbed distribution function with external force F = E + v B and the unperturbed distribution function, which is × the Fermi-Dirac equation (equation 4.4). We consider f(k, r, t) the perturbed function that describes the probability of occupation of an electron at time t and position r in the k-space include between k and k + dk. The Fermi-Dirac unperturbed distribution is noted f0. The BTE expresses the shift from the equilibrium distribution due to perturbation as : df F ∂f ∂f = .∇kf(k) + v.∇rf(k) + coll (4.10) dt ~ ∂t − ∂t | The first term is related to the changes induced by external forces. The second term is the change of distribution related to the concentration of the carriers. The two last terms are the local change of the distribution and the change due to collision term. Considering a total number of carrier constant, df/dt = 0 and the local change of the distribution can be written as: ∂f ∂f F = coll .∇kf(k) v.∇rf(k) (4.11) ∂t ∂t | − ~ − -41- Chapter 4. Development of a Mathematical Approach for the ... a) b) Figure 4.3: Calculation of n and p in the a) intrinsic and b) extrinsic (n-type) case of semiconductors -42- Chapter 4. Development of a Mathematical Approach for the ... In the case of the steady state, the perturbation is considered small and the distribu- tion function deviate slightly to its equilibrium value f0. Then, the common Relaxation Time Approximation (RTA) is used. It defines a k-space dependent relaxation time related to all the scattering process inside the structure. This approximation simplifies the Boltzmann transport equation. But its value is highly significant for the transport solution as it must describe all the relevant scattering processes. The RTA gives: ∂f f f0 coll = − (4.12) ∂t | − τ and (4.11) becomes: ∂f f f0 F = − .∇kf(k) v.∇rf(k) (4.13) ∂t − τ − ~ − In the steady state, without any concentration gradient and considering the relation v = 1/~(∂εk/∂k), the Boltzmann equation can be written as: f(k) f0(k) ∂f0(k) − = F.v (4.14) τ − ∂ε In order to use this expression of Boltzmann transport equation in the thermo- electric efficiency, the above equation needs further simplification. We assume the displacement of carriers in only one direction and hence the force applied in this direc- tion. We also assume isotropy and that the thermal velocity of carriers is greater than any drift velocity. So: 1 ∗ 2 2 2 2 2 2 E = m v and v = vx + vy + vz = 3 vx (4.15) 2 | | | | | | | | | | | | Hence 2 2 2E v = vx = (4.16) | | 3m∗ Changing variable to the energy space, (4.14) becomes: -43- Chapter 4. Development of a Mathematical Approach for the ... f(E) f0(E) ∂f0(E) ∂ξ E ξ ∂T − = v + − (4.17) τ ∂E ∂x T ∂x 4.2 Thermopower and electronic thermal conduc- tivity calculation for a perturbed system and identification of limiting/enhancing parameters In this section, we use all the results of the solid state theory combined with Boltz- mann’s transport equation to express the density of electronic carriers on the valence and the conduction bands. The expression of the densities depends on the expression of the band structure and the density of state of the material studied. Fortunately, Ab Initio calculations of these component are dealt with Density Function Theory (cf. next Chapter). Consequently, the electrical conductivity, Seebeck coefficient and the thermal conductivity by electronic carriers are calculated with a minimum of empirical parameters. These calculations consider the temperature and doping as influencing parameters. The aim of this section is to study of the Thermoelectric figure-of-merit evolves with the variation of external parameters such as temperature, doping and mobilities. 4.2.1 Electrical conductivity Using the expression of Boltzmann transport equation 4.17, we express the physical coefficient that describes the thermopower of thermoelectric materials: the Seebeck coefficient α, the electrical conductivity σ and the thermal conductivity induced by electronic carriers κe. We start by describing the flow of carriers i. We previously said that the density of electronic carriers is: 1 R ∞ electrons: n = V 0 f(E)g(E)dE (4.18) holes: p = 1 R 0 (1 f(E))g(E)dE V −∞ − -44- Chapter 4. Development of a Mathematical Approach for the ... We replace 1 f(E) by h(E) which is the perturbed distribution of holes. This − hole distribution follows Boltzmann equation too. The overall flow of electronic carriers (both electrons and holes) is written: 1 Z 0 Z ∞ i = evhh(E)g(E)dE evef(E)g(E)dE (4.19) V −∞ − 0 with f the perturbed distribution of carrier in the semiconductor band structure. It is replaced by ∂f0/∂E using equation 4.17. Moreover, as there is no flow of heat or charge when f = f0, f f0 can be replaced by f. Same for h. So i becomes: − Z 0 2e ∂h0(E) ∂ξ E ξ ∂T i = ∗ τhEg(E) + − dE 3V mh −∞ ∂E ∂x T ∂x (4.20) Z ∞ 2e ∂f0(E) ∂ξ E ξ ∂T ∗ τeEg(E) + − dE − 3V me 0 ∂E ∂x T ∂x To obtain the electrical conductivity , the temperature gradient is set to zero, ∂T/∂x = 0. The electric field is given by (∂ξ/∂x)e−1 = . By definition: σ = i/ ± so: 2 Z 0 Z ∞ 2e 1 ∂ho(E) 1 ∂f0(E) σ = ∗ τhEg(E) dE + ∗ τeEg(E) dE (4.21) 3V mh −∞ ∂E me 0 ∂E with h0(E) = [1 f0(E)] − Using the definition of the unperturbed Fermi-Dirac distribution, we know: E ξ −1 f0(E) = exp − + 1 kBT (4.22) ξ E −1 h0(E) = exp − + 1 kBT When differentiated: -45- Chapter 4. Development of a Mathematical Approach for the ... exp E−ξ ∂f (E) 1 kBT 0 = ∂E k T h i2 − B exp E−ξ + 1 kBT (4.23) exp ξ−E ∂h (E) 1 kBT 0 = ∂E k T h i2 B exp ξ−E + 1 kBT We include these expressions in equation 4.21, and it gives: ξ−E Z 0 exp 2e τ E kBT σ = h g(E) dE ∗ h i2 3V −∞ mh kBT ξ−E exp k T + 1 B (4.24) E−ξ Z ∞ exp 2e τ E kBT + e g(E) dE 3V m∗ k T h i2 0 e B exp E−ξ + 1 kBT The first term is the hole contribution and the second is the electron contribution to the electrical conductivity. The notion of mobility appears in this equation as µ = eτ/m∗. Consequently the above equation is similar to another definition of the electrical conductivity in semiconductors: σ = µeen + µpep (4.25) 4.2.2 Seebeck coefficient To express the Seebeck coefficient, the current is set to zero in equation 4.19. Z 0 Z 0 2τhe ∂h0 ∂ξ E ξ ∂T ∂h0 0 = ∗ Eg(E) dE + Eg(E) − dE 3mh −∞ ∂E ∂x −∞ kBT ∂x ∂E (4.26) Z ∞ Z ∞ 2τee ∂f0 ∂ξ E ξ ∂T ∂f0 + ∗ Eg(E) dE + Eg(E) − dE 3me 0 ∂E ∂x 0 kBT ∂x ∂E -46- Chapter 4. Development of a Mathematical Approach for the ... Which is equivalent to: Z 0 Z ∞ ∂ξ 2τhe ∂h0 2τee ∂f0 0 = ∗ E g(E) dE + ∗ E g(E) dE ∂x 3mh −∞ ∂E 3me 0 ∂E Z 0 Z ∞ 1 2τhe ∂h0 2τee ∂f0 + ∗ E (ξ E) g(E) dE + ∗ E (E ξ) g(E) dE T 3mh −∞ − ∂E 3me 0 − ∂E (4.27) To simplify, the equation above is written in the following form where A and B are the terms in brackets: ∂ξ B ∂T A + = 0 (4.28) ∂x T ∂x Note that the term A is proportional to the conductivity σ as follow :Ae = V σ. This term does not need to be expanded as it is easily calculated once the conductivity is calculated. By definition, the Seebeck coefficient α = ∂ξ/∂x [e ∂T/∂x]−1. (cf Nolas Thermoelectric handbook) Using this definition in equation 4.28, one gets: 1 B α = (4.29) −eT A To simplify the calculation, we expand B: Z 0 Z 0 2 τhe 2 ∂h0 ∂h0 B = ∗ E g(E) dE ξ E g(E) dE 3 mh −∞ ∂E − −∞ ∂E (4.30) Z ∞ Z ∞ 2 τee 2 ∂f0 ∂f0 + ∗ E g(E) dE ξ E g(E) dE 3 me 0 ∂E − 0 ∂E to finally get: -47- Chapter 4. Development of a Mathematical Approach for the ... ξ−E Z 0 exp 2 1 kBT B = µ E(E ξ) g(E) dE 3 h k T h i2 − −∞ − B exp ξ−E + 1 kBT (4.31) E−ξ Z ∞ exp 2 1 kBT µ E(ξ E) g(E) dE 3 e k T h i2 − 0 − B exp E−ξ + 1 kBT Having the expressions for A and B, the Seebeck coefficient can also be calculated. 4.2.3 Electronic thermal conductivity The electrical thermal conductivity is due to electronic carriers which carry some en- tropy through the material. This type of thermal conductivity is usually neglected compared to the thermal conductivity by the lattice. But, above a certain tempera- ture, this phenomenon cannot be neglected anymore. The aim of this work is to study how the different intrinsic parameters affects the overall thermoelectric efficiency, i.e. the figure of merit. The flow of heat per unit cross-sectional area is: 1 Z 0 1 Z ∞ w = vh (E ξ) h(E) g(E) dE + ve (E ξ) f(E) g(E) dE (4.32) V −∞ − V 0 − The perturbed distributions for electrons and holes are replaced by the differentiated unperturbed distribution accordingly to Boltzmann equation. Z 0 2τh ∂h0 ∂ξ E ξ ∂T w = ∗ E (E ξ) + − g(E) dE −∞ 3V mh − ∂E ∂x T ∂x (4.33) Z ∞ 2τe ∂f0 ∂ξ E ξ ∂T + ∗ E (E ξ) + − g(E) dE 0 3V me − ∂E ∂x T ∂x -48- Chapter 4. Development of a Mathematical Approach for the ... −1 Using the definition of the electric thermal conductivity: κe = w[∂T/∂x] and − developing the above equation, the electric thermal conductivity can be expressed as: Z 0 Z ∞ 2 τh 2 ∂h0 τe 2 ∂f0 κe = ∗ E (E ξ) g(E) dE + ∗ E (E ξ) g(E) dE 3VT −mh −∞ − ∂E me 0 − ∂E ∗ R 0 ∗ R ∞ 2 τh/m E (E ξ) g(E) ∂h0/∂E dE τe/m E (E ξ) g(E) ∂f0/∂E dE + h −∞ − − e 0 − ∗ R 0 ∗ R ∞ 3VT τh/m E∂h0/∂E g(E) dE τe/m E∂f0/∂E g(E) dE h −∞ − e 0 Z 0 Z ∞ τh ∂h0 τe ∂f0 ∗ E (E ξ) g(E) dE + ∗ E (E ξ) g(E) dE × −mh −∞ − ∂E me 0 − ∂E (4.34) 4.2.4 Discussion As a result, using a semi-classical approach combined with the BTE, one can express the physical values of semiconductors which appear in the thermoelectric power. To calculate these expressions, many parameters must be known and sometimes, further approximations must be made. In our study, we use DFT to calculate the DOS and the electronic band structures of Silicon and Germanium. From the band structure, we calculate the effective masses for electrons and holes using the definition m∗ = ~2.(d2E/d2k)−1 and considering the bottom of the conduction band and the top of the valence band respectively for both materials. The volume is the primitive cell volume used in the DFT to calculate the density of state. In the case of undoped semiconductors, the fermi level is considered to remain in the middle of the band gap as a common approximation for intrinsic semiconductors. For extrinsic semiconductors, the type of dopant and their concentration determine the fermi level which set the concentration of the majority carriers. In our calculations, the fermi level is obtained by self-consistently solving the charge neutrality given in equation 4.8. The fermi level determines the majority carriers concentration. The calculated fermi level are represented on figures 4.4 and 4.5, for germanium and silicon -49- Chapter 4. Development of a Mathematical Approach for the ... Fermi level for Germanium as a function of temperature and impurity concentration 0.4 0.3 Conduction Band Edge 0.2 0.1 Phosphorus Doping 0 Ef - Ei (eV) Boron Doping −0.1 −0.2 Valence Band Edge −0.3 1E+012 1E+014 1E+016 1E+018 −0.4 100 200 300 400 500 600 700 800 900 1000 T(K) Figure 4.4: Calculated of the self-consistent fermi level of germanium for phosphorous and boron doping for different concentrations respectively, in the temperature range 100-1000K. The variation of the band gap is taken into account, following the NSM database [76]. We can see that for each case, the extrinsic fermi level reaches the intrinsic fermi level at a certain temperature. This temperature increases with the concentration of the dopants. We conclude that at high temperatures, the thermoelectric behaviour of doped silicon and germanium becomes close to the one of intrinsic equivalent material. Also, the benefit of doping remains in the range of low to medium temperatures, depending on the dopant concentration. The most significant, user-related parameter that appears in the equations of the thermopower is obviously the temperature. Through these equations, we clearly see a direct participation of the temperature. This direct participation comes from the solid state theory which apply a temperature dependent fermi statistic function to a density of state calculated at T = 0K. This solid state approximation has been proven correct. But there is also other indirect participation of the temperature in our calculated thermopower. The first one, just explained above, is the variation of the band gap and the fermi level which control the number of majority carriers. The second one appears in the carriers lifetime: the temperature of the structure creates oscillations of each atoms around their equilibrium position that interact with electronic carriers. These -50- Chapter 4. Development of a Mathematical Approach for the ... Fermi level for Silicon as a function of temperature and impurity concentration 0.5 Conduction Band Edge 0.4 0.3 0.2 Phosphorus Doping 0.1 0 Ef -−0.1 Ei (eV) −0.2 Boron Doping −0.3 −0.4 Valence Band Edge −0.5 1E+012 1E+014 1E+016 1E+018 100 200 300 400 500 600 700 800 900 1000 T(K) Figure 4.5: Calculated of the self-consistent fermi level of silicon for phosphorous and boron doping for different concentrations oscillations can be described as phonons. The electron-phonon interactions are complex and widely studied in physics because of their consequences in thermoelectric [32, 77] or even for superconductors applications. The last significant parameter is the carrier lifetime. Related to the lifetime and effective mass, the carrier mobility describes the easiness for a carrier to travel in the material. The influence of the mobility upon the thermopower is complex to study too. -51- Chapter 4. Development of a Mathematical Approach for the ... 4.3 Calculation results and variation of the ther- moelectric efficiency with temperature In the previous section, we expanded the expression of the electrical conductivity and the seebeck coefficient starting from the solid state description of semiconductors. Us- ing the density of state calculated by DFT and the Boltzmann transport theory, we are able to calculate the carrier density of a perturbed system. However, thermoelec- tric properties greatly depend on the carriers’ lifetime. Unfortunately, although BTE describes the perturbed density of charge in a semiconductor, the electronic carriers’ lifetime remains uncalculated. Indeed, the lifetime is a consequence of electrons scat- tering mechanisms taking place in the matter such as: ionic and non-ionic impurities scattering, stacking default scattering, surface effects, electron-electron scattering and phonon-electron scattering (just to name the most common ones). The mobility is the physical parameter that takes into account all the scattering processes as, by definition: em∗ µ = (4.35) τ 4.3.1 Calculated thermopower for bulk germanium and silicon To overcome the mobility issue, a first approximation in calculating the thermopower is to use empirical expressions of the mobility for each material. Empirical studies of the electrons and holes mobilities are commonly found in the literature for both Silicon and Germanium [77]. Some of these studies provide mathematical interpretation of the variation of mobility with temperature. To overcome the problem of calculation of the electronic carriers’ lifetime, these empirical equations are included in our study. Amongst many possibilities of scattering, we considered scattering of the carriers by the lattice, by impurities and between themselves. With these empirical expressions of the mobility, we calculate the resistivity of in- trinsic silicon and germanium. They are shown on figure 4.6 and 4.7 for germanium and silicon respectively, varying with temperature. For both materials, the resistivity decreases exponentially with temperature which corresponds to an exponential increase of the conductivity. This increase of the conductivity is the consequence of the increase of the carriers density with temperature. It is mathematically explained by the widen- -52- Chapter 4. Development of a Mathematical Approach for the ... Electrical Resistivity of Intrinsic Germanium 12 10 Simulated H.R. Meddins (1969) 10 0 10 D P.P. Debye (1954) 8 10 6 10 4 10 2 10 Rho (Ohm/cm) 0 10 0 −2 0 10 D D −4 10 0 500 1000 1500 Temperature (K) Figure 4.6: Calculated electrical resistivity of intrinsic germanium and experimental values [12,13] ing of the fermi distribution function with temperature. Results in figure 4.6 are in good agreement with experimental results [12,13]. The variation of calculated Seebeck coefficients with temperature are shown on fig- ure 4.8 and 4.9 for intrinsic germanium and silicon, respectively. In both cases, Seebeck coefficients decrease with temperature: from 1200 to 200 µV/K between 100 and 1400 K for germanium and from 5000 to 200 µV/K between 100 and 1000 K for silicon. These results agree with experimental data from the literature [12,14,15]. Silicon shows a higher coefficient that decreases with temperature more than for germanium. This is explained by the width of silicon band gap being higher than the germanium band gap. In equation 4.31, we note that the contribution of positive carriers is opposed to the negative carriers contribution to the seebeck coefficient. Consequently, the possi- -53- Chapter 4. Development of a Mathematical Approach for the ... Electrical Resistivity of Intrinsic Silicon 20 10 Simulated 15 10 10 10 Rho (Ohm/cm) 5 10 0 10 200 400 600 800 1000 1200 1400 Temperature (K) Figure 4.7: Calculated electrical resistivity of intrinsic silicon bility of having a non zero seebeck coefficient in an intrinsic semiconductor relies on the difference of the mobility for positive and negative semiconductors. Indeed, it is known that in an intrinsic semiconductor, the number of electrons in the conduction band is equal to the number of hole in the valence band. If these two types of carriers would undergo the same kind of scattering, meaning they both have the same mobility, the contribution of hole would completely oppose the contribution of electrons. Hence, the most obvious enhancement of the seebeck coefficient is donor or acceptor doping. For silicon and germanium, two possible options are p-type Boron or n-type Phospohorus doping. The type of doping determines the direction of the induced voltage when the temperature gradient is applied. The increase of a majority carrier population compared to the minority carrier diminishes the opposing contributions of the two types of carriers. However, silicon and germanium are considered as the most -54- Chapter 4. Development of a Mathematical Approach for the ... Seebeck Coefficient of Intrinsic Germanium - Simulated 1200 H.R. Meddins (1969) D V.A. Johnson (1953) 1000 800 \ \ \ 600 \ \ S (microV/K) 400 200 0 200 400 600 800 1000 1200 1400 Temperature (K) Figure 4.8: Calculated Seebeck coefficient of intrinsic germanium, experimental data [14] and previous simulation [12] stable thermoelectric materials at high temperatures with a maximum efficiency at around 1200 K (cf. figure 3.1) and the enhancement by doping is less effective at high temperature as the extrinsic material behaves as an intrinsic material (this proven by the fermi level of an extrinsic material reaching the value of an intrinsic semiconductor at high temperatures, figure 4.5 and 4.4). Moreover, the quantity of doping directly acts upon the seebeck coefficient : low doping enhances greatly the coefficient at low temperatures but the enhancement is effective up to a temperature limit. Whereas heavy doping provides a lower enhancement at low temperatures but its effectiveness remains until higher temperatures [14]. Consequently, depending on the temperature range of use of the thermoelectric material, the doping must be optimized. -55- Chapter 4. Development of a Mathematical Approach for the ... Seebeck Coefficient of Intrinsic Silicon Simulated () L. Weber (1991) () () 3 10 () S (microV/K) () () () () 100 200 300 400 500 600 700 800 900 1000 Temperature (K) Figure 4.9: Calculated Seebeck coefficient of intrinsic silicon and experimental data [15] 4.3.2 Calculated electronic thermal conductivity The electronic contribution of the thermal conductivity is calculated according to equa- tion 4.34 and compared to the lattice thermal conductivity found in the literature for both silicon and germanium [24, 78–81]. The thermal conductivity is known to be the most significant limiting parameter of thermoelectric efficiency. In both silicon and germanium, while the lattice thermal conductivity decreases with temperature, the electronic contribution increases with electronic carriers’ density, hence with tempera- ture. Consequently, above a temperature, the electronic thermal conductivity becomes dominant. It is the case of germanium at 700 K according to our calculations. For sili- con, the lattice thermal conductivity is five times higher at 700 K than for germanium and the electronic addition remains lower than the lattice contribution in the range of study. -56- Chapter 4. Development of a Mathematical Approach for the ... Thermal conductivity of Intrinsic Germanium 1.2 by electronic carriers by the lattice 1 0.8 0.6 K(W/cm.K) 0.4 \ \ 0.2 " ' 0 200 400 600 800 1000 1200 1400 Temperature (K) Figure 4.10: Thermal conductivity by the lattice and by the electronic carriers for germanium The electronic contribution of the thermal conductivity depends on the density of electronic carriers. Consequently, an increase in the number of carriers and their mobility will increase this contribution. Here again, doping can be an issue and it needs optimization. -57- Chapter 4. Development of a Mathematical Approach for the ... Thermal conductivity of Intrinsic Silicon 3 - by electronic carriers by the lattice W. Fulkerson (1968) 2.5 I I I 2 I I I \ 1.5 \ \ ""' \ \ K(W/cm.K) 1 0.5 -""'- - --,6_ - 0 100 200 300 400 500 600 700 800 900 1000 Temperature (K) Figure 4.11: Thermal conductivity by the lattice and by the electronic carriers for silicon 4.3.3 Resulting thermoelectric figure of merit for intrinsic ger- manium and silicon Calculated thermoelectric figure of merit for germanium and silicon are displayed in figure 4.12 and 4.13, respectively. For these two intrinsic semiconductors, the values obtained are much lower than the values reported for doped materials, hence these bulk intrinsic material are irrelevant for any direct application. The maximum ZT value for germanium is 0.114 in the temperature range 100 to 1000 K and 0.029 for silicon. The trend of the curve for silicon seems to have an exponential increase whereas the figure of merit for germanium presents an attenuation in the increase around 800 K. Here again, the difference between these two curves comes from the band gap of these two semiconductors. The figure of merit of germanium shows an increase -58- Chapter 4. Development of a Mathematical Approach for the ... Thermoelectric figure-of-merit ZT of Intrinsic Germanium 0.12 0.1 0.08 0.06 ZT 0.04 0.02 0 100 200 300 400 500 600 700 800 900 1000 Temperature (K) Figure 4.12: Thermoelectric figure of merit of intrinsic germanium which seems to be limited around 700 to 800 K. This limitation must come from the significance of the electronic contribution to the thermal conductivity which overcomes the lattice contribution at 700 K. Remember that the thermal conductivity is dividing the thermopower in the equation of the figure of merit, equation 3.7. To sum it up, the figure of merit varies linearly with the electronic conductivity and the square value of the seebeck coefficient whereas it decreases with the thermal conductivity. For intrinsinc material, the conductivity increases with the number of carriers which increases with temperature. The seebeck coefficient decreases with tem- perature, and the thermal conductivity decreases drastically from 100 to 700 K for both material but increase after 700 K for germanium because of the electronic contri- bution. The standard trend for the figure of merit is similar to a bell shape: it increases, reach a peak, and then decreases. This decrease of ZT after reaching a maximum is -59- Chapter 4. Development of a Mathematical Approach for the ... Thermoelectric figure-of-merit ZT of Intrinsic Silicon 0.035 0.03 0.025 0.02 ZT 0.015 0.01 0.005 0 100 200 300 400 500 600 700 800 900 1000 Temperature (K) Figure 4.13: Thermoelectric figure of merit of intrinsic silicon attributed to the increase of the thermal conductivity. But at high temperatures, the recrystallization and the diffusion of atoms are happening, diminishing the electronic conduction. In case of doped material, this drop is also related to intrinsic behaviour of completely ionized extrinsic semiconductors at high temperature. Consequently, the temperature range of use becomes a significant issue and doping must be optimized according to it. -60- Chapter 4. Development of a Mathematical Approach for the ... 4.4 Summary From the basic theory of thermoelectric explained in the previous chapter, and, in the case of semiconductors, the use of solid-state theory, we calculated the TE thermopower and the electronic contribution to the thermal conductivity of silicon and germanium. Form this calculation, and using experimental results for the lattice contribution to the thermal conductivity, we easily express the figure of merit. This approach is not completely ab initio as we also use empirical study of the electronic carriers mobil- ity. Although other work already implemented solid state theory to the calculation of TE property, none considered the effect of both electrons and holes. Most of the studies relied only on doped materials, which is highly relevant for TE applications as they exhibit higher figure of merit. But, it is known that, at high temperatures, all the dopant are ionized and an extrinsic material will start to behave intrinsic, with both types of electronic carriers participating to the conduction. Moreover,silicon and germanium are interesting TE semiconductors because of their stability at high tem- peratures. Consequently, considering all the parameters in the figure of merit, we see that heavy doping does not contribute much to the Seebeck coefficient and the electri- cal conductivity at high temperature but greatly enhance the electronic contribution to the thermal conductivity at high temperatures. We conclude that, depending on the range of use of a TE device, doping must be optimized. Obviously, we also identify the carriers mobility, the form of the density of state and the band gap as a significant parameters that optimize the figure of merit in certain temperatures range. -61- CHAPTER 5 DENSITY FUNCTIONAL THEORY APPLIED TO SILICON/GERMANIUM STRUCTURES 5.1 Density Functional Theory 5.1.1 The quantum problem: the Schr¨odingerequation Relying on Louis de Broglie’s work which states that every particle is linked to a wave, Erwin Schr¨odingerintroduces, in 1927, the idea of a wavefunction which statistically describes the probability of a particle to be in a specific position. With this idea, Schr¨odingerdescribes any particle using the equation: H ψ(xi, t) = E ψ(xi, t) (5.1) | i | i where H is the Hamiltonian operator. Hence, this equation is used to solve the mo- tion of a system. For a given system, for an electron evolving around an atom, a wavefunction ψ describes the probability for this system to be in a given state. To | i solve this system, the equation must be simplified. Consequently, we must find a basis 62 Chapter 5. Density Functional Theory applied to silicon/germanium ... in the Hilbert space that, once the equation is projected upon, simplifies the form of the Hamiltonian. Let’s assume that this orthonormal basis ( β ) is known. The | i wavefunction can be written as follow: X ψ = β ψ β | i h | i | i (5.2) 0 with β β = δββ0 h | i Hence, the projection of the Schr¨odingerequation (equation (5.1)) on the basis gives: P H β β ψ = P E β β ψ β | i h | i β | i h | i P 0 P 0 = β H β ψβ = E β β ψβ ⇒ β h | | i β h | i (5.3) P 0 = β H β ψβ = Eψβ ⇒ β h | | i ˜ = Hψβ = Eψβ ⇒ A well-chosen basis makes the projection H˜ of the matrix H a diagonal matrix. But the Hilbert space does not have a finite dimension which means that the basis ( β ) | i contains an infinity of vectors. When computing the equation, a cut-off must be defined in order to work with only a finite number of basis vectors. The choice of the cut-off is crucial for the convergence and the accuracy of the calculation. Moreover, in a system of N particles where the position and the spin orientation are considered, the eigenvectors have each 6N degrees of freedom. The Density Functional Theory (DFT) overcomes this problem by describing a system of N particles as a problem which is function of the particle density. The DFT relies on two main theorems that simplify the Schr¨odingerequation. Introduced by Hohenberg and Kohn, the first theorem states: in a set of enclosed particles moving in an average background potential Vext(r) , this potential, within a constant, is a unique functional of the particle density n(r) and so is the many-particle ground state. In other words, the potential created by N particles can be expressed as a function of only the density of particles. And as the Hamiltonian depends on this external potential, the ground state of this many particles system also depends only on the density of particles. The second theorem states that, for a closed -63- Chapter 5. Density Functional Theory applied to silicon/germanium ... system, there is a well-defined minimum in the energy: E0 E (n(r)) and the equality ≤ is verified only for one density of particles n0(r) corresponding to the lowest energy that the system can reach. Knowing n0(r) gives the expression of the ground state. If we consider a closed system of N particles described by a density n(r), the energy can be divided in two terms: E(n(r)) = F (n(r)) + Vext(n(r)) (5.4) Where Vext is the external potential and F is the internal potential. The internal potential can then be divided in three terms: F (n(r)) = T (n(r)) + EH (n(r)) + Exc(n(r)) (5.5) where, T is the kinetic energy of the enclosed system, EH the Hartree energy that de- scribes the potential energy inside the system and Exc the exchange-correlation energy. The general expression of the problem can be simplified thanks to the theorem of the minimum. Indeed in a stationary system, all the of the energy introduced above can be differentiated by the density. Consequently, the Hartree and the exchange correlation energy terms can be expressed as the derivative of a potential. ZZ n(r) n(r0) E (n(r)) = drdr0 (5.6) H r r0 | − | Z Vext(n(r)) = vext(r) n(r)dr (5.7) These expressions are included in the equation (5.4) of the total energy of which must be minimised. To look for an extrema, a Lagrange Multiplier λ is used with the following necessary condition: δ Z n(r)dr = 0 (5.8) δn -64- Chapter 5. Density Functional Theory applied to silicon/germanium ... The Lagrange function becomes: δ Z E(n(r)) λ n(r)dr = 0 (5.9) δn − δT δE Z n(r0) + xc + v (r) + dr0 = 0 (5.10) δn δn ext r r0 | − | This formulation of the problem remains complex. One possible way to solve it is to relate it to a simpler and known problem. So equation (5.10) can be written with only two terms in the left part: The kinetic energy and the rest. Es(n(r)) = Ts(n(r)) + Vs(n(r)) (5.11) By analogy, this form of the energy can be related to a system of non interacting single particles moving in an external potential v(r) which verifies: Z Vs(n(r)) = v(r)n(r)dr (5.12) Consequently, minimizing equation (5.11) leads to solve: δT (n(r)) s + v(r) = 0 (5.13) δn This system corresponds to Schr¨odinger’sequation; Hsφi = εiφi where φi are the particle orbitals and εi are their related eigenvalues. To simplify the problem of min- imization of the energy (equation (5.10)) in the general case, the known case of non interacting single particles is used. The analogy between these two enables the use of the particle orbitals. The link between equation (5.10) and equation (5.13) leads to the following results: -65- Chapter 5. Density Functional Theory applied to silicon/germanium ... T (n(r)) = R φ†(r) φ (r)dr i i − ∇ 0 R n(r ) 0 δExc(n(r)) v(r) = vext(r) + 0 dr + |r−r | δn(r) (5.14) N X 2 n(r) = fi φi(r) i | | The total energy E0 for the minimized electron density is calculated. Using the defini- tion of Vext and equation (5.14), we get: δExc(n(r)) Vext(r) = Vs(n0(r)) 2VH (n0(r)) (5.15) − − δn(r) Consequently, using the results for the non-interacting particle Schr¨odingerequation added to the results for the general case, the total energy can be expressed as: N X δExc(n0(r)) E = ε + V (n (r)) E (n (r)) (5.16) 0 i H 0 δn (r) xc 0 i − 0 − Using the above result, the electron density is solved self-consistently (cf. figure 5.1.1) to find n0(r) which is related to the lowest value of total energy possible. -66- Chapter 5. Density Functional Theory applied to silicon/germanium ... Density mixer that mixes ng(r) and nn(r) to obtain a new ng(r) for the next cycle Initial density n (r) start g set Exc and veff (r) Calculate δExc(ng(r)) δng(r) no R ng(r) 0 and |r−r0| dr Solve 1 2 [ + v(r)]φi(r) = εi(r) − 2 ∇ Obtain new density nn(r) Converging criteria P 2 from n(r) = fi φi(r) En Eg < Ec ? | | | − | yes The SCF has converged Figure 5.1: Flow chart of a Self-Consistent Functional (SCF) to calculate the density n0 related to the minimum energy E0 of the system -67- Chapter 5. Density Functional Theory applied to silicon/germanium ... 5.1.2 implementation of DFT within CASTEP Material Studio is a package developed and distributed by Accelrys. This package includes CASTEP and other softwares. CASTEP relies on the Density Functional Theory, which is described in the previous section, with a plane wave basis set. The use of supercells makes CASTEP adapted to the study of periodic system with a crystal structure. This is the reason why we use this software for the study of Silicon and Germanium structures. In this section, we describe how the DFT is implemented in CASTEP. supercell in the case of the fcc structure In the study of silicon and germanium structures for thermoelectric devices, CASTEP calculates the electronic structure of such materials. Silicon and germanium have the same crystal structure and the lattice mismatch is around 4%. In these cases, CASTEP uses supercell to reduce the number of atoms in the structure, and hence the number of particles. Indeed, as the crystal structure is defined by a primitive cell which is repeated periodically in all three spatial directions defined by the vectors of the cell. The supercell can be chosen to be just the primitive cell or can include as many primitive cells as wanted. In the latest case, the supercell vectors are a linear function of the original primitive cell. The use of bigger supercell makes the results more accurate but it also increases the size of the calculations. The periodicity of the crystal structure implies that the potential is periodic. To solve the single-particle Schr¨odingerequation with a periodic potential, CASTEP uses Bloch’s theorem which states that the eigenvalues can be written as a plane wave modulated by a periodic function. The periodic function can be expanded using a discrete set of plane waves as basis whose wave vectors are reciprocal lattice vectors. X i(G+k)r ψki (r) = Cki,Ge (5.17) These coefficients appear in the simplified Khon-Sham equation: -68- Chapter 5. Density Functional Theory applied to silicon/germanium ... X 2 0 0 0 k + G δGG0 + Vion(G G ) + VH (G G ) + Vxc(G G ) Cik+G0 = εiCik+G | | − − − (5.18) The kinetic energy is diagonal, and the various potentials are described in terms of their Fourier transform. So CASTEP uses a Fast Fourier Transform (FFT) to quickly transpose the electron density from the real space to the reciprocal space. k-point sampling It is seen above that the simplified Kohn-Sham equation is solve in the reciprocal space. More precisely in the first Brillouin Zone (BZ). The first BZ in the reciprocal space acts like the primitive cell in the real space and each translation out of the first BZ is equivalent to a translation within the first BZ. Consequently, by describing the k-points inside the volume of the first BZ, we describe the k-points in the whole reciprocal space. The use of Bloch’s theorem enables the description of any wavefunction by considering a limited number of eigenstates inside the first BZ, i.e. for an infinite number of k- points. To overcome this issue, the BZ must be sampled, which means that a small volume of the k-space inside the BZ can be approximated by one k-point. The k-point mesh is a parameter to be optimized: it must be fine enough to obtain accurate results, but if it is too fine, it will be calculation consuming. Note that the k-point meshing depends on the size of the supercell. If the supercell is big, the meshing does not need to be too fine as the reciprocal space size is the inverse of the supercell size. In the case of the study of bulk silicon and germanium, the small primitive cell is taken as the supercell and consequently, the mesh must be fine. Methods to describe the exchange-correlation interactions In most of the cases, the exchange-correlation term is unknown. Except for the free electron gas. But it is possible to use approximations . These approximations are known as the exchange-correlation functionals. The most common used functional is the Local Density Approximation (LDA) which is the simplest approximation based on the free electron gas regime. This func- -69- Chapter 5. Density Functional Theory applied to silicon/germanium ... kz z ky kx y a) x b) Figure 5.2: a) conventional cell of a fcc structure b) first Brillouin zone for fcc tional must be parameterized, and the commonly used approximation comes from Ceperley and Alder [82]. If a system deviates from the free electron gas regime, the LDA results will be inaccurate. A more general approach includes the density gradient in the functional. It is known as the Generalized Gradient Approximation (GGA). It is a semi-local approach. The first GGA developed by J.P. Perdew and Y. Wang in 1991 is called the PW91 [83]. Other parameterizations are found: the Perdew, Burke and Ernzerhof (PBE) [84]. -70- Chapter 5. Density Functional Theory applied to silicon/germanium ... kz ky kx Figure 5.3: 5 5 5 k-point mesh of the first brillouin’s zone of an fcc structure × × 5.2 Electronic description of silicon/germanium sys- tems 5.2.1 Method for converging calculations All simulations based on silicon and germanium structures in this chapter use the DFT implemented in CASTEP. Before launching a simulation, the cut-off energy and the k-point sampling are varied, using a single point energy calculation, to study their effect on the convergence. For both parameters, the lowest value corresponding to a constant final energy (of the single point energy calculation) is taken as a first criterion of convergence. This ensures a converging simulation with less calculations possible. For simple bulk silicon and germanium, the cut-off energy is usually set slightly higher than the cut off energy suggested in the pseudopotential descriptions. For the k-point sampling, as the primitive cell is studied, a fairly fine mesh is needed. Convergence test showed that a 9 9 9 meshing was enough for converged results. Figure 5.2.1 × × gives an example of a 5 5 5 sampling of the first BZ. Depending on whether the × × number of sampling on one axis is odd or even, an offset is included so that the Γ-point is always included in the mesh. -71- Chapter 5. Density Functional Theory applied to silicon/germanium ... 5.2.2 Pseudopotentials and schemes Our calculations only deal with silicon and germanium. Therefore pseudopotentials for only these two materials are needed. The pseudopotential approximation distinguishes two types of electrons: the electrons that are influenced by the environment, on the outer shells, and those which are too close to the core to be influenced. To reduce the weight of calculations, this approximation considers no participation of the core electron in chemical bonding. The distribution of core electron is not changed by the environment, so they’re considered to be frozen. The rest of the electrons are concerned by the valence and conduction. In the case of silicon, the 1s22s22p6 are commonly considered as core electrons and 3s23p2 as valence electrons. Pseudopotentials using this description simulate correctly the lattice parameter within 0.2% error and the bulk modulus within 2% error. The case of germanium is more complex because of the contribution of 3d. To reduce the weight of calculation, some pseudopotentials treat the electronic structure of 1s22s22p63s23p63d10 as core electrons and 4s24p2 as valence electrons. But to in- crease accuracy, 3d are considered as valence electrons, which increase the weight of calculation. Norm-conserving pseudopotentials are usually used to assure the transferability to the environment. Their related cut-off radius influences the compromise between the transferability and the softness of the pseudopotential. The definition of a norm- conserving pseudopotential comes from the same squared amplitude below and above the cut-off radius remains constant [85]. These pseudopotentials need high cut-off energies. To reduce these cut-off energies, and hence the weight of calculations, ultra- soft pseudopotential have been developed transgressing the conservation of the squared amplitude [86]. In our study, several pseudopotentials are studied and compared. The main idea is to work with ultra-soft pseudopotentials when the calculations are too expensive. Norm-conserving pseudopotentials are used when perturbations are applied to the structure (like the case of DFPT). -72- Chapter 5. Density Functional Theory applied to silicon/germanium ... 5.2.3 Calculated Electronic structures Bulk Silicon and Germanium The DFT calculates eigenvalues for each k-point so it is possible to calculate the elec- tronic band structure of semiconductors along the main axis of symmetry. In the case of silicon and germanium which both have a zinc-blende crystal structure, the main valleys of the conduction band are located along the [111] axis, the [001] axis (and equivalent directions) and the Γ-point. Silicon Pseudopotential Ultra- Ultra- Norm- soft soft conserving Experimental Scheme LDA GGA PBE GGA PBE Values Ecut−off (eV) 500 400 700 Lattice 5.375 5.463 5.393 5.431 Constant (A)˚ (1%) (0.6%) (0.7%) Band 0.44 0.58 0.65 1.17 Gap (eV) (62.4%) (50.4%) (44.8%) Bulk 97.03 88.95 94.27 97.6 Modulus (GPa) (0.6%) (8.9%) (3.4%) Young’s 126.7 123.7 132.4 130-188 Modulus (GPa) Poisson 0.282 0.268 0.266 0.064-0.28 Ratio Table 5.1: Calculated parameters for Silicon with DFT Our calculation based on GGA underestimates the band gap up to 50.4% for silicon and by 62.4% when using LDA (cf. table 5.1). This is caused by the derivative discontinuity of the exchange-correlation energy. It has been reported that LDA within the DFT underestimates the band gap up to 40% [87]. -73- Chapter 5. Density Functional Theory applied to silicon/germanium ... Germanium Pseudopotential Norm- Norm- Ultra- conserving conserving soft Experimental Scheme GGA GGA LDA Values Ecut−off (eV) 700 900 300 Lattice 5.480 5.755 5.544 5.658 Constant (A)˚ (3.1%) (1.7%) (2.0%) Band 0.58 0 0.34 0.74 Gap (eV) (21%) (100%) (54.2%) Bulk 85.1 66.5 78.5 75 Modulus (GPa) (13.4%) (11.4%) (4.7%) Young’s 123.1 76.1 108.2 103 Modulus (GPa) (19.5%) (26.2%) (5%) Poisson 0.259 0.309 0.27 0.26 Ratio (0.4%) (18.8%) (0.3%) Table 5.2: Calculated parameters for Germanium using DFT. First norm-conserving pseudopotential includes 3d in core whereas the second norm-conserving pseudopoten- tial take 3d electrons into account. For germanium, the calculation of the band structure is more complex. First, the band gap can largely be underestimated. When using a norm-conserving pseudopoten- tial that considers 3d as non-frozen electron, the band gap difference with experiments reaches 100%, which means a band gap equal to zero. The norm-conserving pseu- dopotential considering 3d as frozen underestimates the band gap by 21% whereas the ultra-soft pseudopotential shows a 54.2% difference with experiments (cf. table 5.2). To overcome the issue of band gap, a scissor operator is used to rigidly shift the con- duction band from the valence band based on experimental values. These experimental values are Eg = 1.17 for silicon and Eg = 0.74 for germanium at 0K [88]. Then, the order of the conduction valleys for germanium (which is normally L < Γ < ∆) is not -74- Chapter 5. Density Functional Theory applied to silicon/germanium ... respected. While the ultra-soft and simple norm-conserving pseudopotentials follows L < ∆ < Γ, the norm-conserving pseudopotential considering 3d shows Γ < L < ∆. We note that , for germanium, the use of the ultra-soft pseudopotential with CA-PZ schemes gives the best results. In the case of silicon, we choose to use the ultra-soft pseudopotential with a PBE schemes. Hence, the electronic properties are calculated using these pseudopotentials and schemes. 25 25 s 20 p sum 20 15 15 10 10 5 5 Energy (eV) 0 energy (eV) 0 −5 −5 −10 −10 −15 −15 W L G X W K 0 0.2 0.4 0.6 0.8 1 1.2 path in First Brillouin Zone DOS (electron/eV) Figure 5.4: Simulated electronic band structure of Silicon and its partial density of state using LDA with an Ultra-soft pseudopotential and Ecut−off = 300eV Figures 5.4 and 5.5 show the calculated band structures for silicon and germanium. Because silicon and germanium crystalize in the same zinc-blende structure and their lattice parameters and electronic configurations of the valence band are similar, their band structure are close to one another. Using the Pauli principle of exclusion and the population analysis using Mulliken formalism [89] over the calculation mesh of the brillouin zone, CASTEP calculates the band structure-associated partial Density Of State (DOS) for one atom. In both pseudopotentials for silicon and germanium, only electrons from s and p from the outer shell contributes to the calculations. In the case of silicon, p electrons mainly contribute to the conduction band whereas both p and s electrons participate in the valence band. For germanium, s and p electrons equally contribute to both valence and conduction band. -75- Chapter 5. Density Functional Theory applied to silicon/germanium ... 30 30 s p sum 25 25 I== I 20 20 15 15 10 10 5 5 energy (eV) Energy (eV) 0 0 −5 −5 −10 −10 −15 −15 W L G X W K 0 1 2 3 path in First Brillouin Zone DOS (electron/eV) Figure 5.5: Simulated electronic band structure of Germanium and its partial density of state using LDA with an Ultra-soft pseudopotential and Ecut−off = 500eV Bulk SixGe1−x Alloys Although the advantage of Silicon and Germanium alloys as thermoelectrics comes from the drop of the thermal conductivity by the lattice, it seems relevant to use the DFT tool to study their electronic properties. In order to calculate alloys properties using only a primitive cell, we use the Virtual Crystal Approximation (VCA). Imple- mented in CASTEP, it describes each atom of a structure as a mixture of elements. Consequently, it is a simple tool to simulate disordered structures at the same cost as ordered structures. However, using the VCA, some ultra-soft pseudopotentials tend to create ghost states, so it is recommended to use norm-conserving pseudoptentials. Bellaiche et al. successfully described the dielectric and piezoelectric properties of prevskites using ultra-soft pseudopotentials combined with VCA [90]. But we used norm-conserving pseudopotentials in the case of silicon and germanium alloys. Obvi- ously, the extreme cases of 100% (silicon) and 0% (germanium) of silicon are calculated for comparison. The other compositions are 50% and 80% silicon. The case of 80% is the most relevant as it shows the lowest thermal conductivity, hence a higher figure of merit for thermoelectric applications. Figure 5.7 show the densities of state for silion, germanium and alloys calculated with norm-conserving pseudopotentials. These DOS are different from the one calcu- lated previously as the pseudopotentials are not the same. In the DOS of germanium, -76- Chapter 5. Density Functional Theory applied to silicon/germanium ... a forbidden band gap appears in the valence band around 9eV which is repeated in − both case of alloys at higher energies, around 8eV . It is relevant to note that the − DOS for both alloys are very close to one another and show a decrease of the density at the top of the valence band compared to pure silicon and germanium. Whereas the conduction bands show the same range of value as the pure materials. The alloys show a higher contribution of s electrons to the conduction band than pure silicon, but it remains lower than the case of germanium. -77- Chapter 5. Density Functional Theory applied to silicon/germanium ... 8 s p sum 7 6 5 4 3 DOS (electron/eV) 2 1 0 −15 −10 −5 0 5 10 energy (eV) a) 8 s p sum 7 6 5 4 3 DOS (electron/eV) 2 1 0 −15 −10 −5 0 5 10 energy (eV) b) Figure 5.6: Densities of State for a) Silicon b) Germanium -78- Chapter 5. Density Functional Theory applied to silicon/germanium ... 8 s p sum 7 6 5 4 3 DOS (electron/eV) 2 1 0 −15 −10 −5 0 5 10 energy (eV) c) 8 s p sum 7 6 5 4 3 DOS (electron/eV) 2 1 0 −15 −10 −5 0 5 10 energy (eV) d) Figure 5.7: Densities of State for c) Si0.8Ge0.2 and d) Si0.5Ge0.5 -79- Chapter 5. Density Functional Theory applied to silicon/germanium ... 5.3 Summary In this chapter, we explain the fundamentals of density functional theory (DFT). Using DFT, we come to the issue of electronic structure simulations and the advantages and drawbacks of these calculations. Except for the band gap value, the case of silicon does not show too many inaccuracies. Whereas the case of germanium is more complex. For germanium, the problem comes from the consideration of the 3d10 electrons in the pseudopotential. In basic pseudopotentials, two types of considerations can be found: either 3d10 electrons are considered as frozen, or they participate the the electronic conduction. While the first consideration simplifies the problem, the second, considered as more accurate, increase the weight of calculations. But it both cases, the inaccuracies are more significant than for silicon. After several test, we select the more accurate pseudopotential and scheme for both materials to calculate the band structure and the density of state. In addition, using the virtual crystal approximation (VCA) we also calculate the electronic properties of silicon germanium alloy for various compositions. It shows DOS shape close to the one of silicon and germanium bulk materials. -80- CHAPTER 6 PHONON-ELECTRON INTERACTIONS: LINEAR DEFORMATION POTENTIAL THEORY IMPLEMENTED IN THE DFT The enhancement of thermoelectric materials is divided into two parts: enhancement of electronic properties or the decrease of thermal conductivity. Although the figure of merit includes these two parts, it is possible to study the thermopower (describing the electronic properties) and the thermal conductivity separately. This parting of the figure of merit is the solution to study only one aspect of the challenges as it is complex to describe the overall phenomenon. However, the electronic and thermal properties are linked. The flux of electrons carries entropy and participates to the thermal conductivity. The flux of phonon interacts with electrons, making the lattice a significant contributor to electron’s mobility. At low temperature, a gradient of temperature is responsible of the phonon drag which will tend to push the electrons to one end of the material. These examples show how significant electron-phonon coupling is for thermoelectric applications. In the previous chapter, we calculated the electronic properties of silicon, germanium and some of their alloys using DFT. Even though, our package does not allow direct electron-phonon coupling calculation, we rely on the use of deformation potentials. 81 Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... 6.1 Theory 6.1.1 principle of the deformation potential theory applied to study electron-phonon scattering In 1950, J. Bardeen and W. Schockley in the Bell Telephone Laboratories introduced the deformation potential theory to describe the interactions between electrons and acoustic phonons [91]. This theory relies on the deformation of the lattice, simulating phonons, causing shifts of the energy bands. In 1955, C. Herring and E. Vogt developed this method for many-valley semiconductors [16]. Later, some work extended this the- ory to acoustic and optical phonons [92, 93]. With the emergence of superlattices, the deformation potential theory was used to describe the effects of strained lattice upon the carriers mobility [7, 17, 32, 94–96]. This theory started to be applied to ab initio calculations in the late seventies with the work of J.B. Renucci who used pseudopo- tential calculations to validate experimental Raman spectrums for silicon [97]. The ab initio calculation approach continued in the study of interactions of electrons with long wavelength acoustic phonons and optic phonons in the case of mutlivalley semiconduc- tors [98, 99]. The possibility of calculating the electron-phonon interaction became a main concern in the study of superconductors [100]. In 1996, X. Gonze developed the calculations that led to the Density Functional Perturbed Theory (DFPT) that is com- monly implemented in most of DFT package to calculate the phonons properties [101]. Since then, the deformation potential theory is commonly used within DFT to study intervalley and intravalley scattering for multivalley semiconductors [102–110]. k0, n0 k, n q q k, n k0, n0 (a) Emission (b) Absorption Figure 6.1: The two electrons-phonon scattering processes The scattering processes studied are the absorption and the emission of a phonon with change of momentum of the electron (cf figure 6.1). In a multivalley semiconduc- -82- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... tor, this change of momentum, depending on its significance, can produce two types of scattering. The first type, intravalley scattering, when the electron changes momen- tum but remains in its valley. The second type, the intervalley scattering, when the electron change its momentum but also jumps in another valley. In the case of silicon and germanium, the intervalley scattering takes place mainly between Γ, ∆ and L. 6.1.2 Form of the scattering probability To the hamiltonian describing a system, according to Fermi’s golden rule (cf. ap- pendix), it is possible to introduce small time dependent perturbations. If we consider a small perturbation that pushes atoms slightly out of their equilibrium positions, this perturbation will scatter electrons from the conduction band. Hence, this perturbation can be considered as vibration of the lattice caused by phonons. Finally, the scattering of electronic carriers by phonons can be described by the probability of scattering given by Fermi’s golden rule. This probability applied to the scattering processes described in figure 6.1 gives: λ 2π 0 0 2 Pnk,n0k±q = n, k ∆Wλq n , k Nq + 1/2 1/2 δ(εk εk0 ~ω) (6.1) ~ |h | | i| × | ∓ | − ± with ∆Wλq the perturbation of the crystal potential due to a phonon q from branch • λ. Nq the Bose-Einstein distribution of phonons. • δ(εk εk0 ~ω) considers only the phonons that verifies the conservation of • − ± energy. 0 0 n, k ∆Wλq n , k is the element of the electron-phonon coupling matrix • h | | i The increasing interest of the electron-phonon coupling matrix has made this cal- culation available in some DFT tools, such as QUANTUM espresso. To simplify our study, we decided to use the deformation potential theory that links element of the matrix to deformation potentials as follow: -83- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... r λ 2V ρωλq 0 Ξnk,n0k±q = n, k ∆Wλq n , k q (6.2) ~ |h | | ± i| In their early work on semiconductors, C. Herring and E. Vogt developed the way to calculate the deformation potential found in equation 6.2 for silicon and germanium [16]. This calculation deals with only the vectors in the k-space that account for scattering between the valleys Γ, ∆ and L. Indeed, the Bardeen-Shockley deformation potential theory defines three independent constants that describes the deformation potentials for each valley that lay on symmetry axes [91]. These three independent constants are the uniaxial shear deformation potential Ξu, the dilatation deformation potential Ξd and Ξp due to shear. The shift of the energy for a given valley i is given by: 6 (i) X (i) δ = Ξs us (6.3) s=1 with us the strain induced by a phonon. Therefore, the strain must be calculated to obtain the deformation potentials. -84- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... 6.2 Stress and strain relationships for deformation potentials calculations We used DFT to calculate the shift of the energy band edges under the effect of induced strain. We used the norm-conserving pseudopotentials for both silicon and germanium. These structures had their geometry optimized while applying stress in [100], [111] and [110] was applied at different pressures. [100] Direction P 0 0 s11 0 0 σ = 0 0 0 ε = P 0 s12 0 0 0 0 0 0 s12 [110] Direction P/2 P/2 0 s11 + s12 s44/2 0 σ = P/2 P/2 0 ε = P s44/2 s11 + s12 0 0 0 0 0 0 2s12 [111] Direction P/3 P/3 P/3 s11 + 2s12 s44/2 s44/2 σ = P/2 P/3 P/3 ε = P s44/2 s11 + 2s12 s44/2 P/3 P/3 P/3 s44/2 s44/2 s11 + 2s12 s11, s12 and s44 are the elastic compliances constants. These constants are calcu- lated using the DFT for silicon, germanium and each alloy studied. Combining these relationships with equation 6.3, the energy shifts for ∆ and L valleys are expressed as follow [94]: v v v T δ = Ξd Tr(ε) + Ξu ai εai (6.4) -85- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... Stress δ/P Valley Valley direction direction ∆ ∆ ∆ [100] Ξd (s11 + 2s12) + Ξu s11 [100] L L L [111][111][111][111] Ξd (s11 + 2s12) + Ξu (s11 + 2s12) Γ Γ [000] Ξd (s11 + 2s12) ∆ ∆ ∆ [100][010] Ξd (s11 + 2s12) + Ξu /2(s11 + s12) [110] L L L [111][111] Ξd (s11 + 2s12) + Ξu (s11 + 2s12 + s44/2) Γ Γ [000] Ξd (s11 + 2s12) ∆ ∆ ∆ [100][010][001] Ξd (s11 + 2s12) + Ξu /3(s11 + 2s12) [111] L L L [111] Ξd (s11 + 2s12) + Ξu (s11 + 2s12 + s44) Γ Γ [000] Ξd (s11 + 2s12) Table 6.1: Relationships between the shift of the band edge energy and the deformation potentials for a given valley, depending on the stress direction applied on the structure [17] and for the valley at Γ: Γ Γ δ = Ξd Tr(ε) (6.5) These are the general expressions to calculate each deformation potential for a given valley v and given perturbation. The explicit expression can be found in the work of C. Herring [16] or more recently in the work of E. Ungersb¨ock [17]. In the following table, we give only the relationships used for our calculations. So the shift of the band edge is measured when applying different pressures on the structure according to the directions given in table 6.1. This shift is differentiated according to pressure. These values obtained are then used to calculate the uniaxial shear and dilatation deformation potential for each valley. For L and ∆ valleys, we v v obtain three equations with two unknown parameters (Ξu and Ξd) while for Γ, there -86- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... Γ are three equations for one unknown parameters (Ξu). As the calculation is not theo- retically perfect, for each valley, the equations are not linked. Therefore, depending on the equations considered, the results are going to change. For L and Γ there are three possibilities for each unknown parameter. So the average value is taken into account. Same for the Γ band edge. 6.3 Calculated deformation potentials for silicon, germanium and silicon germanium alloys 16 14 12 ∆ Ξu 10 10 ∆ Ξd 8 6 ∆ Ξu 4 ∆ Ξd 0.5 0.6 0.7 0.8 0.9 1 Composition x Figure 6.2: Calculated uniaxial shear and dilatation deformation potential of the ∆ valley of silicon germanium alloys according to its composition The deformation potentials have been calculated for different compositions of silicon germanium alloys. A small program was written to execute the whole calculation from -87- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... the simulated band structures. The aim of our work is to analyse a possible effect of the alloy composition upon the electron-phonon interactions. 41 −5.8 −6 −6.2 −6.4 40 −6.6 L −6.8 Ξu L Ξd 39 38 ΞL 1-8--=-1u ΞL ~d 0.5 0.6 0.7 0.8 0.9 1 Composition x Figure 6.3: Calculated uniaxial shear and dilatation deformation deformation potential of the L valley of silicon germanium alloys according to its composition So the same process described in the DFT chapter was applied to a conventional cell with virtual crystal optimization. The DFT was run with a LDA functional and the structures were described with norm conserving pseudopotentials. In the case of germanium, a simple pseudopotential that considers only 4s and 4p is used considering the 3d as core electrons. Except for one potential, calculated results are close to what is found in the lit- L erature. The biggest error comes from Ξu where we obtain a value of 37.7eV while ∆ the literature account for a value between 6.5eV up to 18eV [32]. The value for Ξu found of 9.41eV correspond to what is found by Fischetti [32] and in the literature. For -88- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... 0.25 0.2 0.15 0.1 0.05 Γ Ξu +------0 −0.05 −0.1 −0.15 −0.2 −0.25 0.5 0.6 0.7 0.8 0.9 1 Composition x Figure 6.4: Calculated dilatation deformation potential of the L valley of silicon germanium alloys according to its composition ΞL and Ξ∆, respectively 5.93eV and 10.394eV , seem to reasonably agree with the d d − literature with no big differences. The calculated uniaxial shear deformation poten- L ∆ tials for germanium are 19.63eV for Ξu (L-valley) and 8.25eV for Ξu (∆ Valley). The ∆ L dilatation deformation potentials are 3.82eV for Ξd , 5.16eV for Ξd and 0.86eV for Γ − Ξd . Except for the value of the uniaxial shear deformation potential for silicon which is higher than what is found in the literature, the overall deformation potential are in agreement with other works on the topic. Using the Virtual Crystal Approximation, deformation potentials were also calcu- lated for few silicon germanium alloys: compositions of x = 50%, 75% and 80%. Figure 6.3, 6.2 and 6.4 show the variation of the different potential with the composition. As expected, there is no major influence of the alloy composition upon the electron-phonon -89- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... 2000 .... ------+- .. ------+ 1800 1600 ~------·-·------~ 1400 1200 L-G parallel L-G perpendicular 1000 D-G parallel I=! D-G perpendicular 800 600 • ------Contribution to scattering rate 400 • --- -·-·- -- -..""" 200 0 0.5 0.6 0.7 0.8 0.9 1 Composition x Figure 6.5: Contribution of intervalley scattering for + parallel and perpendicular in L valleys, and parallel and perpendicular in ∆ valley according◦ to Herring’s calculations [16] 4 ∗ scattering process. But slight changes can be seen at 80% for the ∆ valley on figure ∆ ∆ 6.2:an increase of 2eV for Ξu and a decrease of 1 2eV for Ξd . In the case of the L − L valley, figure 6.3, there is a slight decrease of less than 1eV for Ξd through the range L of study while the absolute value of Ξu decrease by 3eV between 50% and 100% of silicon. The deformation potential at Γ does not vary much and stays close to zero (figure 6.4). No conclusion can be made upon the overall electron-phonon scattering as these deformations potentials must be integrated through the first brillouin’s zone and according to the possible scattering allowed by the conservation of the energy. In his study, Herring developed a mathematic tools for this integration [16]. This math- ematical approach leads to equations 49 and 50 of the cited article. We include our -90- Chapter 6. Phonon-electron interactions: Linear Deformation Potential ... calculated deformation potentials into the part of the equation into brackets for the parallel and the perpendicular scattering. The values obtained appear in figure 6.5. This figure clearly shows that the electron-phonon scattering rates are not affected by the alloy composition. These contributions remain within 5% of the value calculated for silicon. Deformation potential theory has been applied for sixty years. With the improve- ment on simulation packages, this theory can be applied to nanoscale structures such as layer superlattices, nano wires, nano wires superlattices and even nanodots su- perlattices. This method shows if there is any improvement on the electron phonon scattering for thermoelectric applications. We conclude there is no enhancement of electronic carriers’ mobility by alloying silicon and germanium. 6.4 Summary In this chapter, we focus on the phonon-electron interaction as it is dominantly re- sponsible of the variation of the electrons mobility. In this interaction, we study more precisely the inter valley scattering of electrons by phonons. Despite the simplicity of measuring the mobility of a sample, the theoretical calculation of the mobility is very complex as it must consider the probability of various scattering mechanisms. In the case of silicon germanium, we apply the deformation potential theory to our DFT cal- culations. Deformation potential theory was developed in 1956 but the improvement of simulations for the past few years makes this method more interesting. We calcu- lated the deformation potentials for silicon and germanium and compare them to the literature. The results agree. We then applied this method to silicon germanium alloys for different composition, using the VCA. because of the minor changes in the defor- mation potentials, we conclude that there is no major change in the electron-phonon scattering in silicon germanium alloys. The interesting point of such a method is that it can be applied to any structure. With enough computing power, one can calculate the electron-phonon interaction to superlattices of even nanowires and nanodots. -91- CHAPTER 7 FABRICATION OF SI/GE HETEROSTRUCTURES FOR THERMOELECTRIC APPLICATIONS Thermoelectrics have found applications in very diverse fields from every day commer- cial use to aerospace. In the past two decades, it has been found that silicon germanium superlattices structures yield a higher figure of merit due to the decrease of the thermal conductivity at interfaces [11]. There are many different methods of deposition for silicon germanium heterostruc- tures. Ultra High Vacuum Chemical Vapour Deposition (UHV-CVD) was already used by A.G. Cullis in 1971 to grow silicon and germanium thin films on (111) orientated silicon substrate [57]. This method was also used by M.Tomitori to grow germanium thin films on silicon substrates [111] and by S.R. Cheng in 2002 to grow five super- lattice of silicon/silicon-germanium of 115nm overall [112]. Pulsed Laser Deposition (PLD) is also another method of deposition of heterostructures [63, 113]. We choose to use radio frequency magnetron sputtering to grow heterostructures silicon and sili- con/germanium layers. 92 Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... power pressure plasma deposition rates (W )(mT orr)(sccm)(A˚.min−1) SiO2 100 3 42.5 Ar/7.5 O2 (15%) 11 Si 100 3 50 Ar 39.6 Ge 100 3 50 Ar 38.1 Table 7.1: Measured deposition rates obtained with sputtering 7.1 Silicon germanium alloy on silicon Thin films sputtering 7.1.1 RF Sputtering RF magnetron sputtering is a physical vapour deposition process for thin films. The aim is to hit a target with particles which ejects atoms from the target. These ejected particles then deposit on a substrate. Under vacuum, a plasma made of argon and/or oxygen is created to bombard either a high purity silicon or germanium target. The ejected particles drop on a silicon substrate of 5 centimetres diameter. The targets are positioned at a 45◦ and 12.5 cm from the substrate. These values were kept for all the depositions. Prior to the deposition, the silicon substrate was RCA1 cleaned. A predeposition stage of 10 minutes was set to clean the targets with cold plasma with a power of 80 W . No deposition was done during this stage. The first layer grown is SiO2 to facilitate electronic measurements of only the deposited layer. These silicon dioxide layers are well studied and grown either by wet oxidation at 1100◦C of a silicon substrate [70] or by deposition on the substrate by sputtering [69, 72]. We choose the second method to avoid removing the substrate out of the chamber, limiting deposition of impurities on the silicon dioxide layer. The deposition rate of Si, Ge and SiO2 are determined in order to control the depositions. The thickness of the SiO2 layer can be roughly determined thanks to the colour chart developed by HTE Labs [114]. Further study will give us a more accurate deposition rate for silicon dioxide of 11 A.˚ min−1 with the deposition parameters given in table (7.1). These values are obtained by a synthesis of results from reflectivity and TEM analysis. -93- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.1.2 Deposition process There are different possibilities for the deposition of alloys thin films. The simplest of all is to sputter from an alloy target which has the desired composition [73]. Another method would be to stick pieces of silicon wafers on a germanium target. Assuming that silicon and germanium have similar deposition rates, the composition of the alloy thin film will correspond to the area that the silicon wafers cover on the germanium target [69,72]. This method is simple but it involves a sticking component that would contaminate the germanium target. Moreover, this kind of deposition will erodes the germanium unevenly which leads to a strong variation of the deposition rate in time. To avoid contamination and to keep the accuracy in the deposition, we choose to alternate the deposition. Our chamber counts two RF guns, so the plasma can be turned on simultaneously on both silicon and germanium targets. With the use of the shutters on the targets, the silicon and germanium can be alternated: one shutter open for few seconds while the other is closed, and then they switch. These cycles are repeated as many times as wanted. By opening one shutter longer than the other, we roughly control the composition of the alloy thin film. The time for each shutter is given in table (7.2) for each sample. Unfortunately, the deposition rates were not known, these depositions time were set with the guess that the silicon deposition rate was higher than the germanium’s one. So we compensated a bit by increasing slightly the opening time for germanium. Further results showed that the deposition rates were similar for silicon and germanium and that the alloy thin film contains more germanium than silicon. -94- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... plasma temperature Sample layer cycle target time (sccm)(◦C) 1 42.5Ar SiO Si 20min 500 2 /7.5O (15%) 1 2 Ge 15s 50Ar 525 SiGe 30 { Si 10s 50Ar 525 1 42.5Ar SiO Si 20min 500 2 /7.5O (15%) 2 2 Ge 10s 50Ar 525 SiGe 30 { Si 10s 50Ar 525 1 42.5Ar SiO Si 20min 500 2 /7.5O (15%) 3 2 Ge 20s 50Ar 650 SiGe 30 { Si 15s 50Ar 650 1 42.5Ar SiO Si 20min 500 2 /7.5O (15%) 4 2 Ge 15s 50Ar 650 SiGe 30 { Si 10s 50Ar 650 Table 7.2: Sputtering parameters for the deposition of SixGe1−x layers on SiO2 with power fixed at 100 W and vacuum at 3 mT orr -95- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2 Characterization of the heterostructure Using the sputter deposition, thin films are obtained. But nothing is known about these layers. Our study focuses on the thickness of film, their possible crystal structure, their composition and the roughness of interfaces. These parameters were found to have effect on the electronic properties and on the thermal properties [31, 38, 45, 47, 50, 53, 115, 116]. A powerful tool to study a layer structure is the X-Ray Diffraction (XRD) under the condition of a crystal structure and a low roughness at interfaces [70, 71]. We also used TEM imaging to validate the XRD measurement. 7.2.1 Study of the phase by X-ray diffraction Experimental details A crystal structure is defined by the way atoms are arranged periodically in all three directions. Hence it is possible to define families of parallel atom planes in the structure. Distances between two neighbour planes from the same family varies between 0.15 A˚ and 15A.˚ These distances depend on the crystal structure itself, and the size of atoms composing the structure. To each family corresponds one distance. An X-ray directed toward a crystal is diffracted by each one of the family of planes every time Bragg’s law is verified (Equation 7.1). Figure 7.1: X-ray diffraction between two planes of the same family To obtain the diffraction, the diffracted waves from one family of plane must have the same phase. When this happens, the angle of diffraction θ is linked to the distance between planes via Bragg’s law. Consequently, with λ known and θ measured, it is possible to obtain the distance d between planes from different family of planes. With -96- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... several d measured from the sample and identifying them with a database, the crystal structure can be deduced. For our experiment, we use the 2θ method which consists in a fixed X-ray source, a θ-rotating stage for the sample and a 2θ-rotating detector as shown on figure 7.2 Figure 7.2: XRD equipment settings for the 2θ measurement The X-Ray Diffraction study is done using the Empyrean xrd developed by PAN- alytical. This X-ray spectrometry relies on copper Kα emission with a wavelength of λ = 1.5418nm which is of the order of the distance d between adjacent crystal planes, enabling diffraction along the angle θ. The diffraction follows Bragg’s Law: d sin θ = nλ (7.1) As the sample is thin, the grazing angle diffraction is selected to study the surface as the penetration is limited (up to 500 nm deep). To obtain the best results, each sample was 5 cm diameters. The machine has an auto-alignement procedure which simplifies the manipulation. It is recommended to use a divergence slit of 1/8◦ for silicon and ◦ ◦ 1/4 for germanium. We chose a divergence slit of 1/8 to study our SixGe1−x layers on top of SiO2 on a silicon wafer. Results and Discussion Samples from 1 to 4, described in table (7.2), are studied by grazing angle diffraction. Figure 7.2.1 show the four scans. -97- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... ) ) ) r 1 0 1 e 1 2 1 f 1 2 3 a 2 ( ( ( O w i e e e i S G G G S XRD Intensity (a.u.) 20 30 40 50 60 70 80 90 100 Position (2θ) Figure 7.3: Grazing angles diffraction of samples 1, 2, 3 and 4 from table 7.2 from bottom to top respectively. It is known and admitted that direct sputtering deposits amorphous layers [66–68]. It is also common to do post-annealing at temperatures from 550◦C to 1000◦C for several hours, considering that the annealing temperature is 525◦C and depends no the concentration of possible dopant [69–73, 111]. No post-annealing is done for our samples, but we control the deposition temperature during the deposition to obtain a crystal structure. This temperature is always greater than the annealing temperature found in the literature. The first two samples are deposited at 525◦C whereas samples 3 and 4 are deposited at 650◦C. The grazing angles X-ray scans all show the silicon substrate peak at around 55◦ and a peak at 22◦ which corresponds to the silicon dioxide layer. The peaks relative to germanium do not appear on the scan for the depositions at 525◦C but appear for the depositions at 650◦C. Even though, these peaks are not well defined, they clearly show the formation of a crystal structure of -98- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... germanium. This result can be explained by the Stranski-Krastanov growth mechanism of germanium [117]. We deduce that micro grains of germanium crystallise in the layer as well as homogenous alloy phase as the silicon peak tends to shift toward the left. The formation of the crystal structure is limited by the silicon dioxide layer as it prevents a preferred growth orientation. Solutions are either post-annealing or an increase the temperature of deposition. Increasing the content of silicon in the alloy layer may also be a solution to obtain a crystal structure as germanium shows an islands growth when deposited on silicon, due to the lattice mismatch. -99- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2.2 Determination of thin films’ thickness by reflectivity Principle of X-Ray Reflectivity X-Ray Reflectivity (XRR) is a tool to probe surfaces and interfaces in a layered struc- ture. The principle of XRR is to measure the reflection of an x-ray on a sample at a grazing angle. Below a critical angle, according to Snell’s law (small angle limit) and because of the difference of the refractive index between the material and its environ- ment, the ray is completely reflected. Snell’s equation: θ3 θ1 n1 θ1 = θ3 n2 (7.2) sin θ1 n2 θ2 = sin θ2 n1 Figure 7.4: Illustration of Snell’s law Above this critical angle, one part of the X-ray is reflected and one part is trans- mitted. The transmitted part goes through the material until it reaches an interface, where a part is reflected and one transmitted, like on figure 7.5. The use of a grazing angle is the significant point in the experiment as it enables the diffraction figure of interfaces several tenth of micrometre apart. From 10nm to 500nm. The reason comes from Bragg’s law, equation 7.1: with a very small angle θ, the distance d must be big for the product dsinθ to equal nλ. Then, the theory of diffraction states that diffraction happens when the difference of length between reflected ray rI and rII is equal to k 2π, with k an integer (cf figure 7.5). Consequently, for each × 2π layer related to a thickness d, there is a reflection every time θ = k d . The measured reflectivity (figure 7.6) hence shows periodical fringes with a periodicity linked to the thickness of the thin film. In the case of two layers on a substrate, the measurement shows two combined periodicities. -100- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Figure 7.5: Reflected and transmitted X-rays in a layered structure From this measurement, knowing the density of the air ( approximately 1 ) it is possible to calculate the density of each layer by using the value of the critical angle for transmission and Snell’s law. It is also possible to calculate the roughness of each interface as the effect of a rough surface will attenuates the amplitude of the periodic peaks. This calculation is implemented in the software and identifies a measurement to a simulation to obtain the roughness and the density of each layer. Figure 7.6: Typical reflectivity measurement of a simple surface (left) and of a layer on a substrate (right) with a thickness of 2π/∆q -101- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Experimental details Reflectivity is based on the reflection of the X-rays on interfaces. This method is widely used to study thin films and multilayer under the condition of a sharp interface. Reflectivity procedure is done with X’pert PRO Materials Research Diffraction system (MRD) which also use Cu X-ray tubes. On this machine, the alignment of the sample is automatic. The acceptance angle is set at 0.8◦ and we use a soller slit 0f 0.04 rad and a divergence slit of 1/32◦. 8 10 SiO2 7 10 6 10 0.08 5 10 Intensity a.u. 4 10 3 10 2 10 0.2 0.4 0.6 0.8 1 1.2 1.4 angle (degree) Figure 7.7: Reflectivity for a SiO2 single layer on a silicon wafer Results and discussion As said before, reflectivity results strongly depend on the quality of the interface. With a rough interface, the fringes that characterize a layer thickness cannot be seen. Figure 7.7 shows the reflectivity scan for a single deposition of SiO2, sputtered for 20 minutes -102- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... on a silicon wafer. The scan shows some periodic interferences, with periodic peaks. The angle between two peaks is around 0.08 ◦. The Fourier transform of the scan returns a value of 27 nm for the thickness of the layer. The simulation made with the analysis software returns a roughness of 0.8 nm and a density of 2.3927 g.mol−1 which is close but lower than 2.65 g.mol−1, the known density of silica. If accurate, the experimental value of the density is closer to one of cristobalite (2.334 g.mol−1) than of silica. Si on SiO2 7 10 6 10 5 10 0.15 Intensity a.u. 4 10 3 10 0.5 1 1.5 2 2.5 angle (degree) Figure 7.8: Reflectivity for a Si on SiO2 layers on a silicon wafer Figure 7.8 show the reflectivity result for a silicon layer grown on a silicon dioxide layer. In this case, only one periodicity can be seen although two periodicities were expected. This may come from the roughness of the top surface which prevents the first reflection of the x-ray. Figure 7.9 shows the reflectivity for Sample 3. The long periods correspond to a 27.0 nm thick SiO2 layer and the short period corresponds to the 60.9 nm thick SixGe1−x -103- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... layer, according to simulations. The roughness is estimated as very sharp between the wafer and the silicon dioxide layer. The simulated roughness of the silicon dioxide layer is 0.72 nm and the one the silicon germanium alloy is 4.4 nm. The density simulated for the silicon germanium alloy is 4.55 g.mol−1, and 2.37 g.mol−1 for silicon dioxide. The latest density is close to the one simulated from the silicon dioxide reflectivity, figure 7.7. The density for the alloy remains between the density of germanium (5.32 g.mol−1) and the density of silicon (2.33 g.mol−1). If the composition of the alloy is to be determined by the densities, it would give Si0.26Ge0.74. We just consider this composition as a rough idea due to the lack of accuracy of the simulated results. 6 10 Sam ple 3 5 10 4 0.01 10 1.16 3 10 Intensity a.u. 2 10 1 10 0 10 0.2 0.4 0.6 0.8 1 1.2 angle (degree) Figure 7.9: Reflectivity measurement of Sample 3. -104- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2.3 Imaging of heterostructures with transmission electronic microscopy Transmission electron microscopy (TEM) was invented in 1931 by Max Knoll and Ernst Ruska (Physic Nobel prize in 1986). It consists in hitting a thin sample with a coherent electron ray and visualizing the diffraction pattern created by the sample’s crystal structure. The resolution is limited by Broglie’s wavelength of electrons which depends on the accelerating voltage. In reality, the resolution is around few Angstroms.˚ Figure 7.10: Schematic figure of a TEM -105- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... So a TEM focus an electron ray produced by an electron gun with magnetic lenses onto a thin sample (cf figure 7.10). A diffracted pattern is projected to the other side of the sample, in the focal plane of the projection lens. This projection lens act as a Fourier transform and projects the sample image on the sample plate. Consequently, two functions are available with the TEM: the image mode and the diffraction mode. In the image mode, depending on the sample thickness, density and composition, electrons are absorbed. By placing the CCD sensors in the imaging plate, an image of the area of the sample hit by the electron ray can be seen. In the diffraction mode, when electrons hit a crystal structure, them will be sent toward directions that depends on the crystal stacking. Figure 7.11: ”Lift-out” method to mill a TEM sample using the FIB The sample must be very thin so the sample preparation is very important. From our heterostructure, we must cut a slab from the surface few micrometres deep and few hundreds nanometres thick. To do so, we use the focused ion beam (FIB) microscope. Prior to the use of the FIB, the sample is coated with a conducting layer of Au and of P t. We use the FIB to mill two square holes close to each other by the desired thickness of the slab on the surface of the sample (cf figure 7.11). Then the slab separating these two holes is cut off the sample by ion milling again and is taken out with a very thin needle under an optical microscope. Static electricity at the tip of the needle gets the slab to stick to the tip. This slab is then put on a very thin graphite grid, which is the sample holder for the TEM. It is called the lift-out method. This method is very delicate to achieve (loss of the ion beam milled slab during the lift out with the needle) but is time efficient compared to the ”H bar milling” which demands a longer milling -106- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... time. Figure 7.12: Grain structure by TEM imaging of Sample 3 Experimental details The Transmission Electron Microscopy was done with the Philips CM200 which allows very high resolution images. This TEM is able to perform Energy Dispersive X-ray spectroscopy (EDX) on localised points of the sample. Sample 3, described in table 7.2, is studied in this part. The preparation starts with the deposition of a conducting layer for the use of a Focused Ion Beam (FIB) equipment to cut a very thin slice of the sample. So a gold and a platinum layers were deposited on the top of the sample. Once a thin slice is cut with the FIB, it is put on a substrate holder. Finally, the sample is ready to go in the TEM. Results and discussion Figure 7.13 shows the deposited structures with the conducting layers on top. The bright side is the silicon wafer, then the darker first layer is the silicon dioxide. The next layer is the silicon germanium alloy, then the gold layer and the platinum layer. The thickness is easily measured: 22.06 nm for SiO2 and 67.46 nm for SixGe1−x. It appears clearly that the interface between the silicon dioxide layer and the silicon wafer is very sharp. The interface between the insulator and the alloy is less sharp but still -107- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Figure 7.13: thickness of SiO2 and SixGe1−x thin films by TEM imaging of Sample 3 good, whereas the top surface of the alloy is very rough. These results validate the reflectivity study of Sample 3. The thickness of the layers also matches the thickness found with reflectivity. On Figure 7.13 as well as on figure 7.12, the texture of the alloy layer is less homogeneous compared to the texture of the silicon dioxide layer. It implies a hint of a micro grains structure of the silicon germanium layer. -108- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2.4 Thin film composition by energy dispersive X-ray spec- trometry The energy dispersive X-ray spectrometry (EDX) consists in focusing the electron beam in the TEM on the sample and study the X-ray emission from the sample. The emitted X-ray depends on the composition of the sample. The emission takes place from the surface of the sample down one micron deep. It implies that the sample preparation by FIB for the TEM is not too thin. Nonetheless, the resolution of the EDX used for our study is in the 10 nm range which enables us to focus on each one of the deposited layer (cf figure 7.14 for the interaction volume). Figure 7.14: Interaction volume of the various electron-sample interactions. The X-ray resolution is around 1 micron The EDX sensor is a Li-doped and polarized silicon monocrystal. The emitted X-ray of energy E0 produce in this crystal sensor a number N of electron-hole pairs which depends of the energy E0. This produces a voltage across a capacitor which is also proportional to the energy E0. An analyser sorts the number N as a function of the electron beam energy. This forms the EDX spectra of the sample. -109- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Results and discussion Figure 7.15: Location of the 5 points where the EDX spectra was measured In our experiment, we studied the spectra of each layer forming our sample: the silicon wafer, the insulator SiO2 layer and the silicon germanium alloy layer. Each measurement point is indicated on figure 7.15. Figure 7.16: EDX spectra of point ’lois 1’ which is the silicon wafer The spectrum for the silicon wafer appears on figure 7.16. We identify two peaks: the big silicon peak and a smaller carbon peak. We know that the silicon wafer contains -110- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 99.99% pure silicon. Consequently, the carbon content comes from the graphite grid from the sample holder. Figure 7.17: Composition of the SiO2 layer at point ’lois2’ Figure 7.17 show the composition of the insulator layer. Here again, we dismiss the carbon composition as the composition of the sample holder. We clearly identify two peaks as silicon and as oxygen which corresponds to the SiO2 layer. However, the silicon peak is twice higher than the oxygen peak. Figure 7.18: Composition of the SixGe1−x layers at point ’lois 3’ Figure 7.18 is the spectra of the silicon germanium alloy layer. Same as previously, -111- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... the carbon peak is related to the sample holder. The oxygen content comes either from oxidation or interdiffusion between the alloy layer and the insulator layer. We also identify one silicon peak and one germanium peak. If intensities of both peaks are compared, the given composition at this point would be Si20Ge80. But the overall composition of the alloy calculated with the deposition rates gives Si40Ge60 for this sample. Consequently, this EDX study highlights a possible variation of the alloy composition along the film which may be caused by the formation of germanium micro grains. Figure 7.19: Composition analysis n◦ 2 of Sample 3 along the arrow Figure 7.19 shows the composition of the structure along the arrow for three dif- ferent locations on the sample. Here again the interface between the insulator and the wafer appears very sharp and the interface between the alloy and the insulator less sharp. The surface of the alloy seems to intermix with gold: gold traces are found in the alloy layer. So is oxygen. This intermixing may have two origins: The roughness of the silicon germanium alloy being significant, the interface between Au and the alloy is -112- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... not sharp. The second reason comes from the EDX accuracy which is around 10 nm. This lack of accuracy prevents the EDX to distinguish sharp interfaces (such as silicon substrate and SiO2 insulator layer). When comparing the three spectra, the ratio of germanium content on silicon con- tent varies roughly from 1 to 1/4 which corresponds to compositions of Si50Ge50 to Si20Ge80. This variation contributes to the hypothesis of the formation of micro grains. -113- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2.5 Band gap engineering with heterostructures UV/Vis/NIR Spectrophotometry consists in studying the reflectance of a sample using light emission from the ultraviolet (UV) through visible range (Vis) to near infra- red (NIR) frequencies. This type of spectrometry measures the magnetic interaction between the incoming wave and the matter. These interactions cause electronic transi- tions and the most common electronic transition in a semiconductor is the jump from the valence to the conduction band. So we use this kind of spectrometry to measure our samples’ band gaps. Figure 7.20: schematic settings of a spectrophotometer -114- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Figure 7.20 show the schematic description of the spectrophotometer. The wave source is made from deuterium and halogen (D2 and W 1) whose light goes through a double holographic grating monochromator (G). S2 is a beam mask to adjust to the sample dimension and F is a depolarizer to correct of inherent instrument polarization. The beam is then split in two to allow a reference measurement at the same time as the sample measurement. The two split beams go through the sample compartment and finally reach high sensitivity photodiodes. 3 2.5 2 2 %)) R ( 1.5 hνF ( − hν 1 0.5 0 0.6 0.8 1 1.2 1.4 1.6 hν (eV) Figure 7.21: Tauc diagram for of Ge on SiO2 (blue), Ge on Si wafer (green) and Si on SiO2 (red). The measured band gaps are, respectively, 0.74eV , 1.02eV and 1.04eV Experimental details Spectrophotometry is conducted with Lambda 950 UV/Vis/NIR Spectrophotometer from PerkinElmer. Spectrophotometry measures reflectance of a sample from light in -115- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... the range of ultraviolet up to near-infrared. Several samples were studied: the first sample is the deposition of a germanium layer on a silicon dioxide, the second is a layer of silicon on silicon dioxide and the third is germanium grown directly on a silicon wafer. Layers of alloy are also studied: Sample 2, 3 and 4 are also studied. Their indirect band gap are measured (cf table 7.2). Results and discussion 1.4 1.2 1 2 %)) 0.8 R ( hνF ( 0.6 − hν 0.4 0.2 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 hν (eV) Figure 7.22: Tauc diagram for of Sample 2 (blue), Sample 3 (red) and Sample 4 (green). The measured band gap are, respectively, 1.04eV , 1.044eV and 1.014eV The reflectance for each sample is measured. To determine the band gap, the Tauc plot of the absorbance is traced. We use the Kubelka-Munk function F to express the absorbance α according to the measured reflectance R. The function is expressed as follow: -116- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... (1 R)2 F (R) = − = α (7.3) 2R Once the absorbance is calculated, the Tauc plot is trace with hν on the x-axis and hν (hνF (R))2 as we measure indirect band gap transition. The tangent of the curve’s − step is trace and the value of the band gap determined. Figure 7.21 show the Tauc plot for germanium and silicon thin layers. The measured band gap for the germanium layer on top of a silicon dioxide layer is 0.74eV , whereas the germanium layer on top of the silicon wafer is 1.02eV . The band gap measured for the layer of silicon on top of a silicon dioxide layer is 1.04eV . Figure 7.22 show the Tauc plot for the alloy thin films. The measured band gap are 1.04eV for Sample 2, 1.044eV for Sample 3 and 1.014eV for Sample 4. These band gap values are close to one another. -117- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... 7.2.6 Atomic Forces Microscopy measurement of the surface roughness Experimental details The atomic force microscopy (AFM) relies on tiny cantilever scanning the surface of a sample. The cantilever is usually made of silicon and its size is approximately 450 µm long, 50 µm wide and few micron thick. The end of the cantilever holds a few micron long tip (cf. Figure 7.23). Figure 7.23: Cantilever tip for atomic force microscopy applications When the tip of the cantilever is approached to the sample surface, the cantilever is deflected by either capillarity or attractive force. This deflection is then measured by a laser reflection on the tip of the cantilever. There are two ways to scan a surface. The first way is the contact method: in this case, the surface is scanned with a constant deflection due to interaction between the tip and the sample. To keep the deflection constant, the microscope has to compensate by keeping a constant distance between the tip and the sample. This compensation is saved and traced with scanning and represents the variation of the sample’s surface height. The second method, called semi-contact method, is to oscillate the cantilever at its resonant frequency (typically several MHz) and to scan the sample surface. The vari- ation of interaction forces between the sample and the AFM tip changes the resonant frequency of the cantilever. This leads to the decrease of the oscillations amplitude. -118- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... In this method, the microscope regulates the height of the tip to obtain the maximum oscillation amplitude, which corresponds to the resonant frequency and hence describes the surface of the sample. 4 3.5 3 2.5 2 1.5 1 0.5 0 Si substrate Ge (low T) Ge (high T) Sample 3 sample 4 Figure 7.24: Measured surface roughness with atomic force microscopy of a bare silicon wafer, of low and high temperature sputtered germanium layer and sample 3 and 4 described in table 7.2 Figure 7.25: AFM scan of a 2 by 2 µm area of sputtered germanium layer deposited at 650 ◦C Experimental details AFM scanning has been done on the bare silicon wafer, on samples 3 and 4 and on two layers of germanium directly deposited by sputtering on a silicon wafer. The first -119- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... germanium layer was deposited at 350 ◦C whereas the second one was deposited at 650 ◦C. Sample 3 and 4 are a silicon germanium alloy layer on a silicon dioxide layer deposited on a silicon substrate (cf table 7.2). For each sample, the AFM scanning was made by the semi-contact method on a 20 by 20 µm area on ten different locations in order to verify the uniformity of the measures. The average roughness was measured for each scan. Because few impurities were present on the substrate before any deposition, amongst the ten locations, the highest value of the roughness was not taken in consideration. As the sample may not have been flat on the sample holder stage, a plane correction was made to each scanning prior to any roughness measurement. Results and discussion In order to have an idea of the initial roughness, we measured the roughness of a bare silicon substrate. The AFM is very accurate and these measures remained below the nanometre. Figure 7.26: AFM scan of a silicon dioxide layer deposited on a silicon wafer We then studied the deposition of germanium for two temperatures, 350 ◦C and 650 ◦C to study the effect of the temperature deposition on the roughness of germanium. The low temperature deposition does not permit annealing of the germanium layer, hence the material is in its amorphous form. In the contrary, the high temperature -120- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... deposition enables the formation of germanium crystal grains (verified with XRD). Figure 7.27: AFM 3D image of a scan of a silicon layer deposited on a silicon dioxide layer In the case of the annealed germanium the measured roughness is twice higher than the roughness of the amorphous layer. Moreover, in the case of the high temperature deposition, islands are formed at the surface, figure 7.25. This island growth is widely reported and explained by the Stranski-Krastanov growth the [117,118]. The silicon dioxide layer studied was sputtered at 500 ◦C. The measured roughness remained below 1.5 nm. Figure 7.26 shows a one by one micron scan of the silicon dioxide layer. This picture clearly shows a repetitive pattern in the deposition which looks like a triangle with the same angles and same orientation. This is caused by the mismatch of the silicon dioxide layer and the silicon substrate [119]. Figure 7.27 and 7.28 show the 3-D scan of 1 x 1 micron area and 2 x 2 microns area of sputtered silicon on silicon dioxide on a silicon substrate. Here again, we find the repeated patterns that are found on the silicon dioxide layer but thicker. The average roughness in this case was around 2 nm. -121- Chapter 7. Fabrication of Si/Ge heterostructures for Thermoelectric ... Figure 7.28: AFM scan of a silicon layer deposited on a silicon dioxide layer 7.3 Summary In this chapter, we study the sputter deposition of silicon germanium alloys and silicon dioxide thin film. We then characterized the depositions with XRD, reflectivity, TEM, EDX, AFM and UV spectrophotometry. The sputtering parameters are diverse, we started with the same parameters found in the literature. Then, we optimized the parameters for a controlled deposition. To avoid targets contamination, the deposition of silicon germanium alloy thin film was done by alternating short period of time of silicon and germanium depositions. With a temperature of deposition of 650 K, we start to obtain crystal structure instead of an completely amorphous layer. By controling the time of the alternated deposition, we control the composition and the thickness of the alloy thin films. The calculated deposition rates are shown in table 7.1. We studied sample 3, and despite rough interface, reflectivity and TEM imaging validated the thickness 67 nm and the composition Si0.4Ge0.6. -122- CHAPTER 8 CONCLUSION Silicon and germanium materials have many advantages for thermoelectric applications. As semiconductors, they are widely studied and technologies to tune their electronic properties is proven for decades. Moreover, when alloyed, these two materials exhibit a particularly low thermal conductivity which is a significant benefit for thermoelectric applications. Adding to it, with new state-of-the-art equipments, silicon and germa- nium spearhead the field of nanotechnologies; with their use in thin films, nano wires and quantum dots superlattices structures. These new nano sized structures display some promising thermoelectric efficiencies. Finally, these two semiconductors show an unmatched stability at high temperatures. In the first part of this thesis, we linked the knowledge of thermoelectrics with the solid state theory, usually applied to semiconductors, to understand and identify the many parameters that control the thermoelectric effect. We also developed a mathe- matical approach to calculate the thermoelectric figure of merit with the fewest input as possible. By using DFT, we successfully near ab initio calculated the figure of merit for bulk silicon and germanium. With minor modifications and approximations, and with access to higher computing power, this approach could be applied to silicon germanium alloys and silicon germanium nano structures. During the development of this approach, we have seen that intrinsic properties to silicon and germanium strongly influence their efficiencies for thermoelectric ap- 123 Chapter 8. Conclusion plication. It is common knowledge for TE improvement, known as ”electron-crystal, phonon-glass” approach, that the Seebeck coefficient and the electronic conductivity must be enhanced whereas the electronic and the lattice contributions to the thermal conductivity must be limited. Behind this knowledge, thanks to the solid state theory, we identify the band gap as the main reason for thermoelectric efficiencies at high temperatures. The example of the narrow band gap bismuth telluride validates this conclusion. Added to doping, a wide band gap sustain the improvement of doping until higher temperatures. This is explained by the fact that an intrinsic material behaves as an intrinsic semiconductor at high temperatures, the fermi level of a doped materials get closer to the one of an intrinsic material. And the effect of dominant majority carriers acts mainly upon the Seebeck coefficient. We also stress that, although heavy doping increases the electronic conductivity and the seebeck coefficient, it also increase the electronic contribution to the thermal conductivity. We demonstrated that this thermal contribution becomes significant around 700◦C for germanium. Consequently, doping must be optimized to obtain the highest efficiency possible. In the search of electronic properties, we use a density function code (DFT) to simulate, from their crystal structures, the band structure and the density of state for silicon, germanium and silicon germanium alloys. These types of calculation depends on the description of the outer shells electrons approximated by pseudopotentials. But, while pseudopotentials for silicon are reliables, the case of germanium can be a chal- lenge. Through this study, we also identify the mobility of electronic carriers as a main parameters in the thermoelectric efficiency. In the Relaxation Time Approximation (RTA) we considered an overall lifetime that accounts for all types of electrons and holes scattering in the matter. It is a complex task to calculate this mobility as it depends on many parameters. It is the reason why we first introduced experimental data in our calculation of the figures of merit. But we identify three main types of scattering process in bulk; the scattering due to the lattice (also known as electron- phonon scattering), the electrons-electrons scattering (dominant at high temperatures) and the surface scattering (dominant at low temperatures). The electron-phonon being the most significant one, many research teams focus their work on its calculation. Despite the fact that the measurement of mobility in a sample is easy, calculating, and hence predicting the mobility is more complex. Thanks to DFT and new calculation -124- Chapter 8. Conclusion tools, the electron-phonon interactions start to show agreement with experimental data. In the search of a possible ab initio calculation of the figure of merit for a semiconductor, using our mathematical approach, we followed C. Herring and E. Vogt deformation potential theory, that we coupled with DFT simulation in order to calculate electron- phonon inter-valley scattering probability. We applied this calculation to different compositions of silicon germanium alloys. The results validate the literature: there is no enhancement of the mobility of electronic carriers by alloying silicon and germanium. But here again, the method could be ap Finally, our experiments focused on the deposition by sputtering of silicon and germanium thin films on a silicon substrate. We successfully defined a set of sputtering parameter to control the composition growth of silicon germanium alloy thin films and silicon dioxide thin films. Without annealing, we obtain crystal grains in the thin films for deposition temperatures of 650 ◦C . With the calculation of the deposition rates for both silicon and germanium, we are able to control the composition and the thickness of the alloy thin film by optimizing the shutter opening times in a target alternated method. This alternated method is chosen to avoid contaminating each target. The measured composition and thickness are both verified with reflectivity and TEM imaging and the agreement is very satisfying. We also verify the band gap of sputtered thin films which are close to the values for the bulk provided that layers are separated by an insulator layer (silicon dioxide) from the substrate. The roughness of each surface is also studied for germanium , silicon dioxide and silicon germanium alloy layers. Due to its composition, the alloy thin film shows an average roughness around 2 nanometres. 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The Schr¨odinger equation is: | i H0 ψn = En ψn (A.1) | i | i To this Hamiltonian is added a ’small’ time-dependent coupling W (t) = αw(t) with α 1 . The system is now described by: H = H0 + αw(t) (A.2) w(t)’s eigenvalues must be smaller than those of H0. The new eigenvectors ψ(t) will be | i time-dependent and their eigenvalues will be Taylor series of α. The new Schr¨odinger equation is: d ψ(t) i~ | i = H ψ(t) (A.3) dt | i At anytime t, ψ(t) can be projected on H0’s basis [ ψn ]n. | i | i X ψ(t) = cn(t) ψn (A.4) | i n | i 141 Appendix A. Fermi’s Golden Rule at t < 0,W (t) = 0 The initial conditions are: at t = 0, system is in state ψi | i and cn(t 0) = δi,n ≤ So (A.3) can be written: X dcn(t) X i ψ = c (t)[H ψ + αw(t) ψ ] (A.5) ~ dt n n 0 n n n | i n | i | i we project (A.5) upon ψp knowing that ψp ψn = δp,n and we obtain: h | h | i dcp(t) X i = c (t)E + α c (t) ψ w(t) ψ (A.6) ~ dt p p n p n n h | | i Note that ψp w(t) ψn = wpn(t) is the matrix element of the perturbation w(t). If h | | i the perturbation has no stationary term, wnn(t) = 0. To solve this, we do a change of iEnt/~ variables bn(t) = cn(t)e which are solutions of equation: dc (t) i p = c (t)E (A.7) ~ dt p p Ep−En We introduce Bohr’s pulsation of transitions from state p n : ωnp = . → ~ So (A.6) is written: dbp(t) X i = α w (t)eiωpn(t)b (t) (A.8) ~ dt pn n n This system of coupled equations (consider every state p) is solved by using a Taylor serie of α( 1) , so: 0 1 2 2 2 bp(t) = bp(t) + αbp(t) + α bp(t) + 0(α ) (A.9) We inject (A.9) in (A.8) and we identify term by term. order 0: db0(t) p = 0 (A.10) dt Without any perturbation (α = 0) bp stays in its initial state ψi . | i -142- Appendix A. Fermi’s Golden Rule order 1: 1 X iωpnt 0 dbp(t)t = α wpn(t)e bn(t) (A.11) n iωpit = αwpi(t)e (A.12) 0 Because bn(t) = δi,n according to initial condictions After integration t Z 0 1 0 iωpit 0 bp(t) = α wpi(t )e dt (A.13) 0 In statistical quantum mechanics, the probability for a system to be in final state ψf | i at t < 0 is : 2 2 2 Pif (t) = ψf ψ(t) = cf (t) = bf (t) (A.14) |h | i| | | | | 2 t α Z 0 0 iωfit 0 2 = 2 wfi(t )e dt (A.15) ~ | 0 | t 1 Z 0 0 iωfit 0 2 Pif (t) = 2 Wfi(t )e dt (A.16) ~ | 0 | t 1 Z 0 2 0 iωfit 0 2 Pif (t) = ψf ψ(t) = 2 Wfi(t )e dt (A.17) |h | i| ~ | 0 | This results expresses the probability of a particle in the initial state ψi to reach the | i final state ψf . This probability is the direct consequence of the small time-dependent | i perturbation applied to H0. Note that it depends on t. Hence scattering time caused by small perturbations can be calculated. This Fermi Golden Rule in widely used to study small perturbations. In our study, this rule enables the calculation of electron scattering rates by long wavelength phonons. -143-