Development of Thermoelectric Materials for Cryogenic Cooling and Study on Magnon and Phonon Heat Transport

Transcription

1 Development of Thermoelectric Materials for Cryogenic Cooling and Study on Magnon and Phonon Heat Transport DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Hyungyu Jin Graduate Program in Mechanical Engineering The Ohio State University 2014 Dissertation Committee: Joseph P. Heremans, Advisor Walter R. Lempert Vish Subramaniam Roberto C. Myers

2 Copyright by Hyungyu Jin 2014

3 Abstract This dissertation consists of three interrelated research themes that encompass the engineering and science of thermal material properties. The primary goal is to contribute to solving the world s energy crisis through solid-state conversion of waste heat into usable electrical energy. To achieve this goal, it is necessary to extend our understanding of underlying physics of different heat carriers in solids: electrons, phonons, and magnons. The three research subjects presented here attempt to address this challenge. The first subject is the development of thermoelectric materials for cryogenic cooling applications, including solid-state cooling for far infrared, x-ray, and γ-ray detectors in space applications. Bismuth and bismuth-antimony alloys were chosen as the materials of study. We aim to improve the thermoelectric efficiency (i.e. figure of merit) of these material systems through the development of new doping approaches and other mechanisms. First, we found that indium, the same trivalent element with bismuth, can dope bismuth p-type through a novel doping mechanism and enhance its figure of merit above 150K by 20% as well. Second, an effort is made to improve the figure of merit of p-type bismuth-antimony. It is found that by modifying the band structure with antimony content and by heavily doping with tin, a condition in which multiple valence bands are simultaneously doped is achieved. This leads to the figure of merit of 0.13 at 240K, which is a new record for this system. ii

4 The second part of the dissertation focuses on thermal conduction and the spin- Seebeck effect in ferromagnetic metallic glasses (Metglas). The recently discovered spin-seebeck effect utilizes electron spins, instead of electron charge, to convert thermal energy to electrical energy. Two key mechanisms that mediate the spin-seebeck effect are the magnon thermal conductivity and phonon-drag. The Metglas system, in which most phonon modes are localized, allows a separation of the former from the latter. Through experiments, we find that although it is possible to distinguish the magnon contribution to the thermal conductivity from others, no evident relation is found between the magnon thermal conductivity and the spin-seebeck signals. On the other hand, spin- Seebeck-like signals are detected in the absence of a normal metal detection layer which has been believed necessary for detection of these signals. This observation is attributed to the intrinsic inverse spin-hall effect in Metglas samples. In the last portion, we provide evidence for a new property of phonons, namely, diamagnetism in phonons, which hints at a new approach for magnetically manipulating phonon heat transfer in materials. Through a high accuracy heat flux potentiometric method, we observe a magnetic field dependence of the lattice thermal conductivity in a diamagnetic semiconductor, InSb; a 15% decrease in the lattice thermal conductivity is observed under a 7 T magnetic field at a temperature of 5.2 K. The effect is observed in the temperature regime where the phonon transport is ballistic, but appreciable phonon-phonon interactions still exist. It is shown that phonon-phonon interactions that involve longitudinal acoustic phonons with low energy are related to the observed effect, and that the external magnetic field mediates these interactions through modifying the anharmonicity of associated phonon modes. iii

5 This dissertation is dedicated to my lovely wife Suhye iv

6 Acknowledgments First of all, I would like to express my sincere appreciation and respect to my advisor, Dr. Joseph Heremans, who has patiently guided me to grow as an independent researcher throughout my entire graduate study. His endless passion and inspiration for research as well as for physics lessons always have been great motivations for me. I will not forget the exciting moments we shared with thrills about new findings. I am sure that I will recall later that working with him was a true joy and a lucky part of my life. I truly think that the completion of my PhD is almost entirely indebted to my dear wife, who consistently showed incredible patience and love for probably the worst husband in the past 4 years. I would also like to thank my family who always cheered and supported me from afar in South Korea. I was lucky to work with wonderful lab partners: Michele Nielsen, Yibin Gao, Sunphil Kim, Steve Boona, Bin He, Mike Adams, Sarah Watzman, Chris Jaworski, and Audrey Chamoire. I would like thank all of them for their help and the fun memories we shared. This work is the result of collaborative efforts with various research groups: Dr. Oscar Restrepo and Dr. Wolfgang Windl at The Ohio State University; Zihao Yang and v

7 Dr. Roberto Myers at The Ohio State University; Gloria Lehr and Dr. Donald Morelli at Michigan State University; and my friend Dr. Bartek Wiendlocha at AGH University of Science and Technology in Krakow, Poland. I would also like to acknowledge the financial support of AFOSR-MURI Cryogenic Peltier Cooling for my project and personal funding. Additionally, I would like to acknowledge the Department of Mechanical and Aerospace Engineering and The Ohio State University for awarding me with a University Fellowship for my first year of graduate school. Finally, I would like to thank God for giving me a chance to take this wonderful journey in my life. vi

8 Vita B.E. Mechanical Engineering, Korea University, Seoul, South Korea 2008 to present... Graduate Research Associate, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA Publications E. G. Evola, M. D. Nielsen, C. M. Jaworski, H. Jin, and J. P. Heremans, Thermoelectric transport in indium and aluminum-doped lead selenide, J. Appl. Phys. 115, (2014). H. Jin, Z. Yang, R. C. Myers, and J. P. Heremans, Spin-Seebeck like signal in ferromagnetic bulk metallic glass without platinum contacts, Solid State Commun. (2014), G. J. Lehr, D. T. Morelli, H. Jin, and J. P. Heremans, The influence of chemical pressure and intermediate valence on the Seebeck coefficient of Yb 1-x Sc x Al 2 alloys, J. Appl. Phys. 114, (2013). H. Jin, C. M. Jaworski and J. P. Heremans, Enhancement in the figure of merit of p-type Bi 100-x Sb x alloys through multiple valence-band doping, Appl. Phys. Lett. 101, (2012). vii

9 C. M. Orovets, A. M. Chamoire, H. Jin, B. Wiendlocha and J. P. Heremans, Lithium as an Interstitial Donor in Bismuth and Bismuth-Antimony Alloys, J. Elec. Mat. 41, 1648 (2012). Fields of Study Major Field: Mechanical Engineering viii

10 Table of Contents Abstract... ii Dedication... iv Acknowledgments... v Vita... viiii List of Tables... xi List of Figures... xii Chapter 1: Introduction Thermoelectric devices and materials for cooling at cyrogenic temperature Thermoelectric refrigerators and figure of merit Peltier cooling at cryogenic temperatures Materials for cryogenic cooling Spin caloritronics and spin Seebeck effect Spin caloritronics Spin Seebeck effect Thermal transport by lattice vibrations Lattice thermal conductivity Lattice vibrations and phonons Debye model and lattice heat capacity Temperature dependence of lattice thermal conductivity Anharmonicity and phonon-phonon interactions ix

11 Chapter 2: Development of Bi and Bi 1-x Sb x based Thermoelectric Materials for Cryogenic Cooling Applications Introduction to bismuth and bismuth-antimony alloys Bismuth Crystal and band structures Thermoelectric properties Bismuth-antimony alloys Single crystals Polycrystalline samples Study on indium doping in single crystalline bismuth Introduction Isovalent doping effect induced by indium in bismuth Transport properties of indium doped bismuth Enhancement in the figure of merit of p-type BiSb alloys through multiple valence-band doping Chapter 3: Study on Magnon Thermal Conduction and Spin Seebeck Effect in Ferromagnetic Bulk Metallic Glass Measurement of magnon thermal conductivity in ferromagnetic bulk metallic glass Spin Seebeck like signal in ferromagnetic bulk metallic glass without platinum contacts Chapter 4: Observation of Phonon Diamagnetism in Non-magnetic Semiconductor Introduction Experimental Results and discussion Error analysis Conclusions References x

12 List of Tables Table 1.1. Classification of various spin transport effects driven by thermal forces Table 2.1. Magnetic field oscillation frequencies [Δ(1/B)] -1, cross sectional areas of Fermi surface A F, Fermi energy E F, and hole concentrations P of the Bi 100-α In α samples obtained from the SdH oscillations Table 2.2. Electron (N), excess hole (P-N), and hole (P) concentrations calculated from the ρ xy (B z ) measurement Table 2.3. Temperature dependence of parameters in the trigonal plane of Bi 99.6 In 0.4 sample obtained by fitting Eq. (2.4) to the experimental data: N, electron concentration; P, hole concentration; µ 1, binary electron mobility; µ 2, bisectrix electron mobility; ν, isotropic hole mobility in the trigonal plane Table 3.1. Properties of Co- and Fe-Metglas. The data are provided by www. metglas.com xi

13 List of Figures Figure 1.1. Schematic illustration of (a) TE power generator and (b) TE heat pump / refrigerator made from n- and p-type semiconductor legs. The yellow arrow indicates the direction of electron (e - ) flow Figure 1.2. COP max of a single-stage Peltier cooler normalized by COP Carnot, plotted against ZT for different temperature differences ΔT. Here, T = T m Figure 1.3. Maximum temperature reduction ΔT max plotted against ZT m for the heat sink at 300K Figure 1.4. Themoelectric properties as a function of carrier concentration Figure 1.5. Minimum temperature reachable with a multi-stage Peltier cooler as a function of the number of stages, assuming a base temperature of 160K and zt = 1 throughout Figure 1.6. Figure of merit (zt) of BiSb alloys as a function of temperature illustrating that T 7/2 dependence is approximately followed below 60K Figure 1.7. Low temperature figure of merit (zt) of various materials as a function of temperature Figure 1.8. Schematic diagram for the spin-polarized band structure of ferromagnetic metals. Local ferromagnetic moments induce the conduction band to split into spin-up and spin-down bands. µ is the chemical potential, and and indicate spin -up and spindown, respectively. Note that µ measured from the band edge is different for the two split bands (µ µ ) Figure 1.9. Thermal spin injection across a ferromagnet (FM) normal metal (NM) interface using the spin-dependent Seebeck effect. The slopes of the chemical potential µ (solid black lines) in FM and NM reflect the charge Seebeck coefficient of the bulk materials. j Q is the heat current, µ and µ are the chemical potentials of spin-up and spin-down electrons, respectively. λ F and λ F denote the spin diffusion lengths in the FM and NM, respectively xii

14 Figure Schematic illustration of the (a) transverse spin Seebeck configuration and (b) inverse spin-hall effect. j s is the spin current, M the magnetization, E ISHE the electric field induced by the inverse spin-hall effect. The blue and red arrows in (b) indicate spin-down and spin-up, respectively Figure Schematic illustration of the longitudinal spin Seebeck configuration. j s is the spin current, M the magnetization, and E ISHE the electric field induced by the inverse spin-hall effect Figure A schematic diagram of interatomic potential for bonding in solids. The potential u between two atoms has a minimum at the equilibrium distance a Figure (a) Spring-ball model representing a one-dimensional chain with one atom per unit cell. (b) Vibration frequency ω as a function of wave vector k. The 1 st Brillouin zone is defined as -π/a < k < π/a as shown Figure Debye approximation. Red curve is the real phonon dispersion within the 1 st Brillouin zone. The blue line indicates the Debye approximation. ω D is the Debye frequency. Note how closely the blue line approximates the red curve at small k, while it becomes significantly different at higher k Figure An example of thermal conductivity of single crystalline solids. Three different temperature regimes can be defined according to dominant scattering mechanisms: (a) T 3 range where boundary scattering on phonons is dominant. (b) T-flat range where defect scattering is dominant. (c) exp(-t) range where phonon-phonon scattering is dominant. The green curve represents the thermal conductivity of the same material, but with a larger size than the one represented by the orange curve Figure Diagram for the thermal expansion of a solid. The energy scale has been chosen such that the potential minimum has zero energy. The blue solid curve shows the interatomic potential u as a function of interatomic distance r. The blue dashed curve indicates the potential of a perfect harmonic oscillator. The red solid line marks the temperature-dependent mean interatomic distance. The mean energy of the oscillator at a temperature T is k B T Figure (a) Normal and (b) umklapp processes. The points are reciprocal lattice points, and G is the nonzero reciprocal lattice vector. The dashed arrow in (b) indicates the transformation of k 1 + k 2 into k 3 by an addition of G Figure 2.1. The rhombohedral unit cell of bismuth. Two atoms are contained in the cell. x-, y-, and z- axes denote the binary, bisectrix, and trigonal (or c-axis) axes, respectively xiii

15 Figure 2.2. The Brillouin zone of bismuth, with principal crystallographic axes labeled as the binary (x), bisectrix (y), and trigonal (z) axes. The hole Fermi surface is an ellipsoid at the T-point, and the electron Fermi surfaces are three ellipsoids at the L-point. Since there are two T-points and six L-points in a Brillouin zone, but each ellipsoid is only half in the first Brillouin zone, there are three electron and one hole Fermi surfaces. ϕ is the tilt angle of the electron ellipsoids with respect to the crystallographic axes Figure 2.3. Band structure of bismuth at T = 0K showing the conduction band at the L- point, valence band at the L-point, and the valence band at the T-point. The dashed line indicates the absolute position of the Fermi energy. E Fe and E Fh denote the Fermi energies of electrons and holes with respect to the band maxima Figure 2.4. Figure of merit (zt) of single crystalline Bi adapted from Ref. [52]. (a) zt obtained from measured thermoelectric properties. (b) Hypothetical z 0 T calculated for the electron-only system by assuming that the compensation from the opposite carrier is removed. // and denote the trigonal axis direction and in plane direction, respectively Figure 2.5. Phase diagram of bismuth-antimony alloys adapted from Ref. [55] Figure 2.6. Schematic band diagram of Bi 100-x Sb x alloys as a function of x at T ~ 0K, adapted from Ref. [54] Figure 2.7. Figures of merit of BiSb alloys along the directions parallel (z 33 ) and perpendicular (z 11 ) to the trigonal axis as a function of Sb concentration, adapted from Ref. [53] Figure 2.8. Figure of merit (zt) of single crystalline and polycrystalline BiSb alloys as a function of Sb concentration at 300K. For single crystalline samples, zt is plotted in trigonal axis direction and binary/bisectrix (trigonal plane) direction Figure 2.9. (a) X-ray diffraction pattern of a single crystalline In doped Bi sample. The measurement was made on the trigonal plane of a cleaved disk from the ingot shown in (b). The highest peak corresponds to (222), a multiple of the trigonal (111) direction. (b) A single crystal ingot grown by the modified horizontal Bridgman method. The ingot was cleaved along the trigonal plane direction after cooling by liquid nitrogen Figure Shubnikov de-haas (SdH) traces for Bi 100-α In α samples studied here at 2K. The amplitude of the SdH oscillations for α=0.09 and α=0.4 have been magnified by 10 2, and the background subtracted for all samples. The inset shows the configuration used for the SdH measurements which we denote ρ xx (B z ) where the first, second, and third indexes indicate the direction of current flow, potential difference, and magnetic field, xiv

16 respectively. Each axis in the configuration corresponds to binary (x), bisectrix (y), and trigonal (z) crystallographic direction, respectively Figure Hall resistivity ρ xy (B z ) versus magnetic field for Bi 100-α In α samples measured at 2K. The points indicate the experimental data, while the lines are added to guide the eye. The symbols are: (green cross) α=0.09, and (blue circle) α=0.4. The inset contains magnification of ρ xy (B z ) at low magnetic field, which shows the transition from the negative slope to the positive slope for both samples Figure Schematic band diagrams indicating positions of the Fermi energy E F for the samples in this study. The dashed line denotes E F of pure Bi Figure Densities of states (DOS) for In doped Bi as a function of electron energy calculated by the supercell calculations. Spin-orbit interactions have been taken into account. The top panel shows the total DOS per atom, and the bottom panel shows DOS of In with partial contributions from different orbitals. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission Figure Charge density in logarithmic scales around a In impurity. The plot reflects the hyperdeep defect state peak in Fig The nearest (Bi(1)) and next nearest (Bi(2)) atoms are labeled. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission Figure Evolution of densities of states (DOS) for In doped Bi with increasing In concentration. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission Figure Temperature dependence of the electrical resistivity in (a) the trigonal axis direction ( ρ //), and (b) the trigonal plane ( ρ ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The solid black line represents the computed curve for Bi 99.6 In 0.4, while the dashed black line corresponds to the computed curve for pure Bi Figure Magnetoresistivity ρ xx (B z ) of Bi 99.6 In 0.4 versus magnetic field measured at 2K. The points indicate the experimental data, while the solid black curve is the fit generated by the method of least squares with regard to Eq. (2.4) Figure Temperature dependence of the Seebeck coefficient in (c) the trigonal axis direction ( S // ), and (d) the trigonal plane ( S ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The solid black line represents the computed curve for Bi 99.6 In 0.4 ; the dashed black line corresponds to the computed curve for pure Bi xv

17 Figure Temperature dependence of the thermal conductivity in (a) the trigonal axis direction ( κ // ), and (b) the trigonal plane (κ ). Figure of merit in (c) the trigonal axis direction ( zt // ), and (d) the trigonal plane ( zt ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In Figure Temperature dependence of the Seebeck coefficient in 7T magnetic field in (a) the trigonal axis direction ( S // ), and (b) the trigonal plane ( S ). The direction of magnetic field is parallel to that of heat flux. The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The dashed lines indicating zero Seebeck coefficient are added to help readers recognize the sign change of the Seebeck coefficient Figure Effects of interstitial indium on electronic structure of bismuth. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission Figure Comparison of the best zts between n-type and p-type BiSb alloys. The n- type zt is plotted using the data in Ref. [63], and p-type zt from Ref. [68] Figure Dependence of the energies of the various band extrema on composition x in Bi 100-x Sb x alloys at T=0K (compiled from Refs. 54, 96-98). At the L-points of the Brillouin zone the symmetric and antisymmetric bands are inverted between elemental Bi and Sb, leading to a Dirac point near x=5 at % fraction. The T-point valence band is the upper valence band of the semimetal Bi, but in elemental Sb, the holes are inside the Brillouin zone in three distorted ellipsoidal pockets near the H-points. This band becomes the upper valence band for x>18. As in most solids, temperature can make band extrema shift by 100 mev between liquid nitrogen and room temperatures, an effect that is secondary in wide-gap semiconductors, but has a very important relative influence on the present diagram Figure Comparison of thermoelectric properties of single crystal Bi Sb 18.2 Sn 0.75 sample between in the trigonal plane and along the trigonal axis directions. (a) thermal conductivity, (b) Thermopower S, (c) electrical resistivity ρ, (d) Figure of merit zt as functions of temperature. Points are experimental data, lines are added to guide the eye Figure (a) Thermal conductivity κ and (b) electrical resistivity ρ of single crystal Bi 100-x Sb x Sn 0.75 samples for x = 11.6, 18.2, 19.5, 22.9, 26.5, 30, and 37 from 2K to 400K. All properties were measured with the heat flux and current in the trigonal plane. Points are experimental data, lines are added to guide the eye Figure Thermopower S of single crystal Bi 100-x Sb x Sn 0.75 samples for x = 11.6, 18.2, 19.5, 22.9, 26.5, 30, and 37 from 2K to 400K. The S was measured with the heat flux and current in the trigonal plane. The insert shows the best figures of merit zts in this xvi

18 study as a function of temperature. Points are experimental data, lines are added to guide the eye Figure Thermoelectric properties of polycrystalline Bi 100-x Sb x Sn 0.75 samples for x = 70, 80, and 90 from 2K to 400K. (a) Thermal conductivity, (b) Seebeck coefficient, and (c) resistivity as functions of temperature. Points are experimental data, lines are added to guide the eye Figure (a) Pisarenko s plot of thermopower versus hole concentration at T=80K for all measured samples in this study. Solid line and dashed line are calculated for the valence bands at the Η-points and the valence band at the T-point, respectively. Points are experimental data. Carrier density of each sample was obtained from Hall measurement. (b), (c), (d) Schematic band diagrams for the three different Sb concentration regimes shown. Relative position of each band and the position of Fermi energies (E F ) are not to scale Figure 3.1. Dispersion relation of magnons for a one-dimensional ferromagnetic chain constructed using Eq. (3.1) Figure 3.2. Experimental setup for thermal conductivity and electrical resistivity measurements. The sample is mounted on the Thermal Transport Option in the PPMS Figure 3.3. Electrical resistivity ρ of Co- and Fe-Metglases as a function of temperature measured in 0 and 7T magnetic fields Figure 3.4. Thermal conductivity κ of Co- and Fe-Metglas samples as a function of temperature. κ e denotes the electronic thermal conductivity calculated using the Wiedemann-Franz law at 0T Figure 3.5. (Upper plots) Thermal conductivity of Co- and Fe-Metglas samples after subtracting electronic contribution (κ LM = κ - κ e ) as a function of temperature in 0 and 7T magnetic fields. (Lower plots) Relative change (%) of κ LM in magnetic fields as a function of magnetic field: Δκ LM (H) = κ LM (H) - κ LM (0T) Figure 3.6. Amount of change in κ LM when the 7T magnetic field is applied. The inset shows a magnification for the lower temperature which exhibits the magnitude of change more clearly Figure 3.7. Experimental setup and measurement conventions. (a) Experimental setup. Magnetic field is aligned along the vertical direction. (b) Schematic illustration of the transverse and longitudinal magnetothermopower geometries: T and the external xvii

19 magnetic field, H, lie in the plane (x-direction) of the sample. (c) Nernst geometry: H is applied along the perpendicular (z-) direction Figure 3.8. Measurements in perpendicular magnetic field. (a) H dependence of the transverse voltage, V xyz, measured between the point contacts in the configuration shown in Fig. 3.7(c). After measuring V xyz close to the hot end, the same sample is flipped upside down to measure V xyz close to the cold end. The shaded area corresponds to a hysteresis loop induced mostly by the ANE. The inset shows magnification at low H in which the hysteretic behavior is observed. Data are taken at sample temperature, T avg = 64K, with temperature difference, ΔT x = 19.8K. (b) T dependence of anomalous Nernst coefficient, α ANE. The ordinary Nernst coefficient, α ONE taken from the hot end is shown below (right ordinate). The orange line for α ONE is a guide to the eye. The black dashed lines are T 3/2 fits for α ANE for 20K< T <80K, and the black solid lines are T 1 fits for T < 20K Figure 3.9. Measurements in parallel magnetic field. (a) Hysteresis loops on the in-plane transverse voltage, V xyx, as a function of H for the hot and cold ends with an applied ΔT x = 10.5K at T avg = 24K. (b) T-dependence of the transverse magnetothermopower (planar Nernst coefficient), α xyx, on different positions of the sample. Orange squares are data taken on the hot side close to the middle of the sample, while sky blue triangles are on the cold side, same distance away from the middle as orange squares. Lines are drawn to guide the eyes. (c) In-plane longitudinal voltage, V xxx, as a function of H with an applied ΔT x = 17.3K at T avg = 49K. Inset: T dependence of the longitudinal magnetothermopower, α xxx. Points are experimental data and line is drawn to guide the eyes. (d) In-plane longitudinal voltage, V xxy, as a function of H (aligned along the y-axis) with an applied ΔT x = 18.5K at T avg = 56.4K Figure Transverse voltage ΔV xyx as a function of temperature difference ΔT x. Red diamonds are data points, and the straight line is a linear fit through the data points. The point at (0,0) is artificially added from an expected extrapolation Figure Angular dependence of V xyx. (a) Dependence on sample tilt angle ϕ in the z- direction. The straight lines indicate the relative change of the slope between different ϕ. (b) Rotation angle θ dependence in the xy-plane (the line follows a cosθ) Figure Measurements on Cu+Pt strips. (a) Transverse voltage V xyx as a function of H for the hot and cold ends with an applied ΔT x = 13.2K at T avg = 72.4K. (b) Rotation angle θ dependence in the xy-plane (the line follows a cosθ) Figure Comparison of α xyx between point contacts and Cu+Pt strips. Points are experimental data, and lines are added to guide the eye xviii

20 Figure 4.1. Experimental setup and measurement protocol. (a) Photo of the InSb tuning fork sample mounted on measurement system. The scale bar indicates the actual length of the sample. (b) Raw traces of the differential thermal conductivity measurement, showing temperature differences between the large and small arms (ΔT LS ) as a function of time, measured in the zero magnetic field. Red curves correspond to ΔT LS with four different heater powers on the small arm (Q S ) while the heater power on the large arm (Q L ) is kept constant. (c) Experimental geometry for differential and absolute thermal conductivity measurements, not to scale. H denotes the magnetic field. ΔT L and ΔT S are the longitudinal temperature differences on the large and small arms, respectively. (d) ΔT DIFF versus Q S obtained from the raw traces in (b). T DIFF = T LS T R where T R is the residual T LS when no heat is applied. Purple crosses are experimental data, and the red cross indicates a value of Q S when ΔT DIFF = 0, obtained by a linear interpolation. The solid line is a linear fit to the data points Figure 4.2. Thermal conductivity ratio κ L /κ S of pristine sample (a) and sample with more dislocations (b) as a function of temperature. The symbols are the experimental data: (blue diamond) 0T, and (red triangle) 7T. The solid lines are T -3 fits to the data points. Inset in (a) shows κ L /κ S converge to unity at higher temperatures. (c) Magnetic field dependence of κ L /κ S at 3K (green diamond) and 4.4K (orange triangle) measured on the sample with more dislocations. All error bars represent the standard deviation of the mean (standard errors) Figure 4.3. (a) Thermal conductivity of the small arm (κ S ) as a function of temperature in 0T (blue diamond) and 7T (red triangle) magnetic fields. The dashed curve indicates the thermal conductivity of the large arm calculated from κ S and κ L /κ S. The inset shows the linear relationship between the temperature difference on the small arm (ΔT S ) and the heater power (Q S ) (Fourier s law), verifying that transport occurs in the linear regime. Error bars represent standard errors. (b) The temperature dependence of the specific heat C p of a piece of the same boule of InSb, showing no effect of the magnetic field in the temperature range of interest. The inset shows C p up to 300K where it starts to saturate approaching the Dulong-Petit value Figure 4.4. Galvanomagnetic transport in the InSb material used. (a) Temperature dependence of the resistivity at zero magnetic field and in a magnetic field of 7T. (b) Hall resistivity versus magnetic field at the temperatures indicated Figure 4.5. Electron concentration and mobility. The temperature dependence of the carrier concentration (a) and mobility (b) calculated from the low-field Hall coefficient. Shubnikov-de Haas oscillations in the low-field resistivity (d) and Hall resistivity (e) xix

21 Figure 4.6. Temperature dependence of the thermoelectric power α of the small arm of the sample used in Fig. 4.1, at zero field and in a 7T longitudinal magnetic field (H // [100]) Figure 4.7. (a) ( κl κs) = κl κs (0T) κl κs (7T) as a function of T -3 for pristine sample (a) and sample with more dislocations (b). Red triangles are the data points. The solid line is the linear fit to the data points, obtained by the method of least squares. The dashed line in (b) implies a saturation of ( κl κs). The error bars represent the standard errors. The inset in (a) shows the Debye approximation in which the entire phonon spectrum is replaced by a linear one Figure 4.8. Temperature differences between large and small arm (ΔT LS ) as a function of time at 10K, and histograms for no-heat, heater-on, and heater-off regimes. n j denotes the number of data points for jth interval, and σ the standard deviation Figure 4.9. Schematic diagrams for magnetic field dependence of Cernox thermometers. (a) Variation of temperature readings as a function of magnetic field H. It is assumed that the heater power ratio Q L / Q S is adjusted such that for one measurement, T L = T 1 and T S = T 2 at 0T, and vice versa for the other measurement. (b) Effect of the magnetic field induced errors e i and e ii on measured temperature differences ΔT LS Figure Experimentally measured magnetic field dependence of Cernox thermometers. (a) Variation of temperature readings as a function of magnetic field H. T R,L and T R,S denote temperature readings when no heat is applied. (b) ΔT DIFF as a function of heater power on the small arm, Q S. ΔT DIFF is the temperature difference between the large and small arms after subtracting the residual temperature difference. Two circled points are determined by processes shown in (c): left and right columns correspond to left and right circled points, respectively, as pointed by arrows xx

22 Chapter 1: Introduction The introduction consists of three parts, each part corresponding to an introduction to each subsequent chapter in order. In the first section, a general introduction to the thermoelectric devices and materials is given, paying special attention to those used for cooling applications at cryogenic temperatures. The second section introduces an emerging field, spin caloritronics, and one of the most important experimental observations in this field, named the spin Seebeck effect. The last section is devoted to introducing the thermal conduction in solids by lattice vibrations, with a focus on the anharmonicity in lattice vibrations. 1.1 Thermoelectric devices and materials for cooling at cryogenic temperature Thermoelectric refrigerators and figure of merit Thermoelectric (TE) devices are solid-state energy converters which can be used as either power generators in which they directly convert waste heat into electricity, or heat pumps and refrigerators in which they pump heat from cold end to hot end when an electrical power is applied. With increasing concern about energy crisis nowadays, there 1

23 is a growing interest in TE devices as one of the promising renewable energy sources. TE devices have many advantages over conventional power generation or refrigeration systems: they are small, lightweight, have no moving parts, and do not release pollutants. 1,2 Figure 1.1 illustrates the two operation modes of TE devices. Commercial TE devices are made of many couples of n- and p-type semiconductor materials joined at a junction on one end. When heat is applied to the junction, heat flux Q flows in parallel through both materials, developing a temperature gradient across each of them (Fig. 1.1(a)). The temperature difference ΔT between the hot and cold ends of each material in turn creates a voltage potential V, and this is called the Seebeck effect. The voltage V is directly proportional to ΔT. Figure 1.1. Schematic illustration of (a) TE power generator and (b) TE heat pump / refrigerator made from n- and p-type semiconductor legs. The yellow arrow indicates the direction of electron (e - ) flow. 2

24 For each material, the proportionality constant is defined as the Seebeck coefficient S, also known as the thermoelectric power or thermopower 3 : S V T (1.1) The Seebeck effect, first observed by Thomas J. Seebeck in , is the basis for TE power generation. The net voltage across the cold ends of the device is equal to the sum of the voltage potentials in the n- and p-type materials, V=V n +V p. If an external load R is connected across the cold ends, the single TE couple device delivers power W=V 2 /R, working as a power generator. Note that many TE couples can be connected in series to make a pile for a higher power output. On the other hand, if an electric current I is applied to flow from the n-type material to the p-type across a junction as shown in Fig. 1.1(b), electron/hole pairs are created in the vicinity of the junction. The formation of the pairs takes energy away from the junction, thus cooling it. The direction of I makes both electrons and holes flow away from the junction in the n-type and p-type materials, respectively. On the opposite ends of the materials, electrons and holes recombine, and this process releases energy and heats the junctions. Therefore, in this example, the TE couple device works as a refrigerator (top junction) or a heat pump (bottom junctions), using the electrical current (flow of electrons) as a working fluid. If the direction of I is reversed, the opposite operation takes place: the top junction heats up while the bottom junctions cool down. 3

25 These refrigeration and heap pump operations of TE devices are based on the Peltier effect, which was discovered by Jean Peltier in Peltier observed that if an electric current is applied across a junction of dissimilar metals or semiconductors, either heating or cooling can occur at the junction. When the direction of the current is reversed, the opposite effect is observed. The Peltier coefficient Π is defined as Π Q (1.2) I where Q is the amount of heat taken into/from the junction (Q=Q n +Q p ). Π and S are related by the Kelvin relation: Π= ST. (1.3) The efficiency of a refrigerator is expressed by the coefficient of performance (COP), which is defined as the ratio of the electrical power P needed to obtain the cooling capacity Q C : COP Q C. (1.4) P The COP cannot exceed the Carnot efficiency set by the laws of thermodynamics. The COP at Carnot efficiency (COP Carnot ) is COP Carnot = T H T C T C (1.5) 4

26 where T C and T H denote the temperatures at the cold and hot ends, respectively. The actual efficiency of any cooling device is often given as a percentage of COP Carnot. The TE refrigerators in the market operate at about 12% of COP Carnot, whereas the efficiency of a conventional compressor-based refrigerator can reach up to 60% depending on its size. The key difference between the two systems is that in the conventional refrigerator, cooling and heat-rejection components can be physically separated so that large temperature differences prevent the heat backflow that degrades efficiency in the TE systems. Therefore, it is easily expected that using materials that poorly conduct heat between the cooling and heat-rejection junctions would improve efficiency of TE refrigerators. As we will see soon, this is one of the important material properties which largely influence the COP of TE refrigerators. Due to the low efficiency, TE refrigerators, also called the Peltier coolers, find their use mostly in niche applications such as car seat coolers/heaters, laser diode coolers, and electronic coolers for picnic baskets. However, there are some cases in which other advantages of Peltier coolers can overcome the shortage of efficiency, e.g. in applications where small size or high reliability is more important than the efficiency. In addition, due to their fast response, Peltier coolers can be used when rapid on-off cycling is required with small temperature differences. Regarding the performance of a Peltier cooler, there are two values of the input current that are of special interest. 5 The current I M that yields the maximum cooling power is 5

27 I Q = ( S S ) T p n C ( R + R ) p n (1.6) where R p and R n are the electrical resistances of the p- and n- branches, respectively. At I Q, the COP is COP Q ( ) 2 C / 2 H C ZT T T =. (1.7) ZT T H C Here, Z is equal to S p Sn 2 ( K p + K n) ( R p + Rn) ( ), where K p and K n are the thermal conductances of the branches. Equation (1.7) shows that the COP Q depends solely on Z and temperatures, and Z is therefore known as the figure of merit of the thermocouple, which represents the efficiency of a n- and p- type pair. Nowadays, the dimensionless figure of merit ZT is more widely used. Z can be expressed in another way that includes only materials properties, after an additional process is taken to match the geometrical factors of the two branches: Z = ( Sp Sn) ( κρ p p) + ( κρ n n) 2 12 / 12 / 2, (1.8) where ρ is the electrical resistivity, and κ is the thermal conductivity of the material. The other value of the current that is of interest is I max which results in the maximum COP: 6

28 I max = ( Sp Sn)( TH TC), (1.9) R ( 1 ) 12 / ( p + Rn) + ZTm 1 where T m is the mean temperature, T m = (T H +T C )/2. At I max, the maximum COP of a Peltier cooler is given by COP max = 12 / ( 1 + ) ( / ) T C ZTm TH T C. (1.10) 12 / ( TH TC ) ( 1+ ZTm) + 1 It is noted that the COP max is also a function of Z and T only. In general, the preferred value of the current lies somewhere between that for COP Q and COP max. Figure 1.2 shows the relationship between the normalized COP max and ZT for different temperature differences ΔT = T H -T C. The ZT of the thermocouple used in currently available Peltier cooling devices is about 1 which corresponds to the COP max of ~10% of COP Carnot as mentioned earlier. To achieve a COP max comparable to the conventional refrigeration systems, it is obvious that a many fold improvement in ZT is required. 7

29 0.5 COPmax / COPCarnot T= 10K 20K 40K ZT Figure 1.2. COP max of a single-stage Peltier cooler normalized by COP Carnot, plotted against ZT for different temperature differences ΔT. Here, T = T m. One of the most important characteristics of a thermocouple is the maximum temperature reduction that can be reached through the Peltier effect. This is given by 1 2 Tmax = ZTC. (1.11) 2 Figure 1.3 shows how ΔT max varies with ZT when T C is kept at 300K. Values of ZT m significantly greater than unity are required for temperature depressions of 100 o C or more. 8

30 T max ZT m Figure 1.3. Maximum temperature reduction ΔT max plotted against ZT m for the heat sink at 300K. Equation (1.7), (1.8), and (1.10) suggest that COP of Peltier coolers strongly depends on the material parameters which make up the thermocouples in the devices. While ZT expresses the efficiency of a thermocouple (a pair of n- and p-type materials), zt represents the efficiency of a single TE material and is defined as zt 2 S σ T (1.12) κ 9

31 where σ is the electrical conductivity (= 1/ρ), and T is the absolute temperature. The numerator of Eq. (1.12) is labeled the thermoelectric power factor P=S 2 σ, and represents the amount of electrical power density (current density times voltage) a TE material can generate/use for a given temperature gradient. It is important to notice that zt cannot be used to calculate the performance of a thermocouple even if its value is known for both branches. For this purpose, the efficiency of a thermocouple, ZT must be used. However, in many cases, the value of ZT is close to the average of n- and p-type zt, so that it is reasonable to select TE materials based on the single material figure of merit zt. Therefore, maximizing zt of TE materials is imperative for broader applications of the Peltier coolers, and is the main goal of research on thermoelectrics. The zt consists of mutually counter-indicated properties of the material, i.e. most mechanisms that improve one property in zt also are detrimental to another. Explicitly, zt can be written as the product of two sets of counter-indicated properties shown below in parentheses: µ 2 zt = ( S n). qt. (1.13) κ In writing Eq. (1.13), we have written the electrical conductivity as: σ = nqµ (1.14) where n is the carrier concentration, q is the elementary charge, and µ is the mobility of the carriers. The mobility itself is defined as the ratio between the magnitude of the 10

32 electron drift velocity in a sample under the influence of an applied electric field Ε and the magnitude of the electric field. The ratio (µ/κ) is counter-indicated because defects and impurities that affect one of these properties usually also affect the other. As discussed earlier, in order to reduce the backflow of heat from the heat-rejecting side, materials with a low κ are desirable in Peltier coolers. Imperfections such as defects and impurities in materials can reduce κ, but they also decrease µ by scattering mobile charge carriers, thus resulting in almost no effect on (µ/κ). The other counter-indicated property is the product (S 2 n): it is a general rule in doped semiconductors and metals that the higher the carrier concentration, the lower the thermopower; this is called the Pisarenko relation. 6 An exemplified carrier concentration dependence of the S,σ, κ and zt is shown in Fig Because S,σ, and κ in Eq. (1.12) are all strongly dependent on the carrier concentration, there is an optimum carrier concentration which yields a maximum zt. Therefore, improving zt is a rather complicated task for which optimizing each counterindicated pair of properties is required. 11

33 intensity S zt σ κ carrier concentration Figure 1.4. Themoelectric properties as a function of carrier concentration Peltier cooling at cryogenic temperatures The cryogenic temperature means the temperature range below 120K. Even at these very low temperatures, there are needs for Peltier cooling using TE devices, especially in space applications. Satellites evolving in space have on-board electronic circuits which need to dissipate heats efficiently for durable operations. Far-infrared, x- ray, and γ-ray detectors should be kept cool for their optimum performance. The operation temperature range for those applications is below 10K up to 40K. As will be discussed below, Peltier cooling in this temperature regime is extremely challenging, mostly due to the temperature dependence of the materials properties and thus that of zt. Historically, cryogenic cooling down to 10K and below has been provided by thermodynamic cycles or by stored cryogens. Although working as intended, these 12

34 conventional methods have shortcomings, such as the system size is massive and the operation cost is highly expensive, thus not suitable for the applications mentioned above. Provided that TE Peltier coolers could achieve the required cooling capability, their compactness, long-term cost-effectiveness, and reliability will certainly make them attractive for these applications. In practical uses, Peltier coolers are stacked in stages, as shown in Fig. 1.5, in which the stages are connected in series both electrically and thermally. 7 This cascaded cooler has an advantage over a single-stage cooler in that it can operate over a large temperature difference without losing cooling performance. 8 This is made possible by using a number of different TE materials, each one being used over the particular temperature interval where it has better properties than other materials. In principle, this cascaded module can achieve any arbitrary ΔT max with infinitely increasing number of stages. However, the performance of the module is limited in practice by contact resistances, such that 6 stages are used for the best available commercial modules. 13

35 Figure 1.5. Minimum temperature reachable with a multi-stage Peltier cooler as a function of the number of stages, assuming a base temperature of 160K and zt = 1 throughout. Figure 1.5 also shows the calculated minimum temperature reachable by a multi-stage cooler as a function of the number of stages. Assumptions were made in the calculation: the base temperature is 160K, zt = 1 for both n- and p-type materials used, and thermal and electrical contact resistances are neglected. A 6-stage cooler could reach below 30K, which is well within the operation temperature range of the space applications. This suggests that if TE materials with zt 1 below 200K are developed, the Peltier coolers can be used for various special applications. 14

36 Besides the contact resistances, efficiency of Peltier coolers is primarily limited by zt of the materials used. Cryogenic cooling imposes a special challenge on TE materials because zt decreases strongly with temperature. Classic Peltier coolers use degenerately-doped narrow-gap semiconductors. Assuming that the dominant scattering mechanism of the charge carriers is due to the acoustic lattice vibrations (acoustic phonons), it was shown 9 that zt T 5/2 to T 7/2 in those semiconductors. Thus, materials that perform well at ambient temperatures become no longer useful at cryogenic temperatures. Figure 1.6 illustrates how rapidly zt decreases as temperature lowers below 60K in case of BiSb alloys Bi 90 Sb Bi 83 Sb 17 Tetradymite n-type zt Tetradymite p-type 0.01 T 7/ T(K) Figure 1.6. Figure of merit (zt) of BiSb alloys as a function of temperature illustrating that T 7/2 dependence is approximately followed below 60K. 15

37 1.1.3 Materials for cryogenic cooling Despite the above mentioned difficulty, there are many material systems that seem to be promising for cryogenic cooling. It was shown 10 that single crystalline FeSb 2 and similar materials with strongly correlated electronic bands show colossal Seebeck coefficient peaks at cryogenic temperatures, resulting in very high power factors at 10 40K. Based on the experimental data, zt ~ 3 is predicted in FeSb 2 if the lattice thermal conductivity can be suppressed lower than 1 Wm -1 K -1. Considering that the intrinsic lattice thermal conductivity of this material is ~500 Wm -1 K -1 at the corresponding temperatures, this is a formidable task. Recently, nanostructuring has been successfully applied to reduce the thermal conductivity of FeSb 2 to less than 1 Wm -1 K Although the Seebeck coefficient of about 125 µv/k is only a small fraction of the single crystals 42,000 µv/k, in the end zt of was achieved which is about 3 times better than the value of the single crystals. Intermetallic compounds that contain 4f electronic levels, e.g. p-type CePd 3 and n-type YbAl 2 were suggested to show promising TE properties at low temperatures. 12 When the Fermi energy is tuned close to the sharp 4f levels, a large enhancement of the Seebeck coefficient is expected. Those metallic compounds typically have a high electronic thermal conductivity due to the large number of carriers, yet the lattice thermal conductivity is still about 30% of the total thermal conductivity. It has been shown 13 that zt of CePd 3 is enhanced by 40% at 100K through substituting Pd with Pt, mainly due to 16

38 reduction of the lattice thermal conductivity caused by the large mass difference between the two elements. For YbAl2 compounds, it has been reported 14 that substitution of smaller Sc for Yb introduces a chemical pressure that alters the Yb valence as a function of composition and temperature. The magnitude of the maximum of the Seebeck coefficient and the temperature at which this maximum occurs can be adjusted by Sc concentration, and the largest absolute value of the Seebeck coefficient was achieved when the average Yb valence was near 2.5. On contrary to those relatively new materials introduced above, BiSb alloys are classical TE materials, which have been studied since the 1950s. Surprisingly, the TE properties of this alloy system have not been improved over the past six decades, yet still shows the best performance below 200K. Chapter 2 will present various attempts used to improve the TE performance of BiSb alloys as well as to uncover new physics behind them. A detailed introduction to this interesting material systems will be given as well. Figure 1.7 summarizes zt of the material systems discussed so far. 17

39 1.00 Bi 91 Sb 9 (H = 0.5 Tesla) 0.10 Bi 90 Sb 10 zt Tetradymite p-type 0.01 YbAl 3 Calculated FeSb T(K) Figure 1.7. Low temperature figure of merit (zt) of various materials as a function of temperature. 18

40 1.2 Spin caloritronics and spin Seebeck effect Spin caloritronics Spintronics is a broad research field which focuses on various phenomena related to the flow of electrical charges and the flow of electron spin (spin current). By adding the spin degree of freedom to conventional charge-based electronic devices or using the spin alone, it is possible to obtain the potential advantages of nonvolatility, increased data processing speed, decreased electric power consumption, and increased integration densities compared to conventional semiconductor devices. 15 Spin caloritronics is an emerging field which is at the interface between non-equilibrium thermal transport phenomena and spin physics, positioning it complementary to the established fields of thermoelectricity and spintronics. 16 More specifically, the main theme of this new field is to investigate the interaction between electron spin and heat currents to obtain fundamental understanding behind various phenomena. In terms of applications, the additional degree of freedom provided by the electron spin and magnetic order can open many new ways to improve the existing energy conversion devices, e.g. TE devices. 17 While there are various thermally-driven transport phenomena associated with spin-dependent currents in the field of spin caloritronics, wherein charge currents play important roles, we will focus on only the phenomena which involve pure spin currents, wherein charge currents are canceled out or do not exist. The latter phenomena include the spin Seebeck effect that initiated the field of spin caloritronics after its discovery in 19

41 Table 1.1 provides a list of representative phenomena which belong to each category. Thermally driven spin transport Spin-dependent currents Pure spin currents Spin-dependent Spin-dependent Spin on conduction Spin waves thermoelectric thermomagnetics electrons (magnons) Spin-dependent Peltier effect Planar Nernst effect Spin-dependent Seebeck effect Magnon thermal conductivity Anomalous Nernst effect Transverse spin Seebeck effect Longitudinal spin Seebeck effect Transverse spin Seebeck effect Table 1.1. Classification of various spin transport effects driven by thermal forces The origin of the pure spin current can be sought from two different sources: 1) spin current induced by mobile conduction electrons, and 2) by excitations of long-range order magnetization (magnons), which originate from the spins of core electrons. The first type of pure spin current results from a cancellation of any associated charge currents. This case can be visualized by thinking of a system where spin-up electrons move in one direction and the exact same number of electrons move in the opposite direction at the same drift velocity, so that no net charge current flows but spin current 20

42 does. As this type of spin current is carried by spin-polarized conduction electrons, it is observed in ferromagnetic conductors and semiconductors in which the populations of spin-up and spin-down conduction electrons spontaneously become different. As shown in Table 1.1, two different effects are based on the spin current originated from the conduction electrons: the spin-dependent Seebeck effect and the transverse spin-seebeck effect. The latter can be also caused by spin currents associated with spin waves. To account for the differences between the two effects, the spin-dependent Seebeck effect will be introduced first, and then the spin Seebeck effect will be discussed. In ferromagnetic metals, an imbalance between the numbers of spin-up and spindown electrons results in a spin splitting of the chemical potential µ measured with respect to the band edges such that µ µ µ where and denote spin-up and spindown, respectively (Fig. 1.8). This in turn gives rise to a difference between the partial Seebeck coefficient S and S. Transport of these electrons is well described by the two current channel model, in which the majority and minority spin carriers form parallel channels with different conductivities σ and σ, leading to the conventional charge conductivity, σ = σ + σ and Seebeck coefficient S ( σ S σ S ) ( σ σ ) = + +. Under a temperature difference ΔT across the material, a charge current density flowing in each channel can be written as j C = σ S ΔT for spin-up, and j C = σ S ΔT for spin-down, and thus the total charge current density is j C = (σ S + σ S )ΔT. The net spin current is obtained from the difference between j C and j C which is proportional to S - S. If we impose a condition where j C = 0, but a heat current j Q 0, e.g. in an open circuit with a temperature gradient across the material, a charge accumulation driven by the 21

43 temperature gradient will occur at the edges of the material which leads to a spin accumulation µ - µ. Then µ - µ relaxes from the edges over the length scale of the spin diffusion length λ. This phenomenon is called the spin-dependent Seebeck effect, since it is caused by the difference in the numbers of spin-up and spin-down electrons, thus by the spin-dependence of the Seebeck coefficient described as S - S. Energy µ Density of states Figure 1.8. Schematic diagram for the spin-polarized band structure of ferromagnetic metals. Local ferromagnetic moments induce the conduction band to split into spin-up and spin-down bands. µ is the chemical potential, and and indicate spin -up and spindown, respectively. Note that µ measured from the band edge is different for the two split bands (µ µ ). A spin-dependent Seebeck effect has been demonstrated in lateral spin-valve structures 19 in which a normal metal is attached to a ferromagnetic metal. A temperature gradient is applied over the interface between the two metal layers while the charge 22

44 current is absent. As illustrated in Fig. 1.9, the spin-dependent Seebeck effect allows spins to accumulate at the interface in the ferromagnetic metal, and those spins can be injected into the normal metal by aid of the heat current. The injected spin current decays over the spin diffusion length of the normal metal and can be detected by means of the voltage difference between the normal metal and a reference ferromagnetic contact. Therefore, this experiment suggests that the spin-dependent Seebeck effect can be utilized to create thermal spin injection across an interface. However, we will see below that the spin-dependent Seebeck effect is not the only method for the thermal spin injection. j Q Ferromagnetic metal Normal metal µ µ λ F λ N Figure 1.9. Thermal spin injection across a ferromagnet (FM) normal metal (NM) interface using the spin-dependent Seebeck effect. The slopes of the chemical potential µ (solid black lines) in FM and NM reflect the charge Seebeck coefficient of the bulk materials. j Q is the heat current, µ and µ are the chemical potentials of spin-up and spin-down electrons, respectively. λ F and λ F denote the spin diffusion lengths in the FM and NM, respectively. 23

45 The second type of pure spin current can be generated by exciting the spins of core electrons in ferromagnetic materials. These spins show a net magnetization with long range order. When a perturbing force is applied, e.g. by a temperature gradient to this spin order, the spins are excited out of thermal equilibrium, and propagate in the form of collective excitations which are called spin waves or magnons. Magnons can carry heat and spin, but not charge. Under a temperature gradient, the heat current that flows by means of magnons carries spin angular momentum opposite to that of the magnetization. 20,21 Since magnons are associated with the spins residing in core electrons, the magnon spin current can be observed in both electrically insulating and electrically conducting ferromagnetic materials. In the latter, the magnon spin current flows in parallel with the charge current, thus it is possible that both types of spin currents exist in these materials. Indeed, the transverse spin Seebeck effect can be induced by either type of spin current, as indicated in Table 1.1. Here, we will briefly discuss a process in which the magnon spin current leads to the spin Seebeck effects. Unlike the spin-dependent Seebeck effect which can occur in a bulk sample, the spin-seebeck effect is generally known to require an interface between a material with spin-polarization, and a normal metal. However, as we will discuss in Chapter 3, spin- Seebeck like signals were also observed directly on a ferromagnetic conductor without a normal metal. 22 In the conventional geometry with an interface, the effect can be explained as a two step process. As a first step, a temperature gradient is applied to the spin-polarized material to create spin currents either through spin-polarized conduction 24

46 electrons, magnons, or both. Here, two different mechanisms are considered to bring the spin-carrying particles out of thermal equilibrium: electron / magnon thermal conductivity and phonon drag. The former reflects a thermodynamic force induced by the temperature gradient, which thermally excites the spin carriers. The latter mechanism includes intense interactions either between electrons and phonons (electron-phonon drag), or magnons and phonons (magnon-phonon drag). Usually, phonons are strongly coupled to the heat baths, and therefore most sensitive to the temperature gradient. Phonons that have proper wavelengths (or frequencies) to interact with electrons or magnons are called coherent phonons, and those phonons intensely interact with the spin carrying particles at a certain temperature range, pushing them out of thermal equilibrium. The temperature gradient in turn generates a gradient in magnetization, which results in a magnetization flux or spin flux. In the second step, the spin current is injected across the interface into the normal metal as a result of the non-equilibrium between the spin-polarized material and the normal metal at the interface. The detailed theory about this process can be found elsewhere. 23 The injection of the spin current across the interface can be driven by two types of processes. The first process is the local thermal spin injection in which the spin current is driven to drift into the normal metal in the same direction with the applied heat current. The longitudinal spin Seebeck effect is observed based on this local thermal spin injection. On the other hand, the transverse spin Seebeck effect uses a very different process, wherein the spin current diffuses into the normal metal in the direction perpendicular to the driving heat current, which we call non-local thermal spin injection. In this case, the normal metal behaves 25

47 as a spin sink that attracts the spin current. This process allows a detection of the spin current over a much larger length scale than the spin-diffusion length of the spinpolarized material Spin Seebeck effect The first spin Seebeck effect (SSE) was discovered on a ferromagnetic metal (FM) Ni 81 Fe 19 film (permalloy) 18, using the transverse spin Seebeck (TSSE) geometry, which is shown in Fig. 1.10(a). This configuration was also used to detect the SSE in ferromagnetic semiconductors, 24 magnetic insulators, 25 and non-magnetic semiconductors. 26 In TSSE measurements, both temperature gradient and magnetic field are applied along the x-direction, and spin current diffuses in the z-direction into the normal metal (NM), which is usually a platinum (Pt) film. The spin current is then converted into an electric field E ISHE along the y-direction by the inverse spin-hall effect (ISHE) 27 in the NM. The ISHE is normally present in NMs that have strong spin-orbit coupling. The E ISHE can be written as 28 E ISHE θ ρ 2 e j A SH = s σ (1.15) where θ SH, ρ, A, e, and σ are the spin-hall angle of the NM, the electrical resistivity, the contact area between FM and NM, the electron charge, and the spin-polarization vector, respectively. σ is parallel to the direction of magnetization (M). θ SH is especially large in 26

48 noble metals with strong spin-orbit interaction, such as Pt. Figure 1.10(b) shows a schematic diagram of ISHE in a NM. When the spin current is injected into the NM, electrons with one spin orientation are preferentially deflected to one side of the NM, resulting in a negative charge accumulation on that side. At the same time, electrons with the other spin orientation are pulled away from the other side and into the direction of the spin current, leading to a positive charge accumulation on that side. This process establishes E ISHE in the direction perpendicular to the spin current. Therefore, the NM serves as a spin current detector in this configuration, and one can detect the TSSE by simply measuring a voltage difference induced by E ISHE in the NM. (a) Transverse configuration (b) Inverse spin-hall effect V z y x NM V E ISHE M E ISHE js js FM E ISHE T j s Figure Schematic illustration of the (a) transverse spin Seebeck configuration and (b) inverse spin-hall effect. j s is the spin current, M the magnetization, E ISHE the electric field induced by the inverse spin-hall effect. The blue and red arrows in (b) indicate spin-down and spin-up, respectively. 27

49 One of the striking features of the TSSE is that the effect persists over a much longer distance (a few millimeters) than the conventional spin diffusion length of ferromagnetic materials (a few nanometers). 24 In order to enable the non-local thermal spin injection over this macro length scale, there must be certain mechanisms that have comparably long characteristic lengths. In addition, the magnitude of the voltage signals has a strong spatial dependence along the x-direction of the sample, including a sign reversal of the measured voltage between hot and cold side. The voltage near the middle of the sample becomes close to zero. Sanders and Walton 29 give a clue in the form of their description for the effective magnon * ( m ) T and phonon * ( p ) T temperature distributions in a magnetic insulator such as yttrium iron garnet (YIG) under a uniform temperature gradient. Their solution to a simple heat-rate equation yields a T T * * sinh( m p) profile with a magnon decay length λ m. Together with a theory given by Xiao et al., 30 this closely matches the observed spatial dependence of the SSE, including the sign reversal. The long-range feature of the TSSE has been experimentally shown in a magnetic insulator (La:YIG/Pt), 25 a ferromagnetic metal (Ni 81 Fe 19 /Pt), 31 and a ferromagnetic semiconductor (GaMnAs/Pt). 24 In a magnetic insulator with weak magnetic damping, such as YIG and La:YIG, the rate at which magnons are equilibrated is relatively slow, and the resulting λ m was shown to reach several millimeters. This is called magnon-mediated SSE in which the long λ m can lead to the long-range SSE. 30,32 It was also shown that λ m decreases rapidly as temperature is decreased. Surprisingly, the long-rage effect still remains at low temperatures where λ m is no longer comparable with the sample length, and the magnitude of the SSE shows a sharp increase at those 28

50 temperatures. The similar low temperature behavior has been observed in Ni 81 Fe 19 /Pt, and GaMnAs/Pt. 33 In these ferromagnetic conductors, the long-range SSE cannot be explained by the magnon-mediated mechanism because the strong magnetic damping in ferromagnetic conductors drastically suppresses λ m. The experimental observations along with a theory proposed by Adachi et al. 34 suggests that the coherent phonons residing in the substrate lead to the observed long-range feature as well as the SSE peak at low temperatures in Ni 81 Fe 19 /Pt and GaMnAs/Pt; this process is called phonon-mediated SSE. The distribution function (or * T m ) of magnons is modulated by the thermally excited, nonequilibrium phonons through the magnon-phonon interaction. This modulation results in the thermal spin injection into the Pt strip. This model also gives an answer to the low temperature behavior of La:YIG/Pt. A modern microscopic theoretical model for the general non-local nature of the TSSE has been recently given by Tikhonov et al. 35 The TSSE configuration has an advantage in that it can remove possible contaminations in the measured SSE voltage signals from other thermomagnetic effects caused by charge currents, such as an anomalous Nernst effect. This is because the temperature gradient and the magnetic field are applied along the same direction. However, it was reported 36 that a mismatch of the thermal conductivity between the ferromagnet film and the substrate can lead to an accidental temperature gradient along the z-direction ( T z ), and the resulting anomalous Nernst effect contaminates the SSE signals. It is also possible that Tz exists when there is a relatively large heat loss through the voltage or thermometry contacts. Therefore, it is important to carefully 29

51 choose materials for the substrate as well as to design the experimental setup so that heat losses are minimized. Using bulk materials without a substrate can also be a way to avoid an undesirable temperature gradient. Overall, assuming a properly designed experimental setup, the TSSE configuration offers an ideal platform for detection of pure SSE signals since parasitic thermomagnetic effects are ruled out. V normal metal ferromagnet E ISHE js z y M T x Figure Schematic illustration of the longitudinal spin Seebeck configuration. j s is the spin current, M the magnetization, and E ISHE the electric field induced by the inverse spin-hall effect. Figure 1.11 shows an experimental configuration for measurements of the longitudinal spin Seebeck effect (LSSE), in which a ferromagnet (F) is covered with a NM. A temperature gradient applied along the z-direction creates a gradient in 30

52 magnetization M, which in turn results in the spin current j s along the same direction through either magnon thermal conductivity or magnon-phonon drag. The j s is injected into the NM and converted into an electric field E ISHE by the ISHE. 37 The injection of j s across the F/NM interface can be qualitatively explained by the electron-magnon exchange interaction at the interface. 38 Experimentally, it has been shown 39 that the quality of the F/NM interface significantly affects the magnitude of the SSE signals. For the LSSE measurement, it is required to use a magnetic insulator since the geometry is exactly the same as that for the conventional Nernst measurement. Thus, if a ferromagnetic metal or semiconductor is used, the measured voltage signals could be dominated by the anomalous Nernst effect which looks exactly like the SSE signals. The most commonly used material for the LSSE measurement is YIG covered with Pt (YIG/Pt). The LSSE has been measured on either bulk YIG (single crystalline or polycrystalline), or single-crystal YIG film grown on a gadolinium gallium garnet (GGG) substrate. While the spin current is driven by the magnon thermal conductivity in this electrically insulating material, it has been shown 28 that the phonon-mediated process through magnon-phonon interaction plays an important role at low temperatures in a single-crystal bulk YIG sample, which dramatically enhances the SSE signal. The YIG film on GGG allows for the study of how the sample thickness effects the SSE signal. It has been demonstrated 39 that the signal increases as the YIG film thickness is increased from 25nm to 150nm, and saturates after that. 31

53 The LSSE configuration has an advantage in that the required sample geometry and measurement setup are relatively simple compared to those of the TSSE measurement. On the other hand, its geometry places a limitation on material choices. The large dependence of SSE signal on the quality of sample and interface demands an establishment of a standardized method for sample preparation for further progress in this field. 32

54 1.3 Thermal transport by lattice vibrations In a solid, heat is conveyed by atomic vibrations. If the solid is an electrical conductor, conduction electrons also contribute to the heat transport. In a magnetic material, an excitation of the long-range magnetic order (magnons) can carry heat. How well a material conducts heat depends on numerous factors: temperature, defects, impurities, sample size, isotopes, etc. Sometimes, the three different heat conducting modes can interact with each other to largely influence the thermal conduction in the material. Here, we will study the heat conduction by lattice vibrations in solids Lattice thermal conductivity Lattice vibrations and phonons In a solid, bonding between atoms is established by a balance between the attractive and repulsive forces between atoms. Different types of attractive force can participate in the bonding, resulting in ionic bonding, covalent bonding, metallic bonding, and van der Waals bonding. On the other hand, a repulsive force is required to keep the atoms from getting too close to each other. Figure 1.12 shows an interatomic potential for bonding in solids. The interatomic potential u is given by 40 A B ur () = n m r r (1.16) 33

55 where r is the interatomic distance, the first term represents the repulsive part, the second term the attractive part, and n > m, meaning the repulsive part prevails only for short distances. Interatomic potential, u (arb. units) attractive part repulsive part resulting potential a Interatomic distance, r Figure A schematic diagram of interatomic potential for bonding in solids. The potential u between two atoms has a minimum at the equilibrium distance a. The interatomic potential u can be expanded for distances x close to the minimum where r = a, using a Taylor series: 2 3 dua ( ) 1 dua ( ) 2 1 dua ( ) 3 u( x) = u( a) + ( x a) + ( x a) + ( x a) dx 2 dx 6 dx (1.17) 34

56 The first term is simply an offset. The second term, which represents the force acting between two atoms, is zero because the slope of u is zero at the equilibrium distance a. The third term states that the u close to the equilibrium depends on the square of the distance change, i.e. the force depends linearly on the distance change. Thus, this term is responsible for the elastic behavior of solids. The fourth and higher order terms are usually neglected, but when they become non-negligible, e.g. a becomes sufficiently large, an anharmonic effect arises, which plays an important role in the heat conduction by lattice vibrations as well as in the thermal expansion of solids as we will discuss later. Lattice vibrations are often simulated using a spring-ball chain model. Here the simplest case is considered: one-dimensional infinite chains of one atom per unit cell. Figure 1.13(a) shows an atomic chain with one atom per unit cell, which can be considered as a one-dimensional atomic lattice with one atom per unit cell and a lattice vector of length a. Then, the reciprocal lattice vector has a length of 2π/a. The atoms are sitting at the lattice sites and connected to each other with springs of a spring constant K. The equation of motion for nth atom is 2 d xn ( 2 n n 1) ( n+ 1 n) M = K x x + K x x (1.18) dt where M is the mass of the vibrating atom, and x n is the displacement of the nth atom from the equilibrium position (or lattice site). A solution for this equation is given in the form of a wave defined on only the lattice sites: x () t e i kan t n ( ω ) = x (1.19) 35

57 where k = 2π/λ is the one-dimensional wave vector of the wave with the given wavelength, and ω and x are the frequency and amplitude of the oscillation, respectively. By substituting this into Eq. (1.18) and solving for ω, it is found that K ka ω ( k) = 2 sin. (1.20) M 2 (a) a+x n -x n-1 a+x n+1 -x n n-1 n n+1 (b) Frequency, ω (arb. units) 0-2π/a 0 1st Brillouin zone k = 2π/λ 2π/a Figure (a) Spring-ball model representing a one-dimensional chain with one atom per unit cell. (b) Vibration frequency ω as a function of wave vector k. The 1 st Brillouin zone is defined as -π/a < k < π/a as shown. 36

58 Equation (1.20) is called a dispersion relation, which describes a relation between a wave vector and a frequency or energy. This dispersion relation is plotted in Fig. 1.13(b). For a small k which corresponds to a long wavelength, the sine function in Eq. (1.20) can be replaced by its argument, and the linear dispersion relation is obtained: K ω = ak = vk (1.21) M where v = ω/k is the phase velocity of the wave. The group velocity of the wave is given by ω/ k. Note that when Eq. (1.21) is valid, the group velocity and the phase velocity are the same. This case corresponds to the propagation of sound waves with a velocity v in a uniform elastic medium, as if there is no atomistic structure. Therefore, in the small k limit, v is equivalent to the sound velocity and obtained simply by taking the slope in the dispersion curve with the limit of k 0. On the other hand, the shortest possible wavelength is given as λ = 2a and k = π/a. For k = ±π/a, the group velocity of the wave is zero, therefore, the solutions of Eq. (1.18) become standing waves. Equation (1.20) suggests that ω(k) is periodic in k with a period of 2π/a, which corresponds to the reciprocal lattice vector. Due to this periodicity and the symmetry with respect to the origin, it is sufficient to consider the dispersion curve in the interval of -π/a < k < π/a. This region is defined as the first Brillouin zone of the lattice. To have a more realistic model than the infinite chain of atoms discussed so far, we will consider a case where a finite number of atoms vibrates in a chain. For this case, 37

59 the appropriate boundary condition is given by M. Born and T. von Karman ( Born-von Karman boundary condition ) for a chain with N atoms: xn+ n() t = xn() t (1.22) which describes a periodic boundary condition where the two remote ends of the chain are joined back together by another of the same springs that connect internal atoms. By substituting Eq. (1.19), it becomes e = e, (1.23) ikan ika( N + n) which requires that e ikna = 1. (1.24) From this, we find a finite number of possible values of k: 2π k = m (1.25) an where m is an integer. Since the vibrations are unaffected by adding multiples of the reciprocal lattice vectors 2π/a to k (Eq. (1.20)), there are just N possible different values for k, each with a unique vibration frequencies ω. These N independent vibration modes are called normal modes. The quantum theory of lattice vibrations 41 teaches us that the energy levels each harmonic oscillator can take are quantized. If we consider vibrations of one atom as one harmonic oscillator, its energy levels are expressed as 38

60 1 E l = l+ ω 2 (1.26) where l = 0, 1, 2,, and ( ) 1/2 ω = K / M. In the case of N harmonic oscillators, each with a different k and ω(k) as we have seen previously, the quantized energy levels should be 1 El ( k) = l+ ω( k) 2. (1.27) Note that in these expressions, k can be considered as a quantum number like l, and it can label different oscillators by taking only discrete values. The combination of k and l defines one vibration state (or normal mode) of the solid. In other word, it is the oscillator k that is excited to the level l. Now, we introduce a quantum mechanical counterpart of normal modes in the classical mechanics, which are called phonons. Although named differently, phonons are exactly equivalent to normal modes. While normal modes represent a wave behavior, phonons have particle-like properties, analogous to photons. The language of phonons is preferred often, as it provides much more convenience in describing properties like thermal conductivity. 39

61 Debye model and lattice heat capacity The lattice heat capacity for a solid was first described through classical statistical mechanics, as derived from the equipartition theorem. For a one-dimensional harmonic oscillator in contact with a heat bath, the mean energy is k B T. For a three-dimensional oscillator, it is 3k B T. Therefore, the heat capacity for one oscillator is simply 3k B, which is independent of both temperature and material. This is called the Dulong-Petit law. While this straightforward model explains the heat capacity of materials at high temperatures quite well, it conflicts with the third law of thermodynamics, which dictates that the heat capacity of all substances must go to zero at absolute zero temperature. Also, it has been observed experimentally that the heat capacity at low temperatures typically shows T 3 dependence. In order to address this discrepancy, Debye suggested a more realistic model for the lattice vibrations at low temperatures. 40

62 ω D ω ω = vk -π/a 0 π/a k Figure Debye approximation. Red curve is the real phonon dispersion within the 1 st Brillouin zone. The blue line indicates the Debye approximation. ω D is the Debye frequency. Note how closely the blue line approximates the red curve at small k, while it becomes significantly different at higher k. The Debye model assumes that the real dispersion curve as shown in Fig. 1.13(b) can be approximated by a linear dispersion ω ( k) = vk for all k in the 1 st Brillouin zone (Fig. 1.14). Obviously, this model is inaccurate for the excitations at higher k and for a unit cell containing more than one atom because it ignores the optical branches completely. Even so, it successfully explains the heat capacity at low temperatures. It is because that at low temperatures, only phonons with low frequencies (or low k values) are excited and become relevant. The heat capacity C given by the Debye model is 41

63 3 ΘD T 4 x xe 2 (1.28) 0 T C = 9nkB dx ΘD x ( e 1) where n is the number of ions per unit volume, and ω kt B = x. Here, D Θ is the Debye temperature, defined by k Θ = ω = vk (1.29) B D D D where ω D = kdv is the Debye frequency. k D is a measure of the inverse interparticle spacing, ω D is a measure of the maximum phonon frequency, and Θ D is a measure of the temperature above which all phonon modes begin to be excited, below which modes begin to be frozen out. Equation (1.28) expresses the heat capacity at all temperatures in terms of Θ D. If we take the low temperature limit such that T Θ D, it can be simplified into 3 2 π T B 5 ΘD 12 C = nk ; T ΘD. (1.30) This is the Debye T 3 law that fits experimental data very well. At the high temperature limit, on the other hand, Eq. (1.28) becomes C = 3 nk ; T Θ (1.31) B D which is the Dulong-Petit law. 42

64 Temperature dependence of lattice thermal conductivity The total thermal conductivity of a non-magnetic solid is written as κ = κe + κl where κ e and κ L are the electronic and lattice thermal conductivities, respectively. The κ e is often calculated using the Wiedemann-Franz law, which states that the ratio of thermal and electrical conductivity is constant for all metals at a given temperature: 2 κe π kb = T = LT σ 3 e 2 (1.32) where L is the Lorenz number. The κ L can be described using the kinetic gas theory and is given as κl= Cvplp= Cvpτ p (1.33) 3 3 where C is the lattice heat capacity, v p the phonon velocity, l p = v p τ p the phonon mean free path, and τ p the relaxation time of phonons. The v p can be replaced with the sound velocity v which is calculated by taking the slope at the origin in the phonon dispersion curve. The τ p is determined by different phonon scattering mechanisms at a given temperature. So far, we have discussed how C and v p can be calculated using the Debye approximation. In this section, mechanisms that affect τ p, and thus l p, will be briefly reviewed. Then, we will discuss how the interrelation between the three parameters in Eq. (1.33) determines the κ L at different temperature regimes. 43

65 (a) (b) (c) κ 1 10 T (K) Figure An example of thermal conductivity of single crystalline solids. Three different temperature regimes can be defined according to dominant scattering mechanisms: (a) T 3 range where boundary scattering on phonons is dominant. (b) T-flat range where defect scattering is dominant. (c) exp(-t) range where phonon-phonon scattering is dominant. The green curve represents the thermal conductivity of the same material, but with a larger size than the one represented by the orange curve. Figure 1.15 shows the typical thermal conductivity of single crystalline solids. Depending on what scattering mechanisms are dominant to limit l p of phonons at a certain temperature regime, the temperature dependence of l p can be determined. In crystals with very few defects, it is possible that phonons propagate long distances at low temperatures (regime (a)) such that l p becomes longer than the sample size. This temperature regime is called Casimir regime. 42,43 Thus, the boundary scattering becomes the dominant scattering mechanism for phonons in the Casimir regime, and l p can be assumed to be constant with the same value as the sample size. Since the heat 44

66 capacity C follows T 3 behavior in the same temperatures, the resulting κ L also increases as temperature is increased, usually following the T 3 law. It is possible to observe a strong size effect in the Casimir regime, in which a sample with a larger size exhibits higher κ L due to the longer l p. Such an example is shown as the green curve in Fig At higher temperatures (regime (c)), the number of phonons increases due to higher excitation energies, and phonons can be scattered from other phonons, thus phonon-phonon scattering becomes dominant. This causes l p and thus κ L to decrease. The temperature dependence of the phonon-phonon scattering probability 1 τ p is given by τ exp( T ), since the number of phonons that can give rise to the phonon-phonon 1 p scattering increases exponentially as temperature increases. Based on the temperature dependence of κ L discussed so far, it is obvious that there must be a temperature regime between (a) and (c) that produces a maximum in κ L. The most influencing scattering mechanism in this temperature range (regime (b)) is scattering by imperfections such as point defects, dislocations, etc. The more imperfections present, the smaller the maximum value of κ L. The maximum peak is typically found at about 1/10 of the Debye temperature. 45

67 1.3.2 Anharmonicity and phonon-phonon interactions Thus far, discussions made have relied on two simplifying assumptions: 1) displacements of atoms from their equilibrium sites are small, and 2) the properties of solids can be calculated accurately by retaining only the second-order term in Eq. (1.17), which corresponds to the harmonic approximation. While the assumption 1) appears to be reasonable in most solids well below the melting point, the assumption 2) brings about incorrect descriptions of many important physical phenomena. For instance, phonon-phonon scattering cannot happen in a situation where all lattice vibrations are purely harmonic. Since the principle of superposition holds for waves propagating in a harmonic solid, the waves can propagate through each other without creation, destruction, or scattering between them. This yields an unrealistic result that κ L can be infinite in a perfect crystal even at relatively high temperatures due to the lack of phonon-phonon scattering, which is the only intrinsic source of thermal resistance. To address this type of problem arising from the harmonic approximation, the anharmonic terms in Eq. (1.17) must be taken into account. This corresponds to the case where the amplitude of the oscillations becomes so large that the cubic and higher-order terms in Eq. (1.17) can no longer be neglected. An important phenomenon that can be explained by the presence of the anharmonicity in lattice vibrations is the thermal expansion of solids, which reflects the temperature dependence of the volume change of a solid. If we only consider the linear expansion of isotropic solids, the coefficient of thermal expansion α can be defined as 46

68 L = α T (1.34) L where L/L is the fractional length change of the solid. α is usually quite small and temperature dependent, vanishing at zero temperature. The thermal expansion of solids can be understood by inspecting Eq. (1.17). While a harmonic potential is expressed in terms of the dominant second-order term, the thermal expansion requires that the first anharmonic term (the cubic term) be important. This cubic term makes the potential asymmetric and thus leads to a change of the equilibrium distances for different temperatures, as shown in Fig At a low temperature (T 1 ), the oscillation takes place between the positions r 1 and r 2. Since the potential is approximately symmetric around its minimum, the average interatomic spacing is equal to the equilibrium distance. For a higher temperature (T 2 ), the oscillation happens between r 3 and r 4. The potential is not symmetric anymore, and the interatomic distances expand slightly on average, leading to the expansion of the whole solid. The interatomic distance follows the red curve in Fig at higher temperatures. 47

69 Interatomic potential, u (arb. units) r 3 r 1 r 2 r 4 k B T 2 k B T 1 Interatomic distance, r Figure Diagram for the thermal expansion of a solid. The energy scale has been chosen such that the potential minimum has zero energy. The blue solid curve shows the interatomic potential u as a function of interatomic distance r. The blue dashed curve indicates the potential of a perfect harmonic oscillator. The red solid line marks the temperature-dependent mean interatomic distance. The mean energy of the oscillator at a temperature T is k B T. From Fig we can learn another important lesson. If we take a first derivative of u, the interatomic force F is obtained ( F = u r). It becomes obvious that F also depends on the interatomic distance r. This suggests that the spring constants K between atoms are not constant. Therefore, the phonon frequencies ω ( ( ) 1/2 ω = K / M ) also vary with r or with volume V. Here, we define the mode- averaged Grüneisen parameter γ as 48

70 (ln ω) γ (1.35) (ln V ) which represents the dependence of average frequency ω for all phonon modes on the crystal volume. The thermal expansion coefficient α can be also expressed in terms of γ: γ c α = v (1.36) 3B where c v is the specific heat, and B is the bulk modulus defined by B= V ( P V) T. Thus, γ is also strongly related to the anharmonicity of solids. Often, it is not unreasonable to think of γ as a parameter that quantifies the anharmonicity of a solid. γ is temperature dependent, and approaches two different constant values as T 0 and for T >Θ D. Equation (1.36) then implies that the temperature dependence of α will follow that of c v, expressed by Eq. (1.30) and (1.31) for two different temperature limits. Returning to the discussion of phonon-phonon interactions, now we can establish a crucial relationship between the anharmonicity and phonon-phonon interactions. When atoms in an anharmonic solid are displaced by a lattice wave, the elastic properties of the medium (solid) vary as seen by another wave traveling along the path of the first, thus resulting in strong interactions. Therefore, it can be stated that the anharmonicity dictates the intensity of phonon-phonon interactions. Since γ quantifies by how much the stiffness of the bonds is affected by r, the phonon-phonon interaction probability is given to be proportional to γ 2 : 49

71 τω ( ) γ 1 2. (1.37) Phonon-phonon interactions can be classified into two different processes: normal and umklapp processes. Figure 1.17(a) shows a normal process in which the total initial and final crystal momenta are strictly equal. The momentum conservation equation can be written as k1+ k2 = k3+ G (1.38) where G is the reciprocal lattice vector. In a normal process, G = 0, and the two colliding wave vectors (k 1 and k 2 ) and the resulting wave vector (k 3 ) are all contained within the 1 st Brillouin zone. This requires that all k 1, k 2, and k 3 be much smaller than the Debye wave vector k D, and their energy be smaller than ωd. Since both energy and momentum are conserved in a normal process, the process is not resistive and allows heat to flow without interruptions. On the other hand, in an umklapp process (Fig. 1.17(b)), the initial and final momenta differ by a nonzero reciprocal lattice vector, meaning G 0 in Eq. (1.38). This happens when the sum of the two colliding vectors (k 1 + k 2 ) becomes too large such that it can no longer be contained in the 1 st Brillouin zone. This k 1 + k 2 is physically equivalent to k 3 inside the 1 st Brillouin zone due to the periodicity of the reciprocal lattice, and it can be mathematically transformed into k 3 by the addition of G. Because the total momentum is not conserved, the umklapp process is resistive and limits heat flow in solids. Therefore, between the two phonon-phonon interactions, the umklapp process is the intrinsic source of thermal resistance that makes the thermal conductivity of perfect crystals finite. 50

72 (a) (b) 1 st Brillouin zone k 1 k 3 k 3 G k 1 + k 2 k 1 k 2 k 2 Figure (a) Normal and (b) umklapp processes. The points are reciprocal lattice points, and G is the nonzero reciprocal lattice vector. The dashed arrow in (b) indicates the transformation of k 1 + k 2 into k 3 by an addition of G. The nature of both phonon-phonon processes suggests the temperature regime at which each process will dominate. While the normal process happens between phonons with low energies, the umklapp process requires at least one phonon involved in the collision event have a sufficiently high energy. Thus, at very low temperatures, only normal processes can occur with appreciable probability while there still can be a certain amount of umklapp process. The probability of umklapp process is directly related to the number of available phonons that can participate in umklapp processes. This number decreases exponentially as temperature lowers, and the probability that the umklapp process takes place ( 1 τ U ) is expressed as 51

73 τ ~ exp( T / T) ; T Θ (1.39) 1 U 0 D at temperatures well below Θ D, where T 0 is a temperature of order Θ D. From this equation, it can be said that umklapp processes are frozen out at sufficiently low temperatures. However, as long as there is still a non-negligible amount of umklapp processes, the thermal conductivity will still be finite. Note that Eq. (1.37) is applicable to both normal and umklapp processes. We can imagine that a small change in γ associated with either normal or umklapp process would largely affect the lattice thermal conductivity, because γ contributes as a squared term in Eq. (1.33). This observation will play an important role in Chapter 4, where we will discuss magnetic field dependence of lattice thermal conductivity. 52

74 Chapter 2: Development of Bi and Bi 1-x Sb x Based Thermoelectric Materials for Cryogenic Cooling Applications 2.1 Introduction to bismuth and bismuth-antimony alloys Bismuth Bismuth (Bi), one of the three elemental semimetals, played a paramount role in the history of metal physics as well as in the early stage of thermoelectrics research. Many of the experimental phenomena currently used to probe the Fermi surface and the band structure of crystalline solids were first discovered on Bi. 44 Such phenomena include quantum oscillations, such as the de Hass-van Alphen effect 45, and the Shubnikov-de Haas effect. 46 This was possible due to the peculiar properties of Bi that distinguish it from other metals. Moreover, many famous thermoelectric, galvano- and thermo-magnetic effects, e.g. the Seebeck, Hall, Ettingshausen and Nernst effects were also first found on Bi in the 19 th century. Despite the long history and extensive studies done on Bi, it seems that this material still holds unveiled mysteries. Recently, an 53

75 electron fractionalization has been observed for the first time in the ultra quantum limit of Bi. 47 Also, an unknown, new electronic state has been found in Bi in the presence of a strong external magnetic field. 48 Besides those exciting new discoveries, there are many challenges remaining in Bi, especially in understanding its transport properties such as the electrical and thermal conductivities and Seebeck coefficient at temperatures above the liquid nitrogen temperature. This difficulty arises mainly from the strong temperature dependence of band structure in this temperature regime. In terms of a thermoelectric material, although elemental Bi has intrinsically less competitive zt values, it provides various possibilities for developing better thermoelectric materials such as bismuthantimony alloys Crystal and band structures Bi has a rhombohedral crystal structure which can be considered as a slightly distorted cubic structure along its body diagonal. 49 Figure 2.1 shows the unit cell of Bi. The x-, y-, and z- axes correspond to the binary, bisectrix, and trigonal (or c-axis) crystallographic axes, respectively. The unit cell contains two atoms located on the trigonal body diagonal at 23.4 and 76.6 percent of its length. 50 Since one Bi atom has five valence electrons, ten valence electrons belong to one unit cell. The distortion in the rhombohedral structure results in a small overlap between the conduction and valence bands. If there was no such overlap, the ten valence electrons per unit cell would fill a whole number of bands. Due to the overlap, a few electrons are accommodated in the 54

76 next-higher band, and this leads to the presence of a small but equal number of electrons and holes in the vicinity of the Fermi energy at all temperatures. This unique characteristic caused by the small distortion classifies Bi as a semimetal. The distortion from cubic symmetry also gives rise to the strong anisotropy in Bi s electronic properties. z y x Figure 2.1. The rhombohedral unit cell of bismuth. Two atoms are contained in the cell. x-, y-, and z- axes denote the binary, bisectrix, and trigonal (or c-axis) axes, respectively. The Brillouin zone of Bi is shown in Fig The surfaces which contain the T- points are regular hexagons, and the six surfaces with the L-points, called pseudohexagonal, are not regular. The electrons occupy the three ellipsoidal Fermi surfaces at the L-points, and the holes occupy one ellipsoidal Fermi surface at the T-points. The electron pockets (Fermi surfaces) are slightly tilted by an angle ϕ from the respective 55

77 crystallographic axes. The volume of each ellipsoid is only about 10-5 of the Brillouin zone. This extremely small Fermi surfaces indicate very low carrier concentrations. trigonal (z) T L L ϕ y y Γ L binary (x) bisectrix (y) Figure 2.2. The Brillouin zone of bismuth, with principal crystallographic axes labeled as the binary (x), bisectrix (y), and trigonal (z) axes. The hole Fermi surface is an ellipsoid at the T-point, and the electron Fermi surfaces are three ellipsoids at the L-point. Since there are two T-points and six L-points in a Brillouin zone, but each ellipsoid is only half in the first Brillouin zone, there are three electron and one hole Fermi surfaces. ϕ is the tilt angle of the electron ellipsoids with respect to the crystallographic axes. Figure 2.3 shows the band structure of Bi near the Fermi energy at 0K. The band overlap induced by the rhombohedral distortion occurs between the conduction band at the L-point and the valence band at the T-point, and the overlap energy is about 37meV. Bi is a good conductor at 0K due to the presence of this band overlap. The small size of 56

78 the overlap energy suggests a small number of carriers. Therefore, although it is metallic, the electrical conductivity of Bi at 0K is smaller than ordinary metals. The small carrier concentration results in several important properties of Bi. First, the carrier concentration becomes very sensitive to thermal excitations, and it indeed increases rapidly above the liquid nitrogen temperature by an order of magnitude up to 300K. This observation is also related to the band overlap, which can be treated as a negative band gap when applying the Fermi-Dirac statistical distribution, as well as the small direct energy gap at the L-point. Second, the early observations of quantum oscillations stem from the small carrier concentration and extremely high carrier mobilities of Bi. Last, the transport properties can be drastically altered by doping from extrinsic impurities. This can be problematic when it comes to experiments that require materials with an extremely high purity, such as measurements of quantum oscillations. On the contrary, the high sensitivity to doping allows for the possibility of an optimization of the carrier concentration to yield the optimum thermoelectric performance. 57

79 L s E Fh E Fe 37meV 13.7meV L a T Figure 2.3. Band structure of bismuth at T = 0K showing the conduction band at the L- point, valence band at the L-point, and the valence band at the T-point. The dashed line indicates the absolute position of the Fermi energy. E Fe and E Fh denote the Fermi energies of electrons and holes with respect to the band maxima. There are a couple of aspects that are noteworthy in the band structure of Bi. The very small direct band gap (~13.7meV) present at the L-points is responsible for nonparabolicity of the interacting bands, as is the case for some narrow-gap semiconductors. This non-parabolicity of the conduction band strongly influences the transport properties. The other interesting characteristic of Bi band structure is its dependence on temperature. While it is usually a good approximation in other solids that the band structure is temperature independent, Vecchi and Dresselhaus 51 showed that it is not the case for Bi. They observed that the band parameters have a strong temperature dependence using magnetooptical measurements up to room temperature. Specifically, the temperature variation of the direct band gap at the L-point (E g ) was found to be expressed as 58

80 E = E + T + T (2.40) g g (mev) where E g0 is the value of E g at 0K (E g0 = 13.7 mev). The effective masses of electrons were also shown to vary with temperature, except for the heaviest mass along the bisectrix direction, which was not measurable. This dramatic temperature dependence of E g and unknown temperature dependence of other band parameters such as the heavy bisectrix mass have hampered quantitative studies on the transport properties above the liquid nitrogen temperature Thermoelectric properties Single crystalline Bi has a strong anisotropy in its carrier mobilities caused by the rhombohedral distortion. From Fig. 2.2, it is noticed that the electron and hole pockets are ellipsoids elongated along one direction. This anisotropic shape of Fermi surfaces is directly related to the anisotropy of carrier mobilities, which in turn depends on the anisotropic effective masses of carriers. The effective masses along the short axes of the ellipsoids are smaller than those along the longer axes. For electrons, the smallest mass is found along the trigonal direction, and thus electrons are fastest in this direction. Both binary and bisectrix axes lay in the trigonal plane perpendicular to the trigonal axis, and since both axes have three-fold rotational symmetry, it is conventional to take the average values between them in transport measurements. Typically, transport measurements are made either in the trigonal axis direction or in the (trigonal) plane direction in this system. 59

81 The electron effective masses in the trigonal plane are much larger than those in the trigonal axis direction, and thereby electrons are slower. On the other hand, the opposite holds for the effective masses of holes which reside in the ellipsoidal pockets at the T- points. Gallo et al. 52 performed a pioneering set of measurements on the transport properties of single crystalline Bi. Their work showed the pronounced anisotropy in the measured electrical resistivity, Seebeck coefficient, and thermal conductivity. As a result, the zt turned out to be highly anisotropic as well: zt ~ 0.39 in the trigonal axis direction, and zt ~ 0.07 in the trigonal plane at 300K (Fig. 2.4(a)). The superior performance in the trigonal axis direction is attributed to: 1) much higher electron mobility in that direction, and 2) less compensation from holes because hole mobility is higher in the plane and so is the compensation. It is noted that the Seebeck coefficients in both directions are negative despite the equal number of electrons and holes. This result suggests that transport in Bi is dominated by electrons in both measurement directions due to higher electron mobility than hole mobility. 60

82 (a) 0.4 (b) (zt) // 1.6 zt 0.2 z 0 T 1.2 (z 0 T) // (zt) 0.4 (z 0 T) T (K) T (K) Figure 2.4. Figure of merit (zt) of single crystalline Bi adapted from Ref. [52]. (a) zt obtained from measured thermoelectric properties. (b) Hypothetical z 0 T calculated for the electron-only system by assuming that the compensation from the opposite carrier is removed. // and denote the trigonal axis direction and in plane direction, respectively. The zt of Bi is limited mainly due to the compensation from the other carriers with opposite sign, arising from the coexisting equal number of electrons and holes. Gallo et al. 52 made an interesting prediction in which they calculated zt in the trigonal axis and in-plane directions, assuming that the band overlap is lifted, so only one type of carrier is present. Figure 2.1(b) shows z 0 T, the calculated hypothetical zt when only electrons exist in Bi. Surprisingly, z 0 T in both directions is considerably improved over the same temperature range, reaching ~1.8 in the trigonal axis direction, and ~1 in the trigonal plane at 300K. This suggests that if any trick is found to lift the intrinsic band overlap, Bi certainly promises excellent thermoelectric performances in the vicinity of room temperature. 61

83 2.1.2 Bismuth-antimony alloys Single crystals When Bi is substituted by antimony (Sb), one of the same group V elements, these two elements form complete solid solutions over the entire composition range. In other word, they become an alloy at any arbitrary composition as shown in Fig This is because Sb has the same rhombohedral crystal structure with similar lattice constants as well as an almost the same electronegativity with Bi. Cucka and Barrett 49 showed that the lattice constants of the rhombohedral lattice of Bi 1-x Sb x alloys satisfy the Vegard s rule when 0 < x < 0.3. The large temperature difference between the liquidus and solidus in the phase diagram (Fig. 2.5) makes growing homogeneous BiSb crystals a challenging task. Because of the low melting point, post-annealing processes do not help to remove inhomogeneity in the microstructure. Among various attempts, a zone leveling technique 53 and a traveling heater method 54 were successfully applied to grow homogeneous single crystalline BiSb alloys. 62

84 Figure 2.5. Phase diagram of bismuth-antimony alloys adapted from Ref. [55]. Due to the aforementioned similarities between Bi and Sb, the overall properties of BiSb alloys are similar to those of Bi. Still, there are striking differences between the two systems. By adding Sb, six additional hole pockets at the H-points are added in the electronic structure. Electrons are located also at the L-points in Sb. The departure from cubic symmetry is more pronounced in Sb than Bi, and therefore the substitution of Bi atoms by Sb atoms drastically modifies the band structure of Bi. Figure 2.6 shows the band structure variations for Bi 1-x Sb x alloys as a function of x at temperatures close to 0K. It is seen that alloying primarily affects three different band parameters: 1) the overlap between L and T bands, 2) the value of the direct energy gap E g at the L-point, and 3) the 63

85 maximum energy of the H band. In the range of 0 < x < 0.04, the E g gradually decreases as x increases, and at x ~ 0.04, a gapless state appears where the dispersion of the L bands becomes linear ( Dirac dispersion ). Beyond this composition, the bonding L s and antibonding L a bands are inverted and the E g increases with increasing x. At x ~ 0.07, the overlap between the valence band at the T-point and the conduction band at the L-point is lifted, and the material becomes a narrow gap semiconductor. In this semiconducting regime, the maximum band gap occurs at x = 0.15 ~ 0.17, with a value of about 30 mev. The valence band maximum at the H-point keeps increasing with increasing x, and a crossover with L-conduction band occurs at x ~ Thus, above this composition, the material again becomes a semimetal. This strong dependence of band structure on composition suggests that the transport properties will dramatically change with Sb concentration. 64

86 Figure 2.6. Schematic band diagram of Bi 100-x Sb x alloys as a function of x at T ~ 0K, adapted from Ref. [54]. Extensive measurements of the transport properties of single crystalline BiSb alloys with different Sb concentrations have been done by Yim and Amith, 53 and Lenoir et al. 56 A significant anisotropy is observed in the measured properties as in Bi, although the degree of magnitude varies according to Sb concentration. Figures of merit (z) of BiSb alloys with different Sb content are shown in Fig. 2.7 along two crystallographic directions. Like in case of Bi, the trigonal axis direction exhibits much better performance than the in-plane direction. It is noted that the anisotropy is more significant at 80K compared to 300K. The maximum z at 80K occurs at about 16% Sb for both 65

87 directions, yielding zt ~ 0.52 in the trigonal axis direction, and zt ~ 0.3 in the trigonal plane direction. Comparing these numbers to those of Bi in Fig. 2.4, we find that the Bi 0.84 Sb 0.16 alloy has much higher zt at low temperatures, and this is mainly due to the presence of the band gap between the L-conduction band and the T-valence band. In the same context, it is observed from Fig. 2.7 that generally higher zt values are obtained for compositions where the alloy becomes semiconducting. Figure 2.7. Figures of merit of BiSb alloys along the directions parallel (z 33 ) and perpendicular (z 11 ) to the trigonal axis as a function of Sb concentration, adapted from Ref. [53]. 66

88 At a fixed composition, the band structure of the BiSb alloy is also strongly dependent on temperature. Vecchi et al. 57 measured temperature dependence of E g and the effective mass of the binary electrons by measuring magnetoreflection spectra. While the strong dependence of those two parameters on temperature was confirmed, the larger number of unknown parameters in BiSb alloys than in Bi causes more complications. In BiSb alloys, for instance, the H-band is added, and the temperature dependence of this band is unknown. Unless temperature dependences of the dominant bands are discovered, interpretations of the transport properties above the liquid nitrogen temperature will remain as challenging tasks Polycrystalline samples In contrary to single crystals, homogeneous polycrystalline BiSb alloys can be easily synthesized using ball-milling and powder sintering. Effects of grain size and annealing temperature on the transport properties in polycrystalline samples have been studied by Martin-Lopez et al. 58 The mechanical strength of polycrystalline samples is proven to be much higher than that of single crystals. 59 Also, polycrystalline samples do not have cleavage planes and therefore do not have the cleaving issue that single crystalline samples have along their trigonal plane direction. The zt of polycrystalline BiSb alloys is usually lower than that of single crystalline samples as the anisotropy in transport properties is averaged out by the random orientations of grains. Figure 2.8 shows zt of polycrystalline BiSb sample 67

89 compared to zts in two different directions of single crystalline sample at 300K. One of the advantages of polycrystalline samples is that the lattice thermal conductivity can be suppressed by reducing the grain sizes. This is the basis for improved thermoelectric performances of many nanostructured materials. 60 Unfortunately, nanostructuring is not effective in BiSb alloys. 61 In BiSb alloys, grain boundaries scatter electrons more than phonons, possibly due to the long mean free path of electrons compared to other materials. 62 As a result, the loss of electrical conductivity overwhelms the reduction in the lattice thermal conductivity. 68

90 trigonal ZT (300K) polycrystal 0.1 binary/bisectric Sb concentration x (at.%) Figure 2.8. Figure of merit (zt) of single crystalline and polycrystalline BiSb alloys as a function of Sb concentration at 300K. For single crystalline samples, zt is plotted in trigonal axis direction and binary/bisectrix (trigonal plane) direction. At the current stage, the ease of synthesis and superior mechanical stability of polycrystalline BiSb alloys are compensated by relatively lower zt compared to single crystals. Therefore, if the zt of polycrystalline BiSb alloys can be improved to comparable values with those of single crystalline alloys, these materials are likely to become very promising thermoelectric materials for low temperature cooling applications. 69

91 2.2 Study on indium doping in single crystalline bismuth Introduction Although elemental bismuth (Bi) has intrigued many interesting experimental studies until recently because of its peculiar electronic properties as a semimetal, 47,48 it has not received much attention from thermoelectric researchers because the intrinsic overlap between the conduction bands at the L-points and the valence band at the T-point in the Brillouin zone prevents it from having a competitive thermoelectric figure of merit (zt). On the other hand, its alloys with antimony (Sb), Bi 100-α Sb α alloys, show promising n-type thermoelectric performance especially below 200K, 53,56,63 suggesting that this alloy system can be used for cryogenic cooling applications with further improvement in zt. While many band structure properties and parameters are still unknown in Bi 100- αsb α alloys, elemental Bi is an excellent platform for studying new dopant impurities because most of its band structure properties as well as transport properties are well understood. Furthermore, elemental Bi and Bi 100-α Sb α alloys share very similar lattice and electronic structures except for the presence of the band gap between the L- conduction band and the T-valence band. Therefore, once the behavior and physics of a new dopant impurity are uncovered in elemental Bi, they can be easily extended to Bi 100- αsb α alloys. 70

92 There are only few electrically active dopants currently known in elemental Bi and Bi 100-α Sb α alloys. Past studies show that Pb 63,64 and Sn 50,65 are acceptor impurities while Te 50,66 and Li 67 are donors. Discovering new electrically active impurities and understanding their properties are of importance in the sense that each impurity can introduce different effects in the same host material. It has been shown recently 81 that certain impurity dopants form resonant levels in a host material, which may lead to improvement of the thermoelectric power at a given carrier concentration. In addition, conventional dopant impurities can be used to dope a host material n- or p-type in order to optimize the carrier concentration so that its zt is maximized. In this section, we study a new impurity dopant in elemental Bi: Indium (In). Seemingly, this group III element is supposed to have almost no influence on the electronic properties of Bi, as it possesses the same number of outermost electrons as Bi. It is shown that due to its unusual influence on the electronic structure of Bi, In actually behaves as an p-type dopant in Bi. Based upon this unusual effect, a novel doping mechanism is proposed which may open a new way to doping studies of semiconducting materials. In the second part of this section, experimental results for the transport properties of In doped Bi samples are presented and discussed in detail. 71

93 2.2.2 Isovalent doping effect induced by indium in bismuth Experimental Bi 100-α In α single crystals with nominal atomic concentrations α = 0, 0.1, 0.5 were grown by a modified horizontal Bridgman technique. 68 Elemental Bi (99.999%), and In (99.999%) were loaded into the quartz ampoules following the stoichiometric ratios and the ampoules were sealed after evacuation down to about 10-6 torr. Then the samples were horizontally placed and melted at 600 C in a box furnace and slowly cooled. X-ray diffraction (XRD) was used to confirm single crystal peaks on oriented disks of the crystals (Fig. 2.9). (a) (222) (b) (111) Figure 2.9. (a) X-ray diffraction pattern of a single crystalline In doped Bi sample. The measurement was made on the trigonal plane of a cleaved disk from the ingot shown in (b). The highest peak corresponds to (222), a multiple of the trigonal (111) direction. (b) A single crystal ingot grown by the modified horizontal Bridgman method. The ingot was cleaved along the trigonal plane direction after cooling by liquid nitrogen. 72

94 The actual In concentrations reported here were first measured by x-ray fluorescence (XRF). Then the amount of In segregation was estimated by detecting the pure In peak in differential scanning calorimetry (DSC) analysis and subtracted to obtain the final In concentration of each sample. The actual concentrations of In were determined to be α = 0, 0.09, 0.4. The crystals were cut into approximately 2.5mm x 1.5mm x 7mm parallelepipeds for Shubnikov de-haas (SdH) and Hall measurements. The parallelepiped samples were cut so that their long axis is parallel to the trigonal plane (xyplane) crystallographic direction. The trigonal axis (z-) direction is perpendicular to the trigonal plane. The SdH oscillations in ρ xx (B z ) [inset in Fig. 2.10] were measured using the AC Transport (ACT) option in a Physical Properties Measurement System (PPMS) by Quantum Design and a Lakeshore 370 AC bridge. The samples were cooled to 2K and ρ xx (B z ) was recorded while sweeping the magnetic field from 0 to 7T at 50 Oe/s. The Hall coefficients R H of the doped samples were determined by measuring Hall resistances in ρ xy (B z ) configuration using the same instruments. The Hall resistances were measured in both positive and negative magnetic fields, and values in one polarity were subtracted from those in the opposite in order to remove the magnetoresistance component, which is an even function of magnetic field. Results and discussion Figure 2.10 shows the traces of SdH oscillations for three single crystalline samples at 2K. It is notable that the amplitude of the oscillations decreases as the 73

95 impurity content increases. This attenuation of the amplitude can be explained by the decrease in the number of turns that an electron makes along a cyclotron orbit before it loses its phase information due to scattering by impurities. Therefore, this result follows the general observation that the amplitude of the SdH oscillations is reduced as the quality of the sample diminishes. 26 In the presence of a strong magnetic field, electrons become less mobile than holes as supported by their threefold higher Dinger temperature. 69 While the SdH frequency of electrons can still be detected for all other magnetic field directions despite their low mobility, the electron frequency disappears when the field is aligned with the trigonal axis possibly because of a resonance between electron and hole frequencies. 47 Nevertheless, since we are mainly interested in hole concentration of the samples in this study, the limitation does not cause any difficulties. 74

96 z, Trigonal ΔV x I x - y, Bisectrix I x + x, Binary B z Figure Shubnikov de-haas (SdH) traces for Bi 100-α In α samples studied here at 2K. The amplitude of the SdH oscillations for α=0.09 and α=0.4 have been magnified by 10 2, and the background subtracted for all samples. The inset shows the configuration used for the SdH measurements which we denote ρ xx (B z ) where the first, second, and third indexes indicate the direction of current flow, potential difference, and magnetic field, respectively. Each axis in the configuration corresponds to binary (x), bisectrix (y), and trigonal (z) crystallographic direction, respectively. By analyzing the SdH traces in Fig. 2.10, the magnetic field oscillation frequencies [Δ(1/B)] -1, the cross sectional area of the Fermi surface A F, the Fermi energy of holes E F, and the hole concentration P of each sample can be calculated. The results are summarized in Table 2.1. It is clear that In introduces excess holes in Bi. Doping efficiency of In ( = P / In concentration) is estimated to be significantly lower than that of 75

97 Sn and Pb. Considering that 0.08 at.% Sn introduces 1.5x10 19 cm -3 holes in Bi, 70 In is about 40 times less efficient than Sn in terms of p-type doping. Since the amount of In segregation is already subtracted by the DSC analysis, this low doping efficiency requires other explanations. We will try to address this issue in the next section where the transport properties of this system will be discussed. Sample [Δ(1/B)] -1 (T) A F (10 16 m -2 ) E F (mev) P (10 17 cm -3 ) α = α = α = Table 2.1. Magnetic field oscillation frequencies [Δ(1/B)] -1, cross sectional areas of Fermi surface A F, Fermi energy E F, and hole concentrations P of the Bi 100-α In α samples obtained from the SdH oscillations. To further elucidate the effect of In doping on the electronic structure of Bi, Hall resistivity ρ xy (B z ) of the In doped samples is shown in Fig Contradictory to the SdH oscillations, ρ xy (B z ) reveals the presence of both electrons and holes in all In doped samples. At low magnetic field (B z 0.1T), ρ xy (B z ) of both samples shows negative slope suggesting that electrons are present [inset in Fig. 2.11]. In this regime, electrons are barely affected by the magnetic field. They are more mobile than holes as they are in 76

98 zero magnetic field, 71 satisfying µµ, while ν 2 B 2 1, where µ 1 and µ 2 are the 2 1 2B 1 electron mobilities taken for each electron ellipsoid along its 1- and 2-axes, respectively, 50 ν is the isotropic hole mobility in the xy-plane. On the other hand, near linearity with positive slope is observed at higher fields where ν 2 B 2 1, which indicates that holes dominate. 100 ρ xy (B z ) (µω m) B (T) ρ xy (B z ) (µω m) T = 2K B (T) Figure Hall resistivity ρ xy (B z ) versus magnetic field for Bi 100-α In α samples measured at 2K. The points indicate the experimental data, while the lines are added to guide the eye. The symbols are: (green cross) α=0.09, and (blue circle) α=0.4. The inset contains magnification of ρ xy (B z ) at low magnetic field, which shows the transition from the negative slope to the positive slope for both samples. 77

99 The slope of each curve corresponds to the Hall coefficient R H of each sample. R H taken at the low field limit yields electron concentration while that taken at the high field limit reflects excess hole concentration: lim R = C / Nq and B 0 H lim R = C /( P N) q, where q is the elementary charge, N is the electron B H concentration and C is the Hall prefactor for the ρ xy (B z ) configuration. C can be determined using the following expression adapted from Ref. [71] and [72]: C P P 2 = 4 µ 1 µ 2 ν µ 1 + µ ν N N 2. (2.41) When B 0, Eq. (2.2) can be reduced to C 4µµ / ( µ µ ) 2 = +. By inserting µ 1 = cm / Vs and 6 2 µ 2 = 3 10 cm / Vs taken from Ref. [71] at 4.2K, C 0.1 is calculated. Here, we assumed that the ratio between µ 1 and µ 2 is not affected by In doping. Additionally, it is observed that variation in C from 4.2K to 10K is negligible. 71 Therefore, the same C 0.1 may be safely used for T = 2K in Fig When B, ρ xy (B z ) becomes linear and thus R H saturates, indicating that the material becomes degenerate. In degenerate semiconductors or semimetals with spherical constant energy surfaces, like the T-hole pockets in Bi, C = N and (P-N) for each In doped sample can be calculated using the obtained R H and C. Table 2.2 shows the results including P. The P values acquired from ρ xy (B z ) are in excellent agreement with those from SdH in Table 2.1, which were obtained independently. 78

100 Sample N (10 17 cm -3 ) P - N (10 17 cm -3 ) P (10 17 cm -3 ) α = α = Table 2.2. Electron (N), excess hole (P-N), and hole (P) concentrations calculated from the ρ xy (B z ) measurement. Figure 2.12 shows E F of the samples, illustrated on a schematic band diagram of Bi. According to Noothoven van Goor, 50 about cm holes are required to empty the conduction band at the L-point of the Brillouin Zone. Since the higher doped α = 0.4 sample has only cm holes, presence of a small number of electrons is expected and has been confirmed by ρ xy (B z ) measurement as discussed earlier. It is estimated that approximately 2 at.% In is required to vacate the conduction band, which is above the solubility limit of In in Bi

101 L s 13.7meV 37meV E 0.09%In F E 0.4%In F L a T Figure Schematic band diagrams indicating positions of the Fermi energy E F for the samples in this study. The dashed line denotes E F of pure Bi. The results of both SdH and Hall measurements consistently confirm that In, which is the same trivalent as Bi, is a p-type dopant in Bi. The conventional doping scheme in semiconductors, in which the doping impurities form a shallow level close to either conduction band minimum or valence band maximum, but do not affect the shape of these bands, cannot explain the doping effect of In in Bi because it requires the impurities to have a different valence from the host elements substituted by them. In the following, ab initio calculations for In doped Bi are presented based on which we introduce a new doping scheme in semiconductors. We will show that this new doping scheme can effectively explain the experimentally observed behavior of In in Bi. 80

102 Theory Figure 2.13 shows the densities of states (DOS) for a 1.4 at.% In doped Bi sample calculated by the supercell calculations by Dr. Bartlomiej Wiendlocha. Figure Densities of states (DOS) for In doped Bi as a function of electron energy calculated by the supercell calculations. Spin-orbit interactions have been taken into account. The top panel shows the total DOS per atom, and the bottom panel shows DOS of In with partial contributions from different orbitals. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission. 81

103 The most striking feature in this figure is the sharp peak in the DOS at around -5 ev, just below the main valence block. We will call this state as a hyperdeep defect state (HDS), following the convention used by Hoang and Mahanti. 75 According to Ref. [75] and [76], certain impurity atoms in a semiconductor may form a pair of defect states, i.e. a filled, electrically inactive HDS lying below the valence band, and a deep (or trap) defect state (DDS), located in the neighborhood of the narrow band gap. Those defect states correspond to the bonding and anti-bonding pairs of orbitals in a molecular analogy. Examples of such impurities are oxygen in GaAs, 76 and In, Ga, and Tl in PbTe. 75 The DDS is called a resonant state in case it is hybridized with the main valence band block, creating a hump in the valence band DOS of host material. Tl in PbTe 77,78 is a well established example which shows a formation of the resonant state by Tl in the valence band of PbTe. The HDS is not expected to affect the transport properties since its energy level is too far from the E F, and it is a highly localized state. This has been shown to be the case for PbTe doped with In 75 and Tl. 75,79 However, we will see that the situation is different in In doped Bi. Detailed features of the HDS can be found from the analysis of the real-space charge distribution around the In impurity shown in Fig Charge density, corresponding to the sharp HDS peak at -5 ev, is projected on the plane along the Bi-In bonds, and this plane contains two Bi atoms from the three nearest neighbors and two from the next-nearest neighbors. The charge density shows a typical s-p hybridization where the spherical 5s In electron cloud at the center is hybridized with the 6p Bi orbitals, creating the s-p bonds. From the reciprocal-space point of view, this local bond would 82

104 correspond to a dispersionless band, and the electronic states are expected to be localized. In addition, it is found that the HDS accommodates about one In and one Bi electron. This one Bi 6p electron comes mainly from the three nearest and three next-nearest neighbors, thus one In 5s electron binds with 1/6 p-electron from each of 6 neighboring Bi atoms. Figure Charge density in logarithmic scales around a In impurity. The plot reflects the hyperdeep defect state peak in Fig The nearest (Bi(1)) and next nearest (Bi(2)) atoms are labeled. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission. 83

105 Unlike the cases of oxygen in GaAs and In, Ga, and Tl in PbTe, only the HDS is formed in In doped Bi; there is no DDS. Furthermore, this HDS does affect the transport properties, even though it is a localized state in the deep energy level, and this unusual behavior distinguishes In doped Bi from the other aforementioned systems. By doing a simple electron count, we can learn what effect In will cause in the electronic structure of Bi. Let us assume that we have 100 Bi atoms. Since Bi is trivalent, the number of electrons in the main valence band block is 300. Now, if we substitute one Bi atom with one In atom, which corresponds to doping with 1 at.% In, we will have 99 Bi atoms and 1 In atom. In is also trivalent, therefore if we think about the classical impurity behavior, it is supposed to provide three electrons to the system, making no differences in the electron counting. However, as we have seen, one In 5s electron forms the HDS by binding with six 1/6 6p electrons (= one 6p electron) from Bi. Thus, In can only give two electrons (one 5s and one 5p) to the main valence band block /6 electrons are provided by the six neighboring Bi atoms, and the other 93 Bi atoms contribute three electrons per atom. The total electron count becomes: x (2 + 5/6) + 93 x 3 = 298, leaving two holes in the valence band block. Therefore, by forming the HDS in the deep energy level, trivalent In behaves as a strong p-type dopant in Bi, donating two holes per In atom. To confirm the p-type doping behavior of In in Bi, additional calculations were performed by Dr. Wiendlocha in the vicinity of E F using the Korringa-Kohn-Rostoker method with the coherent potential approximation (KKR-CPA). 80 The computed DOS for In doped Bi with different In concentrations is shown in Fig The general effect 84

106 is that the E F moves deeper in the valence band, confirming the acceptor behavior of In. Also, lack of a DDS peak near the E F is confirmed. Instead, In creates a broad DOS 'hump', consisting of s- and p-like contributions. Even for the In concentration as small as 0.1%, the full width at the half maximum of the hump is of the order of 500 mev. This is at least 10 times broader than the resonant DOS peak introduced by Tl in PbTe at the same concentration order. 78,79,81 Upon increasing the In concentration, the E F moves to lower energies, again being consistent with the p-type impurity behavior, although modifications of the DOS suggest more than a rigid-band shift. As is a characteristic for a disordered system, smoothing of the DOS is observed when the In concentration is increased. 85

107 Figure Evolution of densities of states (DOS) for In doped Bi with increasing In concentration. The figure has been provided by Dr. Bartlomiej Wiendlocha and used with his permission. 86

108 2.2.3 Transport properties of indium doped bismuth Experimental The same single crystalline Bi 100-α In α samples used for the SdH and Hall measurements were used for transport measurements. Electrical resistivity ρ, thermal conductivity κ, and Seebeck coefficient S were measured simultaneously while slowly sweeping the temperature from 320K to 2K using the Thermal Transport Option (TTO) in a Physical Properties Measurement System (PPMS) by Quantum Design. Magnetoresistivity ρ xx (B z ) was measured at various temperatures with the aid of the AC Transport (ACT) option in the PPMS and a Lakeshore 370 AC bridge. We estimate error on the individual transport properties to be 5%, with the error in S stemming mainly from the size of the thermometry and that in ρ and κ from errors in sample geometry. These geometric errors are canceled out in the measurement of zt, giving an error of 10% in zt. Furthermore, at temperatures above 200K, thermal radiation adversely affects the accuracy of the thermal conductivity measurement due to its static heater and sink nature, thus reporting a higher value than is correct. This will lead to an underestimation of zt above 200K. Results and discussion ρ(t) of Bi, Bi In 0.09, and Bi 99.6 In 0.4 in both trigonal axis direction and trigonal plane are shown in Fig. 2.16(a) and (b), respectively. ρ(t) of the In doped samples shows 87

109 a bump at 60K in both directions, while that of pure Bi displays a metal-like behavior. 82 This bump is more evident for the higher doped samples. It is noted that ρ(t) of the In doped samples shows very different behavior from what has been observed in Sn doped Bi samples, 83 suggesting that those two different acceptors introduce different scattering mechanisms. In Sn doped Bi, the ionized impurity scattering is dominant especially at low temperature, and samples doped with almost the same acceptor concentrations as 0.09% and 0.4% In show ρ(t) which rather resembles that of pure Bi with no bumps. At higher Sn concentrations, ρ(t) flattens out above 200K. On the other hand, ρ(t) of the In doped samples closely follows what has been observed in compensated Bi by Issi et al. 84 In their work, it was shown that a significant amount of neutral impurity scattering is introduced by doping Bi with similar concentrations of Te and Sn simultaneously. Whereas the neutral impurity scattering is insensitive to temperature, it is especially predominant at low temperature where the acoustic phonon scattering is suppressed. By fitting the experimental data with the neutral impurity scattering taken into account, Issi et al. confirmed that the bumps in ρ(t) at low temperatures is a characteristics of the dominant neutral impurity scattering in this system. 88

110 (a) T (K) T (K) (b) ρ // (µω m) 1 1 ρ (µω m) 0 0 Figure Temperature dependence of the electrical resistivity in (a) the trigonal axis direction ( ρ //), and (b) the trigonal plane ( ρ ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The solid black line represents the computed curve for Bi 99.6 In 0.4, while the dashed black line corresponds to the computed curve for pure Bi. Here, we employ the same calculation procedure for the 0.4% In doped sample in the trigonal plane. Unlike the compensated Bi in which the authors assumed the same electron and hole concentrations as pure Bi, the carrier concentrations are unknown in the In doped samples except at the lowest temperatures. The total electrical conductivity σ of the In doped samples can be expressed as σ = σe + σh = Nq + + Pq + µ φ µ r νφ ν r (2.42) where σ e and σ h are the partial electrical conductivities, N and P the carrier concentrations, and µ and ν the mobilities of electrons and holes, respectively. The index ϕ refers to the 89

111 scattering by acoustical phonons, while the index r refers to the scattering due to the added impurity. At 2K, we may assume that σ has reached its residual value σ r, hence 1/ µ φ 1/ µ r, and σ σ r = Nqµ r + Pqνr. Since N and P at 2K are known from the SdH and Hall resistivity measurements (Table 2.2), µ r and ν r can be obtained by fitting an experimental galvanomagnetic coefficient. Saunders et al. 85 showed that several band parameters can be determined simultaneously by measuring only one component of the magnetoresistivity tensor at intermediate magnetic fields, which had been suggested theoretically by Aubrey. 86 The longitudinal magnetoresistivity ρ xx (B z ) of Bi is given as ρxx( Bz ) = σ σ xx 2 2 xx + σxy µ N νp µµ 1 2B 1+ ν B = 2 q 2 µ N ν P µµ 1 2NB ν PB µµ 1 2B 1 ν B µµ 1 2B 1+ ν B 2 (2.43) where µ 1 and µ 2 are the electron mobilities taken for each electron ellipsoid along its 1- and 2-axes respectively, 50 ν is the isotropic hole mobility in the xy-plane, and ( ) µ = µ + µ. 1 2 / 2 90

112 8 ρ xx (B z ) (µω m) 4 T = 2K B (T) Figure Magnetoresistivity ρ xx (B z ) of Bi 99.6 In 0.4 versus magnetic field measured at 2K. The points indicate the experimental data, while the solid black curve is the fit generated by the method of least squares with regard to Eq. (2.4). Figure 2.17 shows the experimental data points of ρ xx (B z ) at 2K and the fitted curve using Eq. (2.4) with the known N and P. The method of least squares was used with µ 1, µ 2, and ν as unknowns for the fitting. Overall, very good agreement between the experimental points and the fitted curve was obtained. It is worthy to note that Noothoven van Goor 50 also used the same procedure and found that the fitting does not work at relatively higher magnetic fields. It is known that Bi exhibits magnetostriction, 87 which contributes additional magnetoresistance essentially in a quadratic form, preventing the magnetoresistance to saturate at high fields. Since Eq. (2.4) does not take the effect of magnetostriction into account, we confine the fitting within intermediate magnetic fields as pointed out by Saunders et al. 85 The obtained µ r and ν r at 2K are assumed, as a first step, to be insensitive to temperature, which will be valid if the neutral impurity scattering dominates. Since we know the temperature dependence of µ ϕ and 91

113 ν ϕ, 71,82 the temperature dependence of combined mobilities can be calculated using Matthiessen s rule as in Eq. (2.3). Then N and P at each temperature are found again by fitting ρ xx (B z ) with the known mobilities. Table 2.3 summarizes the carrier densities as well as the mobilities in the trigonal plane of the 0.4% In doped Bi sample obtained using the above procedure. It was noticed that the fit to ρ xx (B z ) above 100K is not as good as at lower temperatures, possibly because the thermal excitations from the valence band at the L-point were not taken into account. Finally, Eq. (2.3) gives σ at each temperature, which is plotted in Fig. 2.16(b) along with the calculated curve for pure Bi. Both computed curves fit the experimental data reasonably well. This result assures the presence of neutral impurity scattering in the In doped Bi samples, although the origin of it is not obvious. We will discuss about possible origins of the neutral impurity scattering later in this section. 92

114 T N P Mobility (10 4 cm 2 V -1 s -1 ) (K) (10 17 cm -3 ) (10 17 cm -3 ) µ 1 µ 2 ν Table 2.3. Temperature dependence of parameters in the trigonal plane of Bi 99.6 In 0.4 sample obtained by fitting Eq. (2.4) to the experimental data: N, electron concentration; P, hole concentration; µ 1, binary electron mobility; µ 2, bisectrix electron mobility; ν, isotropic hole mobility in the trigonal plane. In pure Bi, S is always negative in both the trigonal axis and trigonal plane directions since mobility of electrons is much higher than that of holes due to their small effective mass. It is also known that zt along the trigonal axis direction is always larger than that in the trigonal plane. 52 While the latter is also the case in the In doped samples, it is interesting that the doped samples show enhanced negative S compared to pure Bi (Fig. 2.18(a) and (b)). When there are both electrons and holes, the total S for each direction can be expressed as 93

115 S = S σ + S σ e e h h σ + σ e h (2.44) where S e and S h are the partial thermopowers of electrons and holes, respectively. Moreover, in a similar way with the Matthiessen s rule, one can combine S for different scattering mechanisms using the Gorter-Nordheim rule, which we apply here for S e and S h : S = Sφ µ φ + Sr µ r. (2.45) 1 µ + 1 µ φ r From Eq. (2.5) and (2.6), an expression for the total S of the In doped samples is found: S = Seφ µ φ + Ser µ r Shφ νφ + Shr νr Nq + Pq ( 1 µ 1 ) 2 ( 1 1 ) 2 φ + µ r νφ + νr. (2.46) 1 1 µ φ + µ r νφ + νr Nq + Pq µµ νν φ r φ r To evaluate the partial thermopowers, the variation of the Fermi energy E F with temperature can be computed from the known effective masses and temperature dependence of N and P. Heremans et al. 88 introduced a pseudo-parabolic model for Bi which takes into account the non-parabolicity of the conduction band at the L-point as well as the temperature dependence of electron effective masses. In the model, the relation between E F and N is given by 16π 12 EF 32 f0 N = ( 2det m ) / γ ( E) de 3 / e. (2.47) 3h 0 E 94

116 Here, m e is the band-edge mass tensor of electrons, ( E) E( 1 EEG) γ = + where E G is the gap in the energy spectrum, and f 0 is the Fermi distribution function. While the model has been successfully applied to explain the behavior of Sn doped Bi samples, 89 the authors found that it does not provide an adequate temperature dependence of E F at high temperature where the materials become partially degenerate. The difficulty lies in the unknown temperature dependence of the heavy electron mass along the bisectrix direction. In the current work, we have realized that the pseudoparabolic model works even at high temperature when the temperature dependence of the electron effective mass is completely ignored while the non-parabolicity is kept. This statement may sound contradictory, since generally the non-parabolicity is expected to accompany the temperature dependence of the effective masses. We suspect that the bisectrix mass decreases with increasing temperature while the binary and trigonal masses increase, resulting in a seemingly null temperature dependence of the mass determinant. Although it is ordinary to consider that the thermal expansion leads to an increase in the energy gap and thus of the effective masses, the anisotropy in the spinorbit coupling in Bi 90 can cause decrease in one mass while the others increase with temperature. Indeed, it was shown 91 that in InSb, InAs and GaAs above 60K, the bandedge effective mass decreases as temperature increases. Therefore, in our calculation, the mass determinant in Eq. (2.8) was assumed to be temperature independent. 95

117 (a) 0 0 (b) S // (µv/k) S (µv/k) T (K) T (K) -80 Figure Temperature dependence of the Seebeck coefficient in (c) the trigonal axis direction ( S // ), and (d) the trigonal plane ( S ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The solid black line represents the computed curve for Bi 99.6 In 0.4 ; the dashed black line corresponds to the computed curve for pure Bi. The partial thermopower of electrons is given by the pseudo-parabolic model as S e ( 5 + ) F ( ) ( λ + + ) F52+ λ 2 2 ( ) ( 3 + ) 12 ( ) ( 5 + λ + + ) 32+ λ ( ) k B λ ηf λ ηf η G = η F (2.48) q λ F ηf λ F ηf η G 2 2 where λ is the scattering parameter, F is the Fermi integral: F r = r η dη (2.49) exp ( η ηf ) 96

118 and η F and η G denote EF / kt B and EG / kt B, respectively. As regards the T-point holes, the dispersion can be effectively described by the parabolic model for which equations are obtained by setting EG in Eq. (2.8) and (2.9): 16π 12 EF f 32 0 P = ( 2 m ) / / det h E de 3. (2.50) 3h 0 E S h ( 5 + λ) F λ ( ηf ) ( 3 + λ 2 ) F12+ λ ( ηf ) k B = η F. (2.51) q With the known temperature dependence of N and P, the temperature dependence of E F for electrons and holes can be found from Eq. (2.8) and (2.11). Those E F s are in turn substituted into Eq. (2.9) and (2.12) to yield the partial thermopowers for electrons and holes, respectively. For pure Bi, λ = -1/2 denoting the acoustical phonon scattering, while λ = 0 is added for the In doped sample to account for the effect from the neutral impurity scattering. The resulting total S calculated by Eq. (2.7) is plotted in Fig. 2.18(b) for both pure Bi and 0.4% In doped sample. The computed curves indicate that the neutral impurity scattering certainly enhances S. While the same trend is observed in the experimental data, the magnitude of enhancement is smaller than that in the calculation. This discrepancy was also observed in the compensated Bi. 84 Figure 2.19(a) and (b) show κ(t) of Bi, Bi In 0.09, and Bi 99.6 In 0.4 in both the trigonal axis direction and the trigonal plane, respectively. Interestingly, there is almost no difference in κ between pure Bi and the In doped samples over the measured 97

119 temperature range even though a certain degree of decrease was expected in the doped samples due to the impurity scattering. There could possibly be a noticeable difference around the dielectric maximum below 10K where the phonon mean free path is most sensitive to the impurity scattering. Unfortunately, the experimental data are incomplete at this temperature regime to draw any conclusions. zt(t) of the samples in both the trigonal axis direction and the trigonal plane is shown in Fig. 2.19(c) and (d), respectively. In both directions, the In doped samples show higher zt than pure Bi mainly due to the enhanced S. Even so, zt in the trigonal plane is still too low to be of interest for practical applications. In the trigonal axis direction, zt of the In doped samples reaches 0.36 at 280K which is a 20% enhancement compared to that of pure Bi. 98

120 (a) T (K) T (K) (b) κ // (W/m K) κ (W/m K) (c) zt // (d) zt T (K) T (K) 0 Figure Temperature dependence of the thermal conductivity in (a) the trigonal axis direction ( κ // ), and (b) the trigonal plane (κ ). Figure of merit in (c) the trigonal axis direction ( zt // ), and (d) the trigonal plane ( zt ). The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. To further investigate the p-type doping effect of In on the transport, S(T) measured in the 7T magnetic field is shown in Fig. 2.20(a) and (b). In the zero magnetic field, S(T) is always negative in both measurement direction (Fig. 2.18) as a result of higher mobility of electrons than that of holes. However, as mentioned earlier in the discussion of SdH, when a magnetic field is applied along the trigonal axis direction, 99

121 electrons rapidly become less mobile, thus increasing the chances of observing holes. Especially since the hole mobility is much higher in the trigonal plane than in the trigonal axis direction, 71 we may expect the hole transport will be more pronounced in the former. (a) S // (µv/k) (b) S (µv/k) T (K) T (K) Figure Temperature dependence of the Seebeck coefficient in 7T magnetic field in (a) the trigonal axis direction ( S // ), and (b) the trigonal plane ( S ). The direction of magnetic field is parallel to that of heat flux. The symbols are the experimental data: (red diamond) pure Bi, (green cross) Bi In 0.09, and (blue circle) Bi 99.6 In 0.4. The dashed lines indicating zero Seebeck coefficient are added to help readers recognize the sign change of the Seebeck coefficient. Figure 2.20(a) and (b) show that this qualitative prediction explains the experimental data reasonably well. In the trigonal axis direction (Fig. 2.20(a)), the electron mobility is still dominant, leading to the mostly negative S in all three samples except below 10K where S becomes positive. The 0.4% In doped sample maintains the 100

122 positive S up to 30K. Note that all samples exhibit the positive phonon drag thermopower peaks at the lowest temperatures, indicating the prevailing hole transport in this temperature regime. On the other hand, more striking p-type behaviors are observed in the trigonal plane (Fig. 2.20(b)). Due to the higher mobility of holes in this direction, S of both In doped samples remains positive over the whole measured temperature range. The 0.4% In doped sample has a maximum at 140K then its S decreases as temperature rises due to the thermal excitation of minority carriers. The same behavior is observed in the 0.09% In doped sample, except that this sample has an additional maximum at a lower temperature. It is plausible that in the lower doped sample, E F hits two different valence bands with different band masses due to the temperature dependence of the band structure. Referring to Fig. 2.12, those two bands are likely to be the T- and L- valence bands. In contrast, the single maximum in the higher doped sample suggests that E F is lying deep in both valence bands such that the effect of temperature is small. S of pure Bi remains negative at most temperatures, while the positive S at the low temperatures is more pronounced compared to that in the other direction; again this is due to the higher hole mobility in this direction. From these observations of S(T) in the 7T magnetic field, we can confirm that In enhances the p-type transport in Bi by introducing excess holes, consistent with the results of SdH and Hall measurements. Another advantage of the measurement in a strong magnetic field ( µ B 1) is that the relaxation time becomes energy independent, and thus S becomes independent of scattering. 66 This allows us to examine the effects of other band parameters independently of the scattering mechanisms. 101

123 Therefore, the effects of the neutral impurity scattering marked in the zero magnetic field disappear in Fig. 2.20(a) and (b), which simplifies the interpretation. Theory Thus far, we have observed two distinct behaviors of In in Bi: 1) as an acceptor dopant and 2) as a neutral impurity scattering center. While there seems to be no direct relations between 1) and 2), the unusually low doping efficiency of In may be related to the presence of neutral impurity scattering. Figure 2.21 shows how the electronic structure of Bi changes when a certain amount of In goes into interstitial sites rather than substituting Bi atoms. When all 0.5 at. % In goes into interstitial sites (middle panel in Fig. 2.21), the E F moves close to the conduction band, which indicates that the interstitial In behaves as an n-type dopant. Thus, it is possible that if some of the inserted In atoms take interstitial sites while the rest substitute Bi atoms, a compensation between n- and p- type doping occurs, leading to the low p-type doping efficiency of In. Whereas the neutral impurity scattering can stem from the compensation effect, there is a possibility that the segregated In matrices which are not dissolved in Bi can also behave as neutral impurity scattering centers. 102

124 Figure Effects of interstitial indium on electronic structure of bismuth. The figure has been provided by Dr. Bartlomiej Wiendlocha, and used with his permission. 103

125 2.3 Enhancement in the figure of merit of p-type BiSb alloys through multiple valence-band doping Introduction N-type Bi 100-x Sb x alloys are the materials with the highest thermoelectric figure of merit (zt 0.5 at 100K 63 ) below 200K. The performance of p-type material is not equivalent: the highest zt of p-type Bi 100-x Sb x alloys reported thus far is about 0.08 at 200K for the composition of Bi 88.5 Sb 7.5 Sn Figure 2.22 shows the best zt values for n- type and p-type Bi 100-x Sb x alloys hitherto. Most of investigations of p-type Bi 100-x Sb x alloys have been made for the semiconducting compositions which correspond to 7-20% Sb, 53,68,92,93 presumably because this range of compositions yields high zt values in n- type materials. Because the nature of the valence bands is quite different from that of the conduction band, here the study of the p-type material is extended to the semi-metallic Sb-rich region, where p-type material has not been studied previously, reaching x=37% Sn in single crystals and 50-90% in polycrystals. We report a 60% improvement of the maximum zt. 104

126 Figure Comparison of the best zts between n-type and p-type BiSb alloys. The n- type zt is plotted using the data in Ref. [63], and p-type zt from Ref. [68]. Contrasting with the experimental observations, Thonhauser et al. 94 predicted that zt=1.4 at room temperature can theoretically be reached if Bi is heavily doped p-type such that its Fermi level would fall about 0.25eV below the upper valence band (VB) edge (3.1 atomic % of Sn, if each Sn atom substituted for a Bi atom was to capture one electron). This shifts the Fermi level into the heavy VBs near the Η-points as well as the VBs at the T- and L-points inside the Brillouin zone so that the hole Fermi surface consists of the T- and L-hole pockets and saddle points near the heavy Η-hole bands. This approach was shown to be successful in PbTe 95 using the hole levels near its Σ-point. In elemental Bi, the approach runs into a difficulty: at temperatures above 220K, Sn loses its activity as an acceptor 89 due to a change in the relative band positions and Sn-impurity 105

127 level with temperature. 88 We address that difficulty here by observing that the Η-band moves up in energy with x in Bi 100-x Sb x alloys at T = 0K (see Fig. 2.23, compiled from 54,96,97 and 98 ; we could not compile a similar figure for higher temperatures because not all the parameters are known above cryogenic temperatures 88 ). Conceptually, the idea is as follows: the motion of the Bi bands with increasing T is mostly due to thermal expansion effects. Because the lattice constant of Bi 100-x Sb x alloys decreases with increasing x, the latter effect can be viewed to the first order as compensating the effect of T. This suggests that one may dope the Η-bands more effectively at high Sb concentration. Fig shows that at about 15-17% Sb, the Η-band crosses over the light VB at the L-point and keeps increasing in energy while the T-band decreases to be located at lower energy than that of the Η-band. Hence, for x>15-17% Sb, the VB nearest to the L-point conduction band becomes the Η-band, and we can expect to dope it by introducing an acceptor impurity. Also, an attempt is made to additionally dope the T- and L- bands, fulfilling the condition of Ref. [94]. The concept is reinforced by the observation 98 that Sn remains as an acceptor at 300K in pure Sb; therefore, we can expect that using high Sb% will stabilize the acceptor behavior of Sn even at relatively high temperatures. As a p-type dopant, Sn is chosen over Pb based on the work of Noguchi et al. 64 where they reported that Sn is a more effective acceptor than Pb in Bi 100-x Sb x alloys. We show experimentally that the concept works, although not well enough to reach zt values near the theoretical prediction. 106

128 E (mev) H 240 mev mev T L a L s Bi x, fraction of Sb Sb 180 mev Figure Dependence of the energies of the various band extrema on composition x in Bi 100-x Sb x alloys at T=0K (compiled from Refs. 54, 96-98). At the L-points of the Brillouin zone the symmetric and antisymmetric bands are inverted between elemental Bi and Sb, leading to a Dirac point near x=5 at % fraction. The T-point valence band is the upper valence band of the semimetal Bi, but in elemental Sb, the holes are inside the Brillouin zone in three distorted ellipsoidal pockets near the H-points. This band becomes the upper valence band for x>18. As in most solids, temperature can make band extrema shift by 100 mev between liquid nitrogen and room temperatures, an effect that is secondary in wide-gap semiconductors, but has a very important relative influence on the present diagram. Bi 100-x Sb x single crystals with 12 x 37 doped with 0.75 at % Sn (corresponding to roughly atoms of Sn per cm 3 ), and polycrystalline samples with x = 50, 70, 80, and 90 at % were prepared for thermoelectric property measurements. The work is organized as follows: first, we compare the transport properties between along the trigonal axis and in the trigonal plane and show that the best performance 107

129 occurs when the fluxes are in-plane. Then we report the complete set of the S, ρ, κ, and the corresponding zt of the single crystalline and polycrystalline samples. Finally, the relation between carrier concentration and thermopower, known as Pisarenko s relation, for all samples is constructed and interpreted based on the Sb concentration dependent band structure of Bi 100-x Sb x alloys. Experimental Seven Bi 100-x-0.75 Sb x Sn 0.75 single crystals were grown by a modified Bridgman technique. Elemental Bi (5N), Sb (6N), and Sn (5N) were loaded into the quartz ampoules following the stoichiometric ratios, and the ampoules were sealed at less than 10-6 torr. Then the samples were melted at 632 C in a tube furnace and slowly pulled out of the furnace at a rate of 0.13mm/hr. Because of the large segregation coefficient in the Bi-Sb phase diagram, it is known to be difficult to grow a homogeneous Bi 100-x Sb x single crystal, and therefore the samples were annealed for 6 months at 255 C. In addition, four polycrystalline Bi 100-x-0.75 Sb x Sn 0.75 samples with x=50, 70, 80, and 90 were prepared by ball-milling and cold pressing followed by 2 weeks sintering at 255 C. Powder XRD confirmed single crystal peaks and polycrystalline peaks as well as shift of the peaks as Sb concentration increases. The actual Sb concentration reported here for the samples was identified by x-ray diffraction using Vegard s law. Then the samples were cut into approximately 2.5 x 1.5 x 7mm 3 parallelepipeds for thermoelectric measurements. For the single crystal samples, the parallelepipeds were obtained in two different 108

130 crystallographic directions: the long axis parallel to the trigonal axis and to the trigonal plane, respectively. ρ(t), κ(t), and S(T) were measured simultaneously in a quasi-static heater-and-sink configuration while slowly sweeping the temperature from 400K to 2K using the Thermal Transport Option (TTO) in a Physical Properties Measurement System (PPMS) by Quantum Design. Hall coefficient of each sample was measured using AC Transport Option in the PPMS at several different temperatures with the magnetic field up to 7T and oriented along the trigonal axis direction of the sample. Because the crystal symmetry of Bi and Bi 100-x Sb x alloys is such 99 that ρ, κ and S are isotropic in the trigonal plane even in the presence of a magnetic field aligned along the trigonal axis, we did not identify the specific in-plane binary or bisectrix crystallographic direction when we made measurements in that plane, and no optimization of zt is possible by further orienting the current and heat flux in the plane. Results and discussion Figure 2.24 shows κ(t), S(T), ρ(t), and zt of the Bi Sb 18.2 Sn 0.75 sample in the trigonal plane and in the trigonal axis direction, respectively. The best zt in n-type Bi 100- xsb x alloys is obtained along the trigonal axis direction 63 due to much higher mobility of electrons than holes in this direction. We show in Fig. 2.24(d) that for p-type material, in contrast, zt is higher in the trigonal plane. In-plane ρ and S are better than their equivalents along the trigonal axis. In spite of the less favorable κ, the resulting zt at 150K is about five times larger in the trigonal plane than perpendicularly to it. This 109

131 effect is also due to the difference in mobility between electrons and holes, which here contributes to zt by two mechanisms. At 4K, the hole mobility in the trigonal plane is ~6 times higher than that along trigonal axis in single crystal bismuth. 71 This is first reflected in the fact that the electrical conductivity (σ) of p-type material in the trigonal plane will show ~6 times larger than along the trigonal axis, as observed experimentally in ρ in Fig. 2.24(c) at the lowest temperatures. But the anisotropy of the hole mobility also affects the thermopower, which is less compensated by the presence of minority electrons in the plane than along the trigonal axis. The total σ e i (i = 1 or 3 for the trigonal plane or the trigonal axis direction, respectively) is expressed as: S e e h h S σ i + S σi i = e h σ i +σi (2.52) where σ e i and σ e i are the partial contribution of electrons and holes to the total electrical conductivity in the corresponding direction, and σ e e i and σ i are the partial thermopower of electrons and holes, respectively. Because the mobility ratio of holes to electrons is the largest in the trigonal plane, the contribution of the σ e i term, which is negative, is minimized, and we consequently can obtain the largest p-type thermopower in the plane (Fig. 2.24(b)). Therefore, due to the superior thermoelectric properties in the trigonal plane, we present the transport results of the samples in this direction only. 110

132 (a) κ (W/mK) (c) (b) S (µv/k) 0.03 (d) ρ (µω m) T (K) T (K) zt Figure Comparison of thermoelectric properties of single crystal Bi Sb 18.2 Sn 0.75 sample between in the trigonal plane and along the trigonal axis directions. (a) thermal conductivity, (b) Thermopower S, (c) electrical resistivity ρ, (d) Figure of merit zt as functions of temperature. Points are experimental data, lines are added to guide the eye. Thermoelectric properties in the trigonal plane of single crystal Bi 100-x Sb x Sn 0.75 samples with 11 x 37 are presented in Fig and κ(t) (Fig. 2.25(a)) of all Sn-doped samples follow the classical behavior of undoped Bi 100-x Sb x alloys, but they have lower values especially at low temperatures. Below 20K, heat transport in Bi 100- xsb x alloys is dominated only by phonons and has a maximum near 10K. Fig. 2.25(a) shows that the amplitude of the peak in κ(t) decreases with increasing Sb concentration, consistent with the effects of alloy scattering on the lattice thermal conductivity (κ L ). 111

133 All samples exhibit similar temperature dependence in ρ(t) (Fig. 2.25(b)), which is typical in heavily doped small band-gap semiconductors. Increasing Sb concentration generally decreases ρ, which is consistent with the increasing band overlap with increasing x (Fig. 2.23). 44 One exception is the sample with 22.9% Sb which has the lowest ρ among the samples despite its intermediate Sb concentration, a fact that is noted without explanation. ρ (µω m) κ (W/mK) (a) (b) 11.6% Sb 18.2% Sb 19.5% Sb 22.9% Sb 26.5% Sb 30% Sb 37% Sb Figure (a) Thermal conductivity κ and (b) electrical resistivity ρ of single crystal Bi 100-x Sb x Sn 0.75 samples for x = 11.6, 18.2, 19.5, 22.9, 26.5, 30, and 37 from 2K to 400K. All properties were measured with the heat flux and current in the trigonal plane. Points are experimental data, lines are added to guide the eye. 112

134 For the samples with relatively low Sb concentration 19.5), (x there is a point where S changes its sign from p-type to n-type below 400K (Fig. 2.26). This turnover point tends to move to a higher temperature as the Sb concentration increases, as expected from the arguments about the Sb compensating for the temperature-dependence of the band structure made in the introduction. Samples with higher Sb concentration (x>19.5) show positive thermopower over the measured temperatures. However, the magnitude of S decreases rapidly as Sb concentration exceeds 30%, indicating a steady increase in carrier concentration with increasing Sb content, which is consistent with the increase in overlap between Η and L-point bands shown in Fig for x>30 and the concomitant increase in hole concentration. The largest p-type thermopower is about +60 µv/k obtained for 11.6% Sb sample at 150K. Last, the inset in Fig shows that the best zt is obtained for the sample with 22.9% Sb at 240K reaching zt=0.13. While this value is admittedly low, it is the best p- type zt reported in this system. According to Fig. 2.23, the Η band crosses over the conduction band at the L-point at about 22% Sb, therefore the alloy turns into semimetal again so that this concentration could be one of the most favorable points for doping the Η-bands as well as the additional T- or L-bands. The transport results for Bi Sb 22.9 Sn 0.75 sample seem to confirm the singularity of this composition. 113

135 11.6% Sb S (µv/k) % Sb 19.5% Sb 22.9% Sb 26.5% Sb 30% Sb 37% Sb zt T (K) T (K) Figure Thermopower S of single crystal Bi 100-x Sb x Sn 0.75 samples for x = 11.6, 18.2, 19.5, 22.9, 26.5, 30, and 37 from 2K to 400K. S was measured with the heat flux and current in the trigonal plane. The insert shows the best figures of merit zts in this study as a function of temperature. Points are experimental data, lines are added to guide the eye. Further data on higher Sb concentration samples (x=70, 80, 90) is shown in Fig Since these samples are synthesized by mechanical alloying, the grain size of the samples is small enough that the phonon mean-free-path is limited by the grain boundaries. For this reason, the typical κ(t) peaks at 2-20K are not observed in these samples (Fig. 2.27(a)). In Fig. 2.27(b), S at 300K are less than half that of pure Sb (about +42 µv/k at 300K 100 ) for all three samples. This is mainly because the samples have more carriers than pure Sb due to Sn doping. Also, ρ(t) (Fig. 2.27(c)) suggests metallic 114

136 behavior of the samples over the measured temperature range. As a result, zt of the high Sb concentration samples remains very low. (a) T (K) T (K) (b) 16 κ (W/mK) (c) S (µv/k) ρ (µω m) % Sb 80% Sb 90% Sb T (K) Figure Thermoelectric properties of polycrystalline Bi 100-x Sb x Sn 0.75 samples for x = 70, 80, and 90 from 2K to 400K. (a) Thermal conductivity, (b) Seebeck coefficient, and (c) resistivity as functions of temperature. Points are experimental data, lines are added to guide the eye. Figure 2.28(a) shows a plot of thermopower as a function of hole concentration (Pisarenko s plot) obtained from the Hall measurements at T=80K for all of the samples 115

137 in this study. The solid line and the dashed line are calculated 89 for the VBs centered at the T-point and at the Η-points of the Brillouin zone, respectively, assuming that the density of states in those bands is not a function of x, and also assuming acoustic phonon scattering. There are three distinctive sample groups in Fig. 2.28(a) classified by their relative position vis-à-vis the solid line calculated for the H-bands. Figures 2.28(b), (c) and (d) show schematic band diagrams for those sample groups. Readers are also advised to refer to Fig in order to follow how the band structure evolves as Sb concentration increases. In the first group (Fig. 2.28(b)), 3 sets of pockets (at L, T and Η- points) of the VB have their extrema located within 25 mev of each-other. Considering that 0.75% Sn corresponds to heavy doping in Bi, 70 there is a possibility that 2 or 3 VBs at different symmetry points in Brillouin zone are doped at the same time even though their extrema are located at slightly different energies: all their extrema are moved above E F. The thermopower S is then given by Eq. (2.13) and is increased over the partial S of the Η-bands since all doped bands have partial thermopowers of the same polarity. With further increase in Sb concentration, the T-valence band edge goes deeper in energy and moves vis-à-vis E F so that only the Η-bands are doped (Fig. 2.28(c)). This picture corresponds to the second group of 30%, 37%, and 50% Sb samples, whose thermopower falls on the solid line in Fig. 2.28(a). Last, in the three samples with the highest Sb concentrations (70%, 80%, and 90%), the energy overlap between the Η-bands and the conduction bands at the L-points is large enough for E F to cross the bottom of the L- conduction bands (Fig. 2.28(d)) and the samples become more compensated. The presence of minority electrons at 80K, in spite of heavy p-type doping with 0.75% Sn, in 116

138 turn decreases S below that of the Η-bands. We conclude that p-type doping is most effective when it involves more than one type of VB (the case of Fig. 2.28(b)), and it is in this category that the best zt s in this study are found (inset in Fig. 2.26). 117

139 (b) L a (c) L a (d) L a E F L s H T 11.6<Sb <26.5 L s H 30<Sb<50 H 70<Sb<90 Figure (a) Pisarenko s plot of thermopower versus hole concentration at T=80K for all measured samples in this study. Solid line and dashed line are calculated for the valence bands at the Η-points and the valence band at the T-point, respectively. Points are experimental data. Carrier density of each sample was obtained from Hall measurement. (b), (c), (d) Schematic band diagrams for the three different Sb concentration regimes shown. Relative position of each band and the position of Fermi energies (E F ) are not to scale. 118

140 Chapter 3: Study on Magnon Thermal Conduction and Spin- Seebeck Effect in Ferromagnetic Bulk Metallic Glass Excited spin waves (magnons) can carry heat in solids, just as electrons and phonons do. Since magnons can exist only in magnetic solids such as ferromagnets and ferrimagnets, heat conduction in those materials is mediated by electrons, magnons, and phonons (if they are conductors), or by magnons and phonons alone (if they are insulators). As discussed in Ch. 1.2, the magnon thermal conductivity can be one mechanism for inducing a spin current, and thus generating the spin Seeebeck effect. In ferromagnetic metallic glasses (Metglas), heat conduction by phonons is limited to that by localized phonon modes due to the amorphous crystal structures. As a result, it is expected that the relative contribution of magnons to the thermal conductivity would be large in a Metglas compared to crystalline solids where most of heat is carried by phonons and electrons. In addition, this system may be considered a good platform to test the dependence of spin Seebeck effect on magnon thermal conductivity since the phonon-drag effect is likely to be absent due to the absence of long-wavelength phonons. 119

141 3.1 Measurement of magnon thermal conductivity in ferromagnetic bulk metallic glass Introduction Similar to phonons, magnons can be considered as propagating waves in a crystal. While one-dimensional phonons can be pictured as a ball-spring system as shown in Ch. 1.3, one-dimensional magnons can be simulated as a propagating wave with a precessing spin with respect to the axis of magnetization. The restoring force for magnons is provided by the magnetic torque. One important difference between magnons and phonons lies in their dispersion relations. For one-dimensional magnons in a ferromagnet, the dispersion relation is given as ω = 4 JS(1 cos ka) (3.53) where ω is the angular frequency, J the exchange energy, S the magnitude of a spin vector, and a the lattice constant. Figure 3.1 illustrates how the dispersion of magnons looks like in one-dimensional systems. At low k-values near the zone center, Eq. (3.1) can be approximated by ( 2JSa ) ω 2 k 2. (3.54) Therefore, magnons have a parabolic dispersion at low k-values like electrons while phonons show a linear dispersion. 120

142 ω 4JS π ka Figure 3.1. Dispersion relation of magnons for a one-dimensional ferromagnetic chain constructed using Eq. (3.1). If we now take the quantum mechanical consideration into account, 101 Eq. (3.2) is modified to ( 2 ) 2 2 ω µ B z (3.55) k = gh + JSa k where μ B is the Bohr magneton, g the Landé g-factor, and H z is the magnetic field along the z-direction perpendicular to the precession plane. It is important to note that the first term in Eq. (3.3) gives rise to an energy gap in the magnon dispersion relation. By adjusting the intensity of applied magnetic field, it is possible to change the magnitude of the energy gap, i.e. by increasing the field intensity we can freeze out magnons. This 121

143 mechanism is the key to measuring the magnon contribution to the thermal conductivity in ferromagnetic solids as we will see below. The thermal conductivity of a Metglas can be written as κ = κe + κl + κm (3.56) where κ e, κ L, and κ M are the electron, phonon, and magnon thermal conductivities, respectively. In metals, κ e generally constitutes a large portion of κ due to the large electrical conductivity. As a first step to evaluate κ M, κ e is calculated by the Wiedemann- Franz law (Eq. (1.32)) and subtracted from κ. When an external magnetic field is applied, the energy gap in the magnon dispersion opens, which results in a decrease of the magnon population, and thus κ M is reduced. If the magnetic field intensity keeps increasing, there will be a point where all magnons are frozen, and we can assume κ M = 0 from that point. κ e is usually not sensitive to the magnetic field in metals, and this can be experimentally verified by measuring the magnetic field dependence of the electrical resistivity. This method has been used to detect κ M in Fe-, Co-, and Ni-based Metglases by several researchers. 102,103 Among those measured, the Fe-based Metglas with composition of Fe 77 Si 10 B 13 showed the largest κ M. 102 Interestingly, Muller and Pompe 104 observed the opposite effect of magnetic field in Fe 80 B 20 whose composition is not much different from that used in Ref. [102]. In their work, it was shown that applying 6T magnetic field increases κ -κ e below 20K, rather than decreasing it. They attributed this effect to the suppression of phonon scattering on magnons. Low energy magnons are less 122

144 likely to contribute to the thermal conduction compared to higher energy magnons. Rather, they can act as scattering centers for more energetic phonons and electrons. Applying a magnetic field tends to freeze out those low energy magnons first, and therefore gives rise to increase of κ e or κ L. Even in this case, it is expected that κ M will be eventually suppressed when a sufficiently high magnetic field is applied, which can now stiffen magnons with higher energies. Those previous studies suggest a complexity of Metglas systems in which various interactions are possible between electrons, phonons, and magnons. In this study, we attempt to measure κ M in two different Metglases: Cobased and Fe-based. Experimental The Co- and Fe-Metglas samples are provided by Metglas, Inc. for research purposes. Table 3.1 summarizes important properties of both Metglases. Sample Composition Relative Permeability Curie temperature (K) Thickness (μm) Co-Metglas Co 85 Si 5 Fe 3 B 3 Mo 2 Ni Fe-Metglas Fe 85 Si 10 B Table 3.1. Properties of Co- and Fe-Metglas. The data are provided by www. metglas.com

145 Figure 3.2 shows the experimental setup used in this study. The samples were cut into 16 x 5mm 2 pieces, and gold-plated copper pads were bonded to four different positions from top to bottom of the samples using silver epoxy. The top and bottom copper pads were thermally and electrically connected to a block with a 2kΩ resistive heater and a heat sink, respectively. The two middle copper pads were connected to Cernox thermometers to measure temperature differences between the pads. Those two copper pads also served as electrodes to detect voltage differences between them in electrical resistivity measurements. All thermal conductivity as well as electrical resistivity measurements were performed using the Thermal Transport Option (TTO) in a Physical Properties Measurement System (PPMS) by Quantum Design. The heat flux and magnetic field H are applied along the vertical direction in the figure as shown in Fig 3.2. heater sample H // ΔT thermometer Figure 3.2. Experimental setup for thermal conductivity and electrical resistivity measurements. The sample is mounted on the Thermal Transport Option in the PPMS. 124

146 Results and discussion The electrical resistivity ρ of Co- and Fe-Metlases is shown in Fig As is expected for metallic systems, ρ shows very weak dependences on temperature and magnetic field. In Co-Metglas, an increase of ρ is observed below 5K, which suggests that electrons are scattered by localized magnetic impurities, known as Kondo effect. 106 This effect disappears when the 7T magnetic field is applied because the magnetic field forces spins of the magnetic impurities to align in the same direction, reducing the number of excited spin states in the impurities. In contrast, the same effect is not observed in Fe-Metglas, possibly due to lack of localized magnetic impurities. The weak magnetic field dependence of ρ allows us to use the Wiedenmann-Franz law to calculate κ e in the 7T magnetic field. It is possible that the weak temperature and magnetic field dependences would lead to deviations in κ e that are well below resolution of the measurement. 125

147 T (K) x10-7 Co-Metglas H = 0T H = 7T ρ (Ω m) 9.10x x x10-6 Fe-Metglas ρ (Ω m) 1.13x10-6 H = 0T H = 7T 1.12x T (K) Figure 3.3. Electrical resistivity ρ of Co- and Fe-Metglases as a function of temperature measured in 0 and 7T magnetic fields. Figure 3.4 shows the thermal conductivity κ of Metglas samples as a function of temperature. The κ e calculated by the Weidemann-Franz law and κ - κ e are also shown. The κ e exhibits an almost linear temperature dependence in both samples because it is dominated by the T 1 term in Eq. (1.32) due to the very week temperature dependence of ρ. Although κ e contributes a considerable amount to κ, it is observed that the contribution 126

148 from κ-κ e is much larger in this system. The normal dielectric maximum in κ is absent in both samples, implying that the long range phonons play a negligible role in heat conduction for these materials. Co-Metglas Fe-Metglas κ 6 κ κ (W / mk) 4 κ - κ e κ (W / mk) 4 κ - κ e 2 κ e 2 κ e T (K) T (K) Figure 3.4. Thermal conductivity κ of Co- and Fe-Metglas samples as a function of temperature. κ e denotes the electronic thermal conductivity calculated using the Wiedemann-Franz law at 0T. Now let s define the sum of lattice and magnon thermal conductivities κ LM = κ - κ e in which we can try to separate κ M from κ L by applying a 7T magnetic field. In Co- Metglas, the magnetic field enhances κ LM above 40K, which is similar with what Muller and Pompe 104 observed (upper plots in Fig. 3.5). Thus, in this temperature regime, κ LM is originally limited by scattering of phonons by magnons, and applying the magnetic field suppresses this scattering by freezing out magnons. At lower temperatures, almost no 127

149 difference is observed, but as we will see below, there are very small reductions of κ LM in the 7T magnetic field. On the other hand, relatively large reductions of κ LM occur in Fe- Megtlas when the magnetic field is applied. Co-Metglas Fe-Metglas 6 6 κ LM = κ κ e (W/mK) 4 2 H = 0T H = 7T κ LM = κ κ e (W/mK) 4 2 H = 0T H = 7T T (K) T (K) 1 2 (%) 0 (%) 0 κ LM (H) κ LM (0) -1-2 T = 10K T = 20K T = 40K κ LM (H) κ LM (0) -2-4 T = 7K T = 20K T = 50K H (Oe) H (Oe) Figure 3.5. (Upper plots) Thermal conductivity of Co- and Fe-Metglas samples after subtracting electronic contribution (κ LM = κ - κ e ) as a function of temperature in 0 and 7T magnetic fields. (Lower plots) Relative change (%) of κ LM in magnetic fields as a function of magnetic field: Δκ LM (H) = κ LM (H) - κ LM (0T). 128

150 At a constant temperature, κ LM generally decreases as the magnetic field intensity is increased in both samples (lower plots in Fig. 3.5). Depending on temperature, κ LM also shows complicated behaviors. For instance, at 40K in Co-Metglas, κ LM tends to increase at low fields, and gradually decreases as the field increases, being smaller than the value at 0T when H > 4T. A similar trend is found in Fe-Metglas at 20K. Those behaviors indicate a competition between phonon scattering on magnons and magnon thermal conductivity κ M. Low energy magnons, which behave as scattering centers for phonons and carry relatively a small amount of heat, are frozen out first with comparatively small magnetic fields. Thus, κ LM tends to increase as κ L becomes larger. At higher fields, high energy magnons start being frozen as the energy gap in the magnon dispersion becomes comparable to their energies. Since these magnons carry much more heat than low energy magnons, a decrease of their population leads to suppression of κ M. When all heat carrying magnons are frozen, it is expected that κ LM will saturate. While the saturation happens at 40K in Co-Metglas and 7K in Fe-Metglas, it is not seen at the other measured temperatures. Therefore, the 7T magnetic field seems not enough to fully suppress κ M at certain temperatures. Figure 3.6 shows the amount of change in κ LM after the 7T magnetic field is applied. For Co-Metglas sample, very small reductions of κ LM are observed below 40K which correspond to about 2% changes. This suggests that κ M contributes about the order of 2% to κ LM. Considering that κ M is not fully saturated at some temperatures with 7T, this number would increase if stronger magnetic fields are used. The sharp increase of 129

151 κ LM (7T) - κ LM (0T) above 40K indicates the suppression of phonon scattering on magnons with the 7T magnetic field. Based on the discussions so far, it is possible that κ LM (7T) - κ LM (0T) starts to decrease at a higher field intensity, i.e. a sufficient field intensity has not been reached to initiate freeze-out of high energy magnons at those higher temperatures. On the other hand, much larger contribution of κ M is observed in Fe-Metglas, reaching about 7% of κ LM at its maximum at 10K. In the measured temperature range, the 7T magnetic field only gives rise to reductions of κ LM, possibly indicating that the field intensity is enough to freeze-out high energy magnons, unlike in Co-Metglas. This observation suggests that the energy gap in the magnon dispersion is likely to be different for Co- and Fe-Metglas when a magnetic field with the same intensity is applied; the data shown in Fig. 3.6 imply that the latter has a larger gap. Co-Metglas Fe-Metglas 0 κ LM (7T) κ LM (0T) (W/mK) κ LM (7T) κ LM (0T) (W/mK) T (K) T (K) Figure 3.6. Amount of change in κ LM when the 7T magnetic field is applied. The inset shows a magnification for the lower temperature which exhibits the magnitude of change more clearly. 130

152 3.2 Spin Seebeck like signal in ferromagnetic bulk metallic glass without platinum contacts Introduction As introduced in Ch. 1.2, the spin-seebeck effect (SSE) is the first of a new class of spin caloritronic effects. It was first discovered in in permalloy, and it represents a new field of studies on the interaction between spin and heat. It has been measured using either a longitudinal (LSSE) 37,107 or a transverse configuration (TSSE) in ferromagnetic metals, 18 semiconductors 24 or insulators, 25 and most recently in systems in which the spin polarization is not due to exchange coupling between local moments, but to magnetic field induced spin polarization of conduction band electrons, i.e. spin-orbit enhanced Zeeman splitting. 26 Possible contaminations of the SSE signals by classical thermomagnetic effects have been reported. Specifically, the geometry of LSSE measurements is also that of the anomalous Nernst effect (ANE). In highly electrically conducting materials, it is not possible to separate ANE from LSSE, yet it was originally thought that pure LSSE were possible on ferromagnetic insulators such as on the Pt/YIG structure. While Huang et al. 108 suggested that interface effects can add parasitic signals to the LSSE, demonstration of the LSSE free from the parasitic interface effects was reported. 109,110 When the applied temperature gradient is not perfectly controlled in a TSSE geometry, ANE can 131

153 contaminate TSSE signals. 36,111 This problem can be avoided by using bulk samples or free-standing thin films. Even with the thermal gradient perfectly controlled and in the absence of ANE, another parasitic signal can contaminate TSSE signals: the anisotropic nature of the magnetothermopower in some ferromagnets can lead to transverse voltages even with the temperature gradient parallel to the magnetic field, which is called the planar Nernst effect (PNE). 112,113 Parasitic ANE and PNE voltages arise in electrically conducting ferromagnets; they can then be transferred to the Pt electrodes used as a spinsensitive detector in SSE measurements, thereby contaminating the pure spin-induced voltages. The amount of contamination is proportional to the amplitude of the parasitic voltage in the conducting ferromagnet, which roughly scales with conductivity. The ANE and PNE signals in GaMnAs were shown 24 to be completely screened out by Pt, but one can expect the parasitic signals to be more important with more metallic ferromagnets. It is necessary to check the magnitude of the PNE and ANE signals when executing SSE measurements, by measuring them independently on the same samples using point contacts. Here we show that reverse contamination can also happen during such measurements: the SSE voltage can contaminate measurements of the PNE, ANE, and ordinary (transverse) Nernst effects (ONE) even in the absence of Pt. This suggests that inverse spin-hall effect (ISHE) can occur in a variety of materials, an observation supported by the previous work on GaMnAs 24 as well as by Saitoh. 114 We report thermomagnetic measurements on non-crystalline Co-based ferromagnetic bulk metallic glasses (Co-Metglas), in geometries used for TSSE measurements, except that we use 132

154 point contacts instead of spin-sensitive metals (Pt). In addition, TSSE measurements are performed on Pt strips deposited right next to point contacts, and discussions are given to the obtained signals in comparison to those obtained on point contacts. The effect of magnon thermal conductivity on TSSE signals is briefly discussed. The Co-Metglas was initially chosen over crystalline metallic ferromagnets because its amorphous nature was expected to minimize the effects of phonon-drag on the SSE signals, and because the material can be measured as free-standing substrate-less foils, a geometry that minimizes the parasitic ANE. 36 Experimental Samples were cut from the same Co-Metglas batch which was used for thermal conductivity measurements in the previous section (refer to Table 3.1). Figure 3.7(a) shows the experimental setup used in this study. The sample was cut into 12 x 4mm 2 size; sapphire heat-sink blocks were attached on the top and bottom to electrically insulate the sample. Gold-plated copper pads were bonded to the top of the sapphire blocks on both ends of the sample in a thermally symmetrical geometry, and connected to a 2kΩ resistive heater and a heat sink, respectively. Point contacts were made to the sample with Ag-epoxy and 25µm diameter copper voltage leads. For TSSE measurements, a 10nm thick Cu layer was deposited on the sample right next to the point contacts and a 15nm thick Pt layer was deposited on the Cu layer. The insertion of the Cu layer is to prevent the proximity effect that may be induced by the ferromagnetic Metglas. 108,

155 By pumping down to a high vacuum, and by the use of gold-plated radiation shields and fine wires contacts, we minimized heat losses and ensured that the temperature gradient is properly aligned. Thermomagnetic and TSSE measurements were performed in the customized TTO option of a Quantum Design PPMS system used for previous TSSE measurements. 24 Cernox thermometers determined the longitudinal temperature difference T x and the average sample temperature. We stepped the heater power and the temperature, while monitoring voltages with a K2182 nanovoltmeter and the various thermometers; values were recorded after ~1hr stabilization time. The static errors (about 5%) in the setup are dominated by uncertainty in sample geometry, the field and temperature-dependent errors by the signal to noise ratio in voltage and the noise and calibration errors in the thermometry. 134

156 (a) (b) point contacts heater Metglas V V H x thermometers y z V xyx, α xyx, V xxx, α xxx (c) ` V ` H Metglas x y z V xyz =V ANE + V ONE, α xyz =α ANE + α ONE Figure 3.7. Experimental setup and measurement conventions. (a) Experimental setup. Magnetic field is aligned along the vertical direction. (b) Schematic illustration of the transverse and longitudinal magnetothermopower geometries: T and the external magnetic field, H, lie in the plane (x-direction) of the sample. (c) Nernst geometry: H is applied along the perpendicular (z-) direction. Figures 3.7(b) and (c) show the measurement configurations. L is the length between the thermometers and longitudinal point contacts along the x-axis, and w is the width between the transverse point contacts along y or width of the Pt strip. We define the measured voltage V xyz using three indices, the first (x) indicating the direction of the heat flux, the second (y) of the measured electric field, and the third (z) of the external magnetic field. Figure 3.7(b) shows the configuration with H applied along the x- 135

157 direction. The in-plane longitudinal voltage is V xxx, yielding the longitudinal magnetothermopower α xxx V / T. The in-plane transverse voltage, V xyx, yields the xxx x transverse magnetothermopower α E / T = L V / 2w T also known as the xyx y x xyx x PNE. Similarly, we define the spin Seebeck coefficient S xy as S E / T = L V / 2w T. After V xyx is measured on the hot side close to the xy y x xyx x heater, we flip the sample upside down to measure V xyx on the cold side by geometrical symmetry. For position dependence measurements of V xyx, another sample is prepared with point contacts and a Cu+Pt layer close to the middle of the sample. Figure 3.7(c) shows the configuration with H applied along the z-direction perpendicularly to the sample plane, used to measure the Nernst effect. The transverse voltage, V xyz, yields the Nernst coefficient xyz ( 1/ xt ) ( dey / dhz ) ( L / w) ( 1/ Tx ) ( dvxyz / dhz ) α =, which can arise either from the ANE or the ONE: α xyz =α ANE + α ONE. Results and discussion Figure 3.8(a) shows V xyz as a function of H measured on both hot and cold sides with an applied ΔT x =19.8K using the setup shown in Fig. 3.7(c). The inset shows the hysteresis; the magnetization saturation field is 11000Oe due to the geometrical demagnetization factor. The two distinct slopes of the ANE and ONE are observed in each curve. 136

158 For Oe < H < 11000Oe (shaded region in Fig. 3.8(a)), V ANE is induced according to the expression VANE = α ANEmˆ T ( mˆ is the unit vector of magnetization). This suggests that V ANE can be generated in two different cases in which T and H were originally intended to align in the same xy-plane as in the TSSE measurements. The first case is when a small T is formed accidentally along the z- direction due to either thermal mismatch between the sample and the substrate, 36,111 or heat drain through the contacts and the voltage leads. However, we will show that this possibility is excluded here by the use of bulk Metglas samples that have no substrate, and in which the heat drain issue is minimized by the use of thin wires, high vacuum, and a gold plated radiation shield. The second case is when there is a z-component of H, as in Fig. 3.7(c) and Fig. 3.8(a). Thus, it should be noted that a slight misalignment of H in the z-direction can introduce an ANE contribution to the transverse voltage measured in the TSSE setup. Further details about the effect of the misalignment will be discussed later. Above the saturation field, the slope of the V xyz curves decreases abruptly. Because the ANE contribution to the signal becomes constant as the magnetization saturates, the small slope can be attributed entirely to a classical ONE, V ONE. Therefore, when an accidental misalignment introduces a z-component of H, it is possible to separate V ANE and V ONE by referring to the saturation field. Figure 3.8(a) also shows that V xyz on the hot side is larger than that on the cold side, but that this difference is mainly due to difference in V ANE, while V ONE is almost same for both sides. We will return to this observation later. 137

159 V xyz (µv) (a) Hot side Cold side H (Oe) H (Oe) α ANE (µv T -1 K -1 ) (b) Hot T 3/2 T 1 Cold 20 0 α ONE (nv T -1 K -1 ) T (K) Figure 3.8. Measurements in perpendicular magnetic field. (a) H dependence of the transverse voltage, V xyz, measured between the point contacts in the configuration shown in Fig. 3.7(c). After measuring V xyz close to the hot end, the same sample is flipped upside down to measure V xyz close to the cold end. The shaded area corresponds to a hysteresis loop induced mostly by the ANE. The inset shows magnification at low H in which the hysteretic behavior is observed. Data are taken at sample temperature T avg = 64K with temperature difference ΔT x = 19.8K. (b) T dependence of anomalous Nernst coefficient, α ANE. The ordinary Nernst coefficient α ONE taken from the hot end is shown below (right ordinate). The orange line for α ONE is a guide to the eye. The black dashed lines are T 3/2 fits for α ANE for 20K< T <80K, and the black solid lines are T 1 fits for T < 20K. Figure 3.8(b) shows the T-dependence of α ANE and α ONE. The α ONE shows a strong dependence on T, and, as expected, its magnitude is much smaller than α ANE. What is 138

160 unexpected is that α ANE on the hot side is larger than that on the cold side over all measurement temperatures, as also seen in Fig. 3.8(a), while the α ONE is not positiondependent. The α ANE on both hot and cold sides decreases as T is lowered, following a T 3/2 law down to about 20K, and a T 1 law below. Because the saturation magnetization is not very temperature dependent here (T/T C < 0.1), the T 3/2 law mimics the temperaturedependence of the concentration of spin waves (the Bloch law 101 ), in a ferromagnetic crystal (albeit that this study concerns an amorphous metal). Because the Nernst coefficient is particularly sensitive to scattering, we suggest that α ANE (20K < T << T C ) is indicative of magnetic scattering of the conduction electrons. The T 1 behavior below 20K is also observed by Miyasato et al. 115 Figure 3.9(a) shows the H dependence of the in-plane transverse voltage, V xyx (the setup in Fig. 3.7(b)). V xyx switches sign with the magnetization in the hysteresis loops. The magnitude of the switch, voltages. We verified that Vxyx, is defined as the difference between the saturation Vxyx varies linearly with T x on the hot-side contact (Fig. 3.10), so that a PNE coefficient α xyx Ey xt can be defined. The sign of Vxyx is the same for the hot and cold sides while the magnitude is significantly larger on the hot side. The T dependence of the PNE α xyx is plotted in Fig. 3.9(b) for four positions along the x- direction of the sample. The magnitude of α xyx decreases as we move from the hot side to the cold side, as observed in the ANE (Fig. 3.8). The temperature dependence of α xyx shown in Fig. 3.9(b) is not sufficiently strong to explain this spatial dependence by the change in local average temperature along the sample. 139

161 V xyx (nv) V xxx (µv) H (Oe) (a) (c) T avg = 24K T x = 10.5K T avg = 49K T x = 17.3K α xxx (µv K -1 ) 0-1 Hot side Cold side T (K) V xyx V xxy (µv) α xyx (nv K -1 ) T (K) (b) (d) T avg = 56.4K T x = 18.5K Hot side Cold side H (Oe) H (Oe) Figure 3.9. Measurements in parallel magnetic field. (a) Hysteresis loops on the in-plane transverse voltage, V xyx, as a function of H for the hot and cold ends with an applied ΔT x = 10.5K at T avg = 24K. (b) T-dependence of the transverse magnetothermopower (planar Nernst coefficient), α xyx, on different positions of the sample. Orange squares are data taken on the hot side close to the middle of the sample, while sky blue triangles are on the cold side, same distance away from the middle as orange squares. Lines are drawn to guide the eyes. (c) In-plane longitudinal voltage, V xxx, as a function of H with an applied ΔT x = 17.3K at T avg = 49K. Inset: T dependence of the longitudinal magnetothermopower, α xxx. Points are experimental data and line is drawn to guide the eyes. (d) In-plane longitudinal voltage, V xxy, as a function of H (aligned along the y-axis) with an applied ΔT x = 18.5K at T avg = 56.4K. Does this observation imply that the hysteretic behavior in Fig. 3.9(a) is a signature of the ANE in our in-plane measurement? We show V xxx as a function of H in Fig. 3.9(c): unlike V xyx, V xxx shows only symmetrical and weak dependence on H. The thermopower α xxx (inset in Fig. 3.9(c)) has the linear T-dependence of a metal, and is 140

162 three orders of magnitude larger than α xyx. It is unlikely that the asymmetric behavior of V xyx is originated from V xxx, although it may still contribute a small error to V xyx through a possible misalignment between the point contacts V xyx (nv) -200 Hot side -300 T avg 64 ± 0.3 K T x (K) Figure Transverse voltage ΔV xyx as a function of temperature difference ΔT x. Red diamonds are data points, and the straight line is a linear fit through the data points. The point at (0,0) is artificially added from an expected extrapolation. Another possible contamination of V xyx and α xyx could arise from the presence of ΔT z which will result in the ANE with H aligned in the xy-plane. If this is the case, the asymmetric signals caused by the ANE should be measurable in the configuration shown in the inset in Fig. 3.9(d). The V xxy plotted in Fig. 3.9(d) is measured between point contacts separated from each other by about 8mm, which covers approximately 67% of the total length of the sample (12mm). The point contacts used for V xyx were located 141

163 about 2.5mm far from one end of the sample, which is within the coverage of the V xxy measurement. Therefore, if asymmetric signals appear in V xxy, it is possible that they may have contributed to V xyx more or less. The measured V xxy is shown in Fig. 3.9(d) to be a symmetric function of H, thus proving our earlier claim that there is no ΔT z in our sample. To further elucidate the relation between V xyx and the ANE, the sample is tilted ±10 degree off the xy-plane, thereby introducing the H z component as shown in Fig. 3.11(a) and allowing detection of the ANE contribution caused by the H z component to V xyx. While Vxyx remains unchanged, the tilts introduce skews in the opposite directions which do not exist when the sample is mounted upright. Considering that the saturation of the magnetization in the z-direction only happens at much higher H (Fig. 3.8(a)), these skews should be attributed to the ANE rather than the ONE. This result suggests that the ANE induced by H z is not responsible for the hysteretic behavior of V xyx. An angular dependence of V xyx is plotted in Fig. 3.11(b), with the sample rotated in the xy-plane to change the direction of the magnetization. The data follow a cosθ function. 142

164 V xyx (nv) (a) φ = +10 o φ = 0 o φ = 10 o H (Oe) H x Metglas ϕ T z V xyx (θ) / V xyx (θ=0) (b) point contact T y x θ H θ (deg) Figure Angular dependence of V xyx. (a) Dependence on sample tilt angle ϕ in the z- direction. The straight lines indicate the relative change of the slope between different ϕ. (b) Rotation angle θ dependence in the xy-plane (the line follows a cosθ). Both SSE and ANE are known to exhibit the hysteretic dependence on H and a cosθ angular dependence. From the results discussed hitherto, it can be safely stated that there is no ANE provoked either by ΔT z or by H z in our Co-Metglas sample. Then, can it be the SSE that gives rise to the above observations? A similar phenomenon was reported by Saitoh, 114 who showed how Fe can give an inverse spin Hall signal in Fe/YIG bilayers. It was also observed in GaMnAs. 24 In the latter work, V xyx measured on point 143

165 contacts resulted in the same H dependence and no sign inversion between the hot and cold sides. Those signals were attributed to a mixture of the SSE and the PNE, and it was suggested that the GaMnAs itself can behave as a spin transducer due to the intrinsic spin-orbit interaction. The same argument could apply here. If the measured V xyx signals are SSE signals contaminated by the PNE owing to misalignments of the point contacts, the PNE should contribute to the background signals with the same sign across the sample. Although we may subtract the PNE contribution by taking an average of the measured V xyx in two magnetic fields of equal amplitude and opposite polarity (the magnetothermopower being an even function of field in amorphous systems 99 ), it should be assumed then that the degree of the misalignment is the same for all measurement positions, which is not likely to be the case. Therefore, an exact assessment of the magnitude of the TSSE signals may not be possible in this system. It is also noted that small signals are still present even when θ = 90 deg and θ = 270 deg in Fig. 3.11(b), which may possibly be attributed to the pure PNE contribution. Lastly, we will discuss results of TSSE measurements on Cu+Pt strips. V xyx measured on Cu+Pt strips are shown in Fig. 3.12(a). The typical SSE-like hysteretic signals are observed on both hot and cold sides, but as for point contacts, there is no sign inversion between hot and cold sides. The magnitude of the signal is larger on the hot side than on the cold side, also akin to the observation on point contacts. An angular dependence measurement of V xyx (Fig. 3.12(b)) in the xy-plane reveals that the data follow a cosθ function closely. Therefore, all of the data consistently suggest that the measured V xyx are spin Seebeck signals, except for the absence of the sign inversion. 144

166 (a) 60 Cu + Pt V xyx (nv) T avg = 72.4K T = 13.2K Hot side Cold side -40 (b) V xyx (θ) / V xyx (θ=0) H (Oe) T avg = 72.4K T = 13.2K Cu+Pt θ (deg) Figure Measurements on Cu+Pt strips. (a) Transverse voltage V xyx as a function of H for the hot and cold ends with an applied ΔT x = 13.2K at T avg = 72.4K. (b) Rotation angle θ dependence in the xy-plane (the line follows a cosθ). Figure 3.13 compares α xyx of point contacts with those of Cu+Pt strips. For comparison purpose, we substituted S xy with α xyx, both being conceptually the same measured quantities. Over the entire measured temperature range, α xyx of point contacts are larger than those of Cu+Pt at the same side of the sample. The temperature dependence is almost identical for both contacts. Based on the observations in Fig

167 and Fig. 3.13, it is suspected that the signals on Cu+Pt strips are of the same origin with those on point contacts. The highly electrically conducting Cu+Pt layers possibly short out the signals, reducing their magnitude. It is also noted that the small residual signals seen on point contacts at θ = 90 deg and θ = 270 deg in Fig. 3.11(b) disappear on Cu+Pt, again perhaps due to the short-out effect. Regarding the effect of magnon thermal conductivity on SSE signals, no obvious correlation is found between the temperature dependences of the magnon thermal conductivity and SSE signals from Fig. 3.6 and Fig The interpretation is limited due to less accurate temperature dependence of magnon thermal conductivity caused by lack of sufficient magnetic field intensity to fully suppress κ M at each measured temperature. 5 α xyx (nv K -1 ) Point Hot Point Cold Pt Hot Pt Cold T (K) Figure Comparison of α xyx between point contacts and Cu+Pt strips. Points are experimental data, and lines are added to guide the eye. 146

168 Chapter 4: Observation of Phonon Diamagnetism in Nonmagnetic Semiconductor 4.1 Introduction As previously mentioned, electrons, phonons and magnons (spin waves) are three elemental excitations in solids known to carry heat. Electrons are subject to a Lorentz force in the presence of an applied external magnetic field (H), giving rise to thermomagnetic effects. 116 Magnon density of states is affected by H, resulting in an H- dependent magnon heat capacity (C) 16 and thermal conductivity (κ). 117 In ferromagnetic crystals lacking inversion symmetry H can induce a Dzyaloginskii-Morya effects that results in a magnon Hall effect. 118 The phonon properties are not considered sensitive to H, although some reports exist of thermomagnetic effects in insulating paramagnetic materials. 119,120 In contrast, we report here a magnetic response of the phonons in an electrically insulating and diamagnetic semiconductor, InSb, where the electronic states near the Fermi energy consists only of s and p-electron shells of In and Sb. We attribute this response to a phonon-specific local diamagnetic moment on the In and Sb atoms. This moment exists only locally when the atomic displacement associated with the 147

169 phonon is frozen in time (the frozen phonon approximation ). Obviously, symmetry arguments impose that both the spatial and the temporal averages of this phonon-induced diamagnetic moment are zero. Yet it exercises a force on the atoms that results in magnetic-field sensitive phonon-phonon interactions and scattering. 4.2 Experimental The starting material for the samples is an InSb boule doped with Te to an electron concentration of 1.3 x cm -3 (measured by Hall effect at temperatures T with 3 K < T <50 K). The electronic transport properties of the material are given later. The samples are cut into ( 23 x 4 x 5 mm) pieces that are further cut in the tuning fork geometry shown in Fig. 4.1(a) and (c). The large arm has a cross section A L 4 x 3 mm, the small arm has A S 4 x 1 mm. The sample was mounted on a heat sink, the small arm equipped with a 120 Ohm heater, and the large arm with four of them. Each arm was equipped with two Cernox thermometers as shown, which were calibrated as a function of temperature. The bottom surface of the sample was indium-soldered to the thick copper block heat sink. All contacts on the sample were made using 6N indium in order to minimize contact thermal resistances as well as thermal stresses during heating/cooling cycles. The operating procedure for the thermal potentiometer, at any given value of the cryostat base temperature, is as follows. We fixed the power Q L on the large arm, and 148

170 stepped the heater power on the small arm (Q S ) while monitoring the temperature difference T LS ( T LS = T L T S ): traces showing T LS as function of time for different values of Q S are shown in Fig. 4.1(b). We then plot the steady-state value of T DIFF ( T DIFF = T LS T R where T R corresponds to the residual T LS when no heat is applied) versus Q S and interpolate the value of Q S that makes ΔT DIFF = 0 (Fig. 4.1(d)). Here the longitudinal temperature differences in both arms are equal (ΔT L = ΔT S ); thus for this value of Q S the thermal conductivity ratio ( A A )( Q Q ) κ κ =, where κ L and κ S L S S L L S denote the thermal conductivities, and A L and A S the cross sectional areas of the large and small arms, respectively. The area ratio ( ) A A is determined accurately from the hightemperature asymptote of QL Q S. S L Thermal conductivity measurements were made using the differential or absolute heater and sink method in a modified Thermal Transport Option in the Physical Properties Measurement System by Quantum Design. Radiative or convective losses were minimized by use of the fully gold-plated cyopumped sample chamber. For temperature readings, Cernox thermometers were calibrated as a function of temperature before the thermal conductivity measurements. For differential thermal conductivity measurements, the temperature difference was measured between the upper thermometers with different heater powers on the small arm while a constant power was used for the large arm. The heater power on the small arm was adjusted such that the measured temperature differences are approximately symmetric with respect to zero. Enough stabilization time was given after heater on/off, to confirm that a saturation has been 149

171 reached in the temperature traces (Fig. 4.1(b)). For absolute thermal conductivity measurements on the small arm, the temperature difference was measured between the upper and lower thermometers with a fixed heater power on each arm. For both measurements, actual temperature differences were obtained by subtracting the residual temperature differences measured with no heat applied from the temperature differences measured after the heater was turned on. Great care was taken for accurate measurements of the residual temperature differences, since at the lowest temperatures, those differences become only an order of magnitude smaller than the temperature differences under applied heat, primarily due to the very high thermal conductance of the sample. For the same reason, although the high purity indium was used for the thermal contacts, the sample temperature deviated substantially from the base temperature set by the instrument when heat was applied. Thus, residual temperature differences were separately measured after adjusting the base temperature to the same temperature at which the sample had been measured with heater on. 150

172 a b 40 no heat heater on heater off T LS (mk) T = 30K H = 0T Constant Q L Step Q S 23 mm Time (s) c Q L T L T DIFF T S Q S d T DIFF (mk) T = 30K H = 0T Q L = 44.5mW H Cernox Q (mw) S Figure 4.1. Experimental setup and measurement protocol. (a) Photo of the InSb tuning fork sample mounted on measurement system. The scale bar indicates the actual length of the sample. (b) Raw traces of the differential thermal conductivity measurement, showing temperature differences between the large and small arms (ΔT LS ) as a function of time, measured in the zero magnetic field. Red curves correspond to ΔT LS with four different heater powers on the small arm (Q S ) while the heater power on the large arm (Q L ) is kept constant. (c) Experimental geometry for differential and absolute thermal conductivity measurements, not to scale. H denotes the magnetic field. ΔT L and ΔT S are the longitudinal temperature differences on the large and small arms, respectively. (d) ΔT DIFF versus Q S obtained from the raw traces in (b). T DIFF = T LS T R where T R is the residual T LS when no heat is applied. Purple crosses are experimental data, and the red cross indicates a value of Q S when ΔT DIFF = 0, obtained by a linear interpolation. The solid line is a linear fit to the data points. 151