Thermal Energy Conversion Utilizing Magnetization Dynamics and Two-Carrier Effects Dissertation Presented in Partial Fulfillment

Thermal Energy Conversion Utilizing Magnetization Dynamics and Two-Carrier Effects

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Sarah June Watzman

Graduate Program in Mechanical Engineering

The Ohio State University

2018

Dissertation Committee

Joseph P. Heremans, Advisor

Nandini Trivedi

Fengyuan Yang

Igor Adamovich

Copyrighted by

Sarah June Watzman

2018 Abstract

This dissertation seeks to contribute to the field of thermoelectrics, here utilizing magnetization dynamics in two-carrier systems, employing unconventional thermoelectric materials. Thermoelectric devices offer fully solid-state conversion of waste heat into usable electric energy or fully solid-state cooling. The goal of this dissertation is to elucidate key transport phenomena in ferromagnetic transition metals and Weyl semimetals in order to positively contribute to the overarching effort of using thermoelectric materials as a clean energy source.

The first subject of this dissertation is magnon drag in Fe, Co, and Ni. Magnon drag is shown to dominate the thermopower of elemental Fe from 2 to 80 K and of elemental Co from

150 to 600 K; it is also shown to contribute to the thermopower of elemental Ni from 50 to 500

K. Two theoretical models are presented for magnon-drag thermopower. One is a hydrodynamic theory based purely on non-relativistic electron-magnon scattering, and the other is based on microscopic spin-motive forces. In spite of their very different origins, the two give similar predictions for pure metals at low temperature, providing a semi-quantitative explanation for the observed thermopower of elemental Fe and Co without adjustable parameters. Magnon- drag may also contribute to the thermopower of Ni, but not dominantly as it does in Fe and Co.

A spin-mixing model is presented, in which spin-up and spin-down electrons are seen as separate carriers in independent conduction channels. This model describes the magnon-drag

ii contribution to the anomalous Nernst effect in Fe, again enabling a semi-quantitative match to the experimental data without fitting parameters. This work suggests that particle non-conserving processes may play an important role in other types of drag phenomena and also gives a predicative theory for improving metals as thermoelectric materials.

The second subject of this dissertation is the Nernst effect in the Weyl semimetal NbP.

Weyl semimetals expand research on topologically protected transport by adding bulk Berry monopoles with linearly dispersing electronic states and topologically robust, gapless surface

Fermi arcs terminating on bulk node projections. Here, the Nernst effect, combining entropy with charge transport, is shown to give a unique signature for the presence of Dirac bands. The

Nernst thermopower of NbP, maximum of 800 V K-1 at 9 T and 109 K, exceeds its conventional thermopower by a hundredfold and is significantly larger than the thermopower of traditional thermoelectric materials. Both the low-field and high-field Nernst coefficients also have pronounced maxima at separate temperatures. A self-consistent theory without adjustable parameters shows that this results from electrochemical potential pinning to the Weyl point energy at temperatures above the maximum temperature of the Nernst coefficients, driven by charge neutrality and Dirac band symmetry. The charge neutrality condition dictates that the

Fermi level shifts with increasing temperature toward the energy that has the minimum density of states. In NbP, the agreement of the Nernst and Seebeck data with a model that assumes this minimum density of states resides at the Dirac points is taken as strong experimental evidence that the trivial (non-Dirac) bands play no role in high-temperature transport.

The third subject of this dissertation, which is a study currently in progress, is the Nernst effect of YbMnBi2, a Weyl semimetal that breaks time-reversal symmetry, in the absence of an

iii externally applied magnetic field. While NbP demonstrates a very large Nernst thermopower, transverse transport in that material also requires a large externally applied magnetic field, making NbP commercially unviable for use in energy generation devices. The breaking of time-reversal symmetry in YbMnBi2 gives rise to a net Berry curvature of its electronic band structure, which functions like an internal magnetic field. Here, the applied temperature gradient and measured voltage are mutually perpendicular to the predicted direction of net Berry curvature in YbMnBi2, producing a large Nernst thermopower in the absence of an externally applied magnetic field. This isothermal Nernst thermopower peaks above 1000

V K-1 at 60 K and is expected to be due directly to the material’s internal Berry curvature. The zT at 60 K is also calculated to be 2.4, exceeding the state-of-the-art zT for thermoelectric materials in the advantageous transverse geometry and eliminating the need for an externally applied magnetic field when using the transverse geometry. Nevertheless, this huge Nernst thermopower has currently only been seen on one sample; repeatability studies are in progress.

Additionally, adiabatic thermomagnetic transport data presently do not align with results seen isothermally, even when parasitic temperature gradients are considered. These confirmation studies and their explanations are left in progress.

Dedication

To my mom.

Acknowledgments

This work would not have been possible without the support of my advisor, Joseph P.

Heremans. Thank you for being excited about new research ideas and for helping me grow into an independent researcher. I sincerely appreciate the enthusiasm with which I was met every day. One of the most important things I have learned from you is the “of course you can measure that” attitude, which forced me to be creative in developing new measurement techniques and approach projects with an open mind. Additionally, I greatly appreciate being

(not so) subtly pushed in the direction of a more fundamental project, especially as an engineer.

I am incredibly thankful for the instigating collaborator in my work in Weyl semimetals -

Claudia Felser. I am so glad you pursued collaboration in the Weyl semimetal field with myself and my group as working with you has been a complete pleasure. I am so appreciative for your positive energy and support since the first day I met you, especially while I was a visitor in your group in Dresden, Germany.

Another instrumental collaboration is that with Nandini Trivedi. I have greatly enjoyed our extensive scientific conversations, trying to understand what is actually going on and how we can explain it. Thank you for offering a theoretical point of view and explanation that pairs with my work, and thank you doing so with such passion for your work. Your support and collaboration has greatly enhanced my graduate career.

I will forever be grateful for my labmates throughout my PhD, especially Arati, Bin, and

Mike. Thank you for teaching me so much in the lab, always listening, and being my friend.

Hyun and Steve – thank you for your patience and assistance as I began my PhD. Koen,

Yuanhua, and Dung – thank you for always being supportive, positive, and patient. Renee – thank you for going above and beyond in everything you do. And for finding that tiny sample of

NbP, thank you Koen, Bin, and Renee for crawling on the floor and sorting through dust with me.

I would like to thank Tim McCormick for collaboration in the Weyl semimetal projects – thank you for all the discussions, physics explanations, and shared weary glances at our advisors.

I would also like to thank the postdocs at the Max Planck Institute for Chemical Physics of

Solids who helped me in every way they could to make my time there as productive as possible, especially Satya Guin, Kaustuv Manna, and Chengyuang Fu. Additionally, I would like to thank

Johannes Gooth for all of his time and discussions regarding not only my data but the future of the field – thank you for bringing so much energy to this project and being supportive of my work.

I would like to acknowledge my committee members, Nandini Trivedi, Fengyuan Yang, and Igor Adamovich. Thank you for your participation in this process, for supporting me, and for always being open to questions and scientific discussion.

I would not have started this PhD process if it hadn’t been for encouragement from Jim

Saunders and Rob Siston. Jim – thank you for identifying that graduate school was definitely for me and offering support throughout my time pursuing my PhD. Rob – thank you for guiding me

vii through the twists and turns of choosing research projects and advisors, muddling through the middle of a PhD, and pursuing an academic career path.

This PhD would not have been possible without the constant support and encouragement of my friends and family – thank you all for listening to more details about research than you ever wanted to hear and encouraging me to keep going. I would especially like to thank Rachel for being a constant, unending source of positivity, insight, and advice, and for listening to all of my interview talks. Joe – thank you for your infinite patience, love, and encouragement, and for always believing in me more than I believe in myself.

Above all else, thank you mom and Josh. I wouldn’t have loved science, stuck with a

PhD, or become the person I am today if it weren’t for you. Thank you for driving my confidence in myself and constantly reminding me that I can do anything I set my mind to throughout my entire life. You are the best.

Content in Chapter 2 is published in the following: “Magnon-drag thermopower and

Nernst coefficient in Fe, Co, and Ni.” S. J. Watzman, R. A. Duine, Y. Tserkovnyak, S. R.

Boona, H. Jin, A. Prakash, Y. Zheng, and J. P. Heremans. Phys. Rev. B 94 (14), 144407 (2016).

Content in Chapter 3 is available as a pre-print and has been accepted for publication in

Physical Review B as a Rapid Communication: “Dirac dispersion generates large Nernst effect in

Weyl semimetals.” S. J. Watzman, T. M. McCormick, C. Shekhar, S.-C. Wu, Y. Sun, A.

American Physical Society.

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Vita

2013 – Present…………………………………………….NSF Graduate Research Fellow University Fellow FAST (Future Academic Scholar Training) Fellow The Ohio State University

2017……………………………………………………………………Visiting Researcher Max Planck Institute for Chemical Physics of Solids Dresden, Germany

2016…………………………………………………………M.S. Mechanical Engineering The Ohio State University

2013……………………………………………………...….B.S. Mechanical Engineering The Ohio State University

2012-2013………………………………………..…….Undergraduate Research Assistant The Ohio State University

2012-2013…………………………………………………………………………….Intern Battelle Memorial Institute

2011…………………………………………………………………………………..Intern GE Energy

Publications

S. J. Watzman, T. M. McCormick, C. Shekhar, S.-C. Wu, Y. Sun, A. Prakash, C. Felser, N. Trivedi, and J. P. Heremans. “Dirac dispersion generates large Nernst effect in Weyl semimetals.” arXiv:1703.04700 (2017). Accepted for publication in Physical Review B (R) (2018).

C. Fu. T. Scaffidi, J. Waissman, Y. Sun, R. Saha, S. J. Watzman, A. K. Srivastava, G. Li, W. Schnelle, P. Werner, M. E. Kamminga, S. Sachdev., S. S. P. Parkin, S. A. Hartnoll, C. Felser, and J. Gooth. “Thermoelectric signatures of the electron-phonon fluid in PtSn4.” arXiv:1802.09468 (2018).

T. M. McCormick, S. J. Watzman, J. P. Heremans, and N. Trivedi. “Fermi arc mediated entropy transport in topological semimetals.” arXiv: 1703.04606 (2017).

K. Vandaele, S. J. Watzman, B. Flebus, A. Prakash, Y. Zheng, S. R. Boona, and J. P. Heremans. “Thermal spin transport and energy conversion.” Materials Today Physics 1. (2017).

U. Stockert, R. D. dos Reis, M. O. Ajeesh, S. J. Watzman, M. Schmidt, C. Shekhar, J. P. Heremans, C. Felser, M. Baenitz, and M. Nicklas. “Thermopower and thermal conductivity in the Weyl semimetal NbP.” Journal of Physics: Condensed Matter 29 (2017).

C. Shekhar, Y. Sun, N. Kumar, J. Gooth, M. Nicklas, S. J. Watzman, K. Manna, V. Suess, O. Young, I. Leermakers, S. N. Guin, T. Foerster, M. Schmidt, L. Muechler, P. Werner, W. Schnelle, U. Zeitler, B. Yan, S. S. P. Parkin, and C. Felser. “Extremely high conductivity observed in the unconventional triple point fermion material MoP.” arXiv:1703.03736 (2017).

S. J. Watzman, R. A. Duine, Y. Tserkovnyak, S. R. Boona, H. Jin, A Prakash, Y. Zheng, and J. P. Heremans. “Magnon-drag thermopower and Nernst coefficient in Fe, Co, and Ni.” Physical Review B 94 (2016).

S. R. Boona, S. J. Watzman, and J. P. Heremans. “Utilizing magnetization dynamics in solid- state thermal energy conversion.” Applied Physics Letters Materials 4 (2016).

J. P. Heremans, H. Jin, Y. Zheng, S. J. Watzman, and A. Prakash. “BiSb and spin-related thermoelectric phenomena.” Proceedings of SPIE 9821, Tri-Technology Device Refrigeration (2016).

Fields of Study

Major Field: Mechanical Engineering

Table of Contents

Abstract ...... ii Dedication ...... v Acknowledgments...... vi Vita ...... ix List of Tables ...... xiv List of Figures ...... xv Chapter 1. Introduction ...... 1 1.1 Introduction to Thermoelectrics ...... 1 1.1.1 Characterizing Thermoelectric Transport ...... 1 1.1.2 Conventional Thermoelectric Materials ...... 4 1.1.3 Thermoelectric Devices ...... 5 1.2 Geometric Considerations in Thermoelectric Transport...... 7 1.2.1 Longitudinal vs. Transverse Geometries ...... 7 1.2.2 Nernst Effect ...... 9 1.2.3 Transverse Devices ...... 11 1.3 Ferromagnetic Transition Metals ...... 12 1.3.1 Metals as Thermoelectric Materials ...... 12 1.3.2 Magnons and Magnon Drag...... 15 1.4 Weyl Semimetals ...... 16 1.4.1 Unique Characteristics of Weyl Semimetals ...... 16 1.4.2 Experimental Discovery of and Emerging Interest in Weyl Semimetals ...... 27 Chapter 2. Magnon-Drag Thermopower and Nernst Effect in Ferromagnetic Transition Metals32 2.1 Motivation ...... 32 2.2 Measurement Technique ...... 33 2.3 Magnon-Drag Thermopower ...... 37 xi

2.3.1 Experimental Results for the Thermopower of Fe, Co, and Ni ...... 37 2.3.2 Hydrodynamic Theory ...... 41 2.3.3 Theory Based on Spin-Motive Forces ...... 43 2.3.4 Comparison of Experiment to Theory ...... 46 2.4 Thermomagnetic Effects ...... 47 2.4.1 Spin-Mixing Theory for the Anomalous Nernst Effect ...... 47 2.4.2 Experimental Results for Nernst Thermopower and the Anomalous Nernst Effect in Fe ...... 51 2.4.3 Longitudinal and Transverse Magneto-Thermopower in Fe ...... 54 2.4.4 Planar Nernst Effects in Fe ...... 56 2.5 Conclusion ...... 58 Chapter 3. Large Nernst Effect in the Weyl Semimetal NbP Generated by Dirac Dispersion ..... 60 3.1 Motivation ...... 60 3.2 Measurement Technique ...... 61 3.2.1 Isothermal Measurement Technique ...... 61 3.2.2 Adiabatic Measurement Technique ...... 65 3.2.3 Discussion of Adiabatic vs. Isothermal Thermomagnetic Measurements ...... 66 3.3 Sample Characterization ...... 69 3.3.1 Electrical Conductivity, Magnetoresistance, and Hall Effect ...... 69 3.3.2 Thermal Conductivity ...... 72 3.3.3 Specific Heat ...... 73 3.3.4 Magnetization ...... 76 3.4 Thermomagnetic Transport in NbP ...... 79 3.4.1 Experimental Results ...... 79 3.4.2 Theoretical Model ...... 84 3.4.3 Comparison of Experiment to Theory ...... 86 3.4.4 Contribution of Trivial Pockets to Transport ...... 88 3.5 Conclusion ...... 89 Chapter 4. Berry Curvature-Induced Huge Anomalous Nernst Effect in the Weyl Semimetal YbMnBi2 ...... 91 4.1 Motivation ...... 91 4.2 Measurement Technique ...... 95 4.2.1 Samples Used in This Study ...... 95 xii

4.2.2 Isothermal Measurement Technique ...... 96 4.2.3 Adiabatic Measurement Technique ...... 101 4.2.3.1 Modified Thermal Transport Puck...... 101 4.2.3.2 PPMS AC Transport Puck and Continuous Flow Cryostat ...... 102 4.3 Sample Characterization ...... 104 4.3.1 Electrical Conductivity, Magnetoresistance, and Hall Effect ...... 104 4.3.2 Thermal Conductivity ...... 108 4.3.3 Magnetization ...... 112 4.4 Thermomagnetic Transport ...... 115 4.4.1 Isothermal Thermomagnetic Transport ...... 115 4.4.2 Adiabatic Thermomagnetic Transport ...... 124 4.5 Device Applications ...... 131 4.6 Outlook ...... 133 Chapter 5. Concluding Remarks ...... 137 References ...... 143

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List of Tables

Table 1: Weyl semimetals and their classifications by type and broken symmetry. Table taken from N. Trivedi, APS March Meeting 2018, Los Angeles (2018)...... 24 Table 2: Shubnikov-de Haas and de Haas-van Alphen quantum oscillations in NbP. Table taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 79

Table 3: YbMnBi2 samples used in thermomagnetic measurements displaying the measurement technique, measurement system, and crystal axes corresponding to the applied and measured quantities ...... 95

Table 4: YbMnBi2 samples used for electrical transport and magnetic measurements displaying the crystal axes corresponding to the applied and measured quantities...... 96

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List of Figures

Figure 1: Schematic of thermoelectric couple in energy generation mode (left) and refrigeration mode (right) for the longitudinal geometry, which is conventionally used for thermoelectric transport. Figure taken from G. J. Snyder, http://thermoelectrics.matsci.northwestern.edu/thermoelectrics/index.html ...... 2 Figure 2: Thermopower, thermal conductivity, and electrical conductivity as functions of carrier concentration in relation to zT as a function of carrier concentration ...... 5 Figure 3: Schematic of a longitudinal thermoelectric module prepared for use in energy generation mode depicting n-type and p-type semiconductors connected electrically in series and thermally in parallel. Figure taken from G. J. Snyder and E. S. Toberer, Nature Materials 7 (2008)...... 6 Figure 4: Photo of a longitudinal Peltier module showing the cascaded structure desired for refrigeration mode. Each level is considered a stage, and each stage has a different heat transfer area. Module manufactured by Marlow Industries, Inc...... 7 Figure 5: Comparison of longitudinal (left) to transverse (right) geometries for thermoelectric transport, in which the longitudinal geometry requires both n-type and p-type materials while the transverse geometry typically requires an externally applied magnetic field. Applied temperature gradient and measured voltage are parallel for the longitudinal geometry and perpendicular for the transverse geometry...... 9 Figure 6: Ettingshausen cooler, a refrigeration device utilizing a transverse geometry with a cascaded structure. Figure taken from D. K. C. MacDonald, Thermoelectricity: An introduction to the principles, Dover Publications, New York (1962)...... 12 Figure 7: zT as a function of temperature in common metallic thermoelectric materials, demonstrating the low zT of metals. Figure taken from K. Vandaele et al., Materials Today Physics 1 (2017)...... 13 Figure 8: Power factor as a function of temperature for ferromagnetic transition metals as compared to conventional thermoelectric materials, demonstrating that ferromagnetic metals can have a large power factor. Figure taken from K. Vandaele et al., Materials Today Physics 1 (2017)...... 14

Figure 9: Schematic depiction of a magnon, where the arrows represent the spins (magnetic moments) in a given electron and the circles represent their precession which in turns creates the magnon wave. Figure taken from C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, Inc., New York (2005)...... 15 Figure 10: Magnon-drag effect demonstrating coupling between phonons and magnons to drag electrons through a ferromagnetic metal. Image credit to Renee Ripley...... 16 Figure 11: Conventional periodic crystal structure with cation, or A+, states in the conduction band and anion, or B-, states in the valence band. In k-space, periodic bands emerge and surface states occur when the periodicity ends at the edge of a real material. Figure taken from J. P. Heremans, US/Japan Workshop on Nanoscale Thermal Transport, Tokyo (2017)...... 18 Figure 12: Energy band structure evolution as a function of k-vectors as the gap closes between the conduction band (CB) and the valence band (VB). Band insulators can be described as semiconductors when the band gap is on the order of 1 eV or smaller. Dirac and Weyl semimetals emerge when the bands overlap at crossing points. Dashed orange lines represent the Fermi level (specifically for an intrinsic semiconductor in the left frame)...... 19 Figure 13: Periodic structure of a solid combined with strong spin-orbit coupling leads to band inversion. Here, the B- states make up the conduction band (CB) while the A+ states make up the valence band (VB). This gives rise to linearly dispersing Dirac bands as surface states, where the velocity of electrons on each surface are oppositely signed, as shown by the surface labels on the Dirac bands. Figure taken from J. P. Heremans, US/Japan Workshop on Nanoscale Thermal Transport, Tokyo (2017)...... 21 Figure 14: Band inversion plus spin-orbit coupling (SOC) giving rise to topological insulators (TI), Dirac semimetals (DSM), and Weyl semimetals (WSM). a. Topological insulators emerge when the energy band crossing points are gapped. Surface states are closed loops. b. Dirac and Weyl semimetals emerge when no gap exists between the energy band crossing points. Surface states are open arcs. When inversion or time-reversal symmetry is broken, a Dirac point splits into a pair of Weyl points. c. Type I Weyl semimetal, where the Fermi surface is zero at the energy band crossing points. d. Type II Weyl semimetal, where the Dirac cone is tilted and the energy band crossing point is the separation between the electron and hole pockets in the Fermi surface. Figure taken from B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. (2017). ... 22 Figure 15: Time-reversal symmetry (left frame), in which a property flips sign when moved in momentum space from a positive k-vector to a negative k-vector. Inversion symmetry (right frame), in which a property maintains the same sign whether it is at a positive or negative k- vector in momentum space. Figure taken from N. Trivedi, APS March Meeting 2018, Los Angeles (2018)...... 23 Figure 16: Weyl semimetal band structure near Weyl points. Orange dotted lines indicate linear, Dirac bands near the Weyl points. Colored arrows depict the orbital character of the wave function in the conduction or valence band respectively, giving rise to a source and sink of Berry curvature indicated in blue...... 25

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Figure 17: Section of k-space, simplified to one pair of Weyl points, depicting a Weyl semimetal with Fermi arcs on both top and bottom surfaces connecting Weyl point projections. Dirac cones are present in the bulk. Figure taken from T. M. McCormick et al, arXiv:1703.04606 (2017). . 27 Figure 18: Weyl semimetal band structure with tilted bands breaking time-reversal symmetry, giving rise to a net Berry curvature due to a different magnitude of chirality at each Weyl point ...... 29 Figure 19: Comparison of traditional to canted antiferromagnetic structures, showing a small effective magnetization when canting is present ...... 30 Figure 20: Geometric configurations for transition metal measurements with crystal axes labeled for the single crystal of Fe ...... 34 Figure 21: Electrical resistivity temperature dependence of transition metal samples. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 36 Figure 22: Thermal conductivity temperature dependence of transition metal samples. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 37 Figure 23: Fe thermopower temperature dependence of single-crystal sample (main frame). Inset shows single-crystal sample and polycrystal. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 38 Figure 24: Co thermopower temperature dependence. Main frame shows negative thermopower on a logarithmic scale while inset shows actual thermopower on a linear scale. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 39 Figure 25: Ni thermopower temperature dependence. Main frame shows negative thermopower on a logarithmic scale while inset shows actual thermopower on a linear scale. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 40 Figure 26: Spin-mixing model for Nernst effect in transition metals depicted as electrical circuit with separate conduction channels for spin-up and spin-down electrons in the presence of a magnetic field...... 49 Figure 27: Nernst thermopower in polycrystalline Fe as a function of magnetic field. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 52 Figure 28: Nernst thermopower in single-crystal Fe as a function of magnetic field. The anomalous Nernst effect extends from about -20 kOe to 20 kOe with the ordinary Nernst effect extending outside this range. Hysteresis is visible and most likely due to domain realignment. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 53 Figure 29: Nernst coefficient temperature dependence of single-crystal Fe. Closed circles represent experimental data and the curve represents the model calculation. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 54

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Figure 30: Transverse magneto-thermopower temperature dependence of single-crystal Fe, normalized by the zero field thermopower. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 56 Figure 31: Planar Nernst thermopower as a function of magnetic field in single-crystal Fe. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 57 Figure 32: Saturated planar Nernst thermopower as a function of temperature in single-crystal Fe. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016)...... 58 Figure 33: NbP mounted isothermally for thermomagnetic transport measurements. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 62 Figure 34: Isothermal sample mounting technique shown schematically with isotherms represented by dashed lines. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 63 Figure 35: NbP sample mounted isothermally and shown on TTO puck for PPMS measurements. Resistive heaters in series are shown in the left frame, thermometer shoes are shown in the middle frame, and the heat sink side is shown in the right frame...... 63 Figure 36: NbP adiabatic mounting technique for measurements in cryostat ...... 66 Figure 37: Temperature dependence of electrical resistivity in NbP. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 70 Figure 38: Transverse magneto-resistivity in NbP as a function of magnetic field. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 71 Figure 39: Hall resistivity in NbP as a function of magnetic field. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 72 Figure 40: Thermal conductivity in NbP as a function of temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 73 Figure 41: Specific heat of NbP as a function of temperature, with experimental data shown as brown dots and fit to a Debye model as blue curve. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 75 Figure 42: Low-temperature specific heat of NbP divided by temperature as a function of temperature squared. Brown circles are experimental data and blue curve is Debye model fit. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 76 Figure 43: Diamagnetic moment of NbP at 5 K with magnetic field applied parallel to <001>. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 78 Figure 44: Diamagnetic moment of NbP at 5 K with magnetic field applied perpendicular to <001>. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 78

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Figure 45: Experimental (a) and theoretical (b) magnetic field dependence of the Nernst thermopower in NbP. Inset in top frame shows Shubnikov-de Haas oscillations in raw Nernst voltage as a function of magnetic field at 4.92 K. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 81 Figure 46: Experimental (a) and theoretical (b) temperature dependence of the Nernst coefficient in NbP. Inset of top frame is conventional thermopower as a function of temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 82 Figure 47: Adiabatic magneto-thermopower (a) and Nernst thermopower (b) in NbP. Adiabatic magneto-thermopower includes a contribution from the thermal Hall effect not reflected in the isothermal thermopower measurements. Adiabatic Nernst thermopower reasonably corresponds to isothermal Nernst thermopower over available temperature and magnetic field range considering two different measurement instruments were used. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 84 Figure 48: Two-dimensional dispersion around a single Dirac cone (left) where, at low temperatures, the electrochemical potential lies in the valence band at an energy o below the Dirac point. The calculated (T) temperature dependence (right) shows it moving towards the energy of the Weyl node with increasing temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017)...... 85 Figure 49: Transverse geometry of the Nernst effect with Berry curvature aligned along the z- axis. When a magnetic field is applied, it will also be applied parallel to the z-axis...... 94

Figure 50: Orientation of YbMnBi2 sample 1, z-axis parallel to Berry curvature. Isotherms are parallel to the y-axis...... 97

Figure 51: Isothermal sample 1 of YbMnBi2 prepared for measurements in the PPMS. The technique shown here was used for both samples 1 and 2, although crystal axes are identified corresponding to sample 1...... 98

Figure 52: Orientation of YbMnBi2 sample 2, z-axis perpendicular to Berry curvature. Isotherms are parallel to the y-axis...... 99

Figure 53: YbMnBi2 isothermal sample 1 prepared for measurement in PPMS on TTO puck. Heat spreader and resistive heater is visible in the left frame, thermometer shoe assemblies are visible in the middle frame, and the heat sink side is visible in the right frame. The technique shown here was used for both samples 1 and 2...... 100

Figure 54: Adiabatic measurement technique used for YbMnBi2 samples 3 and 4. Mount used for both AC puck in PPMS and in cryostat...... 103

Figure 55: Temperature dependence of electrical resistivity in sample 7 of YbMnBi2 at 0 Oe and 90 kOe, Happ parallel to <110>. Data and image courtesy of Kaustuv Manna...... 105

Figure 56: Magnetoresistance as a function of applied magnetic field in sample 7 of YbMnBi2. Data and image courtesy of Kaustuv Manna...... 105

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Figure 57: Transverse (Hall) resistivity as a function of applied magnetic field in sample 7 of YbMnBi2, measured using a 5-wire technique with balance that sets the signal to 0 -m at 0 Oe. Data and image courtesy of Kaustuv Manna...... 106 Figure 58: Transverse (Hall) resistivity as a function of applied magnetic field in sample 6 of YbMnBi2, measured using a 4-wire technique and unbalanced at 0 Oe. For reference, right frame shows resistivity as a function of magnetic field in the same sample. Data and image courtesy of Kaustuv Manna...... 108

Figure 59: Thermal conductivity in YbMnBi2, comparing samples 3, 4, and 5. Inset only shows samples 3 and 4 to emphasize their temperature dependences. Sample 5 is likely to have more heat losses, and thus a larger apparent thermal conductivity, than samples 3 and 4 due to the measurement technique used...... 111

Figure 60: Magnetization as a function of temperature in sample 7 of YbMnBi2 along the <110> crystal axis. Data and image courtesy of Kaustuv Manna...... 112

Figure 61: Magnetization as a function of temperature in sample 7 of YbMnBi2 along the <001> crystal axis. Data and image courtesy of Kaustuv Manna...... 113

Figure 62: Magnetization as a function of applied magnetic field in sample 7 of YbMnBi2 along the <110> crystal axis. Data and image courtesy of Kaustuv Manna...... 114

Figure 63: Magnetization as a function of applied magnetic field in sample 7 of YbMnBi2 along the <001> crystal axis. Data and image courtesy of Kaustuv Manna...... 114

Figure 64: Nernst effect magnetic field sweeps in sample 1 of YbMnBi2 with Happ parallel to Berry curvature direction, the <110> crystal axis ...... 117

Figure 65: Magneto-thermopower magnetic field sweeps in sample 1 of YbMnBi2 with Happ parallel to Berry curvature direction, the <110> crystal axis ...... 118

Figure 66: Nernst effect magnetic field sweeps in sample 2 of YbMnBi2 with Happ parallel to <001> crystal axis. Large peak observed at 0 Oe when z-axis was parallel to <110> crystal axis in sample 1 is no longer observed...... 120 Figure 67: Isothermal thermopower (Nernst and conventional) as a function of temperature for z- axis parallel to <110> and <001>. Large signal is only seen in transverse geometry with z-axis parallel to <110> crystal axis, the isothermal Nernst effect measurement in sample 1. Data taken from field sweep measurements is shown as closed diamonds, and data taken from steady-state measurements is shown as open triangles...... 122

Figure 68: Isothermal Nernst coefficient in YbMnBi2 for z-axis parallel to <110> (sample 1), shown as green diamonds, and z-axis parallel to <001> (sample 2), shown as blue circles. Both data sets are in the high-field limit...... 124

Figure 69: Nernst thermopower at 0 Oe in YbMnBi2, comparing isothermal sample 1 to adiabatic samples 3 and 4. Inset only shows samples 3 and 4 to emphasize their small temperature dependence. Sample 1 is green circles, sample 3 is blue diamonds, and sample 4 is red triangles...... 126

Figure 70: Nernst coefficient in YbMnBi2, comparing isothermal sample 1 to adiabatic samples 3 and 4. Only high-field limit is shown for the isothermal sample due to the sharp peak observed at low field in the magnetic field dependence of the Nernst thermopower...... 128 Figure 71: Magneto-thermopower as a function of temperature in the isothermal sample 1 and the adiabatic samples 3 and 4 of YbMnBi2 ...... 131

Figure 72: zT comparison of YbMnBi2 in a transverse geometry to a commercially available thermoelectric material, the state-of-the-art thermoelectric material, and single-crystal Fe, all in a longitudinal geometry ...... 133 Figure 73: Expected comparison of hysteresis in magnetic field dependence of magnetic moment and Nernst thermopower for Berry curvature-dominated transport. Nernst thermopower, serving as a probe of Berry curvature, shows no sign change depending on the magnetic field sweeping direction, as observed in the isothermal Nernst effect measurements. Magnetization is expected to show an opening of a hysteresis loop and thus a sign change for the remnant moment and coercive field depending on the magnetic field sweeping direction...... 135

xxi

Chapter 1. Introduction

1.1 Introduction to Thermoelectrics

Thermoelectric devices convert either a temperature gradient into a usable voltage output or an electric current into an output temperature gradient. As these devices are fully solid-state, thermoelectric devices offer advantages over other commonly used energy generation or refrigeration devices in that they utilize no moving parts, are robust, and have a high power density making them small and lightweight.1 Due to the increasingly negative impact of non- renewable energy sources on the environment, thermoelectric devices offer a potentially clean alternative in energy generation with waste-heat recovery and refrigeration for their niche applications.

1.1.1 Characterizing Thermoelectric Transport

Employing the principles of semiconductor physics, thermoelectric materials conventionally utilize n-type and p-type semiconductors connected electrically in series and thermally in parallel. Charge carriers accumulate on the cold side of the thermoelectric materials; a charge buildup then causes a repulsive electrostatic force. This induced charge movement causes a current, and the current causes a potential difference. This transport is shown in the schematic of Figure 1, where e- represents electrons in n-type materials and h+ represents holes in p-type materials.2 The left frame of Figure 1 shows this transport in energy

1 generation mode, in which a temperature gradient drives this transport, and the desired output is a usable voltage. The right frame of Figure 1 shows this transport in refrigeration mode, in which an electric current drives transport, with the desired output being a temperature gradient.

In both modes, the temperature gradient and electric flux are parallel to one another, deeming this geometry as longitudinal.

Thermoelectric transport is governed by the Seebeck effect, which states that a temperature difference across a material will induce a potential difference in terms of an electric voltage. This input temperature gradient, T, and measured temperature difference, T, are

2 proportional to the output electric field, E, and voltage, V, via the Seebeck coefficient or thermopower, :3

EV   (1). TT

In refrigeration mode, the Peltier effect describes the transport, stating that a current injected across a junction of dissimilar materials (n-type and p-type semiconductors) results in a heat flow at that junction. The Peltier coefficient, , is the material-dependent proportionality factor between the input electric current, I, and the output heat flow, Q:3

Q  (2). I

The Kelvin relation relates the thermopower to the Peltier coefficient in a given material at a constant temperature, T:3

T (3)

The efficiency of a given thermoelectric material is quantified in terms of zT, the dimensionless thermoelectric figure of merit. Because zT is dimensionless, a larger zT is desirable with ~0.3 becoming useful, ~1 characterizing what is commercially available, and >2 representing the most cutting edge, state-of-the-art materials. zT combines the thermodynamically reversible and desired component, , with the thermodynamically irreversible transport components of electrical conductivity,  (=1/ is the electrical resistivity), and thermal conductivity, . zT is defined at a given temperature, T, as follows:3

 2 zT T (4). 

The numerator of Equation 4 represents the power factor, PF, of a given material:

PF   2 (5).

The power factor characterizes the electrical power density for a given temperature gradient in a thermoelectric material.

1.1.2 Conventional Thermoelectric Materials

Unfortunately, the individual components of zT do not necessarily increase/decrease in an agreeable manner, i.e. thermal conductivity does not monotonically decrease as a function of increasing thermopower. A convenient way to observe the counteracting components of zT is to plot their tendencies as a function of n, the carrier concentration; this is shown in Figure 2. The top frame shows the trends of each individual component of zT as a function of n while the bottom frame shows the trend of zT as a function of n. The middle labels indicating material class (insulator, semiconductor, metal) demonstrate that the range of n for maximum zT falls in the semiconductor regime. Thus, the most commonly used and commercially available thermoelectric materials are semiconductors.

Figure 2: Thermopower, thermal conductivity, and electrical conductivity as functions of carrier concentration in relation to zT as a function of carrier concentration

1.1.3 Thermoelectric Devices

Longitudinal thermoelectric modules are comprised of multiple pairs of n-type and p-type thermoelectric materials connected electrically in series and thermally in parallel, as shown in

Figure 34 for a module designed for energy generation. Heat flow runs parallel to the length of each of the thermoelectric legs, with heat being absorbed on one side and rejected on the

5 opposite side. Current wires are connected across all thermoelectric legs and can be attached to a load to obtain a usable power output.

Figure 3: Schematic of a longitudinal thermoelectric module prepared for use in energy generation mode depicting n-type and p-type semiconductors connected electrically in series and thermally in parallel. Figure taken from G. J. Snyder and E. S. Toberer, Nature Materials 7 (2008).

Peltier modules, those designed for refrigeration and heat pumping, are similar to the schematic of Figure 3 but include a cascaded structure, much like a pyramid, in which modules are stacked on top of one another in stages with different total cross-sectional areas.5 A photograph of a

Peltier module is shown in Figure 4. This cascaded structure is utilized for refrigeration mode in thermoelectric transport because each stage can utilize different thermoelectric materials, where the materials in a given stage are specifically chosen to have better properties over their operational temperature interval. Overall, this increases the temperature drop (or rise) achieved across the entire module.

Figure 4: Photo of a longitudinal Peltier module showing the cascaded structure desired for refrigeration mode. Each level is considered a stage, and each stage has a different heat transfer area. Module manufactured by Marlow Industries, Inc.

1.2 Geometric Considerations in Thermoelectric Transport

1.2.1 Longitudinal vs. Transverse Geometries

Thermoelectric transport conventionally utilizes the longitudinal geometry, shown in

Figure 1 and described in Section 1.1.1. This geometry is considered longitudinal because the temperature gradient and voltage output are parallel to one another. Additionally, the 7 longitudinal geometry requires the use of thermoelectric couples, in which an n-type material and a p-type material make up a thermoelectric pair. Both polarity of materials, and thus both electrons and holes as dominant charge carriers, are necessary in the longitudinal geometry because the two electrodes for the output must be connected in an isothermal plane.6 Electrons in the n-type thermoelectric material move from hot to cold, carrying an electric current parallel to the temperature gradient; holes in the p-type thermoelectric material carry a current antiparallel to the temperature gradient and thus return the current to the hot side. Here, as shown in the left frame of Figure 5, the electrode is connected across the hot side of the n-type and p-type material to obtain a voltage output. Conversely, when the temperature gradient and measured voltage output are perpendicular to one another, as shown in the right frame of Figure

5, only one polarity of charge carrier is necessary.6,7 In this transverse geometry, the two electrodes can be applied in an isothermal plane because the desired resultant electric current runs perpendicular to the applied temperature gradient. Thus, no opposite polarity charge carrier is needed to return the current back to an isothermal plane, making transverse thermoelectric devices significantly simpler than longitudinal ones. Furthermore, output scales extrinsically with material properties in the transverse geometry, implying that a larger voltage can be generated by increasing the dimension parallel to the produced electric field, and a larger temperature gradient can be produced by increasing the dimension parallel to the applied temperature gradient.7 In contrast, in a longitudinal geometry, output scales intrinsically with the material properties – more thermoelectric pairs must be connected electrically in series and thermally in parallel, increasing the complexity and cost of the device to increase the desired voltage output.

Figure 5: Comparison of longitudinal (left) to transverse (right) geometries for thermoelectric transport, in which the longitudinal geometry requires both n-type and p-type materials while the transverse geometry typically requires an externally applied magnetic field. Applied temperature gradient and measured voltage are parallel for the longitudinal geometry and perpendicular for the transverse geometry.

1.2.2 Nernst Effect

While the longitudinal geometry utilizes the Seebeck effect, the transverse geometry most commonly utilizes the Nernst effect. In order to drive charge carrier movement perpendicular to an applied temperature gradient, an extra force is necessary in the transverse geometry. In the

Nernst geometry, this is the Lorentz force and is found in either a materials’ internal magnetization (such as that in a ferromagnet) or in an externally applied magnetic field, Happ||z, that is mutually perpendicular to both the applied temperature gradient and the measured electric field. The applied temperature gradient and magnetic field, which are perpendicular vectors, give a cross product as the Lorentz force in the third perpendicular direction, which is the direction of the output electric field. The Nernst thermopower can be quantified similarly to the conventional thermopower of the Seebeck effect:8

Vy EL  yy (6). xyz T xT x Happ|| z Lx Happ|| z

Here, the subscripts xyz are utilized to indicate the direction of the applied temperature gradient

(x), the direction of the measured electric field (y), and the direction of the applied magnetic field

(z); L indicates the sample length in the direction of its corresponding subscript. The Nernst effect can further be characterized in terms of a Nernst coefficient, Nxyz, which is the derivative of the Nernst thermopower as a function of the applied magnetic field:8

 xyH app|| z  Nxyz  (7). Happ|| z

Nxyz can often hold different values in a zero-field limit as opposed to in a high-field limit, especially in magnetic materials. The low-field limit is called the anomalous Nernst effect while the high-field limit describes the ordinary Nernst effect in ferromagnets. These field regimes are often characteristic of different dominant transport mechanisms, which gives rise to different slopes in xyz as a function of field, and thus different values of Nxyz, between the field regimes.

The Nernst effect is an even function of charge carrier polarity, while the Seebeck effect is an odd function of charge carrier polarity.8 This means that in a given material, electrons and holes deflect in opposite directions in the Nernst effect and in the same direction in the Seebeck effect. Thus, the Nernst thermopower goes as the difference between the intrinsic thermopower of holes and electrons, while the conventional thermopower is the sum of the intrinsic thermopower of both holes and electrons. In most materials where one type of carrier dominates, the Nernst thermopower, xyz, is significantly smaller than the conventional magneto-

10 thermopower, xxz. This typical difference in order of magnitude is due to the two-step process occurring in the Nernst effect, where a thermoelectric effect is followed by charge carrier deflection due to the Lorentz force.8 Contrastingly, in the Seebeck effect, a thermoelectric effect converts a temperature gradient to a voltage output without the need for charge carrier deflection to obtain a signal. Yet in two carrier systems, such as semimetals, the Nernst thermopower is expected to vastly exceed that of the conventional thermopower. Because the Nernst effect is an even function of charger carrier polarity and the intrinsic thermopower of the individual charge carriers are oppositely signed, the total Nernst thermopower in a system is dominated by the sum of magnitudes of the individual intrinsic thermopowers.9

1.2.3 Transverse Devices

The advantages of a transverse geometry in comparison to a longitudinal geometry, discussed in Section 1.2.1, are relevant in device applications. Nernst generators again only require one polarity of material, shown in the right frame of Figure 5, and their output scales extrinsically with device size. Transverse thermoelectric cooling devices, called Ettingshausen devices,7 are also much simpler than their longitudinal counterparts. Figure 6 shows a schematic of an Ettingshausen cooler, which maintains the cascaded structure of Peltier coolers (Figure 4), but is significantly simplified by enforcing the cascade on one constant material, eliminating the need for interconnected stages.3 Nevertheless, the externally applied magnetic fields required in transverse devices to approach the efficiency of longitudinal devices can be large (order of 10 kOe), hindering the commercial viability of Nernst generators and Ettingshausen coolers.

Figure 6: Ettingshausen cooler, a refrigeration device utilizing a transverse geometry with a cascaded structure. Figure taken from D. K. C. MacDonald, Thermoelectricity: An introduction to the principles, Dover Publications, New York (1962).

1.3 Ferromagnetic Transition Metals

1.3.1 Metals as Thermoelectric Materials

As mentioned in Section 1.1.2, conventional thermoelectric materials are semiconductors.

Nevertheless, semiconductors do have limitations: they are brittle, made of uncommon elements

(e.g. Te, Co, Zr, Hf), typically expensive, do not operate well at high temperature (above 500oC), and are hard to form and connect. Contrastingly, metals are strong, widely available, typically cheap, do operate well at high temperature (offering the potential for a higher Carnot efficiency), and are easily formed in net shapes. Yet metals are limited as thermoelectric materials due to a

10 low zT, with the maximum zT ~ 0.3 in CePd2Pt which is prohibitively expensive as a material itself. For more common metals and metallic alloys, such as those in Figure 7, zT rarely exceeds

0.1.11

Figure 7: zT as a function of temperature in common metallic thermoelectric materials, demonstrating the low zT of metals. Figure taken from K. Vandaele et al., Materials Today Physics 1 (2017).

Interestingly, though, the power factor (Equation 5) of the ferromagnetic transition metals Fe,

Co, and Ni does exceed that of the commercially available thermoelectric semiconductor Bi2Te3, as shown in Figure 8.11 This characteristic will be discussed further in Chapter 2.

Figure 8: Power factor as a function of temperature for ferromagnetic transition metals as compared to conventional thermoelectric materials, demonstrating that ferromagnetic metals can have a large power factor. Figure taken from K. Vandaele et al., Materials Today Physics 1 (2017).

Despite their low zT, metals are a current source of study in the field of thermoelectrics due to their material advantages over conventional semiconductors. Previous thermoelectric studies in

Fe12 did reveal an unexpectedly large thermopower, attributing it to magnon drag, a phenomena which will be explained in the following section. Additionally, magnon drag has been theoretically predicted13 to be the dominant contributor to thermopower in ferromagnetic transition metals, and experimental evidence to support this claim is given in Chapter 2.

1.3.2 Magnons and Magnon Drag

Core electrons on the unfilled d-shells of transition metal elements in a magnetic material each have a spin, which is not frozen above 0 K.11 Given a finite amount of temperature, thermal fluctuations can perturb the spins, or magnetic moments, of these core electrons, causing them to organize into a magnon (spin wave) in the direction of the net magnetization.14 Much like a phonon, a magnon is a bosonic quasi-particle having characteristics of both a wave and a particle. Unlike in phonons, in which the wave is vibrational within a crystal lattice,15 magnons are precessions of the cone angle of the spins of neighboring atoms. This collective excitation is shown in Figure 9, where the arrows represent the spins, each rotating at a given cone angle.16

The bottom frame shows the precessions causing a collective wave, a magnon. Magnons themselves have a density of states, specific heat, and thermal conductivity.17

In thermoelectric transport, magnons themselves can drag electrons through a crystal lattice, increasing thermoelectric transport in an advective process. This concept is called magnon drag. As shown in Figure 1018, magnons couple strongly to phonons, which are also

15 sensitive to temperature fluctuations. When a temperature gradient exerts a force on magnons, magnons form a heat flux and a spin flux. The magnon flux can then drag electrons through a crystal lattice.11 Ferromagnetic metals are expected to be particularly prone to magnon-drag effects due to their magnetic ordering in free electrons,11 and they will be the primary source of experimental study relating to magnon-drag thermopower in Chapter 2.

Figure 10: Magnon-drag effect demonstrating coupling between phonons and magnons to drag electrons through a ferromagnetic metal. Image credit to Renee Ripley.

1.4 Weyl Semimetals

1.4.1 Unique Characteristics of Weyl Semimetals

On a quantum-mechanical level, electrons in a crystal lattice are subject to a periodic electrostatic potential due to the periodicity inherent in the locations at which the atomic nuclei can sit within a crystal structure itself, shown in the top section of Figure 1119. High-energy, free

16 electrons move at a potential above that of the square wave representation shown in the top portion of the figure, ultimately forming the conduction band (CB) in energy with cation, or A+, states. Bound electrons sit in the wells, or sections of lowest potential, of the square wave potential forming the valence band (VB) in energy with anion, or B-, states. When these energy bands are depicted in momentum space, or k-space, they both become parabolic, as shown in the bottom portion of Figure 11. At the surfaces of real materials, this periodicity must terminate; this gives rise to extra energy states available to electrons at the surface, called surface states.20

Energy must be conserved during any transport process within a given material; for classical materials, energy is a function of momentum only. Momentum is not coupled to any other parameter in this conventional band structure picture, so surface states are not localized here. In

Weyl semimetals, these surface states are unique and distinct from this classical picture and will be discussed later in this section.

Figure 11: Conventional periodic crystal structure with cation, or A+, states in the conduction band and anion, or B-, states in the valence band. In k-space, periodic bands emerge and surface states occur when the periodicity ends at the edge of a real material. Figure taken from J. P. Heremans, US/Japan Workshop on Nanoscale Thermal Transport, Tokyo (2017).

The energy of the extrema of the conduction band and valence band as a function of their k-vectors, as shown in the bottom portion of Figure 11, maintain an energy gap separating the parabolic bands. The amount of energy in this gap can determine the classification of the material, with Figure 12 labeling material types as the energy gap closes. A band insulator maintains the largest energy gap; when the gap between the bands becomes on the order of 1 eV or smaller, the material is called a semiconductor. Once these bands begin to overlap, having states in both the conduction band and the valence band present at the same energy, a semimetal emerges. A specific case of a semimetal is the Dirac or Weyl semimetal, where the parabolic

18 bands cross with a linear dispersion near the crossing points. A metal, with the Fermi energy deep in the conduction band, is shown for the sake of completion in the far right of Figure 12.

Figure 12: Energy band structure evolution as a function of k-vectors as the gap closes between the conduction band (CB) and the valence band (VB). Band insulators can be described as semiconductors when the band gap is on the order of 1 eV or smaller. Dirac and Weyl semimetals emerge when the bands overlap at crossing points. Dashed orange lines represent the Fermi level (specifically for an intrinsic semiconductor in the left frame).

Spin-orbit coupling can drive the closing of this energy gap and ultimately leads to the band crossing present in the semimetal picture of Figure 12. In the presence of strong spin-orbit coupling, the conduction band and valence band invert, with certain states of the valence band sitting at a higher energy than those of the conduction band.21 Spin-orbit coupling is a relativistic effect in which the orbital angular momentum interacts with the spin angular momentum of the electrons.21 Relativistic effects including spin-orbit coupling add a correction to the atomic energy levels, which can lead to the cation band moving below the energy of the

19 anion band when these corrections are large enough.21 This is depicted in Figure 1319, similar in nature to Figure 11 but with the addition of spin-orbit coupling restricting the energy bands beyond limitations imposed by the periodicity of the crystal lattice itself. In Figure 13, B- states encompass the conduction band, which is at a higher energy than the A+ states in the valence band. The band inversion due to spin-orbit coupling implies that energy is no longer simply a function of momentum alone but also now a function of another parameter, with the Hamiltonian given as:22

Hv Fk σ (8) where vF is the Fermi velocity, k is the momentum, and σ is the other quantity that must be conserved and could be spin, orbital, or sublattice. For the specific case of a topological semimetal, σ is the orbital due to the crossing of the valence band and the conduction band. The linkage of momentum to another parameter induces topological protection since σ in addition to momentum must be conserved everywhere in the material, even at the edges, in order to obey conservation of energy. At the surface where the periodicity terminates, this leads to a unique restriction of energy levels available for electrons to occupy. Energy bands take the form of linear Dirac bands at the surface for the case of the topological insulator shown in Figure 13.

One Dirac band depicts the states on the bottom surface and the other Dirac band depicts the states on the top surface, with each band giving an oppositely signed velocity for the electrons at that corresponding surface.

Figure 13: Periodic structure of a solid combined with strong spin-orbit coupling leads to band inversion. Here, the B- states make up the conduction band (CB) while the A+ states make up the valence band (VB). This gives rise to linearly dispersing Dirac bands as surface states, where the velocity of electrons on each surface are oppositely signed, as shown by the surface labels on the Dirac bands. Figure taken from J. P. Heremans, US/Japan Workshop on Nanoscale Thermal Transport, Tokyo (2017).

The starting point for topological materials, including Weyl semimetals, is band inversion combined with spin-orbit coupling (SOC), with a 3D representation of the energy band inversion shown in the top left part of Figure 14.23 In a topological insulator (TI), Figure 14(a), the band inversion occurs near a high-symmetry point21, inducing a full energy gap23. Metallic states emerge on the surface in the form of a closed-loop Fermi surface.23,24 When the band inversion occurs away from a high-symmetry point, shown in Figure 14(b), the linearly dispersing crossing points have no energy gap although the bulk bands are gapped in 3D momentum space.23 Here, a Dirac semimetal (DSM) or a Weyl semimetal (WSM) emerges.23

Figure 14: Band inversion plus spin-orbit coupling (SOC) giving rise to topological insulators (TI), Dirac semimetals (DSM), and Weyl semimetals (WSM). a. Topological insulators emerge when the energy band crossing points are gapped. Surface states are closed loops. b. Dirac and Weyl semimetals emerge when no gap exists between the energy band crossing points. Surface states are open arcs. When inversion or time-reversal symmetry is broken, a Dirac point splits into a pair of Weyl points. c. Type I Weyl semimetal, where the Fermi surface is zero at the energy band crossing points. d. Type II Weyl semimetal, where the Dirac cone is tilted and the energy band crossing point is the separation between the electron and hole pockets in the Fermi surface. Figure taken from B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. (2017).

In a Dirac semimetal, the crossing points are doubly degenerate; the degeneracy can be lifted, splitting a Dirac point into two Weyl points (far right of Figure 14(b)) by breaking either 22 crystalline inversion symmetry or time-reversal symmetry.23,25 Figure 1522 schematically depicts both time-reversal and inversion symmetry, where the colored arrows represent the quantity σ in k-space. Weyl points always come in opposite chirality pairs, with a minimum of two Weyl points (one pair) when time-reversal symmetry is broken and a minimum of four Weyl points

(two pairs) when inversion symmetry is broken.26,27

Figure 15: Time-reversal symmetry (left frame), in which a property flips sign when moved in momentum space from a positive k-vector to a negative k-vector. Inversion symmetry (right frame), in which a property maintains the same sign whether it is at a positive or negative k- vector in momentum space. Figure taken from N. Trivedi, APS March Meeting 2018, Los Angeles (2018).

One of the most prominent characteristics of a Weyl semimetal is its exotic topological states, where the bulk bands give surface states in the form of open Fermi arcs.23,24 Two types of Weyl semimetals exist: Type I (Figure 14(c)) in which the Fermi energy is close to the Weyl points and the density of states is zero at the Weyl points, and Type II (Figure 14(d)) in which the Weyl cones are tilted in momentum space giving a finite density of states at the energy of the Weyl points.23 Table 122 lists Weyl semimetals by type and which symmetry (inversion or time- reversal) is broken. Chapter 3 will discuss the Type I, inversion symmetry-breaking Weyl semimetal NbP, and Chapter 4 will discuss the Type II, time-reversal symmetry-breaking Weyl semimetal YbMnBi2.

Table 1: Weyl semimetals and their classifications by type and broken symmetry. Table taken from N. Trivedi, APS March Meeting 2018, Los Angeles (2018).

Figure 16 gives a closer look at the band-inverted portion of the band structure in a Weyl semimetal, here in 2D as opposed to the 3D depiction in Figure 14. The energy bands are linear,

Dirac bands near the crossing points, indicated by the orange dotted lines. The band crossing points, or Weyl points, serve as magnetic monopoles in momentum space28 that always come in opposite chirality pairs26,27. At these Weyl points, the conduction and valence bands are entwined, giving a twist in the wave function at the crossing points. This can be further described in terms of Berry curvature, which acts like a magnetic field in momentum space.23

Berry curvature has a singularity at each Weyl point, causing the Weyl points to act like magnetic monopoles.28 The blue plus or minus sign at the Weyl points of Figure 16 indicates the sign of the fixed chirality at the Weyl points, where positive chirality indicates a source of Berry curvature and negative chirality indicates a sink. The sign of the chirality also indicates the sign of the Hamiltonian in Equation 8. This chirality emerges from each Weyl point in a pair having reversed crossing of A+ and B- states and an inherent difference in the wave functions present in each band, indicated by the colored arrows. If an electron is excited in k-space traveling from left to right in Figure 16, the electron will first see A+ states followed by B- states at the left Weyl point but will first see B- states followed by A+ states at the right Weyl point. In the depiction of

Figure 16, the right Weyl point is the source and the left Weyl point the sink of Berry curvature,

24 as shown by the colored arrows and the corresponding chirality signs. Here, there is no net

Berry curvature between the two Weyl points because the magnitude of Berry curvature, and thus the chirality, at each Weyl point is the same due to the symmetry of the inverted bands.

Figure 16: Weyl semimetal band structure near Weyl points. Orange dotted lines indicate linear, Dirac bands near the Weyl points. Colored arrows depict the orbital character of the wave function in the conduction or valence band respectively, giving rise to a source and sink of Berry curvature indicated in blue.

Weyl semimetals are predicted to have unique transport properties due to their multiple pairs of Dirac bands and Weyl points in addition to their chirality. The coupling of the orbital character of the wave function to the momentum, both of which must be conserved, gives rise to exotic, topologically protected surface states in the form of Fermi arcs in surface Brillouin zones that terminate at Berry monopole (Weyl point) projections.24,29 These projections are shown on

25 the surfaces of the k-space depiction in Figure 17, which is simplified to one pair of Weyl points and their corresponding Dirac cones (linear Dirac bands rotated in three dimensions) in the bulk.

The Fermi arcs represent surface states restricted to precise values of k-vectors, while the Dirac cones are bulk states due to Dirac dispersions characterized by velocity.29 Energy is defined as zero at the Weyl points and also at their projections onto the surfaces, restricting surface states to the Fermi arcs.29 In the presence of an externally applied magnetic field parallel to the length of the Dirac cones in Figure 17, electrons undergo a force which pushes them into a conveyor belt type motion:30 an electron appears on the surface of a Weyl semimetal at a projection of a Weyl point, traverses the Fermi arc to the projection of the other Weyl point, conducts through the bulk, traverses the Fermi arc on the opposite surface, conducts through the bulk, and appears back where it started at the original Weyl point’s projection on the original surface. Both the currents through the two Weyl points in the bulk and the currents between the arcs on the upper and lower surfaces are exactly matched, thus no net charge flows. In the absence of inelastic scattering, the effect also consumes no energy.

1.4.2 Experimental Discovery of and Emerging Interest in Weyl Semimetals

Semimetals have long been of interest to researchers due to their unique two-carrier characteristics and band structure, in which both the conduction and valence band edges lie near the Fermi energy, usually crossing one another.9 Applications of two-carrier models to thermal, electrical, and magnetic transport in these materials is highly compelling, specifically in the

Nernst effect.9 Topological materials have also recently become a field of major interest with the material class of Weyl semimetals combining both topological and semi-metallic materials

27 through bulk transport with surface states. Weyl semimetals were first predicted in 1929 by H.

Z. Weyl, giving chiral, massless fermions (Weyl fermions) as one solution to the Dirac equation.31 These Weyl fermions were predicted to exist in wide-ranging candidate materials.29,32,33,34,35 Nevertheless, the existence of Weyl fermions was not discovered until recently. Angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy confirmed their existence experimentally in transition metal monopnictides36,37,38,39,40,41,42,43 and dichalcogenides44,45. Weyl Semimetals can be characterized by measuring the Fermi arc surface states, a huge longitudinal magnetoresistance, and ultrahigh mobility within the same material.46

Boller and Parthe discovered a class of semi-metallic transition metal monopnictides which includes NbP, TaP, NbAs, and TaAs.47 These semimetals break crystalline inversion symmetry but maintain time-reversal symmetry and were among the first class of materials to be experimentally proven as Weyl semimetals. In Chapter 3, thermomagnetic transport is explored in NbP, which has been proven to have an unsaturated, large magnetoresistance and ultrahigh mobility, although it does not have a net Berry curvature.46,48 NbP is unique in its band structure in that it has a hole pocket contributed from quadratic bands and an electron pocked contributed from its linear Weyl bands.46 The Nernst effect is explicitly explored and of major interest due to the semimetallic two-carrier effects, which should give a Nernst signal significantly larger than the thermopower, proving useful for engineering applications for energy generation since the Nernst effect occurs in a transverse geometry.

Weyl semimetals that break time-reversal symmetry potentially offer further unique and interesting transport properties, especially when they contain a net Berry curvature.23,24 When the inverted bands of a Weyl semimetal are no longer symmetric in k-space near the Weyl point

28 but are instead tilted, as shown in Figure 18, a difference in amount of Berry curvature at each

Weyl points exists, giving rise to a net Berry curvature. In this case, an electron traveling in k- space from the left to the right of Figure 18 would first see some amount of A+ then B; once it reaches the other Weyl point, it would see a different amount of B- then A+.

Figure 18: Weyl semimetal band structure with tilted bands breaking time-reversal symmetry, giving rise to a net Berry curvature due to a different magnitude of chirality at each Weyl point

Borisenko et al. experimentally confirmed the existence of a time-reversal symmetry-

49 breaking Weyl semimetal using angle-resolved photoemission spectroscopy in YbMnBi2. In the case of YbMnBi2, a canted antiferromagnetic structure fit the observed band structure and was proven to be the cause of the breaking of time-reversal symmetry and thus the net Berry curvature.49 In comparison to a traditional antiferromagnet, which has its spins aligned

29 antiparallel and no net effective magnetization, a canted antiferromagnet has a small tilt on each of its spins in the same direction yielding a small effective magnetization in the direction of the canting; this is shown in Figure 19.

Figure 19: Comparison of traditional to canted antiferromagnetic structures, showing a small effective magnetization when canting is present

When time-reversal symmetry is obeyed, such as in the classical semimetals like Bi and in transition metal monopnictide Weyl semimetals like NbP, the large Nernst effect is definitively non-existent when no applied magnetic field is offering a source for a Lorentz force, i.e. the Nernst thermopower is 0 V K-1 at 0 Oe for all temperatures in non-magnetic materials.

When time-reversal symmetry is broken and a net Berry curvature exists, though, a Nernst effect is predicted to exist even in the absence of an applied magnetic field.50,51 In Chapter 4, this is precisely what is explored – the Nernst effect in the time-reversal symmetry-breaking Weyl semimetal YbMnBi2 and its potential to be large and finite in the absence of an externally applied magnetic field. The potential for engineering applications here is great as the Nernst effect utilizes the device-advantageous transverse geometry, Weyl semimetals are predicted to

30 have a large Nernst thermopower, and the Berry curvature eliminates the need for a large, externally applied magnetic field.

Chapter 2. Magnon-Drag Thermopower and Nernst Effect in Ferromagnetic Transition Metals

2.1 Motivation

Multicomponent fluids and gases exist abundantly in nature at all scales, with these interactions often giving rise to new and interesting physics. A recent example involving a two- component system of interacting fluids is that of magnons with electrons at an interface between a magnetic insulator and a normal metal, such as that in the spin-Seebeck effect,6 spin-Hall magnetoresistance,52 and the attenuation of magnetization by electric current through the normal metal. Yet magnons also exist in magnetic metals, interacting with electrons throughout the entire bulk, eliminating the need for an interface. Through magnon-drag thermopower, a contribution to thermopower involving a magnonic heat flux dragging electronic charge carriers, one can measure this multicomponent effect.

In a study on ferromagnetic metals, Blatt et al.12 suggested magnon drag as a contributing mechanism to the unexpectedly large thermopower of Fe. Magnon-drag was also suggested as the dominant transport mechanism in the field-dependence of the thermopower of a Permalloy thermopile,53 but no proof or quantitative theory was offered in either study. Grannemann and

Berger54 attribute an 8% variation in the magnetic field dependence of the Peltier coefficient of

13 Ni66Cu34 at 4 K to magnon-drag. Lucassen et al. offer a theory for a contribution to magnon drag based on spin-motive forces, in which a thermal flux of magnons pumps an electronic spin

32 current. The electronic spin current results in a charge current voltage due to the spin- polarization of the charge carrier.

Through this work and other recent work11,55 , models are presented to offer guiding principles for the optimization of metallic thermoelectric alloys. Magnon drag, as a dominant contributor to the thermopower of ferromagnetic transition metals, offers a potential pathway to increase the thermopower and thus zT of metals, which provide many advantages over semiconductors in engineering applications. The capability of measuring magnon drag on a bulk material eliminates the conversion losses at an interface in a ferromagnet-normal metal bilayer present in the spin-Seebeck effect.6 Furthermore, the general relevance of this study is the fact that the two approaches to magnon drag theory are very different in nature – one relying on spin- conserving scattering and the other requiring spin-flip scattering and/or spin-orbit coupling. This suggests that other drag phenomena are subject to the latter process, contributions to which have not yet been considered and might be particularly relevant to transverse effects like the Nernst effect.

2.2 Measurement Technique

Numerous authors have measured and reported the thermopower of Fe12,56 over the temperature range included here, but these measurements were also repeated in this work to extend the study to the thermomagnetic tensor, including transverse effects.

Figure 20: Geometric configurations for transition metal measurements with crystal axes labeled for the single crystal of Fe

Experimental data for the thermomagnetic tensor elements of single-crystal Fe are presented using the configurations shown in Figure 20. The components of the thermopower tensor in a magnetic field are expressed as ABC, where A designates the direction of the applied heat flux jq||A, B designates the direction of the measured electric field EB, and C designates the direction of the applied magnetic field H||C. The zero magnetic field thermopower is denoted simply as . The geometries discussed extensively here include the thermopower or Seebeck effect and the Nernst effect. The thermopower in a longitudinal magnetic field is xxx, in a transverse field is xxz, the Nernst effect is xyz, and the planar Nernst effects (or anisotropic magneto-Seebeck effects) are xyx and xyy. Figure 20 gives the relationship between the crystallographic axes of the Fe single crystal and these directions.

All Fe measurements were completed on a 99.994% pure single crystal from Princeton

Scientific, measuring 7.13 mm x 5.05 mm x 1.07 mm. Two similarly-sized 99.9% pure 34 polycrystalline samples of Co from Alfa Aesar were used for the bulk Co measurements

(referred to as the ingot). A porous bulk polycrystalline sample of Co (referred to as the sintered sample) was prepared using 99.998% pure, -22 mesh particle size powder from Alfa Aesar. The powder was compacted via spark plasma sintering at 250oC for 30 minutes under 50 MPa of uniaxial pressure. The resulting pellet was estimated to be ~50% dense.

The Thermal Transport Option (TTO) on a 70 kOe and 90 kOe Quantum Design Physical

Property Measurement System (PPMS) with customized controls programmed in LabVIEW was used for materials characterization between 1.8 K and 400 K. Copper heat sinks and gold-plated copper leads of width 0.65 mm were used, attached to the samples using silver epoxy. Gold- plated copper assemblies purchased from Quantum Design containing calibrated CernoxTM thermometers and voltage measurement wires were clamped to the leads. A resistive heater assembly was clamped to a gold-plated copper heat spreader, also attached to the sample using silver epoxy. Temperature-dependent steady-state measurements of the thermomagnetic transport tensor, thermal conductivity, and resistivity were taken in fields ranging from -90 kOe to 90 kOe. Thermopower and resistivity data between 400 K and 1000 K for the Co ingot were taken using a Linseis LSR-3 using type S thermocouples and helium as a purge gas.

The measurement of the Nernst effect (xyz) in Fe was conducted with the heat flux along

<100>, the electric field along <010>, and the external magnetic field along <001>. Data for the

Nernst effect and planar Nernst effects was taken at discrete temperatures between 1.8 K and 400

K with magnetic fields sweeping in both directions at multiple magnetic field ramp rates.

Figure 21 shows the temperature dependence of the electrical resistivity, , for the same samples. Figure 22 shows the temperature dependence of the thermal conductivity, , for the same samples. These values are used in a model for the Nernst coefficient later in this chapter.

Figure 21: Electrical resistivity temperature dependence of transition metal samples. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

Figure 22: Thermal conductivity temperature dependence of transition metal samples. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

2.3 Magnon-Drag Thermopower

2.3.1 Experimental Results for the Thermopower of Fe, Co, and Ni

The temperature dependence of the thermopower of single-crystal Fe is shown in Figure

23, Co in Figure 24, and Ni in Figure 25. The insets represent the data on linear scales while the main frames utilize a logarithmic scale (negative thermopower for Co and Ni since they have negative thermopowers). For Fe, red triangles represent a 95% dense sintered polycrystal and black circles represent a single crystal. For Co and Ni, red triangles represent polycrystalline ingots and blue squares represent 50% porous samples. The data for the Ni ingot are taken from literature.57 The dashed black lines give the magnon-drag thermopower as derived from theory;

37 the full black lines are the sum of the magnon-drag and diffusion thermopowers. This data will be discussed extensively in comparison to theory later in this chapter.

Figure 23: Fe thermopower temperature dependence of single-crystal sample (main frame). Inset shows single-crystal sample and polycrystal. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

Figure 24: Co thermopower temperature dependence. Main frame shows negative thermopower on a logarithmic scale while inset shows actual thermopower on a linear scale. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

Figure 25: Ni thermopower temperature dependence. Main frame shows negative thermopower on a logarithmic scale while inset shows actual thermopower on a linear scale. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

The thermopower of Fe is in close agreement with previously published work.12,56 The thermopower of the Co ingot agrees with existing data above 150 K58 but shows a sign reversal near 100 K with a well-defined maximum at 11-14 K. The porous Co sample shows a negative thermopower closely following a T3/2 law up to 400 K with no pronounced extrema. Because the maximum seen in the Co ingot is not seen in the porous Co sample, the positive peak near 11-14

K is attributed to phonon drag. The Umklapp-limited phonon mean-free path in elemental Co is expected to be longer than the grains in the porous sample at this temperature. Thus, the phonon mean-free path is limited by boundary scattering in the porous sample, suppressing phonon drag.

This reasoning is used with Ni as well since the thermopower of the Ni ingot57 has a feature near

20 K that is not present in the porous sample.

2.3.2 Hydrodynamic Theorya

Magnons and electrons are modeled as two interpenetrating fluids in the hydrodynamic theory.54 Electrons and magnons are described by a single parabolic band, assuming Galilean invariance. Furthermore, Umklapp and magnon non-conserving processes are neglected.59

Thus, the sign of the thermopower is determined only by the sign of the charge carriers’ effective charge e, therefore their effective masses determine the sign of the thermopower. For carriers with a positive effective mass (conduction band electrons), e < 0, while e > 0 for charge carriers with a negative effective mass (valence band holes).

Electrons make up the first fluid, and magnons make up the other fluid. The magnonic

2 Cm thermopower is m  , where Cm is the magnon specific heat capacity per unit volume. 3 nem

This is derived by considering magnons as a free, ideal gas with a parabolic dispersion relation

2 and taking the gradient of the ideal gas relation P  U between the pressure P and internal 3 energy density U (in units of energy per volume) in the presence of a temperature gradient.59

The time scales τme and τem parametrize the magnon-electron collision rate.

a Theory presented in this section has been majorly developed by Joseph P. Heremans, Sarah J. Watzman, Yaroslav Tserkovnyak, and Rembert Duine. 41

Under steady-state conditions and for zero electric current, the electric field required to counteract the thermal gradient is determined by a balance of forces. The magnon-drag thermopower then becomes:59

|E | 2C 1  m (9) md |T | 3 n e  e 1 em  m with m defined as the transport mean free time for magnons and ne defined as the number

60 density of electrons. The electronic diffusion thermopower d , is added and given by:

k 2T   B d 3eE F (10).

Here, EF is the Fermi energy. The total electron thermopower, including both the diffusive and magnon-drag contributions, but neglecting electron-phonon drag, is:

   md d (11).

1 m is expected to scale with temperature as m  T because the density of states varies with energy as  .59 At higher temperatures, but low enough to ignore spin-conserving magnon- phonon interactions, scattering is likely to be dominated by spin non-conserving processes

61 1 59 parameterized by the Gilbert damping parameter GD so that m GDT . The electron-

12 1 magnon scattering frequency is expected to scale with temperature as  em  T . Thus,  em

1 1  em decreases with temperature with a higher power of T than  m , and the factor 1 should  m vanish as temperature approaches zero.59

In the limit parameterized by Gilbert damping, the attenuation of the magnon-drag thermopower is expected to have a linear dependence on T. Conversely, in the regime where τm

1 1 is dominated by magnon-phonon scattering,  m would vanish faster than  em due to the rapidly

1  shrinking phase space associated with the linearly-dispersing phonons, and the factor 1 em  m would approach unity.59 This corresponds to the clean case where spin-conserving magnon- phonon scattering of momentum is faster than spin non-conserving processes parameterized by the Gilbert damping, and this is likely to be the case for high-purity elemental metals.59 This is the case which will be explored further in this study.

2.3.3 Theory Based on Spin-Motive Forcesb

In addition to the hydrodynamic contribution, a contribution to magnon-drag thermopower ultimately stems from spin-orbit interactions.13 This contribution is parameterized by a dimensionless material parameter  typically of the order of 0.01-0.1. It arises from the electric current pumped by the dynamic magnetization associated with a magnon heat flux (via the aforementioned spin-motive forces62,63). This electric current density can be combined with   Fourier’s law for magnons, jQ.m   mT , where m is the magnon thermal conductivity. If diffusive magnon transport and a boundary condition of an electrically open circuit in the sample is assumed, the following is obtained:59

b Theory presented in this section has been majorly developed by Rembert A. Duine and Yaroslav Tserkovnyak. 43

   '  p m (12) md s 2e sD where ps is the spin polarization of the electric current, s is the saturation spin density and approximately a-3 (in units of ħ), and D is the spin stiffness. In the simplest microscopic models,64, 65, 66 the sign of this thermopower also depends on the sign of the effective mass.

Based on Landau-Lifschitz-Gilbert phenomenology, Flebus et al.67 pointed out a Berry-phase correction to the above result which amounts to replacing  with *= - 3GD. Therefore, the effective  entering the expressions for the magnon-drag thermopower should be understood as

*. This factor can also affect the sign of the magnon-drag Seebeck coefficient (depending on

68 the ratio of /GD which can be estimated to be of the order of 1 to 10), irrespective of the effective mass considerations.59

To compare Equation 9 and Equation 12, m and Cm are estimated. Assuming that the magnon dispersion is quadratic (i.e. at sufficiently low magnon energies and T < Tc),

3/ 2  T    -3 Cm ~ kB s  , where s ~ a (in units of ħ) is the saturation spin density, T is the temperature,  Tc  kB is the Boltzmann constant, and Tc is the Curie temperature of the ferromagnet, the hydrodynamic formula for magnon-drag thermopower can be rewritten:59

3/2 kB sT 1 md   e n T  (13). ec1 em  m

To estimate the thermopower due to spin-motive forces, the following is utilized:

 T  l T 2   2/ 3 at TT* due to magnon diffusion.69  m ~ kBT s m~ k B T s GD Tc   Tc

2/ 3 59 Using the latter expression and kBTc ~ s D , the following is obtained:

3/2 '  ps kB T md ~  (14). GDeT c

The two magnon-drag contributions to thermopower were obtained using different microscopic physics tools. The hydrodynamic contribution is nonrelativistic and the contribution due to spin-motive forces is based on spin-orbit interactions that are intrinsically relativistic and non-hydrodynamic as they do not conserve magnons. Remarkably, the contributions estimated

s  p in Equations 9 and 12 yield comparable values if ~ 1 and s ~1, which are certainly ne GD reasonable values for pure elemental transition metals, and the last, scattering time-dependent factor in Equation 9 is omitted. However, there are various transport regimes in which the hydrodynamic approach and the spin-motive force approach to magnon-drag thermopowers are clearly distinct; then the spin-motive force approach is the more rigorous one.

2.3.4 Comparison of Experiment to Theory

The data on the thermopower of Fe, Co, and Ni are compared to Equations 9-11 (md dashed line and  full line in Figure 23) in the limit where electron-magnon scattering is assumed to dominate all magnon scattering, allowing the scattering time-dependent prefactor

70 (em < m) to be ignored. The following numerical values are used. Cm is derived from the experimental magnon dispersion relation for Fe,71 Co,72 and Ni73. Below energies of about 4 meV, the magnon dispersions are approximately quadratic (D  2.7  10-22 eV-m2 for Fe, 4.3 

10-22 eV-m2 for Co, and 5.9  10-22 eV-m2 for Ni), which leads to Equation 13. At higher energies (the relevant case for Fe), Cm is calculated from the polynomial fit to the experimental dispersion.71 The total charge carrier concentrations are74 1.7  1023 cm-3 for Fe, 8.9  1022 cm-3 for Co, and 9.2  1022 cm-3 for Ni. Only s- and p-electrons are assumed to contribute to

60 75 transport ; their concentrations are derived from the density of states at the Fermi energy: ne 

21 -3 21 -3 21 -3 nsp = 2.36  10 cm (Fe), 8.1  10 cm (Co), and 3  10 cm (Ni). The sign of the thermopower is derived from the slope of the s- and p-bands’ density of states at EF:  is positive for Fe and negative for Co and Ni. Equation 10 then gives the dashed line representing md in

Figure 23. The diffusion thermopower in Equation 10 can be estimated roughly using the Fermi

75 energy (EF =0.48 eV for Fe, 0.76 eV for Co, 1.9 eV for Ni) for s- and p-electron bands. The band structure involves several pockets of electrons with dominantly s-p and d-character so that these simplified band structure parameters are affected by uncertainties of about a factor of 2.

No adjustable parameter is used to fit the lines in Figure 23. The similarities in magnitude and temperature dependence of  observed at T < 80 K for Fe and T < 400 K for the porous sample

46 of Co are evidence that both models are reasonable. Above those temperatures, the data points

1  fall under the calculated line. This could be due to the increased effect of the factor 1 em ,  m

* to the contribution of the additional term  = - 3GD discussed previously, or to a breakdown of the approximation for the magnon-heat conductivity that was used to estimate the contribution due to spin-motive forces. In addition, the data on the Co ingot show a discontinuity at the face- centered cubic/hexagonal phase transition near 700 K, as reported previously.58 The data on the porous Ni sample do not agree as well with this simple model: the calculated values are about two times larger than the experimental data on the porous sample.

2.4 Thermomagnetic Effects

2.4.1 Spin-Mixing Theory for the Anomalous Nernst Effectc

Neither model for md presented previously accounts for the generation of a skew force.

Electric fields perpendicular to the direction of an applied temperature gradient in the presence of an applied magnetic field in the third perpendicular direction can arise from two other mechanisms. First, in ferromagnetic metals that also have strong spin-orbit interactions and a measureable spin-Hall coefficient, one expects a bulk Spin-Seebeck Effect-like contribution to the Nernst coefficient.76 No measurements of the spin-Hall angle of Fe have been published; an interpolation of the spin-Hall angle measurements in 3d elements77 as a function of their atomic number suggests that Fe has a small spin-Hall angle. Therefore, this contribution is neglected

c Theory presented in this section has been majorly developed by Joseph P. Heremans. 47 here. The second possible mechanism for a magnon-drag contribution to the Nernst coefficient arises from spin-mixing, which was suggested for the resistivity60 and thermopower.78 This model, inspired by a similar model for the phonon-drag contribution to the Nernst Effect,79 is presented here.

Consider two independent spin-up and spin-down conduction electron channels with carrier densities (n and n) at EF, partial electrical conductivities ( and ), mobilities ( and

), Hall coefficients (RH and RH), thermopowers ( and ), and Nernst coefficients (N and

N). Figure 26 depicts this two-channel model as an electrical circuit in the presence of an applied magnetic field and temperature gradient mutually perpendicular to the channel voltages.

Two internal voltages are induced due to the spin-up and spin-down channels which produce two

values of thermopower,  and  . Thus, a drift current subject to the Lorentz force exists.

Figure 26: Spin-mixing model for Nernst effect in transition metals depicted as electrical circuit with separate conduction channels for spin-up and spin-down electrons in the presence of a magnetic field

The total Nernst coefficient is then derived by writing the Onsager relation for each channel, adding the fluxes, and solving the proper boundary relations for the transport coefficients, as is done for multi-carrier transport in semiconductors: 9

NNNRNRRRBRR()    2 2  2      N   HHHHHH    2 2 22 2 ()       RRBHH    

(15)

2 with B=0Happlied in the field range where Fe’s magnetization is saturated. The terms in B are neglected because the mobility is low. The intrinsic Nernst coefficients can reasonably be

59 assumed as zero as well, NN0. Since    and   , the term in ( – ) in

Equation 15 becomes dominant:59

             (16). N  2 ()

Further assumptions include  and  are md and md (Equation 9), and that n  nsp and n

59  nsp. These are proportional to the density of states at the Fermi level, i.e. nsp  Dsp, which

75 are known. The partial conductivities for each channel, up or down (), are  nesp  

 with mobilities sp given as a function of scattering frequencies and effective sp  e msp

2 Cm masses. The partial thermopowers are md  , and Equation 16 can be expressed 3 nesp

n  in terms of the ratios sp sp . Thus, Equation 16 is further reduced to the rrn ,  nsp sp following:59

1 2 C 11rnn r r r m (17) N  23 3nsp e (1 r n r )

where nsp  nsp n sp , and the carrier mobility is derived from the sample’s resistivity . To estimate the mobility ratios, the effective masses are taken as proportional to the density of states

2/3 at the Fermi level to the 2/3 power, mspD sp  . Assuming that s-d scattering dominates, this mechanism is further assumed to be spin-selective, i.e. that the scattering frequency of electrons in the spin-up channel is proportional to the density of states of spin-up d-electron bands:

11 2/3 59  spD d  , so that spDD d  sp  . With the band parameters of Ref. [75], Equation 17 becomes:59

Cm md N 0.052 0.07 (18). nesp

0 3/2 In the low-temperature limit,   T , mdCT m , and Equation 18 predicts that at low temperature NT 3/2 .

2.4.2 Experimental Results for Nernst Thermopower and the Anomalous Nernst Effect in Fe

Prior to the data presented here, only values for the Nernst coefficient near room temperature are reported in the literature. No systematic data as a function of field and temperature have been published previously. Historical references are by Zahn,80 Hall,81

Butler,82 and Smith83.

59 Figure 27 shows the magnetic field dependence of the Nernst thermopower, xyz, in polycrystalline Fe at two representative temperatures. The externally applied magnetic field is applied in magnitudes of up to 90 kOe, sweeping the magnetic field in both directions (positive to negative and negative to positive). The low magnetic field, anomalous Nernst effect regime and the high magnetic field, ordinary Nernst effect regime are both indicated with labels, where the anomalous Nernst effect is clearly evident from -20 kOe to 20 kOe.

Figure 27: Nernst thermopower in polycrystalline Fe as a function of magnetic field. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

The magnetic field dependence of the Nernst thermopower of single-crystal Fe is shown in

Figure 28, with the magnetic field applied parallel to the <100> crystal axis. The field range shown here has a maximum magnitude of 10 kOe, and the ordinary Nernst effect falls outside this range. Hysteresis is visible in the inner loop here, likely due to magnetic domain realignments.

Figure 28: Nernst thermopower in single-crystal Fe as a function of magnetic field. The anomalous Nernst effect extends from about -20 kOe to 20 kOe with the ordinary Nernst effect extending outside this range. Hysteresis is visible and most likely due to domain realignment. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

 xyH app|| z  The Nernst coefficient, N  , as a function of temperature in single-crystal Happ|| z

Fe is shown in Figure 29. It is derived in the anomalous Nernst effect regime and follows the

T3/2 temperature dependence discussed previously, indicating that it is of magnonic origin. A semi-quantitative comparison of the data in Figure 29 with Equation18, using the experimental values for  (Figure 23) and  (Figure 21), gives the solid curve in Figure 29. This procedure again uses no adjustable parameters. The agreement with the data is reasonable up to about 200

K, a higher temperature than for the thermopower, which is expected since the experimental values of thermopower are used in Equation 18 to obtain the solid curve. Above this temperature, the experimental data continue to increase with temperature while Equation 18 53 saturates. The fit can be improved by adding a negative constant contribution to N of -50 nV K-1 T-1.

Figure 29: Nernst coefficient temperature dependence of single-crystal Fe. Closed circles represent experimental data and the curve represents the model calculation. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

2.4.3 Longitudinal and Transverse Magneto-Thermopower in Fe

Measurements of both the temperature dependence of the longitudinal (xxx) and transverse (xxz) magneto-thermopower were completed here. The values for xxx(|Ha,x|  70 kOe) do not deviate measurably from : no longitudinal magneto-thermopower effect is resolved above the error bar of the present measurements, which is limited by the noise floor of 50 nV to about 0.2% on relative measurements. Blatt et al.’s in-field data84 are internally inconsistent

54 with data presented here. Figure 3 in Ref. [84] shows no magnetic field dependence in xxx, consistent with these observations. Nevertheless, Figures 1 and 2 in Ref. [84] do show a difference between xxx(Ha,x = 2 kOe) and xxx(Ha,x = 0 kOe)= , which is not reproduced here.

The transverse magneto-thermopower xxz(Ha,z) is reported as relative values for the change of xxz(Ha,z) vis-à-visxxz(Ha,z=0 kOe) in Figure 30 as a function of Ha,z at various temperatures. The relative effect is a small increase in thermopower, which is not resolved

gH below 100 K. In principle, an applied magnetic field opens an energy gap B in the magnon spectrum of ferromagnets (here, μB is the Bohr magneton and g is the Landé factor, which is about 2 for Fe). In practice, this gap is too small at 70 kOe to have a resolvable effect on Cm and

md above about 10 K, given the accuracy of the measurements. Below 10 K, the thermopower is still dominated by electronic diffusion, and the magnitude of the magnon-drag contribution to the thermopower is too small to resolve its magnetic field dependence. Therefore, the most likely cause for the magneto-thermopower effect in xxz is not related to changes in magnon density but perhaps due to the spin-mixing effects, which were not taken into account during the calculations of the net thermopower.

2.4.4 Planar Nernst Effects in Fe

The planar Nernst effect geometries, reported here for single-crystal Fe, are that of the

xyx and xyy geometries. A non-zero planar Nernst thermopower xyx, exceeding the noise level of 50 nV, is shown in Figure 31 as a function of magnetic field. The signal is an even function of magnetic field and saturates around the field value where the magnetization of the sample saturates. The difference between the zero field value (set to be zero) and the saturation value is plotted as a function of temperature in Figure 32. This value increases rapidly with decreasing temperature below 50 K, but it is non-zero and nearly constant between 50 and 300 K. No signal is detected for xyy except for noise transients at what amounts to the coercive field of the sample in that geometry, such that, for all practical purpose, xyy  0 in these measurements. This is 56 consistent with the observations of Pu et al.85 The planar Nernst effect is associated with the switching of the magnetization of the sample.85 The magnetic field range over which a signal change is observed in xyx does correspond to the field range over which hysteresis is observed in

xyz (Figure 28) and is only a fraction of the extent of the anomalous Nernst effect field range.

This justifies attributing that feature in the anomalous Nernst effect to the switching of a domain in the sample.

Figure 31: Planar Nernst thermopower as a function of magnetic field in single-crystal Fe. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

Figure 32: Saturated planar Nernst thermopower as a function of temperature in single-crystal Fe. Figure taken from S. J. Watzman et al., Phys. Rev. B 94 (14), 144407 (2016).

2.5 Conclusion

This chapter has described both hydrodynamic and relativistic approaches to calculating the magnon-drag thermopower and a spin-mixing model for the magnon-drag Nernst coefficient at magnetic fields above the saturation magnetization. The thermopower theories have shown that, depending on which scattering processes limit the electronic and magnon transport, they can coincide at low temperatures. The theories explain the experimental thermopower of Fe and Co, which have two different polarities, semi-quantitatively without adjustable parameters. The results are less conclusive about the thermopower of Ni. The theories presented also have predictive power, potentially enabling the design of metallic thermoelectric alloys that might become competitive with semiconductor thermoelectrics. Specifically, alloys with a lower

58 concentration of s- and p-electrons than those of elemental Fe, Co, and Ni are expected to have a higher md and therefore thermoelectric figure of merit, zT. Note that such optimization does not require changing the overall concentration of electrons in a metal, which would be a daunting task, but involves the redistribution of free electrons between bands of s-p orbital character and bands of d-orbital character. Furthermore, the contribution due to spin-motive forces may be increased by increasing the ratio of  to the Gilbert damping.

Chapter 3. Large Nernst Effect in the Weyl Semimetal NbP Generated by Dirac Dispersion

3.1 Motivation

The Weyl nodes’ Berry curvature leads to a variety of electronic transport phenomena in

WSMs including negative longitudinal magnetoresistance86, a huge magnetoresistance, and ultrahigh mobility46. Preliminary theoretical studies predict that bulk Weyl nodes51,87 and surface Fermi arcs30 have signatures in magneto-thermal and thermomagnetic transport, beyond that of classical electrical transport in semimetals, where electrons and holes co-exist, as seen with nanostructuring in NbP88. The Seebeck and Hall effects measure the difference between charges carried by both types of carriers, whereas entropy transport and the Nernst effect measure their sum. Additionally, the low-field limit Nernst coefficient measures the Berry curvature directly on the Fermi surface50, contrasting with the Hall conductivity, which averages over the Fermi surface volume (not applicable to NbP, which has no net Berry curvature)89.

Experimental thermomagnetic data in Dirac and Weyl semimetals exists,89,90,91,92 including the magnetic-field dependent adiabatic Seebeck coefficient of NbP,93 but quantitative explanations for the data are lacking. The Nernst effect in Weyl semimetals remains unexplored theoretically and experimentally. This study shows that the symmetry between the electron-like and hole-like

Dirac band portions results in a characteristic temperature- and magnetic-field-dependent Nernst thermopower and a relation between the Nernst and Seebeck coefficients that are unique Dirac band signatures.

The NbP band structure consists of 24 Weyl points and several trivial pockets.46 In the low-temperature limit, the electron or hole charges residing in the trivial bands and/or in chemical defects in the samples, including vacancies or unintentional aliovalent impurities, determine the NbP chemical potential, 0. The temperature-dependent electrochemical (T) moves with increasing temperature, and the electrons/holes in the symmetric Dirac bands start to dominate transport. A theoretical model is used here to determine how sensitive the Nernst effect is to this movement of the electrochemical potential in temperature in conjunction with the space-charge neutrality condition.

The Nernst effect is a decreasing function of temperature in classical semimetals like

Bi94, except in the phonon-drag regime, which is excluded here93. In NbP, and likely other Weyl semimetals, the Nernst effect becomes a non-monotonic function of temperature, demonstrating a unique signature of Dirac band transport.

3.2 Measurement Technique

3.2.1 Isothermal Measurement Technique

A single-crystal sample of NbP, dimensions 2.25 x 1.29 x 0.52 mm, made using methods described in Ref. [46], was characterized in this work. The sample was mounted as shown in

Figure 33, placed on a silicon backing plate to which a temperature gradient along x is applied.

The silicon plate that underlies the sample’s whole length and width is designed to short-circuit the sample thermally so that the temperature gradient established in the silicon also is imposed

95 on the NbP, with yT  0. This is the paramount characteristic of the isothermal mount , which is shown schematically in Figure 34. This structure in turn is mounted in the Thermal Transport

Option (TTO) on a Quantum Design Physical Property Measurement System (PPMS) modified for isothermal Nernst and Seebeck measurements. Figure 35 shows the isothermal sample of

NbP prepared for measurements and mounted on the TTO puck for PPMS measurements. This mount differs fundamentally from the conventional adiabatic mount recommended by the manufacturer and used in Ref. [93].

Figure 33: NbP mounted isothermally for thermomagnetic transport measurements. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 34: Isothermal sample mounting technique shown schematically with isotherms represented by dashed lines. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 35: NbP sample mounted isothermally and shown on TTO puck for PPMS measurements. Resistive heaters in series are shown in the left frame, thermometer shoes are shown in the middle frame, and the heat sink side is shown in the right frame.

The magnetic field was applied parallel to the thin direction (labeled z), corresponding to the <001> crystal axis. The heat flux was applied along the x-axis, while the measurements of electric fields were either along the x-axis for the Seebeck coefficient xxz or along the y-axis for the Nernst thermopower xyz. The (x,y,z) coordinates are orthonormal, but the orientations of x and y in the <001> plane are not identified. The sample was attached to the silicon base plate using GE varnish. Two insulated copper wires were attached on opposite ends of the sample using silver epoxy. Three 120  resistive heaters were attached in series to a copper foil heat spreader, which was then attached to the base plate and sample using silver epoxy. A copper foil heat sink was used with a gold-plated copper plate for attachment to the puck clamp. Due to the small size of the sample, the thermometers themselves could not be attached to the sample.

Gold-plated copper leads for temperature measurements were attached to the bottom of the silicon plate using silver epoxy at the location of the edges of the sample. The silicon base plate and sample are assumed to be isothermal at these points due to the high thermal conductivity of silicon and the fact that the temperature gradient is applied to the silicon itself. Gold-plated copper assemblies purchased from Quantum Design containing calibrated CernoxTM thermometers and voltage measurement wires were attached to the leads; the voltage wires were detached from these assemblies and soldered directly to the copper voltage wires on the sample.

Measurements were taken at discrete temperatures between 2.5 K and 400 K, and magnetic fields were swept in both directions to a maximum magnitude of 9 T. Controls software was programmed using LabVIEW.

3.2.2 Adiabatic Measurement Technique

Adiabatic transport properties were measured on a second sample of single-crystal NbP of the same batch as the isothermal sample, with dimensions of 1.78 mm x 1.67 mm x 0.25 mm.

Measurements were taken in a home-built, continuous-flow cryostat. This instrument was cooled using liquid nitrogen with a maximum externally applied magnetic field of approximately

1.03 T. The magnetic field was applied parallel to the thin direction corresponding to the <001> crystal axis. The heat flux was applied along the x-axis, while the measurements of electric fields were either along the x-axis for the transverse Seebeck coefficient xxz or along the y-axis for the

Nernst thermopower xyz. As in the isothermal mount, the (x,y,z) coordinates are orthonormal and the orientations of x and y in the <001> plane are not identified.

The sample of NbP prepared for isothermal measurements in the cryostat is shown in

Figure 36. In contrast to the sample used in the isothermal mount, which was attached to a silicon base, this sample was attached to a glass base plate with rubber cement. Silicon has a higher thermal conductivity than NbP, while glass has a much lower one. Consequently, the mount used here no longer measures the isothermal magneto-Seebeck and Nernst thermopowers but thermomagnetic properties closer to the adiabatic ones. Two copper-constantan thermocouples were attached to the sample using silver epoxy parallel to the x-axis of the sample, and two copper voltage wires were attached using silver epoxy opposite the thermocouples. A 120  resistive heater was attached to a copper foil heat spreader using silver epoxy, which was then attached to the base plate and sample also using silver epoxy. A copper foil heat sink was placed opposite the heat source, again using silver epoxy to attach it to the

65 sample and base plate. Controls software was programmed using LabVIEW, and Keithley nanovoltmeters and precision current sources were used as instrumentation.

Figure 36: NbP adiabatic mounting technique for measurements in cryostat

3.2.3 Discussion of Adiabatic vs. Isothermal Thermomagnetic Measurements

Experimentally, magneto-Seebeck contributions can contaminate measurements of the

Nernst effect, and Nernst effect contributions can contaminate the magneto-Seebeck measurements. This cross-contamination can occur as a source of contact misalignment, which

66 can be eliminated as a source of error with data post-processing. The other main source of cross- contamination is more intricate and falls in the difference between adiabatic and isothermal

Nernst and magneto-Seebeck effects. This difference is well known in thermoelectric transport theory, has been discussed at length in Ref. [95], and will be the focus here as it is particularly important in measurements of high-mobility Weyl semimetals.

The Onsager relation, using the (x,y,z) axes system previously described with the magnetic field applied parallel to the z-axis, give:

EE EE ET ET jExx LLLLxx xx xy xy xx xy       jELLLLEE  EE  ET ET yy  yx xy yy xx yx yy   (19). QT LLLLTE TE TT TT   xx xx xy xx xy   QT TE TE TT TT   yy LLLLyx yy yx yy  

This equation is solved for jx=jy=0 and the ratios between Ex, Ey, xT and yT to obtain the isothermal Seebeck (xxz, yyz) and Nernst (yxz, xyz) components:

ETx   xxz yxz  x       (20). ETy   xyz yyz  y 

The Nernst and conventional thermopowers are now defined in an isothermal manner, respectively:

EH() EH() yz ; xz (21). xyzTT xxz  xxT 0 yT 0 y

The Nernst coefficient, which is the derivative of the Nernst thermopower with respect to magnetic field, can also be defined isothermally:

xyz Nxyz  (22). H z yzTH0,

If the isothermal condition of yT=0 is not imposed, the adiabatic Seebeck coefficient is obtained:

Ex  yT xxz yxz (23) xxTT  as well as the adiabatic Nernst thermopower:

ETyy xyz yyz (24). xxTT 

The more commonly used adiabatic mount measures the values described in Equations 23 and

24, in which the sides of the sample along z are thermally isolated. In samples in which the electronic contribution to the thermal conductivity is appreciable and subject to a thermal Hall effect, as is the case here, the measurement of the adiabatic Nernst effect (Equation 24) is a mixture of the isothermal xyz and yyz, the longitudinal thermopower along y. Additionally, the measurement of the adiabatic Seebeck effect (Equation 23) contains a contribution of both the isothermal Seebeck xxz and the isothermal Nernst xyz effect. Contrastingly, the isothermal mount shown in Figure 33 imposes the condition that yT=0, allowing for the direct measurement of xxz and xyz. When no external magnetic field is applied, no thermal Hall effect is present, indicating that the isothermal and adiabatic Seebeck coefficients are identical in this case.

Numerical post-processing of the data to alternate between isothermal and adiabatic

95 coefficients is possible via the Heurlinger relations , but they require a measurement of yT and

68 the thermal Hall coefficient, which was not measured in the present experiment due to the small dimensions of the sample. Unique to the case of NbP (and likely other Weyl semimetals) is the existence of a Nernst thermopower significantly larger than the conventional thermopower. This indicates that the contamination of the Nernst thermopower by the Seebeck effect is small, but the contamination of the magneto-Seebeck effect by the Nernst effect is large. To eliminate this contamination, both adiabatic and isothermal thermomagnetic transport was measured on NbP, but only isothermal data will be considered in relation to theory. A more extensive study of the adiabatic magneto-Seebeck effect can be found in Ref. [93].

3.3 Sample Characterization

3.3.1 Electrical Conductivity, Magnetoresistance, and Hall Effect

The temperature dependence of the electrical resistivity of the NbP sample used in this study is shown in Figure 37. The transverse magneto-resistivity and Hall resistivity are shown at representative temperatures as a function of magnetic field in Figure 38 and Figure 39, respectively, with the magnetic field applied parallel to <001> in both cases. These data were taken using the adiabatic sample mount technique described previously. Electrical resistivity is seen to increase with temperature. The transverse magneto-resistivity increases with magnetic field, and the magnetic field dependence becomes stronger as the temperature decreases towards

100 K. A broad maximum in the Hall coefficient can be inferred near 160 K.

Samples used in this study are similar to those of Ref. [46], although a discrepancy in electrical resistivity is seen between the samples. This indicates that samples used in this study are ten times less resistive than those of Ref. [46]. A large contribution to this must come from

69 the geometric uncertainty in the measurements. Due to the small sample size, a geometric error on the order of hundredths of a millimeters results in a noticeable offset in resistivity data.

Measuring the distance between the voltage wires on this sample was not possible to do with a high level of precision because measuring to the order of micrometers could actually have resulted in breaking the wires off the sample.

Figure 37: Temperature dependence of electrical resistivity in NbP. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 38: Transverse magneto-resistivity in NbP as a function of magnetic field. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 39: Hall resistivity in NbP as a function of magnetic field. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

3.3.2 Thermal Conductivity

The temperature dependence of the thermal conductivity in NbP is shown in Figure 40.

This data was taken using the adiabatic sample mounting technique described previously and shows a decrease in thermal conductivity with increasing temperature. This data compares quite well to that of Ref. [93], with both studies using similar single crystals of NbP. Despite the geometric uncertainty described in the previous section, electrical and thermal conductivity, which used the same geometric measurements, are internally consistent.

Figure 40: Thermal conductivity in NbP as a function of temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

3.3.3 Specific Heat

Specific heat data was taken on a single-crystal sample of NbP using the Heat Capacity

Option on a Quantum Design PPMS. Data was taken between 2 K and 400 K, and a 2 fitting method was used per manufacturer’s recommendation. Figure 41 shows the specific heat versus temperature, and the data are consistent with the calculation in the literature96. In Figure 41, the full curve is shown, fitting a Debye model with a Dulong-Petit limit of 49.90.2 mJ mole-1 K-1 and a Debye temperature =5252 K. Data are given as brown points, and the Debye model is given as the blue curve. Figure 42 shows the separation of the specific heat into a linear term and a T3 term following the equation:

C() T T AT 3 (25) where C is the sample specific heat, T is the temperature,  is the coefficient for the linear contribution, and A is the coefficient for the cubic contribution. This plot was made by dividing

C  AT 2 2 the entire equation by the temperature, plotting T as a function of T . The result gives =1.5x10-4 J mole-1 K-2 and A=2.7x10-5 J mole-1 K-4. Because the Dirac dispersion is linear and the density of states goes as the square of energy, DOS(E)  E2, one expects the electronic specific heat of the electrons in the Dirac cone to follow a T3 law, which is indistinguishable from the phonon contribution. Therefore, the -value reported corresponds presumably only to the charge carriers located in the trivial pockets in the zero-K limit since  can be resolved only below 10 K.

The measured value for the electronic specific heat corresponds very well to the density of electronic states at the low-temperature value of the electrochemical potential, 0 = -8.20.7 meV below the Dirac point of the W2 bands, determined from de Hass-van Alphen oscillations.97

-1 Using density functional theory calculations, the DOS(0=-8.2 meV) is 0.062 states eV per formula unit or 2.0x1021 states eV-1 cm-3 corresponding to =1.45 10-4 J mole-1 K-2.97 The

21 -1 -3 -4 -1 DOS(0=0 meV) would have been 2.3 x10 states eV cm corresponding to =1.7 10 J mole

K-2. The states in the W2 Dirac bands contribute very little to this.97

Figure 41: Specific heat of NbP as a function of temperature, with experimental data shown as brown dots and fit to a Debye model as blue curve. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 42: Low-temperature specific heat of NbP divided by temperature as a function of temperature squared. Brown circles are experimental data and blue curve is Debye model fit. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

3.3.4 Magnetization

Magnetization data were measured using the Vibrating System Magnetometer on a

Quantum Design PPMS in an external magnetic field applied either H // <001> or H  <001>, with data shown in Figure 43 and Figure 44, respectively. Two orientations with H  <001> were studied normal to each other, but they were not aligned with the <100> or <010> axes; the results were isotropic in-plane. The sample was attached to the sample holder using GE Varnish.

The magnetic moment of the sample was measured with the externally applied magnetic field sweeping both up and down from negative to positive field and vice-versa between maximum magnitudes of 70 kOe (7 T).

The data indicate that NbP is a diamagnet with an isotropic susceptibility d = -4.00.2

-1 B Oe per NbP formula unit at 5 K, with B the Bohr magneton. Clear de Haas – van Alphen quantum oscillations are visible that were analyzed using a Fourier transform. The observed periods, Fermi surface areas (assuming a circular Fermi surface), and Fermi momenta are displayed in Table 2. With the magnetic field applied parallel to the <001> crystal axis, three periods are present and correspond very well to those found from Shubnikov–de Haas oscillations in the Nernst measurements (explained later in this text) and from Shekhar et al.’s data46. The in-plane axes, perpendicular to the c-axis, show two periods present, and they are similar when comparing the two in-plane axes. The periods, Fermi surface areas, and Fermi momenta are reported in Table 2. The periods with H // <001> also correspond quite well with those of the Shubnikov–de Haas oscillations in the Nernst effect. Stockert et al.93 report more quantum oscillations in heat capacity, magnetization, thermal conductivity, and adiabatic magneto-thermopower that, for H // <001>, all have periods of oscillations comparable to each other and comparable to those reported here, although those samples were different and presumably have slightly different values of 0.

Figure 43: Diamagnetic moment of NbP at 5 K with magnetic field applied parallel to <001>. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Figure 44: Diamagnetic moment of NbP at 5 K with magnetic field applied perpendicular to <001>. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Table 2: Shubnikov-de Haas and de Haas-van Alphen quantum oscillations in NbP. Table taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

3.4 Thermomagnetic Transport in NbP

3.4.1 Experimental Results

The experimental results of the isothermal Nernst effect measurements are shown in

Figure 45(a) (magnetic-field dependence of the Nernst thermopower xyz at discrete temperatures), where xyz is defined as in Equation 21, and Figure 46(a) (temperature dependence of the Nernst coefficient Nxyz), where Nxyz is defined as in Equation 22. xyz is an

79 odd function of Hz with a higher slope near zero magnetic field than at high field. An unsaturated, large Nernst thermopower is observed with a maximum exceeding 800 V K-1 at 9

T, 109 K, which is 2-4 times larger than the maximum thermopower of conventional, commercial thermoelectric semiconductors. The Nernst coefficient Nxyz temperature dependence taken at low field (Hz<2 T) and high field (magnitude between 3 T and 9 T) is non-monotonic, with a maximum around TM50 K for low-field Nxyz and TM90 K for high-field Nxyz. The inset of Figure 46(a) shows the temperature dependence of xxz, which is nearly two orders of magnitude smaller than the Nernst thermopower xyz(9 T) near 100 K, and its absolute value is much smaller than the high-field Nernst effect at all temperatures. The xxz temperature dependence has a broad minimum around TM ranging from 60 to 100 K. No magnetic-field dependence is observed forxxz(Hz) within instrument sensitivity, despite repeated attempts. This result does not contradict prior measurements93 of the adiabatic magneto-Seebeck coefficient

EHyz() xxz  because those reflect a contribution from xyz in the data, as explained xT yT 0 previously. The properties of the present sample’s Fermi surface exclude phonon drag as a

93 possible source for observed non-monotonicity in xxz and xyz at temperatures exceeding 14 K.

Figure 46: Experimental (a) and theoretical (b) temperature dependence of the Nernst coefficient in NbP. Inset of top frame is conventional thermopower as a function of temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

At low temperatures, inset of Figure 45(a), the Nernst thermopower exhibits Shubnikov– de Haas oscillations. The periods observed in Shubnikov-de Haas oscillations and the periods observed in de Haas-van Alphen oscillations found in the magnetization (see Section 3.3.4)

82 correspond very well, within the error bars of the measurements, even though they were obtained on different samples that may have had different defect densities and thus different residual doping levels. These results are also similar to those found by J. Klotz et al.98. The de Haas-van

Alphen oscillation periods for the magnetic field applied parallel to all three axes and the corresponding values for the area of the Fermi surfaces and values for kF calculated assuming that the cross-sections are circles are reported in Table 2. These periods are used in conjunction with the density function theory calculations to derive the position of the electrochemical potential vis-à-vis the Weyl points in the zero-Kelvin limit, which is determined to be 0  -

8.22 meV below the main Weyl bands’ Dirac point.97 This sets the Fermi level in the zero-

Kelvin limit in the valence band, which is consistent with the positive thermopower observed in the inset of Figure 46(a) at T < 25 K.

For the sake of completion, measurements of the adiabatic magneto-thermopower and

Nernst thermopower were conducted, and results are shown in Figure 47. The zero-field thermopowers in both the isothermal and adiabatic mounts reasonably agree, especially considering that two different instruments and samples were used (measurement techniques described in Section 3.2). The isothermal and adiabatic Nernst coefficients in the low-field limit show discrepancies on the order of 10% at 100 K, indicating the correspondence between adiabatic and isothermal Nernst thermopower data is quite within the experimental error.

However, because the Nernst thermopower is much larger than the conventional, Seebeck thermopower, the adiabatic mount gives a very large magnetic field dependence to the latter, which is absent in the isothermal mount. Over the temperature and field range allowed by the instrumentation used with the adiabatic mount, the data reported here agree considerably with

83 those of Stockert et al.93 Thus, all data sets are consistent, and the magnetic-field dependence of the adiabatic Seebeck thermopower can be concluded as a projection of the Nernst coefficient induced by the development of a temperature gradient along y via the thermal Hall effect.

Figure 47: Adiabatic magneto-thermopower (a) and Nernst thermopower (b) in NbP. Adiabatic magneto-thermopower includes a contribution from the thermal Hall effect not reflected in the isothermal thermopower measurements. Adiabatic Nernst thermopower reasonably corresponds to isothermal Nernst thermopower over available temperature and magnetic field range considering two different measurement instruments were used. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

3.4.2 Theoretical Modeld

The theoretical model is based on a Weyl semimetal Hamiltonian, supporting four Weyl nodes in an inversion symmetry-breaking Weyl semimetal.99 The model is solved self-

d Theory presented in this section has been majorly developed by Timothy M. McCormick and Nandini Trivedi. 84 consistently for the chemical potential (T), which moves to the nodal energy with increasing temperature following the scale T , as illustrated in the right side of Figure 48. A picture 0 kB of one Dirac cone is given in the left side of Figure 48. The temperature dependence of the chemical potential is crucial to the temperature dependence of the Nernst effect.

Figure 48: Two-dimensional dispersion around a single Dirac cone (left) where, at low temperatures, the electrochemical potential lies in the valence band at an energy o below the Dirac point. The calculated (T) temperature dependence (right) shows it moving towards the energy of the Weyl node with increasing temperature. Figure taken from S. J. Watzman et al., arXiv: 1703.04700 (2017).

Thermomagnetic tensor elements are calculated using the Boltzmann formalism,51,87 preserving time-reversal symmetry. The calculated Nernst thermopower is given by:97

EE ET EE ET ELLLLy xx xy xy xx xyz 22 (26).  T EE EE x LLxx   xy 

Results are shown in Figure 45(b) as a function of magnetic field and in Figure 46(b) as a function of temperature. The magnetic field dependence shows several salient features, notably a change in monotonicity for different temperatures and distinct slopes at high and low magnetic field. The origin of the non-monotonicity at low temperatures arises from the second term in the

EE numerator of Equation 26. At low temperatures, the conventional Hall response Lxy dominates the Nernst response. At higher temperatures, the Nernst response is given by:97

ET Lxy  xyz  EE (27). Lxx

Two distinct regions are present with different slopes at low field and high field. For reasonable choices of the lattice spacing a, the field scale delineating the two regions matches that found in the experimental results for NbP.97

Analytical solutions can be derived for the temperature dependence of the zero-field

Seebeck thermopower and the low-field Nernst coefficient. These functions quantitatively describe most of the important features seen in the experimental results, including the observed

3 0 97 maxima at TM  , above which Nxyz decreases following a 1/T law.  kB

3.4.3 Comparison of Experiment to Theory

The full model of Equation 26, including the magnetic field and Berry curvature effects, without any adjustable parameters, is compared to the experimental data. The magnetic field

86 scales with the magnetic length LB and the lattice spacing a. The temperature scale is set by the zero-Kelvin chemical potential 0. The scattering time  is assumed to be energy- and

5 -1 temperature-independent; appropriate values of vF2 x 10 m s (determined using values from density functional theory and angle-resolved photoemission spectroscopy) and resistivity (from

Section 3.3.1) yield a value of τ~10-13 s.97 The experimental and theoretical results are shown in

Figure 45 for xyz magnetic-field dependence at discrete temperatures and Figure 46 for Nxyz temperature dependence. The calculated field and temperature dependences show remarkable agreement to experimental results, apart from the sign change that theory predicts for low-field

Nxyz below 40 K. The amplitudes agree within a factor of 4; this is remarkable given the uncertainty of .

0 Figure 46 shows the high-field Nernst effect peaks at TM  in both theory and kB

3 0 experiment, whereas near zero-field Nxyz peaks around TM  in both theory and  kB experiment. The Seebeck coefficient also is calculated to have an absolute value maximum at

3 0 TM  . The xxz and xyz extrema originate from the chemical potential temperature  kB dependence within the Dirac bands. As the chemical potential shifts to the nodal energy with increasing temperature, the effects of the Berry curvature strengthen, maximizing the Nernst effect, shown via experiment and theory. At temperatures above TM, the Fermi function derivative broadens, leading states away from the Weyl nodes to contribute more to transport, lowering the Nernst thermopower and Nernst coefficient.

The two main, unique features seen in both experiment and theory for NbP are: (1) the temperature-dependence of the Nernst effect is non-monotonic with a maximum at TM, and (2) the experimental Nernst thermopower is two orders of magnitude larger than the Seebeck coefficient. Through the qualitative and quantitative comparison of the model to experiment, these two characteristics are seen to be specific to Dirac bands. When electrons and holes coexist in equal density in perfectly symmetric bands, the Seebeck coefficient tends to zero and the Nernst effect, representing the sum of the contributions of electrons and holes9, is large.

Furthermore, the local charge neutrality condition in undoped semiconductors and semimetals dictates that the (T) is located at the energy where the density of states is minimal. When defects or aliovalent impurities add an extrinsic charge carrier density, (T), in the limit of T 0

K, falls in a band at a value 0. As the temperature increases sufficiently such that the thermally- induced intrinsic charge carriers outnumber the extrinsic carriers, (T) tends toward the energy where the density of states is minimal. Evidence in this study suggest that in NbP, this energy is the Dirac point of the carriers in the W2 band, as identified in Ref. [46], which has been defined as the zero for the energy scale, as shown in Figure 48.

3.4.4 Contribution of Trivial Pockets to Transport

The model ignores trivial pockets in the Fermi surface by hypothesis, whereas band structure calculations valid in the 0 K limit show they exist. When these contributions are included in the Nernst and Seebeck effect calculations, the model no longer follows experiment.97 This is experimental evidence that non-Dirac bands do not contribute significantly to transport at temperatures above 0/kB. The behavior of xxz (inset of Figure 46(a)) strengthens 88 this argument. xxz tends to zero, a value reached rigorously in a Dirac band when (T)=0. If trivial pockets contributed to thermopower, xxz(T) would become metallic and increase monotonically with increasing temperature, as it does for Bi.94 The partial thermopowers for the different trivial pockets of the Fermi surface that exist at low temperature can be inferred from the low temperature Fermi energies calculated for these pockets.97 If these pockets had been present at room temperatures, their partial thermopowers would have been of the order of 50-500

V K-1, compared to which the experimental value in the inset of Figure 46(a) is factually zero.

Because an accidental cancellation over a wide temperature range of the contributions of all non-

Dirac bands is highly unlikely, this argument is offered as a reductio ad absurdum proof that the trivial pockets do not contribute.

3.5 Conclusion

In conclusion, thermomagnetic transport in the inversion-symmetry-breaking Weyl semimetal NbP was experimentally and theoretically explored in this chapter. Two regimes in

Nernst thermopower were seen: one for Hz >|3 T|, and the other Hz <|2 T|; both have non- monotonic temperature dependences. The theory explaining these properties quantitatively shows them to be Dirac band transport signatures, with no measureable contributions from trivial pockets above 100 K. Theory and experiment show that as the chemical potential shifts to the

Dirac point energy with increasing temperature, the Nernst effect is maximized. At temperatures above the peak temperature, the Fermi function derivative broadens, leading states away from the Weyl nodes to contribute more to transport, lowering the Nernst thermopower and

-1 coefficient. The Nernst thermopower xyz(9 T, 109 K) exceeds 800 μV K , surpassing the

Seebeck thermopower xxz by two orders of magnitude. This study offers an understanding of the temperature dependence of the electrochemical potential position vis-à-vis the Weyl point and shows a direct connection between the Nernst effect and topology, a potentially robust mechanism for investigating topological states and the chiral anomaly.

Chapter 4. Berry Curvature-Induced Huge Anomalous Nernst Effect in the Weyl e Semimetal YbMnBi2

4.1 Motivation

Weyl semimetals have gained interest due to their unique experimental signatures as contributions from topological transport. Recent work further shows exciting properties found in magneto-thermal transport, specifically in the Nernst effect of NbP (Chapter 3)97, an inversion symmetry-breaking Weyl semimetal with no net Berry curvature, where a perpendicular temperature gradient and magnetic field are applied to a sample, generating a mutually orthogonal voltage. This Nernst thermopower vastly exceeds that of conventional thermoelectric materials in the advantageous transverse geometry,55 but a large magnetic field is required.

When time-reversal symmetry is broken in a Weyl semimetal, a non-zero Berry curvature exists within the electronic band structure. The Weyl nodes act as monopole sources or sinks of this

Berry curvature, which can act as an effective magnetic field residing in reciprocal space.23,24 In turn, the Berry curvature can introduce an anomalous velocity to electron motion that is skew to both the Berry curvature and the electrons’ k-vector. This skew force is predicted theoretically to generate a non-zero thermoelectric power in the direction normal to both the net Berry

e The work presented in this chapter is currently in progress and still under investigation. Because the main result of this chapter (the isothermal Nernst effect) was obtained on only one sample, the content of this chapter has not yet been submitted for publication. This work is in collaboration with Kaustuv Manna, Timothy M. McCormick, Yan Sun, Bin He, Satya N. Guin, Yuanhua Zheng, Johannes Gooth, Chandra Shekhar, Nandini Trivedi, Claudia Felser, and Joseph P. Heremans. 91 curvature integrated over the entire Fermi surface (which is a scalar) and the direction of the applied thermal gradient.50,51 This setup would involve a temperature gradient applied parallel to the x-axis (xT), an electric field measured parallel to the y-axis (Ey), and the Berry curvature projection parallel to the z-axis (z). Furthermore, this anomalous Nernst response is theoretically predicted to be a direct measurement of the integral of the z-component of the Berry curvature over the Fermi surface when time-reversal symmetry is broken. The Nernst response,

ET Lxy , is a function of the integral of the product of the Berry curvature, z, and the entropy density function, s(E):50

푘 푒 푑3푘 퐿퐸푇 = 퐵 ∫ Ω 푠(퐸) (28) 푥푦 ℏ (2휋)3 푧 where kB is the Boltzmann constant, ℏ is the reduced Planck constant, and e is the charge of an electron. The entropy density function is Gaussian-like and centered at the Fermi level:50

 E s( E ) ln(1  e )   Efo (29)

1 1  E where   , EEk F with EF as the Fermi level, and feo  1 as the Fermi- kTB

Dirac distribution. This is distinct from the Hall effect, where the anomalous Hall conductivity,

퐸퐸 퐿푥푦 = 휎푥푦, is a function of the integral of the Berry curvature over all occupied states, thus measuring an average:100

푒2 푑3푘 퐿퐸퐸 = 휎 = ∫ Ω 푓 (k) (30) 푥푦 푥푦 ℏ (2휋)3 푧 표 where fo(k) is the equilibrium Fermi-Dirac distribution. Direct measurements of the integral of

Berry curvature over the Fermi surface at the Fermi level via the Nernst effect further offer

Nernst measurement in time-reversal symmetry-breaking Weyl semimetals as a potentially robust mechanism for investigating topological states and the chiral anomaly.

Borisenko et al. have experimentally confirmed broken time-reversal symmetry in

49 YbMnBi2 via angle-resolved photoemission spectroscopy. Band structure calculations in conjunction with experimental results conclude that YbMnBi2 must contain a canted antiferromagnetic structure.49 Further calculations demonstrate that the net Berry curvature lies

101 along the <110> crystal axis in YbMnBi2. The <110> crystal axis is the direction of the slight antiferromagnetic canting present in YbMnBi2, which is the source of the net Berry curvature.

The study presented here predicts that if the <110> crystal axis is aligned mutually perpendicular to both the direction of an applied temperature gradient and the direction of a measured voltage, a Nernst thermopower will be generated in the absence of a magnetic field with the Berry curvature itself acting as the source of a skew force. This is depicted in Figure 49, where the z- axis is parallel to the direction of net Berry curvature and antiferromagnetic canting, represented by red arrows. When an external magnetic field is applied in the experiments described, it will also be applied parallel to the z-axis.

Figure 49: Transverse geometry of the Nernst effect with Berry curvature aligned along the z- axis. When a magnetic field is applied, it will also be applied parallel to the z-axis.

Here, preliminary experimental evidence is shown for a Berry curvature-induced isothermal Nernst thermopower in the absence of an externally applied magnetic field, resulting

-1 in a peak in zero-field Nernst thermopower of xyz ~ 1000 V K near 60 K. This Nernst thermopower is expected to be a direct measurement of the integral of the material’s Berry curvature over the Fermi surface and due to the Berry curvature itself acting like an internal magnetic field. YbMnBi2 is further characterized through electrical resistivity, magnetoresistance, Hall effect, thermal conductivity, and magnetization measurements. With this data, a transverse thermoelectric figure of merit, zT, is calculated for YbMnBi2, with zT ~ 2.4 at 60 K. This zT is a record on 3 levels: first, it is comparable (within errors bars) to the highest zT of published thermoelectric materials; second, it surpasses the zT of any thermoelectric material in a transverse geometry; third, it surpasses the zT of any thermoelectric material in a transverse geometry without an externally applied magnetic field. Due to the low temperature of the peak zT of YbMnBi2, device applications involve cryogenic cooling in Ettingshausen coolers.

Although this data is repeatable between measurement systems, sample repeatability is still in 94 progress and necessary to confirm these results. Furthermore, adiabatic transport results are reported here which do not compare to the isothermal data in a currently explainable manner.

Thus, this work is left in progress and is currently under further investigation and interpretation.

4.2 Measurement Technique

4.2.1 Samples Used in This Study

Seven different single-crystal samples of YbMnBi2, all synthesized by Kaustuv Manna using a Bi flux technique, were used in this study and will be discussed in this chapter. Table 3 displays a summary of the samples used for thermomagnetic transport measurements, and Table

4 displays the samples used for electrical transport and magnetic measurements; samples will be referenced by their corresponding sample numbers in this chapter. The subscript “app” indicates an applied quantity, and the subscript “meas” indicates a measured quantity. In the rest of this chapter, subscript labeling for transport coefficients will be used in the form of ABC , where the index “A” represents the direction of the applied flux, “B” represents the direction of the measured property, and “C” represents the direction of the applied magnetic field.

Table 3: YbMnBi2 samples used in thermomagnetic measurements displaying the measurement technique, measurement system, and crystal axes corresponding to the applied and measured quantities

Table 4: YbMnBi2 samples used for electrical transport and magnetic measurements displaying the crystal axes corresponding to the applied and measured quantities

4.2.2 Isothermal Measurement Technique

Two single crystals of YbMnBi2 were measured isothermally using the technique described in Section 3.2.1. The single crystals were grown using a flux technique, and both isothermal single crystals are from the same batch. The isothermal crystals are designated as sample 1 and sample 2. Sample 1, with dimensions of 2.58 mm x 1.85 mm x 0.44 mm, was oriented using Laue backscattering x-ray diffraction. The crystal axes identified are labeled in

Figure 50. During measurements, the magnetic field of the PPMS was parallel to the <110> crystal axis, which is the axis predicted to have antiferromagnetic canting and thus a net Berry curvature. A schematic is shown in Figure 50, where the temperature gradient is always applied along the x-axis, 110 , and the Berry curvature is along the z-axis, <110>; when a magnetic field is applied, it is applied parallel to the z-axis. The silicon backing plate, instituting the isothermal measurement condition, forces isotherms along the y-axis, which is parallel to the

<001> crystal axis. This orientation allows for the direction of the predicted net Berry curvature to be mutually perpendicular to both the applied temperature gradient and measured electric field during measurements of the Nernst effect such that contributions from Berry curvature can be determined in the absence and presence of an externally applied magnetic field. This sample, prepared for isothermal measurements on a PPMS, is shown as prepared in Figure 51. A 2 k resistive heater housed in the heater assembly as supplied by Quantum Design was clamped to a

96 gold-plated copper heat spreader, which was attached to a copper heat spreader using silver epoxy. The copper heat spreader was attached to the silicon and the sample using silver epoxy as well. The rest of the sample mounting technique is similar to that for NbP as described in

Section 3.2.1.

Figure 50: Orientation of YbMnBi2 sample 1, z-axis parallel to Berry curvature. Isotherms are parallel to the y-axis.

The second isothermal single crystal, sample 2, schematically shown in Figure 52, has dimensions of 1.73 mm x 1.48 mm x 0.29 mm. This crystal was also oriented using Laue backscattering x-ray diffraction such that the direction of the net Berry curvature would be perpendicular to the applied magnetic field within the PPMS when the field was used in the measurements. Here, the <110> crystal axis is parallel to the direction of the measured electric field during measurements of the Nernst effect, rather than mutually perpendicular to the applied

98 temperature gradient and measured electric field, so that the Berry curvature effects in the Nernst effect akin to those of an externally magnetic field can be isolated between this sample and sample 1 (Figure 50). The silicon backing plate, instituting the isothermal measurement condition, forces isotherms along the y-axis, which is parallel to the <110> crystal axis. This sample was mounted in the same manner as sample 1 as shown in Figure 51.

Figure 52: Orientation of YbMnBi2 sample 2, z-axis perpendicular to Berry curvature. Isotherms are parallel to the y-axis.

Both samples 1 and 2 were prepared for measurements of Nernst thermopower, and sample 1 was also prepared for measurements of the conventional thermopower. For the conventional Seebeck effect, xxz, the temperature gradient was applied and measured parallel to the x-axis, the electric field was measured as a voltage parallel to the x-axis, and the magnetic field was applied parallel to the z-axis. For the Nernst effect, xyz, the temperature gradient was applied and measured parallel to the x-axis, the electric field was measured as a voltage parallel to the y-axis, and the magnetic field was applied parallel to the z-axis. When magnetic fields 99 were swept, they were swept in both directions to a maximum magnitude of 9 T. Controls software was written in LabVIEW. Keithley nanovoltmeters were used for electrical measurements, and a Keithley precision current source was used with the resistive heater. The measurement program used here was the same as that for NbP, described in Section 3.2.1.

Sample 1 (Figure 51), prepared on the TTO puck for measurements in the PPMS, is shown in

Figure 53.

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4.2.3 Adiabatic Measurement Technique

4.2.3.1 Modified Thermal Transport Puck

A third single crystal of YbMnBi2, made from the same batch as samples 1 and 2, was measured using a modified setup on a TTO puck for the PPMS using the adiabatic method described and depicted by Figure 1(a) in Ref. [93]. This setup is referred to as the “mini puck” and utilizes chip CernoxesTM rather than the thermometry assemblies provided by Quantum

Design. Although the chip CernoxesTM were calibrated, their offset from the chamber temperature at room temperature was on the order of 10 K; Nernst thermopower and thermal conductivity measurements only take into account a relative temperature difference, so this offset only serves as a problem for determining the base temperature of the sample, not the coefficients themselves. This setup also utilizes a 10 k resistive heater rather than that supplied in the heater assembly from Quantum Design. Controls software was programmed in LabVIEW and similar to that used in the isothermal technique.

Sample 5 was measured using the mini puck, and its dimensions are 1.82 mm x 0.48 mm x 0.41 mm with the axes aligned as shown in Figure 50 (although the sample was not mounted on a silicon backing plate). The temperature gradient was applied and measured parallel to the x- axis, the electric field induced by the Nernst effect was measured parallel to the y-axis, and the magnetic field was applied parallel to the z-axis, which is also the predicted direction of net

Berry curvature. Thus, an adiabatic Nernst thermopower, xyz, was determined from this measurement technique. Because the technique used here was adiabatic, the magneto-thermal conductivity (xxz) was measured using this technique as well.

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4.2.3.2 PPMS AC Transport Puck and Continuous Flow Cryostat

Two single crystals of YbMnBi2, both made in separate batches from each other and the

YbMnBi2 crystals in the previous sections, were measured adiabatically using the AC transport puck on a PPMS and in a home-built Janis continuous flow cryostat. These crystals are referred to as samples 3 and 4. The dimensions of sample 3 are 1.97 mm x 0.30 mm x 0.49 mm; the dimensions of sample 4 are 1.89 mm x 0.45 mm x 0.49 mm. The axes were aligned as shown in

Figure 50 (although the sample was not mounted on a silicon backing plate), with the temperature gradient applied parallel to the x-axis and the magnetic field applied parallel to the z- axis. Each sample had a 120  resistive heater attached to a brass heat spreader using silver epoxy, which was then attached to the edge of the sample. The opposite edge of the sample along the x-axis was attached to a brass heat sink, also using silver epoxy. Copper-constantan thermocouples were soldered together then attached to the samples using silver epoxy, with three thermocouples attached to each sample: two parallel to the x-axis to measure the induced temperature gradient and one transverse thermocouple aligned along the y-axis cold side to measure the transverse temperature gradient (thermal Hall effect). The copper wires from the thermocouples were used for voltage measurements parallel to the x-axis and the y-axis. This entire setup was attached to an alumina plate using silvery epoxy. The alumina plate contained solder joints for the wires from the sample to connect to insulated copper wires, which were used for electrical connections to either the AC transport puck or the cryostat cold finger. The alumina plate was attached to a copper foil “bracket” using silver epoxy so that the sample assembly itself could be appropriately aligned to the magnetic field direction of both instruments.

This setup is shown in Figure 54.

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Figure 54: Adiabatic measurement technique used for YbMnBi2 samples 3 and 4. Mount used for both AC puck in PPMS and in cryostat.

For the PPMS measurements, the sample was attached to the AC transport puck of a

Quantum Design Physical Property Measurement System (PPMS). All electronics were wired through a home-built breakout box to a Keithley electronic channel switchbox, where voltages

(both sample and thermocouple) were measured using a Keithley nanovoltmeter and the heater current was controlled by a Keithley current source. Controls software was programmed in

LabVIEW. Control was accomplished similarly in the home-built Janis continuous flow cryostat. The heat flux was applied along the x-axis, while the measurements of electric fields were either along the x-axis for the Seebeck coefficient in a transverse field, xxz, or along the y- axis for the Nernst thermopower, xyz. Additionally, the temperature gradient parallel to the y- 103 axis was measured for the thermal Hall effect. Because the technique used here was adiabatic, the magneto-thermal conductivity, xxz, was measured as well.

Both instruments showed similar results over their respective magnetic field and temperature ranges. Because the PPMS has a larger range for both magnetic field and temperature, PPMS data will be shown in the following sections for samples 3 and 4.

4.3 Sample Characterization

4.3.1 Electrical Conductivity, Magnetoresistance, and Hall Effectf

The temperature dependence of the electrical resistivity at zero-field and with an applied field of 90 kOe parallel to the <110> crystal axis is shown in Figure 55 for sample 7, indicating that the sample is more resistive in a magnetic field parallel to <110> than it is at 0 Oe. The magnetic field dependence of the resistivity in the same orientation is shown as magnetoresistance in Figure 56, displaying an increase in magnetoresistance with decreasing temperature and a maxima, creating a non-monotonicity, near 50 K, 60 kOe.

f The data presented in this section was measured and analyzed by K. Manna. It is presented here to give a full transport story for YbMnBi2. 104

Figure 55: Temperature dependence of electrical resistivity in sample 7 of YbMnBi2 at 0 Oe and 90 kOe, Happ parallel to <110>. Data and image courtesy of Kaustuv Manna.

Figure 56: Magnetoresistance as a function of applied magnetic field in sample 7 of YbMnBi2. Data and image courtesy of Kaustuv Manna.

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The Hall effect in sample 7 is shown in Figure 57 as the magnetic field dependence of the transverse resistivity, xy. The magnetic field is applied parallel to the z-axis, which is the <110> crystal axis here. Data was taken use a 5-wire balance technique, which utilizes both transverse and longitudinal voltage measurement wires in an attempt to eliminate a 0 Oe contribution from

xx stemming from voltage wire misalignment; this involves forcing the signal to 0 -m at 0 Oe, which may not be rigorously true in this case due to the existence of a net Berry curvature along the <110> crystal axis. No anomaly is observed near the Neel temperature, TN ~ 290 K, or the expected spin-canting temperature, TS ~ 240 K. No hysteretic effect is seen in the Hall effect, although the magnetic field dependence is asymmetric at all temperatures measured and shows a peak near 0 Oe. This sample has a large even-in-field contribution to its Hall resistivity, the origin of which is unknown.

Because YbMnBi2 is predicted to have a net Berry curvature along its <110> crystal axis, the possibility arises for a non-zero Hall resistivity at 0 Oe under conditions in which the applied current and measured voltage are mutually perpendicular to the <110> axis where the Berry curvature could contribute a skew force in the direction of the measured voltage. This is the case of data taken in Figure 57. Thus, the Hall effect was also measured using a 4-wire method without balancing the signal to 0 -m at 0 Oe in order to effectively ascertain the magnitude of

xy with the electric current applied and voltage measured both mutually perpendicular to the

<110> crystal axis. Results are shown in the left frame of Figure 58 for sample 6; the crystal measured here has a low antilocalization effect and is a separate crystal than that of Figure 57.

Here, an offset from 0 -m at 0 Oe is observed, although the 0 Oe values at each temperature correlate closely to the resistivity values observed at 0 Oe at the same temperatures in the same crystal; resistivity data is shown in the right frame of Figure 58 for sample 6. The non-zero xy values at 0 Oe are potentially due to a wire offset, picking up a contribution from xx at 0 Oe, as this signal was not forcibly removed by balance the signal at 0 Oe. Nevertheless, the magnetic field dependence of xy in sample 6 (left frame of Figure 58) appears incredibly different than that of sample 7 (Figure 57), which could be due to a difference in sample quality because two separate samples were used or a difference in measurement technique since the 5-wire balance method was used for sample 7 while a 4-wire unbalanced method was used for sample 6.

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Figure 58: Transverse (Hall) resistivity as a function of applied magnetic field in sample 6 of YbMnBi2, measured using a 4-wire technique and unbalanced at 0 Oe. For reference, right frame shows resistivity as a function of magnetic field in the same sample. Data and image courtesy of Kaustuv Manna.

4.3.2 Thermal Conductivity

The temperature dependence of the magneto-thermal conductivity, xxz, in YbMnBi2 is shown in Figure 59. Data is displayed for samples 3, 4, and 5, although no magnetic field values were measured for sample 5. No thermal conductivity data is shown for samples 1 and 2, the isothermally measured samples, as thermally short-circuiting the samples externally influences the heat flow, nullifying the meaning of the measurement. For +90 kOe and -90 kOe data, xxz values were averaged since the magneto-thermal conductivity was symmetric about 0 Oe. Data from sample 3 is shown in blue, from sample 4 in red, and from sample 5 in green. Open circles represent 0 Oe data and plus signs represent the averaged data at +90 kOe and -90 kOe.

Samples 3 and 4 show similar trends in magneto-thermal conductivity, namely that the 0

Oe values and the |90 kOe| values are similar within a sample above approximately 100 K, although the samples deviate from one another by nearly an order of magnitude. Data shown 108 here is from the AC transport puck used in a PPMS, as described in Section 4.2.3.2, although data from the cryostat on the same samples was very comparable. The inset of Figure 59 displays only the data for samples 3 and 4 so that the temperature dependence of xxz is more apparent than when compared to that of sample 5 in the main frame. The thermal conductivity in sample 5 is an additional order of magnitude larger than that of sample 3, with a pronounced minima near 100 K, above and below which the thermal conductivity increases. Obtaining thermal conductivity data below 50 K in the mini puck setup for sample 5 was not possible due to the base temperature of the sample increasing significantly in order to obtain a reasonable and resolvable temperature gradient.

Thermal conductivity tends to be a robust indicator of sample quality in that heat flow is sensitively inhibited by defects and impurities. Samples 3 and 4 were both measured in similar manners, each in two different instruments which gave similar results for each sample. The samples are from separate batches, but their trends of xxz in temperature are similar even though they deviate by an order of magnitude. These results indicate that the quality of the two samples is most likely different.

The temperature dependence of the thermal conductivity of sample 5 further deviates from both other adiabatic samples (3 and 4) and even shows a different trend as a function of temperature. Lack of accuracy in the mini puck setup, mentioned in Section 4.2.3.1, is a potential cause for this. At room temperature, the calibrated CernoxesTM measured temperatures offset by up to 10 K from the PPMS chamber temperature. Although the temperature measurement used in the calculation of xxz is differential, this offset is worrisome and further leads to a measurement of the residual temperature difference at an inaccurate temperature.

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Furthermore, the CernoxesTM were hung by fishing line to a relatively large brass frame – if one

CernoxTM is closer to the frame than the other, a temperature offset could occur in one CernoxTM but not the other. This setup is shown in the inset of Figure 1(a) in Ref. [93]. Thus, heat losses are likely to be larger in the mini puck setup used for sample 5, in comparison to the setup used for samples 3 and 4, which could ultimately lead to a higher apparent thermal conductivity in sample 5. Nevertheless, sample 5 is from the same batch as samples 1 and 2, the two isothermal samples, and is considered in this study since sample quality is similar within a given batch. As mentioned previously, thermal conductivity could not be measured on the isothermal samples

(samples 1 and 2) due to the nature of the isothermal mount and a desire to keep the samples in tact rather than attempt to remove them from their silicon base plates, potentially cracking or cleaving the single crystals.

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4.3.3 Magnetizationg

Magnetization measurements were taken cooling in zero-field and cooling in a 1000 Oe magnetic field, with the magnetic field applied parallel to the <110> crystal axis in Figure 60 and parallel to the <001> crystal axis in Figure 61. Both data sets were taken on sample 7. All data sets demonstrate a distinct change in slope near TS ~ 240 K, which is predicted to be the spin- canting temperature (although the signal is too low to be captured via neutron scattering experiments). At TS, both zero field-cooled data sets show a small maximum, and a sharp increase in slope with decreasing temperature is observed in the field-cooled data. The Curie tails present below 20 K on all four data sets are likely due to magnetic impurities in the sample mount.

Figure 60: Magnetization as a function of temperature in sample 7 of YbMnBi2 along the <110> crystal axis. Data and image courtesy of Kaustuv Manna. g The data presented in this section was measured and analyzed by K. Manna. It is presented here to give a full transport story for YbMnBi2. 112

Figure 61: Magnetization as a function of temperature in sample 7 of YbMnBi2 along the <001> crystal axis. Data and image courtesy of Kaustuv Manna.

The magnetic field dependence of the magnetization of YbMnBi2 is shown in Figure 62 for Happ||<110> and in Figure 63 for Happ||<001>; both data sets are taken on sample 7. At 2 K, well within the Curie tails of Figure 60 and Figure 61, the magnetization follows a Brillouin function, suggesting the presence of magnetic impurities in or around the sample. In a magnetic field and below room temperature, the magnetic moment with Happ||<110> is over two times greater than that for Happ||<001>. Small hysteresis loop opening is observed near 0 Oe, although this is potentially also a measurement artifact due to a magnetic impurity in or around the sample. The magnetic moment of YbMnBi2 near zero-field and its consequential hysteresis loop in this field regime remains under study.

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Figure 62: Magnetization as a function of applied magnetic field in sample 7 of YbMnBi2 along the <110> crystal axis. Data and image courtesy of Kaustuv Manna.

Figure 63: Magnetization as a function of applied magnetic field in sample 7 of YbMnBi2 along the <001> crystal axis. Data and image courtesy of Kaustuv Manna.

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4.4 Thermomagnetic Transport

4.4.1 Isothermal Thermomagnetic Transport

Results of the magnetic field dependence of the isothermal Nernst thermopower, xyz, at set temperatures for the magnetic field applied parallel to the direction of Berry curvature, the

<110> axis as depicted in Figure 50, are shown in Figure 64 for sample 1. This data has distinct and unique features in comparison to that typically expected in materials without a net Berry curvature. First, the data clearly shows a sharp peak near 0 Oe, a result not conventionally found in the zero-field limit of the Nernst effect. In ferromagnetic materials, an anomalous and an ordinary Nernst effect regime are both expected, such as that seen in Fe (discussed in Chapter

2)59, with the former occurring in the zero-field limit and the latter occurring in the high-field limit. Additionally, hysteresis is expected in the zero-field, anomalous Nernst effect due to the remnant magnetization and coercive field of the ferromagnet itself, such as that seen in Fe in

Figure 28. In non-magnetic Weyl semimetals, these field regimes have also been seen, such as that in NbP97 (discussed in Chapter 3) and in TaAs102. In these materials, different slopes in the different field regimes are expected, but the magnetic field dependence of the Nernst thermopower is monotonic over a given field regime. Here, the sharp peak seen at temperatures below room temperature indicates non-monotonicity in the zero-field limit and looks similar to the zero-field limit peaks seen in the Hall resistivity, xy, for sample 7 in Figure 57. Furthermore, no hysteresis is seen between the magnetic field sweeping directions, even in the zero-field limit, indicating that the mechanism causing this unique transport is robust and unrelated to a hysteretic magnetic effect.

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Moreover, this data shows a large and finite value for xyz at all measured temperatures at

-1 zero field. Conventionally in a non-ferromagnetic material, the value of xyz must be 0 V K at

0 Oe since there is no source of a skew force via the Lorentz force in the absence of an externally applied magnetic field. Here, the Berry curvature direction is aligned mutually perpendicular to the directions of both the applied temperature gradient and the measured electric field. The finite value of xyz at 0 Oe offers evidence for not only the existence of a net Berry curvature along the z-axis, <110>, of YbMnBi2 but also its ability to act as an “intrinsic magnetic field” in k-space and thus a source of a skew force along the y-axis, pushing electron movement perpendicular to the applied temperature gradient parallel to the x-axis. The large magnitude of xyz at 0 Oe, exceeding 1000 V K-1 at 59.44 K, indicates the efficiency with which a net Berry curvature contributes a skew force to the Nernst effect in YbMnBi2. Although confirmation studies are ongoing, the sharp peak in xyz’s magnetic field dependence, the finite and large value of xyz at 0

-1 Oe, and the entire curve offset in xyz from 0 V K at 0 Oe are expected to be direct results of a net Berry curvature existing along the <110> crystal axis in YbMnBi2.

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Figure 64: Nernst effect magnetic field sweeps in sample 1 of YbMnBi2 with Happ parallel to Berry curvature direction, the <110> crystal axis

For comparison’s sake, the same single crystal of YbMnBi2 was measured in the magneto-thermopower geometry to obtain the magnetic field dependence of xxz, with results shown in Figure 65. The Seebeck coefficient, xxz at 0 Oe, is clearly negative for the temperatures shown. Additionally, the thermopowers obtained from this longitudinal geometry are up to three orders of magnitude smaller than those obtained from the transverse geometry of the Nernst effect. Although the peak seen in the zero-field limit of the Nernst effect is present in the magneto-Seebeck effect as well (although inverted in sign due to the sign convention of the

Seebeck effect compared to that of the Nernst effect), this is potentially simply a parasitic effect.

Because the Nernst thermopower is significantly larger than the conventional thermopower, a

117 slight voltage wire misalignment in the y-direction would result in picking up a contribution from

xyz in xxz, contaminating the xxz signal. Data here is noisier than that of Figure 64, especially at low temperature in the 21.95 K curve. This curve has much more noise due to the increased thermal conductivity in this temperature range, indicated by data from sample 5 (made from the same batch) and shown in Figure 59, making it more difficult to develop a temperature gradient large enough to produce a longitudinal Seebeck voltage.

Figure 65: Magneto-thermopower magnetic field sweeps in sample 1 of YbMnBi2 with Happ parallel to Berry curvature direction, the <110> crystal axis

The Nernst effect was also measured in a single crystal of YbMnBi2 with the direction of the predicted net Berry curvature not aligned to the direction of the externally applied magnetic

118 field; here, the magnetic field was aligned to the <001> crystal axis as shown in Figure 52.

Results for sample 2 are shown in Figure 66. At all temperatures and fields, xyz(z||<001>)

(sample 1) is much smaller than xyz(z||<110>) (sample 2). Furthermore, the large peak seen in the zero-field limit of xyz(z||<110>) is no longer apparent in xyz(z||<001>). Although some temperatures indicate that xyz(z||<001>) has a finite value at 0 Oe, this value is small and within the error of the measurement. Similarly to the low temperature data of Figure 65, data here has an increased noise level, especially in the 74.23 K and 19.75 K curves. Again, the increase in thermal conductivity in this temperature range makes it difficult to develop a sufficient temperature gradient to produce a Nernst voltage. Noteworthy, though, is the fact that the scale of the axes is over 200 times smaller than those in Figure 64, making noise more apparent here.

119

Values of the thermopowers at 0 Oe were extracted from Figure 64, Figure 65, and

Figure 66. These values are plotted in comparison to each other and steady-state, zero-field measurements of the same effects in Figure 67. The label “field sweeps” indicates data at 0 Oe extracted from measurements in which the magnetic field was swept in both directions, with the positive field sweep value averaged with the negative field sweep value. The label “steady state” indicates data at 0 Oe from measurements in which the magnetic field was constant and set to 0

Oe throughout the entire measurement. As seen in Figure 67, both steady state and field sweep

120 data at 0 Oe for each measurement set (xyz, z||<110> in sample 1; xxz, z||<110> in sample 1; and

xyz, z||<001> in sample 1) are consistent.

Data in Figure 67 offers further evidence for Berry curvature being the cause of the large peak and zero-field value of xyz(z||<110>) in sample 1 – the effect disappears in the longitudinal geometry of xxz(z||<110>) in sample 1 and the transverse geometry of xyz(z||<001>) in sample

2. Since Berry curvature is predicted to act like an intrinsic magnetic field50,51, it can contribute to transport via a skew force. When the direction of net Berry curvature is parallel to the direction of either the applied temperature gradient or the measured transverse field, the Berry curvature exerts no skew force on the charge carriers. Thus, Berry curvature is not predicted to contribute to thermoelectric transport in this manner when the measured electric field is parallel to the measured temperature gradient (regardless of the direction of the Berry curvature), which is the geometry of xxz(z||<110>) and the magneto-Seebeck effect. This prediction is proven true as the 0 Oe value of xxz(z||<110>) is significantly smaller than that of xyz(z||<110>); it is also

-1 comparable to 0 V K at all temperatures when plotted against xyz(z||<110>). When a transverse geometry is maintained but the direction of net Berry curvature is no longer mutually perpendicular to the direction of the applied temperature gradient and measured transverse electric field, as in the measurement of xyz(z||<001>) in sample 2, the 0 Oe Nernst thermopower is again predicted to disappear as there is no longer a source of a skew force to push charge carrier movement perpendicular to a temperature gradient since a mutually perpendicular magnetic field or Berry curvature is no longer present. This prediction is also proven true as the

0 Oe value of xyz(z||<001>) is significantly smaller than that of xyz(z||<110>) and comparable to

-1 0 V K at all temperatures when plotted against xyz(z||<110>). Although both xxz(z||<110>) 121 and xyz(z||<001>) do show a temperature dependence as indicated by Figure 65 and Figure 66, respectively, their temperature dependences of the 0 Oe values are not as distinct as those in

xyz(z||<110>), making them nearly unnoticeable in Figure 67.

Figure 67: Isothermal thermopower (Nernst and conventional) as a function of temperature for z- axis parallel to <110> and <001>. Large signal is only seen in transverse geometry with z-axis parallel to <110> crystal axis, the isothermal Nernst effect measurement in sample 1. Data taken from field sweep measurements is shown as closed diamonds, and data taken from steady-state measurements is shown as open triangles. 122

A complete study of the Nernst effect involves analysis of the Nernst coefficient, which is found in Figure 68 for Happ parallel to both the <110> (sample 1) and <001> (sample 2) crystal axes. Both data sets plot the temperature dependence of Nxyz in the high-field limit (60 kOe <

|Happ | < 90 kOe); Nxyz in the zero-field limit is not calculated due to the large peak in

97 xyz(z||<110>) creating a discontinuity over this magnetic field range. Unlike in NbP (Chapter

3), Nxyz for the z-axis aligned to <110> shows two peaks: a positive peak near 50 K, 26

-1 -1 -1 -1 V K T and a negative peak near 160 K, -4V K T . The positive peak in Nxyz correlates closely to the temperature of the large peak in the 0 Oe value of xyz(z||<110>). Nxyz for the z- axis parallel to <001> shows mostly no temperature dependence and is extremely small in comparison.

123

4.4.2 Adiabatic Thermomagnetic Transport

The results of the adiabatic measurements described in Section 4.2.3.2 were consistent between the AC transport puck in the PPMS and measurements taken in the cryostat. Because the PPMS has a larger magnetic field range and temperature range, data from the PPMS will be shown in this section.

Figure 69 compares the temperature dependence of xyz(z||<110>) at 0 Oe between the isothermal sample 1 and the adiabatic samples 3 and 4; the inset displays only data from the adiabatic samples to make visible the slight temperature dependence of the data. Data from 124 sample 1 is shown in green, data from sample 3 is shown in blue, and data from sample 4 is shown in red. Both adiabatic samples 3 and 4 have a zero-field Nernst thermopower under 10

-1 V K below 200 K; at 200 K, the samples depart with xyz at 0 Oe in sample 3 becoming negative and in sample 4 becoming positive. For all temperatures, the magnitude of both of the adiabatic zero-field Nernst thermopowers is smaller than the isothermal Nernst thermopower of sample 1, not displaying the huge peak seen in the isothermal data.

125

The temperature dependence of the Nernst coefficient is shown in Figure 70 for both adiabatic samples 3 and 4 and the isothermal sample 1. Although the isothermal magnetic field dependence of xyz in sample 1 had a sharp peak near 0 Oe, making an isothermal zero-field limit

Nxyz meaningless, the adiabatic samples did show a more conventional magnetic field dependence of xyz in that they did not have this sharp peak; thus, the low-field limit of the 126 adiabatic Nxyz is plotted here for samples 3 and 4. Open circles represent the low-field limit (|H|

< 5 kOe) and plus signs represent the high-field limit (|H| > 60 kOe). Data from sample 1 is shown in green, data from sample 3 is shown in blue, and data from sample 4 is shown in red.

The high-field limit of the adiabatic Nxyz for sample 3 is slightly larger than that of sample 4, which is consistently near 0 V K-1 T-1 for all temperatures. The low-field limit of the adiabatic

-1 -1 Nxyz for sample 3 is significantly larger than that of sample 4, peaking near 15 V K T at 150

-1 -1 K while Nxyz for sample 4 remains below 5 V K T at all temperatures. Comparatively, the isothermal high-field Nernst coefficient for sample 1 is at least an order of magnitude greater than the both adiabatic high-field Nernst coefficients up to approximately 100 K, where it becomes negative then returns to larger, positive values above 300 K. At this time, this observation remains unexplained.

127

Because YbMnBi2 is calculated to have a net Berry curvature below the ordering temperature for the canting of the Mn ions, which was aligned to the z-axis in certain measurements described in this section, the straight-forward comparison used between adiabatic and isothermal Nernst effect data in NbP97 (Chapter 3), which does not have a net Berry

128 curvature, does not hold here. Namely, the adiabatic and isothermal Nernst thermopowers and

Nernst coefficients are not similar within reasonable measurement error, while they were for

NbP.97 Here, they vastly deviate from one another. Predicted results in comparing adiabatic to isothermal Nernst effect data included either seeing strong similarities between the data sets or seeing large differences in the data sets for which a strong, Berry curvature-induced thermal Hall effect could be the root cause. No thermal Hall effect was observed in the adiabatic samples 3 and 4 within the sensitivity of the measurement systems, despite repeated attempts. The thermal

Hall effect was shorted out by design in the isothermal setup of sample 1. This deviation in results between isothermal and adiabatic Nernst effect measurements is presently still being explored and is left currently unexplained. A point to make here, though, is that Weyl semimetals have been seen to be very sensitive to impurities, especially in thermal transport.97

Samples 1, 3, and 4 did come from different batches and are the samples on which the Nernst effect, with z-axis parallel to the <110> crystal axis, was measured. Thus, residual doping levels and sample quality could be different among the samples. A similar discrepancy was found in the Hall effect data as well, offering potential explanation for the contrast in data from Figure 57 and Figure 58 beyond the difference in the measurement technique used (5-wire with zero-field balance and 4-wire without zero-field balance) since the samples were from separate batches.

The temperature dependence of the magneto-thermopower in YbMnBi2 is compared between both adiabatic samples 3 and 4 and the isothermal sample 1 at 0 Oe, +90 kOe, and -90 kOe in Figure 71. Open circles represent 0 Oe data, plus signs represent +90 kOe data, diamonds represent -90 kOe data; green is the isothermal sample 1, blue is adiabatic sample 3, and red is adiabatic sample 4. The +90 kOe and -90 kOe values were not averaged as the data was not fully

129 symmetric, although the +90 kOe and -90 kOe thermopowers for both adiabatic samples are comparable within the sample data sets themselves. All values of xxz for all samples are negative at all temperatures. The isothermal sample 1 does have an apparent extrema in all three data sets, near 50 K for +90 kOe and -90 kOe and near 200 K for 0 Oe.

-1 xxz should rigorously approach 0 V K at 0 K for all magnetic field values, although this trend is not evident in the isothermal data of sample 1. Sample 3 and sample 4 appear to monotonically approach 0 V K-1 at 0 K. The low-temperature peak seen only in the isothermal data of sample 1 is indicative of phonon drag. Wire misalignment, consequently picking up a signal from the Nernst effect, can be eliminated as a cause here since the +90 kOe and -90 kOe are considerably similar. If wire misalignment were the case, xyz would not contribute to xxz in the same manner at positive and negative magnetic fields due to its anisotropy of xyz as a function of magnetic field seen in Figure 64. Additionally, wire misalignment would contribute consistently at all temperatures while phonon drag is a temperature-dependent effect. Because phonon drag only appears to contribute in the isothermal sample 1 (no low-temperature peak is apparent in either adiabatic sample 3 or 4), the sample quality of the isothermal sample is predicted to be higher than that of the adiabatic samples because it does pick up a contribution indicative of phonon drag.

130

Figure 71: Magneto-thermopower as a function of temperature in the isothermal sample 1 and the adiabatic samples 3 and 4 of YbMnBi2

4.5 Device Applications

In order to characterize YbMnBi2 for potential applications in transverse thermoelectric devices, calculating the temperature dependence of zT is necessary. As described in Chapter 1, zT is conventionally calculated at 0 Oe in a longitudinal geometry. Since YbMnBi2 excels in the advantageous transverse geometry without the need for a magnetic field, zT will be calculated at

0 Oe using the Nernst thermopower. Nernst thermopower is obtained from Figure 67, using the isothermal 0 Oe values extracted from magnetic field sweeps with magnetic field applied parallel to the <110> crystal axis in sample 1; resistivity is assumed to be isotropic in plane, xx = yy, such that values of xx at 0 Oe in sample 7 are used from Figure 55; thermal conductivity is obtained from Figure 59 using data from sample 5 since this sample is from the same batch as 131 sample 1. Worth noting here is that Nernst thermopower values were obtained isothermally while resistivity and thermal conductivity values were obtained adiabatically.

The temperature dependence of the transverse, 0 Oe zT of YbMnBi2 is plotted in Figure

103 72. This zT is compared to that of commercially available Bi2Te3; the state-of-the-art PbTe doped with 2 mol% Na and nanostructured with 4 mol% SrTe, which has the highest published

104 59 zT; and single-crystal Fe. The transverse, 0 Oe zT of YbMnBi2 has a maximum value of 2.4

104 at 60 K, making it comparable to the maximum published zT . Furthermore, YbMnBi2 shows the maximum zT for any transverse geometry, especially those eliminating the need for an externally applied magnetic field. Therefore, YbMnBi2 not only compares to state-of-the-art zT, but it also utilizes the device-advantageous transverse geometry6 while eliminating the conventional need for a large, externally applied magnetic field.

132

Due to these exciting results, YbMnBi2 has been suggested as a candidate material for use in Ettingshausen cooling devices.105 Cooling applications are most appropriate here as the zT peaks in the cryogenic temperature range. As stated previously but worth repeating, data in this chapter are still under investigation and confirmation studies, especially in relation to zT.

4.6 Outlook

This project is currently a work in progress as confirmation studies are currently ongoing.

Foremost, the excitingly large Nernst thermopower exceeding 1000 V K-1 at 60 K has only been observed in one isothermal sample, sample 1. Although this data was repeatable after temperature cycling and repeatable in multiple measurement systems, the results needs to be 133 confirmed for sample repeatability. Furthermore, the comparison of isothermal to adiabatic data described in Chapter 3 for NbP was not corroborated with the adiabatic samples (samples 3, 4, and 5) of YbMnBi2 presently used. Namely, the results in the absence of an externally applied magnetic field were not the same for the adiabatic and isothermal samples. This could reasonably be the case if a measurable thermal Hall effect had been observed, which in the adiabatic samples (samples 3, 4, and 5) could potentially compensate the zero-field Nernst thermopower seen in the isothermal sample (sample 1). The quality of the adiabatic samples is in question due to the order of magnitude difference between their thermal conductivities and the absence of the low-temperature phonon-drag peak observed only in the isothermal sample, both of which are typically good indicators of decreased sample quality. As confirmation studies are currently in progress, sample quality is being considered along with new physics yet to be explained.

Determining if the observed hysteresis loop opening in the magnetic field dependence of the magnetization, specifically when the magnetic field is applied parallel to the <110> crystal axis (Figure 62), is important in determining the effects of Berry curvature in transport within

YbMnBi2, specifically in the large peak near 0 Oe of the Nernst thermopower, xyz, observed in

Figure 64 for sample 1. If a ferromagnetic hysteresis loop in the magnetic field dependence of the magnetic moment is observed, such as that schematically shown in the left frame of Figure

73, the remnant moment is likely the cause of a large 0 Oe Nernst thermopower. The low-field

(anomalous) Nernst effect is then typically due to the remnant magnetization within a given material and therefore the sign of xyz is dependent on the field sweeping direction, which holds

106,107 for the crystal symmetry of YbMnBi2 . However, the low-field Nernst effect would then

134 also have a field dependence similar to that of the left frame of Figure 73, which is actually the case for Fe, a ferromagnet, with data shown in Figure 28. Experimentally (Figure 64), the curve looks similar to the right frame of Figure 73, showing no sign dependence on the magnetic field sweeping direction, which excludes simple ferromagnetism as the cause of the fact that xyz(0

Oe) does not equal 0 V K-1 when the <110> crystal axis is parallel to the z-axis. Thus, if the observed data for YbMnBi2 follows both trends shown schematically in Figure 73, Berry curvature is likely the cause of the large Nernst thermopower observed in the absence of an externally applied magnetic field.

Figure 73: Expected comparison of hysteresis in magnetic field dependence of magnetic moment and Nernst thermopower for Berry curvature-dominated transport. Nernst thermopower, serving as a probe of Berry curvature, shows no sign change depending on the magnetic field sweeping direction, as observed in the isothermal Nernst effect measurements. Magnetization is expected to show an opening of a hysteresis loop and thus a sign change for the remnant moment and coercive field depending on the magnetic field sweeping direction.

135

A point to be made when discussing the zT presented here in Figure 72 is the contradicting methods in which the different components of data were collected. Foremost, the

Nernst thermopower used in this calculation was explicitly the large isothermal Nernst thermopower at zero-field, measured on sample 1. By definition, thermal conductivity cannot be measured isothermally, so the thermal conductivity used to calculate zT was adiabatic and came from sample 5. Thus, the Nernst thermopower, electrical resistivity, and thermal conductivity data were all taken on three separate, but similar, samples (sample 1, 7, and 5, respectively).

Preliminary resistivity data was later taken on the isothermal sample used in the Nernst measurements (sample 1), and it did correlate within a factor of two with the resistivity data used in the zT calculations. However, the data used in the zT calculations is that from the separate sample as that data was more extensive and proven to be repeatable. In order to obtain thermal conductivity measurements on the sample used in the Nernst measurements, the pristine single crystal of YbMnBi2 would have to be removed from its silicon substrate. Removing GE Varnish is difficult; doing so would ultimately risk the sample’s existence, which is why this has been disregarded as an option currently.

All of this being said, the data shown here is still exciting and worthy of further study with these precautions. When YbMnBi2 is evaluated as a thermoelectric material, an appropriate manner for calculating zT, taking isothermal and adiabatic parameters into account, will need to be determined.

136

Chapter 5. Concluding Remarks

The field of thermoelectrics was born in 1821 when Thomas J. Seebeck first discovered the thermoelectric, or appropriately-named Seebeck, effect. Since then, scientists working in this field have been driven to improve the efficiency of thermoelectric materials, ultimately hoping to achieve a zT to make thermoelectric devices competitive as renewable energy sources. The field has focused mainly on semiconductors and has recently struggled to achieve significant zT improvements in materials that are economically practical. Here, two alternative approaches have been offered in metals and Weyl semimetals, both utilizing magnetization dynamics (either within the material or applied externally) and two-carrier effects.

The first material class explored in this dissertation was ferromagnetic transition metals, which are stronger, made of more common materials, typically cheaper, and easier to form into net shapes than semiconductors. Despite these advantages, metals are known for having a low thermopower and thus zT. When F. J. Blat originally studied the thermopower of single-crystal

Fe, he attributed its uncharacteristically large thermopower to magnon transport but did not experimentally prove his explanation.12 This work mapped out the thermomagnetic transport tensor of single-crystal Fe and studied the thermoelectric effect in both Co and Ni, including a porosity study. Two theoretical models were developed as well – one based on the ideal gas law and Newtonian mechanics, while the other focused on spin dynamics and relativistic quantum mechanics. Under reasonable conditions, both theories converged and followed the temperature

137 dependence of single-crystal Fe’s thermopower quite well. The strong agreement between experiment and theory led to the conclusion that magnon-drag dominates the thermopower of single-crystal Fe and Co, especially at low temperatures. This magnon-drag model was extended to the Nernst effect, which looked at spin-up and spin-down electrons as separate particles contributing in separate conduction channels (a two-carrier approach), and magnon drag was again found to contribute to transport even in the transverse geometry of the Nernst effect.

Thermopower data in dense samples of both Co and Ni indicated a phonon-drag contribution to thermopower, which motivated a porosity study. The temperature dependence of thermopower was compared between dense and porous samples in Fe, Co, and Ni. The phonon-drag peaks present in dense samples of Co and Ni disappeared in the porous samples, and the thermopower of both samples more closely matched that of the model in the porous samples. This implies that magnon drag is a less sensitive transport mechanism than phonon drag and that it is robust against defects.

The model developed has predictive power in terms of increasing the magnon-drag thermopower, and thus total thermopower and zT, in ferromagnetic transition metals. Magnon- drag thermopower is proportional to the magnon specific heat, which comes from the magnon structure of the material, and is inversely proportional to the number of carriers. If a metallic alloy is allowed to maintain this magnon structure with a decreased number of carriers, magnon- drag thermopower could be increased, ultimately increasing the thermoelectric efficiency of metallic materials.

This dissertation then moved to explore thermomagnetic transport in Weyl semimetals, a solid state analogy to the physics predicted by H. Z. Weyl31 in 1929. Weyl semimetals were

138 recently experimentally realized in TaAs24 in 2015. Weyl semimetals were predicted to have a large Nernst effect, in part due to semimetals intrinsically being two-carrier systems9 and in part due to their unique band structure51. The Nernst effect utilizes the device-advantageous transverse geometry6 so emphasis was placed on it. Work began with the inversion symmetry- breaking Weyl semimetal NbP, which is part of the same family of transition metal monopnictides as TaAs. The Nernst effect and the Seebeck effect were explored both isothermally and adiabatically. The Nernst thermopower was seen to be unsaturated at the maximum magnitude of applied magnetic field, 9 T, exceeding 800 V K-1 at 109 K.

Furthermore, the low-field and high-field Nernst coefficients were both non-monotonic, peaking near 50 K in the low-field regime and near 90 K in the high-field regime. The conventional thermopower from the Seebeck effect had an extrema at -8 V K-1 near 100 K, nearly two orders of magnitude less than the maximum of the Nernst thermopower, as expected in a two-carrier system such as a semimetal.

A theoretical model for an inversion symmetry-breaking Weyl semimetal was developed from the Weyl semimetal Hamiltonian99 with no adjustable parameters. This model showed strong qualitative comparison to experiment; quantitatively, the two agreed within a factor of 4, which is a good correlation considering no fitting parameters were applied to the model. More noteworthy, though, was the explanation provided by theory for the peak in the Nernst thermopower. The chemical potential, having its own temperature dependence, moves from the valence band to the Weyl point as temperature increases near 100 K, at which point it gets pinned at the energy of the Weyl point which is also the energy of the minimum density of states.

Thus, the peak in Nernst thermopower near 100 K correlates with the temperature at which the

139 chemical potential reaches the energy of the Weyl point. Using density functional theory and experimental inputs from resistivity and magnetization measurements, the Fermi level was determined to be 0  -8.22 meV below the main Weyl point in the zero-Kelvin limit.

Furthermore, the trivial pockets known to exist in NbP were determined to not contribute to this transport. The model that strongly correlated to experimental data did not include their contributions; when contributions from trivial pockets were included, the model deviated from experiment quite notably. Thus, via a reductio ad absurdum argument, the trivial pockets do not majorly contribute to transport here, with contributions from the linear Dirac bands dominating.

An important discussion sparked from the transport study in NbP is that surrounding adiabatic vs. isothermal measurement techniques, both of which were completed on NbP. When the thermal conductivity of a given material is strongly dependent on the external magnetic field, or a material is expected to exhibit a large thermal Hall effect, adiabatic thermal transport measurements will become contaminated in the presence of a magnetic field.95 This is the case for Weyl semimetals, as shown in the large difference between the adiabatic and isothermal magneto-Seebeck effect in NbP. This work suggests studying thermomagnetic transport in Weyl semimetals isothermally in order to properly isolate the thermomagnetic coefficients and to not contaminate their signals.

Although the maximum Nernst thermopower in NbP exceeds that of conventional semiconductor thermoelectric materials by a factor of 2-4 in the advantageous transverse geometry, the Nernst effect here required a large externally applied magnetic field. Predictions have shown that in Weyl semimetals that break time-reversal symmetry rather than inversion symmetry, an externally applied magnetic field may not be necessary to achieve transverse

140 thermoelectric transport. The breaking of time-reversal symmetry gives rise to a net Berry curvature in the absence of an externally applied magnetic field, which acts like a magnetic field

50,51 that is intrinsic to the material. YbMnBi2 has been predicted as a time-reversal symmetry- breaking Weyl semimetal,49 and transverse transport in this material was explored in this work.

YbMnBi2 exhibits a huge isothermal Nernst thermopower in the absence of magnetic field, exceeding 1000 V K-1 near 60 K when the applied temperature gradient and measured electric field are mutually perpendicular to the direction of the net Berry curvature. In the longitudinal geometry and when the temperature gradient and electric field are perpendicular to each other but not the direction of net Berry curvature, the effect goes away.

This work is still under confirmation studies for sample repeatability (only one isothermal sample has been measured) and comparison to adiabatic data (which has been taken on two separate samples and does not compare to isothermal data in the same manner that data from the two measurement techniques compared in NbP). Nevertheless, this result is outstanding if it can be confirmed as true and indeed due directly to the Berry curvature of the material.

Preliminary zT calculations, where thermal conductivity was measured adiabatically on a sample from the same batch as the isothermal sample, show a peak zT of 2.4 at 60 K. This zT exceeds that of the state-of-the-art104 thermoelectric material while utilizing the advantageous transverse geometry in the absence of an externally applied magnetic field. Due to the low temperature of the peak zT, applications here will most likely be found in cryogenic cooling in an

Ettingshausen cooler.

The field of Weyl semimetals has moved incredibly quickly since the first experimental signature of a Weyl fermion was seen in 2015, but there is much exciting work still to be done.

141

Weyl physics was first predicted to be seen in high-energy physics, and only within the past year has the condensed matter field made ties back to particle physics within this class of materials, which has also recently been proven to show experimental signatures of electron hydrodynamics .88,108 The relationship and connection between the two fields can lead to further understanding of the exotic transport found within Weyl semimetals and will hopefully have important implications for applications in devices. Because the field is so young, Weyl semimetals have not made their way into engineering applications as of yet, but they do show promising signatures making them worthy of further development.

142

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