Downloadable in Supplementary Materials of Ref[190], and the Training Process Is Performed Using the QUIP Package

THERMAL CONDUCTIVITY OF COMPLEX CRYSTALS, HIGH TEMPERATURE MATERIALS AND TWO DIMENSIONAL LAYERED MATERIALS

By XIN QIAN B.S. Huazhong University of Science and Technology, 2014

A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy Department of Mechanical Engineering 2019

This thesis entitled: Thermal Conductivity of Complex Crystals, High Temperature Materials and Two Dimensional Layered Materials written by Xin Qian has been approved for the Department of Mechanical Engineering

Prof. Ronggui Yang, Chair

Prof. Baowen Li

Date:

The final copy of this thesis has been examined by the signatories, and we find

that both the content and the form meet acceptable presentation standards

of scholarly work in the above mentioned discipline.

ABSTRACT

Xin Qian (Ph.D, Mechanical Engineering)

Thermal Conductivity of Complex Crystals, High Temperature Materials and Two Dimensional

Layered Materials

Thesis directed by Professor Ronggui Yang

Thermal conductivity is a critical property for designing novel functional materials for engineering applications. For applications demanding efficient thermal management like power electronics and batteries, thermal conductivity is a key parameter affecting thermal designs, stability and performances of the devices. Thermal conductivity is also the critical material metrics for applications like thermal barrier coatings (TBCs) in gas turbines and thermoelectrics (TE).

Therefore, thermal conductivities of various functional materials have been investigated in the past decade, but most of the materials are simple and isotropic crystals at low temperature. This is because the first-principles calculation is limited to simple crystals at ground state and few experimental methods are only capable of measuring thermal conductivity along a single direction.

The objective of this thesis is to develop first-principles based atomistic modeling tools to study thermal conductivity and phonon properties of complex crystals, high temperature materials, as well as and ultrafast laser based pump-probe techniques to characterize anisotropic thermal conductivity of layered two-dimensional materials.

In the first part of this thesis, an integrated density functional theory and molecular dynamics

(DFT-MD) method is developed to model the thermal conductivity and phonon properties of hybrid organic-inorganic crystals, a special kind of complex crystals integrating both organic molecules and inorganic frameworks. This DFT-MD method first develops an empirical potential

iii field from first-principles DFT calculations, then predicts thermal conductivity using MD simulation. We applied this method to predict thermal conductivities of II-VI based hybrid crystals and organometal halide perovskites. An ultralow thermal conductivity (0.6 W/mK) is predicted in the perovskite CH3NH3PbI3, agreeing well with experimental measurements.

In the second part, instead of using empirical functional forms, a data driven machine learning algorithm is used to develop high-fidelity potential field for phonon modeling. We demonstrated that the machine learning based potential is a powerful tool for modeling phonons at high temperature, even for dynamically unstable high-temperature phases, which is a challenging problem for both empirical potential based MD and static first-principles calculations. Using a simple machine learning algorithm called Gaussian process regression, we developed potential field that can effectively capture the stabilization of BCC phase of Zirconium at 1188 K, which is predicted to be unstable using static first-principles calculations.

In the third part, a varied laser spot size technique based on time-domain thermoreflectance

(TDTR) is developed to characterize anisotropic thermal conductivity. This method is applied to measure both the thermal conductivity parallel to the basal planes as well as the through-plane thermal conductivity of transition metal dichalcogenides, a group of layered two dimensional materials. Interestingly, the through-plane thermal conductivity is observed to decrease with the increasing heating frequency (modulation frequency of the pump laser) from 0.6 to 10 MHz, due to the non-equilibrium transport between different phonon modes. A two channel thermal model is developed to capture the non-equilibrium transport and to derive the thermal conductivity at local equilibrium. This finding suggest that in electronic devices working at a few GHz, the material could tend to become much more thermally insulating than steady state, raising great challenges for near junction thermal management.

ACKNOWLEGEMENT

When taking a look back at this my PhD journey, indeed I wish to thank various people for their support and company. First and Foremost, I would like to offer my deep and sincere gratitude to my advisor Prof. Ronggui Yang for his continuous support during my PhD journey, for his vision and enthusiasm in scientific research and his immense knowledge. From him I learned how to probe and think critically of scientific problems, which will be invaluable for my future career.

In addition to my advisor, I would like to thank the rest of my thesis committee: Prof. Baowen

Li, Prof. David Marshall, Prof. Margret Murnane and Prof. Kurt Maute, for their encouragement and insightful comments and being supportive during the last five years.

I thank my fellow labmates and friends in our group for their encouragement and company:

Xiaokun Gu, Dongliang Zhao, Puqing Jiang, Xinpeng Zhao, Rongfu Wen, Shanshan Xu, Ablimit

Aili and Tianzhu Fan. I owe Xiaokun deeply since he always remained available and supportive whenever I have difficulties and problems in my research. Xiaokun is like my “second advisor” especially in my first two years. From him I learned not only research skills but also perseverance and patience. I enjoyed discussing scientific problems with him so much that we remained in frequent touch till today. Thanks to Donliang and Xinpeng for their support, knowledge and discussions, and of course, their sense of humor which brought me lots of happiness. I thank Dr.

Puqing Jiang for his help on the pump-probe experiments and lots of collaborated work with him.

I thank Prof. Xiaobo Yin in our department and Prof. Jun Liu at North Carolina State University for various scientific discussions and their advice and help on bringing back the pump-probe setup when the laser system was down. During the visiting time of Prof. Congliang Huang and Prof. Xu

Ji, the time I spent with them is helpful and joyful. Thanks to my friends: Junling Long, Haoran

Jiang, Lu Ma, Duanfeng Gao, Baochen Wu, Lili Feng and so on for their encouragement and

v support during the past five years. I thank the fellows playing soccer with me who are great resources to refresh myself during the weekends. I will miss the time spent with you guys.

Finally, I would like to thank my parents and grandparents for their unconditional love and understanding and support throughout my life, without which I can achieve nothing. (I would not forget to thank my girlfriend who never appeared in my entire life, allowing me to fully dedicate myself to research.)

Table of Contents CHAPTER I INTRODUCTION ...... 1

I.1 Advances and Challenges in Modeling and Characterizing Thermal Conductivity ...... 1

I.2 Thermal Transport in Hybrid Organic-Inorganic Complex Crystals ...... 4

I.3 Phonon and Thermal Properties of High Temperature Materials ...... 8

I.4 Thermal Transport in Two-Dimensional Layered Materials ...... 9

I.5 Objectives of this Thesis...... 11

I.6 Organization of this Thesis ...... 13

CHAPTER II THERMAL CONDUCTIVITY MODELING OF HYBRID ORGANIC-

INORGANIC CRYSTALS ...... 15

II.1 Introduction ...... 15

II.2 Simulation Strategy ...... 17

II.3 II-VI based Hybrid Organic-Inorganic Crystals ...... 23

II.4 Organometal Halide Perovskites ...... 34

II.5 Summary of this Chapter ...... 41

CHAPTER III MACHINE LEARNING DRIVEN ATOMISTIC MODELING ON PHONON

DIPSERSION STABILITY OF ZIRCONIUM ...... 42

III.1 Introduction ...... 42

III.2 Methodology of Building Machine Learning Potential ...... 46

III.2.1 Fitting Potential Energy Surface using GAP Method ...... 46

III.2.2 Generation of Training Database ...... 51

III.3 Results and Discussions ...... 55

III.4 Summary of this Chapter ...... 62

vii

CHAPTER IV PROBING ANISOTROPIC AND NON-EQUILIBRIUM THERMAL

TRANSPORT IN TRANSITION METAL DICHALCOGENIDES ...... 63

IV.1 Introduction...... 63

IV.2 Varied Spot Size Approach for Measuring Anisotropic Thermal Conductivity ...... 65

IV.3 Non-equilibrium Phonon Transport in Transition Metal Dichalcogenides ...... 72

IV.4 Summary of this Chapter ...... 87

CHAPTER V CONCLUSIONS AND FUTURE WORK ...... 88

BIBLIOGRAPHY ...... 90

APPENDIX ...... 111

Appendix I Equilibrium Molecular Dynamics for Thermal Conductivity Calculation ...... 111

Appendix II Detailed Data Reduction and Error Analysis of TDTR Measurement...... 115

viii

TABLES

Table II.1. Lattice parameters (Å) of α-ZnTe(en)0.5...... 30

Table II.2. The group velocities (m/s) of ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5 by solving the anisotropic wave propagation equation using the Christoffel package.[156] ...... 32

Table III.1. Hyper parameters for GAP with SOAP kernels...... 51

Table III.2 Detailed parameters for DFT calculations to generate training databases...... 54

FIGURES

Figure I.1. Schematic illustrating that hybrid organic-inorganic materials can be classified into three groups according the bonding strength between organic and inorganic components and their feature sizes...... 7

Figure II.1. Computation strategy for obtaining thermal conductivity and phonon properties using first-principles based atomistic simulations...... 22

Figure II.2. Atomic structures of the II-VI based hybrid organic-inorganic crystals. The three black arrows denote the crystal orientations. The x-axis is along the direction where the organic- inorganic layers are stacked together. Parallel to the ZnTe monolayers, the y-axis is along the atomic ridges of the ZnTe monolayers, and z-axis is in the perpendicular direction. (a) The unit cell of β-ZnTe(en)0.5. (b) The unit cell of α-ZnTe(en)0.5. (c) From up to down: the folding of Zn- Te hexagonal rings along the short diagonals in the β-phase, the upper view of β -ZnTe monolayers and the side view of the β -ZnTe monolayers along the blue arrow. (d) From up to down: the folding of Zn-Te hexagonal rings along the longest diagonals in the α-phase, the upper view of α- ZnTe monolayers and the side view of the α-ZnTe monolayers along the blue arrow. (e) Atomic structure of ZnTe(pda)0.5. (f) Atomic structure of ZnTe(ptda)0.5...... 25

Figure II.3. Construction of the potential field for α-ZnTe(en)0.5. The potential field is divided into three parts: inorganic potential EI, organic potential EO and the organic-inorganic coupling EI-O. For each part, the potential can be further divided into different interaction terms to describe bond stretching, angular bending, dihedral bending, and the van der Waals coupling...... 26

Figure II.4. DFT potential energy surface sampled by displacing the Te atom along the Zn-Te bond is fitted by the empirical potential field. The stretching of Zn-Te bond shown in the figure is not to scale...... 29

Figure II.5. (a) Elastic constants of α-ZnTe(en)0.5 calculated from MD and DFT simulations. (b) The strains corresponding to the elastic constants are indicate by arrows...... 30

Figure II.6. Comparison of elastic constants in α-ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5. . 32

Figure II.7. Comparison of thermal conductivity in α-ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5...... 33

Figure II.8. The crystal structure of (a) the tetragonal and (b) the pseudocubic phases of MAPbI3

(MASnI3) projected in (010) and (001) plane from left to right correspondingly. The a and b axes of the both phases are defined along the two directions with smaller lattice constants, and the c axis is defined along the direction with the largest lattice constant. The legend on the top indicates different elements depicted in the figure...... 36

Figure II.9. Comparison between the energy surfaces from DFT calculations and the empirical potential which fits the local curvature of the DFT energy surface. The rotational axis of MA ion is in the [001] direction...... 37

Figure II.10. Temperature-dependent thermal conductivity of MAPbI3 by MD simulations compared with the experimental results by A. Pisoni et al.[164] and direct AIMD simulation by Yue et al.[171] ...... 40

Figure III.1 (a-b). Comparison of (a) energy and (b) inter-atomic forces between GAP and AIMD calculations of the HCP-Zr. (c-d). Comparison of (c) energy and (d) force components between GAP and AIMD calculations of the BCC-Zr...... 55

Figure III.2 (a) Equation of state (energy v.s. volume) of HCP-Zr and BCC-Zr calculated by GAP and DFT. (b) Symmetry-irreducible elastic constants of HCP-Zr (left panel) and BCC-Zr (right panel). The experimental elastic constants of HCP-Zr is from ref. [204] (c) Phonon dispersion of HCP-Zr. INS measurement data is taken from ref. [27] (d) Phonon dispersion of BCC-Zr. INS measurement data is taken from ref. [173] ...... 57

Figure III.3 (a) PES along eigenvectors at high symmetry point N. Q1 and Q2 correspond to dimensionless normal coordinate of the two TA modes with the order of increasing frequency. (b)

PES along the Q1 direction with Q2 = 0. (c) PES along the Q2 direction with Q1 = 0...... 59

Figure III.4 (a-b) SED of HCP-Zr at (a) 100 K and (b) 300 K. (c) SED of bcc-Zr at 1188 K. (d) SED as a function of phonon frequency at q = (0.3,0,0). The dashed lines indicate the frequency measured by INS in ref. [173] at 1188 K...... 61

Figure IV.1. (a) A schematic of the TDTR system implemented in our lab. Abbreviations are listed as follows. EOM, electric-optical modulator; PBS/NPBS: polarized/non-polarized beam splitter;

λ/2: half wave plate. (b) Schematic for measuring kz using a large spot size and a high modulation frequency of TDTR measurements. (c) Schematic for measuring kz using a small spot size and a low modulation frequency of TDTR measurements...... 67

Figure IV.2. Sensitivity ratio map Skz/Skr for the choices of modulation frequency and laser spot radius...... 69

Figure IV.3. Measuring (a) cross-plane and (b) in-plane thermal conductivity of a (0001) ZnO using varied laser spot radii...... 70

Figure IV.4. Measured kr and kz of HOPG, h-BN, ZnO, TiO2 and silica compared with the literature values. Literature values are from Schmidt et al.[246], Sichel et al., [247] Wu et al.,[9] Thurber and Mante, [29] and Sugawara[248]...... 70

Figure IV.5. (a) In-plane thermal conductivity for SI and n-type 4H-SiC compared with the first- principles calculation by Protik et al.,[249] and the steady-state measurement by Morelli et al. [250] (b) The cross-plane thermal conductivity for SI and n-type 4H-SiC compared with the calculation by Protik et al. [249] and the laser flash analysis measurement by Wei et al. [251] ...... 71

Figure IV.6. (a) In-plane thermal conductivity for SI 6H-SiC compared with the first-principles calculation by Protik et al.,[249] the steady-state measurement by Morelli et al.[252] and the radiation thermometry by Burgemeister et al.[253] (b) The cross-plane thermal conductivity for SI 6H-SiC compared with the calculation by Burgemeister et al.[253], Protik et al. ,[249] and Nilsson et al. [254] ...... 71

Figure IV.7. TDTR experimental data (symbols) along with the fittings from a thermal model (solid, red lines) and 20% bounds on the fitted through-plane (dash, blue lines) and in-plane (dash- dot, green lines) thermal conductivity values. The data were taken as MoS2 measured at room temperature, using two different laser spot sizes (w0 = 6 μm and w0 = 24 μm) at a modulation frequency 1 MHz. The thermal conductivities obtained from the fits are 4.4 ± 0.45 W m-1 K-1 along the through-plane and 80 ± 17 W m-1 K-1 along the in-plane direction, respectively...... 73

xii

Figure IV.8. Apparent cross-plane thermal conductivity and interface thermal conductance GA of

Al/MoS2 and Al/WSe2 systems from TDTR measurements at 300 K and 100 K as a function of modulation frequency. Symbols represent TDTR measurements and the solid lines indicate predicted results using the two-channel model. The dash lines indicate the thermal conductivity and interface conductance at the local thermal equilibrium limit. The unit of g is W/m3K...... 77

Figure IV.9. Schematic illustrating heat flow pathways from Al transducer across the interface into the TMD substrates. A non-equilibrium thermal resistance RNE characterizes the energy exchange between the two groups of phonons in the substrate with a temperature difference. The effect of

RNE on the apparent thermal conductivity kz and interface conductance GA depends on the relative length of the non-equilibrium distance dNE (which denotes the exponential decay of the temperature difference), as compared to the thermal penetration depth dp (which denotes the exponential decay of the average temperature profile)...... 78

Figure IV.10. Measured in-plane and through-plane thermal conductivities (solid symbols) of

MoS2, WS2, MoSe2 and WSe2 as a function of temperature, compared with literature values, both numerically and experimentally. The solid curves are calculated in-plane and through-plane thermal conductivity of natural, bulk MX2 from Ref. [103]. The dash curves are through-plane thermal conductivity of natural WS2 and WSe2 with boundary scattering length of 150 nm from Ref. [103]. The dash-dot curves are the calculated in-plane and through-plane thermal conductivity of natural bulk MoS2 from Ref. [24]. The measurements from literature are synthetic MoS2 by

Pisoni et al.[101], natural MoS2 crystal by Liu et al.[102], synthetic WS2 by Pisoni et al.[263], single crystal WSe2 by Chiritescu et al.[223], and single crystal MoS2 and WSe2 by Murato et al.[34] ...... 80

Figure IV.11. (a) TDTR signal measured on 76 nm Al on WSe1.2Te0.8 using root-mean-square spot radius w = 8.4 μm and modulation frequency f0 = 2.16 MHz. The TDTR signal is dominantly sensitive to kz. The best-fit cross-plane thermal conductivity is kz = 0.42 W/mK. (b) Frequency dependent kz of WSe2, WSe1.2Te0.8, WTe1.6Se0.4 and WTe2. (c) Measuring in-plane thermal conductivity of WTe1.2Se0.8 sample using w = 4.2 μm. The best-fit kr = 10.1 W/mK...... 82

Figure IV.12. (a) In-plane and (b) cross-plane thermal conductivity of WSe2(1-x)Te2x measured by

TDTR. The thermal conductivity of pristine WSe2 (x=0) and WTe2 (x=1) are compared with first-

xiii principles calculations by Lindroth et al. [103] and Liu et al., [268] and measurement by Jiang et al.,[265] Chiritescu et al.,[223] Murato et al. [34], Zhou et al.,[266] Brixner et al.[269] and Jana et al.[270] ...... 84

Figure IV.13. Temperature dependent (a) in-plane thermal conductivity and (b) cross-plane thermal conductivity of 2H WSe2 and WSe1.2Te0.8 (x= 0.4), and temperature dependent (c) in-plane thermal conductivity and (d) cross-plane thermal conductivity of WTe2 and WSe0.4Te1.6 (x= 0.8)...... 86

xiv

CHAPTER I INTRODUCTION

I.1 Advances and Challenges in Modeling and Characterizing Thermal Conductivity

Thermal conductivity ( k ) is a transport coefficient correlating the heat flux 퐪 and the temperature gradient ∇T through the Fourier’s law 퐪 = −k∇T . Understanding of thermal conductivity is of critical importance for the performance and stability of lots of technological applications. With the miniaturization of electronic devices to sub-10-nm, effective heat removal is becoming more and more challenging. Thermal management is also important in energy applications like light emitting diodes (LEDs), photovoltaics and electrochemical batteries, since the heat accumulation would result in compromised stability, efficiency and safety. In other applications like thermal barrier coatings (TBCs) in gas turbines and thermoelectrics (TE), thermal conductivity is the key material metrics affecting the efficiency. Thermal conductivity is therefore a critical parameter for designing and discovery of multifunctional materials. Due to the importance of thermal conductivity, the community of nanoscale heat transfer have been conducted research on lots of functional materials from simple crystals like diamond[1], Si[2],

Ge[3], GaAs[4], GaN[5] to complex crystals like Ba8Ga16Sn30 clathrate-I[6], skutterudites Co4Sb12

[7], Mn11Si19[8], from isotropic materials to anisotropic materials like ZnO[9], Ga2O3[10] and so on. The fruitful research on thermal transport properties happens thanks to the development of both modeling and characterizing techniques.

On the modeling side, the major advancement in the past decade is first-principles phonon calculation. Frist-principles calculation predicts thermal conductivity by solving the Boltzmann transport equation (BTE) using detailed phonon scattering rate derived from density functional

1 theory (DFT).[11] It is well-known that first-principles calculations do not require any fitting parameters and have high accuracy of predicting thermal conductivity of a wide range of materials including silicon, [11-13] GaN[14] and two-dimensional layered materials such as graphene,[15-

18] silicene,[18, 19] black phosphorus[20-23] and many transition metal dichalcogenides [24-26].

However, there still exist two major problems challenging for first-principles calculations. First, first-principles calculations are still limited to simple crystals with only a few atoms in the unit cell. Since the computational cost of first-principles calculations increases dramatically with the number of atoms (N) in the unit cell with the scaling law of N3, it remains challenging to apply this method to model thermal conductivity of complex crystals whose unit cell usually contains dozens of atoms. The other challenge is that DFT solves Kohn-Sham equation at the ground state, which limits its applicability to model phonon and thermal properties at high temperature, especially for the high temperature phase of for materials like the body centered cubic structure of group IV metals,[27] CmCm phase of SnSe[28] and so on. If density functional theory (DFT) is directly used to predict phonon dispersions, there would be soft phonon modes with imaginary frequencies indicating that these structures are unstable despite that they still appear as stable phases at high temperature. With these soft phonon modes, DFT would fail to calculate any physical values of thermal properties like free energy, specific heat and thermal conductivity. An alternative way to solve these two challenges is molecular dynamics (MD) simulation. Since MD intrinsically includes thermal vibrations of atoms and is much more efficient than DFT, it is a promising method for modeling thermal transport in complex crystals and high temperature materials. However, MD suffers from the limited number of potential fields available in the literature, as well as the relative low accuracy of the potential fields. To solve these two challenges,

2 this thesis will integrate density functional theory (DFT) and molecular dynamics (MD) for modeling thermal and phonon properties of complex crystals and high temperature materials.

One the experimental side, few experimental techniques are capable of characterizing anisotropic thermal conductivity, although anisotropy can be found in a wide range of functional materials with low symmetry of crystal systems, anisotropic interatomic strengths like van der

Waals layered materials and nanostructures like superlattices. Traditionally, anisotropic thermal conductivities are measured using the steady-heat-flow method, in which many samples are cut with different orientations and then measured. [29, 30] These techniques, however, requires the samples to be large enough to accommodate at least two thermocouples to measure the temperature gradient, rendering it not suitable to measure the through-plane thermal conductivity of thin films.

Over the last three decades, significant progress has been made in using the 3-omega[31] and pump-probe thermoreflectance method[2, 32, 33] for measuring thermal conductivity of small samples. However, one significant limitation of the 3-omega method is that it requires not only complicated nano-fabrication of the metal strips but also usually a large and flat sample surface to accommodate them. In comparison, the thermoreflectance method is more flexible, requiring only an optically smooth area of less than 100 × 100 µm2. Most of the past works used the thermoreflectance method to measure the through-plane thermal conductivity and the interface thermal conductance,[34-36] since the laser heat is deposited on the sample surface. In order to measure anisotropic thermal conductivity, this thesis will develop a technique using different laser spot size to measure anisotropic thermal conductivity along both the through-plane direction and the in-plane direction.

This thesis contains three parts in response to the challenges for modeling and characterizing thermal conductivity addressed above. In the first part, we address the challenge of high

3 computational cost for predicting thermal conductivity of complex crystals. To mitigate the computational demand, we develop empirical potential field from DFT for modeling thermal conductivity of a family of complex crystals called hybrid organic-inorganic crystals, then MD simulations are performed for predicting thermal conductivity of these hybrid crystals. Although empirical potentials are computationally efficient, their transferability to different temperatures and atomic structures is limited due to the existence of empirical functional forms with fitting parameters. Once the functional forms are not chosen appropriately, the accuracy of the empirical potential will be significantly affected. This also requires reformulating the functionals in the potential field when there is phase change in the material, which limits the transferability. In the second part, we address the problem of limited accuracy and transferability of empirical potential, and use machine learning regression algorithms to develop potential fields for modeling phonon properties of high temperature materials with temperature dependent phase changes. In the third part, we develop a varied spot size technique based time-domain thermoreflectance for measuring anisotropic thermal properties of one of the best-known family of anisotropic materials, the layered two-dimensional materials.

I.2 Thermal Transport in Hybrid Organic-Inorganic Complex Crystals

Hybrid organic-inorganic crystals (HOICs) are a family of crystals with complex atomic structures and also a subset of the materials family called hybrid organic-inorganic materials

(HOIMs), which are synthesized by combining the organic and inorganic components over length scales from a few angstroms to ~100 nanometers.[37, 38] The most appealing feature of HOIMs is that they can be conveniently designed for properties that are difficult to achieve in either organic or inorganic materials alone. For example, superb electronic properties can be derived from the inorganic component while still maintaining structural flexibility due to the blending with organic

4 components.[38] In the past decade, a variety of novel HOIMs have been synthesized with promising applications like light emitting diodes[39, 40], flexible solar cells,[41-47] and flexible thermoelectrics,[48, 49] to name a few. However, thermal transport in HOIMs is not well understood although thermal conductivity greatly affects the thermal stability and the performance of the hybrid material-based devices. [50] Due to the structural diversity and the wide range of organic- inorganic coupling strength in HOIMs, it would be helpful to categorize the HOIMs into different groups for a better understanding of their mechanical and thermal properties. In Figure I.1, we illustrate that the organic-inorganic hybrid materials can be categorized into three groups (hybrid organic-inorganic nanocomposites, superlattices and crystals) according to the organic-inorganic coupling strength and the feature size of organic and inorganic components. The first group of

HOIM is the hybrid organic-inorganic composites whose organic and inorganic components are coupled by weak van der Waals forces or hydrogen bonds. While organic-inorganic composite materials have been studied for many decades, [51-54] significant efforts have been devoted to understanding the thermal and mechanical properties of nanocomposites when the feature size of the organic/inorganic structural units in these composites is on the order of a few nanometers. In these hybrid nanocomposites with distinguishable organic-inorganic interfaces, such as carbon nanotube-polymer composites[55, 56], self-assembled materials[57-61] and nanocrystal arrays [62-64] and super-atomic crystals,[65] thermal transport is determined by organic component, inorganic component, and most dominantly the phonon-interface scattering.[63] However, the organic- inorganic interfaces are not present in hybrid organic-inorganic crystals, when the blending of organic and inorganic constituents happens at the atomic scale.[66] In contrast to the hybrid nanocomposites with either amorphous organic or inorganic structures, the hybrid organic- inorganic crystals possess long range periodicity. A few examples of hybrid crystals are the family

5 of group II-VI element based hybrid crystals [67] and the hybrid organometal halide perovskites[47,

68-71]. The dominant coupling mechanisms between the organic and inorganic constituents in these hybrid crystals are either covalent bonds (II-VI based hybrid crystals) or ionic bonds (hybrid perovskites). The third group of HOIMs is the hybrid organic-inorganic superlattices, which lies between the hybrid crystals and the hybrid nanocomposites, in terms of both the length scale of feature size and the bonding strength between the organic and inorganic components, as shown in

Figure I.1. The most notable examples of hybrid organic-inorganic superlattices are the atomic/molecular layer deposited (ALD/MLD) organic-inorganic superlattice[72, 73] and the organic intercalated superlattice[48, 49]. In these superlattices, the organic-inorganic coupling strength can be in a wide range, from the strong covalent bonding in ALD/MLD superlattices to the electrostatic forces in the organic-intercalated superlattices. The feature size of the organic and inorganic component can also vary from sub-nanometer to a few nanometers. Although a few experimental research found that the hybrid crystals and superlattices usually have low thermal conductivity,[66, 72] it is still not well understood how phonons transfer their energy in these hybrid crystals and superlattices, and how the organic components affect the thermal transport.

To answer these questions, the key is to develop a modeling tool to simulation phonon transport in these HOICs. However, the structural complexity makes phonon simulation of HOICs very challenging. Since the unit cell of HOICs is usually large with dozens of atoms, the computational cost for performing full quantum-mechanical calculations (e.g. density functional theory (DFT) calculations) are often too expensive. An alternative way is to use the efficient molecular dynamics

(MD) simulation which describes lattice vibrations by solving the Newton’s equation of motion.

Unfortunately, this MD method requires an empirical potential field for calculating interatomic forces, but there exist no potential field in the literature for modeling thermal transport in HOICs.

This work seeks to integrate the DFT calculation and MD simulation, so that the thermal conductivity of HOICs can be calculated accurately and efficiently.

I.3 Phonon and Thermal Properties of High Temperature Materials

Temperature dependent thermal properties of materials are important for lots of high temperature applications including thermal barrier coatings and nuclear claddings, since the performance is significantly affected by phase stabilities. Prediction of these macroscopic thermal properties depends on accurately description of atomic vibrational dynamics, which is quantified as harmonic and anharmonic force constants. Although recent progress in first-principles calculation has enabled prediction of thermal properties routinely for many materials, it has been one of the long-standing challenges in material physics to model the vibrational spectra at high temperature. Conventional first-principles calculations are performed by perturbing the ground- state and assumed that the force constants are temperature-independent,[74, 75] but such assumption is questionable at high temperatures (>1000K), especially for the materials with phase changes whose high temperature phase is not at ground-state. One classical example is the body centered cubic of group-IV metals (Ti, Zr and Hf) which appears as stable phases as high temperatures, yet these bcc structures are predicted to be dynamically unstable using first- principles calculations at ground state, due to the existence soft phonon modes with imaginary frequencies in the phonon dispersion.[76, 77]

In 1955, Hooton realized that atoms vibrate in an effective potential due to their nonstationary neighbors, and the potential energy surface (PES) is stochastically sampled around the most probable position which is not necessarily a local minima.[78] They then renormalized the soft phonon modes by an effective harmonic potential that is temperature-dependent. Along this line, the problem of dynamical instability is addressed by a self-consistent approach under the harmonic approximation,[77] which starts with the phonon dispersion at static limit as an initial guess and iteratively solve the eigenmodes of the dynamical equation. However, several recent studies

8 suggest that care must be taken for strongly anharmonic crystals where the PES should be expanded to the third and even the fourth order.[79-81] Therefore, the accuracy of the force constants could be significantly affected by the artificial truncation of the Taylor expansion of the

PES.[80] On the other hand, Classical molecular dynamics can naturally incorporate the phonon anharmonicity of arbitrary order without truncating the Taylor expansion of the PES, but it suffers from the inaccuracy of the empirical potential field as limited by the fitting with the empirical functional forms.[82-84] To solve this challenge, this thesis will utilize machine learning regression, a much more adaptive way for developing high-fidelity potential field to fit the ab- initio potential energy surfaces, and then MD will be used for modeling phonon properties at high temperature.

I.4 Thermal Transport in Two-Dimensional Layered Materials

Since the discovery of graphene in 2004, [85, 86] two dimensional (2D) materials like boron nitride (BN), silicene, transition metal dichalcogenides (TMDs), black phosphorus and so on attracted intensive research interests due to the unique physical properties and potential applications in electronics,[87-89] photonics,[90, 91] electrochemical batteries,[92-95] thermoelectrics[48, 49] and many other fields. Many of these 2D materials showed even superior properties than graphene. For example, for monolayer TMDs like MoS2 have a direct bandgap that is desired for electronics and photonics, while such bandgap is absent in monolayer graphene.

Therefore, 2D materials provide an opportunity for the next generation of transistors to deal with the demise of Moore’s law of silicon based semiconductors. Understanding thermal transport properties of 2D materials, therefore, becomes crucial, because they affect the stability and performances of 2D materials based devices.

The bulk form of 2D materials usually have layered structures with the atomically thin 2D monolayers stacked together. The interatomic coupling in these layered materials are highly anisotropic. Within each monolayer, the atoms are bonded through strong covalent bonds, while the inter-layer coupling is usually the weak van der Waals interaction. The presence of van der

Waals gaps in these layered materials results in interesting physical properties and potential applications. For example, the gaps in layered materials provides the passage for ionic intercalation, which can be used for electrochemical battieries, supercapacitors. Since the intercalation is a dynamical process with structural changes, the properties of the 2D materials should also be meditated by these intercalants. These process inspired a lots of research trying to manipulate optical, electronic, and thermoelectric properties of these layered materials. It is also interesting to explore how the intercalation process affects the thermal conductivity of 2D materials. On the other hand, the layered structure of these materials also give rise to strong anisotropy in thermal transport properties. One famous, the in-plane thermal of graphite is as high as 2000 W/mK, yet the cross-plane thermal conductivity is only 6 W/mK, three orders of magnitude lower than the in- plane thermal conductivity. Such anisotropy also imposes great challenges to accurately characterize the thermal conductivity in different directions.

Unfortunately, even the thermal conductivity of the layered 2D materials in bulk form are far from well known. For example, the thermal conductivity of TMDs in the literature is far from consistent. For example, the few layer MoS2 scatters over the range 13-81 W/mK,[96-100] all much lower than the first principles calculation 138 W/mK.[25] The thermal conductivity of even bulk layered MoS2 is widely scattered. For example, in-plane thermal conductivity of synthetic

MoS2 (16 W/mK)[101] is one order of magnitude lower than the geologically mined MoS2 (110

W/mK).[102] The cross-plane thermal conductivity measured by pump-probe thermoreflectance

(2.0 W/mK)[34] also compares poorly with the first principles calculations (4.8 W/mK).[103] The measured thermal conductivity of 2D materials might also be strongly affected by the heat source.

For example, the optothermal Raman measures the thermal responses to laser heating using the frequency shift of the Raman peaks. However, it is demonstrated that the laser light is not a thermalizing heat source and different phonon modes are out of thermal equilibrium. As a result, the thermal conductivity is severely underestimated. Therefore, it remains challenging to accurately determine the anisotropic thermal conductivity of 2D layered materials.

I.5 Objectives of this Thesis

The main objectives of this thesis is to develop a computational and experimental techniques to understand the thermal transport in HOICs and 2D layered materials. The computational tool is developed to predict the thermal conductivity of HOICs, and to find out how the phonon transport is affected by the organic components. To address the computation challenges described in section

I.2, the computation tool that integrates the first-principles calculation with the MD simulation is developed. Using first principles calculation to sample the potential surface of the atoms around the equilibrium position, we developed empirical potential fields to fit the first-principles potential surface. The thermal conductivity and phonon properties of HOICs are then calculated using MD simulations based on the developed potential field. This simulation strategy is applied to model thermal conductivity of hybrid organic-inorganic crystals including the II-VI based hybrid crystals and hybrid organic-inorganic perovskites. As will be discussed in this thesis, we predicted an ultralow thermal conductivity (0.6 W/mK) of organometal halide MAPbI3 crystal, even lower than glasses (~ 1.0 W/mK) due to the short phonon lifetimes. In addition, the organic molecules are also predicted to have significant tuning effect on the thermal conductivity, showing the complex hybrid crystals have large degrees of freedom for tunable thermal properties. However, such

11 derived empirical potential field still involves artificial assigned functionals with fitting parameters, which is the limiting factor preventing empirical potential to achieve accuracy close to first- principles calculations. In addition, such assignment of functional forms could no longer be appropriate if the material has complex phase change behavior.

To further solve these problems of empirical potential field, we implemented machine learning regression techniques to build high-fidelity potential field for modeling phonons at high temperature (> 1000 K), which does not involve any artificial assignment of empirical functional forms or fitting parameters. We examine whether such machine learning potential field could be applied to calculate phonon properties at high temperatures, especially for materials with phase change behavior. For example, the body centered cubic (BCC) phase of Zirconium (Zr) is predicted to be dynamically unstable at static limit (0 K), showing soft phonon modes by first-principles calculation, while it became stable phases at high temperatures (> 1188 K). We will develop a machine learning driven potential field that can accurately capture the renormalization of soft phonon modes to a positive frequency above the phase change temperature, which indicates the stabilization of the BCC structure of Zr.

To systematically study the anisotropic thermal conductivity of 2D layered materials, we first develop a technique using a varied laser spot size based on the pump-probe time-domain thermoreflectance (TDTR) measurement. The size of the laser spot is varied to control the direction of the heat flow in the samples, so that the TDTR is sensitive to thermal conductivity along different directions. To be more specific, when the spot size is large, the heat flow is quasi-one- dimensional along the cross-plane direction, and TDTR can measure cross-plane thermal conductivity separately in this case. When the laser is tightly focused to small length scale, the heat flow is three dimensional and TDTR can then measure the in-plane thermal conductivity. We

12 systematically analyzed the TDTR signals and quantified the criterion for selecting appropriate spot sizes for measuring the anisotropic thermal conductivities. This technique is applied to measure a series of anisotropic crystals including graphite, SiC, ZnO and TiO2. This varied spot size TDTR technique is then applied to study the anisotropic thermal conductivity of TMDs including MoS2, WS2, MoSe2 and WSe2 as well as the TMD ternary alloys WSe2(1-x)Te2x.

Interestingly, we discovered that the measured cross-plane thermal conductivity depends on the heating frequency (modulation frequency of the pump laser). This phenomenon is explained by the non-equilibrium between the high frequency phonon modes and the low frequency phonon modes. A two temperature diffusive model is developed to capture such non-equilibrium phonon transport and extract the thermal conductivity at thermal equilibrium.

I.6 Organization of this Thesis

In CHAPTER I, the motivation and the objectives of this thesis for the phonon transport in

HOICs and 2D layered materials are discussed. In CHAPTER II, we develop the integrated DFT and MD simulation method for modeling phonon transport in HOICs. We studied the II-VI based

HOIC with alternative organic and inorganic layers, and the organometal halide perovskites.

CHAPTER III, we applied the machine learning algorithm to develop high-fidelity potential field for modeling the phonon dispersion at high temperatures. Using a simple machine learning algorithm called Gaussian process regression, we developed potential field that can effectively capture the stabilization of BCC phase of Zirconium at 1188 K, which is predicted to be unstable using static first-principles calculations. Using MD simulations at high temperature, we have successfully observed that the soft transverse acoustic (TA) phonons of BCC-Zr is renormalized to a positive frequency. In CHAPTER IV, we develop the varied spot size technique for measuring the anisotropic thermal conductivity and apply this technique to measure the anisotropic thermal

13 conductivity of TMD and TMD alloys. Instead of being a constant, the thermal conductivity of

TMD and TMD alloys is observed to depend on the heating frequency we applied to the sample.

This phenomenon is explained by the non-equilibrium between the low frequency phonons and high frequency phonons, and a two-channel diffusive model is developed to extract the thermal conductivity at equilibrium limit. In CHAPTER V, a summary of the thesis is presented and future work is proposed based on the discussions in the thesis.

CHAPTER II THERMAL CONDUCTIVITY MODELING OF HYBRID

ORGANIC-INORGANIC CRYSTALS

II.1 Introduction

In this chapter, we discuss the modeling of thermal transport in HOICs. As discussed in chapter

I, the absence of modeling tool for phonon simulations is responsible for the lack of understanding of thermal transport in HOICs. The challenge of performing phonon simulations arises from the structural complexities of these HOICs, despite the extraordinary progress made over the past two decades. [104-109] The most distinguished examples are first-principles-based simulation methods like Boltzmann transport equation (BTE) or ab initio molecular dynamics (AIMD), taking great advantage of the progress in computational power.[11, 13, 14, 74] Without fitting parameters, such first-principles-based methods has been used to model the thermal conductivity of inorganic bulk crystals such as diamond, [110, 111] silicon,[11-13] GaN[14] and two-dimensional materials such as graphene,[15-18] silicene,[18, 19] black phosphorus[20, 21, 23, 112] and many transition metal dichalcogenides.[24-26] Not only great agreements between theoretical calculations and measurement results on the thermal conductivity of materials have been found, but also these tools serve significantly speed up the materials discovery in both extremely high/low thermal conductivity and multifunctional thermal materials. [11-13, 26, 74, 110, 111, 113]

However, much less effort has been devoted to study phonon transport in hybrid organic- inorganic crystals and superlattices, not catching up the pace on the synthesis efforts for multifunctional thermal materials. This is understandable because there are significant computational challenges due to the structural complexity of hybrid organic-inorganic crystals and superlattices. For example, hybrid crystals usually have very large unit cells containing dozens of atoms, in comparison with only one to a few atoms in inorganic crystals. In addition the organic

15 components usually have internal degrees of freedom of motion (for example, the organic ions in hybrid perovskite can rotate) instead of just vibrating around the local equilibrium.[114-116] To capture such dynamical disorder caused by the internal degrees of freedom of motion, a very large supercell is required to simulate phonon transport instead of just one unit cell with periodic boundary conditions. In superlattices, the organic components might even be amorphous. The dynamical disorder and lack of periodicity in the hybrid crystals and superlattices hinder the implementation of first-principles-based simulation methods like BTE or AIMD. In the first- principles-based BTE approach, thermal conductivity is calculated using phonon properties including phonon dispersion and detailed phonon-phonon scattering rates. However, the computation time increases dramatically with the number of atoms in a unit cell, making it a great challenge for hybrid materials. Similarly, the computational time required for direct AIMD simulations is not affordable to capture the size effect on the thermal conductivity of hybrid materials.[117]

Alternatively, classical molecular dynamics (MD) simulation which traces atom movements according to Newton’s law of motion are computationally efficient. [118] However, it suffers from the lack of appropriate potential fields that govern interatomic interactions. Most of the potential fields in literature are not designed for hybrid organic-inorganic crystals and superlattices. For example, the most commonly used potential fields including the Tersoff potential,[84, 119, 120]

Stilinger-Weber potential[84, 121] and bond order potential[122-125] were developed for inorganic materials. However, the structure of the inorganic component in the hybrid materials can be very different from the original inorganic structure as some of the bonds are now connected to organic components, making it inappropriate to directly apply the potential fields in literature.[126]

Another challenge is the interaction between the organic and the inorganic components, although

16 the organic components can be reasonably well described using the potential fields such as the class II force field[127, 128] and consistent valence force field (CVFF) [76]. To address these challenges, we present in this chapter, a strategy for calculating thermal conductivity of HOICs using the ab-initio based simulations. The predicted thermal conductivity of II-VI based organic- inorganic semiconductors, organometal perovskites are studied and compared with experimental data when available. This work sets the methodology for modeling thermal conductivity of materials with complex atomic structures from the first principles, which could be a good tutorial for both graduate students and experienced researchers who are interested in thermal energy transport in emerging hybrid materials.

II.2 Simulation Strategy

Figure II.1 summarizes the workflow for modeling the thermal conductivity of organic- inorganic hybrid crystals and superlattices by integrating the first-principles density functional theory (DFT) calculations and the classical molecular dynamics (MD) simulations.

First-principle calculations are used to construct the potential fields. The very first step to construct the potential field is to assign functional forms. In HOIMs, the bonding nature of the organic and inorganic parts could be very different. Therefore, the potential field EV is separated into three parts: the potential for the organic part EO, the potential for the inorganic part EI and the organic-inorganic coupling EI−O:

I O I−O EV = E + E + E (II.1)

Based on the bonding characteristics, a functional form is then assigned. For example, the Lennard-

Jones potential is assigned to a van der Waals bond. Such a separation of the potential into three parts helps to greatly reduce the amount of work for developing the potential fields, by allowing to use some of the existing potential fields as parts for EI, Eo, and EI−O. For example, the class

II force fields [127, 128] or the consistent valance force field (CVFF)[76] can be used to describe the covalent bonding within the organic ligands.

After assigning the functional forms to a potential field, a set of unknown parameters 퐏 in the functional form need to be determined. The parameters 퐏 essentially control the shape of the empirical potential surface and they are obtained by minimizing the difference between the empirical potential surface EV and the DFT energy surface EDFT:

2 W(퐏) = ∑|EDFT({퐫퐢}) − EV(퐏, {퐫퐢})| (II.2) {퐫퐢} using optimization algorithms like quasi-Newton algorithm[129] or genetic algorithm[130], where

퐫i denotes the atomic coordinate configurations.

The unknown parameters P can also be obtained by the force matching method. Instead of the static sampling of energy surface, an ab-initio molecular dynamics (AIMD) simulation is performed, and the atomic trajectory 퐫i(t) and the interatomic forces 퐅AIMD(퐫i(t)) are recorded.

For any set of parameters, the empirical potential field would also predict a set of interatomic

forces 퐅V(퐏; ri) = −∇퐫iEV(퐏; ri). Then the unknown parameters 퐏 can be obtained by minimizing the difference between the atomic forces from AIMD calculation and that calculated from the empirical potential field:

Nt N 1 2 W(퐏) = ∑ ∑|퐅V(퐏; ri(t)) − 퐅AIMD(퐫i(t))| (II.3) NtN t=1 i=1 In most cases, both the energy surface fitting by Eq. (II.3) and the force matching using Eq. (II.4) can be used to obtain the parameters with subtle difference, but the force matching method is suggested when there are internal degrees of freedom. One example is the hybrid perovskite

CH3NH3PbI3 (MAPbI3) whose organic ions CH3NH3 (MA) can rotate at high temperature, therefore a local sampling by small displacements would fail at large rotational degrees and overestimate the rotational energy barrier (see detailed discussion in next section). After parametrizing the potential field, a validation process need to be performed before calculating the thermal conductivity. To ensure proper fitting of the potential fields is achieved, the physical properties such as lattice constants and elastic properties that can be more easily calculated from both MD simulations and DFT calculations are often calculated and compared, or even compared with measurements when available.

After a satisfying potential field is found, the thermal conductivity can be calculated rather straightforwardly using the classical molecular dynamics. The most commonly used classical MD methods to derive thermal conductivity can be classified into the non-equilibrium molecular dynamics (NEMD)[131-134] and equilibrium molecular dynamics (EMD)[135-137]. One of the mostly used NEMD method is the direct method, which uses a hot reservoir and a cold reservoir to create a steady state temperature gradient ∇T, and the heat flux q is calculated. Then the thermal conductivity is calculated by Fourier’s law: k = −〈q/∇T〉 , where 〈 ⋅ 〉 denotes the ensemble average. The direct method, however, usually requires a large simulation cell to establish a reproducible diffusive temperature profile.[132] In addition, the direct method only calculates thermal conductivity along the temperature gradient, making the computation costly when the

19 thermal conductivity of a material is anisotropic, as multiple simulation runs need to be performed for thermal conductivity in different directions. Similar to the laser flash analysis measurement,

[138] a transient method called the approach-to-equilibrium molecular dynamics (AEMD)[134] uses a pulsed heating reservoir and monitors the temperature decay as a function of time. The thermal conductivity is then extracted using the characteristic decay time. This method, however, is also limited to isotropic materials. Different from the direct method and the AEMD method which drive the system out of thermal equilibrium, EMD calculates the thermal conductivity at thermal equilibrium using the Green – Kubo relation: [139]

V ∞ kα = 2 ∫ 〈Jα(0) ⋅ Jα(t)〉dt (II.4) 3kBT 0 where α denotes the directions in the Cartesian coordinates, 〈Jα(0) ⋅ Jα(t)〉 is the heat current autocorrelation function (HCACF), V is the volume of the simulation cell, kB is the Boltzmann constant and T is the temperature. A great advantage of EMD is that the thermal conductivity along different directions can be calculated in a single EMD run, and therefore EMD is suggested for calculating the thermal conductivity of hybrid crystals and superlattices. Noting that extracting thermal conductivity from MD simulations requires a series of convergence tests with respect to simulation cell size, integration time, etc. Readers who are interested in MD simulations for thermal conductivity are referred to literature.[132, 135, 137, 140-143].

In addition to the thermal conductivity, the MD simulation can also provide phonon properties, by projecting the atomic movements in real space into the momentum-frequency space where the phonons transport and scatter with each other. The most commonly used phonon modal analysis based on MD simulations include the normal mode analysis (NMA)[109, 143] and spectral energy density (SED) [144, 145]. The NMA projects the atomic vibrations into normal coordinates, and the phonon lifetime is extracted from the autocorrelation of the modal vibrational energy.

Therefore, NMA requires eigenvectors as input which are usually determined from lattice dynamics calculation at 0 K. However, in some materials with strong anharmonicity, the eigenvector (phonon dispersion) at high temperature would shift away significantly from the 0K lattice dynamics calculation.[144] The advantage of SED is that this method does not require predetermined eigenvectors. The SED analysis directly calculates the vibrational energy distribution in the frequency-momentum space. The SED ϕ(퐪, ω) is defined as:

2 τ 1 0 퐥 퐥 ϕ(퐪, ω) = ∑ ∑ mb |∫ ∑ vα ( , t) ⋅ exp [i퐪 ⋅ 퐫 ( ) − iωt] dt| (II.5) 4πτ0NT b 0 α b 0 퐥 퐥 where 퐪 is wavevector and ω is the phonon frequency, N is the number of unit cells, v ( , t) T α b denotes the velocity component of the b-th basis atom in the 퐥-th unit cell along the α-th direction,

퐥 퐫 ( ) denotes the position of the l-th unit cell, m is the atomic mass of the b-th basis atom in the 0 b unit cell. In the (훋, ω) space, the location of SED peaks (bright colors) indicates the phonon dispersion and the bandwidth of the branches are related to the phonon lifetime. The smaller the broadening of the SED peak, the longer the phonon lifetime. [48], 31 The phonon lifetime can be extracted by fitting the SED peaks using the Lorentzian function:

I ϕ(ω) = 2 2 (II.6) 1 + (ω − ω0) /Γ

where I is the peak magnitude, ω0 is the frequency at the peak center and Γ is the bandwidth. The phonon lifetime can therefore be obtained by τ(퐪, ω) = 1/2Γ.[109]

In the following sections, we discuss the implementation of this integrated DFT-MD simulation strategy for predicting the thermal conductivity of II-VI based hybrid crystals[126] and the organometal halide perovskites.[146]

Figure II.1. Computation strategy for obtaining thermal conductivity and phonon properties using first-principles based atomistic simulations.

II.3 II-VI based Hybrid Organic-Inorganic Crystals

The II-VI based hybrid organic-inorganic crystal family has the chemical formula XY(L)0.5 where X is a group II-B element (Zn, Cd and Hg), Y is a group VI-A element (S, Se and Te), and

L denotes the organic ligand. The II-VI based hybrid crystals form alternating organic-inorganic layered structures, which is a quantum-well designed for white light emitting diodes applications.[39] Among this family, the β-ZnTe(en)0.5 is the first discovered member in 2000 as the first example of crystalline hybrid organic-inorganic network formed by strong covalent bonds.[67] The hybrid crystal β-ZnTe(en)0.5 has a layered structure of alternating ZnTe monolayers connected by the organic ethylenediamine ((CH2NH2)2, hereafter denoted as (en) ligands through covalent bonds as shown Figure II.2a. Besides the β-ZnTe(en)0.5, there are also other polymorphs including the α-ZnTe(en)0.5 (Figure II.2b) whose inorganic ZnTe monolayers has a different structure from the β- phase. Different from the β- phase in which the ZnTe hexagons are folded along the shorter diagonals (Figure II.2c), the α-ZnTe monolayers are formed by folding the ZnTe hexagons along the longest diagonals (Figure II.2d). Polymorphs like ZnTe(pda)0.5 and

ZnTe(ptda)0.5 can form by replacing the (en) ligand in α-ZnTe(en)0.5 to longer organic molecules like H2N-(CH2)3-NH2 (pda) and H2N-(CH2)5-NH2 (ptda), as shown in Figure II.2e-f. Details of thermal conductivity modeling in the β-ZnTe(en)0.5 can be found in ref. [126]. In this part, α-

ZnTe(en)0.5 is first discussed in detail as an example to review the implementation of the integrated

DFT-MD simulation methodology, then the thermal conductivity of other polymorphs with longer organic ligands like ZnTe(pda)0.5 and ZnTe(ptda)0.5 which were not published before, are discussed here to understand the effect of organic component on the thermal conductivity of the hybrid crystals.

As discussed in section III, the interatomic potential EV of α -ZnTe(en)0.5 can be divided into the three parts: the inorganic potential EI within the ZnTe monolayers, the organic potential EO within the organic ligand layers, and the coupling EI−O between the organic and inorganic constituents. Functional forms need to be assigned to different interaction terms. Appropriate choice of functional forms is essential for the success of this strategy. For thermal conductivity calculations of solids with strong chemical bonds, the potential does not need to be reactive (i.e. capable of describing the bond breaking process) because the atoms usually vibrates locally around the equilibrium position.[147, 148] Therefore, simple functional forms with minimal number of empirical parameters are preferred for faster computation.

I O I−O For α-ZnTe(en)0.5, the potential E , E and E can be further divided into different terms accounting for pairwise, angular and dihedral deformations. A list of typical deformations included in the potential field are shown in Figure II.3. For example, the inorganic potential energy can be

I I divided into pairwise interaction EP and angular interaction EA:

I I I I I E = EP + EA = ∑ ϵp(rp) + ∑ ϵa(θa) ⏟p ⏟a (II.7) I EI EP A

I where ϵp(rp) is the bonding energy of the atomic pair p (e.g. Zn-Te), depending on the atomic

I distance of the atomic pair rp, and ϵa(θa) is energy of an angle θa formed by an atomic triplet (e.g.

Zn-Te-Zn). Morse potential is chosen to describe the pairwise energy of nearest neighbors (Zn-

Te bonds) and the second nearest neighbors within the inorganic part:

I 2 ϵp(r) = D(exp[−αm(r − r0)] − 1) (II.8)

where r0 is the bonding length at equilibrium. The characteristic bonding energy D and the characteristic potential well width α are the empirical parameters to be determined. For three-body angular interaction, a cosine-squared form is used:

I 2 ϵa(θ) = K(cos θ − cos θ0) (II.9)

where θ0 is the equilibrium angle, and the angular stiffness K is the empirical parameter. Morse potential and the cosine-squared angular potential has been proven accurate and is commonly used to describe other covalent bonded systems like metal-organic-framework[149], bismuth telluride[150] and skutterudites[151].

For the organic part EO, the COMPASS potential[128] is directly applied for the pairwise, angular and dihedral deformations within each organic molecule. The inter-molecular coupling is described by the Lennard-Jones potential: [152]

σ 12 σ 6 ϵ = 4ϵ (( ) − ( ) ) (II.10) LJ r r

Finally, the organic-inorganic coupling potential EI−O is also separated into pairwise deformation

I−O I−O energy EP and angular deformation energy EP :

I−O I−O I−O E = EP + EA (II.11)

I−O where EP includes the Zn-N bonding and the second nearest neighbor interactions (Zn-C and

I−O Te-N) and EA includes the angular deformation energies of the triplets Zn-N-C, Zn-N-H and Te-

I−O I−O Zn-N. Similarly, Morse potential and cosine-squared potential is used for EP and EA , respectively. After the assigning the functional forms to the potential, a set of unknown empirical parameters 퐏 in the functional form is then obtained by fitting the energy surface obtained from the DFT calculation. Figure II.4 shows an example of the DFT energy surface sampling and fitting process. The DFT energy surface sampled by displacing a Te atom along the Zn-Te bond. By varying the parameters 퐏 , the shape of the empirical potential surface would also change accordingly. Quasi-Newton algorithm[129] is used to adjust the parameters 퐏 until the difference between the DFT energy surface and the empirical potential surface reaches minimum. The paramters 퐏 at the best fit is then adopted for the optimized potential field.

The developed potential field is then validated by comparing the elastic constants that can be easily calculated by both DFT simulations and MD simulations using the empirical potential field.

The elastic constants are second-derivatives of the potential surface against different deformations

1 ∂2E Cij = , where ei and ej are components of the strain tensor in Voigt notation,[153] and V is V ∂ei ∂ej the volume of the simulation cell. Table II.1 shows great agreement of the lattice constants and

Figure II.5 shows the agreement of elastic constants between the calculations from MD using the empirical potential field and the DFT calculations.

Table II.1. Lattice parameters (Å) of α-ZnTe(en)0.5.

Parameters DFT MD Experiment

x 7.241 7.230a 7.075 7.061b Lattice y 7.025 7.023a 7.055 6.927b Constants z 17.815 17.715a 17.813 17.524b

a. Ref. [[154]], b. Ref. [[155]], c. Subscripts of elastic constants, 1-3 denotes x, y and z axis

correspondingly.

Figure II.5. (a) Elastic constants of α-ZnTe(en)0.5 calculated from MD and DFT simulations. (b)

The strains corresponding to the elastic constants are indicate by arrows.

Using the similar method, the potential fields for polymorphs with longer organic chains

ZnTe(pda)0.5 and ZnTe(ptda)0.5 are also developed. Figure II.6 compares the elastic constants of

α-ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5. Interestingly, the normal stiffness C11 along the stacking direction (x-axis) increased when the length of the organic ligand increases. When the same normal strain is applied along the x-axis, more C-C-C triplets are involved in the deformation when the organic ligand becomes longer. Therefore, the energy change associated with the applied strain increases with the organic ligand length, resulting in a larger C11 for the polymorphs with longer organic ligands. However, the shearing stiffness C55 and C66 decreased dramatically due to the increased degrees of freedom of dihedral deformations in the organic ligands. When the crystal is imposed with shearing deformations parallel to the ZnTe monolayer (corresponding to

C55 and C66), the major deformation occurs in the dihedrals including the C-C-C-C or C-C-C-N, yet these dihedrals have very small stiffness. As a result, the crystals are more flexible to shearing deformations when the organic ligand becomes longer.

Figure II.7 shows the anisotropic thermal conductivity of α-ZnTe(en)0.5, ZnTe(pda)0.5 and

ZnTe(ptda)0.5. The length of the organic ligands affects thermal conductivity anisotropically. The thermal conductivity parallel to the ZnTe monolayers (ky and kz) decreased when the organic chains becomes longer. Due to the decreased shearing stiffness C55 and C66, the group velocity of

TA phonons would also decrease along the y- and z- axis, resulting in the reduced ky and kz (see

Table II.2). The thermal conductivity along the stacking direction kx, however, has no significant changes, because the increased LA group velocity and the decreased TA group velocity affect kx counteractively.

Table II.2. The group velocities (m/s) of ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5 by solving the anisotropic wave propagation equation using the Christoffel package.[156]

x y z

en pda ptda en pda ptda en pda ptda

LA 3449.7 4537.3 5406.5 2962.2 3322.8 3685.7 3857.9 4055.7 3941

TA1 1242.5 1192.2 622.1 1547.6 1518.3 1737.4 1547.6 1518.3 1737.4

TA2 1141.9 909.9 568.8 1252.5 1192.2 622.1 1141.9 909.9 568.8

Figure II.6. Comparison of elastic constants in α-ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5.

Figure II.7. Comparison of thermal conductivity in α-ZnTe(en)0.5, ZnTe(pda)0.5 and ZnTe(ptda)0.5.

II.4 Organometal Halide Perovskites

The organometal halide perovskite is a family of hybrid organic-inorganic crystals with chemical formula XYZ3, where the X site is occupied by organic ions (e.g. CH3NH3, hereafter

MA), the Y site is a metal element (e.g. Sn, Pb etc.) and the Z site is a halide element (Cl, Br and

I).[157] These hybrid perovskites have attracted intensive research interests in photovoltaics[45,

47, 158-161] and recently in thermoelectrics[162, 163]. However, thermal stability remains a problem for photovoltaics based on these hybrid perovskites.[164] A phase change from the tetragonal phase to the pseudocubic phase would occur under continuous solar radiation (see

Figure II.8).[157] A good understanding in thermal conductivity of hybrid perovskites is therefore of great importance.

Different from the covalently bonded hybrid crystals as described in Section IV.A, the organic- inorganic coupling in MAPbI3 and MASnI3 is ionic, and the Buckingham potential and Coulombic potential are therefore used to describe EI−O: [165]

r q q I−O ij C i j E = ∑ A exp (− ) − 6 + (II.12) ρ r rij i,j ij where i is the index of an inorganic atom (Pb or I) and j is the index of an organic atom (C, N or

H). A , ρ and C are unknown parameters to be determined, and q is the atomic charge. To accurately determine the Coulombic energy, Bader analysis based on DFT simulation is performed to calculate the atomic charges,[166] see details in ref [146]. Functional forms for EI and EO are also given in ref [146].

However, the method of matching the local energy surface as described in part A becomes inappropriate for modeling the material systems with internal degrees of freedom, because the non- zero temperature would play an important role in affecting the rotational dynamics of MA ions. At

34 low temperature, the atom can only “feel” the potential surface very close to the static equilibrium position, but at the elevated temperature the internal rotational degrees of freedom are introduced and the potential surface far away from the equilibrium position becomes important. The DFT calculations by imposing displacements to the MA ions which has large internal rotation become very large to ensure that the potential surface along different rotation paths are effectively sampled.

To solve this problem, ab-initio molecular dynamics (AIMD) is used to sample the potential surface randomly by the random atomic vibration. The unknown parameters P of the potential field is then obtained by matching the AIMD forces for minimal W(퐏) in Eq. (II.3). Figure II.9 shows the potential surface associated with the rotation of the MA ion around the c-axis. If only a small portion of the potential well is sampled, the empirical potential surface would dramatically overestimate the rotational energy barrier (~0.54 eV by DFT). At 400 K, the thermal excitation energy ~0.414 eV (kBT times 12 atoms in a unit cell) is comparable with the rotational energy barrier, making the rotations of MA ions non-negligible. By matching the AIMD forces at 400 K, the rotational barrier can be reasonably represented by the empirical potential field (~0.57 eV).

After the parameters P in the functional form are determined, the potential is verified by comparing the vibrational density of states from MD to the DFT calculation[167], see details in Ref [149].

Figure II.8. The crystal structure of (a) the tetragonal and (b) the pseudocubic phases of MAPbI3

With the developed potential field, the thermal conductivity of MAPbI3 and MASnI3 are then calculated using the classical MD simulations, as shown in Figure II.10a. Despite the detailed curvature discrepancy of the rotational potential surface between the fitting and the DFT calculations, the thermal conductivity of MAPbI3 still shows good agreement with experiment.

This is because the rotational modes of MA ions have negligible contributions to thermal conductivity.[146] However, accurately representing the rotational barrier is important. If the rotational barrier of MA ions is severely overestimated, then these rotational modes are constrained at room temperature. As a result, the phonon-phonon scattering rate involving the MA rotations would be underestimated and thereby the thermal conductivity would be overestimated.[168]

Both the MAPbI3 and MASnI3 have very low thermal conductivity in the tetragonal phase, but the thermal conductivity of MAPbI3 (0.59 W/m·K) is slightly lower than MASnI3 (0.69 W/m·K) at around 300 K. This is because the Sn-I bond is shorter and stiffer than the Pb-I bond, resulting in a higher acoustic velocity in MASnI3.[69] In both materials, a sharp increase in the thermal conductivity is observed during the traFnsition from the tetragonal phase to the pseudocubic phase.

However, the thermal conductivity of the pseudocubic MASnI3 is smaller than MAPbI3. This is because the third order force constant in the Sn-I bond is ~20% larger than that in the Pb-I bond, which results in the stronger anharmonicity. With the lower thermal conductivity and the much smaller electric resistivity in the pseudocubic MASnI3,[163] it is expected that replacing the Pb atoms with Sn atoms would significantly improve the thermoelectric performance.

Figure II.10b compares the computational results of thermal conductivity in MAPbI3 existed in the literature and the measurement by Pisoni el. al.[164] Among the computational studies, Hata et al.[168] used a similar ab-initio based force matching method, but their potential overestimate the results above 50 K. The possible reason is that all bonds are assumed harmonic in their simulations, which significantly underestimate the anharmonicity. Another noteworthy potential field for thermal conductivity modeling of MAPbI3 is the MYP potential with purely pairwise interaction,[114] which is adopted for MD simulation by Wang et al[169] and Caddeo et al.[170]

This MYP potential field, however, has relatively low accuracy of thermal conductivity depending on the simulation method (EMD[169] or AEMD[170]). Considering MYP potential is primarily designed to reproduce atomic structures for each phase of MAPbI3 with one set of parameters, the thermal conductivity accuracy for each phase might be compromised. Recently the direct non- equilibrium ab-initio molecular dynamics (NEAIMD) simulation without any fitting parameters has been proposed by Yue et al[171] to model thermal transport in the hybrid organic-inorganic

38 perovskites. However, NEAIMD can only be used to model a system with very small number of atoms (a few hundred) due to the high computational cost, which limits its application to materials with low thermal conductivity. Similar to Qian’s MD simulation results,[146] the NEAIMD also predicted the increased thermal conductivity during the tetragonal-to-pseudocubic phase transition occurs in MAPbI3 at 330 K. By analyzing the phonon properties from AIMD, they observed that the modes above 25 THz are purely organic vibrations with negligible group velocities. The phonon modes above 25 THz make small contribution to the thermal conductivity despite their relatively long lifetime. Phonons modes below 5 THz contribute to majority of the thermal conductivity, which are vibrations of Pb-I and movements of the MA ions as a whole.

Figure II.10. Temperature-dependent thermal conductivity of MAPbI3 by MD simulations compared with the experimental results by A. Pisoni et al.[164] and direct AIMD simulation by

Yue et al.[171]

II.5 Summary of this Chapter

In summary, based on our recent work, this article systematically presents an integrated first- principles-driven computational strategy for modeling the thermal conductivity of emerging

HOICs. This novel simulation strategy avoids both the computational challenges in direct first- principles simulation and the limits of classical molecular dynamics simulation. By either fitting the energy surface or interatomic forces from the ab-initio based first-principles simulations, empirical potential fields have been successfully developed for a range of emerging hybrid organic-inorganic crystals and superlattices, including organic-inorganic II-VI based hybrid crystals, the organometal halide perovskites, and the organic-intercalated TiS2 superlattice. With the potential fields developed, the classical molecular dynamics simulation is then used to predict the anisotropic thermal conductivity of these novel materials which are of great technical interests including photovoltaics, thermoelectrics and light emitting diodes.

When implementing the DFT-MD method described in this work, the readers should be aware that establishing the potential fields by matching ab-initio potential surface or inter-atomic forces are material-specific and non-transferable. The potential field should be re-constructed when the atomic structure is changed, even with the same atomic species. For example, the potential fields need to be constructed separately for the tetragonal and pseudocubic phases of CH3NH3PbI3. This work provides a guidance for modeling thermal conductivity of materials with complex atomic structures from the first principles, which could be a good tutorial for both graduate students and experienced researchers who are interested in thermal energy transport in emerging hybrid materials.

CHAPTER III MACHINE LEARNING DRIVEN ATOMISTIC MODELING

ON PHONON DIPSERSION STABILITY OF ZIRCONIUM

III.1 Introduction

Understanding temperature-dependent thermal properties of materials is important for a lot of high temperature applications, such as thermal barrier coatings, nuclear applications and high temperature thermoelectrics. Prediction of macroscopic thermal properties depend on the microscopic description of vibrational dynamics of the atoms in the solids, which is primarily characterized by phonon dispersions. Although recent progress in first-principles calculation has enabled prediction of thermal properties routinely for many materials, it has been one of the long- standing challenges in material physics to model the vibrational spectra for materials that are dynamically unstable. Conventionally, lattice dynamics calculations are performed at the static limit (0 K) using the finite displacement method [75] or density functional perturbation theory[172], but these methods failed to explain why the dynamically unstable structures can emerge at high temperatures. For example, SnSe in the CmCm phase is one of thermoelectric materials with best figure of merit ZT at the high temperature (~1000 K). However, the CmCm structure displays soft phonon modes with imaginary frequencies in the phonon dispersion at the static limit. For these soft phonons, the harmonic force constants are negative, which means that the inter-atomic forces no longer pull the atoms back to the equilibrium position but push them away once the atoms are displaced from the equilibrium position. Clearly, the existence of soft phonons is a sign of lattice instability, but the static lattice dynamics failed to explain why the

CmCm phase of SnSe is stable at high temperature. Another example is the body centered cubic

(BCC) structure for group IV metals like Ti, Zr and Hf. They all have soft phonons at the static limit but become stable phases at high temperature.[173, 174] In 1955, Hooton realized that

42 atoms vibrate in an effective potential due to their nonstationary neighbors, and the potential energy surface (PES) is stochastically sampled around the most probable position which is not necessarily a local minima.[78] They then renormalized the soft phonon modes by an effective harmonic potential that is temperature-dependent. Along this line, the problem of dynamical instability is addressed by a self-consistent approach under the harmonic approximation,[77] which starts with the phonon dispersion at static limit as an initial guess and iteratively solve the eigenmodes of the dynamical equation. However, several recent studies suggest that care must be taken for strongly anharmonic crystals where the PES should be expanded to the third and even the fourth order.[79-81] Therefore, the accuracy of the force constants could be significantly affected by the artificial truncation of the Taylor expansion of the PES.[80] On the other hand,

Classical molecular dynamics can naturally incorporate the phonon anharmonicity of arbitrary order without truncating the Taylor expansion of the PES, but it suffers from the inaccuracy of the empirical potential field as limited by the fitting with the empirical functional forms.[82-

84]_ENREF_8

To overcome the challenges of both the first-principles lattice dynamics and the molecular dynamics simulations using empirical potential, machine learning (ML) based regression algorithms provide an elegant solution to reconstruct the ab-initio PES. Instead of decomposing the PES to simple empirical functional forms, the ML algorithm is totally data-driven, which fits the PES by “learning” the correlation between the atomic configurations and the resulting energy from the ab-initio data.[175] Since the ML algorithm does not assume any form of functions when fitting the ab-initio PES, it does suffer from the error caused by artificially truncating the Taylor expansions of the PES. In principle, the ML algorithm includes all orders of anharmonic terms in the PES. Such data-driven feature of ML algorithms also resulted in a significantly improved

43 accuracy of the ML-based potential compared with the empirical potentials, because it bypasses the difficulty of decomposing the high dimensional PES to simple functional forms when fitting for empirical potentials. Due to these advantages, machine learning algorithms including artificial neural networks[176], Gaussian process regression, [177] and others[178] have been successfully used to model the thermal and mechanical properties in simple crystals such as Si, [179-

181]_ENREF_15 GaN,[180] and graphene,[182] as well as complex atomistic structures and processes, such as the amorphous carbon,[183] lithium ion transport in electrode materials, [184,

185] and _ENREF_19 phase-change material GeTe[186].

Since machine learning algorithms addressed both the problem of truncating expansions of

PES in first-principles calculations and the inaccuracy problem of the empirical potentials, it could be a promising tool to capture the lattice dynamics above 0 K by fitting the PES at elevated temperatures. This paper is therefore focused on modeling the phonon renormalization using ML- driven potential in Zirconium (Zr) crystal, one of the most classic example of dynamical instability.

Zr and its alloys are indeed widely used as cladding materials in nuclear reactors.[187] At room temperature, Zr takes the hexagonal closed packed (HCP) phase and transitions into a body centered cubic (BCC) phase at higher temperature, which is dynamically unstable at 0 K.[77] Since phase stability is usually required to prevent structural failures in nuclear applications, understanding the temperature dependent vibrational dynamics of elemental Zr is critical. Recently,

Zong et al. successfully reproduced the phase diagram of Zr using a potential developed by kernel ridge regression algorithm,[188] indicating that ML could be a promising tool to model lattice dynamics of dynamically unstable crystals. However, their potential has limited accuracy for predicting phonon dispersion of both HCP and BCC Zr, with discrepancy of optical phonon frequency as large as 2 THz at the Brillouin zone center.[188] This is probably because their

44 machine learning potential was developed to reproduce phase diagram based on a multi-phase- learning strategy. The training database therefore contains multi-phase structures with regions of phase space beyond thermal vibrations, which is unnecessary for modeling phonons. As a result, the accuracy of phonon dispersions could be compromised.[83] It remains unexplored whether such ML potential can be applied to study the temperature-induced renormalization of the soft phonon modes in dynamically unstable structures.

In this Chapter, we focused on modeling the temperature effect on phonon dispersions using

ML potential. Gaussian approximation potential (GAP) model[180, 189] based on the Gaussian

Process Regression algorithm[177] is used to fit the PES of both HCP-Zr and BCC-Zr. For each phase of Zr, we developed a GAP model which accurately reproduced the energies and interatomic forces, the equation of state and the elastic constants derived from first-principles calculations. We observed that the instability of the BCC Zr at the static limit originates from the double-well shape of the PES, and the BCC structure corresponds to the local maxima of the PES. The high temperature BCC structure is stabilized by a stochastic average due to atomic vibrations over the two low symmetry minima separated by a low potential barrier. The phonon renormalization of the BCC-Zr can therefore be captured by performing molecular dynamics (MD) simulations which stochastically samples the PES. Using spectral energy density analysis, we have successfully observed that the soft transverse acoustic (TA) phonons of BCC-Zr is renormalized to ~ 1 THz at

1188 K.

III.2 Methodology of Building Machine Learning Potential

Here we briefly review the formalism to use the GAP method for fitting PES and the symmetry invariant descriptors for characterizing the atomic configurations in Section 2.A. We then discuss the details for generating the database from the first-principles calculations including total energies, inter-atomic forces and virial stresses for training the machine learning based GAP model in

Section III.2.2. The training databases are downloadable in supplementary materials of ref[190], and the training process is performed using the QUIP package. [191]

III.2.1 Fitting Potential Energy Surface using GAP Method

To construct the machine learning-driven potential using GAP, the total energy of the simulation cell is decomposed into the contributions from each individual atom:

E = ∑ ε(퐪i) (III.1) i where ε(퐪i) is the contribution of energy from atom i, and 퐪i is the descriptor vector that characterizes the local chemical environment of atom i, i.e. the configurations of atoms in the neighborhood of atom i. The local energy contribution ε(퐪) is given by a linear combination of the kernel functions:

ε(퐪i) = ∑ αjK(퐪i, 퐪j) = ∑ Kijαj (III.2) j j where the summation over j includes all the atomic configurations in the first-principles database.

The kernel function Kij = K(퐪i, 퐪j) is a nonlinear function that quantifies the degree of similarity between the chemical environments described by 퐪i and 퐪j. The vector 훂 = (α1, α2, … , αj, … ) are the unknown coefficients to be determined using the first-principles data. Here we discuss first

46 the determination of the unknown coefficient vector 훂, which is also called as “training process”, and then briefly discuss the specification of the kernel function K and descriptors 퐪i. Detailed derivations can be found in refs [192, 193]_ENREF_24.

The database for building the GAP potential is collected into the vector 퐲, which contains the results from the first-principles calculations including total energies, inter-atomic forces and virial stresses. Another vector 훆 is introduced to denote the set of local atomic energies with components

T εj = ε(퐪j). Then a linear operator 퐋 can be introduced to correlate 퐲 and 훆 through 퐲 = 퐋 훆. The operator 퐋 is then constructed as follows. If the data entry yi in vector 퐲 is the total energy of a

T certain atomic configuration, then (퐋 )ij is 1 if the local energy ε(퐪j) of atom j should be included

T into the summation to find total energy as shown in Eq. (III.1), otherwise (L )ij is 0. If the data yi

T ∂ is a component of interatomic forces or stresses, then (퐋 )ij are differential operators with ∂xj respect to atomic coordinate xj. Using the linear operator 퐋, the covariance matrix 퐊DD can be constructed to quantify the similarity correlation between any pair of data points in the vector 퐲 as:

T 퐊DD = 퐋 퐊NN퐋 (III.3) where the subscript D and N denotes the length of 퐲 and 훆, respectively. 퐊NN is the covariance matrix for the joint covariance matrix for energies with elements (퐊NN)ij = K(퐪i, 퐪j) corresponding to the atomic configurations in 훆. However, computing the full covariance matrix

5 퐊NN is expensive since N can easily approach 10 when the forces and virial stresses are included in the database. Therefore, a sparsification method [192] is used to reduce the computational cost.

Instead of computing the full matrix 퐊NN, a representative set containing M atoms (M ≪ N) are chosen from the full set of N atoms randomly, so that the computational cost is reduced by dealing with a much smaller covariance matrix 퐊MN between the representative set and the full set and the

T covariance 퐊MM of the representative set. Then the unknown coefficients 훂 = (α1, α2, … , αM) is calculated as a linear combination of the input data 퐲, which is derived from Bayesian probability formula:

−1 T T −1 −1 훂 = (퐊MM + 퐊MN퐋횲 퐋 퐊MN) 퐊MN퐋횲 퐲 (III.4)

2 where 횲 is a diagonal matrix with diagonal elements the squared uncertainties (σv) of the input data due to convergence parameters in ab-initio calculations, see Table III.1.

We now discuss the formalism for the kernel functions K(퐪, 퐪′) and the descriptor vector 퐪.

The descriptor vector 퐪 is used to characterize the structural features of atomic configurations in the neighborhood of a certain atom (later referred as local chemical environments), which is usually referred as the chemical environment. The descriptor of a dimer molecule is simply the bond length between the two atoms. However, in condensed matter systems like crystals, one needs to deal with the many body feature of atomic interactions, which makes the choice of descriptor much more difficult. One of the most intuitive choice of descriptor for solids is the list of atomic

N N positions {퐫i}i=1 . However, {퐫i}i=1 is not a good descriptor, because it fails to uniquely characterize certain atomic configurations. For example, one can simply generate a complete different list by changing the order of atoms in the list, or imposing arbitrary rotations/translations to the coordinates, while the new list and the old list corresponds to the same atomic structure. A good descriptor should therefore be invariant to permutation, translation and rotation operations.[189] Recently, Bartok et al. derived the so called SOAP descriptor[194] that can be used to uniquely characterize and differentiate chemical environments, which is chosen as the descriptor in this work. Since the nonlocal metallic bonds in Zr crystals are intrinsically many- body interactions, the many-body SOAP descriptor becomes the natural choice. In SOAP, the

48 chemical environment of an atom i is represented by the density of neighboring atoms, which is smoothed by a Gaussian function:

2 |퐫−퐫 | − ij 2σ2 ρi(퐫) = ∑ e a fcut(|퐫ij|) (III.5) j where 퐫ij = 퐫i − 퐫j is the vector connecting atom i and its neighboring atom j, σa is corresponding to “size” of atom. The function fcut is a smooth cut-off function:

1, r < rcut − d 1 r − r + d f (r) = { [1 + cos (π cut )] , r − d < r ≤ r (III.6) cut 2 d cut cut 0, r > rcut where rcut is the cutoff radius, and d is the cutoff transition width where the fcut smoothly decays from 1 to 0. Obviously, ρi only depends on the relative coordinate 퐫ij thus invariant to translations, and the summation over j ensured permutation invariance of ρi. To ensure the rotational invariance, the atomic density distribution ρi is further expanded to a set of orthonormal radial basis functions gn(r) and spherical harmonics Ylm:

l 퐫 ρ (퐫) = ∑ ∑ ∑ ci g (|퐫|)Y ( ) (III.7) i nlm n lm |퐫| n

The components in descriptor vector 퐪i are then calculated as the power spectrum of the expansion

i coefficients cnlm:

i ∗ i (퐪i)nn′l = ∑(cnlm) cn′lm (III.8) m After specifying the descriptors, the kernel functions are constructed by inner products of descriptor vectors:

퐪 ⋅ 퐪 ζ 2 i j Kij = σw | | (III.9) |퐪i| ⋅ |퐪j| where the exponent ζ is a positive integer to improve the sensitivity to different local atomic

2 environments, and σw is an overall scaling parameter. From Eq. (III.5)-(III.9), the hyper parameters (σv, σa, σw, rcut, d, ζ, nmax, lmax) are summarized in Table III.1 for constructing the descriptors and the kernel functions. Here we choose typical values of σa, σw, d, rcut, ζ in the literature. [193, 195]_ENREF_25 The expansion cutoff nmax, lmax are chosen so that a converged phonon dispersion can be obtained with the tolerance in frequency of 0.01 THz.

In summary, the procedure of fitting PES works as follows. The data from first-principles calculations are collected into the vector 퐲 first and the coefficient vector 훂 is then calculated using

Eq. (III.4). The kernel functions used to generate covariance matrices 퐊MN, 퐊MM are specified as

Eq. (III.5)-(III.9). After obtaining the coefficient vector 훂, total energies of an arbitrary atomic configuration 퐪 can be calculated using Eq. (III.1)-(III.2), which completes the Gaussian process regression process. In the following part, we are going to discuss the details for generating the training database, i.e., the vector 퐲 using the first-principles calculations.

Table III.1. Hyper parameters for GAP with SOAP kernels.

rcut 5.0 Å

d 1.0 Å

σv for energy 0.001 eV/atom

σv for forces 0.05 eV/Å

σv for virial stress 0.05 eV/atom

σw 1.0 eV

σa 0.5 Å

ζ 4

nmax 12

lmax 12

III.2.2 Generation of Training Database

Since the purpose of this work is to model the temperature effect on phonon dispersion of Zr, the database should be constructed with specific emphasis on the phase space region around equilibrium that is approachable by thermal vibrations. The developed potential is expected to accurately fit the curvature of ab-initio PES at equilibrium. In addition to the curvature at the static limit, the thermal vibrations would sample a wider region of the PES in the phase space, which is essentially the physical origin for phonon dispersion renormalization. Therefore, the training database should not only include responses to perturbations of the equilibrium structure such as

51 strains and atomic displacements, but also snapshots of thermal vibrations at high temperatures. In order to avoid the potential fitting unnecessary phase space regions beyond thermal vibrations, we separately train the potential for each phase (HCP and BCC) of Zr studied in this work to ensure the accuracy of phonon dispersions. For both HCP and BCC Zr, the databases are constructed as follows.

Database 1 is used to train the GAP model in the descriptor space around the equilibrium geometry and the mechanical response to bulk strains. Self-consistent field (SCF) calculations are performed with different strain tensors with distortion parameters up to 4% imposed on the simulation cell. The symmetry-irreducible strain tensors for the hexagonal lattice and the cubic lattice are specified in ref. [196] and ref. [197] , respectively. The size of the simulation cells for

HCP-Zr and BCC-Zr are specified in Table III.2. Database 1 includes forces on atoms, total energies and virial stress on the simulation cell.

Database 2 is used to teach the GAP model with harmonic and anharmonic force constants at different volumetric strains. First, simulation cells of HCP-Zr and BCC-Zr are constructed with uniform strains on each lattice constant from -4% to 4% with the step of 1%. In each supercell with strains, small displacements (0.03 Å) are imposed to the irreducible atoms according to the space groups using the Phonopy package[198] and ShengBTE package[199]. SCF calculations are then performed for each perturbed supercell with strains and displacements, so that the total energies, forces, and virial stresses at the perturbed states are recorded.

Database 3 provides the information of chemical environments and PES above 0 K. ab-initio molecular dynamics (AIMD) simulations were performed at different temperatures to generate snapshots of atomic configurations for both BCC and HCP Zr. At each temperature, 1000

52 snapshots of atomic configurations are generated with AIMD using a time step of 1 femtosecond.

Total energy and forces are used as training data quantities.

All the training data in the databases is generated by the density functional theory (DFT) based first-principles calculations using the Vienna Ab-initio Simulation Package (VASP).[200, 201]

Since the goal is to capture the effect of temperature on phonon dispersion (renormalization), the training database should include DFT data at both 0 K and at finite temperatures. All DFT calculations are performed using PBE functional[202] with projector augmented wave (PAW) method.[200, 201] For all DFT simulations, the cutoff energy is chosen as 300 eV.[77] For both

HCP phase and BCC phase of Zr, the following databases were generated to train the GAP model with chemical environments. Table III.2 summarizes the detailed parameters in DFT and AIMD calculations, including total number of atoms in all AIMD snapshots and DFT calculations (N), number of representative set of atoms (M), temperature T, dimensions of supercells, convergence threshold of SCF calculations (EDIFF tag in VASP package).

Table III.2 Detailed parameters for DFT calculations to generate training databases.

HCP Zr

N M T (K) Supercell k-mesh EDIFF

Database 1 1350 20 0 3×3×2 7×7×7 1e-10

Database 2 4266 65 0 3×3×2 7×7×7 1e-10

Database 3 72000 750 100, 300 3×3×2 3×3×3 1e-6

BCC Zr

N M T (K) Supercell k-mesh EDIFF

Database 1 1458 20 0 3×3×3 7×7×7 1e-10

Database 2 2214 45 0 3×3×3 7×7×7 1e-10

Database 3 54000 750 100, 300, 1200 3×3×3 3×3×3 1e-6

III.3 Results and Discussions

Figure III.1 (a-b). Comparison of (a) energy and (b) inter-atomic forces between GAP and AIMD calculations of the HCP-Zr. (c-d). Comparison of (c) energy and (d) force components between

GAP and AIMD calculations of the BCC-Zr.

This section discusses the application of GAP to model the phonon renormalization in Zr at elevated temperature. Before that, the accuracy of the GAP model to reproduce DFT calculations should be examined. As shown in Figure III.1, the GAP prediction of total energies and components of forces (Fix, Fiy, Fiz of atom i along three Cartesian axes) are compared with the original AIMD simulation, which are corresponding to 200 equally spaced snapshots randomly selected from the 1000 AIMD snapshots at 300 K. The GAP model is observed to reproduce the

55 energies from AIMD calculation with the root mean squared error (RMSE) of 0.0002 eV/atom for the HCP phase and 0.0003 eV/atom for the BCC phase. The RMSE of the atomic forces between

GAP model and AIMD simulations is 0.025 eV/Å for the HCP phase and 0.053 eV/Å for the BCC phase. The comparisons indicate good fitting of the ab-initio PES and its derivatives.

In addition to accurately reproduce the training observables (energies and forces), the GAP model is also expected to reproduce the thermal and mechanical properties of the Zr crystals.

Figure III.2a shows the equation of state E = E(V) and Figure III.2b shows the symmetry- irreducible elastic constants Cij for both hcp and BCC-Zr. Excellent agreement is achieved in the equation of state as well as the elastic constants. The instability of the BCC-Zr is manifested in the elastic constants. For a crystal to be energetically stable, the Born criteria requires the Cij tensor to be positive-definite. In the case of BCC structure, the stability criteria requires C11, C12 and C44 to satisfy C11 − C12 > 0, C44 > 0 and C11 + 2C12 > 0.[203] Clearly the criteria C11 − C12 > 0 is not satisfied as shown in the right panel of Figure III.2b. Besides the elastic constants, we also compare the phonon dispersions predicted by the GAP model of both HCP-Zr and BCC-Zr at the static limit with the inelastic neutron scattering (INS) measurements[27, 173]_ENREF_35 and the

DFT calculations, as shown in Figure III.2c and Figure III.2d. For the HCP phase, there is only small difference in the phonon dispersion, while larger discrepancy is observed for the soft modes

(plotted as imaginary frequencies) of the BCC phase, which is likely due to the larger RMSE of energy and forces in for the BCC phase as shown in Figure III.1b and Figure III.1d.

HCP-Zr. INS measurement data is taken from ref. [27] (d) Phonon dispersion of BCC-Zr. INS measurement data is taken from ref. [173]

To illustrate the origin of the soft phonon modes, the PES is plotted in the normal coordinates for the two lowest modes at the high symmetry point N in the Brillouin zone. In order to obtain the shape of the PES around the equilibrium position, small displacements are imposed along the eigenvectors for the lowest soft TA mode with a scaling factor Q1 and the second lowest TA mode

57 with a scaling factor Q2 and the PES as a function of scaled coordinates E = E(Q1, Q2) is plotted as Figure III.3a. It is clear that the PES shows a double-well shape. The dynamic instability of the

BCC structure originates from the fact that the equilibrium state (Q1, Q2) = (0,0) is a saddle point of the PES. Along the Q1 direction, the equilibrium state is the local maxima of the double-well as shown in Figure III.3b, while it is the local minima along the Q2 direction. As a result of the

2 ∂ E 2 negative local curvature 2 < 0, the eigenvalue for the lowest TA mode ω is also negative when ∂Q1 the lattice dynamics simulations are performed at the static limit, so that the imaginary phonon frequency is observed in Figure III.2d. At high temperature, the normal mode oscillator is hopping between the two potential wells, and the equilibrium position corresponding to the BCC structure is indeed the dynamical average between the two local minima. In addition, due to the complicated multi-minimum shape of the PES of the BCC phase, the fluctuations of AIMD energy and forces could also be larger compared with the stable HCP phase even at the same temperature, which results in the larger RMSE when reproducing the AIMD energies and forces.

Figure III.3 (a) PES along eigenvectors at high symmetry point N. Q1 and Q2 correspond to dimensionless normal coordinate of the two TA modes with the order of increasing frequency. (b)

PES along the Q1 direction with Q2 = 0. (c) PES along the Q2 direction with Q1 = 0.

With the idea that the BCC structure is stabilized through dynamical average of the low- symmetry minima of the PES, the phonon dispersion should be renormalized to real frequency values at high temperature when the PES is dynamically sampled. MD simulations are therefore performed to stochastically sample the PES, using the machine learning driven GAP potentialas we have developed above. Phonon dispersion is then calculated by SED analysis[144, 145] which maps the vibrational energy distribution in wave-vector space and frequency domain (퐪, ω). Here

59 the SED distribution is calculated by summing the Fourier transform of the amplitudes of vibrational velocities:

2 1 τ0 ϕ(퐪, ω) = ∑ ∑ mb |∫ ∑ u̇ α(퐑, b, t) ⋅ exp(i퐪 ⋅ 퐑 − iωt) dt| (III.10) 4πτ0Ncells α=x,y,z b 0 퐑 where Ncells is the total number of unit cells, 퐑 is the lattice vector and b is the index of basis atoms in the unit cell, u̇ α(퐑, b, t) is the velocity component along the α = (x, y, z) axis of the atom

(퐑,b) at time t. The quantity dt (=1 fs) is the time step between neighboring MD snapshots, and

τ0, the total integration time is selected as 1 ns, and longer τ0 is found not to affect the SED distributions. For the HCP phase, SED along the Γ − A direction and the Γ − M direction are calculated, using supercells containing 3×3×50 primitive cells and 50×3×3 primitive cells, respectively. For the BCC phase, SED is extracted along the Γ − N path using a supercell containing 50×3×3 primitive cells. Figure III.4a-b shows the SED of the HCP-Zr at 100 K and

300 K. For the HCP phase, the most pronounced effect of non-zero temperature is the broadening of the SED lines due to stronger phonon scattering at higher temperature. Figure III.4c shows the phonon dispersion of the BCC-Zr at 1188 K. The SED analysis has successfully captured the renormalization of the soft TA mode in BCC-Zr which is now renormalized to ~1 THz at 1188 K.

Figure III.4d shows the SED as a function of frequency at 퐪=(0.3, 0, 0) along the Γ − N direction.

The broad SED peak observed near 1 THz is agreeing well with INS experiments.[173]

Figure III.4 (a-b) SED of HCP-Zr at (a) 100 K and (b) 300 K. (c) SED of bcc-Zr at 1188 K. (d)

SED as a function of phonon frequency at q = (0.3,0,0). The dashed lines indicate the frequency measured by INS in ref. [173] at 1188 K.

III.4 Summary of this Chapter

In summary, we studied the temperature effect on phonon dispersions of the HCP phase and the dynamically unstable BCC phase of Zr, using molecular dynamics simulation with machine learning-driven Gaussian approximation potential. The GAP model accurately reproduces energies and interatomic forces corresponding to the atomic configurations of the AIMD snapshots as well as the mechanical properties of Zr. The dynamical instability of BCC Zr is captured by the GAP model with the soft phonon modes in the dispersion relationship as well as the non-positive- definite elastic constant tensor. The instability of the BCC structure is observed to originate from the double-well shape of the PES, and the BCC phase becomes stable at high temperature as a result of dynamical average as the normal mode oscillators hopping between the two local minima of the PES. The stabilization of BCC Zr is captured by examining the phonon dispersion at high temperature using MD simulations and SED analysis. In addition to the broadening effect at elevated temperature, the SED analysis also captures the phonon renormalization of the soft TA mode in BCC crystal, with the frequency renormalized to ~ 1THz at 1188 K, agreeing well with the INS experiments. This work for the first time approaches the problem of phonon renormalization in dynamically unstable crystals using molecular dynamics, showing that machine learning-driven potential is a promising tool for modeling high temperature lattice dynamics and thermal properties.

CHAPTER IV PROBING ANISOTROPIC AND NON-EQUILIBRIUM THERMAL TRANSPORT IN TRANSITION METAL DICHALCOGENIDES

IV.1 Introduction

Transition metal dichalcogenides (TMDs) have attracted intensive research interest in recent years due to their many intriguing properties derived from the layered atomic structures, including the thickness-tunable electronic bandgaps,[205, 206] strong photoluminescence,[207-209] and highly anisotropic elastic modulus.[210-212] In contrast to the extensive studies on their optical, electrical, electrochemical, and mechanical properties,[213-216] experimental measurements on the thermal properties of TMDs are relatively few. Despite the important role in affecting the lifetime and stability of a wide range of TMD-enabled devices in electronics,[87, 216, 217] optoelectronics[217-219] and electrochemical energy storage,[220-222] a thorough understanding of the thermal conductivity of TMDs is still lacking. Up to date, there existed only a few experimental studies on the thermal conductivity of TMDs, but with very inconsistent values. For example, the in-plane (basal-plane) thermal conductivity of synthesized MoS2 was measured[101]

-1 -1 -1 to be 16 W m K , one order of magnitude lower than the geologically-mined MoS2 (110 W m

-1 K ).[102] In the through-plane direction, the thermal conductivity of WSe2 shows a two-order-of- magnitude variance due to the random stacking of WSe2 layers.[223] Besides the inconsistency in the measured thermal conductivity values, large discrepancies also exist between the measurements and the theoretical calculations. For example, the in-plane thermal conductivity of

-1 -1 few-layer MoS2 measured by optothermal Raman method are in the range 13 – 81 W m K ,[96-

100] all much lower than the first-principles predictions of 138 W m-1 K-1.[24] Clearly, there is an urgent need for a systematic experimental study on the thermal conductivity of TMDs that could be used to verify the theoretical calculations and to elucidate the underlying physics.

Besides pure TMDs, alloys of layered 2D TMDs further opened of the opportunities for tunable properties.[89, 224] For example, electronic bandgaps can be continuously changed in several

TMD alloy systems including MoS2(1-x)Se2x, [225, 226] WxMo1-xS2[227, 228] and WxMo1- xSe2.[229] With intrinsically large power factor of TMDs,[230-232] alloying layered TMDs could potentially lead to highly efficient thermoelectrics where the thermal conductivity is effectively reduced due to the mass-disorder scattering. [233, 234] Along this line, Gu et al.[26] performed a first-principles study on MoS2(1-x)Se2x monolayers, and predicted an order-of-magnitude reduction in the thermal conductivity of these alloys as compared with those of the pristine MoS2 and MoSe2.

However, there yet exists any experimental work on anisotropic thermal transport in layered TMD alloys, especially as a function of alloy compositions.

Intrigued by the unique electronic structures[235-237] and the abundant phase transition[238,

239] in MoTe2 and WTe2, MoTe2 or WTe2 based TMD alloys have also attracted intensive research, especially on manipulating the physical properties through controlling the phase transitions. For example, the topological electronic states of WxMo1-xTe2 can be effectively manipulated by the alloys composition.[240] The hysteresis of the cross-plane thermal conductivity during the 1T’- to-Td phase transition of WxMo1-xTe2 is also observed, and the phase change point can be effectively tuned to room temperature,[241] which makes it a promising material for phase change memory devices.[242] Depending on the composition of WTe2, WSe2(1-x)Te2x also exhibits a 2H- to-Td phase transition, which can be used for bandgap tuning.[243] Such phase transition in ternary

TMD alloys could also be a potential way for tuning and optimizing thermoelectric performances,

[244] but how such a phase transition would affect the thermal transport properties has not been studied.

In this chapter, we discuss the characterization of anisotropic thermal conductivity of TMD crystals and TMD alloys. In section IV.2, we briefly discuss the time-domain thermoreflectance

(TDTR) measurement and the development varied spot size approach for measuring anisotropic thermal conductivity. We use highly ordered pyrolytic graphite (HOPG), ZnO, TiO2 and SiC with well-accepted thermal conductivity for the validation of this technique. We then apply this varied spot size approach to measure both the in-plane and cross-plane thermal conductivity of TMD crystals and TMD alloys in section IV.3. Interestingly the thermal conductivity is observed to depend on modulation frequency. Such frequency dependent thermal conductivity is attributed to the non-equilibrium transport between the high frequency and low frequency phonons. A two- channel diffusive model is developed to describe such non-equilibrium transport feature and to extract the thermal conductivity at equilibrium limit.

IV.2 Varied Spot Size Approach for Measuring Anisotropic Thermal Conductivity

The time-domain thermoreflectance (TDTR) method is a robust and powerful technique that can measure thermal properties of a wide variety of materials.[2, 32, 245] Figure IV.1 shows a schematic of the TDTR system implemented in our lab. This system splits the femtosecond laser into a pump beam and a probe beam. The pump beam is modulated by an electric-optical modulator

(EOM) to create a periodic thermal excitation on the surface of TMD alloy samples. The TMD alloy samples are first deposited with an aluminum metal thin film (70 ~ 100 nm). This Al transducer film absorbs the pump beam and its reflectance would change linearly with the surface temperature rise. The probe beam is delayed by a translational delay stage to monitor the surface temperature change. The surface reflectance change is measured by detecting the reflected probe beam from the sample using a photodetector.

There are two important length scales in the TDTR experiments that determine the heat flow direction in the SiC substrate and hence the different sensitivities to the in-plane thermal conductivity kr and the cross-plane thermal conductivity kz. The first length scale is the size of

2 the Gaussian laser spot, defined as the root-mean-square average of the 1/e radii of the pump (w0)

2 2 and the probe (w1) as: w = √(w0 + w1 )/2. The other important length scale is the thermal penetration length dP,α = √kα/Cπf0 , where the subscript α(= r, z) denotes the direction in cylindrical coordinates, k is the thermal conductivity, and C is the volumetric heat capacity. Since

TDTR measures the surface temperature rise within the RMS radius of the laser spot, whether the

TDTR signal is sensitive to kr depends on how large the laser spot radius w is compared to the in- plane thermal diffusion length dP,r . If the spot radius w is much larger than the in-plane penetration length dP,r, the in-plane temperature gradient is negligible and the heat flow can be regarded as one-dimensional along the cross-plane direction. In this case the surface temperature rise should be dominantly determined by kz of the sample. On the other hand, when the laser spot is tightly focused, the heat flow in the sample becomes three-dimensional, and the surface temperature rise is also affected by kr.

Figure IV.1. (a) A schematic of the TDTR system implemented in our lab. Abbreviations are listed as follows. EOM, electric-optical modulator; PBS/NPBS: polarized/non-polarized beam splitter;

Based on the discussion above, we realized that kz and kr can be separately measured by using different laser spot size. We use ZnO as an example for illustrating measuring kr and kz using different laser spot sizes. ZnO is a hexagonal crystal with kz = 62 W/mK along the [001] direction and kr = 44 W/mK parallel to the (001) crystal plane estimated from first principles calculations.

[9] We take the ratio between the in-phase (Vin) and the out-of-phase (Vout) voltages from the lock- in outputs, R = −Vin/Vout as the measured signals that are fitted with a multi-layer heat conduction model[32] to extract the thermal properties. Choosing appropriate experimental parameters (spot radius w and modulation frequency f0) is critical to ensure the accuracy of kz and kr. We therefore define the sensitivity parameter to quantify how sensitively the TDTR signal is affected by different parameters (e.g. kr and kz). The sensitivity parameter is defined as:

Vin d ln (− V ) S = out (IV.1) p d ln p where p is the parameter of interest. If the sensitivity Sp equals 2, then a 10% change in p would result in a 20% change in the signal −Vin/Vout. Since we are measuring both kr and kz, the error propagation between kr and kz determines the measurement accuracy. The ratio Skz/Skr also has its physical meaning. For example, if Skz/Skr equals 2, then a 10% uncertainty of kz would result in a 20% error in kr. We plot the sensitivity ratio Skz/Skr map with respect to the experimental parameters w and f0 in Figure IV.2. To determine kz independent of kr, the Skz should be at least

Skz one order of magnitude greater than Skr. Therefore, we require > 10 for measuring kz. The Skr laser spot radius and modulation frequency can be selected from the region on upper right corner in Figure IV.2. After kz is measured, kr needs to be determined. Since typical error of kz

Skz determined by TDTR is 10%~ 15%, we need < 2 so that the error in kr is within 30%. Skr

As shown Figure IV.3, when the laser spot radius is w = 16 μm, the TDTR signal is determined by kz independent of kr. We therefore first determine kz = 55 ± 5.6 W/mK. After kz is already known, we tightly focus the laser to w = 4 μm to measure kr. The extracted in-plane thermal conductivity kr is determined as 43 ± 7 W/mK.

Using the similar method, we measured kr and kz of a series of anisotropic crystals with different ratio of anisotropy at room temperature, including HOPG, hexagonal boron nitride (h-

BN), TiO2 and silica, as shown in Figure IV.4. We also apply this technique to measure the temperature dependent thermal conductivity of 4H and 6H SiC, as shown in Figure IV.5 and Figure

IV.6. All these measurements agree well with the literature values.

Figure IV.2. Sensitivity ratio map Skz/Skr for the choices of modulation frequency and laser spot radius.

Figure IV.3. Measuring (a) cross-plane and (b) in-plane thermal conductivity of a (0001) ZnO using varied laser spot radii.

Figure IV.4. Measured kr and kz of HOPG, h-BN, ZnO, TiO2 and silica compared with the literature values. Literature values are from Schmidt et al.[246], Sichel et al., [247] Wu et al.,[9] Thurber and

Mante, [29] and Sugawara[248].

Figure IV.5. (a) In-plane thermal conductivity for SI and n-type 4H-SiC compared with the first- principles calculation by Protik et al.,[249] and the steady-state measurement by Morelli et al. [250]

(b) The cross-plane thermal conductivity for SI and n-type 4H-SiC compared with the calculation by Protik et al. [249] and the laser flash analysis measurement by Wei et al. [251]

6H-SiC compared with the calculation by Burgemeister et al.[253], Protik et al. ,[249] and Nilsson et al. [254]

IV.3 Non-equilibrium Phonon Transport in Transition Metal Dichalcogenides

In this section, we discuss the measurement of anisotropic thermal conductivity and the non- equilibrium transport phenomenon in single crystals and alloys of TMDs. We first discuss the measurement details and the anisotropic transport phenomenon of single crystalline TMDs, then we discuss the thermal conductivity affected by the atomic disorder introduced in TMD alloys

WSe2(1-x)Te2x.

TMD Single Crystals. Using the varied spot size TDTR approach, we measure the thermal conductivity of single crystals and alloys of TMDs. An example is shown in Figure IV.7, in which the kr and kz of MoS2 is measured at 1MHz. At 1 MHz, kz is first determined as 4.4 W/mK using a spot radius of 24 μm, and kr is the measured as 80 W/mK using a spot radius of 6 μm. However, when we performed the same experiment on MoS2 described above at a different frequency of 2

-1 -1 MHz, we obtained a smaller through plane-thermal conductivity kz = 4.1 ± 0.4 W m K , although

-1 -1 the in-plane thermal conductivity remained the same kr= 80 ± 22 W m K . The through-plane

-1 -1 thermal conductivity of MoS2 further reduces to 3.3 ± 0.5 W m K when the modulation frequency is increased to 10 MHz. Similar frequency dependence in kz was also observed in the other TMD samples, see Figure IV.8 for examples of MoS2 and WSe2. We have also checked whether the frequency dependence kz is caused by the choice of spot size. We have therefore performed the cross-plane measurement at 10 MHz using different spot radius (6 μm and 11 μm), and the same kz is obtained. Based on this result, we refer that the measured kz at lower modulation frequency would also not depend on laser spot size, because the heat region

(penetration length) at lower modulation frequency is even larger than that at 10 MHz, which by no means should introduce any ballistic effect caused by laser spot size if we did not observe spot

72 size dependent kz at 10 MHz. We are thus confident that the frequency-dependent kz we have observed in the TMD samples are not due to the choice of laser spot sizes. More interestingly, we also consistently observed a frequency dependence in the interface thermal conductance G, with

~30% increase from 1 MHz to 10 MHz shown in Figure IV.8c-d. We have verified that such frequency dependences in kz and G are not due to any systematic error by measuring sapphire and silica as calibrations of our TDTR system; none of these measurements shows any frequency dependence in either kz of the substrate or the interface conductance, just as expected.[255, 256]

2.5 K 20% z K 20% 2 r w =6 m 0

out 1.5

- w =24 m 0 1

MoS , 300 K, f=1 MHz 2 100 1000 10000 t (ps) d

Figure IV.7. TDTR experimental data (symbols) along with the fittings from a thermal model

(solid, red lines) and 20% bounds on the fitted through-plane (dash, blue lines) and in-plane (dash- dot, green lines) thermal conductivity values. The data were taken as MoS2 measured at room temperature, using two different laser spot sizes (w0 = 6 μm and w0 = 24 μm) at a modulation frequency 1 MHz. The thermal conductivities obtained from the fits are 4.4 ± 0.45 W m-1 K-1 along the through-plane and 80 ± 17 W m-1 K-1 along the in-plane direction, respectively.

We attribute the frequency dependence in both kz and G of TMDs to the non-equilibrium thermal transport between the low-frequency and the high-frequency phonons in the substrate;[257-260] see our discussion below. In the layered TMDs, the existence of the phonon bandgap[25] allows us to group the low-frequency phonons (below the phonon bandgap) and the high-frequency phonons (above the phonon bandgap) into two different heat conduction channels.

The low-frequency phonons contribute to the majority of the through-plane thermal conductivity due to their long mean-free-paths, while the high-frequency phonons contribute little to the thermal conductivity due to their near-zero group velocities. When the thermalized phonons from the transducer layer penetrate into the TMD substrate, the heat carried by the low-frequency phonons

(denoted as channel 1) quickly dissipates away due to the relatively high group velocities. On the other hand, a significant amount of heat accumulates in the high-frequency-phonon channel

(denoted as channel 2), due to their large heat capacity and small thermal conductivity.

Temperature difference thus exists between the two channels under thermal excitations, especially when the coupling between the two channels is weak. Therefore, an extra non-equilibrium thermal resistance RNE arises as the heat flows from the high-temperature channel to the low-temperature channel; see Figure IV.9 for a schematic illustration. For each channel, there is a heat conduction equation describing its own temperature profile with a coupling term for the energy exchange between the two channels:

∂T k ∂ ∂T ∂2T C 1 = r1 (r 1) + k 1 + g(T − T ) 1 ∂t r ∂r ∂r z1 ∂z2 2 1 (IV.2) ∂T k ∂ ∂T ∂2T C 2 = r2 (r 2) + k 2 + g(T − T ) 2 ∂t r ∂r ∂r z2 ∂z2 1 2

-3 -1 Here g is the coupling constant between the two phonon channels, with the unit W m K , kz1 and kz2 are the thermal conductivity of the two channels. The detailed solution of the two-channel diffusive model can be found in Appendix A. The temperature difference between the two channels

−1/2 will exponentially decay with a length scale estimated as dNE = (g/kz1 + g/kz2) , within which the non-equilibrium between the two channels is not negligible.[257] The ratio of the non- equilibrium length dNE over the thermal penetration depth dp would determine how the non- equilibrium thermal resistance RNE affects the apparent thermal conductivity kz and the interface conductance GA derived using the one-channel model.[32] When dNE is much shorter than dp, non-equilibrium only happens near the interface. In this case, the result of using a one-channel model assuming local thermal equilibrium is to combine the non-equilibrium thermal resistance into the effective interfacial thermal conductance, resulting in a lower apparent thermal conductance GA at a low modulation frequency. On the other hand, when dNE is comparable to dp, non-equilibrium exists throughout the whole thermally excited region. In this case, the non- equilibrium thermal resistance manifests in both GA and kz. Such predictions based on this non- equilibrium picture is consistent with the frequency-dependence trend of kz and GA observed in our TDTR experiments.

To obtain the thermal conductivity at equilibrium limit, we employed the diffusive two-channel heat transfer model[257] to analyze our measured data at multiple modulation frequencies. We choose a large spot size (w > 20 μm) for the measurements so that the obtained signals are independent of the in-plane thermal conductivity kr. In the two-channel model, we group the low- frequency phonons beneath the phonon band gap to channel 1, and the high-frequency phonons above the band gap to channel 2 (See division of phonon channels in Appendix C).[257] In MoS2,

-3 -1 -3 -1 the heat capacity for channel 1 is C1 = 0.72 MJ m K and for channel 2 is C2 = 1.17 MJ m K

75 at room temperature. We take the through-plane thermal conductivities of the two channels kz1 and kz2, interface thermal conductance G1 and G2, and the coupling constant between the two channels g as the free fitting parameters, determined using the simplex optimization algorithm[261] by fitting the measurement signals obtained from 1 to 10 MHz simultaneously. Converged results have been achieved with different initial values for iterations. The parameters achieving the best

2 fit for MoS2 are kz1 = 3.52 ± 0.30 W/mK, kz2 = 1.23 ± 0.11 W/mK, G1 = 11.6 ± 1.75 MW/m K,

2 14 3 G2 = 76.4 ± 7.7 MW/m K and g = (0.10 ± 0.04) ×10 W/m K at room temperature. Therefore, the equilibrium through-plane thermal conductivity of MoS2 is kz = kz1 + kz2 = 4.75 ± 0.32 W/mK.

The uncertainty analysis can be found in Appendix AII.2.

To quantitatively demonstrate that the frequency-dependent GA and kz,A are caused by the non-equilibrium and the finite coupling strength between the low-frequency phonons and the high- frequency phonons, we calculated GA and kz,A of MoS2 and WSe2 as a function of modulation frequency using the two-channel model, at 300 K and 100 K, as shown in Figure IV.8. The predicted frequency-dependent GA and kz,A are calculated by fitting the signals generated by the two-channel model with the one-channel model. The model predictions compare very well with our measurements for both GA and kz,A , thus supporting our statement that the frequency dependence in GA and kz,A originates from the non-equilibrium thermal transport near the interface.

Figure IV.8. Apparent cross-plane thermal conductivity and interface thermal conductance GA of

We summarize our measured in-plane and through-plane thermal conductivity of TMDs in

Figure IV.10 as a function of temperature, along with other experimental and numerical values in literature for comparison. In general, our measured in-plane thermal conductivity values compare very well with the first-principles predictions by Lindroth and Erhart[103] for all the four layered compounds over the temperature range of 80 – 300 K, while all the measured through-plane

78 thermal conductivity values are lower than the calculations, with a peak at around 150 K. Such a temperature dependence in the through-plane thermal conductivities suggests that there could be some boundary scattering of phonons in the through-plane direction, although our samples are bulk crystals instead of few-micron thin films. This additional boundary scattering in the through- plane direction, however, should not be alarming for the layered materials, as Chiritescu et al.[223] have demonstrated that the random stacking of WSe2 layers could result in almost two orders of magnitude variation in the through-plane thermal conductivity. In bulk TMD crystals, either geologically mined or chemically synthesized, there could inevitably be some planar defects, such as the stacking faults and the subtle variations in layer spacing,[262] which impose scattering on the phonon transport in the through-plane direction.

We could use an effective boundary scattering length to represent the strength of boundary scattering for TMDs in the through-plane direction, although we note that the nature of boundary scattering for TMDs in the through-plane direction is more complicated than a single boundary scattering length. To estimate the effective boundary scattering length, we compare our measurements of through-plane thermal conductivity of WS2 and WSe2 with the first-principles calculations[103] that incorporated a boundary scattering length of 150 nm, shown as the dash lines in Figure 4. Our measured values are still higher than the calculations with 150 nm boundary scattering length, suggesting that the boundary scattering length should be larger than 150 nm for our samples. However, the strength of boundary scattering in the through-plane direction could be specific to different samples due to its complicated nature. We also note that our through-plane measurements are consistently higher than the thermal conductivity of MoS2 and WSe2 at room temperature reported by other TDTR measurements by Liu et al.[102], Chiritescu et al.[223] and

Murato et al.[34] Since they all measured at 10 MHz and failed to consider the non-equilibrium

79 thermal transport in the through-plane thermal conductivity of TMDs, their reported values are underestimated.

1000 MoS WS WSe 2 2 MoSe2 2

Liu et al. Pisoni et al. 100 ) Pisoni et al.

-1

-1 In-plane In-plane In-plane In-plane Through-plane Through-plane Through-plane Through-plane

W m

( 10

K Chiritescu et al.

Murato et al. Liu et al. 1 Pisoni et al. Murato et al.

30 100 300 30 100 300 30 100 300 30 100 300 T (K) T (K) T (K) T (K)

Figure IV.10. Measured in-plane and through-plane thermal conductivities (solid symbols) of

Ref. [103]. The dash-dot curves are the calculated in-plane and through-plane thermal conductivity of natural bulk MoS2 from Ref. [24]. The measurements from literature are synthetic MoS2 by

Pisoni et al.[101], natural MoS2 crystal by Liu et al.[102], synthetic WS2 by Pisoni et al.[263], single crystal WSe2 by Chiritescu et al.[223], and single crystal MoS2 and WSe2 by Murato et al.[34]

The in-plane thermal conductivity values from our variable spot-size TDTR measurements for

MoS2, WS2, MoSe2 and WSe2 agree very well with the first-principles calculations by Lindroth and Erhart [103] which has the following decreasing order: WS2 > MoS2 > WSe2 > MoSe2. At room temperature, the measured in-plane thermal conductivities are 82 W/mK for MoS2, 120

W/mK for WS2, 35 W/mK for MoSe2 and 42 W/mK for WSe2, respectively. The higher thermal conductivity in WS2 than MoS2 observed in our measurements is due to the wider phonon bandgap in WS2, caused by the larger atomic mass difference between W and S. Due to the large phonon bandgap, the phonon-phonon scatterings are suppressed, yielding a higher thermal conductivity.

This is consistent with some recent first-principles calculations.[25, 103, 264] The thermal conductivity of the selenides (MoSe2 and WSe2) are much smaller than those of the sulfides (MoS2 and WS2), because replacing the S atoms with the heavier Se atoms would suppress the all the phonon frequencies (reduce velocity).[103]

However, discrepancies exist between our in-plane thermal conductivity measurements and some other experiments. Our measured in-plane thermal conductivity of MoS2 is ~35% lower than the value reported by Liu et al.,[102] measured using beam-offset TDTR and MOKE. This large discrepancy could be due to the large uncertainties from the spot size and the reference phase in the beam-offset TDTR and MOKE measurements, especially when measured at low frequencies.

Our measurements of in-plane thermal conductivity agree well with the steady-state measurements by Pisoni et al. for WS2,[263] but not for MoS2.[101] The reason, as they have suggested, is that their WS2 was not contaminated while their MoS2 samples could be unintentionally doped during the growing process. Our measurements of through-plane thermal conductivity of WS2, however, are consistently higher than the steady-state measurements by Pisoni et al.[263], although they exhibit the same temperature dependence. As it is very difficult to grow WS2 crystals to be thick

81 enough for the steady-state measurement, they instead used silver paste to pile different WS2 crystals into a stacked structure. The additional thermal resistance introduced by the silver paste is responsible for their underestimated through-plane thermal conductivity.

TMD Alloys. Using the similar varied spot size method, we also measure the anisotropic thermal conductivity TMD alloys WSe2(1-x)Te2x. As shown in Figure IV.11, for each alloys, we first use a large spot radius for measuring cross-plane transport properties so that the signal has negligible sensitivity to in-plane thermal conductivity. By scanning over the frequency from 0.3

MHz to 10 MHz, kz of the TMD samples is also observed to be dependent on modulation frequency, similar to pristine TMDs. The cross-plane thermal conductivity at equilibrium limit is similarly extracted using two-channel diffusive model.[257]

Figure IV.12 summarizes the composition dependent thermal conductivity of WSe2xTe2(1-x) at room temperature. As a validation, we compared the thermal conductivity of the pure WSe2 and

WTe2 with the results available in the literature. For WSe2, our measurements for both kr = 40

W/mK and kz = 2.45 W/mK are consistent with our previous work[265] and agrees well with first principles calculation by Lindroth et al. [103] However, the measured kz is significantly higher than the TDTR measurements by Chiritescu et al. [223] and Murato et al., [34] probably due to the neglection of non-equilibrium transport in their work. For WTe2, our measurement for the in-plane thermal conductivity (kr = 13.5 W/mK) agrees well with the measurement by Zhou et al. (kr =

15 W/mK).[266] Interestingly, the kr of WTe2 measured by TDTR is even higher than the first principles calculation of phononic thermal conductivity. There are two possible reasons for such discrepancy. First of all, the electrons might have non-negligible contribution to the total thermal conductivity measured by the TDTR since WTe2 is a semi-metal,[267] while it was completely neglected by the first principle calculations of lattice thermal conductivity. The other possible reason is likely due to the computational error. WTe2 has a rather complicated atomic structure.

However, in the calculation by Liu et al.,[268] the authors used a relative small supercell (2×2×1 unit cells) for calculating the force constants to mitigate the large computation cost, which might lead to an underestimated thermal conductivity.[103] In the cross-plane direction, our measurement is kz = 1.34 W/mK agrees well with both the first principles calculations[268] and measurements by others. [266, 269, 270] For both the in-plane and the cross-plane direction, the thermal conductivity is reduced as the composition fraction x in WSe2(1-x)Te2x approaches 0.5. We also observe clearly a sharp change as x increases from 0.4 to 0.6. This is due to the phase change from the 2H to Td phase as x increases from 0.4 to 0.6.

Figure IV.12. (a) In-plane and (b) cross-plane thermal conductivity of WSe2(1-x)Te2x measured by

TDTR. The thermal conductivity of pristine WSe2 (x=0) and WTe2 (x=1) are compared with first- principles calculations by Lindroth et al. [103] and Liu et al., [268] and measurement by Jiang et al.,[265] Chiritescu et al.,[223] Murato et al. [34], Zhou et al.,[266] Brixner et al.[269] and Jana et al.[270]

In Figure IV.13, we study the temperature dependent kr and kz of the 2H WSe1.2Te0.8 (x=0.4) and Td WSe0.4Te1.6 (x=0.8) compared with the pure 2H WSe2 and Td WTe2, respectively. In the entire temperature range from 80 K to 300 K, both kr and kz of 2H WSe1.2Te0.8 are greatly reduced compared with the pure WSe2 due to the alloy scattering, and similar reduction is also observed in

WSe0.4Te1.6 in the Td phase compared with WTe2. As shown in Figure IV.13a and Figure IV.13c, the kr(T) curve flattens below 150 K for both 2H WSe1.2Te0.8 and Td WSe0.4Te1.6, while kr kept increasing with temperature for the pristine WSe2 and WTe2. Since the alloy scattering due to mass disorder is temperature independent,[271] it becomes more dominant at lower temperature when intrinsic three phonon scattering is much weaker at cryogenic temperature, contributing to the weaker temperature dependence of kr below 150 K. It’s interesting to note that the kz(T) curve of

WSe0.4Te1.6 is much flatter than the kz(T) curve WTe2, as shown Figure IV.13d. Since WSe0.4Te1.6

84 has randomly distributed Se defects, the distribution pattern of defects is expected to be very different in each monolayer, which breaks the periodicity in the cross-plane direction. As a result of such atomic disorder, the vibration modes are greatly localized in the cross-plane direction, resulting in the lower kz value and a much flatter kz(T) curve than the prinstine TMD crystal. For the same reason, the kz(T) curve of WSe1.2Te0.8 showed a similar shape compared to WSe0.4Te1.6.

However, the temperature dependence of kz of WSe2 is very different from WTe2. Instead of increasing with decreasing temperature, the kz(T) curve of WSe2 showed a peak near 150 K.

Suggested by the Lindroth et al.[103] and Jiang et al.,[265] there are staking faults that induce boundary scattering on the length scale of 150 nm, resulting in such a peak in the kz(T) curve and much lower kz compared with first principles calculations. However, such peak in the kz(T) curve is absent in the WTe2. Such different temperature dependent behavior of kz can be explained by the different distribution of phonon mean free paths (MFPs) in WTe2 and WSe2. As suggested by the first principles calculations, half of kz of WTe2 is contributed by phonons with MFPs between

200 nm to 1 μm, but 50% of kz of WSe2 is contributed by phonons with MFPs above 1 μm. Due

85 to the much longer MFPs in WSe2 than WTe2, the stacking defects are expected to have a much more pronounced effect on kz in WSe2 than WTe2.

IV.4 Summary of this Chapter

In summary, we developed the varied spot size TDTR technique for probing anisotropic thermal conductivity. We validated this technique by measuring a series of anisotropic crystals including HOPG, h-BN, 4H and 6H SiC, ZnO and TiO2. We then applied this technique to study the anisotropic thermal conductivity of pristine TMD crystals (MoS2, MoSe2, WS2 and WSe2) and

TMD alloys WSe2(1-x)Te2x. Using the TDTR measurement, we observed that the cross-plane thermal conductivity depends on modulation frequency of the pump beam, due to the non- equilibrium transport between different phonon modes in the cross-plane direction. A two-channel model is used to extract the cross-plane thermal conductivity at the near thermal equilibrium limit.

The in-plane thermal conductivity can then be determined using a tightly focused laser spot. The measured in-plane thermal conductivities compare very well with the recent first-principles calculations[103] for all the four pure TMD compounds, while our through-plane thermal conductivities are consistently lower than the calculations. The lower through-plane thermal conductivity than calculations along with the temperature dependence suggest that there could be boundary scattering of phonons in the through-plane direction of TMDs. For TMD alloys, both the in-plane and cross-plane thermal conductivity is reduced at higher alloy mixing level as the composition fraction x in WSe2(1-x)Te2x approaches 0.5. We also found that the temperature dependence behavior of thermal conductivity for the TMD alloys becomes weaker compared with the pristine 2H WSe2 and Td WTe2 due to the atomic disorder. This work serves as an important starting point for exploring phonon transport physics in layered two-dimensional alloys.

CHAPTER V CONCLUSIONS AND FUTURE WORK

In this thesis, we developed first-principles-based atomistic simulation tools as well as ultrafast pump-probe techniques to study the thermal conductivity and phononic properties of complex crystals and two dimensional layered materials. In the first part, empirical potential fields are developed to fit first-principles potential energy surface near equilibrium positions of atoms, and these potential fields are then used as input for molecular dynamics simulations to predict thermal conductivity of hybrid organic-inorganic crystals. However, the development of these empirical potential still involves artificial assignment of functionals with fitting parameters, which limits its further application and accuracy for materials with complex phase behavior. In the second part of this thesis, we further addressed problems of empirical potential mentioned above by implementing machine learning regression techniques to develop potential fields, which does not involve any artificial assignment of empirical functional forms or fitting parameters. Using

Zirconium as an example, we demonstrated that machine learning potential could be a powerful to model phononic properties, especially for materials dynamically unstable phase at high temperature, which is a challenging problem for empirical potential based MD and first-principles calculations. In the third part of the thesis, the anisotropic thermal conductivity of 2D layered materials is systematically studied. It is observed that the through-plane thermal conductivity depends on modulation frequency, due to the nonequilibrium transport between phonon groups.

The findings in this thesis could guide the discovery of complex crystals and two-dimensional layered materials with desirable thermal transport properties. There are still a few issues to be addressed in the future.

The machine learning driven atomistic simulation of this thesis is still limited to a crystalline

Zr. One future direction is to further implement this technique to study thermal transport in more

88 complex systems. For example, thermal barrier coating (TBC) materials like Yittria-stabilized-

Zirconia (YSZ) and rare-earth niobates contains high concentrations of oxygen vacancy and mass disorders. In these TBC materials, first-principles calculations based on perturbation theory to the phonon picture can no longer apply due to the high concentration of atomic disorder. On the other hand, developing high-fidelity empirical potential field in these disordered materials is also nearly impossible, because the potential energy in disordered systems cannot no longer be easily decomposed to simple functional forms. The machine learning driven atomistic simulation we described in CHAPTER III is a promising way to solve such dilemma and to accurately predict thermal conductivity of disordered TBC materials. Another interesting direction is to further implement machine learning to study thermal transport beyond atomic scale. For example, we can train machine learning models to conduct high-throughput prediction to look for the optimal thermal conductivity of nanocomposites and compound materials for thermal electrics and insulation materials.

Thermal conductivity of two-dimensional layered materials is found to decrease with increasing heating frequency in MHz range, due to the non-equilibrium phonon transport near the interface between the sample and the Al transducer. This finding suggest that electronic materials could become much more thermally insulating under the working condition of electronic devices, because the transistors work at a much higher frequency (GHz) than the modulation frequency

(MHz) in pump-probe experiments. To study thermal transport physics at such highly transient conditions, development of multiscale modeling combining first-principles phonon calculations and device-level modeling, as well as high-resolution experimental techniques mapping the device thermal profile remains to be done in the future.

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APPENDIX

Appendix I Equilibrium Molecular Dynamics for Thermal Conductivity Calculation

We provide the details of EMD simulation in this part using the orthorhombic crystal β-ZnTe(en)0.5 as an example. EMD simulations are performed to calculate the thermal conductivity of β-

ZnTe(en)0.5 using LAMMPS simulation package.[272] Then trajectories of all the atoms are calculated by integrating the Newton’s equation of motion by Verlet algorithm. The time step to integrate Newton’s equation of motion is 1 fs. The thermal conductivity kα in the α direction (x, y or z) is written using the Green-Kubo (GK) relation [139]:

V ∞ kα = 2 ∫ 〈Jα(0)Jα(t)〉dt (A1) kBT 0 where the bracket 〈⋅〉 denotes the ensemble and time average, Jα is the component of heat flux 퐉 in the α direction, and 〈Jα(0)Jα(t)〉 is the heat current auto-correlation function (HCACF). The heat flux 퐉 is defined as:

1 d 퐉 = ∑ 퐫 E V dt i i (A2) i where V is the volume of the system, 퐫i and Ei are the position and total energy of atom i, respectively.

Anisotropic thermal conductivity of β-ZnTe(en)0.5 is calculated over a temperature range from

250 to 500 K. At the beginning of MD simulation, the atom position and lattice constants are adjusted until the potential energy of the system is minimized. This structure optimization is performed to make sure that the atoms are at the equilibrium position in MD simulation because the potential field does not perfectly match the energy surface from DFT calculation due to the

111 limitation of the functional forms assigned. The system is then thermalized in NVT ensemble

(canonical ensemble) to stabilize the temperature. During this process, the temperature fluctuation gradually decayed and minimized at a difference of 3 K, which takes 200 ps. The system is then switched to NVE ensemble (microcanonical ensemble) to run for 4 ns to sample the HCACF as a function of time. The HCACF is sampled every 2 fs until it reaches the maximum correlation time

20 ps. This sampling process is repeated until the 4 ns simulation time is reached and HCACF is averaged over all sampled results. The HCACF vanishes and thermal conductivity tends to converge when correlation time is longer than 10 ps (see Figure A I.1).

In order to establish the diffusive phonon transport process to avoid the artificial size effect due to the limited size of the computational domain, the size of the simulation cells should be large enough so that a converged thermal conductivity value corresponding to the bulk limit could be obtained.[142] Figure A I.2 shows the effect of simulation domain size on the calculated thermal conductivity of β-ZnTe(en)0.5 at 250 K. We calculated the thermal conductivity over two sets of simulation cells with different shapes to make sure that the shape of the simulation cell has no significant effect on the convergence of thermal conductivities. We first calculated the size effect of thermal conductivity with simulation cells containing n × 3n × 4n unit cells in the x, y and z direction respectively. The ratio between number of unit cells in the x, y and z direction is 1:3:4 to keep the length of simulation cell in each direction approximately equal. Five runs with different initial atom velocities are performed for each system size to obtain a better ensemble average of thermal conductivity. The thermal conductivity values show convergence when n ≥ 4. We have also tested the convergence of thermal conductivity on simulation cells containing the equal number of unit cells in each direction, the convergence of thermal conductivity is reached when there are 15×15×15 unit cells in each direction. The converged thermal conductivities at 250K are

112 of 3.0%, -3.6% and -2.8% difference in the x, y and z direction respectively, compared with the thermal conductivity data with 4×12×16 unit cells. Thus we consider the shape of the simulation cell does not have a significant effect on thermal conductivity. However a simulation cell with n ×

3n × 4n unit cells can significantly shorten the computational time. Since the thermal conductivity are expected to be lower at higher temperature with a smaller phonon mean free paths, a simulation domain size of 5×15×20 unit cells should be large enough for all the MD calculations for temperature higher than 250 K. Once the converged cell size is determined, it is used to perform

EMD simulations at different temperatures.

Figure A I.1 (a) Decay of normalized HCACF of β-ZnTe(en)0.5 and (b) thermal conductivity obtained by integrating HCACF at 300K.

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Figure A I.2. Size Effect of thermal conductivity at 250K. Two sets of simulation cell sizes are plotted in the figure to test the effect of simulation cell shape on the convergence of thermal conductivity. The solid line denotes the simulation cells containing n × 3n × 4n unit cells, and the dash line represents the simulation cells containing 5n × 5n × 5n unit cells, in the x, y and z direction respectively.

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Appendix II Detailed Data Reduction and Error Analysis of TDTR Measurement

AII.1 Two-channel Diffusive Model

We implement the two-channel heat transfer model[257] to consider the non-equilibrium thermal transport between the low-frequency and the high-frequency phonons. The governing equations in cylindrical coordinates (r, z) are:

∂T k ∂ ∂T ∂2T C 1 = r1 (r 1) + k 1 + g(T − T ) 1 ∂t r ∂r ∂r z1 ∂z2 2 1 (A3) ∂T k ∂ ∂T ∂2T C 2 = r2 (r 2) + k 2 + g(T − T ) 2 ∂t r ∂r ∂r z2 ∂z2 1 2 where the subscript 1 and 2 denotes the two channels, C, kr and kz are volumetric heat capacity, in-plane thermal conductivity and cross-plane thermal conductivity, and g is the coupling constant between the two heat transfer channels.

By applying the Fourier and Hankel transforms to the time variable t and the radial coordinate r accordingly, the two-channel heat conduction equations can be simplified to:

g 2 α1 − d θ1 kz1 θ1 2 [ ] = [ ] [ ] (A4) dz θ2 g θ2 − α2 kz2

2 iωCj+krjx +g where θ is the temperature in the Hankel-Fourier space, and αj = (j = 1,2), and x is kzj the Hankel transform variable. This linear equation can be solved by a eigenvalue problem θi =

λz 2 v1 u1 Be . We note λi and X = [ ] as the eigenvalues and eigenvector matrix for the matrix, v2 u2 respectively:

115

g α1 − kz1 [ g ] − α2 kz2

Therefore, in a single layer, the temperature and the heat flux q at the top and bottom of the surface can be written as:

θ1 θ1 θ θ [ 2] = [R] [ 2] (A5) q1 q1 q2 z=d q2 z=0

With the transfer matrix [R]

[R] = [N][M] (A6)

And

v e− u e− v e+ u e+ 1 1 1 2 1 1 1 2 v e− u e− v e+ u e+ [N] = 2 1 2 2 2 1 2 2 v γ e− u γ e− + + (A7) 1 11 1 1 12 2 −v1γ11e1 −u1γ12e2 − − + + [v2γ21e1 u2γ22e2 −v2γ21e1 −u2γ22e2 ]

u2 −u1 u2/γ11 −u1/γ21 1 −v2 v1 −v /γ v /γ [M] = [ 2 12 1 22 ] (A8) v1u2 − v2u1 u2 −u1 −u2/γ11 u1/γ21 −v2 v1 v2/γ12 −v1/γ22

± where ej = exp(±λjd) and d is the thickness of the layer, γmn = kzmλn.

For interface d → 0 and C1 = C2 → 0, the transfer matrix becomes:

(A9)

116 where Gmn is the interface conductance between channel m and channel n, and the coefficients are:

CG22 −DG21 G22 −G21 1 −CG DG −G G [R] = × [ 12 11 12 11] (A10) G11G22 − G12G21 β −β −EG22 −EG21 −β β −FG12 FG11 And:

C = G11 + G21

D = G12 + G22

E = G11 + G12 (A11)

F = G22 + G21

β = EG22G21 + FG11G12 For a sample containing N layers (including the interfaces as a layer), the temperature at the surface and the bottom can be calculated by multiplying all the transfer matrices:

θ1 N−1 θ1 θ1 θ2 θ2 θ2 [ ] = ∏[R]N−n [ ] = [Z] [ ] (A12) q1 q1 q1 n=0 q2 b q2 t q2 t

With adiabatic boundary condition q1b = q2b = 0, and the assumption that the thermoreflectance is only contributed by the lattice temperature (T2) in the transducer layer, the Green’s function of the two-channel heat conduction equation can be solved as:

Z32Z43 − Z42Z33 풢(ω, x) = − (A13) Z32Z41 − Z42Z31 The thermoreflectance signal in the frequency domain is therefore calculated by integrating over the Gaussian profile of the laser beam:

117

∞ 2 2 w0 + w1 H(ω) = A ∫ 풢(ω, x) exp (− x2) xdx (A14) 0 8 where w0 and w1 is the radius of pump and probe spot, and the coefficient A is the amplitude of absorbed heat and could be neglected when the ratio between in-phase and out-of-phase signal is processed as the measured signal.

To fit the obtained experiment signal −Vin/Vout, we use the nonlinear regression method[261] to minimize the cost function, which is defined as:

2 ( ) W 퐔 = ∑ ∑[RExp(τi, f0j) − F(τi, f0j, 퐔, 퐏)] (A15) i j where RExp(τi, f0j) is the ratio −Vin/Vout measured experimentally at the delay time τi and the modulation frequency f0j . F is the solution of the two-channel heat conduction model which

T predicts the signal −Vin/Vout . The vector 퐔 = [kz1, kz2, G21, G22, g] is the set of unknown parameters that need to be determined. The vector 퐏 is the vector of control parameters including laser spot radius, thickness, heat capacity and thermal conductivity of the transducer, and heat capacity of the sample. The simplex algorithm[261] is used to seek the minimum of the cost function by varying the values of 퐔 iteratively, until the change in 퐔 and W is both smaller than

0.1%.

At the local equilibrium limit, the cross-plane thermal conductivity is simply calculated as the summation over the contribution from each channel:

eq kz = kz1 + kz2 (A16)

After the cross-plane transport properties [kz1, kz2, G21, G22, g] are determined, the unknown

T parameter is set to be 퐔 = [kr1, kr2] and the cross-plane transport properties are grouped into the

118 vector of control variables 퐏 which are fixed during the nonlinear regression. kr1, kr2 are then extracted by minimizing Eq. (A15). The in-plane thermal conductivity at the near equilibrium limit is similarly calculated by the summation of the contributions from both channels:

eq kr = kr1 + kr2 (A17)

AII.2 Uncertainty Analysis

Since we are extracting multiple parameters from the nonlinear regression, the uncertainties of the multiple fitting parameters are estimated using the method based on Jacobi matrices developed by

Yang et al.[273] At the best fit, the gradient of the cost function W in the unknown variable space

퐔 should be zero:

M N ퟎ ∂f 0 ∑ ∑ 2 (REXP(τi, ω0j) − f(퐔 , 퐏, τi, ω0j)) ⋅ ( ) = 0, ∀ul ∈ 퐔 (A18) ∂ul ퟎ j=1 i=1 퐔 where ul is the l-th component of 퐔. We can do a Taylor expansion at the first order at a neighboring point (퐔∗, 퐏∗):

∗ ∗ f(퐔 , 퐏 , τi, ω0j) p ∂f(퐔ퟎ, 퐏∗, τ , ω ) ퟎ ∗ i 0j ∗ 0 = f(퐔 , 퐏 , τi, ω0j) + ∑ (un − un) ∂un n=1 (A19) q ∂f(퐔ퟎ, 퐏∗, τ , ω ) i 0j ∗ + ∑ (pm − pm) ∂pm m=1 where pm is the m-th component of the vector 퐏. Substituting the above equation into Eq. (A18), we can obtain:

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M N p ∂f 퐔ퟎ, 퐏∗, τ , ω ∗ ∗ ( i 0j) ∗ 0 ∑ ∑ (REXP(τi, ω0j) − f(퐔 , 퐏 , τi, ω0j) − ∑ (un − un) ∂un j=1 i=1 n=1 q (A20) ∂f 퐔ퟎ, 퐏∗, τ , ω ∂f τ , ω ( i 0j) ∗ 0 ( i 0j) − ∑ (pm − pm) ) ⋅ ( ) = 0 ∂pm ∂ul 0 m=1 ul

We can simplify the above equation using a matrix format:

M M 퐓 퐓 ∗ ퟎ [∑ 퐉퐔(j)(퐑EXP(ω0j) − 퐅(ω0j))] − [∑ 퐉퐔(ω0j)퐉퐟,퐔(ω0j)] (퐔 − 퐔 ) j=1 j=1

M (A21) 퐓 ∗ − [∑ 퐉퐔(ω0j)퐉퐏(ω0j)] (퐏 − 퐏) = 0 j=1

T where 퐑EXP(ω0j) = (REXP(τ1, ω0j), REXP(τ2, ω0j), … , REXP(τN, ω0j)) is a column vector of the signal measured at ω0j, and 퐅 is the column vector of signal by model calculation, the matrices

퐉퐔 and 퐉퐏 are Jacobi matrices of the thermal model:

∂f(τ ) ∂f(τ ) ∂f(τ1) 1 1 ⋯ ∂u1 ∂u2 ∂up ∂f(τ2) ∂f(τ2) ∂f(τ2) ⋯ 퐉퐔(ω0j) = ∂u1 ∂u2 ∂up (A22)

⋮ ⋮ ⋱ ⋮ ∂f(τ ) ∂f(τ ) ∂f(τ ) N N … N ( ∂u1 ∂u2 ∂up ) ퟎ 퐔 ,퐏,ω0j

And

∂f(τ ) ∂f(τ ) ∂f(τ1) 1 1 ⋯ ∂pt1 ∂pt2 ∂ptq ∂f(τ2) ∂f(τ2) ∂f(τ2) ⋯ 퐉퐏(ω0j) = ∂pt1 ∂pt2 ∂ptq (A23)

⋮ ⋮ ⋱ ⋮ ∂f(τ ) ∂f(τ ) ∂f(τ ) N N … N ( ∂pt1 ∂pt2 ∂ptq ) ퟎ 퐔 ,퐏,ω0j

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After some arrangements, Eq. (A21) can be written as:

M 퐓 ∗ ퟎ ∗ ∑ 퐉퐔(ω0j)(퐑퐄퐗퐏(ω0j) − 퐅) − 횺퐔(퐔 − 퐔 ) − 횺퐔퐏(퐏 − 퐏) = 0 (A24) j=1 where

M 퐓 횺퐔 = ∑ 퐉퐔(ω0j)퐉퐔(ω0j) j=1 (A25) M 퐓 횺퐔퐏 = ∑ 퐉퐔(ω0j)퐉퐏(ω0j) j=1

We can then express 퐔∗ − 퐔ퟎ as a linear function of 퐏∗ − 퐏 as:

M ∗ ퟎ −ퟏ 퐓 −ퟏ ∗ 퐔 − 퐔 = 횺퐔 ∑ 퐉퐔(ω0j)(퐑퐄퐗퐏(ω0j) − 퐅) + 횺퐔 횺퐔퐏(퐏 − 퐏) (A26) j=1

We then take covariance on both sides:

M −ퟏ 퐓 −ퟏ −ퟏ 퐓 −ퟏ var[퐔] = 횺퐔 [∑ 퐉퐔(ω0j)var[퐑퐄퐗퐏(ω0j)]퐉퐔(ω0j)] 횺퐔 + 횺퐔 횺퐔퐏var[퐏]횺퐔퐏횺퐔 (A27) j=1

The above equation is the error propagation formula, which is a summation of two terms. The first term is the uncertainty from the experimental noises, which contributes only < 2% of the uncertainty for our measurements of TMDs. The major error comes from the second term, which is the propagation of the error from the control variables. The term

−1 T −1 횺퐔 [∑j 퐉퐔(f0j)var[퐑Exp(f0j)]퐉퐔(f0j)]횺퐔 in describes the uncertainty contributed by the noise of the signal, where the var[퐑Exp(f0j)] represents the noise of the signal at the frequency f0j, which is obtained by calculating the variance of the signal −Vin/Vout among five individual measurements at each sampled delay time τ0j. The experimental noise only contributes to less than

121

5% of the uncertainty, and the major error comes from the uncertainties of the control variables.

T If 퐔 is a vector of N elements U = [u1, u2, … , uN] , the covariance matrix var[퐔] is a N × N matrix with the following form:

σ2 cov[u , u ] ⋯ cov[u , u ] u1 1 2 1 N 2 cov[u , u ] σ ⋯ cov[u2, uN] var[퐔] = 2 1 u2 (A28) ⋮ ⋮ ⋱ ⋮ 2 [cov[uN, u1] cov[uN, u2] ⋯ σuN ] where cov[ui, uj] is the covariance between ui and uj and it is identical to cov[uj, ui] so that the var[퐔] is a symmetrical matrix. The covariance cov[ui, uj] denotes the correlation between the two variables ui and uj. If cov[ui, uj] is zero, then ui and uj are independent. When determining the confidence interval of the multiple parameters, it is also necessary to consider the covariance.

The confidence interval for multiple parameters are determined by a quadratic surface in the parameters space:

0 T −1 0 2 (퐔 − 퐔 ) (var[퐔]) (퐔 − 퐔 ) = χN(P = 0.95) (A29)

0 2 where 퐔 denotes the best-fit parameters. χN(P) is the N-th order quantile function,[274] and P =

eq eq 0.95 is the probability of the confidence interval. To estimate the error of kz and kr , we first plot the confidence interval projected to the sub-space of the entire parameter space using the following equation respectively:

0 0 −1 0 0 T ([kα1, kr2] − [kα1, kα2])(var[kα1, kα2]) ([kα1, kα2] − [kα1, kα2]) = χ2(0.95) (A30) where α = (r, z) denotes the direction.

The confidence intervals are generally ellipses as shown in Figure A II.1. Based on the shape of

eq eq the confidence interval, the upper and lower limit of kz and kr can be obtained as showed in

eq Figure A II.1. For example, when determining kz , the line tangential to the ellipse with the

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eq eq equation kz1 + kz2 = kz are drawn, and we can determine the uncertainty of kz from the intersect with the two axes. When performing the data analysis, the input uncertainties (2σ) of the input parameters are estimated as follows: 10% for the thermal conductivity of Al, 3% for the heat capacity of Al and the substrate, 4% for the Al thickness, and 3% for the laser spot size.

Figure A II.1. Schematic of the confidence intervals (ellipses with blue solid lines) and the determination of the thermal conductivity at the near equilibrium limit from the confidence interval

(dashed black lines). The red dot shows the best fit value. (a) confidence interval of kr1 and kr2. (b) confidence interval for kz1 and kz2 for WSe1.2Te0.8.

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AII.3 Division of the high-frequency and low frequency channels.

Figure A II.2. Phonon dispersions of (a) 2H WSe2, (b) 2H WTe2 and (c) 2H WSe1.2Te0.8, (d) Td

WSe2, (e) Td WTe2 and (f) Td WSe0.4Te1.6. The horizontal purple lines show the cutoff frequencies that divide phonons into the low frequency channel and high frequency channels.

We divide the phonons in the TMD alloys according to the phonon dispersions obtained from VCA.

For the alloys 2H phase, the heat capacity is divided into two channels according to the bandgap in the phonon dispersion:

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∂ C = ∑ ℏω n (T) 1 ∂T 퐪s 퐪s ω퐪s<ωc (A31) ∂ C = ∑ ℏω n (T) 2 ∂T 퐪s 퐪s ω퐪s>ωc where ωc is the cutoff frequency. Based on the physical picture discussed in Section S2, the cutoff frequency ωc should be set to include all acoustic phonons with large group velocities. Therefore, we set ωc to the maximum frequency of LA/ZO branches at Brillouin zone boundaries, as shown in Figure A II.2. After setting the cutoff frequency, the heat capacities can be divided into the two channels, as shown in Figure A II.3a-b.

Figure A II.3. (a) Division of the low-frequency and the high frequency channels according to the phonon bandgap in 2H WSe1.2Te0.8. (b) Division of the low-frequency and the high frequency channels for Td WSe0.4Te1.6.

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