Thermal transport of chalcogenides

Thermal transport of chalcogenides

11

Meng An1, Han Meng2, Tengfei Luo3 and Nuo Yang2 1 College of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an, China, 2State Key Laboratory of Coal Combustion and Nano Interface Center for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, China, 3Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States

11.1

Introduction

11.1.1 Basic theory of heat conduction 11.1.1.1 Phonons Phonons are the quantized energy levels of lattice vibrations, in analogy with the photo, which is the quantum of the electromagnetic wave [1]. Heat conduction processes involve phonons of all allowable energies, as determined by the phonon dispersion relation of each material. There are typically three acoustic branches including two transverse acoustic branches and one longitudinal acoustic branch, and 3(N-1) optical branches, where N is the number of atoms at each lattice point.

11.1.1.2 Phonon dispersion Phonon dispersion is the relationship ωp(k) between the lattice vibration frequency ωp and a phonon wave vector k. Phonon dispersion relation can be determined by the inelastic scattering of neutrons with emission or absorption of phonons. The group velocity is calculated by vp ðkÞ 5

@ωðkÞ @k

The group velocity is the velocity of the propagation of the phonon wave packet, or the velocity of the propagation of energy in the medium.

Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00008-7 © 2020 Elsevier Ltd. All rights reserved.

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11.1.1.3 Gru¨neisen parameter The Gru¨neisen parameter describes the overall effect of the volume change of a crystal on vibration properties, which is expressed as γG 5

V ðdP=dTÞV CV

Where V is the volume of a crystal, Cv is the heat capacity at the constant volume, and (dP/dT) is the pressure change due to temperature vibration at the constant volume.

11.1.1.4 Phonon density of states The phonon density of states G(ω) measures the number
of phonon modes of a selected frequency ω(k, p) in a given frequency interval ω 2 12 Δω; ω 1 12Δω . ÐN The phonon density of states is normalized 0 dω GðωÞ 5 1. The phonon density of states spreads from zero to the maximal phonon frequency existing in a given crystal. In simple crystals, due to a large mass difference of the constituents the vibrations could be separated to a low-frequency phonon band, caused mainly by oscillations of heavy atoms, and to a high-frequency band, occupied by light atoms, Thus such bands could be separated by a frequency gap. To describe the vibration of a specific atom μ moving a partial phonon density of states Gu, Ð N along i-direction, 1 (ω) is introduced. It is dω G ðωÞ 5 , where r is the number of degree of freeu;i i r 0 dom in the normalized to primitive unit cell.

11.1.1.5 Fourier Law The capability of a material to conduct heat energy can be described by thermal conductivity through a macroscopic expression [24], i.e. Fourier’s law, j 5 2 κrT Where j is the local heat flux and represents the amount of heat energy that flows through a unit area per unit time, and rT is the temperature gradient along the heat flux direction. It is widely accepted in non-metals, in which heat is mainly carried by phonons.

11.1.1.6 Phonon Boltzmann transport equation (BTE) In nonmetal crystalline materials, the superposition of phonon waves leads to wave pockets carrying energy at the group velocity. These wave pockets can be treated as quasi-particles, and phonon transport in a crystal is similar to that of gas molecules inside a container. Thus, the phonon gas model is generally used to describe the lattice thermal transport: phonons is treated as a quasi-particles careering a certain

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amount of energy and transporting in a certain direction with group velocity v. Under this framework, the transport of phonons can be described by the phonon Boltzamnn transport equation (BTE): [1,5,6]   @fλ dfλ 1 vg;λUrfλ 5 (11.1) @t dt scattering where f is the probability distribution function for a phonon mode labeled by λ, vg,λ is the phonon group velocity, and the right-hand side of Eq. (11.1) corresponds to the changes affected in the distribution function due to collisions which act to restore equilibrium. This scattering term that incorporates different scattering mechanisms. Under small perturbation, the nonequilibrium distribution function can be written as fλ 5 fλ0 1 f 0λ , where fλ0 is the equilibrium BoseEinstein distribution function and f 0λ is a temperature-independent small perturbation. If we further assume that the temperature gradient rT is small, the λterm rfλ can be linearized and written as rTUð@fλ0 =@TÞ. At steady state Eq. (11.1) becomes vg;λU rT

  fλ0 dfλ 5 @T dt scattering

(11.2)

Single-mode relaxation time approximation (SMRTA) is commonly used to solve this equation, in which a relaxation time τλ is assigned to each phonon mode, and the scattering term can be written as 

dfλ dt



 52 scattering

fλ 2 fλ0 τλ

 (11.3)

Different phonon scattering mechanisms, including phonon-phonon (p-p) scattering, phonon-impurity (p-i) scattering, and phonon-boundary (p-b) scattering etc., can be incorporated into the relaxation time through Matthiessen’s rule: 1 1 1 1 5 1 1 τλ τp2p τp2i τp2b

(11.4)

When applying the expression of heat current and Fourier’s law, the thermal conductivity can be evaluated as X καβ 5 c v v τ (11.5) λ λ λα λβ λ where cλ is the heat capacity (per unit volume) of mode λ and vλα is the group velocity of the phonon mode λ along the α direction.

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Chalcogenide

11.1.2 The structure characteristics of chalcogenides 11.1.2.1 The chemical composition of chalcogenides In this chapter, we review the thermal transport properties of chalcogenides. The thermal conductivity is strong related with the chemical composition of materials. For chalcogenides, Chalcogen, or the oxygen family, consist of the elements S, Se, and Te. The name is chalcos (ore)  gen (formation) [7]. Chalcogenides are a kind of compounds as shown in Fig. 11.1 including one or more chalcogen elements e.g. sulfur (S), selenium (Se) and tellurium (Te) (MaXb, M is an element of Group IV, Group III, Group VI or transition metal, and X: S, Se, Te). These chalcogenides contain an uncommonly wide range of crystal structures, which exhibit promising applications areas related to thermal properties due to an indirect- to direct-bandgap transfromation with the a reduction in the materials thickness from bulk to monolayer, such as thermoelectric materials [814], semiconducting materials [1517] and batteries [18,19]. Thus, the understanding of thermal properties of chalcogenides are of vital importance for solving the challenge of thermal management. More importantly, the emergency of graphene-like two-dimensional (2D) (such as MoS2, a transition metal layer sandwiched by two chalcogenide atomic layers [20,21]) and lower-dimensional chalcogenide materials provide perfect candidate materials to investigate the fundamental questions [22,23] of thermal conduction shown in the following: 1. Are the thermal conductivity independent of the characteristic size (dimension effect or size effect)? 2. In practical applications, these semiconductor devices based on chalcogenides are placed on substrate. How do the substrates affect thermal conductivity of chalcogenides? 3. In addition, imperfections such as defects, vacancies, and grain boundaries inevitably occur during sample preparations. How do these imperfections and mechanical strains modulate the lattice thermal conductivity? 4. The 2D chalcogenides can be functionalized and intercalated. How can the thermal conductivity of chalcogenides be manipulated by functionalization and intercalation?. Slack formula [24,25] suggested that the thermal conductivity usually is determined by four factors, including (1) average atomic mass, (2) interatomic bonding, (3) crystal structure and (4) size of anharmonicity. The first three factors determines from harmonic properties. This expression can estimate thermal conductivity κs 5 A

M n1=3 δθ3 Tγ 2

Figure 11.1 The chemical composition of chalcogenides.

(11.6)

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where A is the numerical coefficient that equals 3.13106, M is the average mass of the basis atom, n is the number of atoms in the primitive unit cell, T is temperature and γ is the Gru¨neisen parameter. The average volume per atom is denoted by δ3 and θ is the Debye temperature, defined as ¯hωD/kB, in which ωD denotes the maximum vibrational frequency of a given model in a crystal. For thermal conductivity in W/m-K, mass in amu, and δ in Angstroms. Based on the Slack derivation, the low Debye temperature and heavy mean atomic mass and complex crystal structure can result in low lattice thermal conductivity [26]. we take transition metal dichalcogenides (TMDs) as an example to analyze their thermal conductivity. Taking the general chemical formula of TMDs to be XY2, thermal conductivity is roughly proportional to the thermal average ,ω2 . of phonon frequency. Since in a diatomic linear chain the maximum frequency ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωD ~ M121 1 M221 (where M1 and M2 are the mass of the constituent species), increasing either mass should, if the stiffness of the new compound is equal to or smaller than that of the old compound, decrease the ωD and hence thermal conductivity.

11.2

Geometrical effect

Intensive experimental and theoretical efforts have also been directed toward the effect of domain size on thermal properties.

11.2.1 Dimensional effect Transitional metal chalcogenides are an important family of chalcogenides materials that have received significant interests in recent years as they have a promising potential for diverse applications ranging from use in electronics, sustainable photovoltaic to industrial lubricants. Due to their inherent 2D layered structure (taking MoS2 as an example shown in Fig. 11.2), these materials behave significant anisotropy of in-plane (i.e., basal plane) vs. out-of-plane (i.e. cross-plane) physical properties due to differences in the nature of the atomic interactions. Within a molecular layer (in-plane), these interactions are characterized by covalent bonding, while the individual layers (cross-plane) are held together by significantly weaker van der Waals forces. Experimentally, the measured in-plane thermal conductivity values in bulk natural MoS2 crystal are about 100 W/m-K [2729], while the cross-plane thermal conductivity value are more than one order of magnitude smaller, ranging from 2.0 6 0.3 W/m-K to 4.75 6 0.32 W/m-K. The Raman optothermal measurements indicated that the room temperature thermal conductivity of tantalum diselenide (2H-TaSe2) obtained via mechanical exfoliation of crystals grown by chemical vapor transport, is 16 W/m-K [30]. However, its thermal conductivity of 45-nm thick films is 9 W/m-K. The thermal conductivity of the exfoliated thin films of 2H-TaSe2 is dominated by phonon contributions and reduced substantially compared with the bulk value.

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Chalcogenide

Figure 11.2 Schematic illustration of the different dimensional MoS2 systems with the top view and side view: bulk MoS2, single-layer MoS2 sheet, and (8, 8) armchair MoS2 nanotube.

Lead chalcogenides, the widely studied thermoelectric (TE) materials, are one kind of the best TE semiconductors for power generation at intermediate temperatures [31]. Therefore, their lattice thermal conductivities have attracted numerous attention due to that is the only materials properties that can be manipulated independently to enhance thermoelectric efficiency. For PbX (PbX, X 5 S, Se, Te), their thermal conductivity are 1.66, 1.01 and 1.91 W/m-K for PbS, PbSe, and PbTe at room temperature, respectively [32]. Another first-principles calculations [33] successfully reproduced the phonon dispersion relations of PbSe and PbTe along the high symmetry lines of the experiments [34] shown in Fig. 11.3A. The calculated lattice thermal conductivity matches well with the experimental results above 400 K [35, 36] (Fig. 11. 3B). By comparison of the thermal conductivity PbSe and PbTe, the author also revealed that the optical phonons not only contribute to thermal transport around 20%, but they provide strong scattering channels for acoustic phonons. This physical insight can advance the development of reducing thermal conductivity. Recently, germanium chalcogenides (GeTe, GeSe, and GeS) from IV-VI family have been increasingly attracting researchers’ attention as promising candidates for the replacement lead-based thermoelectric materials due to the toxic nature of leadbased chalcogenides [37]. In order to gain a deep insight into the thermal conductivity and other related properties of GeTe, it is essential to understand more about the chemical bonding, crystal structure. GeTe possesses a thermal conductivity of 2.6 W/m-K at room temperature. Due to the requirement of low thermal conductivity in thermoelectric fields, large number strategies have been used to reduce thermal conductivity, such as twinning, presence of secondary phase and doping [38]. Jana et al. reported an ultralow lattice thermal conductivity 0.4 W/m-K in highquality crystalline ingots of InTe, in the temperature range of 300650 K. This is due to the presence of strongly anharmonic phonons originate from rattling

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Figure 11.3 (A-B) Phonon dispersion for PbSe and PbTe: red line-calculated results and black dots-experimental results [34]. (C-D) Temperature-dependent lattice thermal conductivity of PbSe and PbTe, red line: calculated results; black crosses: experimental data [35,36]. Reproduced with permisssion [33]. Copyright 2012, American Physical Society.

vibrations of In1 cations (along the z-axis) within the columnar ionic substructure, which couple with the heat-carrying acoustic modes and lead to an ultralow thermal conductivity [39]. When the dimension of chalcogenides decreases from bulk to single layer sheet (shown in Fig. 11.2), the thermal conductivity exhibits many interesting properties due to the increased phonon scattering originating from the large surface area-tovolume ratio. The thermal conductivity of bulk, bilayer, and single layer MoS2 are 75.37 6 2.16, 84.26 6 3.24, and 132.68 6 4.67 W/m-K [57]. The DFT-calculated lattice thermal conductivities of MoS2 sheet are about 54 W/m-K along zigzag and 57 W/m-K along armchair directions at 300 K, which suggests insignificant difference between the two directions. It is also found that the calculated thermal conductivity also slighted decreased with decreasing width [40]. More interestingly, folding effect significantly reduce thermal conductivity of over 60% relative to the flat single layer due to the increased contribution of anharmonic phonon scattering introduced by the folding. Meanwhile, the broken symmetry results in highly distorted bonds, driving the low energy-high group velocity modes to higher energies and reducing viable transverse modes along the folding axis [41].

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Chalcogenide

In addition to 2D sheet of chalcogenides, its low-dimensional structures i.e. nanoribbons and nanotube, have shown wide application potential in nanoelectronics, optoelectronic device and thermoelectric applications. In such applications, thermal conductivity directly affects the life span and performance of chalcogenide-based devices. The thermal conductivity of MoS2 nanoribbons can be modulated by several geometries parameters, such as chirality, width, length and the type of edge [42]. Though the first-principle calculations, the thermal conductivity of MoS2 nanoribbons shows a non-monotonic dependence on crystal chirality   within the chirality’s from 0 to 30 . The thermal conductivity of MoS2 nanoribbons reaches a local maximum thermal conductivity at 19.1 . Moreover, the thermal conductivity can be decreased by increasing the edge roughness due to the largely degraded longitudinal phonons [43]. For MoS2 nanoribbons, the width can modulate electronic transport. In contrast, thermal conductivity is not senstive with ribbon width. The experimental measurement and numerical simulation have revealed that the thermal conductivity of MoS2 nanoribbons decreases with weak dependence on the ribbon width and the type of edge [40, 43, 44] which are explained by the local heat flux analysis and phonon scattering mechanisms [44]. The similar trend has been demonstrated in the experimental measurement of confocal micro-Raman spectroscopy. The 2 μm and 1 μm ribbons exhibit thermal conductivity of 31.2 6 2.5 and 29.6 6 1.1 W/m-K, respectively. Both values were further decreased to 27.7 6 1.9 and 25.8 6 3.7 W/m-K for system length 500 nm and 250 nm [40]. The electron beam self-heating technique is applied to measure the thermal conductivity of MoS2 nanoribbon and its value is around 30 W/m-K [58]. The thermal conductivity of single-wall and multi-wall MoS2 nanotubes has been studied by mean of molecular simulations and experimental measurement [4547]. The thermal conductivity, 16 W/m-K for L 5 10 nm at room temperature of single-wall MoS2 nanotube is two orders of magnitude smaller than that of carbon nanotubes. Moreover, the chirality, temperature, length, diameter and strain dependences of thermal conductivity are also discussed. Interestingly, it is found that thermal conductivity of armchair nanotube slightly decreases within a small range of strain. In contrast, thermal conductivity of zigzag nanotube exhibits a significant decreasing trend with strain. Such chirality-dependent strain effect is identified and originates from the different sensitivity of phonon group velocity with strain [46]. In addition, thermal conductivity of the multi-walled MoS2 nanotubes at room temperature was estimated to be in the range of 4.8 6 0.1 to 11.2 6 0.2 W/m-K based on the measurement of the temperature-dependent Raman signals [45].

11.2.2 Length dependence Several theoretical calculations attempted to reveal the intrinsic thermal conductivity of both monolayer and few-layer MoS2 [4953]. In comparison, the experimental measurement is very rare and leaving the intrinsic thermal conductivity of MoS2 almost unclear. The experimental study of thermal transport in few-layer MoS2 prepared by chemical vapor deposition method has been reported by Sahoo et al.

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Figure 11.4 (A) The calculated thermal conductivity of MoS2 at 300 K as a function of sample size [62]. (B) 1/κ(L) as a function of 1/L, where the solid line represents a quadratic fit and the dashed line represents a linear fit to the MD data with L 5 200 to 400 nm [47]. Reproduced with permisssion [47]. Copyright 2014, American Physical Society. Reproduced with permisssion [62]. Copyright 2019, American Physical Society.

The thermal conductivity of an 11-layer sample was measured to be about 52 W/m-K at room temperature [54]. Lately, a more detailed study of temperature and laser-power dependent Raman characterization on monolayer MoS2 exfoliated from naturally occurring bulk materials yielded a thermal conductivity of 34.5 6 4 W/m-K at room temperature [55]. Jo et al [56]. utilized the micro-bridge method to measure the basal-plane thermal conductivity of MoS2 with 4-layer and 6-layer thickness across a wide temperature range. The results showed that thermal conductivity of monolayer and few-layer MoS2 are lower than that of bulk MoS2. In addition to the suspended sample, a relative higher thermal conductivity 62 W/ m-K was observed in supported monolayer MoS2 [57]. As to the thickness dependence for thermal conductivity of MoS2, a theoretical calculation has shown a decreasing trend from monolayer to three layers due to the smaller group velocity for different phonon modes and higher phonon scattering rate induced by the changes of the anharmonic force constant [58]. The measured room temperature thermal conductivity of MoS2 is around 30 W/m-K by means of electron beam selfbeam self-heating technique [59]. In addition to MoS2, other kinds of TMDs are also studied. For example, thermal conductivity of monolayer WS2 has been measured with the optothermal Raman technique, which is comparable to monolayer MoS2 in Yan’s work [54]. The thermal conductivity of a 45 nm-thick TaSe2 sample was measured as 9 W/m-K [30]. Moreover, an ultralow cross-plane thermal conductivity (0.05 W/m-K) was observed in disordered, layered WSe2 sheets by TDTR technique [60]. Thermal conductivity of 5 nm thickness polycrystalline MoS2 utilizing 2-laser Raman thermometry, is 0.73 6 0.25 W/m-K. Yan et al. [54] utilized frequency domain thermoreflectance to study the cross-plane thermal transport in mechanically exfoliated MoS2 samples supported on SiO2 and the substrate. It is observed a significant improvement in heat transport across monolayer MoS2 as compared to few layer

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Chalcogenide

MoS2. Thermal conductivity of monolayer MoS2 is 34.5 6 4 W/m-K at room temperature obtained from temperature-dependent Raman spectroscopy, which agrees well with that of the first-principles lattice dynamics simulations [55]. As for the cross-plane thermal conductivity of MoS2 film, thermal conductivity of MoS2 obtained from NEMD tends to a convergent value when the film thickness is beyond about 40 nm. The analysis of cross-plane phonon MFPs suggests that phonons with MFPs below 40 nm contribute 90% of the MoS2 cross-plane thermal conductivity at room temperature [61]. The thermal conductivity of polycrystalline MoS2 with 5 nm thick is investigated by means of the combined 2-laser Raman thermometry and theoretical method [62].

11.2.3 Single-layer sheet As a family of novel two-dimensional (2D) materials beyond graphene, monolayer TMDs and Si-based chalcogenides exhibits unique physical properties. For TMDs, there are two polymorphs for monolayer TMDs: 1T phase with D3d point group and 2H phase with D3h point group shown in Fig. 11. 5. Monolayer MoX2 (X 5 S, Se, Te) with 2H phase have extensively studied as a representive materials of TMDs. Therefore, in this chapter, we mainly discuss the thermal transport of 2H MoX2, denoted as MoX2. For monlayer Si-based chalcogenides, the structure of puckered SiX (X 5 S, Se, Te) is shown in Fig. 11. 5. The corresponding phonon dispersions are presented in Fig. 11. 5. For MoX2 and SiX, the anion (S, Se and Te), the same trend can be found that the maximum frequencies of acoustic as well as optical branches are shifted downwards due to the inverse relationship of frequency to overall mass, with the change of anion inducing the largest shift due to the heaviest mass of Te. Unlike the ultra-high thermal conductivity in graphene, the low thermal conductivity of TMDs have been reported in both theoretical and experimental studies. At room temperature, the thermal conductivity of single-layer MoS2 is about 108 W/m-K for a 10-μm-long sample obtained using the Boltzmann transport equation (BTE), with the third-order anharmonic force constants obtained from quantum-mechanical density functional theory (DFT) calculations [63]. With a similar method, Gu et al. found that the in-plane thermal conductivity of 10-10-μmlong samples in layered, naturally occurring MoS2 monotonically reduces from 138 to 98 W/m-K when the thickness increases from one to three layers. The BTE approach is widely used in predicting the thermal conductivity of materials, which has its limitation that it is assumed that the higher-order phonon-phonon interactions are unimportant and expensive computational cost. Its thermal conductivity is about 23.3 W/m, obtained by solving the nonequilibrium Green’s function (NEGF) [64], Compared to experimental value of 34.5 W/m [55]. The thermal conductivity of monolayer WSe2 is 3.935 W/m, one order of magnitude lower than that of MoS2. The is due to the ultralow Debye frequency and heavy atom mass in WSe2 [65]. Similar to the substrate effect on thermal conductivity of graphene, the flexural acoustic phonons are damped. Through optothermal Raman technique, the measured thermal conductivity of suspended monolayer MoS2 and MoSe2 are 84 and 59 W/m-K at room temperature, respectively, while

Figure 11.5 Some typical atomic structures phonon dispersion curve of 2D TMDs and Si-based chalcogenides sheet. They are MoS2, MoSe2, MoTe2, SiS, SiSe, SiTe, respectively.

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Chalcogenide

the thermal conductivity of that supported on SiO2 substrate decrease to 55 and 24 W/m-K [57]. Its experimental thermal conductivity of monolayer MoS2 is much lower. Liu et al. investigated thermal conductivity of monolayer MoS2 sheet and nanoribbon. In addition to low thermal conductivity, the length dependence is also quite different from that of carbon-based materials. The thermal conductivity of monolayer MoS2 is calculated to be as high as 116.8 W/m-K [44]. Phonon transport of monolayer WSe2 is found to have an ultralow thermal conductivity due to the ultralow Debye frequency and heavy atom mass. The room temperature thermal conductivity for a typical sample size of 1 m is 3.935 W/m-K, which is one order of magnitude lower than that of MoS2. In addition, it is also found that the ZA phonons have the dominant contribution to thermal conductivity, and the relative contribution is almost 80% at room temperature, which is remarkably higher than that for monolayer MoS2 [65]. As a member of the 2D chalcogenides, MoS2 has a unique sandwich structure and a natural thickness-dependent energy gap. This unique property makes MoS2 a promising candidate material for transistor as an alternative to graphene. They calculate the lattice thermal conductivity of monolayer MoS2 by an iterative solution of the phonon BTE with the help of first-principle force constants. The lattice thermal conductivity is reduced by about 10% with the introduction of isotopes. The diffusion-limited MFP of monolayer MoS2 is longer than 7 μm at 300 K [52]. Peierls-Boltzmann transport equation (PBTE) suggested that the thermal conductivity of single-layer MoS2 could even be higher than 70 W/m-K when the size of sample is larger than 1 μm [50]. Individual phonons thermalize independently under SMRTA, without collisions repopulating them. The simultaneous interaction of all phonon populations, however, can be important, and such collective behavior can lead to the emergence of composite excitation as the leading heat carriers. Such collective behavior is driven by the dominance of normal (that is, heat-flux conserving) phonon scattering events, which allow the phonon gas to conserve to a large extent its momentum before other resistive scattering mechanisms can dissipate away the heat. In thermal transport, the bonding strength, mass of the basis atoms and frequency gap plays an important role in determining their phonon dispersion relations, which in term determines the related group velocity and thermal properties [63]. In a smaller system, the difference between full iterative solution of PBTE and SMRTA methods with the systems smaller than 30 nm is negligible (less than 5%) due to the dominating role of phonon-boundary scattering over phonon-phonon scattering. However, SMRTA cannot distinguish the resistive Umklapp process and the momentum-conserving normal process, which does not directly provide the resistance to heat flow. The under-prediction of SMRTA becomes distinguishable when the scattering due to normal process is strong. The difference between SMRTA and the iterative solutions of PBTE for all single layer TMDs are larger than 10% when the sample size is 1 μm. Among the four monolayer 2H TMDs, WS2 has the highest thermal conductivity of 142 W/m-K at room temperature and then followed by MoS2 (103 W/m-K), MoSe2 (54 W/m-K), and WSe2 (53 W/m-K). The MoS2 sheet

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suspended on QUANTIFOIL exhibits thermal conductivity of 38.3 6 3.8 W/m-K, which is comparable to above reports about multilayer MoS2 sheet [54,56]. The thermal transport of MoS2/graphene heterostructure is studied. It is found that the thermal conductivity can be tuned by interlayer coupling, environment temperature, and interlayer overlap. Interestingly, the highest thermal conductivity at room temperature is achieved as more than 5 times of that of monolayer MoS2 [66]. In addition to the multilayer heterostructure, the in-plane MoS2-graphene heterostructure is widely investigated. The interfaces are connected via strong covalent bonds between Mo and C atoms, were energetically stable. Interestingly, the interfacial thermal conductance was high and comparable to those of covalently bonded graphene-metal interfaces. Each interfacial Mo-C bond served as an independent thermal channel, enabling modulation of the interfacial thermal conductance by controlling the Mo vacancy concentration at the interface [67].

11.2.4 Discussion on the overall trend from single-layer to bulk The thermal conductivity of layered 2D chalcogenide materials was reported either to be substantially lower than that of their bulk counterparts or to increase with sample thickness. This observation seems to support the argument that the classical size effect in conventional 3D materials, which assumes that boundary scattering reduces the thermal conductivity, also occur in layered 2D crystals (Table 11.1). Recent experiments showed a quite different trend for the layer thicknessdependent thermal conductivity of MoS2 by different research groups, due to differences in sample quality and experimental conditions, thermal conductivity of MoS2 increases with the number of layers (Fig. 11.6). A recent first-principles-based PBTE study [58] showed that basal-plane thermal conductivity of 10-μm-long Table 11.1 Thermal conductivity of single-layer MoS2. Experimental simulations

TC (W/m-K)

Method

Layer number

Cited work

Exp.

34.5

Raman spectroscopy

Single-layer

Exp.

52

Raman spectroscopy

Few layer

Exp.

85100

Pump-probe

Bulk

Theory

26.2

Klemens’ formula

Single-layer

Simulation

100

PBTEs

Single-layer

ACS Nano 8, 986 (2014). J. Phys. Chem. C 117, 9042 (2013) J. Appl. Phys. 116, 233107 (2014). Appl. Phys. Lett. 105, 103902 (2014). Appl. Phys. Lett. 105, 131903 (2014).

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Figure 11.6 Thermal conductivity as a function of the number of layers for MoS2 at room temperature and zero pressure. The source of reference data: Gu [58]; Fan [48]; Luo [87]; Bae [68]; Yarali [88]; Yan [55]; Li [40]; Zhang [69]; Jo [56]; Aiyiti [70]; Sahoo [54]; Liu [67]; Jiang [97]; Zhu [28].

samples reduces monotonically from 138 to 98 W/m-K for naturally occurring MoS2 when its thickness increases from one layer to three layers, and thermal conductivity of trilayer MoS2 approaches that of bulk MoS2. The reduction is attributed to both the change of phonon dispersion and thickness-induced anharmonicity. Phonon scattering for ZA mode in bilayer MoS2 is found to be substantially larger than that of monolayer MoS2, which is attributed to the fact that of single-layer MoS2, which is attributed to the fact the additional layer breaks the mirror symmetry. The measured thermal conductivity of MoS2 are presented, but the experimental results show no clear thickness dependence.

11.3

Extrinsic thermal conductivity of chalcogenide

11.3.1 Strain effect Strain engineering as a mechanical means that can regulate geometric shapes and morphologies of materials and structures down to the nanoscale is particularly attractive because of their direct coordination with the phononic mechanism of thermal transport, where mechanical strain can be introduced by external load or force. Depending on the direction of external force, it could include either tensile or compressive strain. In fact, during the synthesis and processing of nanoscale materials and structures, it is difficult to ensure a strain-free environment and nanostructured materials will commonly have residual strain. On the other hand, there exists a series of techniques to regulate the strain of materials, such as bending the flexible

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substrate, elongating the substrate, and local thermal expansion of the substrate. It is known that strain has large effects on the electronic and optical properties of nanostructure materials. For example, with about 9% cross-plane compressive strain, MoS2 exhibits a semiconductor to metal transition [71,72], with an electrical conductivity enhancement from 0.03 to 18 S/m [73,74]. Following the formation of Peierls-Boltzamnn, Bhowmick and Shenoy [75] introduced a power-law scaling of thermal conductivity on phonon frequency  αγ ω (11.5) κ 5 AT 21 ω0 where ω and ω0 are the phonon peak frequencies with and without strain, A is the constant, α is a material-dependent positive constant, and γ is the Gru¨neisen parameter. From Raman spectroscopy measurement, a redshift was observed in phonon peaks in monolayer MoS2 under tensile strain [85]. Under the same experiment, the extracted Gru¨neisen parameter for MoS2 monolayer is about 1.06. The redshift in phonon peaks and positive Gru¨neisen parameter lead to the same trend with MD simulation results that thermal conductivity of monolayer MoS2 monotonically decreases with tensile strain [77]. Ding et. al found that for defect-free and defective MoS2, the reduction of thermal conductivity results from the softened phonon vibrations and the decreased of both group velocity and specific heat under a tensile strain and the increased phonon scatterings due to the out-of-plane deformation of monolayer structure under a compressive strain [58]. In adidition, when a moderate biaxial tensile strain 2-4% is applied, the thermal conductivity of single-layer MoS2 can be reduced by 10-20%. The effect of tensile strain is more obvious than that of compressive strain. When the system size of MoS2 sample varies, the reduction rate of thermal conductivity is size-dependent due to different dominant phonon scattering mechanisms [80]. In addtion, the thermal conductivity of other 2D chalcognides also exhibits a similar decreasing trend under a tensile strain. For example, The lattice thermal conductivity of the strained ZrS2 monolayer is much smaller than that of the unstrained system. Specifically, the thermal conductivity decreases from 3.29 W/m-K to 1.99 W/m-K, a 40% reduction when the strain of 6% is applied at 300 K. In such circumstance, the phonon dispersion of transverse and longitudinal acoustic (TA and LA) modes become softened, while the out-of-plane acoustic (ZA) mode is slightly stiffened. Based on the dispersion, the phonon softening leads to the reduced group velocity, which will be a source of the reduction in the lattice thermal conductivity compared with the unstrained ZrS2 monolayer, the relaxation times of the strained system are generally increased [78]. Shafique et al. studied the effect strain on lattice thermal conductivity of monolayer 2H-MoTe2 by solving BTE based on the first principles calculations. It is found that the monolayer 2H-MoTe2 is more sensitive to strain compared with that of ZrS2, and the lattice thermal conductivity is reduced by approximately 2.5 times by applying 8% biaxial tensile [88]. These diverse dependence of thermal conductivity on strain for different 2D chalcognides materials can be attributed to different strain dependece of

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heat capacity, group velocity and phonon scatterings rate with different crystal structures in accordance with the simple kinetic theory. Besides the in-plane strain effect on thermal conductivity of chalcogenides discussed above, the out-of-plane strain also can modulate thermal conductivity of layered chalcognides. Van der Waals layered chalcogenides are very sensitive to the strain due to the weak nature of van der Waals interactions. Under 9% crossplane compressive strain created by hydrostatic pressure in a diamond anvil cell shown in Fig. 11.7, it is observed the increasement of the cross-plane thermal conductivity in multilayer MoS2. This enhancement is due to the greatly strengthened interlayer interaction and heavily modified phonon dispersion along crossplane direction [81]. In experiment, the picosecond transient thermoflectance integrated with DAC device are used to study strain-tuned cross-plane thermal conductivity in bulk MoS2 over 9% cross-plane strain. Specifically, it is observed

Figure 11.7 Experimental setup, total, and electronic thermal conductivity under high pressure. (A) Schematic of thermal conductivity measurement with a diamond anvil cell integrated with a ps-TTR system. (B) Experimental data and fitting of ps-TTR measurements at two selected pressures, with 20% confidence interval shown. (C) Extracted cross-plane thermal conductivity (both lattice and electronic) as a function of pressure. The red curve is included only as a guide to the eye. Semiconducting and intermediate regions are labeled based on Ref [29]. (D) Electronic thermal conductivity of MoS2 against pressure, determined from measured electronic conductivity via the Wiedemann-Franz law. Three regions of the semiconductor to metal transition are labeled [81]. Reproduced with permisssion [81]. Copyright 2019, American Physical Society.

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roughly a 7 3 increase of cross-plane thermal conductivity, from 3.5 W/m-K at ambient pressure to about 25 W/m-K at 9% strain. First-principle and electrical conductivity measurements reveal that this drastic change originate from the substantially strengthened interlayer force and heavily modified phonon dispersions along the cross-plane direction. The group velocity of coherent longitudinal acoustic phonons, increase by a factor of 1.6 at 9% strain due to phonon stiffening, while their lifetimes decrease due to the phonon unbundling effect [81].

11.3.2 Effect of atomic disorder and defect Most of the phonon scattering mechanisms are prevailing toward a certain range of phonon frequencies, point defect scattering targets high frequency phonons, 2D interfacial scattering, grain boundaries, or fine nanoprecipitates dominantly scatter low frequency phonons. To achieve the minimum thermal conductivity of chalcogenides-based thermoelectric materials, many phonon scattering mechanisms are applied to suppress the entire scale hierarchical phonon transport in practical applications. For example, the lattice disorder and large number of grain boundary are present in fabricated thin film samples. The cross-plane thermal conductivity of bulk polycrystalline MoS2 is around 1.1-5.8 W/m-K [82] and the in-plane thermal conductivity is 85-110 W/m-K [27]. Generally, heat in dielectric materials mainly is transported by phonons. The thermal conductivity of layered two-dimensional TMDs alloys plays a critical role in the reliability and functionality of TMDs-enabled devices. In thermoelectric areas, the thermal conductivities of chalcogenides are manipulated based on different promising strategies, such as introduction of doping and defect. The strategy of alloying/doping engineering aim to introduce atomic disorder in the crystal lattices. Alloying/doping leads to the mass contrast between foreign atoms and regular lattice sites, which is generally considered as point defects, indeed enhancing the phonon scattering rate. Frequency dependence of the point defect phonon scattering relaxation time is given by the following equation [83]: τ21 PD 5

Vω4 Γ 5 Aω4 4πv3s

Where V is the average volume per atom, is the phonon frequency, Γ is the disorder scattering parameter which denotes Γ 5 ΓMF 1 ΓSF. The subscripts are mass fluctuation term and strain field term, respectively. Therefore, the higher the mass and size mismatched between the host and the foreign atom, the higher the phonon scattering and lower thermal conductivity. Aiyiti et. al. introduced the defects by mid oxgen plasma and observed the oxotic crystllineamorphous transition evidenced by typical characteristic of crystalline and amorphous phases of the samples shown in Fig. 11. 8A and 8B. In the typical crystalline phase, thermal conductivity increases due to the activation of more phonon modes and further decreases due to the increased Umklapp scattering

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Figure 11.8 Characterization of samples in crystalline and amorphous phase with TEM and thermal measurements. (A) HRTEM image and diffraction pattern of the intrinsic sample indicative of crystalline phase. (B) HRETEM image and diffraction pattern of the tailored sample indicative of amorphous phase. (C) the derived thermal conductivity of the samples as a function of temperature. (D) the measured thermal conductivity in the range of 20300 K after the oxygen plasma process. The blue dash lines denotes the theoretical low limit of amorphous thermal conductivity in MoS2. Reproduced with permission [84]. Copyright 2018, Royal society of Chemistry.

with temperature (Fig. 11. 8C). In contrast, thermal conductivity of MoS2 sample after oxygen plasma treatment is two orders of magnitude smaller than that of pristine samples (Fig. 11. 8D). Mo, as the heavier atom, contributes to the eigenvectors of low frequency phonon modes more than the S atom. Therefore, the largest thermal conductivity reduction when Mo isotopes are introduced is B 20%, which is larger than the 7%B 10% achieved using natural isotopes for the sample length [82]. In TMDs, the isotopes of metal atoms generally have a larger impact on the thermal conductivity than the chalcogen isotopes. This is due to the fact that Mo or W are heavier than S and thus contribute more to the eigenvectors of low-frequency acoustic phonon modes. For example, S isotopes can influence optical phonons from 8 to 14 THz, while Mo isotopes impact acoustic phonons in the frequency range of 27 THz that dominate thermal transport at room temperature [85].

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In phonon transport, the phonon-point defect scattering, such as isotope, vacancy, and substitution can suppress thermal conductivity. Wang et al [86]. studied the point effect (sulfur vacancies and oxygen substitution to sulfur) effect on thermal conductivity of MoS2 nanoribbon. They found that the suppression of thermal conductivity by vacancies is stronger than that by substitution. Wu et al [87]. use molecular dynamics simulations with first-principles force constants to study the isotope effect on thermal transport of single layer MoS2. It is found the isotope scattering in MoS2 strongly scatter phonons with intermediate frequency, in which compared with S isotopes, the Mo isotopes have stronger impact on thermal conductivity. Milad et al [88]. explored the effect the lattice defects on thermal conductivity of the suspended MoS2 monolayer grown by chemical vapor deposition (CVD). The measured room temperature thermal conductivity are 30 6 3.3 and 35.5 6 3 W/ m-K for two samples, which are more than two times smaller than that of their exfoliated counterpart. In addition, they also explore the effect of lattice vacancies and substitution tungsten (W) doping on the thermal transport of the suspended MoSe2 monolayers grown by chemical vapor deposition (CVD). The results suggest that a Se vacancy concentration of 4% results in thermal conductivity reduction up to 72% [89]. Using a semi-ab initio method, they have computed the thermal conductivity of MoS2, WS2, MoTe2. The results suggests that for a TMDs XY2 where one constituent species is fixed, more pronounced charged to the thermal conductivity will be changed by changing the masses of Y rather than X [90]. The thermal properties of 2D TMDs can be tailored through isotope engineering. Monolayer crystals of MoS2 were synthesized with isotopically pure Mo and Mo by chemical vapor deposition employing isotopically enriched molybdenum oxide precursors. The in-plane thermal conductivity of the MoS2 monolayers, measured using a non-destructive, optothermal Raman technique, is found to be enhanced by 50% compared with the MoS2 synthesized using mixed Mo isotopes from naturally occurring molybdenum oxide. The boost of thermal conductivity in isotopically pure MoS2 monolayers is attributed to the combined effects of reduced isotopic disorder and a reduction in defect-related scattering, consistent with observed stronger photoluminescence and longer exciton lifetime. The in-plane thermal conductivity of the suspended 100MoS2 and 50% 100MoS2 monolayers are 61.6 6 6.0 W/m-K and 52.8 6 2.4 W/m-K [91], respectively, showing a B50% and a B30% enhancement compared with the NatMoS2 (40.8 6 0.8 W/m-K) shown in Fig. 11.9. Both of these values may be underestimated due to the assumption that all phonon modes by Raman thermal measurement are in equilibrium. The in-plane thermal conductivity of 92MoS2, 96MoS2, 100MoS2 and NatMoS are calculated using self-consistent Boltzmann transport equation (BTE) by considering scattering mechanisms, such as boundary scatterings and isotopes scatterings. These conductivities of four samples do not show significant difference (B100 W/m-K at 1 μm). Qian et al. investigated the temperature-dependent anisotropic thermal conductivity of the phase-transition 2D TMDs alloys WSe2(1-x)Te2x in both the in-plane direction and the cross-plane direction using time-domain thermoreflectance measurement [92]. In addition, the thermal conductivity of lead selenide (PbSe) and

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Figure 11.9 (A) A sketch of different Mo isotoped-MoS2. (B) The sample length-dependent thermal conductivity of natural MoS2, 92MoS2, 96MoS2 and 100MoS2. Reproduced with permisssion [91]. Copyright 2019, American Chemical Society.

lead telluride (PbTe) and their alloys (PbTe1-xSex) are investigated by firstprinciples calculations [33]. Compared with the thermal conductivity of PbSe and PbTe, they found that the contribution of optical phonons is not negligible based on their direct contribution and strong scattering channels for acoustic phonons. Besides, the nanostructure of less than B10 nm is effective to reduce thermal conductivity of pure PbSe and PbTe. Importantly, the alloying is a relatively effective way to reduce the lattice thermal conductivity.

11.3.3 Anisotropy The thermal anisotropy of layered chalcogenides-based materials is fundamentally associated with their crystal structures. In most materials, the thermal conductivity along the cross-plane direction is much lower than the thermal conductivity within plane direction primarily because of the much weaker van der Waals interactions between layers compared to the stronger covalent bonds between atoms in plane. For layered van der Waals chalcogenides, heat transport is strongly anisotropic, featuring high thermal conductivity in plane and low conductivity across the layers. For example, the reported in-plane thermal conductivity of MoS2 ranges from 35 to 85 W/m-K [27,54,56], more than 10 3 higher than the cross-plane thermal conductivity (24.5 W/m-K) [27,88,89] shown in Fig. 11.10. For polycrystalline MoS2 thin films (50150 nm thick), the thermal conductivity was found to be approximately 1.5 W/m-K in-plane and 0.25 W/m-K out-of-plane [90], which demonstrates the importance of thermal boundary scattering as the limiting factor for thermal conductivity in nano-crystalline MoS2 thin films. They demonstrate that 2D nanoplates of vertically grown MoS2 can have anomalous thermal anisotropy, in which in-plane thermal conductivity 0.83 W/m-K at 300 K is B1 order of magnitude lower than out-of-plane thermal conductivity about 9.2 W/m-K at 300 K. Lattice dynamics analysis reveals that this anomalous thermal

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Figure 11.10 Measured in-plane and out-of-plane thermal conductivity (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 out-ofplane thermal conductivity of natural, bulk MX2 from Ref. [84]. The dashed curved are outof-plane thermal conductivity of natural WS2 and WSe2 with boundary scattering length of 150 nm from Ref. [84]. The dashed-dotted curves are the calculated in-plane and out-ofplane thermal conductivity of natural bulk MoS2 from Ref. [57]. Measurement from literature are synthetic MoS2 by Pisoni et al [85]. natural MoS2 crystal by Liu et al. [27]. synthetic WS2 by Pisoni et al [86]. single crystal WSe2 by Chiritescu et al. [59]. and single crystal MoS2 and WSe2 by Murato et al. [87]. Reproduced with permisssion [88]. Copyright 2017, Wiley.

anisotropy can be attributed to the anisotropic phonon dispersion relations and the anisotropic phonon group velocity along different directions. The low in-plane thermal conductivity to the weak phonon coupling near the x-y plane interface [91]. The thermal conductivity anisotropy can be modulated by two orders of magnitude by mean of lithium intercalation and cross-plane strain. Specifically, the inplane and out-of-plane thermal conductivity can be tuned over one and two orders of magnitude, respectively. For LiMoS2, lithium intercalation leads to a seven-fold reduction of in-plane thermal conductivity, and two-fold reduction in cross-plane thermal conductivity. The two-fold reduction of PBTE calculation is consistent with the experimental measurement [28]. The effect of lithium intercalation on inplane phonon modes to reduce the frequency range of acoustic modes with significant group velocity, as well as to reduce the overall lifetime and mean free path. The two effect contribute to the consequence of major reduction from 93.7 to 12.2 W/m-K. However, for pristine MoS2, strain reduces the MFPs of ZA modes through two different mechanisms. In details [92]. Thermal conductivity of molybdenum disulfide can be modified by electrochemical intercalation. Distinct behavior for thin film with vertically aligned basal planes and natural bulk crystals with basal planes aligned parallel to the surface is observed by Gaohua, et al. The change of thermal conductivity correlates with the lithiation-induced structural and compositional disorder. The ratio of the in-plane to

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through-plane thermal conductivity of bulk crystal is enhanced by the disorder [28]. These results suggest that stacking disorder and mixture of phases is an effective mechanism to modify the anisotropic thermal conductivity of 2D materials. The thermal conductivity of LixMoS2 samples with different degrees of electrochemical interaction of lithium ions were measured by time-domain thermoreflectance (TDTR). We show that lithium ion intercalation has drastically different effects on thermal transport in these different forms of MoS2 due to the differences in crystalline orientation and initial structural disorder. Our most striking observation is that the thermal anisotropy ratio in bulk LixMoS2 crystals increases from 52 (x 5 0) to 110 (x 5 0.34) as a result of lithiation-induced stacking disorder and phase transitions. In addition, Na-intercalating in molybdenum disulfide leads to slightly structural modification and reduced thermal conductivity, which results from the enhanced anharmonicity of low-frequency phonons and the increased several quasi-local modes of vibration in the range of moderate frequency, which increases the scattering channel of phonon-phonon interactions [93]. The intercalates could influence dramatically the structural properties between layers, while slightly within layers, thus Na-intercalating could make more influence on the in-plane phonon transport than on that along interlayer (cross-plane) direction.

11.4

Fundamental insight into thermal transport

Understanding thermal transport properties of chalcogenides is critical to the development of better chalcogenides-based thermoelectric and phase-change materials. In particularly, the perspective from lattice dynamics and chemical bonding point have been extensively investigated. Based on previous studies, we will review several strategies of yielding anharmonicity including lone pair electron, resonant bonding and rattling mode.

11.4.1 Resonant bonding Resonant bonding [9497] has been appreciated as an important feature in some electron-deficient chalcogenides (i.e., compounds that contain one of the chalcogen elements S, Se, or Te, such as lead chalcogenides). Resonant bonding can be understood as resonance or hybridization between different electronic configurations: three valence p-electron alternate their occupancy of six available covalent bonds that exist between a given atom and its octahedral neigh-bonds that exist between a given atom and its octahedral neighbors. The established of resonant bonding can significantly delocalize the electrons and shrink the band gap, leading to soft optical phonons. Such behavior reduces thermal conductivity through two mechanisms: strong anharmonic scattering and a large scattering phase-space volume [94]. For PbTe of a typical lead chalcogenides, due to the fact that sp-hybridization is small and the s-band is lower than the p-band by 1.5 eV, it is considered only

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p-electrons for valence states and each atom has three valence electrons on average. Given PbTe’s octahedral structure and its three valence electrons per atom, the choice of bond occupation is not unique. This induces hybridization between all the possible choices for the three electrons forming the six bonds. This description of resonant bonding is based on IV-VI compounds, but the resonant bonding exists in even more complicated materials. In general, the unsaturated covalent bonding by p-electrons with rocksalt-like crystal structure can be regarded as a resonant bonding. The main feature of resonant bonding is the long-ranged interaction and energy density distribution due to the resonant bonding. For example, for the harmonic force constants of typical chalcogenides PbTe, it is obviously found that these compounds is the presence of long-ranged interactions along the ,100. direction of rocksalt structure. The long-ranged interactions of resonant bonding, different from monotonically decreasing of the long-ranged Coulomb interaction, are that fourthnearest neighbor interactions are stronger than second-or third-nearest neighbor interactions due to the long-ranged electronic polarizability. Moreover, compared with ground-state electron density distribution of NaCl, it is clearly observed that PbTe has largely delocalized electron density distribution due to the resonant bonding. In PbTe, the valence p-electrons form highly directed networks of resonant bonds that lead to long-range interatomic interactions along particular directions, which in turn causes soften transverse optical (TO) phonons and large lattice anharmonicity, both contributing to a low thermal conductivity, which was predicted and experimentally observed. By controlling resonant bonding in chalcogenides, the thermal transport properties can be modified for various applications in extreme conditions. The resonance character can be weaken by increasing hybridization and iconicity. Strong hybridization increases the number of covalent bond and iconicity tends to localize the electrons. Therefore, the material with weak hybridization and iconicity (some state-of-the-art thermoelectric materials such as PbTe, SnTe, PbSe and SnSe), which possess stronger resonant bonding [96]. In addition, resonant bonding can also be tuned by various means, including thermal excitations, changes in composition, and large hydrostatic-like pressure. Recently, by means of Synchrotron X-ray diffraction and density functional theory, Xu et al. found that at high pressure orthorhombic lattice of GeSe appears to become more symmetric and the Born effective charge has significantly increased, indicating that resonant bonding has been established. In contrast, the resonant bonding is partially weakened in PbSe at high pressure due to the discontinuity of chemical bonds along a certain lattice [95].

11.4.2 Lone pair electron Anharmonicity can be significantly amplified by the presence of stereochemically active lone pair electrons (LPEs). A lone electron pair effect refers to a pair of charge electrons that are not shared with another atom and is also called a nonbonding pair [96]. The lone pair is formally from the s-valence electron pair (s2), which tends to be more and more difficult to remove from the metal as we move

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down the respective group in the p-block elements. For example in group 13, the lone pair becomes increasingly stable as the element become heavier. Thus it is more stable configuration for the heaviest. The large stability of the s2 pair in the heavier elements of the main group is attributed to relativistic effects that contract the size of the s-orbital and bring its elements closer to the nucleus. The Cu-Sb-Se ternary system presents a unique opportunity to study the effect of LEPs on thermal conductivity [98]. In CuSbSe2 and Cu3SbSe3, however, Sb has the same coordination yet the average Se-Sb-Se angle is quite different. Wang and Libau studied this effect, and found that the change in X-Sb-X bond angle (where X denotes a chalcogen atom) correlates to the stereochemical activity of the LEP, or the delocalization of the Sb 5 s LEP away from the Sb nucleus (shown in Fig. 11.11). The morphology of LEP is directly related to lattice anharmonicity, and propensity of a given crystal to exhibit. The interaction of lone-pair electrons with

Figure 11.11 (A) Schematic representation of the local atomic environment of Sb in Cu3SbSe4, Cu3SbSe3, and CuSbSe2. Shaded lines represent Sb-Se bonds, dashed lines illustrate the approximate morphology of the Sb lone-pair 5 s electron oribial. (B) Temperature of the lattice thermal conductivity of Cu3SbSe4, Cu3SbSe3, and CuSbSe2. Reproduced with permisssion [97]. Copyright 2011, American Physical Society.

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neighboring atoms can produce minimum lattice thermal conductivity in group VA chalcogenide compounds. Both the morphology of the lone-pair electron orbital and the coordination environment of the group VA atom affect the extent to which the LEPs induce anharmonicity in the crystal lattice. The s2 lone pair behaves in unique ways depending on the local coordination environment. It can either stereochemically express itself by occupying its own distinct space around the metal atom or it can effectively from view. The relationship between anharmonicity and stereochemically active LPEs is that there exist a nonlinear repulsive electrostatic force between LPEs and neighboring bonds, which lowers the lattice symmetry and hinders lattice vibration [98,99]. A microscopic understanding of the lone-pair s2 electrons and the mechanisms responsible for enhanced anharmonicity is missing. Our focus is on the role of the stereochemically active lone-pair s2 electrons, can have different thermal transport properties.

11.4.3 Rattling modes The “Phonon-glass and electron-crystal” is an ideal strategy for high performance thermoelectric (TE) energy generator, where electron transport through the regular crystal lattice freely, while the phonon are seriously scattered. Rattling model refers to large amplitude vibrations of specific atoms or atom clusters in materials, in which phonons are seriously scattered. The typical ratting models are in clathrates and skutterudites with cage-like structure [100]. In these materials, guest atoms reside in cage-like structures. The large space and weak bonding make the guest atoms vibrate with larger displacement and different frequency compared with atoms in the host framework. All this kind of vibration yields additional scattering of phonons depending on the structure of the cage and the guest atoms induced. In term of phonon dispersion, the rattling model results in a downward shift of acoustic branch due to an avoided crossing between transverse optical branch and longitudinal acoustic branch. It is noteworthy that the coupling between the optical branch and the acoustic branch is different from that in resonant bonding. In resonant bonding, the coupling happened mainly due to the softening of the optical branch, where the acoustic branch in rattling model is also soften together with optical branch [94]. The potential energy of rattling atom shows a flatter vibration around the equilibrium position, which means that the rattling atoms vibrate with larger amplitude compared with the normal atoms when the potential energy is fixed. Moreover, the difference in the amplitude is amplified with increasing temperature [101]. It is noteworthy that the coupling between the optical branch and the acoustic branch is different from that in resonant bonding. In resonant bonding, the coupling happened mainly due to the softening of the optical branch, whereas the acoustic branch in ratting model is also soften together with optical branch. Another signal is the large atomic displacement parameter (ADP). ADP can be obtained from the inelastic neutron scattering (INS) and the refinement of rattling atom (weak bonded) and normal atoms (strong bonded) as a function of ADP in crystal lattice. The potential energy of rattling atom exhibits a flatter vibration around the equilibrium position, which is

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when the potential energy is fixed the ratting atoms vibrate with larger amplitude compared with the normal atoms. In these compounds, a guest atom rattles within oversized structural cages and scatters the heat-carrying acoustic phonons, thereby significantly lowering thermal conductivity. The exploration of new materials with intrinsically low along with a microscopic understanding of the underlying correlations among bonding, lattice dynamics, and phonon transport is fundamentally important towards designing promising thermoelectric materials. The presence of strongly anharmonic phonon originating from rattling vibration of In1 cations (along the z-axis) within the columnar ionic substructure, which couple with the heat-carrying acoustic modes and lead to an ultralow. By intentional p-type doping through creation of Indeficiencies, the power factor. In the nominal InTe sample, which is significantly higher than that of pristine InTe.

11.5

Conclusion and outlook

In this chapter, we summarized the recent development the emerging theory and many remarkable achievements and progress on phonon and thermal properties of chalcogenides materials including lead chalcogenides, transition metal chalcogenides, etc. In order to satisfy the diverse application of chalcogenides, several strategies of engineering thermal conductivity including isotopes, defects, strain and disorder are discussed. Finally, three fundamental physical mechanisms of thermal transport are concluded. Therefore, we hope this chapter can provide some inspirations to the researchers and will be beneficial for further progress in this field of thermal transport properties of chalcogenides.

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