Accepted Manuscript. First-principles Debye-Callaway approach to lattice thermal conductivity. Yongsheng Zhang

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1 Accepted Manuscript First-principles Debye-Callaway approach to lattice thermal conductivity Yongsheng Zhang PII: S (16) DOI: /j.jmat Reference: JMAT 66 To appear in: Journal of Materiomics Received Date: 4 May 2016 Revised Date: 21 June 2016 Accepted Date: 22 June 2016 Please cite this article as: Zhang Y, First-principles Debye-Callaway approach to lattice thermal conductivity, Journal of Materiomics (2016), doi: /j.jmat This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Gaphical Abstract_Research Article First-principles Debye-Callaway approach to lattice thermal conductivity Yongsheng Zhang The calculated lattice thermal conductivities using the first-principles Debye-Callaway approach are in reasonable agreement with the experimental results. Soft acoustic modes with low Θ and v but large γ play vital roles in the determination of low lattice thermal conductivity of thermoelectric materials. The first-principles Debye-Callaway approach is proved as an effective tool to calculate the lattice thermal conductivity. The results calculated by this approach can be compared to the experimental results and used to predict the low lattice thermal conductivity of the compounds prior to the experiments. 1

3 First-principles Debye-Callaway approach to lattice thermal conductivity Yongsheng Zhang 1, 2 * 1 Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei , China 2 University of Science and Technology of China, Hefei , China Å Abstract Lattice thermal conductivity (κ) is one of the most important thermoelectric parameters in determining the energy conversion efficiency of thermoelectric materials. However, thermal conductivity calculations are time-consuming in solving the Boltzmann transport equation or performing the molecular dynamic simulations. This paper reviews the first-principles Debye-Callaway approach, in which the Debye-Callaway model input parameters (i.e., the Debye temperature Θ, the phonon velocity v and the Grüneisen parameter γ) can be directly determined from the first-principles calculations of the vibrational properties of compounds within the quasiharmonic approximation. The first-principles Debye-Callaway approach for three experimentally well studied compounds (i.e., Cu 3 SbSe 4, Cu 3 SbSe 3 and SnSe) was proposed. The theoretically calculated κ using the approach is in agreement with the experimental value. Soft acoustic modes with low Θ and v but large γ have an effect on the determination of low lattice thermal conductivity thermoelectric materials. The first-principles Debye-Callaway approach has been proved as an effective tool to calculate κ without the experimental inputs. The results calculated by this approach can be used to to predict the performance of low lattice thermal conductivity compounds. Keywords: DFT; phonon; Debye-Callaway; thermoelectric materials * Corresponding author. address: yshzhang@theory.issp.ac.cn. (Yongsheng Zhang). 1

4 1. Introduction Thermoelectric materials play vital roles in solving the impending energy crisis due to directly converting heat into electricity. [1][2] The conversion efficiency is characterized by the thermoelectric figure of merit ZT = S 2 σt /κ, where S, σ and κ are the Seebeck coefficient, the electrical conductivity and the thermal conductivity (consisting of the electronic (κ e ) contribution and the dominate lattice vibration (κ L ) contributions, respectively). In the ZT formula, increasing S andσ or decreasing κ can enhance the figure of merit. However, the Seebeck coefficient (S), the electrical conductivity (σ) and the electronic thermal conductivity (κ e ) are coupled with each other through the carrier concentration and effective mass. Only the lattice thermal conductivity (κ L ) is an independent thermoelectric parameter. Thus, decreasing the lattice thermal conductivity is one of the most attractive aspects in the thermoelectric community. The most popular methodology is to introduce phonon scattering regions, such as by the nanostructured materials [4][5][6][7], which efficiently reduce thermal conductivity by enhancing phonon scattering. Unfortunately, the nanostructured materials scatter electrons as well and decrease the electrical conductivity. For a long time, an intense anharmonicity has been proved to induce a low lattice thermal conductivity. Thus, ordered crystal structures with an intense anharmonicity (such as AgSbTe 2 [8], Cu 3 SbSe 3 [9][10], Cu 12 Sb 4 S 13 [11], and Cu 12 Sb 4 Se 13 [12]) could decrease the lattice thermal conductivity intrinsically without affecting the electrical conductivity. Zhao et al. [13] discovered a high-performance thermoelectric material, SnSe with ZT of ~2.6 at ~930K. The high thermoelectric figure of merit is due to the ultralow lattice thermal conductivity attribute to an intense anharmonicity. Therefore, it is important to efficiently evaluate the lattice thermal conductivity of compounds for seeking high performance thermoelectric materials. The lattice thermal conductivity (κ L or κ for simplicity) is the heat transfer ability through a sample that has a temperature difference between its two ends (see Fig. 1). In principle, a direct method can be used to determine the thermal conductivity by simulating the temperature gradient from the nonequilibrium molecular dynamic (MD) simulations at a given heat current [14][15]. Another MD simulated thermal conductivity is based on the Green-Kubo method using the equilibrium MD 2

5 simulations [16]. However, the MD simulations need a large unit cell to consider the finite size effect and a long simulation time to converge the autocorrelation function. The lattice thermal conductivity can be also achieved by solving the phonon Boltzmann transport equation [17], but it is not trivial to solve the equation since a large amount of linear equations should be properly considered. Those linear coefficients are dependent on the interatomic interactions, such as the second- and third-order force constants. Density functional theory (DFT) is a useful tool to accurately describe the interatomic interactions in many materials. [18][19] Thus, based on the DFT calculations, a full iterative solution to the phonon Boltzmann transport equation is developed, such as the ShengBTE software package [20]. However, obtaining the third-order force constants used to describe the anharmonic is time consuming. Even though the recently developed compressive sensing lattice dynamics (CSLD) approach [21] significantly saves the calculation time, the computational burden is still heavy. An alternative way to calculate the lattice thermal conductivity is to use the Debye-Callaway model [22] within the relaxation time approach. In the model, the relaxation time due to the phonon-phonon scattering can be written as a function of the Debye temperature (Θ), phonon velocity (v) and the Grüneisen parameter (γ). The simple Debye-Callaway model is computationally feasible methodology used in the experimental community to estimate the lattice thermal conductivity based on the experimentally measured parameters (Θ, v and γ) [23][24][25][26]. However, the question is what if there has a compound without the experimentally measured parameters and how to obtain the lattice thermal conductivity of the compound theoretically. In the mini-review paper, we will present the first-principles Debye-Callaway approach to calculate the lattice thermal conductivity based on the first-principles determined parameters (Θ, v and γ) within the quasiharmonic approximation. We applied the methodology to different compounds (i.e., Cu 3 SbSe 3, Cu 3 SbSe 4 and SnSe), and the theoretically simulated lattice thermal conductivity is in agreement with the experimental results. Furthermore, we predicted a new compound (i.e., Cu 12 Sb 4 Se 13 ) with a ultralow lattice thermal conductivity. The first-principles Debye-Callaway model has been proved as a useful and feasible computational tool to determine the lattice thermal conductivity of thermoelectric materials. 2. Theoretical Lattice Thermal Conductivity Simulation Approaches 3

6 2.1 Molecular Dynamic Simulating Lattice Thermal Conductivity The thermal conductivity (κ) characterizes the ability of heat conduct through a sample that has a temperature difference between its two ends (the hot region with T H and the cold region with T L, T H > T L ) (see Fig. 1). Based on the Fourier law (or the law of heat conduction), the heat transfer can be written as, J α = κ αβ ( T ) β (1) β where J α, κ αβ and T are the α-component of heat current, the second-order tensor (αβ) of thermal conductivity and the temperature gradient, respectively. Molecular dynamic (MD) method is a powerful tool for simulating material properties at finite temperatures, such as the transport properties [27][28]. The atomic positions and momenta of a sample can be evolved by numerically integrating the Newtonian equations of motion. The heat current (J) is generated by the approach E k developed by Ikeshoji [29]: J =, where E k and t are the suitably chosen t kinetic energy and the time step, respectively. The temperature gradient ( T) can be drawn from MD simulations by monitoring the momenta of atoms along the heat flux. Thus, for a known heat current (J), the thermal conductivity (κ) can be evaluated using Eq. (1). However, using the direct method to simulate T, we need the nonequilibrium, steady-state MD approach, which is hard to implement in the first-principles codes. An alternative way to simulate the lattice thermal conductivity using MD is based on the Green-Kubo method [30][31]. From the fluctuation-dissipation theorem, describing a relationship between the fluctuations of the heat current vector (J) to the thermal conductivity, the second-order tensor of lattice thermal conductivity is given by 1 iω 2 dt J (0) J ( t) e t α β BT 0 κ αβ = (2) Vk d J = E i i ri (3) dt where α/β indicates the α/β direction and J α ( 0) J β( t) is the heat current autocorrelation function. E i and r i are the total energy and atomic position of atom i. 4

7 The equilibrium MD approach, implanting in most DFT codes, is suitable for the autocorrelation function evaluation, and the corresponding lattice thermal conductivity can be estimated using Eq. (2). Using MD to evaluate the lattice thermal conductivity is straightforward [32][33], but the computational burden using the first-principles MD is heavy. To obtain the accurate lattice thermal conductivity, the finite size effect in the MD simulations should be considered and a large simulation cell is necessary, since a small cell cannot accurately reproduce all phonon scattering processes. Moreover, convergence of the autocorrelation function requires a large number of ensembles or a long MD simulation time (~10ns) Solving Phonon Boltzmann Transport Equation for Lattice Thermal Conductivity The lattice thermal conductivity can also be determined by the phonon Boltzmann transport equation. The heat current (J α ) is a function of phonon distribution (f λ ). At thermal equilibrium, the phonon distribution obeys the Bose-Einstein statistics, f 0 (ω λ ), ω λ the frequencies of the whole phonon (λ) spectrum. A temperature gradient ( T) (see Fig. 1) drives the distribution function (f λ ) from f 0. Based on the Boltzmann transport equation (BTE) [17], the phonon distribution includes two parts, i.e., the df diffusion due to T and the phonon scattering. In the steady state ( λ = 0 ), we can dt write, fλ fλ - v λ T + = 0 (4) T T diffusion scattering where v λ is the phonon velocities. For the small temperature gradient, f λ of two- and three-phonon processes can be expressed as [17][35][36] 0 df 0 df f0 ( ω λ ) τ λ( v λ + λ ) T = f0( ωλ ) F T (5) dt dt 0 fλ = λ where 0 τ λ is the relaxation time calculated from the perturbation theory. In the relaxation time approximation, λ =0. 0 λ and τ λ are the functions of three-phonon 5

8 scattering rate and λ specifically depends on F λ. The anharmonic interatomic force constants (IFCs) are required to calculate the three-phonon scattering rate [20][37][38]. The final lattice thermal conductivity can be written as [20][37], κ αβ = 1 2 α β f + λ 0( f0 1)( h ωλ ) v λ F 2 λ (6) k B T ΩN where Ω and N are the volume of the unit cell and the number of q points in the Brillouin zone (BZ). Unfortunately, it is time-consuming to obtain the lattice thermal conductivity by solving Eq. (5), which results from (1) the computational expensive third-order IFCs for determining the anharmonic IFCs; (2) solving a large amount linear equations. In the ShengBTE code [20], Eq. (5) is solved iteratively starting with RTA, λ =0. This method has predicted the lattice thermal conductivity of many compounds,[21] [38][39]. To decrease the computational burden of third-order IFCs, Zhou et. al [21] developed a compressive sensing lattice dynamics (CSLD) approach, which uses the compressive sensing method to effectively select the physically important terms in the lattice dynamics model and determine the anharmonic IFCs. This approach can obtain the accurate lattice thermal conductivity of different systems, such as Si, NaCl and Cu 12 Sb 4 S 13. To accurately determine the anharmonic IFCs, a very large supercell is needed to consider long range interactions, especially in the polarized systems. 3. First-principles Debye-Callaway Approach 3.1. Debye-Callaway Model for Lattice Thermal Conductivity Calculations For the nonmetallic materials, such as semiconductors, the acoustic modes play an important role in the heat transfer by three phonon interactions [40][41]. In the Debye-Callaway model [23][42], the lattice thermal conductivity is contributed by three acoustic modes, and κ is written as a sum over one longitudinal (κ LA ) and two transverse (κ TA and κ TA ) acoustic phonon branches, κ = κ LA + κ TA + κ TA. (7) In the relaxation-time approximation, the thermal conductivity is a function of phonon scattering rates (1/τ c ) or the relaxation time (τ c ), which indicates the time that the phonon distribution is restored to the equilibrium distribution. The heat transfer involves three phonon interactions, which gives rise to two kinds of phonon process, 6

9 i.e., the Normal phonon process and the Umklapp phonon process. Usually, the Umklapp process is considered to give rise to the thermal resistance due to the reversed phonon flux in directions. However, in the Debye-Callaway model, the normal process is taken into account as well due the ability to redistributing momentum and energy among phonons. Therefore, in a perfect order crystal (without impurities, defects, etc.), the total phonon scattering rate [43][44] (1/τ c ) involves the contributions from normal phonon scattering (1/τ N ) and the Umklapp phonon-phonon scattering (1/τ U ), which means that the normal and the Umklapp phonon-phonon processes dominate the phonon scattering (1/τ c = 1/τ N + 1/τ U ). Thus, the partial conductivities κ i (i corresponds to LA, TA or TA modes) are given by, θ / T i i 4 x τc( x ) x e 2 θi / T [ dx ] i 4 x i x τc( x ) x e 0 τ N( e 1) κ i = C it { dx + (8) x 2 θi / T 3 i 4 x 0 ( e 1) τc( x ) x e dx i i x 2 τ τ ( e 1) hω where Θ i is the longitudinal (transverse) Debye temperature, x =, k T k =, h is the Planck constant, k B is the Boltzmann constant, ω is the 2π h v 4 B C i 2 3 i phonon frequency, and v i is the longitudinal or transverse acoustic phonon velocity. We can write the normal and the Umklapp phonon scattering as[23][45], 1 = LA τ ( x ) τ τ N TA / TA ' N 1 ( x ) k γ M 3 2 B LA 2 5 h v LA k V k B ( ) x T h γ V k 4 2 B TA / ' B 5 = TA xt (9) 3 5 Mh v h TA / TA ' hγ 1 2 k B = ( ) x T e Θ i 2 U( x ) Mv i Θi h i / 3T 0 where γ, V and M are the Grüneisen parameter, the volume per atom and the average mass of an atom in the crystal, respectively. The anharmonicity scattering is proportional to γ 2, indicating a similar quadratic dependence on Grüneisen parameter as the forms in Refs. [23][40][46]. The Grüneisen parameter characterizes the relationship between phonon frequency and volume change and can be defined as [47] N U (10) B 7

10 V ωi γ i. (11) ω V i From Eqs. 9 and 10, we can see the lattice thermal conductivity or the phonon scattering is a function of the Debye temperature (Θ), phonon velocity (v) and Grüneisen parameter (γ). The Debye-Callaway model is such an important model that it has been widely used in the experimental works [21][38][39]. Using the experimentally measured Debye temperature (Θ), phonon velocity (v) and Grüneisen parameter (γ), researchers can simulate the lattice thermal conductivity and figure out the essential contributions to the lattice thermal conductivity by comparing the results with the experimental measurements. However, such experimental parameter dependence restricts the ability to screen for the low lattice thermal conductivity materials from the theoretical point of view. We notice that the three Debye-Callaway needed parameters (i.e., Θ, v and γ) can be drawn from phonon dispersions. We will show how the qusiharmonic approximation produces those parameters and simulates the corresponding theoretical lattice thermal conductivity. For the relaxation time approximation (RTA) based Debye-Callaway approximation, it over-resists phonon scattering processes [37]. Thus, the RTA based methods always underestimate the lattice thermal conductivity (κ) Density functional Theory and Phonon Calculations We perform the DFT calculations using the Vienna Ab Initio Simulation Package (VASP) code with the projector augmented wave (PAW) scheme [48], and the generalized gradient approximation of Perdew, Burke and Ernzerhof [49] (GGA-PBE) for the electronic exchange-correlation functional. The energy cutoff for the plane wave expansion is 800 ev and 500eV for the Cu-Sb-Se system and SnSe, respectively. We treat 3d 10 4s 1, 5s 2 5p 3, 2s 2 2p 4 and 4d 10 5s 2 5p 2 as valence electrons in Cu, Sb, Se and Sn atoms, respectively. The Brillouin zones (BZ) are sampled by the Monkhorst-Pack [50] k-point meshes for all compounds with meshes chosen to give a roughly constant density of k-points (30 3 ) for all compounds. Atomic positions and unit cell vectors are relaxed until all the forces and components of the stress tensor are below 0.01 ev/ and 0.2 kbar, respectively. The vibrational properties are calculated using the supercell (64, 54, 58 and 112 atoms in Cu 3 SbSe 4, Cu 3 SbSe 3, Cu 12 Sb 3 Se 13 and SnSe 8

11 supercells, respectively) force constant method [51] by ATAT[52]. In the quasi-harmonic DFT phonon calculations, the system volume is isotropically expanded by few percents (+6% and +2% for Cu-Sb-Se and SnSe, respectively) from the DFT relaxed volume. Once we have the theoretically calculated phonon dispersions, the three Debye-Callaway needed parameters (i.e., Θ, v and γ) can be estimated by using Θ = ω D /k B (ω D is the largest acoustic frequency in each direction), the slope of the acoustic phonon dispersion around Γ point and frequency change with volume (see Eq. 11), respectively. There are two different ways to define the Debye temperature using the whole phonon spectrum or the acoustic branches only [40]. Using the whole spectrum, we can define the Debye temperature as the temperature, which the highest-frequency mode (not just the acoustic mode) is excited. However, in the Debye-Callaway model, which only considers the acoustic contributions, the Debye temperature defined in the model is the highest frequency of the acoustic branches. Thus, the Debye temperature used in the Debye-Callaway approach is usually (much) smaller than that defined using the whole spectrum, e.g. from the theoretically calculated phonon dispersions of PbTe [53], the Debye temperature from acoustic braches is 128 K, which is smaller than that from the whole spectrum (i.e., 160 K). Using the first-principles determined parameters for the Debye-Callaway approach, we can calculate the lattice thermal conductivity without experimental inputs. 4. Results and Discussion 4.1. Validation of the First-principles Debye-Callaway Approach: Cu 3 SbSe 4 and Cu 3 SbSe 4 We first apply the fist-principles Debye-Callaway model to two experimentally well studied two compounds, i.e., Cu 3 SbSe 4 and Cu 3 SbSe 3 [10]. The two compounds have the similar stoichiometry but different geometries (the zincblende-based tetragonal Cu 3 SbSe 4 and orthorhombic Cu 3 SbSe 3 in Fig. 2). Even though the two compounds share the same elements and the similar stoichiometry, experimental measurements (see Fig. 4) show that Cu 3 SbSe 4 does exhibit a typical thermal conductivity behavior (κ decreases with increasing temperature), whereas Cu 3 SbSe 3 exhibits anomalously a low and nearly temperature independent lattice thermal conductivity. Understanding the distinct lattice thermal conductivity behavior of the two compounds is beneficial 9

12 for the future low lattice thermal conductivity material search. Fig. 3 shows the phonon dispersions of Cu 3 SbSe 4 and Cu 3 SbSe 3. Since the acoustic modes play an important role in the thermal conductivity, we highlight these branches with different colors in the plot. From the phonon dispersions, we can find that the boundary frequency of Cu 3 SbSe 3 is 30 cm 1, which is smaller than that of Cu 3 SbSe 4, i.e., 60 cm 1. The low frequency indicates that Cu 3 SbSe 3 has softer interatomic bonds and lower acoustic modes than Cu 3 SbSe 4. The corresponding Debye temperature and phonon velocity (see Table I) can be calculated based on the phonon dispersions. Cu 3 SbSe 3 has lower Debye temperatures and phonon velocities than Cu 3 SbSe 4. To estimate the lattice thermal conductivity using the Debye-Callaway model, we still need the Grüneisen parameters. We carry out the quasi-harmonic phonon calculations to obtain the phonon dispersions at a different volume and the Grüneisen dispersion can be calculated using Eq. 11 (see Fig. 3). Most of all the Grüneisen parameters of Cu 3 SbSe 4 are negative. The negative Grüneisen parameter is coming from the tetrahedral local geometry in the Cu 3 SbSe 4 crystal structure, which is a common behavior in many tetrahedral semiconductors, such as Si, Ge and GaAs [54][55]. Xu et. al [55] described the Grüneisen parameter of diamond structure using a simple nearest-neighbor model, which is suitable for the zinic-blend structure as 2 4 CB 0d 0 well. In the model, the Grüneisen parameter can be written as γ =, 3 2 3mω where C is the number of bonds bent by the phonon distortion, B 0 is the bulk modulus, d 0 is the equilibrium bond length and ω 0 is the restoring force on the atoms. The first and second parts of the equation are the contributions from the noncentral angular part and central force parts, respectively. The sign of the Gruneisen parameter is the competition of the two parts. For Si, Ge, GaAs and the zinc-blend Cu 3 SbSe 4, the second part has a dominant effect. The average Grüneisen parameter (γ) of each acoustic dispersion (see Table 1) is 2 calculated by the method described [23]: γ = < γ >. Cu 3 SbSe 3 has much 0 larger Grüneisen parameters than Cu 3 SbSe 4, especially for the transverse models. Since the Grüneisen parameter characterizes the anharmonicity of a compound, the large Grüneisen parameter in Cu 3 SbSe 3 indicates more intense anharmonicity than Cu 3 SbSe 4. The largest Grüneisen parameters within the Cu 3 SbSe 3 BZ is along the Γ 10

13 T direction, 7 at the T point. From analyzing eigenvectors of the mode, we find that this mode involves in-phase Sb vibrations along the [100] direction (see Fig. 2). For the Cu 3 SbSe 3 stoichiometry, the nominal valence state of Sb is 3+, which indicates two additional non-bonding or lone-pair electrons at the Sb sites. The lone-pair electrons just sit along the Sb vibrational direction ([100]). The electrostatic repulsion between the lone pair electrons and the bonding charges in Sb-Se bonds weaken the restoring force for this mode, which softens phonon frequencies and possesses low thermal conductivity. Using the first-principles determined Debye-Callaway input parameters (i.e., Θ, v and γ), we calculate the lattice thermal conductivity of the two compounds. We can see a reasonable agreement between the theoretical calculation and experimental measurement (see Fig. 4). The deviation between the experimental data and the calculated curves is approximately 20%, which is quite satisfactory given the approximations inherent in the Debye-Callaway formalism [40]. The above work validates that the first-principles Debye-Callaway approach is a useful theoretical tool in the investigation of the lattice thermal conductivity of materials High Figure of Merit SnSe After validating the first-principles Debye-Callaway approach, we could apply the approach to other thermoelectric materials for lattice thermal conductivity calculations. With the combination of experimental measurements and theoretical calculations, we discovered a high figure of merit (ZT = 2.6 at 930 K) compound, SnSe (see Fig. 5) [13]. The high ZT value is due to the ultralow lattice thermal conductivity along the three crystal directions (a, b and c). The experimentally measured thermal conductivity trend is κ a < κ c < κ b (see Fig. 7). However, the physical reason behind the ultralow lattice thermal conductivity should be scrutinized. SnSe undergoes phase transition from Pnma to Cmcm at ~ K and the high ZT occurs at the high-temperature phase (i.e., SnSe-Cmcm, in Fig. 5). Usually, the lattice thermal conductivity decreases with increasing temperature, roughly showing κ 1/T behavior [40][41][56]. It is not surprising that thermoelectric materials have a low lattice thermal conductivity at high temperatures (the ultralow lattice thermal conductivity of SnS-Cmcm at ~900K). However, compounds (i.e., the low-temperature SnSe-Pnma phase, in Fig. 5) with a low κ (i.e., <1 W/mK) even at 11

14 room temperature (300 K) should be carefully checked. Thus, even though the SnSe-Cmcm phase shows a ultralow lattice thermal conductivity, we only consider the lattice thermal conductivity calculations of the low-temperature phase (i.e., SnSe-Pnma). From the theoretically calculated phonon dispersions and the Grüneisen parameters of SnSe (see Fig. 6), the acoustic modes along the three directions (x, y, z or a, b, c) are soft and the corresponding average Grüneisen parameters are large (see Table 1), which suggests the strong anharmonicity along all three directions. Moreover, the acoustic modes along the x direction are even softer than those along the y and z directions. The averaged Grüneisen parameters along the x direction are larger than those along the other two directions (see Table 1), which indicates that the x or a direction has the most intense anharmonicity and lowest lattice thermal conductivity. The large Grüneisen parameters or intense anharmonicity of SnSe is a consequence of the soft bonding in its special crystal structure (see Fig. 5): a zigzag like geometry in the b c plane and a layered slab along the a direction. Such geometry patterns provide a good stress buffer to scattering phonon transport. Moreover, using the inelastic neutron scattering and first-principles calculations, Li et. al [57] confirmed this strong phonon anisotropic and ionic-potential anharmonicity, which is due to the instability of Sn-Se bonds formed by the long-range resonant interactions between Sn 5s lone pair electrons and Se-4p orbitals. Using the first-principles determined Grüneisen parameters, Debye temperatures and phonon velocities, we calculated the lattice thermal conductivities along three directions (see Fig. 7). We find that the calculated lattice thermal conductivities are in reasonable agreement with the experimental results. The a axis has the lowest lattice thermal conductivity, the theoretically predicted thermal conductivity trend is κ a < κ c < κ b and the lattice thermal conductivity along b and c are similar. The largest discrepancy between the theoretical curve and experimental data is along the a direction, which might be due to the van der Waals (vdw) interactions between Sn-Se slabs. Such weak interactions cannot be captured in the traditional DFT-PBE calculations. Clearly, the lack of extra binding interactions in the calculations could further enforce anharmonicity and lower the lattice thermal conductivity. In addition, Guo et. al. [58] calculated the lattice thermal conductivity based on the phonon BTE considering all modes and vdw. Their theoretically calculated κ is greater than those 12

15 from the experimental measurements, which could possible result from (a) the value of Lorenz number V 2 K 2, which can well vary from 1 to V 2 K 2 ; (b) the values of the thermal diffusivity, which depends on the details of a fit to time-dependent reflectivity curves (the instrumental probably span 5% or more); (c) the sample thickness, the homogeneity of that thickness, and the sample density, and (d) it is difficult to measure precisely along one crystallographic axis. Since the lattice thermal conductivity of SnSe is rather low and its phonon mean free path is so short, it is nearly impossible to further decrease its lattice thermal conductivity by nanostructuring. Instead of decreasing the lattice thermal conductivity, Zhao et. al. [59] enhanced the ZT values in a low temperature range (i.e., K) to by boosting the power of factor in hole-doped SnSe crystals Theoretically Predicted a Novel Thermoelectric Material: Cu 12 Sb 3 Se 13 The three compounds above (i.e., Cu 3 SbSe 4, Cu 3 SbSe 3 and SnSe) have been well investigated experimentally. We next want to use the first-principles Debye-Callaway approach to predict the lattice thermal conductivity of experimentally not discovered compounds [12]. Ternary Cu-Sb-Se compounds contain Earth-abundant and nontoxic elements and they are attractive materials for numerous applications, such as thermoelectric materials, optical materials and 2D electronic materials. To search for novel compounds in the Cu-Sb-Se system, we use the cluster expansion (CE) method and prototype structures from the Cu-Sb-S system to predict new possible Cu-Sb-Se ternary phases (details in Ref. [12]). A high-temperature stable compound, cubic Cu 12 Sb 4 Se 13 with the I-43m space group (see Fig. 8), is discovered theoretically. From phonon calculations (see Fig. 9), we notice that the cubic structure is dynamically unstable. It involves out-of-plane displacements of the three-fold coordinated Cu atoms (see Fig. 8). Such a structural instability was also observed in the experimentally determined Cu 13 Sb 4 S 13 compound. The boundary acoustic frequencies of Cu 12 Sb 4 Se 13 are as low as those of Cu 3 SbSe 3 (see Fig. 3). The theoretical Debye temperature (Θ), phonon velocity (v) and Grüneisen parameter of Cu 12 Sb 4 Se 13 are calculated based on the phonon dispersions. We can see that the compound has the large Grüneisen parameters (especially for the transverse modes) in the Cu-Sb-Se phases (see Table 1), which indicates strong anharmonicity and low thermal conductivity. From the Grüneisen dispersions of Cu 12 Sb 4 Se 13, we can see that the acoustic modes along high symmetry directions (Z-Γ and T-Γ) in the BZ have very 13

16 large Grüneisen parameters (>10), indicating that the branches are very anharmonic and interact strongly with other acoustic phonons. The vibration of these modes involve Sb movement away from Sb-Se bonds, such as along the [110] direction (see Fig. 8), roughly along which the Sb lone-pair electrons are also oriented. As we discussed in the Cu 3 SbSe 3 case, the interactions between the lone pair and the three Sb-Se bonds are sensitive to the Sb-Se bond angles and result in large TA Grüneisen parameters in Cu 12 Sb 4 Se 13. We therefore use the first-principles determined parameters of Cu 12 Sb 4 Se 13 (see Table 1) to parameterize the Debye-Callaway model and calculate its lattice thermal conductivity. The resulting thermal conductivity of Cu 12 Sb 4 Se 13 is extremely low (see Fig. 4), which is lower than that of all other Cu-Sb-Se compounds. The theoretically calculated lattice thermal conductivity of Cu 12 Sb 4 Se 13 at 300 K is 0.1 Wm 1 K 1. The Debye-Callaway approximation based on the RTA underestimates the lattice thermal conductivity [37]. Considering the smallest error bar (~0.28 Wm -1 K -1 ) in the Cu 3 SbSe 4 and Cu 3 SbSe 3 calculations, we can estimate that the lattice thermal conductivity of Cu 12 Sb 4 Se 13 at 300 K should be less than ~0.38 Wm -1 K -1, which is still an extremely small value. Also, the lattice thermal conductivity trend of the three compounds (i.e., Cu 3 SbSe 4, Cu 3 SbSe 3 and Cu 12 Sb 4 Se 13 ) determined by the Debye-Callaway approach should be reasonable. We can find that the theoretically calculated lattice thermal conductivity of Cu 12 Sb 4 Se 13 is even lower than that of Cu 3 SbSe 3 (see Fig. 4). In addition, the electronic structures of the compound are investigated by plotting its band structures, which exhibits favorable band features for good thermoelectric performance, such as valence band degeneracy, high effective masses, etc.. The good electric structures and low thermal conductivity indicate that Cu 12 Sb 4 Se 13 should be a promising thermoelectric material. [13] 5. Conclusions We developed a first-principles Debye-Callaway approach, in which the Debye-Callaway model needed parameters (i.e., the Debye temperature Θ, the phonon velocity v and the Grüneisen parameter γ) could be directly determined from the first-principles calculations of the vibrational properties of compounds within the quasiharmonic approximation. The soft acoustic modes with low Θ and v but large γ 14

17 played vital roles in the determination of low lattice thermal conductivity of thermoelectric materials. The lone pair electrons in the compounds had the electrostatic repulsion with the bonding charges, which weakened the restoring force and softened the corresponding phonon modes. Thus, the lone pair electrons had the essential contributions to the intense anharmonicity and large Grüneisen parameters. Searching compounds with lone pair electrons could discover intrinsically the ultralow lattice thermal conductivity candidates. From the crystal structures analysis of Cu 3 SbSe 3, SnSe and Cu 12 Sb 3 Se 13, the low lattice thermal conductivity candidates could possess (1) nonsymmetrical local binding environment; and (2) unexpected crystal structures, such as layered, isotropic or zigzag geometries. For the experimentally well studied compounds (i.e., Cu 3 SbSe 4, Cu 3 SbSe 3 and SnSe), the theoretically calculated lattice thermal conductivities using the first-principles Debye-Callaway approach were in reasonable agreement with the experimental results. The deviation between the theoretical data and experimental results was due to the optical mode contributions. The Debye-Callaway model did not take into account optical mode. However, from the phonon dispersions, especially for SnSe, the optical modes had low frequencies and interact with the acoustic modes. These optical modes could contribute 20%-50% contributions to the heat transfer as discussed by Slack [40]. To obtain better agreement, the optical contributions should be taken into account in the future works. For an experimentally not discovered or synthesized compound (Cu 12 Sb 3 Se 13 ), we estimated its lattice thermal conductivity and predicted it as a promising thermoelectric material. The first-principles Debye-Callaway approach was proved as an effective tool to calculate the lattice thermal conductivity without experimental inputs and without too much of computational burden as in solving the phonon Boltzmann transport equation or running MD simulations. The results from this approach could not only be compared to the experimental measurements for explanations, but also be used to predict the low lattice thermal conductivity compounds prior to the experiments. Acknowledgments The author acknowledges the financial support from the National Natural Science Foundation of China (No ), the Major/Innovative Program of Development Foundation of Hefei Center for Physical Science and Technology (No. 2014FXCX001) 15

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23 List of Figure/Table Captions Fig. 1 Schematic view of heat current (the red empty arrow, J) through a sample that has a temperature difference between its two ends [the hot (red) region with T H and the cold (blue) region with T L ]. The linear gradient of color from red to blue indicates the temperature gradient.. Fig. 2 Crystal structures of Cu 3 SbSe 4 (I42m, a exp =b exp =5.66, c exp =11.28 ) and Cu 3 SbSe 3 (Pnma, a exp =7.99, b exp =10.61, c exp =6.84 ). The red arrows on Sb atoms in Cu 3 SbSe 3 represent atomic displacements responsible for the anomalously high Grüneisen parameters of the transverse acoustic phonon branch in Figure 3. Cu=blue, Sb=brown, Se=green, Sn=gray. Fig. 3 Phonon dispersions and corresponding acoustic Grüneisen parameters of Cu 3 SbSe 4 and Cu 3 SbSe 3. The big circle lines and small dotted lines in the dispersions of Cu 3 SbSe 4 and Cu 3 SbSe 3 represent the phonon dispersions at the DFT equilibrium volume and the expansion cell (+6%), respectively. The red, green and blue lines highlight TA, TA' and LA modes, respectively. The inset figures are the first Brillouin Zones of Cu 3 SbSe 4 and Cu 3 SbSe 3 with high symmetry points (red points) we considered in our phonon dispersion calculations. [9] Fig. 4 Theoretical lattice thermal conductivity of Cu 3 SbSe 4 (black line), Cu 3 SbSe 3 (red line) and Cu 12 Sb 4 Se 13 (blue line) using their corresponding Grüneisen parameters (γ), Debye temperatures (Θ) and phonon velocities (v) in Table 1. The black and red filled circles are experimental measurements [10] of Cu 3 SbSe 4 and Cu 3 SbSe 3, respectively. ( [9][12]) Fig. 5 Crystal structures of the low temperature phase SnSe-Pnma (a exp =11.49, b exp =4.44, c exp =4.135 ) and the high temperature phase SnSe-Cmcm (a exp =4.31, b exp =11.71, c exp =4.31 ). Sn=grey and Se=green. 21

24 Fig. 6 Phonon dispersions and corresponding acoustic Grüneisen parameters of SnSe. The inset figure is the first Brillouin Zone of SnSe with high symmetry points (red points) we considered in our phonon dispersion calculations. [13] Fig. 7 Theoretical lattice thermal conductivity of SnSe along the a (black line), b (red line) and c (blue line) directions. The black square, red circle and blue triangle lines are experimental measurements along the a, b and c directions, respectively. Fig. 8 Crystal structure of theoretically predicted Cu 12 Sb 4 Se 13 (I-43m, a=b=c=10.9 ) The red arrows on Sb atoms in Cu 12 Sb 4 Se 13 represent atomic displacements responsible for the anomalously high Grüneisen parameters of the transverse acoustic phonon branch in Figure 9. The structure is thermodynamically unstable at low temperatures. The Cu atom at the center of the CuSe 3 plane shifts along one of the blue arrows after phonon stabilizations. [12] Fig. 9 Phonon dispersions and corresponding acoustic Grüneisen parameters of the theoretically predicted Cu 12 Sb 4 Se 13. The inset figure is the first Brillouin Zone of Cu 12 Sb 4 Se 13 with high symmetry points (red points) we considered in our phonon dispersion calculations. [12] Table 1 Average transverse (TA/TA ) and longitudinal (LA) Grüneisen parameters ( TA / TA '/ LA γ ), Debye temperatures ( Θ TA / TA '/ LA ) and phonon velocities ( v TA / TA '/ LA ) in TA TA Cu 3 SbSe 4, Cu 3 SbSe 3 and Cu 12 Sb 4 Se 13. For SnSe, we give the three parameters averaged over the three acoustic branches. The Grüneisen parameters, Debye temperatures and phonon velocities are averaged by the weight of the high symmetry points (Fig. 3). [9][12][13] LA Prof. Yongsheng Zhang is currently working at Institute of Solid State Physics, CAS. He was awarded PhD degree in computational material science, Fritz-Haber-Institute, MPG, Germany in 2008, and conducted postdoctoral researches at Fritz-Haber-Institute and Northwestern University. He is the author or co-author of 30 peer-reviewed papers. His primary scientific interests cover using first-principles simulation tools to materials for alternative energies and sustainability. TA TA LA 22

25 Fig. 1 Schematic view of heat current (the red empty arrow, J) through a sample that has a temperature difference between its two ends [the hot (red) region with T H and the cold (blue) region with T L ]. The linear gradient of color from red to blue indicates the temperature gradient. Fig. 2 Crystal structures of Cu 3 SbSe 4 (I42m, a exp =b exp =5.66, c exp =11.28 ) and Cu 3 SbSe 3 (Pnma, a exp =7.99, b exp =10.61, c exp =6.84 ). The red arrows on Sb atoms in Cu 3 SbSe 3 represent atomic displacements responsible for the anomalously high Grüneisen parameters of the transverse acoustic phonon branch in Figure 3. 23

26 Cu=blue, Sb=brown, Se=green, Sn=gray. Fig. 3 Phonon dispersions and corresponding acoustic Grüneisen parameters of Cu 3 SbSe 4 and Cu 3 SbSe 3. The big circle lines and small dotted lines in the dispersions of Cu 3 SbSe 4 and Cu 3 SbSe 3 represent the phonon dispersions at the DFT equilibrium volume and the expansion cell (+6%), respectively. The red, green and blue lines highlight TA, TA' and LA modes, respectively. The inset figures are the first Brillouin Zones of Cu 3 SbSe 4 and Cu 3 SbSe 3 with high symmetry points (red points) we considered in our phonon dispersion calculations. [9] 24

27 Fig. 4 Theoretical lattice thermal conductivity of Cu 3 SbSe 4 (black line), Cu 3 SbSe 3 (red line) and Cu 12 Sb 4 Se 13 (blue line) using their corresponding Grüneisen parameters (γ), Debye temperatures (Θ) and phonon velocities (v) in Table 1. The black and red filled circles are experimental measurements [10] of Cu 3 SbSe 4 and Cu 3 SbSe 3, respectively. [9][12] Fig. 5 Crystal structures of the low temperature phase SnSe-Pnma (a exp =11.49, b exp =4.44, c exp =4.135 ) and the high temperature phase SnSe-Cmcm (a exp =4.31, b exp =11.71, c exp =4.31 ). Sn=grey and Se=green. 25

28 Fig. 6 Phonon dispersions and corresponding acoustic Grüneisen parameters of SnSe. The inset figure is the first Brillouin Zone of SnSe with high symmetry points (red points) we considered in our phonon dispersion calculations. [13] 26

29 Fig. 7 Theoretical lattice thermal conductivity of SnSe along the a (black line), b (red line) and c (blue line) directions. The black square, red circle and blue triangle lines are experimental measurements along the a, b and c directions, respectively. Fig. 8 Crystal structure of theoretically predicted Cu 12 Sb 4 Se 13 (I-43m, a=b=c=10.9 ) The red arrows on Sb atoms in Cu 12 Sb 4 Se 13 represent atomic displacements responsible for the anomalously high Grüneisen parameters of the transverse acoustic phonon branch in Figure 9. The structure is thermodynamically unstable at low temperatures. The Cu atom at the center of the CuSe 3 plane shifts along one of the blue arrows after phonon stabilizations. [12] 27

30 Fig. 9 Phonon dispersions and corresponding acoustic Grüneisen parameters of the theoretically predicted Cu 12 Sb 4 Se 13. The inset figure is the first Brillouin Zone of Cu 12 Sb 4 Se 13 with high symmetry points (red points) we considered in our phonon dispersion calculations. [12] 28