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J Materiomics 2 (2016) 237e247www.ceramsoc.com/en/ www.journals.elsevier.com/journal-of-materiomics/

First-principles DebyeeCallaway approach to lattice thermal conductivity

Yongsheng Zhang a,b,*a Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China

b University of Science and Technology of China, Hefei 230026, China

Received 4 May 2016; revised 21 June 2016; accepted 22 June 2016

Available online 28 June 2016

Abstract

Lattice thermal conductivity (k) is one of the most important thermoelectric parameters in determining the energy conversion efficiency ofthermoelectric materials. However, thermal conductivity calculations are time-consuming in solving the Boltzmann transport equation orperforming the molecular dynamic simulations. This paper reviews the first-principles DebyeeCallaway approach, in which theDebyeeCallaway model input parameters (i.e., the Debye temperature Q, the phonon velocity v and the Gruneisen parameter g) can be directlydetermined from the first-principles calculations of the vibrational properties of compounds within the quasiharmonic approximation. The first-principles DebyeeCallaway approach for three experimentally well studied compounds (i.e., Cu3SbSe4, Cu3SbSe3 and SnSe) was proposed. Thetheoretically calculated k using the approach is in agreement with the experimental value. Soft acoustic modes with low Q and v but large g havean effect on the determination of low lattice thermal conductivity thermoelectric materials. The first-principles DebyeeCallaway approach hasbeen proved as an effective tool to calculate k without the experimental inputs. The results calculated by this approach can be used to predict theperformance of low lattice thermal conductivity compounds.© 2016 The Chinese Ceramic Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: DFT; Phonon; DebyeeCallaway; Thermoelectric materials

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2382. Theoretical lattice thermal conductivity simulation approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

2.1. Molecular dynamic simulating lattice thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2382.2. Solving phonon Boltzmann transport equation for lattice thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3. First-principles DebyeeCallaway approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.1. DebyeeCallaway model for lattice thermal conductivity calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.2. Density functional theory and phonon calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

4. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.1. Validation of the first-principles DebyeeCallaway approach: Cu3SbSe4 and Cu3SbSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.2. High figure of merit SnSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.3. Theoretically predicted a novel thermoelectric material: Cu12Sb3Se13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

* Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China.

E-mail address: yshzhang@theory.issp.ac.cn.

Peer review under responsibility of The Chinese Ceramic Society.

http://dx.doi.org/10.1016/j.jmat.2016.06.004

2352-8478/© 2016 The Chinese Ceramic Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http

creativecommons.org/licenses/by-nc-nd/4.0/).

://

238 Y. Zhang / J Materiomics 2 (2016) 237e247

1. Introduction

Thermoelectric materials play vital roles in solving theimpending energy crisis due to directly converting heat intoelectricity [1,2]. The conversion efficiency is characterized bythe thermoelectric figure of merit ZT ¼ S2sT/k, where S, sand k are the Seebeck coefficient, the electrical conductivityand the thermal conductivity (consisting of the electronic (ke)contribution and the dominate lattice vibration (kL) contri-butions, respectively). In the ZT formula, increasing S ands ordecreasing k can enhance the figure of merit. However, theSeebeck coefficient (S ), the electrical conductivity (s) and theelectronic thermal conductivity (ke) are coupled with eachother through the carrier concentration and effective mass.Only the lattice thermal conductivity (kL) is an independentthermoelectric parameter. Thus, decreasing the lattice thermalconductivity is one of the most attractive aspects in thethermoelectric community. The most popular methodology isto introduce phonon scattering regions, such as by the nano-structured materials [3e7], which efficiently reduce thermalconductivity by enhancing phonon scattering. Unfortunately,the nanostructured materials scatter electrons as well anddecrease the electrical conductivity. For a long time, anintense anharmonicity has been proved to induce a low latticethermal conductivity. Thus, ordered crystal structures with anintense anharmonicity (such as AgSbTe2 [8], Cu3SbSe3[9,10], Cu12Sb4S13 [11], and Cu12Sb4Se13 [12]) coulddecrease the lattice thermal conductivity intrinsically withoutaffecting the electrical conductivity. Zhao et al. [13] discov-ered a high-performance thermoelectric material, SnSewith ZT of ~2.6 at ~930 K. The high thermoelectric figure ofmerit is due to the ultralow lattice thermal conductivityattribute to an intense anharmonicity. Therefore, it is impor-tant to efficiently evaluate the lattice thermal conductivity ofcompounds for seeking high performance thermoelectricmaterials.

The lattice thermal conductivity (kL or k for simplicity) isthe heat transfer ability through a sample that has a tempera-ture difference between its two ends (see Fig. 1). In principle,a direct method can be used to determine the thermal con-ductivity by simulating the temperature gradient from thenonequilibrium molecular dynamic (MD) simulations at agiven heat current [14,15]. Another MD simulated thermalconductivity is based on the GreeneKubo method using the

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 TH and the cold (blue) region with TL]. The linear gradient of color

from red to blue indicates the temperature gradient.

equilibrium MD simulations [16]. However, the MD simula-tions need a large unit cell to consider the finite size effect anda long simulation time to converge the autocorrelation func-tion. The lattice thermal conductivity can be also achieved bysolving the phonon Boltzmann transport equation [17], but it isnot trivial to solve the equation since a large amount of linearequations should be properly considered. Those linear co-efficients are dependent on the interatomic interactions, suchas the second- and third-order force constants. Density func-tional theory (DFT) is a useful tool to accurately describe theinteratomic interactions in many materials [18,19]. Thus,based on the DFT calculations, a full iterative solution to thephonon Boltzmann transport equation is developed, such asthe ShengBTE software package [20]. However, obtaining thethird-order force constants used to describe the anharmonic istime consuming. Even though the recently developedcompressive sensing lattice dynamics (CSLD) approach [21]significantly saves the calculation time, the computationalburden is still heavy. An alternative way to calculate the latticethermal conductivity is to use the DebyeeCallaway model[22] within the relaxation time approach. In the model, therelaxation time due to the phononephonon scattering can bewritten as a function of the Debye temperature (Q), phononvelocity (v) and the Gruneisen parameter (g). The simpleDebyeeCallaway model is computationally feasible method-ology used in the experimental community to estimate thelattice thermal conductivity based on the experimentallymeasured parameters (Q, v and g) [23e26]. However, thequestion is what if there has a compound without the experi-mentally measured parameters and how to obtain the latticethermal conductivity of the compound theoretically. In themini-review paper, we will present the first-principlesDebyeeCallaway approach to calculate the lattice thermalconductivity based on the first-principles determined param-eters (Q, v and g) within the quasiharmonic approximation.We applied the methodology to different compounds (i.e.,Cu3SbSe3, Cu3SbSe4 and SnSe), and the theoretically simu-lated lattice thermal conductivity is in agreement with theexperimental results. Furthermore, we predicted a new com-pound (i.e., Cu12Sb4Se13) with a ultralow lattice thermalconductivity. The first-principles DebyeeCallaway model hasbeen proved as a useful and feasible computational tool todetermine the lattice thermal conductivity of thermoelectricmaterials.

2. Theoretical lattice thermal conductivity simulationapproaches

2.1. Molecular dynamic simulating lattice thermalconductivity

The thermal conductivity (k) characterizes the ability ofheat conduct through a sample that has a temperature differ-ence between its two ends (the hot region with TH and the coldregion with TL, TH > TL) (see Fig. 1). Based on the Fourier law(or the law of heat conduction), the heat transfer can be writtenas,

239Y. Zhang / J Materiomics 2 (2016) 237e247

Ja ¼Xb

kabðVTÞb ð1Þ

where Ja, kab and VT are the a-component of heat current, thesecond-order tensor (ab) of thermal conductivity and thetemperature gradient, respectively.

Molecular dynamic (MD) method is a powerful tool forsimulating material properties at finite temperatures, such asthe transport properties [27,28]. The atomic positions andmomenta of a sample can be evolved by numerically inte-grating the Newtonian equations of motion. The heat current(J ) is generated by the approach developed by Ikeshoji [29]:J ¼ DEk=Dt, where DEk and Dt are the suitably chosen kineticenergy and the time step, respectively. The temperaturegradient (VT ) can be drawn from MD simulations by moni-toring the momenta of atoms along the heat flux. Thus, for aknown heat current (J ), the thermal conductivity (k) can beevaluated using Eq. (1). However, using the direct method tosimulate VT, we need the nonequilibrium, steady-state MDapproach, which is hard to implement in the first-principlescodes.

An alternative way to simulate the lattice thermal conduc-tivity using MD is based on the GreeneKubo method [30,31].From the fluctuationedissipation theorem, describing a rela-tionship between the fluctuations of the heat current vector (J )to the thermal conductivity, the second-order tensor of latticethermal conductivity is given by

kab ¼ 1

VkBT2

Z∞

0

dt⟨Jað0ÞJbðtÞ⟩eiut ð2Þ

J ¼ d

dt

Xi

Eiri ð3Þ

where a/b indicates the a/b direction and ⟨Jað0ÞJbðtÞ⟩ is theheat current autocorrelation function. Ei and ri are the totalenergy and atomic position of atom i. The equilibrium MDapproach, implanting in most DFT codes, is suitable for theautocorrelation function evaluation, and the correspondinglattice thermal conductivity can be estimated using Eq. (2).

Using MD to evaluate the lattice thermal conductivity isstraightforward [32,33], but the computational burden usingthe first-principles MD is heavy. To obtain the accurate latticethermal conductivity, the finite size effect in the MD simula-tions should be considered and a large simulation cell isnecessary, since a small cell cannot accurately reproduce allphonon scattering processes. Moreover, convergence of theautocorrelation function requires a large number of ensemblesor a long MD simulation time (~10 ns).

2.2. Solving phonon Boltzmann transport equation forlattice thermal conductivity

The lattice thermal conductivity can also be determined bythe phonon Boltzmann transport equation. The heat current(Ja) is a function of phonon distribution ( fl). At thermal

equilibrium, the phonon distribution obeys the BoseeEinsteinstatistics, f0(ul), ul the frequencies of the whole phonon (l)spectrum. A temperature gradient (VT ) (see Fig. 1) drives thedistribution function ( fl) from f0. Based on the Boltzmanntransport equation (BTE) [17], the phonon distribution in-cludes two parts, i.e., the diffusion due to VT and the phononscattering. In the steady state ððdfl=dtÞ ¼ 0Þ, we can write,

�vl$VT þ vflvT

����diffusion

þ vflvT

����scattering

¼ 0 ð4Þ

where vl is the phonon velocities. For the small temperaturegradient, fl of two- and three-phonon processes can beexpressed as [17,34e36].

fl ¼ f0ðulÞ � t0lðvl þDlÞ$VT df0dT

¼ f0ðulÞ �Fl$VTdf0dT

ð5Þ

where tl0 is the relaxation time calculated from the perturba-

tion theory. In the relaxation time approximation, Dl ¼ 0. Dl

and tl0 are the functions of three-phonon scattering rate and Dl

specifically depends on Fl. The anharmonic interatomic forceconstants (IFCs) are required to calculate the three-phononscattering rate [20,37,38]. The final lattice thermal conduc-tivity can be written as [20,37],

kab ¼ 1

kBT2UN

Xl

f0ðf0 þ 1ÞðZulÞ2valFbl ð6Þ

where U and N are the volume of the unit cell and the numberof q points in the Brillouin zone (BZ). Unfortunately, it istime-consuming to obtain the lattice thermal conductivity bysolving Eq. (5), which results from (1) the computationalexpensive third-order IFCs for determining the anharmonicIFCs; (2) solving a large amount linear equations. In theShengBTE code [20], Eq. (5) is solved iteratively starting withRTA, Dl ¼ 0. This method has predicted the lattice thermalconductivity of many compounds [21,38,39]. To decrease thecomputational burden of third-order IFCs, Zhou et al. [21]developed a compressive sensing lattice dynamics (CSLD)approach, which uses the compressive sensing method toeffectively select the physically important terms in the latticedynamics model and determine the anharmonic IFCs. Thisapproach can obtain the accurate lattice thermal conductivityof different systems, such as Si, NaCl and Cu12Sb4S13. Toaccurately determine the anharmonic IFCs, a very largesupercell is needed to consider long range interactions, espe-cially in the polarized systems.

3. First-principles DebyeeCallaway approach

3.1. DebyeeCallaway model for lattice thermalconductivity calculations

For the nonmetallic materials, such as semiconductors, theacoustic modes play an important role in the heat transfer bythree phonon interactions [40,41]. In the DebyeeCallawaymodel [23,42], the lattice thermal conductivity is contributed

240 Y. Zhang / J Materiomics 2 (2016) 237e247

by three acoustic modes, and k is written as a sum over onelongitudinal (kLA) and two transverse (kTA and kTA0) acousticphonon branches,

k¼ kLA þ kTA þ kTA’: ð7ÞIn the relaxation-time approximation, the thermal conduc-

tivity is a function of phonon scattering rates (1/tc) or therelaxation time (tc), which indicates the time that the phonondistribution is restored to the equilibrium distribution. Theheat transfer involves three phonon interactions, which givesrise to two kinds of phonon process, i.e., the Normal phononprocess and the Umklapp phonon process. Usually, theUmklapp process is considered to give rise to the thermalresistance due to the reversed phonon flux in directions.However, in the DebyeeCallaway model, the normal processis taken into account as well due to the ability to redistributingmomentum and energy among phonons. Therefore, in a per-fect order crystal (without impurities, defects, etc.), the totalphonon scattering rate [43,44] (1/tc) involves the contributionsfrom normal phonon scattering (1/tN) and the Umklapp pho-nonephonon scattering (1/tU), which means that the normaland the Umklapp phononephonon processes dominate thephonon scattering (1/tc ¼ 1/tN þ 1/tU). Thus, the partialconductivities ki (i corresponds to LA, TA or TA0 modes) aregiven by,

ki ¼ 1

3CiT

3

8>>>>>>><>>>>>>>:

Zqi=T

0

ticðxÞx4exðex � 1Þ2 dxþ

24Z qi=T

0

ticðxÞx4extiNðex � 1Þ2 dx

35

2

Z qi=T

0

ticðxÞx4extiNt

iUðex � 1Þ2 dx

ð8Þ

where Qi is the longitudinal (transverse) Debye temperature,x ¼ ðZu=kBTÞ, Ci ¼ ðk4B=2p2Z3viÞ, Z is the Planck constant,kB is the Boltzmann constant, u is the phonon frequency,and vi is the longitudinal or transverse acoustic phononvelocity.

We can write the normal and the Umklapp phonon scat-tering as [23,45],

1

tLAN ðxÞ ¼k3Bg

2LAV

MZ2v5LA

�kBZ

�2

x2T5

1

tTA=TA’N ðxÞ

¼ k4Bg2TA=TA’V

MZ3v5TA=TA’

kBZxT5 ð9Þ

1

tiUðxÞ¼ Zg2

Mv2iQi

�kBZ

�2

x2T3e�Qi=3T ð10Þ

where g, V and M are the Gruneisen parameter, the volume peratom and the average mass of an atom in the crystal, respec-tively. The anharmonicity scattering is proportional to g2,indicating a similar quadratic dependence on Gruneisenparameter as the forms in Refs. [23,40,46]. The Gruneisen

parameter characterizes the relationship between phonon fre-quency and volume change and can be defined as [47].

gi �V

ui

vui

vV: ð11Þ

From Eqs. (9) and (10), we can see the lattice thermalconductivity or the phonon scattering is a function of theDebye temperature (Q), phonon velocity (v) and Gruneisenparameter (g). The DebyeeCallaway model is such animportant model that it has been widely used in the experi-mental works [21,38,39]. Using the experimentally measuredDebye temperature (Q), phonon velocity (v) and Gruneisenparameter (g), researchers can simulate the lattice thermalconductivity and figure out the essential contributions to thelattice thermal conductivity by comparing the results with theexperimental measurements. However, such experimentalparameter dependence restricts the ability to screen for the lowlattice thermal conductivity materials from the theoreticalpoint of view. We notice that the three DebyeeCallawayneeded parameters (i.e., Q, v and g) can be drawn fromphonon dispersions. We will show how the quasiharmonicapproximation produces those parameters and simulates thecorresponding theoretical lattice thermal conductivity. For therelaxation time approximation (RTA) based DebyeeCallawayapproximation, it over-resists phonon scattering processes[37]. Thus, the RTA based methods always underestimate thelattice thermal conductivity (k).

3.2. Density functional theory and phonon calculations

We perform the DFT calculations using the Vienna AbInitio Simulation Package (VASP) code with the projectoraugmented wave (PAW) scheme [48], and the generalizedgradient approximation of Perdew, Burke and Ernzerhof [49](GGA-PBE) for the electronic exchange-correlation func-tional. The energy cutoff for the plane wave expansion is800 eV and 500 eV for the CueSbeSe system and SnSe,respectively. We treat 3d104s1, 5s25p3, 2s22p4 and 4d105s25p2

as valence electrons in Cu, Sb, Se and Sn atoms, respectively.The Brillouin zones are sampled by the MonkhorstePack [50]k-point meshes for all compounds with meshes chosen to givea roughly constant density of k-points (30 Å3) for all com-pounds. Atomic positions and unit cell vectors are relaxeduntil all the forces and components of the stress tensor arebelow 0.01 eV/Å and 0.2 kbar, respectively. The vibrationalproperties are calculated using the supercell (64, 54, 58 and112 atoms in Cu3SbSe4, Cu3SbSe3, Cu12Sb3Se13 and SnSesupercells, respectively) force constant method [51] by ATAT[52]. In the quasi-harmonic DFT phonon calculations, thesystem volume is isotropically expanded by few percents(þ6% and þ2% for CueSbeSe and SnSe, respectively) fromthe DFT relaxed volume. Once we have the theoreticallycalculated phonon dispersions, the three DebyeeCallawayneeded parameters (i.e., Q, v and g) can be estimated by usingQ ¼ uD/kB (uD is the largest acoustic frequency in each di-rection), the slope of the acoustic phonon dispersion around G

point and frequency change with volume (see Eq. (11)),

241Y. Zhang / J Materiomics 2 (2016) 237e247

respectively. There are two different ways to define the Debyetemperature using the whole phonon spectrum or the acousticbranches only [40]. Using the whole spectrum, we can definethe Debye temperature as the temperature, which the highest-frequency mode (not just the acoustic mode) is excited.However, in the DebyeeCallaway model, which only con-siders the acoustic contributions, the Debye temperaturedefined in the model is the highest frequency of the acousticbranches. Thus, the Debye temperature used in theDebyeeCallaway approach is usually (much) smaller than thatdefined using the whole spectrum, e.g. from the theoreticallycalculated phonon dispersions of PbTe [53], the Debye tem-perature from acoustic branches is 128 K, which is smallerthan that from the whole spectrum (i.e., 160 K). Using thefirst-principles determined parameters for the DebyeeCall-away approach, we can calculate the lattice thermal conduc-tivity without experimental inputs.

4. Results and discussion

4.1. Validation of the first-principles DebyeeCallawayapproach: Cu3SbSe4 and Cu3SbSe4

Fig. 2. Crystal structures of Cu3SbSe4 (I42m, aexp ¼ bexp ¼ 5.66 Å,cexp ¼ 11.28 Å) and Cu3SbSe3 (Pnma, aexp ¼ 7.99 Å, bexp ¼ 10.61 Å,cexp ¼ 6.84 Å). The red arrows on Sb atoms in Cu3SbSe3 represent atomic

displacements responsible for the anomalously high Gruneisen parameters of

the transverse acoustic phonon branch in Fig. 3. Cu ¼ blue, Sb ¼ brown,

Se ¼ green, Sn ¼ gray.

We first apply the first-principles DebyeeCallaway modelto two experimentally well studied two compounds, i.e.,Cu3SbSe4 and Cu3SbSe3 [10]. The two compounds have thesimilar stoichiometry but different geometries (the zincblende-based tetragonal Cu3SbSe4 and orthorhombic Cu3SbSe3 inFig.e 2). Even though the two compounds share the same el-ements and the similar stoichiometry, experimental measure-ments (see Fig. 4) show that Cu3SbSe4 does exhibit a typicalthermal conductivity behavior (k decreases with increasingtemperature), whereas Cu3SbSe3 exhibits anomalously a lowand nearly temperature independent lattice thermal conduc-tivity. Understanding the distinct lattice thermal conductivitybehavior of the two compounds is beneficial for the future lowlattice thermal conductivity material search.

Fig. 3 shows the phonon dispersions of Cu3SbSe4 andCu3SbSe3. Since the acoustic modes play an important rolein the thermal conductivity, we highlight these branches withdifferent colors in the plot. From the phonon dispersions, wecan find that the boundary frequency of Cu3SbSe3 is~30 cm�1, which is smaller than that of Cu3SbSe4, i.e.,~60 cm�1. The low frequency indicates that Cu3SbSe3has softer interatomic bonds and lower acoustic modes thanCu3SbSe4. The corresponding Debye temperature andphonon velocity (see Table 1) can be calculated based on thephonon dispersions. Cu3SbSe3 has lower Debye temperaturesand phonon velocities than Cu3SbSe4. To estimate the latticethermal conductivity using the DebyeeCallaway model, westill need the Gruneisen parameters. We carry out the quasi-harmonic phonon calculations to obtain the phonon disper-sions at a different volume and the Gruneisen dispersion canbe calculated using Eq. (11) (see Fig. 3). Most of all theGruneisen parameters of Cu3SbSe4 are negative. The nega-tive Gruneisen parameter is coming from the tetrahedral localgeometry in the Cu3SbSe4 crystal structure, which is a

common behavior in many tetrahedral semiconductors, suchas Si, Ge and GaAs [54,55]. Xu et al. [55] described theGruneisen parameter of diamond structure using a simplenearest-neighbor model, which is suitable for the zinc-blendstructure as well. In the model, the Gruneisen parameter canbe written as g ¼ 2

3 � ð4CB0d0=ffiffiffi3

pmu2

0Þ, where C is thenumber of bonds bent by the phonon distortion, B0 is the bulkmodulus, d0 is the equilibrium bond length and u0 is therestoring force on the atoms. The first and second parts of theequation are the contributions from the noncentral angularpart and central force parts, respectively. The sign of theGruneisen parameter is the competition of the two parts. ForSi, Ge, GaAs and the zinc-blend Cu3SbSe4, the second parthas a dominant effect.

The average Gruneisen parameter (g) of each acousticdispersion (see Table 1) is calculated by the method described[23]: g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<g2 >

p. Cu3SbSe3 has much larger Gruneisen

parameters than Cu3SbSe4, especially for the transversemodels. Since the Gruneisen parameter characterizes theanharmonicity of a compound, the large Gruneisen parameterin Cu3SbSe3 indicates more intense anharmonicity thanCu3SbSe4. The largest Gruneisen parameters within theCu3SbSe3 BZ are along the GeT direction, ~7 at the T point.From analyzing eigenvectors of the mode, we find that thismode involves in-phase Sb vibrations along the [100] direction(see Fig. 2). For the Cu3SbSe3 stoichiometry, the nominalvalence state of Sb is 3þ, which indicates two additional non-bonding or lone-pair electrons at the Sb sites. The lone-pairelectrons just sit along the Sb vibrational direction ([100]).The electrostatic repulsion between the lone pair electrons andthe bonding charges in SbeSe bonds weaken the restoringforce for this mode, which softens phonon frequencies andpossesses low thermal conductivity.

Using the first-principles determined DebyeeCallawayinput parameters (i.e., Q, v and g), we calculate the lattice

Fig. 3. Phonon dispersions and corresponding acoustic Gruneisen parameters of Cu3SbSe4 and Cu3SbSe3. The big circle lines and small dotted lines in the

dispersions of Cu3SbSe4 and Cu3SbSe3 represent the phonon dispersions at the DFT equilibrium volume and the expansion cell (þ6%), respectively. The red, green

and blue lines highlight TA, TA0 and LA modes, respectively. The inset figures are the first Brillouin Zones of Cu3SbSe4 and Cu3SbSe3 with high symmetry points

(red points) we considered in our phonon dispersion calculations [9].

Fig. 4. Theoretical lattice thermal conductivity of Cu3SbSe4 (black line),

Cu3SbSe3 (red line) and Cu12Sb4Se13 (blue line) using their corresponding

Gruneisen parameters (g), Debye temperatures (Q) and phonon velocities (v)

in Table 1. The black and red filled circles are experimental measurements [10]

of Cu3SbSe4 and Cu3SbSe3, respectively [9,12].

Table 1

Average transverse (TA/TA0) and longitudinal (LA) Gruneisen parameters (gTA=TA

Cu3SbSe4, Cu3SbSe3 and Cu12Sb4Se13. For SnSe, we give the three parameters a

temperatures and phonon velocities are averaged by the weight of the high symm

System gTA gTA’ gLA QTA (K) QTA

Cu3SbSe4 1.27 1.14 1.26 60 65

Cu3SbSe3 3.03 2.92 1.26 33 34

Cu12Sb4Se13 9.48 8.50 4.65 30 30

System ga gb gc Qa (K) Qb (

SnSe 4.1 2.1 2.3 24 65

242 Y. Zhang / J Materiomics 2 (2016) 237e247

thermal conductivity of the two compounds. We can see areasonable agreement between the theoretical calculation andexperimental measurement (see Fig. 4). The deviation betweenthe experimental data and the calculated curves is approxi-mately 20%, which is quite satisfactory given the approxi-mations inherent in the DebyeeCallaway formalism [40]. Theabove work validates that the first-principles DebyeeCallawayapproach is a useful theoretical tool in the investigation of thelattice thermal conductivity of materials.

4.2. High figure of merit SnSe

After validating the first-principles DebyeeCallawayapproach, we could apply the approach to other thermoelectricmaterials for lattice thermal conductivity calculations. Withthe combination of experimental measurements and theoret-ical calculations, we discovered a high figure of merit(ZT ¼ ~2.6 at ~930 K) compound, SnSe (see Fig. 5) [13]. Thehigh ZT value is due to the ultralow lattice thermal

0=LA), Debye temperatures (QTA=TA0=LA) and phonon velocities (vTA=TA0=LA ) in

veraged over the three acoustic branches. The Gruneisen parameters, Debye

etry points [9,12,13].

0 (K) QLA (K) vTA (m/s) vTA0 (m/s) vLA (m/s)

78 1485 1699 3643

36 1072 1344 3014

39 1256 1398 2961

K) Qc (K) va (m/s) vb (m/s) vc (m/s)

58 1422 2002 1814

Fig. 5. Crystal structures of the low temperature phase SnSeePnma

(aexp ¼ 11.49 Å, bexp ¼ 4.44 Å, cexp ¼ 4.135 Å) and the high temperature

phase SnSeeCmcm (aexp ¼ 4.31 Å, bexp ¼ 11.71 Å, cexp ¼ 4.31 Å). Sn ¼ gray

and Se ¼ green.

Fig. 6. Phonon dispersions and corresponding acoustic Gruneisen parameters

of SnSe. The inset figure is the first Brillouin Zone of SnSe with high sym-

metry 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 di-

rections, respectively.

243Y. Zhang / J Materiomics 2 (2016) 237e247

conductivity along the three crystal directions (a, b and c). Theexperimentally measured thermal conductivity trend iska < kc < kb (see Fig. 7). However, the physical reason behindthe ultralow lattice thermal conductivity should be scrutinized.

SnSe undergoes phase transition from Pnma to Cmcmat ~750e800 K and the high ZT occurs at the high-temperature phase (i.e., SnSeeCmcm, in Fig. 5). Usually,

the lattice thermal conductivity decreases with increasingtemperature, roughly showing kf 1/T behavior [40,41,56]. Itis not surprising that thermoelectric materials have a low lat-tice thermal conductivity at high temperatures (the ultralowlattice thermal conductivity of SnSeCmcm at ~900 K).However, compounds (i.e., the low-temperature SnSeePnmaphase, in Fig. 5) with a low k (i.e., <1 W/m K) even at roomtemperature (300 K) should be carefully checked. Thus, eventhough the SnSeeCmcm phase shows a ultralow lattice ther-mal conductivity, we only consider the lattice thermal con-ductivity calculations of the low-temperature phase (i.e.,SnSeePnma). From the theoretically calculated phonon dis-persions and the Gruneisen 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 Gruneisen parametersare large (see Table 1), which suggests the strong anharmo-nicity along all three directions. Moreover, the acoustic modesalong the x direction are even softer than those along the y andz directions. The averaged Gruneisen parameters along the xdirection are larger than those along the other two directions(see Table 1), which indicates that the x or a direction has themost intense anharmonicity and lowest lattice thermalconductivity.

The large Gruneisen parameters or intense anharmonicityof SnSe is a consequence of the soft bonding in its specialcrystal structure (see Fig. 5): a zigzag like geometry in the becplane and a layered slab along the a direction. Such geometrypatterns provide a good stress buffer to scattering phonontransport. Moreover, using the inelastic neutron scattering andfirst-principles calculations, Li et al. [57] confirmed this strongphonon anisotropic and ionic-potential anharmonicity, whichis due to the instability of SneSe bonds formed by the long-range resonant interactions between Sn 5s lone pair elec-trons and Se-4p orbitals.

Using the first-principles determined Gruneisen parame-ters, Debye temperatures and phonon velocities, we

Fig. 8. Crystal structure of theoretically predicted Cu12Sb4Se13 (I-43m,

a ¼ b ¼ c ¼ 10.9 Å) The red arrows on Sb atoms in Cu12Sb4Se13 represent

atomic displacements responsible for the anomalously high Gruneisen pa-

rameters of the transverse acoustic phonon branch in Fig. 9. The structure is

thermodynamically unstable at low temperatures. The Cu atom at the center of

the CuSe3 plane shifts along one of the blue arrows after phonon stabilizations

[12].

244 Y. Zhang / J Materiomics 2 (2016) 237e247

calculated the lattice thermal conductivities along three di-rections (see Fig. 7). We find that the calculated latticethermal conductivities are in reasonable agreement with theexperimental results. The a axis has the lowest lattice thermalconductivity, the theoretically predicted thermal conductivitytrend is ka < kc < kb and the lattice thermal conductivityalong b and c are similar. The largest discrepancy betweenthe theoretical curve and experimental data is along the adirection, which might be due to the van der Waals (vdW)interactions between SneSe slabs. Such weak interactionscannot be captured in the traditional DFT-PBE calculations.Clearly, the lack of extra binding interactions in the calcu-lations could further enforce anharmonicity and lower thelattice thermal conductivity. In addition, Guo et al. [58]calculated the lattice thermal conductivity based on thephonon BTE considering all modes and vdW. Their theo-retically calculated k is greater than those from the experi-mental measurements, which could possibly result from (a)the value of Lorenz number 1.5 � 10�8V2 K2, which can wellvary from 1 to 2.4 � 10�8V2 K2; (b) the values of the thermaldiffusivity, which depends on the details of a fit to time-dependent reflectivity curves (the instrumental probablyspan 5% or more); (c) the sample thickness, the homogeneityof that thickness, and the sample density, and (d) it is difficultto measure precisely along one crystallographic axis. Sincethe lattice thermal conductivity of SnSe is rather low and itsphonon mean free path is so short, it is nearly impossible tofurther decrease its lattice thermal conductivity by nano-structuring. Instead of decreasing the lattice thermal con-ductivity, Zhao et al. [59] enhanced the ZT values in a lowtemperature range (i.e., 300e773 K) to 0.7e2.0 by boostingthe power of factor in hole-doped SnSe crystals.

4.3. Theoretically predicted a novel thermoelectricmaterial: Cu12Sb3Se13

Fig. 9. Phonon dispersions and corresponding acoustic Gruneisen parameters

of the theoretically predicted Cu12Sb4Se13. The inset figure is the first Bril-

louin Zone of Cu12Sb4Se13 with high symmetry points (red points) we

considered in our phonon dispersion calculations [12].

The three compounds above (i.e., Cu3SbSe4, Cu3SbSe3 andSnSe) have been well investigated experimentally. We nextwant to use the first-principles DebyeeCallaway approach topredict the lattice thermal conductivity of experimentally notdiscovered compounds [12]. Ternary CueSbeSe compoundscontain Earth-abundant and nontoxic elements and they areattractive materials for numerous applications, such as ther-moelectric materials, optical materials and 2D electronicmaterials. To search for novel compounds in the CueSbeSesystem, we use the cluster expansion (CE) method and pro-totype structures from the CueSbeS system to predict newpossible CueSbeSe ternary phases (details in Ref. [12]). Ahigh-temperature stable compound, cubic Cu12Sb4Se13 withthe I-43m space group (see Fig. 8), is discovered theoretically.From phonon calculations (see Fig. 9), we notice that thecubic 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 inthe experimentally determined Cu13Sb4S13 compound. Theboundary acoustic frequencies of Cu12Sb4Se13 are as low asthose of Cu3SbSe3 (see Fig. 3). The theoretical Debye

temperature (Q), phonon velocity (v) and Gruneisen param-eter of Cu12Sb4Se13 are calculated based on the phonon dis-persions. We can see that the compound has the largeGruneisen parameters (especially for the transverse modes) inthe CueSbeSe phases (see Table 1), which indicates stronganharmonicity and low thermal conductivity. From the Gru-neisen dispersions of Cu12Sb4Se13, we can see that the

245Y. Zhang / J Materiomics 2 (2016) 237e247

acoustic modes along high symmetry directions (ZeG andTeG) in the BZ have very large Gruneisen parameters (>10),indicating that the branches are very anharmonic and interactstrongly with other acoustic phonons. The vibration of thesemodes involves Sb movement away from SbeSe bonds, suchas along the [110] direction (see Fig. 8), roughly along whichthe Sb lone-pair electrons are also oriented. As we discussedin the Cu3SbSe3 case, the interactions between the lone pairand the three SbeSe bonds are sensitive to the SbeSe bondangles and result in large TA Gruneisen parameters inCu12Sb4Se13.

We therefore use the first-principles determined parametersof Cu12Sb4Se13 (see Table 1) to parameterize theDebyeeCallaway model and calculate its lattice thermalconductivity. The resulting thermal conductivity ofCu12Sb4Se13 is extremely low (see Fig. 4), which is lower thanthat of all other CueSbeSe compounds. The theoreticallycalculated lattice thermal conductivity of Cu12Sb4Se13 at300 K is ~0.1 W m�1 K�1. The DebyeeCallaway approxi-mation based on the RTA underestimates the lattice thermalconductivity [37]. Considering the smallest error bar(~0.28 W m�1 K�1) in the Cu3SbSe4 and Cu3SbSe3 calcula-tions, we can estimate that the lattice thermal conductivity ofCu12Sb4Se13 at 300 K should be less than ~0.38 W m�1 K�1,which is still an extremely small value. Also, the latticethermal conductivity trend of the three compounds (i.e.,Cu3SbSe4, Cu3SbSe3 and Cu12Sb4Se13) determined by theDebyeeCallaway approach should be reasonable. We can findthat the theoretically calculated lattice thermal conductivity ofCu12Sb4Se13 is even lower than that of Cu3SbSe3 (see Fig. 4).

In addition, the electronic structures of the compound areinvestigated by plotting its band structures, whichexhibit favorable band features for good thermoelectric per-formance, such as valence band degeneracy, high effectivemasses, etc. The good electric structures and low thermalconductivity indicate that Cu12Sb4Se13 should be a promisingthermoelectric material [13].

5. Conclusions

We developed a first-principles DebyeeCallaway approach,in which the DebyeeCallaway model needed parameters (i.e.,the Debye temperature Q, the phonon velocity v and theGruneisen parameter g) could be directly determined from thefirst-principles calculations of the vibrational properties ofcompounds within the quasiharmonic approximation. The softacoustic modes with low Q and v but large g played vital rolesin the determination of low lattice thermal conductivity ofthermoelectric materials. The lone pair electrons in the com-pounds had the electrostatic repulsion with the bondingcharges, which weakened the restoring force and softened thecorresponding phonon modes. Thus, the lone pair electrons hadthe essential contributions to the intense anharmonicity andlarge Gruneisen parameters. Searching compounds with

lone pair electrons could discover intrinsically the ultralowlattice thermal conductivity candidates. From the crystalstructures analysis of Cu3SbSe3, SnSe and Cu12Sb3Se13, thelow lattice thermal conductivity candidates could possess (1)nonsymmetrical local binding environment; and (2) unex-pected crystal structures, such as layered, isotropic or zigzaggeometries.

For the experimentally well studied compounds (i.e.,Cu3SbSe4, Cu3SbSe3 and SnSe), the theoretically calculatedlattice thermal conductivities using the first-principlesDebyeeCallaway approach were in reasonable agreementwith the experimental results. The deviation between thetheoretical data and experimental results was due to the op-tical mode contributions. The DebyeeCallaway model didnot take into account optical mode. However, from thephonon dispersions, especially for SnSe, the optical modeshad low frequencies and interact with the acoustic modes.These optical modes could contribute 20%e50% contribu-tions to the heat transfer as discussed by Slack [40]. To obtainbetter agreement, the optical contributions should be takeninto account in the future works. For an experimentally notdiscovered or synthesized compound (Cu12Sb3Se13), weestimated its lattice thermal conductivity and predicted it as apromising thermoelectric material. The first-principlesDebyeeCallaway approach was proved as an effective toolto calculate the lattice thermal conductivity without experi-mental inputs and without too much of computational burdenas in solving the phonon Boltzmann transport equation orrunning MD simulations. The results from this approachcould not only be compared to the experimental measure-ments for explanations, but also be used to predict the lowlattice thermal conductivity compounds prior to theexperiments.

Acknowledgments

The author acknowledges the financial support from theNational Natural Science Foundation of China (No.11474283), the Major/Innovative Program of DevelopmentFoundation of Hefei Center for Physical Science and Tech-nology (No. 2014FXCX001) and one Hundred Person Projectof the Chinese Academy of Sciences (No. Y54N251241).

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Prof. Yongsheng Zhang is currently working at

Institute of Solid State Physics, CAS. He was awar-

ded 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.