Anharmonic phonon effects on linear thermal expansion of trigonal selenide and Chee Kwan Gan1, a) and Ching Hua Lee1 Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632 (Dated: 15 March 2018 (b: CMS version)) We adopted and extended an efficient Gr¨uneisenformalism to study the phonon anharmonicity and linear thermal expansion coefficients (TECs) of trigonal bismuth selenide (Bi2Se3) and antimony telluride (Sb2Te3). Anharmonicity of the systems is studied via extensive calculation of Gr¨uneisenparameters that exploit symmetry-preserving deformations. Consistent with experimental findings, a large anisotropy between the TECs in the a and c directions is found. The larger anharmonicity inherent in Sb2Te3, as compared to Bi2Se3 is offset by the volumetric effect, resulting in comparable temperature dependence of their linear TECs. The Debye temperatures deduced from our first-principles data also agree very well with the existing tabulated values. The highly efficient methodology developed in this work, applied for the first time to study the linear TECs of two trigonal thermoelectric systems, opens up exciting opportunities to address the anharmonic effects in other thermoelectrics and other low-symmetry materials.

PACS numbers: 63.20.D-, 65.40.-b, 65.40.De Keywords: Bismuth selenide, antimony telluride, phonon calculations, thermal expansion, topological insula- tors, thermoelectric materials, Gr¨uneisenparameter, Debye temperature

I. INTRODUCTION per, we adopted and extended an efficient Gr¨uneisenap- 13–16 proach by to study Bi2Se3 and Sb2Te3 with a min- imal set of relatively expensive (compared to standard Bismuth selenide (Bi2Se3) and antimony telluride density-functional total-energy calculations) phonon cal- (Sb2Te3) belong to a large family of metal dichalco- genides that hosts excellent thermoelectric materials1 culations. Through it, we were able to perform a sys- and topological insulators2–4. As paradigmatic examples tematic investigation on the anharmonicity of these two materials with relatively low symmetry, and make consis- of materials that simultaneously host enigmatic 3D Z2 topological states, these two materials have been exten- tent comparisons between some of their important ther- sively studied experimentally5–8 and theoretically9,10 due mal properties such as linear TECs. to their technological importance and fundamental inter- est. The linear and volumetric thermal expansion coef- II. METHODOLOGY ficients (TECs) of Bi2Se3 and Sb2Te3 have been deter- mined experimentally5 where a high anisotropy is found between linear TECs in the a and c directions for these The trigonal Bi2Se3 and Sb2Te3 belong to the sym- two systems. morphic space group R3m (No. 166). There are three For engineering applications of these materials, good inequivalent atoms: an Sb atom occupies 6c(0, 0, µ) site, device performance hinges on a solid understanding of a Te atom occupies 3a(0, 0, 0) site, and a second Te thermal expansion behavior because phonon dynamics is atom occupies 6c(0, 0, ν) site. This gives a total of 15 intimately affected by temperature-induced defor- atoms in the conventional hexagonal unit cell. How- mations. As found in11,12, knowledge of the linear ther- ever, in order to reduce the amount of computing time, mal expansion and phonon anharmonicity can be cap- we use a primitive rhombohedral cell of five atoms that tured through phonon frequency lineshifts through the is three times smaller than the conventional hexago- Gr¨uneisenparameters. Such calculations of the ther- nal cell. The rhombohedral cell length ar and angle mal expansion properties are commonly performed using αr can be deduced from the hexagonal lattice param- a quasi-harmonic approximation (QHA), which involves eters ah and ch, and vice√ versa. The relations are: many phonon calculations on many possible combina- arXiv:1803.05693v1 [cond-mat.mtrl-sci] 15 Mar 2018 ah = 2ar sin(αr/2), ch = ar 3 + 6 cos αr. On the other p 2 2 2 tions of lattice parameters. But due to its complexity, hand, ar = (ah/3) η + 3, cos αr = (2η − 3)/(2η + 6) the QHA is efficient only when dealing with highly sym- where η = ch/ah. metric systems such as cubic lattice structures. However, We perform density-functional theory (DFT) calcu- many technological important crystals are not cubic, and lations within the local density approximation as im- other more efficient approaches are necessary. In this pa- plemented in the plane-wave basis suite QUANTUM ESPRESSO17 (QE), with wavefunction and density cut- offs of 60 and 480 Rydberg, respectively. A 10 × 10 × 10 Monkhorst-Pack mesh is used for the k-point sampling. a)Electronic mail: [email protected] The pseudopotentials for Bi, Se, Sb, and Te are gen- 2 erated using the pslibrary.1.0.0 that is based on the plicit dependence of α’s and Ii’s on temperature T is 18 Rappe-Rabe-Kaxiras-Joannopoulos scheme. We relax suppressed in Eq. 1. We will discuss more about Ii later. the structures fully before carrying out the phonon calcu- The Cij are the elastic constants. The linear TECs are lations. For Bi2Se3, we obtain (a, c) = (4.110, 27.900) A.˚ inversely proportional to the volume Ω of primitive cell 19 This is in good agreement with the experimental re- at equilibrium. We note that Bi2Se3 has a smaller Ω than 3 3 sult of (4.143, 28.636) A.˚ For Sb2Te3, we obtain (a, c) = Sb2Te3 (i.e., 136.05 A˚ vs 152.87 A˚ ). From a series of (4.244, 29.399) A,˚ which is in good agreement with the symmetry-preserving deformations with strain parame- experimental5 result of (4.242, 30.191) A.˚ ters ranging from −0.01 to 0.01, the elastic constants According to the Gr¨uneisen approach13–16,20,21, the are deduced from parabolic fits to the energy-strain22 linear TECs in the a and c directions, denoted as αa(T ) curves. For Bi2Se3, C11 + C12 = 121.74, C13 = 30.18, and αc(T ), respectively, are given by and C33 = 54.45 GPa. For Sb2Te3, C11 + C12 = 110.73, C = 32.16, and C = 60.97 GPa. We note that the       13 33 αa 1 C33 −C13 I1 expression for TECs in Eq. 1 is identical to the hexagonal = (1) 16 αc ΩD −2C13 [C11 + C12] I3 case since a trigonal cell can be perfectly embedded in a hexagonal cell. 2 where D = (C11 + C12)C33 − 2C13. For clarity, the ex-

8 (a) (b) Bi2Se3 Bi2Se3 4 γ 0 8 (c) Sb2Te3 (d) Sb2Te3 4 γ 0

-4 Γ Γ B1 L X/Q FP1 g(γ) B1 L X/Q FP1 g(γ)

FIG. 1. The Gr¨uneisenparameters for Bi2Se3 due to (a) an xy biaxial strain and (b) a z uniaxial strain. The corresponding results for Sb2Te3 are shown in (c) and (d), respectively. The label and coordinates of the k points are taken from Ref. [23]. The densities of Gr¨uneisenparameters, g(γ), shown on the right side of each figure are obtained with a sampling of 30 × 30 × 30 k points.

Central to our Gr¨uneisenformalism is the tempera- further define the the density of phonon states weighted ture dependent heat capacity weighted by the Gr¨uneisen by Gr¨uneisenparameter, Γi(ν), given by parameter, Ω X Z Γ (ν) = dk δ(ν − ν )γ (3) Ω X Z i 3 λk i,λk I (T ) = dk γ c(ν ,T ) (2) (2π) BZ i (2π)3 i,λk λk λ λ BZ R νmax such that Ii(T ) = dν Γi(ν)c(ν, T ). νmin (νmax) νmin where the integral is over the first Brillouin zone is the minimum (maximum) frequency in the phonon −1 −1 (BZ). Here γi,λk = −n ν ∂νλk/∂i are the mode- λk spectrum. The functions Γi(ν) provide a deeper under- dependent and deformation-dependent Gr¨uneisen pa- standing about Ii(T ) since it isolates the anharmonicity- rameters, which measures the rate of change of the dependent contributions from the harmonic specific phonon frequency νλk (of mode index λ and wavevec- heat capacity c(ν, T ), which has a well-known univer- tor k) with respect to the strain parameter i. n equals 15 sal form . Finally we note that Ii(T ) is related to to 1 (2) for a uniaxial (biaxial) strain. The specific 24 the macroscopic Gr¨uneisen parameters, γm,i(T ) by heat contributed by a phonon mode of frequency ν is the relation γ (T ) = I (T )/C (T ) where C (T ) = 2 m,i i v v c(ν, T ) = kB(r/ sinh r) , r = hν/2kBT . kB and h are the Ω P R dk c(ν ,T ) is the specific heat at constant Boltzmann and Planck constants, respectively. To keep (2π)3 λ BZ λk track of the origin of anharmonicity more precisely, we volume. Therefore γm,i(T ) can be interpreted as an av- erage over Gr¨uneisenparameters weighted by the mode 3 dependent heat capacity. Its physical meaning is clear- 0.4 ) (a) Bi Se -1 2 3 Debye est in the large-T limit, where γm,i reduces to a simple DFT arithmetic average of all Gr¨uneisenparameters in the BZ 0.2 since the heat capacities for each mode approaches unity ρ(ν) (1/cm (in units of kB) in this limit. 0 To calculate the Gr¨uneisenparameters resulted from a 0.6 Bi Se ) (b) 2 3 21 Γ1 deformation of the crystal due to an xy biaxial strain, -1 Γ 0.4 3 a strain-parameter set of (1, 1, 0, 0, 0, 0) (in Voigt’s no- tation) is used, where the rhombohedral cell has a new 0.2 0 p 2 2 2 Γ(ν) (1/cm lattice parameters ar = ar [η + 3(1 + 1) ]/(η + 3) 0 2 2 2 2 0 and cos αr = [2η − 3(1 + 1) ]/[2η + 6(1 + 1) ]. For 0 50 100 150 200 -1 a z uniaxial strain, we use the strain-parameter set of ν (cm ) 0 0.4 (0, 0, 3, 0, 0, 0), where the rhombohedral cell has ar =

) Sb Te p 2 2 2 0 2 (c) 2 3 -1 0.3 ar [η (1 + 3) + 3]/(η + 3) and cos αr = [2η (1 + Debye 2 2 2 3) − 3]/[2η (1 + 3) + 6]. Importantly, these two defor- 0.2 DFT mations preserve the space group of the crystal so that we 0.1 can use the QE symmetry switch of IBRAV=5. We use ρ(ν) (1/cm 0 small strains of e1 = ±0.25 % and e3 = ±0.5 % for the 0.8 calculation Gr¨uneisenparameters using finite-differences. ) (d) Sb2Te3 For phonon calculations under the QE implementation, -1 0.6 Γ1 Γ we use a q mesh of 5 × 5 × 5, which is equivalent to a 0.4 3 5 × 5 × 5 supercell25 for the determination of interatomic

Γ(ν) (1/cm 0.2 force constants. 0 0 50 100 150 200 -1 ν (cm ) III. RESULTS FIG. 2. (a) The phonon density of states (from DFT and De- The Gr¨uneisen parameters along the representative bye approximation), ρ(ν). (b) The phonon density of states weighted by Gr¨uneisenparameter, Γi(ν), due to an xy biaxial high-symmetry directions for Bi2Se3 and Sb2Te3 due to strain and a z uniaxial strain for the Bi2Se3. The correspond- an xy biaxial strain are shown in Fig. 1(a) and (c), respec- ing results are shown in (c) and (d) for Sb2Te3. tively. Similarly, the results due to a z uniaxial strain are shown in Fig. 1(b) and (d), respectively. The densities of Gr¨uneisenparameters (displayed on the right side of each −1 the xy biaxial strain of Sb2Te3 at about 38 cm . subfigure) show that most Gr¨uneisen parameters range Since the temperature dependence of TECs is intri- between 0 to 4, with a dominant peak centered around cately related to that of the heat capacity at constant 1. There is a small population of negative Gr¨uneisen volume, which is typically characterized by the Debye 14 parameters, which may lead to negative linear TECs . temperature, here we suggest a simple approach to ex- For the above discussion, the temperature dependence tract the effective Debye temperature. From phonon cal- of the TECs hinges on the integrated quantities Ii(T ), culations based on density-functional perturbation the- which could be calculated from a direct summation ory (DFPT), we could obtain very accurate phonon den- over BZ or through an integration over frequency ν of sity of states and hence heat capacity as a function of the product of heat capacity c(ν, T ) and the density of temperature [see Fig. 3(a) and (d)]. We propose to fit phonon states weighted by Gr¨uneisenparameter, Γi(ν). the DFT heat capacity data with that obtained from a We find the second approach is more illuminating. The Debye model approximation by minimizing the absolute density of phonon states for Bi Se in Fig. 2(a) shows 2 3 error as a function of a cutoff frequency νc, there is a phonon gap of 89 to 94 cm−1 which is due mainly to a large mass difference between Bi (atomic 1 Z ∞ d(ν ) = dT [CD(ν ,T ) − CDFT (T )]2 (4) mass of 208.98) and Se (atomic mass of 78.97). Such c 2 v c v (3NkB) 0 phonon gap is not seen for Sb2Te3 [shown in 2(c)] since Sb (atomic mass of 121.76) and Te (atomic mass of 127.60) where the integrand is the square of the difference be- are consecutive elements in the periodic table. Γi(ν) tween of the heat capacities from DFT and from the for Bi2Se3 are shown in Fig. 2(b) for both xy biaxial Debye model. N = 5 is the number of atoms in the and z uniaxial strains, where large Gr¨uneisen parameters primitive cell in current systems. According to this are associated with frequencies of about 100 cm−1. For scheme, the Debye temperature will be naturally defined Sb2Te3, large Gr¨uneisenparameters are associated with as θD = hνc/kB. The heat capacity evaluated according −1 D R νc frequencies of about 90 cm .Γi(ν) shown in Fig. 2(b) to the Debye model is Cv (νc,T ) = 0 ρD(ν)c(ν, T )dν and (d) also indicate that effect of negative Gr¨uneisen and the density of phonon states under the Debye ap- 2 parameters are negligible for all frequencies except for proximation is ρD(ν) = Aν for 0 ≤ ν ≤ νc and zero 4

3 20

B (a) Bi Se otherwise (A = 9N/νc ). We note that a similar scheme 2 3 for finding the Debye temperature as a function of tem- 10 (T)/k DFT v perature has been proposed in Ref. [26]. The best cutoff C Debye −1 0 2 frequencies are 133 and 119 cm for Bi2Se3 and Sb2Te3, B 15 (b) Bi2Se3 m respectively. This translates to θD of 191 K and 172 K, γ

(T)/k I 1 i 1 I I respectively. These values are in the correct order and 0 3

27 ) agree well with the literature values of 182 and 160 K, -1 (c) Bi Se K 20 2 3 respectively. It is interesting to see that even though the -6 phonon densities of states from the Debye approximation 10 and DFT differ significantly [see Fig. 2(a) and (c)], the a c heat capacities between DFT and Debye approximation a (exp) c (exp) agree remarkably well with each other [see Fig. 3(a) and Linear TEC (10 0 (d)], which demonstrates the robustness of Debye model 0 100 200 300 Temperature (K) to describe the heat capacity. 20 B (d) Sb2Te3 With the Γi(ν) data, we calculate the integrated quan- 10 (T)/k DFT tities Ii(T ) as shown in Fig. 3(b) and (e) (solid lines), for v

C Debye Bi2Se3 and Sb2Te3, respectively. These are positive func- 0 2

B (e) Sb Te tions, which eliminate the occurrence of negative linear 20 2 3 m γ I1 (T)/k

i 10 1

TECs. For Sb2Te3, Ii(T ) for large-T limit coincides for- I I3 tuitously for xy biaxial and z uniaxial strains. We also 0 ) 30 -1 (f) Sb Te show the values of the macroscopic Gr¨uneisenparame- K 2 3 -6 ters, γm,i(T ) in Fig. 3(b) and (e) (dashed lines). Bi2Se3 20 has a large-T limit of γm of 1.27 and 1.09 for the xy biax- 10 a ial and z uniaxial strains. For Sb2Te3, the large-T limit c a (exp) of γm is 1.36 for both xy biaxial and z uniaxial strains, c (exp) 28 Linear TEC (10 0 which is in good agreement with a reported result of 0 100 200 300 1.40. Therefore it is concluded that Sb2Te3 has a higher Temperature (K) phonon anharmonicity than Bi2Se3 based on the macro- scopic Gr¨uneisenparameters. FIG. 3. The temperature dependence of (a) Cv, (b) Ii (solid lines) and γ (dashed lines), and (c) the linear TECs of The linear TECs for Bi Se in the a and c directions m,i 2 3 Bi2Se3. The respective data shown in (d), (e), and (f) are for are shown in Fig. 3(c). We observe very good agreement Sb2Te3. between theory and experiment for αc up to 60 K, be- yond which the experiment data shows a dip between 60 and 100 K and raises slowly after 100 K. The the- IV. SUMMARY oretical values for αa are underestimated below 60 K but a good agreement with experiment is observed be- tween 60 and 180 K. The linear TECs for Sb2Te3 in In summary, we have performed density-functional the- Fig. 3(f) show a reasonable agreement between theory ory (DFT) calculations to study the phonon anharmonic- and experiment for both αa and αc for temperature be- ity of two trigonal systems Bi2Se3 and Sb2Te3. Build- low 80 K. Finally we note that the theoretical large-T ing upon previous computational approaches, we devised limit of αa for Bi2Se3 and Sb2Te3 are the fortuitously an efficient Gr¨uneisen approach in calculating the linear −6 −1 the same (11.3 × 10 K ). The large-T limit of αc for thermal expansion coefficients (TECs). The symmetry of −6 −1 −6 −1 Bi2Se3 (17.4 × 10 K ) and Sb2Te3 (17.3 × 10 K ) the crystals are fully utilized to reduce the comparatively are also very similar. Since the elastic constants for expensive phonon calculations (compared to standard both materials are rather similar, from Eq. 1 we reason DFT total-energy calculations) to a minimal set. Even that the slightly larger anharmonicity found in Sb2Te3 though the main aim of the paper is to study the linear is somewhat compensated by its slightly larger primitive TECs of the systems, many intermediate quantities such cell volume, which results in very similar temperature de- as density of phonon states, heat capacity, Debye temper- pendence of linear TECs for Bi2Se3 and Sb2Te3. Finally, ature, mode-dependent Gr¨uneisenparameter, density of we note that the complicated temperature dependence of Gr¨uneisenparameters, density of phonon states weighted linear TECs in experiments was argued to be attributed by Gr¨uneisen parameter, and macroscopic Gr¨uneisenpa- to higher-order anharmonic effects and the breaking of rameter have been carefully analyzed to shed light on the van der Waals bonds between two Se-Se (or Te-Te) the temperature dependence of linear TECs of Bi2Se3 5 layers at elevated temperatures. We expect the use of and Sb2Te3. Reasonably good agreement between the- quasi-harmonic approximation (QHA) may improve the ory and experiment for linear TECs has been demon- prediction of the Gr¨uneisen formalism at higher temper- strated. With the demonstrated accuracy and efficiency atures, however, we do not have enough computational of the method, we are confident that a wide applicability resources for a full QHA treatment for both crystals. of our approach to other thermoelectrics or even other 5 classes of low-symmetry materials. We hope our results 10D. Bessas, I. Sergueev, H.-C. Wille, J. Perbon, D. Ebling, and will encourage the inclusion of our method in accelerated R. P. Hermann, Phys. Rev. B 86, 224301 (2012). 11 materials search packages. J. Lin, L. Guo, Q. Huang, Y. Jia, K. Li, X. Lai, and X. Chen, Phys. Rev. B 83, 125430 (2011). The raw/processed data required to reproduce these 12Y. Kim, X. Chen, J. Shi, I. Miotkowski, Y. P. Chen, P. A. findings cannot be shared at this time as the data also Sharma, A. L. L. Sharma, M. A. Hekmaty, Z. Jiang, and forms part of an ongoing study. D. Smirnov, Appl. Phys. Lett. 100, 071907 (2012). 13P. K. Schelling and P. Keblinski, Phys. Rev. B 68, 035425 (2003). 14N. Mounet and N. Marzari, Phys. Rev. B 71, 205214 (2005). 15C. K. Gan, J. R. Soh, and Y. Liu, Phys. Rev. B 92, 235202 (2015). V. ACKNOWLEDGMENTS 16C. K. Gan and Y. Y. F. Liu, Phys. Rev. B 94, 134303 (2016). 17P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- We gratefully thank the National Supercomputing cioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, Center (NSCC), Singapore and A*STAR Computational G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, Resource Center (ACRC), Singapore for computing re- M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Maz- zarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, sources. S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Mat- ter 21, 395502 (2009). 18 1G. J. Snyder and E. S. Toberer, Nature Mater. 7, 105 (2008). A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, 2 Phys. Rev. B 41, 1227 (1990). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 19 3H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, S. Nakajima, J. Phys. Chem. Solids 24, 479 (1963). 20P. Pavone, K. Karch, O. Sch¨utt,W. Windl, D. Strauch, P. Gian- Nature Phys. 5, 438 (2009). 4W. Zhang, R. Yu, H.-J. Zhang, X. Dai, and Z. Fang, Nature nozzi, and S. Baroni, Phys. Rev. B 48, 3156 (1993). 21C. H. Lee and C. K. Gan, Phys. Rev. B 96, 035105 (2017). Phys. 12, 065013 (2010). 22 5X. Chen, H. D. Zhou, A. Kiswandhi, Y. P. Chen, P. A. Sharma, A. Dal Corso, J. Phys.: Condens. Matter 28, 075401 (2016). 23W. Setyawan and S. Curtarolo, Comput. Mater. Sci. 49, 299 A. L. L. Sharma, M. A. Hekmaty, D. Smirnov, and Z. Jiang, Appl. Phys. Lett. 99, 261912 (2011). (2010). 24N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders 6P. Dutta, D. Bhoi, A. Midya, N. Khan, P. Mandal, S. S. Samatham, and V. Ganesan, Appl. Phys. Lett. 100, 251912 College Publishing, New York, 1976). 25Y. Liu, K. T. E. Chua, T. C. Sum, and C. K. Gan, Phys. Chem. (2012). 7Y. Tian, S. Jia, R. J. Cava, R. Zhong, J. Schneeloch, G. Gu, and Chem. Phys. 16, 345 (2014). 26 K. S. Burch, Phys. Rev. B 95, 094104 (2017). T. Tohei, A. Kuwabara, F. Oba, and I. Tanaka, Phys. Rev. B 8D. Das, S. Das, P. Singha, K. Malik, A. K. Deb, A. Bhat- 73, 064304 (2006). 27 tacharyya, V. A. Kulbachinski, R. Basu, S. Dhara, S. Bandy- O. Madelung, : Data Handbook (Springer, opadhyay, and A. Banerjee, Phys. Rev. B 96, 064116 (2017). Berlin, 2004). 28 9G. C. Sosso, S. Caravati, and M. Bernasconi, J. Phys.: Condens. R. P. Stoffel, V. L. Deringer, R. E. Simon, R. P. Hermann, and Matter 21, 095410 (2009). R. Dronskowski, J. Phys.: Condens. Matter 27, 085402 (2015).