Thermal Properties and Phonon Dispersion of Bi 2 Te 3 and CsBi 4 Te 6 from First-Principles Calculations

The narrow-gap semiconductor CsBi4Te6 is a promising material for low temperature thermoelectric applications. Its thermoelectric property is significantly better than the well-explored, highperformance thermoelectric material Bi2Te3 and related alloys. In this work, the thermal expansion and the heat capacity at constant pressure of CsBi4Te6 are determined within the quasiharmonic approximation within the density functional theory. Comparisons are made with available experimental data, as well as with calculated and measured data for Bi2Te3. The phonon band structures and the partial density of states are also investigated, and we find that both CsBi4Te6 and Bi2Te3 exhibit localized phonon states at low frequencies. At high temperatures, the decrease of the volume expansion with temperature indicates the potential of a good thermal conductivity in this temperature region.


Introduction
In the recent years, thermoelectric (TE) materials have been studied extensively due to the advances in the material synthesis and an improved device performance [1] [2].Special attention has been paid on searching for new compounds, alloys, and/or nanostructures with higher thermoelectric performance.The efficiency of the thermoelectric materials can be evaluated from the figure of merit ZT = (S 2 /ρκ)•T where T is the absolute tem-perature; S is the Seebeck coefficient; ρ is the electrical resistivity; and κ is the thermal conductivity.κ has contribution from the electronic κ e and the lattice thermal κ L conductivities [3].The power factor S 2 /ρ defines the characterized electrical properties.A good thermoelectric material shall typically exhibit low thermal conductivity and a large power factor.In the past years, many research groups have reported enhanced ZT in superlattices such as the Bi 2 Te 3 /Sb 2 Te 3 systems, where the superlattice structures reduce the lattice thermal conductivity.Also, novel bulk and alloy compounds, such as antimony slivery telluride and its alloys with skutterudites, have shown improved ZT value which indicates that the materials can be suitable for thermoelectric applications.Bi 2 Te 3 is already a well-established thermoelectric material at room temperature.Incorporating Cs in Bi 2 Te 3 yields a somewhat more complex electronic structure, and this CsBi 4 Te 6 compound is a potentially thermoelectric material with ZT max = 0.8 at T = −23˚C, which thus is suitable for low temperatures.
All factors related to an optimized ZT are strongly influenced by the crystal structure, the electronic band structure, and the actual carrier concentration of the material.For the considered compounds (i.e., Bi 2 Te 3 and CsBi 4 Te 6 ) several investigations of the electronic structure and the electronic conductivity have been reported; see for instance Refs.[4]- [8].The electronic part κ e of the thermal conductivity can be calculated from the electronic structure through the Wiedemann-Franz relation κ e = L 0 T/ρ (where L 0 is the Lorenz number) but the corresponding lattice part κ L cannot be calculated that easily.Analyzing the thermal properties makes it possible to at least better understand and describe the lattice part κ L of thermal conductivity.In this study, we have therefore theoretically studied the thermal properties of Bi 2 Te 3 and CsBi 4 Te 6 .We have computed the thermal expansions, the heat capacities at constant pressure, and the isothermal bulk moduli at finite temperatures; this can serve as a help to understand the underlying mechanism for the low κ L for these two compounds.The computational study is based on the density functional theory (DFT) within the quasi harmonic approximation (QHA), which is known to provide reasonable good description of the thermal properties below the melting point of bulk materials [9]- [11].The phonon frequencies in the first Brillouin zone are calculated by means of the density functional perturbation theory (DFPT).Recently, QHA based on DFPT has successfully been employed for several related materials, such as Ti 3 SiC 2 , Al 3 Mg, Al 3 Sc, and GaN [12]- [14].

Theoretical Background
The most fundamental thermal properties of solids can be determined from the phonon dispersion ω q,v (for wave vector q of the vth mode) and the corresponding phonon density of states (DOS) as a function of frequency.The Helmholtz free energy at the temperature T and for a constant volume V is given by , where E 0 (V) is the ground state total energy at T = 0 K, F ph (V,T) is the vibration free energy from the phonon contribution, and F el (V,T) is the free energy from the electronic excitations.From the phonon frequencies, the temperature dependent vibrational heat capacity C V at constant volume is determined through where k B is the Boltzmann's constant.The thermal properties at constant pressure are analyzed from the free energy F(V,T).For a given temperature T, the equilibrium volume V 0 is determined by minimizing the Gibbs free energy G(T,p) with respect to volume.This is utilized to further analyze the thermal properties, such as the thermal expansion ΔV/V 0 .The heat capacity at constant pressure is obtained from the derivative of G(T,p) as Here, S(T,p) is the entropy of the system and V(T,p) is the equilibrium volume at a specific pressure p and temperature T.Moreover, the thermal expansion coefficient is given by ( ) ( ) ( ) , and the bulk modulus at zero pressure is given by ( ) ( )

Computational Details
The computational study is based on the first-principles DFT approach as implement in the VASP program package [15] [16], employing the projector augmented wave method (PAW) and using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [17].We fully relax the structure parameter and volume V 0 of the primitive unit cell with a convergence of 10 −5 eV/cell for the total energy, and 10 −4 eV/Å for the forces on each atom.The energy cutoff was 500 eV.The k-space integration was performed with the tetrahedron method, involving a Γ-centered 10 × 10 × 10 k-mesh for Bi 2 Te 3 and corresponding 4 × 4 × 4 k-mesh for the larger CsBi 4 Te 6 compound.QHA was employed to compute the thermal properties at constant pressure.The thermodynamic functions were fitted to the integral form of Vinet's equation of state (EOS) at zero pressure [18].The Helmholtz free energy and the Gibbs free energy were obtained from the minimum values of the thermodynamic functions at finite temperatures, whereupon the equilibrium volume and the bulk moduli were obtained through the EOS.The heat capacity C p (see, Equation ( 2)) was determined by a numerical differentiation ∂V/∂T and by polynomial fitting for both C V and S.
When calculating the phonon dispersion, we have employed the supercell approach and the force-constant method.The real space force constants of the supercells were calculated by the DFPT, whereupon the phonon modes were calculated from the force constants using the PHONOPY package [19].Here, the phonon dispersions and the phonon DOS were calculated with a 2 × 2 × 2 supercell for Bi 2 Te 3 and a 1 × 1 × 2 supercell for CsBi 4 Te 6 , which implies 40 atoms and 88 atoms, respectively.In those calculations, 41 × 41 × 41 Monkhorst-Pack grids were used which is expected to be sufficient to avoid the mean relative error of the DOS.crystal structure, and therefore the structure is strongly anisotropic.CsBi 4 Te 6 can be regarded as a reduced structure of Bi 2 Te 3 .From comparing the crystal structure of CsBi 4 Te 6 and Bi 4 Te 6 = 2(Bi 2 Te 3 ) one finds that the additional electron per two formula units of Bi 2 Te 3 implies a complete reorganization of the Bi 2 Te 3 framework.Thereby, the extra valence electrons in CsBi 4 Te 6 localize on the Bi atoms which leads to a new formation along the a-axis with Bi-Bi bonds.Our calculated length of this Bi-Bi bond in CsBi 4 Te 6 is 3.23 Å, which is thus close to the bond length of Bi-Te(2) in Bi 2 Te 3 .

Thermal Expansion, Bulk Modulus, and Heat Capacities
Table 1 summarizes the volume expansion ΔV/V 0 , thermal expansion coefficient α, as well as the heat capacities C p and C V of Bi 2 Te 3 and CsBi 4 Te 6 ; we present the results for the temperatures T = 300 and 600 K.The temperature dependence of the volume expansion for T = 0 -900 K are shown in Figure 2. The volume expansion is defined as ΔV/V 0 , with ΔV = V − V 0 and where V 0 is the corresponding volume at T = 300 K, and by definition ΔV/V 0 is negative below this 300 K.The volume expansions of the two considered compounds have almost the same linear increase at low temperature (in the region 50 -300 K), but this consistency disappeared for higher temperatures.This is obvious for temperatures above 400 K where Bi 2 Te 3 has somewhat larger volume expansion than CsBi 4 Te 6 .
Figure 3 displays the thermal expansion coefficient ( ) of Bi 2 Te 3 and CsBi 4 Te 6 .The results reveal that the thermal expansion increases considerably with increasing temperatures in the low temperature region below 170 K.In this region the two compounds have almost equivalent thermal expansion, which is in agreement with similar volume expansions for low temperatures.Moreover, the expansion coefficient reaches a maximum value of roughly 55 × 10 −5 K −1 for both Bi 2 Te 3 (maximum at T ~ 300 K) and CsBi 4 Te 6 (at T ~ 150 K).For higher Table 1.The volume expansion ΔV/V 0 , the thermal expansion coefficient α, and the heat capacities C p and C V of Bi 2 Te 3 and CsBi 4 Te 6 at the temperatures T = 300 and 600 K.The unit J•mol −1 •K −1 for the heat capacities refers to formula unit cell: 40 atoms for Bi 2 Te 3 and 88 atoms for CsBi 4 Te 6 .temperatures, the thermal expansion of Bi 2 Te 3 is significantly larger than that of CsBi 4 Te 6 .Moreover, whereas the expansion coefficient of Bi 2 Te 3 tends to be rather stable at ~(50 -55) × 10 −5 K −1 for high temperatures, the corresponding coefficient of CsBi 4 Te 6 drops almost linearly to about half its maximum value, that is, from ~57 × 10 −5 K −1 to ~28 × 10 −5 K −1 at T = 900 K.
It is noticeable that for many similar compounds the thermal expansion coefficient is increasing with increasing temperature.However, for Bi 2 Te 3 we thus find a rather constant (and slightly decreasing) expansion coefficient, and for CsBi 4 Te 6 , we observe a strong decrease of the expansion coefficient in the high temperature region.This is a direct consequence of the decrease of the volume expansion slope for large T for CsBi 4 Te 6 ; see Figure 2.
The bulk modulus is determined from the EOS calculation, and the resulting values for T = 0 K are B 0 = 47.8GPa for Bi 2 Te 3 and 37.8 GPa for CsBi 4 Te 6 .Thus, we find that the bulk modulus of Bi 2 Te 3 is about 25% larger than that of CsBi 4 Te 6 .
The heat capacities C V and C p are investigated directly from the phonon frequency dispersion using the QHA approach, and the resulting C V and C p for Bi 2 Te 3 and CsBi 4 Te 6 are presented in Figure 4. We find that the two compounds have very similar heat capacities.C p is roughly 3% -4% larger than C V at T = 300 K and 4% -7% larger at T = 600 K (Table 1).Moreover, C p and C V for both Bi 2 Te 3 and CsBi 4 Te 6 obey the law of T 3 behavior at low temperatures.At high temperatures however, C V reaches a constant value which is approximately given by the classic equipartition law where N is the number of atoms of the considered system.Here, N = 5 for Bi 2 Te 3 and 11 for CsBi 4 Te 6 , yielding 130.7 respectively, in the classical limit.At ambient pressure and at room temperature T = 300 K, the calculated value of C p for Bi 2 Te 3 is 126 J•mol −1 •K −1 (Table 1) which agree with the experimental data 126 J•mol −1 •K −1 [20] [21].We find also that the calculated results fit very well with the experimental data [20] [21] in the whole low temperature region apart from the measure data point for the lowest temperature; see Figure 4.The corresponding calculated C p value at T = 300 K for CsBi 4 Te 6 is 280 J•mol −1 •K −1 .This is roughly twice as large value compared with Bi 2 Te 3 , and the reason is that the unit cell of CsBi 4 Te 6 contains roughly twice as many atoms (88 atoms) as in the unit cell of Bi 2 Te 3 (40 atoms) and the mol −1 describes formula unit cell.In the units of J•kg −1 •K −1 , the corresponding value is C p = 391 J•kg −1 •K −1 for Bi 2 Te 3 and 400 J•kg −1 •K −1 for CsBi 4 Te 6 .

Phonon Dispersion and Phonon Density of States
The dispersion curves for Bi 2 Te 3 and CsBi 4 Te 6 are shown along the high symmetry directions in their respective Brillouin zones (Figure 5).For Bi 2 Te 3 , the atom-resolved DOS reveals that the phonon states in the lower energy The acoustic modes in Bi 2 Te 3 are rather disperse up to 1.12 THz and they depend primarily the Bi atoms, while the acoustic modes in CsBi 4 Te 6 are disperse up to 0.76 THz and involve mainly contribution from the Cs atoms.It has been discussed that the low frequency phonons as a function of temperature play an important role in the thermal expansion [22].
In Bi 2 Te 3 , the phonon dispersion with frequencies lower than 1.7 THz is a mixture between acoustic and optical modes, and these phonons contribute significantly to the thermal expansion below 300 K. CsBi 4 Te 6 on the other hand, shows relatively delocalized states in the whole phonon dispersion curve because the Cs atom is rather different from Bi.The differences in the phonon vibration modes are mainly due to the different crystal symmetry and the distribution of atom mass in Bi 2 Te 3 and CsBi 4 Te 6 , which also lead to the different in the thermal expansions as shown in Figure 3.

Conclusion
In this work, the thermal properties and the phonon dispersions of Bi 2 Te 3 and CsBi 4 Te 6 have been calculated, employing the DFT and the DFPT within the quasi-harmonic approximation.The volume expansions of these two compounds have similar linear increase for temperatures below 300 K, and Bi 2 Te 3 has slightly larger volume expansion than CsBi 4 Te 6 for temperatures above 300 K.However, both compounds show a decrease of the volume expansion in the high temperature region.For Bi 2 Te 3 the calculated value of C p is 126 J•mol −1 •K −1 at ambient pressure and room temperature which supports the experimental data.From the calculated phonon dispersion and phonon DOS, we conclude that CsBi 4 Te 6 has relatively delocalized states in the phonon dispersion curve due to the Cs atomic mass which is between those of Bi and Te.

Bi 2 Figure 1 .
Figure 1.Crystal structure of (a) hexagonal layered Bi 2 Te 3 and (b) CsBi 4 Te 6 .The layered structure of CsBi 4 Te 6 is composed of anionic, infinitely long [Bi 4 Te 6 ] − blocks where the Cs + ions are located between two anionic blocks.The main Bi-Bi bond in CsBi 4 Te 6 is indicated by the arrow.

Figure 2 .
Figure 2. Relative volume expansion ΔV/V 0 as a function of temperature T, where V 0 is the corresponding volume at T = 300 K (figure caption).

Figure 3 .
Figure 3. Thermal expansion coefficient α of Bi 2 Te 3 and CsBi 4 Te 6 as function of temperature.The dashed line indicates the temperature T = 230 K where the expansion coefficients are equal for the two compounds.

Figure 4 .Figure 5 .
Figure 4.The heat capacity at constant pressure C p and the heat capacity at constant volume C V as functions of temperature.The curves of C V follow roughly the T 3 -law at low temperature and tend to be fairly constant at higher temperatures.Here, the unit J•mol −1 •K −1 refers to formula unit cell, and due to a larger unit cell the values of the heat capacity of CsBi 4 Te 6 is about 2.2 times larger than of Bi 2 Te 3 .