**Journal of Applied Mathematics and Physics**

Vol.03 No.12(2015), Article ID:61692,8 pages

10.4236/jamp.2015.312180

Thermal Properties and Phonon Dispersion of Bi_{2}Te_{3} and CsBi_{4}Te_{6} from First-Principles Calculations

Shen Li^{1}, Clas Persson^{1,2}

^{1}Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm, Sweden

^{2}Department of Physics, University of Oslo, Oslo, Norway

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 29 October 2015; accepted 1 December 2015; published 4 December 2015

ABSTRACT

The narrow-gap semiconductor CsBi_{4}Te_{6} is a promising material for low temperature thermoelectric applications. Its thermoelectric property is significantly better than the well-explored, high- performance thermoelectric material Bi_{2}Te_{3 }and related alloys. In this work, the thermal expansion and the heat capacity at constant pressure of CsBi_{4}Te_{6} 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 Bi_{2}Te_{3}. The phonon band structures and the partial density of states are also investigated, and we find that both CsBi_{4}Te_{6 }and Bi_{2}Te_{3} 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.

**Keywords:**

Quasi Harmonic Approximation, Thermal Expansion, Heat Capacity, Phonon Dispersion

1. 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}/rk)∙T where T is the absolute temperature; S is the Seebeck coefficient; r is the electrical resistivity; and k is the thermal conductivity. k has contribution from the electronic k_{e} and the lattice thermal k_{L} conductivities [3] . The power factor S^{2}/r 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 k_{e} of the thermal conductivity can be calculated from the electronic structure through the Wiedemann-Franz relation k_{e} = L_{0}T/r (where L_{0} is the Lorenz number) but the corresponding lattice part k_{L} cannot be calculated that easily. Analyzing the thermal properties makes it possible to at least better understand and describe the lattice part k_{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 k_{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] .

2. Computational Method

2.1. 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

, (1)

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

. (2)

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.

2.2. 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-Ern- zerhof (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.

3. Results

3.1. Crystal Structure

Bi_{2}Te_{3} is a semiconductor with a narrow band gap. Although its primitive unit cell has rhombohedral symmetry with the space group, the crystal structure is usually described by hexagonal coordinates. With a hexagonal unit cell [see Figure 1(a)], there are five layers consisting of Te(1)-Bi-Te(2)-Bi-Te(1) chains along the hexagonal axis. From our relaxation, we find that the calculated average bond length of Te-Te, Bi-Te(1), and Bi-Te(2) are 3.60, 3.06, and 3.22 Å, respectively. The crystalline structure of CsBi_{4}Te_{6} is related to that of Bi_{4}Te_{6}, however it crystallizes with the space group C2/m; see Figure 1(b). The layered structure of CsBi_{4}Te_{6} is composed by infinitely long anionic [Bi_{4}Te_{6}]^{−} blocks with the Cs^{+} ions reside between the anionic layers. The [Bi_{4}Te_{6}]^{−} blocks can be seen as fragments of NaCl-like lattices, and each block is a two-Bi octahedral thick and four-Bi octahedral wide in the ac-plane while infinitely along the b-axis. The compound can be considered as a one-dimensional like

(a)(b)

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.

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}.

3.2. Thermal Expansion, Bulk Modulus, and Heat Capacities

Table 1 summarizes the volume expansion ΔV/V_{0}, thermal expansion coefficient a, 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 a, 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}.

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. 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.

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} = 47.8 GPa 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
and 281.2 J∙mol^{−1}∙K^{−1}, 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}.

3.3. 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

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}.

(a) (b)

Figure 5. Phonon band structure and its corresponding total and atomic-resolved DOS of (a) Bi_{2}Te_{3} and (b) CsBi_{4}Te_{6}. The difference in the frequency at the Brillouin zone edge (1.12 THz for Bi_{2}Te_{3} whereas 0.76 THz for CsBi_{4}Te_{6}) is due to the Cs atoms in CsBi_{4}Te_{6} which have atomic mass between that of Bi and Te. The CsBi_{4}Te_{6} structure implies also the presence of Bi-Bi bond (see Figure 1).

region compose mainly of Bi-like states, while Te-like states contribute more in the higher energy region since the atomic mass of Te is significantly lighter than that of Bi.

When comparing the phonon dispersions of atomic-resolved DOS of Bi_{2}Te_{3} [Figure 5(a)] with CsBi_{4}Te_{6} [Figure 5(b)], it is clear the shapes of the dispersions and DOS of Bi_{2}Te_{3} and CsBi_{4}Te_{6} shows both similarities and differences. The flat regions of phonon dispersion curves in Bi_{2}Te_{3} lead to two main peaks in the atom-re- solved DOS indicating localizations of states that behave as the “atomic states” for Bi and Te atoms, respectively. Similar atom-like characters were also found in the atom-resolved DOS of CsBi_{4}Te_{6} for the Cs and Te atoms, whereas the Bi atoms show more delocalization in CsBi_{4}Te_{6} because the Bi-Bi bonds are influenced by the Cs^{+}.

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.

4. 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.

Acknowledgements

This work is supported by the European EM-ECW scholarship program Tandem, the Swedish Energy Agency, and the Swedish Research Council. C.P. acknowledges support from the Research Council of Norway (contracts No. 228854 and 221469). We acknowledge access to the high-performance computing resources at the NSC and HPC2N centers through SNIC and Matter network.

Cite this paper

ShenLi,ClasPersson, (2015) Thermal Properties and Phonon Dispersion of Bi_{2}Te_{3} and CsBi_{4}Te_{6} from First-Principles Calculations. *Journal of Applied Mathematics and Physics*,**03**,1563-1570. doi: 10.4236/jamp.2015.312180

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