The narrow-gap semiconductor CsBi 4Te 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 2Te 3 and related alloys. In this work, the thermal expansion and the heat capacity at constant pressure of CsBi 4Te 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 2Te 3. The phonon band structures and the partial density of states are also investigated, and we find that both CsBi 4Te 6 and Bi 2Te 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.
In the recent years, thermoelectric (TE) materials have been studied extensively due to the advances in the material synthesis and an improved device performance [
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., Bi2Te3 and CsBi4Te6) several investigations of the electronic structure and the electronic conductivity have been reported; see for instance Refs. [
The most fundamental thermal properties of solids can be determined from the phonon dispersion ωq,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 kB 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 V0 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/V0. 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
The computational study is based on the first-principles DFT approach as implement in the VASP program package [
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 [
Bi2Te3 is a semiconductor with a narrow band gap. Although its primitive unit cell has rhombohedral symmetry with the space group
crystal structure, and therefore the structure is strongly anisotropic. CsBi4Te6 can be regarded as a reduced structure of Bi2Te3. From comparing the crystal structure of CsBi4Te6 and Bi4Te6 = 2(Bi2Te3) one finds that the additional electron per two formula units of Bi2Te3 implies a complete reorganization of the Bi2Te3 framework. Thereby, the extra valence electrons in CsBi4Te6 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 CsBi4Te6 is 3.23 Å, which is thus close to the bond length of Bi-Te(2) in Bi2Te3.
Bi2Te3 | CsBi4Te6 | |||
---|---|---|---|---|
300 K | 600 K | 300 K | 600 K | |
ΔV/V0 | 0 | 0.017 | 0 | 0.014 |
a [10−5 K−1] | 55 | 54 | 52 | 39 |
Cp [J∙mol−1∙K−1] | 126 | 131 | 280 | 285 |
CV [J∙mol−1∙K−1] | 121 | 123 | 271 | 273 |
temperatures, the thermal expansion of Bi2Te3 is significantly larger than that of CsBi4Te6. Moreover, whereas the expansion coefficient of Bi2Te3 tends to be rather stable at ~(50 - 55) × 10−5 K−1 for high temperatures, the corresponding coefficient of CsBi4Te6 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 Bi2Te3 we thus find a rather constant (and slightly decreasing) expansion coefficient, and for CsBi4Te6, 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 CsBi4Te6; see
The bulk modulus is determined from the EOS calculation, and the resulting values for T = 0 K are B0 = 47.8 GPa for Bi2Te3 and 37.8 GPa for CsBi4Te6. Thus, we find that the bulk modulus of Bi2Te3 is about 25% larger than that of CsBi4Te6.
The heat capacities CV and Cp are investigated directly from the phonon frequency dispersion using the QHA approach, and the resulting CV and Cp for Bi2Te3 and CsBi4Te6 are presented in
The dispersion curves for Bi2Te3 and CsBi4Te6 are shown along the high symmetry directions in their respective Brillouin zones (
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 Bi2Te3 [
The acoustic modes in Bi2Te3 are rather disperse up to 1.12 THz and they depend primarily the Bi atoms, while the acoustic modes in CsBi4Te6 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 [
In Bi2Te3, 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. CsBi4Te6 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 Bi2Te3 and CsBi4Te6, which also lead to the different in the thermal expansions as shown in
In this work, the thermal properties and the phonon dispersions of Bi2Te3 and CsBi4Te6 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 Bi2Te3 has slightly larger volume expansion than CsBi4Te6 for temperatures above 300 K. However, both compounds show a decrease of the volume expansion in the high temperature region. For Bi2Te3 the calculated value of Cp 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 CsBi4Te6 has relatively delocalized states in the phonon dispersion curve due to the Cs atomic mass which is between those of Bi and Te.
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.
ShenLi,ClasPersson, (2015) Thermal Properties and Phonon Dispersion of Bi2Te3 and CsBi4Te6 from First-Principles Calculations. Journal of Applied Mathematics and Physics,03,1563-1570. doi: 10.4236/jamp.2015.312180