Prediction for electronic, vibrational and thermoelectric properties of chalcopyrite AgX(X=In,Ga)Te2: PBE + U approach

The electronic, vibrational and thermoelectric transport characteristics of AgInTe2 and AgGaTe2 with chalcopyrite structure have been investigated. The electronic structures are calculated using the density-functional theory within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof functional considering the Hubbard-U exchange correlation. The band-gaps of AgInTe2 and AgGaTe2 are much larger than previous standard GGA functional results and agree well with the existing experimental data. The effective mass of the hole and the shape of density of states near the edge of the valence band indicate AgInTe2 and AgGaTe2 are considerable p-type thermoelectric materials. An analysis of lattice dynamics shows the low thermal conductivities of AgInTe2 and AgGaTe2. The thermoelectric transport properties' dependence on carrier concentration for p-type AgInTe2 and AgGaTe2 in a wide range of temperatures has been studied in detail. The results show that p-type AgInTe2 and AgGaTe2 at 800 K can achieve the merit values of 0.91 and 1.38 at about 2.12 × 1020 cm−3 and 1.97 × 1020 cm−3 carrier concentrations, respectively. This indicates p-type AgGaTe2 is a potential thermoelectric material at high temperature.


Introduction
Thermoelectric material, which can be used in thermoelectric devices to convert waste heat into electricity without any 400 eV. The total energy convergence threshold on each atom is lower than 1 × 10 −6 eV. The maximal force on each ion is less than 5 meV Å −1 for relaxation both atomic positions and cell parameters. The phonon transports have been investigated in the form of harmonic approximation and supercell approach. We perform 2 × 2 × 2 supercell including 128 atoms and 0.01 Å atomic displacement distance to calculate the second-order force constants by using VASP code. Based the calculated force constants, the phonon dispersion relations are computed along high-symmetry points in the first Brillouin zone by Phonopy code [29]. The BoltzTraP code [30] has been employed to obtain the thermoelectric transport functions such as Seebeck coefficients (S) and electronic conduction with respect to scattering time (σ /τ ) using first-principles data. This method is based on the Boltzmann transport theory in the form of constant scattering time approximation (CSTA). To help obtain correct transport properties, much denser kmesh (19 × 19 × 19 in primitive cell) is used to ensure the accuracy of self-consistent energy values. The relaxation time has been adjusted from existing experimental data to obtain the electronic conduction (σ ).

Results and discussion
The ternary chalcopyrite structure compound AgX(In,Ga)Te 2 is crystallized in the tetragonal phase with space group I42d (no. 122). The crystal structure of AgX(In,Ga)Te 2 is shown in figure 1 using VESTA software [31]. There are four formula unit atoms in each unit cell. Each Ag or In/Ga atom connects with four Te atoms to form a diamond-like structure. The optimized structural constants for AgInTe 2 are a = b = 6.56 Å and c = 12.65 Å, which are in substantial agreement with previously reported experimental results of a = b = 6.44 Å, c = 12.64 Å [32] and theoretical values [33]. The optimized lattice constants (a = b = 6.39 Å, c = 11.82 Å) for AgGaTe 2 are in excellent agreement with experimental values within a 1% mismatch [18].
The Seebeck coefficient is directly proportional to the effective mass of carriers and slope of the DOS near the Fermi level [34,35]. The chalcopyrite structure semiconductors have a direct band-gap at the Γ point [36][37][38][39]. The shapes of the calculated band structures are similar to the previous DFT results and expected band-gap [40] and thus no more considered in this paper. In this paper, we give insight into the effective masses of holes and electrons at the valence band maximum (VBM) and conduction band minimum (CBM). The effective masses of holes and electrons at the Γ point along different crystallographic directions have been calculated using: (1/m * ) = (1/h 2 )(∂ 2 E/∂k 2 ), whereh, E and k are the reduced Planck constant, energy eigenvalue and wavevector, respectively. For example, the effective masses of holes and electrons for AgX(In,Ga)Te 2 along the c-direction can be obtained from the VBM and CBM of band structure along Z(0.5, 0.5, −0.5)-Γ (0, 0, 0)-Z(0.5, 0.5, −0.5) path. The effective masses of holes and electrons for AgX(In,Ga)Te 2 along the a-and c-directions are listed in table 1. The data from table 1 indicate the effective masses of holes and electrons for the chalcopyrite structure compound AgX(In,Ga)Te 2 are anisotropic. We find the effective masses of holes and electrons along the c-direction are the same, while they are different along the a-direction. In addition, we noted that the effective mass of    holes is larger than the effective mass of electrons along the a-direction. Hence, a large Seebeck coefficient for p-type AgX(In,Ga)Te 2 along the a-direction is anticipated. The partial and total DOS for AgX(In,Ga)Te 2 are shown in figure 2. We can see that the calculated band-gap energies using PBE + U approach for AgInTe 2 and AgGaTe 2 are 0.99 eV and 1.04 eV, respectively. Our calculated results agree well with previous experimental data [14,20], which inspires confidence in the accuracy of our calculation. Moreover, the Ag-4d and Te-5p states are the main contributors to VBM. The CBM is chiefly determined by the mixture between Te-5p and In-5s or Ga-4s states. Meanwhile, as can be clearly seen from figure 2, DOS near the edge of the valence band is slightly steeper than that near the edge of the conduction band. The steep DOS leads to large Seebeck coefficient [41], which indicates AgX(In,Ga)Te 2 may be possible p-type thermoelectric materials. Now, we turn to the lattice dynamical properties of the chalcopyrite structure compound AgX(In,Ga)Te 2 . The vibrational spectrum curves together with the corresponding projected phonon DOS for AgX(In,Ga)Te 2 are plotted in figure 3. As there are eight atoms in a primitive cell, the phonon dispersion curves involve 24 phonon modes. The positive phonon frequencies suggest AgInTe 2 and AgGaTe 2 are dynamically stable. The phonon dispersion relations for AgInTe 2 and AgGaTe 2 are similar to each other. Both phonon dispersion relations of the two compounds have frequency gaps and consist of two groups of bands. The high-frequency optical modes (up to 145 cm −1 and 173 cm −1 , respectively, for AgInTe 2 and AgGaTe 2 ) are determined by the contributions of In/Ga and Te atoms. The low frequencies up to the frequency gap are mixed by the Ag, Te and In/Ga atoms. Compared with flat optic phonon modes, highly dispersive acoustic phonon modes have larger group velocities. Moreover, there are some low-frequency (below 50 cm −1 ) optic phonon modes mixing with acoustic phonon modes. The mixture of optic and acoustic phonon modes promotes the reduction of thermal conductivity. We find that the phonon DOS of AgGaTe 2 is a little wider than that of AgInTe 2 in the low-frequency range of up to 50 cm −1 and the Ga atom contributes to moving towards a lower frequency because of the lighter mass. In addition, the minimum frequency of the optic mode for AgGaTe 2 is 17.  frequency of the corresponding mode for AgInTe 2 is 22.7 cm −1 , which suggests that AgGaTe 2 has smaller thermal conductivity than AgInTe 2 . The thermal conductivities for AgInTe 2 and AgGaTe 2 [16] yielded by experiments confirm what we had anticipated. The Boltzmann transport theory in the form of CSTA can directly yield Seebeck coefficients. The Seebeck coefficients as a function of chemical potential for AgInTe 2 at 300 K along the a-and c-directions are shown in figure 4. The Seebeck coefficients display anisotropy for the p-type AgInTe 2 compound and show isotropy for the n-type one. Additionally, the Seebeck coefficient for the p-type AgInTe 2 in the a-direction is larger than that in the c-direction, which mainly stems from the fact that the effective mass of holes in the a-direction is significantly larger than that in the c-direction as mentioned above. Furthermore, the Seebeck coefficient of the p-type AgInTe 2 along the c-direction is larger than that of the n-type one, due to the fact that DOS near VBM increases sharply with an energy decrease as indicated in figure 2a.
In the form of CSTA, the electronic conduction σ cannot be independently calculated directly from the electronic structure by using the Boltzmann transport theory. In order to obtain σ , the information of the relaxation time τ is needed. We fit the theoretical total σ /τ with experimentally measured σ at a fixed temperature and carrier concentration to determine the behaviour of τ . We take advantage of the experimental data (σ = 2132.2 S m −1 , S = 390 µV K −1 ) at 700 K [42] to fit the relaxation time. We use the average S and σ /τ along the a-, b-and c-directions to compare with experimental data, because the experimental data [42] are tested for arbitrary direction. The corresponding carrier concentration is 2.02 × 10 19   compound is A = 4.44 × 10 −6 . Thus, the relaxation time τ as an expression of T and n for the AgGaTe 2 compound is presented as τ = 4.44 × 10 −6 T −1 n −1/3 . The relaxation time τ for the AgInTe 2 compound is obtained using the same approach as that of AgGaTe 2 because of their similar crystal structures. We now discuss thermal conductivity. Thermal conductivity is contributed by both the electron and the lattice. The lattice thermal conductivity is the main contributor (over 98%) and is determined mostly by the lattice structure [43]. We assume the thermal conductivity only depends on temperature, which is widely used in thermoelectric materials [44,45]. Thermal conductivity is adopted by fitting the experimentally measured data at different temperatures: k = A + (B/T), where A and B are fitted constants. Figure 5 plots the temperature dependence of fitting thermal conductivity (solid lines) and the experimentally measured data [16] (scattered dots) for AgInTe 2 and AgGaTe 2 compounds. The thermal conductivities for AgInTe 2 and AgGaTe 2 compounds decrease greatly as the temperature increases. The temperature dependence of thermal conductivity for AgInTe 2 and AgGaTe 2 compounds can be obtained by using the formulae k =     high temperatures. We discover the bipolar effect of AgInTe 2 is a little more evident than that of AgGaTe 2 . The reason is that the smaller band-gap of AgInTe 2 makes electronic thermal excitation a little easier than that of AgGaTe 2 . As can be seen, the figure of merit ZT heavily depends on carrier concentration and temperature. The bipolar effects are clearly visible, which cause the ZT at 800 K to be lower than the ZT at 700 K at low carrier concentrations. The maximum ZT values for AgInTe 2  approximately 2.12 × 10 20 cm −3 and 1.97 × 10 20 cm −3 carrier concentrations, respectively. This indicates AgGaTe 2 is a potential high-temperature thermoelectric material for use in thermoelectric devices.

Conclusion
We have presented the electronic, vibrational and thermoelectric transport properties of AgInTe 2 and AgGaTe 2 in the form of the first-principles theory. The electronic structures are obtained using the PBE + U approach and the results agree well with existing experimental results. The calculated effective mass shows that AgInTe 2 and AgGaTe 2 are possible p-type thermoelectric materials and thermoelectric properties are anisotropic in different crystallographic directions. The results of vibrational transport properties calculated using harmonic approximation anticipate the low thermal conductivity of AgInTe 2 and AgGaTe 2 . Thermoelectric properties including Seebeck coefficients and figures of merit for AgInTe 2 and AgGaTe 2 are obtained using the Boltzmann transport theory in the form of CSTA based on firstprinciples electronic structures and previous experimental results. The bipolar effects at low doping levels and high temperatures for AgInTe 2 and AgGaTe 2 are clearly visible and have a significant influence on the thermoelectric properties. When we compare the thermoelectric properties between AgInTe 2 and AgGaTe 2 , the ZT value of AgGaTe 2 at 800 K is 1.38 at a doping level of about 1.97 × 10 20 cm −3 , which confirms AgGaTe 2 is a potential thermoelectric material at a high temperature.