Electric and Thermoelectric Properties of Cu12Sb4S13 Tetrahedrite with Impurities

The high figure of merit and earth abundance of Cu12Sb4S13 thermoelectric materials have recently attracted many attentions toward these type of complex compounds. Intrinsic low thermal conductivity, as well as tunable electronic transport properties, make them suitable for thermoelectric power generation. In this study, we perform a comparative theoretical study on the substituted compounds, primarily at the Cu site including known tetrahedrite Cu12Sb4S13, by means of first-principles calculations. The density functional theory of electric structure is applied to investigate the result of substitution.


Introduction
The efficiency of a thermoelectric (TE) material defined by a dimensionless figure of merit, ZT = S 2 σT κ = P T κ , wherein σ is the electrical conductivity, P is power factor, S Seebeck coefficient and κ is the thermal conductivity. The κ includes both lattice contribution κ L and electronic κ e contribution κ L , i.e. 5 κ = κ e + κ L . The enhanced ZT determines the productivity of a thermoelectric material. The preeminent of the known TE materials have a value of ZT to be of the order of unity at ambient condition. [1] The ZT equation implies that finding materials with ZT more than one is an open challenge, since it is required to satisfy the conflicting tie of high P = S 2 σ resembling an in- 10 sulator and simultaneously behave as a decent conductor like a metal. High ZT indicates that a good TE material should have high electrical conductivity σ and low thermal conductivity κ leading to large phonon scattering and small electron scattering. In last couple of years, a lot of efforts have been made to study thermoelectric materials and develop new methods to enhance 15 the value of the ZT . Multiple reports have been published by various groups with a focus on studying atomic structure and electronic behavior leading to low thermal conductivity [2,3]. In addition, several studies performed on band structure engineering to improve Seebeck coefficient and electric conductivity and implementing nanostructure technology for decreasing the lattice thermal 20 conductivity [4,5]. There has been some research on the effect of Kondo correlations on the thermopower [6,7,8,9,10,11,12].
Recently, several studies focused on multiple-filled skutterudites [13,14]. Xun Shi et al, [14] reported that there are a few doped samples which show an improved figure of merit of 1.7 at around 850 K. This ZT is one of the highest 25 value reported in skutterudites type of TE materials. Furthermore, Biswas et al., [15] found high-performance bulk thermoelectrics ZT 2.2 value at 923 K for p-type PbTe-SrTe doped with Na. This high value of ZT is assigned to the hierarchical structure which enhances the phonon scattering.
Several restrictions such as toxicity and scarcity of the elements limit de- In this work, we study and compare the electronic structure of substituted tetrahedrite with other group of tetrahedrites substituted by some other elements. The electronic structure was obtained by implementing a DFT calculation of Wien2k program [21] and the BoltzTrap package based on the Boltzmann 50 transport theory. [22] It has already been reported that the bands crossing the Fermi level are primarily consist up of d-orbitals of copper atoms. [23,24,25] Hence, in order to determine the substitution impact of a equal amount of doping for electronic structure calculations, equal amount of d-orbital occupancy is considered for parent compound of Cu 12 Sb 4 S 13 .

Methodology
First principle density functional theory (DFT) calculations were carried out in the WIEN2k. [21] by a technique that uses Full Potential Linear Augmented plane wave (FPLAPW) as well as Plane wave self consistent field (Pwscf). The FPLAPW is implemented to investigate the transport and electronic properties of the tetrahedrites while the bond lengths and structure optimizations were done by using Pwscf method. The Generalized Gradient Approximation (GGA) of Perdew Burke Ernzerhof (PBE) [26] was used to attain the total energy by self consistently solving the Kohn-Sham equations.
A variable cell optimization carried out by implementing conjugate gradient 65 method as employed in Pwscf. A 9 × 9 × 9 k points mesh used in the reciprocal space of the first Brillouin zone based on Monkhorst-Pack scheme[27] to guarantee a decent convergence. A plane wave cut off energy of 100 eV is applied.
The FP-LAPW method is employed as described in the WIEN2k code [21].
The standard local scheme of the exchange-correlation functional (LDA or GGA) 70 used in the first principles calculations commonly underestimate the band gap and they can't predict precisely the localized electrons in d orbitals and sometimes in f orbitals, in DFT calculations involving transition metals [28]. To resolve this problem, we have implemented the onsite Coulomb repulsion U combined with Generalized Gradient Approximation (GGA) (GGA+U). Further-75 more, several physical properties such as thermopower (S) and electrical conductivity σ and electron relaxation time τ have calculated utilizing BOLTZTRAP [22] package with various k points up to 5000 k-points. In our calculation we used the Constant Scattering Time (τ ) Approximation (CSTA) along with the Rigid Band Approximation (RBA) [29] . The equations used in the calculations for 80 electric part of thermal conductivity (κ), electrical conductivity (σ), and Seebeck coefficient (S), obey the following relations: where ε is is energy, T is the temperature, µ is the chemical potential, τ is a relaxation time and e is the electron charge .
tem will not modify its band structure, the chemical potential will be adjusted.
This could be utilized to approximate semiconductors that are not heavily doped in order to calculate some of the the transport properties theoretically. [30,31,32,33] Nevertheless, adding some specific types of dopant will severely alter the nature of electronic structure below and above the gap leading to resonant 90 states [34,35] in which case the RBA approximation fail to predict precisely. [36] According to constant scattering time approximation (CSTA), the scattering time for an electron regardless of its energy relies only on temperature and concentration of electrons. [37] The CSTA approximation has effectively predicated the thermoelectric quantities in several different materials which were in good 95 agreement with experimental reports. [38,39] 3. Result and discussion

Ground state properties
The In order to better estimate electron relaxation time τ electron and lattice thermal conductivity, it is required to separated into following two terms where, the first term T σS 2 κe can be replaced by a factor α and the second term 1 1+ κ L κe can be replaced by another factor β. DFT calculation for electrons will produce α, while since β includes τ e and κ L , it cannot derived from DFT calculations for electrons. To resolve this problem, the equation is being used. Where, v L is the phonon group velocity, C L the specific heat of 140 the lattice, and τ L the phonon relaxation time.
The correlation between κ e and σ is expressed by Wiedeman-Franz law which is being used to reduce ZT form Where, κ e is being replaced from κ e = LT σ = 2.44 × 10 −2 mV 2 /K 2 . Considering this assumption, DFT calculation for Seebeck S coefficient and thermal 145 and electrical conductivities will result in ZT for doped tetrahedrite materials. Figure 2 shows the calculated ZT , S and κe LσT for studied materials. The ZT is highest for Cu 10 Zn 2 Sb 4 S 13 as S.
The comparison with experimental data [20] shows that calculated ZT and S are slightly smaller than the observed values. However, the increasing tendency 150 for S is in good agreement. Similar to S increasing trend in Lorentz number observed which partially pulls off the effect on ZT .

Conclusions
The structural and electronic transport properties of three doped tetrahedrite are studied using density functional theory. The Seebeck S coefficient 155 calculated using the BoltzTrap program is positive for all three tetrahedrite compounds studied here. Electronic structure calculations show that all the investigated compounds are indirect band gap semiconductors, in good agreement with earlier reports. We further calculated the thermoelectric properties of the these compounds and compared with the available experimental reports. The 160 calculations show all the investigated compounds to be very good thermoelectric materials for p-type doping.

Acknowledgment
We would like to thank the Sharif University of Technology for financial support under Grant No. G960204.