First-principles study of electronic transport and structural properties of ${\mathrm{Cu}}_{12}{\mathrm{Sb}}_4{\mathrm{S}}_{13}$ in its high-temperature phase

We present an ab initio study of the structural and electronic transport properties of tetrahedrite, ${\mathrm{Cu}}_{12}{\mathrm{Sb}}_4{\mathrm{S}}_{13}$, in its high-temperature phase. We show how this complex compound can be seen as the outcome of an ordered arrangement of S-vacancies in a semiconducting fematinite-like structure (${\mathrm{Cu}}_3{\mathrm{SbS}}_4$). Our calculations confirm that the S-vacancies are the natural doping mechanism in this thermoelectric compound and reveal a similar local chemical environment around crystallographically inequivalent Cu atoms, shedding light on the debate on x-ray photoelectron spectroscopy measurements in this compound. To access the electrical transport properties as a function of temperature we use the Kubo-Greenwood formula applied to snapshots of first-principles molecular dynamics simulations. This approach is essential to effectively account for the interaction between electrons and lattice vibrations in such a complex crystal structure where a strong anharmonicity plays a key role in stabilizing the high-temperature phase. Our results show that the Seebeck coefficient is in good agreement with experiments and the phonon-limited electrical resistivity displays a temperature trend that compares well with a wide range of experimental data. The predicted lower bound for the resistivity turns out to be remarkably low for a pristine mineral in the Cu-Sb-S system but not too far from the lowest experimental data reported in literature. The Lorenz number turns out to be substantially lower than what is expected from the free-electron value in the Wiedemann-Franz law, thus providing an accurate way to estimate the electronic and lattice contributions to the thermal conductivity in experiments, of great significance in this very low thermal conductivity crystalline material.

Figure 1

Tetrahedrite cubic conventional cell. S atoms are yellow, Cu atoms are blue, and Sb atoms are orange.

Figure 2

(a) $2\times 2\times 1$ supercell of fematinite, ${\mathrm{Cu}}_3{\mathrm{SbS}}_4$. (b) Structure obtained by swapping Cu and Sb atoms within the green (001) plane. (c) Structure in which six S atoms are removed (white atoms). (d) Relaxed structure obtained starting from (c). Color scheme for the atoms as in Fig. 1.

Figure 3

Comparison between the DOS of the symmetric structure in Fig. 1 (solid blue line) and the snapshot averaged DOS at 300 K (orange dashed line) and at 600 K (red dotted line). $E_{\text{F}}$ is the Fermi level.

Figure 4

Resistivity as a function of temperature. The blue squares (green circles) are the K-G (BTE-CRT) results. The black symbols are the experimental data: the down triangles are from Ref. [2], the up triangles from Ref. [1], the diamonds from Ref. [3], the stars from Ref. [4], and the plus symbols from Ref. [5]. The error bars in the K-G results are smaller than the symbols used.

Figure 5

Seebeck coefficient as a function of temperature. The symbols are as in Fig. 4.

Figure 6

Lorenz number as a function of temperature. The blue squares (green circles) are the K-G (BTE-CRT) results; the red dashed line is free-electron value in the Wiedemann-Franz law ($L_0=2.44\times {10}^{-8}\mathrm{W}\mathrm{\Omega }{\mathrm{K}}^{-2}$).