Liquidlike Cu atom diffusion in weakly ionic compounds ${\mathrm{Cu}}_2\mathrm{S}$ and ${\mathrm{Cu}}_2\mathrm{Se}$

Copper sulphide (${\mathrm{Cu}}_2\mathrm{S}$) and copper selenide (${\mathrm{Cu}}_2\mathrm{Se}$) are known to exist in solid (S/Se)-liquid (Cu) hybrid phases which exhibit favorable thermoelectrics properties. The diffusion characteristics and its mechanism in these systems are therefore of significant interest. In this paper, we analyze these properties through examining the atomic radial distributions, mean-square displacements, and velocity autocorrelations obtained from ab initio molecular dynamics simulations. Exceptionally high Cu diffusion coefficients with values over ${10}^{-5}{\mathrm{cm}}^2$/s are obtained, indicating the unexpected liquidlike behavior of Cu in these weakly ionic compounds. The diffusion mechanism obtained through analysis of the Cu atomic trajectories is found to be at variance with the previously proposed Chudley-Elliott jump diffusion model. In addition, tunability of these diffusion coefficients via small changes in the stoichiometry, namely, Cu deficiencies, is demonstrated. The higher number of low-frequency acoustic phonon modes associated with Se sublattices, compared to those with S sublattices, correlates well with the experimentally observed thermal conductivity difference between ${\mathrm{Cu}}_2\mathrm{S}$ and ${\mathrm{Cu}}_2\mathrm{Se}$.

Figure 1

(a) and (b) highlight the preferred Cu symmetry sites relative to the tetrahedron formed by closest S or Se atoms. Cu (blue spheres) atoms are shown at all preferred symmetry sites. (c) and (d) show the cubic structure of the high temperature phases of ${\mathrm{Cu}}_2\mathrm{S}$ and ${\mathrm{Cu}}_2\mathrm{Se}$, respectively. S (yellow spheres) and Se (green spheres) atoms in both structures occupy fixed fcc sites. This figure was illustrated with VESTA [35].

Figure 2

Partial radial distribution functions: (a) $g_{\text{SS}}\left(g_{\text{SeSe}}\right)$, (b) $g_{\text{CuCu}}$, and (c) $g_{\text{SCu}}\left(g_{\text{SeCu}}\right)$ in ${\mathrm{Cu}}_{2-x}\mathrm{S}$ (${\mathrm{Cu}}_{2-x}\mathrm{Se}$), where $x=0$ and 0.03. Partial static structure factors: (d) $S_{\text{SS}}\left(S_{\text{SeSe}}\right)$, (e) $S_{\text{CuCu}}$ and (f) $S_{\text{SCu}}\left(S_{\text{SeCu}}\right)$ in ${\mathrm{Cu}}_{2-x}\mathrm{S}$ (${\mathrm{Cu}}_{2-x}\mathrm{Se}$) where $x=0$ and 0.03. Results are obtained from ab initio molecular dynamics simulations at 900 K.

Figure 3

(a) Total radial distribution functions $g\left(r\right)$ and (b) total static structure factors $S\left(k\right)$ of ${\mathrm{Cu}}_{2-x}\mathrm{S}$ and ${\mathrm{Cu}}_{2-x}\mathrm{Se}(x=0$ and 0.03) from ab initio molecular dynamics simulations at 900 K.

Figure 4

Mean-square displacements $\langle \mathrm{\Delta }{r}^2\left(t\right)\rangle$ of (a) S in ${\mathrm{Cu}}_{2-x}\mathrm{S}$, (b) Se in ${\mathrm{Cu}}_{2-x}\mathrm{Se}$, (c) Cu in ${\mathrm{Cu}}_{2-x}\mathrm{S}$, and (d) Cu in ${\mathrm{Cu}}_{2-x}\mathrm{Se}$ as a function of time from ab initio molecular dynamics simulations at 900 K. Note the scale differences between the top and bottom panels. In (c) and (d), the best linear fits are indicated by solid lines.

Figure 5

Normalized velocity autocorrelation functions of (a) S/Se and (b) Cu in ${\mathrm{Cu}}_{2-x}\mathrm{S}$ and ${\mathrm{Cu}}_{2-x}\mathrm{Se}(x=0$ and 0.03). Power spectra of (c) S/Se and (d) Cu in ${\mathrm{Cu}}_{2-x}\mathrm{S}$ and ${\mathrm{Cu}}_{2-x}\mathrm{Se}(x=0$ and 0.03), obtained by Fourier transforming the velocity autocorrelation functions in (a) and (b) respectively. Each velocity autocorrelation functions is normalized by its respective value at $\tau =0$, while each power spectrum is normalized by its respective area. Results shown here are obtained from ab initio molecular dynamics simulations at 900 K. Error bars are too small to be distinguished.

Figure 6

Trajectories (red lines) of three representative Cu atoms in ${\mathrm{Cu}}_{1.97}\mathrm{Se}$ shown in three planar projections for the simulation run of 70 ps. The blue circles indicate the spheres tangential to the faces of the tetrahedral cages formed by fcc Se, with a diameter of 1.68 A and centered at 8c sites. The solid regular (inverted) triangles indicate the beginning (ending) locations of the Cu atoms.

Figure 7

Partial phonon density of states (DOS) of monoclinic ${\mathrm{Cu}}_2\mathrm{S}$.