Solidlike to liquidlike behavior of Cu diffusion in superionic ${\mathrm{Cu}}_2\mathrm{X}$ $(X=\mathrm{S}, \mathrm{Se})$: An inelastic neutron scattering and ab initio molecular dynamics investigation

${\mathrm{Cu}}_2\mathrm{Se}$ and ${\mathrm{Cu}}_2\mathrm{S}$ are excellent model systems of superionic conductors with large diffusion coefficients that have been reported to exhibit different solidlike and liquidlike Cu-ion diffusion. In this paper, we clarify the atomic dynamics of these compounds with temperature-dependent ab initio molecular dynamics (AIMD) simulations and inelastic neutron scattering experiments. Using the dynamical structure factor and Van Hove correlation function, we interrogate the jump time, hopping length distribution, and associated diffusion coefficients. In cubic ${\mathrm{Cu}}_2\mathrm{Se}$ at 500 K, we find solidlike diffusion with Cu jump lengths matching well the first-neighbor Cu-Cu distance of ∼3 Å in the crystal, and clearly defined optic phonons involving Cu vibrations. Above 700 K, the jump-length distribution becomes a broad maximum centered around 4 Å, spanning the first and second neighbor lattice distances, and a concurrent broadening of the Cu-phonon density of states. Further, above 900 K, the Cu diffusion becomes close to liquidlike, with distributions of Cu atoms continuously connecting crystal sites, while the vibrational modes involving Cu motions are highly damped, though still not fully overdamped as in a liquid. At low temperatures, the solidlike diffusion is consistent with previous x-ray diffraction and quasielastic neutron scattering experiments, while the higher-temperature observation of the liquidlike diffusion is in agreement with previous AIMD simulations. We also report AIMD simulations in ${\mathrm{Cu}}_2\mathrm{S}$ in the hexagonal and cubic superionic phases, and observe nearly liquidlike diffusion above ∼500 K. The calculated ionic conductivity is in fair agreement with reported experimental values.

Figure 1

The crystal structure of (a) ${\mathrm{Cu}}_2\mathrm{Se}$ and (b) ${\mathrm{Cu}}_2\mathrm{S}$ in the high-temperature cubic phase ($Fm\text{−}3m$). The Cu occupancies are (a) ${\mathrm{Cu}}_2\mathrm{Se}$- $8c$ (0.72), $4b$ (0.0), $32f$ (0.07); and (b) ${\mathrm{Cu}}_2\mathrm{S}$- $8c$ (0.26), $4b$ (0.09), $192l$ (0.03). The Cu occupancies are shown by partially colored large and small spheres, while fully occupied S/Se are shown by green spheres (see legends). (c) The crystal structure of superionic hexagonal phase of ${\mathrm{Cu}}_2\mathrm{S}$. Cu partially occupies $2b$, $4f$, and $6g$ Wyckoff sites, while S fully occupies the $2c$ Wyckoff site.

Figure 2

(a) Temperature dependence of neutron-weighted PDOS, $g^{\left(n\right)}\left(E\right)$, of cubic ${\mathrm{Cu}}_{1.95}\mathrm{Se}$ from INS measurements. (b) AIMD calculated partial contributions of Cu and Se to $g^{\left(n\right)}$ ($E$) and the total at 500 K.

Figure 3

The AIMD calculated temperature-dependent total PDOS, $g$($E$) in superionic-cubic phase of (a) ${\mathrm{Cu}}_2\mathrm{Se}$ and (b) ${\mathrm{Cu}}_2\mathrm{S}$. The presence of nonzero PDOS at zero energy corresponds to Cu diffusion.

Figure 4

(a)–(c) The calculated MSD of Cu atoms in ${\mathrm{Cu}}_2\mathrm{Se}$ and ${\mathrm{Cu}}_2\mathrm{S}$ at different temperatures using AIMD simulations performed in superionic phases. (d)–(f) Calculated temperature dependence of diffusion coefficient ($D$) estimated from MSD slope and estimated activation barrier $E_a$ for Cu migration.

Figure 5

The calculated Cu self–Van Hove correlation function $g_s\left(r,t\right)$ of (a) ${\mathrm{Cu}}_2\mathrm{Se}$ superionic-cubic phase at 500, 700, and 900 K; (b) ${\mathrm{Cu}}_2\mathrm{S}$ superionic phases at 500 K (hexagonal), 750, and 900 K (cubic).

Figure 6

The calculated self–van Hove correlation functions (black lines) at selected time intervals. The red lines show the least-squared fits comprising individual Lorentzian profiles (green lines); the results are given in Table 2.

Figure 7

The calculated Cu trajectories of a few representative Cu's in superionic phases of ${\mathrm{Cu}}_{2}X$ ($X=\mathrm{S}$, Se). The Cu trajectories are shown by the red dotted line, while the average Cu and $X$ positions are shown by spheres. Details of the average Cu and $X$ sites are shown in the legends. Cu trajectory in cubic ${\mathrm{Cu}}_2\mathrm{Se}$ (top panel) and cubic ${\mathrm{Cu}}_2\mathrm{S}$ (middle panel) shows hops between tetrahedral sites ($8c$) via an octahedral site ($4b$), and also direct jumps. Cu trajectory in hexagonal ${\mathrm{Cu}}_2\mathrm{S}$ (bottom panel) shows hops from $2b$ to $4f$ sites. In addition, all the trajectories show localized diffusion around the respective sites.

Figure 8

(a), (d) The calculated incoherent $\left[S_{\mathrm{inc}}^{\mathrm{Cu}}\left(Q,E\right)\right]$ dynamical structure factors in superionic cubic phases of ${\mathrm{Cu}}_{2}X$ ($X=\mathrm{S}$, Se). (b), (e) The estimated $Q$ dependence of the narrower Lorentzian HWHM from the incoherent $S_{\mathrm{inc}}^{\mathrm{Cu}}\left(Q,E\right)$ in superionic phases of ${\mathrm{Cu}}_2\mathrm{Se}$ (${\mathrm{Cu}}_2\mathrm{S}$) at $T=500$ K (500 K) and 700 K (750 K) is shown by the black and red markers, respectively. The solid lines in (b), (d) are the CE model fit to respective HWHM and (c), (f) the estimated $Q$ dependence of the broader Lorentzian HWHM from incoherent $S_{\mathrm{inc}}^{\mathrm{Cu}}\left(Q,E\right)$ in ${\mathrm{Cu}}_2\mathrm{Se}$ and ${\mathrm{Cu}}_2\mathrm{S}$.