Exploring diffusion behavior of superionic materials using machine-learning interatomic potentials

Superionic materials possess mobile atoms with liquidlike behavior in the rigid frameworks of other atoms. Theoretically, the diffusion behavior of the mobile atoms is usually probed by ab initio molecular dynamics simulations where enormous computing resources are requested for a complete thorough study. Thus, only limited cases are investigated without providing the most critical quantity, such as the diffusion barrier. To address this shortcoming, we perform molecular dynamics simulations based on machine-learning interatomic potentials, fitted from ab initio molecular dynamics simulations, to have complete studies for ${\mathrm{Ag}}_2\mathrm{S}, {\mathrm{Ag}}_8{\mathrm{SiTe}}_6, {\mathrm{Cu}}_2\mathrm{S}$, and ${\mathrm{Zn}}_{3.6+x}{\mathrm{Sb}}_3$ systems. Our results indicate that the Arrhenius equation can describe very well the diffusion behaviors of the studied superionic systems where the activation barriers range from 0.09–0.22 eV. The small diffusion barrier provides the fundamental origin for the liquid behaviors of superionic materials.

Figure 1

The mean-square displacements of (a) Ag and (b) S atoms of ${\mathrm{Ag}}_2\mathrm{S}$ at temperatures ranging from 450–800 K.

Figure 2

(a) The constant S density surface (at 20% of maximum density) viewed from [100] direction; (b), (c), and (d) The constant Ag density surface (at 50% of maximum) viewed from [100], [110], and [111] directions for ${\mathrm{Ag}}_2\mathrm{S}$ at the 750 K. The constant density surfaces of ADDF were plotted using vesta program [32].

Figure 3

The diffusion barrier of Ag atoms for ${\mathrm{Ag}}_2\mathrm{S}$ system obtained by fitting of the Arrhenius equation.

Figure 4

The probability density function (PDF) curves of root-mean-square displacements of Ag atoms in the unit of Å (a) or one RMSD unit (b) for the ${\mathrm{Ag}}_2\mathrm{S}$ system.

Figure 5

The atomic density distribution functions (a), (b) of Ag atoms and (c), (d) of Te atoms from ${\mathrm{Ag}}_8{\mathrm{SiTe}}_6$ system (3840 atoms) at 300 K viewed from [100] and [111] direction. The results are obtained by summarizing four individual MD runs of 20 ns and applying the experimental suggested symmetry of $F\overline{4}3m$. The constant density surfaces of ADDF were plotted using the vesta program [32].

Figure 6

The atomic density distribution functions (a), (b) of Ag atoms and (c), (d) of Te atoms from ${\mathrm{Ag}}_8{\mathrm{SiTe}}_6$ system (30720 atoms) at 500 K viewed from [100] and [111] direction. The results are obtained by summarizing one MD run of 1 ns and applying the experimental suggested symmetry of $F\overline{4}3m$. The constant density surfaces of ADDF were plotted using the vesta program [32].

Figure 7

The probability density function (PDF) curves of root-mean-square displacements of Ag atoms for the ${\mathrm{Ag}}_8{\mathrm{SiTe}}_6$ system in unit of one RMSD unit for (a) $225\sim 375\mathrm{K}$ and (b) $400\sim 550\mathrm{K}$ at MD simulation time 250 ps; shown in (c) and (d) are the RMSD of Ag atoms at 225 and 550 K for different MD simulation times at 0.25, 0.5, 1 ns.

Figure 8

The diffusion barrier of Ag atoms for ${\mathrm{Ag}}_8{\mathrm{SiTe}}_6$ system obtained by fitting of the Arrhenius equation.

Figure 9

(a), (b) The temperature dependence of MSD; and the diffusion barrier of Cu atoms for (c) $\beta \text{−}{\mathrm{Cu}}_2\mathrm{S}$ and (d) $\gamma \text{−}{\mathrm{Cu}}_2\mathrm{S}$ phases obtained by fitting of Arrhenius equation.

Figure 10

The probability density function (PDF) curves of root-mean-square displacements of Cu atoms for the ${\mathrm{Cu}}_2\mathrm{S}$ system in unit of one RMSD unit for $\beta$ phase (a) $400\sim 750\mathrm{K}$ and $\gamma$ phase (b) $750\sim 1000\mathrm{K}$ at MD simulation time 250 ps.

Figure 11

The atomic density distribution functions of Zn atoms at 350 K. The results are obtained by summarizing one MD run of 500 ps and applying the experimental suggested symmetry of R-3c.

Figure 12

The diffusion barrier of Zn atoms in ${\mathrm{Zn}}_{3.6+x}{\mathrm{Sb}}_3$ ($x=0.05\sim 0.2$) obtained by fitting of Arrhenius equation.

Figure 13

The probability density function (PDF) curves of root-mean-square displacements of Zn atoms for the ${\mathrm{Zn}}_{3.8}\mathrm{Sb}3$ system for (a) $300\sim 650\mathrm{K}$ at 250 ps and (b) 650 K at 0.5–3.5 ns in the step of 0.5 ns.