Modeling the high-pressure solid and liquid phases of tin from deep potentials with ab initio accuracy

Constructing an accurate atomistic model for the high-pressure phases of tin (Sn) is challenging because the properties of Sn are sensitive to pressures. We develop machine-learning-based deep potentials for Sn with pressures ranging from 0 to 50 GPa and temperatures ranging from 0 to 2000 K. In particular, we find the deep potential, which is obtained by training the ab initio data from density functional theory calculations with the state-of-the-art SCAN exchange-correlation functional, is suitable to characterize high-pressure phases of Sn. We systematically validate several structural and elastic properties of the $\alpha$ (diamond structure), $\beta$, bct, and bcc structures of Sn, as well as the structural and dynamic properties of liquid Sn. The thermodynamics integration method is further utilized to compute the free energies of the $\alpha , \beta$, bct, and liquid phases, from which the deep potential successfully predicts the phase diagram of Sn including the existence of the triple-point that qualitatively agrees with the experiment.

Figure 1

Equation of state of the $\alpha$-Sn, $\beta$-Sn, bct, and bcc phases as computed by the DP-SCAN model and the DFT method with the SCAN XC functional.

Figure 2

Phonon dispersion of (a) $\alpha$-Sn and (b) $\beta$-Sn from DFT calculations with the SCAN functional (green dotted lines) and the DP-SCAN model (blue dashed lines). The experimental data for the $\alpha$-Sn and $\beta$-Sn phases (red dots) are from Refs. [71, 72], respectively. The coordinates of special $k$ points are listed in Tables S3 and S4 of Ref. [58].

Figure 3

Radial distribution functions $g\left(r\right)$ of 432-atom liquid Sn systems as obtained from molecular simulations using the DP-SCAN model (blue solid lines). The experimental data [74] (red dotted lines) are shown for comparison. The temperatures are set to 573, 773, 1073, 1673, and 1873 K. The pressure is set to zero.

Figure 4

Comparison of the predicted liquid densities from the DP-SCAN model, the RB [30], Vella [31], Etesami [32], and Ko [33] empirical force fields, and the experimental data [75]. Three different pressures (0, 10, and 20 GPa) are adopted in simulations using the DP-SCAN model.

Figure 5

Self-diffusion coefficients $D$ of liquid Sn calculated from different system sizes (128–2000 atoms) and at different pressures, including (a) 0, (b) 10, and (c) 20 GPa. The temperature ranges from 573 to 1873 K. $L$ is the cell length of the cubic cell. The extrapolation data of diffusion coefficients are listed as $1/L$ approaches zero.

Figure 6

Comparison of the predicted self-diffusion coefficients of Sn from the DP-SCAN model and the empirical force fields (RB, Vella, Etesami, and Ko), as well as the experimental data (Bruson and Gerl [80], and Itami et al. [81]).

Figure 7

Phase diagram of Sn with temperatures ranging from 0 to 2000 K and pressures ranging from 0 to 20 GPa from (a) experiments [12] and (b) the DP-SCAN model. Four phases of Sn are presented including the $\alpha$-Sn, $\beta$-Sn, bct-Sn, and liquid phases. The coexistence points on the phase diagram of (b) are labeled as A (5 GPa, 741 K), B (20 GPa, 531K), C (20 GPa, 1387 K), and D (10 GPa, 853.5 K). In addition, (c) shows the phase diagram with temperatures ranging from 0 to 1000 K and pressures ranging from 0 to 3 GPa from the DP-PBE model.

Figure 8

Radial distribution functions $g\left(r\right)$ of Sn at different temperature and pressure conditions (listed in Table 2). Results of $g\left(r\right)$ for the $\beta$-Sn, bct-Sn, and liquid Sn are shown with the pressure being (a) 5, (b) 10 GPa and (c) 15 GPa. The gray vertical lines depict the coordination numbers (CNs) of [(a) and (b)] a $\beta$-Sn structure and (c) a bct-Sn structure. The densities of the solid phases are 7.54 (a), 8.06 (b), and 8.50 $g/c{m}^3$ in (c).