Lattice dynamics and structural phase stability of group IV elemental solids with the r$^2$SCAN functional

TL;DR

This study benchmarks the r$^2$SCAN meta-GGA against SCAN, PBE, and HSE for the diamond-structure group IV solids C, Si, Ge, and Sn, focusing on elastic properties, lattice dynamics, and structural phase transitions. It finds that r$^2$SCAN closely reproduces SCAN for elastic constants and phonon dispersions while offering improved numerical stability, making it attractive for high-throughput lattice-dynamics; however, it overestimates the α↔β transition energetics and transition pressures for Ge and Sn, unlike SCAN. PBE remains underbinding and misrepresents Ge and Sn electronic structure, whereas HSE provides the best phonon accuracy among the tested functionals. The results suggest r$^2$SCAN is a compelling, efficient alternative to SCAN for lattice-dynamics studies, with caveats for phase-transition predictions that require higher-level benchmarks or broader testing across materials.

Abstract

The strongly constrained and appropriately normed (SCAN) meta-GGA functional is a milestone achievement of electronic structure theory. Recently, a revised and restored form (r$^2$SCAN) has been suggested as a replacement for SCAN in high-throughput applications. Here, we assess the accuracy and reliability of the r$^2$SCAN meta-GGA functional for the group IV elemental solids carbon (C), silicon (Si), germanium (Ge), and tin (Sn). We show that the r$^2$SCAN functional agrees closely with its parent functional SCAN for elastic constants, bulk moduli, and phonon dispersions, but the numerical stability of r$^2$SCAN is superior. Both meta-GGA functionals outperform standard GGA (Perdew-Burke-Ernzerhof) in terms of accuracy and approach the level of common hybrid functionals (Heyd-Scuseria-Ernzerhof). However, we find that r$^2$SCAN performs much worse than SCAN for the $α\leftrightarrow β$ phase transition of both Ge and Sn, yielding larger phase energy differences and transition pressures.

Figures (9)

• Figure 1: Comparison of the (left) $\alpha$-Sn and (right) $\beta$-Sn crystal structures Momma2011. The $\beta$-Sn structure can be obtained from the $\alpha$-Sn structure (diamond, shown here in a body-centered tetragonal unit cell) by applying a tetragonal compression along (001) Yin1982aMoll1995Mehl2021. The two phases coincide at $c/a = \sqrt{2} \approx 1.414$, while the ideal $\beta$ phase corresponds to sixfold coordination at $c/a = \sqrt{4/15} \approx 0.516$Christensen1993.
• Figure 2: Phonon dispersion relations for 128-atom supercells of group IV elemental solids in the diamond phase: (a) C, (b) Si, (c) Ge, and (d) Sn. Phonon dispersions calculated with the PBE, r$^2$SCAN, and SCAN functionals are given as lines, whereas symbols indicate experimental data Warren1967Kulda2002Kulda1994Nilsson1971Nilsson1972Dolling1963Price1971.
• Figure 3: Energy–volume curves for the $\alpha$ and $\beta$ phases of (a) Si, (b) Ge, and (c) Sn, calculated with PBE, r$^2$SCAN, and SCAN. Energies are referenced to the minimum of the $\alpha$-phase curve for each element and functional. The solid lines are Birch–Murnaghan fits to the data. The dash–dotted lines show the common tangent to each pair of $E(V)$ curves; its negative slope equals the transition pressure $P_{\rm t}$ reported in Table \ref{['tab:transition_pressures']}.
• Figure 4: Charge density differences $\Delta n(\mathbf{r}) = n_{\mathrm{r^2SCAN}}(\mathbf{r}) - n_{\mathrm{SCAN}}(\mathbf{r})$ for the $\alpha$ phase of (a) Si, (b) Ge, and (c) Sn; and for the $\beta$ phase of (d) Si, (e) Ge, and (f) Sn. Top panels show the $\alpha$ phases in the $(0\bar{1}1)$ plane, and bottom panels show the $\beta$ phases in the (110) plane. Experimental geometries are used where available (see Tables \ref{['tab:lattice_constants']} and \ref{['tab:transition_pressures']}), while SCAN geometries are used for $\beta$-Si and $\beta$-Ge. Blue circles indicate the positions of the atomic nuclei, and the nearest-neighbor bonds are indicated by blue lines. Dark areas indicate positive charge density differences ($n_{\rm r^2SCAN}>n_{\rm SCAN}$), and light areas indicate negative charge density differences ($n_{\rm r^2SCAN}<n_{\rm SCAN}$). Contour lines are drawn in steps of 0.5 electrons per cell volume Momma2011. Negative contours lines are indicated in white.
• Figure S1: Convergence with respect to the plane-wave energy cutoff, varied from $1.2\times\texttt{ENMAX}$ to $2.0\times\texttt{ENMAX}$, where ENMAX is the default cutoff of the respective pseudopotentials. All convergence curves are shown for the three functionals PBE, r$^2$SCAN, and SCAN, and the labels A and B in the legend refer to the numerical setups A and B listed in Table \ref{['tab:numerical setup']}. Panel (a) shows the bulk modulus $B_0^{\mathrm{EOS}}$ obtained from Birch-Murnaghan equation-of-state fits [Eq. \ref{['eq:BM EOS']}]. Panel (b) shows the bulk modulus $B_0^{\mathrm{elastic}}$ evaluated from the elastic constants using Eq. (\ref{['eq:bulk-with-C']}) of the main text. $B_0^{\mathrm{ref}}$ in panels (a) and (b) denotes a reference bulk modulus obtained from an equation-of-state fit at a very high cutoff (4$\times$ENMAX). Panel (c) shows the convergence of the $\Gamma$-point phonon frequency. In all panels, the black vertical line marks the cutoff $1.8\times\texttt{ENMAX}$, which is the value adopted for the final elastic-constant calculations (Tables \ref{['tab:bulk moduli']} and \ref{['tab:elastic_constants_EXPT']} in the main text) and phonon-frequency calculations (Table \ref{['tab:vibrational_properties_combined']} in the main text).