Electride behavior at high pressure in silicon and other elements in solid and liquid phases

TL;DR

The study addresses identifying high-pressure electrides in elemental solids and liquids, including silicon, using ab initio MD and DFT-based topological analysis. It defines a quantitative electride criterion set: interstitial charge basins with ELF maxima above $ELF^* = 0.7$, average pocket charge $c^* = 0.9$ e, and a negative Laplacian $\nabla^2 \rho(oldsymbol{r}_0) \le -L^*$ with $L^* = 10^{-3}$ e/bohr^5. Across Si, Na, K, and Mg, various high-pressure structures exhibit electride behavior, with Si fcc becoming electride above ~400 GPa, while other elements show electrides in select phases and often under heating; pocket charges remain robust under thermal disorder, and XRD patterns show pockets modulate peak intensities by ~20% rather than introducing new peaks. The work provides a practical benchmark for identifying electrides under extreme conditions and links electronic localization to measurable XRD signatures, enabling future experimental validation.

Abstract

Electrides are materials in which some of the electrons are localized at the interstitial sites rather than around the atoms or along atomic bonds. Most elemental electrides are either alkali metals or alkaline-earth metals because of their low ionization potential. In this work, we report that elemental silicon becomes an electride at pressures exceeding 400 GPa. With {\it ab initio} molecular dynamics (MD) simulations, we study this behavior for silicon, sodium, potassium, and magnesium at high pressure and temperature. We performed simulations for liquids and ten crystal structures. Charge density and electron localization functions (ELF) are analyzed for representative configurations extracted from the MD trajectories. By analyzing a variety of electride structures, we suggest the following quantitative thresholds for the ELF and charge density in each interstitial site to classify high-pressure electrides: (1) the maximum ELF value should be greater than 0.7, (2) there should be at least 0.9 electrons near the ELF basin, and (3) the Laplacian charge density, $\nabla^2 ρ(\mathbf{r}_0)$, should be negative with magnitude greater than $10^{-3}\ e/\mathrm{bohr}^5$. Finally, we compute X-ray diffraction patterns to determine the degree to which they are affected by the electride formation. Overall, this framework could become a benchmark for future theoretical and experimental studies on electrides.

Figures (12)

• Figure 1: Top: ELF isosurface at a value of 0.7 is shown in yellow for the fcc structure (left) at 500 GPa and for the bcc structure (right) of Si at 3500 GPa. Si bcc has 4 yellow dots on the faces of the cell, each counted as a pocket. While the pockets at 0 K seem connected by the green isosurface, that is due to setting the ELF isovalue at ELF = 0.7. Bottom: ELF contour plot along the (110) plane of the fcc unit cell with the [111] direction highlighted as a black line (left) along which the values of ELF are shown on the right. The ELF basins (pockets) correspond to locations where the ELF reaches a local maximum, ${\rm ELF}_{\rm max}>0.7$, but whose value is lower than at atomic sites.
• Figure 2: Left and middle: Unit cell of the cI16 structure of sodium at 200 GPa with Na atoms colored in green, critical points at non-nuclear sites in black, and ELF isosurface in yellow. Right: a line profile along the [001] crystalline direction starting at the point (0, 1/2, 0) in blue (along type A maxima), and starting from the point (0.34288,0.34288,0) in red (along type B maxima). At 500 K, the ELF values at both type A and B maxima barely change, but they can decrease strongly in between type B maxima, indicating more delocalization between these pockets.
• Figure 3: Electronic localization function (ELF) in Mg (left) and Si (right) in the fcc structure. In Mg (1000 GPa), charge basins contain a charge of 1.7 electrons and are located at octahedral positions along the cube edges, while in Si (500 GPa), the charge basins are inside the cube in the tetrahedral sites and contain two electrons.
• Figure 4: Pressure-induced localization of electrons in simple-hexagonal (top) and simple-cubic (bottom) Mg. All panels depict the isosurface defined by the ELF value of 0.7.
• Figure 5: Silicon: (a) The average ELF maximum value of each pocket, (b) the average number of electrons per pocket, (c) the average number of pockets per atom, and (d) the average number of electrons in charge pockets per atom vs pressure for Si (starting at 200 GPa). The different shading represents the different stable structures in those pressure regimes. The transparent points represent the T = 0K data, the filled-in shapes matching their backgrounds represent the heated solids, and the red circles represent the melted structures (i.e. liquids). The standard error bars for most data points are smaller than the markers. e-f) 3D ELF (isovalue $=$ 0.7) of Si (blue atoms) from top to bottom of fcc and bcc, respectively. e) Si fcc unit cell at 0 K (left), and the heated super cell (right). f) The Si bcc unit cell at 0 K and heated super cell at 2000 K. There were 4 uniform pockets on each face of the cell at 0 K, represented by the 4 yellow dots. This equates to 6 pockets per atom for Si bcc at 0 K. While the pockets at 0 K seem connected by the green isosurface, that is due to setting the ELF isovalue at ELF = 0.7. At 2000 K, some of those pockets merge, going from 6 pockets per atom to around 2 (plot c). The ELF pockets on the faces of the bcc cells have colors representing the relative ELF intensities, increasing from blue to green to red.