Computer-Assisted Inverse Design of Inorganic Electrides

Electrides are intrinsic electron-rich materials enabling applications as excellent electron emitters, superior catalysts, and strong reducing agents. There are a number of organic electrides; however, their instability at room temperature and sensitivity to moisture are bottlenecks for their practical uses. Known inorganic electrides are rare, but they appear to have greater thermal stability at ambient conditions and are thus better characterized for application. Here, we develop a computer-assisted inverse-design method for searching for a large variety of inorganic electrides unbiased by any known electride structures. It uses the intrinsic property of interstitial electron localization of electrides as the global variable function for swarm intelligence structure searches. We construct two rules of thumb on the design of inorganic electrides pointing to electron-rich ionic systems and low electronegativity of the cationic elements involved. By screening 99 such binary compounds through large-scale computer simulations, we identify 24 stable and 65 metastable new inorganic electrides that show distinct three-, two-, and zero-dimensional conductive properties, among which 18 are existing compounds that have not been pointed to as electrides. Our work reveals the rich abundance of inorganic electrides by providing 33 hitherto unexpected structure prototypes of electrides, of which 19 are not in the known structure databases.

Popular Summary

Electrides are crystalline compounds that are rich with electrons. These materials have attracted great attention because they can act as excellent electron emitters, superior catalysts, and strong reducing agents. Organic electrides, in which organic molecules help isolate electrons from positive ions in the crystal, are not stable at room temperature and are sensitive to moisture. Inorganic electrides appear to be better suited to practical applications; however, the search for new materials is challenging. So far, only three kinds of inorganic electrides have been reported. We develop a computer algorithm that can efficiently search for a variety of inorganic electrides and identify 89 new candidates.

Our technique relies on the principle of inverse design, which starts with a desired property (in our case, a specific electronic trait of electrides) and then searches for materials that have that property. This lets us search for possible inorganic electrides unbiased by preconceived notions of electride structures. The algorithm starts by generating random crystal structures, and then iteratively tweaks those structures, optimizing the desired electronic property. Among the 89 new compounds identified, 18 are existing compounds that have not yet been classified as electrides while the remaining ones have a chance of being synthesized.

This technique presents a significant advancement toward automated, intelligent design of inorganic electrides. While only a few inorganic electrides are currently known, and their practical applications are limited, we believe our methodology can substantially change this situation.

Figure 1

Computational details of calculation of $\omega$ and inverse-design scheme for electrides. (a) A step-by-step process of calculating $V_{\text{inter}}$ in a tetragonal ${\mathrm{Sr}}_2\mathrm{C}$ crystal. In the ELF map with an isosurface of 0.75, the regions for localized electrons are shown as yellow. Exclusion of ionic cores areas for localized electrons from the crystal as illustrated by Sr and C atoms (top panel). Exclusion of $\mathrm{C}─\mathrm{C}$ covalent bonding areas from the crystal (middle panel). The volume of the interstitial areas where the electrons are localized ($V_{\text{inter}}$, shown as yellow in ELF map in bottom panel). (b) Flow chart of the calypso module for searching for electrides with the inverse-design method.

Figure 2

Benchmarking to two known electrides: ${\mathrm{Ca}}_2\mathrm{N}$ at ambient pressure and transparent Na at 320 GPa. $\omega$ versus energy maps of ${\mathrm{Ca}}_2\mathrm{N}$ at ambient pressure (a) and transparent Na at 320 GPa (b). Symbols represent structures produced by the calypso run within 30 generations. The experimental anti-${\mathrm{CdCl}}_2$ and $hP4$ electride structures for ${\mathrm{Ca}}_2\mathrm{N}$ and Na are shown as red stars in the maps. (c) Evolution of the optimum $\omega$ versus generations for ${\mathrm{Ca}}_2\mathrm{N}$ using $\omega$ (red line) and total energy (blue line) as global variable functions, respectively. The $x$ axis is the $n$th generation in an inverse-design simulation by calypso.

Figure 3

Tests for the design principles. (a) ELF of ${\mathrm{Ca}}_2\mathrm{N}$ (left-hand panel) on the ${(110)}_R$ plane parallel to the hexagonal $c$ axis. Upon removal of one excess electron per ${\mathrm{Ca}}_2\mathrm{N}$, the system loses electron localization (right-hand panel). Formation probabilities of electrides (ratios of electride phases to all structures generated by calypso code in a simulation, characterizing the probability of the system to form electrides) in metal nitrides (b) and ${\mathrm{Ca}}_{2}X$ compounds (c). Metal and $X$ elements are listed on the $x$ axis and the formation probabilities of electrides are shown on the $y$ axis, respectively. The detailed testing metal nitrides and ${\mathrm{Ca}}_{2}X$ compounds with their formation probabilities in (b) and (c) are listed in Tables S1 and S2 of Supplemental Material [39].

Figure 4

Inverse-design results for binary electrides. (a) Stability map of $A_{2}B$ (top) and $AB$ (bottom) ($A$, electron donor; $B$, electron acceptor) electrides. $A$ and $B$ elements are shown in horizontal and vertical directions, respectively. Stable electrides are pointed to the electride structures with negative formation energies in their lowest-energy states, i.e., the thermodynamically most stable states, while metastable electrides are those either not in the lowest-energy states or having positive formation energy. Blue and red squares indicate stable and metastable existing compounds, respectively. Green and yellow circles indicate stable and metastable compounds yet to be synthesized, respectively. (b) Type map of $A_{2}B$ (top) and $AB$ (bottom) electrides. Triangles, crosses, and diamonds indicate 3D, 2D, and 0D electrides, respectively.

Figure 5

Structural and electronic properties of ${\mathrm{Ca}}_2\mathrm{C}$ and ${\mathrm{CaC}}_2$. (a) Crystal structure (left-hand panel) and ELF (right-hand panel) on the ${(0\overline{1}2)}_R$ plane of ${\mathrm{CaC}}_2$. (b) Crystal structure (left-hand panel) and ELF (right-hand panel) on the ${(012)}_R$ plane of ${\mathrm{Ca}}_2\mathrm{C}$. Excess electrons localize in the interstitial regions indicated by dashed circles. (c) Electronic properties of ${\mathrm{Ca}}_2\mathrm{C}$: band structure and projected density of states (PDOS) (left-hand panel), and partial electron density for region near $E_F$ ($|E|<0.05\mathrm{eV}$) on the ${(012)}_R$ plane (right-hand panel). The dashed line indicates the Fermi energy ($E_F$).

Figure 6

Structural and electronic properties of LaCl. (a) 3D ELF of LaCl with an isosurface value of 0.7. (b) 2D ELF (top panel) on the ${(\overline{1}\overline{1}0)}_R$ plane and partial electron density for the region near $E_F$ ($|E|<0.05\mathrm{eV}$) (bottom panel) on the ${(\overline{1}\overline{1}0)}_R$ plane. (c) Band structure and electron density of state of LaCl. The dashed line indicates the Fermi energy ($E_F$). (d) The high-symmetry lines (red) are in the first Brillouin zone.

Figure 7

Semiconducting ${\mathrm{Be}}_2\mathrm{N}$ as a 0D electride. (a) 3D ELF of ${\mathrm{Be}}_2\mathrm{N}$ (left-hand panel) with an isosurface value of 0.83. Sixfold and fivefold coordinated Be atoms are shown as structural units in the left-hand panel. The 2D ELF on the ${(110)}_R$ plane in the right-hand panel shows the isolated interstitial localization regions consistent with the 3D ELF. (b) Band structure and PDOS of ${\mathrm{Be}}_2\mathrm{N}$. The HSE06 functional [47, 48] corrects the DFT band gap of 1.51 eV to 1.96 eV. The dashed line indicates the Fermi energy ($E_F$).