Application of topological quantum chemistry in electrides

The recently developed theory of topological quantum chemistry (TQC) has built a close connection between band representations in momentum space and orbital characters in real space. It provides an effective way to diagnose topological materials, leading to the discovery of lots of topological materials after the screening of all known nonmagnetic compounds. On the other hand, it can also efficiently reveal spatial orbital characters, including average charge centers and site-symmetry characters. By using TQC theory with the computed irreducible representations in the first-principles calculations, we demonstrate that the electrides with excess electrons serving as anions at vacancies can be well identified by analyzing band representations (BRs), which cannot be expressed as a sum of atomic-orbital-induced band representations (aBRs). In fact, the floating bands (formed by the excess electrons) belong to the BRs induced from the “pseudo-orbitals” centered at vacancies. In other words, the electrides are proved to be unconventional ionic crystals, where a set of occupied bands is not a sum of aBRs, but necessarily contains a BR from vacancies. The TQC theory provides a promising avenue to pursue more electride candidates in ionic crystals.

Figure 1

(a) Crystal structure and (d) ($\overline{1}10$) lattice plane of ${\mathrm{Ca}}_2\mathrm{As}$. (b) The mBJ band structure of ${\mathrm{Ca}}_2\mathrm{As}$ with spectral weights of Ca-$d$ and As-$p$ orbitals represented by the size of the red and blue circles, respectively. (c) The orbital-resolved DOS of ${\mathrm{Ca}}_2\mathrm{As}$. (e) Electron localization function of the total electron density of ${\mathrm{Ca}}_2\mathrm{As}$, and (f) PED for the states in the energy range $-1.3$ to 0 eV on the ($\overline{1}10$) plane of ${\mathrm{Ca}}_2\mathrm{As}$. Hereafter, the corresponding vacancy sites (i.e., WKS $2b$ for ${\mathrm{Ca}}_2\mathrm{As}$) are depicted by the blue-dashed circle.

Figure 2

(a)–(c) The band structures of ${\mathrm{Ca}}_2\mathrm{N}$, LaCl, and C12A7, respectively. The spectral weight of N $p$ orbitals for ${\mathrm{Ca}}_2\mathrm{N}$ is denoted by the size of the red circles. (d)–(f) The calculated PED of the blue-colored bands for ${\mathrm{Ca}}_2\mathrm{N}$, LaCl, and C12A7 in the energy ranges of [$-1.5$ eV, 1 eV], [$-1.5$ eV, $-0.5$ eV] and [$-1$ eV, 1 eV], respectively.

Figure 3

(a) Band structures of ${\mathrm{Y}}_2\mathrm{C}$ with generalized gradient approximation. (c) Band structures of ${\mathrm{Ba}}_2\mathrm{Bi}$ with mBJ modification. The irreps $L1+$, X2–, P1 replaced by “?” can be, respectively, assigned to the irrep L2–, $X1+$, P3 marked by blue filled triangles after solving the BR decomposition. The braces show the BRs produced by the bands in the corresponding energy ranges. The PED of the states for (b) ${\mathrm{Y}}_2\mathrm{C}$and (d) ${\mathrm{Ba}}_2\mathrm{Bi}$ in the energy ranges [$-1.5$ eV, 0 eV] and [$-1$ eV, 0.5 eV], respectively.

Figure 4

The band structures of (a) ${\mathrm{Ca}}_5{\mathrm{P}}_3$, (b) ${\mathrm{Ca}}_5{\mathrm{P}}_3\mathrm{H}$, (d) NaBaO, and (e) NaBaOH. The floating bands (blue-solid lines) in ${\mathrm{Ca}}_5{\mathrm{P}}_3$ and NaBaO denote electronic states of excess electrons, whose eBRs can be well determined by the obtained irreps. To facilitate the comparison, the irreps of particular bands are marked out, and the floating bands that have “disappeared” for ${\mathrm{Ca}}_5{\mathrm{P}}_3\mathrm{H}$ are depicted by blue-dashed lines in (b). (c),(f) The crystal structures of ${\mathrm{Ca}}_5{\mathrm{P}}_3\mathrm{H}$ and NaBaOH, respectively.

Figure 5

The calculated band structures and PED of the floating bands (blue-colored lines) for (a),(b) ${\mathrm{K}}_2\mathrm{O}$ and (c),(d) ${\mathrm{Sr}}_6{\mathrm{GaN}}_5$. The PED of the states for ${\mathrm{K}}_2\mathrm{O}$ and ${\mathrm{Sr}}_6{\mathrm{GaN}}_5$ in the energy ranges [1 eV, 4 eV] and [0 eV, 2 eV], respectively. The Fermi levels of ${\mathrm{K}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ and ${\mathrm{Sr}}_6{\mathrm{Ga}}_{0.5}{\mathrm{Ge}}_{0.5}{\mathrm{N}}_5$ are indicated by red-dashed lines, which are obtained by the virtual crystal approximation as implemented in vasp.