First-principles design of halide-reduced electrides: Magnetism and topological phases

We propose a design scheme for potential electrides derived from conventional materials. Starting with rare-earth-based ternary halides, we exclude halogen anions and perform global structure optimization to obtain thermodynamically stable or metastable phases but having an excess of electrons confined inside interstitial cavities. Then, spin-polarized interstitial states are induced by chemical substitution with magnetic lanthanides. To demonstrate the capability of our approach, we test with 11 ternary halides and successfully predict 30 stable and metastable phases of nonmagnetic electrides subject to three different stoichiometric categories, and successively 28 magnetic electrides via chemical substitution with Gd. 56 out of these 58 designed electrides are discovered for the first time. Two electride systems, the monoclinic $A\mathrm{C}$ ($A=$ La, Gd) and the orthorhombic $A_2\mathrm{Ge}$ ($A=$ Y, Gd), are thoroughly studied to exemplify the set of predicted crystals. Interestingly, both systems turn out to be topological nodal line electrides in the absence of spin-orbit coupling and manifest spin-polarized interstitial states in the case of $A=$ Gd. Our work establishes a novel computational approach of functional electrides design and highlights the magnetism and topological phases embedded in electrides.

Figure 1

Flowchart of designing halide-derived electrides.

Figure 2

A statistical overview of predicted materials. (a)–(f) Convex hull diagrams of binary compounds $A_{n}B$ ($A=$ Y, La, Gd; $B=$ C, Ge). Electride candidates are colored by red, and distinguished by the attached numbers that represent their energy rankings. Notice that the number 1 means the corresponding phase has the lowest formation energy merely within the predicted electride structures; there could be a nonelectride structure that is more stable. Gray dots stand for the other (meta)stable structures predicted by the CSP, which show no evidence of electrides or have relatively high formation energy. No gray dots in diagrams of the Gd systems, since the Gd-based structures are obtained by chemical substitution instead of the CSP. (g) Stability (measured by $E_{\text{hull}}$) distribution of 58 designed electrides. (h) New stable polymorphs ($E_{\text{hull}}=0$) of each stoichiometric category discovered by the CSP. Nonelectride systems are also presented.

Figure 3

Electronic properties from the bulk calculation of LaC. (a) Primitive cell of LaC. Yellow bubbles represent the ELF. For a clear visualization, the ELF around atoms is not displayed. (b) The Brillouin zone of LaC. The high-symmetry points and the path chosen are highlighted in red. The $y$-direction projected Brillouin zone is indicated too. (c) Projected band structure of LaC along the path denoted in (b). The Fermi energy is set 0. As the color changes from red to blue, the contribution of interlayer states diminishes. (d) Nodal line of LaC in the reciprocal space. The left and right panels are viewed from x and y directions, respectively. (e) The Zak phase integrated along the y direction. Green and white separately stand for $\pi$ and 0 values. High-symmetry points in this reduced reciprocal space are also shown. (f), (g) Real branches of wave functions with the corresponding positions indicated in (c). The yellow and cyan mean positive and negative parts, respectively; red and blue stand for their cross sections.

Figure 4

Electronic structure of a 15-layer (010) LaC slab. (a) Electronic band structure of the LaC slab. High-symmetry points in the reduced Brillouin zone are given by Figs. 3 or 3. Energy is measured from the Fermi level. The red indicates a large proportion of floating electronic states that localize on the top and bottom of the slab. (b) The squared wave function of a surface state localized around the top layer. The corresponding point in the band structure is marked by a green circle in (a).

Figure 5

Electronic properties of GdC. (a) Primitive cell of GdC. Partial charge densities around the Fermi level ($-0.05\sim 0.05$ eV) are shown for spin up (left) and spin down (right). Yellow bubbles represent the partial charge density, the green and blue are the cross sections. (b) Electronic band structure of FM GdC for spin up (top) and spin down (bottom). The high-symmetry points are given in (c). Arrows are indicative of the degeneracies between the highest VB and the lowest CB. (c) Nodal lines of FM GdC for the spin-up (left) and spin-down (right) channels. The Brillouin zone together with high-symmetry points are depicted too. (d) The Zak phase for spin up (left) and spin down (right), integrated along the y direction, i.e., $\mathrm{\Gamma }\text{−}X$ direction. Green and white mean $\pi$ and 0 values, respectively.

Figure 6

Electronic properties from the bulk calculation of ${\mathrm{Y}}_2\mathrm{Ge}$. (a) Primitive cell of ${\mathrm{Y}}_2\mathrm{Ge}$. Yellow bubbles are ELF. The isosurface value in the bottom panel is smaller than those in the top and middle panels. Connected interstitial states are highlighted by a dashed red circle in the bottom panel. (b) Projected band structure of ${\mathrm{Y}}_2\mathrm{Ge}$ with the high-symmetry points denoted in (c). The Fermi energy is set 0. As the color changes from red to blue, the contribution of interstitial states decays. Some gapless points are indicated by arrows. (c) Nodal lines of ${\mathrm{Y}}_2\mathrm{Ge}$ accompanied by the Brillouin zone. Two inequivalent groups of nodal lines are labels as NL1 and NL2, respectively. Spinless Weyl points are also displayed, with the orange and magenta representing the monopole charge $+1$ and $-1$ separately. (d), (e) show the real branches of the wave functions at the positions indicated in (b); the positive part is in yellow and the corresponding cross sections are colored by red; the cyan and blue are for the negative part.

Figure 7

Electronic properties of ${\mathrm{Gd}}_2\mathrm{Ge}$. (a) Primitive cell of ${\mathrm{Gd}}_2\mathrm{Ge}$. Partial charge densities around the Fermi level ($-0.05\sim 0.05$ eV) are presented for spin up (left) and spin down (right). The yellow bubbles represent the partial charge density, and the green means the cross section. (b) Electronic band structure of FM GdC for the spin-up (top) and spin-down (bottom) parts. The high-symmetry points are indicated in (c). Degenerate points between the highest VB and lowest CB are marked by arrows. (c) Nodal lines of FM GdC for spin up (left) and spin down (right). The Brillouin zone, together with high-symmetry points, are depicted too. The blue or green distinguishes nodal lines that are distributed on the vertical or horizontal mirror plane. Weyl points are also displayed, with the orange and magenta representing the monopole and antimonopole, respectively.

Figure 8

Primitive cells of the initial compounds that serve as electride hosts. Boxes classify the stoichiometric categories. All the compounds are subject to halides, with one exception ${\mathrm{La}}_2{\mathrm{C}}_2{\mathrm{H}}_2$ (a hydride) that makes no difference after taking away the hydrogens.

Figure 9

Primitive cells of newly discovered crystals that correspond to the ground polymorphs in accordance with our calculations. The left three are shown to be potential electrides, with their ELF distributions displayed.

Figure 10

Phonon DOS of (a) LaC and (b) ${\mathrm{Y}}_2\mathrm{Ge}$, respectively.

Figure 11

Magnetic ground states of (a) GdC and (b) ${\mathrm{Gd}}_2\mathrm{Ge}$, respectively. The red arrow stands for the direction of each spin.

Figure 12

Electronic band structures computed using the PBE pure functional and the HSE06 hybrid functional, indicated by solid and dashed lines, respectively. Degenerate points relating to nodal lines are marked by arrows. (a) is for ${\mathrm{Y}}_2\mathrm{Ge}$ and (b) is for spin down of ${\mathrm{Gd}}_2\mathrm{Ge}$.