Multi‐Dimensional Topological Fermions in Electrides

Topological electrides have attracted extensive attention in various fields, for example, electrocatalysis, spintronics, electron emitters, etc., due to their non‐trivial topological surface states and unique electronic properties. It is well known that topologically protected nontrivial surface states are not broken by external perturbations and further exhibit high carrier mobility and high electron density on some specific surfaces. In addition, electrides usually possess a lower work function due to the presence of approximately loose excess electrons. In this case, topological electrides not only build a bridge between topological materials and electrides, but also couple various excellent properties of these two materials. Since the concept of topological electrides was first proposed, several novel types of topological electrides have been reported in the last few years. Therefore, it is necessary to give a comprehensive review of these topological electrides. In this review, the history of the development of topological electrides and their current status is systematically summarized. In addition, relevant insights into the challenges and opportunities facing topological materials are provided.


Electrides
Unlike conventional ionic, covalent, and intermetallic compounds, electrides belong to the typical electron-rich materials. Specifically, due to the presence of different-dimensional-cavities in the crystal structure, the excess electrons of the electrides can be transferred to the interstitial positions and act as anions. [1][2][3] In general, electrides can be classified as organic and inorganic DOI: 10.1002/apxr.202200119 based on their atomic composition. In fact, the electrides originated from the formation of "solvated electron" in ammonia solution of alkali metals at the earliest. [4] Subsequently, some compounds with similar structure were reported, such as Mott insulator Cs + (15crown-5) 2 ·e − , Cs + (18-crown-6) 2 ·e −, Li + (cryptand-2.1.1)e − etc., [5][6][7] and defined as electrides. In other words, electrides originate from organic compounds. However, researchers found that the chemical stability of organic electrides is extremely poor, namely organic electrides are easily oxidized and hydrolyzed, which greatly limits the study of their physical and chemical properties. [1][2][3][4][5][6][7] Therefore, it is extremely important to explore stable inorganic electrides.
After that, Zhang et al. [19] supposed whether there existed a 1D channel to carry the excess electrons? They gave a positive answer, namely that, the metal Ti reduced the free O 2− at the four boundaries of La 8 Sr 2 (SiO 4 ) 6 O 2 at high temperature, causing that the excess electrons can be transferred to the 1D channels, forming the typical 1D electride, namely La 8 Sr 2 (SiO 4 ) 6 :4e − . Subsequently, Wang et al. [20] identified a series of Sr-P phases through structural search. Based on DFT calculations, it was found that Sr 5 P 3 and Sr 8 P 5 are typical 1D and 0D electrides. Furthermore, the predicted Sr 5 P 3 compound was successfully synthesized. By measuring its electrical transport properties and UPS spectra, it was found that Sr 5 P 3 is the first reported 1D inorganic electride with the characteristics of a Mott insulator, which is in sharp contrast to the 2D electrides with metallic feature. Correspondingly, the electrides form a fundamental frame from 0D to 2D, as shown in Figure 1a-c.

Judging Index of Electrides
Well known, electrides belong to typical electron-rich materials. On the contrary, not all electron-rich materials possess the properties of electrides, because the formation of electrides needs to meet three conditions [30] : 1) except for some special highpressure electrides, the overall valence of the compounds should be greater than zero, in order to provide the necessary excess electrons; 2) the crystal structure should exist a large interstitial position to prevent excess electrons from bonding with orbital electrons of nearby atom; 3) the cations should occupy around the interstitial positions, so that the cavity can capture excess electrons. Then, due to the rapid development of theoretical calculation, currently electrides usually adopt four judgment indexes : [29][30][31]38] 1) electron local function (ELF); 2) partial charge density (PCD) near Fermi level; 3) density of states (DOS) near Fermi level; 4) topological quantum chemical theory (TQC). These indexes provide a strong support for the determination of electrides.

Applications of Electrides
As we stated at the beginning, inorganic electrides usually possess a high conductivity and a low work function. Due to these excellent properties, inorganic electrides are widely used in high-performance catalysts, electron emitters, spintronic devices, nonlinear optics, high-pressure superconductors, high electron mobility, ion batteries, low-temperature superconductivity etc. [1][2][3][11][12][13][14][39][40][41] In the context of these broad applications, inorganic electrides are the most representative in promoting ammonia synthesis. Next, we will fully present the high advantages of inorganic electrides in ammonia synthesis.
It is well known that some nano-metallic particles (e.g., metal Ru) are usually used as catalysts in the process of ammonia syn- Figure 2. a) The crystal structure of 1D Y 5 Si 3 /Ru, the curve of time and ammonia production in 1D Y 5 Si 3 /Ru. Adapted with permission. [40] Copyright 2016, American Chemical Society. b) Schematic diagram of ammonia synthesis of electrides LaRuSi loaded Ru (LaRuSi/Ru). Working principle: First, the reversible hydrogen storage ability could suppress hydrogen poisoning during ammonia synthesis. Second, electrides with the lower work function can provide excess electrons for Ru to facilitate the cleavage of N 2 . Adapted with permission. [43] Copyright 2022, American Chemical Society.
thesis, but hydrogen poisoning is common in this process, which greatly reduces the production and rate of ammonia. [1][2][3]13,14,42,43] With the gradual understanding of inorganic electrides, researchers found that when inorganic electrides are loaded with nano-metallic particles, they not only solve the bottleneck of hydrogen poisoning, but also further reduce the barrier of nitrogen cleavage and the reaction temperature (e.g., Ca 2 N/Ru ≈ 200°C [44] ) in the process of ammonia synthesis. [13,14,[42][43][44] The main principles are as follows: 1) inorganic electrides usually possess high hydrogen storage capacity due to the existence of large lattice cavities. More importantly, inorganic electrides show excellent reversible hydrogen storage capacity, which has also been confirmed in numerous experiments and theoretical analysis, such as 0D electrides C12A7:4e − and LaXSi (X = Co, Fe, Mn), 1D electride Sr 3 CrN 3 :e − . [13,14,42,45] 2) Inorganic electrides usually have a strong ability to supply electrons due to their low work function. Therefore, when the inorganic electrides are loaded with nano-metal particles, they will provide a large number of free electrons to the surface of the nano-metal particles. These favorable conditions can not only reduce the barrier of nitrogen cleavage, but also accelerate the rate of nitrogen cleavage, thus greatly improving the efficiency of ammonia synthesis. Among them, C12A7:4e − /Ru, Ca 2 N:e − /Ru and 1D electride Y 5 Si 3 /Ru are three typical examples [13,14,44,40] (see Figure 2a).
Remarkably, researchers found that when inorganic electrides are loaded with some nano metal particles, there are usually two problems: 1) It is difficult to experimentally control the growth of nano-metal particles on the surface of inorganic electrides; 2) The nano-metal particles easily accumulate on the surface of inorganic electrides. To solve these two problems, researchers have developed some independent inorganic electrides, which can still address the problems faced by nano metal particles in the process of ammonia synthesis, such as 0D electrides LaXSi (X = Co, Sc, Ru) [43,46,47] (see Figure 2b). In short, compared to traditional transition metals, inorganic electrides show an absolute advantage in industrial ammonia synthesis.
It is well known that high carrier mobility, high electronic densities near Fermi level can facilitate electron transfer during the electrocatalytic process. [48,49] Benefiting from linear band crossings and the topological surface states, topological semimetals commonly possess high carrier mobility. The Fermi arc surface states can induce high electronic density of states on the surface which then improve the catalytic performance. At present, some work has reported that topological semimetals show important application prospects in ammonia synthesis, such as Dirac semimetals Ti/NbB 2 , [50] nodal line Cu 2 Si. [51] Whether topological states are introduced into electrides will further promote the synthesis of ammonia remains to be studied.

Topological Materials
Before discussing topological semimetals, let us briefly describe the differences between conventional insulators, topological insulators, and topological semimetals from electronic band theory. First, for conventional insulators, there is a large band gap between the conduction band and the valence band, which makes it difficult for the valence electrons to jump to the conduction band. Therefore, conventional insulators do not conduct electricity either on the surface/boundary or inside the material. For topological insulators, their electronic structure shows the same characteristics as that of conventional insulators, while there are two opposite spin channels on the surfaces or boundaries of topological insulators. These two channels facilitate the unimpeded transfer of electrons from the valence band to the conduction band, leading to the peculiar property that the surface or boundary of the material is electrically conducting but the interior is not. However, topological semimetals are substantially different from conventional insulators and topological insulators, originating from the nature metallic feature of these electronic structures. Furthermore, these bands form different types of crossing points near the Fermi level.

Nodal Point Semimetals
In quantum field theory, fermions usually exist in three categories: Majorana fermions, Weyl fermions, and Dirac fermions. [68] Remarkably, the discovery of Weyl semimetals laid the foundation for the development of topological semimetals. In 2011, Wan et al. [55] first discovered the ternary compound Y 2 Ir 2 O 7 can host several crossing points near the Fermi level. These crossing points are double degenerate points formed by two linear single bands, denoted as Weyl points (WPs). Furthermore, they found that these WPs possessed two special properties, namely opposite chirality, and topologically protected Fermi arc surface states, as shown in Figure 3d. Subsequently, based on the dispersion of the degenerate bands, the WPs can further exhibit type-I, type-II, critical-type, and type-III dispersion. [69][70][71][72][73][74][75] In this case, the researchers proved theoretically that the dispersion types of WPs determine the morphology of Fermi surfaces. [73][74][75] Specifically, type-I Weyl semimetals show a point-like Fermi surface.
The Fermi surface of the type-II WPs shows that electron and hole-pockets can coexist at the same energy level. The Fermi surface of the type-III WPs only can hold one type of carrier, namely electron-or hole-pockets, due to the saddle-shaped bands. The above results cause that these WPs show different properties in light absorption and Landau energy level effect etc. [72,[74][75][76] For Dirac semimetals, [59,[77][78][79][80][81] four single bands or two double degenerate bands form a fourfold crossing point in the lowenergy region, denoted as Dirac point (DP). At present, a large amount of Dirac semimetals has been predicted theoretically and confirmed experimentally, where the most two typical examples are Na 3 Bi, Cd 3 As 2 . [77,78] In addition, similar to the WPs, the DPs can also show different types of dispersion. Remarkably, topologically protected surface states are typical features of topological semimetals. However, theoretical studies have confirmed that the Fermi arc surface states of Dirac semimetals are not protected. In some cases, it is possible to disappear and become a Fermi loop. [82] In other words, Dirac semimetals may exhibit Fermi arc surface state, but it is fragile. More interestingly, Dirac semimetals usually exhibit peculiar transport properties due to the existence of massless Dirac fermions. [83,84] With the gradual improvement of the theoretical analysis, researchers [68] found that this classification of fermions was incomplete and found some crystal-symmetry-protected band crossings, namely triple degenerate nodal points (TDNPs), sixfold degenerate points (SDPs) and eightfold degenerate points (EDPs; see Figure 3a). It is worth noting that, compared to WPs/DPs, the high-fold fermions with chiral feature possess a long Fermi arc surface state. [85][86][87][88] Importantly, this unique surface feature has been successfully observed in some materials (e.g., CoSi, PbSb 2 ) by angle resolved photoemission spectroscopy (ARPES). [85,88] Interestingly, these long Fermi arc surface states provide favorable conditions for hydrogen evolution reactions (HER). [89,90] Overall, these band crossings greatly enrich the classification of nodal point semimetals.

Nodal Surface Semimetals
For nodal surface semimetals, [64,67,[102][103][104][105][106][107] it is formed by the connection of countless 1D NLs (see Figure 3c). In 2016, Weng's team [67] first reported the presence of 2D topological state in BaMX 3 family, namely nodal surface (NS). Also, this NS is protected by a nonsymmorphic crystal symmetry. Then, Wu et al. [64] systematically classified the NS semimetals into two categories by symmetry analysis. Specifically, the first category is accidental band degeneracy due to the band inversion. The second category is the forced band degeneracy resulting from the joint operation of time reversal symmetry (T) and twofold screw-rotational symmetry (S 2i ), namely (TS 2i ) 2 = e −ik i = −1. Remarkably, the spinorbit coupling (SOC) effect may destroy these two types of NSs, but it is possible to retain them in the second category. Overall, this theoretical work provides guidance for screening actual NS semimetals, such as XTiO 2 (X = Li, Na, K, Rb), Ti 3 Al family, antiferromagnetic electride Ba 4 Al 5 etc. [103][104][105][106] In addition, some materials can contain multiple topological states (e.g., NLs and NSs, WPs and DPs, TDNPs, and DPs ect.) at the same time. [104][105][106][107] These results provide a good platform for studying the potential entanglement between various topological states.

Topological Electrides
Due to the existence of excess electrons, electrides are easier to form band inversion near the Fermi level. In this case, topological electride becomes a bridge connecting electrides and topological materials, as shown in Figure 4. In other words, this novel type of quantum material possesses the excellent properties of electrides and topological materials at the same time, and further exhibit four unique features. First, the band crossings near the Fermi level are mainly contributed by excess electrons from lattice cavities of different dimensions rather than orbital electrons. [107] Second, the intrinsic SOC effect of the topological electrides is greatly weakened due to the approximately s-orbital feature of the excess electrons. [106,107] Third, 2D topological electrides can appear as "floating-surface-states" on some specific surfaces. [108] Fourth, the excess electrons can induce spin polarization and further form unique magnetic topological fermions. [106] By investigating the related literature, we find that 22 related works have reported on the topological electrides.  Next, we systematically review them from the dimension of the excess electrons. Besides, due to the similar properties in some topological electrides, we only summarize six related works, where detailed material descriptions are given in Table 1.

0D Topological Electride
The 0D topological electride refers to the excess electrons of 0D cage form the topological states of different dimensions in space of momentum. In this review, taking two typical 0D topological electrides as examples, namely Ca 3 Pb and Sr 2 Bi family, [107,111] Figure 5a,b.
Before concluding the discussion of 0D topological electrides, Meng et al [109] recently discovered new physicochemical proper-ties in the typical 0D elecctride 12CaO·Al 2 O 3 (C12A7:4e − ). Specifically, based on DFT calculations and symmetry analysis, they found that the electride C12A7:4e − were a new topological electride phase. It possessed a sixfold and fourfold degenerate point near the Fermi level. In addition, these two topological states formed the long Fermi arc surface states on the (001) surface. By comparing the catalytic performance under different hole doping and strain, they proved that the excellent catalytic performance in C12A7:4e − originated from the topological state.

0D Topological Nodal Point Electride
The concept of topological electrides was first proposed in 2018 by four research groups, [107,108,116,121] including 0D electride Ca 3 Pb reported by Zhang et al. [107] They found that the lattice center [1b site (0.5, 0.5, 0.5)] of electride Ca 3 Pb could host excess electrons, which was further confirmed by electron local function (ELF; Figure 5a). From the calculated electronic band structures, it was found that there were two different dispersion (namely type-I and type-II) triple nodal points (TNPs) on k-paths R-Г-X (Figure 5a). Based on the partial charge density (PCD) at |E−E F | < 0.1 eV, the TNPs were mainly contributed by excess electrons. Then, due to the presence of PT symmetries, the TNPs transformed into two fourfold degenerate points (namely Dirac points), when the SOC was considered (Figure 5a). Besides, Zhang et al. [107] also proposed three necessary conditions for the formation of electrides. First, the total valence of the electrides should be greater than zero. Second, the crystal structure should have a large enough cavity. Third, the electron-losingcations should be localized around the cavity. Overall, this work provides guidance for the future development of topological electrides.

0D Topological Nodal Line Electride
It is well known that the intrinsic properties of nodal lines are extremely weakened in most existing nodal line semimetals, because nodal lines are usually affected by SOC effects. Zhang et al. [111] found that the excess electrons of electrides possess a feature of s-orbital electrons, which led to a large weakening of the SOC effect even in the presence of heavy elements. In this case, they proposed an ideal class of topological nodal line electrides. Specifically, based on the design of electrides, Zhang et al. found that binary electrides formed from the second and fifth main groups can hold such nodal lines, namely A 2 B (A = Ca, Sr, Ba; B = As, Sb, Bi) ( Figure 5b). Specifically, the crossing points contributed by the excess electrons eventually formed the nodal lines around X and P points of Brillouin zone (BZ), respectively. Furthermore, drumhead-like surface bands were clearly observed. The above results also agree well with a work of Hirayama et al. in 2008. [108]

1D Topological Electrides
Different from 0D topological electrides, 1D topological electrides refer to the topological states near the Fermi level are mainly contributed by the excess electrons of 1D channel. In addition, this novel type of topological electrides with a large cavity show a wide application prospect, such as gas storage, ion transport, and metal intercalation. [1][2][3] Next, we review the two typical 1D topological electrides Cs 3 O and Ba/Sr 3 CrN 3 . [116,117]

1D Topological Nodal Line Electrides
In 2008, Park et al. [116] first reported the presence of Dirac nodal lines in 1D electride Cs 3 O and Ba 3 N though materials database searches (see Figure 6a). Specifically, similar to -TiCl 3 structure, the cations Cs + /Ba 2+ and the anions O 2−/ N 3− form 1D nanorods along the z-axis in Cs 3 O and Ba 3 N compounds. The only difference is that the cation Ti 4+ of TiCl 3 is not located around the cavity, but the cations Cs + and Ba 2+ of electrides Cs 3 Figure 6a. This work implies that compared with traditional materials, electrides are easier to form band inversion at the Fermi level.

1D Topological Nodal Surface Electrides
It is well known that finding electrides of partially filled d-shell transition metals is a challenge. [36] In 2019, an experimental work and a purely theoretical work [36,117] simultaneously demonstrated that the compounds Ba 3 CrN 3 and Sr 3 CrN 3 containing the transition-group-metal are a typical 1D electrides. Specifically, the valence states of Ba/Sr, Cr, and N in compounds Ba/Sr 3 CrN 3 are +2, +4, and −3, following the form of [Ba/Sr 3 CrN 3 ] + :e − . The electron local function and charge difference density prove that the d-orbital electrons of the cation Cr 4+ are transferred to the four 1D channels (see Figure 6b). In addition, the excess electrons of 1D channel form a double degenerate band (namely nodal line) along the k-paths Г-M-K-Г. According to DFT calculation, this nodal line belongs to a part of the nodal surface on k z = ± planes (see Figure 6b). Based on Wu et al.'s work, [64] space group of Ba/Sr 3 CrN 3 belongs to No. 176, which satisfies the symmetric operation of (TS 2i ) 2 = e −ik i = −1, thus forming a forced nodal surface, which agree well the results of DFT calculation. Furthermore, they found that the Dirac nodal surface of electrides Sr/Ba 3 CrN 3 causes that the frequency, intensity, and damping are independent of the carrier density.

2D Topological Electrides
The 2D topological electrides mean that the topological states near the Fermi level are contributed by the excess electrons of the interlayer. Compared to 0D and 1D topological electrides, the formation conditions of 2D topological electrides are relatively complicated. First, the crystal structure of the compound should satisfy the layered structure. Second, the layer spacing should be greater than 3 Å (>3 Å). Third, the cations should be located on both sides of the interlayer. [108] In addition, since the excess electrons are approximately non-nuclear bound, the electrons of the interlayer can be considered as a 2D electron gas. Next, we give a detailed review of 2D topological nodal line electrides Y 2 C and Gd 2 C. [121,123]

2D Topological Nodal Line Electrides
The 2D topological electride was first proposed in Y 2 N compound by Huang et al. [121] The crystal structure of Y 2 C is a typically layered structure (see Figure 7a), and its electride properties have been demonstrated in previous work. [108] The excess electrons of Figure 7. a) The crystal structure, the Brillouin zone, and the corresponding projection surfaces of electride Y 2 C. Adapted with permission. [121] Copyright 2018, American Chemical Society. b) The electronic band structures without SOC and the related partial charge density near the Fermi level of electride Y 2 C. Adapted with permission. [121] Copyright 2018, American Chemical Society. c) The drumhead surface states of nodal line in electride Y 2 C. Adapted with permission. [121] Copyright 2018, American Chemical Society. d) The crystal structure, the electron local function, and the Brillouin zone of electride Gd 2 C. Adapted with permission. [123] Copyright 2020, American Physical Society. e) The electronic band structures without SOC, the partial charge density, and the positions of nodal lines of electride Gd 2 C. Adapted with permission. [123] Copyright 2020, American Physical Society.
the interlayer form a closed node loop around the L-point (see Figure 7b). In addition, they found that the electride Y 2 C possesses topologically protected drumhead surface states via Z 2 invariance. Remarkably, spin-polarized topological surface states have been further confirmed by ARPES (see Figure 7c). For Gd 2 C compound, [123] its crystal structure is similar to that of the electrides Ca 2 N/Y 2 C. In addition, it is the first proposed intrinsic ferromagnetic (FM) topological electride (Figure 7d). The intrinsic magnetism of electride Gd 2 C originates from the rare-earth element Gd, which was fully confirmed by later experiments. [127] From the calculated the electronic band structure, the spin-up and spin-down bands form several crossing points near the Fermi level, which eventually forms two nodal loops around the L and Z-points under the mirror symmetry operations, as shown in Figure 7e. Therefore, this work provides a new platform to study the interactions of electrides, topological fermions, and magnetism.

The Prospect of Topological Electrides
Before concluding the discussion, we provide some new views on the future development of topological electrides. Although the basic framework of topological electrides has been formed, there are still some challenges and problems.
First, topological electrides are mainly concentrated in nonmagnetic systems. Magnetic topological electrides, especially fer-romagnetism with 100% spin polarization (namely half-metal), have not been reported. For topological electride half-metals, one spin channel is an insulator, the other is a metal. More importantly, the bands passing through the Fermi level are mainly contributed by excess electrons. These bands further form various topological states under the protection of multiple symmetric operations, which will have important application prospects in spintronics.
Second, at present, researchers are not limited to traditional topological insulators, but to explore higher-order topological insulators. The most typical characteristic of high-order topological insulators is that the charge centers and atomic positions do not coincide. [128,129] The electrides with semiconductor feature is likely to have natural high-order topological insulator properties Third, topological electrides possess rich physical and chemical properties, such as topologically-protected surface states, low work function, high conductivity, high electron mobility etc. [1][2][3] These properties are important indicators of various catalysts. Topological electride catalysts are still in the initial stage and have important development potential in the future.

Conclusion
In this review, we review the development history from 0D electrides to 2D electrides and summarize the basic classification of topological states in 0D nodal points, 1D nodal lines, and 2D nodal surfaces. Eventually, a bridge between the topological materials and the electrides is proposed namely topological electrides. Since the concept of topological electrides was proposed in 2018, a total of 21 related works have been reported on topological electrides. Topological electrides are classified into 0D, 1D, and 2D topological electrides by the dimension of the excess electrons. Finally, these excess electrons of different dimensions form multiple types of topological states, such as Weyl points, triple nodal points, Dirac points, nodal loops, and nodal surfaces. This review can help researchers to understand the background, the course of development, and the most recent progress in topological electrides.