Exploration for Two-Dimensional Electrides via Database Screening and Ab Initio Calculation

Takeshi Inoshita, Sehoon Jeong, Noriaki Hamada, and Hideo Hosono National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan Materials and Structures Laboratory, Tokyo Institute of Technology, Nagatsuta, Kanagawa 226-8503, Japan Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan (Received 9 November 2013; revised manuscript received 10 April 2014; published 4 August 2014)


I. INTRODUCTION
Electrides are ionic crystals in which electrons serve as anions, occupying the space normally occupied by anions [1,2].Since these electrons are not bound to any ions, electrides may be viewed as excess-electron materials where there is an intrinsic excess of electrons based on the formal valences of the constituent elements.Pioneered by Dye and co-workers [3], research on electrides has progressed steadily, but, until recently, all the known electrides were zero or one dimensional [4].(Here, the dimension refers to that of the electron-confining space and therefore that of the electronic structure.)Most of these electrides are extremely sensitive to the ambient atmosphere, but the first electride that was air resistant, thermally stable, and chemically inert was synthesized by Matsuishi et al. [5], paving the way for the application of electrides.
In view of the overwhelming importance of twodimensional (2D) materials in electronic devices and also in the conceptual development of condensed-matter physics and chemistry, it is natural to wonder whether there are electrides with a 2D electronic structure.Research in this direction has continued [6,7], and recently, Lee et al. [8] demonstrated that dicalcium nitride (Ca 2 N) is a 2D electride in which conduction electrons are confined between calcium layers with a concentration in good agreement with the chemical formula ½Ca 2 N þ e − .The 2D nature of its electronic structure was confirmed by both transport measurements and a band calculation.Moreover, Ca 2 N was found to have a very small work function (2.6-3.5 eV, depending on the surface orientation) [8], making it attractive for use as an electron emitter [9].Shortly afterward, Walsh and Scanlon calculated the electronic structures of Sr 2 N and Ba 2 N and showed that they are also 2D electrides that are very similar to Ca 2 N [10].Now, the question arises as to whether there are 2D electrides other than M 2 N (M ¼ Ca, Sr, Ba).The present paper answers this question in the affirmative through a systematic screening of crystal structure databases combined with ab initio band calculations.

II. SCREENING CONDITIONS
Searching for a new material requires screening conditions at the starting point.We employ the following conditions, or working hypotheses.(the cation) is selected from groups 1A, 2A, 3A, 3B, and 4B, and X (the anion) is selected from groups 4B to 7B of the periodic table (Fig. 1).Transition metals are excluded because they can take multiple valences.(2) The structure is layered with interlayer spacing > 3.0 Å. (3) The ions on both sides of the interlayer gap are cations.(4) The sum (S) of the oxidation numbers (O) of the constituent ions > 0.
The second condition states roughly that the cation layers are separated by a van der Waals gap.The third condition follows from the fact that the electrostatic potential in the space between cation layers (the cationcation gap) is much deeper than inside an anion-anion gap [10], which we confirm through our calculation.Conditions (2) and (3) may be replaced by a more elaborate condition such as one based on the ionic radius and direct cationcation distance, but our results show that the simple conditions (2) and (3) suffice.The fourth condition ensures that there are excess electrons from the viewpoint of the standard valence: For example, GaAs has S ¼ 0 (no excess electron) with OðGaÞ ¼ 3 and OðAsÞ ¼ −3, while Ca 2 N has S ¼ 1 (one excess electron) with OðCaÞ ¼ 2 and OðNÞ ¼ −3.

III. DATABASE SCREENING
Currently, there are two large-scale inorganic crystal structure databases in the world: MatNavi [11] and the Inorganic Crystal Structure Database (ICSD) [12].We search both databases and identify five materials (Y 2 C, Tb 2 C, Dy 2 C, Ho 2 C) in addition to the previously known M 2 N (M ¼ Ca, Sr, Ba) that satisfies all the screening conditions (Table I).We then search literature databases for M 2 C, where M is a lanthanide other than Tb, Dy, and Ho, and find a recent article on Gd 2 C [13].Er 2 C is mentioned TABLE I. Materials satisfying all the screening conditions.Here, S denotes the sum of the oxidation numbers of the constituent elements in a primitive rhombohedral unit cell (M 2 X).The lattice constants (in the hexagonal setting) and magnetic moments per lanthanide ion obtained in the present calculation are compared with the experimental values.The values before and after a slash represent the values obtained with U − J ¼ 6 and 9 eV (with J fixed at 0.7 eV), respectively.The magnetic moment m is obtained by dividing the total magnetization per unit cell by the number of lanthanide ions therein.briefly in Ref. [14] as having the same structure.We include both these materials in our candidate list (Table I).(Because no detailed structural data are available for Er 2 C, we assume the structure of Ho 2 C.) Initially, we also considered Sc 2 C, which met the search conditions, as a candidate, but later rejected it because its entry in the ICSD misrepresented the material: The quoted structure pertains not to pure Sc 2 C but to Sc 2 B 1.1 C 3.2 .
A remarkable feature of Table I is that all the materials have the same anti-CdCl 2 structure [space group R 3m (No. 166), with three atoms in the primitive rhombohedral unit cell] shown in Fig. 2(a).Each layer of this structure is made of ions of a single species arranged in a 2D hexagonal close-packed structure, leaving little space inside the layer for electrons to escape into.
Also note that M 2 N (M being the divalent cation) has one excess electron per unit cell (S ¼ 1) while M 0 2 C (M 0 being the trivalent cation) has two excess electrons (S ¼ 2).This difference in S results in different band-filling and electronic structures near the Fermi level, as discussed below.

IV. ELECTRONIC STRUCTURE CALCULATIONS A. Method
We calculate the electronic structures of all the materials listed in Table I and inspect their charge distributions.The calculations are based on density-functional theory using projector-augmented-wave potentials, as implemented in the Vienna ab initio simulation package (VASP) [23][24][25][26].Exchange and correlation are treated in the generalizedgradient approximation (GGA) as parametrized by Perdew, Burke, and Ernzerhof [27].The structures are relaxed until the force acting on each ion becomes less than 0.02 eV=Å.The plane waves used to expand the wave functions are cut off at 800 eV, and integration over the rhombohedral Brillouin zone (BZ) is performed using a 9 × 9 × 9 Γ-centered sampling mesh.
Figure 2(b) shows the BZ for which we present our results, with the k path highlighted by solid lines.The notation follows Ref. [22].(B 1 is equivalent to B, and X is equivalent to Q by symmetry.)The understanding of band diagrams in the following will be facilitated by noting that the BZ resembles the hexagonal BZ and tends to the hexagonally shaped 2D zone in Fig. 2(c) in the 2D limit (i.e., in the limit of infinite separation between layer units).Thus, the Γ-L-B 1 =B-Z (Γ-X=Q-F-P 1 -Z) segment may be roughly identified as the (Throughout this paper, the 0 of energy is taken to be the Fermi level.)The conduction band, with its bottom located about 1.7 eV below E F , can be seen to be a 2D band confined within the gap between the Ca layers, demonstrating that Ca 2 N is an electride.Although the band dispersion is free-electron-like, the states near E F are weakly hybridized with nitrogen p states, as seen in the PED plot.
The band structure of Y 2 C near E F [Fig. 3(b)] is similar to that of Ca 2 N and shares the same 2D anionic, interlayer character.While the conduction band of Ca 2 N is halffilled, it is almost fully occupied in Y 2 C and is connected to the higher-energy bands without a gap.The bands overlap slightly near E F , resulting in a semimetal-like electronic structure.A 2D empty lattice calculation reveals that these bands originate from a single freeelectron (parabolic) band folded back into the 2D BZ of Fig. 2(c).The flattening of the bands near E F makes the conduction-band effective mass of Y 2 C heavier than that of Ca 2 N. The band filling clearly indicates that there is one excess electron in the unit cell of Ca 2 N, in contrast to two excess electrons in the unit cell of Y 2 C, justifying our hypothesis (4).To facilitate understanding of electron localization inside the gap between the cation layers, we average the PED in the layer plane to obtain the 1D partial electron density ρ p ðzÞ.The result is compared for Ca 2 N and Y 2 C in Fig. 4, where both the PED (thin line) and the total valenceelectron density ρ v ðzÞ (thick line) are plotted.While ρ v ðzÞ has peaks at ionic layers, ρ p ðzÞ exhibits a sharp peak at the center of each cation-cation gap and drops rapidly to near 0 as it approaches the cation layers, indicating strong 2D confinement.
Similar localization of conduction electrons in the interlayer gap between cations can also be seen in Y 2 C [Fig.4(b)].However, the PED is nearly flat across the gap and extends partly into the adjacent Y 3þ layers, which is consistent with the larger effective mass [the flat dispersion around E F in Fig. 3(b)].
We examine the variation of the electron confinement with energy and find that the confinement becomes stronger with decreasing energy.The rightmost plot for Y 2 C [Fig.3(b)] shows the PED in the energy range −0.35 eV < E < −0.25 eV.It can be seen that the 2D confinement in this case is stronger than that at E F , suggesting that greater 2D mobility of the interlayer electrons may be achieved by lowering E F by hole doping.
The band structures of other nonmagnetic 2D electrides Sr 2 N and Ba 2 N [10] are shown in Appendix A for the convenience of readers.

C. Results for magnetic electrides
To take account of the strong on-site Coulomb energy and large spin of localized f electrons in lanthanides, we use the spin-polarized GGA þ U method to calculate the electronic structure of lanthanide carbides.
The band structures for the two spin channels of Gd 2 C calculated with U − J ¼ 6 eV and J ¼ 0.7 eV [28] are shown in the left panels of Fig. 5. Aside from the strong spin polarization that lifts the spin-down bands and lowers the spin-up bands, the overall band structures are similar to those of the nonmagnetic 2D electrides (Y 2 C, in particular).
There is, however, a striking new feature in the PED (right panels of Fig. 5): The spin-up states near E F are confined in the cation layer, whereas the spin-down states are confined in the gap between the cation layers to form giving a weak ferrimagnetic character to the magnetism of Gd 2 C. It was recently reported that Gd 2 C is a roomtemperature ferromagnet with a Curie temperature of 351 K.The magnetic moment measured per Gd atom is 7.26 [13], which is in reasonably good agreement with our calculation (7.73-7.76,depending on U; see Table I).It remains to be seen how the spatial spin texture of this material is reflected in its magnetization behavior, transport properties, and so forth.
Figure 7 shows the band structure and PED of Er 2 C, and Fig. 8 illustrates its layer-averaged PED and partial magnetization.The similarity with Gd 2 C is clear, although the 2D confinement is weaker and the PED is not as smooth as that of Gd 2 C.This difference may be a result of the smaller spin splitting of Er 2 C. For the band structures of magnetic 2D electrides not discussed in this section, see Appendix A.
The above results are obtained for U − J ¼ 6 eV and J ¼ 0.7 eV.To check their validity, we also carry out calculations for U − J ¼ 3 and 9 eV, keeping J fixed at 0.7 eV, and confirm that the results near E F are essentially the same (Appendix B).The effect of spin-orbit interactions is found to be negligible (Appendix C).
To summarize, all the lanthanide carbides in Table I are 2D magnetic electrides, in which the electrons near E F in one spin channel float between M 2 C layers, while the electrons in the other spin channel reside on these layers.

V. CONCLUSIONS
2D electrides are a novel class of low-dimensional electronic systems, where 2D electrons near E F exist mainly in the space between ionic layers and are expected to have very high mobility ("2D electron gas in bulk").These natural "modulation-doped" systems will stimulate research in low-dimensional condensed-matter physics, just as modulation doping has catalyzed a new avenue of semiconductor physics in past decades, leading to the discovery of, for example, integer and fractional quantum Hall effects.We have already initiated the synthesis and transport measurements of some of the materials reported in the present paper [29].
2D electrides are also likely to have an impact on the research in chemistry.In view of the excellent activity of Ru-loaded C 12 A 7 (a zero-dimensional electride) in the synthesis of ammonia from N 2 and H 2 [30], intriguing chemical interactions may take place between these anionic electrons and various external molecules because electrons in 2D electrides can come in direct contact with such molecules.
The Materials Genome Initiative, launched in 2011 by the U.S. government, has revealed to materials researchers the significance of a data-driven approach for accelerating the discovery of new materials [31][32][33][34][35].The present work, in which six new 2D electrides were identified, is a clear     [1] J. L. Dye, Electrons as Anions, Science 301, 607 (2003).
[2] Some metals (e.g., alkali metals) are also predicted to be electrides under extremely high pressures.See, for example, FIG. 1. Elements considered in the screening for 2D electrides.

FIG. 2
FIG. 2. (a) Crystal structure (anti-CdCl 2 structure) of Ca 2 N. The large and small spheres denote Ca and N ions, respectively.A conventional unit cell in the hexagonal setting (lattice vectors ã, b, and c), containing three primitive rhombohedral cells, is shown by solid lines.(b) BZ for the rhombohedral Bravais lattice and the k path (solid lines) used to plot the band structure in the present paper [22].In the 2D limit, the BZ in (b) approaches the hexagonally shaped zone in (c).
Figure 3(a) presents the calculated results for Ca 2 N. The band structures are shown in the left panel while the

FIG. 3 .
FIG. 3. Calculated energy bands (left) and PED isosurfaces (right) for (a) Ca 2 N and (b) Y 2 C. The PEDs are calculated for states in the energy range −0.05 eV < E < 0.05 eV except for the rightmost PED for Y 2 C, which corresponds to an energy range that is 0.3 eV lower, i.e., −0.35 eV < E < −0.25 eV.The PEDs are shown for the conventional unit cell [Fig.2(a)], and the large (small) spheres denote the cations (anions).See also Fig. 4, which presents the variation of the layer-averaged PED in the c direction.

FIG. 4 .
FIG. 4. PED for the energy range from −0.05 to 0.05 eV (solid line) and total valence-electron density (dotted line) for (a) Ca 2 N and (b) Y 2 C averaged over the ab plane (perpendicular to the c axis) in the conventional unit cell [Fig.2(a)].
testimony to the effectiveness of combining large-scale databases and ab initio calculations.

Figures 9 -
Figures 9-13 present the band structures of 2D electrides, excepting those discussed already in the main text.Note the close resemblance of Sr 2 N (Fig.9) and Ca 2 N [Fig.3(a)].The overall band structures of lanthanide dicarbides are similar to each other, and their spin-down interlayer bands [e.g., Fig.11(b)] have dispersions resembling that of Ba 2 N (Fig.10).

Figures 15 (
Figures 15(a) and 15(b) present the band structures of Gd 2 C calculated with and without the spin-orbit interaction, respectively.Note the insensitivity of the interlayer bands near E F to the interaction.This insensitivity may reflect the small amplitude of the interlayer states at the lanthanide sites.