Interacting Electrons in Two-Dimensional Electride Ca2N

We investigated the role of correlation effects in the formation of the spectral and magnetic properties of two-dimensional electride Ca2N. Using the combination of Density Functional Theory with Dynamical Mean-Field Theory (DFT+DMFT), we found that Coulomb interactions between the electrons of the electride states cause a strong renormalization and shift of the energy bands near the Fermi level. Besides, the electronic correlations lead to a competition between the Stoner-type ferromagnetic instability and the antiferromagnetic interactions of the localized moments. The observed patterns should be inherent to other isostructural electrides. ■ INTRODUCTION The term “electrides” coined by James Dye describes a rather broad class of materials that have electrons confined in the interstices of the crystal structure. The topology and dimensions of these cavities determine primarily the properties of anionic electrons, including the mobility and degree of localization, especially if the energy bands of these excess electrons cross the Fermi level. These electrons can participate in complex interactions with each other and with cations. As a result, electrides exhibit diverse properties, providing a rich field for experiments and a challenge for theoretical research. A high density of anionic electrons and low binding energy make such materials very promising for a variety of practical applications, particularly as efficient electron emitters and high-performance catalysts. When a new class of low-dimensional electronic systems with a layered structure was discovered, the first among them, dicalcium nitride, immediately attracted the attention of researchers. Recently, the existence of several twodimensional electrides isostructural to Ca2N has been confirmed, including Sr2N, Ba2N, Sc2C, Gd2C, Tb2C, Dy2C, Ho2C, and Y2C. Dicalcium nitride is a two-dimensional electride (Q2DE) with the chemical formula [Ca2N] +·e− and has an anti-CdCl2 crystal structure with an R3̅m space group. The anionic electrons in this system are confined in the interlayer voids between the Ca layers. It was shown that Ca2N undergoes a metal-to-semiconductor transition under pressure closely associated with a change in the degree of localization of anionic electrons responsible for the transport properties of an electride. An intrinsic negative in-plane dispersion of the anionic plasmon (collective oscillations of the density of the free electron gas), revealed using the first-principles time-dependent density functional theory calculations, is in striking contrast with what could be expected for a homogeneous electron gas. The temperature dependence of the resistivity of Ca2N is quadratic, which indicates the presence of Coulomb interactions between the electrons responsible for the transport properties. The measurements of the anisotropic magnetoresistance make it possible to argue that the electron−electron interaction in Ca2N is stronger than in metallic Ca because the electrons do not propagate over the entire crystal but are localized in a much tighter space. The experimental data demonstrate a large enhancement of the effective electron mass, 1.9−2.5me, in Ca2N. The band structure calculations for this system show that a single band with a width of 2.5 eV which crosses the Fermi level EF consists mainly of nonatomic electronic states centered between the [Ca2N] + layers with small contributions from the atomic orbitals of Ca and N. This indicates that the electrical conductivity of Ca2N is due to the electrons enclosed in the space between the layers. The experimental work has shown that the results of DFT calculations do not correctly describe the positions of the energy bands that do not coincide well with the angle-resolved photoemission spectroscopy Received: May 21, 2021 Revised: June 20, 2021 Published: July 9, 2021 Article pubs.acs.org/JPCC © 2021 American Chemical Society 15724 https://doi.org/10.1021/acs.jpcc.1c04485 J. Phys. Chem. C 2021, 125, 15724−15729 D ow nl oa de d vi a SK O L K O V O I N ST S C IE N C E & T E C H N O L O G Y o n A ug us t 2 4, 2 02 1 at 1 0: 29 :0 8 (U T C ). Se e ht tp s: //p ub s. ac s. or g/ sh ar in gg ui de lin es f or o pt io ns o n ho w to le gi tim at el y sh ar e pu bl is he d ar tic le s. (ARPES) results. Thus, the available data indicate that Ca2N can behave as a strongly correlated material and it is important to understand how the Coulomb correlations affect the physical properties of Ca2N, as well as other isostructural electrides. Unfortunately, there is no direct experimental evidence of the presence of magnetic moments in Ca2N. Since correlations are expected on electride states which are localized between atomic layers, and their wave functions are a combination (superposition) of atomic ones, the magnetic properties and behavior of electrons should be different from what we expect for well-localized dor f-electrons. In its turn, it makes it difficult to apply standard experimental methods to estimate the magnitude of local magnetic moments. In the present paper we evaluate the strength of the Coulomb interactions of anionic electrons and investigate the influence of correlation effects on the spectral and magnetic properties of the two-dimensional electride Ca2N. ■ METHODS In the present study we used the combination of DFT and DMFT methods which provides the most unambiguous way to investigate the electronic structure. The DFT calculations were carried out with VASP and Quantum ESPRESSO packages, using the PBE exchange-correlation functional. The results of the latter were used to construct the Hamiltonian in the Wannier function (WF) basis using the Wannier90 package for extraction the noninteracting GGA Hamiltonian HGGA in the real space and transformation to the reciprocal space. The VASP package was used for the estimation of the Coulomb U parameter using a localized basis of atomic orbitals by the linear response approach. Coulomb correlations were taken into account for the constructed Hamiltonian within the DFT+DMFT approach. This method allows taking into account many-body effects such as Coulomb correlations within the dynamical mean-field theory (DMFT). On the first step an effective Hamiltonian HDFT was constructed on the basis of the Wannier functions using realistic noninteracting band structure ε(k)⃗ obtained within the DFT for the compound under consideration. Finally, full many-body Hamiltonian which is solved within the DFT+DMFT has the form: ∑ ̂ = ̂ − ̂ + ′ ̂ ̂ ′ σ σ σ σ σ σ ′ ′ ′ ′ H H H U n n 1 2 i m m m m i m i m DFT DC , , , , , , , , , , (1) Here Um,m′ ′ is the Coulomb interaction matrix and n̂i,m,σ is the occupation number operator for the correlated electride electrons with orbital and spin indices i,m,σ at the ith site. The elements of Um,m′ ′ matrix were parametrized by the on-site Hubbard parameter U and Hund’s intra-atomic exchange JH according to the procedure described in ref 35. Since the electride state is nondegenerate and has only two spin−orbitals JH was set to 0 eV in the present calculations for the sake of simplicity. The DFT+DMFT calculations were performed for the inverse temperature values β from 2 to 90 eV−1, where β = 1/ kBT, kB is the Boltzmann constant, and T is the absolute temperature. We used the simplified fully localized limit form for double-counting correction: ĤDC = U̅(N − /2)I ̂ in a selfconsistent DMFT loop, where N is the total self-consistent number of electrons on the electride site obtained within the DFT+DMFT, U̅ is the average Coulomb parameter for the electride shell, and I ̂ is the identity operator. The continuoustime quantum Monte Carlo hybridization expansion solver from the AMULET package was applied to solve the effective DMFT quantum impurity problem. The analytical continuation of the self-energy dependence on the real frequencies was obtained by using the Pade ́ approximation method. ■ RESULTS AND DISCUSSION To describe the electronic states localized between the atomic layers, we used the maximally localized Wannier functions (MLWFs). Four MLWFs were constructed within the energy window spanned by one energy band crossing the Fermi level and three lower-energy bands (Figure 1a). The first obtained MLWF is centered in the crystal void at position (0.5, 0.5, 0.5), has the symmetry of an s-orbital (Figure 1b), and represents the electride state. The contribution of the first MLWF to the band structure is shown in Figure 1a. The other three MLWFs have a symmetry of p-orbitals, are centered on the nitrogen atoms, and describe three fully occupied energy bands in the interval [−3.5;−1.2] eV. The noninteracting Hamiltonian constructed in the basis of the obtained Wannier functions was solved in the framework of the DMFT method. The dependences of the self-energy on the Matsubara frequencies and Green’s functions on the imaginary time τ, obtained in DMFT calculations, are presented in Figure 2. The value of the Green’s function at the point β/2 on the imaginary time domain indicates the metallic character of conductivity at intermediate values of U and a tendency to transition to the semimetallic state at U = 4 eV caused by the splitting of the half-filled double degenerate electride band into a completely filled and an empty one. The self-energy function for U = 4 eV tends to diverge at zero, indicating a sharp increase in the localization of the electronic states at the electride site (Figure 2). The analytical continuation of the self-energy dependence on the real frequencies and evaluated the spectral functions is shown in Figure 3. For a small value of U = 1 eV, the spectral function of the electride states resembles the DOS obtained using DFT. An increase of U up to 2.5 eV leads to a renormalization of the spectral weight and narrowing of the band around the Fermi level. Well-pronounced upper and lower Hubbard bands appear at +1.8 eV and −1.0 eV, respectively. A further increase of the Coulomb interaction Figure 1.