Atomic-orbital-free intrinsic ferromagnetism in electrenes

Atomic orbitals play fundamental roles in the modern theory of magnetism, not only providing local moments via their partial occupation, but also offering exchange interactions through their direct or indirect hybridization. Here, we report atomic-orbital-free intrinsic ferromagnetism in monolayer electrides or electrenes, in which excess electrons act as anions. Taking the honeycomb ${\mathrm{LaBr}}_2$ (${\mathrm{La}}^{3+}{\mathrm{Br}}_2{}^{-}·e^{-}$) as an example, our first-principles calculations, in combination with a low-energy effective model in the basis of the Wannier function and Anderson's superexchange theory, reveal that the excess electron is localized at the center of the hexagon, which leads to the spontaneous formation of a local moment due to the strong Stoner instability of the associated state near the Fermi level ($E_f$). The large off-site Coulomb interaction and extended tails of both wave and Wannier functions indicate a significant spatial extension of the anionic electron state in ${\mathrm{LaBr}}_2$. The overlap of the long tails mediates an extended ferromagnetic direct exchange to second-nearest neighbors (up to 7–8 Å), in sharp contrast to the conventional direct exchange which is short ranged due to the overlap of atomic orbitals with limited spatial extension. The dual nature, localization and extension, of the anionic electron state results in a unique magnetic mechanism in such atomic-orbital-free intrinsic two-dimensional ferromagnets.

Figure 1

(a) ${\mathrm{LaBr}}_2$ (${\mathrm{La}}^{3+}{\mathrm{Br}}_2{}^{-}·e^{-}$) ML in the $H$-phase ${\mathrm{MoS}}_2$ structure with ELF maps (isosurface $\text{value}=0.75$) in yellow. The dashed diamond denotes the unit cell. Only the ELF on interstitial sites is shown for clarity. The anionic electron in the geometric space, labeled as X, is at the center of each hexagon. The red arrows are the schematic representation of the hopping paths between anionic electrons, and the calculated hopping parameters are $t_{01}=12.50$ meV, $t_{02}=53.18$ meV, and $t_{03}=-9.11$ meV, respectively. Band structure and density of states of ML ${\mathrm{LaBr}}_2$ (b) without and (c) with spin polarization. The yellow band and DOS in (b) are from the one-band tight-binding model.

Figure 2

(a) Radial distributions of the normalized wave function of states of ${\mathrm{Cr}}_{3d}, {\mathrm{I}}_{5p}$, and the anionic electron. Schematics of strain-induced changes in the localization of wave functions and their overlap (upper panel) as well as the calculated spin density (lower panel) of (b) strain-free ML ${\mathrm{LaBr}}_2$, (c) ML ${\mathrm{LaBr}}_2$ with $10\%$ tensile strain, and (d) ML ${\mathrm{LaBr}}_2$ with $10\%$ compressive strain. The isovalue is $0.001e/{\AA }^3$. (e) The strain dependence of the exchange energy, which is calculated by the total energy difference between the AFM and FM configurations.

Figure 3

The maximally localized Wannier function (a) and its colored contour plot in the plane passing through the La atoms (b) of the anionic electron state of ${\mathrm{LaBr}}_2$. The isovalue of MLWF is set to 2.0. The localized $s$-symmetric body and its extended tails of Wannier function are indicated by black arrows.

Figure 4

(a) The temperature dependence of magnetization without (square) and with (circle) $0.1e$/u.c. hole doping. A cooling field of 0.01 T is applied along the (100) direction. The inset shows the doping concentration dependence of the exchange energy. (b) The simulated magnetic field dependence of magnetization at the temperature of 50 K.