Superconductivity in monolayer ${\mathrm{Ba}}_2\mathrm{N}$ electride: First-principles study

The exploration of superconductivity in low-dimensional materials has attracted intensive attention for decades. Based on first-principles electronic structure calculations, we have systematically investigated the electronic and superconducting properties of the two-dimensional electride ${\mathrm{Ba}}_2\mathrm{N}$ in the monolayer limit. Our results show that monolayer ${\mathrm{Ba}}_2\mathrm{N}$ has a low work function of 3.0 eV and a predicted superconducting transition temperature ($T_c$) of 3.4 K. The superconductivity can be further improved with the tensile strain, which results from the increase of density of states at the Fermi level as well as the enhanced coupling between inner-layer electrons and phonons. Remarkably, at the $4\%$ tensile strain, the acoustic branches have noticeable softening at the $K$ point of the Brillouin zone and the superconducting $T_c$ can reach 10.8 K. The effect of lattice strain on the electron transfer from the superficial region to the inner-layer region of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ may also apply to other electride materials and influence their physical properties.

Figure 1

(a) Top and side views of monolayer ${\mathrm{Ba}}_2\mathrm{N}$. The blue and red balls represent Ba and N atoms, respectively. The unit cell is marked by the black solid line. The lower right corner shows the two-dimensional (2D) Brillouin zone of the unit cell. (b) Cleavage energy of ${\mathrm{Ba}}_2\mathrm{N}$ calculated with and without the van der Waals correction. The inset shows the schematic picture for the cleavage process. (c) Work function of monolayer ${\mathrm{Ba}}_2\mathrm{N}$. The red dotted line labels the Fermi level. The atomic positions are marked by the color bars on the horizontal axis.

Figure 2

(a) Band structure, (b) density of states (DOS), and (c) Fermi surface of monolayer ${\mathrm{Ba}}_2\mathrm{N}$. (d) Top and side views of the electron localization function maps for monolayer ${\mathrm{Ba}}_2\mathrm{N}$ with an isosurface value of 0.5. The dashed circles represent the positions of pseudoatoms.

Figure 3

(a) Phonon dispersion of monolayer ${\mathrm{Ba}}_2\mathrm{N}$. The sizes of red dots are proportional to the phonon linewidth ${\gamma }_{\mathbf{q}\nu }$. (b) Phonon density of states (PHDOS). (c) Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. (d) Vibrational patterns of the optical phonon modes with the frequency of 55 ${\mathrm{cm}}^{-1}$ around the $\mathrm{\Gamma }$ point. The length of the arrows indicates the amplitude of the atomic motion.

Figure 4

(a) Total DOS, (b) PDOS from Ba $5d$ orbitals, and (c) PDOS from N $2p$ orbitals under different biaxial tensile strains. (d) Partial charge densities in the energy range of $-0.1$ to 0 eV at different strains with an isosurface value of 0.0005 e/Å${}^3$.

Figure 5

(a) DOS at the Fermi level $N\left(0\right)$ (red) and the logarithmic average frequency ${\omega }_{log}$ (blue) of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ at different strains. (b) Calculated superconducting transition temperature $T_c$ (red) and EPC constant $\lambda$ (blue) of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ at different strains.

Figure 6

(a) Phonon dispersion of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ at $4\%$ tensile strain. The sizes of red dots are proportional to the phonon linewidth ${\gamma }_{\mathbf{q}\nu }$. (b) PHDOS. (c) Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. The vibration patterns of (d) the optical phonon modes with the frequency of 49 ${\mathrm{cm}}^{-1}$ around the $\mathrm{\Gamma }$ point and (e) the softened acoustic phonon modes around the frequency of 24 ${\mathrm{cm}}^{-1}$ at the $K$ point. The length of arrows indicates the amplitude of atomic motions.

Figure 7

Anisotropic superconducting gap of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ as a function of temperature.

Figure 8

Phonon dispersion of monolayer ${\mathrm{Ba}}_2\mathrm{N}$ at $5\%$ tensile strain.

Figure 9

Relationship between superconducting $T_c$ and applied tensile strain for monolayer ${\mathrm{Ba}}_2\mathrm{N}$ with the ${\mu }^{*}$ values varying from 0.10 to 0.13.