Emergence of intrinsic superconductivity in monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$

Two-dimensional (2D) materials possessing intrinsic superconductivity with high transition temperature and unconventional pairing are highly desired, but their realizations are few and far between. Recently, ${\mathrm{W}}_2{\mathrm{N}}_3$ nanosheets down to three layers were successfully prepared [Yu et al., Adv. Mater. 30, 1805655 (2018)]. By performing solid ab initio calculations based on the anisotropic Migdal-Eliashberg theory, we predict that monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$ is an unexplored intrinsic (without the assistance of external gating, strain, or a special substrate) 2D superconductor with large electron-phonon coupling (EPC) and high critical temperature $T_c=38\mathrm{K}$ accompanied with a single and broad superconducting gap ∼7.5 meV. The ratio between the gap and the critical temperature is much larger than the value derived from Bardeen-Cooper-Schrieffer (BCS) theory, further confirming the strong coupling feature of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. The extremely strong EPC originates from the large deformation potential of low-frequency acoustic phonons rather than Fermi-surface nesting. Due to the partially filled $d$ orbitals, electron-electron correlation leads to remarkable enhancement of EPC based on frozen phonon analysis. Here, $T_c$ can be further enhanced via hydrogen passivation based on the McMillian-Allen-Dynes formula. In addition, the symmetry-restricted spin-orbit coupling (SOC) brings forth exotic type-I Ising pairing whose in-plane upper critical field is far beyond the Pauli paramagnetic limit. Our predictions provide a fascinating and highly feasible platform for realizing high-temperature 2D superconductivity and studying the interplay between electron-phonon coupling, electron-electron correlation, and SOCs.

Figure 1

(a) Top and side view of crystal structure of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. The filled rhombus denotes a primitive unit cell, the brown and blue balls represent tungsten and nitrogen atoms, respectively. Orbital projected band structures on (b) W atom, (c) inner layer N atom, and (d) outer layer N atom with orbital contributions proportional to the size of the colored dots. (e) Projected density of states (DOS).

Figure 2

(a) Phonon dispersion decorated with red dots proportional to the partial electron-phonon coupling strength ${\lambda }_{\mathbf{q}\upsilon }$. (b) Phonon density of states (PHDOS) (black solid line), isotropic Eliashberg spectral functions ${\alpha }^{2}F\left(\omega \right)$ (pink shaded region) and cumulative frequency-dependent coupling λ(ω) (blue dashed line). (c) Schematic illustration of the vibrational pattern in real space with large ${\lambda }_{\mathbf{q}\upsilon }$ for α, β, γ, and δ modes. The length of the arrows is proportional to the amplitude of vibration. (d) Fermi surface and the major nesting wave vectors (color arrows) connecting the strong nesting pieces of Fermi surface. Calculated band structures of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$ in a 6 × 6 × 1 supercell (e) before and (f) after being distorted by the ZA phonon at $1/2M$ of the primitive cell. The band is plotted from the Γ point to $1/2M$ of the 6 × 6 × 1 supercell Brillouin zone. (g) Total DOS of the 6 × 6 × 1 supercell.

Figure 3

The blue lines represent the band structures perturbed by (a) and (b) the $E^{\prime }$ phonon at the Γ point and (c) and (d) the ZA phonon at $1/2K$ with the frozen phonon calculations. (a) and (b) are calculated with the primitive cell, and (c) and (d) are calculated with a 4 × 1 × 1 supercell. The black dashed lines denote the bands of undistorted structures. The maximum displacement of the atom is 0.04 Å in the calculations. (e) Electronic structure with (red) and without (black) SOC included. Spin texture at the Fermi level in monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. The color in (f) represents the out-of-plane spin polarization $\langle S_z\rangle$.

Figure 4

Anisotropical effect in monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. (a) Comparison of band structure obtained by density functional theory (DFT) (red lines) and interpolation with maximally localized Wannier functions (MLWFs) (black dots). (b) Distribution of the anisotropical electron-phonon coupling (EPC) strength ${\lambda }_{\mathbf{k}}$. (c) Corresponding quasiparticle density of states (DOS) for four representative temperatures. Quasiparticle DOS in the superconducting state, relative to the DOS in the normal state $N_s\left(\omega \right)/N_N\left(\omega \right)$, as a function of frequency. The superconducting DOS is scaled to coincide with normal DOS, which can facilitate comparing with experimental detection. (d) Energy distribution of the anisotropic superconducting gap as a function of temperature.

Figure 5

(a) Band structures of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3\text{−}{\mathrm{H}}_2$ by density functional theory (DFT) (black lines) and interpolation with maximally localized Wannier functions (MLWFs) (red asterisk). (b) Projected density of states (DOS). (c) Phonon dispersion decorated with red dots proportional to the partial electron-phonon coupling strength ${\lambda }_{\mathbf{q}\upsilon }$. (d) Phonon DOS (PHDOS) (black solid line), ${\alpha }^{2}F\left(\omega \right)$ (pink shaded region) and λ(ω) (blue dashed line). (e) Fermi surface and the major nesting wave vectors (color arrows) connecting the strong nesting pieces of Fermi surface.

Figure 6

Phonon dispersion of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$ with 16 × 16 × 1 k mesh and 8 × 8 × 1 q mesh and 20 × 20 × 1 k mesh and 10 × 10 × 1 q mesh.

Figure 7

Spatial decay of (a) electronic Hamiltonian ||$H$($R$)||, (b) dynamical matrix ||$D$($R$)|| and (c) electron-phonon matrix elements $g$(Re,0) and in the Wannier representation.

Figure 8

Distribution of the (a) superconducting gap at $T=10\mathrm{K}$ and (b) anistropical electron-phonon coupling strength with various uniform k and q meshes.

Figure 9

Spin texture at the Fermi level in monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. The color in (a) and (b) represents the in-plane spin polarization along $\langle S_x\rangle$ and $\langle S_y\rangle$.

Figure 10

(a) and (d) Electronic band structure. (b) and (e) Phonon dispersion decorated with red dots proportional to the partial electron-phonon coupling strength ${\lambda }_{\mathbf{q}\upsilon }$. (c) and (f) Phonon density of states, isotropic Eliashberg spectral functions ${\alpha }^{2}F\left(\omega \right)$, and cumulative frequency-dependent coupling λ(ω). (a)–(c) 0.1 e/cell doped monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. (d)–(f) 0.1 h/cell doped monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. Dashed lines in (a), (d) and (b), (e) represent the band structure and phonon spectra of pristine monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$.

Figure 11

(a) and (d) Electronic band structure. (b) and (e) Phonon dispersion decorated with red dots proportional to the partial electron-phonon coupling strength ${\lambda }_{\mathbf{q}\upsilon }$. (c) and (f) Phonon density of states, isotropic Eliashberg spectral functions ${\alpha }^{2}F\left(\omega \right)$, and cumulative frequency-dependent coupling λ(ω). (a)–(c) 0.5% tensile strained. (d)–(f) 2.0% compressive strained monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. Dashed lines in (a), (d) and (b), (e) represent the band structure and phonon spectra of pristine monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$.

Figure 12

Variation of total potential energy $E_0$ with time during ab initio molecular simulation of monolayer ${\mathrm{W}}_2{\mathrm{N}}_3\text{−}{\mathrm{H}}_2$ at 300 K and the top and side view of the structure at 6 and 12 ps.

Figure 13

Phonon dispersion of (a) pristine, (b) $\varepsilon =-0.5\%$, (c) $\varepsilon =-2\%$ strained monolayer ${\mathrm{W}}_2{\mathrm{N}}_3$. Solid black lines and blue dashed lines are calculated with Methfessel-Paxton smearing of 0.02 and 0.008 Ry, respectively.