Evolution of multigap superconductivity in the atomically thin limit: Strain-enhanced three-gap superconductivity in monolayer ${\mathrm{MgB}}_2$

Starting from first principles, we show the formation and evolution of superconducting gaps in ${\mathrm{MgB}}_2$ at its ultrathin limit. Atomically thin ${\mathrm{MgB}}_2$ is distinctly different from bulk ${\mathrm{MgB}}_2$ in that surface states become comparable in electronic density to the bulklike $\sigma$ and $\pi$ bands. Combining the ab initio electron-phonon coupling with the anisotropic Eliashberg equations, we show that monolayer ${\mathrm{MgB}}_2$ develops three distinct superconducting gaps, on completely separate parts of the Fermi surface due to the emergent surface contribution. These gaps hybridize nontrivially with every extra monolayer added to the film owing to the opening of additional coupling channels. Furthermore, we reveal that the three-gap superconductivity in monolayer ${\mathrm{MgB}}_2$ is robust over the entire temperature range that stretches up to a considerably high critical temperature of 20 K. The latter can be boosted to $>50\mathrm{K}$ under biaxial tensile strain of $\sim 4\%$, which is an enhancement that is stronger than in any other graphene-related superconductor known to date.

Figure 1

The superconducting spectrum of one-ML ${\mathrm{MgB}}_2$, calculated by anisotropic Eliashberg theory with ab initio input. (a) The distribution of the three superconducting gaps $\mathrm{\Delta }({\mathbf{k}}_{\mathrm{F}},T)$ on the Fermi surface: $\pi , S$ (for surface), and $\sigma$, at $T=1$ K. (b) The density of states in the superconducting state at $T=1$ K, showing three distinct peaks corresponding to the three gaps. (c) The evolution of the gap spectrum with temperature, including the gap averages. The calculation shows that one-ML ${\mathrm{MgB}}_2$ has $T_{\mathrm{c}}\cong 20$ K.

Figure 2

(a),(b) The distribution of the superconducting gap spectrum of two-ML and four-ML ${\mathrm{MgB}}_2$, respectively, on the Fermi surface, calculated from anisotropic Eliashberg theory with ab initio input. Both are anisotropic two-gap superconductors, with surface condensates $S$ and $S$' hybridized with the $\pi$ condensate. (c) The density of states in the superconducting state for two and four MLs, calculated at $T=1$ K, showing the overall two-gap nature as well as the anisotropy of the gap spectrum. The critical temperatures found for two- and four-ML ${\mathrm{MgB}}_2$ are 23 K and 27 K, respectively.

Figure 3

The overall e-ph coupling $\lambda \left(\mathbf{q}\right)={\sum }_{\nu }{\lambda }_{\nu }\left(\mathbf{q}\right)$ (i.e., summed over all phonon modes) as a function of phonon wave vectors $\mathbf{q}$ for (a) one-ML, (b) two-ML, and (c) four-ML ${\mathrm{MgB}}_2$.

Figure 4

Phonons and electron-phonon coupling of biaxially strained one-ML ${\mathrm{MgB}}_2$ calculated using DFPT. (a) The phonon dispersion for strains of $-4.5\%, +0\%$, and $+4.5\%$. Increasing strain leads to lower phonon frequencies. (b) The $E_{2g}$ phonon mode of the B atoms that gives the strongest contribution to the electron-phonon coupling. (c) The isotropic Eliashberg function under different strains, ${\alpha }^{2}F\left(\omega \right)={\langle {\langle {\alpha }^{2}F(\mathbf{k}{\mathbf{k}}^{\prime },\omega )\rangle }_{{\mathbf{k}}_{\mathrm{F}}^{\prime }}\rangle }_{{\mathbf{k}}_{\mathrm{F}}}$ (i.e., the double Fermi surface average). The peaks originating from the $E_{2g}$ mode are indicated by arrows. The resulting electron-phonon coupling $\lambda$ is shown as the inset.

Figure 5

The superconducting spectrum of a biaxially strained one-ML ${\mathrm{MgB}}_2$. (a) The distribution of the superconducting gap for $+4.5\%$ tensile strain as a function of temperature, displaying the same three gaps ($\pi , S$, and $\sigma$) as in the unstrained case (Fig. 1). The calculation shows an enhancement of the critical temperature to $T_{\mathrm{c}}=53$ K. (b) The maximum value of the superconducting gap, ${\mathrm{\Delta }}_{\max }$, as a function of temperature and strain. Superconductivity depletes upon compression and is strongly boosted with tensile strain. (c) $T_{\mathrm{c}}$ as a function of the film thickness and as a function of strain for a one-ML ${\mathrm{MgB}}_2$. The bulk value, $T_{\mathrm{c}}=39$ K, is shown for comparison.