Superconducting States in Pseudo-Landau-Levels of Strained Graphene

We describe the formation of superconducting states in graphene in the presence of pseudo-Landau-levels induced by strain, when time reversal symmetry is preserved. We show that superconductivity in strained graphene is quantum critical when the pseudo-Landau-levels are completely filled, whereas at partial fillings superconductivity survives at weak coupling. In the weak coupling limit, the critical temperature scales linearly with the coupling strength and shows a sequence of quantum critical points as a function of the filling factor that can be accessed experimentally. We argue that superconductivity can be induced by electron-phonon coupling and that the transition temperature can be controlled with the amount of strain and with the filling fraction of the Landau levels.

Figure 1

Dependence of the chemical potential $\mu$ in units of ${\omega }_c$ with the filling factor $\nu$. Red (light) curve: normal state $\Delta =0$. Solid black curves: superconducting state, with $\Delta /{\omega }_c$ ranging from 0 [red (light) line)] to 1. All curves are plotted at $T/{\omega }_c=0.02$. Inset: Detail of the zeroth LL ($n=0$). The red (light) curve is analytically described by Eq. (10), while solid black ones at $\Delta /{\omega }_c\ll 1$ are described by Eq. (9) in the $T\to 0$ limit.

Figure 2

Critical temperature $T_c/{\omega }_c$ versus the dimensionless coupling strength $x\equiv |U|g{N}_{\varphi }{\ell }_B/(\sqrt{2}v)$. (a) Red (light) curve: $\nu =2$, where the transition is quantum critical below $x_0^c\approx 0.025$. Solid black curves: $\nu =0$, 8, 24 and $\nu =40$, from right to left. (b) $\nu =0$, 1.2, 1.6, 1.98, 1.9998, and 2, from left to right. (c) $\nu =4$, 5.2, 5.65.98, 5.9998, and 6, from left to right. At partial filling of the LL, $T_c\propto x$ in the $x\to 0$ limit.

Figure 3

Phase diagram $T_c/{\omega }_c$ as a function of the filling factor $\nu$ for different coupling strengths $x$ (solid curves). Blue (dark gray) region: $x<x_1^c\approx 0.022$. Medium gray region: $x_1^c<x<x_0^c\approx 0.025$. Light gray region: $x>x_0^c$. The critical temperature drops to zero at integer filling factors ${\nu }_{\mathrm{I}}(n)=g[n+(1/2)]$ whenever $x<x_n^c$, where $x_n^c$ is the critical coupling of the ${\nu }_I(n)$ state.