Superconductivity in engineered two-dimensional electron gases

We consider Kohn-Luttinger mechanism for superconductivity in a two-dimensional electron gas confined to a narrow well between two grounded metallic planes with two occupied subbands with Fermi momenta $k_{FL}>k_{FS}$. On the basis of a perturbative analysis, we conclude that non-$s$-wave superconductivity emerges even when the bands are parabolic. We analyze the conditions that maximize $T_c$ as a function of the distance to the metallic planes, the ratio $k_{FL}/k_{FS}$, and $r_s$, which measures the strength of Coulomb correlations. The largest attraction is in $p$-wave and $d$-wave channels, of which $p$ wave is typically the strongest. For $r_s=O\left(1\right)$ we estimate that the dimensionless coupling $\lambda \approx {10}^{-1}$, but it likely continues increasing for larger $r_s$ (where we lose theoretical control).

Figure 1

Schematic picture of the geometry of the two-component 2DEG considered in this paper.

Figure 2

(a) and (b) The coupling constants in the $p$-wave and $d$-wave channels as functions of the ratio of Fermi momenta $k_{FS}/k_{FL}<1$ for $\alpha =1$ and three different values of $r_s$. Negative value of the coupling means attraction. The range of $k_{FS}/k_{FL}$, where the attraction holds, is larger in the $p$-wave channel, and the magnitude of the attraction is generally larger for the $p$-wave channel. However, except near $k_{FS}/k_{FL}=0.25$, where the attraction is the strongest, the couplings in $p$-wave and $d$-wave channels are comparable in strength. At larger $r_s$, the coupling in the $p$-wave channel gets larger even for $k_{FS}/k_{FL}\sim 0.25$ (see Fig. 3).

Figure 3

(a) and (b) The dimensionless pairing coupling constant $\lambda$ in $p$-wave and $d$-wave channels as functions of $r_s$ for $\alpha =1$ and three different values of $k_{FS}/k_{FL}=0.25$, 0.5, and 0.75, and (c) and (d) as functions of $\alpha$ for $k_{FS}/k_{FL}=0.5$ and two different $r_s=1$ and $r_s=3$. Observe that the coupling constants in both channels become weakly dependent on $\alpha$ once $r_s$ gets larger, and that the attraction in the $d$-wave channel holds at larger $r_s$ only for $k_{FS}/k_{FL}=0.25$.

Figure 4

The coupling constants in $p$-wave and $d$-wave channels as functions of $k_{FS}/k_{FL}$ in the formal limit $r_s\to \infty$. The coupling in the $p$-wave channel is attractive, and its magnitude increases as the ratio $k_{FS}/k_{FL}$ decreases. The coupling in the $d$-wave channel is repulsive for all values of $k_{FS}/k_{FL}$. When $r_s$ is large but finite, there is an attraction in the $d$-wave channel, but only for small $k_{FS}/k_{FL}$.