Photoinduced superconductivity in semiconductors

We show that optically pumped semiconductors can exhibit superconductivity. We illustrate this phenomenon in the case of a two-band semiconductor tunnel-coupled to broad-band reservoirs and driven by a continuous wave laser. More realistically, we also show that superconductivity can be induced in a two-band semiconductor interacting with a broad-spectrum light source. We furthermore discuss the case of a three-band model in which the middle band replaces the broad-band reservoirs as the source of dissipation. In all three cases, we derive the simple conditions on the band structure, electron-electron interaction, and hybridization to the reservoirs that enable superconductivity. We compute the finite superconducting pairing and argue that the mechanism can be induced through both attractive and repulsive interactions and is robust to high temperatures.

Figure 1

Energy levels and laser. (a) A continuous wave laser drives transitions between the lower (1) and upper (2) bands. Both bands are coupled to reservoirs. The chemical potentials of the reservoirs $\mu$ is set in the gap. (b) In the rotating frame, the laser induces an avoided band crossing (with splitting $\sim \left|\Omega \right|)$. (c) This creates an effective resonant surface ${\mathcal{S}}_{{\omega }_0}$ consisting of the set of momenta ${\mathbf{k}}_0$ for which the laser resonantly connects the two bands.

Figure 2

Nonequilibrium population deviation due to driving and dissipation. The laser causes an electron in the valance band to transition into the conduction band. The figure illustrates particular examples when the two rates ${\Gamma }_{1,2}$ differ substantially. In (a), the rate ${\Gamma }_1\gg {\Gamma }_2$, so the reservoirs fill the hole in the valance band much faster than the electron in the conduction band can relax back; the state is blocked, and one has $n_{\mathbf{k}}^{11}+n_{\mathbf{k}}^{22}-1\sim +1$. In (b), the rate ${\Gamma }_2\gg {\Gamma }_1$, so the reservoirs remove the electron in the conduction band much faster than it can relax back; one is left with two holes, and one has $n_{\mathbf{k}}^{11}+n_{\mathbf{k}}^{22}-1\sim -1$ in this example.

Figure 3

Examples of how to choose optimal conditions. One must seek points in the Brillouin zone where two bands have opposite velocities. The transitions are depicted by the vertical dashed line, whose length determines the laser frequency ${\omega }_0$. Notice that the transition of choice does not need to be between two consecutive bands, as is the case depicted in (a). In (b), the transition of choice is between two consecutive bands. The horizontal dashed line demarcates the position of the chemical potential, which can be chosen by doping or gating. The topologies of the band structures were sketched to resemble the bands in Si (a) and in GaAs (b).

Figure 4

(a) Time evolution of the order parameter $\Delta$ for three representative values of the coupling constant: $V=5>V_c=2.14$, $V=2.3\sim V_c$, and $V=1<V_c$. The straight lines correspond to the analytic expressions in Eq. (23). Here, $V_c$ is computed using Eq. (22) (corrected to account for cutoff effects). (b) Perfect matching between the steady-state anomalous correlator ${\tilde{s}}_{\mathbf{k}}^{21}$ as given by Eq. (10) (straight lines) and the anomalous correlator ${\tilde{s}}_{\mathbf{k}}^{21}$ obtained numerically after the time dynamics have converged (circles). The color coding is the same as in (a). $({\kappa }_{+}=-50$, $v_{-}=10$, $\Omega =0.5$, $\Gamma ={10}^{-2}$, and ${\gamma }_2={10}^{-3}.)$

Figure 5

(a) Time evolution of the superconducting pairing $\Delta$ for a scenario where ${\gamma }_1\sim {\gamma }_2$. The straight line corresponds to the steady-state value computed with Eq. (23). The discrepancy between analytics and numerics is because the parameters for the numeric integration are outside the limits of the approximations used in Sec. 2b. (b) Perfect matching between the steady-state anomalous correlator ${\tilde{s}}_{\mathbf{k}}^{21}$ as given by Eq. (10) (straight lines) and the anomalous correlator ${\tilde{s}}_{\mathbf{k}}^{21}$ obtained numerically after the time dynamics have converged (circles). $({\kappa }_{+}=-50$, $v_{-}=10$, $\Omega =0.5$, $\Gamma ={10}^{-2}$, ${\gamma }_1=0.8$, and $V=5.)$

Figure 6

(a) Time evolution of the order parameter $\Delta$ for $\Omega =\Gamma ={10}^{-2}$. (b) Time evolution of the order parameter $\Delta$ for $\Omega =\Gamma /5=0.2{10}^{-2}$. $({\kappa }_{+}=-50$, $v_{-}=10$, $\Gamma ={10}^{-2}$, ${\gamma }_2={10}^{-4}$, $V=20)$.

Figure 7

Optical pumping. The upper band (1) of a two-band semiconductor is populated with a single broad band optical pump. The chemical potential $\mu$ is tuned halfway between the two bands.

Figure 8

Different available optical pumping mechanisms in a three band semiconductor. (a) Deplete the population of the bottom band into a third reservoir band. (b) Populate the top band from a third band.

Figure 9

Feynman diagrams. Series of ring diagrams that contribute to the action $\tilde{S}\left[\Delta \right]$. Each line corresponds to a propagator $G^{A/R/K}$. The terms $1/n$ are symmetry factors for the diagrams.

Figure 10

Energy levels and laser. The upper band of a three-band semiconductor is populated with a single laser drive pumping from the lower band. The chemical potential $\mu$ is set between the lower and middle (reservoir) bands.