Unconventional Multiband Superconductivity in Bulk ${\mathrm{SrTiO}}_3$ and ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ Interfaces

Although discovered many decades ago, superconductivity in doped ${\mathrm{SrTiO}}_3$ remains a topic of intense research. Recent experiments revealed that, upon increasing the carrier concentration, multiple bands cross the Fermi level, signaling the onset of Lifshitz transitions. Interestingly, $T_c$ was observed to be suppressed across the Lifshitz transition of oxygen-deficient ${\mathrm{SrTiO}}_3$; a similar behavior was also observed in gated ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interfaces. Such a behavior is difficult to explain in the clean theory of two-band superconductivity, as the additional electronic states provided by the second band should enhance $T_c$. Here, we show that this unexpected behavior can be explained by the strong pair-breaking effect promoted by disorder, which takes place if the interband pairing interaction is subleading and repulsive. A consequence of this scenario is that, upon moving away from the Lifshitz transition, the two-band superconducting state changes from opposite-sign gaps to same-sign gaps.

Figure 1

(a) Illustration of the two-band model used in this work, where ${ϵ}_0$ is the energy separation between the bottom of the two electronlike bands, ${\mathrm{\Omega }}_0\gg {ϵ}_0$ is the interaction cutoff, and $\mu$ is the tunable chemical potential. Panels (b) and (c) reproduce the experimental $T_c$ reported in Refs. [7] and [20], respectively. In bulk STO (b), the carrier concentration $n$ is changed by chemical doping, and Lifshitz transitions take place at the critical values $n_{c1}$ and $n_{c2}$ (see inset). In LAO/STO (c), the occupation number is controlled by the gate voltage $V_G$ and a Lifshitz transition happens at $V_c$. Panels (d) and (e) show the calculated $T_c$ as function of the number of electrons $N$ for a clean two-band superconductor with (d) 3D bands and (e) 2D bands. The parameters are the same as in Figs. 2 and 3. $N_c$ indicates the Lifshitz transition.

Figure 2

SC transition temperature $T_c$ as function of the occupation number $N$ for 2D bands. A Lifshitz transition takes place at $N_c$. Different pointlike impurity scattering rates ${\tau }^{-1}$ are shown for an interband pairing interaction ${\lambda }_{12}$ that is either (a) repulsive or (b) attractive.

Figure 3

SC transition temperature $T_c$ as function of the occupation number $N$ for 3D bands for different scattering rates ${\tau }^{-1}$. Similarly to Fig. 2, a Lifshitz transition takes place at $N_c$. Both repulsive (a) and attractive (b) interband pairing interactions ${\lambda }_{12}$ are shown.

Figure 4

Ratio between the two isotropic SC gaps in bands 1 and 2 (${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, respectively) across the Lifshitz transition at $N=N_c$. These plots refer to the 3D bands case shown in Fig. 3. For a sufficiently large impurity scattering rate, the relative sign of the two SC gaps change for $N>N_c$, signaling a crossover from an $s^{+-}$ SC state to an $s^{++}$ one.