Two-band superconductivity in doped SrTiO${}_3$ films and interfaces

We investigate the possibility of multiband superconductivity in SrTiO${}_3$ films and interfaces using a two-dimensional two-band model. In the undoped compound, one of the bands is occupied whereas the other is empty. As the chemical potential shifts due to doping by negative charge carriers or application of an electric field, the second band becomes occupied and gives rise to a strong enhancement of the transition temperature and a sharp feature in the gap functions, which is manifested in the local density of states spectrum. By comparing our results with tunneling experiments in Nb-doped SrTiO${}_3$, we find that intraband pairing dominates over interband pairing, unlike other known multiband superconductors. Given the similarities with the value of the transition temperature and with the band structure of LaAlO${}_3$/SrTiO${}_3$ heterostructures, we speculate that the superconductivity observed in SrTiO${}_3$ interfaces may be similar in nature to that of bulk SrTiO${}_3$, involving multiple bands with distinct electronic occupations.

Figure 1

A general illustration of the two-band model as applied to bulk STO (left panel) and LAO/STO (right panel). For STO, the $d_{xz/yz}$ (band 1) crosses the Fermi level and the bottom of the $d_{xy}$ (band 2) is located above the Fermi level by ${\mu }_c-\mu$. In the case of LAO/STO interface, orbital reconstruction switches the position of the bands.

Figure 2

(a), (b) Superconducting transition temperature $T_c$ and (c), (d) the $T$ $=$ 0 ratio of the gap as a function of chemical potential for various values of ${\lambda }_{12}$ and ${\lambda }_{22}$. The light gray dotted line refers to ${\Delta }_2$/${\Delta }_1$ $=$ 1. Note the gap of the second band is finite at all doping, but it is significantly smaller for $\mu <{\mu }_c$. In all panels, ${\lambda }_{11}=0.14$.

Figure 3

(a) Superconducting gap as a function of chemical potential and temperature, displaying the standard BCS temperature dependence. (b) Calculated LDOS for the two-band model as function of increasing chemical potential and energy. (c)–(e) The chemical potential levels with respect to the two bands (left) and the calculated LDOS for the two-band model for $\mu$/${\mu }_c$ $=$ (c) 0.5, (d) 1.01, and (e) 1.5 and for the set of parameters ${\lambda }_{11}=0.14$, ${\lambda }_{22}=0.13$, and ${\lambda }_{12}=0.02.$

Figure 4

Calculations for the 3D two-band model as function of chemical potential for ${\lambda }_{11}$ $=$ 0.14 and ${\lambda }_{22}$ $=$ 0.13. We present $T_c$ (a) and ${\Delta }_2$/${\Delta }_1$ (b) for ${\lambda }_{12}$ $=$ 0.02, 0.04, and 0.08. The light gray dotted line in (b) across ${\Delta }_2$/${\Delta }_1$ $=$ 1 denotes the point at which the second gap crosses the first gap. The individual energy gaps (${\Delta }_i$) for ${\lambda }_{12}$ $=$ 0.02 are shown in (c).

Figure 5

Calculated LDOS for the weakly coupled 3D two-band model as a function of $\omega /{\mu }_c$ for $\mu$/${\mu }_c$ $=$ 0.5 (solid black line), 0.8 (dashed red line), 1.2 (dotted blue line), 1.5 (dashed-dotted green line). Here, we use the parameters ${\lambda }_{11}$ $=$ 0.14, ${\lambda }_{22}$ $=$ 0.13, ${\lambda }_{12}$ $=$ 0.02, and $\kappa$ $=$ 0.4.