Multiple nodeless superconducting gaps in optimally doped ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$

We present a study of thermal conductivity in superconducting ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$, sufficiently doped to be near its maximum critical temperature. The bulk critical temperature, determined by the jump in specific heat, occurs at a significantly lower temperature than the resistive $T_c$. Thermal conductivity, dominated by the electron contribution, deviates from its normal-state magnitude at bulk $T_c$, following a Bardeen-Rickayzen-Tewordt behavior, which is expected for thermal transport by Bogoliubov excitations. The absence of a T-linear term at very low temperatures rules out the presence of nodal quasiparticles. On the other hand, the field dependence of thermal conductivity points to the existence of at least two distinct superconducting gaps. We conclude that optimally doped strontium titanate is a multigap nodeless superconductor.

Figure 1

(a) Variation of resistive critical transition in Nb-doped ${\mathrm{SrTiO}}_3$ with carrier concentration (determined by the Hall coefficient). Above the critical doping, marked by a vertical line, three bands are occupied, as sketched in the inset. The symbol representing the sample subject to the in-depth study in this work is surrounded by a red ellipse. (b) The temperature dependence of heat capacity divided by temperature (open squares), compared with thermal conductivity divided by temperature (solid circles) and electrical resistivity (solid diamonds). The dotted-dashed vertical lines mark the end of the resistive and the beginning of the bulk superconducting transitions.

Figure 2

(a) In agreement with the Wiedemann-Franz law, the thermal conductivity divided by temperature $\kappa /T$ (solid circles) and the Lorenz number divided by resistivity $L_0/\rho$ (solid squares) of the normal state ($T>T_c$) extrapolate both to the same value. (b) $\kappa /T$ as a function of $T^2$ for different magnetic fields. At zero magnetic field, extrapolating $\kappa /T$ to zero temperature leads to a vanishing intercept.

Figure 3

(a) Zero-field thermal conductivity of the superconducting state (open circles) as a function of temperature. Electronic and lattice components of thermal conductivity in normal ($B=0.6$ T) and superconducting ($B=0$) states are shown as solid lines. In the normal state, the Wiedemann-Franz law allows an unambiguous determination of ${\kappa }_{\text{ph}}^N$ and ${\kappa }_e^N$. In the superconducting state, ${\kappa }_{\text{ph}}^s$ and ${\kappa }_e^s$ were estimated by assuming that the phonon mean free path at $T_c$ linearly increases to a saturated maximum of 150 $\mu \mathrm{m}$ at $T=0.1$ K. (b) The best fit of ${\kappa }_e^s$ to a BRT function yielding the gap and the $T_c$ as parameters.

Figure 4

(a) The field dependence of thermal conductivity at different temperatures reveals a shoulder at a field $H^{*}$ below $H_{c2}$. This is more clearly see in the upper inset. The lower inset compares the field dependence of thermal conductivity in Nb-doped ${\mathrm{SrTiO}}_3$, a multigap (${\mathrm{NbSe}}_2$) and two single-gap (Nb and ${\mathrm{V}}_3\mathrm{Si}$) superconductors (see Refs. [8, 21]). (b) The two field scales extracted from $\kappa \left(H\right)$ compared to $H_{c2}\left(\rho \right)$, the magnetic field at which resistivity vanishes. In the region filled with horizontal lines the resistivity vanishes, but bulk electrons are still normal.