Single-Gap Superconductivity and Dome of Superfluid Density in Nb-Doped ${\mathrm{SrTi}\text{O}}_3$

${\mathrm{SrTiO}}_3$ exhibits a superconducting dome upon doping with Nb, with a maximum critical temperature $T_c\approx 0.4\mathrm{K}$. Using microwave stripline resonators at frequencies from 2 to 23 GHz and temperatures down to 0.02 K, we probe the low-energy optical response of superconducting ${\mathrm{SrTiO}}_3$ with a charge carrier concentration from 0.3 to $2.2\times {10}^{20}{\mathrm{cm}}^{-3}$, covering the majority of the superconducting dome. We find single-gap electrodynamics even though several electronic bands are superconducting. This is explained by a single energy gap $2\mathrm{\Delta }$ due to gap homogenization over the Fermi surface consistent with the low level of defect scattering in Nb-doped ${\mathrm{SrTiO}}_3$. Furthermore, we determine $T_c$, $2\mathrm{\Delta }$, and the superfluid density as a function of charge carrier concentration, and all three quantities exhibit the characteristic dome shape.

Figure 1

Frequency dependence of the (a) real and (b) imaginary parts of the complex optical conductivity $\sigma ={\sigma }_1+\mathrm{i}{\sigma }_2$ in superconducting Nb-doped ${\mathrm{SrTiO}}_3$ ($n_{\text{Hall}}=2.0\times {10}^{20}{\mathrm{cm}}^{-3}$, $T_c\approx 0.28\mathrm{K}$), normalized to the normal-state conductivity ${\sigma }_{\mathrm{n}}$, plotted for different temperatures. Kinks in $\sigma (\omega )$ mark the spectroscopic gap $2\mathrm{\Delta }$. The solid lines represent Mattis-Bardeen fits carried out simultaneously to ${\sigma }_1(\omega )$ and ${\sigma }_2(\omega )$ with the spectroscopic gap $2\mathrm{\Delta }$ as the only free parameter. Details on the data treatment are discussed in the Supplemental Material [40]. The inset shows a schematic drawing of the stripline resonator used to make the measurements.

Figure 2

Temperature dependence of ${\sigma }_1$, measured at various fixed frequencies for the sample shown in Fig. 1. Kinks in ${\sigma }_1(T)$, marked by the lines projecting down to the frequency-temperature plane, indicate the temperature at which the microwave photon energy $\hslash \omega$ equals $2\mathrm{\Delta }$. Dashed lines help to clarify the kinks. The grey line in the frequency-temperature plane shows the single-gap BCS prediction for $2\mathrm{\Delta }(T)$.

Figure 3

(a) Temperature and doping dependence of the spectroscopic gap $2\mathrm{\Delta }$: open symbols denote values from Mattis-Bardeen fits as shown in Fig. 1; closed symbols are obtained from ${\sigma }_1(T)$ as shown in Fig. 2. Solid lines indicate fits to a single-gap BCS temperature dependence in which the zero-temperature gap, $2{\mathrm{\Delta }}_0$, is the only adjustable parameter. (b) BCS gap ratio $2{\mathrm{\Delta }}_0/k_B{T}_c$ as a function of carrier density: closed symbols are from microwave spectroscopy; open symbols are from tunneling spectroscopy on thin films [13]; dashed line denotes the BCS value, $2{\mathrm{\Delta }}_0/k_B{T}_c=3.528$.

Figure 4

(a) Temperature dependence of the absolute superfluid density ${\rho }_s$ for all five charge-carrier concentrations, measured at $\omega /2\pi \approx 2\mathrm{GHz}$. As described in the Supplemental Material [40], solid lines are fits to single-gap BCS superfluid density in the presence of disorder, with the absolute scale for the superfluid density taken from infrared measurements of the Drude weight [66]; scattering rate ${\mathrm{\Gamma }}_{\mathrm{sf}}$ and $T_c$ are the only two adjustable fit parameters. (b) Domes of $T_c$, $2{\mathrm{\Delta }}_0$ and zero-temperature superfluid density ${\rho }_{s0}$ as a function of the charge carrier concentration determined from Hall effect measurements. The shaded area denotes the confidence band on ${\rho }_{s0}$, including a 10% scale uncertainty arising from the resistivity measurements used to calibrate the surface impedance. (c) The superfluid fit parameter ${\mathrm{\Gamma }}_{\mathrm{sf}}$ is compared to independent determinations of scattering rate from infrared spectroscopy ${\mathrm{\Gamma }}_{\mathrm{IR}}$ [6], resistivity ${\mathrm{\Gamma }}_{\rho }$, and Hall effect ${\mathrm{\Gamma }}_{\text{Hall}}$ assuming effective masses $m^{*}=1.3,2,4m_e$. The shaded area denotes the confidence band on ${\mathrm{\Gamma }}_{\mathrm{sf}}$.