Superfluid density and carrier concentration across a superconducting dome: The case of strontium titanate

We present a study of the lower critical field, $H_{c1}$, of ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ as a function of carrier concentration with the aim of quantifying the superfluid density. At low carrier concentration (i.e., the underdoped side), superfluid density and the carrier concentration in the normal state are equal within experimental margin. A significant deviation between the two numbers starts at optimal doping and gradually increases with doping. The inverse of the penetration depth and the critical temperature follow parallel evolutions as in the case of cuprate superconductors. In the overdoped regime, the zero-temperature superfluid density becomes much lower than the normal-state carrier density before vanishing all together. We show that the density mismatch and the clean-to-dirty crossover are concomitant. Our results imply that the discrepancy between normal and superconducting densities is expected whenever the superconducting gap becomes small enough to put the system in the dirty limit. A quantitative test of the dirty BCS theory is not straightforward, due to the multiplicity of the bands in superconducting strontium titanate.

Figure 1

(a) Typical field dependence of the Hall resistivity of a microprobe with a superconducting sample above. The measured signal, directly proportional to the local magnetic field, suddenly rises when the first vortex penetrates the sample. The sharpness indicates the absence of surface-barrier effects or gradual vortex leak from sample corners [27]. The inset shows, in false colors, a picture of the array of Hall microprobes used for this experiment and a scheme of how a sample is mounted on the array of probes. (b) Raw data for the $n_H=2.1\times {10}^{20}{\mathrm{cm}}^{-3}$ sample at different temperatures.

Figure 2

Upper panels show magnetization $M$ as a function of magnetic field at different temperatures for the six samples. Lower panels show the extracted penetration fields $H_p$ for opposite orientations of magnetic field by taking the crossing point of $M\left(H\right)$ at a given temperature with the dashed lines shown in the panels. Black dashed lines are empirical fits to $H_p\left(T\right)=H_p\left(0\right)(1-{(T/T_c)}^{\alpha })$.

Figure 3

(a) Lower critical field vs temperature for all six samples deduced from $H_p$ and geometrical correction of the demagnetization field. (b) Doping dependence of the critical temperature determined by Hall probe magnetometry (blue disks) and thermal conductivity (black squares) [33]; the blue line is a guide to the eye. (c) Doping dependence of $H_{c1}$; the blue line is a guide to the eye.

Figure 4

Upper panels show the temperature dependence of the difference in signal between one probe under the sample and one probe far from it at various fixed fields and for three dopings. The difference is proportional to the magnetic susceptibility $\xi$. The lower panel shows the temperature dependence of the upper critical fields $H_{c2}$ extracted from the upper panels. The lines are WHH fits [38] to the data points.

Figure 5

(a) Doping dependence of the penetration depth plotted as ${\lambda }^{-2}$ vs $n_H$. It exhibits a domelike structure comparable to $T_c$, as in the case of cuprates [6]. (b) Penetration depth plotted as a function of $n_H/m^{★}$ in three different superconductors with $m^{★}$ being unitless. The values are detailed in Table 2. The dashed lines represent the expectation of Eq. (5), assuming $n_H=n_s$. The penetration depth in cuprates is the $ab$ plane one ${\lambda }_{ab}$.

Figure 6

(a) Evolution of superfluid density as a function of carrier density. The straight line represents $n_S=n_H$. (b) The ratio of superfluid density over normal-state carrier density (red discs) compared to the product of the scattering time and the superconducting gap (blue squares) as a function of doping. Lines are guides to the eye. (c) The evolution of superfluid density in Nb (blue squares [43, 54, 55]) and in Nb-doped ${\mathrm{SrTiO}}_3$ (red discs). The continuous black line is the analytic solution of the Ferrell Glover Tinkham sum rule Eq. (7).