Fermi Surface of the Most Dilute Superconductor

The origin of superconductivity in bulk ${\mathrm{SrTiO}}_3$ is a mystery since the nonmonotonous variation of the critical transition with carrier concentration defies the expectations of the crudest version of the BCS theory. Here, employing the Nernst effect, an extremely sensitive probe of tiny bulk Fermi surfaces, we show that, down to concentrations as low as $5.5\times {10}^{17}{\mathrm{cm}}^{-3}$, the system has both a sharp Fermi surface and a superconducting ground state. The most dilute superconductor currently known therefore has a metallic normal state with a Fermi energy as little as 1.1 meV on top of a band gap as large as 3 eV. The occurrence of a superconducting instability in an extremely small, single-component, and barely anisotropic Fermi surface implies strong constraints for the identification of the pairing mechanism.

Popular Summary

The origin of superconductivity in bulk $\mathrm{SrTi}{\mathrm{O}}_3$ (strontium titanate) is a mystery several decades old. Pure $\mathrm{SrTi}{\mathrm{O}}_3$ is an insulator even at zero temperature. But, it becomes a superconductor upon the addition of small amounts of mobile electrons ($n$ doping) achieved by either dilute substitution of strontium by lanthanum, or titanium by niobium, or removal of a very small fraction of oxygen atoms. Is there a threshold in the mobile-electron density for the emergence of the superconductivity? If yes, does the doped system preceding the superconductor behave as a normal metal? Ultimately, does the standard BCS theory for superconductivity hold for $\mathrm{SrTi}{\mathrm{O}}_3$? In this experimental paper, we find that $\mathrm{SrTi}{\mathrm{O}}_3$ is in fact a superconductor with the lowest mobile-charge density currently known in all superconductors—in other words, the most dilute superconductor—and is also a new and interesting candidate for non-BCS-type unconventional superconductors.

Our experimental approach is to employ a probe that is extremely sensitive to features of tiny Fermi surfaces: the so-called Nernst effect—the generation of a transverse electric field under the application of a longitudinal temperature gradient and a perpendicular magnetic field. As the strength of the magnetic field is varied, the Nernst signal changes and is effectively a map of the Fermi surface of the material probed. From this map, the velocity, the effective mass, and the density of mobile electrons can all be quantitatively determined. We have thus been able to obtain the following results: (a) $\mathrm{SrTi}{\mathrm{O}}_3$ remains superconducting at an extremely low doping of mobile electrons—2 times lower than previously thought—that corresponds to the removal of one in ${10}^5$ oxygen atoms. (b) The normal state of this superconductor has a well-defined Fermi surface, indicating that even minute doping is enough to put the system on the metallic side of the metal-insulator transition. This is a consequence of the anomalously large dielectric coefficient of $\mathrm{SrTi}{\mathrm{O}}_3$. (c) The mobile electrons in the superconducting $\mathrm{SrTi}{\mathrm{O}}_3$ are too slow and too far apart compared to those in conventional superconductors. This poses a serious challenge for the standard BCS pairing scenario, which relies on phonon-induced electron-electron attraction.

These results, which beg for a fundamental theoretical explanation, add new motivations and new information for solving the decade-old puzzle. The work should also be important to the currently very active research on the oxide interface between $\mathrm{SrTi}{\mathrm{O}}_3$ and another oxide insulator, lanthanum aluminate ($\mathrm{LaAl}{\mathrm{O}}_3$).

Figure 1

Top: The experimental setup for measuring Nernst and Seebeck coefficients of bulk chemically doped ${\mathrm{SrTiO}}_3$. Bottom: As the carrier density is reduced in ${\mathrm{SrTiO}}_3$, the Nernst signal shows oscillations with larger amplitude and longer periodicity. In each panel, the vertical red bar represents a constant scale of $0.7\mu \mathrm{V}/\mathrm{K}$.

Figure 2

Nernst signal as a function of the inverse of the magnetic field for two carrier concentrations. The quantum oscillations display a complex structure with multiple periodicities. The inset shows a sketch of the position of Fermi energy relative to the two lowest bands at the $\Gamma$ point.

Figure 3

Nernst quantum oscillations as a function of (a) the magnetic field and (b) the inverse of the magnetic field at different temperatures for the sample with a carrier density of $5.5\times {10}^{17}{\mathrm{cm}}^{-3}$. There is a single periodicity, with the lowest peak displaying Zeeman splitting. (c) The variation of the Nernst coefficient ($\nu =S_{xy}/B$) divided by temperature, with the inverse of the magnetic field at different temperatures. (d) The amplitude of oscillations in $\nu /T$ for different magnetic fields and temperatures. The solid line represents the damping expected in the Lifshitz-Kosevitch theory for an effective mass of $1.83m_e$.

Figure 4

Top: The Shubnikov–de Haas effect is visible in the magnetoresistance of the lowest doped ${\mathrm{SrTiO}}_3$ sample. Bottom: The effective mass obtained from the resistivity data is close to the one obtained by the analysis of the Nernst data.

Figure 5

(a) The temperature dependence of the zero-field Seebeck coefficient for three different carrier concentrations. (b) The temperature dependence of $S/T$. As is seen in the figure, the diffusive component of thermopower becomes larger with underdoping, which is indicative of a decrease in the Fermi energy. (c),(d) Superconducting resistive transition for two different carrier concentrations in the presence of a magnetic field. (e) Temperature dependence of the upper critical field near $T_c$ (defined as the temperature at which resistivity drops by half) for three carrier concentrations.

Figure 6

(a) Variation of the critical temperature with carrier concentration according to this work and Ref. 11. The width of the superconducting transition in each case is represented by error bars. (b) Evolution of detected frequencies as a function of doping. For carrier concentrations larger than ${10}^{18}{\mathrm{cm}}^{-3}$, the oscillations display more than one frequency. The frequency and carrier density reported in Ref. 16 are also shown. (c) Variation of four relevant length scales, namely, the electronic mean free path (mfp), the average Fermi wavelength ${\lambda }_F$ of the electrons of the main band, the superconducting coherence length $\xi$, and the interelectron distance.