Electronic band sculpted by oxygen vacancies and indispensable for dilute superconductivity

Dilute superconductivity survives in bulk strontium titanate when the Fermi temperature falls well below the Debye temperature. Here, we show that the onset of the superconducting dome is dopant dependent. When mobile electrons are introduced by removing oxygen atoms, the superconducting transition survives down to $2\times {10}^{17}{\mathrm{cm}}^{-3}$, but when they are brought by substituting Ti with Nb, the threshold density for superconductivity is an order of magnitude higher. Our study of quantum oscillations reveals a difference in the band dispersion between the dilute metals made by these doping routes and our band calculations demonstrate that the rigid band approximation does not hold when mobile electrons are introduced by oxygen vacancies. We identify the band sculpted by these vacancies as the exclusive locus of superconducting instability in the ultradilute limit.

Figure 1

Different thresholds for emergence of superconductivity. (a) Temperature dependence of electrical resistivity (${\rho }_{xx}$) in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ and ${\mathrm{SrTiO}}_{3-\delta }$ (orange) sample with similar carrier density $n_H\simeq 1.6\times {10}^{18}{\mathrm{cm}}^{-3}$. Neither the thin film nor the single crystalline ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ shows a superconducting transition at this carrier density. (b) Doping evolution of the superconducting critical temperature ($T_c$). Solid symbols represent samples studied in the present work. Open symbols represent what was reported previously [2, 22, 31].

Figure 2

Distinct Fermi surfaces. (a) Magnetoresistance ($\frac{\mathrm{\Delta }{\rho }_{xx}}{{\rho }_0}$) vs B in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ (orange) and ${\mathrm{SrTiO}}_{3-\delta }$ (blue) with the same Hall carrier density ${\mathrm{n}}_H=1.65\times {10}^{18}{\mathrm{cm}}^{-3}$ at $T=120$ mK. (b) Traces of the quantum oscillations in $\frac{{\partial }^2{\rho }_{xx}}{\partial B^2}$ as function of ${\mathrm{B}}^{-1}$ for both samples. (c) Amplitude of the Fourier transform (FT) vs frequency (F) deduced from (b). The amplitude was normalized by the peak value and shift for clarity. In both samples, two frequencies (and their harmonics) can be identified, respectively labeled ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$.

Figure 3

Doping evolution of the Fermi surface of ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ and ${\mathrm{SrTiO}}_{3-\delta }$. (a) Frequencies of the quantum oscillations of the two subbands ${\mathrm{F}}_1$ (circles) and ${\mathrm{F}}_2$ (squares) vs the Hall carrier density (${\mathrm{n}}_H$) in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$, in ${\mathrm{SrTiO}}_{3-\delta }$ [29] and in ${\mathrm{Sr}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$ [34]. The Lifshitz transition occurs at different threshold densities. (b) Ratio of the carrier density extracted from the Hall coefficient to the carrier density deduced from quantum oscillation frequency assuming two spherical pockets. A difference persists in all samples. This implies distinct geometries for the Fermi surface in the two cases.

Figure 4

Quantum oscillations in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ with $n_H=4.4\times {10}^{18}{\mathrm{cm}}^{-3}$. (a) $\frac{\mathrm{\Delta }{\rho }_{xx}}{{\rho }_0}$ vs B at different temperatures from 80 mK to 2.1 K. (b) $\frac{{\partial }^2{\rho }_{xx}}{\partial B^2}$ as a function of ${\mathrm{B}}^{-1}$ displaying quantum oscillations. (c) FT of the data at $T=80$ mK.

Figure 5

Effect of impurities on the crystal structure and electron density. (a) Crystal structure with a Nb impurity (green octahedron). (b) Same for an O vacancy (black ball) preserving the tetragonal symmetry. (c) The electron charge distribution around the vacancy; here the red balls are O and the blue-gray ones Ti and Sr are green. The black ball indicates the approximate location of the vacancy. The (yellow) isosurfaces are drawn at the level of electron density of $0.05e/{\AA }^3$.

Figure 6

Nonrelativistic band states in ${\mathrm{Sr}}_{64}{\mathrm{Ti}}_{63}{\mathrm{NbO}}_{192}$ (a) and in ${\mathrm{Sr}}_{64}{\mathrm{Ti}}_{64}{\mathrm{O}}_{191}$ (b), (c), (d). In all cases, the structure was fully optimized within DFT. Only in the case of Nb substitution is the dispersion virtually identical to the pristine case. In (b), the oxygen vacancy (v-T) keeps the tetragonal symmetry of the lattice. In (c) and (d), the oxygen vacancy (v-O) makes the system orthorhombic and as seen in (d) the dispersion along the $x$ and $y$ axes are no longer identical. The horizontal axis refers to distances in the momentum space along different orientations expressed in reciprocal lattice units.

Figure 7

Effect of a tetragonal v-O vacancy on the ligand crystal field. Relevant Ti $d$ and O $p$ wave functions for the wave vector k corresponding to the $\mathrm{\Gamma }$ point, in the real space around a Ti site. In absence of O vacancy, the hybridization of all $t_{2g}$ orbitals with the O states cancels out by symmetry. Introducing a vacancy does not change this for the $xy$ orbital (top) but the cancellation is broken by the vacancy for the $xz$ and $yz$ (not shown) orbitals (bottom). As a consequence, in the momentum space, the corresponding bands at the $\mathrm{\Gamma }$ point shift upward.