Fermi Surface and Superconductivity in Low-Density High-Mobility $\mathit{\delta }$-Doped ${\mathbf{SrTiO}}_3$

The electronic structure of low-density $n$-type ${\mathrm{SrTiO}}_3$ $\delta$-doped heterostructures is investigated by angular dependent Shubnikov–de Haas oscillations. In addition to a controllable crossover from a three- to two-dimensional Fermi surface, clear beating patterns for decreasing dopant layer thicknesses are found. These indicate the lifting of the degeneracy of the conduction band due to subband quantization in the two-dimensional limit. Analysis of the temperature-dependent oscillations shows that similar effective masses are found for all components, associated with the splitting of the light electron pocket. The dimensionality crossover in the superconducting state is found to be distinct from the normal state, resulting in a rich phase diagram as a function of dopant layer thickness.

Figure 1

(a) Shubnikov–de Haas (SdH) oscillations with varying $d_{\mathrm{NSTO}}$; $T=100\mathrm{mK}$. (b) Fourier transformation (FT) data of (a). Data sets in (a) and (b) have been offset vertically and scaled for clarity.

Figure 2

SdH oscillations at $T=100\mathrm{mK}$, for various $\theta$, and the positions of the peaks in the FT spectra, vs $\sin \theta$, for two samples: (a),(b) $d_{\mathrm{NSTO}}=37\mathrm{nm}$, (c),(d) $d_{\mathrm{NSTO}}=124\mathrm{nm}$. Solid symbols in (b) and (d) show the raw FT frequencies, while open symbols in (b) are scaled by $\sin \theta$. Dotted lines are guides to the eye.

Figure 3

(a) SdH oscillations for $d_{\mathrm{NSTO}}=37\mathrm{nm}$ (2D) at various $T$ in the range $0.6\le T\le 2.0\mathrm{K}$. (b) FT of data in (a). Arrows indicate the different peaks labeled $\alpha$ (red), $\beta$ (blue), and $\gamma$ (green), respectively. Inset: Amplitude variation of each peak versus $T$. Dotted lines are fits to Eq. (1). (c) IFT result for the $\alpha$ peak. Frequency window is indicated by the dotted box in (b). Inset: Amplitude variation with $T$ of the peak indicated by the black arrow in the main graph. Dotted line is the theoretical fit.

Figure 4

Schematic diagram of the STO Fermi surface for (a) $d_{\mathrm{NSTO}}=124\mathrm{nm}$ (3D) and (b) $d_{\mathrm{NSTO}}=37\mathrm{nm}$ (2D). The extremum of the Fermi surface in the [001] direction is shown with a thick line for each electron pocket.

Figure 5

Low-temperature Hall mobility $\mu$ and superconducting transition temperature $T_c$ vs $d_{\mathrm{NSTO}}$. Error bars for $T_c$ denote the 10%–90% width of the superconducting transition. Regions are labeled as follows: I, insulating; 2DM, 2D metallic; 3DM, 3D metallic; 2DSC, 2D superconducting; 3DSC, 3D superconducting. Solid lines are guides to the eye, dotted lines show estimates of the 3D-2D dimensional crossover for the normal and superconducting states. ${\xi }_{\mathrm{GL}}$ denotes the Ginzburg-Landau coherence length.