Two-dimensional electron gas in $\delta$-doped ${\text{SrTiO}}_3$

It is shown that Shubnikov-de Haas oscillations in ${\text{SrTiO}}_3$ $\delta$-doped with La can be understood as arising from a two-dimensional electron gas of one subband immersed in the space charge layer. Despite the inherent complexity of a subband that is derived from four $d$-band states near the conduction-band minimum of ${\text{SrTiO}}_3$, the quantum oscillations can be modeled quantitatively by recognizing that the magnetic field $\left(B\right)$ induced effective spin splitting and Landau-level splitting are comparable. The oscillations are not strictly periodic in $1/B$, which can be understood as caused by a weak dependence of the electron density on the magnetic field in the subband that produces the observed oscillations.

Figure 1

(Color online) (a) Shubnikov-de Haas oscillations at $T=1.8\text{K}$ as a function of the inverse of the perpendicular component of the magnetic field. The inset shows the $\theta$ dependence of a Shubnikov-de Haas oscillation minimum position in a magnetoresistance vs $1/B$ plot (relative to its position at $\theta =0°$), confirming the two-dimensional electron gas behavior. For comparison, the solid line shows the ideal $\text{cos}\theta$ dependence. (b) Shubnikov-de Haas oscillations as a function of the inverse of the magnetic field at $\theta =0°$ and $T=0.41\text{K}$. The inset shows the Fourier transform of the magnetoresistance data, showing two harmonically related frequencies (27 and 54 T).

Figure 2

(Color online) (a) Experimental Shubnikov-de Haas oscillations as a function of the inverse of the magnetic field at temperatures between 0.41 and 3 K. The amplitude of the oscillations decreases with increasing temperature. (b) Calculated Shubnikov-de Haas oscillations at temperatures between 0.41 and 3 K using a single, spin-split subband model as described by Eq. (1). The values for the parameters in Eq. (1) were obtained by fitting to the low-temperature data, using $R_0$, $g$, $T_D$, and $\Delta 1/B$ as fit parameters. The effective mass used in the fits was determined from the experimental temperature dependence.