Doping-induced dimensional crossover and thermopower burst in Nb-doped SrTiO${}_3$ superlattices

Using advanced ab initio calculations, we describe the formation and confinement of a two-dimensional electron gas in short-period ($\simeq$4-nm) Nb-doped SrTiO${}_3$ superlattices as a function of Nb doping. We predict complete two-dimensional confinement for doping concentrations higher than 70$\%$. In agreement with previous observations, we find a large thermopower enhancement at room temperature. However, this effect is primarily determined by dilution of the mobile charge over a multitude of weakly occupied bands. As a general rule, we conclude that thermopower in similar heterostructures will be more enhanced by weak, rather than tight, spatial confinement.

Figure 1

Seebeck calculated by BBT approach for STO bulk. Left: calculations for $\tau$ given in Eq. (1) with $\lambda =3/2$; right: calculation with $\lambda =0$ (i.e., constant $\tau$). Top panels refer to doping concentrations reported in Ref. 23, bottom panels to the concentrations reported in Ref. 25. To facilitate the comparison with these experiments, some experimental data (star symbols) extracted by hand from the figures of the original articles are also included.

Figure 2

Left: measured inverse Hall resistivity ${\left(e{R}_H\right)}^{-1}$ for two STO bulk samples, a lightly doped sample I (top) and a heavily doped sample II (bottom). Blue square symbols show ${\left(e{R}_H\right)}^{-1}$ as measured, and blue solid line the carrier density $n_{\text{3D}}\left(T\right)=r_H/\left(e{R}_H\right)$ obtained rescaling the measured Hall resistivity by the calculated Hall factor $r_H$. Middle panels, red lines: ${\left(e{R}_H\right)}^{-1}$ calculated for a range of fixed densities spanning the experimental doping range: from $1.8\times {10}^{19}$ to $2.3\times {10}^{19}$ cm${}^{-3}$ with incremental steps of $0.1\times {10}^{19}$ for sample I (top); from $2.3\times {10}^{20}$ to $3.0\times {10}^{20}$ cm${}^{-3}$ with increments of $0.1\times {10}^{20}$ for sample II (bottom). Middle panels, blue circles: ${\left(e{R}_H\right)}^{-1}$ calculated for the variable charge density $n_{\text{3D}}\left(T\right)$ given by the solid line in the left panel, to be directly compared with the measured values (blue squares) to the left. Right panels: Hall factors $r_H=\left(e{R}_H\right)n_{\text{3D}}$ obtained rescaling $e{R}_H$ calculated at fixed $n_{\text{3D}}$ (red curves of middle panels) with these densities. Clearly $r_H$ is weakly dependent on $n_{\text{3D}}$, but strongly $T$ dependent. The average $r_H$ over these densities is used to rescale the measured Hall resistivity and determines $n_{\text{3D}}\left(T\right)$ in the left panels.

Figure 3

Nb- and Ti-projected DOS of conduction $t_{2g}$ states in the STO${}_9$/Nb-STO${}_1$ superlattice at 25$\%$, 50$\%$, and 100$\%$ Nb concentration (gray shaded lines, $d_{xy}$; red lines, $d_{xz}+d_{yz}$). Top panel is the doped layer, lowest panel the STO layer farthest from the doped side. Dashed lines: Fermi energy (energy zero: valence band top).

Figure 4

Top: bands of the STO${}_9$/Nb-STO${}_1$ SL at 25$\%$, 50$\%$, and 100$\%$ doping. Dashed lines are Fermi energies; energy zero is placed at the valence band top. The character of the three lowest bands is labeled. The conduction bands of $d_{xz},d_{yx}$ character along $\Gamma -Z=\left[001\right]$ are highlighted in violet: with increasing Nb doping, a gap opens between the flat lowest branch and the higher downfolded bulklike sections. Bottom: enlargement of the bands around ${\epsilon }_F$.

Figure 5

Bloch-Boltzmann-calculated planar Seebeck coefficient $S$ vs $T$ for STO bulk (left) and the STO${}_9$/Nb-STO${}_1$ SL (right) at various Nb-doping concentrations. For the bulk, calculations are actually done at 25$\%$ doping, where data at other dopings are calculated in a rigid-band approach, which is acceptable in the bulk.

Figure 6

Top: total DOS for the 25$\%$ and 50$\%$ Nb-doped STO${}_9$/Nb-STO${}_1$ SL and for the 25$\%$-doped bulk. For the former, two different DOS are shown: $n_1$, normalized on a one-layer volume, and $n_{10}$, normalized on the whole SL volume. Center: planar logarithmic conductivity ($\sigma$) at $T=300$ K. Bottom: planar Seebeck coefficient at $T=300$ K for the same systems. Red lines indicate Fermi energies for 25$\%$ doping (dashed lines) and 50$\%$ doping (dotted lines).

Figure 7

Left: Thermopower as a function of Nb-doping concentration calculated using the multiband effective-mass model with doping-dependent band parameters, for STO bulk (black line) and the STO${}_9$/Nb-STO${}_1$ SL (green line) at $T=300$ K. For the latter, contributions from the three lowest $t_{2g}$ bands (red) and the remaining 27 $t_{2g}$ bands (blue line) are also shown. The vertical dashed line separates regions of 3D (low-doping) and 2D (high-doping) carrier regime (see text). Inset: model parameters interpolation as a function of doping (see text).

Figure 8

Left: Thermopower as a function of Nb-doping concentration calculated using the multiband effective-mass model with doping-dependent band parameters, for STO bulk (black line) and the STO${}_9$/Nb-STO${}_1$ SL (green line) at $T=300$ K. For the latter, contributions from the three lowest $t_{2g}$ bands (red line) and the remaining 27 $t_{2g}$ bands (blue line) are also shown. Results of different panels only differ for the value of the power parameter $\lambda$ in Eq. (1).