Thickness dependence of electron-electron interactions in topological $p-n$ junctions

Electron-electron interactions in topological $p-n$ junctions consisting of vertically stacked topological insulators are investigated. $n$-type ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and $p$-type ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ of varying relative thicknesses are deposited using molecular beam epitaxy and their electronic properties measured using low-temperature transport. The screening factor is observed to decrease with increasing sample thickness, a finding which is corroborated by semiclassical Boltzmann theory. The number of two-dimensional states determined from electron-electron interactions is larger compared to the number obtained from weak antilocalization, in line with earlier experiments using single layers.

Figure 1

Comparison of the number of 2D channels from WAL ($n_{\mathrm{WAL}}$) and EEI ($n_{\mathrm{EEI}}$) as a function of the layer thickness. The values are taken from literature, with the references given in brackets. The bars indicate the spread between $n_{\mathrm{EEI}}$ (top) and $n_{\mathrm{WAL}}$ (bottom). Squares indicate experiments where $n_{\mathrm{EEI}}=n_{\mathrm{WAL}}$. The widths of the bars are proportional to the screening factor $F$ (see scale bar at the bottom right).

Figure 2

(a)–(d) Sheet resistance $R_{\mathrm{s}}$ dependance on temperature for four different samples. The arrows indicate the transition temperature $T^{*}$. Inset in (a): Transition temperature $T^{*}$ dependence on thickness of the ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ layer.

Figure 3

(a)–(d) Longitudinal conductivity ${\sigma }_{\mathrm{xx}}$ of four different samples at low temperature for three different perpendicular magnetic fields. Using a logarithmic scale for the temperature, the linear regions are fitted using Eq. (1) (straight lines). The magnetic field leads to a change of slope, from which the screening and number of 2D channels can be derived.

Figure 4

(a) Difference of conductivity correction $\delta \sigma$ between 5 K and base temperature as a function of the ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ thickness. (b) Change of the slope $f$ with an external, perpendicular magnetic field, as shown in Fig. 3. (c) The screening factor $F$ calculated from $f=n(1-3/4*F)$, assuming the number of 2D states $n$ is 1 (black squares) or 2 (red circles). The screening is negative for $n=1$ and between 0 and 1 for $n=2$, supporting the presence of more than one 2D channel. (d) Number of 2D channels $\alpha$ from WAL, obtained as described in the text. A value of 0.5 corresponds to one 2D channel.