Lateral $p\text{−}n$ Junction in an Inverted $\mathrm{InAs}/\mathrm{GaSb}$ Double Quantum Well

We present transport measurements on a lateral $p\text{−}n$ junction in an inverted $\mathrm{InAs}/\mathrm{GaSb}$ double quantum well at zero and nonzero perpendicular magnetic fields. At a zero magnetic field, the junction exhibits diodelike behavior in accordance with the presence of a hybridization gap. With an increasing magnetic field, we explore the quantum Hall regime where spin-polarized edge states with the same chirality are either reflected or transmitted at the junction, whereas those of opposite chirality undergo a mixing process, leading to full equilibration along the width of the junction independent of spin. These results lay the foundations for using $p\text{−}n$ junctions in $\mathrm{InAs}/\mathrm{GaSb}$ double quantum wells to probe the transition between the topological quantum spin Hall and quantum Hall states.

Figure 1

(a) Schematic representation of the sample layout (not to scale). A constant voltage $V_{\mathrm{in}}$ drives current $I$ through the sample and results in the junction voltages $V_{\mathit{u}}$ and $V_{\mathit{l}}$ along the upper and lower sample edges, respectively. The voltage $V_{\text{tg},l}$ ($V_{\text{tg},r}$) applied to the left (right) top gate tunes the charge carrier density below it. The inset shows a cross section through the gate stack. (b) Dependence of the junction resistance $R_{\mathit{l}}=V_{\mathit{l}}/I$ on $V_{\text{tg},l}$ and $V_{\text{tg},r}$ at a perpendicular magnetic field $B_{\perp }=0$. The dashed lines divide the phase diagram into four regions $ij$, $i,j=n$, $p$, where $i$ ($j$) is the charge carrier polarity under the left (right) gate. On the diagonal dotted line and at the positions denoted by symbols, $I\text{−}V$ traces of the junction were measured, see Fig. 2. The arrows are the same as in (c). (c) Line cuts through the color map in (b) at $V_{\text{tg},l}=0$ (left panel) and $V_{\text{tg},r}=0$ (right panel). The inset depicts a simplified band structure, with dashed lines indicating the position of the Fermi energy $E_{\mathit{F}}$ corresponding to features seen in the resistance, as marked by arrows.

Figure 2

(a) $I\text{−}V$ traces of the junction at the points identified by symbols in Fig. 1. The left $y$ axis is the current $I(V_{\mathit{l}})$; the right is the differential conductance $\mathit{d}I(V_{\mathit{l}})/\mathit{d}{V}_{\mathit{l}}$ calculated numerically from $I(V_{\mathit{l}})$. The traces were captured by sweeping the dc bias voltage $V_{\mathrm{in}}$ and recording $V_{\mathit{l}}$ and $I$. (b) Step height in $\mathit{d}I(V_{\mathit{l}})/\mathit{d}{V}_{\mathit{l}}$ along the diagonal dotted line from Fig. 1. The vertical dashed line demarcates the transition between $np$ and $pn$ regions. The points corresponding to traces from the upper left and upper right panels in (a) are additionally highlighted. (c) Example diode $I\text{−}V$ characteristic obtained in the $pn$ region at the $+$ point together with a fit to the data in the forward bias direction, $V_{\text{diode}}>0$, of the form $I=A[\exp (V_{\text{diode}}/B)-1]$. The inset pictures the equivalent circuit employed.

Figure 3

(a) Dependence of the junction resistance $R_{\mathit{l}}$ on the filling factors below the left and right gates, ${\nu }_{\mathit{l}}$ and ${\nu }_{\mathit{r}}$, respectively, for $B_{\perp }=5\mathrm{T}$, with positive filling factors for electrons and negative filling factors for holes. The dashed lines divide the phase diagram into four regions $ij$, $i,j=n$, $p$. Along the dotted diagonal line, ${\nu }_{\mathit{l}}={\nu }_{\mathit{r}}$. The dashed-dotted line highlights the staircase pattern seen in the $pp$ and $nn$ regions. (b) Same as (a), but for the resistance $R_{\mathit{u}}=V_{\mathit{u}}/I$ along the upper sample edge. (c) Line cut through the color map in (a) at ${\nu }_{\mathit{l}}=2$. The dotted lines indicate the expected positions of resistance plateaus, as elaborated upon in the main text. The corresponding filling factors ${\nu }_{\mathit{r}}$ are repeated for clarity next to the marked plateaus. The vertical dashed line demarcates the transition between $np$ (${\nu }_{\mathit{r}}<0$) and $nn$ (${\nu }_{\mathit{r}}>0$) regions. (d) Similar to (c), but at ${\nu }_{\mathit{r}}=-2$.

Figure 4

(a)–(d) Schematic illustrations of edge state dynamics at the $p\text{−}n$ junction in the QH regime for different charge carrier configurations. Electron and hole edge states are colored differently for clarity. The accompanying insets display which regions of the color maps in Figs. 3 and 3 correspond to the respective illustration. The expected resistances $R_{\mathit{u}}$ and $R_{\mathit{l}}$ are also provided, assuming perfect reflection and transmission at the junction ($pp$, $nn$ regions) and complete mode mixing ($pn$, $np$ regions), as explained in the main text and Eqs. (1) and (2). The shaded areas in (c) and (d) symbolize the edge state mixing process along the junction width.

Figure 5

Comparison between experiment (left panel) and theory (right panel) for the conductance quantization of $G_{\mathit{l}}=1/R_{\mathit{l}}$ in the $np$ region for $B_{\perp }=5\mathrm{T}$ plotted using a common color scale. The left panel is derived from Fig. 3 by inverting $R_{\mathit{l}}$. The right panel is produced by assuming complete mode mixing between spin-polarized edge states, Eq. (2).