Voltage-tunable Majorana bound states in time-reversal symmetric bilayer quantum spin Hall hybrid systems

We investigate hybrid structures based on a bilayer quantum spin Hall system in proximity to an $s$-wave superconductor as a platform to mimic time-reversal symmetric topological superconductors. In this bilayer setup, the induced pairing can be of intra- or interlayer type, and domain walls of those different types of pairing potentials host Kramers partners (time-reversal conjugate pairs) of Majorana bound states. Interestingly, we discover that such topological interfaces providing Majorana bound states can also be achieved in an otherwise homogeneous system by a spatially dependent interlayer gate voltage. This gate voltage causes the relative electron densities of the two layers to vary accordingly, which suppresses the interlayer pairing in regions with strong gate voltage. We identify particular transport signatures (zero-bias anomalies) in a five-terminal setup that are clearly related to the presence of Kramers pairs of Majorana bound states.

Figure 1

Schematic of the bilayer hybrid system. It shows a bilayer consisting of two identical TIs (green slabs). The helical edge states (with the same helicity) of both TIs are in proximity to an $s$-wave superconductor (orange, grounded) and are subject partly to an interlayer gate voltage $V={\mu }_1-{\mu }_2$ to control the chemical potentials of the two TI layers. Intralayer (${\mathrm{\Delta }}_{1,2}$) as well as interlayer (${\mathrm{\Delta }}_c$) pairings are possible.

Figure 2

${\text{N-S-S}}^{\prime }\text{-N}$ junction of a bilayer topological insulator (TI1 and TI2). (a) In region S $(l_1\le x\le 0)$, we only consider intralayer superconducting pairing ${\mathrm{\Delta }}_{1,2}$, whereas in region ${\text{S}}^{\prime } (0<x\le l_2)$ we assume a dominating interlayer pairing ${\mathrm{\Delta }}_c$. (b) In both superconducting regions S and ${\text{S}}^{\prime }$, we assume ${\mathrm{\Delta }}_c^2>{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2$. Here, the difference between regions S and ${\text{S}}^{\prime }$ is a steplike variation of the relative density of the layers, i.e., a variation of ${\mu }_{1,2}$. The N sides are characterized by gapless helical edge states illustrated by arrows and spins in the figure. Full lines correspond to electron states and dashed lines to hole states (in a Bogoliubov–de Gennes picture).

Figure 3

Illustration of the effective dispersion relation according to Eq. (8) (thick green and blue lines) combined with the spectrum of an infinite (in the $y$-direction) ribbon of width $N_x=50$ sites of a double HgTe quantum well in the QSH phase, where we set the nonzero parameters $A/\mathrm{\Delta }=B/\mathrm{\Delta }=m/\mathrm{\Delta }=1.0$, while we assumed ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }$. The pairing terms ${\mathrm{\Delta }}_{1,2}$ open a conventional superconducting gap, while its counterterm ${\mathrm{\Delta }}_c$ competes with them, which can result in a gap closing. The gap closing happens at ${\mathrm{\Delta }}_c^2={\mathrm{\Delta }}_1{\mathrm{\Delta }}_2$. The color change of the bands (green vs blue) after the reopening of the gap signals a band inversion.

Figure 4

Illustration of the dispersion relation according to the effective model in Eq. (11) in combination with ribbon spectra of double wells with SC pairings and varying strength of the interlayer gate voltage. For illustrative reasons, we chose the nonzero parameters $A/\mathrm{\Delta }=B/\mathrm{\Delta }=m/\mathrm{\Delta }=1.0$. The interlayer voltage varies from $V=0.0,\mathrm{\Delta },2\sqrt{{\mathrm{\Delta }}_c^2-{\mathrm{\Delta }}^2},5\mathrm{\Delta }$ as shown atop the panels. In all plots, we assume ${\mathrm{\Delta }}_c=2\mathrm{\Delta }$. We can realize the gap closing at $V_c=\pm 2\sqrt{{\mathrm{\Delta }}_c^2-{\mathrm{\Delta }}^2}$ and its reopening for $\left|V\right|=|{\mu }_1-{\mu }_2|>|V_c|$. We illustrate the band inversion also by the colors of the bands.

Figure 5

Normalized absolute square of a bound state at zero excitation energy (i.e., a MBS) at the interface between S and ${\text{S}}^{\prime }$ in Fig. 2 for three different values of ${\mathrm{\Delta }}_{\text{c}}$. Evidently, the MBS emerges for ${\mathrm{\Delta }}_c^2>{\mathrm{\Delta }}^2$. For the particular choice ${\mathrm{\Delta }}_c=2\mathrm{\Delta }$, we observe a perfectly symmetric MBS because then the magnitudes of the gaps in the regions S and ${\text{S}}^{\prime }$ are equal. The parameters are chosen such that ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }=1/\lambda , l_1=-10\lambda$, and $l_2=10\lambda$. The realistic size of $\lambda$ varies from hundreds of nanometers to micrometers.

Figure 6

Normalized absolute square of a bound state at zero excitation energy (i.e., a MBS) at the interface between S and ${\text{S}}^{\prime }$ illustrated in Fig. 2. Both superconducting regions have the same choice of pairing ${\mathrm{\Delta }}_c=2\mathrm{\Delta }$ but they differ by the choice of interlayer voltage $V={\mu }_1-{\mu }_2$. In region II we set $V=0$, whereas in region III we vary $V$ from $2\sqrt{3}\mathrm{\Delta }$ to $10\mathrm{\Delta }$. Further parameter choices are ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }=1/\lambda , l_1=-10\lambda$, and $l_2=10\lambda$. Typical parameter settings are discussed in Sec. 6 [33, 34, 35, 36, 37].

Figure 7

Local (nonlocal) differential conductance $d{I}_1/d{V}_1$ ($d{I}_1/d{V}_2$) as a function of $e{V}_1/\mathrm{\Delta }$ ($e{V}_2/\mathrm{\Delta }$) for different values of ${\mathrm{\Delta }}_{\text{c}}$. A zero-bias dip (peak) arises exactly at the transport phase transition (see the text) where ${\mathrm{\Delta }}_c=(1+\frac{|l_1|}{l_2})\mathrm{\Delta }$. For higher values of ${\mathrm{\Delta }}_c$, both differential conductances reach the values of $e^2/h$. We chose the parameters such that ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }, l_1=-2\lambda$, and $l_2=3\lambda$.

Figure 8

Scattering probabilities for different values of $V$. We clearly observe the emergence of zero-bias peaks or dips (in $R_{\text{LAR}}$ and $R_{\text{CAR}}$) for interlayer voltages characterized by $V>V_c$, while the cotunneling probabilities ($T_{\text{EC}}$ and $T_{\text{CEC}}$) are negligible within the subgap regime. We set the other parameters to ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }, {\mathrm{\Delta }}_c=2\mathrm{\Delta }, l_1=-2\lambda$, and $l_2=8\lambda$. Here, $V_c=2\sqrt{3}\mathrm{\Delta }$.

Figure 9

Local (nonlocal) differential conductance $d{I}_1/d{V}_1$ ($d{I}_1/d{V}_2$) as function of $e{V}_1/\mathrm{\Delta }$ ($e{V}_2/\mathrm{\Delta }$) for different values of $V$ (in units of $\mathrm{\Delta }$). A zero-bias dip (peak) arises only for $V>V_c$ (green dashed line). In contrast to Fig. 7, we now find a dip instead of a peak at zero bias in the presence of MBSs. We set the other parameters to ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }, {\mathrm{\Delta }}_c=2\mathrm{\Delta }, l_1=-2\lambda$, and $l_2=8\lambda$. Here, $V_c=2\sqrt{3}\mathrm{\Delta }$.

Figure 10

Normalized absolute square of a bound state at zero energy at the interface in Fig. 2 for three different sizes of region III, while we assumed ${\mathrm{\Delta }}_c=2\mathrm{\Delta }$ for ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }$ and $V=0$. In addition, we show the corresponding local conductance $d{I}_1/d{V}_1$. The bound state gets shifted to the left interface with decreasing length, while the conductance also vanishes. We used for illustrative reasons the size of region II $|l_1|=2\lambda$, and we do not display the free propagating waves in the normal regions.

Figure 11

Real part (blue, dashed line) of the wave vectors at $E=0$, in dependence on the symmetric interlayer voltage $V$ in units of ${\mathrm{\Delta }}_c=2\mathrm{\Delta }$. One sees a splitting of the vectors at $V=2{\mathrm{\Delta }}_{\text{c}}$. Imaginary part (red, solid line) of the wave vectors at $E=0$. For values $V\ge 2{\mathrm{\Delta }}_{\text{c}}$, the imaginary part stays constant.