Two-Dimensional Platform for Networks of Majorana Bound States

We model theoretically a two-dimensional electron gas (2DEG) covered by a superconductor and demonstrate that topological superconducting channels are formed when stripes of the superconducting layer are removed. As a consequence, Majorana bound states (MBSs) are created at the ends of the stripes. We calculate the topological invariant and energy gap of a single stripe, using realistic values for an InAs 2DEG proximitized by an epitaxial Al layer. We show that the topological gap is enhanced when the structure is made asymmetric. This can be achieved either by imposing a phase difference (by driving a supercurrent or using a magnetic-flux loop) over the strip or by replacing one superconductor by a metallic gate. Both strategies also enable control over the MBS splitting, thereby facilitating braiding and readout schemes based on controlled fusion of MBSs. Finally, we outline how a network of Majorana stripes can be designed.

Figure 1

Illustration of a stripe hosting MBSs in a 2DEG-based platform. Panel (a) shows the device with the superconducting layer removed along a stripe, and point contacts formed at the ends to facilitate tunneling spectroscopy of the MBSs shown in (b), where the probability density [38] $P(n_x,n_y)$ of the MBS wave function is plotted. The calculation is done on a lattice with $N_x=260$ and $N_y=160$ sites, and the parameters are $E_Z=200\mu \mathrm{eV}$, ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=180\mu \mathrm{eV}$, $\alpha =1.42\times {10}^{-4}c$, $\mu =0$, $L_L=L_R=1\mu \mathrm{m}$, $L_M=250\mathrm{nm}$, $L_Y=4\mu \mathrm{m}$, $m^{*}=0.023m_e$, and ${\phi }_L={\phi }_R=0$.

Figure 2

Topological phase transition with increasing Zeeman field $E_Z$. The top pictograms illustrate the two cases studied here: on the left a symmetric device ($L_L=L_R$, class BDI) and on the right an asymmetric top-gated device ($L_R=0$, class $D$). The upper panels show the topological invariant $W_{\mathbb{Z}}$ in (a) and $W_{{\mathbb{Z}}_2}$ in (d). The middle panels (b) and (e) depict the 50 lowest eigenenergies of $H_{\text{eff}}(x,y)$ [Eq. (4)]. Higher excited states form a quasicontinuum in the light-blue shaded areas. Close-ups of the midgap-mode energies are shown in the lower panels (c) and (f). All parameters are as in Fig. 1.

Figure 3

Enhancing the topological energy gap defined by $E_g\equiv \underset{n,k_y}{\min }|E_n(k_y)|$, where $E_n(k_y)$ are the eigenenergies of $H_{\text{eff}}(x,-i{\partial }_y\to k_y)$ [see Eq. (4)]. The gap is shown as a function of the Zeeman energy $E_Z$ and (a) the spin-orbit velocity $\alpha$, (b) the electrochemical potential $\mu$, (c),(d) the width of the stripe $L_M$ for a symmetric and asymmetric device, respectively, and (e) the phase difference ${\phi }_L=-{\phi }_R=\mathrm{\Delta }\phi /2$. We use $N_x=300$ lattice points in (a), (b), and (e) but a finer resolution of $N_x=3/2(L_L+L_M+L_R)[\mathrm{nm}]$ for (c) and (d). If not varied, all parameters are as in Fig. 1. The color scale is cut off at $50\mu \mathrm{eV}$ to enhance the contrast inside the regimes with a single MBS (we inserted the number of MBSs, marked phase-separation lines with white arrows, and indicated the parameters used in all other plots by white lines). We use absolute units, since various energy scales ($\mathrm{\Gamma }$, $\mathrm{\Delta }$, $\mu$, $m^{*}{\alpha }^2$) are in the same range and different combinations of them can govern the physics in the different regimes plotted.

Figure 4

Electrically controlled manipulation of MBSs in 2DEG devices. (a) Sketch of a MBS stripe whose width is controlled by a linear voltage drop tuned by a top gate ($G$) with (b) the corresponding low-energy spectrum. Below, we show the probability density $P(n_x,n_y)$ for the state closest to zero energy when the MBSs are (c) coupled and (d) uncoupled. We use $N_x=N_y=150$ lattice points; all other parameters are as in Fig. 1. (e) By segmenting a MBS stripe into two parts using a gate ($T$), one could implement a qubit consisting of four MBSs (denoted by crosses). The intraisland coupling of the MBSs could be tuned by fusion gates ($F1$) and ($F2$) and the interisland coupling by the tunneling gate ($T$). For readout, the two gates ($P1$) and ($P2$) allow for coupling to two detectors ($D1$) and ($D2$). (f) Braiding setup with two gates ($T1$) and ($T2$) controlling the tunnel coupling between the three center MBSs.