Coupling and braiding Majorana bound states in networks defined in two-dimensional electron gases with proximity-induced superconductivity

Two-dimensional electron gases with strong spin-orbit coupling covered by a superconducting layer offer a flexible and potentially scalable platform for Majorana networks. We predict Majorana bound states (MBSs) to appear for experimentally achievable parameters and realistic gate potentials in two designs: either underneath a narrow stripe of a superconducting layer (S stripes) or where a narrow stripe has been removed from a uniform layer (N stripes). The coupling of the MBSs can be tuned for both types in a wide range ($<1\mathrm{neV}$ to $>10\mu \mathrm{eV}$) using gates placed adjacent to the stripes. For both types, we numerically compute the local density of states for two parallel Majorana-stripe ends as well as Majorana trijunctions formed in a tuning-fork geometry. The MBS coupling between parallel Majorana stripes can be suppressed below 1 neV for potential barriers in the meV range for separations of about 200 nm. We further show that the MBS couplings in a trijunction can be gate controlled in a range similar to the intrastripe coupling while maintaining a sizable gap to the excited states (tens of $\mu \mathrm{eV}$). Altogether, this suggests that braiding can carried out on a time scale of 10–100 ns.

Figure 1

Two routes to realize topological superconducting channels in 2DEG-superconductor heterostructures. S stripe (left panels): MBSs are formed underneath a stripe of a superconductor (SC) on top of the 2DEG. A depletion gate (G) turns the 2DEG outside the stripe insulating (I) and confines the states in the transverse direction. N stripe (right panels): MBSs are confined in a normal-conducting 2DEG stripe (N) sandwiched between two 2DEG regions with proximity-induced superconductivity. For both types, MBSs occur at the ends of the stripes as indicated by crosses (the other end of the stripe is not shown here). Panels (a) and (b) show device sketches from above and (c) and (d) from the side. The electrostatic confinement potential $\varphi \left(x\right)$ for the electrons (red) is shown in (e) and (f) alongside the tunnel coupling $\mathrm{\Gamma }\left(x\right)$ to the superconducting top layer (blue).

Figure 2

Basic elements of more complex Majorana devices in 2DEGs: (a) two parallel S-stripe ends, (b) two parallel N-stripe ends, (c) trijunction of S stripes, and (d) trijunction of N stripes. All devices have a finite extension in the $x$ direction transverse to the stripes, $\left|x\right|<W_F/2+W_S+W_B$, and where fading extensions are shown, the stripes extend infinitely along the $y$ direction. The gate-induced depletion potential in the 2DEG is denoted by ${\varphi }_D$ and the control potentials for the junctions are denoted by ${\varphi }_{L/R}$.

Figure 3

Majorana bound states and topological phase transition in a semi-infinite S stripe. The device is sketched in (a), and the potential profile (red) is sketched in (b) alongside the tunnel coupling to the superconductor (blue). The local density of states at energy $E=0$ is shown in (c) for $E_Z=0.6\text{meV}$ and $L_S=600\mathrm{nm}$. The total density of states is shown in (d) for $L_S=300\text{nm}$. The potential barrier height is ${\varphi }_D=1.5\text{meV}$ for both (c) and (d). The dependence of the lowest excited state energy $E_g$ on the potential barrier height ${\varphi }_D$ is depicted in (e). In all plots, we use $W_S=200\text{nm}$ and $W_B=500\text{nm}$ [we show only a restricted part in the $x$ direction in (c)]. The material parameters are $\mathrm{\Gamma }=180\mu \text{eV}$, $\mathrm{\Delta }=235\mu \text{eV}$, $m^{*}=0.023m_e$, $E_{SO}=m^{*}{\alpha }^2/2=118.5\mu \text{eV}$. We further chose $\mu =0$, a broadening $\eta =1\text{neV}$, and a lattice constant of $d=10\text{nm}$.

Figure 4

Gate tunability of the MBS wave function overlap in finite S stripes. Panels (a) and (b) show the energy spectrum as a function of the barrier potential ${\varphi }_D$ adjacent to the S stripe [Fig. 3] on a linear and a logarithmic scale, respectively. In (c)–(e), we show the probability density $P(n_x,n_y)$ for the energy eigenstate closest to zero energy for different barrier heights ${\varphi }_D$ as indicated in (a) and (b). The stripe length is $L_S=2\mu \text{m}$ and all other parameters are as in Fig. 3.

Figure 5

Majorana bound states and topological phase transition in a semi-infinite N stripe. The device is sketched in (a), and the potential profile (red) is sketched in (b) alongside the tunnel coupling to the superconductor (blue). The local density of states at energy $E=0$ is shown in (c) for $E_Z=0.2\text{meV}$ and $L_S=600\mathrm{nm}$. The total density of states is shown in (d) for $L_S=300\text{nm}$. The green arrows indicate the positions of the gap closings (see text). All parameters are as in Fig. 3.

Figure 6

Suppression of MBS coupling in parallel S stripes with potential-barrier height ${\varphi }_D$ and stripe distance $W_F$. The device is sketched in (a) and the electrostatic confinement potential $\varphi \left(x\right)$ (red) is sketched alongside the tunnel coupling $\mathrm{\Gamma }\left(x\right)$ to the superconductor (blue) in (b). We show the the local density of states $\rho$ in (c) and (d) for indicated energies $E$ and barrier heights ${\varphi }_D$. The total density of states ${\rho }_{\text{tot}}$ is mapped out in (e) and the energy $E_M$ of the first peak of ${\rho }_{\text{tot}}$ is depicted in (f) and (h). The energy $E_g$ of the onset of the energy continuum is shown in (g). The Zeeman energy is $E_Z=0.6\text{meV}$, and the geometrical dimensions are, unless stated otherwise, $L_S=600\text{nm}$ [(c) and (d)] / $200\text{nm}$ [(e)–(h)], $W_S=200\text{nm}$, $W_B=500\text{nm}$, and $W_F=200\text{nm}$. We use $\eta =1\text{neV}$ except for (f), where $\eta =0.1\text{neV}$. All other parameters are as in Fig. 3.

Figure 7

Suppression of MBS coupling in parallel N stripes with potential-barrier height ${\varphi }_D$ in between and stripe distance $W_F$. The device is sketched in (a) and the electrostatic confinement potential $\varphi \left(x\right)$ (red) is sketched alongside the tunnel coupling $\mathrm{\Gamma }\left(x\right)$ to the superconductor (blue) in (b). We show the local density of states $\rho$ in (c) and (d) for indicated energies $E$ and barrier heights ${\varphi }_D$. The total density of states ${\rho }_{\text{tot}}$ is mapped out as a function of both these parameters in (e). The energy $E_M$ of the lowest peak of ${\rho }_{\text{tot}}$ is given in (f) and (h) and we show the energy $E_g$ of the onset of the energy continuum in (g). We use the same parameters as in Fig. 6 except for a different Zeeman energy of $E_Z=0.2\text{meV}$.

Figure 8

Controlling the coupling of three MBSs in a tuning-fork trijunction formed from S stripes. The device is sketched in (a). The total density of states ${\rho }_{tot}\left(E\right)$ is shown in (b) for fixed depletion potential ${\varphi }_R=1.5\text{meV}$ at the right junction as a function of the potential difference ${\varphi }_L$ at the left junction. We compute ${\rho }_{\text{tot}}$ for energies in steps of $0.1\mu \text{eV}$ ($1\mu \text{eV}$) below (above) $10\mu \text{eV}$ and for potential values ${\varphi }_L$ in steps of $0.05\text{meV}$. This finite resolution leads to the steplike changes of the subgap peak position. In (c), we show the energy $E_M$ of the first peak of ${\rho }_{tot}\left(E\right)$ at nonzero energy for different values of ${\varphi }_R$ as indicated. Panels (d)–(g) show for different coupling configurations (sketched in top part) the local density of states $\rho (E,n_x,n_y)$ in the central region (bottom part) for different potential barrier heights: (d) and (e) ${\varphi }_L=0\text{meV}$, ${\varphi }_R=5\text{meV}$, (f) ${\varphi }_L={\varphi }_R=5\text{meV}$, and (g) ${\varphi }_L={\varphi }_R=0\text{meV}$. The energies $E$ are indicated in the figure. For illustrational purposes, we chose the maximal values of ${\varphi }_L$ and ${\varphi }_R$ to be larger than in (b) and (c). The finite-energy peak in (e) is thus shifted to a smaller value (which could be increased by lowering ${\varphi }_R$). In all plots, the depleted areas are at a potential ${\varphi }_D=5\text{meV}$, the Zeeman energy is $E_Z=0.6\text{meV}$, and the dimensions of the device are given by $L_S=100\text{nm}$, $L_C=200\text{nm}$, $W_F=200\text{nm}$, $W_S=200\text{nm}$, $W_B=500\text{nm}$. All other parameters are as in Fig. 3.

Figure 9

Controlling the coupling of three MBSs in a tuning-fork trijunction formed from N stripes. The device is sketched in (a). The total density of states ${\rho }_{tot}\left(E\right)$ is shown in (b) for potential barrier height ${\varphi }_R=1.5\text{meV}$ at the right junction as a function of the potential ${\varphi }_L$ applied to the left junction. In (c), we show the energy $E_M$ of the first peak of ${\rho }_{tot}\left(E\right)$ at nonzero energy for different values of ${\varphi }_R$ as indicated. Panels (d)–(g) show for different coupling configurations (sketched in top part) the local density of states $\rho (E,n_x,n_y)$ in the central region (bottom part) for different potential barrier heights: (d) and (e) ${\varphi }_L=0\text{meV}$, ${\varphi }_R=5\text{meV}$, (f) ${\varphi }_L={\varphi }_R=5\text{meV}$, and (g) ${\varphi }_L={\varphi }_R=0\text{meV}$. The energies $E$ are indicated in the figure. We use different maximal values for ${\varphi }_L$ and ${\varphi }_R$ as compared to (b) and (c) for illustrational purposes. We chose ${\varphi }_D=5\text{meV}$, $E_Z=0.2\text{meV},W_F=250\mathrm{nm}$, and all other parameters are as in Fig. 8.

Figure 10

Gate-controlled adiabatic braiding scheme. The steps shown in (a)–(g) exchange the MBSs 1 and 2 located initially at the lower ends of the two upper stripes. The crosses indicate MBSs at zero energy, while the circles and lines indicate fermionic modes at finite energy. Note that coupling three MBSs results in one fermionic mode at finite energy and one MBS at zero energy, which is delocalized between the positions of the three MBSs when uncoupled [as in (b), (d), and (f)]. Even though the protocol is illustrated here for N stripes, the protocol can be carried out in the same way also for S-stripe trijunctions.

Figure 11

Effective vs actual stripe width. The plot is a cut through the lower part of Fig. 3 at $y=100\text{nm}$, here shown for the entire $x$ range used in the calculation. The actual stripe width $W_S=200\text{nm}$ and the effective stripe width ${\tilde{W}}_S=W_S+2\lambda \approx W_S+\sqrt{2/m^{*}{\varphi }_D}\approx 266\text{nm}$ are indicated.

Figure 12

Gate tunability of the MBS wave function overlap in finite S stripes. Panels (a) and (b) show the resulting energy spectrum as a function of the barrier potential ${\varphi }_D$ adjacent to the S stripe [see Fig. 3] on a linear and a logarithmic scale, respectively. In (c)–(e), we show the probability density $P(n_x,n_y)$ for the energy eigenstate closest to zero energy for different barrier heights ${\varphi }_D$ as indicated in (a) and (b). Except for $W_S=160\text{nm}$, all parameters are as in Fig. 4.

Figure 13

Influence of electrostatic potential profile on energy spectrum. (a) Sketch of the cross section of an S-stripe device, consisting of a top gate (G), separated from the superconducting stripe (SC) and the semiconductor heterostructure (SM) by an oxide layer (OX) whose width is neglected. (b) Electrostatic potential $\varphi (x,z)$, Eq. (C7), induced by a split-gate at potential ${\varphi }_G=1.5\text{meV}$ and a grounded superconducting stripe. (c)–(f), top panels: Horizontal cuts (blue) through the potential landscape in (b) along the dashed lines at different distances $-z$ to the gate. In our calculations, the electrochemical potential (green) $\mu =\varphi (x=0,-z)$ is adjusted to the bottom of the potential well created by the gates. (c)–(f), bottom panels: Total density of states ${\rho }_{\text{tot}}$ as a function of energy and Zeeman energy. Except for the modified potential profile and adjusted value of $\mu$, all parameters are the same as in Fig. 3.