Majorana fermions in a tunable semiconductor device

The experimental realization of Majorana fermions presents an important problem due to their non-Abelian nature and potential exploitation for topological quantum computation. Very recently Sau et al. [Phys. Rev. Lett. 104, 040502 (2010)] demonstrated that a topological superconducting phase supporting Majorana fermions can be realized using surprisingly conventional building blocks: a semiconductor quantum well coupled to an $s$-wave superconductor and a ferromagnetic insulator. Here we propose an alternative setup, wherein a topological superconducting phase is driven by applying an in-plane magnetic field to a (110)-grown semiconductor coupled only to an $s$-wave superconductor. This device offers a number of advantages, notably a simpler architecture and the ability to tune across a quantum phase transition into the topological superconducting state while still largely avoiding unwanted orbital effects. Experimental feasibility of both setups is discussed in some detail.

Figure 1

(a) Setup proposed by Sau et al. (Ref. 14) for realizing a topological superconducting phase supporting Majorana fermions in a semiconductor quantum well with Rashba spin-orbit coupling. The $s$-wave superconductor generates the pairing field in the well via the proximity effect while the ferromagnetic insulator induces the Zeeman field required to drive the topological phase. (b) Alternative setup proposed here. We show that a (110)-grown quantum well with both Rashba and Dresselhaus spin-orbit coupling can be driven into a topological superconducting state by applying an in-plane magnetic field. The advantages of this setup are that the Zeeman field is tunable, orbital effects are expected to be minimal, and the device is simpler since it does not require a good interface with a ferromagnetic insulator.

Figure 2

(Color online) Excitation gap $E_g$ normalized by $\Delta$ in the proximity-induced superconducting state of a Rashba-coupled quantum well adjacent to a ferromagnetic insulator. In (a), the chemical potential is chosen to be $\mu =0$. For $\Delta /V_z<1$ the system realizes a topological superconducting phase supporting a single Majorana mode at a vortex core while for $\Delta /V_z>1$ an ordinary superconducting state emerges. In the topological phase, the gap is maximized when $\Delta /V_z=1/2$ and $m{\alpha }^2/V_z⪢1$, where it is given by $E_g=V_z/2$. In contrast, the gap vanishes as $m{\alpha }^2/V_z\to 0$ because the effective $p$-wave pair field at the Fermi momentum vanishes in this limit. In (b), we have taken $m{\alpha }^2/V_z=0.1$ to illustrate that $V_z$ can exceed $m{\alpha }^2$ by more than an order of magnitude and still yield a sizable gap in the topological superconducting phase.

Figure 3

(Color online) Excitation gap ${\mathcal{E}}_g$ normalized by $\Delta$ in the proximity-induced superconducting state of a (110) quantum well, with both Rashba and Dresselhaus spin-orbit coupling, in a parallel magnetic field. In (a) we set $\mu =0$, $m{\lambda }_D^2/V_y=2$, and $\Delta /V_y=0.66$, and illustrate the dependence of the gap on the Rashba coupling anisotropy $\gamma$ as well as ${\lambda }_R/{\lambda }_D$. When $\gamma =0$ and ${\lambda }_R/{\lambda }_D=1$, the problem maps onto the Rashba-only model considered by Sau et al. (Ref. 14). Remarkably, the gap survives unaltered here even in the physically relevant case with $\gamma$ of order one, provided the Rashba coupling is reduced. In (b) and (c), we focus on the realistic case with $\gamma =1$ to illustrate the stability of the topological phase in more detail. We take $\mu =0$ and $m{\lambda }_D^2/V_y=2$ in (b) and allow $\Delta /V_y$ as well as ${\lambda }_R/{\lambda }_D$ to vary. In (c), we fix $\Delta /V_y=0.66$ and $m{\lambda }_D^2/V_y=2$, allowing $\mu /V_y$ and ${\lambda }_R/{\lambda }_D$ to vary.