Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures

We propose and analyze theoretically an experimental setup for detecting the elusive Majorana particle in semiconductor-superconductor heterostructures. The experimental system consists of one-dimensional semiconductor wire with strong spin-orbit Rashba interaction embedded into a superconducting quantum interference device. We show that the energy spectra of the Andreev bound states at the junction are qualitatively different in topologically trivial (i.e., not containing any Majorana) and nontrivial phases having an even and odd number of crossings at zero energy, respectively. The measurement of the supercurrent through the junction allows one to discern topologically distinct phases and observe a topological phase transition by simply changing the in-plane magnetic field or the gate voltage. The observation of this phase transition will be a direct demonstration of the existence of Majorana particles.

Figure 1

(a) Top view of SM/SC heterostructure embedded into small-inductance SC loop. (b) Side view of the SM/SC heterostructure. The nanowire can be top gated to control chemical potential. Here we assume $L\ll \xi$ and $L_1\gg \xi$ with $\xi$ being the SC coherence length. (c) Proposed readout scheme for the Andreev energy levels. Inductively coupled rf-driven tank circuit allows time-resolved measuring of the effective state-dependent Josephson inductance [19].

Figure 2

Andreev energy spectrum in SM/SC heterostructure for the junction with $\tilde{L}\to 0$. (a) Energy spectrum in TP trivial (dashed line: ${\tilde{V}}_x=0.75$) and nontrivial (solid line: ${\tilde{V}}_x=1.25$) states. The two TP distinct phases differ by having even and odd number of crossings, respectively. (b) Schematic plot of the Josephson current through the junction carried by Andreev states: light (red) and dark (blue) lines describe Josephson current in TP trivial and nontrivial phases, respectively. (c) and (d) The evolution of Andreev energy spectrum with chemical potential. (c) The spectrum in TP nontrivial phase. The dashed (red) line is a fit to $\pm \cos (\phi /2)$ function. (d) The spectrum in TP trivial phase. There is no crossing at $\phi =\pi$.

Figure 3

(a) Andreev spectrum for a finite-size junction $\tilde{L}=3$ in a TP nontrivial phase. Here $\tilde{\mu }=0$, $\tilde{\Delta }=1$, ${\tilde{V}}_x=2$, and $U_0/\alpha =1$. (b) Andreev spectrum in TP trivial phase for $\tilde{\mu }=5$, $\tilde{\Delta }=1$, ${\tilde{V}}_x=2$, $\tilde{L}\ll 1$, and $U_0/\alpha =1$.