Detecting Majorana bound states by nanomechanics

We propose a nanomechanical detection scheme for Majorana bound states, which have been predicted to exist at the edges of a one-dimensional topological superconductor, implemented, for instance, using a semiconducting wire placed on top of an $s$-wave superconductor. The detector makes use of an oscillating electrode, which can be realized using a doubly clamped metallic beam, tunnel coupled to one edge of the topological superconductor. We find that a measurement of the nonlinear differential conductance provides the necessary information to uniquely identify Majorana bound states.

Figure 1

Schematic of the setup. A TS is assumed to be realized as a 1D semiconducting wire on top of a grounded $s$-wave superconductor (SC). The wire can host MBS at its left (${\gamma }_L$) and right (${\gamma }_R$) edges. We assume that one of the edges is tunnel coupled to a movable, doubly clamped beam (at bias voltage $V$). The gate electrodes ($V_{g_1}$–$V_{g_4}$) can be used to increase or decrease the overlap $\xi$ of the MBS by changing the effective length $L$ of the TS.

Figure 2

Differential conductance for $\xi /\Omega =0.2$ in the presence of MBS for the case $t_{\mathrm{x}}=0$ (without oscillator) as well as for the case $t_{\mathrm{x}}\ne 0$ for different values of $\langle n\rangle$. The satellite peaks have been enlarged for better visibility.

Figure 3

Schematic of energetically allowed tunnel processes for an oscillator in the ground state. Panels (a)–(f) depict the processes stemming from the tunnel terms $c^{†}\psi$, $c^{†}{\psi }^{†}$, $a^{†}{c}^{†}\psi$, $a^{†}c\psi$, $a^{†}c{\psi }^{†}$, and $a^{†}{c}^{†}{\psi }^{†}$, respectively, where $\widehat{x}=a^{†}+a$. Black dots (circles) represent electrons (holes) and the condensate is depicted as the blue bubble.

Figure 4

Differential conductance for $\xi \simeq 0$ at $\mathit{eV}\approx \Omega$ for a single RL coupled to an oscillator with $\langle n\rangle =0$ (solid black line) and $\langle n\rangle =1$ (dotted gray line). In the former case, the oscillator can enhance transport only for $\mathit{eV}>\Omega$. This is different in the presence of MBS. Then, an oscillator with $\langle n\rangle =0$ gives rise to a positive $d\langle I\rangle /dV$ for $\mathit{eV}>\Omega$ and $\mathit{eV}<\Omega$ (dashed-dotted blue line). We also show in dashed red the $d\langle I\rangle /dV$ for $\langle n\rangle =1$ in the MBS situation. The insets depict the decisive tunnel processes for the $\langle n\rangle =0$ case.