All-electrical spectroscopy of topological phases in semiconductor-superconductor heterostructures

Semiconductors in the proximity of superconductors have been proposed to support phases hosting Majorana bound states. When the systems undergo a topological phase transition towards the Majorana phase, the spectral gap closes, then reopens, and the quasiparticle band spin polarization is inverted. We focus on two paradigmatic semiconductor-superconductor heterostructures and propose an all-electrical spectroscopic probe sensitive to the spin inversion at the topological transition. Our proposal relies on the indirect coupling of a time-dependent electric field to the electronic spin due to the strong Rashba spin-orbit coupling in the semiconductor. We analyze within linear response theory the dynamical correlation functions and demonstrate that some components of the susceptibility can be used to detect the nontrivial topological phases.

Figure 1

The present paper studies the (a) 1D and (b) the 2D models proposed in Refs. [4, 5], respectively, Ref. [6], as physical platforms supporting Majorana bound states. Additional gates with time-oscillating voltage are attached to the semiconductor (in red), allowing to modulate the spin-orbit coupling. The response of the systems to the alternating electrical field discriminates near the topological transition the nontrivial phases.

Figure 2

Spin polarization $\langle {\sigma }_x\rangle$ of the electronic bands of the 1D model, close to the topological transition, in the (a) trivial phase and (b) in the nontrivial phase. The spectral gap has the same magnitude in both cases, $|E_g|=0.04\mathrm{\Delta }$ and $\mu =0$.

Figure 3

Imaginary part of the susceptibility ${\chi }_d^{″}\left(\omega \right)$ as a function of frequency for (a) the 1D model and (b) the 2D model. The spectral gap has the same magnitude in all cases $|E_g|=0.04\mathrm{\Delta }$, but, according to the sign of $E_g$, the systems may be either in a topologically trivial phase (T, blue), or a nontrivial one (NT, red). The response shows different behavior depending on the phase. The solid lines are obtained by numerical integration in Eq. (10), while the dashed lines plot the analytical results in Eqs. (16) and (18). The panels share the legend.

Figure 4

Imaginary part of the susceptibility ${\chi }_d^{″}\left(\omega \right)$ as a function of the spectral gap $E_g=E_{\mathrm{Z}}-\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$ for (a) the 1D model and (b) the 2D model. The frequency is fixed at $\hslash \omega =0.16\mathrm{\Delta }$. The susceptibility is computed either from the analytical expressions (16) and (18) (red line), or numerically from Eq. (10) (blue line). The panels share the legend.

Figure 5

Susceptibility ${\chi }_d^{″}$, in units of $\mathrm{\Delta }/{\alpha }^2$, in the (a) 1D model and, in units ${\mathrm{\Delta }}^2/{\alpha }^3$, in the (b) 2D model as a function of chemical potential $\mu$ and Zeeman energy $E_{\mathrm{Z}}$. The response changes sign at the topological phase transition (green dashed line) $E_g=0$ or $E_{\mathrm{Z}}^2={\mathrm{\Delta }}^2+{\mu }^2$. The frequency is tuned at the resonance condition $\hslash \omega =2|E_g|$.

Figure 6

Spin polarization in the lowest-energy bands of the 1D model (2) for the modified system parameters used in the finite system simulations, $t=1\mathrm{meV}$, $\alpha =0.1\mathrm{nm}·\mathrm{meV}$, $\mathrm{\Delta }=0.25\mathrm{meV}$, and $\mu =0$. The spectral gap is (a) $E_g=-0.04\mathrm{\Delta }$, in the trivial phase, and (b) $E_g=0.04\mathrm{\Delta }$, in the nontrivial phase.

Figure 7

Imaginary part of the disorder-averaged Rashba susceptibility in the discretized 1D nanowire Eq. (2) on 1200 sites, and for 100 disorder realizations at each frequency. The lines corresponding to $\sigma =0$ are obtained in the infinite clean system. (a) Rashba susceptibility as a function of frequency in the topologically nontrivial phase $E_g=0.04\mathrm{\Delta }$ (red lines) and trivial phase $E_g=-0.04\mathrm{\Delta }$ (blue lines). The dashed vertical line at $2|E_g|$ marks the excitation threshold in the clean system. (b) Rashba susceptibility near the transition, as a function of the gap $E_g$ at fixed frequency $\hslash \omega =0.08\mathrm{\Delta }$.

Figure 8

Imaginary part of the disordered-averaged Rashba susceptibility for a tight-binding model which discretizes Eq. (3) on a $120\times 120$ site periodic patch for two disorder strengths, with 100 disorder realizations at each frequency, and in an infinite system ($\sigma =0$). (a) Susceptibility as a function of frequency in the topologically nontrivial phase ($E_g=0.04\mathrm{\Delta }$, red curves) and trivial phase ($E_g=-0.04\mathrm{\Delta }$, blue curves). The vertical line marks the excitation threshold $2|E_g|$. (b) Susceptibility as a function of the spectral gap at fixed driving frequency $\hslash \omega =0.04\mathrm{\Delta }$.

Figure 9

Imaginary susceptibility ${\chi }_{1\mu }^{″}$ in one dimension, in units $1/\alpha$, and ${\chi }_{2\mu }^{″}$ in two dimensions, in units $\mathrm{\Delta }/{\alpha }^2$. (a, c) Susceptibility ${\chi }_{1\mu }^{″}$, respectively ${\chi }_{2\mu }^{″}$, at the resonance condition $\hslash \omega =2|E_g|$, when the chemical potential $\mu$ varies and $E_{\mathrm{Z}}=1.04\mathrm{\Delta }$. At finite $\left|\mu \right|$, a dip in the response marks the topological transition. (b) A zoom for the 1D case at a topological transition for a fixed $\mu =0.4\mathrm{\Delta }$. The system crosses the topological transition when varying the gap $E_g$, by changing the Zeeman field strength. (d) Susceptibility ${\chi }_2^{″}$ as a function of frequency behaves similarly in the topologically nontrivial phase (blue line, $\mathrm{NT}$, at $E_g=0.04\mathrm{\Delta }$) and in the trivial phase (red line, $\mathrm{T}$, at $E_g=-0.04\mathrm{\Delta }$) at $\mu =0$. Similar results are presented in the inset for ${\chi }_1^{″}$. In contrast to the Rashba susceptibility there is no sign change in response at the topological transition.