Detecting Topological Phase Transition in Superconductor-Semiconductor Hybrids by Electronic Raman Spectroscopy

TL;DR

This work develops a bulk-probe framework for detecting the topological phase transition in superconductor–semiconductor hybrids by examining the dynamical density-density response, including Coulomb backflow, and its Raman-active signatures. The authors show that the field evolution of the renormalized density response reveals the bulk gap closing at $B_c=\sqrt{\mu^2+\Delta^2}$ and subsequent reopening in the topological phase, even when trivial Andreev bound states obscure local conductance signals. In the normal state, gapless plasmons signaling a Lifshitz transition at $B_L=\mu$ appear, softening as the field approaches the transition. The results indicate that bulk Raman spectroscopy can serve as a robust spectroscopic tool to detect the TPT in superconducting nanowires, with the Higgs and phase modes of the parent superconductor contributing modestly to the signal and the wire-dominated response providing a clear experimental handle.

Abstract

In superconductor-semiconductor hybrids, applying a magnetic field closes a trivial bulk gap and causes a topological phase transition (TPT), resulting in the emergence of Majorana zero modes at both ends of the wires. However, trivial Andreev bound states formed at the interface with metallic leads mimic the local Majorana properties, making it difficult to detect the TPT through local conductance measurements. In this work, we investigate the detection of the TPT by exploiting the static and dynamical density response of the hybrid system. In particular, we demonstrate that the dynamical renormalized responses, the density response including the effect of Coulomb interactions, reveal the characteristic electronic structure and detect the TPT, which we then show to produce strong intensities of Raman scattering. Furthermore, we find that gapless plasmons emerge in the normal state, signaling the bulk Lifshitz transition. Our results thus predict that the bulk response of superconducting nanowires is a powerful spectroscopic approach to detect the bulk topological phase transition.

Figures (13)

• Figure 1: (a) Schematics of our setup: Superconducting wire with Rashba spin-orbit coupling, called the Rashba superconductor, under a Zeeman field $B$. Incident light (red) from a laser (green) reaches the Rashba superconductor, where it is scattered and then reaches the detector (yellow). (b) Energy dispersion in the normal state at $\delta B=-0.18~{\rm T}$, the Lifshitz transition $\delta B = 0$ ($B=B_{\rm L}$), and $\delta B=0.18~{\rm T}$, where we set $\Delta = 0$. (c) Energy dispersion in the superconducting state at $\delta B=-0.18~{\rm T}$, the TPT $\delta B = 0$ ($B=B_{\rm c}$), and $\delta B=0.18~{\rm T}$, where $k^{\rm N}_{{\rm F},+}$ ($k^{\rm N}_{{\rm F},-}$) denotes the Fermi wavenumber of the inner (outer) band in the normal state and $\Delta_{1}$ ($\Delta_{2}$) represents the excitation gap at $k=k^{+}_{\rm F}$ ($k=k^{-}_{\rm F}$).
• Figure 2: Static dielectric function, $\epsilon(q)$, in the superconducting state (thick red/blue curves) and in the normal state (thin black/green curves) at $\delta B \equiv B-B_{\rm c}=-0.36~{\rm T}$ (a), $\delta B=-0.01~{\rm T}$ ($B=B_{\rm L}$) (b), $\delta B=0$ ($B=B_{\rm c}$) (c), and $\delta B=0.36~{\rm T}$ (d), where we set $T=0.01T_{\rm c}$. The thick blue and red (thin black and green) curves correspond to $\epsilon(q)$ in the superconducting (normal) state with the radius $a=1~{\rm nm}$ and $10~{\rm nm}$, respectively. (e) Field dependence of $\epsilon(q)$ in the superconducting state. The dotted lines show the $B$-dependences of the characteristic wavenumbers, $2k^{\rm N}_{{\rm F},\pm}$ and $k^{\rm N}_{{\rm F},+}+k^{\rm N}_{{\rm F},-}$.
• Figure 3: (a-c) Density response $-{\rm Im}\tilde{\chi}_{\rho\rho}(q,\omega)$ in the normal state $\Delta=0$ at $\delta B = -0.18~{\rm T}$ (a), $0$ (b), and $0.18~{\rm T}$ (c), where $\delta B = 0$ corresponds to the Lifshitz transition since here $\Delta=0$. (d-f) Three particle-hole continua are denoted by processes I, II, and III, where I (II) is for intraband excitations in the inner (outer) band, while process III corresponds to the interband excitations. We take $a=10~{\rm nm}$ and the other parameters are the same as those in Fig. \ref{['fig:dd']}.
• Figure 4: (a-c) Renormalized density response $-{\rm Im}\tilde{\chi}_{\rho\rho}(q,\omega)$ in the superconducting state as a function of $q$: (a) $\delta B = -0.18~{\rm T}$, (b) $0$ ($B=B_{\rm c}$), and (c) $0.18~{\rm T}$. The green (magenta) curve denotes $\omega^+_{\rm pair}$ ($\omega^-_{\rm pair}$), which is the lowest pair excitation energies within the inner and outer bands, respectively [see Fig. \ref{['fig:setup0']}(b)]. (d) Bare density response, $-{\rm Im}\chi_{\rho\rho}(q,\omega)$ and (e) renormalized density response $-{\rm Im}\tilde{\chi}_{\rho\rho}(q,\omega)$ as a function of $B$ at $q=1.0\times 10^{-4}~{\rm nm}^{-1}$. In (d,e), the thin dashed and dotted-dashed lines correspond to the field dependence of the superconducting gaps, $2\Delta_1$ and $2\Delta_2$, respectively. We remark that $\tilde{\chi}_{\rho\rho}$ coincides with the Raman response function, $\tilde{\chi}_{\gamma\gamma}$, at $\gamma = 1$. Here we set $a=10~{\rm nm}$. (f) Quasiparticle excitation gaps in the inner and outer Fermi surfaces, $2\Delta_{1,2}$, as a function of $B$. See Fig. \ref{['fig:setup0']}(c) for the definition of $\Delta_{1,2}$.
• Figure 5: (a) Three processes that influence the density response of the semiconducting wire. Process I corresponds to $\tilde{\chi}_{\rho\rho}$ in Eq. \ref{['eq:chi_rpa']}. Process II involves the tunneling of pair excitations and Higgs bosons driven by an incident light into the wire, while process III is mediated by collective excitations in the parent superconductor. (b,c) Density response functions through three processes: (b) $\delta B=-0.18~{\rm T}$, (c) $0$, and (d) $0.18~{\rm T}$ at $q=1.0\times 10^{-4}~{\rm nm}^{-1}$. Here we take the tunneling energy as $\Gamma={\mu}=0.2~{\rm meV}$. We also choose the gap and the chemical potential of the parent superconductor as $\Delta_{\rm sc}=1~{\rm eV}$ and $\mu_{\rm sc}=0.2~{\rm meV}$, respectively (see Sec. \ref{['sec:CM']}).