Decays of Majorana or Andreev Oscillations Induced by Steplike Spin-Orbit Coupling

The Majorana zero mode in the semiconductor-superconductor nanowire is one of the promising candidates for topological quantum computing. Recently, in islands of nanowires, subgap-state energies have been experimentally observed to oscillate as a function of the magnetic field, showing a signature of overlapped Majorana bound states. However, the oscillation amplitude either dies away after an overshoot or decays, sharply opposite to the theoretically predicted enhanced oscillations for Majorana bound states. We reveal that a steplike distribution of spin-orbit coupling in realistic devices can induce the decaying Majorana oscillations, resulting from the coupling-induced energy repulsion between the quasiparticle spectra on the two sides of the step. This steplike spin-orbit coupling can also lead to decaying oscillations in the spectrum of the Andreev bound states. For Coulomb-blockade peaks mediated by the Majorana bound states, the peak spacings have been predicted to correlate with peak heights by a $\pi /2$ phase shift, which was ambiguous in recent experiments and may be explained by the steplike spin-orbit coupling. Our work will inspire more works to reexamine effects of the nonuniform spin-orbit coupling, which is generally present in experimental devices.

Figure 1

(a) Schematic of the semiconductor-superconductor nanowire island [31, 32, 33, 34, 35, 36], its two ends may host a pair of Majorana bound states (MBSs). [(b)-(d)] The red and black curves are adapted from Ref. [31]. The MBSs can hybridize. The hybridization energy $E_0$ in the experiments oscillates with decaying amplitude and increasing period as a function of the magnetic field $B$. However, opposite to the experiments, Majorana theory predicts that $E_0$ oscillates with increasing amplitude as a function of the $B$-induced Zeeman energy $V_Z$ [29]. By considering the steplike spin-orbit coupling $\alpha (x)$ in Fig. 2 (see the parameters listed in Sec. SI of Ref. [51]), we find that the oscillation patterns of $E_0$, both the decay in amplitude and increase in period, can be well captured by the blue curves. See Fig. 4 for the relations between $⟨S_{e/o}⟩$ and $E_0$.

Figure 2

Why Majorana oscillations decay. (a) A nanowire is decoupled at $x_L$ into two parts with different spin-orbit coupling described by $\alpha (x)$. [(b) and (c)] The energy spectra of the left and right parts, respectively. (d) The coupling $V_{ee/eh}$ between the lowest-energy spectra in (b) and (c) (red and black dashed) repels their lower energies to form the blue solid spectrum, qualitatively consistent with the oscillation pattern in Fig. 1. (e) $V_{ee/eh}$ increase with increasing $V_Z=g_{\mathrm{eff}}{\mu }_{B}B/2$ since the wave functions move towards the wire ends [see top of (a)], so they suppress the enhanced oscillations in (b) into decaying oscillations with increasing periods [blue solid in (d)]. [(f) and (g)] Majorana wave functions ${\psi }_A=(1/\sqrt{2})({\psi }_{E_0}+{\psi }_{-E_0})$ and ${\psi }_B=(i/\sqrt{2})({\psi }_{E_0}-{\psi }_{-E_0})$, with ${\psi }_{\pm E_0}$ the lowest-energy wave functions of the entire wire, at $V_Z$ indicated in (d). The parameters are $m^{*}=0.026m_e$, $\mathrm{\Delta }=0.25\mathrm{meV}$, ${\alpha }_0=0.04\mathrm{eV}\AA$, $A=0.4\mathrm{eV}\AA$, and $\mu =0$.

Figure 3

(a) The steplike spin-orbit coupling $\alpha (x)$ and smoothly varying chemical potential $\mu (x)$. (b) $x_L=0$ means a uniform spin-orbit coupling, for which, the near-zero-energy Andreev bound states persist for a wide range of $V_Z$ before the topological phase transition point at which the superconducting gap nearly closes and reopens. (c) At $V_Z$ marked by the triangle in (b), the projections of the lowest-energy wave functions on the Majorana basis are partially separated. [(d) and (e)] In the presence of the steplike spin-orbit coupling with different $x_L$, decaying oscillations also exist in the spectrum of the Andreev bound states (trivial regime). The parameters except for $\mu (x)$ are the same as those in Fig. 2.

Figure 4

(a) The Coulomb blockade peaks of the conductance as a function of the gate voltage $V_{\mathrm{G}}$. The transition from the odd to even parity ($o\to e$) differs from $e\to o$ in the height of peak ($G_{o\to e}$, $G_{e\to o}$) and spacing between two neighboring peaks ($S_e$, $S_o$). The experimental measured $\pm E_0$ are extracted from the peak spacings $\pm E_0=\eta ⟨S_{e(o)}⟩-E_C$ [corresponding to the red and blue data in Figs. 1], where $⟨S_e⟩$ and $⟨S_o⟩$ are the ensemble-averaged peak spacings, $E_C$ the charging energy, and $\eta =2E_C/(⟨S_e⟩+⟨S_o⟩)$ (Sec. SVI of the Supplemental Material [51]). Lowest-energy spectrum $E_0$ (blue curves) and peak height ratio $\mathrm{\Lambda }=G_{e\to o}/(G_{e\to o}+G_{o\to e})$ (orange curves) as functions of $V_Z$ with (b) constant spin-orbit coupling $\alpha =0.16\mathrm{eV}\AA$ and (c),(d) smoothed steplike spin-orbit coupling described by Eq. (2) with (c) $x_L=0.43\mu \mathrm{m}$, $x_R=2\mu \mathrm{m}$, ${\alpha }_0=0.03\mathrm{eV}\AA$, and $A=0.44\mathrm{eV}\AA$, and (d) $x_L=0.34\mu \mathrm{m}$, $x_R=1.8\mu \mathrm{m}$, ${\alpha }_0=0.02\mathrm{eV}\AA$, and $A=0.5\mathrm{eV}\AA$. Other parameters are $L=2.3\mu \mathrm{m}$, ${\lambda }_L={\lambda }_R=0.05\mu \mathrm{m}$, $\mu =0$, and ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$. Here the steps of spin-orbit coupling are smoothed by using finite ${\lambda }_{L/R}$.