Interedge backscattering in time-reversal symmetric quantum spin Hall Josephson junctions

Abstract

Using standard tight-binding methods, we investigate a novel backscattering mechanism taking place on quantum spin Hall N'SNSN' Josephson junctions in the presence of time-reversal symmetry. This extended geometry allows for the interplay between two types of Andreev bound states (ABS): the usual phase-dependent ABS localized at the edges of the central SNS junction \emph{and} phase-independent ABS localized at the edges of the N'S regions. Crucially, the latter arise at discrete energies $E_n$ and mediate a backscattering process between opposite edges on the SNS junction, yielding gap openings when both types of ABS are coherently coupled. In this scenario, a 4$π$-periodic ABS decouples from the rest of the 2$π$-periodic spectrum, yielding several observable consequences: Firstly, we show that the $4π$-periodic spectrum can be probed by means of the Shapiro experiment even in the presence of dynamical transitions between the ABS and the quasicontinuum. Secondly, the presence of this backscattering mechanism distorts the superconducting quantum interference (SQI) pattern within the length scale, determined by the ratio between $4π$- and $2π$-periodic supercurrent contributions. Finally, we propose to use a magnetic flux to tune $E_n$ to zero, resulting in the selective lifting of the fractional Josephson effect.

Figures (8)

• Figure 1: Panel (a): Sketch of the extended quantum spin Hall Josephson junction. Here, the partial cover of the QSH bar (light blue) by superconducting leads (black) defines the extended N'SNSN' Josephson junction. Helical edge states are represented by blue and red curves. Inset: Andreev spectra as a function of the phase difference $\phi$ of the disconnected parts N'S (left), SNS (center) and SN' (right) regions. Panel (b): DOS for a N'SNSN' Josephson junction with $L_s=0.38\,\mu$m and $\xi_s=0.19\,\mu$m, $L_{N'}=0.5\,\mu$m. Avoided crossings coincide with the position of the energy levels at $E\approx 0.25\Delta_0$ and $0.65\Delta_0$.
• Figure 2: Panel (a): Schematic ABS spectrum that we use to approximate the full tight-binding spectrum shown in Fig. \ref{['fig:setup']}(b). The solid (dashed) line represents the states below (above) zero energy. The two insets show the gap $\delta_{1}$ between the $4\pi$- and $2\pi$-periodic ABS and the gap $\delta_{2}$ between the $2\pi$-periodic ABS and the quasiparticle continuum. Panel (b)-(e): The phase dynamics of a single edge for different values of the spectral gaps $\bar{\delta}_{1,2}$ compared to the phase velocities $v_\phi$ of the particles. Landau-Zener transitions do not occur for $\bar{\delta}_{1/2}\ll v_\phi$. In panel (c) we schematically represent the occupied quasiparticle continuum below zero energy and the unoccupied quasiparticle continuum above zero energy.
• Figure 3: DC Voltage $V$ as a function of the applied $I_{\mathrm{dc}}$ for various values of the Landau--Zener gaps $\bar{\delta}_{1,2}$ relative to the phase velocity $v_\phi$. First row: $\bar{\delta}_1\gg v_\phi$ remains fixed while $\bar{\delta}_2$ varies from $\bar{\delta}_2\gg v_\phi$ in panel (a) to $\bar{\delta}_2\ll v_\phi$ in panel (d). Then also $\bar{\delta}_1$ decreases from $\bar{\delta}_1\gg v_\phi$ in panel (d) to $\bar{\delta}_1\ll v_\phi$ in panel (g). Second row: $\bar{\delta}_2\gg v_\phi$ remains fixed and $\bar{\delta}_1$ varies from $\bar{\delta}_1\gg v_\phi$ in panel (h) to $\bar{\delta}_1\ll v_\phi$ in panel (k). Then also $\bar{\delta}_2$ decreases from $\bar{\delta}_2\gg v_\phi$ in panel (k) to $\bar{\delta}_2\ll v_\phi$ in panel (g).
• Figure 4: The SQI pattern for Josephson junctions with the same parameters as in Fig. \ref{['fig:setup']}(b) but for (a) different $L_{N'}/a=7, 20,70,120,170,220$ and (b) in the presence of disorder with increasing disorder strengths $\lambda$ for $L_{N'}/a=100$. For better visibility, the SQI curves are shifted upwards by a constant $\Delta I_\text{c}=I_{c,max}$. Panel (c): critical current vs magnetic flux. Panel (d): integrand of Eq. \ref{['eq:statcurr']} as a function of energy and $\Phi/\Phi_0$, for $\phi=\phi_\text{c}$ fulfilling $I_\text{c}=I(\phi_\text{c})$, with the same parameters as in Fig. \ref{['fig:setup']}(b). Panel (e): Andreev bound spectrum as a function of $\phi$, with $\Phi$ fulfilling the condition \ref{['eq.condition']}, where the phase independent ABS hybridizes at zero energy with the phase-dependent ABS. We set $L_s=0.33\,\mu$m to make the gaps more visible.
• Figure 5: Panel (a): critical current versus magnetic flux for (a) different lengths of the superconductor $L_S/a=180, 150, 120, 90, 60, 30$ and (b) different lengths of the external parts. We fix $L_{N’,L}/a=120$ and increase the length of the right part to $L_{N',R}/a=140,160,180,200,220$. For visibility, the SQI curves are shifted upwards a constant $\Delta I_\text{c}=I_{c,max}$. Panel (c): The Andreev bound state spectrum for the full BHZ Hamiltonian with both spin degrees of freedom.