Majorana edge and end states in planar Josephson junctions

TL;DR

The paper addresses how Majorana bound states arise and localize in proximitized planar Josephson junctions with in-plane magnetic fields and Rashba/Dresselhaus SOCs. It develops a BdG-based theoretical framework and a topological gap character to categorize end-like versus edge-like MSs, demonstrating that transitions between these states can be driven by the superconducting phase difference and Zeeman energy in both phase-biased and phase-unbiased settings. The main findings show that edge-like MSs can robustly extend along the sample edges and function as interconnects between adjacent junctions, while end-like MSs localize at junction ends; the presence and strength of the topological gap depend on SOC composition and orientation. These results provide design principles for controllable Majorana networks in planar JJs and highlight how disorder can modulate the topological protection, with implications for scalable topological qubit architectures and interconnect strategies.

Abstract

We theoretically investigate the localization properties of Majorana states (MSs) in proximitized, planar Josephson Junctions (JJs) oriented along different crystallographic orientations and in the presence of an in-plane magnetic field and Rashba and Dresselhaus spin-orbit couplings. We show that two types of MSs may emerge when the junction transits into the topological superconducting state. In one case, referred to as end-like MSs, the Majorana quasiparticles are mainly localized inside the normal region at the opposite ends of the junction. In contrast, edge-like MSs extend along the opposite edges of the system, perpendicular to the junction channel. We show how the MSs can transit from end-like to edge-like and vice versa by tuning the magnetic field strength and/or the superconducting phase difference across the junction. In the case of phase-unbiased JJs the transition may occur as the ground state phase difference self-adjusts its value when the Zeeman field is varied. We propose exploiting the extended nature of edge-like MSs as effective interconnects enabling the coupling between topological states in adjacent planar JJs. The impact of electrostatic disorder on the MSs is also analyzed.

Figures (10)

• Figure 1: (a) Schematic of a JJ consisting of a noncentrosymmetric semiconductor 2DEG (blue) in contact with two superconducting (S) leads (green). The $\hat{x}$ and $\hat{y}$ axes define the coordinate system in the junction's reference frame. A top gate (not shown) over the normal (N) region can be used to modulate the Rashba SOC strength PhysRevLett.126.036802mayer2020gate. (b) Relevant angles in the junction coordinate system: $\varphi_B$ defines the direction of the in-plane magnetic field ($\mathbf{B}$) with respect to the $\hat{x}$ axis, while $\theta_c$ determines the orientation of the junction reference frame with respect to the semiconductor's [100] crystallographic axis.
• Figure 2: (a) Topological gap character ($\tilde{\Delta}$) as a function of the Zeeman energy $E_Z$ and the superconducting phase difference ($\phi$) across an Al/HgTe JJ with only Rashba SOC ($\theta_{so}=0$). The junction and magnetic field orientations are set to $\theta_c=0$ and $\varphi_B=\pi/2$, respectively. The green solid line represents the path of the ground-state phase ($\phi_{GS}$) as the Zeeman energy is varied. The vertical dashed line marks a possible transition between a TS state supporting a zero-phase edge-like MS (cyan triangle) and one supporting an end-like MS (magenta dot) during which $E_Z$ is kept constant, while $\phi$ is tuned. (b)-(d) Probability density (normalized to its maximum value) of the MSs corresponding to the $E_Z$ and $\phi$ values marked in (a) by the cyan triangle (edge-like MS), magenta square (end-like MS), and magenta dot (end-like MS), respectively. (e)-(g) Energy spectra as a function of the Zeeman energy for $\phi=0$, $\phi=\pi/2$, and $\phi=\pi$, respectively. Red-solid and dashed-blue lines represent states that evolve into MSs as $E_Z$ is varied. Vertical dashed lines indicate the boundaries of the first topological region in which only a single pair of MSs (red solid lines) exists.
• Figure 3: (a) Topological gap character ($\tilde{\Delta}$) as a function of the Zeeman energy $E_Z$ and the superconducting phase difference ($\phi$) across an Al/InSb JJ, where the Rashba SOC has been tuned to a negligibly small value and only Dresselhaus SOC is relevant ($\theta_{so}=\pi/2$). The junction and magnetic field orientations are set to $\theta_c=0=\varphi_B=0$. The green solid line represents the path of the ground-state phase ($\phi_{GS}$) as the Zeeman energy is varied. The vertical dashed line marks a possible transition between a TS state supporting a zero-phase edge-like MS (cyan triangle) and one supporting an end-like Majorana state (magenta dot) during which $E_Z$ is kept constant, while $\phi$ is tuned. (b)-(d) Probability density (normalized to its maximum value) of the MSs corresponding to the $E_Z$ and $\phi$ values marked in (a) by the cyan triangle (edge-like MS), cyan square (edge-like MS), and magenta dot (end-like MS), respectively. (e)-(g) Energy spectra as a function of the Zeeman energy for $\phi=0$, $\phi=\pi/2$, and $\phi=\pi$, respectively. Red-solid and dashed-blue lines represent states that evolve into MSs as $E_Z$ is varied. Vertical dashed lines indicate the boundaries of the first topological region in which only a single pair of MSs (red solid lines) exists. The deviation of the edge-like MS energies (red solid lines) from zero in (f) results from the wavefunction overlap between edge-like MSs on opposite edges [see (c)].
• Figure 4: (a) Topological gap character ($\tilde{\Delta}$) as a function of the Zeeman energy $E_Z$ and the superconducting phase difference ($\phi$) across an Al/InSb JJ, with equal Rashba and Dresselhaus SOC strengths ($\theta_{so}=\pi/4$). The junction and magnetic field orientations are set to $\theta_c=3\pi/4$ and $\varphi_B=\pi/2$, respectively. The green solid line represents the path of the ground-state phase ($\phi_{GS}$) as the Zeeman energy is varied. The vertical (horizontal) dashed line marks a possible transition between a TS state supporting a zero-phase edge-like MS (cyan triangle/dot) and one supporting an end Majorana state (magenta square) during which $E_Z$ ($\phi$) is kept constant while $\phi$ ($E_Z$) is tuned. A transition between edge-like (e.g., cyan dot) and end-like (e.g., magenta cross) MSs can also be achieved by solely tuning $E_Z$, as the value of $\phi$ self-adjusts and follows the path of the ground-state phase (green solid line). (b)-(d) Probability density (normalized to its maximum value) of the MSs corresponding to the $E_Z$ and $\phi$ values marked in (a) by the cyan triangle (edge-like MS), magenta square (end-like MS), and cyan dot (edge-like MS), respectively. (e)-(g) Energy spectra as a function of the Zeeman energy for $\phi=0$, $\phi=\pi/2$, and $\phi=3\pi/5$, respectively. Red-solid and dashed-blue lines represent states that evolve into MSs as $E_Z$ is varied. Vertical dashed lines indicate the boundaries of the first topological region in which only a single pair of MSs (red solid lines) exists.
• Figure 5: (a) Topological gap character ($\tilde{\Delta}$) as a function of the Zeeman energy $E_Z$ and the superconducting phase difference ($\phi$) across an Al/InSb JJ, with Rashba SOC strength about 2.4 times greater than the Dresselhaus SOC strength ($\theta_{so}=\pi/8$). The junction and magnetic field orientations are set to $\theta_c=3\pi/4$ and $\varphi_B=\pi/2$, respectively. (b)-(d) Probability density (normalized to its maximum value) of the MSs corresponding to the $E_Z$ and $\phi$ values marked in (a) by the cyan triangle (edge-like MS), magenta square (end-like MS), and magenta dot (end-like MS), respectively. (e) Energy spectrum along the path indicated by the vertical dashed line in (a), where $E_Z=0.23$ meV and $\phi$ is varied from 0 to $2\pi$. The symbols in (e) indicated the energy of the MSs whose probability densities are plotted in (b)-(d).