Planar Josephson junction devices with narrow superconducting strips: Topological properties and optimization

TL;DR

This paper analyzes planar semiconductor–superconductor Josephson junctions with narrow superconducting films to optimize the topological superconducting phase and Majorana modes. By combining recursive Green's function methods with a static, proximity-renormalized effective Hamiltonian, it maps the topological phase diagram and shows that reducing the SC film width to about $W_{SC}\in[100,200]$ nm can yield topological gaps up to roughly $0.40\,Δ_0$ in clean systems. The study also examines how gate potentials, outside SM regions, and junction depletion influence topology, revealing a nontrivial dependence of the gap on geometry and screening effects and identifying an optimal regime where the gap is maximized. The work proposes a stepwise optimization strategy (theory baseline, theory–experiment feedback, and realistic device realization) toward robust planar JJ devices hosting Majorana zero modes, with the potential for enhanced disorder robustness relative to wide-SC structures.

Abstract

We study the low-energy physics of planar Josephson junction structures realized in a quasi-two dimensional semiconductor system proximity-coupled to narrow superconducting films. Using both a recursive Green's function approach and an effective Hamiltonian approximation, we investigate the topological superconducting phase predicted to emerge in this type of system. We first characterize the effects associated with varying the electrostatic potentials applied within the unproximitized semiconductor regions. We then address the problem of optimizing the width of the superconductor films and identifying the optimal regimes characterized by large topological gap values. We find that structures with narrow superconducting films of widths ranging between about $100~$nm and $200~$nm can support topological superconducting phases with gaps up to $40\%$ of the parent superconducting gap, significantly larger than those characterizing the corresponding wide-superconductor structures. This work represents the first component of a proposed comprehensive strategy to address this optimization problem in planar Josephson junction structures and realize robust topological devices.

Figures (29)

• Figure 1: Schematic representation of the planar SM-SC hybrid structure: (a) lateral view; (b) top view. A 2D electron gas hosted by a SM quantum well (orange) is proximity coupled to two thin SC films (blue) of width $W_{SC}$, forming an infinitely long Josephson junction of width $W_{J}$. Unproximitized SM regions (of widths $W_L$ and $W_R$) are present outside the SC films. The electrostatic potential in the regions not covered by the SCs is controlled by top gates (gray). An external magnetic field $B$ is applied in the $x-y$ plane (typically parallel to the junction, i.e., in the $x$ direction).
• Figure 2: Top: Density of states as a function of Zeeman field (applied parallel to the junction) and energy for a JJ structure with chemical potential $\mu=31.4~$meV and junction potential $V_J=25~$meV. The SC width is $W_{SC}=150~$nm and the outside SM regions are depleted, $V_L=V_R=100~$meV. The vanishing of the quasiparticle gap (at $\Gamma_x\approx 0.35~$meV and $\Gamma_x\approx 2.4~$meV) is associated with topological quantum phase transitions. For $\Gamma_x\gtrsim 1~$meV the topological gap is nearly zero (on the order of a few $\muup$eV). Middle: Zeeman field dependence of the spectral function at $k=0$. Bottom: Quasiparticle gap (black) and $k=0$ modes (blue lines) calculated using the effective Hamiltonian corresponding to the static approximation (see main text).
• Figure 3: Topological phase diagram as a function of Zeeman field ($\Gamma_x$) and chemical potential for a JJ device with $W_{SC}=150~$nm. The phase boundaries correspond to the vanishing of the (bulk) quasiparticle gap at $k=0$. The white areas are topologically trivial, while the red regions correspond to a topological superconducting phase. The gate potential on the junction is $V_J$ = 25 meV and the gate potentials outside the SC regions are $V_L = V_R = 100~$meV. The relative phase difference between the two narrow SCs is (a) $\phi=0$ and (b) $\phi=\pi$. The horizontal cut at $\mu=31.4~$meV (marked by a green line) corresponds to the low-energy spectra in Fig. \ref{['Fig2']}, while the DOS along the cut at $\mu=36.4~$meV is shown in Fig. \ref{['Fig7']}(a).
• Figure 4: Evolution of a section of the topological phase diagram in Fig. \ref{['Fig3']} as function of the angle $\theta$ between the applied Zeeman field and the direction parallel to the junction (i.e. the $x$-direction). The lines correspond to a finite spectral weight at zero energy and $k=0$, i.e., they indicate the vanishing of the quasiparticle gap at $k=0$. Note that within each "loop" the $\mathbb{Z}_2$ topological invariant $\nu$ given by Eq. (\ref{['nu']}) is nontrivial (i.e., $\nu=-1$, see Fig. \ref{['Fig6']}). However, the corresponding low-energy spectrum may be gapless (see Figs. \ref{['Fig7']} and \ref{['Fig8']}).
• Figure 5: Phase diagram as a function of Zeeman field, $\Gamma_y$, applied perpendicular to the junction and chemical potential. The gate potential in the junction region is $V_J = 25~$meV and the gate potential outside the SC regions is $V_L= V_R = 100~$meV. The relative phase difference between the two narrow SCs is (a) $\phi=0$ and (b) $\phi=\pi$. The light-colored lines correspond to a finite spectral weight at zero energy and $k=0$. The density of states along the horizontal cuts marked by green lines in (a), which correspond to $\mu = 36.4~$meV and $\mu=31.4~$meV, is shown in Fig. \ref{['Fig7']}, panels (b) and (c), respectively.