Signatures of Topological Superconductivity and Josephson Diode Effects on the Magnetocurrent-Phase Relation of Planar Josephson Junctions

Abstract

We present a theoretical study of proximitized planar Josephson junctions (JJs) with Rashba spin-orbit coupling (SOC) subject to an in-plane magnetic field and demonstrate that the magneto-current-phase relation (magneto-CPR) provides a powerful and unified probe of their microscopic and topological properties. By analyzing the full phase and Zeeman-field dependence of the supercurrent, we show that magneto-CPR measurements allow one to reconstruct the ground-state phase that minimizes the system's free energy in the absence of phase bias. This reconstructed phase generally displays 0-pi-like transitions as a function of the Zeeman energy, and we demonstrate that the magnitudes of the associated phase jumps provide quantitative information about the Rashba SOC. We further show that the mixed phase-field response encoded in the magneto-CPR enables the extraction of the second mixed spin susceptibility, which serves as a sensitive diagnostic of gap closings and can be used to construct a superconducting topological phase diagram in terms of the relative topological gap. In addition, the magneto-CPR yields the field dependence of the forward and reverse critical currents, allowing one to characterize the Josephson diode effect and its connection to the Zeeman field, Rashba SOC, and junction transparency. Our results establish magneto-CPR measurements as a versatile spectroscopic tool that can be used to extract key system parameters and provide evidence of topological superconducting phases in planar JJs.

Figures (7)

• Figure 1: Schematic of a planar JJ composed of a semiconductor 2DEG (blue, bottom layer) in contact with two superconducting (S) leads (green, top layers). The yellow and red arrows indicate the direction of the magnetic field and current, respectively.
• Figure 2: Top: Ground-state phase difference as a function of Zeeman energy for different values of the Rashba SOC strength. (a) Results from the TB numerical simulations. (b) Analytical results computed using Eq. (\ref{['eqn:gs-phase-app']}). Bottom: Rashba SOC momentum strength [$k_{so}/k_F=\alpha m^\ast/(k_F \hbar^2)$] as a function of the GS-phase jump size. Solid line and symbols correspond to the analytical model and TB simulations, respectively. The TB simulations were performed for a junction with $W_N=96$ nm and $W_S=240$ nm. Other system parameters are as specified in Sec. \ref{['ss:numerical']}.
• Figure 3: (a) Magneto-CPR normalized to its maximum absolute value. The black dashed line traces the GS phase difference as a function of the Zeeman energy $E_Z$. Shaded (unshaded) regions denote trivial (topological superconducting) states with topological charge $Q_{\rm top}=1$ ($Q_{\rm top}=-1$). The red dot marks a transition from the trivial to the TS state upon increasing the Zeeman field in a phase-unbiased junction. (b) Topological gap as a function of the Zeeman field and the superconducting phase difference. (c) Second mixed susceptibility (solid line), normalized to its maximum value, and topological charge (dashed line) along the GS-phase path as functions of the Zeeman field. The second mixed susceptibility exhibits sharp sign changes at topological transitions, coincident with sign reversals of $Q_{\rm top}$. (d) Second mixed susceptibility, normalized to its maximum absolute value, as a function of Zeeman energy and the phase difference. The color scale has been truncated to improve contrast and better resolve fine features. Its behavior closely tracks the topological gap, exhibiting rapid sign changes at topological transitions and vanishing in regions with a larger gap. The TB simulations were performed for a junction with $\alpha = 16$ meV nm, $W_N=96$ nm, and $W_{S}=40$ nm.
• Figure 4: Same as in Fig. \ref{['fig:i-chi-1']}, but for a junction with $\alpha = 16$ meV nm and $W_{S}=88$ nm.
• Figure 5: (a) Zeeman-field dependence of the forward (dashed line) and reverse (solid line) critical currents obtained from the simplified analytical model for a junction with $\alpha = 4$ meV nm. (b) Same as panel (a), but for $\alpha = 16$ meV nm. (c) Diode quality factor as a function of the Zeeman field. Solid and dashed lines were obtained from the results in (a) and (b), respectively.