Upper critical in-plane magnetic field in quasi-2D layered superconductors

TL;DR

This work develops a general, analytically tractable framework to analyze in-plane upper critical fields $H_{c2}(T)$ in quasi-2D layered superconductors with proximity-induced Ising and Rashba SOC, orbital depairing, and Zeeman effects, for both spin-singlet and spin-triplet pairings. Starting from a BBG-based microscopic model, the authors derive an effective low-energy 2×2 Hamiltonian and solve the linearized gap equation to obtain $H_{c2}(T)$, revealing distinct singlet versus triplet behaviors and how SOC components tune Pauli-limit violation via the PVR. Applying the framework to four BBG/WSe$_2$ experiments yields Ising-dominated SOC, negligible Rashba SOC, modest orbital coupling, and a consistently enhanced g-factor ($g>2$), suggesting interaction- or projection-driven renormalization beyond noninteracting theory. Overall, the methodology constrains pairing symmetry through in-plane field responses and is readily extendable to other van der Waals superconductors with similar low-energy physics.

Abstract

The study of the interplay of applied external magnetic field and superconductivity has been invigorated by recent works on Bernal bilayer and rhombohedral multilayer graphene. These studies, with and without proximitized spin-orbit coupling, have opened up a new frontier in the exploration of unconventional superconductors as they offer a unique platform to investigate superconductivity with high degree of in-plane magnetic field resilience and even magnetic field-induced superconductivity. Here, we present a framework for analyzing the upper critical in-plane magnetic field data in multilayer superconductors. Our framework relies on an analytically tractable superconducting pairing model that captures the normal state phenomenology of these systems and applies it to calculate the relationship between the upper critical field $H_{c2}$ and the corresponding critical temperature $T_{c}$. We study the $H_{c2}-T_{c}$ critical curve as a function of experimental parameters (Ising and Rashba spin-orbit coupling) and depairing mechanisms (Zeeman and orbital coupling) for both spin-singlet and spin-triplet pairing. By applying our framework to analyze four recent Bernal bilayer graphene-WSe$_2$ experiments [1-4], we identify an apparent discrepancy between fitted and measured spin-orbit parameters, which we propose can be explained by an enhancement of the Landé g factor in the Bernal bilayer graphene experiments.

Figures (2)

• Figure 1: a: Equipotential lines of BBG shows three low-density pockets arising from trigonal warping together with approximate parametrization of the relevant pocket momenta (see text). The inset shows one such (idealized to a circular shape) pocket spin-split with expected spin textures (here the out-of-plane component) under the in-plane magnetic field and SOCs. b: The ratio (PVR) of the upper critical field to the Pauli limiting value (see text for definition). We fit the experimental data extracted from iZhang2023_Enhanced_SC_proximitized_BBG (blue), iiZhang2025-ty (red), iiiHolleis2025_Nemacity_BBG (yellow), and ivTingxin_Li2024_BBG_WSe2 (purple). The fitted parameters are attached in Table \ref{['tab:fitting result']}. c: Upper critical field for spin singlet 2D superconductor with Ising and Rashba SOCs for a range of parameters. Ising SOC shows an enhancement to $H_{c2}$, while the Rashba SOC are less relevant to the upper critical field except at the zero temperature limit.
• Figure 2: a-c: The upper critical field in spin-triplet superconductor with varying Ising SOC when order parameter along x, y, and z direction; d-f: The upper critical field in spin-triplet superconductor with varying Rashba SOC when order parameter along x, y, and z direction. In a-f the superconducting and normal regions as defined by the critical PVR curve are denoted with "S" and "N" letters respectively. The color of the letters corresponds to the relevant curve shown in the panel.