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Semiclassical theory for proximity-induced superconducting systems with spin-orbit coupling

Zhen-Cheng Liao, Cong Xiao, Zhi Wang, Qian Niu

TL;DR

This work develops a semiclassical framework for superconducting quasiparticles in proximity-induced systems with spin-orbit coupling, revealing a rich structure of phase-space Berry curvatures that govern quasiparticle dynamics and thermal responses.Using a field-variational approach, the authors derive two key thermo-spin effects: a thermal Edelstein response driven by mixed $(k,m)$ Berry curvature and a spin-Nernst response arising from momentum-space Berry curvature, both tied to the superconducting gap structure and band geometry.Applied to a minimal 2D Ferromagnetic Rashba model, the theory demonstrates substantial Berry-curvature–driven thermo-spin signals across topological and geometrical regimes, including in topologically trivial states where the curvature concentrates near electron Fermi surfaces.The results illuminate how proximity-induced superconductivity with SOC entangles pairing and Bloch-band geometry to produce measurable geometric spin responses, providing a framework for superconducting Berry-curvature physics in engineered heterostructures.

Abstract

We develop a semiclassical theory of superconducting quasiparticles for proximity-induced superconducting systems, where spin-orbit coupling plays a critical role in shaping the quasiparticle dynamics. We reveal the structure of superconducting Berry curvatures in such systems, and derived the superconducting Berry curvature induced thermal Edelstein effect and spin Nernst effect. We calculate these two thermo-spin responses with model systems where Rashba spin-orbit coupling, proximity induced superconductivity, and ferromagnetic order are coexisting.

Semiclassical theory for proximity-induced superconducting systems with spin-orbit coupling

TL;DR

This work develops a semiclassical framework for superconducting quasiparticles in proximity-induced systems with spin-orbit coupling, revealing a rich structure of phase-space Berry curvatures that govern quasiparticle dynamics and thermal responses.Using a field-variational approach, the authors derive two key thermo-spin effects: a thermal Edelstein response driven by mixed $(k,m)$ Berry curvature and a spin-Nernst response arising from momentum-space Berry curvature, both tied to the superconducting gap structure and band geometry.Applied to a minimal 2D Ferromagnetic Rashba model, the theory demonstrates substantial Berry-curvature–driven thermo-spin signals across topological and geometrical regimes, including in topologically trivial states where the curvature concentrates near electron Fermi surfaces.The results illuminate how proximity-induced superconductivity with SOC entangles pairing and Bloch-band geometry to produce measurable geometric spin responses, providing a framework for superconducting Berry-curvature physics in engineered heterostructures.

Abstract

We develop a semiclassical theory of superconducting quasiparticles for proximity-induced superconducting systems, where spin-orbit coupling plays a critical role in shaping the quasiparticle dynamics. We reveal the structure of superconducting Berry curvatures in such systems, and derived the superconducting Berry curvature induced thermal Edelstein effect and spin Nernst effect. We calculate these two thermo-spin responses with model systems where Rashba spin-orbit coupling, proximity induced superconductivity, and ferromagnetic order are coexisting.
Paper Structure (20 sections, 111 equations, 3 figures)

This paper contains 20 sections, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Semiclassical approach
  3. Local Hamiltonian
  4. Quasiparticle wavepacket
  5. Equations of motion
  6. Thermal responses
  7. Thermo-spintronic responses of Berry curvature origins
  8. Thermal Edelstein effect
  9. Spin Nernst effect
  10. Analysis of Berry curvature
  11. Berry curvature around the electron Fermi surface
  12. Two-band electron system
  13. Ferromagnetic Rashba model
  14. Model calculation
  15. Berry curvature
  16. ...and 5 more sections

Figures (3)

  • Figure 1: (a) The electron spectrum for the tight-binding ferromagnetic Rashba model. The dashed lines labeled with a-d represent different chemical potentials. (b) Quasiparticle spectrum when the chemical potential cuts both two electron bands as the dashed line c shown in (a). The color denotes the sign of the effective charge of the quasiparticle. The model parameters are taken as $\alpha_R/t=0.2$, $V_z/t=0.15$ and $\Delta/t=0.05$.
  • Figure 2: Distribution of Momentum-space superconducting Berry curvatures for four typical states with distinct chemical potentials, which are illustrated in Fig. \ref{['fig:spectrum']}(a). The Chern numbers are indicated explicitly for each figure.
  • Figure 3: (a) Thermal Edelstein coefficient and (b) spin Nernst conductivity as a function of temperature. The four lines correspond to the four typical model parameters shown in Fig. \ref{['Fig:BcFerro']}. Here, $\chi_0=\hbar k_B/2ta$, $\alpha_0=k_B$.