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Fluctuation effect on Nonlinear Transport and Nernst-Ettingshausen Response in Two-Dimensional Superconductors under electric and magnetic field

Tran Ky Vi, Bui Duc Tinh, Ngo Quang Duc, Chu Gia Bao, Le Viet Hoang, Le Xuan The Tai, Nguyen Viet Hung

Abstract

In this paper, we present a unified theoretical study of fluctuation-dominated transport and transverse thermoelectric response in two-dimensional superconducting films subjected to out-of-plane magnetic fields and electric-field drive. Our approach is based on the time-dependent Ginzburg-Landau equation with Langevin thermal noise, in which interaction effects of fluctuating Cooper pairs are incorporated self-consistently at the Gaussian (Hartree) level. We derive closed-form expressions for the fluctuation-induced Cooper-pair density, the renormalized resistance $R(T,B_\perp)$, and the nonlinear current response $J(E,B_\perp)$, explicitly accounting for the feedback of the electric field on the fluctuation spectrum. A central result is the emergence of an intrinsic S-shaped nonlinear $J$-$E$ (or $I$-$V$) characteristic, featuring a negative-differential segment and multivalued solutions under voltage control. Within this framework, we introduce a physically transparent procedure to identify characteristic instability scales, such as the magnetic field $B^{\ast}$ (or equivalently $B_χ$), which marks the terminal point of the S-shaped instability where the nonlinear response becomes single-valued. In parallel, we analyze the off-diagonal Peltier coefficient $α_{xy}$ as a direct probe of the transverse thermoelectric response of superconducting fluctuations. The theory is validated through systematic comparisons with recent experimental measurements of multi-field $R(T)$ curves, nonlinear $I$-$V$ characteristics, and $α_{xy}$ data across a broad range of thin-film superconducting materials.

Fluctuation effect on Nonlinear Transport and Nernst-Ettingshausen Response in Two-Dimensional Superconductors under electric and magnetic field

Abstract

In this paper, we present a unified theoretical study of fluctuation-dominated transport and transverse thermoelectric response in two-dimensional superconducting films subjected to out-of-plane magnetic fields and electric-field drive. Our approach is based on the time-dependent Ginzburg-Landau equation with Langevin thermal noise, in which interaction effects of fluctuating Cooper pairs are incorporated self-consistently at the Gaussian (Hartree) level. We derive closed-form expressions for the fluctuation-induced Cooper-pair density, the renormalized resistance , and the nonlinear current response , explicitly accounting for the feedback of the electric field on the fluctuation spectrum. A central result is the emergence of an intrinsic S-shaped nonlinear - (or -) characteristic, featuring a negative-differential segment and multivalued solutions under voltage control. Within this framework, we introduce a physically transparent procedure to identify characteristic instability scales, such as the magnetic field (or equivalently ), which marks the terminal point of the S-shaped instability where the nonlinear response becomes single-valued. In parallel, we analyze the off-diagonal Peltier coefficient as a direct probe of the transverse thermoelectric response of superconducting fluctuations. The theory is validated through systematic comparisons with recent experimental measurements of multi-field curves, nonlinear - characteristics, and data across a broad range of thin-film superconducting materials.

Paper Structure

This paper contains 15 sections, 36 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The quasi two-dimensional TDGL model with Langevin thermal noise
  3. The quasi two-dimensional description
  4. Dimensionless variables and characteristic scales
  5. Linearization and self-consistent Gaussian approximation
  6. Fluctuation Cooper-pair density and transport properties
  7. Fluctuation Cooper-pair density
  8. Electric current and renormalized resistance
  9. Heat current density and the transverse thermoelectric coefficient
  10. Numerical results and comparison with experiment
  11. Fluctuation-controlled resistive transitions under out-of-plane magnetic fields
  12. S-shaped $I$--$V$ characteristics as a historical and theoretical hallmark of nonequilibrium superconductivity
  13. Transverse thermoelectric response and vortex entropy: from resistive broadening to $\alpha_{xy}$
  14. Discussion
  15. Conclusions

Figures (11)

  • Figure 1: Schematic of an effectively two-dimensional superconducting film sample in an out-of-plane magnetic field $\mathbf{B}=B\hat{z}$, with in-plane electric field $\mathbf{E}$ driving transport and vortex-like excitations confined to the $xy$ plane.
  • Figure 2: Electrical resistance $R(T)$ of gated monolayer MoS$_2$ under out-of-plane magnetic fields, reproduced from Ref. lu2015evidence. The field-driven shift and systematic broadening of the transition reflect strong superconducting fluctuations in two dimensions.
  • Figure 3: Resistance $R(T)$ of atomically thin 2M-WS$_2$ under out-of-plane magnetic fields, reproduced from Ref. zhang2023spin. The continuous crossover highlights the robustness of fluctuation-dominated transport in low-dimensional systems with strong spin--orbit coupling.
  • Figure 4: Electrical resistance $R(T)$ of a disordered superconducting thin film under out-of-plane magnetic fields, reproduced from Ref. ienaga2024broadened. Symbols denote experimental data, while solid lines represent theoretical fits obtained within the self-consistent Gaussian TDGL approach.
  • Figure 5: Resistance $R(T)$ of underdoped infinite-layer nickelate superconducting thin films under out-of-plane magnetic fields, reproduced from Ref. shao2025anomalous. The broadened transitions emphasize the universality of fluctuation-driven transport beyond cuprate-based systems.
  • ...and 6 more figures