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

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 $B_{\chi }$), 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 ${\alpha }_{xy}$ 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 ${\alpha }_{xy}$ data across a broad range of thin-film superconducting materials.