Temperature-Induced Superconductivity Enhancement under Large Exchange Field

Abstract

Through a comprehensive free energy analysis, we demonstrate that finite temperature can simultaneously weaken superconductivity and mitigate spin polarization induced depairing, leading to potential non-monotonic temperature-dependent behaviors in superconductors subjected to large exchange fields. Remarkably, superconductivity can be counterintuitively enhanced by temperature when the Zeeman energy exceeds the superconducting order parameter, owing to the competition between thermal and magnetic effects. We propose that multiband effect offers one possible microscopic route for this temperature-induced enhancement and demonstrate it explicitly within a two-band superconducting model. A detailed parameter analysis identifies the conditions under which this phenomenon emerges, suggesting that temperature-enhanced superconductivity may be observable in materials such as MgB2 and FeSe through transport and tunneling measurements.

Figures (15)

• Figure 1: Free energy analysis for one-band s-wave superconductors. (a) Landscapes of $f_{ns}(x)$ at $t=0$ for $h=0.0$, $0.7h_p$, $0.999h_p$, and $1.2h_p$. (b,c) Corresponding profiles of $\mathrm{d}f_{h}(x)$, $\mathrm{d}f_{t}(x)$, $\mathrm{d}f_{h}(x)+\mathrm{d}f_{t}(x)$, and $\mathrm{d}f_{ht}(x)$ for (b) $h=0.10$, $t=0.15$ and (c) $h=0.70$, $t=0.15$. (d) Temperature evolution map of $\mathrm{d}f_{ht}(x)-\mathrm{d}f_{h}(x)$ at $h=0.70$. The solid black line marks the condition $\mathrm{d}f_{ht}(x)=\mathrm{d}f_{h}(x)$, orange points $x_{\mathrm{stable}}$ denote the equilibrium order parameter, and the color scale represents the amplitude of $\mathrm{d}f_{ht}(x)-\mathrm{d}f_{h}(x)$, corresponding to the negative of the energy saved by temperature effects. Contour lines of $\mathrm{d}f_{ht}(x)-\mathrm{d}f_{h}(x)$ illustrate the distribution of the energy saving. Regions where $\mathrm{d}f_{ht}(x)-\mathrm{d}f_{h}(x)\ge 0$ are shown as transparent but remain traceable through their positive contour levels.
• Figure 2: Superconductivity under large exchange fields for one-band and two-band s-wave systems. (a,b) $h$-$t$ phase diagrams for (a) one-band and (b) two-band case with specific parameters. The first-order and second-order transition boundaries (FOT and SOT) are denoted by orange and blue solid lines, respectively. Color scales indicate the magnitude of the order parameter $x$, and contour lines highlight the monotonic and non-monotonic superconducting features. The same color scales and line conventions are used in the following figures. (c,d) Temperature dependence of the order parameter $x(t)$ at various exchange fields for (c) one-band and (d) two-band systems.
• Figure 3: $\alpha$-$\gamma$ parameter dependence of superconductivity enhancement. (a,b) $h$-$t$ phase diagrams for two-band systems with (a) large $\alpha$ and small $\gamma$, and (b) small $\alpha$ and large $\gamma$. (c,d) Maps of (c) $h_{c,max}/h_{c0}$ and (d) $t_{max}$ in the $\alpha$-$\gamma$ parameter space. The color scales in (c) and (d) represent the amplitudes of $h_{c,max}/h_{c0}$ and $t_{max}$, respectively. The parameter points corresponding to (a) and (b) are indicated by orange markers.
• Figure 4: $\alpha$-$\gamma$ parameter dependence of the relation between $x(h_{c0}^-)$ and $h_{c0}$ at zero temperature. (a) Map of $x(h_{c0}^-)$. (b) Order parameter profiles $x(h)$ for different $\alpha$ and $\gamma$ values at zero temperature. Each curve is vertically offset by $1$ for clarity, and the drop near $h_{c0}$ is marked by double-sided arrows. (c,d) Maps of (c) $h_{c0}-x(h_{c0}^-)$ and (d) $h_{c0}-\alpha x(h_{c0}^-)$. The color scales in (a), (c), and (d) represent the amplitudes of $x(h_{c0}^-)$, $h_{c0}-x(h_{c0}^-)$, and $h_{c0}-\alpha x(h_{c0}^-)$, respectively. Contour lines of $h_{c0}-\alpha x(h_{c0}^-)$ below the boundary defined by $x(h_{c0}^-)=1$ are omitted for clarity.
• Figure S1: Comparison between the weak-coupling approximation and finite-coupling calculations for the field dependence of $\Delta_i(h)$ and $\Delta_2(h)/\Delta_1(h)$. (a-d) Results obtained from the full free energy [Eq. (\ref{['fx1x2']})] for different coupling strengths: (a) $\mathrm{PF}=3$, (b) $\mathrm{PF}=5$, (c) $\mathrm{PF}=10$, and (d) $\mathrm{PF}=20$. The corresponding weak-coupling results from Eq. (\ref{['fx_approx']}) are also shown in panels (a-d) for comparison.