Universal crossover in surface superconductivity: Impact of varying Debye energy

TL;DR

This work demonstrates a universal crossover in interference-induced surface superconductivity within a 1D attractive Hubbard model at half-filling, driven by the coupling strength g and modulated by the Debye energy ℏω_D. Using self-consistent Bogoliubov–de Gennes theory with a Debye cutoff, the authors map how T_cs and T_cb evolve and show that the surface enhancement τ=(T_cs-T_cb)/T_cb reaches up to ~0.70 at a g* that depends on ℏω_D. The study also reveals substantial surface–bulk differences in the zero-temperature gap ratios Δ_s0/k_B T_cs versus Δ_b0/k_B T_cb, especially in the weak-coupling regime, and argues for the relevance of these effects to higher T_c in narrow-band systems. The findings are expected to generalize to higher dimensions and may inform experiments in arrays of coupled chains and corner superconductivity.

Abstract

Recently, interference-induced surface superconductivity (SC) has been predicted within an attractive Hubbard model with $s$-wave pairing, prompting intensive studies of its properties. The most notable finding is that the surface critical temperature $T_{cs}$ can be significantly enhanced relative to the bulk critical temperature $T_{cb}$. In this work, considering a $1D$ attractive Hubbard model for the half-filling level, we investigate how this enhancement is affected by variations in the Debye energy $\hbarω_D$, which controls the number of states contributing to the pair potential and, in turn, influences the critical temperature. Our study reveals a universal crossover of the surface SC from the weak- to strong-coupling regime, regardless of the specific value of the Debye energy. The location of this crossover is marked by the maximum of $τ= (T_{cs} - T_{cb})/T_{cb}$, which depends strongly on $\hbarω_D$. At its maximum, $τ$ can increase up to nearly $70\%$. Additionally, we examine the evolution of the ratio $Δ_{s0}/k_B T_{cs}$ along the crossover, where $Δ_{s0}$ is the zero-temperature pair potential near the surface (the chain ends), and demonstrate that this ratio can significantly deviate from $Δ_{b0}/k_B T_{cb}$, where $Δ_{b0}$ is the zero-temperature bulk pair potential (in the chain center). Our findings may offer valuable insights into the search for higher critical temperatures in narrow-band superconductors.

Figures (6)

• Figure 1: The pair potential $\Delta(i)$ versus the site index $i$, as calculated for $g=1.3$, $1.6$, $3.2$, with $\hbar\omega_D=1.5$ (a-c), $10$ (d-f). The blue curves represent data for $T=0$, while the green and red ones are for $T=0.87\,T_{cb}$ and $1.0\,T_{cb}$, respectively.
• Figure 2: The surface (at the chain ends; curves with red stars) and bulk (in the chain center; curves with blue squares) local pair potentials, $\Delta_s = \Delta(i=1,,301)$ and $\Delta_b = \Delta(i=151)$, are shown as functions of temperature $T$ for $g=1.3$, $1.6$, and $3.2$, with $\hbar\omega_D=1.5$ (a-c) and $\hbar\omega_D=10$ (d-f). The surface and bulk critical temperatures ($T_{cs}$ and $T_{cb}$), where $\Delta_s$ and $\Delta_b$ respectively drop to zero, are highlighted in panel (c).
• Figure 3: $T_{cs}$ and $T_{cb}$ as functions of the coupling strength $g$ for $\hbar\omega_D=1.5$ (a) and $10$ (b). The results of $T_{cs}$ and $T_{cb}$ are given by red-starred and blue-squared curves, respectively. The black curves with solid squares represent the BCS critical temperature $T_{c,{\rm BCS}}$.
• Figure 4: The surface SC enhancement parameter $\tau \tau=(T_{cs}-T_{cb})/T_{cb}$ is presented as a function of the coupling strength $g$ (a, b) and $1/g$ (c, d) for $\hbar\omega_D=1.5$ and $10$. Panels (c, d) employ a base-10 logarithmic scale to plot the data for $\tau$. The location of the $\tau$-maximum $g=g*$ can be considered as the crossover point that distinguishes between the weak-coupling and strong-coupling behaviors of $\tau$ as a function of $g$.
• Figure 5: The $\hbar\omega_D$-dependence of the characteristics of the universal crossover in the surface SC, with $g^*$ (curves with red stars) and $\tau_{\rm max}$ (curves with blue spheres) plotted as functions of $\hbar\omega_D$.