Universal crossover in surface superconductivity: Impact of varying Debye energy

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 $\hbar\omega_D$, 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 $\tau = (T_{cs} - T_{cb})/T_{cb}$, which depends strongly on $\hbar\omega_D$. At its maximum, $\tau$ can increase up to nearly $70\%$. Additionally, we examine the evolution of the ratio $\Delta_{s0}/k_B T_{cs}$ along the crossover, where $\Delta_{s0}$ is the zero-temperature pair potential near the surface (the chain ends), and demonstrate that this ratio can significantly deviate from $\Delta_{b0}/k_B T_{cb}$, where $\Delta_{b0}$ 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.