Berezinskii-Kosterlitz-Thouless transition with enhanced phase stiffness in $d$-wave strongly coupled two-dimensional superconductors

TL;DR

This work addresses how $d$-wave gap symmetry influences the Berezinskii-Kosterlitz-Thouless transition and phase stiffness in strongly coupled two-dimensional superconductors, with emphasis on Van Hove singularities and nodal structure. The authors employ a mean-field BCS treatment on a 2D tight-binding lattice for both $s$- and $d$-wave gaps and determine the BKT transition via $T_{BKT} = (\pi/2)\rho_s(T_{BKT})$, where $\rho_s$ includes diamagnetic and paramagnetic contributions. A key finding is that near the BCS–BEC crossover and around the Van Hove singularity, the $d$-wave channel can yield a larger $T_{BKT}$ than $s$-wave due to a two-component condensate consisting of cold nodal regions (BCS-like) and hot BE-like regions, which stabilizes phase coherence. The study provides a framework for achieving higher $T_{BKT}$ in engineered 2D superconductors and is applicable to multi-band systems with at least one $d$-wave partial condensate, offering insights into high-$T_c$ cuprates and related materials.

Abstract

We reveal the key role of the $d$-wave symmetry of the superconducting gap in strongly coupled two-dimensional superconductors in determining the properties of the Berezinskii-Kosterlitz-Thouless (BKT) transition, associated with a sizable enhancement of the phase stiffness compared to nodeless-gap superconductors. The enhanced stiffness originates from extended regions of vanishing gap around the nodal lines of the Brillouin zone (BZ). Our study, based on mean-field and BKT theory, presents a comparative analysis of $s$-wave and $d$-wave scenarios, highlighting the features of the latter that boost the stiffness and the BKT transition temperature (T$_{BKT}$). The comparison focuses on two quantities: the mean-field critical temperature and the maximum superconducting gap related to the pairing strengths. We present a phase diagram showing the scaling of T$_{BKT}$ with respect to the mean-field critical temperature across the BCS-BEC crossover and the evolution of the pseudogap. We also present a zero-temperature phase-stiffness intensity map over the Brillouin zone, displaying a two-component structure consisting of low- and high-stiffness regions whose extent depends on microscopic parameters. These results identify the nodal gap structure of strongly coupled two-dimensional superconductors as a key mechanism enabling enhanced stiffness and elevated T$_{BKT}$ compared to their $s$-wave counterparts.

Figures (12)

• Figure 1: Brillouin zone and schematic Fermi surface of a $d$-wave cuprate superconductor close to optimal doping, with hot region (red patches) and cold region (blue patches).
• Figure 2: The tight binding energy dispersion with nearest and next-nearest neighbor hopping parameters $t = 0.15$ eV and $t^{\prime}$= 0.04 eV. The chemical potential ($\mu$) is fixed near the VHs, at -4t$^{\prime}$.
• Figure 3: The pairing strength ($\lambda$) as a function of the maximum value of the superconducting gap scaled to $t$ = 0.15 eV.
• Figure 4: The phase stiffness at zero temperature as a function of the maximum superconducting gap in units of $t$ = 0.15 eV.
• Figure 5: T$_{\text{BKT}}$ as a function of the maximum superconducting gap in units of $t$ = 0.15 eV, around VHs. (a)-(c) Chemical potential $\mu$ is fixed at VHs, VHs+35 meV, and VHs-35 meV, respectively.