Coexistence of superconductivity and excitonic pairing in a doped-biased double-layer system

TL;DR

This work investigates a doped-biased bilayer square-lattice system with intralayer phonon modulations, focusing on the coexistence of spin-singlet excitonic pairing and spin-triplet superconductivity within a generalized Hubbard framework. By deriving a self-consistent mean-field theory that couples density, excitonic, and electron-phonon channels via Hubbard-Stratonovich transformations and a Gorkov formalism, the authors obtain coupled equations for the chemical potential, interlayer density imbalance, and both order parameters. Numerical solutions reveal robust coexistence of excitonic and spin-triplet superconducting order over a broad temperature range, with Tc peaking near half-filling and enhanced by stronger electron-phonon coupling; the superconducting energy scale is about two orders of magnitude larger than the excitonic one. The results yield phase diagrams of Tc versus doping and highlight the potential relevance to high-$T_C$ cuprates as well as applicability to other bilayer systems, suggesting a phonon-dominated mechanism for unconventional pairing.

Abstract

The subject of the present study is the double-layer square-lattice system with the intralayer phonon modulations. We investigate the superconducting and excitonic pairings, as well as their coexistence, as functions of various physical parameters in the system. These parameters include temperature, intralayer and interlayer Coulomb interactions, the electron-phonon coupling parameter, doping and the applied electric field. The existence of superconductivity is demonstrated by considering the influence of intralayer phonons on the total charge density leading to the modification of the total energy of the electrons. Our results provides insights into the long-standing problem of the mechanism of superconductivity in high-$T_c$ cuprate superconductors.

Figures (10)

• Figure 1: (Color online) Panel (a): Schematic representation of the metallic double-layer system with the applied electric field potential $V$. The upper layer is gated with an electric field potential of $V/2$, while the lower layer is gated with $-V/2$. The intralayer nearest-neighbor hopping amplitude ($t_0$), next-nearest neighbor hopping amplitude (nnn $t_1$), and interlayer hopping amplitude ($t_{\perp}$) are indicated in the picture. Panel (b): Factorization of a single-particle electron excitation using two simultaneous single-particle excitations, as described in Eq.(\ref{['Equation_7']}).
• Figure 2: (Color online) As a complement to Fig. \ref{['fig:Fig_1']}, we illustrate the metallic double-layer system with excitonic pair formations (depicted in yellow and red) and superconducting pair formations (represented by the opaque squares on the top and bottom layers). On the left side of the figure, the spin-triplet superconducting pairs and spin-singlet excitonic pairs are shown.
• Figure 3: (Color online) The numerical solution of the system of self-consistent equations in Eq.(\ref{['Equation_54']}). The temperature dependence is shown for the chemical potential $\mu$ (see panel (a)) and the average interlayer charge density difference $\delta{\bar{n}}$ (see panel (b)). A large value of electron-phonon interaction parameter is considered with $\lambda_{\rm eff}=0.45t_{0}=0.135$ eV. The intralayer local Hubbard interaction parameter is set to $U=3t_{0}$, and the external gate potential is fixed at $V=t_{0}$. The hopping amplitudes are set to $t_{1}=0.4t_{0}$ and $t_{\perp}=0.02t_{0}$, respectively. The interlayer Hubbard interaction potential is $W=0.1t_{0}$. The calculations are performed for different values of the inverse filling coefficient $\kappa$.
• Figure 4: (Color online) The numerical solution of the system of self-consistent equations in Eq.(\ref{['Equation_54']}). The temperature dependence is shown for the excitonic order parameter $\Delta$ (see panel (a)) and the superconducting order parameter $\Delta_{s}$ (see panel (b)). A large values of electron-phonon interaction parameter is considered with $\lambda_{\rm eff}=0.45t_{0}$. The intralayer local Hubbard interaction parameter is set to $U=3t_{0}$, and the external gate potential is fixed at $V=t_{0}$. The hopping amplitudes are set to $t_{1}=0.4t_{0}$ and $t_{\perp}=0.02t_{0}$, respectively. The interlayer Hubbard interaction potential is $W=0.1t_{0}$. The calculations have been done for different values of the inverse filling coefficient $\kappa$.
• Figure 5: (Color online) The critical temperatures of the spin-triplet superconducting phase transition in a system of two metallic layers as a function of electron doping. The following values of the interaction parameters were chosen when evaluating the curve: electron-phonon interaction parameter: $\lambda_{\rm eff}=0.45t_{0}$, intralayer Hubbard interaction $U=3t_0$, interlayer Hubbard interaction parameter $W=0.1t_0$, and the external interlayer gate voltage $V=t_{0}$.The next-nearest neighbor (nnn) hopping was set at $t_{1}=0.4t_{0}$, and the interlayer hopping amplitude was fixed at $t_{\perp}=0.02t_{0}$.