Correlated nature of hybrid $s$-wave superconducting and Rashba lattices

We elucidate the electronic state of a two-dimensional (2D) Rashba square lattice proximitized to a square monolayer $s$-wave superconductor, analyzing the role played by dynamical electron-electron interactions. The $\text{2D}+\text{2D}$ proximity effect induces sharp Bogoliubov and low-energy Andreev-reflected bound states, suppressing the $s$-wave gap globally. Dynamical correlations strongly renormalize the Bogoliubov quasiparticles and the Andreev levels, evolving the spin-resolved Andreev linewidths into a single bound state. We explore the channel- and spin-resolved spectral functions and analyze the reconstructed superconducting state, showing that the Rashba spin-orbit coupling drives strong channel differentiation. The mutual interplay between electron-electron and spin-orbit interactions, with proximity-induced electron pairing, lead us to introduce a generally applicable mechanism for designing Majorana fermions in 2D superconducting structures.

Figure 1

Half-filled, locally hybridized bilayer lattice system. Circles represent electrons, with those of opposite spins occupying the same lattice site [24]. The arrows indicate the on-site hybridization $V$ between the two-dimensional layers and the intralayer hopping $t$. In this $\text{2D}+\text{2D}$ layer structure all intra- and interlayer hopping elements are assumed to be equal.

Figure 2

Local density of states (LDOS) for the hybrid ($V=0.5$) $\text{2D}+\text{2D}$ Hubbard model in the paramagnetic (non-spin-split) phase at half filling. Notice the emergent Hubbard bands with increasing the on-site Coulomb interaction $U$, and the zero frequency pinning of the LDOS to the bare ($U=0.0$) value. The inset shows for comparison the bare two- (2D) and three-dimensional (3D) LDOS with the hybrid $\text{2D}+\text{2D}$ case.

Figure 3

Evolution of the two-channel spectral functions across the superconducting (SC) transition in the absence of Rashba spin splitting. The upper left-hand inset displays the $d$-channel LDOS for two pair-field values, showing conventional Bogoliubov quasiparticles above the $s$-wave SC gap. The upper right-hand inset shows the changes in the bare $\text{2D}+\text{2D}$ LDOS across the SC transition and the emergence of sharp Andreev bound states in the true $d$- and the proximitized $c$-channel superconductor. The main panels display the changes in the SC spectral functions with increasing $U$, showing how Bogoliubov- and Andreev-level linewidths are reshaped by local dynamical correlations.

Figure 4

$\text{2D}+\text{2D}$ spin-resolved spectral functions in the presence of Rashba spin splitting. The lower left-hand inset displays the 2D LDOS with lifted spectral degeneracy for fixed spin-orbit coupling ($t_{\text{SO}}$), showing particle-hole asymmetry and the emergence of van Hove singularities at the band edges. The right-hand insets display the combined effect of $\text{2D}+\text{2D}$ hybridization and spin-orbit coupling, showing less pronounced spin splitting within the bare $d$ channel. The main panels reveal the changes in the SC spectral functions with increasing $U$, showing how the spin-resolved Andreev levels in the $d$ channel evolve towards a single peak. (The $\uparrow$-spin normal state LDOS are also shown for comparison in the main panels).

Figure 5

Evolution of the total LDOS across the SC transition for $U=2.0$ (main panels) [$U=1.5$ (insets)] and two $t_{\text{SO}}$ values. Notice the enhancement of the single Andreev peak for $t_{\text{SO}}=0.125$ as compared to the Rashba free case within the normal $\left[{\rho }_d\left(\omega \right)\right]$ SC component, inducing a sharp low-energy line shape in the anomalous $\left[F_d\left(\omega \right)\right]$ component of the $d$-channel spectral functions. Also noteworthy is the $t_{\text{SO}}$ dependence of the $c$ channel.