Superconducting monolayer deposited on substrate: Effects of the spin-orbit coupling induced by proximity effects

Spin-orbit coupling can lead to exotic states of matter and unexpected behavior of the system properties. In this paper, we investigate the influence of spin-orbit coupling induced by proximity effects on a monolayer of superconductor (with $s$-wave or $d$-wave pairing) placed on an insulating bulk. We show that the critical temperatures $T_c$ of the superconducting states can be tuned by the spin-orbit coupling both in the case of on-site and intersite pairing. Moreover, we discuss a possibility of changing the location of the maximal $T_c$ from the half filling into the underdoped or overdoped regimes.

Figure 1

Schematic representation of the analyzed interface (left) and its effective model (right). We assume the existence of a two-dimensional superconducting (SC) monolayer placed on the surface of an insulator with strong spin-orbit coupling (SOC). We study the superconducting layer with half-spin particles (red and blue balls denote opposite spins), which can move between sites of the lattice (e.g., between the nearest-neighbors with hopping amplitude $t$). The on-site ($U_{ii}$) or intersite ($U_{ij}$) attractive (negative) interactions (schematically indicated by yellow areas) are sources of singlet pairing. An effective spin-orbit-coupling interaction ($V_{\mathrm{SO}}$) in the layer is introduced by proximity effects from the bulk SOC material.

Figure 2

The lower and upper Rashba bands for $V_{\mathrm{SO}}=2t$ in the absence of the next-nearest-neighbor hopping ($t^{\prime }=0$) and superconductivity (${\mathrm{\Delta }}_0=0$) in the first Brillouin zone for a system on the square lattice.

Figure 3

Critical temperature as a function of the chemical potential $\mu$ for several spin-orbit coupling strengths $V_{\mathrm{SO}}$ (as labeled) for a square lattice in the absence of the next-nearest-neighbor hopping ($t^{\prime }=0$). Two types of superconductivity are considered: (a) $s$-wave and (b) $d$-wave.

Figure 4

Critical temperature as a function of spin-orbit coupling $V_{\mathrm{SO}}$ for several chemical-potential values $\mu$ (as labeled) in the absence of the next-nearest-neighbor hopping ($t^{\prime }=0$). Two types of superconductivity are considered: (a) $s$-wave and (b) $d$-wave.

Figure 5

Critical temperature as a function of the chemical potential $\mu$ for several spin-orbit coupling values $V_{\mathrm{SO}}$ (as labeled) for a square lattice in the presence of the next-nearest-neighbor hopping term ($t^{\prime }=-0.1t$). Two types of superconductivity are considered: (a) $s$-wave and (b) $d$-wave. The half filling condition corresponds to $\mu =-0.4t$.

Figure 6

Critical temperature as a function of spin-orbit coupling $V_{\mathrm{SO}}$ for several chemical-potential values $\mu$ (as labeled) in the presence of NNN hopping ($t^{\prime }=-0.1t$). Two types of superconductivity are considered: (a), (b) $s$-wave and (c), (d) $d$-wave. The half filling condition corresponds to $\mu =-0.4t$. Panels (a) and (c) present curves for an underdoped system, whereas panels (b) and (d) are obtained in the overdoped regime.

Figure 7

Noninteracting total density of states for (a) $t^{\prime }=0$ and (b) $t^{\prime }=-0.1t$ and for several values of $V_{\mathrm{SO}}$ (as labeled). The Fermi level is located at $E=-4t^{\prime }$ for the half filling. The shape of the noninteracting density of states is not dependent on $\mu$. Only the Fermi level $E_F$ depends on $\mu$ ($E_F=\mu -4t^{\prime }$).

Figure 8

(a) The band structure along high-symmetry points $\mathrm{\Gamma }$–X–M–$\mathrm{\Gamma }$ (see left inset for locations of the points) for $t^{\prime }=0$ (solid lines) and $t^{\prime }=-0.25t$ (dashed lines): the electron band structure (for $V_{\mathrm{SO}}=0$) is represented by the black lines, whereas the lower and the upper Rashba bands for $V_{\mathrm{SO}}=1t$ are indicated by red and green lines, respectively. The lower panels show the total $\left[\rho \right(E)$, black line] and partial $[{\rho }_{\mp }\left(E\right)$, red and green lines for the lower and the upper Rashba bands] densities of states for (b) $t^{\prime }=0$ and (c) $t^{\prime }=-0.25t$, with spin-orbit coupling $V_{\mathrm{SO}}=1t$.

Figure 9

Schematic illustration of the influence of the SO coupling on the band structure. (a) In the presence of the SO the spin degeneracy is lifted and two Rashba bands $|{\mathrm{\Psi }}_{\mathit{k},\alpha }\rangle$ with different indexes ($\alpha =-,+$) of pseudospin (helical spin texture) arise (the arrows mark the pseudospin directions). The lower $|{\mathrm{\Psi }}_{\mathit{k},-}\rangle$ and upper $|{\mathrm{\Psi }}_{\mathit{k},+}\rangle$ Rashba bands are denoted by red and blue lines, respectively. Examples of the intraband [panel (b)] and interband [panel (c)] pairings are shown.