Enhancement of superconducting $T_c$ due to the spin-orbit interaction

We calculate the superconducting $T_c$ for a system which experiences Rashba spin-orbit interactions. Contrary to the usual case where the electron-electron interaction is assumed to be wave vector independent, where superconductivity is suppressed by the spin-orbit interaction (except for a small region at low electron or hole densities), we find an enhancement of the superconducting transition temperature when we include a correlated hopping interaction between electrons. This interaction originates in the expansion of atomic orbitals due to electron-electron repulsion and gives rise to superconductivity only at high electron (low hole) densities. When superconductivity results from this interaction it is enhanced by spin-orbit coupling, in spite of a suppression of the density of states. The degree of electron-hole asymmetry about the Fermi surface is also enhanced.

Figure 1

Free particle Rashba spectrum on a square lattice with $V_{\mathrm{SO}}=0.4t$. The blue and orange bands represent the $s=-1$ and $s=+1$ helicity bands, respectively. The dashed lines in the bottom figure show the locations of Van Hove singularities.

Figure 2

Noninteracting single-particle Rashba density of states on a square lattice for the lower helicity band (orange), upper helicity band (blue), and the two combined (green). Here we have set $V_{\mathrm{SO}}=0.4t$.

Figure 3

Low-temperature gap dependence on kinetic energy for various values of $U$. Here we have set $\mathrm{\Delta }t=4.5t,n=1.875,k_{B}T=0.01t$. The solid lines correspond to $V_{\mathrm{SO}}=0.5$. The dashed lines show the result for $V_{\mathrm{SO}}=0$. A slightly larger value for the absolute value of the slope indicates that $V_{\mathrm{SO}}$ increases the asymmetry around the Fermi level.

Figure 4

Critical temperature as a function of electron density for various values of the spin-orbit coupling with $\mathrm{\Delta }t=0$: $U=-t$ (top left), $U=-2t$ (top right), $U=-3t$ (bottom left), and $U=-4t$ (bottom right). By particle-hole symmetry, the plot above half filling is a reflection of this plot, i.e., $T_c(2-n)=T_c\left(n\right)$ for $0<n<1$.

Figure 5

Critical temperature as a function of electron density for various values of the spin-orbit coupling with $\mathrm{\Delta }t=4.5t$: $U=90t$ (left), $U=115t$ (right).

Figure 6

Critical temperature as a function of electron density for various values of the spin-orbit coupling with $U=75t$: $\mathrm{\Delta }t=3.5t$ (left), $\mathrm{\Delta }t=4t$ (right).

Figure 7

Maximum critical temperature as a function of spin-orbit coupling with $U=90t,\mathrm{\Delta }t=4.5t$.

Figure 8

Critical temperature $E_{\min }$ and the corresponding gap ratio at low temperature as a function of spin-orbit coupling. Here we have set $U=90t,\mathrm{\Delta }t=4.5t,n=1.875,k_{B}T=0.01t$. The dashed line shows the conventional BCS value.