Manifestations of spin-orbit coupling in a cuprate superconductor

Exciting new work on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) shows the presence of nontrivial spin-orbit coupling effects as seen in spin-resolved angle-resolved photoemission spectroscopy data [K. Gotlieb et al., Science 362, 1271 (2018)]. Motivated by these observations we consider how the picture of spin-orbit coupling through local inversion symmetry breaking might be observed in cuprate superconductors. Furthermore, we examine two spin-orbit driven effects, the spin-Hall effect and the Edelstein effect, focusing on the details of their realizations within both the normal and superconducting states.

Figure 1

The Fermi surfaces for the two bands with spin aligned along the spin-orbit coupling vector $\mathit{\lambda }\left(\mathbf{k}\right)$. For the opposite helicity the two bands are exchanged. The spin texture for the band parallel to the spin-orbit vector is shown by the arrows.

Figure 2

A schematic depiction of the spin-Hall effect. The charge current $\mathbf{j}$ is split into spin-up (red) and spin-down (blue) components, which accumulate on opposing boundaries. This can be interpreted as a charge current inducing a transverse spin current ${\mathbf{j}}_S^z$.

Figure 3

The total dc spin-Hall conductivity ${\sigma }_{xy}^z$ as a function of temperature for different values of the spin-orbit coupling strength $\lambda$. The conductivity is roughly constant in the normal phase. We therefore normalize all results by the value of the spin-Hall conductivity in the normal phase for the corresponding choice of $\lambda$. The spin-Hall conductivity, however, exhibits a decrease with the onset of superconductivity. This can be attributed to a finite energy for reorienting spins associated with the formation of singletlike Cooper pairs in the superconducting phase. In these calculations, we have chosen $T_c=0.016t_1$.

Figure 4

A schematic depiction of the Edelstein effect. The charge current $\mathbf{j}$ induces a uniform transverse spin polarization $\mathbf{s}={\widehat{\alpha }}_{\mathbf{EE}}\mathbf{j}$ throughout the bulk of the material.

Figure 5

The Edelstein susceptibility ${\alpha }_{\text{EE}}$ relating the layer-staggered polarization to the transverse current via ${\tilde{s}}_x={\alpha }_{\text{EE}}{j}_y$ as depicted schematically in Fig. 4. The coefficient $\alpha$ is plotted for a range of spin-orbit coupling strengths $\lambda$ with each line normalized by the (roughly) constant normal state Edelstein response. There is a marked change in the magnitude of the effect coinciding with the onset of superconductivity.

Figure 6

The components of the static spin susceptibility ${\chi }_{ij}(0,\mathbf{0})$ as a function of temperature for the layer spin-orbit coupled model. The diagonal in-plane components have a noticeable decrease below $T_c$ but do not go exactly to 0 at zero temperature. This behavior is consistent with the decrease normally attributed to spin-singlet superconductivity in the cuprates.