Spin-orbit induced equilibrium spin currents in materials

The existence of pure spin currents in absence of any driving external field is commonly considered an exotic phenomenon appearing only in quantum materials, such as topological insulators. We demonstrate instead that equilibrium spin currents are a rather general property of materials with non-negligible spin-orbit coupling (SOC). Equilibrium spin currents can be present at the surfaces of a slab. Yet, we also propose the existence of global equilibrium spin currents, which are net bulk spin currents along specific crystallographic directions of solid-state materials. Equilibrium spin currents are allowed by symmetry in a very broad class of systems having gyrotropic point groups. The physics behind equilibrium spin currents is uncovered by making an analogy between electronic systems with SOC and non-Abelian gauge theories. The electron spin can be seen as analogous to the color degree of freedom in $\mathrm{SU}\left(2\right)$ gauge theories and equilibrium spin currents can then be identified with diamagnetic color currents appearing as the response to a effective non-Abelian magnetic field generated by the SOC. Equilibrium spin currents are not associated with spin transport and accumulation, but they should nonetheless be carefully taken into account when computing transport spin currents. We provide quantitative estimates of equilibrium spin currents for a number of different systems, specifically the Au(111) and Ag(111) metallic surfaces presenting Rashba-type surface states, nitride semiconducting nanostructures, and bulk materials, such as the prototypical gyrotropic medium tellurium. In doing so, we also point out the limitations of model approaches showing that first-principles calculations are needed to obtain reliable predictions. We therefore use density functional theory computing the so-called bond currents, which represent a powerful tool to deeply understand the relation between equilibrium currents, electronic structure, and crystal point group.

Figure 1

Left: top view of the Au(001), Au(011), and Au(111) surfaces. To carry out the calculations we use rectangular $2\times 2$ supercells in the surface $xy$ plane. Right: calculated ESC pseudotensor as defined in Eq. (19). The components are expressed in meV per surface unit cell. The surface unit cells are represented by the dashed blue lines. The components $I_z^x$, $I_z^y$, and $I_z^z$ are zero because $z$ is the direction normal to the surface and there can not be current flowing into the vacuum. Aside from that, the structure of the ESC pseudotensor can be determined by analyzing the surface symmetry, as demonstrated in Appendix pp3 for Au(111) and in Appendix pp7 for Au(011) and Au(001). The mirror reflection lines of the surface point groups are represented as red lines (see also Appendixes pp3 and pp7).

Figure 2

Band structure of Au(111) (left) and Ag(111) (right). The red and green lines are the fitted Rashba bands.

Figure 3

Left: InP(001)-oriented slabs with and without the same top and bottom terminations. The large gray spheres and small yellow spheres are, respectively, the In and P atoms. The calculations are carried out for $2\times 2$ supercells in the $xy$ plane. The surface unit cell is contained inside the blue dashed rectangle. The mirror reflection lines are in red (see Appendix pp7). Right: corresponding ESC pseudotensors as defined in Eq. (19) and expressed in meV per unit cell. The components $I_z^x$, $I_z^y$, and $I_z^z$ are zero because $z$ is the normal direction to the slab surfaces. The symmetry analysis of the ESC pseudotensors is presented in Appendix pp7.

Figure 4

Layer-resolved components of the ESC for the InP(001)-oriented slabs (units meV). Top: 21-layer slab. Bottom: 20-layer slab. At each surface, the various ESC components are related by the surface symmetry operations as demonstrated in Appendix pp7.

Figure 5

Left: InP(011)-oriented slab. The large gray spheres and small yellow spheres are, respectively, the In and P atoms. The calculations are carried out for a $2\times 2$ supercell in the $xy$ plane. The surface unit cell is delimited by the blue dashed rectangle. The mirror reflection lines are in red (see Appendix pp7). The bulk InP unit cell, which is composed of two parallel atomic layers along (001), is shown inside the magenta dashed rectangle. Right: corresponding ESC pseudotensor as defined in Eq. (19) and expressed in meV per unit cell. The components $I_z^x$, $I_z^y$, and $I_z^z$ are zero because $z$ is the normal direction to the slab surfaces. The structure of the ESC pseudotensor can be further understood based on the symmetry analysis in Appendix pp7.

Figure 6

Layer-resolved ESC for the InP(011)-oriented slab (units meV). $I_y^y$ is zero for all atomic layers. The magenta diamonds correspond to the layer-resolved $I_x^z$ components for the bulk InP unit cell (which is shown inside the magenta dashed rectangle in the Fig. 5).

Figure 7

Left: top view of the InN $2\times 2\times 2$ rectangular supercell used in the calculations. The large and small spheres are, respectively, the In and N atoms. The unit cell is delimited by the dashed lines. The $x$, $y$, and $z$ axes are, respectively, along the (0001), the $\left(1\overline{1}00\right)$, and the $\left(11\overline{2}0\right)$ directions. There are three mirror planes (red lines) and three glide planes (green lines) in the ${\mathbb{C}}_{6v}$ point group. The structure of the ESC pseudotensor can be fully understood based on the corresponding symmetry operations like in the case of the Au(111) and of the InP(001) surfaces. Left: ESC pseudotensor in meV per unit cell.

Figure 8

Top view (left) and side view (right) of the rectangular supercell of right-handed Te used in the calculations. In the left panel the rhombohedral unit cell is delimited by the blue dashed line. The rotation axes $R_1$, $R_2$, and $R_3$ are represented as red dashed lines. The right panel also shows the three planes, which are normal to the $C$ axis (dashed black line) and through which ${\mathbf{I}}_z$ is calculated.

Figure 9

Procedure to calculate the bond currents for a periodic system. The model simulation cell contains the atom 1 inside a square unit cell. We perform an inverse Fourier transform of the Hamiltonian matrix, of the the overlap matrix, and of the (energy) density matrix to a real-space representation. Once this is done, the $a$-spin component bond current ${\mathcal{I}}_{1^{\prime }1}^a$ between 1 and its equivalent atom $1^{\prime }$ can be easily calculated. In (a) the global $a$-spin current through the surface $A$ is equal to ${\mathcal{I}}_{1^{\prime }1}^a$. In (b) we assume that nonzero bond currents extend to second-nearest-neighbor atoms. Therefore, we need to use the $2\times 1$ supercell in the calculation. The global current is given by the sum of ${\mathcal{I}}_{1^{\prime }1}^a$, ${\mathcal{I}}_{1^{\prime }2}^{\alpha }$, and ${\mathcal{I}}_{2^{\prime }1}^a$.

Figure 10

ESC $I_S\equiv I_y^x=-I_x^y$ as a function of the SOC rescaling parameter $\alpha$ for the Au(111) and the Ag(111) surfaces.

Figure 11

Spin-$x$ bond currents from a P atom (labeled ${\mathrm{P}}_1$) in bulk InP to its surrounding nearest-neighbor P atoms. The large gray spheres and small yellow spheres are, respectively, the In and P atoms. An arrow entering (leaving) ${\mathrm{P}}_1$ means that the bond current is negative (positive). The bond currents have all the same modulus.

Figure 12

Spin-$x$ bond currents, which connect N atoms in bulk InP. Left: bond currents relevant for $I_x^x$. Right: bond currents relevant for $I_y^x$. Bond currents represented in different colors have different magnitudes.

Figure 13

Transformations of the momentum vector, of the spin pseudovector, and therefore of the ESC components under various symmetry operations. The momentum and the spin are, respectively, represented as a black thin arrow and a rounded tridimensional arrow. Mirror reflection lines are painted in red. ${180}^{\circ }$ rotations in (i) and (n) are represented as thin dashed lines terminating with an arrow.

Figure 14

Top-down symmetry transformations for the InP(001)-oriented slab [(a) and (b)] and the InP(011)-oriented slab [(c), (d), and (e)].