Rashba–Edelstein effect
[8] Indeed, a structural inversion symmetry breaking (i.e., a structural inversion asymmetry (SIA)) causes the Rashba effect to occur: this effect breaks the spin degeneracy of the energy bands and it causes the spin polarization being locked to the momentum in each branch of the dispersion relation.
[2] If a charge current flows in these spin-polarized surface states, it generates a spin accumulation.
[8] The reverse process is called the inverse Rashba–Edelstein effect and it converts a spin accumulation into a bidimensional charge current, resulting in a 2D spin-to-charge conversion.
[1][8] In both cases, the material surface displays the spin polarization locked to the momentum, meaning that these two quantities are univocally linked and orthogonal one to the other (this is clearly visible from the Fermi countours).
[1][8][10][11] It is worth noticing that also a bulk inversion asymmetry could be present, which would result in the Dresselhaus effect.
[1] In fact, if, in addition to the spatial inversion asymmetry or to the topological insulator band structure, also a bulk inversion asymmetry is present, the spin and momentum are still locked but their relative orientation is not straightforwardly determinable (since also the orientation of the charge current with respect to the crystallographic axes plays a relevant role).
[1][2][11] Indeed, when a charge current flows in the surface states of the topological insulator, it can also be seen as a well-defined momentum shift
[1] This unbalance, according to the structure of the topological insulator band dispersion relation, produces a spin accumulation in the investigated material, i.e., a charge-to-spin conversion occurs.
[1][15] For what concerns the Rashba–Edelstein effect, the spin-split dispersion relation consists in two bands displaced along the k-axis due to a structural inversion asymmetry (SIA), accordingly to the Rashba effect (i.e., these bands show a linear splitting in k due to the spin-orbit coupling[10][16]).
This results in two Fermi countours, which are concentric at equilibrium, both displaying spin-momentum locking but with opposite helicity.
[10] If the system is driven in an out-of-equilibrium condition by injecting a charge current, the two disk displace one from the other and a net spin accumulation arises.
[1][4] Experimentally speaking, the Rashba–Edelstein effect occurs if a charge current is electrically injected inside the topological insulator, for instance by means of two electrodes where a potential difference is applied.
The resulting spin accumulation can be probed in several ways, one of them is by employing the magneto optical Kerr effect (MOKE).
[11] This is how the 2D spin-to-charge conversion occurs in these materials and this could allow topological insulators to be exploited as spin detectors.
For what concerns the inverse Rashba–Edelstein effect, the process is very similar despite the presence of four energy branches, with spin-momentum locking, in the dispersion relation and two consequent Fermi countours with opposite helicity.
[1][8] In this case, the two Fermi countours, when a spin accumulation is generated inside the material, will be displaced one from the other, generating a charge current, at variance with the equilibrium case in which the two Fermi countours are concentric and no net momentum unbalance nor spin accumulation are present.
[1][8] In both cases, the asymmetry in the charge and spin current dimensions results in a ratio which dimensionally has the units of a length: this is the origin of the name of this efficiency parameter.
[1] Analytically, the value of the bidimensional charge current density can be computed employing the Boltzmann equation and considering the action of an electric field
[1] For what regards the Edelstein and its inverse effects, the conversion efficiency is:[1] This parameter is conventionally positive for a Fermi contour with a counterclockwise helicity.
