The Rashba effect, also called Bychkov–Rashba effect, is a momentum-dependent splitting of spin bands in bulk crystals and low-dimensional condensed matter systems (such as heterostructures and surface states). The splitting is a combined effect of spin–orbit interaction and asymmetry of the crystal potential, in particular in the direction perpendicular to the two-dimensional plane (as applied to surfaces and heterostructures). The effect is named after Emmanuel Rashba, who discovered it with Valentin I. Sheka in 1959 for three-dimensional systems and afterward with Yurii A. Bychkov in 1984 for two-dimensional systems. The Rashba effect drives a wide variety of physical phenomena, such as operating electron spins by electric fields, despite being only a small correction to the band structure of the two-dimensional metallic state. An example of a physical phenomenon that can be explained by Rashba model is the anisotropic magnetoresistance (AMR). Additionally, superconductors with large Rashba splitting are suggested as possible realizations of the elusive Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, Majorana fermions and topological p-wave superconductors. A momentum dependent pseudospin-orbit coupling has also been realized in cold atom systems.
Hamiltonian The Rashba effect is most easily seen in the simple model Hamiltonian known as the Rashba Hamiltonian
H R = α ( z ^ × p ) ⋅ σ {\displaystyle H_{\rm {R}}=\alpha ({\hat {z}}\times \mathbf {p} )\cdot {\boldsymbol {\sigma }}} , where α {\displaystyle \alpha } is the Rashba coupling, p {\displaystyle \mathbf {p} } is the momentum and σ {\displaystyle {\boldsymbol {\sigma }}} is the Pauli matrix vector. This is identical to the two-dimensional version of the Dirac Hamiltonian, but with a 90 degree rotation of the spins. The Rashba model in solids can be derived in the framework of the k·p perturbation theory or from the point of view of a tight binding approximation. However, the specifics of these methods are considered tedious and many prefer an intuitive toy model that gives qualitatively the same physics (quantitatively it gives a poor estimation of the coupling α {\textstyle \alpha } ). Here we will introduce the intuitive toy model approach followed by a sketch of a more accurate derivation.
Naive derivation The Rashba effect arises from the breaking of inversion symmetry in the direction perpendicular to a two-dimensional electron system. To illustrate this qualitatively, we add an electric field that breaks this symmetry:
E = E 0 z ^ {\displaystyle \mathbf {E} =E_{0}{\hat {z}}}
Due to relativistic corrections, an electron moving with velocity v {\displaystyle \mathbf {v} } in an electric field E {\displaystyle \mathbf {E} } experiences an effective magnetic field in its rest frame, given by
B = − v × E c 2 {\displaystyle \mathbf {B} =-{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}}
where c {\displaystyle c} is the speed of light. This magnetic field couples to the electron spin through the spin–orbit interaction as
H S O = g μ B 2 c 2 ( v × E ) ⋅ σ {\displaystyle H_{\mathrm {SO} }={\frac {g\mu _{\mathrm {B} }}{2c^{2}}}(\mathbf {v} \times \mathbf {E} )\cdot {\boldsymbol {\sigma }}}
where σ {\displaystyle {\boldsymbol {\sigma }}} are the Pauli matrices and − g μ B σ / 2 {\displaystyle -g\mu _{\mathrm {B} }{\boldsymbol {\sigma }}/2} represents the electron magnetic moment. Within this simplified "toy" model, the resulting Rashba Hamiltonian can be written as
H R = α R ( z ^ × p ) ⋅ σ {\displaystyle H_{\mathrm {R} }=\alpha _{\mathrm {R} }({\hat {z}}\times \mathbf {p} )\cdot {\boldsymbol {\sigma }}}
with a coupling strength
… excerpt ends here. Continue reading the full article.
