Question

what is Rashba effect .
and give Naive derivation of Rashba Hamiltonian.


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Answer 1

Answer:

Here,the complete explanation of Rashba Hamilotation-

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) similar to the splitting of particles and anti-particles in the Dirac Hamiltonian. 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). This effect is named in honour of Emmanuel Rashba, who discovered it with Valentin I. Sheka in 1959[1] for three-dimensional systems and afterward with Yurii A. Bychkov in 1984 for two-dimensional systems.

Remarkably, this effect can drive a wide variety of novel physical phenomena, especially operating electron spins by electric fields, even when it is 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.

Lately, a momentum dependent pseudospin-orbit coupling has been realized in cold atom systems.

Answer 2

The Rashna effect is a momentum-dependent splitting of spin bands in two-dimensional condensed matter systems similar to the splitting of particles and anti-particles in the Dirac Hamiltonian.

let's see,

$H_{E}=-E_{0}z,$

Due to relativistic corrections an electron moving with velocity v in the electric field will experience an effective magnetic field B

${\mathbf {B}}=-({\mathbf {v}}\times {\mathbf {E}})/c^{2},$

where c is the speed of light. This magnetic field couples to the electron spin

$H_{{SO}}={\frac {g\mu _{B}}{2c^{2}}}({\mathbf {v}}\times {\mathbf {E}})\cdot {\mathbf {\sigma }},$

where - $g\mu _{B}{\mathbf {\sigma }}/2$

is the magnetic moment of the electron.

Within this toy model, the Rashba Hamiltonian is given by

$H_{{R}}=\alpha ({\boldsymbol {\sigma }}\times {\mathbf {p}})\cdot {\hat {z}},$

where $\alpha ={\frac {g\mu _{B}E_{0}}{2mc^{2}}}$