Angelescu polynomials

The Rashba effect, or Rashba-Dresselhaus effect, is a momentum-dependent splitting of spin bands in two-dimensional condensed matter systems (heterostructures and surface states) similar to the splitting of particles and anti-particles in the Dirac Hamiltonian. The splitting is a combined effect of atomic spin-orbit coupling and asymmetry of the potential in the direction perpendicular to the two-dimensional plane. This effect is named in honour of Emmanuel Rashba who discovered it.

Remarkably, this effect can drive a wide variety of novel physical phenomena even when it is a small correction to the band structure of the two-dimensional metallic state.

Additionally, superconductors with large Rashba splitting are suggested as possible realizations of the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state and the longed topological p-wave superconductor.

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

The Rashba Hamiltonian

The Rashba effect is most easily seen in the simple model Hamiltonian known as the Rashba Hamiltonian

$H_R=\alpha (\mathit{\sigma }\times \mathbf{p})\cdot \hat{z}$,

where $\alpha$ is the Rashba coupling, $\mathit{p}$ is the momentum and $\mathit{\sigma }$ is the Pauli matrix vector. This is nothing but a two-dimensional version of the Dirac Hamiltonian (with a 90 degrees rotation of the spins).

The Rashba model in solids can be derived in the framework of the $\mathit{k}\cdot \mathit{p}$ 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 qualitativley the same physics (quantitatively it gives a poor estimation of the coupling $\alpha$). Here we will introduce the intuitive toy model approach followed by a sketch of a more accurate derivation.

Naive derivation of the Rashba Hamiltonian

The Rashba effect is a direct result of inversion symmetry breaking in the direction perpendicular to the two-dimensional plane. Therefore, let us add to the Hamiltonian a term that breaks this symmetry in the form of an electric field

$H_E=-E_{0}z$,

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

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

where $-g{\mu }_B\mathit{\sigma }/2$ is the magnetic moment of the electron.

Within this toy model, the Rashba Hamiltonian is given by

$H_R=\alpha (\mathit{\sigma }\times \mathbf{p})\cdot \hat{z}$,

where $\alpha =\frac{g{\mu }_B{E}_0}{2m{c}^2}$. However, while this "toy model" is superficially convincing, the Ehrenfest theorem seems to suggest that since the electronic motion in the $\hat{z}$ direction is that of a bound state that confines it to the 2D surface, the time-averaged electric field (i.e., including that of the potential that binds it to the 2D surface) that the electron experiences must be zero! When applied to the toy model, this argument seems to rule out the Rashba effect (and caused much controversy prior to its experimental confirmation), but turns out to be subtly-incorrect when applied to a more realistic model.

Estimation of the Rashba coupling in a realistic system - the tight binding approach

In this section we will sketch a method to estimate the coupling constant $\alpha$ from microscopics using a tight-binding model. Typically, the itinerant electrons that form the two-dimensional electron gas (2DEG) originate in atomic s and p orbitals. For the sake of simplicity let consider holes in the $p_z$ band. In this picture electrons fill the all the $p$ states except for a few holes near the $\mathrm{\Gamma }$ point.

The necessary ingredients to get Rashba splitting are atomic spin-orbit coupling

$H_{SO}={\mathrm{\Delta }}_{SO}\mathbf{L}\otimes \mathit{\sigma }$,

and an asymmetric potential in the direction perpendicular to the 2D surface

$H_E=E_{0}z$,

The main effect of the symmetry breaking potential is to open a band gap ${\mathrm{\Delta }}_{BG}$ between the isotropic $p_z$ and the $p_x$, $p_y$ bands. The secondary effect of this potential is that it hybridizes the $p_z$ with the $p_x$ and $p_y$ bands. This hybridization can be understood with in a tight-binding approximation. The hopping element from a $p_z$ state at site i with spin $\sigma$ to a $p_x$ or $p_y$ state at site j with spin ${\sigma }^{\prime }$ is given by

$t_{ij;\sigma {\sigma }^{\prime }}^{x,y}=⟨p_z,i;\sigma \left|H\right|p_{x,y},j;{\sigma }^{\prime }⟩$,

where $H$ is the total Hamiltonian. In the absence of a symmetry breaking field, i.e. $H_E=0$, the hopping element vanishes due to symmetry. However, if $H_E\ne 0$ then the hopping element is finite. For example, the nearest neighbor hopping element is

$t_{\sigma {\sigma }^{\prime }}^{x,y}=E_0⟨p_z,i;\sigma \left|z\right|p_{x,y},i+1_{x,y};{\sigma }^{\prime }⟩=t_0\mathrm{s}\mathrm{g}\mathrm{n}(1_{x,y}){\delta }_{\sigma {\sigma }^{\prime }}$,

where $1_{x,y}$ stands for unit distance in the $x,y$ direction respectively and ${\delta }_{\sigma {\sigma }^{\prime }}$ is Kronecker's delta.

The Rashba effect can be understood as a second order perturbation theory in which a spin-up hole, for example, jumps from a $|p_z,i;\uparrow ⟩$ state to a $|p_{x,y},i+1_{x,y};\uparrow ⟩$ with amplitude $t_0$ then uses the spin=orbit coupling to flip spin and go back down to the $|p_z,i+1_{x,y};\downarrow ⟩$ with amplitude ${\mathrm{\Delta }}_{SO}$. Note that overall the hole hopped one site and flipped spin. The energy denominator in this perturbative picture is of course ${\mathrm{\Delta }}_{BG}$ such that all together we have

$\alpha \approx \frac{a{t}_0{\mathrm{\Delta }}_{SO}}{{\mathrm{\Delta }}_{BG}}$,

where $a$ is the interionic distance. This result is typically two orders of magnitude larger than the naive result derived in the previous section.

Application

Spintronics - Electronic devices are based on the ability to manipulate the electrons charge by the means of electric fields. Similarly, devices can be based on the manipulation of the spin degree of freedom. The Rashba effect allows to manipulate the spin by the same means, that is, without the aid of a magnetic field. Such devices have many advantages over their electronic counterparts.

Topological Quantum computation - Lately it has been suggested that the Rashba effect can be used to realize a p-wave superconductor. Such a superconductor has very special edge-states which are known as Majorana bound states. The non-locality immunizes them to local scattering and henceforth they are predicted to have long coherence times. Decoherence is one of the largest barriers on the way to realize a full scale quantum computer and these immune states are therefore considered good candidates for a quantum bit.

The Dresselhaus spin orbit coupling

The Rashba spin-orbit coupling is a direct result of a symmetry breaking field in the direction perpendicular to the 2D surface. Such systems are classified as systems that lack inversion symmetry. A similar effect, known as the Dresselhaus spin orbit coupling arises in systems that lack symmetry under reflection about a plane that contains at least one lattice site in the underlying crystal. Such crystals are known as noncentrosymmetric or systems that lack an inversion center.

Footnotes

• 1. AMR in most common magnetic materials was reviewed by McGuire and Potter. A more recent work focused on the possibility of Rashba-effect-induced AMR and some extensions and corrections were given later.

References

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