Rashba effect

 ( Semiconductors ) 

The Rashba effect, also called Bychkov–Rashba effect, is a momentum-dependent splitting of spin bands in bulk More specifically, uniaxial noncentrosymmetric crystals. and low-dimensional condensed matter systems (such as and surface states) similar to the splitting of and in the Dirac equation 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 1959E. I. Rashba and V. I. Sheka, Fiz. Tverd. Tela – Collected Papers (Leningrad), v.II, 162-176 (1959) (in Russian), English translation: Supplemental Material to the paper by G. Bihlmayer, O. Rader, and R. Winkler, Focus on the Rashba effect, New J. Phys. 17, 050202 (2015), http://iopscience.iop.org/1367-2630/17/5/050202/media/njp050202_suppdata.pdf. for three-dimensional systems and afterward with Yurii A. Bychkov in 1984 for two-dimensional systems.Yu. A. Bychkov and E. I. Rashba, Properties of a 2D electron gas with a lifted spectrum degeneracy, Sov. Phys. - JETP Lett. 39, 78-81 (1984)G. Bihlmayer, O. Rader and R. Winkler, Focus on the Rashba effect, New J. Phys. 17, 050202 (2015)

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

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

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

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

$$

H_{\rm R}=\alpha(\hat{z}\times\mathbf{p})\cdot \boldsymbol{\sigma} , where $\backslash alpha$ is the Rashba coupling, $\backslash mathbf\; p$ is the momentum and $\backslash boldsymbol\; \backslash sigma$ is the Pauli matrix vector. This is nothing but a two-dimensional version of the Dirac Hamiltonian (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 $\backslash alpha$). Here we will introduce the intuitive toy model approach followed by a sketch of a more accurate derivation.

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Naive derivation 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_{\rm E}= - E_0 e 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 in a spin-orbit term

$$

H_{\mathrm{SO}}=\frac{g\mu_{\rm B}}{2c^2}(\mathbf{v}\times\mathbf{E})\cdot \boldsymbol{\sigma} , where $-g\backslash mu\_\{\backslash rm\; B\}\; \backslash mathbf\{\backslash sigma\}/2$ is the electron magnetic moment.

Within this toy model, the Rashba Hamiltonian is given by

$$

H_{\mathrm{R}} = -\alpha_{\rm R}(\hat{z} \times \mathbf{p})\cdot \boldsymbol{\sigma}, where $\backslash alpha\_\{\backslash rm\; R\}\; =\; -\backslash frac\{g\backslash mu\_\{\backslash rm\; B\}E\_0\}\{2mc^2\}$. However, while this "toy model" is superficially attractive, the Ehrenfest theorem seems to suggest that since the electronic motion in the $\backslash hat\{z\}$ direction is that of a bound state that confines it to the 2D surface, the space-averaged electric field (i.e., including that of the potential that binds it to the 2D surface) that the electron experiences must be zero given the connection between the time derivative of spatially averaged momentum, which vanishes as a bound state, and the spatial derivative of potential, which gives the electric field! 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. While the above naive derivation provides correct analytical form of the Rashba Hamiltonian, it is inconsistent because the effect comes from mixing energy bands (interband matrix elements) rather from intraband term of the naive model. A consistent approach explains the large magnitude of the effect by using a different denominator: instead of the Paul Dirac gap of $mc^2$ of the naive model, which is of the order of MeV, the consistent approach includes a combination of splittings in the energy bands in a crystal that have an energy scale of eV, as described in the next section.

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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 $\backslash alpha$ from microscopics using a tight-binding model. Typically, the itinerant electrons that form the two-dimensional electron gas (2DEG) originate in atomic and orbitals. For the sake of simplicity consider holes in the $p\_z$ band.Typically in semiconductors the Rashba splitting is considered for the band around the $\backslash Gamma\_6$ point. In the discussion above we consider only the mixing of the anti-bonding bands. However, the induced Rashba splitting is simply given by the hybridization between and bands. Therefore, this discussion is actually all one needs to understand the Rashba splitting at near the $\backslash Gamma\_6$ point. In this picture electrons fill all the states except for a few holes near the $\backslash Gamma$ point.

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

$$

H_{\mathrm{SO}}=\Delta_{\mathrm{SO}} \mathbf{L} \otimes \boldsymbol{\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 $\backslash Delta\_\{\backslash mathrm\{BG\}\}$ 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 within a tight-binding approximation. The hopping element from a $p\_z$ state at site $i$ with spin $\backslash sigma$ to a $p\_\{x\}$ or $p\_\{y\}$ state at site j with spin $\backslash sigma\text{'}$ is given by

$$

t_{ij;\sigma \sigma'}^{x,y}=\langle p_z,i;\sigma | H | p_{x,y},j ;\sigma'\rangle , 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 $H\_E\backslash ne\; 0$ then the hopping element is finite. For example, the nearest neighbor hopping element is

$$

t_{\sigma \sigma'} ^{x,y}=E_0 \langle p_z,i ;\sigma| z | p_{x,y},i+1_{x,y} ;\sigma'\rangle = t_0 \,\mathrm{sgn}(1_{x,y}) \delta_{\sigma \sigma'}, where $1\_\{x,y\}$ stands for unit distance in the $x,y$ direction respectively and $\backslash delta\_\{\backslash sigma\; \backslash sigma\text{'}\}$ is Kronecker delta.

The Rashba effect can be understood as a second order perturbation theory in which a spin-up hole, for example, jumps from a $|p\_z,i;\backslash uparrow\backslash rangle$ state to a $|p\_\{x,y\},i+1\_\{x,y\};\backslash uparrow\backslash rangle$ with amplitude $t\_0$ then uses the spin–orbit coupling to flip spin and go back down to the $|p\_z,i+1\_\{x,y\};\backslash downarrow\backslash rangle$ with amplitude $\backslash Delta\_\{\backslash mathrm\{SO\}\}$. Note that overall the hole hopped one site and flipped spin. The energy denominator in this perturbative picture is of course $\backslash Delta\_\{\backslash mathrm\{BG\}\}$ such that all together we have

$$

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

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Application Spintronics - Electronic devices are based on the ability to manipulate the electrons position by 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. Rashba Effect in Spintronic Devices

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 fermion. The non-locality immunizes them to local scattering and hence 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 qubit.

Discovery of the giant Rashba effect with $\backslash alpha$ of about 5 eV•Å in bulk crystals such as BiTeI, ferroelectric GeTe, and in a number of low-dimensional systems bears a promise of creating devices operating electrons spins at nanoscale and possessing short operational times.

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Comparison with Dresselhaus spin–orbit coupling The Rashba spin-orbit coupling is typical for systems with uniaxial symmetry, e.g., for hexagonal crystals of CdS and CdSe for which it was originally foundE. I. Rashba and V. I. Sheka, Fiz. Tverd. Tela - Collected Papers (Leningrad), v.II, 162-176 (1959) (in Russian), English translation: Supplemental Material to the paper by G. Bihlmayer, O. Rader, and R. Winkler, Focus on the Rashba effect, New J. Phys. 17, 050202 (2015). and perovskites, and also for heterostructures where it develops as a result of a symmetry breaking field in the direction perpendicular to the 2D surface. All these systems lack inversion symmetry. A similar effect, known as the Dresselhaus spin orbit coupling arises in cubic crystals of A_IIIB_V type lacking inversion symmetry and in manufactured from them.

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See also

• Electric dipole spin resonance
• Orbital Rashba effect

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Footnotes

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Further reading

• (2025). 9781441910394, Springer. ISBN 9781441910394
• (2025). 9783642105524, Springer. ISBN 9783642105524
• A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, New perspectives for Rashba spin–orbit coupling, Nature Materials 14, 871-882 (2015), http://www.nature.com/nmat/journal/v14/n9/pdf/nmat4360.pdf, stacks.iop.org/NJP/17/050202/mmedia
• http://blog.physicsworld.com/2015/06/02/breathing-new-life-into-the-rashba-effect/
• 

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External links

Categories: Semiconductors, Quantum Magnetism, Spintronics

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