# Rashba effect

The Rashba 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.[1][2] Both the Rashba and Dresselhaus effects are concepts of the PhySH Physics Subject Headlines scheme.

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[3] and the longed for topological p-wave superconductor. [4] [5]

Lately, a momentum dependent pseudospin-orbit coupling has been realized in cold atom systems.[6]

## The Rashba Hamiltonian

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

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

where ${\displaystyle \alpha }$ is the Rashba coupling, ${\displaystyle {\boldsymbol {p}}}$ is the momentum and ${\displaystyle {\boldsymbol {\sigma }}}$ 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 k·p perturbation theory [7] or from the point of view of a tight binding approximation.[8] 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 ${\displaystyle \alpha }$). 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

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

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

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

where ${\displaystyle c}$ is the speed of light. This magnetic field couples to the electron spin

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

where ${\displaystyle -g\mu _{B}{\mathbf {\sigma } }/2}$ is the magnetic moment of the electron.

Within this toy model, the Rashba Hamiltonian is given by

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

where ${\displaystyle \alpha ={\frac {g\mu _{B}E_{0}}{2mc^{2}}}}$. However, while this "toy model" is superficially convincing, the Ehrenfest theorem seems to suggest that since the electronic motion in the ${\displaystyle {\hat {z}}}$ 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.[9]

## 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 ${\displaystyle \alpha }$ 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's consider holes in the ${\displaystyle p_{z}}$ band.[10] In this picture electrons fill all the ${\displaystyle p}$ states except for a few holes near the ${\displaystyle \Gamma }$ point.

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

${\displaystyle H_{SO}=\Delta _{SO}{\mathbf {L}}\otimes {\boldsymbol {\sigma }}}$,

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

${\displaystyle H_{E}=E_{0}\,z}$,

The main effect of the symmetry breaking potential is to open a band gap ${\displaystyle \Delta _{BG}}$ between the isotropic ${\displaystyle p_{z}}$ and the ${\displaystyle p_{x}}$, ${\displaystyle p_{y}}$ bands. The secondary effect of this potential is that it hybridizes the ${\displaystyle p_{z}}$ with the ${\displaystyle p_{x}}$ and ${\displaystyle p_{y}}$ bands. This hybridization can be understood within a tight-binding approximation. The hopping element from a ${\displaystyle p_{z}}$ state at site i with spin ${\displaystyle \sigma }$ to a ${\displaystyle p_{x}}$ or ${\displaystyle p_{y}}$ state at site j with spin ${\displaystyle \sigma '}$ is given by

${\displaystyle t_{ij;\sigma \sigma '}^{x,y}=\langle p_{z},i;\sigma |H|p_{x,y},j;\sigma '\rangle }$,

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

${\displaystyle t_{\sigma \sigma '}^{x,y}=E_{0}\langle p_{z},i;\sigma |z|p_{x,y},i+1_{x,y};\sigma '\rangle =t_{0}\,\mathrm {sgn} (1_{x,y})\delta _{\sigma \sigma '}}$,

where ${\displaystyle 1_{x,y}}$ stands for unit distance in the ${\displaystyle x,y}$ direction respectively and ${\displaystyle \delta _{\sigma \sigma '}}$ 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 ${\displaystyle |p_{z},i;\uparrow \rangle }$ state to a ${\displaystyle |p_{x,y},i+1_{x,y};\uparrow \rangle }$ with amplitude ${\displaystyle t_{0}}$ then uses the spin-orbit coupling to flip spin and go back down to the ${\displaystyle |p_{z},i+1_{x,y};\downarrow \rangle }$ with amplitude ${\displaystyle \Delta _{SO}}$. Note that overall the hole hopped one site and flipped spin. The energy denominator in this perturbative picture is of course ${\displaystyle \Delta _{BG}}$ such that all together we have

${\displaystyle \alpha \approx {a\,t_{0}\,\Delta _{SO} \over \Delta _{BG}}}$,

where ${\displaystyle a}$ 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 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.[11][12]

Topological quantum computation - Lately it has been suggested that the Rashba effect can be used to realize a p-wave superconductor.[4][5] 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.

Discovery of giant Rashba effect in bulk crystals such as BiTeI[13] and ferroelectric GeTe[14] and in a number of low-dimensional systems bears a promise of creating devices operating electrons spins at nanoscale and possessing short operational times.

## The 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 found,[15] and for heterostructures where it develops as a result of a symmetry breaking field in the direction perpendicular to the 2D surface.[16] All these systems lack inversion symmetry. A similar effect, known as the Dresselhaus spin orbit coupling[17] arises in cubic crystals of A${\displaystyle _{III}}$B${\displaystyle _{V}}$ type lacking inversion symmetry and in quantum wells manufactured from them.

## Footnotes

1. AMR in most common magnetic materials was reviewed by McGuire and Potter.[18] A more recent work[19] focused on the possibility of Rashba-effect-induced AMR and some extensions and corrections were given later.[20]

## References

1. ^ G. Bihlmayer, O. Rader and R. Winkler, Focus on the Rashba effect , New J. Phys. 17, 050202 (2015)
2. ^ Han Woong Yeom and Marco Grioni, eds, Special issue on electron spectroscopy for Rashba spin-orbit interaction, Journal of Electron Spectroscopy and Related Phenomena, 201, 1-126 (2015)
3. ^ Agterberg, Daniel (2003). "Anisotropic magnetoresistance of spin-orbit coupled carriers scattered from polarized magnetic impurities". Physica C. 387 (1-2): 13–16. Bibcode:2003PhyC..387...13A. doi:10.1016/S0921-4534(03)00634-8.
4. ^ a b Sato, Masatoshi & Fujimoto, Satoshi (2009). "Topological phases of noncentrosymmetric superconductors: Edge states, Majorana fermions, and non-Abelian statistics". Phys. Rev. B. 79: 094504. arXiv:. Bibcode:2009PhRvB..79i4504S. doi:10.1103/PhysRevB.79.094504.
5. ^ a b V. Mourik, K. Zuo1, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhoven (2012). "Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices". Science Express. 1222360: 1003–1007. arXiv:. Bibcode:2012Sci...336.1003M. doi:10.1126/science.1222360.
6. ^ Lin, Y.-J.; K. Jiménez-García; I. B. Spielman (2011). "Spin-orbit-coupled Bose-Einstein condensates". Nature. 471: 83–86. arXiv:. Bibcode:2011Natur.471...83L. doi:10.1038/nature09887.
7. ^ Winkler, Ronald. Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (PDF). New-York: Springer Tracts in Modern Physics.
8. ^ L. Petersena & P. Hedegård (2000). "A simple tight-binding model of spin–orbit splitting of sp-derived surface states". Surface Science. 459 (1-2): 49–56. Bibcode:2000SurSc.459...49P. doi:10.1016/S0039-6028(00)00441-6.
9. ^ P. Pfeffer & W. Zawadzki (1999). "Spin splitting of conduction subbands in III-V heterostructures due to inversion asymmetry". Physical Review B. 59: R5312–5315. Bibcode:1999PhRvB..59.5312P. doi:10.1103/PhysRevB.59.R5312.
10. ^ Typically in semiconductors the Rashba splitting is considered for the s band around the ${\displaystyle \Gamma _{6}}$ point. In the discussion above we consider only the mixing of the anti-bonding p bands. However, the induced Rashba splitting is simply given by the hybridization between p and s bands. Therefore, this discussion is actually all one needs to understand the Rashba splitting at near the ${\displaystyle \Gamma _{6}}$ point.
11. ^ D. Bercioux and P. Lucignano, Quantum transport in Rashba spin–orbit materials: a review, Rep. Prog. Phys. 78, 106001 (2015); https://arxiv.org/abs/1502.00570.
12. ^ Rashba Effect in Spintronic Devices
13. ^ K.Ishizaka, K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi,S. Shin, H. Miyahara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame, M. Taniguchi, R. Arita, N. Nagaosa, K. Kobayashi, Y. Murakami, R. Kumai, Y. Kaneko, Y. Onose, and Y. Tokura, "Giant Rashba-type spin splitting in bulk BiTeI", Nature materials 10, 521-526 (2011).
14. ^ D. Di Sante, P. Barone, R. Bertacco, and S. Picozzi, Electric Control of the Giant Rashba Effect in Bulk GeTe, Advanced Materials, 25, 509-513 (2013).
15. ^ E. 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).
16. ^ 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)
17. ^ G. Dresselhaus, Spin-Orbit Coupling Effects in Zinc Blende Structures, Phys. Rev. 100, 580 (1955)
18. ^ McGuire, T.; R. Potter (1975). "Anisotropic magnetoresistance in ferromagnetic 3d alloys". IEEE Transactions on Magnetics. 11 (4): 1018–1038. Bibcode:1975ITM....11.1018M. doi:10.1109/TMAG.1975.1058782.
19. ^ Schliemann, John; Daniel Loss (2003). "Anisotropic transport in a two-dimensional electron gas in the presence of spin-orbit coupling". Physical Review B. 68: 165311. arXiv:. Bibcode:2003PhRvB..68p5311S. doi:10.1103/physrevb.68.165311.
20. ^ Trushin, Maxim; et al. (2009). "Anisotropic magnetoresistance of spin-orbit coupled carriers scattered from polarized magnetic impurities". Physical Review B. 80: 134405. arXiv:. Bibcode:2009PhRvB..80b4405I. doi:10.1103/PhysRevB.80.024405.