Rashba effect

Not to be confused with the Rashba–Edelstein effect, which describes the conversion of a bidimensional charge current into a spin accumulation.

The Rashba effect, also called Bychkov–Rashba effect, is a momentum-dependent splitting of spin bands in bulk crystals^{[note 1]} 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.^[2][3][4]

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).^{[note 2][5][6][7]}

Additionally, superconductors with large Rashba splitting are suggested as possible realizations of the elusive Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state,^[8] Majorana fermions and topological p-wave superconductors.^[9][10]

Lately, a momentum dependent pseudospin-orbit coupling has been realized in cold atom systems.^[11]

Hamiltonian

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

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

where ${\displaystyle \alpha }$  is the Rashba coupling, ${\displaystyle \mathbf {p} }$  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 degree rotation of the spins).

The Rashba model in solids can be derived in the framework of the k·p perturbation theory^[12] or from the point of view of a tight binding approximation.^[13] 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 ${\textstyle \alpha }$ ). Here we will introduce the intuitive toy model approach followed by a sketch of a more accurate derivation.

Naive derivation

The Rashba effect 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

${\displaystyle H_{\rm {E}}=-E_{0}ez}$ .

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 in a spin-orbit term

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

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

Within this toy model, the Rashba Hamiltonian is given by

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

where ${\displaystyle \alpha _{\rm {R}}=-{\frac {g\mu _{\rm {B}}E_{0}}{2mc}}}$ . However, while this "toy model" is superficially attractive, 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 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.^[14] 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 Dirac gap of ${\displaystyle mc^{2}}$  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.

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 consider holes in the ${\displaystyle p_{z}}$  band.^[15] In this picture electrons fill all the p states except for a few holes near the ${\displaystyle \Gamma }$  point.

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

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

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 ${\displaystyle \Delta _{\mathrm {BG} }}$  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 ${\displaystyle i}$  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 ${\displaystyle \Delta _{\mathrm {SO} }}$ . Note that overall the hole hopped one site and flipped spin. The energy denominator in this perturbative picture is of course ${\displaystyle \Delta _{\mathrm {BG} }}$  such that all together we have

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

where ${\displaystyle a}$  is the interionic distance. This result is typically several 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.^[16][17]

Topological quantum computation - Lately it has been suggested that the Rashba effect can be used to realize a p-wave superconductor.^[9][10] Such a superconductor has very special edge-states which are known as Majorana bound states. 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 quantum bit.

Discovery of the giant Rashba effect with ${\displaystyle \alpha }$  of about 5 eV•Å in bulk crystals such as BiTeI,^[18] ferroelectric GeTe,^[19] and in a number of low-dimensional systems bears a promise of creating devices operating electrons spins at nanoscale and possessing short operational times.

Comparison with Dresselhaus spin–orbit coupling

Main article: Dresselhaus effect

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^[20] 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.^[2] All these systems lack inversion symmetry. A similar effect, known as the Dresselhaus spin orbit coupling^[21] arises in cubic crystals of A_IIIB_V type lacking inversion symmetry and in quantum wells manufactured from them.

See also

• Electric dipole spin resonance

Footnotes

1. More specifically, uniaxial noncentrosymmetric crystals.
2. AMR in most common magnetic materials was reviewed by McGuire & Potter 1975. A more recent work (Schliemann & Loss 2003) focused on the possibility of Rashba-effect-induced AMR and some extensions and corrections were given later (Trushin et al. 2009).

References

1. 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), http://iopscience.iop.org/1367-2630/17/5/050202/media/njp050202_suppdata.pdf.
2. 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)
3. G. Bihlmayer, O. Rader and R. Winkler, Focus on the Rashba effect , New J. Phys. 17, 050202 (2015)
4. Yeom, Han Woong; Grioni, Marco, eds. (May 2015). "Special issue on electron spectroscopy for Rashba spin-orbit interaction". Journal of Electron Spectroscopy and Related Phenomena. 201: 1–126. doi:10.1016/j.elspec.2014.10.005. ISSN 0368-2048. Retrieved 28 January 2019.
5. McGuire, T.; Potter, R. (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.
6. Schliemann, John; Loss, Daniel (2003). "Anisotropic transport in a two-dimensional electron gas in the presence of spin-orbit coupling". Physical Review B. 68 (16): 165311. arXiv:cond-mat/0306528. Bibcode:2003PhRvB..68p5311S. doi:10.1103/physrevb.68.165311. S2CID 119093889.
7. Trushin, Maxim; Výborný, Karel; Moraczewski, Peter; Kovalev, Alexey A.; Schliemann, John; Jungwirth, T. (2009). "Anisotropic magnetoresistance of spin-orbit coupled carriers scattered from polarized magnetic impurities". Physical Review B. 80 (13): 134405. arXiv:0904.3785. Bibcode:2009PhRvB..80m4405T. doi:10.1103/PhysRevB.80.134405. S2CID 41048255.
8. 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.
9. Sato, Masatoshi & Fujimoto, Satoshi (2009). "Topological phases of noncentrosymmetric superconductors: Edge states, Majorana fermions, and non-Abelian statistics". Phys. Rev. B. 79 (9): 094504. arXiv:0811.3864. Bibcode:2009PhRvB..79i4504S. doi:10.1103/PhysRevB.79.094504. S2CID 119182379.
10. 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 (6084): 1003–1007. arXiv:1204.2792. Bibcode:2012Sci...336.1003M. doi:10.1126/science.1222360. PMID 22499805. S2CID 18447180.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
11. Lin, Y.-J.; K. Jiménez-García; I. B. Spielman (2011). "Spin-orbit-coupled Bose-Einstein condensates". Nature. 471 (7336): 83–86. arXiv:1103.3522. Bibcode:2011Natur.471...83L. doi:10.1038/nature09887. PMID 21368828. S2CID 4329549.
12. Winkler, Ronald. Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (PDF). New-York: Springer Tracts in Modern Physics.
13. 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.
14. P. Pfeffer & W. Zawadzki (1999). "Spin splitting of conduction subbands in III-V heterostructures due to inversion asymmetry". Physical Review B. 59 (8): R5312-5315. Bibcode:1999PhRvB..59.5312P. doi:10.1103/PhysRevB.59.R5312.
15. 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.
16. Bercioux, Dario; Lucignano, Procolo (2015-09-25). "Quantum transport in Rashba spin–orbit materials: a review". Reports on Progress in Physics. 78 (10): 106001. arXiv:1502.00570. Bibcode:2015RPPh...78j6001B. doi:10.1088/0034-4885/78/10/106001. ISSN 0034-4885. PMID 26406280. S2CID 38172286.
17. Rashba Effect in Spintronic Devices
18. Ishizaka, K.; Bahramy, M. S.; Murakawa, H.; Sakano, M.; Shimojima, T.; et al. (2011-06-19). "Giant Rashba-type spin splitting in bulk BiTeI". Nature Materials. Springer Science and Business Media LLC. 10 (7): 521–526. Bibcode:2011NatMa..10..521I. doi:10.1038/nmat3051. ISSN 1476-1122. PMID 21685900.
19. Di Sante, Domenico; Barone, Paolo; Bertacco, Riccardo; Picozzi, Silvia (2012-10-16). "Electric Control of the Giant Rashba Effect in Bulk GeTe". Advanced Materials. Wiley. 25 (4): 509–513. doi:10.1002/adma.201203199. ISSN 0935-9648. PMID 23070981. S2CID 33251068.
20. 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).
21. Dresselhaus, G. (1955-10-15). "Spin-Orbit Coupling Effects in Zinc Blende Structures". Physical Review. American Physical Society (APS). 100 (2): 580–586. Bibcode:1955PhRv..100..580D. doi:10.1103/physrev.100.580. ISSN 0031-899X.

Further reading

• Chu, Junhao; Sher, Arden (2009). Device Physics of Narrow Gap Semiconductors. Springer. pp. 328–334. ISBN 978-1-4419-1039-4.
• Heitmann, Detlef (2010). Quantum Materials, Lateral Semiconductor Nanostructures, Hybrid Systems and Nanocrystals. Springer. pp. 307–309. ISBN 978-3-642-10552-4.
• 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/
• E. I. Rashba and V. I. Sheka, Electric-Dipole Spin-Resonances, in: Landau Level Spectroscopy, (North Holland, Amsterdam) 1991, p. 131; https://arxiv.org/abs/1812.01721
• Rashba, Emmanuel I (2005). "Spin Dynamics and Spin Transport". Journal of Superconductivity. 18 (2): 137–144. arXiv:cond-mat/0408119. Bibcode:2005JSup...18..137R. doi:10.1007/s10948-005-3349-8. S2CID 55016414.

External links

• Ulrich Zuelicke (30 Nov – 1 Dec 2009). (PDF). Institute of Fundamental Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology Massey University, Palmerston North, New Zealand. Archived from the original on 2012-03-31. Retrieved 2011-09-02.{{cite web}}: CS1 maint: bot: original URL status unknown (link)

• "Finding the beat: New discovery settles a long-standing debate about photovoltaic materials". DOE, Ames Laboratory, Division of Materials Sciences. April 7, 2020.