Origin of the -orbit interaction

B E v s

+ E

Mott 1927 1 1 In a frame associated with the : B = E × v = E × p c mc 2  ˆ µB  i  Zeeman energy in the SO field: H = σ i (E × pˆ ) = − 2 2 σ i (∇V × ∇) mc 2m c 1/c expansion of the Dirac equation

pˆ 2 pˆ 4 2    ˆ 2 ˆ H = + eV + 2 2 + 2 2 ∇ V + 2 2 σi(∇V × p) 2m 8mc 8mc 4mc non-relativistic K.E. correction Darwin term SOI All three corection terms are of the same order

Smaller by a factor of 2 (“Thomson factor of two”, 1926) Reason: non-inertial frame

Landau& Lifshits IV: Berestetskii, Lifshits, Pitaevskii Quantum Electrodynamics, Ch. 33

2 SO coupling Ze2 / c ( )

2 2 (Ze / c) Si14 Ga31 Ge32 As33 Bi83 0.01 0.051 0.055 0.058 0.38

Ga32

Ga33

Si14 Ge32 Bi83 Fine structure of atomic levels

Hydrogen

2s Runge-Lentz (non-relativistic Coulomb) 2s1/2 +spin degeneracy Johnson-Lippman (relativistic Coulomb) degeneracy lifted by radiation corrections (Lamb shift) Two types of SOI in solids

1) Symmetry-independent: exists in all types of crystals stem from SOI in atomic orbitals

2) Symmetry-dependent: exists only in crystals without inversion symmetry a) Dresselhaus interaction (bulk): Bulk-Induced-Assymetry (BIA) b) Bychkov-Rashba (surface): Surface-Induced-Asymmetry (SIA) Example of symmetry-independent SOI: SO-split-off valence bands Winkler, Ch. 3

I

E: j = 1 / 2; jz = ±1 / 2

no I

2 2 2 ⎡(ZGa + ZAs ) / 2 ⎤ ⎛ ZGe ⎞ ⎛ 32⎞ ⎢ ⎥ = ⎜ ⎟ = ⎜ ⎟ = 5.2 ⎣ ZSi ⎦ ⎝ ZSi ⎠ ⎝ 14 ⎠

ΔGaAs / ΔSi = .34/0.044 ≈ 7.7

ΔGaAs / ΔGe = .34/.029 ≈ 1.2 Non-centrosymmetric crystals 7 space group: Oh (non-symmorphic)

factor group Oh = Td × I

S. Sque (Exter) [111] Diamond (C): Si, Ge 3a

3a / 4

[111] 3a

2 3a / 4 space group: T d (symmorphic)

point group: Td (methane CH4 ) Zincblende (ZnS): GaAs, GaP, InAs, InSb, ZnSe, CdTe … Additional band splitting in non-centrosymmetric crystals

Kramers theorem: if time-reversal symmetry is not broken, all eigenstates are at least doubly degenerate if ψ is a solution, ψ * is also solution

Kramers doublets εs (k), s = ±1 (not necessary spin projection!)

Time reversal symmetry: k-k,t-t εs (k) = ε−s (−k)

No SOI: ε (k) = ε (−k) regardless of the inversion symmetry

With SOI: i) If a crystal is centrosymmetric

εs (k) = ε−s (−k) = ε−s (k) ⇒ εs (k) = ε−s (k) t→−t k→−k ii) If a crystal is non-centrosymmetric,

εs (k) ≠ ε−s (k) B=0 “spin” splitting Symmetry-dependent SOI: Dresselhaus band splitting in non-centrosymmetric crystals Surface-induced asymmetry: Rashba interaction nˆ z>0 and z<0 half-spaces are not equivalent: k nˆ has a specified direction  Three vectors: k,nˆ,σ E. I. Rashba and V. I. Sheka: Fiz. Tverd. Tela 3 (1961) 1735; ibid. 1863; Sov. Phys. Solid State 3 (1961) 1257; ibid. 1357.

Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984); How to form a scalar? Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys. 17 (1984) 6039. k 2  k 2 ˆ ˆ x y H R = + αni(σ × k) = + α (σ ky − σ kx ) 2m 2m t → −t : k → −k,σ → −σ

⎛ 2 ⎞ 2 k / 2m α (ky + ikx ) k H ⎜ ⎟ R = ⇒ ε± = ± αk ⎜ α k − ik k 2 / 2m ⎟ 2m ⎝ ( y x ) ⎠ Rashba states ε Δ R ε F k

2 k 1 ⎛ 1 ⎞ ε± = ± αk ψ ± = ⎜ iφ ⎟ 2m 2 ⎝ ie k ⎠ 1 S = ψ σ zψ = 0 z,± 2 ± ± 1 x 1 1 y 1 Sx,± = ψ ±σ ψ ± = ± sinφk ; Sy,± = ψ ±σ ψ ± =  cosφk 2 2 2 2 k S 0 i ± = Conductance of Rashba wires

BSO

k 2 2D : Hˆ = + α σ x k − σ yk R 2m ( y x ) k 2 1D : Hˆ = x − ασ yk R 2m x Quay et al. Nature Physics 6, 336 - 339 (2010) Hole wire Dresselhaus+Rashba Dresselhaus Hamiltonian

G. Dresselhaus: Phys. Rev. 100(2), 580–586 (1955)

zincblende (AIIIBV) lattice   H D = DΩ(k)iσ  Ω = k k 2 − k 2 ,k k 2 − k 2 ,k k 2 − k 2 ( x ( y z ) y ( z x ) z ( x y )) (001) surface (interface) of a AIIIBV semiconductor k = 0, k 2 ~ 1 / d 2 ⎛ z z ⎞  2 2 2 2 2 Ω = ⎜ kxk y − kx k z ,ky kz − kyk x ,0⎟ = kz (−kx ,ky ,0) ⎝   ⎠ cubic→0 cubic→0 y x H DS = β (σ ky − σ kx ) in general: Rashba+Dresselhaus k 2 k 2 H = + α σ x k − σ yk + β σ yk − σ x k = + σ x αk − βk + σ y αk + βk 2m ( y x ) ( y x ) 2m ( y x ) ( x y ) Datta-Das Spin Transistor

 k 2 gµ  2α H = + B σiB (k); B k = k × n ≡ Rashba field 2m 2 R R ( ) gµ B even number of π rotations: ON odd number of π rotations: OFF Loss-DiVincenzo Quantum Computer Phys. Rev. A 57, 120 (1998)

GaAs: strong spin-orbit coupling; strong hyperfine interaction (Extrinsic) Spin-Hall Effect

FSO F J SO Js

Spin Hall Effect: the regular current (J) drives a spin current (Js) across the bar resulting in a spin accumulation at the edges.

Skew scattering e More spin up are deflected to the right than to the left (and viceversa for spin down) impurity

Side Jump

For a given deflection, spin up and e spin down electrons make a side- jump in opposite directions. Intrinsic Spin-Hall Effect No observations as of yet unbounded 2D: magnetoelectric effect [V. M. Edelstein, Solid State Comm. 73, 233 (1990). Can an electric field produce n magnetization? B Drift momentum k R k = eEτ M x  k 2 gµ  2α I H = + B σiB (k); B k = k × n ≡ Rashba field 2m 2 R R ( ) gµ B Current induces 2α 2αeEτ M = µ By = k = steady Rashba field y B R g x g Impurities are only necessary to maintain steady state

(but forgetting about them leads to incorrect results--”universal spin-Hall conductivity” Murakami et al. Science 301, 1348 (2003) Sinova et al. Phys. Rev. Lett. 92, 126603 (2004) References

1. M. I. Dyakonov, ed. Spin Physics in Semiiconductors, Spinger 2008 http://www.springerlink.com.lp.hscl.ufl.edu/content/g7x071/#section=215878&page=1 available through UF Library

Ch. 1 M.I. Dyakonov, Basics of Semiconductor and Spin Physics. Ch. 8 M. I. Dyakonov and A. V. Khaetskii, Spin Hall Effect

2. R. Winkler, Spin-Orbit Coupling Effects in 2D Electron and Hole System, Springer 2003. 3. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, Springer 1999 4. J. Fabian et al. Semiconductor Spintronics, Acta Physica Slovaca 57, 565 (2007) arXiv:0711.1461 5. I. Zutic, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and Applications, Rev. Mod. Phys. 76, 323 (2004) 6. M. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Applications to the Physics of Condensed Matter, Springer 2008.