( 1) 1 ( [1] ) [] ( ) (AC) [3] [4, 5, 6] 3 (i) AC (ii) (iii) 3 AC [3, 7] [4, 5, 6] 1.1 ( e; e>0) Ze r v [ 1(a)] v [ 1(a )] B = μ 0 4π Zer v r 3 = μ 0 4π 1 Ze l m r 3,
μ 0 l = mr v ( l s ) s μ s = μ B s [μ B = e /(m) g ] μ s B = μ 0 Ze 4π m ( l s) r3 1 H SO = μ 0 Ze 4π m ( l s). (1) r3 E = 1 4πε 0 Ze r r r, U ( F = ee = U) H SO = e m s (p E) = c 4m σ (p U) () c c =1/ μ 0 ε 0 σ : s =(1/)σ. () [8] Z : () mc 1MeV [8] 1. ( ) ( ) GaAs, InAs p ( l =1) (1) p H SO l s l, s l z, s z j = l + s l s = 1 [(l + s) l s ] = 1 (j l s ). H SO j j z p j = l + s =3/ (j z = ±3/, ±1/), j = l s =1/ (j z = ±1/) 1 1/ ( ) [L. H. Tomas (196)] [ : (, 1974)]
(a) (a') +Ze e e +Ze (b) (b') + + + e + + = J J B 1: (a) : (a ) (b) ± : (b ) k p [9] Γ [k =(0, 0, 0) ] j =3/ j =1/ split-off Γ j z = ±3/ eavy-ole j z = ±1/ ligt-ole s (l =0) 1.3 ( ) ( ) ( ) ( ) m (GaAs 0.067 ) GaAs k p () ( [7] ) H SO = P [ 1 3 E 0 ] 1 (E 0 +Δ 0 ) 1 σ (p U). (3) P E 0 Δ 0 j =3/ j =1/ ( ) ( ) (Bloc ), Scrödinger (Bloc ) 3
E E 0 Δ 0 k HH LH split-off band : GaAs eavy-ole (HH) ligt-ole (LH) split-off 3 InGaAs, InAs : InGaAs/GaAs (Rasba) ( ) xy z H RSO = ᾱ σ (p ẑ) =ᾱ (p yσ x p x σ y ) (4) k α /k F, k α = mα (m k F Fermi ) InGaAs α =(3 4) 10 11 evm, Δ R k F α =15 0 mev k α /k F =Δ R /(4E F ) 0.1 [10, 11] 1(b) ± z x x [ 1(b )] y [ (4) ] y x : (Dresselaus) [1] III-V 3 x, y, z 4
[100], [010], [001] H (3D) DSO = γ ] [p x (p y p z)σ x + p y (p z p x)σ y + p z (p x p y)σ z [001] z H DSO = γ β = γ p z, p z =0 [ ] p x (p y p z )σ x + p y ( p z p x )σ y = β ( p xσ x + p y σ y )+(p 3 ). (5) GaAs [13] InGaAs 1.4 Datta Das [14] 3(a) (FET) ( ) 3 x y (4) gμ B s B eff = γ s B eff [g g (GaAs g 0.4); γ = gμ B / μ s = γ s ] s γ B eff = ᾱ (p ẑ), ds dt = μ s B eff = γ s B eff x k x 4 y ω = γ B eff =αk x / x L Δθ = αk x Δt = αk x L ( k x /m) = αml. (6) ω ( k x /m) k x Δθ k x (6) α Δθ 3 3 : ( ) ( ) ( ) 4 y 1 ( ) [14] 5
(a) z y x (b) E (c) E k k + k k 3: (a) Datta Das ( ) ( ) ( ) (b) 1 k ± (c) : E k,± = k /(m) ± gμ B B/. x Hamiltonian H = p x m ᾱ p xσ y. ψ k,± = e ikx χ (y) ±, E k,± = k αk = m m (k k α) kα m. χ (y) ± y : σ yχ (y) ± = ±χ (y) ±. E k,± 3(b) (i) k ± k + k α = k + k α ( k 0 ). (ii) 1 E k,± k k=k± (iii) Kramers ( ) 5 : E k,+ = E k,. [E k,± = k /(m) ± gμ B B/] 3(c) 5 J. J. Sakurai: ( 1989) 4 6
(i) (iii) x χ (x) + x =0 χ (x) + = 1 [χ (y) + + iχ (y) ] χ (x) = 1 [iχ (y) + + χ (y) ] ψ = 1 [e ik +x χ (y) + + ieik x χ (y) ] = 1 e ik 0x [e ikαx χ (y) + + ie ikα χ (y) ] = e ik 0x [cos(k α x)χ (x) + + sin(k α x)χ (x) ]. x = L k α (AC) (AB) B A (B = rota) 4(a) AB Aaronov Caser AB [15] ( ) ( ) (AC) [ ] AB AB AC AB AC.1 AB 4(a) A B Φ B(r) A(r) Hamiltonian H = 1 m [p + ea(r)] + V (r). (7) C i (i =1, ) A =0 ψ (0) i (r) A 0 [ ψ i (r) =ψ (0) i (r) exp i ē ] r A(r ) dr (8) r A (C i ) 7
(a) A Φ C 1 B (b) ϕ C (c) A B B eff 4: (a) A B Φ (b) (c) (B eff p ẑ) AC Hamiltonian (7) ψ i (r) 6 : (p + ea)ψ i (r) =( i + ea)ψ i (r) =[ i ψ (0) i (r)]e i ē A dr, etc. B ψ(r B )=ψ 1 (r B )+ψ (r B ), ψ(r B ) = ψ 1 + ψ + cos(θ 1 θ +Δφ) θ 1, θ ψ (0) 1 (r B), ψ (0) (r B) Δφ = ē r B A(r) dr + ē r B A(r) dr r A (C 1 ) r A (C ) = ē A(r) dr = ē rota ds = ē B ds =π Φ. Φ 0 3 Φ Φ 0 = /e (quantum flux) Φ Φ 0 C 1 C ( ψ 1 = ψ, θ 1 = θ ) Landauer [16] [ G = e 1 + cos (π Φ )] (9) Φ0 6 (8) B = rota =0, χ(r) A = rχ (8) r 8
. AC AB (4) p p + ea [ () [8]] 4(a) [ 4(b)] 7 r ϕ p x = i 1 r sin ϕ ϕ, p y = i 1 r cos ϕ ϕ. A =(Br/)e ϕ =(Φ/πr)e ϕ (e ϕ ϕ Φ =πr ) Hamiltonian H = = ( mr i ϕ + Φ ) + α ( Φ 0 r (cos ϕσ x + sin ϕσ y ) i ϕ + Φ ) Φ 0 iα(cos ϕσ y sin ϕσ x ) 1 (10) r [ mr i ϕ + Φ + mrα ] Φ 0 (cos ϕσ x + sin ϕσ y ) mα. (11) [ (10) r / r 1/r [17] H 8 ] Hamiltonian (11) i ϕ + Φ + mrα Φ 0 (cos ϕσ x + sin ϕσ y )= i ϕ + Φ + mrα ( 0 e iϕ Φ 0 e iϕ 0 ψ(ϕ +π) =ψ(ϕ) ψ = 1 ( π C1 e inϕ C e i(n+1)ϕ Hamiltonian (11) E n,± = ψ n,+ = ψ n, = ) ( mr n + Φ ) Φ + ΦAC ± mα 0 π, (1) ( ) 1 cos(θα /)e i(n 1)ϕ π sin(θ α /)e inϕ, ( 1 π sin(θα /)e inϕ cos(θ α /)e i(n+1)ϕ 7 4(a) z (χ (z) ± ) A ψ1(rb) = A 1χ (z) + + B1χ(z), ψ(rb) =Aχ(z) + + Bχ(z) A1/B1 = A/B AC [C 1χ (z) + + Cχ(z), ] A1/B1 = A/B B 8 ). ) 9
± = π 1 Φ AC (mrα ) +1, (13) tan θ α = mrα/ (π/ <θ α π; α =0 θ α = π) (α =0) e ikx ( x = rϕ) k /(m). πrk =πn, AB πrk πφ/φ 0 =πn. n n +Φ/Φ 0 (1) 9 ( [16] ) 10 (1) Φ AC ± [ 4(c) (B eff p ẑ) ] (±) AC 4(a) (mrα ) G = e 1 cos π +1. (14) α AC [3, 18, 19] AB [0] AC Qian Su [1] 11 non-abelian AC [] (007 1 ) G. E. W. Bauer [1] : 43, 1 (008). [] : 18 ( 1993). [3] T. Bergsten: 4, 331 (007). 9 Kramers : E n,+ = E n, 10 Kramers : E n,+ = E n,. 11 [1] Aaronov-Anandan (AA) AC (13) AA π(1 cos θ α) 10
[4] : 39, 7 (004). [5] : 6, (007). [6] : 4, 873 (007). [7] : 40, 189 (005). [8] : (II) ( 1969). [9] : ( ) ( 1991). [10] D. Grundler, Pys. Rev. Lett. 84, 6074 (000). [11] Y. Sato, T. Kita, S. Gozu and S. Yamada, J. Appl. Pys. 89, 8017 (001). [1] G. Dresselaus: Pys. Rev. 100, 580 (1955). [13] J. B. Miller, D. M. Zumbül, C. M. Marcus, Y. B. Lyanda-Geller, D. Goldaber-Gordon, K. Campman and A. C. Gossard, Pys. Rev. Lett. 90, 76807 (003). [14] S. Datta and B. Das, Appl. Pys. Lett. 56, 665 (1990). [15] Y. Aaronov and A. Caser, Pys. Rev. Lett. 53, 319 (1984). [16] : ( 003). [17] F. E. Meijer, A. F. Morpurgo and T. M. Klapwijk, Pys. Rev. B 66, 33107 (00). [18] M. König et al., Pys. Rev. Lett. 96, 76804 (006). [19] T. Bergsten, T. Kobayasi, Y. Sekine and J. Nitta, Pys. Rev. Lett. 97, 196803 (006). [0] : 4, 1 (007). [1] T. Z. Qian and Z. B. Su, Pys. Rev. Lett. 7, 311 (1994). [] N. Hatano, R. Sirasaki and H. Nakamura, Pys. Rev. A 75, 03107 (007). 11