( 2) 3.4 5 (b),(c) [ 5 (a)] [ 5 (b)] [ 5 (c)] (extrinsic) skew scattering side jump [] [2, 3] (intrinsic) 2 Sinova 2 heavy-hole light-hole ( [4, 5, 6] ) Sinova Sinova 3. () 3 3 Ṽ = V (r)+ σ [p V (r)] λ h = V (r)+v (r)l s (5) V (r) ( ) 2.3 (3) : λ = P 2 [ ] 3 (E 0 +Δ 0 ) 2 E 2 0
(a) E B + + + + e (b) (c) E 5: (a) (b) (c) (a) (b), (c) λ = h 2 /(4m 2 0 c2 )(m 0, c ) V (r) = λ 2 r V (r) r λ >0 V (r) V (r) (Rashba Dresselhaus ) [ ] h2 2m Δ+Ṽ ψ = Eψ, (6) dv dr E = h2 k 2 2m, (m Δ 3 ) l s l 2, s 2 l l [2, 3, 7] (V =0) z l e ikz = e ikr cos θ = (2l +)i l j l (kr)p l (cos θ) (7) l=0 (Rayleigh ). Legendre P l (cos θ) l ( z l z =0) eikr cos θ ϕ l z 0 j l (kr) Bessel V (r) l l (6) R l (r) [ d 2 dr 2 + 2 d l(l +) r dr r 2 2m ] h 2 V (r)+k2 R l (r) =0 2
R l (r) r R l (r) eiδ l kr sin(kr lπ/2+δ l) δ l δ l [j l (kr) sin(kr lπ/2)/(kr)] 2 : V δ l <π/2 V <0( ) δ l > 0( ) V > 0( ) δ l < 0( ) (θ, ϕ) l f(θ, ϕ) = (2l +) e2iδ l P l (cos θ). 2ik l=0 V =0 ϕ σ(θ, ϕ) = f(θ, ϕ) 2, σ tot = = 4π k 2 σ(θ, ϕ) sin θdθdϕ (2l + ) sin 2 δ l l=0 l δ l =0, δ l = ±π/2 l 2l s =(l + s) 2 l 2 s 2 (s 2 =4/3) j = l + s l z, s z (5) l s P l (cos θ) l s = 2 (l +s + l s + )+l z s z (l ± = l x ± il y, s ± = s x ± is y l z, s z ) l =0(S ) l l sp l (cos θ) = 2 [ s e iϕ + s + e iϕ ]Pl (cos θ) (8) = i( sin ϕs x + cos ϕs y )Pl (cos θ). (9) (8) l ± (l, l z =0) (l, l z = ±) Pl (cos θ)e±iϕ (Pl Legendre ) j z = l z + s z (s z =/2) (s z = /2) z e iϕ e iϕ n = ẑ =(0, 0, ) n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) ν = n n n n =( sin ϕ, cos ϕ, 0) (9) iν spl (cos θ). 2 l 3
A (7) f(θ, ϕ) = A +2Bν s, (20) A = 2ik [(l + )(e2iδ+ l ) + l(e 2iδ l )]P l (cos θ), B = l=0 l= 2k (e2iδ+ l e 2iδ l )P l (cos θ). δ l ± l j = l ± /2 : 2 V (r)+(/2)v (r)[j(j +) l(l +) 3/4] ( A) V (r) V (r) j = l +/2 j = l /2 S δ 0 + = δ 0 δ 0. n ν P = A + B 2 A B 2 A + B 2 + A B 2 ν = 2Re(AB ) A 2 + B 2 ν (2) Re(AB ) > 0 z x y x y GaAs [8] Engel [] Si-doped GaAs ( n =3 0 6 cm 3 ) V (r) = e2 4πεr e qsr /q s 9nm. Boltzmann a ka : δ l (ka) 2l+ l [7] k F =(3π 2 n) /3 0.096nm k F /q s 3.2 (2) 2 2 (xy ) 3 r, θ 2 z Ṽ = V (r)+v (r)l z s z. (22) s z =/2 l z > 0 l z < 0 s z = /2 l z = m (m =0, ±, ±2, ) ( B m m ) xy x e ikx = e ikr cos θ = i m J m (kr)e imθ, (23) m= 4
(a) 2DEG 2DEG (b) a r a r V 0 V 0 (c) 2 2 ~m / r r 6: (a) ( ) 2 (2DEG) (b) 2 V 0 < 0 (c) A m 2 /r 2 J m (kr) Bessel s z = ±/2 m R m ± (r) ( B) r R ± m (r) 2 πkr eiδ± m cos(kr mπ/2 π/4+δ ± m ) (24) δ ± m [ J m(kr) 2/(πkr) cos(kr mπ/2 π/4)] δ ± m = δ m, S (m =0) δ + 0 = δ 0 δ 0. s z = ±/2 f ± (θ) = A ± B, (25) A = i + 2πk (e2iδ 0 ) + m= i [ ] (e 2iδ+ m ) + (e 2iδ m ) cos mθ, 2πk B = (e 2iδ+ m e 2iδ m ) sin mθ, 2πk m= θ z P z = f + 2 f 2 f + 2 + f 2 = 2Re(AB ) A 2 + B 2 (26) 5
+ sin 2 δ m + δ 2 δ 2 + δ 0 δ δ Pz (θ= π/2) 0 0.6 0 0.2 5 9 k' / k = + V 0 /E 7: A(V 0 < 0, ka =0.5, λ/a 2 =0.0) S (m =0) P (m = ±) D (m = ±2) sin 2 δ ± m. k /k = + V 0 /E ( B ) θ = π/2 P z 3 6 (a) 2 6 (b) 2 (A) (B) V (r) =V 0 θ(a r), V (r) =V 0 [ (r/a) 2 ]θ(a r). a>0, θ(a r) [θ(x) =(x>0), 0 (x <0)] s z = ±/2 m (A) V 0 [θ(a r) ± m(λ/a)δ(r a)], (B) V 0 [ (r/a) 2 ± 2mλ/a 2] θ(a r). (B) (B ) V 0 [ ± 2mλ/a 2] θ(a r) A, B δ m ± ( B) (V 0 < 0) A m sin 2 δ m ± 7 ka =0.5 ( a, 2π/k 0 00nm ), λ/a 2 =0.0 ( 3 [9] 6
(a) E (b) ky k x ky B eff Δk kx kx 8: (a) InGaAs/GaAs (2 ) (b) Sinova k Δk ) 4 V 0 S P (m = ±) D (m = ±2) ( 6 (c))[7] S m 2 /r 2 P D m 7 (26) y (θ = π/2) P z (θ = π/2 D ) P ( λ/a 2 =0.00 0% ) B B 3.3 3. 3.2? Bloch ψ n,k (r) =u n,k (r)e ik r, n k u n,k (r) k 2 [0, 4] 2 4 [] GaAs λ =5.3Å 2 λ/a 2 0 4. InGaAs Rashba α λ V 0 /a 0meV nm, V 0 0meV λ/a 2 0 2. 7
3.3. Sinova 2 ( InGaAs ) Hamiltonian H = p2 2m + ᾱ σ (p ẑ). (27) h ψ k,±(r) = e ik r χ k,±, (28) E k,± = h2 k 2 h2 αk = 2m 2m (k k α) 2 h2 kα 2 2m, (29) χ k,± k ẑ : σ k ẑ = k ẑ σ k ẑ, σ k ẑχ k,± = ±χ k,±. χ k,+ 8 (a) k =(k x,k y ).4 Hamiltonian (27) 2 Zeeman γ hb eff =2α(k ẑ) k E x k : h k = i[k,h ( e)e x x]= ee x ˆx (30) hδk =( e)e x ˆxΔt ( ) B eff (k +Δk) B eff (k) : h ds dt = s B(t), B(t) = γ hb eff (t) [ 8 (b)] z ( k ) [0] s z = h d B 2 B 2 dt. B ΔB eff (t) B eff (k) ( k ) y 5 d B dt s z = =2α(ẑ k) k k = 2α( e)e x k y h k h 2α( e)e x k y 2(2αk) 2 h k = ( e)k y 4αk 3 E x, dk j s,y = (2π) 2 hsz(k) hk y m = ( e) h2 6παm (k F,+ k F, )E x 5 y v y = hky [y, H ( e)exx] = + ασx 2 i h m 8
(a) E k (b) HH LH ky ky Δk k z kx kz kx 9: (a) GaAs 3 Γ heavy-hole (HH) (λ = ±3/2) heavy-hole (LH) (λ = ±/2) λ =3/2, /2 S k x -k y (b) ( ) k Δk λ S k F,± 2 k F,+ k F, =2k α =2mα/ h 2 σ sh = j s,y = e E x 8π universal Sinova jspin z = h 2 {s z, v} [0] [] Chalaev [2] vertex σ sh! σ sh [3] 3.3.2 GaAs (3 ).2 (Γ ) j =3/2 j =/2 2 j =3/2 Γ j z = ±3/2 heavy-hole j z = ±/2 light-hole 2 heavy-hole light-hole Luttinger Hamiltonian Spherical [( H = h2 γ + 5 ) ] 2m 2 γ 2 k 2 2γ 2 (k S) 2, (3) S 3/2 6 γ, γ 2 Hamiltonian 6 p (l =) s =/2 k 9
(3) S k λ = S k/k (helicity ) λ = ±3/2 (heavy-hole ) λ = ±/2 E H (k) = h2 2m (γ 2γ 2 )k 2 E L (k) = h2 2m (γ +2γ 2 )k 2 h2 2m H k 2 h2 2m L k 2 (light-hole ) 9 (a) S k 8 (a) E x h k = ee x ˆx 9 (b) heavy-hole λ =3/2 λ heavy-hole light-hole k [4] Hamiltonian (3) U(k) =e iθsy e iϕsz S k S z (θ, ϕ k ) 7 ee x x H = U(k)(H ee x x)u (k) ( γ + 5 2 γ 2 2γ 2 Sz 2 = h2 k 2 2m ) [ ee x x + iu(k) ] U (k). k x k S ( ) S z 8 y k z ẏ = hk y λ ē m eff h k 3 E x. (32) (m eff m H m L ) λ =3/2, /2 λ = 3/2, /2 σ sh = j s,y = e E x 2π (3kH F kl F ), k H,L F heavy-hole, light-hole (3 Sinova 2 σ sh ) Wunderlich [5] 7 S k z ϕ y θ z 8 (32) i hẏ =[y, H] S z (32) r k [4, 5, 6, 4] 0
Sinova vertex [6] [4, 5, 6] A 3. (5) l (l ) l P l (cos θ)χ ± (χ ± σ z ) j = l + s (j, j z ) P l (cos θ)χ ± j = l ± /2 2 P l (cos θ)χ + Clebsch-Gordan 4π ( P l (cos θ)χ + = l +ψl+/2,/2 ) lψ 2l + l /2,/2, ψ j=l±/2,jz=/2 z (5) V (r)+ [ 2 V (r) j(j +) l(l +) 3 ] Ṽ l,± (r). 4 ψ l±/2,/2 Ṽl,±(r) δ ± l P l (cos θ)χ + f l,+ f l,+ = ( l e 2iδ+ l 4π + ψ 2ik l+/2,/2 ) l e2iδ l ψ 2ik l /2,/2. (33) Clebsch-Gordan ψ l±/2,/2 (l, l z =0;s z =/2) (l, l z =;s z = /2) ψ l+/2,/2 = ψ l /2,/2 = (33) [ l +Pl (cos θ)χ + ] Pl (cos θ)e iϕ χ, 4π l + [ lp l (cos θ)χ + ] Pl (cos θ)e iϕ χ. 4π l f l,+ = 2ik [(l + )(e2iδ+ l ) + l(e 2iδ l )]P l (cos θ)χ + 2ik (e2iδ+ l e 2iδ l )P l (cos θ)e iϕ χ (34) P l (cos θ)χ f l, (34) f l,± = 2ik [(l + )(e2iδ+ l ) + l(e 2iδ l )]P l (cos θ)χ ± 2ik (e2iδ+ l e 2iδ l )P l (cos θ)e iϕ χ. (35) l f l,± (20)
B 2 xy x (23) ( s z = ±/2 ± ) r ψ e ikx + f(θ) ei(kr+π/4) r 9 (?) σ(θ) = f(θ) 2 3 (23) m S S m =+i 2πkf m = e 2iδm [f m f(θ) : f(θ) = m f me imθ ] ψ f(θ) = m= m= 2 πkr im e iδm cos(kr mπ/2 π/4+δ m )e imθ, e 2iδm i 2πk eimθ. (36) σ tot = 2π 0 = 2 πk σ tot = σ(θ)dθ m= sin 2 δ m, 2 πk Imf(0) 2 (3 ) Aharonov-Bohm [7] (22) s z = ±/2 δ m ± m R± m (r) [ d 2 dr 2 + ] d r dr m2 r 2 2m h 2 Ṽ m ± (r)+k2 R m ± (r) =0, Ṽ ± m (r) =V (r) ± mv (r)/2. E = h 2 k 2 /(2m ). R m ± = R m, δ m ± = δ m (36) (25) A δ m ± = δ m m 0 r >a Ṽ =0 R m ± (r) = C J m (kr)+c 2 Y m (kr) (37) 2 πkr [C cos(kr mπ/2 π/4) + C 2 sin(kr mπ/2 π/4)], (38) 9 2 π/4 ( σ(θ) ) 2
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