note4.dvi

10 016 6 0 4 (quantum wire) 4.1 4.1.1.6.1, 4.1(a) V Q N dep ( ) 4.1(b) w σ E z (d) E z (d) = σ [ ( ) ( )] x w/ x+w/ π+arctan arctan πǫǫ 0 d d (4.1) à ƒq [ƒg w ó R w d V( x) QŽŸŒ³ džq x (a) (b) 4.1 (a) (b) 10-1

3 m 500nm 4. (a) InAs(111)B - - MBE [111] InAs (b) GaN-InGaN - (a) (b) (4.1) d η DEG V sg = eηe z (d) V sg (x) 4.1(b) *1 4.1(b) z x 1 4.1(b) MBE T 4.1. 4.(a) InAs(111)B - - MBE [111] InAs 4.(b) GaN InGaN - (Carbon nanotube) 4. 1 4..1 pn pn p n p n *1 10-

(quantum coherence length) * l φ scattering centers ( quantum entanglement) ψ = ψ 1 +ψ = ψ 1 + ψ + ψ 1 ψ cosθ χ ψ 1 ψ 1 χ 1, ψ ψ χ ψ 1 ψ cosθ χ 1 χ χ 1 χ ψ χ (maximally entangled state) ( F) ( (environment) ) χ 1 χ *3 (monochromaticity) E F E = k B T τ π fτ = π Eτ/h = πk B Tτ/h π τ c τ c = h k B T l = Dτ l th l th = hd k B T (4.) * *3 ( ) 10-3

(thermal diffusion length) v F l th = hv F k B T (thermal length) l φ l φ l φ ( ) l φ (characteristic length) (magnetic B l B = h/eb l B l φ (4.3) 4.. Landauer 0 (Kubo formula) [1]Landauer Landauer L L S R ev L 1 1 (particle reservoir) µ L µ R k j(k) L e/l J J = kr k L j(k) = e L v g = e de(k) L dk j(k) L π dk = e h µl (4.4) µ R de = e h (µ L µ R ) = e h V (4.5) G = J G = e h G q R 1 q. (4.6) (quantum conductance) e /h R q k x 1 e/ x 10-4

y ƒq [ƒg E 1 E E 3 ŽŸŒ³ džq ƒq [ƒg ó R w Veff ( x ) E 3 V( y) E E 1 x (a) (b) 4.3 (a) (Quantum Point Contact, QPC, ) (b) QPC E 1,,3 (4.9) V eff (x) E/ k x k = π E = ev J = e E x k = e h V (4.7) (quantum wire, QW) (quantum point contact, QPC) (two-dimensional electron gas, DEG) QPC 4.3(a) DEG 4.1.1 4.3(b) x E = E kx +E ky x y E kx E ky ( ) 4.1(b) y W ϕ n (y) = cos(nπy/w) (n ) sin(nπy/w) (n ) x y ψ(x,y) = ϕ n (y)φ(x) ( Hψ(x,y) = m = ϕ n (y) m x + y ) ϕ n (y)φ(x) ( x + ( nπ W ) ) φ(x) = Eϕ n (y)φ(x) (4.8) (4.8) x ( ) V eff (n,x) = nπ (4.9) m W(x) 10-5

4.3(b) y n V eff (n,x) E tot = E kx (n,x)+v eff (n,x) (4.10) n (conductance channel) (8.1) d f = 1 E kx (n,x) E F k xf QPC (4.9) W(x) 4.4(a) (AFM) 4.4(b) V g QPC G( e ) ) G V g e /h G e /h (4.6) 4.5 QPC n e /h(n ) ( ) (AFM) QPC QPC 4.5(b) n = n = 3 n = 1 1 y n *4. d C ` ± x (e / h) 14 1 10 8 6 4 0-1.0-0.8 ƒq [ƒg dˆ³ (V) -0.6 (a) (b) 4.4 (a)qpc Al- GaAs/GaAs (b) QPC 30 mk *4 3 1 3 1 ǫ 0 1/ ǫ ǫ 0 10-6

T j ƒq [ƒg ƒq [ƒg x A V y (a) (b) 4.5 (a)qpc (SPM) QPC SPM (b) QPC ( ) n ch = 3 ( Topinka et al., Science 89, 33 (000) ) QPC E F E F n D nd 4.6(b) **(c) i j T ij e /h T = 1 {T ij } G G = e h T ij (4.11) i,j 10-7

inlet outlet i j (a) (b) (c) 4.6 (a) (b) (c) (4.11) (Landauer formula for -terminal conductance) 4..3 S ( ) (electron waveguide) (S ) (scattering matrix, S-matrix) 4.7(b) a 1 (k) a (k) b 1 (k) b ( ) b1 (k) = S b (k) ( ) ( )( ) a1 (k) rl t = R a1 (k) a (k) t L r R a (k) A i (k) t L,R r L,R (phase shift) T L,R R L,R (4.1) T L,R = t L,r = 1 R L,R = 1 r L,R (4.13) S T (4.1) QPC G = n e /h n (4.1) a 1 (k) S ( ) A 1 A a 1 a M T S B 1 B b 1 b (a) (b) 4.7 (a)t- M T (b) S- S 10-8

ψ ai (k F ) a i (k) = v Fi ψ ai (k F ) (4.14) (b i ) t t = T (4.14) S ( ) T S 8 S ( ( b1 r (A) L t (A) R b ) = S A ( a1 a ) = t (A) L r (A) R ) (a1 a ), ( b3 b 4 ) ( ) ( a3 r (B) = S B = L a 4 t (B) L t (B) R r (B) R ) (a3 a 4 ) (4.15) a 1 a a 3 a 4 S A S B b 1 b b 3 b 4 S AB = r(a) L +t (A) R r(b) L t (B) L ( I r (A) ( I r (A) R r(b) L ) 1t R r(b) (A) L L ) 1t (A) L b = a 3, a = b 3 (4.16) S S S AB ( t (A) R r (B) R +t(b) L ) 1t L r(a) (B) R R I r (B) ( I r (A) R r(b) L ) 1r (A) R t(b) R. (4.17) (4.17) (1,1) ( ) 1 I r (A) R r(b) (A) L = I +r R r(b) L +(r(a) R r(b) L ) +(r (A) R r(b) L )3 + (4.18) A B S (4.1) a b a 3 b 3 b 1 a 1 S a j b j T S T S (10.18) (4.18) 1 I a 1 r (A) r ( ) S (wire 10-9

connection) S T ( ) ( ) S 4..4 S (unitarity) (10.18) a = Sa (Onsagar reciprocity) S B S(B) = t S( B) (S mn (B) = S nm ( B)) (4.19) Schrödinger Schrödinger [ (i +ea) m ] +V ψ = Eψ (4.0) A A [ ] (i +ea) +V ψ = Eψ m {ψ ( B)} = {ψ(b)} (4.1) ψ(b) ψ ( B) ({ } ) ψ(b) Schrödinger (4.0) Sc{a b}(a S b ) Sc{a(B) b(b)} {ψ(b)}, (4.) i.e., b(b) = S(B)a(B) (4.3) (4.3) b (B) = S (B)a (B). (4.4) exp(±ikr) *5 Sc(b (B) a (B)) {ψ (B)} (4.5) B B Sc{b ( B) a ( B)} {ψ ( B)} = {ψ(b)} (4.6) i.e. a ( B) = S(B)b ( B) (4.7) (4.7) b (B) = S 1 ( B)a (B) (4.8) (4.4) S (B) = S 1 ( B) = S ( B) ( unitarity SS = S S = I) S(B) = t S( B) (4.9) *5 (4.0) Schrödinger iωt ikr 10-10

(ρ xx ) ( ) ρ xx (B) = ρ xx ( B) (4.30) 4..5 Landauer-Büttiker S Landauer-Büttiker p q 4.8 p µ p = ev p p J p p p J p = e h [T q p µ p T p q µ q ] (4.31) q T p q q ev q q J q 1 1 ev 1 J 1 Ž J ev T pq T p q (p q), T pp q pt q p T J = t (J 1,J, ) µ = t (µ 1,µ, ) ( ) J = e h T µ J p p p ev p V q = µ q e, J p = q G pq e h T p q [G qp V p G pq V q ] (4.3) 4.8 LB J q = 0 (4.33) q [G qp G pq ] = 0 (4.34) B Onsager q G qp (B) = G pq ( B) (4.35) 10-11

S-matrix Onsager 4 V 4 = 0 J 1 J = G 1 +G 13 +G 14 G 1 G 13 G 1 G 1 +G 3 +G 4 G 3 V 1 V (4.36) G 31 G 3 G 31 +G 3 +G 34 V 3 J 3 (Casimir) J 1 = J 3, J = J 4 (4.37) J = 0 1 3 4 V ij V i V j ( ) ( )( ) J1 α11 α = 1 V13 (4.38) J α V 4 α 1 α 11 = G q [ T 11 S 1 (T 14 + T 1 )(T 41 + T 1 )] α 1 = G q S 1 (T 1 T 34 T 14 T 3 ) α 1 = G q S 1 (T 1 T 43 T 3 T 41 ) α = G q [ T S 1 (T 1 T 3 )(T 3 + T 1 )] (4.39a) (4.39b) (4.39c) (4.39d) S = T 1 + T 14 + T 3 + T 34 = T 1 + T 41 + T 3 + T 43 (4.40) (4.38) (4.37) V 1 V V 3 B B Landauer-Büttiker 4 (4.35) (4.38) α 11 (B) = α 11 ( B), α (B) = α ( B), α 1 (B) = α 1 ( B) (4.41) 13: 4: LB R 13,4 R 13,4 = V V 4 J 1 = α 1 α 11 α α 1 α 1 (4.4) Onsager R (4.30) R 4,13 = α 1 α 11 α α 1 α 1 (4.43) (4.41) T km T ln T kn T lm R mn,kl = R q, D R D q(α 11 α α 1 α 1 )S (4.44) 10-1

R mn,kl (B) = R kl,mn ( B) (4.45) B B [1] ( 007). [] S. Datta, ElectronTransport in Mesoscopic Systems (Cambridge Univ. Press, 1995). [3] ( 00) F ψ = ( A + B )/ φ = ( 1 + )/ ψ φ Φ Φ = ψ φ = ( A 1 + A + B 1 + B )/ Φ = A + B 1 (F.1) ψ A φ (F.1) (maximally entangled state) ( ) (entanglement entropy) H A H B A = d A i=1 b i η i B = d B j=1 c j ξ j A B = d A,d B i,j=1 b i c j η i ξ j ψ AB = d A,d B i,j c ij η i ξ j (F.) 10-13

ψ AB (Schmidt decomposition) { η i } { ξ j } { u i } { v i } d ψ AB = d k u k v k, k=1 d d k = 1 (d = min(d A,d B )) k=1 (F.3) { u i v i } d k A B ρ A = k d k u k u k, ρ B = k d k v k v k (F.4) ρ A ρ B d S(ρ A ) = S(ρ B ) d klog (d k) k=1 (F.5) S (entanglement entropy) (von Neumann entropy) { φ, φ } { χ, χ } 1 ψ s = 1 ( φ χ φ χ ) (F.6) ρ = 1 ( ) 1 0, S(ρ) = 1 (F.7) 0 1 Ψ = 1 ( ψ 1 χ 1 + ψ χ ) (F.8) χ 1 χ = 0 S(ρ) = 1 χ 1 = χ Ψ = ( ψ 1 + ψ ) χ S(ρ) = 0 10-14