Andreev Josephson Night Club

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1 Andreev Josephson Night Club

2 Cooper Hamiltonian BCS Hamiltonian BCS Bogoliubov-de Gennes Andreev Andreev Josephson I - d d Andreev Josephson

3 H K Onnes Meissener-Ochenfeld Josephson Bardeen- Cooper-Schrieffer (BCS 2 Cooper BCS 1 BCS Bogoliubov-de Gennes Andreev 4 Andreev 2 Fermi Gas Cooper 3 Hamiltonian BSC BSC Hamiltonian BSC Hamiltonian Gap 4 NS Andreev Andreev Josephson 4 s 5 Andreev Josephson 6 spin-triplet Josephson 7 Andreev Josephson (N Andreev 1 s 8 1

4 Fermion Boson Fermion Boson Al Cu Pb ħ/2 e Fermion Hamiltonian H = ħ2 2 2m (2.1 Hamiltonian Hamiltonian 1 ψ k (r = e ik r, V vol (2.2 ϵ k = ħ2 k 2 2m, (2.3 V vol 1 cm 3 Avogadro N A N A Fermion Fermi ϵ k 1 ħ/2( ħ/2( 2 N A µ F = ħ 2 kf 2 /(2m Fermi Fermi Fermi v F = ħk F /m k F 10 8 cm/s cm/s 1 7 Fermi Fermi 10 4 Kelvin 300 Kelvin 1 3/5µ F Fermi 1 Fermi 1 2

5 Hamiltonian ( ( ϵ + δϵ 0 H =H 0 + V, H 0 =, V = 0 ϵ 0 t t 0. (2.4 H 0 2 Hamiltonian V 2 1 CDW SDW 2.1 Fe Ising Hamiltonian N H = J s z i s z i+1 h s z i, (2.5 i i J > 0 h 0 h N z = 2 s z i ( 1 ( (a i = 1 i = 2 J i = 2 i = 3 i = 2 i = 3 J Hamiltonian h 2 h h (b (b Hamiltonian s z i s s s s z i = s + (s z i s (2.6 3

6 Superconducting Elements H Li Na K Be Mg Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr He B C N O F Ne Al Si P S Cl Ar Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Fr Ba Ra La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Ac Th Pa U Np Pu Al Superconducting Si Superconducting under high pressure or thin film Rb Mettalic but not yet found to be superconducting F Nonmetallic elements Ni Elements with magnetic order He Superfluid 2.1: N.W.Ashcroft and N.D.Mermin, Solid State Physics Sec.34 4

7 i = (a (b (c 2.2: 1 Ising (a (b (c 5

8 N H = J [ s + (s z i s ] [ s + ( s z i+1 s ] N h s z i, (2.7 J i=1 N i=1 = (zj s + h [ ( s s z i+1 + s z i s 2 ] h i=1 N s z i, (2.8 i=1 N s z i + JN s 2, (2.9 i=1 =H MF (2.10 Hamiltonian H MF Hamiltonian (2.5 H MF i s H MF Z =Tre H MF/T, (2.11 =e JN s N i=1 [ e (zj s +h/t + e (zj s +h/t ], (2.12 =e JN s [2 cosh {(zj s + h/t }] N. (2.13 T Boltzmann k B 1 s h = 0 s s = Tr N i=1 sz i e H MF/T, (2.14 NZ = T ln Z N h, (2.15 h 0 = tanh [(zj s /T ]. (2.16 s s s self-consistent (2.16 s 2.3(a (b (2.16 s = 0 2.2(a s 0 T c zj T < T c (2.16 s 2.3(c H MF T = T c 2 2 T = T C 2 ( s 6

9 Low temperatures (a <s> High temperatures (b <s> tanh( zj<s>/t tanh( zj<s>/t < s > 1.0 < s > < S > / < S > (c T / T C 2.3: (a T < T c (2.16 (b T > T c = zj (2.16 (c (2.16 s 7

10 T < T c 2.2(b 2.2(c (b (c Hamiltonian (2.5 T = 0 F = E T S (2.17 T = 0 F E 1 self-consistent 2.3 Cooper (a e e 1 Coulomb (a (10 8 cm / s (b (10 4 cm/s

11 +e +e +e +e 1 +e +e (a 2 e electron 1 +e +e +e +e (b e 2 2.4: 2 9

12 Coulomb 2 Landau Fermi 2 2 Cooper Cooper 2 Fermion Boson Boson Bose Cooper Cooper Cooper Cooper Cooper Bose ψ Hamiltonian H ψe iα H α Schrödinger N A Cooper 1 N A Cooper Cooper

13 Cooper Meissener Josephson 11

14 図 2.5: 通常金属中の電子の位相の様子 (上 と超伝導体中の Cooper ペア の位相の様子 (下 12

15 3 Hamiltonian BCS Hamiltonian BSC Hamiltonian Bogoliubov-de Gennes 2 Hamiltonian Hamiltonian Schrödinger Schrödinger Hamiltonian 2 H 1 = dr Ψ σ(r ( ħ2 2 2m µ F Ψ σ (r, (3.1 σ=, Ψ σ(r r σ Ψ σ (r r spin σ Fermion { } Ψ σ (r, Ψ σ (r =Ψ σ(rψ σ + (r + Ψ σ (r Ψ σ (r = δ σ,σ δ(r r, (3.2 {Ψ σ (r, Ψ σ (r } + =0 (3.3 2 Hamiltonian Fermi Ψ σ (r r σ ( ħ 2 2 /(2m µ F Ψ σ(r Hamiltonian r σ Hamiltonian 1 Ψ σ(r = c eik r k,σ (3.4 V vol k Fourier c k,σ k σ 13

16 (3.1 Fourier H 1 = dr ( c eik r k,σ ħ2 2 2m µ F c k,σe ik r, (3.5 σ=, k,k = dr c k,σ c k,σξ k e i(k k r, (3.6 k,k σ=, = k,σ ξ k c k,σ c k,σ, (3.7 ξ k = ħ 2 k 2 /2m µ F Fermi ( ħ2 2 2m µ F e ik r =ξ k e ik r, (3.8 1 dr e i(k k r =δ k,k (3.9 V vol 1 Hamiltonian 2 (3.7 Hamiltonian k σ Hamiltonian k, σ Hamiltonian c k, c k, Hamiltonian (3.1 Hamiltonian Fourier H 2 = dr Ψ σ(r ( ħ2 D 2 2m + V (r µ F Ψ σ (r, (3.10 σ=, D = i e A. (3.11 ħc V (r Schrödinger ( ħ2 D 2 2m + V (r µ F f ν,σ (r =E ν f ν,σ (r, (3.12 dr fν,σ(rf ν,σ (r =δ ν,ν δ σ,σ, (3.13 f ν,σ (rfν,σ (r =δ(r r δ σ,σ, (3.14 ν E ν f ν,σ (r unitary (3.10 H 2 = = σ σ=, Ψ σ (r = ν c ν,σ f ν,σ (r (3.15 dr c ν,σc ν,σfν,σ(r ( ħ2 D 2 2m + V (r µ F f ν,σ(r, (3.16 ν,ν E ν c ν,σc ν,σ, (3.17 ν 14

17 1 2 Hamiltonian Schrödinger 1 2 Hamiltonian BCS Hamiltonian Hamiltonian H = dr Ψ σ(r ( ħ2 D 2 2m + V (r µ F σ σ,σ Ψ σ (r dr dr Ψ σ(rψ σ (r g(r r Ψ σ (r Ψ σ (r, ( ( g(r r 2 r r 2 2 r r σ σ g(r r 2 4 spin-singlet (Cooper Cooper Hamiltonian H = dr Ψ σ(rh 0 (rψ σ (r σ + 1 dr dr Ψ 2 σ(rψ σ(r g(r r Ψ σ (r Ψ σ (r, (3.19 σ h 0 (r = ħ2 D 2 2m + V (r µ F, (3.20 σ σ = σ = σ = σ = (r, r = g(r r Ψ (r Ψ (r, (3.21 (r, r = g(r r Ψ (rψ (r, (3.22 Ψ (rψ (r Cooper Cooper spin-singlet (r, r = (r, r ] Ψ (rψ (r [Ψ g + (rψ (r, (3.23 g ] Ψ (rψ (r [Ψ g + (rψ (r +, (3.24 g Ψ (r Ψ (r [ g + Ψ (r Ψ (r ], (3.25 g Ψ (r Ψ (r g + [ Ψ (r Ψ (r + g ], (

18 Hamiltonian (3.19 [ ] 1 H BCS = dr Ψ σ(rh 0 (rψ σ (r dr dr (r, r 2 g(r r σ ] dr dr [ (r, r Ψ (rψ (r + (r, r Ψ (r Ψ (r, (3.27 BCS Hamiltonian ( [ ] K = dr Ψ (rh 0(rΨ (r + Ψ (rh 0(rΨ (r, (3.28 [ } ] = dr Ψ (rh 0(rΨ (r + {h 0(rΨ (r Ψ (r, (3.29 { }] = dr dr [Ψ (rδ(r r h 0 (r Ψ (r + Ψ (r δ(r r h 0(r Ψ (r, (3.30 dr f 1 (rd 2 f 2 (r = dr { } D 2 f 1 (r f 2 (r (3.31 ( ( P = = dr = dr dr ] dr [ (r, r Ψ (rψ (r + (r, r Ψ (r Ψ (r, (3.32 ] dr [Ψ (r (r, r Ψ (r + Ψ (r (r, r Ψ (r, (3.33 ] dr [Ψ (r (r, r Ψ (r + Ψ (r (r, r Ψ (r. ( r r (3.27 ] H BCS = dr dr [Ψ [ (r, Ψ δ(r r h 0 (r (r, r (r (r, r δ(r r h 0(r dr dr (r, r 2 g(r r ] [ Ψ (r Ψ (r ] (3.35 [ dr δ(r r h 0 (r (r, r (r, r δ(r r h 0(r ] [ u λ (r v λ (r ] = E λ [ u λ (r v λ (r ], (3.36 E λ (u λ, v λ T (3.36 [ ] [ ] [ ] dr δ(r r h 0 (r (r, r vλ (r vλ = E (r (r, r δ(r r h 0(r u λ λ (r u λ (r, (

19 ( vλ, u λ T E λ [ ] [ ] dr [u λ(r, vλ(r] u λ (r vλ = dr [ v λ (r, u λ (r] (r v λ (r u λ (r = δ λ,λ, (3.38 [ ] u λ (r dr [ v λ (r, u λ (r] = 0, (3.39 v λ (r [ ] [ ] u λ (r [u λ(r, vλ(r vλ ] + (r v λ λ (r u λ (r [ v λ (r, u λ (r ] = δ(r r ˆσ 0, (3.40 ( 1 0 ˆσ 0 =. ( (3.27 ( Ψ (r Ψ (r = [( u λ (r v λ λ (r α λ, + ( v λ (r u λ (r α λ, ] (3.42 Bogoliubov α λ,σ Bogoliubov Cooper BCS Hamiltonian H BCS = dr dr [ α λ, (u λ(r, vλ(r + α λ, ( v λ (r, u λ (r] λ,λ [ ] [( ( ] δ(r r h 0 (r (r, r u λ (r vλ α (r, r δ(r r h 0(r v λ (r λ, + (r α u λ (r λ, dr dr (r, r 2 g(r r, (3.43 = dr [ ] α λ, (u λ(r, vλ(r + α λ, ( v λ (r, u λ (r λ,λ [ ( ( ] u λ (r vλ E λ α λ, E (r λ v λ (r u λ (r α λ, dr dr (r, r 2 g(r r, (3.44 = E λ α λ,σ α λ,σ Eλ dr dr (r, r 2 g(r r, (3.45 λ σ λ Hamiltonian (3.10 Schrödinger BCS Hamiltonian (3.27 (3.36 (3.36 Bogoliubov-de Gennes (BdG Green [ ] dr δ(r r h 0 (r (r, r Ĝ(r, r = δ(r r ˆσ (r, r δ(r r h 0(r 0, (

20 Gor kov Gintzburg-Landau Eilenberger Usadel 3.3 BCS BCS Hamiltonian Hamiltonian BCS Hamiltonian ( Hamiltonian (3.18 N N BCS Hamiltonian ( BCS Hamiltonian (3.27 N N N + 2 N 2 BCS Hamiltonian (3.27 ˆN ˆφ [ ˆN, ˆφ] = i 2 coherent BCS Hamiltonian BCS Hamiltonian (3.27 Hamiltonian Bogoliubov (3.45 Bogoliubov Hamiltonian BdG ( u(r 2 v(r D = iea/cħ D 3.4 BdG (3.36 Cooper 2 R = (r r /2 r r = r r Fourier (r, r = (R, r r = 1 k (Re ik rr. (3.47 V vol Fourier R k Cooper spin-singlet k = k s d s k = e iφ (3.48 k 18

21 (3.47 (r r = 1 e iφ e ik rr = e iφ δ(r r, (3.49 V vol k 2 s d 5 d x 2 y 2 d xy k = e iφ ( k x 2 k y, 2 d x 2 y2 symmetry, (3.50 k = e iφ (2 k x ky, d xy symmetry, (3.51 k x =k x /k F, ky = k y /k F, ( Fermi 3.1 s d Fermi 1 4 (b k x k y (c k x k y 45 CuO 2 a 3.5 s BdG (3.36 (3.49 [ ] [ ] [ ] h 0 (r e iφ u(r u(r = E, (3.53 e iφ h 0(r v(r v(r φ V (r = 0 A = 0 [ ] [ ] u(r u k 1 = e ik r (3.54 v(r Vvol v k [ ] [ ξ k e iφ e iφ ξ k u k v k ] = E [ u k v k ], (3.55 E = ± E k, E k = ξk 2 + 2, ( u k = 1 + ξ k 1, v k = 1 ξ k e iφ, ( E k 2 E k Bogoliubov ( [( Ψ (r Ψ (r = 1 Vvol k u k v k e ik r α k, + ( v k u k e ik r α k,, ] (

22 ky (a s k x ky (b d 2 2 x y + + k x // I ky + (c d xy k x // I + 3.1: 2 Fermi (a s (b d x2 y 2 (c d xy 20

23 Ising self-consistent (2.16 (3.21 s e iφ = g 0 Ψ (rψ (r, (3.59 (3.21 g(r r = g 0 δ(r r (3.59 (3.58 e iφ 1 =g 0 (u k α k, v V kα k, (v k α k, + u k α k, e i(k k r, (3.60 vol k,k 1 =g 0 e i(k k r u k vk V α k, α k, + u ku k α k, α k, vol k,k v kv k α k, α k, v ku k α k, α k,, ( α k,σα k,σ =δ k,k δ σ,σ f(e k, (3.62 f(e k = 1 ( ( Ek 1 tanh, Fermi distribution function ( T αk,σ α k,σ =0, (3.64 e iφ 1 =g 0 u k vk (1 2f(E k, (3.65 V vol k 1 ( 1 Ek 1 =g 0 tanh, (3.66 V vol 2E k 2T k u k v k = eiφ 2E k (3.67 (3.66 self-consistent g 0 > 0 T = 0 tanh(e k /2T = =g 0, (3.68 V vol 2E k k ħωd =g 0 dξ 0 g 0 N(0 ħωd 0 =g 0 N(0 ln N(ξ ξ2 + 2, (3.69 dξ ξ2 + 2, (3.70 ħω D + (ħωd 2 + 1, (3.71 g 0 N(0 ln(2ħω D /, (

24 N(ξ = 1 δ(ξ ξ k (3.73 V vol Fermi N(0 phonon Debye ħω D cut-off 1 x dx x2 + c = ln + x2 + c 2 ( k T = 0 (T = 0 0 = 2ħω D e 1/g 0N(0 (3.75 g 0 N(0 1 g 0 N(0 T c = 0 (3.66 ħωd ( dξ ξ 1 =g 0 N(0 0 ξ tanh, (3.76 2T c ( ( ħωd ħωd =g 0 N(0 ln tanh g 0 N(0 dξ ln(ξ cosh 2 (ξ, (3.77 2T c 2T c 0 [ ( ] ħωd g 0 N(0 ln + ln(4γ 0 /π, (3.78 2T c ( 2ħωD γ 0 =g 0 N(0 ln. (3.79 T c π 2 1 ħω D T c dx ln(x cosh 2 (x = ln(4γ 0 /π, (3.80 ln γ Eular constant, (3.81 T c = 2ħω Dγ 0 e 1/g0N(0 = γ 0 π π 0 ( = 3.5T c BCS Gap

25 0 ( T / T C 3.2: 23

26 4 Bogoliubov-de Gennes 4.1 Andreev NS Andreev Andreev 2 s NS ( 4.1(a BdG [ ] [ ] [ ] h 0 (r Θ(xe iφ u(r u(r = E, (4.1 Θ(xe iφ h 0 (r v(r v(r h 0 (r = ħ 2 2m µ F + V 0 δ(x, (4.2 (3.47 Cooper 2 R e iφ Θ(x y W Bogoliubov-de Gennes NS x = 0 δ- = 0 BdG 2 Schrödinger potential h 0 (ru(r =Eu(r, (4.3 h 0 (rv(r =Ev(r, (4.4 ikx eipy e (4.5 W ( 1 ikx eipy E = ξ k e, electron (4.6 0 W ( 0 ikx eipy E = ξ k, e, hole, (4.7 1 W ξ k = ħ2 k 2 2m + ϵ p µ F, (4.8 ϵ p = ħ2 p 2 2m, (4.9 p = 2πn, n = 0, ±1, ±2,..., (4.10 W 24

27 V 0 δ(x Normal conductor Superconductor 0 e iϕ x=0 (a x E k E k k k p 0 k µ F +ε y (b 4.1: 2 s NS (a BdG (b 25

28 4.1(b y 1 p k = 0 µ F + ϵ p k Fermi k = k p (k 2 p + p 2 = k 2 F = 2mµ F /ħ 2 ( BdG ( u(r v(r = ( u k v k e iφ ikx eipy e (4.11 W ( ξ k e iφ e iφ ξ k ( u k v k e iφ = E ( u k v k e iφ (4.12 E = ± E k, E k = ξk 2 + 2, ( u k = 1 + ξ k, ( E k 1 v k = 1 ξ k, ( E k 4.1(b 2 Fermi k = ±k p 2 NS k 4.2(a ( 4.2(a r ee ( 4.2(a r he Andreev Andreev x v e k = 1 ħ kξ k = ħk m v h k = 1 ħ k( ξ k = ħk m ħp ˆx + ŷ, m electron, (4.16 ħp ˆx ŷ, m hole, (4.17 Andreev x x Andreev [( ( ( ] Ψ N 1 (r = e ik+x 1 + e ik+x r ee 0 + e ik x r he e ipy, ( W k ± =k p 1 ± E µ p, (4.19 µ p =µ F ϵ p (

29 x µ p µ F Andreev 4.2(a Andreev 4.2(b [( ( ] Ψ S u (r = e iq+x t ee ve iφ + e iq x t he e ipy, (4.21 ve iφ u W ( 1 u(v = 1 + ( Ω, Ω = E2 2 E 2, (4.22 E E = E q ξ q = ± Ω, (4.23 q ± =k p 1 ± Ω µ p, ( E > (4.21 E < Ω = i 2 E 2 E < µ p < µ F (4.24 q ± k p ± i k p 2 E 2, (4.25 2µ p (4.21 x (4.23 (4.24 E = E q (4.23 (4.14 (4.15 (4.21 E = 0 Andreev 1 Cooper 2 Cooper 1 E Pippard ξ 0 = ħv F /π (4.25 E = 0 Imq ± = k p = 2µ p ħ 2 k p /m (4.26 k p Fermi k F π NS Cooper BdG Coopre 27

30 Andreev Josephson 4.2(c Cooper (4.18 (4.21 Normal conductor E k Superconductor E k r ee r he t ee k t he 0 k p k (a 0 (b e iϕ e i ϕ (c e i ϕ e i ϕ : BdG (a Andreev (b-c (b (c V 0 δ(x 2 r ee Andreev r he E µ F k ± q ± k p 1 x = 0 Ψ S (0, y = Ψ N (0, y, (

31 ( ( r ee r he = ( u v e iφ v e iφ u ( t ee t he, ( δ- BdG (4.1 γ < xγ γ 0 γ lim dx [ ħ2 2 ] γ 0 γ 2m u(r µ F u(r + V 0 δ(xu(r + e iφ v(r = lim γ [ ħ 2 2 ] lim dx γ 0 2m v(r + µ F v(r V 0 δ(xv(r + e iφ u(r = lim γ γ γ 0 γ γ γ 0 γ dx Eu(r, (4.29 dxev(r. ( δ- ( ħ2 d 2m dx u(r d x=0+ dx u(r + V 0 u(0, y =0, (4.31 x=0 ( ħ2 d 2m dx v(r d x=0+ dx v(r + V 0 v(0, y =0, (4.32 ħ2 2m x=0 ( d dx ΨS (r d x=0 dx ΨN (r + V 0 Ψ S (0, y = 0, (4.33 x=0 ( k 0 kˆσ 3 ( r ee r he = ( u ve iφ ve iφ u ( kˆσ3 + 2iz 0 ( t ee t he, (4.34 k = k p /k F z 0 = mv 0 /ħ 2 k F ˆσ 3 Pauli 3 (4.28 (4.34 Andreev r ee 2r n Ω = (2 t n 2 Ω + t n 2 E, (4.35 r he t n 2 e iφ = (2 t n 2 Ω + t n 2 E, (4.36 r eh t n 2 e iφ = (2 t n 2 Ω + t n 2 E, (4.37 t n = k, k + iz 0 r n = iz 0. k + iz 0 (4.38 t n r n Andreev r eh 4.2(b (c e iφ e iφ Andreev Andreev E < r ee 2 + r he 2 = 1, (

32 z 0 1 r ee 2 1 Andreev r he 0 Cooper (z 0 = 0 r ee = 0 Andreev 1 r he =e i arctan( 2 E 2 /E e iφ, (4.40 r eh =e i arctan( 2 E 2 /E e iφ, (4.41 NS G NS = di dv = 2e2 ( 1 r ee 2 + r he 2 ev h p ev =E, (4.42 Blonder-Tinkham-Klapwijk Takane-Ebisawa p 4.3 G N 4 G NS / G N 2 Z=0 Z=1 Z= ev / 0 4.3: NS z z 0 = 0 E < 2 (4.39 (4.40 G NS = 2e2 h p 2 r he 2 = 2e2 h 2N c (

33 G N = (2e 2 /hn c 2 Andreev 1 1 Cooper 2 E > Andreev G NS /G N 1 (z 0 = 5 sub gap E < Andreev z 0 STM STM z 0 STS 4.2 Andreev NS incident electron(k p, p, (4.44 r ee : ( k p, p, (4.45 r he : (k p, p, (4.46 t ee : (k p, p, (4.47 t he : ( k p, p, (4.48 y (4.16 ( Andreev Fermi Cooper 4.4 X NS Josephson Andreev vk e = 1 [ ] ħk ħ ħp ke k = ˆx + m m ŷ /E k, (4.49 [ ] ħk vk h ħp = ˆx + m m ŷ /E k, (4.50 ( NS NS 31

34 Normal metal Superconductor h e e h 2 e h e 1 4.4: NS 32

35 Andreev Lorentz F = ma = ev B, (4.51 Lorentz (4.51 m e Lorentz NS 1 NS 2 2 NS Aharonov-Bohm Cooper Cooper BdG 1 BdG Cooper BdG E < Cooper Cooper BdG Cooper Cooper Cooper (3.21 Cooper Cooper Cooper BdG Cooper Cooper BdG 4.3 Josephson Josephson Cooper Josephson Josephson Andreev Josephson Josephson SNS Bogoliubov

36 s Josephson Furusaki-Tsukada J = e T ħ p ω n [ r he r eh], (4.52 Ω ħ = k B = 1 k B Boltzmann Josephson Superconductor E k Normal E k Superconductor E k k x k k k ϕ L ϕ R r he r eh 4.5: Josephson Josephson 2 Andreev 2 Andreev r he r eh r he Andreev Cooper 4.5 r eh Andreev Cooper Josephson Josephson E iω n = i(2n + 1πT n T 2 Josephson Josephson Andreev Landauer (4.52 ( δ- 2 Josephson BdG Andreev ( s (

37 Superconductor Superconductor α β A B ϕ L C D ϕ R 4.6: SIS 35

38 BdG [ h 0 (r (x (x h 0 (r ] [ u(r v(r ] = E [ u(r v(r ], (4.53 h 0 (r = ħ 2 2m µ F + V 0 δ(x, (4.54 e iφ L x < 0, (x = (4.55 e iφ R x > 0, α β Ψ L (r =ˆΦ L [( ˆΦ L = ( u v e iφ L/2 0 e ikpx α + 0 e iφ L/2 ( v u e ikpx β + ( u v e ikpx A + ( v u e ikpx B ] e ipy W, (4.56, (4.57 C D Ψ R (r =ˆΦ R [( ˆΦ R = ( u v e iφ R/2 0 e ik px C + 0 e iφ R/2 ( v u e ik px D δ- ] e ipy W, (4.58, (4.59 Ψ L (0, y = Ψ R (0, y,, (4.60 ( ħ2 d 2m dx ΨR (r d x=0 dx ΨL (r + V 0 Ψ R (0, y = 0, (4.61 x=0 (4.60 [( ( ] ( α A C ˆΦÛ + =Û0, (4.62 β B D ( ( ] α ˆΦÛ [ kˆσ 3 kˆσ A ] 3 = [Û ( C kˆσ3 + 2iz 0 Û, (4.63 β B D ( e ˆΦ i(φ L φ R /2 0 =, ( e i(φ L φ R /2 ( u v Û =, (4.65 v u 36

39 C D ( ( A = 1Ξ0 iz 0 ( k iz 0 (u 2 v 2 2 k 2 uv ( 1 cos φ + i sin φ(u 2 v 2 B k 2 uv ( 1 cos φ i sin φ(u 2 v 2 iz 0 ( k + iz 0 (u 2 v 2 2 ( α, (4.66 β ( ( r ee r eh α =, (4.67 r he r hh β Ξ 0 =z 2 0(u 2 v k 2 ( 1 4u 2 v 2 cos 2 (φ/2, (4.68 φ =φ L φ R, Josephson J = e ħ sin φ p (4.69 uv = i, (4.70 2ω n ω u 2 v 2 2 = n + 2, (4.71 ω n T t n 2 2 ω n ωn 2 + ( 2 1 t n 2 sin 2. (4.72 (φ/2 T 1 ω 2 ω n n + y 2 = 1 ( y 2y tanh 2T (4.73 ω n J = e ħ sin φ t n 2 tanh 1 t n 2 sin 2 (φ/2. (4.74 p 2 1 t n 2 sin 2 (φ/2 2T (z 0 1 (R N R 1 N = G N = 2e2 2πħ t n 2, (4.75 p J = π ( ( 0 tanh sin φ, (4.76 2eR N 0 2T 0 (4.76 Ambegaokar- Baratoff Josephson 3.2 Josephson 4.7 Josephson T = T c T = 0 (z 0 = 0 constriction (R N R 1 N = G N = 2e2 N c 2πħ 37 (4.77

40 J C [ 0 / 2eR N ] T / T C 4.7: SIS Josephson J = π 0 er N ( ( cos(φ/2 sin(φ/2 tanh (4.78 2T 0 Kulik-O melyanchuck T = 0 Josephson sin(φ/2 φ = ±π sin(φ 4.8 Josephson T/T c = sin(φ/2 φ = ±π sin φ Josephson 38

41 J [ 0 / 2eR N ] T / T c = : S-Constriction-S Josephson 39

42 5 I - d d Cooper 2 2 BCS Fermi Fermi ( ZES ( ZES ZES Fermi 5.1 d Andreev 3.1 k = ( k x 2 k y 2 2 k x ky s d x2 y 2 d xy k x = k x k F k y = k y k F Fermi x y k F Fermi s d Fermi 1 4 (b k x k y (c k x k y 45 CuO 2 a x a (b a 45 (c (5.1 d NS 5.1 NS α a NS (a α = 0 (b α = π/4 3 Andreev s 5.1 (t ee (t he 40

43 Normal conductor E k Superconductor E k r ee r he t ee k t he k p k k + + (a α = 0 (b α = π / 4 5.1: NS α = 0 d (a α = π/4 d (b 41

44 Fermi (k p p ( k p, p ( x 4 ± (±kp,p (5.2 2 E [( ( ] Ψ S u + (r = e ikpx t ee χ v + e ikpx t he e ipy, (5.3 χ +v + u W ( 1 u ± (v ± = 1 + ( Ω ±, (5.4 2 E χ ± =e i(φ+ϕ ±, (5.5 e iϕ± sgn( ±, (5.6 Ω ± = E 2 ± 2, (5.7 (5.5 φ [( ( ( ] Ψ N 1 (r = e ikpx 1 + e ikpx r ee 0 + e ikpx r he e ipy ( W 3 v 0 δ(x 2 r ee Andreev r he 4 α = 0 ± = 0 ( k 2 p 2, (5.9 r ee 2r n Ω + = (2 t n 2 Ω + + t n 2 E, (5.10 r he t n 2 + e iφ = (2 t n 2 Ω + + t n 2 E, (5.11 s (4.35 (4.36 α = π/4 ± = ± 2 k p, (5.12 r ee 2r n E = (2 t n 2 E + t n 2, (5.13 Ω + r he t n 2 + e iφ = (2 t n 2 E + t n 2, (5.14 Ω + E = 0 z 0 r ee r he BTK NS 5.2 α = 0 42

45 5 4 (di / dv / G N 3 2 = 0 1 / ev / 0 5.2: -d α a 43

46 α = π/ (A YBCO a 0 90 α = 0 45 α = π/4 STS 5.3(B (A (B 5.3: (A Y (B YBCO STS (110 ZES α = 0 ( ZES ZES Pippard ξ 0 44

47 D(E, x z2 0 2 e x/ξ 0 z 4 0 E2 + 2 (5.15 Lorentz Fermi ZES BdG (k p, p Fermi (E = 0 δ- N 1st order S iχ + * N 3rd order S iχ + * t N * t N t N * r N r N r N * iχ * + iχ * + N 2nd order S iχ * + r N * t N iχ iχ t N * r N iχ + * r N * t N iχ 5.4: Andreev r N 2 1 NS NS t N = k, (5.16 k + iz 0 r N = iz 0 /( k x + iz 0, (5.17 t N r N Andreev t N Pippard + Andreev t N Andreev χ + = e i(φ+ϕ + 45

48 (4.40 e i arctan( 2 E 2 /E E = 0 i φ ϕ + r he (1 = t N ( iχ + t N, ( r N iχ = ie i(φ+ϕ r N 1 r he (2 = t N ( iχ + r N ( iχ r N ( iχ + t N, (5.19 Andreev m=0 r N 2 e i(ϕ ϕ + (5.20 Andreev [ ] 2 r he 2 = t N 4 (1 t N 2 m e i(ϕ m ϕ +, (5.21 t N 4 = ( (1 t N 2 e i(ϕ ϕ + 2 E = 0 r ee + r he 2 = 1 (4.42 NS (5.21 4e 2 /h k y d α = π/4 + e i(ϕ ϕ + = 1 Andreev 1 di dv = 2e2 E=eV =0 h 2N c, (5.23 s e i(ϕ ϕ+ = 1 (5.21 di dv = 2e2 2 t N 4 E=eV =0 h (2 t N 2 2, (5.24 k y 2e2 h 4N c 15 1 z 4 0 for z 0 1, (5.25 z 4 0 ( ZES ZES exp[i(ϕ ϕ + ] = 1 + < 0. (5.26 E = 0 ZES Andreev r he 2 r ee 2 46

49 E = 0 E 0 Andreev e i arctan( 2 E 2 /E 5.4 Andreev ZES s + id r N 2 t N 2 (5.26 ZES ZES ZES d s + id 5.3 Josephson 2 d Josephson Furusaki-Tsukada (4.52 d J = e ħ p T [ + r he ] r eh, (5.27 Ω ω + Ω n 5.5 J 1 sin φ J 1 J 0 = π 0 /2eR J R J α L α R a (α L α R = (0, 0 s 4.7 s 1 (α L α R = (π/4, π/4 1/T J = e k y ( tn cos(φ/2 tn sin(φ/2 tanh, (5.28 2T = 2 (T k p BCS t N 2 1/T T 0 t N 2 ZES d ZES α 47

50 S I S α L α R 5.5: 2 d α L α R a (α L α R = (π/8, π/8 (α L α R = (π/8, π/8 0 0 π ZES 48

51 1: H =H 0 + V, (5.29 ( ϵ + δϵ 0 H 0 =, ( ϵ ( 0 t V =. (5.31 t 0 H 0 2 Hamiltonian V 2 H 0 ( ( for E = ϵ + δϵ, (5.32 for E = ϵ, (5.33. δϵ t t/δϵ ( 1 for E = ϵ + δϵ + t 2 /δϵ, (5.34 t/δϵ ( t/δϵ for E = ϵ t 2 /δϵ, ( t 2 /δϵ δϵ t δϵ/2t 1 1 ( 1 + δϵ/4t 1 for E = ϵ + t + δϵ/2, ( δϵ/2t ( 1 δϵ/4t 1 for E = ϵ t + δϵ/2, ( δϵ/2t δϵ t, δϵ = 0 ϵ 2 2t,

52 2: Hamiltonian H = ν (ϵ ν µa νa ν, (5.38 {a ν, a ν } + =a ν a ν + a ν a ν = δ ν,ν, (5.39 {a ν, a ν } + =a ν a ν + a ν a ν = 0, (5.40 (5.41 A A 2 a ν 0 = 0 (5.42 a ν ν 0 a ν i a νj 0 a ν i a νj 0 = 0 (5.43. ν ν = a ν o ν a ν i a νj ν = 0 a ν a ν i a νj a ν 0, (5.44 = 0 ( δ ν,νi a ( ν i a ν δν,νj a νa νj 0, (5.45 = 0 δ ν,νi δ ν,νj a ν i a ν δ ν,νj a νa νj δ ν,νi + a ν i a ν a νa νj 0, (5.46 =δ ν,ν i δ ν,νj ( = 1 ν 1 ν 2 ν 1. ˆn ν = a νa ν ν 0, 1, ν n ν,. Fermi N F = Π ν:(ϵν 0a ν 0 (5.48 Fermi. 1 F = a ν 1 a ν 2 a ν N 0 (

53 a ν i a νj F (5.50 F a ν j a ν i a νj F = ( 1 N a ν i a ν 1 a ν 2 a ν N a νj 0 = 0 (5.51 F = a ν 1 a ν j a ν N 0 (5.52 a ν j a ν i a νj F =a ν i ( 1 j 1 a ν 1 (1 a ν j a νj a ν N 0, (5.53 =a ν i ( 1 j 1 a ν 1 a ν j 1 a ν j+1 a ν N 0. (5.54 i j F F. i = j a ν i a νi F =a ν 1 a ν i 1 a ν i a ν i+1 a ν N 0, (5.55 = F (5.56 F a ν i a νi F = 1 (5.57 a ν i a νi ν i n νi. a νi a νj a ν i a ν j Hamiltonian [ ] Z =Tr e ( ν ϵνnν/t, (5.58 =Π ν (1 + e ϵ ν/t (5.59.Tr ν (n ν = (n ν = 1 2. a ν i a νj a ν i a νj = Tr [ ] e ( ν ϵ νn ν /T n νi δ νi,ν j, (5.60 [ ( Z ] Πν νi 1 + e ϵ ν/t e ϵν 1 /T δ νi,ν = j [Πν ( ], ( e ϵ ν /T = e ϵ ν i /T 1 + e δ ϵν i /T ν i,ν j, ( = e ϵ ν /T i + 1 δ ν i,ν j = 1 [ ( ϵνi ] 1 tanh δ νi,ν 2 2T j (5.63 Fermi 51

54 3 Laundauer delta. 2 Schrödinger [ ħ2 2 ] 2m µ F + V 0 δ(x ϕ(x, y = Eϕ(x, y ( y. ikx eipy ϕ(x, y =e, (5.65 W E k,p = ħ2 ( k 2 + p 2, (5.66 2m p = 2πn, n = 0, ±1, ±2,..., (5.67 W k 5.6,n = 0,n = ±1... y n 1 2 Fermi x k p evanescent evanescent n = 0, ±1, ±2, ±3, ±4 9, n 5 evanscent y p Fermi ϕ L p (x, y =e ik px eipy W + p r p,pe ik p x eip y W, (5.68 ϕ R p (x, y = p t p,pe ik p x eip y W, (5.69 Fermi (k p, p k 2 p + p 2 = k 2 F = 2mµ F /ħ 2 p y p. t p,p, r p,p p p ˆt, ˆr 9. 1 ϕ L p (0, y = ϕ R p (0, y (5.70 e ip y / W y W y dy ei(p p W + e i(p p y W r p,p = dy e i(p p y t p,p W W. (5.71 p p

55 y W x x=0 E k p 5 k n=0 5.6: 2 - W. 53

56 p p W 0 dy ei(p p y W = δ p,p ( r p,p = t p,p. ( δ- Schrödinger γ < xγ γ 0 γ ( ] lim dx [ ħ2 2 γ γ 0 2m x y 2 µ F + V 0 δ(x ϕ(x, y = lim dx Eϕ(x, y, (5.74 γ 0 γ x,δ- [ ħ2 ϕ(x, y 2m x ] ϕ(x, y + V 0 ϕ(0, y = 0 (5.75 x=0+ x x=0+ e ip y / W y k kr p,p =( k + 2iz 0 t p,p, (5.76 z 0 = mv 0 ħ 2 k F, (5.77 k = k p k F, (5.78 k Fermi x (5.73, (5.76. t p,p = γ k k + iz 0, (5.79 r p,p = iz 0 k + iz 0, (5.80 t p,p 2 + r p,p 2 = 1, (5.81. y p p p. G N = 2e2 h Tr [ˆt ˆt ], (5.82 = 2e2 tp,p 2, (5.83 h p,p p,p. Landauer. t p,p = δ p,p G N = 2e2 h p = 2e2 h W kf dp, (5.84 2π k F = 2e2 h N c, (5.85 N c = W k F π. (

57 N c Fermi. δ- G N = 2e2 k 2 h z 2, (5.87 p 0 + k 2 ( z 2 0 = 2e2 h N ln +1 1 c 1 + z z , (5.88 z 0 1 G N = 2e2 h 2 3z 2 0 ( Laudauer

58 .. 17 M. Tinkham, Introduction to Superconductivity 2nd Ed. McGraw-Hill (1996. J. R. Schrieffer, Theory of Superconductivity, Addison-Wesley (1983. P. G. de-gennes, Superconductivity of Metals and Alloys, Benjamin (1966. A. A. Abrikosov, L. P. Gor kov, I. E. Dzyloshinskii,. A. A. Abrikosov, Fundamentals of the Theory of Metals North-Holland ( Andreev 28, 795 ( , 721 ( , 433 (1999; 3 ; 24, 14 (1999. (1999,, Andreev. 56

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e No. 1 1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e I X e Cs Ba F Ra Hf Ta W Re Os I Rf Db Sg Bh

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