013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8 5 11 5.1................................. 11 6 16 7 18 A 18 1
1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l + 1 c i c j V ijkl c k c l 1 c i c j V ijkl c k c l H mf cc c c ij = V ijkl c k c l c i c j = c j c i () ij = ji H mf c = (c, c ) T H mf = 1 c ( ϵ (4) ) ϵ c 1 4 c i c j ij 1 4 ij c ic j + 1 Tr ϵ (5) ϵ ij = t ij µδ ij (3)
1. A A CAC 1 ( ) 0 1 C = K (6) 1 0 K ( ϵ ) ϵ ( ) ϵ ϵ (7) 1.3 H mf ( ) ( ) ( ) c u v γ c = v u γ (8) c i = u iα γ α + v iα γ α (9) c i = u iαγ α + v iαγ α (10) uu + vv = 1, uv T + vu T = 0 (11) γ {γ α, γ β } = 0 {γ α, γ β } = δ αβ (11) c γ 3
±E α +E α (, u iα,, v iα, )T H = α E α γ αγ α + H c (1) H c = 1 Tr (ϵ E) 1 4 c i c j ij 1 4 ij c ic j (13) u ϵu + v T u + u v v T ϵ v = E (14) u ϵv + v T v + u u v T ϵ u = 0 (15) c i c i c i µ γ γ γ γ α 0 = 0, α H c γ γ αγ β = δ αβ f(e α ), f(e) = 1 e E/T 1 = 0 H mf (16) H mf = 1 ( ) ϵ 0 c 0 ϵ c + 1 Tr ϵ (17) ϵ ξ α x α E α = ξ α (ξ α > 0) E α = +ξ α ( ) xα 0 γ α = (u 1 ) αi c i (ξ α < 0) E α = ξ α ( ) 0 x α γ α = (v 1 ) αi c i 0 γ α 0 4
1.4 H mf (10) ij = V ijkl c k c l = V ijkl [u kα v lα (1 f(e α )) + v kα u lα f(e α )] (18) () ij = V ijkl u kα v lα tanh E α T u, v (19) 1.5 H mf = 1 c ( ϵ ) ϵ c (0) H c H mf = 1 E α (γ αγ α γ α γ α) = α α E α γ αγ α (1) (0) 1/ 1/ ( ) ϵ H BdG = ϵ H BdG H BdG O = c i O ijc j O = 1 ( ) O 0 c 0 O c (3) 5 ()
O BdG = ( O 0 ) 0 O 1/ N = i c i c i = i (c i c i c i c i )/ N = ( ) 1 0 0 1 i ie N ie = 1 i ih N ih = 1 (4) (5) (11) H c = 1 ( Tr ϵ E + u v + u T v tanh E ) T F F = T Tr ln (6) ( 1 + e E/T ) + H c (7) S = F T [ = Tr [f ln f + (1 f) ln(1 f)] Tr f(e) E ] H c T T (8) U = α E α f(e α ) + H c (9) C = U T [( E = Tr 4T E 4T 1 ) E T sech E T E T tanh E T + 1 T ( u v tanh E T )] (30) 6
( E T = ReTr u ) T v (31) 3 ij = V ijkl c k c l i V ijkl = i j H k l c i U ij c j i = c i 0 U ii c i 0 = U ii i V ijkl U ii U jj V i j k l U kk U ll ij U ii U jj i j U U T (3) O ij = c i c j O UOU ( )U U R = e is θ/ s θ s y s i s y = s T i R T = s y R s y (33) R R T = R s y R s y (34) = d µ s µ is y d R R T = Rd µ s µ R is y d µ s µ d 0 is y Rd 0 R is y = d 0 is y (35) d sis y Rd sr is y = d s is y (36) d d d 0 d i ( ) ϵ H = ϵ (37) 7
U = diag(1, is y ) H UHU ( ) ϵ dµ s H µ d µs µ s y ϵ s y d µ s y ϵ s y ϵ ( ) O 0 O 0 s y O s y (38) (39) s i ( ) si 0 s i = 0 s i (40) 4 G ij (τ) = T τ c i (τ)c j (41) G c(τ) c(τ) = e τh ce τh c (τ) = e τh c e τh *1 F ij (τ) = T τ c i (τ)c j (4) ij = V ijkl F kl τ = 0 τ = +0 F = F (+0) F F ij (τ) = F ji ( τ) (43) *1 c(τ) c (τ) 8
τ F ij (τ) Fij(τ) = T τ c j (τ)c i (44) G F τ c i = [c i, H] = t ij c j + V ijkl c j c kc l (45) τ G ij (τ) = δ ij δ(τ) + t ik G kj (τ) V iklm T τ c k (τ)c l(τ)c m (τ)c j τ F ij (τ) = t ik F kj (τ) V iklm T τ c k (τ)c l(τ)c m (τ)c j (46) (47) τ G(τ) = δ(τ) + tg(τ) + F (τ) (48) τ F (τ) = tf (τ) G ( τ) (49) G, F 1 τ ( ) ( ) 1 G(τ) F (τ) τ t F (τ) G = ( τ) τ + t δ(τ) (50) G(τ) = T n F (τ) = T n e iϵ nτ G(iϵ n ) (51) e iϵ nτ F (iϵ n ) (5) ( ) ( ) 1 G(iϵn ) F (iϵ n ) iϵn t F ( iϵ n ) G = (iϵ n ) iϵ n + t (53) F iϵ n 9
( ) 1 iϵn t 0 ĝ(iϵ n ) = 0 iϵ n + t = ( ) g(iϵn ) 0 = 0 g (iϵ n ) ( (iϵn t) 1 0 ) 0 (iϵ n + t ) 1 (54) (55) ( Ĝ = ĝ + ĝ ˆ ĝ g + = g g ) g g g + (56) ij = V ijkl T n g kp (iϵ n )g ql(iϵ n ) pq (57) U (U tu) αβ = ξ α δ αβ (58) g ij (iϵ n ) = U iα U αj iϵ n ξ α (59) T n g kp (iϵ n )gql(iϵ n ) = U kα U αpu lβ U βqt 1 1 iϵ n n ξ α iϵ n ξ β ( = U kα U αpu lβ U 1 1 βq tanh ξ α ξ α + ξ β T + tanh ξ ) β T (60) αβ = V αβγδ ξ α + ξ β tanh ξ γ T γδ (61) αβ = U αi U βj ij (6) V αβγδ = U αi U βj V ijkl U kγ U lβ (63) 10
a = A ab b (64) a = (α, β), b = (γ, δ) A 5 5.1 1/ ŝ/ ŝ = c s ss is c is = 1 c s ss is c is 1 iss iss c st ss is iss c is (65) ŝ ( s 0 ) 0 s T (66) s i s = (s x, s y, s z ) M = gµ B s/ g g µ B = eħ/(m e c) * m e e > 0 5.1.1 B M H Z = M B χ µν (ω) = i dte iωt θ(t) [M µ (t), M ν ] (67) M µ (t) = e iht M µ e iht Q µν (iν n ) = 1/T 0 dτe iν nτ T τ M µ ( iτ)m ν (68) * µ B = 9.74 10 4 J/T = 0.0579meV/T 0.67K/T 11
χ(ω) = Q(ω + i0) Q Q µν (iν n ) = g µ B 4 H mf α = E α α αβ f(e α ) f(e β ) iν n + E α E β α s µ β β s ν α (69) 5.1. χ P = χ µµ (0) H = ks ϵ k c ks c ks (70) χ P = g µ B 4 f(ϵ k ) f(ϵ k ) ϵ k ϵ k kk ss + i0 ks s µ k s k s s µ ks (71) ks s µ k s = δ kk s s µ s s µ = 1 χ P = g µ B k lim k k f(ϵ k ) f(ϵ k ) ϵ k ϵ k + i0 = g µ B k ( f(ϵ ) k) ϵ k (7) T 0 χ P = g µ B D 0 (73) D 0 5.1.3 s c = (c k, c k, c k, c k )T BdG ( ) ξk H(k) = ξ k sτ 0 1 (74)
χ = g µ B k f(e k ) E k = g µ B 4T 0 ded(e) sech E T (75) f(e) = (e E/T + 1) 1 D(E) D(0) = 0 T T c T E χ g µ B T 0 ded(e)e E/T g µ e /T B T (76) 5.1.4 E(k) k, k = d i=1 k i D(E) E d 1 (77) d E(k) k, d 1 k = i=1 k i D(E) E d (78) d n D(E) E d d n 1 * 3 χ ded(e) f 0 E T d d n 1 (80) *3 0 E α de e E/T + 1 = ( 1) n dee α e (n+1)e/t = α!t α+1 n=0 0 n=0 ( 1) n (n + 1) α+1 (79) 13
5.1.5 NMR M i = χ i H i z z T 1 S s H HF = γs s *4 1/ B( z ) I H N = g N µ N I z B µ N = eħ/(m N ) (I z = 1/) H HF I z = 1/ T 1 1 T 1 = π ħ γ 4 ρ α 1/, α S s + 1/, β δ(ω + ϵ α ϵ β ) (83) αβ I z, α z I z α ϵ α (ρ α ) β ω = g N µ N B δ(ω) = (π) 1 dteiωt *4 s s = V ψ (0)sψ(0) = kk c k sc k (81) s = d D x ψ (x)sψ(x) = k c k sc k (8) 14
1 = πγ T 1 ħ = γ 4ħ ρ α α s + β β s α δ(ω + ϵ α ϵ β ) αβ dte iωt s + (t)s (84) s + (t) = e iht s + e iht. H HF Im χ + (ω) = π g µ B 4 e ϵα/t e ϵ β/t α s + β β s α δ(ω + ϵ α ϵ β ) Z αβ = π g µ B 4 (1 e ω/t ) αβ e ϵ α/t Z α s + β β s α δ(ω + ϵ α ϵ β ) (85) 1 T 1 = γ ħg µ B [n(ω) + 1] Im χ + (ω) (86) n(ω) = (e ω/t 1) 1 T 1 NMR Im χ + (ω) = πg µ B 1 T lim T 1 ω 0 ω Im χ + (ω) (87) = πg µ B πg µ B ω [f(ϵ k ) f(ϵ k )]δ(ω + ϵ k ϵ k ) kk [f(ϵ k ) f(ϵ k + ω)]d(ϵ k + ω) k ( dϵ f(ϵ) ) D (ϵ) (88) ϵ 15
T ϵ F Im χ + (ω) πg µ B ωd (ϵ F ) (89) 1 T 1 T = πγ D (ϵ F ) (90) T 1 T (Korringa) 6 L = i t 1 m + b + c 4 (91) b, c, m m e iα ϕ A ϕ ϕ = ϕ + t χ (9) A A = A + χ (93) 16
e iẽχ (94) D = iẽa (95) t D t = t iẽϕ (96) L = id t + D D + b + c 4 + L em (97) L em = (E B )/ E B = e iθ ( L ẽ ϕ ) tθ ẽ ( A θ ) + L em (98) ẽ m ẽ θ t θ, θ à = A θ ẽ ϕ = ϕ tθ ẽ (99) (100) θ/ẽ L = ẽ ϕ ẽ m à + L em (101) à *5 µ à µ = 0 ( t M ) à = 0 (103) *5 ( µ ( µ A ν ) + ) A ν L = 0 (10) 17
M = (ẽ /( m)) 1/ ω = k + M A θ B = (0, 0, B z ) (103) z B z = M B z z > 0 B z e Mz (104) (M > 0 ) M 7 A E(k) = (ξ(k) + ) 1/ D(ω) D(ω) = k δ(ω E(k)) = k δ (ω ) ξ (k) + (105) D(ω) = D( ω) ω > 0 E(k) g(e(k)) = dωd(ω)g(ω) (106) = 0 k D N (ω) D(ω) =0 = k δ (ω ξ(k) ) (107) D(ω) = dϵd N (ϵ)δ (ω ) ϵ + (108) 18
x = ϵ + > D(ω) = = x ( ) dx x D N x δ(ω x) ω ( ) ω D N ω θ(ω ) (109) ω ( ) D(ω) = Re ω D N ω (110) D(ω) D N ω Re (111) ω D N D N (0) d ( (k) cos θ) D(ω) dω ω Re (11) D N S D ω (Ω) S D D (Ω) = (k) ξ(k)=0 () d D k D N (ω) = V δ(ω ξ(k) ) (π) D = V S D dω (π) D dkk D 1 δ(ω ξ(k) ) S D 0 dω = D N (ω; Ω) (113) S D D N (ω; Ω) D N (ω; Ω) = V S D (π) D kd 1 F (ω; Ω) 0 dkδ(ω ξ(k) ) (114) k k : k = k F (ω; Ω), ξ(k F (ω; Ω), Ω) = ω k g(k) d D k dω V (π) D g(k) = dωd N (ω; Ω)g(k F (ω; Ω), Ω) (115) S D 19
dω ( ) D(ω) = D N ω S ω (k F (ω; Ω), Ω); Ω Re D ω (k F (ω; Ω), Ω) (116) k F k F (ω; Ω) k F (Ω) dω ω D(ω) = D N (Ω) Re S D ω (k F (Ω), Ω) (117) D N (Ω) D N (0, Ω) k F (Ω) = k F D N ( ω (k F (ω; Ω), Ω); Ω) = D N D(ω) dω ω = Re (118) D N S D ω (Ω) (Ω) = (k F, Ω) (θ, ϕ) = cos θ (θ = 0, π) (θ = π/) ( ) D(ω) D N = π = ω, = ω ( ω Arccsc D( )/D N = π/ dθ sin θ Re ω ω cos θ Re Arccot( ω 1) πω ω < 0 ), ω > (119) (θ, ϕ) = sin θ (10) ( ) D(ω) D N = ω ln ω + ω (11) 0
ω = ω D(ω) D N ω (1) [1].,, 17., 1993. BCS []., 011.. [3],.., 000. [4]..., 00. 1