講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K

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1 2 2 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ takada@issp.u-tokyo.ac.jp

2 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B = (1 + 4πχ)H = (1) χ = 1/4π H λ σ 1: BCS flux quantization Josephson

3 B ds = Φ = nφ n π hc (n =, ±1, ±2, ) (2) e Φ gauss cm Deaver Fairbank Doll Näbauer nm (SIS) (I V ) dc ac dc ac I V ω = 2eV/ h ac Shapiro step ω I V V = hω/2e Landau Ginzburg (GL) GL Ψ GL Ψ T T c F Ψ a b F F =F n + dr { a Ψ(r) 2 + b 2 Ψ(r) 4 + h2 Ψ(r) 2m r ie hc A(r)Ψ(r) 2 1 ( ) } 2 + rota(r) 8π F n T > T c Ψ A(r) m e F (3)

4 A(r) Ψ GL T > T c Ψ = a a = a (T c T ) T c Ψ GL 1957 Bardeen Cooper Schrieffer BCS Ψ (f-sum rule) 2: BCS BCS

5 T c T c GL a b m e T c BCS H FP BCS T c 2 BCS BCS GL BCS GL h = k B = c = BCS BCS BCS H BCS H BCS H BCS = H + H g pσ ξ p c pσc pσ g q Φ + q Φ q (4) ξ p g Φ q Φ q p c p c p+q (5) g gδ(r) g

6 ξ p Θ D ξ p+q Θ D Θ D Θ D E F E F 1 4 K Θ D 1 2 K Θ D /E F.1 ξ p Hartree-Fock ξ p H H BCS H g g g + H BCS H ext F e iωt Φ + q (6) H ext Φ q F H ext Φ q Φ q = F e iωt D R (q, ω) (7) D R (q, ω) D R (q, ω) = i dt e iωt +t [e ih BCS t Φ q e ih BCS t, Φ + q ] (8) D(q, iω n ) = 1/T dτ e iωnτ e H BCS τ Φ q e H BCS τ Φ + q (9) D(q, iω n ) ω n = 2πnT (> ) iω n ω + i + D G G ~ = s + J G G 3: D(q, iω n ) G pσ (iω p ) J 3 g g

7 G pσ (iω p ) H G pσ(iω p ) = 1/(iω p ξ p ) J g g D(q, iω n ) D(q, iω n ) = Π s (q, iω n ) + Π s (q, iω n ) g D(q, iω n ) (1) H Π s (q, iω n ) Π s (q, iω n ) = T G p+q (iω p + iω n )G p ( iω p ) (11) ω p p f(x) Π s (q, iω n ) = p f( ξ p ) f(ξ p+q ) ξ p + ξ p+q iω n (12) Π R s(q, ω) D R (q, ω) = ΠR s(q, ω) 1 gπ R s(q, ω) (13) D R (q, ω) ξ p = ξ p q ω Π R s(, ) Π R s(, ) = p 1 2f(ξ p ) ΘD = N(ξ)dξ 1 2f(ξ) 2ξ p Θ D 2ξ (14) N(ξ) E F Θ D Θ D ξ Θ D N() (14) ΘD Π R s(, )=N() ( ) dξ ξ ξ tanh =N() 2T ΘD /2T dx tanh x x T Θ D Π R s(, ) = N()(Θ D /2T ) T Θ D Θ D /2T ( Π R s(, )=N() ln Θ D 2T dx ln x ) cosh 2 x ( 2e γ =N() ln π γ (γ = ) ) Θ D T (13) D R (, ) N()Θ D /2T T D R (, ) = 1 [ ] T c (17) g gn() ln(t c /T ) g 2 N() T T c (15) (16)

8 T T c + + D R (, ) T c T c 2eγ π Θ D e 1/λ = Θ D e 1/λ, λ gn() (18) λ <.5 T c Θ D (16) D R (, ) (7) Φ = F D R (, ) D R (, ) F = ω = T = T c Φ = Φ = p c p c p p c p c p = c p c p T = T c T c c p c p = c p c p = q ω n Π s (q, iω n ) Π s (, ) ( Π s (q, iω n ) Π s (, ) = N() π 8T ω n 7ζ(3) 48 vf 2 ) π 2 T 2 q2 ζ(3) = n=1 n 3 = 1.22 v F Π R s(q, ω) (13) T T c (19) D R (q, ω) = 1 1 g 2 N() (T T c )/T c + [7ζ(3)vF 2 ]/(48π2 Tc 2 )q 2 i(π/8t c )ω (2) D R (q, ) q = q 2 GL a ω GL TDGL 2.2 BCS T c (18) g Θ D N() T c T c N() N() N() N() T c N() (18)

9 T c N() λ Θ D g T c H BCS T c T c H ep H ep = ξ p c pσc pσ + 1 V c (q)c 2 p+qσc p qσ c p σ c pσ pσ q pσ p σ + ω qλ b qλ b qλ+ g λ (q)c p+qσc pσ (b qλ +b qλ ) (21) qλ pσ q λ ξ p (4) V c (q) 4πe 2 /ε q 2 ε λ v I (r R I ) H ei I RI R I + δr I H ei δh ei δh ei = drψ σ(r) [ v I (r RI δr I ) v I (r RI) ] ψ σ (r) I σ = drψ σ(r) I v I (r R I ) δr I ψ σ (r) (22) R=R I σ δh ei δr I H ep ξ p ω qλ g λ (q) p q (21) q q q = ξ p T τ c pσ (τ)c pσ T τ c p (τ)c p SU(2) ( ) Ψ p = c p c p, Ψ p = (c p c p ) (23)

10 (21) H ep H ep = p + qλ ξ p Ψ pτ 3 Ψ p q ω qλ b qλ b qλ + p V c (q)(ψ p+qτ 3 Ψ p )(Ψ p q τ 3Ψ p ) p,p g λ (q)ψ p+qτ 3 Ψ p (b qλ +b qλ ) (24) q λ τ i (i = 1, 2, 3) τ G p (τ) G p (τ) T τ Ψ p (τ)ψ p ( ) T τ c p (τ)c p = T τc p (τ)c p T τ c p (τ)c p T τc p (τ)c p (25) ω p 1/T G p (iω p ) = dτ e iωpτ T τ Ψ p (τ)ψ p (26) G p (iω p ) G p(iω p ) G p(iω p ) = 1/(iω p τ ξ p τ 3 ) G p (iω p ) G p (iω p ) = G p(iω p ) + G p(iω p )Σ p (iω p )G p (iω p ) (27) Σ p (iω p ) ω q Σ p (iω p ) = T τ 3 G p+q (iω p + iω q )Λ p+q,p (iω p + iω q, iω p ) ω q q V ee (q, iω q ) (28) 3 Λ p,p(iω p, iω p ) 1/T 1/T Λ p,p(iω p, iω p )=G p (iω p ) 1 dτ e iω p τ dτ e i(ω p ω p )τ T τ Ψ p (τ)ρ p p (τ )Ψ p G p (iω p ) 1 (29) (29) ρ q ρ q = p Ψ p τ 3 Ψ p+q V ee (q, iω q ) V ee (q, iω q )=V c (q)+ λ g λ (q) 2 2ω qλ V (iω q ) 2 ωqλ 2 c (q)+v ph (q, iω q ) (3) D E D Λ p,p(iω p, iω p )

11 E Γ p,p(iω p, iω p ) (28) Γ p,p(iω p, iω p ) Σ p (iω p ) = T τ 3 G p+q (iω p +iω q )Γ p+q,p (iω p +iω q, iω p ) ω q q Ṽee(q, iω q ) Ṽee(q, iω q ) (31) Ṽ ee (q, iω q ) V ee (q, iω q ) 1 + V ee (q, iω q )Π(q, iω q ) Π(q, iω q ) Γ p,p(iω p, iω p ) Π(q, iω q ) = T [ Tr τ 3 G p (iω p )Γ p,p+q (iω p, iω p + iω q ) ω p p ] G p+q (iω p + iω q ) (32) (33) G p (iω p ) (31)-(33) L. Hedin G p (iω p ) G p ( iω p ) = G p (iω p ) Σ p ( iω p ) = Σ p (iω p ) Σ p (iω p )=[1 Z p (iω p )]iω p τ +χ p (iω p )τ 3 +ϕ p (iω p )τ 1 +ψ p (iω p )τ 2 (34) Z p (iω p ) χ p (iω p ) iω p iω p Z p (iω p ) χ p (iω p ) ϕ p (iω p ) ψ p (iω p ) ϕ p ( iω p ) = ϕ p(iω p ) ψ p ( iω p ) = ψ p(iω p ) ϕ p (iω p ) ψ p (iω p ) iω p p p Z p (iω p ) χ p (iω p ) ϕ p (iω p ) ψ p (iω p ) (27) G p (iω p ) = 1 G p(iω p ) 1 Σ p (iω p ) = Z p(iω p )iω p τ +[ξ p +χ p (iω p )]τ 3 +ϕ p (iω p )τ 1 +ψ p (iω p )τ 2 [Z p (iω p )iω p ] 2 [ξ p +χ p (iω p )] 2 ϕ p (iω p ) 2 ψ p (iω p ) 2 (35) τ i τ j +τ j τ i =2τ δ ij i, j =1, 2, 3 T c T c 2

12 (35) ϕ p (iω p ) ψ p (iω p ) ϕ p (iω p ) ω G R p (ω) dω G p (iω p ) = π ImG R p (ω ) iω p ω (36) Ṽee(q, iω q ) Ṽ ee R (q, Ω) Ṽ ee (q, iω q ) = V c (q) dω π 2Ω (iω q ) 2 Ω 2 ImṼ R ee (q, Ω) (37) 2.3 Eliashberg T c Θ D E F Θ D /E F 1 (31) Hartree-Fock-Gor kov Γ p+q,p (iω p + iω q, iω p ) = τ 3 (38) (3) (32) V c V ph Ṽee(q, iω q ) V c (q) Ṽ ee (q, iω q ) = 1+V c (q)π(q, iω q ) 1 V ph (q, iω q ) + [1+V c (q)π(q, iω q )] 2 1+ V ph(q, iω q )Π(q, iω q ) 1+V c (q)π(q, iω q ) (39) ξ p (39) 1

13 Ṽee(q, iω q ) E F Θ D Π(q, ) q q Ṽee(q, iω q ) Ṽee(iω q ) (37) Ṽ ee (iω q ) = 1 N() dω α 2 F (Ω) 2Ω (iω q ) 2 Ω 2 (4) N() 1 α 2 F (Ω) (3) V ph (q, iω q ) λ ω qλ = ω Ω Ω 2 = ω 2 2ω g λ (q) 2 Π(q, ) 1 + V c (q)π(q, ) α 2 F (Ω) α 2 g λ (q) 2 ω F (Ω) = N() δ(ω Ω [1 + V c (q)π(q, )] 2 ) (42) Ω Ṽee(iω q ) (38) (31) Σ p (iω p ) q p (= p+q) p G p (iω p +iω q ) Θ D /E F 1 p p dξ p ( dξ ) (14) ( ω c, ω c ) N(ξ ) N() χ p (iω p ) = Z p (iω p ) ϕ p (iω p ) p s Z(iω p ) ϕ(iω p ) G p (iω p ) = N() p ωc dξ Z(iω p )iω p τ + ξ τ 3 +ϕ(iω p )τ 1 ω c [Z(iω p )iω p ] 2 ξ 2 ( tan 1 ω c ( = 2N() iτ + ϕ(iω p ) ) τ 1 Z(iω p )ω p Z(iω p )ω p ω p = ω p +ω q ω c Θ D T c ω c (43) tan 1 (ω c /Z(iω p )ω p ) π/2 ω p ω p < ω c θ(x) θ(ω c ω p ) T c ω c T c ) (41) (43)

14 Σ p (iω p ) (34) Z(iω p ) ϕ(iω p ) Z(iω p ) Z(iω p ) = 1 + π ω p T ωp λ(p p) η p (ω c ) (44) n λ(n) λ(n) = dω α 2 F (Ω) η p (ω c ) η p (ω c ) = 2 π tan 1 ( 2Ω Ω 2 + (2πT n) 2 (45) ω c Z(iω p )ω p ϕ(iω p ) (44) Z(iω p ) (iω p ) ϕ(iω p )/Z(iω p ) (iω p ) (iω p ) = π Z(iω p ) T ω p ) λ(p p) (iω p ) ω p (46) η p (ω c ) (47) T c (iω p ) (39) (41) (42) µ c V c (q) µ c = N() 1 + V c (q)π(q, ) ω p < E F ω p < E F µ c (4) Ṽee(iω q ) Z(iω p ) (iω p ) µ c λ(p p)η p (ω c ) λ(p p)η p (ω c ) µ c η p (E F ) Z(iω p ) (44) µ c ω p η p (E F ) ω p µ c (iω p ) (47) (iω p ) = π Z(iω p ) T ω p (48) (iω p ) [λ(p p)η p (ω c ) µ c η p (E F )] (49) ω p ω c < ω p < E F µ c (iω p ) (49) ω p < ω c (iω p ) = π Z(iω p ) T ω p (iω p ) [λ(p p) µ c ]η p (ω c ) ω p π µ c Z(iω p ) T ω p θ(e F ω p )θ( ω p ω c ) ω p (5)

15 ω c < ω p < E F Z(iω p ) = 1 (49) = µ c πt ωp (iω p ) η p (ω c ) ω p µ c πt ωp θ(e F ω p )θ( ω p ω c ) ω p (51) (51) (5) (49) (iω p ) = π Z(iω p ) T ω p (iω p ) [λ(p p) µ ]η p (ω c ) (52) ω p (52) µ /[ µ = µ c 1 + µ c πt ] θ(e F ω p )θ( ω p ω c ) (53) ω ωp p F (x) 2πT ω 2 ω p=ω 1 F (ω p )= ω 2 ω 1 dx F (x) (53) µ = µ c /[1 + µ c ln(e F /ω c )] (48) V c (q) q 2 µ c N()/Π(, ) Π(, ) = 2N() µ c.5 E F /ω c =.1.1 µ = µ c µ () ωc 1 E 1 F 2.4 (44) Z(iω p ) (52) T c α 2 F (Ω) α 2 F (Ω) µ µ (22) { nk } n k 1 st BZ nk ξ nk

16 g λ (q) nk n k g λ (n k ; nk) q k k 1 st BZ α 2 F (Ω) (42) α 2 F (Ω)= nk n k λ g λ(n k ; nk) 2 δ(ω ω k kλ)δ(ξ nk )δ(ξ n k ) nk δ(ξ nk) N() g λ (n k ; nk) ω qλ (24) (42) ω qλ γ qλ α 2 F (Ω) α 2 F (Ω) = 1 2πN() qλ (54) γ qλ ω qλ δ(ω ω qλ ) (55) ω qλ γ qλ γ qλ ω qλ γ qλ =2π nn k g λ (n k ; nk) 2 [f(ξ nk ) f(ξ nk +ω qλ )]δ(ξ n k ξ nk ω qλ ) 2πω qλ g λ (n k ; nk) 2 δ(ξ nk )δ(ξ n k ) (56) nn k γ qλ (55) α 2 F (Ω) α 2 F (Ω) T T c α 2 F (Ω) µ α 2 F (Ω) T c α 2 F (Ω) µ T c T c McMillan T c T c = Θ [ ] D 1.45 exp 1.4(1 + λ) (57) λ µ (1 +.62λ) λ = λ() (45) α 2 F (Ω) α 2 F (Ω) Ω Ω = 2 λ dω α 2 F (Ω) (58)

17 Θ D /1.45 Ω /1.2 λ > 2 (57) T c T c Allen-Dynes T c T c T c M T c T c M α α T c M α = (57) µ = α =.5 λ M λ λ g 2 / Ω (2.77) g 2 (M Ω ) 1 λ 1/M Ω 2 M µ (53) Ω α T c MgB 2 T c Θ D /E F 1 T c Θ D /E F 1 µ µ T c T c µ Θ D /E F 1

18 µ µ = Θ D /E F 1 T c T c T c Θ D /E F 1 4 E F n s /m T c T c /E F.1.5 T c =.4E F T B 2 T B /E F.218 T c 4: E F T c [1] T c /Θ D.1.5 T c /E F T c /E F = (T c /Θ D )(Θ D /E F ) A 3 C 6 T c 3K T c /E F =.4 ω.2ev ω /E F 1

19 λ λ i α 2 F (Ω) λ i 2.6 G W (21) H ep Θ D /E F µ (32) Ṽee(q, iω q ) (39) G W GW GW GWΓ GWΓ GW G W T c (31) Γ p+q,p (iω p +iω q, iω p ) Γ p+q,p (iω p +iω q, iω p ) G p (iω p ) iω q Γ p+q,p (iω p +iω q,iω p ) q Γ p+q,p (iω p +iω q, iω p ) = G p+q (iω p +iω q ) 1 τ 3 τ 3 G p (iω p ) 1 (59) (59)

20 Γ(iω p, iω p ) = ω p Z(iω p ) ω pz(iω p ) ω p ω p τ 3 (6) Γ(iω p, iω p ) (44) (47) λ(p p) λ(p p)[ω p Z(iω p ) ω p Z(iω p )]/[ω p ω p ] GISC Gauge-Invariant Self-Consistent [2] GISC µ = α 2 F (Ω) (42) α 2 F (Ω) = λ 2 Ω δ(ω Ω ) (61) λ =.5 Ω /E F 1 T c 5 Ω /E F GISC Ω /E F T c 2 GISC G Z(iω p ) = 1 G T c 5: (61) α 2 F (Ω) Ω T c. GISC G [3]. Θ D /E F GW G W G W (31) Σ p (iω p ) (34) G p+q (iω p +iω q ) (35) τ 3 τ 1 T = T c ϕ p (iω p ) ϕ p (iω p ) = T ωp p p ϕ p (iω p ) (iω p ) 2 ξ 2 p Ṽ ee (p p, iω p iω p ) (62) G Z p (iω p ) = 1 χ p (iω p ) = ϕ p (iω p ) (62) 2

21 T c T c iω p T c ϕ p (iω p ) iω p (62) ω ϕ p (iω p ) ϕ R p (ω) iω p ω + i + (36) (37) (62) ϕ R p (ω)= dω π p dω + π ImṼ ee R (p p, Ω) [ ( [f( ω )+n(ω)] ( + [f(ω )+n(ω)] [ ϕ R ] p Im (ω ) { [1 2f(ω )]V ω 2 ξp 2 +i + c ( p p ) 1 ω+ω+ω +i ω Ω+ω +i 1 + ω+ω ω +i + ) ω Ω ω +i + )] } (63) n(ω) p [ ] dω p 2 ξ p π Im ϕ R p (ω) ω 2 ξp 2 + i + (64) (63) ω 2 ξp 2 + i + ω (, ) p = { p [1 2f( ξp ) ][ ] V c ( p p 2 R )+ 2 ξ p p π dωimṽ ee (p p, Ω) Ω+ ξ p + ξ p 2 R + dω ImṼee (p p, Ω) [ f( ξ p )+n(ω) ] π [ 1 Ω+ ξ p + ξ p + θ( ξ p Ω) Ω+ ξ p + ξ p + θ( ξ ] } p +Ω) (65) Ω ξ p + ξ p (65) Ω p = ( ) p ξp tanh K p,p (66) 2ξ p p 2T c BCS BCS K p,p K p,p V c ( p p ) + = 2 π R ImṼee (p p, Ω) dω Ω + ξ p + ξ p 2 π dω ξ p + ξ p Ω 2 + ( ξ p + ξ p ) Ṽee(p p, iω) (67) 2

22 T c (66) [4] (66) (67) ξ p Ṽee(q, iω) T c p (66) s T c p (66) p K p,p (66) T c (62) T c T c (62) (66) T c (62) 2.7 n m ε(q, iω) ξ p Ṽee(q, iω) ξ p = p2 2m E F, Ṽ ee (q, iω) = 4πe 2 q 2 ε(q, iω) (68) ε(q, iω) ω 2 t ε(q, iω) = ε + (ε ε ) ωt 2 + Ω + 4πe2 Π(q, iω) (69) 2 q2 ε ω t ω t ω l ω t = ε /ε ω l G W Π(q, iω) RPA Ω = Ω ω p (69) T c 6 m ε /ε ω l T c n T c m /ε 2 5 1K SrTiO 3

23 6: T c n [5]. T c n SrTiO 3 [5] 6 n T c T c 4 7 T c T c /E F.4.4 7: T c [6]. T c /E F.4

24 .4 T c /E F.4 (68)-(69) Ṽee(q, iω) µ T c SCDFT DFT n(r) F xc n(r) n(r) V xc (r) F xc [n(r)] V xc (r) i ξ i n(r) n(r) F xc [n(r)] n(r) n(r) n σ (r) F xc [n σ (r)] n(r) χ(r, r ) [ ψ (r)ψ (r ) ] F xc [n(r), χ(r, r )]

25 ] [ 2 2 +V KS(r) µ u i (r)+ dr s (r, r )v i (r )=E i u i (r) ] [ 2 2 +V KS(r) µ v i (r)+ dr s(r, r )u i (r )=E i v i (r) (7) Bogoliubov-de Gennes V KS (r) V xc (r) = δf xc [n, χ]/δn(r) s (r, r ) F s [n, χ] F s [n, χ] s (r, r ) = χ(r, r ) r r δf xc[n, χ] δχ (r, r ) = δf s[n, χ] δχ (r, r ) dr dr χ(r, r ) 2 r r (71) + F xc [n, χ] (72) 3.2 T c (7) u i (r) v i (r) i ξ i φ i (r) u i (r) = u i φ i (r) v i (r) = v i φ i (r) E i = ±ξ i χ(r, r ) χ(r, r ) = ψ (r)ψ (r ) = ( ) i ξi tanh φ i (r)φ i (r ) 2ξ i i 2T c i χ i φ i (r)φ i (r ) (73) r, r i i dr dr φ i (r) s (r, r )φ i (r ) = δf s[n, χ] δχ i i 2 (71) (73) s (r, r ) = i (74) δf s [n, χ] φ δχ i (r)φ i (r ). (75) i T c χ F s [n, χ] χ = F s [n, χ] χ χ = δf s [n, χ] δχ i = j δ 2 F s [n, χ] δχ i δχ j χ j χ= j K ij χ j (76)

26 2 K ij (76) (74) χ j (73) T c i = ( ) j ξj tanh K ij (77) 2ξ j j 2T c BCS s (r, r ) V KS (r) i i ξ i K ij F xc [n, χ] µ χ(r, r ) χ(r, r ) T c G W (77) G W (66) K ij G W SCDFT µ 3.3 G W (67) SCDFT K ij Gross α 2 F (Ω) K ij RPA 25 Massidda SCDFT Tc exp = 1.18K T c =.9K Tc exp = 7.19K T c = 6.9K Li Ca MgB 2 T exp c = 4.2K T c = 35.1K CaC 6 T exp c = 11.5K T c = 9.4K 1% µ µ K ij µ T c.4e F T c Ω E F K ij

27 3.4 K ij DFT V xc (r) LDA K ij G W K ij (67) K p,p (67) Ω µ (67) K p,p K ij K ij = 2 π dω ξ i + ξ j Ω 2 + ( ξ i + ξ j ) Ṽij(iΩ) (78) 2 Ṽij(iΩ) i i (i, i ) (j, j ) RPA n(r) K ij n(r) LDA LDA (78) (78) (8) D R (q, ω) H BCS H Φ q (5) T c D R (, ) 3 D R (q, ω) Π R s J D R = Π R s /(1 + J Π R s ) 8(a) D R (q, ω) (11) Π s ΠR s D R = 1 + g Π R s (79) g g J + 1 Π R s 1 Π R s = 1 D R 1 Π R s (8) DFT TDDFT Π Π Π Π

28 (a) G G (b) i j D = + g ~ s G G i * g ~ j* 8: (a) Π s (b) g ij. f xc (8) g f xc (79) T c 1 + g Π R s = Π R s BCS G W K ij (78) RPA Ṽij 8(b) g ij (8) g ij K ij g g = Ṽ g ij ξ BCS T c ξ T c /E F =.361/p F ξ p F a T c /E F 1 4 ξ a T c /E F.4 ξ a A 3 C 6 ξ 2a ξ ξ g N D R DN R Π R s ΠR s,n (8) 2 N g g N g lim N g N ξ N N g N g g = g N g [7, 8] g (78) Ṽij(iΩ) g Ω Ω K ij = g ij

29 4 (1)1911 T c = 4.2K MgB 2 T c = 4.2K (2) T c HgBa 2 Ca 2 Cu 3 O 8 133K (3)199 T c Cs 3 C 6 38K (4)26 T c SmFeAsO 55K SCDFT (1) T c (3) SCDFT SCDFT (2) (4) T c P. B.Allen and Mitrović, in Solid State Physics 37, 1 (1982) p. 221 (212) 29 (25 29 [1] Y. J. Uemura, Physica B , 1 (26). [2] Y. Takada, J. Phys. Chem. Solids 54 (1993) [3] Y. Takada, in Condensed Matter Theories Vol. 1 (Nova), p. 255 (1995). [4] Y. Takada, J. Phys. Soc. Jpn. 45, 786 (1978). [5] Y. Takada, J. Phys. Soc. Jpn. 49, 1267 (198). [6] Y. Takada, J. Phys. Soc. Jpn. 61, 3849 (1992). [7] Y. Takada, J. Phys. Soc. Jpn. 65, 1544 (1996). [8] Y. Takada, Int. J. Mod. Phys. B 21, 3138 (27).

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