Bogoliubov-de Gennes

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1 Bogoliubov-de Gennes 7 Bogoliubov-de Gennes Bogoliubov H = H 0 + H = Ψ rh 0 rψ r +, Ψ rψ r g r r Ψ r Ψ r Ψ r r g r r r r h 0 h 0 h 0 = h i e m hc A + V r µ 3 Bogoliubov BCS BCS Ψ rψ r, Ψ rψ r 4 Cooper Hartree Fock Ψ rψ r Ψ r Ψ r Ψ rψ r = Ψ rψ r + Ψ rψ r Ψ rψ r Ψ rψ r = Ψ rψ r + Ψ rψ r Ψ rψ r 6 5

2 H H H = Ψ rψ r + Ψ rψ r Ψ rψ r, g, r r Ψ rψ r Ψ rψ r Ψ rψ r 7 = g, r r Ψ rψ r Ψ rψ r g, r r Ψ rψ r Ψ rψ r, g, r r Ψ rψ r Ψ rψ r g, r r Ψ rψ r Ψ rψ r Ψ rψ r Ψ rψ r 8 H Ψ r g, r r Ψ rψ r Ψ r, + Ψ r g, r r Ψ rψ r Ψ r 9 Ψr Ψ r Ψ r 0 H Ψ r Ψ r g, r r Ψ rψ r g, r r Ψ rψ r g, r r Ψ rψ r g, r r Ψ rψ r Ψ r Ψ r r, r g, r r Ψ rψ r ˆ r, r r, r r, r r, r r, r 3 Ψ r ˆ r, r Ψ T r 4 H H H rr r r H ˆ r, r ˆ r, r = Ψ T r ˆ r, r Ψr 5 g, r r Ψ rψ r g, r r Ψ rψ r g, r r Ψ rψ r g, r r Ψ rψ r g, r r = g,r r H H H = Ψ T r ˆ r, r Ψr + Ψ r ˆ r, r Ψ T r 7 ˆ r, r = ˆ r, r g r r 6

3 3 Schrodinger H H 0 h 0 H 0 H 0 H 0 = Ψ rh 0 Ψ r 8 h 0 Schrodinger h 0 f ν, r = E ν f ν, r 9 fν,rf ν, r = δ ν,ν δ, 0 f ν, rfν, r = δr r δ, ν E ν f ν, r Ψ r Ψ r = ν c ν, f ν, r H 0 c ν, ν H 0 H 0 = c ν,c ν,fν,rh 0 f ν,r 3 ν,ν = c ν,c ν,fν,re ν f ν,r 4 ν,ν = c ν,c ν, E ν 5 ν c ν,c ν,e ν = c ν,c ν,e ν fν,rf ν,r 6 ν ν,ν = c ν,c ν,e ν f ν, r f ν,r 7 ν,ν = c ν,c ν,h 0 f ν, r f ν,r 8 ν,ν = Ψ r h 0Ψ r 9 = Ψ rh 0Ψ r 30 3

4 H 0 = Ψ rh 0 Ψ r + Ψ rh 0 Ψ r 3 = Ψ rh 0 Ψ r Ψ rh 0Ψ r 3 = δ, δr r Ψ rh 0 r Ψ r δ, δr r Ψ rh 0r Ψ r,, 33 H 0 H 0 = Ψ rh 0 r δr r ˆ 0Ψr ˆ 0 Ψ T rh 0r ˆ 0 δr r Ψ T r 34 4 H 0 H Ψr Φ Ψ T r Ψ r = Ψ r Ψ r Ψ r 734 H = Φr h 0 r δr r ˆ 0 ˆ r, r Φr ˆ 37 r, r h 0 r δr r ˆ Bogoliubov Schrodinger 9 Schrodinger h 0 r δr r ˆ 0 ˆ r, r f ν r f ν r = E ˆ r, r h 0 r δr r ˆ 0 g ν r ν g ν r f ν r g ν r 9 r r ȟ0 ˇ ˇ ȟ0r fν g ν = E ν fν g ν

5 r f ν, g ν Hφ ν = E νφν 40 ȟ0 ˇ H 4 ˇ ȟ0 fν φ ν 4 g ν E ν φ ν H 3 φ ν φ µ = δ νµ 43 f ν r g νr f µ r g µ r f µ r fν r fν r gν r gν r f µ r g µ r g µ r = δ νµ 44 = δ νµ 45 ν = µ fν r + g ν r = 46 H h 0 r δr r f ν, r + r, r g ν, r = E ν f ν, r 47 h 0 r δr r f ν, r + r, r g ν, r = E ν f ν, r 48,r, r f ν, r h 0 r δr r g ν, r = E ν g ν, r 49,r, r f ν, r h 0 r δr r g ν, r = E ν g ν, r 50 h 0 r δr r fν, r r, r gν,r = E ν fν, r 5, r, r fν,r + h 0 r δr r gν, r = E ν gν, r 5 3 5

6 E n E n f g g f E n f ν, r, f ν, r, g ν, r, g ν, r E n gν, r, g ν, r, f ν, r, f ν, r gν, r, g ν, r, f ν, r, f ν, r 5 h 0 r δr r g ν, r + r, r f ν,r = E n g ν, r 53 ar f ν r g ar = ν r ν c f ν r ν g ν r + d gν r ν fν r g ν r fν r Φr Ψ r f ν r g Φr = Ψ r Ψ r = ν r Ψ ν α f ν r ν g ν r + β gν r ν fν r 55 r g ν r fν r α ν f ν, r, f ν, r, g ν, r, g ν, r β ν gν, r, g ν, r, f ν, r, f ν, r Ψ r Ψ r = α ν f ν r + β ν gνr 56 ν Ψ r = Ψ r = ν α ν g ν r + β ν f νr 58 = ν α ν g νr + β νf ν r 59 β ν = α ν 60 Φr = ν αν a ν+ r + α ν a ν r 6 f ν r a ν+ r f ν r g ν r g ν r gν r a ν r = gν r fν r fν r

7 6 Bogoliubov-de Gennes 6 H = = ν,ν ν α ν a ν+r + α ν a h 0 r δr r ˆ 0 ˆ r, r ν r ˆ r, r h 0 r δr r ˆ 0 α ν a ν+r + α ν a h 0 r δr r ˆ 0 ˆ r, r ν r ˆ r, r h 0 r δr r ˆ 0 ν α ν a ν +r + α ν a ν r 64 α ν a ν +r + α ν a ν r = α ν a ν+r + α ν a ν r α ν E ν a ν +r α ν E ν a ν r ν,ν H = α ν a ν+r + α ν a ν r α ν E ν a ν +r α ν E ν a ν r ν,ν = Eν α να ν E ν α ν α ν ν α ν 5 H = ν E ν α να ν 69 h 0 r δr r ˆ 0 ˆ r, r ˆ r, r h 0 r δr r ˆ 0 f ν r g ν r = E ν f ν r g ν r 70 Bogoliubov-de Gennes BdG h 0 r δr r ˆ 0 ˆ r, r u ν r u ν r = E ˆ r, r h 0 r δr r ˆ 0 v ν r ν v ν r 7 s BdG h 0 r r u ν r u ν r = E ν r h 0 r v ν r v ν r 7 Aneev Josephson asano/pdf/book.pdf

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