19 /

19 /

1 1.1............................... 1. Andreev............................... 3 1.3..................................... 3 1.4..................................... 4 / 5.1...................................... 5. DBdG /....... 7.3............................ 8.4............................ 15.5................................. 4.6 Andreev........................... 8 3 / 3 3.1 DBdG /....... 3 3........................................... 33 4 36 37 Dirac-Bogoliubov-de Gennes DBdG.............. 37 1

1 1.1 1911 H.K.Onnes (Meissner ) (Josephson ) 1957 Bardeen Cooper Schrieffer BCS Cooper 1959 Bogoliubov de Gennes Cooper BCS 1 Bogoliubov-de Gennes BdG BdG BdG Bogoliubov-de Gennes / Andreev 1.1 Andreev Cooper Cooper Andreev Andreev BdG Josephson π /

E 1.1: Andreev 1. Andreev 1.1 / Andreev / Andreev Beenaer 3 1.4 Andreev (a) / Andreev (b) / Andreev Andreev (a) (b) e h e h x y 1.: (a) / Andreev (b) / Andreev 1.3 / / 3

Andreev 1.4 / 3 / 4 DBdG 4

/ /.1. DBdG.3.4 DBdG.5.6 Andreev.1 4, 5.1(a) 6 A B σ π 1.1(b) 6 Γ 6 K K 6 M Fermi π π 1 K K Fermi E = E F = K K. K Γ M K π.1: (a) (b) 1 5

.: π K K Dirac K ) γ (ˆ σ F K (r) = ɛf K (r) (.1) K γ (ˆ σ ) F K (r) = ɛf K (r) (.) F (r) A B ( ) ( ) F K FA K F (r) =, F K A K (r) = (.3) FB K FB K σ = (σ x, σ y ) ( ) ( 1 i σ x =, σ y = 1 i ) (.4) ˆ γ ɛ F K (r) F K (r) exp (i r) (.5) ɛ (±) () = ±γ x + y (.6) K K Dirac v = γ (.7) 6

6, 7 8. DBdG /.3 x y x > x > x < x > / x < y x.3: / N S Dirac-Bogoliubov-de Gennes DBdG DBdG H + H + u v = E u v (.8) H + = iσ µ + U = i (σ x x + σ y y ) µ + U (.9) { e iφ if x > (r) = if x < (.1) 7

{ U if x > U (r) = (.11) if x < U U µ, (.1).3 DBdG H + u u = E (.13) H + v v (u, v) exp (i x x + i y y) µ E ( x i y ) ( x + i y ) µ E = (.14) E = µ ± x + y = µ ± = E± e (.15) µ E ( x i y ) ( x + i y ) µ E = (.16) E = µ ± = E h ± (.17).4 E = E+ e = 1 e i xx+i y y u 1 u 1 e ia (.18) E = E e u 1 u = 1 1 e ia e i xx+i y y (.19) e ia = x + i y x + y (.) 8

E µ F F µ.4: 1. E > µ E = E+ e = µ + E + µ = x + y (.1) x + y = E + µ > (.) ( ) E + µ x = y > (.3) q y > (.4) E + µ ( ) q 1 E + µ cos α (.5) E + µ sin α q E + µ x = = E + µ e ia = x + i y x + y = E+µ = e iα = + iq + q = E+µ (.6) cos α > (.7) E+µ ( E+µ E+µ cos α + i sin α = cos α + i sin α 9 E+µ cos α + i sin α ) cos α + ( ) E+µ sin α (.8)

= 1 u 1 u 1 e iα x = = E + µ + iq + q e ia = x + i y = x + y = E+µ E+µ cos α + i E+µ sin α e ix+iqy (.9) cos α < (.3) = cos α + i sin α = e iα (.31) = 1 u 1 u 1 e iα e ix+iqy (.3). E < µ E = E e = µ E + µ = x + y (.33) x + y = E + µ > (.34) ( ) E + µ x = y > (.35) x = = E + µ e ia = x + i y x + y E+µ = cos α < (.36) = + iq + q cos α + i E+µ sin α = cos α + i sin α E+µ = e iα (.37) 1

= 1 u 1 u 1 e iα x = = E + µ e ia = x + i y x + y = E+µ + iq = + q E+µ e ix+iqy (.38) cos α > (.39) E+µ cos α + i sin α = cos α + i sin α = e iα (.4) = 1 u 1 u Ψ e+ N = 1 1 e iα e ix+iqy (.41) (1)E > µ ()E < µ e iα/ e iα/ Ψ e N = 1 e iα/ e iα/ eix+iqy (.4) e ix+iqy (.43) E = E+ h v 1 v = 1 e ia 1 e i xx+iqy (.44) E = E h v 1 v = 1 1 e ia e i xx+iqy (.45) e ia = x + i y x + y (.46) 11

1. E > µ E = E+ h = µ + E µ = x + y (.47) x + y = E µ > (.48) ( ) E µ x = y > (.49) E µ ( ) q 1 E µ cos α (.5) E µ sin α q E µ x = = E µ = iq + q E µ (.51) cos α > (.5) e ia = x i y x + y E µ = cos α i E µ sin α = cos α i sin α = e iα (.53) = 1 v 1 v e iα / e iα / e i x+iqy (.54) x = = E µ e ia = x i y x + y E µ = = iq + q E µ cos α < (.55) cos α + i E µ sin α = cos α i sin α = e iα (.56) 1

= 1 v 1 v e iα / e iα / e i x+iqy (.57). E < µ E = E h = µ E µ = x + y (.58) x + y = E µ > (.59) ( ) E µ x = y > (.6) x = = E µ = + iq + q E µ cos α < (.61) e ia = x + i y x + y E µ = cos α + i E µ sin α = cos α i sin α = e iα (.6) = 1 v 1 v e iα / e iα / e i x+iqy (.63) x = = E µ e ia = x + i y x + y = + iq + q E µ cos α > (.64) = E µ cos α + i E µ sin α = cos α i sin α = e iα (.65) 13

= 1 e i x+iqy v 1 v e iα / e iα / (.66) (1)E > µ ()E < µ Ψ h+ N = 1 x+iqy e iα / ei (.67) e iα / Ψ h N = 1 e iα / e iα / e i x+iqy (.68) Ψ e+ N = 1 e iα/ e iα/ eix+iqy (.69) Ψ e N = 1 e iα/ e iα/ e ix+iqy (.7) Ψ h+ N = 1 Ψ h N = 1 e iα / e iα / e iα / e iα / ei x+iqy e i x+iqy (.71) (.7) 14

5.78 5.15 q < q c = E + µ (.73) q < q c = E µ (.74) q = E + µ sin α (.75) Andreev ( ) E µ α < α c = arcsin E + µ α c (.76).4 DBdG H + H + u v = E u v (.77) H + = iσ µ + U = i (σ x x + σ y y ) µ + U (.78) (u, v) exp (i x x + i y y) u = (u 1, u ) v = (v 1, v ) µ + U E ( x i y ) ( x + i y ) µ + U E = (.79) µ U E ( x i y ) ( x + i y ) µ U E E S ± = E = + (µ U ± ) E S ± (.8) + ξ ±, ξ ± = ± (µ + U ) (.81).5 U µ, E = E S E = ES a e ( ) ia E S + ξ b ( ) c E S + ξ e ia (.8) d 15

E U + µ >>.5: (1) E < Ω E (.83) ( ) E Ω = i 1 (.84) E cos β (.85) 1 < E/ < 1 < β < π sin β > ( ) E Ω = i 1 = i sin β (.86) E = + ξ (.87) ( ) E ξ = ± E = ±i 1 = ±i sin β = (µ + U ) (.88) 16

= µ + U ± i sin β (.89) + ( ) µ + x U ± i sin β = q = (µ + U ) sin β () (µ + U ) () q ± i (µ + U ) sin β () q ± i (µ + U ) sin β () (.9) U y q (.91) (µ + U ) () q (.9) κ (µ + U ) () sin β (.93) x = ( 1 ± i ) (µ + U ) () sin β (.94) x ± i (µ + U ) () sin β = ± iκ (.95) sin γ q µ + U, π < γ < π (.96) = µ + U ( q 1 µ + U ) = µ + U cos γ (.97) x κ 17

1. ( + iκ, q) κ e ia = x + i y x + y = = + iq + q = µ+u + i ( µ+u ) + ( µ+u q ) q µ+u cos γ + i sin γ cos γ + sin γ = eiγ (.98) a e ( ) ia E S + ξ e b ( ) iγ (E + i sin β) c E S + ξ e ia (E + i sin β) e iγ d (.99) a + b + c + d = ( E + sin β ) + ( ) E = + sin β + 1 = 4 (.1) ( ) a e iγ E b c 1 + i sin β ( ) E + i sin β e iγ e iφ = 1 d e iφ e iγ e iβ e iβ e iγ e iφ e iφ e iβ Ψ e+ S = 1 e iγ+iβ e iφ ei x κx+iqy e iγ iφ (.11) (.1). ( iκ, q) κ e ia = + iq + q = cos γ + i sin γ = e iγ (.13) 18

a e iγ e iβ b c 1 e iβ e iγ e iφ d e iφ e iβ e iβ iγ Ψ e S = 1 e iφ e iφ iγ e i x+κx+iqy (.14) (.15) 3. ( + iκ, q) κ e ia = iq + q = cos γ i sin γ = e iγ (.16) ( ) a e iγ E b c 1 i sin β ( ) e iγ e iβ E i sin β e iγ e iφ = 1 e iβ e iγ e iφ d e iφ Ψ h+ S = 1 e iφ e iβ e iβ iγ e iφ e iφ iγ e i x κx+iqy (.17) (.18) 4. ( iκ, q) κ e ia = iq + q = cos γ i sin γ = e iγ (.19) 19

a e iγ e iβ b c 1 e iβ e iγ e iφ d e iφ e iβ Ψ h S = 1 e iγ iβ e iφ e i x+κx+iqy e iγ iφ (.11) (.111) () E > E = E S = + ( µ U ) (.11) ( µ U ) = E > (.113) = ± E + µ + U (.114) + ( x + y µ + U ± ) E = (.115) x = ( µ + U ) y ± (µ + U ) E () + E () = ± (µ + U ) E () + E () (.116) 1 ± (µ + U ) E () = ± (µ + U ) E () (.117) ( ) E cosh δ δ > ( ) E δ = arcosh (.118)

cosh δ sinh δ = 1 (.119) E = ( E ) 1 = sinh δ (.1) = + (µ + U ) sinh δ () (.11) cosh δ = cos (iδ) (.1) sinh δ = i sinh (iδ) (.13) iδ = β (.14) cosh δ = cos β (.15) sinh δ = i sinh β (.16) = + (µ + U ) sinh δ () = + i (µ + U ) sin β () = + iκ (.17) E < E < x κ 1. ( + iκ, q) κ e ia = x + i y x + y = = + iq + q = µ+u + i ( µ+u ) + ( µ+u q ) q µ+u cos γ + i sin γ cos γ + sin γ = eiγ (.18) 1

E a b c 1 ( + sinh ) δ E + sinh δ e iγ e iφ = 1 d e iφ e iγ e iβ 1 e iβ e iγ e iφ e iφ e iγ e iβ Ψ e+ S = 1 e iγ+iβ e iφ ei x κx+iqy e iγ iφ cosh δ + sinh δ (cosh δ + sinh δ) e iγ e iφ e iφ e iγ (.19) (.13). ( iκ, q) κ e ia = + iq + q = cos γ + i sin γ = e iγ (.131) a e iγ e iβ b c 1 e iβ e iγ e iφ d e iφ e iβ e iβ iγ Ψ e S = 1 e iφ e iφ iγ e i x+κx+iqy (.13) (.133) 3. ( + iκ, q) κ e ia = iq + q = cos γ i sin γ = e iγ (.134)

a e iγ (cos β i sin β) e iγ e iβ b c 1 (cos β i sin β) e iγ e iφ = 1 e iβ e iγ e iφ d e iφ e iφ e iβ e iβ iγ Ψ h+ S = 1 e iφ e iφ iγ e i x κx+iqy (.135) (.136) 4. ( iκ, q) κ e ia = iq + q = cos γ i sin γ = e iγ (.137) a e iγ e iβ b c 1 e iβ e iγ e iφ d e iφ e iβ Ψ h S = 1 e iγ iβ e iφ e i x+κx+iqy e iγ iφ (.138) (.139) U µ, γ = arcsin q/ (U + µ) e iβ Ψ e+ S = 1 e iβ e iφ ei x κx+iqy (.14) e iφ 3

Ψ h+ S = 1 e iβ e iβ e iφ e iφ e i x κx+iqy (.141).5 / Andreev ψ N = Ψ e+ N + rψe N + r AΨ h N (.14) U µ, E = E S ψ S = aψ e+ S + bψh+ S (.143) x = Ψ e+ N + rψe N + r AΨ h N = aψe+ S + bψh+ S (.144) Ψ h+ N + r Ψ h N + r AΨ e S = a Ψ e+ S + b Ψ h+ S (.145) Andreev { e iφ X 1 cos α if α < α c r A = (.146) if α > α c α c = arcsin ( ) E µ E + µ (.147) ( ) ( ) α r = ix cos 1 + α α α β sin i sin β sin (.148) { r A e iφ X 1 cos α if α < α c = (.149) if α > α c 4

( ) ( ) α r = ix cos 1 + α α α β sin + i sin β sin ( ) ( ) α α α + α X = cos β cos + i sin β cos (.15) (.151) x vx+ e = 1 E+ e = v + q = v E + µ = v cos α (.15) vx e = 1 E e = v = v cos α (.153) + q vx+ h = 1 E+ h vx h = 1 E h = v + q = v E µ = v cos α (.154) = v + q = v cos α (.155) ( ) ( ) vx e r = α cos r = r = + α α α ix 1 β sin i sin β sin (.156) r A = r = v e x+ vx h vx+ e vx h vx+ h r A = { cos α cos α r A = e iφ X 1 cos α cos α if α < α c if α > α c (.157) ( ) ( ) α r = r = ix cos 1 + α α α β sin + i sin β sin r A = vx e vx+ h r A = ( ) r r A R = r A r (.158) cos α cos α r A = e iφ r A (.159) (.16) E < unitary ( RR ) = δ nm nm E > unitary ( ) q α = arcsin (.161) E + µ ( ) q α = arcsin E µ µ µ 5 (.16)

1. µ ( q α = arcsin µ ), α = arcsin ( ) q µ (.163) α = α (.164). µ α = arcsin ( ) q, α = arcsin E ( ) q E (.165) α = α (.166) µ α = α Andreev µ α = α Andreev.6.7 E E U + µ >>, E r µ r A F F.6: µ ( Andreev ) Andreev r A (E, α) = r A (E, α) = e iφ cos α (E/ ) cos α + ζ, if µ (.167) e iφ cos α E/ + ζ cos α, if µ (.168) 6

E E U + µ >> r r A µ F F.7: µ ( Andreev ) r (E, α) = iζ sin α (E/ ) cos α + ζ, if µ (.169) r (E, α) = i (E/ ) sin α E/ + ζ cos α, if µ (.17) ( ) E 1 if E > ζ = ( ) (.171) E i 1 if E < E < r + r A = 1 α = r A = 1 Blonder-Tinham-Klapwij BTK I αc V = g ( (V ) 1 r (ev, α) + r A (ev, α) ) cos αdα, (.17) g (V ) = 4e h N (ev ), (µ + E) W N (E) =, α c = arcsin π ( ) E µ E + µ (.173) g N W y µ µ α c = arcsin (1) = π (.174) α < π (.175) 7

µ ev/ x < 1 I V 1 g (V ) = = π/ { 4 ( 1 r (ev, α) + r A (ev, α) ) cos αdα if x = 3 1 + ( 1 1 ) 1 ln x 1 if < x < 1 x x x x+1 (.176) ev/ x 1 I V 1 g (V ) = = π/ ( 1 r (ev, α) + r A (ev, α) ) cos αdα { if x = 1 π x 1 + x 1 ln x x x+1 if x > 1 x 1 (.177) µ ev/ x < 1 I V 1 g (V ) = = π/ ( 1 r (ev, α) + r A (ev, α) ) cos αdα { if x = 1 x 1 if < x < 1 (.178) 1 x + 1 x (1 x ) 1 x ln 1 x +1 ev/ x 1 I V 1 g (V ) = = π/ ( 1 r (ev, α) + r A (ev, α) ) cos αdα 4 if x = 1 3 ( ) πχ + 4χ πχ 3 + 4χ χ 1 arcsin if x > 1 χ 1 χ (.179) χ = x x 1 (.18).8.6 Andreev E = E e ± = µ ± (.181) E = E h ± = µ ± (.18) 8

( V ) 1 g I V µ << µ >> ( ) ev.8: µ Andreev µ Andreev 1. y vy e+ = 1 E e E+µ + q = v q x + q = v sin α = v sin α (.183) v e y v h+ y v h y = 1 E e q = v q = 1 E+ h q = v q = 1 E h q = v q E+µ E+µ x + q = v E+µ E µ x + q = v E µ E µ x + q = v E µ sin α = v sin α (.184) sin α = v sin α (.185) sin α = v sin α (.186).5 y 1. µ α = α Andreev hole y Andreev. µ α = α Andreev hole y Andreev α α Andreev Andreev 9

3 / 3 / / 3.1 DBdG / 3.1 / x < x < y x 3.1: / F S DBdG DBdG V ex (r) H + + V ex (r) H + + V ex (r) u v = E u v (3.1) H + = iσ µ + U = i (σ x x + σ y y ) µ + U (3.) V ex (r) = { if x > V ex if x < 3 (3.3)

(r) U (r) { e iφ if x > (r) = if x < U (r) = { U if x > if x < (3.4) (3.5) U µ, (3.6) (u, v) exp (i x x + i y y) µ E + V ex ( x i y ) ( x + i y ) µ E + V ex = (3.7) E = E e ± = µ + V ex ± (3.8) µ E + V ex ( x i y ) ( x + i y ) µ E + V ex = (3.9) E = E h ± = µ + V ex ± (3.1) 3. E µ +V ex µ +V ex 3.: Andreev V ex E E V ex (3.11) 31

α,, α, α α = arcsin = arcsin q E + µ V ex q E µ V ex, = E + µ V ex cos α, = E µ V ex cos α (3.1) / Andreev N ( ) ( ) α r = ix cos 1 + α α α β sin i sin β sin (3.13) { e iφ X 1 cos α cos α r A = if α < α c if α > α c (3.14) ( ) ( ) α α α + α X = cos β cos + i sin β cos (3.15) α c ( ) E µ Vex, α c = arcsin (3.16) E + µ V ex N (E) = µ + E V ex W π DBdG H + V ex (r) H + V ex (r) u v = E u v (3.17) E = E e ± = µ V ex ± (3.18) E = E h ± = µ V ex ± (3.19) 3.3 E E + V ex (3.) α,, α, q α = arcsin, = E + µ + V ex cos α E + µ + V ex α q = arcsin, = E µ + V ex cos α (3.1) E µ + V ex 3

E µ V ex µ V ex 3.3: Andreev φ φ + π (3.) φ Andreev { e i(φ+π) X 1 cos α cos α r A = if α < α c (3.3) if α > α c N N + (E) = µ + E + V ex W π α c ( ) E µ + Vex, α c+ = arcsin (3.4) E + µ + V ex 3. BTK I V = e h N s (ev ) s=±1 αc N s (E) = µ + E + sv ex W π ( 1 r (ev, α) + r A (ev, α) ) cos αdα, (3.5), α cs = arcsin ( ) E µ + svex E + µ + sv ex (3.6) s = ±1 Fermi µ 1. K K Fermi Fermi s = ±1 α α α = α (3.7) 33

.6 Andreev α cs α cs = arcsin (1) = π (3.8) I V = e h W π ( ev + V ex + ev V ex ) N s (E) = µ + E + sv ex W π π/, α cs = arcsin ( 1 r (ev, α) + r A (ev, α) ) cos αdα, ( ) E µ + svex E + µ + sv ex (3.9) (3.3) α = α / ev/ x < 1 π/ ( 1 r (ev, α) + r A (ev, α) ) cos αdα = { if x = 1 x 1 if < x < 1 1 x + 1 x (1 x ) 1 x ln 1 x +1 (3.31) ev/ x 1 π/ = ( 1 r (ev, α) + r A (ev, α) ) cos αdα 4 if x = 1 3 ( ) πχ + 4χ πχ 3 + 4χ χ 1 arcsin if x > 1 χ 1 χ (3.3) χ = x x 1 (3.33) ev > V ex ev + V ex + ev V ex = ev (3.34) < V ex ev + V ex + ev V ex = V ex (3.35) 34

/ < V ex Andreev 9 / V ex V ex = µ / (3.35) I V V ex (3.36) 35

4 / Fermi µ α α α = α Andreev < V ex µ 36

Dirac-Bogoliubov-de Gennes DBdG Fermi K K ψ (r, µ, α) = 1 } {F µ (r, α) e ik r + G µ (r, α) e ik r = 1 { Fµ (r, α) e ik r + G µ (r, α) e ik r} (1) ψ (r, ν, α) = 1 {F ν (r, α) e ik r + G ν (r, α) e ik r } () µ, ν = A, B (3) K = K K K ( ) F A (r, α) ˆF (r, α) = (4) F B (r, α) Ĝ (r, α) = ( G A (r, α) G B (r, α) ) (5) α, β =, (6) { Fµ (r, α), F ν (r, β) } + = δ µ,ν δ (r r ) δ α,β (7) {F µ (r, α), F ν (r, β)} + = (8) { Gµ (r, α), G ν (r, β) } + = δ µ,ν δ (r r ) δ α,β (9) {G µ (r, α), G ν (r, β)} + = (1) {F µ (r, α), G ν (r, β)} + = (11) { Fµ (r, α), G ν (r, β) } + = (1) δ µ,ν δ (r r ) = δ (r r ) (13) 37

{ ψµ (r, α), ψ ν (r, β) } + = δ µ,ν δ (r r ) δ α,β (14) {ψ µ (r, α), ψ ν (r, β)} + = (15) K K Hamiltonian Dirac Hamiltonian Ĥ + = iσ D µ = i (σ x D x + σ y D y ) µ (16) Ĥ = iσ D µ = i (σ x D x σ y D y ) µ (17) ± K K D = i e A ~c v µ Fermi Schrödinger Ĥ + Ĥ Ĥ + Ĥ ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) = E ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) V Hamiltonian H 1 = dr ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) Ĥ + + V Ĥ + V Ĥ + V Ĥ V dr ˆF ) (r, α) (Ĥ+ + V ˆF (r, α) = dr ˆF (r, α) ( iσ D) ˆF (r, α) + 1 dr ˆF (r, α) (σ D) ˆF (r, α) = dr dr F A (r, α) F B (r, α) δ (r r ) ( ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) (18) (19) dr ˆF (r, α) ( µ + V ) ˆF (r, α) () D x id y D x + id y ) F A (r, α) F B (r, α) r = dr dr δ (r r ) (D x + id y ) r F B (r, α) F A (r, α) + (D x id y ) r F A (r, α) F B (r, α) = dr {(D x + id y ) F A (r, α)} F B (r, α) + {(D x id y ) F B (r, α)} F A (r, α) = dr F A (r, α) ( ) Dx idy F B (r, α) + F B (r, α) ( ) Dx + idy F A (r, α) = dr ˆF { T T (r, α) (σ D ) ˆF (r, α)} (1) 38

dr ˆF (r, α) ( µ + V ) ˆF (r, α) = dr ˆF { T T (r, α) ( µ + V ) ˆF (r, α)} () dr ˆF ) (r, α) (Ĥ+ + V ˆF (r, α) = = dr ˆF { } T T (r, α) iσ D ( µ + V ) ˆF (r, α) dr ˆF ( ) { T T (r, α) Ĥ + V ˆF (r, α)} (3) ) drĝ (r, α) (Ĥ + V Ĝ (r, α) = ( ) {Ĝ T drĝt (r, α) Ĥ V (r, α)} (4) H 1 = 1 dr ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T H + + V H + V H + V H V H+ V H V H+ + V H + V ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T (5) H = 1 dr α,β dr ψ α (r) ψ β (r ) g (r r ) ψ β (r ) ψ α (r) (6) spin-singlet g (r r ) = gδ (r r ) = gδ µ,ν δ (r r ) (7) g 39

H = 1 = g α drψ (r, µ, α) ψ (r, µ, ᾱ) gψ (r, µ, ᾱ) ψ (r, µ, α) dr{ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) = g +ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, )} dr { ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) } (8) = g ψ (r, µ, ) ψ (r, µ, ) (9) = g ψ (r, µ, ) ψ (r, µ, ) (3) ψ (r, µ, ) ψ (r, µ, ) g + ψ (r, µ, ) ψ (r, µ, ) g + ψ (r, µ, ) ψ (r, µ, ) g ψ (r, µ, ) ψ (r, µ, ) g (31) (3)... 1 ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) ψ (r, µ, ) = g ψ (r, µ, ) ψ (r, µ, ) + g ψ (r, µ, ) ψ (r, µ, ) g (33) H = dr { ψ (r, µ, ) ψ (r, µ, ) + ψ (r, µ, ) ψ (r, µ, ) } = 1 dr { F µ (r, ) e ik r + G µ (r, ) e ik r} { F µ (r, ) e ik r + G µ (r, ) e ik r} + { F µ (r, ) e ik r + G µ (r, ) e ik r} { F µ (r, ) e ik r + G µ (r, ) e ik r} (34) = 1 dr {F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, )} + { F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, ) } + { F µ (r, ) F µ (r, ) e ik r + G µ (r, ) G µ (r, ) e ik r} + { F µ (r, ) F µ (r, ) e ik r + G µ (r, ) G µ (r, ) e ik r} (35) F F, GG 4

H = 1 dr {F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, )} = 1 + { F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, ) } (36) dr {F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, ) G µ (r, ) F µ (r, ) F µ (r, ) G µ (r, ) } + {F µ (r, ) G µ (r, ) + G µ (r, ) F µ (r, ) = 1 G µ (r, ) F µ (r, ) F µ (r, ) G µ (r, ) } (37) dr ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T (38) Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T Hamiltonian H = H 1 + H = 1 dr ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T H + + V H + V H + V H V H+ V H V H+ + V H + V ˆF (r, ) Ĝ (r, ) ˆF (r, ) Ĝ (r, ) ˆF (r, ) T Ĝ (r, ) T ˆF (r, ) T Ĝ (r, ) T (39) (1,1) (1,8) (8,1) (8,8) BCS Hamiltonian 41

H BCS = dr ˆF (r, ) Ĝ (r, ) T H + + V ˆF (r, ) H + V Ĝ (r, ) T (4) ˆF (r, ) ˆf (r, ) = e i r (41) Ĝ (r, ) ĝ (r, ) Hamiltonian H + V = iσ D + µ + V iσ D + µ + V (4) D = (43) H + V iσ + µ + V = H + + V (44) BdG H + + V u u = E H + + V v v (45) u = φ A φ B +, v = φ A φ B (46) φ A B K K Andreev (,) (,7) (7,) (7,7) BCS Hamiltonian H BCS = dr Ĝ (r, ) ˆF (r, ) T H + V Ĝ (r, ) H+ + V ˆF (r, ) T (47) Ĝ (r, ) ˆF (r, ) = ĝ (r, ) ˆf (r, ) e i r (48) 4

H+ + V = iσ D + µ + V iσ D + µ + V (49) H+ + V H + V (5) BdG H + V u u = E H + V v v (51) u = φ A φ B, v = φ A φ B + (5) K K Andreev BdG H ± + V u u = E (53) H ± + V v v BdG H ± V u u = E (54) H ± V v v BdG H ± Dirac Hamiltonian Dirac-Bogoliubov-de Gennes DBdG K K 43

44

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