NS interface. This effect is fully included here, even though Ψ does not appear explicitly in the expressions which follow. The reason is that the order parameter quantifies the degree of phase coherence between electrons and holes, but does not itself affect the dynamics of the quasiparticles. (The BdG equation (21.12) contains ∆ not Ψ.)
21.3. N-S INTERFACE 415 (The mode-index n has been suppressed for simplicity of notation.) The reflected and transmitted wave has vector of coefficients c out N ≡ ( c - e (N 1 ) , c + e (N 2 ) , c + h (N 1 ) , c - h (N 2 ) ) . (21.23) The ˆ s -matrix s N of the normal region relates these two vectors, c out N = ˆ s N c in N . (21.24) Because the normal region does not couple electrons and holes, this matrix has the block- diagonal form ˆ s N ( ε ) = ˆ s 0 ( ε ) 0 0 ˆ s 0 ( - ε ) * , ˆ s 0 ≡ r 11 t 12 t 21 r 22 . (21.25) Here ˆ s 0 is the unitary ˆ s -matrix associated with the single-electron Hamiltonian H 0 . The reflection and transmission matrices r ( ε ) and t ( ε ) are N × N matrices, N ( ε ) being the number of propagating modes at energy ε . (We assume for simplicity that the number of modes in leads N 1 and N 2 is the same.) The matrix s 0 is unitary (ˆ s † 0 ˆ s 0 = 1) and satisfies the symmetry relation ˆ s 0 ( ε, B ) ij = ˆ s 0 ( ε, - B ) ji . Andreev reflection For energies 0 < ε < ∆ 0 there are no propagating modes in the superconductor. We can then define an ˆ s -matrix for Andreev reflection at the NS interface which relates the vector of coefficients ( c - e (N 2 ) , c + h (N 2 ) ) to ( c + e (N 2 ) , c - h (N 2 ) ) . The elements of this ˆ s -matrix can be obtained by matching the wavefunctions (21.19) at x = 0 to the decaying solutions in S of the BdG equation. If terms of order ∆ 0 / F are neglected (the so-called Andreev approximation, the result is simply c - e (N 2 ) = α e i φ c - h (N 2 ) , c + h (N 2 ) = α e - i φ c + e (N 2 ) , (21.26) where α ≡ exp[ - i arccos( ε/ ∆ 0 )]. Andreev reflection transforms an electron mode into a hole mode, without change of mode index. The transformation is accompanied by a phase shift, which consists of two parts: 1. A phase shift - arccos( ε/ ∆ 0 ) due to the penetration of the wavefunction into the superconductor. 2. A phase shift equal to plus or minus the phase of the pair potential in the supercon- ductor ( plus for reflection from hole to electron, minus for the reverse process).
416 CHAPTER 21. MESOSCOPIC SUPERCONDUCTIVITY General relations We can combine the 2 N linear relations (21.26) with the 4 N relations (21.24) to obtain a set of 2 N linear relations between the incident wave in lead N 1 and the reflected wave in the same lead: c - e (N 1 ) = ˆ s ee c + e (N 1 ) + ˆ s eh c - h (N 1 ) , c + h (N 1 ) = ˆ s he c + e (N 1 ) + ˆ s hh c - h (N 1 ) . (21.27) The four N × N matrices ˆ s ee , ˆ s hh , ˆ s eh , and ˆ s he form together the scattering matrix ˆ s of the whole system for energies 0 < ε < ∆ 0 . An electron incident in lead N 1 is reflected either as an electron (with scattering amplitudes ˆ s ee ) or as a hole (with scattering amplitudes ˆ s he ).
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