Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

134 C.W.J. Beenakker
2 Andreev Reflection
The quantum mechanical description of Andreev reflection starts from a pair
of Schrödinger equations for electron and hole wave functions u(r) andv(r),
coupled by the pair potential ∆(r). These socalled Bogoliubov-De Gennes
(BdG) equations [11] take the form
� � � �
u u
HBG = E , (1)
v v
�
H ∆(r)
HBG =
∆∗ (r) −H∗ �
. (2)
The Hamiltonian H =(p + eA) 2 /2m + V − EF is the single-electron Hamiltonian
in the field of a vector potential A(r) and electrostatic potential V (r).
The excitation energy E is measured relative to the Fermi energy EF .If(u, v)
is an eigenfunction with eigenvalue E, then (−v∗ ,u∗ ) is also an eigenfunction,
with eigenvalue −E. The complete set of eigenvalues thus lies symmetrically
around zero. The quasiparticle excitation spectrum consists of all positive E.
In a uniform system with ∆(r) ≡ ∆, A(r) ≡ 0, V (r) ≡ 0, the solution of
the BdG equations is
E = � (� 2 k 2 /2m − EF ) 2 + ∆ 2� 1/2 , (3)
u(r) =(2E) −1/2 � E + � 2 k 2 /2m − EF
� 1/2 e ik·r , (4)
� 1/2 e ik·r . (5)
v(r) =(2E) −1/2 � E − � 2 k 2 /2m + EF
The excitation spectrum is continuous, with excitation gap ∆. The eigenfunctions
(u, v) are plane waves characterized by a wavevector k. The coefficients
of the plane waves are the two coherence factors of the BCS (Bardeen-Cooper-
Schrieffer) theory.
At an interface between a normal metal and a superconductor the pairing
interaction drops to zero over atomic distances at the normal side. (We assume
non-interacting electrons in the normal region.) Therefore, ∆(r) ≡ 0inthe
normal region. At the superconducting side of the NS interface, ∆(r) recovers
its bulk value ∆ only at some distance from the interface. This suppression of
∆(r) is neglected in the step-function model
�
∆ if r ∈ S,
∆(r) =
(6)
0 if r ∈ N.
The step-function pair potential is also referred to in the literature as a “rigid
boundary condition” [12]. It greatly simplifies the analysis of the problem
without changing the results in any qualitative way.
Since we will only be considering a single superconductor, the phase of
the superconducting order parameter is irrelevant and we may take ∆ real.
We refer to [13] for a tutorial introduction to mesoscopic Josephson junctions,
such as a quantum dot connected to two superconductors, and to [14] fora
comprehensive review of the current-phase relation in Josephson junctions.