shot noise in mesoscopic conductors - Low Temperature Laboratory

86 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
4. Shot noise in hybrid normal and superconducting structures
The dissipationless current (supercurrent) in superconductors is a property of a ground state, and
therefore is noiseless } it is not accompanied by any #uctuations. However, noise appears if the
superconductor is in contact with piece(s) of normal metals. This section is devoted to the
description of shot noise in these hybrid structures, which exhibit a variety of interesting phenomena.
Following the point of view of the previous sections, we present here a description which
is based on an extension of the scattering approach to hybrid structures.
4.1. Shot noise of normal-superconductor interfaces
4.1.1. Simple NS interface, scattering theory and general expressions
We consider "rst an interface of normal metal and superconductor (NS). If the applied voltage is
below the superconducting gap , the only mechanism of charge transport is Andreev re#ection at
the NS interface: an electron with energy E approaching the interface from the normal side is
converted into a hole with energy !E. The velocity of the hole is directed back from the interface
to a normal metal. The missing charge 2e on the normal side appears as a new Cooper pair on the
superconducting side. There is, of course, also a reverse process, when a Cooper pair recombines
with a hole in the normal conductor, and creates an electron. At equilibrium, both processes have
the same probability, and there is thus no net current. However, if a voltage is applied, a "nite
current #ows across the NS interface.
The scattering theory which we described in Section 2 has to be extended to take into account
the Andreev scattering processes [167]. We give here only a sketch of the derivation; a more
detailed description, as well as a comprehensive list of references, may be found in Refs. [72,168].
To set up a scattering problem, we consider the following geometry (Fig. 27): the boundary between
normal and superconducting parts is assumed to be sharp, and elastic scattering happens inside the
normal metal (shaded region) only. The scattering region is separated from the NS interface by an
ideal region 2, which is much longer than the wavelength, and thus we may there use asymptotic
expressions for the wave functions. This spatial separation of scattering from the interface is
arti"cial. It is not really necessary; our consideration leading to Eq. (159) and (160) does not rely on
it. Furthermore, for simplicity, we assume that the number of transverse channels in the normal
lead 1 and in the intermediate normal portion 2 is the same.
We proceed in much the same way as in Section 2 and de"ne the annihilation operators in
region 1 asymptotically far from the scattering area, a( , which annihilate electrons incoming on
Fig. 27. A simpli"ed model of an NS interface adopted for the scattering description. Scattering is assumed to happen in
the shaded area inside the normal metal, which separates the ideal normal parts 1 and 2. Andreev re#ection is happening
strictly at the interface separating 2 and the superconductor.