shot noise in mesoscopic conductors - Low Temperature Laboratory

12 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
2. Scattering theory of thermal and shot noise
2.1. Introduction
In this section we present a theory of thermal and shot noise for fully phase-coherent mesoscopic
conductors. The discussion is based on the scattering approach to electrical conductance. This
approach, as we will show, is conceptually simple and transparent. A phase-coherent description is
needed if we consider an individual sample, like an Aharonov}Bohm ring, or a quantum point
contact. Often, however, we are interested in characterizing not a single sample but rather an
ensemble of samples in which individual members di!er only in the microscopic arrangement of
impurities or small variations in the shape. The ensemble-averaged conductance is typically, up to
a small correction, determined by a classical expression like a Drude conductance formula.
Similarly, noise spectra, after ensemble averaging, are, up to small corrections, determined by
purely classical expressions. In these case, there is no need to keep information about phases of
wave functions, and shot noise expressions may be obtained by classical methods. Nevertheless, the
generality of the scattering approach and its conceptual clarity, make it the desired starting point of
a discussion of noise in electrical conductors.
Below we emphasize a discussion based on second quantization. This permits a concise
treatment of the many-particle problem. Rather than introducing the Pauli principle by hand, in
this approach it is a consequence of the underlying symmetry of the wave functions. It lends itself
to a discussion of the e!ects related to the quantum mechanical indistinguishability of identical
particles. In fact, it is an interesting question to what extent we can directly probe the fact that
exchange of particles leaves the wave function invariant up to a sign. Thus, an important part of
our discussion will focus on exchange ewects in current}current correlation spectra.
We start this section with a review of #uctuations in idealized one- and two-particle scattering
problems. This simple discussion highlights the connection between symmetry of the wave
functions (the Pauli principle) and the #uctuation properties. It introduces in a simple manner some
of the basic concepts and it will be interesting to compare the results of the one- and two-particle
scattering problems with the many-particle problem which we face in mesoscopic conductors.
2.2. The Pauli principle
The investigation of the noise properties of a system is interesting because it is fundamentally
connected with the statistical properties of the entities which generate the noise. We are concerned
with systems which contain a large number of indistinguishable particles. The fact that in quantum
mechanics we cannot label di!erent particles implies that the wave function must be invariant, up
to a phase, if two particles are exchanged. The invariance of the wave function under exchange
of two particles implies that we deal with wave functions which are either symmetric or antisymmetric
under particle exchange. (In strictly two-dimensional systems more exotic possibilities
are permitted). These symmetry statements are known as the Pauli principle. Systems with
symmetric (antisymmetric) wave functions are described by Bose}Einstein (Fermi}Dirac) statistics,
respectively.
Prior to the discussion of the noise properties in electrical conductors, which is our central
subject, in this subsection we illustrate in a simple manner the fundamental connection between the