shot noise in mesoscopic conductors - Low Temperature Laboratory
shot noise in mesoscopic conductors - Low Temperature Laboratory
shot noise in mesoscopic conductors - Low Temperature Laboratory
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12 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166<br />
2. Scatter<strong>in</strong>g theory of thermal and <strong>shot</strong> <strong>noise</strong><br />
2.1. Introduction<br />
In this section we present a theory of thermal and <strong>shot</strong> <strong>noise</strong> for fully phase-coherent <strong>mesoscopic</strong><br />
<strong>conductors</strong>. The discussion is based on the scatter<strong>in</strong>g approach to electrical conductance. This<br />
approach, as we will show, is conceptually simple and transparent. A phase-coherent description is<br />
needed if we consider an <strong>in</strong>dividual sample, like an Aharonov}Bohm r<strong>in</strong>g, or a quantum po<strong>in</strong>t<br />
contact. Often, however, we are <strong>in</strong>terested <strong>in</strong> characteriz<strong>in</strong>g not a s<strong>in</strong>gle sample but rather an<br />
ensemble of samples <strong>in</strong> which <strong>in</strong>dividual members di!er only <strong>in</strong> the microscopic arrangement of<br />
impurities or small variations <strong>in</strong> the shape. The ensemble-averaged conductance is typically, up to<br />
a small correction, determ<strong>in</strong>ed by a classical expression like a Drude conductance formula.<br />
Similarly, <strong>noise</strong> spectra, after ensemble averag<strong>in</strong>g, are, up to small corrections, determ<strong>in</strong>ed by<br />
purely classical expressions. In these case, there is no need to keep <strong>in</strong>formation about phases of<br />
wave functions, and <strong>shot</strong> <strong>noise</strong> expressions may be obta<strong>in</strong>ed by classical methods. Nevertheless, the<br />
generality of the scatter<strong>in</strong>g approach and its conceptual clarity, make it the desired start<strong>in</strong>g po<strong>in</strong>t of<br />
a discussion of <strong>noise</strong> <strong>in</strong> electrical <strong>conductors</strong>.<br />
Below we emphasize a discussion based on second quantization. This permits a concise<br />
treatment of the many-particle problem. Rather than <strong>in</strong>troduc<strong>in</strong>g the Pauli pr<strong>in</strong>ciple by hand, <strong>in</strong><br />
this approach it is a consequence of the underly<strong>in</strong>g symmetry of the wave functions. It lends itself<br />
to a discussion of the e!ects related to the quantum mechanical <strong>in</strong>dist<strong>in</strong>guishability of identical<br />
particles. In fact, it is an <strong>in</strong>terest<strong>in</strong>g question to what extent we can directly probe the fact that<br />
exchange of particles leaves the wave function <strong>in</strong>variant up to a sign. Thus, an important part of<br />
our discussion will focus on exchange ewects <strong>in</strong> current}current correlation spectra.<br />
We start this section with a review of #uctuations <strong>in</strong> idealized one- and two-particle scatter<strong>in</strong>g<br />
problems. This simple discussion highlights the connection between symmetry of the wave<br />
functions (the Pauli pr<strong>in</strong>ciple) and the #uctuation properties. It <strong>in</strong>troduces <strong>in</strong> a simple manner some<br />
of the basic concepts and it will be <strong>in</strong>terest<strong>in</strong>g to compare the results of the one- and two-particle<br />
scatter<strong>in</strong>g problems with the many-particle problem which we face <strong>in</strong> <strong>mesoscopic</strong> <strong>conductors</strong>.<br />
2.2. The Pauli pr<strong>in</strong>ciple<br />
The <strong>in</strong>vestigation of the <strong>noise</strong> properties of a system is <strong>in</strong>terest<strong>in</strong>g because it is fundamentally<br />
connected with the statistical properties of the entities which generate the <strong>noise</strong>. We are concerned<br />
with systems which conta<strong>in</strong> a large number of <strong>in</strong>dist<strong>in</strong>guishable particles. The fact that <strong>in</strong> quantum<br />
mechanics we cannot label di!erent particles implies that the wave function must be <strong>in</strong>variant, up<br />
to a phase, if two particles are exchanged. The <strong>in</strong>variance of the wave function under exchange<br />
of two particles implies that we deal with wave functions which are either symmetric or antisymmetric<br />
under particle exchange. (In strictly two-dimensional systems more exotic possibilities<br />
are permitted). These symmetry statements are known as the Pauli pr<strong>in</strong>ciple. Systems with<br />
symmetric (antisymmetric) wave functions are described by Bose}E<strong>in</strong>ste<strong>in</strong> (Fermi}Dirac) statistics,<br />
respectively.<br />
Prior to the discussion of the <strong>noise</strong> properties <strong>in</strong> electrical <strong>conductors</strong>, which is our central<br />
subject, <strong>in</strong> this subsection we illustrate <strong>in</strong> a simple manner the fundamental connection between the