shot noise in mesoscopic conductors - Low Temperature Laboratory

10 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
The transmitted and re#ected current are correlated,
S "!2
e
dE ¹fRf. (10)
2
In the limit that either ¹ is very small or f is small, the factor (1!¹f ) in Eq. (8) can be replaced
by one. In this limit, since the average current through the barrier is I"(e/2)dE ¹f, the
spectrum, Eq. (8), is Schottky's result [5] for shot noise,
S "2eI . (11)
Schottky's result corresponds to the uncorrelated arrival of particles with a distribution function of
time intervals between arrival times which is Poissonian, P(t)" exp(!t/), with being the
mean time interval between carriers. Alternatively, Eq. (11) is also referred to in the literature as the
Poisson value of shot noise.
The result, Eq. (8), is markedly di!erent from Eq. (11) since it contains, in comparison to
Schottky's expression, the extra factor (1!¹f ). This factor has the consequence that the shot noise
(8) is always smaller than the Poisson value. For truly ballistic systems (¹"1) the shot noise even
vanishes in the zero-temperature limit. As the temperature increases, in such a conductor (¹"1)
there is shot noise due to the #uctuation in the incident beam arising from the thermal #uctuations.
Eventually, at high temperatures the factor 1!f can be replaced by 1, and the ballistic conductor
exhibits Poisson noise, in accordance with Schottky's formula, Eq. (11). The full Poisson noise
given by Schottky's formula is also reached for a scatterer with very small transparency ¹;1. We
emphasize that the above statements refer to the transmitted current. In the limit ¹;1 the
re#ected current remains nearly noiseless up to high temperatures when (1!Rf ) can be replaced
by 1. We also remark that even though electron motion in vacuum tubes (the Schottky problem
[5]) is often referred to as ballistic, it is in fact a problem in which carriers have been emitted by
a source into vacuum either through thermal activation over or by tunneling through a barrier with
very small transparency.
Our discussion makes it clear that out of equilibrium, and at "nite temperatures, the noise
described by Eq. (8) contains the e!ect of both the #uctuations in the incident carrier beam as well
as the partition noise. In a transport state, noise in mesoscopic conductors has two distinct sources
which manifest themselves in the #uctuations of the occupation numbers of states: (i) thermal
#uctuations; (ii) partition noise due to the discrete nature of carriers.
Both the thermal and shot noise at low frequencies and low voltages re#ect in many situations
independent quasi-particle transport. Electrons are, however, interacting entities and both the
#uctuations at "nite frequencies and the #uctuation properties far from equilibrium require in
general a discussion of the role of the long-range Coulomb interaction. A quasi-particle picture is
no longer su$cient and collective properties of the electron system come into play.
Note that this terminology, common in mesoscopic physics, is di!erent from that used in the older literature [6],
where shot noise (due to random injection of particles into a system) and partition noise (due to random division of the
particle stream between di!erent electrodes, or by potential barriers) are two distinct independent sources of #uctuations.