shot noise in mesoscopic conductors - Low Temperature Laboratory

non-thermalized electrons, the high-frequency noise (< ) inside the conductor grows even for
zero temperature, S(x)J(/ )[x(¸!x)]. At the contacts this noise power turns to zero, and
one obtains S ,S(x"0)"eI. Note that this value is the same as for a double-barrier
structure with symmetric capacitances. The di!erence with the result S "2eI originates from
the fact that in this mode the charge pile-up in the contact is forbidden. In addition, the frequencydependent
noise in the same model is studied numerically by Naveh for electron}electron [144]
and electron}phonon [145] interactions.
6.5. Shot noise in non-degenerate conductors
6.5.1. Diwusive conductors
Recently, GonzaH lez et al. [258], motivated by the diversity of proofs for the noise suppression in
metallic di!usive wires, performed numerical simulations of shot noise in non-degenerate di!usive
conductors. They have taken the interaction e!ects into account via the Poisson equation. For the
disorder a simple model with an energy-independent scattering time was assumed. They found
that the noise suppression factor for this system, within the error bars, equals and in three- and
two-dimensional systems, respectively. A subsequent work [259] gives the suppression factor 0.7
for d"1.
An analytic theory of shot noise in these conductors was proposed by Beenakker [260], and
subsequently by Nagaev [261] and Schomerus et al. [262,263]. The general conclusion is as
follows. Shot noise in non-degenerate di!usive conductors is non-universal in the sense that it
depends on the details of the disorder (the energy dependence of the elastic scattering time) and the
geometry of the sample. In the particular case, when the elastic relaxation time is energy
independent (corresponding to the simulations by GonzaH lez et al. [258,259]), the suppression
factors are close to , and 0.7, but not precisely equal to these values. Below we give a brief sketch
of the derivation, following Ref. [262].
We start from the Boltzmann equation (203) and add Langevin sources on the right-hand side.
Three points now require special attention: (i) for a non-degenerate gas fM ;1, and thus the factors
(1!fM ) in the collision integral (204) can be taken equal to 1; (ii) since carriers are now distributed
over a wide energy range, all the quantities must be taken energy-dependent rather than restricted
to the Fermi surface. In particular, the velocity is v(E)"(2E/m), and the energy dependence of
the mean-free path matters, (iii) the screening length is energy dependent, and the system generally
may not be regarded as charge neutral. We employ again the -approximation for the electronimpurity
collision integral. For a while, we assume the relaxation time to be energy independent.
We also neglect all kinds of inelastic scattering. Separating the distribution function into symmetric
and asymmetric parts (216), and introducing the (energy-resolved) charge and current densities,
(r, E, t)"e(E) f (r, E, t) ,
j(r, E, t)" 1
d ev(E)(E) f (r, E, t) ,
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 125
we obtain two equations, analogous to Eq. (217),
) j#eE(r, t)R j"0 (233)