shot noise in mesoscopic conductors - Low Temperature Laboratory

128 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
A similar problem, a ballistic degenerate conductor in the presence of a nearby gate, was posed
by Naveh et al. [269].
We conclude this subsection by recalling that the e!ect of interactions on noise arises not only in
long ballistic structures but already in samples which are e!ectively zero dimensional, like resonant
tunneling diodes [78] or in quantum point contacts [98]. The added complication in extended
structures arises from the long-range nature of the Coulomb interaction.
6.6. Boltzmann}Langevin method for shot noise suppression in chaotic cavities
with diwusive boundary scattering
Now we turn to the classical derivation of the -shot noise suppression in chaotic cavities. In
standard cavities, which are regular objects, the chaotic dynamics arises due to the complicated
shape of a surface. Thus, scattering at the surface is deterministic and in an individual ensemble
member scattering along the surface of the cavity is noiseless. Thus, it is not obvious how to apply
the Boltzmann}Langevin equation.
However, recently a model of a random billiard } a circular billiard with di!usive boundary
scattering } was proposed [270,271] to emulate the behavior of chaotic cavities. It turned out
that the model can be relatively easily dealt with, and Refs. [270,271] used it to study spectral and
eigenfunction properties of closed systems. Ref. [99] suggests that the same model may be used to
study the transport properties of the open chaotic cavities and presents the theory of shot noise
based on the Boltzmann}Langevin approach.
We consider a circular cavity of radius R connected to the two reservoirs via ideal leads; the
angular positions of the leads are ! /2(( # /2 (left) and ! /2(( /2 (right),
see Fig. 34; is the polar angle. The contacts are assumed to be narrow, ;1, though the
numbers of the transverse channels, N "p R /, are still assumed to be large compared to
1. Inside the cavity, motion is ballistic, and the average distribution function fM (r, n) obeys the
equation
nfM (r, n)"0 . (242)
At the surface (denoted by ) we can choose a di!usive boundary condition: the distribution
function of the particles backscattered from the surface is constant (independent of n) and "xed by
the condition of current conservation,
fM (r, n)" (Nn) fM (r, n)dn, Nn(0 , (243)
Nn where r3, N is the outward normal to the surface, and dn"1. Furthermore, we assume that the
electrons coming from the leads are described by the equilibrium distribution functions, and are
Earlier, a similar model was numerically implemented to study spectral statistics in closed square billiards [272,273].
This is the simplest possible boundary condition of this kind. For a review, see Ref. [274]. We expect similar results
for any other di!usive boundary condition.