shot noise in mesoscopic conductors - Low Temperature Laboratory

68 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
Quasi-elastic scattering. In contrast to inelastic scattering, dephasing processes leave the energy
essentially invariant. To simulate a scattering process which destroys phase but leaves the energy
invariant we now have to consider a special voltage probe. We require that the additional electrode
conserves not only the total current, but also the current in each small energy interval [90]. Such
a voltage probe will not give rise to energy relaxation and dissipates no energy.
From the condition that the current in each energy interval vanishes we "nd that the distribution
function in the reservoir of the voltage probe is given by
f (E)"
G f (E)#G f (E)
, (120)
G #G
where f and f are the Fermi functions at the reservoirs 1 and 2, and the conductances are
G "R
, G "R.
We have assumed again that there is no direct transmission between 1 and
2. Straightforward calculation gives for the Fano factor [90]
F" R F #R F #R R #R R . (121)
(R #R )
We analyze now this result for various situations. First, we see that for a ballistic wire divided by
a dephasing electrode into two parts, F "F "0, shot noise does not vanish (unlike Eq. (119)).
We obtain F"R R (R #R ). Thus for a ballistic system, which is ideally noiseless, dephasing
leads to the appearance of shot noise.
For the strongly biased resonant double-barrier structure, we have F "F "1, and obtain
again the result (78), which is thus insensitive to dephasing.
For metallic di!usive wires, F "F ", Eq. (121) yields for the ensemble-averaged Fano factor
F" independent on the location of the dephasing voltage probe, i.e. independent of the ratio of
R and R . Thus, our consideration indicates that the noise suppression factor for metallic di!usive
wires is also insensitive to dephasing, at least when the dephasing is local (in any point of the
sample). This result hints that the Fano factor of an ensemble of metallic di!usive wires is not
sensitive to dephasing even if the latter is uniformly distributed. Indeed, de Jong and Beenakker
[82] checked this by coupling locally a dephasing reservoir to each point of the sample. Already at
the intermediate stage their formulae coincide with those obtained classically by Nagaev [75]. This
proves that introducing dephasing with "ctitious, energy-conserving voltage probes is in the limit
of complete dephasing equivalent to the Boltzmann}Langevin approach. That inelastic scattering,
and not dephasing, is responsible for the crossover to the macroscopic regime, has been recognized
by Shimizu and Ueda [133].
The e!ect of phase breaking on the shot noise in chaotic cavities was investigated by van Langen
and one of the authors [115]. For a chaotic cavity connected to reservoirs via quantum point
contacts with N and N open quantum channels, the Fano factors vanish F "F "0, and since
R "/eN and R "/eN , the resulting Fano factor is given by Eq. (96), i.e. it is identical
with the result that is obtained from a completely phase coherent, quantum mechanical calculation.
Thus for chaotic cavities, like for metallic di!usive wires, phase breaking has no e!ect on the
ensemble-averaged noise power.
Electron heating. This is the third kind of inelastic scattering, which implies that energy can be
exchanged between electrons. Only the total energy of the electron subsystem is conserved.