shot noise in mesoscopic conductors - Low Temperature Laboratory

one "rst goes from the non-interacting regime F" to the hot-electron regime, F"3/4. For
even longer wires electron}phonon collisions play a role, and the Fano factor decreases down
to zero.
6.4. Frequency dependence of shot noise
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 123
While discussing the classical approach to the shot noise suppression, we explicitly assumed that
the sample is locally charge neutral: charge pile-up is not allowed in any volume of any size. As
a result, we obtained white (frequency independent) shot noise.
In reality, however, there is always a "nite (though small) screening radius, which in the case
when the system is locally three-dimensional has the form "(4e). As Naveh et al. [140]
and Turlakov [257] point out, if one of the dimensions of a disordered sample becomes comparable
with , the pile-up of the charge may modify the frequency dependence of noise, though it leaves
the zero-frequency noise power unchanged. If all the dimensions of the sample exceed the screening
radius (which is typically the case for metallic, and often also for semiconducting) mesoscopic
systems, the charge pile-up inside the sample is negligible, and noise stays frequency independent
until at least the plasma frequency, which in three-dimensional structures is very high.
The situation is di!erent if the sample is capacitively connected to an external gate. As we have
seen in the framework of the scattering approach (Section 3), the fact that the sample is now
charged, leaves the zero-frequency noise unchanged, but strongly a!ects the frequency dependence
of the shot noise. An advantage of the Boltzmann}Langevin approach is that it can treat these
e!ects analytically, calculating the potential distribution inside the sample and making use of it to
treat the current #uctuations. The general program is as follows. Instead of the charge neutrality
condition, one uses the full Poisson equation, relating potential and density #uctuations inside the
sample. In their turn, density #uctuations are related to the current #uctuations via the continuity
equation. Finally, one expresses the current #uctuations via those of the potential and the Langevin
sources. Thus, the system of coupled partial di!erential equations with appropriate boundary
conditions needs to be solved. The solution is strongly geometry dependent and has not been
written down for an arbitrary geometry. Particular cases, with a simple geometry, were considered
by Naveh et al. [140,142], Nagaev [141,143], and Naveh [144,145]. Without even attempting to
give derivations, we describe here the main results, referring the reader to these papers for more
details.
A conductor in proximity to a gate can be charged vis-a` -vis the gate. We can view the conductor
and the gate as the two plates of a capacitor. In the limit where the screening length is much
larger than the wire radius, only a one-dimensional theory is needed. It is this atypical situation
which is considered here. For a wire of cross-section A and a geometrical capacitance c per unit
length its low-frequency dynamics is characterized by the electrochemical capacitance
c"c#(eA)
which is the parallel addition of the geometrical capacitance and the
quantum capacitance (eA). Here we have assumed that the potential is uniform both along the
wire and more importantly also in the transverse direction of the wire. Any charge accumulated in
the wire can dynamically relax via the reservoirs connected to the wire and via the external circuit
For a discussion of electrochemical capacitance and ac conductance see Refs. [157,165]; see also Section 3.