shot noise in mesoscopic conductors - Low Temperature Laboratory

where the quantity is expressed through the isotropic part of the average distribution function fM ,
(r)" dEfM (r, E)[1!fM (r, E)] . (220)
The distribution fM obeys the equation
DfM (r, E)#IM [ fM !ln ) fM ]"0 . (221)
The standard boundary conditions for the distribution function are the following. Let ¸ denote
the area of contact n (14n4N), and the rest of the surface of the sample. At contact n, the
non-equilibrium distribution function fM is determined by the equilibrium Fermi distribution
function in the reservoir n, fM "f (E), whereas away from the contacts the current perpendicular
to the surface must vanish and thus N ) fM "0, where N is the outward normal to the surface.
Eqs. (218) and (219) can be used to "nd the current}current #uctuations if the non-equilibrium
carrier distribution is known. Thus we proceed "rst to "nd the non-equilibrium distribution
function, solving Eq. (221). We follow then Ref. [114] and "nd the characteristic potentials ,
which on the ensemble average obey the Poisson equation "0, with the boundary conditions
" , N ) "0 .
In terms of the characteristic potentials the electrostatic potential is [136]
(r)" (r)< ,
where < is the voltage applied to the reservoir n. Note that (r)"1 at every space point as
a consequence of the invariance of the electrical properties of the conductor under an arbitrary
overall voltage shift. With the help of the characteristic potentials, the conductance matrix (which
we, as before, de"ne as I "G < , I being the current through ¸ directed into the sample), we
obtain
G " dr
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 117
(the integration is carried out over the whole sample). The conductances are independent of the
electrical (non-equilibrium) potential inside the conductor. To see this one can rewrite Eq. (222) in
terms of a surface integral.
For an arbitrary conductor the electrostatic potential can be expanded as (r)" (r)< ; Ref. [136] calls the
coe$cients characteristic potentials. We remark that in the absence of inelastic processes, the average distribution
function can be written as a linear combination of the equilibrium reservoir functions f , fM (r)"
(r) f . Refs.
[110,111] call the coe$cients injectivities. In the di!usive metallic conductor of interest here the characteristic
potentials and injectivities are the same functions up to a factor given by the local density of states . Such an
equivalence does not hold, for instance, in systems composed of di!erent metallic di!usive conductors, and in general the
characteristic potentials and injectivities may have a quite di!erent functional form. Here the use of the characteristic
potentials has the advantage that it takes e!ectively the local charge neutrality into account.
(222)