shot noise in mesoscopic conductors - Low Temperature Laboratory

132 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
directions within the cavity if carriers conserve their energy (no inelastic scattering). On the
ensemble average the interior of the cavity can be treated as an additional dephasing voltage probe
(see Section 2.7). The interior acts as a dephasing probe since we assume that there is no inelastic
scattering at the probe. Consequently at such a probe the current in each energy interval is
conserved [90]. Thus we must consider the energy-resolved current.
Let us denote the current at contact n in an energy interval dE by J (E). The total current at the
contact is I "J dE. In terms of the distribution function of the reservoirs and the cavity the
energy-resolved current is
J (E)"eG ( f !f ) , (251)
where G "eN /(2) is the (Sharvin) conductance of the nth contact, and N "p = / is the
number of transverse channels. For energy-conserving carrier motion the sum of all currents in
each energy interval must vanish (Section 2.7). This requirement immediately gives Eq. (250).
Using the distribution function, Eq. (250), gives for the conductance matrix
G "( ! )G . (252)
This conductance matrix is symmetric, and for the two-terminal case becomes G "
(e/2)(N N /(N #N )), as expected.
Now we turn to the shot noise. The #uctuation of the current through the contact n is written as
I "!
ep dn dr dE(nN )f (r, n, E, t) , (253)
2
where and N denote the surface of the contact n and the outward normal to this contact. Here
f is the #uctuating part of the distribution function, and for further progress we must specify how
these #uctuations are correlated.
The terms with nN (0 describe #uctuations of the distribution functions of the equilibrium
reservoirs, f (E). These functions #uctuate due to partial occupation of states (equilibrium noise);
the #uctuations of course vanish for k ¹"0. The equal time correlator of these equilibrium
#uctuations quite generally is (see e.g. Ref. [275])
f (r, n, E, t)f (r, n, E, t)"(r!r)(n!n)(E!E)
fM (r, n, E, t)[1!fM (r, n, E, t)] , (254)
where in the reservoirs fM "f (E). In particular, the cross-correlations are completely suppressed.
On the other hand, the terms with nN '0 describe #uctuations of the distribution function
inside the cavity. These non-equilibrium #uctuations resemble Eq. (254) very much. Indeed, in the
absence of random scattering the only source of noise are the #uctuations of the occupation
numbers. Furthermore, in the chaotic cavity the cross-correlations should be suppressed because of
multiple random scattering inside the cavity. Thus, we assume that Eq. (254) is valid for #uctuations
of the non-equilibrium state of the cavity, where the function f (E) (250) plays the role of
fM (r, n, E, t). In contrast to the true equilibrium state, these #uctuations persist even for zero
temperature, since the average distribution function (250) di!ers from both zero and one.
Furthermore, for tOt the correlator obeys the kinetic equation, (R #v n)f (t)f (t)"0
[275]. We obtain the following formula:
f (r, n, E, t)f (r, n, E, t)"
[r!r!v n(t!t)](n!n)(E!E) f (1!f ) , (255)