shot noise in mesoscopic conductors - Low Temperature Laboratory

where
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 115
G( p, p, r, t)" ( p!p) dp[JM ( p, p, r, t)#JM ( p, p, r, t)]
! [JM ( p, p, r, t)#JM ( p, p, r, t)] . (210)
Note that the sum rule
dp G( p, p, r, t)" dp G( p, p, r, t)"0 (211)
is ful"lled. This sum rule states that the #uctuations only redistribute the electrons over di!erent
states, but do not change the total number of particles.
These equations can be further simpli"ed in the important case (which is the main interest in this
Section) when all the quantities are sharply peaked around the Fermi energy. Instead of the
momentum p, we introduce then the energy E and the direction of the momentum n"p/p. The
velocity and the density of states are assumed to be constant and equal to v and , respectively.
We write
=I ( p, p, r)"(E!E)=(n,
n, r) ,
where = is the probability of scattering from the state n to the state n per unit time at the space
point r. Furthermore, we will be interested only in the stationary regime, i.e. the averages fM and
JM (not the #uctuating parts) are assumed to be time independent. Eliminating the electric "eld E
by the substitution EPE!e(r), with being the potential, we write the Boltzmann}Langevin
equation in the form
(R #v n) f (r, n, E, t)"I[ f ]#I [ f ]#(r, n, E, t) , (212)
where the Langevin sources are zero on average and are correlated as follows:
(r, n, E, t)(r, n, E, t)" 1
(r!r)(t!t)(E!E)G(n, n, r, E) , (213)
G(n, n, r, E), dn[(n!n)!(n!n)]
[=(n, n, r) fM (1!fM )#=(n, n, r) fM (1!fM )] . (214)
We used the notations fM ,fM (r, n, E) and fM ,fM (r, n, E). The expression for the current density,
j(r, t)"ev dn dE nf (r, n, E, t) (215)
(with the normalization dn"1), completes the system of equations used in the Boltzmann}
Langevin method. We remark that in this formulation the local electric potential does not appear
explicitly: for systems with charged carriers such as electric conductors the electric "eld is coupled
to the (#uctuating) charge density via the Poisson equation. We will return to this point when we
discuss situations in which a treatment of this coupling is essential.