shot noise in mesoscopic conductors - Low Temperature Laboratory

Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 21
k "[2m(E!E )] (the summation only includes channels with real k ), and the velocity
of carriers v (E)"k /m in the nth transverse channel.
After some algebra, the expression for the current can be cast into the form
IK (z, t)"
e
4
1
dE dE e
v (E)v (E)
[v (E)#v (E)][exp[i(k (E)!k (E))z]a( (E)a( (E)
! exp[i(k (E)!k (E))z]bK (E)bK (E)]
# [v (E)!v (E)][exp[!i(k (E)#k (E))z]a( (E)bK (E)
! exp[i(k (E)#k (E))z]bK (E)a( (E)] . (31)
This expression is cumbersome, and, in addition, depends explicitly on the coordinate z. However,
it can be considerably simpli"ed. The key point is that for all observable quantities (average
current, noise, or higher moments of the current distribution) the energies E and E in Eq. (31) either
coincide, or are close to each other. On the other hand, the velocities v (E) vary with energy quite
slowly, typically on the scale of the Fermi energy. Therefore, one can neglect their energy
dependence, and reduce the expression (31) to a much simpler form,
IK (t)"
e
2
dE dE e [a( (E)a( (E)!bK (E)bK (E)] . (32)
Note that n( (E)"a( (E)a( (E) is the operator of the occupation number of the incident carriers in
lead L in channel n. Similarly, n( (E)"bK (E)bK (E) is the operator of the occupation number of
the out-going carriers in lead L in channel n. Setting E"E# and carrying out the integral
over gives
IK (t)"
e
2
dE [n( (E, t)!n( (E, t)] . (33)
Here n( (E, t) are the time-dependent occupation numbers for the left and right moving carriers at
energy E. Thus, Eq. (33) states that the current at time t is simply determined by the di!erence
in occupation number between the left and right movers in each channel. We made use of this
intuitively appealing result already in the introduction. Using Eq. (29) we can express the current in
terms of the a( and a( operators alone,
IK (t)"
e
2
dE dE ea( (E)A
(¸; E, E)a( (E) . (34)
Here the indices and label the reservoirs and may assume values L or R. The matrix A is de"ned
as
A
(¸; E, E)" ! s (E)s (E) , (35)
_ _
A discussion of the limitations of Eq. (32) is given in Ref. [10].