shot noise in mesoscopic conductors - Low Temperature Laboratory

28 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
Poissonian limit given by Eq. (58) was one of the aspects of noise in mesoscopic systems which
triggered many of the subsequent theoretical and experimental works. A convenient measure of
sub-Poissonian shot noise is the Fano factor F which is the ratio of the actual shot noise and the
Poisson noise that would be measured if the system produced noise due to single independent
electrons,
F"S /S . (59)
For energy-independent transmission and/or in the linear regime the Fano factor is
F" ¹ (1!¹ )
. (60)
¹
The Fano factor assumes values between zero (all channels are transparent) and one (Poissonian
noise). In particular, for one channel it becomes (1!¹).
Unlike the conductance, which can be expressed in terms of (transmission) probabilities independent
of the choice of basis, the shot noise, even for the two terminal conductors considered here,
cannot be expressed in terms of probabilities. The trace of Eq. (56) is a sum over k, l, m, n of terms
rH r tH t , which by themselves are not real-valued if mOn (in contrast to Eq. (41) for the
conductance). This is a signature that carriers from di!erent quantum channels interfere and must
remain indistinguishable. It is very interesting to examine whether it is possible to "nd experimental
arrangements which directly probe such exchange interference e!ects, and we return to this
question later on. In the remaining part of this subsection we will use the eigen-channel basis which
o!ers the most compact representation of the results.
The general result for the noise power of the current #uctuations in a two-terminal conductor is
S" e
dE¹ (E)[ f (1Gf )#f (1Gf )]$¹ (E)[1!¹ (E)]( f !f ) . (61)
Here the "rst two terms are the equilibrium noise contributions, and the third term, which changes
sign if we change statistics from fermions to bosons, is the non-equilibrium or shot noise
contribution to the power spectrum. Note that this term is second order in the distribution
function. At high energies, in the range where both the Fermi and Bose distribution function are
well approximated by a Maxwell}Boltzmann distribution, it is negligible compared to the equilibrium
noise described by the "rst two terms. According to Eq. (61) the shot noise term enhances the
noise power compared to the equilibrium noise for fermions but diminishes the noise power for
bosons.
In the practically important case, when the scale of the energy dependence of transmission
coe$cients ¹ (E) is much larger than both the temperature and applied voltage, these quantities in
Eq. (61) may be replaced by their values taken at the Fermi energy. We obtain then (only fermions
are considered henceforth)
S" e
2k e<
¹ ¹#e