shot noise in mesoscopic conductors - Low Temperature Laboratory

26 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
(where as before the trace is taken over transverse channel indices, and N is the number of
channels in the lead ), we "nd
S "
ek ¹
dE Rf
! RE [2N !Tr(s s #s s )] . (53)
This is the equilibrium, or Nyquist}Johnson noise. In the approach discussed here it is a consequence
of the thermal #uctuations of occupation numbers in the reservoirs. Comparing Eqs. (45)
and (53), we see that
S "2k ¹(G #G ) . (54)
This is the manifestation of the #uctuation-dissipation theorem: equilibrium #uctuations are
proportional to the corresponding generalized susceptibility, in this case to the conductance. For
the time-reversal case (no magnetic "eld) the conductance matrix is symmetric, and Eq. (54) takes
the form
S "4k ¹G ,
which is familiar for the two-terminal case, S"4k ¹G, with G being the conductance. From
Eq. (54) we see that the #uctuation spectrum of the mean squared current at a contact is positive
(since G '0) but that the current}current correlations of the #uctuations at di!erent probes are
negative (since G (0). The sign of the equilibrium current}current #uctuations is independent
of statistics: Intensity}intensity #uctuations for a system of bosons in which the electron reservoirs
are replaced by black-body radiators are also negative. We thus see that equilibrium noise
does not provide any information of the system beyond that already known from conductance
measurements.
Nevertheless, the equilibrium noise is important, if only to calibrate experiments and as a simple
test for theoretical discussions. Experimentally, a careful study of thermal noise in a multi-terminal
structure (a quantum Hall bar with a constriction) was recently performed by Henny et al. [12].
Within the experimental accuracy, the results agree with the theoretical predictions.
2.4.2. Shot noise: zero temperature
We now consider noise in a system of fermions in a transport state. In the zero temperature limit
the Fermi distribution in each reservoir is a step function f (E)"( !E). Using this we can
rewrite Eq. (52) as
S "
e
2 dE Tr[s s s s ] f (E)[1!f (E)]#f (E)[1!f (E)] . (55)
We are now prepared to make two general statements. First, correlations of the current at the same
lead, S , are positive. This is easy to see, since their signs are determined by positively de"ned
quantities Tr[s s s s ]. The second statement is that the correlations at diwerent leads, S with
For bosons at zero temperature one needs to take into account Bose condensation e!ects.
These quantities are called `noise conductancesa in Ref. [9].