shot noise in mesoscopic conductors - Low Temperature Laboratory

Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 71
the number of charges in the system). Thus, a complete solution of the above system of equations is
usually a hopeless task without some serious approximations. The theoretical task is to choose
idealizations and approximations which are compatible with the basic conservation laws expressed
by the above equations. For instance, we might want to describe interactions in terms of
an e!ective (screened) interaction instead of the full long-range Coulomb interaction. It
is then necessary to ensure that such an e!ective interaction indeed leads to the conservation of
current [135,136].
Three frequency-dependent types of noise spectra should be distinguished: (i) "nite-frequency
noise at equilibrium or in the presence of dc voltage, (ii) zero-frequency noise in the presence of an
ac voltage; the resulting spectrum depends on the frequency of the ac-voltage, (iii) "nite-frequency
noise in the presence of an ac voltage; this quantity depends on two frequencies. Here we are
interested mostly in the "rst type of noise spectra; the second one is only addressed in Section 3.3.
We re-iterate that, generally, one cannot "nd the ac conductance and the current #uctuations from
a non-interacting model. Even the "nite frequency current}current correlations (noise) at equilibrium
or in the presence of a dc voltage source, which are of primary interest in this section, cannot
be treated without taking account interactions. A simple way to see this is to note that due to
the #uctuation-dissipation theorem, the equilibrium correlation of currents in the leads and
at "nite frequency, S (), is related to the corresponding element of the conductance matrix,
S ()"2k ¹[G ()#GH ()]. The latter is the response of the average current in the lead to
the ac voltage applied to the lead , and is generally interaction-sensitive. Thus, calculation of the
quantity S () also requires a treatment of interactions to ensure current conservation.
We can now be more speci"c and make the same point by looking at Eq. (51) which represents
the #uctuations of the particle current at "nite frequency. Indeed, for "0 the current conservation
S "0 is guaranteed by the unitarity of the scattering matrix: the matrix A(, E, E)
(Eq. (44)) contains a product of two scattering matrices taken at the same energy, and therefore it
obeys the property A(, E, E)"0. On the other hand, for "nite frequency the same matrix
A should be evaluated at two di!erent energies E and E#, and contains now a product of two
scattering matrices taken at diwerent energies. These scattering matrices generally do not obey the
property s (E)s (E#)" , and the current conservation is not ful"lled: S ()O0.
Physically, this lack of conservation means that there is charge pile-up inside the sample, which
gives rise to displacement currents. These displacement currents restore current conservation,
and thus need to be taken into account. It is exactly at this stage that a treatment of interactions is
required. Some progress in this direction is reviewed in this section later on.
It is sometimes thought that there are situations when displacement currents are not important.
Indeed, the argument goes, there is always a certain energy scale , which determines the energy
dependence of the scattering matrices. This energy scale is set by the level width (tunneling rate)
for resonant tunnel barriers, the Thouless energy E "D/¸ for metallic di!usive wires (D and
¸ are the di!usion coe$cient and length of the wire, respectively), and the inverse Ehrenfest time
(i.e. the time for which an electron loses memory about its initial position in phase space) for chaotic
cavities. The scattering matrices may be thought as energy independent for energies below .
Then for frequencies below we have s (E)s (E#)& , and the unitarity of the
scattering matrix assures current conservation, S "0. However, there are time-scales which
are not set by the carrier kinetics, like RC-times which re#ect a collective charge response of the
system. In fact, from the few examples for which the ac conductance has been examined, we know