shot noise in mesoscopic conductors - Low Temperature Laboratory

64 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
scattering theory of electrical transport within such a generalized scattering matrix approach has
not been worked out. It is, however, possible to make progress even within the scattering approach
used so far: to treat phase breaking theoretically we often proceed by inventing a Hamiltonian
system with many degrees of freedom while we are interested in the behavior of only a subsystem.
Similarly, it is possible to arrive at an approach which describes inelastic transitions and phase
breaking by "rst considering a completely phase-coherent conductor with one or a continuum of
additional voltage probes which are purely "ctitious [125,47]. The additional "ctitious voltage
probes act as dephasers on the actual conductor of interest. This approach has been widely used to
investigate the e!ect of dephasing on conductance. We refer the reader here only to a few early
works [125}127,47]. In this subsection we illustrate the application of these ideas to noise. Other
approaches, based on Green's function techniques, have also been invoked to derive results for
strongly correlated systems (see Section 7). Furthermore, on the purely classical level, it proved to
be rather simple to extend the #uctuating Boltzmann equation approach to include interactions.
For the results on interaction and noise in double-barrier resonant tunneling structures and
metallic di!usive conductors the reader is addressed to Sections 5 and 6, respectively. The
approach which uses voltage probes as dephasers is interesting because of its conceptual clarity
and because of its close relation to experiments: the e!ect of additional voltage probes can easily be
tested experimentally with the help of gates which permit to switch o! or on a connection to
a voltage probe (see e.g. Ref. [128]).
Voltage probes as dephasers. Consider a mesoscopic conductor connected to N (real) contacts. To
introduce inelastic scattering, we attach a number M of purely "ctitious voltage probes to this
conductor. The entire conductor with its N#M contacts is phase coherent and exhibits the noise
of a purely phase coherent conductor. However, elimination of the M "ctitious voltage probes
leads to an e!ective conduction problem for which the conductance and the noise depend on
inelastic scattering processes [19,74,129}131]. Depending on the properties of the "ctitious voltage
probes, three di!erent types of inelastic scattering can be realized, which de Jong and Beenakker
[90] classify as `quasi-elastic scatteringa (phase breaking), `electron heatinga, and `inelastic
scatteringa. Now we describe these types of probes separately. This division corresponds to the
distinction of ( , , and . We emphasize that only a microscopic theory can give explicit
expressions for these times. What the approach based on "ctitious voltage probes can do is to "nd
the functional dependence of the conductance or the noise on these times.
The results for interaction e!ects in double-barrier structures seem to be well established by now.
In contrast, for di!usive metallic wires with interactions the situation is less clear. For discussion,
the reader is addressed to Section 6.
In this subsection, we assume that the system is charge neutral, i.e. there is no pile-up of
charge. This charge neutrality is normally provided by Coulomb interactions, which thus play an
important role. If this is not the case, one can get di!erent results, like for resonant tunneling
quantum wells with charging (Section 5) or quantum dots in the Coulomb blockade regime
(Section 7).
Inelastic scattering. We begin the discussion with the strongest scattering processes which lead to
carrier energy relaxation and consequently also energy dissipation. Physically, this may correspond
to electron}phonon scattering. To simulate this process, we consider a two-terminal structure