shot noise in mesoscopic conductors - Low Temperature Laboratory

investigated experimentally. A self-consistent spectrum for photon-assisted noise spectra can
probably be developed along the lines of Ref. [153].
3.4. Noise of a capacitor
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 79
Now we turn to the problems where the energy dependence of the scattering matrices is essential.
Rather than trying to give the general solution (which is only available for the case when the
potential inside the system is spatially uniform [156,157]), we provide a number of examples which
could serve as a basis for further investigations of "nite frequency noise.
The simple case of shot noise in a ballistic wire was studied by Kuhn et al. [158,159]. They
found signatures of the inverse #ight time v /¸. However, the interactions are taken into account
only implicitly via what the authors call `quantum generalization of the Ramo}Shockley
theorema. We do not know in which situations this approach is correct, and it certainly cannot be
correct universally. Thus, even this simple case cannot be considered as solved and needs further
consideration.
We start from the simplest system } a mesoscopic capacitor (Fig. 25a), which is connected via
two leads to equilibrium reservoirs. Instead of the full Poisson equation interactions are described
with the help of a geometrical capacitance C. There is no transmission from the left to the right
plate, and therefore there is no dc current from one reservoir to the other. Moreover, this system
does not exhibit any noise even at "nite frequency if the scattering matrix is energy independent.
Indeed, if only the matrices s and s are non-zero, we obtain from Eq. (51)
e
S()"
2 dE Tr [1!s (E)s (E#)]
[1!s (E#)s (E)]f (E)(1!f (E#)) . (135)
Here we used the superscript (0) to indicate that the #uctuations of the particle current, and not the
total current, are discussed. If the scattering matrices are energy independent, Eq. (135) is
identically zero due to unitarity. Another way to make the same point is to note that since S "0,
the only way to conserve current would be S "0.
Before proceeding to solve this problem, we remark that Eq. (135) describes an equilibrium
#uctuation spectrum and via the #uctuation dissipation theorem S
()"
2g () coth (/2k ¹) is related to the real part of a conductance g () given by
g ()"
e
dE Tr[1!s (E)s (E#)] f (E)!f (E#)
. (136)
Fig. 25. A mesoscopic capacitor (a); a mesoscopic conductor vis-à-vis a gate (b).