shot noise in mesoscopic conductors - Low Temperature Laboratory

32 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
which two pairs of indices are identical, R or R "!R . With these resistances we can
now generalize the familiar Johnson}Nyquist relation S "4k ¹R for two-probe conductors, to
the case of a multi-probe conductor. For the correlation of a voltage di!erence < !< measured
between contacts and with a voltage #uctuation < !< measured between contacts and ,
Eq. (67) leads to
(< !< )(< !< )"2k ¹(R #R ) . (69)
The mean-squared voltage #uctuations " and " are determined by the two-terminal
resistances R of the multi-probe conductor. The correlations of voltage #uctuations (in the case
when all four indices di!er) are related to symmetrized four-probe resistances.
If shot noise is generated, for instance, by a current incident at contact and taken out at contact
(in a zero external impedance loop) and with all other contacts connected to an in"nite
impedance circuit, the voltage #uctuations are [22]
(< !< )(< !< )" R R S , (70)
where S is the noise power spectrum of the current correlations at contacts and , and is an
arbitrary index.
These examples demonstrate that the #uctuations in a conductor are in general a complicated
expression of the noise power spectrum determined for the zero-impedance case, the resistances
(or far from equilibrium the di!erential resistances) and the external impedance (matrix). These
considerations are of importance since in experiments it is the voltage #uctuations which are
actually measured and which eventually are converted to current #uctuations.
2.6. Applications
In this subsection, we give some simple applications of the general formulae derived above, and
illustrate them with experimental results. We consider only the zero-frequency limit. As we
explained in the Introduction, we do not intend to give here a review of all results concerning
a speci"c system. Instead, we focus on the application of the scattering approach. For results
derived for these systems with other methods, the reader is addressed to Table 1.
2.6.1. Tunnel barriers
For a tunnel barrier, which can be realized, for example, as a layer of insulator separating two
normal metal electrodes, all the transmission coe$cients ¹ are small, ¹ ;1 for any n. Separating
terms linear in ¹ in Eq. (62) and taking into account the de"nition of the Poisson noise, Eq. (58),
we obtain
S" e<
coth e<
2k ¹
¹ "coth e<
2k ¹ S
. (71)
At a given temperature, Eq. (71) describes the crossover from thermal noise at voltages e