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