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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10 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166<br />
The transmitted and re#ected current are correlated,<br />
S "!2<br />
e<br />
dE ¹fRf. (10)<br />
2<br />
In the limit that either ¹ is very small or f is small, the factor (1!¹f ) <strong>in</strong> Eq. (8) can be replaced<br />
by one. In this limit, s<strong>in</strong>ce the average current through the barrier is I"(e/2)dE ¹f, the<br />
spectrum, Eq. (8), is Schottky's result [5] for <strong>shot</strong> <strong>noise</strong>,<br />
S "2eI . (11)<br />
Schottky's result corresponds to the uncorrelated arrival of particles with a distribution function of<br />
time <strong>in</strong>tervals between arrival times which is Poissonian, P(t)" exp(!t/), with be<strong>in</strong>g the<br />
mean time <strong>in</strong>terval between carriers. Alternatively, Eq. (11) is also referred to <strong>in</strong> the literature as the<br />
Poisson value of <strong>shot</strong> <strong>noise</strong>.<br />
The result, Eq. (8), is markedly di!erent from Eq. (11) s<strong>in</strong>ce it conta<strong>in</strong>s, <strong>in</strong> comparison to<br />
Schottky's expression, the extra factor (1!¹f ). This factor has the consequence that the <strong>shot</strong> <strong>noise</strong><br />
(8) is always smaller than the Poisson value. For truly ballistic systems (¹"1) the <strong>shot</strong> <strong>noise</strong> even<br />
vanishes <strong>in</strong> the zero-temperature limit. As the temperature <strong>in</strong>creases, <strong>in</strong> such a conductor (¹"1)<br />
there is <strong>shot</strong> <strong>noise</strong> due to the #uctuation <strong>in</strong> the <strong>in</strong>cident beam aris<strong>in</strong>g from the thermal #uctuations.<br />
Eventually, at high temperatures the factor 1!f can be replaced by 1, and the ballistic conductor<br />
exhibits Poisson <strong>noise</strong>, <strong>in</strong> accordance with Schottky's formula, Eq. (11). The full Poisson <strong>noise</strong><br />
given by Schottky's formula is also reached for a scatterer with very small transparency ¹;1. We<br />
emphasize that the above statements refer to the transmitted current. In the limit ¹;1 the<br />
re#ected current rema<strong>in</strong>s nearly <strong>noise</strong>less up to high temperatures when (1!Rf ) can be replaced<br />
by 1. We also remark that even though electron motion <strong>in</strong> vacuum tubes (the Schottky problem<br />
[5]) is often referred to as ballistic, it is <strong>in</strong> fact a problem <strong>in</strong> which carriers have been emitted by<br />
a source <strong>in</strong>to vacuum either through thermal activation over or by tunnel<strong>in</strong>g through a barrier with<br />
very small transparency.<br />
Our discussion makes it clear that out of equilibrium, and at "nite temperatures, the <strong>noise</strong><br />
described by Eq. (8) conta<strong>in</strong>s the e!ect of both the #uctuations <strong>in</strong> the <strong>in</strong>cident carrier beam as well<br />
as the partition <strong>noise</strong>. In a transport state, <strong>noise</strong> <strong>in</strong> <strong>mesoscopic</strong> <strong>conductors</strong> has two dist<strong>in</strong>ct sources<br />
which manifest themselves <strong>in</strong> the #uctuations of the occupation numbers of states: (i) thermal<br />
#uctuations; (ii) partition <strong>noise</strong> due to the discrete nature of carriers.<br />
Both the thermal and <strong>shot</strong> <strong>noise</strong> at low frequencies and low voltages re#ect <strong>in</strong> many situations<br />
<strong>in</strong>dependent quasi-particle transport. Electrons are, however, <strong>in</strong>teract<strong>in</strong>g entities and both the<br />
#uctuations at "nite frequencies and the #uctuation properties far from equilibrium require <strong>in</strong><br />
general a discussion of the role of the long-range Coulomb <strong>in</strong>teraction. A quasi-particle picture is<br />
no longer su$cient and collective properties of the electron system come <strong>in</strong>to play.<br />
Note that this term<strong>in</strong>ology, common <strong>in</strong> <strong>mesoscopic</strong> physics, is di!erent from that used <strong>in</strong> the older literature [6],<br />
where <strong>shot</strong> <strong>noise</strong> (due to random <strong>in</strong>jection of particles <strong>in</strong>to a system) and partition <strong>noise</strong> (due to random division of the<br />
particle stream between di!erent electrodes, or by potential barriers) are two dist<strong>in</strong>ct <strong>in</strong>dependent sources of #uctuations.