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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50 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166<br />
Calculat<strong>in</strong>g the averages<br />
¹" 1<br />
2N ¹I<br />
<br />
<br />
and<br />
¹(1!¹)" 1<br />
8N ¹I (2!¹I ),<br />
<br />
<br />
we "nd the Fano factor,<br />
F" ¹(1!¹)<br />
¹<br />
"1<br />
¹I (2!¹I )<br />
. (98)<br />
4 ¹I<br />
<br />
In particular, if all the transmission probabilities ¹I are the same and equal to ¹I , we obta<strong>in</strong><br />
<br />
F"(2!¹I )/4. This expression reproduces the limit<strong>in</strong>g cases F" for ¹I "1 (no barriers } open<br />
<br />
quantum cavity) and F" for ¹I P0 (double-barrier suppression <strong>in</strong> a symmetric system). Thus,<br />
<br />
Eq. (98) describes the crossover between the behavior characteristic for an open cavity and the<br />
situation when the barriers are so high that the dynamics <strong>in</strong>side the cavity does not play a role.<br />
2.6.6. Edge channels <strong>in</strong> the quantum Hall ewect regime<br />
Now we turn to the description of e!ects which are <strong>in</strong>herently multi-term<strong>in</strong>al. The calculation of<br />
the scatter<strong>in</strong>g matrix is <strong>in</strong> general a di$cult problem. However, <strong>in</strong> some special situations the<br />
scatter<strong>in</strong>g matrix can be deduced immediately even for multi-term<strong>in</strong>al <strong>conductors</strong>.<br />
We consider a four-term<strong>in</strong>al conductor (Fig. 13) made by pattern<strong>in</strong>g a two-dimensional electron<br />
gas. The conductor is brought <strong>in</strong>to the quantum Hall regime by a strong transverse magnetic "eld.<br />
In a region with <strong>in</strong>teger "ll<strong>in</strong>g of Landau levels the only extended states at the Fermi energy which<br />
connect contacts [102] are edge states, the quantum mechanical equivalent of classical skipp<strong>in</strong>g<br />
orbits. S<strong>in</strong>ce the net current at a contact is determ<strong>in</strong>ed by the states near the Fermi surface,<br />
transport <strong>in</strong> such a system can be described by consider<strong>in</strong>g the edge states. Note that this fact<br />
makes no statement on the spatial distribution of the current density. In particular, a description<br />
based on edge states does not mean that the current density vanishes away from the edges. This<br />
po<strong>in</strong>t which has caused considerable confusion and generated a number of publications is well<br />
understood, and we refer the reader here only to one particularly perceptive discussion [103]. Edge<br />
states are uni-directional; if the sample is wide enough, backscatter<strong>in</strong>g from one edge state to<br />
another one is suppressed [102]. In the plateau regime of the <strong>in</strong>teger quantum Hall e!ect, the<br />
number of edge channels is equal to the number of "lled Landau levels. For the discussion given<br />
here, we assume, for simplicity, that we have only one edge state. In a quantum Hall conductor<br />
wide enough so that there is no backscatter<strong>in</strong>g, there is no <strong>shot</strong> <strong>noise</strong> [18]. Hence, we <strong>in</strong>troduce<br />
a constriction (Fig. 13) and allow scatter<strong>in</strong>g between di!erent edge states at the constriction [18]:<br />
the probability of scatter<strong>in</strong>g from contact 4 to the contact 3 is ¹, while that from 4 to 1 is 1!¹.In<br />
We do not give a microscopic description of edge states. Coulomb e!ects <strong>in</strong> the <strong>in</strong>teger quantum Hall e!ect regime<br />
lead to a spatial decomposition <strong>in</strong>to compressible and <strong>in</strong>compressible regions. Edge channels <strong>in</strong> the fractional quantum<br />
Hall e!ect regime will be discussed <strong>in</strong> Section 7.