shot noise in mesoscopic conductors - Low Temperature Laboratory

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