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 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.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 51 Fig. 13. Four-probe quantum Hall conductor. Solid lines indicate edge channels; dashed lines show the additional scattering probability ¹ through the quantum point contact. the following, we will focus on the situation when the chemical potentials of all the four reservoirs are arranged so that " ", " "#e
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We note here, however, that there i
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The noise power (201) is strongly v
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Fig. 32. Experimental results of Ku
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where Ya.M. Blanter, M. Bu( ttiker
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where the quantity is expressed th
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6.3. Interaction ewects Ya.M. Blant
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one "rst goes from the non-interact
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non-thermalized electrons, the high
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Fig. 34. Geometry of the chaotic ca
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which describes strictly ballistic
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approximately e/C. Between the peak
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Boltzmann}Langevin approach (see Se
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thermal to shot noise. It is, howev
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Note added in proof Ya.M. Blanter,
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and Lee et al. [340] obtain in this
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shot noise in mesoscopic conductors - Low Temperature Laboratory

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