shot noise in mesoscopic conductors - Low Temperature Laboratory

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44 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 Fig. 10. Distribution function of transmission coe$cients (87) for ¸/l"10. the distribution function of transmission coe$cients: P(¹)" l 1 2¸ ¹1!¹ , ¹ (¹(1, ¹ "4 exp(!2¸/l) , (87) and P(¹)"0 otherwise. As discussed, it has a bimodal form: almost open and almost close channels are preferred. The dependence P(¹) is illustrated in Fig. 10. The distribution function P(¹) must be used now to average expressions (40) and (57) over impurity con"gurations. Direct calculation con"rms Eq. (86), and, thus, the distribution function (87) yields the Drude-Sommerfeld formula (85) for the average conductance. Furthermore, we obtain ¹(1!¹)" l 3¸ , which implies that the zero-temperature shot noise power is S" e< N l 3 ¸ "1 3 S . (88) The shot noise suppression factor for metallic di!usive wires is equal to F"1/3. The remarkable feature is that this result is universal: As long as the geometry of the wire is quasi-one-dimensional and l;¸;¸ (metallic di!usive regime), the Fano factor does not depend on the degree of disorder, the number of transverse channels, and any other individual features of the sample. This result was "rst obtained by Beenakker and one of the authors [74] using the approach described above. Independently, Nagaev [75] derived the same suppression factor by using a classical theory based on a Boltzmann equation with Langevin sources. This theory and subsequent developments are described in Section 6. It seems that the question whether the Fano factor depends on the type of disorder has never been addressed. In all cases disorder is assumed to be Gaussian white noise, i.e. ;(r);(r)J(r!r).
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 45 Later on, the suppression of shot noise became a subject of a number of microscopic derivations. Altshuler et al. [76] recovered the Fano factor by direct microscopic calculation using the Green's function technique. Nazarov [77] proved that the distribution (87) holds for an arbitrary (not necessarily quasi-one-dimensional) geometry; thus, the 1/3-suppression is `superuniversala. He used a slightly di!erent technique, expressing scattering matrices through Green's functions and then performing disorder averages. The same technique, in more elaborated form, was used in Ref. [78], which also obtains the -suppression. It is clear that the quantum-mechanical theories of Refs. [74,76}78] are equivalent for the quasi-one-dimensional geometry, since they deal with disorder averages basically in the same way. On the other hand, their equivalence to the classical consideration of Ref. [75] is less obvious. Experimentally, shot noise in metallic di!usive wires was investigated by Liefrink et al. [79], who observed that it is suppressed with respect to the Poisson value. The suppression factor in this experiment lies between 0.2 and 0.4 (depending on gate voltage). More precise experiments were performed by Steinbach et al. [80] who analyzed silver wires of di!erent length. In the shortest wires examined they found a shot noise slightly larger than and explained this larger value as due to electron}electron interaction. A very accurate measurement of the noise suppression was performed by Henny et al. [81]. Special care was taken to avoid electron heating e!ects by attaching very large reservoirs to the wire. Their results are displayed in Fig. 11. Localized regime. In quasi-one-dimensional wires with length ¸
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For completeness, we mention that i
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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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