shot noise in mesoscopic conductors - Low Temperature Laboratory

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68 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 Quasi-elastic scattering. In contrast to inelastic scattering, dephasing processes leave the energy essentially invariant. To simulate a scattering process which destroys phase but leaves the energy invariant we now have to consider a special voltage probe. We require that the additional electrode conserves not only the total current, but also the current in each small energy interval [90]. Such a voltage probe will not give rise to energy relaxation and dissipates no energy. From the condition that the current in each energy interval vanishes we "nd that the distribution function in the reservoir of the voltage probe is given by f (E)" G f (E)#G f (E) , (120) G #G where f and f are the Fermi functions at the reservoirs 1 and 2, and the conductances are G "R , G "R. We have assumed again that there is no direct transmission between 1 and 2. Straightforward calculation gives for the Fano factor [90] F" R F #R F #R R #R R . (121) (R #R ) We analyze now this result for various situations. First, we see that for a ballistic wire divided by a dephasing electrode into two parts, F "F "0, shot noise does not vanish (unlike Eq. (119)). We obtain F"R R (R #R ). Thus for a ballistic system, which is ideally noiseless, dephasing leads to the appearance of shot noise. For the strongly biased resonant double-barrier structure, we have F "F "1, and obtain again the result (78), which is thus insensitive to dephasing. For metallic di!usive wires, F "F ", Eq. (121) yields for the ensemble-averaged Fano factor F" independent on the location of the dephasing voltage probe, i.e. independent of the ratio of R and R . Thus, our consideration indicates that the noise suppression factor for metallic di!usive wires is also insensitive to dephasing, at least when the dephasing is local (in any point of the sample). This result hints that the Fano factor of an ensemble of metallic di!usive wires is not sensitive to dephasing even if the latter is uniformly distributed. Indeed, de Jong and Beenakker [82] checked this by coupling locally a dephasing reservoir to each point of the sample. Already at the intermediate stage their formulae coincide with those obtained classically by Nagaev [75]. This proves that introducing dephasing with "ctitious, energy-conserving voltage probes is in the limit of complete dephasing equivalent to the Boltzmann}Langevin approach. That inelastic scattering, and not dephasing, is responsible for the crossover to the macroscopic regime, has been recognized by Shimizu and Ueda [133]. The e!ect of phase breaking on the shot noise in chaotic cavities was investigated by van Langen and one of the authors [115]. For a chaotic cavity connected to reservoirs via quantum point contacts with N and N open quantum channels, the Fano factors vanish F "F "0, and since R "/eN and R "/eN , the resulting Fano factor is given by Eq. (96), i.e. it is identical with the result that is obtained from a completely phase coherent, quantum mechanical calculation. Thus for chaotic cavities, like for metallic di!usive wires, phase breaking has no e!ect on the ensemble-averaged noise power. Electron heating. This is the third kind of inelastic scattering, which implies that energy can be exchanged between electrons. Only the total energy of the electron subsystem is conserved.
Physically, this corresponds to electron}electron scattering. Within the voltage probe approach, it is taken into account by including the reservoir 3, with chemical potential determined to obtain zero (instantaneous) electrical current and a temperature ¹ , which is generally di!erent from the lattice temperature (or the temperature of the reservoirs 1 and 2), to obtain zero (instantaneous) energy #ux. For a detailed discussion we refer the reader to the paper by de Jong and Beenakker [82], here we only mention the result for two identical di!usive conductors at zero temperature. The Fano factor is in this case F+0.38, which is higher than the -suppression for the noninteracting case. We will see in Section 6 that the classical theory also predicts shot noise enhancement for the case of electron heating. Intermediate summary. Here are the conclusions one can draw from the simple consideration we presented above. Dephasing processes do not renormalize the ensemble-averaged shot noise power (apart from weak localization corrections, which are destroyed by dephasing). In particular, this statement applies to metallic di!usive wires, chaotic cavities, and resonant double-barrier structures. Inelastic scattering renormalizes even the ensemble-averaged shot noise power: a macroscopic sample exhibits no shot noise. An exception is the resonant double-barrier structure, subject to a bias large compared to the resonant level width. Under this condition neither the conductance nor the shot noise of a double barrier are a!ected. As demonstrated for metallic di!usive wires electron heating enhances noise. The last statement implies the following scenario for noise in metallic di!usive wires [80]. There exist three inelastic lengths, responsible for dephasing (¸ ), electron heating (¸ ) and inelastic scattering (¸ ). We expect ¸ (¸ (¸ . Indeed, requirements for dephasing (inelastic scattering) are stronger (weaker) than those for electron heating. Then for the wires with length ¸;¸ the Fano factor equals and is not a!ected by inelastic processes; for ¸ ;¸;¸ it is above , and for ¸
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8. Concluding remarks, future prosp
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text; they are usually extensions o
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such reviews are not readily availa
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Ya.M. Blanter, M. Bu( ttiker / Phys
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The above considerations are, of co
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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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