shot noise in mesoscopic conductors - Low Temperature Laboratory

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56 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 Fig. 15. Four-terminal conductors for Hanbury Brown}Twiss exchange e!ects (a), quantum point contact geometry (b). Terminals are numbered by digits. The dashed line in (a) indicates a phase-sensitive trajectory which contributes to exchange terms in Eq. (107). Arrows in (b) indicate non-zero transmission probabilities. We investigate now the general expression for the four-terminal phase-sensitive HBT e!ect (107) for various systems. Our concern will be the sign and relative magnitude of the exchange contribution S"S !S !S . Disordered systems. Naively, one might assume that in a disordered medium the phase accumulated along the trajectory, indicated by the dashed line in Fig. 15a, is random. Then the phasesensitive exchange contribution would be zero after being averaged over disorder. Thus, this view implies S "S #S . A quantitative analysis of these questions was provided by the authors of this review in Ref. [78]. In this work the scattering matrices in Eq. (108) are expressed through Green's functions to which disorder averaging was applied using the diagram technique. The key result found in Ref. [78] is that the naive picture mentioned above, according to which one might expect no exchange e!ects after disorder averaging, is completely wrong. Exchange ewects survive disorder averaging. The reason can be understood if the principal diagrams (which contain four di!usion propagators) are translated back into the language of electron trajectories. One sees then that the typical trajectory does not look like the dashed line in Fig. 15a. Instead, it looks like a collection of dashed lines shown in Fig. 16a: the electron di!uses from contact 1 to some intermediate point 5 in the bulk of the sample (eventually, the result is integrated over the coordinate of point 5), then it di!uses from 5 to 2 and back from 2 to 5 precisely along the same trajectory, and so on, until it returns from 5 to 1 along the same di!usive trajectory as it started. Thus, there is no phase enclosed by the trajectory. This explains why the exchange contribution survives averaging over disorder; apparently, there is a classical contribution to the exchange correlations which requires knowledge only of Fermi statistics, but no information about phases of scattering matrices. Indeed, a classical theory of ensemble-averaged exchange e!ects was subsequently proposed by Sukhorukov and Loss [113,114]. Once we determined that exchange correction S exists in di!usive conductors, we must evaluate its sign and relative magnitude. We only describe the results qualitatively; details can be
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 57 Fig. 16. Examples of four-terminal disordered conductors. Disordered area is shaded. Dashed line denotes di!usive motion between its ends. found in Refs. [78,114]. Two speci"c geometries have been investigated: the disordered box (Fig. 16a) and the disordered cross (Fig. 16b). For the box, the exchange correction S is negative, i.e. exchange suppresses noise (S (S #S ). The e!ect is quite considerable: The correction is of the same order of magnitude as the classical contributions in S and S , and is suppressed only by a numerical factor. For the cross, the exchange contribution is positive } exchange enhances noise } but the magnitude is by powers of l/¸ smaller than S and S . Here l and ¸ are the mean free path and the length of the disordered arms, respectively. Thus, neither sign nor magnitude of the exchange e!ects is predetermined in di!usive systems: they are geometry and disorder dependent, and the only limitation is S '0. Gramespacher and one of the authors [110,111] considered a geometry of a disordered wire (along the axis z) between the contacts 1 (z"0) and 3 (z"¸), coupled locally at the points z and z to the contacts 2 and 4, respectively, via high tunnel barriers (these latter can be viewed as scanning tunneling microscope tips) and evaluated Eq. (109). It was found that the exchange e!ect is positive in the case, i.e. it enhances noise, irrespectively of the position of the contacts 2 and 4. For the particular case when both tunnel contacts are situated symmetrically around the center of the wire at a distance d, z"(¸!d)/2 and z"(¸#d)/2, the relative strength of the exchange term is S " S 1 32#d ¸ !2 d ¸ , and reaches its maximum for d"¸/4. We see that the exchange e!ect in this case generally has the same order of magnitude as the classical terms S and S . Chaotic cavities. A similar problem in chaotic cavities was addressed in Ref. [115] (see also Ref. [116]). Similarly to disordered systems, it was discovered that exchange e!ects survive on average. An additional feature is however that the exchange e!ects in chaotic cavities are universal. Ref. [111] also considers a three-terminal structure (a (disordered) wire with a single STM tip attached to it). The #uctuations of the current through the tip are in this case proportional to the local distribution function of electrons at the coupling point, see Eq. (249) below.
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8. Concluding remarks, future prosp
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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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