shot noise in mesoscopic conductors - Low Temperature Laboratory

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52 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 where f and f are Fermi distribution functions in the reservoirs 1 and 3, respectively. This result is remarkable since it vanishes for ¹"1 at any temperature: The correlation function (101) is always `shot-noise-likea. One more necessary remark is that all the shot noise components in this example are actually expressed only through absolute values of the scattering matrix elements: phases are not important for noise in this simple edge channel problem. Early experiments on noise in quantum Hall systems were oriented to other sources of noise (see e.g. Refs. [104,105]), and are not discussed here. Shot noise in the quantum Hall regime was studied by Washburn et al. [106], who measured voltage #uctuations in a six-terminal geometry with a constriction, which, in principle, allows for direct comparison with the above theory. They obtained results in two magnetic "elds, corresponding to the "lling factors "1 and "4. Although their results were dominated by 1/f-noise, Washburn et al. were able to "nd that shot noise is very much reduced below the Poisson value, and the order of magnitude corresponds to theoretical results. 2.6.7. Hanbury Brown}Twiss ewects with edge channels The conductor of Fig. 13 is an electrical analog of the scattering of photons at a half-silvered mirror (see Fig. 1). Like in the table top experiment of Hanbury Brown and Twiss [7,8], there is the possibility of two sources which send particles to an object (here the quantum point contact) permitting scattering into transmitted and re#ected channels which can be detected separately. Bose statistical e!ects have been exploited by Hanbury Brown and Twiss [7] to measure the diameter of stars. The electrical geometry of Fig. 13 was implemented by Henny et al. [12], and the power spectrum of the current correlation between contact 1 (re#ected current) and contact 3 (transmitted current) was measured in a situation where current is incident from contact 4 only. Contact 2 was closed such that e!ectively only a three-terminal structure resulted. For the edgechannel situation considered here, in the zero-temperature limit, this does not a!ect the correlation between transmitted and re#ected current. (At "nite temperature, the presence of the fourth contact would even be advantageous, as it avoids, as described above, the `contaminationa due to thermal noise of the correlation function of re#ected and transmitted currents, see Eq. (101)). The experiment by Henny et al. "nds good agreement with the predictions of Ref. [18], i.e. Eq. (100). With experimental accuracy the current correlation S is negative and equal in magnitude to the mean-square current #uctuations S "S in the transmitted and re#ected beam. The experiment "nds thus complete anti-correlation. This outcome is related to the Fermi statistics only indirectly: If the incident carrier stream is noiseless, current conservation alone leads to Eq. (100). As pointed out by Henny et al. [12], the experiment is in essence a demonstration that in Fermi systems the incident carrier stream is noiseless. The pioneering character of the experiment by Henny et al. [12] and an experiment by Oliver et al. [107] which we discuss below lies in the demonstration of the possibility of measuring current}current correlation in electrical conductors [108]. Henny et al. [12] measured not only the shot noise but used the four-terminal geometry of Fig. 13 to provide an elegant and interesting demonstration of the #uctuation-dissipation relation, Eq. (54). Now it is interesting to ask, what happens if this complete population of the available states is destroyed (the incident carrier stream is not noiseless any more). This can be achieved [355] by inserting an additional quantum point contact in the path of the incident carrier beam (see Fig. 14).
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 53 Fig. 14. Three-probe geometry illustrating the experiment by Oberholzer et al. [355]. We denote the transmission and re#ection probability of this "rst quantum point contact by and "1!, and the transmission and re#ection probability of the second quantum point contact by ¹ and R"1!¹ as above. The scattering matrix of this system has the form !i 0 s" (¹) !iR !i(¹) . (102) !i(R) ¹ !(R) The phases in this experiment play no role and here have been chosen to ensure the unitarity of the scattering matrix. In the zero-temperature limit with a voltage di!erence < between contact 1 and contacts 2 and 3 (which are at the same potential) the noise power spectra are S" e< !¹ !R !¹ ¹(1!¹) !¹R . (103) !R !¹R R(1!R) The correlation function between transmitted and re#ected beams S "S "!R¹ is proportional to the square of the transmission probability in the "rst quantum point contact. For "1 the incident beam is completely "lled, and the results of Henny et al. [12] are recovered. In the opposite limit, as tends to zero, almost all states in the incident carrier stream are empty, and the anti-correlation between transmitted and re#ected beams also tends to zero. 2.6.8. Three-terminal structures in zero magnetic xeld A current}current correlation was also measured in an experiment by Oliver et al. [107] in a three-probe structure in zero magnetic "eld. This experiment follows more closely the suggestion of Martin and Landauer [21] to consider the current}current correlations in a Y-structure. Ref. [21], like Ref. [18], analyzes the noise power spectrum in the zero-frequency limit. Early experiments on a three-probe structure by Kurdak et al. [109] were dominated by 1/f-noise and did not show any e!ect. Here the following remark is appropriate. Strictly speaking, the Hanbury Brown}Twiss (HBT) e!ect is a coincidence measurement. In the optical experiment the intensity #uctuation dI (t) is
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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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