shot noise in mesoscopic conductors - Low Temperature Laboratory

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72 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 that it is the collective times which matter. Displacement currents can be neglected only, if we can assure that these collective times are much shorter than any of the kinetic time-scales discussed above. Furthermore, there are problems which can only be treated by taking interactions into account, even at arbitrarily low frequency: later on in this section we will discuss the RC-times of mesoscopic conductors capacitively coupled to a gate. The noise induced into a nearby capacitor is proportional to the square of the frequency. Naive discussions which do not consider the energy dependence of the scattering matrix cannot predict such currents. Let us at "rst consider the range of frequencies that are much smaller than any inverse kinetic time scale and smaller than any inverse collective response time. This case is in some sense trivial, since the energy dependence of the scattering matrices is neglected, and the system is now not probed on the scale . As a consequence there is no novel information on the system compared to a zero-frequency noise measurement. In this case, as we will see, the entire frequency dependence of the noise is due to the frequency dependence of the Fermi functions. However, it is the lowfrequency measurements which are more easily carried out, and therefore there is some justi"cation to discuss noise spectra in this frequency interval. The rest of the section is organized as follows. Section 3.2 treats #uctuations of the particle current of independent electrons, either at equilibrium or in the presence of a dc bias, in the regime when the scattering matrices can be assumed to be frequency-independent. Section 3.3 generalizes the same notions for noise caused by an ac bias. Afterwards, we relax the approximation of the energy independence of scattering matrices, and in Sections 3.4 and 3.5 consider two simple examples. In both of them Coulomb interactions prove to be important. Though the theory of ac noise is far from being completed, we hope that these examples, representing the results available by now in the literature, can stimulate further research in this direction. In this section, we only review the quantum-mechanical description of frequency-dependent noise, based on the scattering approach. Alternatively, the frequency dependence of shot noise in di!usive conductors may be studied based on the classical Boltzmann}Langevin approach [140}145]. These developments are described in Section 6. 3.2. Low-frequency noise for independent electrons: at equilibrium and in the presence of dc transport 3.2.1. General consideration This subsection is devoted to low-frequency noise, in a regime where the scattering matrices are energy independent. We take the frequency, temperature and the voltage all below , and below any frequencies associated with the collective response of the structure. For simplicity, we only consider the two-terminal case, "e
Our starting point is Eq. (51), which in this form is given in Ref. [146]. Taking the scattering matrices to be energy independent, we write S(),S ()" e 2 ¹ dE[ f (E, )#f (E, )] with the abbreviation Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 73 # ¹ (1!¹ ) dE[ f (E, )#f (E, )] , (125) f (E, )"f (E)[1!f (E#)]#[1!f (E)] f (E#) . (126) Here the ¹ 's are, as before (energy independent) transmission coe$cients. Performing the integration, we obtain S()" e 22 coth 2k ¹ #e< ¹# (#e
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Ya.M. Blanter, M. BuK ttiker / Phys
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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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and the initial state of our two-pa
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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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