shot noise in mesoscopic conductors - Low Temperature Laboratory

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124 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 which connects the wire and the gate. For a zero-external impedance circuit this relaxation generates a charge relaxation resistance R (see the discussion in Section 3) which for a metallic di!usive conductor is of the order of the sample resistance R"¸(A). With these speci"cations we expect that a metallic di!usive wire in proximity of a gate is characterized by a frequency "1/R C which is given by "A/(c¸)#/(e¸). (Refs. [140,142] express in terms of a generalized di!usion constant D"D#A/c using the Einstein relation "eD, such that "D/¸ has the form of a Thouless energy.) For ; , noise measured at the contacts to the wire is dominated by the white-noise zero-frequency contribution (the Fano factor equals for independent electrons or 3/4 for hot electrons). For frequencies higher than the spectrum measured at the contacts of the wire starts to depend on the details of the system, and for in"nite frequency the Fano factor tends to a constant value, which may lie above as well as below the non-interacting value. This is because the zero-temperature quantum noise SJ cannot be obtained by classical means: thus, all the results of these subsections are applicable only outside of the regime when this source of noise is important. In particular, for zero temperature this means (e
non-thermalized electrons, the high-frequency noise (< ) inside the conductor grows even for zero temperature, S(x)J(/ )[x(¸!x)]. At the contacts this noise power turns to zero, and one obtains S ,S(x"0)"eI. Note that this value is the same as for a double-barrier structure with symmetric capacitances. The di!erence with the result S "2eI originates from the fact that in this mode the charge pile-up in the contact is forbidden. In addition, the frequencydependent noise in the same model is studied numerically by Naveh for electron}electron [144] and electron}phonon [145] interactions. 6.5. Shot noise in non-degenerate conductors 6.5.1. Diwusive conductors Recently, GonzaH lez et al. [258], motivated by the diversity of proofs for the noise suppression in metallic di!usive wires, performed numerical simulations of shot noise in non-degenerate di!usive conductors. They have taken the interaction e!ects into account via the Poisson equation. For the disorder a simple model with an energy-independent scattering time was assumed. They found that the noise suppression factor for this system, within the error bars, equals and in three- and two-dimensional systems, respectively. A subsequent work [259] gives the suppression factor 0.7 for d"1. An analytic theory of shot noise in these conductors was proposed by Beenakker [260], and subsequently by Nagaev [261] and Schomerus et al. [262,263]. The general conclusion is as follows. Shot noise in non-degenerate di!usive conductors is non-universal in the sense that it depends on the details of the disorder (the energy dependence of the elastic scattering time) and the geometry of the sample. In the particular case, when the elastic relaxation time is energy independent (corresponding to the simulations by GonzaH lez et al. [258,259]), the suppression factors are close to , and 0.7, but not precisely equal to these values. Below we give a brief sketch of the derivation, following Ref. [262]. We start from the Boltzmann equation (203) and add Langevin sources on the right-hand side. Three points now require special attention: (i) for a non-degenerate gas fM ;1, and thus the factors (1!fM ) in the collision integral (204) can be taken equal to 1; (ii) since carriers are now distributed over a wide energy range, all the quantities must be taken energy-dependent rather than restricted to the Fermi surface. In particular, the velocity is v(E)"(2E/m), and the energy dependence of the mean-free path matters, (iii) the screening length is energy dependent, and the system generally may not be regarded as charge neutral. We employ again the -approximation for the electronimpurity collision integral. For a while, we assume the relaxation time to be energy independent. We also neglect all kinds of inelastic scattering. Separating the distribution function into symmetric and asymmetric parts (216), and introducing the (energy-resolved) charge and current densities, (r, E, t)"e(E) f (r, E, t) , j(r, E, t)" 1 d ev(E)(E) f (r, E, t) , Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 125 we obtain two equations, analogous to Eq. (217), ) j#eE(r, t)R j"0 (233)
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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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2.3.3. Multi-terminal case We consi
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Note that in the absence of time-de
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equilibrium and shot noise. For low
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and S" e A e< dE ¹(E )[1!¹
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arrier, situated in the region !w/2
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we include weak localization e!ects
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Edge channels in the quantum Hall e
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Fig. 19. (a) Geometry of a ring thr
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a manifestation of interference e!e
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Physically, this corresponds to ele
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