shot noise in mesoscopic conductors - Low Temperature Laboratory

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152 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 For quantum wells the noise suppression is F"S/2eI"( # )/( # ). The suppression is universal for metallic di!usive wires (F" ) and chaotic cavities (F" ). Far from equilibrium, in the vicinity of instability points the shot noise can exceed the Poisson value. In the limit of low transmission, the shot noise is Poissonian and measures the charge of transmitting particles. In particular, for normal metal}superconductor interfaces this charge equals 2e, whereas in SNS systems it is greatly enhanced due to multiple Andreev re#ections. In the limit of low re#ection, the shot noise may be understood as Poissonian noise of re#ected particles; in this way, the charge of quasiparticles in the fractional quantum Hall e!ect is measured. For carriers with Fermi statistics, in multi-terminal systems the zero-frequency correlations of currents at di!erent terminals are always negative. For Bose statistics, they may under certain circumstances become positive. These cross-correlations may be used to probe the statistics of quasiparticles. The ensemble-averaged shot noise may be described both quantum-mechanically (scattering approach; Green's function technique) and classically (master equation; Langevin and Boltzmann}Langevin approach; minimal correlation approach). Where they can be compared, classical and quantum-mechanical descriptions provide the same results. Classical methods, of course, fail to describe genuinely quantum phenomena like, e.g. the quantum Hall e!ect. As a function of frequency, the noise crosses over from the shot noise to the equilibrium noise SJ. This is only valid when the frequency is low as compared with the inner energy scales of the system and inverse times of the collective response. For higher frequencies, the noise is sensitive to all these scales. However, the current conservation for frequency-dependent noise is not automatically provided, and generally is not achieved in non-interacting systems. Inelastic scattering may enhance or suppress noise, depending on its nature. In particular, in macroscopic systems shot noise is always suppressed down to zero by inelastic (usually, electron}phonon) scattering. When interactions are strong, shot noise is usually Poissonian, like in the Coulomb blockade plateau regime. We expect that the future development of the "eld of the shot noise will be mainly along the following directions: Within the "eld of mesoscopic electrical systems: (i) experimental developments; (ii) frequency dependence; and (iii) shot noise in strongly correlated systems; and more generally (iv) shot noise in disordered and chaotic quantum optical systems, shot noise measurements of phonons (and, possibly, of other quasiparticles). Acknowledgements We have pro"ted from discussions and a number of speci"c comments by Pascal Cedraschi, Thomas Gramespacher, and Andrew M. Martin. Some parts of this Review were written at the Aspen Center for Physics (Y.M.B); at the Max-Planck-Institut fuK r Physik Komplexer Systeme, Dresden (Y.M.B.); at the Centro Stefano Franscini, Ascona (Y.M.B. and M.B.); and at the UniversiteH de Montpellier II (Y.M.B.). We thank these institutions for hospitality and support. This work was supported by the Swiss National Science Foundation via the Nanoscience program.
Note added in proof Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 153 Since the submission of this Review, a number of articles on shot noise have come to our attention. We give below a very brief account of them, merely to maintain the completeness of our reference list. Experimentally, the doubling of the e!ective charge in shot noise at interfaces between a di!usive metal and a superconductor was observed by Jehl et al. [357] and Kozhevnikov et al. [358]. Photon-assisted tunneling at such an interface was investigated by Kozhevnikov et al. [359]. Theoretically, the following issues are addressed. Lesovik [360] re-examines the expression for shot noise for non-interacting electrons, and "nds that the contribution of non-parabolicity of the spectrum in the vicinity of the Fermi surface leave the "nite-frequency noise "nite even in the limit of ideal transmission ¹"1. Gurevich and Muradov [361] study shot noise in the Coulomb drag current, which appears in a ballistic nanowire coupled to a nearby nanowire. Agam et al. [362] investigate shot noise in chaotic cavities not performing an ensemble average, and "nd a suppression below F"1/4 if the electron escape time is "nite. Monte-Carlo simulations of shot noise in non-degenerate double-barrier diodes are performed by Reklaitis and Reggiani [363]. Bulashenko et al. [364] provide an analytical theory of noise in ballistic interacting conductors with an arbitrary distribution function of injected electrons. Turlakov [365] investigates frequency-dependent Nyquist noise in disordered conductors and reports a structure at the Maxwell frequency "4, being the conductivity, which is yet one more collective response frequency, as we discussed in the main body of the Review. Tanaka et al. [366] study shot noise at NS interfaces where the superconductor has a d-symmetry, for di!erent orientations of the interface, and taking into account the spatial dependence of the pair potential. Korotkov and Likharev [367] calculate noise for hopping transport in interacting conductors; the noise is less suppressed than for Coulomb blockade tunnel junctions arrays, allegedly due to redistribution of the electron density. Stopa [368] performs numerical simulations to study #uctuations in arrays of tunnel junctions. Green and Das apply Boltzmann kinetic theory to treat non-linear noise in conductors with screening (emphasizing the conformity with the Fermi liquid) [369] and study non-linear noise in metallic di!usive conductors [370]. Andreev and Kamenev [371] investigate charge counting statistics for the case when the scattering matrix is time-dependent and apply their theory to the counting statistics of adiabatic charge pumping. Appendix A. Counting statistics and optical analogies The question which naturally originates after consideration of the shot noise is the following: Can we obtain some information about the higher moments of the current? Since, as we have seen, the shot noise at zero frequency contains more information about the transmission channels than the average conductance, the studies of the higher moments may reveal even more information. Also, we have seen that in the classical theories of shot noise the distribution of the Langevin sources (elementary currents) is commonly assumed to be Gaussian, in order to provide the equivalence between the Langevin and Fokker}Planck equations [213]. An independent analysis of the higher moments of the current can reveal whether this equivalence in fact exists, and thus how credible the classical theory is.
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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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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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Our starting point is Eq. (51), whi
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introducing the transmission coe$ci
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investigated experimentally. A self
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and the non-equilibrium resistance
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the sample. These electrons are des
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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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