shot noise in mesoscopic conductors - Low Temperature Laboratory

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12 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 2. Scattering theory of thermal and shot noise 2.1. Introduction In this section we present a theory of thermal and shot noise for fully phase-coherent mesoscopic conductors. The discussion is based on the scattering approach to electrical conductance. This approach, as we will show, is conceptually simple and transparent. A phase-coherent description is needed if we consider an individual sample, like an Aharonov}Bohm ring, or a quantum point contact. Often, however, we are interested in characterizing not a single sample but rather an ensemble of samples in which individual members di!er only in the microscopic arrangement of impurities or small variations in the shape. The ensemble-averaged conductance is typically, up to a small correction, determined by a classical expression like a Drude conductance formula. Similarly, noise spectra, after ensemble averaging, are, up to small corrections, determined by purely classical expressions. In these case, there is no need to keep information about phases of wave functions, and shot noise expressions may be obtained by classical methods. Nevertheless, the generality of the scattering approach and its conceptual clarity, make it the desired starting point of a discussion of noise in electrical conductors. Below we emphasize a discussion based on second quantization. This permits a concise treatment of the many-particle problem. Rather than introducing the Pauli principle by hand, in this approach it is a consequence of the underlying symmetry of the wave functions. It lends itself to a discussion of the e!ects related to the quantum mechanical indistinguishability of identical particles. In fact, it is an interesting question to what extent we can directly probe the fact that exchange of particles leaves the wave function invariant up to a sign. Thus, an important part of our discussion will focus on exchange ewects in current}current correlation spectra. We start this section with a review of #uctuations in idealized one- and two-particle scattering problems. This simple discussion highlights the connection between symmetry of the wave functions (the Pauli principle) and the #uctuation properties. It introduces in a simple manner some of the basic concepts and it will be interesting to compare the results of the one- and two-particle scattering problems with the many-particle problem which we face in mesoscopic conductors. 2.2. The Pauli principle The investigation of the noise properties of a system is interesting because it is fundamentally connected with the statistical properties of the entities which generate the noise. We are concerned with systems which contain a large number of indistinguishable particles. The fact that in quantum mechanics we cannot label di!erent particles implies that the wave function must be invariant, up to a phase, if two particles are exchanged. The invariance of the wave function under exchange of two particles implies that we deal with wave functions which are either symmetric or antisymmetric under particle exchange. (In strictly two-dimensional systems more exotic possibilities are permitted). These symmetry statements are known as the Pauli principle. Systems with symmetric (antisymmetric) wave functions are described by Bose}Einstein (Fermi}Dirac) statistics, respectively. Prior to the discussion of the noise properties in electrical conductors, which is our central subject, in this subsection we illustrate in a simple manner the fundamental connection between the
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 13 symmetry of the wave function and the statistical properties of scattering experiments. We deal with open systems similarly to a scattering experiment in which particles are incident on a target at which they are scattered. The simplest case in which the symmetry of the wave function matters is the case of two identical particles. Here we present a discussion of idealized two-particle scattering experiments. We consider the arrangement shown in Fig. 1, which contains two sources 1 and 2 which can emit particles and two detectors 3 and 4 which respond ideally with a signal each time a particle enters a detector. An arrangement similar to that shown in Fig. 1 is used in optical experiments. In this "eld experiments which invoke one or two incoming particle streams (photons) and two detectors are known as Hanbury Brown}Twiss experiments [7], after the pioneering experiments carried out by these two researchers to investigate the statistical properties of light. In Fig. 1 the scattering is, much as in an optical table top experiment, provided by a half-silvered mirror (beam-splitter), which permits transmission from the input-channel 1 through the mirror with probability amplitude s "t to the detector at arm 4 and generates re#ected particles with amplitude s "r into detector 3. We assume that particles from source 2 are scattered likewise and have probability amplitudes s "t and s "r. The elements s , when written as a matrix, form the scattering matrix s. The elements of the scattering matrix satisfy r#t"1 and trH#rtH"0, stating that the scattering matrix is unitary. A simple example of a scattering matrix often employed to describe scattering at a mirror in optics is r"!i/2 and t"1/2. We are interested in describing various input states emanating from the two sources. To be interesting, these input states contain one or two particles. We could describe these states in terms of Slater determinants, but it is more elegant to employ the second quantization approach. The incident states are described by annihilation operators a( or creation operators a( in arm i, i"1, 2. The outgoing states are, in turn, described by annihilation operators bK and creation operators bK , i"3, 4. The operators of the input states and output states are not independent but are related Fig. 1. An arrangement of a scattering experiment with two sources (1 and 2) and two detectors (3 and 4).
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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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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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