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86 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 4. Shot noise in hybrid normal and superconducting structures The dissipationless current (supercurrent) in superconductors is a property of a ground state, and therefore is noiseless } it is not accompanied by any #uctuations. However, noise appears if the superconductor is in contact with piece(s) of normal metals. This section is devoted to the description of shot noise in these hybrid structures, which exhibit a variety of interesting phenomena. Following the point of view of the previous sections, we present here a description which is based on an extension of the scattering approach to hybrid structures. 4.1. Shot noise of normal-superconductor interfaces 4.1.1. Simple NS interface, scattering theory and general expressions We consider "rst an interface of normal metal and superconductor (NS). If the applied voltage is below the superconducting gap , the only mechanism of charge transport is Andreev re#ection at the NS interface: an electron with energy E approaching the interface from the normal side is converted into a hole with energy !E. The velocity of the hole is directed back from the interface to a normal metal. The missing charge 2e on the normal side appears as a new Cooper pair on the superconducting side. There is, of course, also a reverse process, when a Cooper pair recombines with a hole in the normal conductor, and creates an electron. At equilibrium, both processes have the same probability, and there is thus no net current. However, if a voltage is applied, a "nite current #ows across the NS interface. The scattering theory which we described in Section 2 has to be extended to take into account the Andreev scattering processes [167]. We give here only a sketch of the derivation; a more detailed description, as well as a comprehensive list of references, may be found in Refs. [72,168]. To set up a scattering problem, we consider the following geometry (Fig. 27): the boundary between normal and superconducting parts is assumed to be sharp, and elastic scattering happens inside the normal metal (shaded region) only. The scattering region is separated from the NS interface by an ideal region 2, which is much longer than the wavelength, and thus we may there use asymptotic expressions for the wave functions. This spatial separation of scattering from the interface is arti"cial. It is not really necessary; our consideration leading to Eq. (159) and (160) does not rely on it. Furthermore, for simplicity, we assume that the number of transverse channels in the normal lead 1 and in the intermediate normal portion 2 is the same. We proceed in much the same way as in Section 2 and de"ne the annihilation operators in region 1 asymptotically far from the scattering area, a( , which annihilate electrons incoming on Fig. 27. A simpli"ed model of an NS interface adopted for the scattering description. Scattering is assumed to happen in the shaded area inside the normal metal, which separates the ideal normal parts 1 and 2. Andreev re#ection is happening strictly at the interface separating 2 and the superconductor.
the sample. These electrons are described by wave functions (r ) exp(ik z) with unit incident amplitude, where the coordinate z is directed towards the superconductor (from the left to the right in Fig. 27), the index n labels the transverse channels. Here we neglected the energy dependence of the wave vector, anticipating the fact that only energies close to the Fermi surface will play a role in transport. Similarly, the operator bK annihilates electrons in the outgoing states in the region 1, (r ) exp(!ik z). For holes, wede"ne an annihilation operator in the incoming states in the region 1 as a( , and the corresponding wave function is (r ) exp(!ik z). Note that though this wave function is identical to that for outgoing electrons, it corresponds to the incoming state with energy !E. The velocity of these holes is directed towards the interface. The annihilation operator for holes in the outgoing states, bK , is associated with the wave function (r ) exp(ik z). Creation operators for electrons and holes are de"ned in the same way. Thus, the di!erence with the scattering theory for normal conductors is that we now have an extra index, which assumes values e, h and discriminates between electrons and holes. The electron and hole operators for the outgoing states are related to the electron and hole operators of the incoming states via the scattering matrix, bK bK "s a( a( , s s s s a( , (156) a( where the element s gives the outgoing electron current amplitude in response to an incoming electron current amplitude, s gives the outgoing hole current amplitude in response to an incoming electron current amplitude, etc. The generalized current operator (32) for electrons and holes in region 1 is IK (t)" e 2 dE dE e Tr [a( (E)a( (E)!a( (E)a( (E) !bK (E)bK (E)#bK (E)bK (E)] , (157) or, equivalently, Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 87 IK (t)" e 2 dE dE e Tr [a( (E)A (E, E)a( (E)] , (158) where we have again introduced electron}hole indices and , and the trace is taken over channel indices. The matrix A is given by A(E, E)"!s(E)s(!E), " 1 0 0 !1 , with the matrix discriminating between electron and holes. Introducing the distribution functions for electrons f (E)"[exp[(E!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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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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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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