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Volumn 64, Issue 1, 2001, Pages

Quantum interference and the formation of the proximity effect in chaotic normal-metal/superconducting structures

Author keywords

[No Author keywords available]

Indexed keywords

METAL;

EID: 0034902934     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.64.014512     Document Type: Article
Times cited : (46)

References (44)
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    • Golubov, A.1    Kuprianov, M.Y.2
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    • Muzykantskii, B.A.1    Khmelnitskii, D.E.2
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    • Zh. Éksp. Teor. Fiz. 46, 1823 (1965) [
    • A.F. Andreev, Zh. Éksp. Teor. Fiz. 46, 1823 (1965) [ Sov. Phys. JETP 19, 1228 (1965)].
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  • 28
    • 85038342185 scopus 로고    scopus 로고
    • The conversion of electrons to holes is accompanied by a phase shift of (Formula presented) and the reciprocal conversion of holes into electrons by (Formula presented). A trajectory containing a twofold conversion, electron (Formula presented) hole (Formula presented) electron, therefore comes with a total phase shift of (Formula presented)
    • The conversion of electrons to holes is accompanied by a phase shift of (Formula presented) and the reciprocal conversion of holes into electrons by (Formula presented). A trajectory containing a twofold conversion, electron (Formula presented) hole (Formula presented) electron, therefore comes with a total phase shift of (Formula presented).
  • 29
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    • Pis’ma Zh. Éksp. Teor. Fiz. 61 830 (1995)
    • A. Golubov and M.Y. Kuprianov, Pis’ma Zh. Éksp. Teor. Fiz. 61 830 (1995) [JETP Lett. 61, 851 (1995)].
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    • Golubov, A.1    Kuprianov, M.Y.2
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    • Pis’ma Zh. Éksp. Teor. Fiz. 67, 21 (1998)
    • A. Altland, B.D. Simons, and D. Taras-Semchuk, Pis’ma Zh. Éksp. Teor. Fiz. 67, 21 (1998) [JETP Lett. 67, 22 (1998)].
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    • Altland, A.1    Simons, B.D.2    Taras-Semchuk, D.3
  • 32
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    • Notice that, in the context of superconductor problems, there is a profound difference between the trajectory oriented semiclassical approach and the quasiclassical approach. Whereas the former is based on a summation over classical trajectories of fixed energy, the latter essentially amounts to an expansion of the Gorkov equations to leading order in (Formula presented). Although both approaches use (Formula presented) as a small expansion parameter, they are fundamentally different. See Ref. 43 for a detailed discussion of this point
    • Notice that, in the context of superconductor problems, there is a profound difference between the trajectory oriented semiclassical approach and the quasiclassical approach. Whereas the former is based on a summation over classical trajectories of fixed energy, the latter essentially amounts to an expansion of the Gorkov equations to leading order in (Formula presented). Although both approaches use (Formula presented) as a small expansion parameter, they are fundamentally different. See Ref. 43 for a detailed discussion of this point.
  • 33
    • 85038276848 scopus 로고    scopus 로고
    • Under quantum dot conditions (Formula presented), the defining equation for the diffusion mode (diffuson), (Formula presented), is solved by (Formula presented) (see, e.g., Ref. 44
    • Under quantum dot conditions (Formula presented), the defining equation for the diffusion mode (diffuson), (Formula presented), is solved by (Formula presented) (see, e.g., Ref. 44).
  • 35
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    • The quasiclassical Green function is related to the impurity-averaged Gorkov Green function G through (Formula presented), where (Formula presented)and (Formula presented) is the microscopic Gorkov Green function. Here (Formula presented). The origin of the only approximate equality between the quasiclassical (Formula presented) and the p-integrated exact Wigner transform of the Gorkov Green function (Formula presented) is discussed in Appendix A
    • The quasiclassical Green function is related to the impurity-averaged Gorkov Green function G through (Formula presented), where (Formula presented)and (Formula presented) is the microscopic Gorkov Green function. Here (Formula presented). The origin of the only approximate equality between the quasiclassical (Formula presented) and the p-integrated exact Wigner transform of the Gorkov Green function (Formula presented) is discussed in Appendix A.
  • 36
    • 0009905110 scopus 로고    scopus 로고
    • I. V. Lerner J. Keating D. E. Khmelnitskii Plenum, New York
    • A. Altland and B. D. Simons, in Supersymmetry and Trace Formulae, edited by I. V. Lerner, J. Keating, and D. E. Khmelnitskii (Plenum, New York, 1999).
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    • We note that this modeling scheme is almost certainly incomplete as it suggests that a given realization of a chaotic system autoregularizes itself. In contrast, other approaches to the problem (cf., e.g., Ref. 39) indicate that some minimal averaging over a stochastic set of parameters is needed to stabilize the very formulation of a quasiclassical Green function approach
    • We note that this modeling scheme is almost certainly incomplete as it suggests that a given realization of a chaotic system autoregularizes itself. In contrast, other approaches to the problem (cf., e.g., Ref. 39) indicate that some minimal averaging over a stochastic set of parameters is needed to stabilize the very formulation of a quasiclassical Green function approach.
  • 42
    • 85038327351 scopus 로고    scopus 로고
    • At this stage the derivation of the clean (Formula presented) model becomes problematic: There is no reason for assuming that the relevant fluctuations around the mean field are smooth. In fact, the singular phase space structure of the eigenfunctions of the Liouville operator suggests that the opposite is the case. That we are nevertheless permitted to make a smooth ansatz has to do with the fact that the subsequent disorder average will project out all field configurations of discontinuous type. Notice that the problem outlined above equally affects the pure quasiclassical formalism where identical construction steps are performed in a nonfunctional integral context
    • At this stage the derivation of the clean (Formula presented) model becomes problematic: There is no reason for assuming that the relevant fluctuations around the mean field are smooth. In fact, the singular phase space structure of the eigenfunctions of the Liouville operator suggests that the opposite is the case. That we are nevertheless permitted to make a smooth ansatz has to do with the fact that the subsequent disorder average will project out all field configurations of discontinuous type. Notice that the problem outlined above equally affects the pure quasiclassical formalism where identical construction steps are performed in a nonfunctional integral context.


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