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the semiclassical argument is given in the Introduction
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A clear experimental signature is given in A. M. Chang, H. U. Baranger, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett. 73, 2111 (1994).
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for earlier experimental work see C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992).
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C. M. Marcus, R. M. Westervelt, P. F. Hopkins and A. C. Gossard, Chaos 3, 643 (1993).
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M. J. Berry, J. A. Katine, C. M. Marcus, R. M. Westervelt and A. C. Gossard, Surf. Sci.ibid. 305, 495 (1994).
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M. W. Keller, O. Millo, A. Mittal, D. E. Prober and R. Sacks, Surf. Sci.ibid. 305, 501 (1994).
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Some authors have gone as far as stating that WL is not a semiclassical phenomenon: see, e.g., M. B. Hastings, A. D. Stone and H. U. Baranger, Phys. Rev. B 50, 8230 (1994).
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For finite ω there are three modifications of this formula: (a) both terms are multiplied by (Formula presented); (b) the energy integration variable is shifted, so that &εtilde; +ℏω replaces ε in the Fermi functions [for example, the symmetric term has &εtilde; =0, i.e., ((Formula presented) (E-ℏω/2)-(Formula presented) (E+ℏω/2)]/ℏω replaces -(Formula presented)(E)); (c) an additional term, equal to the principal part of -(Formula presented) ∫(d&εtilde; /π)[tcos(&εtilde; /ℏ)/&εtilde; ]|P[(Formula presented) [E-(&εtilde; +ℏω)/2]-(Formula presented) [E +(&εtilde; +ℏω)/2]|P]/(&εtilde;
-
see, e.g., H. U. Baranger and A. D. Stone, Phys. Rev. B 40, 8169 (1989). For finite ω there are three modifications of this formula: (a) both terms are multiplied by (Formula presented); (b) the energy integration variable is shifted, so that &εtilde;+ℏω replaces ε in the Fermi functions [for example, the symmetric term has &εtilde;=0, i.e., ((Formula presented) (E-ℏω/2)-(Formula presented) (E+ℏω/2)]/ℏω replaces -(Formula presented)(E)); (c) an additional term, equal to the principal part of -(Formula presented) ∫(d&εtilde;/π)[tcos(&εtilde;/ℏ)/&εtilde;]|P[(Formula presented) [E-(&εtilde;+ℏω)/2]-(Formula presented) [E +(&εtilde;+ℏω)/2]|P]/(&εtilde;
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36149041879
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+ℏω), appears (apart from these, there is always the term of F(E, t) which cancels with the diamagnetic term, and reads -(i/ℏω)∫(dε/2π)cos(εt/ℏ)[(Formula presented) (E-ε/2) -(Formula presented) (E+ε/2)]/ε). As long as the whole volume of the sample is integrated over, including possible edge states, there is no need to invoke additional non-Fermi-surface terms [cf., e.g., P. Streda, J. Phys. C 15, L717 (1982)].
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J. Phys. C
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Cf. I. B. Levinson Zh. Eksp. Teor, Fiz. 95, 2175 (1989) [Sov. Phys. JETP 68, 1257 (1989)];
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