Quantum‐chaotic scattering effects in semiconductor microstructures

For reviews see M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1991) and M.-J. Giannoni, A. Voros, and J. Zinn-Justin (editors). Chaos and Quantum Physics (North-Holland, New York, 1991).

R. A.

Jalabert

,

H. U.

Baranger

, and

A. D.

Stone

, , ().

For reviews of mesoscopic physics see C. W. J. Beenakker and H. van Houten in Solid State Physics, edited by H. Ehrenreich and D. Turnbull (Academic, New York, 1991), Vol. 44, pp. 1–228;

B. L. Altshuler, P. A. Lee, and R. A. Webb (editors), Mesoscopic Phenomena in Solids (North-Holland, New York, 1991);

and

S.

Washburn

and

R. A.

Webb

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For reviews of weak-localization in disordered metals see

P. A.

Lee

and

T. V.

Ramakrishnan

, , ();

and

G.

Bergmann

, , ().

P. A.

Lee

and

A. D.

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P. A.

Lee

,

A. D.

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, and

H.

Fukuyama

, , ().

B. L.

Altshuler

, , ()

[

B. L.

Altshuler

, , ()].

B. L.

Altshuler

and

B. I.

Shklovskii

, , ().

N.

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,

Y.

Imry

, and

U.

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C. M.

Marcus

,

A. J.

Rimberg

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R. M.

Westervelt

,

P. F.

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A. C.

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U. W. Keller, O. Millo, A. Mittal, D. E. Prober, and R. N. Sacks, Surf. Sci. in press (1993).

M. J. Berry, J. A. Katine, C. M. Marcus, R. M. Westervelt, and A. C. Gossard, Surf. Sci., in press (1993).

D.

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K.

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A.

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R.

Bergmann

,

H.

Schweizer

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K.

von Klitzing

, and

G.

Weimann

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C. M.

Marcus

,

R. M.

Westervelt

,

P. F.

Hopkins

, and

A. C.

Gossard

, , ().

C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Surf. Sci., in press (1993).

C. M.

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R. M.

Westervelt

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P. F.

Hopkins

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A. C.

Gossard

, , ().

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rhinehart, and Winston, New York, 1976).

R.

Landauer

, , ().

M.

Büttiker

, , ().

D. S.

Fisher

and

P. A.

Lee

, , ().

A. D.

Stone

and

A.

Szafer

, , ().

H. U.

Baranger

and

A. D.

Stone

, , ().

M. V.

Berry

and

J. P.

Keating

, , ();

M.

Sieber

and

F.

Steiner

, , ();

G.

Tanner

,

P.

Scherer

,

E. B.

Bogomolny

,

B.

Eckhardt

, and

D.

Wintgen

, , ();

E. B.

Bogomolny

, , ();

R.

Aurich

,

C.

Mattheis

,

M.

Sieber

, and

F.

Steiner

, , ();

J. P.

Keating

, , ().

M. V.

Berry

, , ().

O.

Bohigas

,

M.-J.

Giannoni

, and

C.

Schmit

, , ();

O. Bohigas, in Ref. 1, pp. 87–200.

A.

Holle

,

J.

Main

,

G.

Wiebusch

,

H.

Rottke

, and

K. H.

Welge

, , ().

For a review of quantum chaotic scattering see U. Smilansky in Ref. 1, pp. 371–441.

R. A.

Jalabert

,

A. D.

Stone

, and

Y.

Alhassid

, , ();

A. D. Stone, R. A. Jalabert, and Y. Alhassid, in Transport Phenomena in Mesoscopic Systems, edited by H. Fukuyama and T. Ando, Springer Series in Solid-State Science (Springer-Verlag, Berlin, 1992), Vol. 109.

A. D.

Stone

and

H.

Bruus

, , ();

A. D.

Stone

and

H.

Bruus

, , in press ().

T.

Ericson

, , ().

R.

Blümel

and

U.

Smilansky

, , ();

R.

Blümel

and

U.

Smilansky

, , ();

R.

Blümel

and

U.

Smilansky

, , ().

H. U.

Baranger

,

D. P.

DiVincenzo

,

R. A.

Jalabert

, and

A. D.

Stone

, , ().

H. U.

Baranger

,

R. A.

Jalabert

, and

A. D.

Stone

, , ().

H. U. Baranger, R. A. Jalabert, and A. D. Stone (unpublished).

See, e.g.,

H. U.

Baranger

, , ().

J. H. Jensen (preprint, 1993).

W. H.

Miller

, , ().

See, e.g., E. H. Heller in Ref. 1, pp. 547–663.

For reviews of classical chaotic scattering see T, Tél in Direction in Chaos, Vol 3, edited by H. B. Lin (World Scientific, Singapore, 1990), pp. 149–211 and U. Smilansky in Ref. 1, pp. 371–441.

R.

Jensen

, , ().

M. L.

Roukes

and

O. L.

Alerhand

, , ().

P. A.

Mello

and

A. D.

Stone

, , ().

C. H.

Lewenkopf

and

H. A.

Weidenmüller

, , ().

E.

Doron

,

U.

Smilansky

, and

A.

Frenkel

, , ();

E.

Doron

,

U.

Smilansky

, and

A.

Frenkel

, , ().

R.

Blümel

and

U.

Smilansky

, , ().

It is important to average over an infinite window in k; see Eq. (15).

There are several subtle aspects of this argument which we address briefly. (1) The paths involved in the sum change as a function of k because the boundary conditions on the angles change. This effect is, however, not essential and can be removed by either averaging using a discrete set of $kn = 2nk0$ for which paths always remain in the considered set (though new paths enter, of course), or by inserting a delta-function when introducing the integral over angles and performing the integrals more $carefully.33$ (2) Because we are considering the part of R which is diagonal in mode index, the number of paths contributing to the sum increases only linearly with k, and there is no interference between paths which start contributing at some large $k2$ and those which were already contributing at $k1<k2.$ For instance, the paths contributing to $Σn evenRnn(2k)$ are exactly the same as those contributing to $Σall nRnn(k)$ while the paths contributing to $Σn oddRnn(2k)$ are new. Thus $δRD$ is a simpler quantity than the full two-point correlation function, considered, for instance, in the statistics of energy levels in Ref. 23. It is this simplicity which allows an exact evaluation. (3) We assume, of course, that the semiclassical expressions for $rnm$ and $Rnm$ $converge.35$ Further, we assume that the expression for $δRD$ converges absolutely in order to exchange the limit $q→∞$ and the sum over paths.

M. V.

Berry

and

M.

Robnik

, , ().

W. A.

Lin

,

J. B.

Delos

, and

R. V.

Jensen

, , ().

Y.-C.

Lai

,

R.

Blümel

,

E.

Ott

, and

C.

Grebogi

, , () and references therein.

V.

Bauer

and

G. F.

Bertsch

, , ().

O.

Legrand

and

D.

Sornette

, , ();

O.

Legrand

and

D.

Sornette

, , ().

R. B. S.

Oakeshott

and

A.

MacKinnon

, , ().

P.-G. DeGennes, Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1966), pp. 255–259.

C. W. J.

Beenakker

and

H.

van Houten

, , ().

D. Ullmo, K. Richter, and R. A. Jalabert (unpublished).

M. Hastings, A. D. Stone, and H. U. Baranger (unpublished).

S.

Iida

,

H. A.

Weidenmüller

, and

J. A.

Zuk

, , ();

S.

Iida

,

H. A.

Weidenmüller

, and

J. A.

Zuk

, , ().

P. A.

Mello

,

P.

Pereyra

, and

T. H.

Seligman

, , () and references therein.

For a review see A. D. Stone, P. A. Mello, K. Muttalib, and J.-L. Pichard in Ref. 3.

N. Argaman, E. Doron, J. Keating, A. Kitaev, M. Sieber, and U. Smilansky (preprint).

M. J. Berry (private communication, 1993).

M. C.

Gutzwiller

, , ().

P.

Gaspard

and

S. A.

Rice

, , , ().

Two details of our power spectrum analysis deserve comment. (1) The singularities in the transmission coefficients at the threshold for a mode in the $leads34$ produce power at high frequency. Most of the time we work within a given mode (away from threshold) in order to minimize this effect; however, the high-frequency power produced does not hinder our getting the characteristic scale of the fluctuations as in Figs. 7 and 9. (2) We do see low-frequency peaks in the power spectra which, however, are eliminated by averaging in the data shown here in order to reveal the high-frequency behavior more clearly. The connection between the peaks in the power spectrum and periodic orbits in the cavity is not clear (numerically) at this time.

S.

Feng

,

C. L.

Kane

,

P. A.

Lee

, and

A. D.

Stone

, , ().