9780198525899

Quantum Chaos and Quantum Dots

by ;
  • ISBN13:

    9780198525899

  • ISBN10:

    0198525893

  • Format: Hardcover
  • Copyright: 2004-02-12
  • Publisher: Oxford University Press

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Summary

Dynamics of billiard balls and their role in physics have received wide attention. Billiards can nowadays be created as quantum dots in the microscopic world enabling one to envisage the so-called quantum chaos, (i.e.: quantum manifestation of chaos of billiard balls). In fact, owing to recent progress in advanced technology, nanoscale quantum dots, such as chaotic stadium and antidot lattices analogous to the Sinai Billiard, can be fabricated at the interface of semiconductor heterojunctions. This book begins ite exploration of the effect of chaotic electron dynamics on ballistic quantum transport in quantum dots with a puzzling experiment on resistance fluctuations for stadium and circle dots. Throughout the text, major attention is paid to the semiclassical theory which makes it possible to interpret quantum phenomena in the language of the classical world. Chapters one to four are concerned with the elementary statistical methods (curvature, Lyapunov exponent, Kolmogorov-Sinai entropy and escape rate), which are needed for a semiclassical description of transport in quantum dots. Chapters five to ten discuss the topical subjects in the field, including the ballistic weak localization, Altshuler-Aronov-Spivak oscillation, partial time-reversal symmetry, persistent current, Arnold diffusion and Coulomb blockade.

Author Biography


Katsuhiro Nakamura is Professor of Applied Physics at Osaka City University in Japan. Takahisa Harayama is Senior Researcher at ATR Adaptive Communications Research Laboratories, Kyoto, in Japan.

Table of Contents

Quantum Chaos and Billiards
1(10)
Birth of the Physics of Billiards
1(2)
What is Quantum Chaos?
3(1)
Resistance of Quantum Dots
4(3)
Dynamics in Billiards and Semiclassical Theory
7(4)
Quantum Transport and Chaos in Billiards
11(7)
Quantum Theory of Conductance
11(1)
Semiclassical Approximation and Stationary-Phase Method
12(1)
Semiclassical Green Function and Transmission Amplitude
13(2)
Autocorrelation Function
15(1)
Conductance and Area Distribution
15(3)
Motion of a Billiard Ball
18(23)
Expanding Wavefront and Lyapunov Exponent
18(9)
Birkhoff Coordinates and the Repeller
27(4)
Kolmogorov--Sinai Entropy and Escape Rate
31(5)
Area Distribution
36(5)
Semiclassical Theory of Conductance Fluctuations
41(19)
Quantum Billiards with Lead Wires
41(2)
Semiclassical Green Function
43(9)
Transmission Coefficients
52(3)
Conductance Fluctuations
55(5)
Semiclassical Quantization and Thermodynamics of Mesoscopic Systems
60(19)
Semiclassical Quantization of Chaos and Regular Orbits
60(12)
Berry-Tabor's Trace Formula
64(2)
Gutzwiller's Trace Formula
66(6)
Thermodynamics of Mesoscopic Systems
72(7)
Grand Canonical Ensemble
72(4)
Canonical Ensemble
76(3)
Orbital Diamagnetism and Persistent Current
79(19)
Historical Background
79(2)
Orbital Diamagnetism in the Light of Nonlinear Dynamics
81(5)
Semiclassical Orbital Diamagnetism in 3-d Billiards
86(8)
Integrable (Spherical Shell) Billiards
89(4)
Fully Chaotic Billiards
93(1)
Semiclassical Persistent Current in 3-d Shell Billiards
94(4)
Quantum Interference in Single Open Billiards
98(32)
Chaos and Quantum Transport
98(3)
Ballistic Weak Localization (WL)
101(4)
Criticism against the Semiclassical Theory of Ballistic WL
105(3)
Ballistic AAS Oscillation
108(3)
Effects of Small-Angle Induced Diffraction
111(9)
Partial Time-Reversal Symmetry and Ballistic Weak-Localization Correction
112(3)
Semiclassical Derivation of Universal Conductance Fluctuations
115(5)
Self-Similar Magneto-Conductance Fluctuations
120(10)
Harmonic Saddles as the Origin of Self-Similarity
122(5)
Scaling Properties
127(3)
Linear Response Theory in the Semiclassical Regime
130(15)
Realization of Sinai Billiards
130(2)
Semiclassical Shubnikov--de Haas Oscillation
132(6)
Semiclassical Kubo Formula in Antidot Superlattices
138(4)
Drude Conductivity
139(1)
Quantum Correction
140(2)
Effect of Finite Temperature and Spin
142(1)
Interpretation of Experiments
143(2)
Orbit Bifurcations, Arnold Diffusion, and Coulomb Blockade
145(29)
Orbit Bifurcations in Triangular Antidot Lattices
145(12)
Semiclassical Conductivity and Orbit Bifurcations
149(2)
Quantum Correction without Orbit Bifurcations
151(3)
Orbit Bifurcations and Anomalous Resistivity Fluctuations
154(3)
Arnold Diffusion and Negative Magneto-Resistance
157(9)
Semiclassical Conductance for Open Three-Dimensional Billiards
157(2)
Completely or Partially Broken-Ergodic 3-d Billiards
159(2)
Effects of Symmetry-Breaking Weak Magnetic Field
161(5)
Semiclassical Theory of Coulomb Blockade
166(8)
Peak Height and Wavefunction
168(1)
Peak Height Distribution
169(5)
Nonadiabatic Transitions, Energy Diffusion and Generalized Friction
174(15)
What is Energy Diffusion?
174(1)
What is Level Statistics?
175(3)
Energy Diffusion: Landau--Zener regime
178(3)
Energy Diffusion: Linear-Response Regime
181(2)
Frictional Force due to Nonadiabatic Transition
183(3)
Future Prospects
186(3)
References 189(6)
Index 195

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