Quantum and Semi-classical Percolation and Breakdown in Disordered Solids

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Asok K. Sen, Kamal K. Bardhan, Bikas K. Chakrabarti
Springer Science & Business Media, Mar 20, 2009 - Science - 326 pages
This lecture notes in physics volume mainly focuses on the semi classical and qu- tum aspects of percolation and breakdown in disordered, composite or granular s- tems. The main reason for this undertaking has been the fact that, of late, there have been a lot of (theoretical) work on quantum percolation, but there is not even a (single) published review on the topic (and, of course, no book). Also, there are many theoretical and experimental studies on the nonlinear current-voltage characteristics both away from, as well as one approaches, an electrical breakdown in composite materials. Some of the results are quite intriguing and may broadly be explained utilising a semi classical (if not, fully quantum mechanical) tunnelling between - cron or nano-sized metallic islands dispersed separated by thin insulating layers, or in other words, between the dangling ends of small percolation clusters. There have also been several (theoretical) studies of Zener breakdown in Mott or Anderson in- lators. Again, there is no review available, connecting them in any coherent fashion. A compendium volume connecting these experimental and theoretical studies should be unique and very timely, and hence this volume. The book is organised as follows. For completeness, we have started with a short and concise introduction on classical percolation. In the ?rst chapter, D. Stauffer reviews the scaling theory of classical percolation emphasizing (biased) diffusion, without any quantum effects. The next chapter by A. K.
 

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Contents

Classical Percolation
1
2 Methods
3
3 Quantities and Exponents
6
Incipient Infinite Cluster
9
5 Simple Renormalisation Group
12
6 Diffusion and Percolation
14
7 Summary
18
Nonlinear Response Semiclassical Percolation and Breakdown in the RRTN Model
21
3 Localization Effects in Quantum Percolation
144
4 Percolative Effects in Advanced Materials
154
5 Conclusions
159
References
160
Quantum Percolation in the Quantum Hall Regime
163
2 The Quantized Hall Effect and Classical Percolation
165
3 Network Models at the IQHE
168
4 HartreeFock Approach to the IQHE
184

2 The Origin of the RRTN Model and Its Percolative Aspects
27
3 Nonlinear SteadyState IV Characteristics
35
4 Periodic Driving and ACResponse in the RRTNCRC Model
46
5 VRH and Low Temperature Conduction in the RRTN
55
6 Slow Powerlaw Dynamics FarfromEquilibrium
59
7 Aspects of Reversible Breakdown in the RRTN Model
65
8 Dynamical Characteristics of Breakdown
73
9 Summary and Further Works
77
References
78
Quantum Transmittance Through Random Media
83
2 One Parameter Scaling Theory of Localization
84
3 Transport Mechanisms in Disordered Media
85
4 Some Models of Disordered Systems
87
5 Some Earlier Studies on the Quantum Percolation Model
89
6 The Vector Recursion Method and Its Applications
92
7 Conclusions
105
References
106
Quantum Percolation in Two Dimensions
108
2 Resonances and Phase Variations in Ordered Limit
116
3 TimeIndependent Schrodinger Equation for Finite Disorder
120
Sending a Wave Packet Through a 2D Cluster
125
5 Summary
132
References
133
Quantum Percolation in Disordered Structures
135
2 Local Distribution Approach
137
5 Conclusions
189
References
190
Percolative Quantum Transport in Manganites
195
2 Standard Quantum Percolation
196
Phenomenology and Model
198
4 Percolative Effects in a One Band Model of Phase Competition
205
5 Percolation in Two Band Models with ElectronPhonon Coupling
217
6 Connection with Quantum Percolation and Resistor Networks
221
7 Conclusions
225
Classical and Quantum Breakdown in Disordered Materials
227
2 Analysis of the Fuse Problem
229
3 Dielectric Breakdown Problem
238
4 Zener Breakdown in Anderson Insulators
247
5 Conclusions
249
Nonequilibrium Quantum Breakdown in a Strongly Correlated Electron System
251
2 Nonadiabatic Evolution and Pair Creation of Carriers
256
References
283
Percolation in Quantum Computation and Communication
286
2 Percolation and Quantum Computing
290
3 Quantum Repeater Networks for Quantum Communication
307
4 Summary and Open Problems
315
References
317
Index
321
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