Low-dimensional Nanoscale Systems on Discrete Spaces

by Papp, Erhardt; Micu, Codrutza

• ISBN13: 9789812706386
• ISBN10: 9812706380
• Format: Hardcover
• Copyright: 2007-07-30
• Publisher: World Scientific Pub Co Inc

Summary

The area of low-dimensional quantum systems on discrete spaces is a rapidly growing research field lying at the interface between quantum theoretical developments, like discrete and q-difference equations, and tight binding superlattice models in solid-state physics. Systems on discrete spaces are promising candidates for applications in several areas. Indeed, the dynamic localization of electrons on the 1D lattice under the influence of an external electric field serves to describe time-dependent transport in quantum wires, linear optical absorption spectra, and the generation of higher harmonics Odd-even parity effects and the flux dependent oscillations of total persistent currents in discretized rings can also be invoked. Technological developments are then provided by conductance calculations characterizing 1D conductors, junctions between rings and leads or rings and dots, and by quantum LC-circuits. Accordingly, the issues presented in this book are important starting points for the design of novel nanodevices.

Table of Contents

Preface	p. vii
Lattice Structures and Discretizations	p. 1
Discrete derivatives	p. 1
The Jackson derivative	p. 3
The q-integral	p. 6
Generalized q-hypergeometric functions	p. 7
The discrete space-time: a short retrospect	p. 9
Quick inspection of q-deformed Schrodinger equations	p. 13
Orthogonal polynomials of hypergeometric type on the discrete space	p. 14
Periodic Quasiperiodic and Confinement Potentials	p. 17
Short derivation of the Bloch-theorem	p. 17
The derivation of energy-band structures	p. 19
Direct and reciprocal lattices	p. 22
Quasiperiodic potentials	p. 25
A shorthand presentation of the elliptic Lame-equation	p. 27
Quantum dot potentials	p. 28
Quantum ring potentials	p. 31
Persistent currents and magnetizations	p. 32
The derivation of the total persistent current for electrons on the 1D ring at T = 0	p. 35
Circular currents	p. 37
Time Discretization Schemes	p. 41
Discretized time evolutions of coordinate and momentum observables	p. 42
Time independent Hamiltonians of hyperbolic type	p. 43
Time independent Hamiltonians of elliptic type	p. 45
The derivation of matrix elements	p. 46
Finite difference Liouville-von Neumann equations and "elementary" time scales	p. 48
The q-exponential function approach to the q-deformation of time evolution	p. 50
Alternative realizations of discrete time evolutions and stationary solutions	p. 55
Discrete Schrodinger Equations. Typical Examples	p. 57
The isotropic harmonic oscillator on the lattice	p. 58
Hopping particle in a linear potential	p. 61
The Coulomb potential on the Bethe-lattice	p. 65
The discrete s-wave description of the Coulomb-problem	p. 66
The Maryland class of potentials	p. 69
The relativistic quasipotential approach to the Coulomb-problem	p. 73
The infinite square well	p. 75
Other discrete systems	p. 76
Discrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl Algebras	p. 79
Lie algebraic approach to the discretization of differential equations	p. 80
Describing exactly and quasi-exactly solvable systems	p. 82
The discrete analog of the harmonic oscillator	p. 84
Applying the factorization method	p. 87
The discrete analog of the radial Coulomb-problem	p. 89
The discrete analog of the isotropic harmonic oscillator	p. 93
Realizations of Heisenberg-Weyl commutation relations	p. 95
Hopping Hamiltonians. Electrons in Electric Field	p. 99
Periodic and fixed boundary conditions	p. 101
Density of states and Lyapunov exponents	p. 103
The localization length: an illustrative example	p. 105
Delocalization effects	p. 107
The influence of a time dependent electric field	p. 108
Discretized time and dynamic localization	p. 111
Extrapolations towards more general modulations	p. 114
The derivation of the exact wavefunction revisited	p. 116
Time discretization approach to the minimum of the MSD	p. 118
Other methods to the derivation of the DLC	p. 120
Rectangular wave fields and other generalizations	p. 122
Wannier-Stark ladders	p. 125
Quasi-energy approach to DLC's	p. 126
The quasi-energy description of dc-ac fields	p. 129
Establishing currents in terms of the Boltzmann equation	p. 131
Tight Binding Descriptions in the Presence of the Magnetic Field	p. 133
The influence of the nearest and next nearest neighbors	p. 134
Transition to the wavevector representation	p. 136
The secular equation	p. 138
The Q = 2 integral quantum Hall effect	p. 140
Duality properties	p. 142
Tight binding descriptions with inter-band couplings	p. 143
Concrete single-band equations and classical realizations	p. 147
The Harper-Equation and Electrons on the 1D Ring	p. 151
The usual derivation of the Harper-equation	p. 152
The transfer matrix	p. 153
The derivation of [Delta]-dependent energy polynomials	p. 155
Deriving [Delta]-dependent DOS-evaluations	p. 157
Numerical DOS-studies	p. 160
Thermodynamic and transport properties	p. 161
The 1D ring threaded by a time dependent magnetic flux	p. 167
The tight binding description of electrons on the 1D ring	p. 170
The persistent current for the electrons on the 1D discretized ring at T = 0	p. 172
The q-Symmetrized Harper Equation	p. 175
The derivation of the generalized qShe	p. 175
The three term recurrence relation	p. 178
Symmetry properties	p. 181
The SL[subscript q] (2)-symmetry of the q She	p. 184
Magnetic translations	p. 188
The SU[subscript q](2)-symmetry of the usual Harper Hamiltonian	p. 190
Commutation relations concerning magnetic translation operators and the Hamiltonian	p. 192
Quantum Oscillations and Interference Effects in Nanodevices	p. 195
The derivation of generalized formulae to the total persistent current in terms of Fourier-series	p. 196
The discretized Aharonov-Bohm ring with attached leads	p. 199
Quantum wire attached to a chain of quantum dots	p. 207
Quantum oscillations in multichain nanorings	p. 210
Quantum LC-circuits with a time-dependent external source	p. 215
Dynamic localization effects in L-ring circuits	p. 219
Double quantum dot systems attached to leads	p. 220
Conclusions	p. 225
Further perspectives	p. 228
Dealing with polynomials of a discrete variable	p. 231
The functional Bethe-ansatz solution	p. 237
Bibliography	p. 241
Index	p. 259
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