Low-dimensional nanoscale systems on discrete spaces / Erhardt Papp, Codrutza Micu.

Format:

Book

Author/Creator:

Papp, E.

Contributor:

Micu, Codrutza.

Language:

English

Subjects (All):

Quantum theory.

Schrödinger equation.

Nanoelectromechanical systems.

Physical Description:

xiii, 262 pages : illustrations ; 24 cm

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, [2007]

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. Book jacket.

Contents:

1 Lattice Structures and Discretizations 1

1.1 Discrete derivatives 1

1.2 The Jackson derivative 3

1.3 The q-integral 6

1.4 Generalized q-hypergeometric functions 7

1.5 The discrete space-time: a short retrospect 9

1.6 Quick inspection of q-deformed Schrodinger equations 13

1.7 Orthogonal polynomials of hypergeometric type on the discrete space 14

2 Periodic Quasiperiodic and Confinement Potentials 17

2.1 Short derivation of the Bloch-theorem 17

2.2 The derivation of energy-band structures 19

2.3 Direct and reciprocal lattices 22

2.4 Quasiperiodic potentials 25

2.5 A shorthand presentation of the elliptic Lame-equation 27

2.6 Quantum dot potentials 28

2.7 Quantum ring potentials 31

2.8 Persistent currents and magnetizations 32

2.9 The derivation of the total persistent current for electrons on the 1D ring at T = 0 35

2.10 Circular currents 37

3 Time Discretization Schemes 41

3.1 Discretized time evolutions of coordinate and momentum observables 42

3.2 Time independent Hamiltonians of hyperbolic type 43

3.3 Time independent Hamiltonians of elliptic type 45

3.4 The derivation of matrix elements 46

3.5 Finite difference Liouville-von Neumann equations and "elementary" time scales 48

3.6 The q-exponential function approach to the q-deformation of time evolution 50

3.7 Alternative realizations of discrete time evolutions and stationary solutions 55

4 Discrete Schrodinger Equations. Typical Examples 57

4.1 The isotropic harmonic oscillator on the lattice 58

4.2 Hopping particle in a linear potential 61

4.3 The Coulomb potential on the Bethe-lattice 65

4.4 The discrete s-wave description of the Coulomb-problem 66

4.5 The Maryland class of potentials 69

4.6 The relativistic quasipotential approach to the Coulomb-problem 73

4.7 The infinite square well 75

4.8 Other discrete systems 76

5 Discrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl Algebras 79

5.1 Lie algebraic approach to the discretization of differential equations 80

5.2 Describing exactly and quasi-exactly solvable systems 82

5.3 The discrete analog of the harmonic oscillator 84

5.4 Applying the factorization method 87

5.5 The discrete analog of the radial Coulomb-problem 89

5.6 The discrete analog of the isotropic harmonic oscillator 93

5.7 Realizations of Heisenberg-Weyl commutation relations 95

6 Hopping Hamiltonians. Electrons in Electric Field 99

6.1 Periodic and fixed boundary conditions 101

6.2 Density of states and Lyapunov exponents 103

6.3 The localization length: an illustrative example 105

6.4 Delocalization effects 107

6.5 The influence of a time dependent electric field 108

6.6 Discretized time and dynamic localization 111

6.7 Extrapolations towards more general modulations 114

6.8 The derivation of the exact wavefunction revisited 116

6.9 Time discretization approach to the minimum of the MSD 118

6.10 Other methods to the derivation of the DLC 120

6.11 Rectangular wave fields and other generalizations 122

6.12 Wannier-Stark ladders 125

6.13 Quasi-energy approach to DLC's 126

6.14 The quasi-energy description of dc-ac fields 129

6.15 Establishing currents in terms of the Boltzmann equation 131

7 Tight Binding Descriptions in the Presence of the Magnetic Field 133

7.1 The influence of the nearest and next nearest neighbors 134

7.2 Transition to the wavevector representation 136

7.3 The secular equation 138

7.4 The Q = 2 integral quantum Hall effect 140

7.5 Duality properties 142

7.6 Tight binding descriptions with inter-band couplings 143

7.7 Concrete single-band equations and classical realizations 147

8 The Harper-Equation and Electrons on the 1D Ring 151

8.1 The usual derivation of the Harper-equation 152

8.2 The transfer matrix 153

8.3 The derivation of [Delta]-dependent energy polynomials 155

8.4 Deriving [Delta]-dependent DOS-evaluations 157

8.5 Numerical DOS-studies 160

8.6 Thermodynamic and transport properties 161

8.7 The 1D ring threaded by a time dependent magnetic flux 167

8.8 The tight binding description of electrons on the 1D ring 170

8.9 The persistent current for the electrons on the 1D discretized ring at T = 0 172

9 The q-Symmetrized Harper Equation 175

9.1 The derivation of the generalized qShe 175

9.2 The three term recurrence relation 178

9.3 Symmetry properties 181

9.4 The SL[subscript q] (2)-symmetry of the q She 184

9.5 Magnetic translations 188

9.6 The SU[subscript q](2)-symmetry of the usual Harper Hamiltonian 190

9.7 Commutation relations concerning magnetic translation operators and the Hamiltonian 192

10 Quantum Oscillations and Interference Effects in Nanodevices 195

10.1 The derivation of generalized formulae to the total persistent current in terms of Fourier-series 196

10.2 The discretized Aharonov-Bohm ring with attached leads 199

10.3 Quantum wire attached to a chain of quantum dots 207

10.4 Quantum oscillations in multichain nanorings 210

10.5 Quantum LC-circuits with a time-dependent external source 215

10.6 Dynamic localization effects in L-ring circuits 219

10.7 Double quantum dot systems attached to leads 220

11.1 Further perspectives 228

Appendix A Dealing with polynomials of a discrete variable 231

Appendix B The functional Bethe-ansatz solution 237.

Notes:

Includes bibliographical references (pages 241-257) and index.

ISBN:

9812706380

9789812706386

OCLC:

141384808