Electronic conduction in linear quantum systems [Elektronische Ressource] : coherent transport and the effects of decoherence / von Matías Zilly

Sujets

• Physik

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2010
• Nombre de lectures: 25
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Electronic conduction in linear quantum systems:
Coherent transport and the effects of decoherence
Von der Fakultat fur Physik¨ ¨
der Universitat Duisburg-Essen¨
zur Erlangung des Grades
Dr. rer. nat.
genehmigte Dissertation
von
Mat´ıas Zilly
aus Mulheim an der Ruhr¨Tag der Disputation: 16.04.2010
Referent: Prof. Dr. Dietrich E. Wolf
Korreferent: Prof. Dr. Juan Carlos CuevasFur¨ Mar´ıa Bel´enAbstract. Coherent quantum transport in linear and quasi-linear tight-binding mod-
els and the influence of decoherence are studied. For the coherent transport descrip-
tion, Green functions and surface Green functions of semi-infinite systems are calcu-
lated and the transmission through defects and finite tight-binding chains with and with-
out diagonal disorder are examined. A statistical model based on the division of a
large system into coherent subsystems and decoherence regions is analyzed. While on
the total system level, classical rate equations interrelate electron energy distribution
functions assigned to the decoherence regions, the transition rates themselves are cal-
culated using quantum transport formalism. Thus a two-scale approach is used. For
contact Fermi energies within the tight-binding band of the system without disorder,
ohmic large scale behavior is observed for any finite density of decoherence regions. If
the Fermi energy is outside the band, and for disordered systems, critical decoherence
densities are defined. Above the critical densities, material-specific resistivities can be
defined. Applying the statistical model for the effects of decoherence on DNA double
strands, experimental findings for base-sequence dependent conductance are reproduced.Contents
Abstract v
1 Introduction and Outline 1
2 Quantum Transport 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 The spectral function A . . . . . . . . . . . . . . . . . . . . . . . . . 6
+2.1.2 The Green functions G and G . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 The Green function and the total density of states (DOS) . . . . . . 6
n2.1.4 The correlation function G . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Open quantum-mechanical systems: Σ, Γ, and ρ . . . . . . . . . . . . . . . 7
2.3 Transport formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Properties of the Green function . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 The Lehmann or spectral representation . . . . . . . . . . . . . . . . 12
2.4.2 The Dyson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.3 The Kramers-Kronig relations for the Green function. . . . . . . . . 13
3 Coherent Transport in Linear Systems 15
3.1 The infinite linear tight-binding chain . . . . . . . . . . . . . . . . . . . . . 15
3.2 The half-infinite linear chain. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Surface Green function of the half-infinite chain . . . . . . . . . . . . 17
3.2.2 An alternative way of calculation: direct back-transformation . . . . 20
3.3 Transmission through a single defect . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Symmetrical coupling to left and right contact . . . . . . . . . . . . 20
3.3.2 Different couplings to left and right contact . . . . . . . . . . . . . . 23
3.4 The alternating chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 The double chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.1 Hamiltonian and its diagonalization . . . . . . . . . . . . . . . . . . 25
3.5.2 Surface Green function of the half-infinite double chain . . . . . . . 27
3.6 The triple chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 The n-tuple chain and the 2d tight-binding grid . . . . . . . . . . . . . . . . 32
3.7.1 Surface Green functions for n-tuple chains . . . . . . . . . . . . . . . 33
3.7.2 Transmission through defects: symmetry effects . . . . . . . . . . . . 35
3.8 The DNA chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8.1 DNA-chain bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8.2 DNA surface Green function . . . . . . . . . . . . . . . . . . . . . . 42
3.9 Transmission of a finite chain . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.9.1 The wide-band limit contact . . . . . . . . . . . . . . . . . . . . . . 44
3.9.2 Transmission of the finite chain without disorder: periodicity in
length and tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 44viii Contents
3.9.3 Transmission of the finite chain with onsite disorder: localization . . 52
3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Statistical Model for the Effects of Decoherence on Electron Transport 57
4.1 Butti¨ ker probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Our approach for modelling the effects of decoherence . . . . . . . . . . . . 59
4.2.1 Decoherence regions . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 Phase coherence length . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.3 The decoherence regions as reservoirs: our η parameter . . . . . . . 61
4.2.4 The rate-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.5 The statistical approach . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Application of the model to a linear tight-binding system . . . . . . . . . . 63
4.3.1 Model for the decoherence regions: decoherence sites and parameterη 63
4.3.2 Model for contacts and contact coupling . . . . . . . . . . . . . . . . 64
4.3.3 The decoherence length: parameter p . . . . . . . . . . . . . . . . . 64
4.3.4 The effective channels and their transmissions . . . . . . . . . . . . . 64
4.3.5 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.6 Results for the ordered chain . . . . . . . . . . . . . . . . . . . . . . 67
4.3.7 Results for a linear chain with onsite disorder . . . . . . . . . . . . . 82
4.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Model for Decoherence Effects Applied on DNA Double Strands 91
5.1 Structure of DNA double strands . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Electronic properties of DNA . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1 Direct measurements of conductance of DNA double strands . . . . 93
5.2.2 Tight-binding models for DNA . . . . . . . . . . . . . . . . . . . . . 94
5.2.3 The extended ladder model . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Application of the decoherence model for DNA double strands . . . . . . . 97
5.3.1 Decoherence region 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.2 Decoherence region 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.3 Decoherence region 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.4 Contact model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.5 Contact model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.6 Contact model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.7 Conductance calculations for different base sequences . . . . . . . . 102
5.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Conclusions and Outlook 107
7 Summary 109
A Recommended Literature 111
A.1 Solid state physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.2 Green functions and many-particle physics . . . . . . . . . . . . . . . . . . . 112
A.3 Nanostructures and nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . 112
A.4 Mesoscopic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.5 Quantum transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.6 Biochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Contents ix
B Green functions in transport theory 115
B.1 Definition and basic properties of time-independent Green functions . . . . 115
B.2 Non-equilibrium Green functions: the Keldysh formalism . . . . . . . . . . 117
B.2.1 Perturbation theory in equilibrium . . . . . . . . . . . . . . . . . . . 117
B.2.2 Perturbation in non-equilibrium . . . . . . . . . . . . . . . . 123
B.2.3 Properties of the Keldysh Green functions . . . . . . . . . . . . . . . 125
B.2.4 Derivation of the current formula . . . . . . . . . . . . . . . . . . . . 126
C Derivation of equations (4.48) and (4.49) 129
Bibliography 133
Zusammenfassung 145
Danksagung 147
Erkl¨arung 149
Lebenslauf 151