Ratchet phenomena in quantum dissipative systems with spin orbit interactions [Elektronische Ressource] / vorgelegt von Sergey Smirnov

Sujets

• Physik

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 34
• Poids de l'ouvrage: 3 Mo

Extrait

Ratchet phenomena in quantum dissipative
systems with spin-orbit interactions
D I S S E R T A T I O N
zur Erlangung des
DOKTORGRADES DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
der Naturwissenschaftlichen Fakult¨at II - Physik
der Universit¨at Regensburg
vorgelegt von
SERGEYSMIRNOV ausSKHODNYA,
MOSKAUREGION,RUSSLAND
im September 2009Ratchet phenomena in quantum dissipative
systems with spin-orbit interactions
−e
−v
S−S
−e v
D I S S E R T A T I O N
Promotionsgesuch eingereicht am: 30.06.2009
Die Arbeit wurde angeleitet von: Prof. Dr. Milena Grifoni
Pr¨ufungsausschuß:
Vorsitz: Prof. Dr. Sergey Ganichev
Erstgutachten: Prof. Dr. Milena Grifoni
Zweitgutachten: Prof. Dr. Klaus Richter
Prof. Dr. Vladimir Braun
Datum des Promotionskolloquiums: 25.09.2009Contents
1 Introduction 7
1.1 Particle ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Ratchets in classical mechanics . . . . . . . . . . . . . . . . . . 9
1.1.3 Ratchets in quantum mechanics . . . . . . . . . . . . . . . . . . 12
1.2 Spin ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 Spin ratchet concept . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 Spin current definitions . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.3 Coherent spin ratchets . . . . . . . . . . . . . . . . . . . . . . . 29
1.2.4 Dissipative spin ratchets and this thesis . . . . . . . . . . . . . . 33
2 Rashba spin-orbit interaction 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Rashba effect in a 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Persistent spin helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Eigenenergies and eigenstates of a 2DEG with RSOI. . . . . . . . . . . 45
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Energy spectrum of a spin-orbit superlattice 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 A periodic structure with RSOI . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 A truly 1D periodic structure . . . . . . . . . . . . . . . . . . . 51
3.2.2 Influence of a transverse potential and RSOI . . . . . . . . . . . 52
3.3 Harmonic confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Eigenenergies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 A periodic structure with V(z) = 0 . . . . . . . . . . . . . . . . . . . . 58
3.5 Magnetic field and orbit-orbit coupling . . . . . . . . . . . . . . . . . . 61
3.5.1 An in-plane transverse static magnetic field . . . . . . . . . . . 61
3.5.2 Effects of orbit-orbit coupling for the case of a harmonic con-
finement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Materials of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 | CONTENTS
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Tight-binding model: discrete variable basis 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Diagonalization of the σˆ operator . . . . . . . . . . . . . . . . . . . . . 65z
4.3 Diagonalization of the xˆ operator . . . . . . . . . . . . . . . . . . . . . 66
4.3.1 Matrix structure . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2 Eigenvalue structure . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 σ-DVR basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Tight-binding model of a spin-orbit superlattice in the σ-DVR basis . . 69
4.5.1 Hamiltonian in the σ-DVR basis . . . . . . . . . . . . . . . . . . 70
4.5.2 Approximations and the effective tight-binding model . . . . . . 70
4.5.3 An additional relation between some hopping matrix elements . 72
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Quantum dissipative spin-orbit ratchet effects 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Electrically driven quantum dissipative spin ratchet . . . . . . . . . . . 75
5.2.1 Isolated system . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 Interaction with an external environment . . . . . . . . . . . . . 76
5.2.3 External driving. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.4 Charge and spin currents . . . . . . . . . . . . . . . . . . . . . . 77
5.2.5 Ratchet transport: averaged charge and spin currents . . . . . . 81
5.2.6 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.7 Calculation of the charge and spin ratchet currents . . . . . . . 83
5.2.8 Role of the spin current definition . . . . . . . . . . . . . . . . . 85
5.2.9 Spin ratchet effect: analytical analysis . . . . . . . . . . . . . . . 87
5.2.10 Spin ratchet effect: numerical analysis . . . . . . . . . . . . . . . 88
5.3 Quantum dissipative charge-spin ratchet: role of magnetic driving . . . 94
5.3.1 Driving Hamiltonian and the σ-DVR basis: transition rates . . . 95
5.3.2 Derivation of the charge and spin ratchet currents . . . . . . . . 96
5.3.3 Charge and spin ratchet effects: analytical analysis . . . . . . . 96
5.3.4 Charge and spin ratchet effects: numerical analysis . . . . . . . 97
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Conclusion - Zusammenfassung 101
7 Acknowledgement - Dankesch¨on 103Chapter 1
Introduction
In this introductory chapter the general concept of a particle ratchet mechanism is
introduced. The role of the spatial and time asymmetry is emphasized. The imple-
mentation of this concept in classical and quantum systems is described. The spin
ratchet concept is presented as a natural generalization of the particle ratchet mecha-
nism. Coherent and dissipative spin ratchets are discussed as two different realizations
of the spin ratchet concept. The focus and the structure of the thesis are outlined at
the end of the chapter.
1.1 Particle ratchets
1.1.1 General concepts
The particle ratchet mechanism is defined for a system driven by time-dependent
external forces with zero time average (also called unbiased forces) and consists in
directedparticletransportundersomeasymmetryeitherinthesystemorinthedriving
forces. If the particles involved in the ratchet transport have non-zero charge, the
particle ratchet mechanism is also referred to as the charge ratchet mechanism.
Role of the space asymmetry
The term ”ratchet” was historically borrowed from the asymmetric toothed wheel. It
was believed that in a system driven by unbiased forces directed transport in a given
direction was only possible if the system did not have the inversion center in space.
A simple example is a windmill. Its asymmetric sails are constructed in such a way
that unbiased flow of air is converted into a directed rotation which is subsequently
converted into useful work. The particle ratchets with space inversion asymmetry are
also called space ratchets.
Role of the time asymmetry
It turns out that even when a system has the inversion center, directed particle trans-
portcanstillbeinducedbyunbiasedforcesiftheseforcesareasymmetricintime.The8 | 1.1 Particle ratchets
asymmetric periodic potential
+ unbiased external force: F cos( t)Ω
DIRECTED
QUANTUM TRANSPORT
Ω t = (2n+1) π Ω t = 2 π n
Figure 1.1: The space rocked ratchet based on quantum mechanical tunneling. The two rocking
situations at times t = π(2n+1)/Ω and t = 2πn/Ω, n = 0,±1,±2,..., induce tunneling processes
which have different rates in opposite directions.
particle ratchets of this type are usually called time ratchets. Although these class of
particle ratchets will not be in the focus of the present thesis, it will be useful below
in this introductory chapter to mention some relevant examples in order to later make
comparison with the space ratchets.
Rocked ratchets
The ratchet mechanism arising when an external unbiased force rocks the periodic
potentialiscalledtherockedratchetmechanism.Thistypeofratchetsisconsideredin
the present thesis within the quantum mechanical context (see Fig. 1.1). The rocked
ratchets should be distinguished from other kinds of ratchets like temperature ratch-
ets or flashing ratchets where the ratchet mechanism appears through either periodic
variation of the bath temperature or switching off and on the periodic potential, re-
spectively.
Mechanics
The concept of the particle ratchet mechanism and the roles of space and time asym-
metry are general for both classical and quantum mechanics. In particular, space and
time ratchet effects can be found in both classical and quantum systems. However, the
underlying physical details are much different in classical and quantum cases. There-
fore it is convenient to consider the classical and quantum ratchet effects separately.
We start with the ratchets based on classical mechanics and then discuss the quantum
mechanical ratchets