Spin-orbit coupling effects in two-electron coupled quantum dots [Elektronische Ressource] / vorgelegt von Fabio Baruffa
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Spin-orbit coupling effects in two-electron coupled quantum dots [Elektronische Ressource] / vorgelegt von Fabio Baruffa

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Spin-orbit coupling effects intwo-electron coupled quantum dotsDissertationzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)derhaftlichen Fakultät II-Physikder Universität Regensburgvorgelegt vonFabio Baruffaaus Napoli, ItalienMai 2010The PhD thesis was submitted on May 2010The colloquium took place on 16th of July 2010Prof. Dr. S. Ganichev ChairmanProf. Dr. J. Fabian 1st RefereeBoardofexaminers:Prof. Dr. M. Grifoni 2ndProf. Dr. G. Bali ExaminerContentsAcknowledgments ixIntroduction 11. Overview of Quantum Computation 51.1. Classical unit of information: the bit . . . . . . . . . . . . . . . . . . 51.2. Quantum unit of the qubit . . . . . . . . . . . . . . . . . 61.3. Logical quantum gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1. Single qubit gate . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2. Single qubit dynamics . . . . . . . . . . . . . . . . . . . . . . 111.3.3. Two qubit gate . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4. Quantum parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5. DiVincenzo criteria for Quantum Computation . . . . . . . . . . . . . 161.6. Quantum dots as qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 172. Quantum Dots (QD) 192.1. Two dimensional electron gas (2-DEG) . . . . . . . . . . . . . . . . . 192.2. Quantum dot fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 202.3. Confinement potential . . . . . . . . .

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Published 01 January 2010
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Spin-orbit coupling effects in
two-electron coupled quantum dots
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
derhaftlichen Fakultät II-Physik
der Universität Regensburg
vorgelegt von
Fabio Baruffa
aus Napoli, Italien
Mai 2010The PhD thesis was submitted on May 2010
The colloquium took place on 16th of July 2010
Prof. Dr. S. Ganichev Chairman
Prof. Dr. J. Fabian 1st Referee
Boardofexaminers:
Prof. Dr. M. Grifoni 2nd
Prof. Dr. G. Bali ExaminerContents
Acknowledgments ix
Introduction 1
1. Overview of Quantum Computation 5
1.1. Classical unit of information: the bit . . . . . . . . . . . . . . . . . . 5
1.2. Quantum unit of the qubit . . . . . . . . . . . . . . . . . 6
1.3. Logical quantum gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1. Single qubit gate . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2. Single qubit dynamics . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3. Two qubit gate . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4. Quantum parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5. DiVincenzo criteria for Quantum Computation . . . . . . . . . . . . . 16
1.6. Quantum dots as qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. Quantum Dots (QD) 19
2.1. Two dimensional electron gas (2-DEG) . . . . . . . . . . . . . . . . . 19
2.2. Quantum dot fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3. Confinement potential . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4. Coulomb blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5. Single shot read-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1. Energy selective read-out . . . . . . . . . . . . . . . . . . . . . 25
2.5.2. Tunneling rate selective read-out . . . . . . . . . . . . . . . . 26
2.6. Gate Operation on Quantum Dots. . . . . . . . . . . . . . . . . . . . 27
3. Single Electron Quantum Dots 31
3.1. Effective mass approximation . . . . . . . . . . . . . . . . . . . . . . 31
3.2. Energy dispersion of GaAs semiconductor . . . . . . . . . . . . . . . 34
3.2.1. Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3. Fock-Darwin spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4. Double-dot states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1. LCAO approximation . . . . . . . . . . . . . . . . . . . . . . . 40
vContents
3.4.2. Tunneling Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 41
4. Two Electron Quantum Dots 43
4.1. Two Electron Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2. Approximative methods for the exchange coupling . . . . . . . . . . . 45
4.2.1. Heitler-London method . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2. Hund-Mullikand . . . . . . . . . . . . . . . . . . . . . . 48
4.2.3. Variational method . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.4. Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . 53
4.3. Unitarily transformed model . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1. Orbital wavefunctions symmetry . . . . . . . . . . . . . . . . . 56
4.3.2. Effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 59
′4.3.3.e Hamiltonian H in zero field . . . . . . . . . . . . . 62ex
4.4. Single Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1. Spin-orbitcorrectionstotheenergyspectruminfinitemagnetic
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5. Double Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5.1. Heitler-London approximation . . . . . . . . . . . . . . . . . . 68
4.5.2. Spin-orbit correction to the energy spectrum in zero magnetic
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.3. Finite magnetic field . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.4. Cubic Dresselhaus contributions . . . . . . . . . . . . . . . . . 77
4.6. Comparison between models for anisotropic exchange . . . . . . . . . 79
4.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5. Numerical Method 85
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2. Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3. Lanczos diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4. Basis for the two electron diagonalization . . . . . . . . . . . . . . . . 91
5.4.1. Relative strengths of parts of the Hamiltonian . . . . . . . . . 92
5.4.2. First approach . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4.3. Secondh . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5. Symmetries of the two electron states . . . . . . . . . . . . . . . . . . 97
5.6. Spin as an additional degree of freedom . . . . . . . . . . . . . . . . . 98
5.6.1. Number of needed Coulomb elements . . . . . . . . . . . . . . 99
5.6.2. Finite width along growth direction (z) . . . . . . . . . . . . . 99
5.7. Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.8. Matrix elements of single particle operators . . . . . . . . . . . . . . . 102
viContents
5.9. Coulomb integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.9.1. Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . 103
5.9.2. Correction factor for the DFT . . . . . . . . . . . . . . . . . . 105
5.9.3. Numerical precision in the DFT method . . . . . . . . . . . . 105
5.9.4. Numerical formula . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10.Precision of the numerical Coulomb integral . . . . . . . . . . . . . . 110
5.11.Spin-orbit matrix elements . . . . . . . . . . . . . . . . . . . . . . . . 114
5.12.Spin-relaxation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Conclusions 117
A.Coulomb integrals 119
B. Spin matrices 123
C. Detailed algorithm 125
C.1. Spin basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.2. States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.3. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.4. Formula for spin-orbit matrix elements . . . . . . . . . . . . . . . . . 130
Bibliography 133
viiAcknowledgments
ThesefewlinesIwouldliketodedicatetoallthosepeoplewho,inonewayoranother,
have contributed to my scientific and life experience in Regensburg.
First of all I would sincerely thank Prof. Jaroslav Fabian, for his continuous support.
Without his help and patience I would not be able to go forward and understand the
beauty of the research in physics.
The next thank goes to Dr. Peter Stano. He has taught me how to attack a problem
and find the right combination between the numerics and the analytical work.
I wish to thanks all members of the Spintronics group. I have learnt many things
from each of you: Sergej Konschuh, Martin Raith, Benedikt Scharf, Martin Gmitra
and Alex Matos-Abiague.
Always present to give me advices and to make me feel less distant to mine own
city are my parents Antonia and Bruno, my brother Dario and my girlfriend Mela-
nia...most of this work is dedicated to them.
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