UNIVERSITY OF CALIFORNIASanta Barbara

Optical control and detection of spin

coherence in semiconductor

nanostructures

A dissertation submitted in partial satisfactionof the requirements for the degree of

Doctor of Philosophy

in

Physics

by

Jesse A. Berezovsky

Committee in charge:

Professor David D. Awschalom, Chairperson

Professor Andrew Cleland

Professor Leon Balents

December 2007

UMI Number: 3291324

32913242008

UMI MicroformCopyright

All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road

P.O. Box 1346 Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company.

The dissertation of Jesse A. Berezovsky is approved:

Committee Chairperson

December 2007

Optical control and detection of spin coherence in

semiconductor nanostructures

Copyright c© 2007

by

Jesse A. Berezovsky

iii

Acknowledgments

There are many people who deserve credit for making my experience

at Santa Barbara what it was: educational, productive, exciting, and fun.

Foremost, I couldn’t have been more fortunate than to have David as my

advisor. Not only does he provide an unparalleled environment in which

to learn and work, but also takes his role as advisor seriously, shepherding

us through the “valley of the shadow of death” (or grad school, as it is also

known).

In David’s lab, I have had the fortune to work with a number of tal-

ented collaborators. Over the last few years, I have worked closely with

Maiken Mikkelsen on the single spin work. The collaboration was produc-

tive under normal circumstances, but when the data was really flowing,

like the individual Voltron robots joining together to make a giant robot,

we transformed into an unstoppable 24-hour-a-day data-taking machine.

This work also could not have been done without the materials expertise

of Nick Stoltz or the theoretical stylings of Oliver Gywat. Before that, on

the nanocrystal work, Min Ouyang initiated me into the esoteric secrets

of chemistry, and taught me not to fear the effects of pyridine on one’s

reproductive organs. David Steuerman and Yong-Qing Li then joined me

on the nanocrystal work, of which there was a large quantity even if it

wasn’t always successful. Of course, we wouldn’t have been able to un-

derstand the results without the theory efforts of both Oliver and Florian

Meier. And right after I joined the group, I was lucky to jump into work

with Jason Stephens and Roland Kawakami on imaging nuclear spins in

semiconductors.

Along, with Jason and Roland, I am grateful to my elders in the lab,

iv

who showed me the ropes and taught me most of what I know about

doing lab work. In particular, Ryan Epstein (bouncing his latest quantum

computer design off me, or organizing a chair race in the hallway), Yui

Kato (trying to explain Group Theory to me, or calling an impromptu Beer

Meeting), and Martino Poggio and Roberto Myers (spewing vitriol at each

other about the correct sign of the g-factor or something, then suddenly

going off sailing together). Also, a number of postdocs came through the

lab, sharing their unique expertise: Ronald Hanson, Alex Holleitner, and

Sai Ghosh, in addition to those already mentioned.

Other labmates who were my own contemporaries, provided valuable

insight and support (both scientific and moral). Felix Mendoza has been

my comrade-in-arms since prospective student visiting day. He can always

be counted on to solve problems – ranging from malfunctioning door locks

to broken snorkels. Hadrian Knotz has provided a calm presence in the

lab, somehow managing to keep all the lab computers running smoothly.

And whenever I felt a need to talk to someone about physics, Nate Stern

was there for me.

I wish good luck to all of the newer additions to the lab, whose antics

in and out of the lab have provided much entertainment.

The various staff members who keep the whole ship afloat deserve many

thanks. Holly Woo and the other CNSI staff have been tireless in their

efforts to counteract my great negligence in financial and bureaucratic

matters. Mike Deal and the facilities staff have always discharged their

building-maintenance duties admirably. Jeff Dutter and the rest of the

machine shop guys have been an amazing resource – particularly, Mike

Wrocklage who oversees the student machine shop with a Buddha-like

calm. And finally, I’d like to acknowledge the cleanroom staff who manage

v

to keep that show going.

Of course, my years in Santa Barbara have not been all work. There

have been a number of people outside the lab that have made the experi-

ence lots of fun. Through numerous parties, various barbecues, countless

Wednesday Wine Nights, and an infinite number of happy hours, Felix

Mendoza, Dave Wood, Corrinne Mills, Jean-Luc Fraikin, Chris McKen-

ney, Melvin McLaurin, and various others have kept my whistle wet, my

belly stuffed with good food, and my mental state sane. Also, the first

year study group, consisting of Sara Hastings-Simon, Felix Mendoza, Juan

Hodelin and myself, dulled the pain of coursework and provided lots of

great memories – the interstices of an evening’s work were always packed

with fun. And of course, our first-year coursework would not have been

nearly as successful if it weren’t for the assistance Dr. Skipper, who never

failed to slake our thirst for knowledge.

vi

Vitæ

Education

2002 B.S., Physics, University of Minnesota.

2005 M.A., Physics, University of California, Santa Barbara.

2007 Ph.D., Physics, University of California, Santa Barbara.

Publications

“Manipulation of spins and coherence in semiconductors,” N. P. Stern,

J. Berezovsky, S. Ghosh and D. D. Awschalom, in Handbook on Magnetism

and Advanced Magnetic Materials, H. Kronmuller and S. Parkin, eds., John

Wiley & Sons (2007).

“Spin coherence in semiconductors,” J. Berezovsky, W. H. Lau, S.

Ghosh, J. Stephens, N. P. Stern, and D. D. Awschalom, in Manipulat-

ing Quantum Coherence in Solid State Systems, M. E. Flatte and I. Tifrea,

eds., Springer (2007).

“Nondestructive optical measurements of a single electron spin in a

quantum dot,” J. Berezovsky, M. H. Mikkelsen, O. Gywat, N. G. Stoltz,

L. A. Coldren, and D. D. Awschalom, Science 314, 1916 (2006).

“Initialization and read-out of spins in coupled core-shell quantum

dots,” J. Berezovsky, O. Gywat, F. Meier, D. Battaglia, X. Peng, and

D. D. Awschalom, Nature Physics 2, 831 (2006).

“Cavity enhanced Faraday rotation of semiconductor quantum dots,”

vii

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and

D. D. Awschalom. Appl. Phys. Lett. 88, 193126 (2006).

“Spin dynamics and level structure of quantum-dot quantum wells,” J.

Berezovsky, M. Ouyang, F. Meier, D. D. Awschalom, D. Battaglia, and X.

Peng. Phys. Rev. B 71, 081309(R) (2005).

“Spintronics: Semiconductors, molecules, and quantum information,”

Y. Kato, J. Berezovsky, and D. D. Awschalom. IEDM Technical Digest,

IEEE International, p.537 (2004).

“Optically patterned nuclear doughnuts in GaAs/MnAs heterostruc-

tures,” J. Stephens, J. Berezovsky, R. K. Kawakami, A. C. Gossard, and

D. D. Awschalom. Appl. Phys. Lett. 85, 1184 (2004).

“Spin accumulation in forward-biased MnAs/GaAs Schottky diodes,”

J. Stephens, J. Berezovsky, J. P. McGuire, L. J. Sham, A. C. Gossard, and

D. D. Awschalom. Phys. Rev. Lett. 93, 097602 (2004).

“Spatial imaging of magnetically patterned nuclear spins in GaAs,” J.

Stephens, R. K. Kawakami, J. Berezovsky, M. Hanson, D. P. Shepherd, A.

C. Gossard, and D. D. Awschalom. Phys. Rev. B 68, 041307 (2003).

“Imaging of spin dynamics in closure domain and vortex structures,” J.

P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell.

Phys. Rev. B 67, 020403(R) (2003).

“Spatially resolved dynamics of localized spin-wave modes in ferromag-

netic wires,” J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and

P. A. Crowell. Phys. Rev. Lett. 89, 277201 (2002).

viii

“Time-domain ferromagnetic resonance in epitaxial thin films,” D. M.

Engebretson, J. Berezovsky, J. P. Park, L. C. Chen, C. J. Palmstrøm, and

P. A. Crowell. J. Appl. Phys. 91, 8040 (2002).

“Control of magnetic anisotropy in Fe1−xCox films on vicinal GaAs and

Sc1−yEryAs surfaces,” A. F. Isakovic, J. Berezovsky, P. A. Crowell, L. C.

Chen, D. M. Carr, B. D. Schultz, and C. J. Palmstrøm. J. Appl. Phys.

89, 6674 (2001).

“Epitaxial ferromagnetic metal/GaAs(100) heterostructures,” L. C. Chen,

J. W. Dong, B. D. Schultz, C. J. Palmstrøm, J. Berezovsky, A. Isakovic,

P. A. Crowell, and N. Tabat. J. Vac. Sci. Technol. B 18, 2057 (2000).

Fields of study

Major field: Physics

Optical control and detection of spin coherence in semiconductor

nanostructures

Professor David D. Awschalom

ix

Abstract

Optical control and detection of spin coherence in

semiconductor nanostructures

by

Jesse A. Berezovsky

Understanding the coherent dynamics of electron spins in quantum dots

(QDs) is important for potential applications in solid-state, spin-based elec-

tronics and quantum information processing. Here, results are presented

focusing on optical initialization, manipulation, and readout of spin co-

herence in various semiconductor nanostructures. Layered semiconduc-

tor nanocrystals are fabricated containing a spherical “quantum shell” in

which electrons and holes are confined. As in a planar quantum well, the

quantized energy levels and g-factors are found to depend on the shell

thickness. Taking this idea a step further, nanocrystals with a concentric,

tunnel-coupled core and shell are investigated. Based on the energy and

g-factor dependences in these structures, spins can be selectively initial-

ized into, and read out from, states in the core and shell. In contrast to

these two ensemble measurements, we next turn to measurements of sin-

gle electron spins in single QDs. First, we demonstrate the detection of a

single electron spin in a QD using a nondestructive, continuously averaged

magneto-optical Kerr rotation (KR) measurement. This continuous sin-

gle QD KR technique is then extended into the time domain using pulsed

x

pump and probe lasers, allowing the observation of the coherent evolution

of an electron spin state with nanosecond temporal resolution. By sweep-

ing the delay between the pump and probe, the dynamics of the spin in

the QD are mapped out in time, providing a direct measurement of the

electron g-factor and spin lifetime. Finally, this time-resolved single spin

measurement is used to observe ultrafast coherent manipulation of the spin

in the QD using an off-resonant optical pulse. Via the optical Stark effect,

this optical pulse coherently rotates the spin state through angles up to π

radians, on picosecond timescales.

xi

Contents

Chapter 1 Introduction 1

1.1 Spin: A brief history . . . . . . . . . . . . . . . . . . . . . 1

1.2 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Quantum information processing . . . . . . . . . . . . . . 8

Chapter 2 Background 11

2.1 Spins in semiconductors . . . . . . . . . . . . . . . . . . . 11

2.2 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Optical properties of spins in semiconductors . . . . . . . . 29

2.4 Faraday rotation: theory . . . . . . . . . . . . . . . . . . . 36

2.5 Faraday rotation: experiment . . . . . . . . . . . . . . . . 41

Chapter 3 Quantum Shells in Semiconductor Nanocrystals 45

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Sample structure and characterization . . . . . . . . . . . 49

3.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Spin dynamics in quantum shells . . . . . . . . . . . . . . 53

3.5 Energy levels in quantum shells . . . . . . . . . . . . . . . 56

3.6 Theoretical description . . . . . . . . . . . . . . . . . . . . 59

xii

Chapter 4 Coupled Shells in Layered Colloidal Nanocrystals 68

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Sample structure and characterization . . . . . . . . . . . 69

4.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Theoretical description . . . . . . . . . . . . . . . . . . . . 76

4.5 Transient absorption and luminescence measurements . . . 80

4.6 Time-resolved Faraday rotation spectroscopy . . . . . . . . 84

4.7 Analysis of spin dynamics and core-shell coupling . . . . . 89

4.8 Nanocrystal QDs in an optical cavity . . . . . . . . . . . . 94

Chapter 5 Non-destructive Measurement of a Single Electron

Spin 101

5.1 Motivation and Background . . . . . . . . . . . . . . . . . 101

5.2 Sample structure . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Characterization: PL and Hanle measurements . . . . . . . 117

5.6 Single spin Kerr rotation . . . . . . . . . . . . . . . . . . . 124

Chapter 6 Coherent Dynamics of a Single Spin 139

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2 Experimental scheme . . . . . . . . . . . . . . . . . . . . . 141

6.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . 145

6.4 Time-resolved single spin measurements . . . . . . . . . . 148

Chapter 7 Ultrafast Manipulation of Single Spin Coherence156

7.1 Motivation and Background . . . . . . . . . . . . . . . . . 156

7.2 Experimental scheme . . . . . . . . . . . . . . . . . . . . . 159

xiii

7.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . 167

7.4 Ultrafast optical spin manipulation . . . . . . . . . . . . . 171

7.5 Further exploration and control measurements . . . . . . . 174

Chapter 8 Conclusion 183

Appendix A Sample structure and processing 187

Appendix B Details of single spin detection and manipula-

tion 191

B.1 Optical path . . . . . . . . . . . . . . . . . . . . . . . . . . 193

B.2 Measurement control scheme . . . . . . . . . . . . . . . . . 196

B.3 Odds and ends . . . . . . . . . . . . . . . . . . . . . . . . 197

Appendix C Other theoretical views of Faraday rotation 200

xiv

List of figures

1.1 The Datta-Das spin transistor. . . . . . . . . . . . . . . . . 6

2.1 The Bloch sphere. . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Typical direct-gap band diagram. . . . . . . . . . . . . . . 14

2.3 Spin decoherence: a random walk. . . . . . . . . . . . . . . 20

2.4 Schematic of confining potential, energy levels, and wave-

functions in a quantum dot. . . . . . . . . . . . . . . . . . 26

2.5 Types of quantum dots. . . . . . . . . . . . . . . . . . . . 28

2.6 Selection rules for interband optical transitions. . . . . . . 32

2.7 Schematic of a Hanle measurement setup. . . . . . . . . . 34

2.8 Illustration of the Hanle effect. . . . . . . . . . . . . . . . . 35

2.9 Diagram of the Faraday effect. . . . . . . . . . . . . . . . . 37

2.10 Theoretical Faraday rotation spectra. . . . . . . . . . . . . 40

2.11 Typical time-resolved Faraday rotation setup. . . . . . . . 42

2.12 Sensitive polarization detection with a balanced photodiode

bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Cutaway schematic of layered nanocrystals. . . . . . . . . . 46

3.2 Photoluminescence of quantum shell samples. . . . . . . . 50

3.3 Setup for quantum shell TRFR. . . . . . . . . . . . . . . . 51

xv

3.4 Spin precession in quantum shells. . . . . . . . . . . . . . . 53

3.5 Temperature dependence of the quantum shell spin lifetime. 55

3.6 Precession frequency and g-factors as a function of shell

thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Quantum shell Faraday rotation spectra. . . . . . . . . . . 58

3.8 Calculated electron and hole energy levels in quantum shells. 60

3.9 Calculated electron and hole wavefunctions in a quantum

shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.10 Calculated FR spectra of quantum shells. . . . . . . . . . . 64

4.1 Diagram of a coupled core-shell nanocrystal. . . . . . . . . 70

4.2 Photoluminescence of coupled core-shell nanocrystals. . . . 71

4.3 Coupled core-shell PL as a function of core and shell thickness. 73

4.4 Setup for TRFR and TA measurements on coupled core-shell

nanocrystals. . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 Calculated core and shell wavefunctions. . . . . . . . . . . 78

4.6 Measured and calculated energy levels as a function of core

and shell thickness. . . . . . . . . . . . . . . . . . . . . . . 79

4.7 Transient absorption measurements on coupled core-shell

nanocrystals. . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Time-resolved photoluminescence of coupled core-shell nanocrys-

tals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.9 Time-resolved Faraday rotation on coupled core-shell nanocrys-

tals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.10 Three precession frequencies in coupled-core shell nanocrys-

tals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.11 Faraday rotation spectra of coupled core-shell nanocrystals. 89

xvi

4.12 Coupled wavefunctions determined using the Hubbard model 92

4.13 Diagram of nanocrystals embedded in an optical cavity . . 97

4.14 Optical characterization of cavity structures. . . . . . . . . 98

4.15 Cavity-enhanced time-resolved Faraday rotation. . . . . . . 99

5.1 Previously demonstrated schemes for single spin detection. 102

5.2 Single versus ensemble Kerr rotation spectrum. . . . . . . 106

5.3 Schematic of the sample for single spin measurements. . . 108

5.4 STM image and cartoon of interface fluctuation quantum

dots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.5 Characterization of the single spin sample. . . . . . . . . . 110

5.6 Setup for single spin detection. . . . . . . . . . . . . . . . . 114

5.7 Photoluminescence of a single QD vs. bias voltage, and

polarization thereof. . . . . . . . . . . . . . . . . . . . . . 118

5.8 Pump power dependence of various single QD PL lines. . . 119

5.9 Lowest energy optical transitions in a quantum dot. . . . . 120

5.10 Mechanisms for single spin initialization. . . . . . . . . . . 121

5.11 Single quantum dot Hanle measurements. . . . . . . . . . . 123

5.12 Single quantum dot Kerr rotation. . . . . . . . . . . . . . . 125

5.13 Single spin KR feature as a function of bias voltage. . . . . 127

5.14 Single spin detection in other quantum dots. . . . . . . . . 128

5.15 Energy of single spin Kerr rotation feature compared to tran-

sition energies. . . . . . . . . . . . . . . . . . . . . . . . . 129

5.16 Single spin Kerr rotation Hanle measurements. . . . . . . . 132

5.17 Analysis of single spin measurements as a function of bias

voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.18 A proposal to use Faraday rotation to couple single spins. . 137

xvii

6.1 Setup for time-resolved single spin measurements. . . . . . 142

6.2 Temporal profile of the pump and probe pulses. . . . . . . 143

6.3 Illustration of spin misalignment leading to dynamic nuclear

polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.4 Coherent precession of a single spin. . . . . . . . . . . . . . 149

6.5 Single spin precession as a function of magnetic field. . . . 151

6.6 Single spin dynamics at zero magnetic field. . . . . . . . . 153

6.7 The effects of nuclear polarization on single spin precession. 154

7.1 Illustration of the optical Stark effect. . . . . . . . . . . . . 157

7.2 Setup for Stark tipping measurements. . . . . . . . . . . . 160

7.3 Energy scales and relevant optical spectra for Stark tipping

measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.4 Single spin Kerr rotation vs. delay and probe energy with

and without the tipping pulse. . . . . . . . . . . . . . . . . 165

7.5 Sequence of rotations in the Stark tipping model. . . . . . 167

7.6 Coherent rotation of a single electron spin. . . . . . . . . . 172

7.7 Dependence of Stark tipping on tipping pulse intensity and

detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.8 Strength of the Stark effect as a function of detuning. . . . 176

7.9 Comparison of the Stark effect with tipping pulses of oppo-

site helicity. . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.10 Comparison of measurement with circularly and linearly po-

larized pump. . . . . . . . . . . . . . . . . . . . . . . . . . 180

B.1 Setup for single spin measurement and control. . . . . . . . 192

C.1 Classical view of Faraday rotation. . . . . . . . . . . . . . 201

xviii

Chapter 1

Introduction

1.1 Spin: A brief history

Spin, the intrinsic angular momentum of a particle, was first described the-

oretically by George Uhlenbeck and Samuel Goudsmit in 1925, and formal-

ized by Wolfgang Pauli in 1926. Experimentally, however, spin phenomena

have been observed and put to practical use for much longer. The earli-

est known spin-based device is most likely the magnetic compass. Here,

a freely rotating needle is constructed out of a material in which electron

spins align with each other under their mutual exchange interaction. This

leads to a macroscopic spin polarization in the needle (ferromagnetism),

causing the needle to align with the Earth’s magnetic field due to the Zee-

man energy of a spin in a magnetic field (see Eq. 2.3). Written records

describing these phenomena exist dating back to the 4th century B.C.

1

However, the earliest known use of a magnetic compass far predates the

appearance of humans on earth, and can be found not in written records,

but in the fossil record.

The oldest known magnetic compass is found in the Gunflint Iron For-

mation in Minnesota and Ontario, a region rich in fossils dating back

approximately 2 billion years. Included among this ancient zoo of mi-

croscopic organisms are the remains of creatures known as magnetotactic

bacteria1 [1, 2]. The distinguishing feature of these bacteria are one or

more microscopic, ferromagnetic needles used to orient the creature with

respect to the earth’s magnetic field, aiding in the search for the optimal

oxygen concentration. By following magnetic field lines in their aquatic

environment, this search is reduced from three dimensions to one.

Some 2 billion years after these bacteria were immortalized in stone,

the now-ubiquitous hairless chimps began to catch on. There is some ev-

idence that the Olmec people of Mesoamerica may have invented a type

of compass earlier than 1000 B.C. However, the first written description of

ferromagnetism is found in a 4th century B.C. Chinese text entitled, “Book

of the Devil Valley Master” [3, 4]: “Thus, one should at the beginning of

1It should be noted that the age of 2 billion years is merely one theory. Young Earth

creationists hold that magnetotactic bacteria were created less than 10,000 years ago.

2

cognition first perceive oneself and then other people. Then the mutual

cognition becomes as clear as the vision of two-eyed fish and as the light

following the shadows. Then one will miss nothing upon examination of

the word, like the magnet draws the needle and the tongue the fried bone

to itself.” 2

Over the next twenty-three hundred years or so, the knowledge of mag-

netism spread across the globe. Clever minds devised new uses for the

phenomenon, and refined old ones, ranging from the electric motor, to the

dynamo, to the posting of notes on a refrigerator. Despite its bountiful

technological applications, at the beginning of the twentieth century, the

physical origins of magnetism were still unclear.

As quantum mechanics was being developed in the 1920s, great strides

were made in understanding atomic spectra by quantizing the orbital mo-

mentum of electrons around the atomic nucleus. However, results such as

the Stern-Gerlach experiment, and unexplained splittings in atomic spec-

tra (the “anomalous Zeeman effect”, and hyperfine splitting) indicated that

there were extra quantum degrees of freedom not being taken into account.

2Translated from the Chinese into German by Kimm (1927) [3], and from the German

into English by F. Mendoza (2007). There is some debate about the dating of this

manuscript (see [4]), though other sources mentioning magnetism from the same period

also exist.

3

A natural candidate for this unknown quantity was the angular mo-

mentum of a particle. The idea was at first considered to be impossible.

Given the known upper bound on the radius of the electron, the angular

velocity of the electron would need to be impossibly high to provide the

observed splittings. Nevertheless, Uhlenbeck and Goudsmit published the

idea in 1925. Despite its apparent impossibility, the idea of “spin” nicely

explained the observations. Originally a skeptic, Pauli warmed to the idea

and ran with it, redefining spin not as an actual rotation of a particle, but

as an angular momentum intrinsic to the particle, just as charge or mass

are intrinsic properties. He then went on to develop a formalism for dealing

with spin in (non-relativistic) quantum mechanics (see, for example, [5]).

Once this theoretical framework was in place, the experimental study

of spin physics could now proceed hand-in-hand with theory, instead of

the pure phenomenology of the previous millennia. Throughout the rest

of the twentieth century numerous advances were made, such as a detailed

understanding of magnetic materials, nuclear spin physics, and spin reso-

nance phenomena. These discoveries led to revolutionary technologies such

as magnetic resonance imaging (MRI) and magnetic data storage (tapes,

hard drives).

The reservoir of interesting spin phenomena is still far from dry. Re-

4

cent advances in materials, electronics, and low-temperature technologies

have brought new untapped wells of spin physics within reach. But how

will we know where to sink a new well? Like a magnetotactic bacterium

following its magnetic needle, we will use the promise of future spin-based

technologies as our divining rod to lead us to new and interesting physics.

1.2 Spintronics

One promising area of application for spin physics is in the field of spin-

based electronics, or “spintronics”. Here, the idea is to integrate magnetic

functionality with more traditional, electronic devices. Virtually all elec-

tronic devices are based on the distribution and flow of electrons within a

material. These electrons have spin of course, but in general, the spin is

ignored as far as the functionality of the device is concerned.

Recently, the first spintronic devices have successfully integrated mag-

netic and electronic functionality. The flow of spin-polarized electrons

through layered magnetic structures has revolutionized hard-drive storage

technology through the phenomenon of giant magnetoresistance (GMR).

Also, new types of non-volatile memory (magnetic RAM: MRAM) are on

the market that integrate magnetic memory and electronic devices in a

single architecture. Future refinements of this idea, possibly using spin

5

ferromagnetic

spin injectorferromagnetic

spin analyzergate

Vg

semiconductor channel I

Figure 1.1: The Datta-Das spin transistor. Spin polarized electrons are injected from

a ferromagnetic source into a semiconductor channel, where they are manipulated using

a gate voltage, and then read out using a ferromagnetic drain contact.

torque devices [6], may play a significant role in consumer electronics.

Both MRAM and GMR devices are based on metallic magnetic mate-

rials. But it is also interesting to consider the potential of semiconductor

spintronic devices. Semiconducting materials are the backbone of modern

electronics, and offer a wide range of functionality, controllable through

combinations of material composition and doping. The further integration

of semiconductor devices with magnetic properties offers another dimen-

sion of potential applications.

The canonical semiconductor spintronic device is the “spin transis-

tor” as proposed by Supriyo Datta and Biswajit Das in 1990, shown in

Fig. 1.1 [7]. While this may or may not be a practical or useful device, it

captures some essential features of a spintronic device: generation, manip-

6

ulation, and detection of spin polarization. The Datta-Das spin transistor

consists of a semiconductor channel with ferromagnetic source and drain

contacts at either end. When a voltage is applied across the source and

drain, spin-polarized electrons flow from the source contact into the semi-

conductor. After traversing the semiconductor channel, the electrons flow

into the drain contact. The conductivity across the semiconductor-drain

interface will depend on the spin polarization of the current, relative to the

spin-dependent density of states in the drain contact. Therefore, by ma-

nipulating the spin of the electrons as they flow through the channel (here,

effected in some way by a gate electrode), the source-drain conductivity

can be modulated.

Most, if not all, of the components of the Datta-Das spin transistor have

been demonstrated in individual devices. However, putting them together

in a commercially viable device has so far proved elusive. Clearly, more

research is needed to find a practical use for spins in semiconductor devices.

Just as the American Plains Indians put the scrotum of the buffalo to good

use3, so too should we strive to use every part of the electron.

3For ceremonial rattles [8]

7

1.3 Quantum information processing

In the typical view of spintronic devices, the spin is thought of classically,

as a vector in 3-space. That is, quantum effects such as coherence or

entanglement are ignored. But if these effects can be understood and

controlled, a number of powerful applications may become possible.

In general, exploiting uniquely quantum effects in spins or other sys-

tems goes by the name of quantum information processing. This field can

be divided into two categories: quantum computing, and quantum com-

munication.

Quantum computing is predicted to offer exponential speedup of certain

computational problems, effectively solving problems, such as factoring

large numbers, that are currently impossible for classical computers. In

a classical computer, information is stored and processed in bits, each

of which can take on two values (“0” or “1”). Extending this idea into

the quantum realm, a quantum bit, or “qubit”, can exist in a coherent

superposition of “0” and “1” and furthermore, an ensemble of qubits can

exist in a superposition, entangled with one another. These superpositions,

in effect, allow many computations to be performed in parallel, leading to

the improvement in computing power.

In order to build a qubit, one needs a two-state quantum system that

8

can be initialized into a particular state, and subsequently measured. One

then needs to construct a system in which “quantum gates” can be applied

to one or more qubits, in analogy to the logical gates that act on classical

bits. Additionally, the system must be scalable up to large numbers of

qubits, and the qubits must maintain their coherence over sufficiently long

timescales such that a large number of gate operations can be performed

within the coherence time.

An electron spin is a popular candidate for a qubit, since it is a natural

two-state system. Electron spins in semiconductors have received much

attention for quantum information applications because 1. semiconductor

processing technology should make the scaling to large systems easier, and

2. electron spins in semiconductors have been found to have long coherence

times relative to the expected times for gate operations. In recent years, a

number of schemes have been demonstrated for achieving the requirements

of state initialization, readout, and control for spin qubits (see Chapters

5-7). Nevertheless, there is still a long way to go before these elements can

be put together in a functioning quantum computer. This is good news for

a physicist, and bad news for an engineer.

Quantum communication involves the transmission of quantum infor-

mation from one place to another. This has applications in secure commu-

9

nication (cryptography) and in teleportation of quantum states. Quantum

communication necessitates a “flying qubit” – a carrier of quantum infor-

mation that can be moved from place to place. Though spins and other

qubit candidates such as single atoms can be moved over micron-scale dis-

tances within the coherence time, the only practical qubits for long-distance

quantum communication are photon-based. Photons make ideal carriers

of quantum information because they travel fast, and they have very long

coherence times. The flip side to the long coherence time, is that photons

interact very weakly with each other - and a strong controllable interaction

is a desired feature for quantum information processing. This leads one to

consider a hybrid system with stationary qubits used for quantum com-

puting at either end, and flying qubits communicating between the two.

This requires a way of converting stationary qubits to flying qubits. Fortu-

nately, spins in semiconductors couple to photons in a variety of ways (as

discussed below), making this an intriguing platform for a potential hybrid

quantum computing/communication system.

10

Chapter 2

Background

2.1 Spins in semiconductors

According to quantum mechanics, angular momentum can be described by

two quantum numbers: total angular momentum L, and the projection of

angular momentum on the (say) z-axis: Lz. These quantum numbers are

derived from eigenvalues of the commuting operators L2

and Lz: L2|ψ〉 =

L(L + 1)h2|ψ〉 and Lz|ψ〉 = Lzh|ψ〉, where L2 = L2x + L2

y + L2z, and Lα

is the angular momentum operator along the α-direction. L can take on

half-integer values, and for a given L, Lz can take on values Lz = −L,−L+

1, ..., L.

In the case of a particle’s spin, the total angular momentum S is fixed,

and the projection of the spin can take on 2S + 1 values, from −S to S.

An electron has total spin S = 1/2, and Sz = ±1/2. Therefore, there are

11

two eigenstates one with S = 1/2 and Sz = +1/2 denoted |↑〉 and the

other with S = 1/2 and Sz = −1/2 denoted |↓〉. A general spin state of an

electron is then given by

|ψ〉 = α|↑〉 + β|↓〉 (2.1)

where α and β are complex numbers such that |α|2 + |β|2 = 1.

In this basis of states, it is convenient to write down the matrix forms

of the operators Sα:

Sx =h

2

0 1

1 0

Sy =h

2

0 −i

i 0

Sz =h

2

1 0

0 −1

,

where

α

β

represents the state given by Eq. 2.1.

These matrices without the factor of h/2 are known as Pauli matrices,

and the vector S =(Sx, Sy, Sz

)is the spin operator for the electron spin-

1/2.

Note that there are four degrees of freedom in the two complex coeffi-

cients α and β in Eq. 2.1. However, the normalization requirement removes

one of these degrees of freedom, and another can be ignored as an overall

phase. Thus there are only two degrees of freedom that we care about, and

Eq. 2.1 can be rewritten

|ψ〉 = cosθ

2|↑〉 + eiφ sin

θ

2|↓〉. (2.2)

12

)cos,sinsin,sin(cos θθφθφ∝Sv

↓+↑

↓+↑ i

↓−↑

↑

↓

↓−↑ i

φθ

x

yz

Figure 2.1: The Bloch sphere. The vectors pointing to the north and south poles of

the Bloch sphere represent the “up” and “down” eigenstates, with the rest of the sphere

representing superpositions of “up” and “down”.

The two parameters θ and φ can be thought of as the polar and az-

imuthal angles defining a point on a sphere. This is known as the Bloch

sphere (Fig. 2.1), and turns out to be a very useful way of picturing a

spin-1/2 state. The usefulness of this picture can be seen by looking at

the expectation values of the spin in the x, y, and z directions. Using the

matrix forms of the Sα operators given above, it is easy to show that the

corresponding expectation values are

〈Sx〉 =h

2cosφ sin θ 〈Sy〉 =

h

2sinφ sin θ 〈Sz〉 =

h

2cos θ.

These expectation values are equivalent to the x, y, and z components

13

0k

heavy holes

J=3/2, Jz=±3/2

light holes

J=3/2, Jz=±1/2

split-off holes

J=1/2, Jz=±1/2

Conduction band

‘s-like’: L=0

Eg

∆

E

Valence band

‘p-like’: L=1

electrons

S=1/2, Sz=±1/2

Figure 2.2: Typical direct-gap band diagram. Near the zone center, the dispersion

of the energy bands in a direct-gap semiconductor can be approximated as parabolas.

The conduction band (red) has zero orbital angular momentum (L = 0). The valence

bands (black) have L = 1, which splits the band into three subbands: heavy holes, light

holes, and split-off holes. Eg is the band-gap, and ∆ is the spin-orbit coupling energy.

of the Bloch vector, as shown in Fig. 2.1. Therefore, it is correct in some

sense to think of the spin as actually “pointing” along the vector on the

Bloch sphere. This one-to-one correspondence between the quantum state

and the intuitive picture of a classical angular momentum vector is ap-

parently just a coincidence. For spin other than 1/2, there is no such

direct correspondence. But since here we are typically interested in elec-

tron spins, the Bloch sphere provides a useful and intuitive way of thinking

about quantum spin states.

14

The results described below will focus on electron spins in semiconduct-

ing materials. In a semiconductor, a completely filled electronic band (the

valence band) is separated from an empty band (the conduction band) by

an energy gap, Eg on the order of 1 eV. Fig. 2.2 shows the typical dispersion

of the conduction and valence bands in a direct gap semiconductor (where

the valence band maximum and conduction band minimum occur at the

same quasi-momentum k). In general, the dispersion has more complex

structure as a function of k, but sufficiently close to the conduction band

minimum (and valence band maximum) we can approximate the curvature

as parabolic. Fortunately, we will be mainly interested in electrons residing

in this region.

In all of the semiconductors considered here, the valence band is derived

from the filled p-orbitals of the constituent atoms, and the conduction

band from the unfilled s-orbitals. At the center of the Brillouin zone (k =

0), an electron in the conduction band has s-symmetry (orbital angular

momentum L = 0) and an electron in the valence band has p-symmetry

(L = 1). Away from the zone center, these pure s and p states are not

eigenstates. Matrix elements of the operator k·p mix states between bands,

yielding some p character in the conduction band and some s character in

the valence band. Near the zone center, one can often ignore this mixing

15

to first order (the conduction band states are “s-like” and the valence

band states are “p-like”). For a detailed discussion of semiconductor band

structure, see Ref. [9].

Since the conduction band states are s-like, there is no orbital angular

momentum, and the total angular momentum of an electron is just the spin:

S = 1/2. However, the situation in the valence band is more complicated.

Here, the electrons have orbital angular momentum, L = 1 in addition to

their spin angular momentum. The eigenstates of the valence band now

must be described by the quantum numbers J and Jz, derived from the

eigenvalues of the total angular momentum operator squared J2 = (L+S)2,

and the projection Jz, respectively. According to the rules for addition

of angular momentum, J can be either |L + S| or |L − S| and Jz can

take on values −J,−J + 1, ..., J . Therefore, in the present case, there

are six angular momentum subbands in the valence band with (J, Jz) =

(32,±3

2), (3

2,±1

2), (1

2,±1

2).

Due to the interaction of the spin and orbital angular momentum in

the valence band (spin-orbit coupling), the energies and dispersions of the

different subbands are altered. The two J = 1/2 bands are split from the

four J = 3/2 bands by the spin-orbit interaction energy ∆. Furthermore,

the curvature of the valence subband and conduction band dispersions

16

are altered by the k · p matrix elements between the different bands, as

mentioned above. These curvatures are interpreted as an “effective mass”

for electrons in the different bands. Fig. 2.2 shows the conduction and

valence band dispersions near the zone center. Since the valence subbands

have negative curvature, it is more convenient to refer to holes (the absence

of an electron) with positive mass. The broadest hole subband has J = 3/2

and Jz = ±3/2, and is known as the heavy hole band, because of its larger

effective mass. The subband with J = 3/2 and Jz = ±1/2 is referred to

as the light hole band, and is degenerate with the heavy hole band at the

zone center. Finally, the subband with J = 1/2 and Jz = ±1/2 is known

as the split-off hole band.

A spin in a magnetic field, B has a contribution to its energy from the

Zeeman Hamiltonian:

HZ =gµB

hB · S, (2.3)

where µB = 9.274 × 10−24 J/T is the Bohr magneton. For an electron in

vacuum, the electron g-factor is approximately 2. However, the presence

of the spin-orbit interaction modifies this quantity. Even though, there is

no orbital angular momentum in the conduction band to first order, the

k · p matrix elements between the conduction and valence bands shift the

effective g-factor of electrons in the conduction band to values less than 2.

17

The g-factor can be calculated to be

g = g0 −2

3

Ep∆

Eg(Eg + ∆), (2.4)

where g0 is the bare electron g-factor, Eg is the bandgap, ∆ is the spin-

orbit energy, and Ep is a parameter specifying the strength of the interband

coupling. For example, electrons in the conduction band of GaAs have

g = −0.44. The g-factor of holes is further complicated by the orbital

angular momentum in the valence band. However, the results below are

concerned with conduction band spins, so hole spin will be disregarded for

the most part.

The result of the Zeeman effect on an electron spin is clearly seen by

choosing the z-axis to be along the magnetic field. Now

HZ =1

2gµBBz

1 0

0 −1

. (2.5)

The spin eigenstates |↑〉 and |↓〉 are split by the Zeeman energy ∆E =

±12gµBBz. If the spin is not in an eigenstate, then it evolves in time,

depending on the Zeeman splitting. For a spin in the state given by Eq. 2.2

at t = 0, the state evolves according to (again, ignoring the overall phase)

|ψ(t)〉 = cosθ

2|↑〉 + ei(ωLt+φ) sin

θ

2|↓〉, (2.6)

where hωL = gµBBz is the Zeeman splitting. ωL is known as the Larmor

frequency. In the Bloch sphere picture, this corresponds to the spin vector

18

precessing about the z-axis at the Larmor frequency:

~S = (cos(ωLt+ φ) sin θ, sin(ωLt+ φ) sin θ, cos θ) . (2.7)

This phenomenon is referred to as Larmor precession.

For a spin perfectly isolated from the environment, a spin in a static

magnetic field would obey the dynamics of Eq. 2.6 forever. In reality,

there are a number of effects that limit the lifetime of an electron spin in

a semiconductor. These effects can be divided into two categories: those

that affect φ, and those that affect θ in Eq. 2.6. The randomization of θ

is referred to longitudinal spin relaxation, and is characterized by a time

T1. The loss of the relative phase information φ is referred to as transverse

spin decoherence, occurring in time T2. The results described below focus

on measurements of spin coherence, so we will primarily be concerned with

mechanisms affecting T2. Such mechanisms can often be described as an

effective magnetic field that fluctuates randomly in time. This leads to

random fluctuations in the precession axis, and precession frequency. Over

time, these fluctuations induce a random walk of the spin away from its

unperturbed state, as shown schematically in Fig. 2.3. The result of these

effects is typically a reduction in the spin polarization over time, with the

form exp(−t/T2).

One source for this effective field is spin-orbit coupling, with Hamil-

19

time

effBr

Sr

Figure 2.3: Spin decoherence: a random walk. A spin ~S in an initial state (represented

by the leftmost blue arrow) is influenced by a randomly fluctuating effective magnetic

field, ~Beff (black arrows). As the spin precesses around the random effective field, it

undergoes a random walk away from its initial state.

tonian HSO = h/(4m20c

2)(k × ∇V ) · S, were V is (for the case of bulk

semiconductors) the potential seen by the electron from the lattice ions.

By comparison with the Zeeman Hamiltonian, it is clear that this can

be seen as a k-dependent effective magnetic field acting on the spin. In

most semiconductors, the momentum scattering time for electrons is very

short (∼ 10 fs). That is, the electron’s quasimomentum, k is randomly

changed every ∼ 10 fs, and the spin feels an effective field fluctuating on this

timescale. This spin decoherence mechanism is known as the Dyakonov-

Perel mechanism. In the valence band, the spin-orbit coupling is strong,

and therefore the hole spin lifetime is often very short (on the same or-

der as the momentum scattering time). This short lifetime means that

20

in many cases, the hole spins can be ignored. In the conduction band,

spin-orbit coupling is only present due to the k ·p terms coupling the con-

duction and valence bands. Therefore, conduction band spin lifetimes can

be significantly longer (exceeding 100 ns in bulk n-type GaAs [10]).

Additionally, spin-orbit coupling can cause spin relaxation or decoher-

ence in the conduction band more directly. Even though the spin-orbit

coupling in the conduction band is small, just as in the valence band, S

and Sz are no longer good quantum numbers. That is to say, |↑〉 and |↓〉

are not eigenstates – there is a little mixing due to spin-orbit coupling.

This mixing can often be ignored, but it means that interactions such as

between the electron and phonons can induce spin-flips. This is known as

the Elliot-Yafet mechanism.

Another source of a randomly fluctuating effective field is the spin of

the atomic nuclei in the semiconductor. As is discussed below, the hyper-

fine interaction governing the electron-nuclear spin interaction also has the

form of an effective magnetic field, seen by the electron spin. This inter-

action is proportional to the electron wavefunction squared at the position

of the nucleus – typically each electron interacts with a number of nuclei.

If the electron has significant overlap with N randomly oriented nuclear

spins, then there will be a randomly oriented net nuclear spin polariza-

21

tion proportional to√N . Due to dipole-dipole interactions, nuclear spins

fluctuate with timescales on the order of hundreds of microseconds [11].

This timescale is typically longer than the spin decoherence time caused

by other (e.g. spin-orbit-related) mechanisms. Therefore, the nuclear spin

polarization is essentially constant within the lifetime of an individual spin,

and the precession axis and frequency are modified accordingly. However,

in most measurements, the spin is reinitialized and measured repeatedly,

or equivalently, an ensemble of spins is measured. In this case, each spin

experiences a different nuclear spin polarization, and the cumulative effect

is a reduced observed spin lifetime. Unlike the decoherence mechanisms

discussed above, this nuclear dephasing mechanism is not predicted to re-

sult in a simple exponential decay, but instead has Gaussian and power-law

terms [12, 13]. This type of mechanism is referred to dephasing, as opposed

to decoherence, since the coherence of individual spins is not affected. The

timescale for decoherence plus dephasing is often referred to as the inho-

mogeneous or effective transverse spin lifetime, T ∗2 . In principle, the effects

of dephasing can be circumvented, as will be mentioned again in Chapter

7.

As mentioned in the preceding paragraph, electron spins aren’t the only

spins present in a semiconductor. Many nuclear species present in common

22

semiconductors have non-zero nuclear spin. The contact hyperfine Hamil-

tonian governs the electron-nuclear spin interaction: H = AI · S, where I

is the spin operator for the nuclear spin, analogous to the electron spin op-

erator S, though not necessarily spin 1/2. The constant A is proportional

to the overlap integral of the electron and nuclear wavefunction.

An important aspect of the hyperfine interaction is that it allows elec-

tron and nuclear spins to flip each other. This can be seen by rewriting

the Hamiltonian in terms of raising and lowering operators: H = AI · S =

A[IzSz + (I+S− + I−S+)/2] where I± = Ix ± iIy and S± = Sx ± iSy. Here,

S±, I± act as raising/lowering operators in that they change the quantum

number Iz or Sz by ±1. For example, if the electron spin is in a state

with Sz = −1/2 and the nuclear spin is in a state with Sz = 3/2, the term

with the raising/lowering operators would yield a state with Sz = 1/2 and

Iz = 1/2. In this way, electron and nuclear spins flip each other.

In the experiments described below, a non-equilibrium electron spin po-

larization will be generated in a semiconductor structure. As a result, this

leads to a net polarization of the nuclear spin parallel to the electron spin

polarization. The component of the nuclear spin perpendicular to any ap-

plied magnetic field will precess, similar to the electron spins, and therefore

there will be no steady-state nuclear spin polarization. However, if there is

23

a component of the spin along the magnetic field, than this non-precessing

component can build up over time. Since the longitudinal relaxation time

for nuclear spins can be quite long, this steady state nuclear polarization

can take seconds or minutes to reach equilibrium. This phenomenon is

known as dynamic nuclear polarization (DNP).

As mentioned above, it is also evident from the hyperfine Hamiltonian

that the nuclear spin can be thought of as an effective field acting on the

electron spins. If the nuclear spins are thought of in a mean field picture,

then the effect of the nuclei on an electron spin can be expressed as an

effective magnetic field

Bn =h

gµBAI, (2.8)

where I is the mean nuclear spin polarization, and A is appropriately

weighted to reflect the strength of the interaction. This effective field can

be quite large. In GaAs, the effective field for 100% nuclear spin polar-

ization has been estimated to be Bn ≈ 6 T [11]. This effect can readily

be observed in a number of measurements, such as time-resolved Faraday

rotation (Chapter 6) or Hanle measurements [14].

24

2.2 Quantum dots

When electrons are confined within a semiconductor structure with one or

more dimensions smaller than the extent of the bulk electron wavefunction,

the electronic properties are drastically modified. For confinement in one,

two, or three dimensions, this type of structure is known as a quantum

well, quantum wire, or quantum dot respectively. Like a particle-in-a-box,

the energy levels of electrons in a quantum dot (QD) are quantized into

discrete levels, as illustrated in Fig. 2.4. Instead of a continuous band of

conduction band states, the energy eigenstates are now spatially localized

within the QD, and separated by an energy that increases with increasing

confinement. The lowest conduction band state is blue-shifted from the

conduction band minimum due to this confinement.

When the temperature is low enough such that kBT < ∆E, where ∆E

is the QD energy level spacing, then the quantized nature of the energy

levels becomes apparent. For temperatures around 4 K, this requires a QD

size on the order of 100 nm. In this regime, QDs will exhibit an atom-like

spectrum of absorption and emission lines.

The spin physics are also altered by quantum confinement. The blue-

shift of the conduction band ground state effectively increases the gap

between the conduction and valence band states, reducing the effect of the

25

conduction band

valence band

E QD

Figure 2.4: Schematic of confining potential, energy levels, and wavefunctions in a

quantum dot. A quantum dot is formed in a semiconductor at a local energy mini-

mum. Quantized energy levels and particle-in-a-box wavefunctions are illustrated. The

confining potential exists in three dimensions, though only one dimension is shown.

k · p matrix elements that couple the two bands. This primarily has the

effect of shifting the electron g-factor towards the bare electron g-factor,

g = 2. Modifying Eq. 2.4 to account for this energy shift, E0,

g = g0 −2

3

Ep∆

(Eg + E0)(Eg + E0 + ∆). (2.9)

Additionally, momentum scattering is now suppressed and along with it,

the various spin-orbit-related decoherence mechanisms that are caused by

26

momentum scattering. However, since the electron wavefunction is now

more spatially concentrated, it interacts more strongly with the nuclear

spins, making nuclear spin dephasing much more significant.

There are a number of ways of physically realizing semiconductor QDs.

These fall into three categories: 1. Chemically synthesized QDs, 2. QDs

grown by molecular beam epitaxy (MBE), and 3. QDs defined by selec-

tively depleting a two-dimensional electron gas (2DEG). Figure 2.5 shows

examples of each type.

Chemically synthesized QDs can by made in a number of ways, though

primarily through colloidal chemistry methods. In this technique, a flask

containing a solution of, for example, Cd and Se ions is heated and cooled in

a specifically controlled protocol to nucleate CdSe nanocrystals and then

grow them to the desired size. CdSe is one common choice of material,

though many materials have been used with this method including II-VI,

III-V, and oxide semiconductors. The result of this process is a large

number of nanocrystal quantum dots in solution with a diameter tunable

from about 1 - 10 nm (Fig. 2.5 a).

One drawback to chemically synthesized QDs is that they live in a hard-

to-control chemical environment. (On the other hand, this could be seen

as a bonus – the chemical environment leads to interesting phenomena and

27

5 nm

(a)

InAs

GaAs

(b)

(c)

Figure 2.5: Types of quantum dots. (a) High-resolution transmission electron micro-

graph of a CdSe nanocrystal QD, showing the regular lattice of Cd and Se atoms (X.

Peng et al., J. Am. Chem. Soc. 119 2007). (b) Cross-sectional transmission electron

micrograph (55 × 55 nm) of a vertical stack of self-assembled InAs QDs (P. M. Koen-

raad et al., Physica E 17 2003). (c) Electron micrograph of a gate-defined 2DEG QD

structure, dashed circle is the QD region (J. M. Elzerman et al., Nature 430 2004).

is another knob to turn.) Nevertheless, it is often nice to simplify things a

bit, and better isolate the system being studied. This can be achieved by

fabricating QDs that are embedded in an epitaxial semiconductor mate-

rial. One method is to fabricate a quantum well out of GaAs, with AlGaAs

barriers. Under appropriate growth conditions, the GaAs/AlGaAs inter-

faces will have roughness of ± 1 atomic layer, with lateral length scales on

the order of 100 nm. These fluctuations in the thickness of the QW form

28

localized potential minima that act as QDs. Another type of MBE grown

QD is grown using strain-induced self-assembly. If InAs is deposited on a

GaAs surface under the right conditions, the InAs will not form a smooth

layer, but instead will bead up into droplets with dimensions of tens of

nanometers. After these droplets have formed, GaAs is grown over the top

of the InAs (Fig. 2.5 b). Since the bandgap of InAs is lower than that of

GaAs, these droplets then serve as QDs.

Finally, a 2DEG (essentially an electron-doped QW with high electron

mobility) can be grown via MBE. The two-dimensional electrons in this

structure can then be further confined by applying a bias to electrical

gates deposited on the surface of the sample. These gates deplete the

nearby electrons in the 2DEG, creating a potential minimum surrounded

by depleted regions (Fig. 2.5 c). This forms a QD well-suited for electrical

contacting – a small gap in the gates creates a tunable tunneling barrier

to an electrical lead.

2.3 Optical properties of spins in

semiconductors

The connection between spins in semiconductors and light is interesting for

practical applications, experimental techniques, and new physical phenom-

29

ena: An example of an application might be the hybrid solid state/optical

platform for quantum information processing described above; Several op-

tical experimental techniques for probing spins in semiconductors will be

described below; and an example of an interesting physical phenomenon

would be the off-resonant interaction between a single spin and light de-

scribed in Chapter 7.

Abraham Lincoln told a story about “an Eastern monarch [who] once

charged his wise men to invent him a sentence, to be ever in view, and which

should be true and appropriate in all times and situations.” If they were

truly wise, they would have said, “Write down the Schrodinger equation,

and solve it.” For an electron interacting with an electromagnetic field,

the Schrodinger equation for the electron wavefunction ψ is

[1

2m

(p +

e

cA

)2

+ V (r)

]

ψ = Eψ (2.10)

where A is the vector potential of the electromagnetic field. For a plane

wave with frequency ω and polarization along ~ǫ, we can take

A = (c/ω)~ǫE0 exp(i(ωt− k · r)) + c.c.. (2.11)

Treating the A-dependent terms as a perturbation and making the usual

“dipole approximation” that k · r is small, then to leading order we have a

30

time-dependent perturbation

V ′ =eE0~ǫ · pmω

(eiωt + e−iωt

). (2.12)

Using time-dependent perturbation theory, it is not too hard to show

that this perturbation induces transitions from an initial state |i〉 to a final

state |f〉 with probability proportional to

(eE0

mω

)2

|〈f |~ǫ · p|i〉|2. (2.13)

For electrons in a semiconductor, the initial states |i〉 are states in any

of the valence subbands, and the final states |f〉 are in the conduction

band. Because of the symmetry of the momentum operator in Eq. 2.13,

there are certain selection rules for these interband transitions. Namely,

for circularly polarized light, ~ǫ = (x± iy)/√

2, the momentum operator in

the matrix element in Eq. 2.13 transforms like a spherical tensor of rank 1,

with q = ±1. This means that these matrix elements are non-zero only for

initial and final states with angular momentum projection Lz differing by

±1. Here, the z-axis is defined by the propagation direction of the light.

This can also be seen in terms of photons: the absorption of a circularly

polarized photon transfers its angular momentum Lz = ±h to the spin.

Looking at Fig. 2.2, it is clear that these selection rules determine

which transitions can take place between various valence subbands, and

31

S=1/2:

J=3/2:

J=1/2: Jz=-1/2Jz=+1/2

Jz=-3/2Jz=-1/2Jz=+1/2Jz=+3/2

Sz=+1/2 Sz=-1/2

31

2 2

1

3

Figure 2.6: Optically allowed transitions from the valence band (heavy and light holes

with J = 3/2 and split-off holes with J = 1/2) to the conduction band (with S = 1/2).

Red and blue indicate left and right circularly polarized light, and the thickness of the

arrow indicates the strength of the transition.

the conduction band. Specifically, the allowed transitions from the heavy

hole, light hole, and split-off hole bands are shown in Fig. 2.6 for right and

left circularly polarized light.

If the matrix elements are actually calculated [9], it is found that the

different transitions shown in Fig. 2.6 have different matrix elements. That

is, some transitions are more likely than others, for the same light intensity.

These relative transition strengths are also shown in the figure: the heavy

hole transition is three times as strong as the light hole transition, and the

split-off hole transition is twice as strong as the light hole transition. This

provides a useful way to generate a net spin polarization in the conduction

band of a semiconductor. If light is incident on the semiconductor at an

32

energy E, such that Eg < E < ∆ + Eg, then only transitions from the

heavy hole or light hole states can be excited. If this light is circularly

polarized, then only transitions from the, say, Jz = +3/2 heavy hole and

Jz = +1/2 light hole are allowed. Since the heavy hole transition is three

times as likely as the light hole transition, the result is three times as many

conduction band spins with Sz = +1/2 than with Sz = −1/2. This results

in a net spin polarization in the conduction band of 50%.

The situation can be improved further if the semiconductor is strained,

or if the electrons and holes are confined in one or more dimensions. In

this case, the degeneracy of the heavy and light hole bands is lifted, and

transitions can be pumped from the heavy hole band only, resulting in

ideally 100% conduction band spin polarization.

The inverse of this optical spin injection provides a means for detecting

the spin polarization of electrons in a semiconductor. When an electron

and hole recombine, light is emitted with circular polarization reflecting

the spin state of the electron and hole. Thus by measuring the degree

of circular polarization of this luminescence, one can measure the spin

polarization of the electrons and holes.

Using the polarization of the luminescence as a probe of the spin polar-

ization, one can get some information about the spin dynamics in a mag-

33

laser

polarizersλ/4-plates

circ. pol.

PL

To

spectrometer

sample

B

Figure 2.7: Schematic of a Hanle measurement setup. Above-bandgap laser light is

circularly polarized using a quarter-wave plate. This light excites spin polarized elec-

trons into the conduction band of the sample. The PL is collected, and its polarization

is analyzed using a quarter waveplate and a polarizer. The angle between injection and

detection may be made smaller than shown here.

netic field. This type of experiment is known as a Hanle measurement, and

the setup is shown schematically in Fig. 2.7. A circularly polarized laser is

incident on the sample in the +x-direction, serving to inject spin-polarized

electrons and holes. A magnetic field Bz is applied in the z-direction.

The polarization of the subsequent photoluminescence (PL) collected back

along the x-direction reveals the steady-state spin polarization along the

measurement direction.

This steady-state spin polarization can be calculated by taking an ini-

tial electron spin polarization S0 at t = 0 along the +x-direction. (For

simplicity, we will assume that the hole spin lifetime is very short and

can be ignored.) The spin then precesses as given by Eq. 2.6, and the x

34

increasing magnetic field, Bz

Sr

xS

x

y

z

Figure 2.8: Illustration of the Hanle effect. At zero magnetic field, the spin decays

without precessing (left). As the magnetic field increases, the spin precesses through

a larger angle before decaying. As the maximum precession angle increases, the time-

averaged spin polarization decreases, as indicated by the red arrows.

component of the spin as a function of time is given by

Sx(t) =

0 t < 0

S0 cos(ωLt) exp(−t/T ∗2 ) t ≥ 0

. (2.14)

As before, ωL = gµBB/h is the Larmor precession frequency, and for gener-

ality both spin decoherence and dephasing are included in the spin lifetime

T ∗2 . The steady state spin polarization Sx is then found by integrating

Eq. 2.14 from t = (−∞,∞):

Sx ∝∫ ∞

0S0 cos(ωLt) exp(−t/T ∗

2 ) dt = S0

(1/T ∗

2

ω2L + (1/T ∗

2 )2

)

. (2.15)

Thus as a function of magnetic field (proportional to ωL), the mea-

sured PL polarization sweeps out a Lorentzian function with width B1/2 =

h/(gµBT∗2 ). This effect is illustrated in Fig. 2.8. If the g-factor is known,

35

then this measurement reveals the effective transverse spin lifetime, T ∗2 .

The analysis of the Hanle measurement can be done more rigorously and

quantitatively by setting up and solving rate equations for spin injection,

decay, and recombination [14]. The theory can be extended to the case

of nuclear polarization, non-exponential decay, doped semiconductors, etc.

This was the technique used for much of the initial exploration of spin

physics in semiconductors. However, there is only so much information

that can be extracted from a Hanle-type measurement. In contrast, the

Faraday rotation measurements described in the next section provide a

more powerful probe of semiconductor spin dynamics.

2.4 Faraday rotation: theory

The Faraday effect provides a useful optical probe of spin polarization in

semiconductors. This effect can be observed as a rotation of the plane

of polarization of linearly polarized light as it is transmitted through a

material (Fig. 2.9). The angle of this rotation is dependent on the spin

polarization in the material. The Kerr effect is directly analogous, but

refers to reflection of the light off of the sample, instead of transmission.

The effect arises from the spin-dependence of the index of refraction for

right and left circularly polarized light (σ+ or σ− polarization). This spin-

36

Sr

Sx

xF S∝θ

Figure 2.9: Diagram of the Faraday effect. As linearly polarized light is transmitted

through the sample, its plane of polarization is rotated through an angle, θF , propor-

tional to the projection of the spin polarization in the material along the light propa-

gation direction.

dependent change in the index of refraction results in a different phase shift

in the light for σ+ or σ− polarized light transmitted through the sample.

Following Eq. 2.11, the vector potential for σ± polarized light propagating

in the z-direction is given by

A± = A0(~x± i~y) exp[i(ωt− k · r ± θF )] + c.c., (2.16)

where θF is the phase shift caused by the Faraday effect, and k = |k|~z.

Linearly polarized light can be written as the sum of σ+ and σ− polariza-

tions: (~x + i~y) + (~x − i~y) = 2~x. After transmission through the sample,

linearly polarized light is given by the vector potential

A+ + A− = 2A0(cos θF~x− sin θF~y) exp[i(ωt− k · r)] + c.c.. (2.17)

This is the vector potential for light propagating in the z-direction with

37

linear polarization rotated through an angle θF with respect to the x-axis.

There are several ways to see where this phase shift comes from. Here,

I will sketch a quantum mechanical derivation with the radiation treated

classically. Appendix C outlines two other derivations: a classical view,

and a fully quantized picture.

Following Ref. [15], we start with the Schrodinger equation for an elec-

tron in an electromagnetic field, given in Eq. 2.10. Again, using the dipole

approximation and time-dependent perturbation theory, we can calculate

the dielectric function ǫ(E) for the electron, in terms of the available initial

and final states. The imaginary part of√ǫ yields the absorption spectrum,

and the real part the index of refraction as a function of energy. For a

single set of initial and final states, the index of refraction for σ± polarized

light is found to be

n± ∝ |〈c|px ± ipy|v〉|2E − Ec,v

(E −Ec,v)2 + Γ2c,v

, (2.18)

where |c〉 and |v〉 are conduction and valence band states, and Ec,v and

Γc,v are the energy and linewidth of the transition from |v〉 to |c〉.

Since the velocity of light in a material is given by c/n, the phase shift

for circularly polarized light caused by transmission through a material

with index n is ∆φ = Lωn/c, where L is the path length through the

38

material. From this we have the Faraday rotation angle

θF (E) =LE

hc(n+ − n−)

= CE∑

c,v

(|〈c|p+|v〉|2 − |〈c|p−|v〉|2

) E − Ec,v

(E − Ec,v)2 + Γ2c,v

,(2.19)

where p± = px ± ipy, C is a constant that depends on the geometry of

the experiment, and we have summed over all valence band to conduction

band transitions.

Note that if the momentum matrix elements are equal for right and

left circular polarization, then the Faraday rotation angle is zero. This is

where the spin dependence comes in. If there is an electron in the conduc-

tion band in a particular spin state, then the transition to this state will be

forbidden by the Pauli exclusion principle. As discussed above, the momen-

tum matrix elements for a given circular polarization couple more strongly

to one conduction band spin state than the other (Fig. 2.6). Therefore,

the magnitude of these matrix elements for right and left circular polariza-

tion depend on the spin polarization in the conduction band. Figure 2.10

shows the predicted Faraday rotation spectrum from Eq. 2.19 for a single

interband transition, and integrated over a typical density of states with

a spin-polarized electron population given by the Fermi distribution near

the conduction band edge.

It should be noted that Faraday rotation can also arise when the am-

39

FRDOS

Energy (a.u.)

Far

aday r

ota

tion (

a.u.)

Filled spin-

up states

(a)

(b)

Figure 2.10: Theoretical Faraday rotation spectra. (a) Faraday rotation spectrum for

a single interband transition. (b) The Faraday rotation spectrum integrated over an

ensemble of states. The dashed red line is (qualitatively) the density of states (DOS)

near the absorption edge, and the blue shaded region indicates the occupied electron

states for a given Fermi energy and temperature.

plitude of the matrix elements are equal, but the energy of the transitions

are spin-split (e.g. by a magnetic field). In the measurements described

here, the magnetic field is sufficiently small that this effect is negligible.

40

2.5 Faraday rotation: experiment

Since the initial interest in Faraday rotation in semiconductors in the

1960’s, a number of experimental techniques have been developed for this

type of measurement. A typical scheme is to use a pump laser to optically

inject spins into the conduction band, and a probe laser to measure Fara-

day rotation (or Kerr rotation). In this type of pump/probe spectroscopy,

the two lasers may have the same or different energies.

If the pump and probe lasers are continuous wave (cw) then this pro-

vides information about the steady-state spin polarization, similar to the

Hanle measurement described above. However, Faraday rotation measure-

ments can be extended into the time domain to provide a more direct look

at the spin dynamics.

In a time-resolved Faraday rotation experiment, pulsed pump and probe

lasers are used, shown schematically in Fig. 2.11. Often, the pump and

probe lasers are derived from a single mode-locked Titanium:Sapphire

(Ti:Sa) laser. Such a laser typically outputs an optical pulse every 13 ns

with duration ∼ 150 fs. The pump pulse is circularly polarized, and excites

spin polarized electrons into the conduction band, as discussed above. The

probe pulse can be made to arrive at the sample with a fixed time delay,

∆t, by controlling the relative path length traveled by the two pulses. This

41

Ti:Sapphire

∆t

Circ.

pol

lin.

pol

pump

probe

~150 fs

sample

to diode

bridge

(not to scale)

Figure 2.11: Typical time-resolved Faraday rotation setup. A mode-locked

Ti:Sapphire laser produces a train of pulses with ∼ 150 fs duration. The pulse train

is split into a circularly polarized pump and linearly polarized probe. The delay, ∆t,

between pulses is controlled with a delay line in one path. The pump excites spins in

the sample, and the probe measures the spin a time ∆t later, via the Faraday effect.

can be achieved by reflecting one of the pulses off of a delay line – that

is, a translatable mirror (or retroreflector). Given the speed of light in air

(about 1 ft./ns), changing the mirror position by one foot changes ∆t by

about 2 ns. After transmission through (or reflection off of) the sample,

the Faraday (or Kerr) rotation of the linear polarization of the probe pulse

is measured. This provides a snapshot of the spin polarization precisely ∆t

after the arrival of the pump pulse. By varying the delay, ∆t, the evolution

of the spin dynamics can be mapped out in time.

The polarization rotation of the probe beam is detected with a balanced

photodiode bridge. As shown in Fig. 2.12, the polarization of the probe

42

θF

λ/2 plate

22.5º

45º-θF

Glan-thomson

(polarizing)

beamsplitter

I0

I0( ) 210 FI θ+

( ) 210 FI θ−

photodiodes:

A B

-

bridge

circuit

FBAout IIV θ∝−∝

Figure 2.12: Sensitive polarization detection with a balanced photodiode bridge. In-

coming light (from the right) is linearly polarized at a small angle, θF to the vertical.

A λ/2 plate reflects the polarization about its axis at 22.5, vertical and horizontal

components are split, and the difference is measured using the diode bridge circuit.

beam, initially vertical, is rotated through an angle θF . The probe beam is

then passed through a half-wave plate which reflects the polarization about

its axis at 22.5, resulting in the polarization at an angle θ = π/4 − θF .

Next, the probe is passed through a polarizing beamsplitter, which sepa-

rates the horizontally and vertically polarized components of the light. As-

suming θF is small (typically on the order of milliradians or less), then the

horizontal and vertical components have intensity IH(V ) ≈ I0(1 ± θF )/√

2

where the incoming probe beam has intensity I0. These two beams are

then focused onto a pair of photodiodes, and the difference between the

two photocurrents is measured by the diode bridge circuit. (For a detailed

discussion of diode bridge circuits, see Kato’s thesis [16]). This difference

43

signal SH−V ∝ IH − IV =√

2I0θF , and is thus proportional to the Faraday

rotation angle.

Since the Faraday rotation angles are typically small, the pump and

probe beams are often modulated at frequencies ranging from Hz to kHz

and lock-in detection is used to isolate the desired signal and to reduce

noise (specific details will be given for the particular experiments described

below). The bandwidth of the diode bridge must exceed these modulation

frequencies, though for noise reduction purposes it is desirable to keep

the bandwidth as low as possible. In particular, the bandwidth will be

much lower than the bandwidth of the probe pulses (which have bandwidth

as high as 1/100fs = 10 THz). This is the great advantage of this sort

of stroboscopic technique – it allows measurements with very high time

resolution with low bandwidth detection.

The experiments described below all make use of some variation on the

theme of time-resolved Faraday rotation. More experimental details will

be given throughout.

44

Chapter 3

Quantum Shells in

Semiconductor Nanocrystals

3.1 Motivation

In the biochemical process used by magnetotactic bacteria to build their

compass (see Chapter 1), magnetic particles are nucleated and grown from

metallic precursor chemicals [2]. Once the particle has reached the desired

size (on the order of 10s of nanometers) the growth is arrested. This

ancient, natural process is remarkably similar to the methods recently

developed by human scientists to grow both magnetic and non-magnetic

nanocrystals through colloidal chemistry.

Here we are focusing on semiconductor nanocrystals which act as quan-

tum dots (QDs), confining electrons on nanometer length scales. Nanocrys-

45

ZnS

CdSe(a)

V

r

e

h

V

r

e

h

CdS

CdSe

(b)

Figure 3.1: Cutaway schematic of layered nanocrystals. (a) A low-bandgap core sur-

rounded by a high-bandgap shell confines electrons and holes to the interior, protecting

them from the environment. (b) A high-bandgap core, a low-bandgap shell, surrounded

by a high-bandgap cap confines electrons and holes to the spherical shell layer. The di-

agrams at the right illustrate the conduction and valence band profiles on a nanocrystal

in the radial direction, with schematic electron and hole wavefunctions.

tal QDs are interesting systems for a number of reasons. First, the strong

confinement of the electrons and holes means that the confinement energy

exceeds kBT even at room temperature. Additionally, the simple chemical

process used to fabricate these nanocrystals (discussed below) allows great

flexibility in choice of materials, the chemical environment of the QDs, the

size, and even the shape of the nanocrystals.

Furthermore, not only can nanocrystal QDs be made of a variety of

46

semiconducting materials, but they can have multiple layers of different

materials within a single nanocrystal. By taking advantage of the dif-

ferent bandgaps of different semiconductors, this allows one to tailor the

confining potential for the electron in an individual nanostructure. For

example, in a core-shell structure with a low bandgap core, surrounded

by a larger bandgap shell, the electron will sit mainly in the core and

thus be protected from the environment. Perhaps more interestingly, a

high-bandgap core can be surrounded by a low-bandgap shell, and then a

second high-bandgap shell. Such structures, referred to as quantum-dot

quantum wells (QDQWs), are essentially a quantum well in a spherical ge-

ometry, within a single nanocrystal. A cutaway schematic of such a struc-

ture is shown in Fig. 3.1. Both core-shell quantum dots [17, 18, 19] and

QDQWs [20, 21, 22, 23, 24, 25] have been synthesized during the past years.

Although both CdS/HgS/CdS [21, 26, 22, 27, 28] and CdS/CdSe/CdS [25]

QDQWs were well characterized by photoluminescence (PL) and absorp-

tion spectroscopy, a detailed investigation of the quantum size levels was

not initially possible, mainly because the PL peak is spectroscopically

broad and the absorption spectrum shows a featureless increase. This

is due to the distribution of size and shape among the ensemble of QDs,

and therefore the distribution of QD confinement energies. In the work

47

described below, the spin dynamics in these systems is addressed using the

Faraday effect, which also provides a more sensitive measurement of the

energy level structure.

The following sections describe time-resolved Faraday rotation (TRFR)

for CdS/CdSe/CdS QDQWs with varying CdSe quantum well width (nCdSe =

1− 5 monolayers). The spin lifetime is of order 2− 3 ns even at room tem-

perature. The QDQWs exhibit g-factors that vary with quantum well

width. TRFR is not only a unique experimental probe for the spin dynam-

ics, but also a sensitive spectroscopic technique. In contrast to absorption

spectra, the amplitude of the TRFR signal as a function of probe energy

exhibits three distinct resonances close to the absorption edge, because

optical transitions to the lowest conduction band level are probed selec-

tively. From the level scheme and dielectric response functions evaluated

with k · p calculations [27, 29], we show that, while several peaks are ex-

pected to emerge in the TRFR signal, the large number of resonances with

comparable spectral weight is not reproduced by the quantum size levels

of a spherical QDQW. One possible mechanism that explains the exper-

imental data is broken spherical symmetry, which mixes different valence

band multiplets.

48

3.2 Sample structure and characterization

Colloidal QDQWs with varying width of the CdSe quantum well were syn-

thesized by a successive ion layer adsorption and reaction (SILAR) tech-

nique to produce nanocrystals with accurate control over the quantum well

width [19, 25]. First, a CdS (Eg = 2.48 eV) core is grown using standard

colloidal chemistry. Above a critical temperature (T >∼ 350C) nanocrys-

tals are nucleated in a solution of Cd- and S-containing precursor molecules.

The temperature is then lowered below the nucleation temperature, and

the nanocrystals then begin to grow. The size of the nanocrystals can be

monitored in situ using the optical absorption. Once the nanocrystals have

grown to the desired size, the growth is stopped by lowering the tempera-

ture further. In this work, a 3.4 nm core diameter was used. Subsequent

layers are then grown by immersing the nanocrystals in alternating cation

or anion containing solutions, adding one monolayer at a time. In this

way, the thickness and composition of the nanocrystals can be controlled

with atomic precision. Samples were made with 1 to 5 layers of CdSe

(Eg = 1.74), followed by a 1.6-nm thick CdS cap. A schematic represen-

tation of the structure is shown in Fig. 3.1, along with a diagram of the

conduction and valence band profiles in the radial direction. Electrons and

holes will be confined to the CdSe shell, as indicated in the figure. The

49

Wavelength (nm)

PL

Inte

nsi

ty (

norm

aliz

ed)

1 2 3 4 5 CdSe layers

500 600 700

0

(a)

(b)

Figure 3.2: Photoluminescence of quantum shell samples. (a) Photograph of single

shell samples under UV illumination with 1 (right) to 5 (left) CdSe layers. (b) PL

spectra of the same samples. The spectra have been normalized for clarity.

QDQWs were dissolved in toluene and all measurements were carried out

on a large ensemble of QDQWs in solution at room temperature unless

otherwise specified.

A photograph of the samples used in these experiments is shown in

Fig. 3.2, illuminated by a handheld UV lamp. As the shell thickness is

50

Ti:Sapphire

∆t

Circ.

Pol

(PEM)

lin.

pol

pump

~150 fs

sampleto diode

bridge

(not to scale)

Regenerative

amplifier

~150 fs

OPA2 OPA1

chopper

permanent

magnets

monochromator

Figure 3.3: Setup for quantum shell TRFR. Two optical parametric amplifiers (OPAs)

provide synchronized, independently tunable pump and probe pulses. The pump has its

circular polarization modulated with a photo-elastic modulator (PEM), and the linearly

polarized probe is modulated with a chopper. The time delay between arrival of the

pump and probe at the sample is controlled by a delay line, and the Faraday rotation

of the probe is measured by a photodiode bridge. A monochromator can be inserted in

the probe path to enhance the spectral resolution.

increased, a redshift of the luminescence can be clearly seen. The corre-

sponding PL spectra are shown in the figure.

3.3 Experimental setup

The experimental setup is shown in Fig. 3.3. A regeneratively amplified

Ti:Sapphire laser (Coherent RegA) outputting ∼ 200 fs duration pulses at

51

a repetition rate of 250 kHz was used to drive two optical parametric am-

plifiers (OPAs). These OPAs, outputting equally short pulses with wave-

length independently tunable from about 500-700 nm, are used as pump

and probe lasers. The pump and probe pulses were both focused to an

overlapped spot within the QDQW solution. The wavelength of the pump

laser was fixed at λpump = 505 nm (Epump = 2.46 eV). This pulse train was

circularly polarized, exciting spin polarized electrons into the conduction

band states of the QDQWs. Relaxation of the electron and hole to the

lowest exciton state presumably occurs on a picosecond time-scale, as in

similar systems such as CdS/HgS/CdS QDQWs [30]. The linearly polar-

ized probe pulse then passes through the QDQW solution a time ∆t later,

where ∆t is set using a mechanical delay line in the pump beam path. The

Faraday effect causes the polarization of the probe pulse to be rotated by

an angle, θF , proportional to the component of the net spin polarization

along the probe beam direction (as described in Section 2.4). By recording

θF for varying ∆t, we detect the time evolution of the optically injected

electron spins in the QDQWs.

52

∆t (ns)

0 1 2 3

θ F(a

.u.)

FT

pow

er

(a.u

.)

0 5 10Frequency (GHz)

(a)

Bapp=3 kG

(b)

Figure 3.4: Spin precession in quantum shells. (a) Faraday rotation as a function of

pump-probe delay, ∆t, in a 3 kG magnetic field. The red arrow indicates the amplitude

of the FR, as plotted in the FR spectra below. (b) Fourier transform of the data in (a).

The apparent rise at short delay is actually due to the second frequency component,

visible as the high frequency shoulder in the FT.

3.4 Spin dynamics in quantum shells

Two permanent magnets with adjustable separation were used to apply a

magnetic field, Bapp, to the sample perpendicular to the pump and probe

direction. Spins that were initially polarized along the pump beam pre-

cess around the magnetic field at the Larmor frequency, νL = gµBBapp/h

where g is the electron g-factor, µB the Bohr magneton, and h the Planck

constant. Figure 3.4 shows typical data from a sample with a quantum

well width of nCdSe = 3 monolayers and Bapp = 0.3 T. The inset shows

the Fourier transform (FT) power spectrum of the time-domain data. A

second precession frequency was observed, as indicated both by the small

53

shoulder in the FT spectrum and the beating in the time-resolved data.

While the origin of this second frequency is unclear in the present case,

similar behavior has been observed in CdSe nanocrystals [31, 32, 33, 34].

This second frequency may be associated with charging of defect states at

the surface of the nanocrystals. Dangling bonds at the nanocrystal surface

give rise to electron states within the bandgap. These states have been

observed to randomly charge and discharge leading to blinking of the PL

and shifts in the PL energy. Measurements of spin dynamics in electro-

chemically gated nanocrystal QDs have shown a dependence of the relative

prominence of the two precession frequencies on the QD charging [35].

There is also a non-oscillating component to the TRFR signal which

was also seen in previous measurements on CdSe nanocrystals [31]. In some

samples, particularly for nCdSe = 5, the magnitude of the non-oscillating

component is comparable to that of the oscillating component. However,

for the purposes of this work we focus only on the oscillating component

(indicated by the arrow in Fig. 3.4). The effective transverse spin lifetime,

T ∗2 , was of order 2 or 3 ns for all samples measured. Remarkably, the spin

lifetime was essentially temperature-independent between room tempera-

ture and 5 K, as shown in Fig. 3.5. (Low temperature measurements were

performed on QDQWs embedded in a polyvinyl butyral (PVB) matrix in

54

0 100 2000

1

2

Temperature (K)

Sp

in l

ifet

ime,

T2

*(n

s)

Figure 3.5: Temperature dependence of the quantum shell spin lifetime. Between 5

and 200 K, the measured spin lifetime, T ∗

2 is essentially constant around 1.5 ns. There

might even be a slight increase in spin lifetime at higher temperature. This data is

from QDQWs in a PVB matrix, but the measured lifetime is consistent with the room

temperature spin lifetime in solution.

a magneto-optical cryostat, as further described in Chapter 4.)

We have performed TRFR measurements as a function of Bapp on sam-

ples with CdSe quantum well width from nCdSe = 1 − 5 monolayers. In

all cases, the results show either one or two precession frequencies that

increase linearly with Bapp. The component with higher frequency always

had a substantially smaller amplitude and will not be addressed further

here. The top panel of Fig. 3.6 shows the FT of the spin precession for

55

shells with nCdSe = 1, 3, and 5 monolayers, with a clear shift in precession

frequency with shell thickness. The inset to Fig. 3.6 shows the linearly

increasing main precession frequency as a function of Bapp for nCdSe = 1, 3,

and 5 monolayers. The measured g-factor for each sample is shown in the

lower panel of Fig. 3.6 (circles) in comparison with the theoretical values

(crosses) obtained from an weighted average of the CdSe and CdS g-factors

(see below). Within the experimental error, the g-factor did not show any

dependence on temperature from 5 K to room temperature or on the probe

wavelength.

3.5 Energy levels in quantum shells

In order to investigate the QDQW energy levels, we have measured the

dependence of the TRFR amplitude on probe wavelength in the samples

with nCdSe = 3, 4, and 5. The probe beam coming out of the OPA typically

had a full width at half maximum (FWHM) of ∼ 10 nm. A monochromator

after the sample (as shown in Fig. 3.3) was used to select a 2 nm FWHM

slice of the probe beam. The measurement was performed by setting the

probe OPA near the desired wavelength, then setting the monochroma-

tor to the desired wavelength within the OPA spectrum. A TRFR scan

was performed at this probe wavelength, and then the monochromator

56

135 CdSe layers

Frequency (GHz)

FT

pow

er (

a.u.)

0 5 10

Bapp=3kG

0

0.6

1.2

1.8

1 2 3 4 5

g-f

acto

r

experiment

theory

CdSe monolayers

8

00 4Bapp (kG)

ν(G

Hz)

Figure 3.6: Precession frequency and g-factors as a function of shell thickness. Top:

Fourier transforms of the TRFR data for samples with shell thickness of 1, 3, and 5

CdSe monolayers, in a magnetic field of 3 kG. Bottom: quantum shell g-factor as a

function of shell thickness. Circles are the measured values, and crosses are calculated

as discussed below. The inset shows the precession frequency as a function of magnetic

field for 1, 3, and 5 CdSe monolayers.

wavelength was adjusted, and the next scan performed. Once the desired

wavelength was outside of the probe OPA spectrum, then the probe OPA

wavelength was adjusted. In this way, a large range of probe wavelength

57

Far

aday

rota

tio

n a

mp

litu

de

(a.u

.)

Op

tica

l ab

sorp

tio

n (

a.u

.)550 600 650 700Probe wavelength (nm)

5 monolayer

4 monolayer

3 monolayer

0

0

0

Figure 3.7: Quantum shell Faraday rotation spectra. Faraday rotation amplitude as

a function of probe wavelength for samples with 3, 4, and 5 CdSe monolayers. Also

shown is the optical absorption for each sample. The FR amplitude was defined as

the difference between the local maximum and minimum closest to ∆t = 500 ps in the

TRFR data.

was swept out.

Figure 3.7 shows the TRFR oscillation amplitude as a function of probe

wavelength for the different samples, together with optical absorption data.

The amplitude was determined by taking the difference between the max-

58

imum and minimum signal for scans like the one shown in Fig. 3.4. While

the absorption signal only shows a featureless staircase-like behavior with

no distinct resonances, the amplitude of the TRFR signal exhibits several

pronounced resonances close to the absorption edge. The results in Fig. 3.7

show that TRFR does not only provide information on the spin dynam-

ics, but also is a more sensitive spectroscopic technique than absorption

spectroscopy and allows one to identify individual exciton transitions in

QDQWs.

3.6 Theoretical description

These results were modeled theoretically in Meier et al. [36]. In that work,

the conduction and valence band level scheme of spherical QDQWs is cal-

culated with k · p theory [27, 29], using a two-band description for the

conduction band and the four-band Luttinger Hamiltonian in the spheri-

cal approximation for the valence band. The radial potential for electrons

and holes is determined by the offset of the CdS conduction and valence

band edge relative to CdSe, 0.32 eV and 0.42 eV, respectively [37]. The

electron and hole wavefunctions are then given by an envelope function,

which is a solution to the Schrodinger equation in the given potential,

multiplied by the underlying Bloch wavefunctions.

59

(a)(b)

Figure 3.8: Calculated electron and hole energy levels in quantum shells. Low-lying

electron (a) and hole (b) energy levels as a function of shell thickness. Energies are

given relative to the CdSe band-gap. The unlabeled hole states are 1D5/2, 1P5/2, and

1D7/2 from the bottom.

The calculated energies of the lowest conduction and valence band

states are shown in Fig. 3.8. Different valence band multiplets are de-

noted by LF [38, 39], where L is the smallest angular momentum of the

envelope wave function and F = |L±J | the total angular momentum. (The

valence band eigenstates are superpositions of envelope functions with an-

gular momentum L and L + 2 [36].) Figure 3.9 shows the radial wave

function envelope of the conduction band ground state 1Se (solid line) and

of 1S3/2 (dashed lines) for nCdSe = 3. Both wavefunctions show some degree

of localization within the spherical quantum well. Because of the larger va-

lence band mass, the valence band states are much better localized in the

quantum well. The valence band ground state, 1P3/2, has a p-type envelope

60

r1=1.7nm

r2-r1=(0.43nm)nCdSe

r3-r2=1.6nm

1Se

1S3/2

Figure 3.9: Calculated electron and hole wavefunctions in a quantum shell. The radial

component of the electron (red) and hole (black) wavefunctions in the 1Se and 1S3/2

states. The dashed line is the L = 0 component, and the dot-dashed line is the L = 2

component of the hole wavefunction.

wave function, which is consistent with a dark exciton ground state. This

characteristic of a p-type envelope wavefunction for the lowest energy hole

state is often found in nanocrystal QDs, and leads to very long radiative

lifetimes (> 10 ns at room temperature).

From the energy and wavefunction of the conduction band ground state

1Se, the electron g-factor is calculated as weighted average over the CdSe

and CdS g-factors, g = 2 − 2Ep∆so/3(Ep + ∆so + E1Se)(Ep + E1Se

) (see

Eq. 2.9). The numerical values obtained with standard parameters [40]

61

for the Kane interband energy Ep, energy gap Eg, and spin-orbit energy

∆so are shown in Fig. 3.6 (crosses). The quantitative discrepancy to the

experimental values is attributed to the fact that the two-band model un-

derestimates the conduction band energies of larger nanocrystals [31].

From the single-particle spectrum, we evaluate the amplitude of the

TRFR signal as a function of probe energy, θF (E), which is proportional to

the difference of the dynamic dielectric response functions for σ± circularly

polarized light, as in Eq. 2.19 [41, 42, 43]. The conduction band electron

with Sz = 1/2 created by the pump pulse relaxes rapidly to 1Se, such that

θF (E) is determined by optical transitions to the unoccupied 1Se state,

|1Se; ↓〉, [44]

θF (E) = CE∑

σ=±1;|Φv〉

σ |〈1Se; ↓ |px + σipy|Φv〉|2 (3.1)

× E −EX,v

(E −EX,v)2 + Γ2v

.

The sum extends over all valence band states |Φv〉, EX,v (Γv) denotes the

energy (linewidth) of the 1Se-Φv exciton transition, and C is a constant.

Equation 3.1 implies that only transitions to the conduction band ground

state contribute to θF (E). The transition matrix element is finite for S3/2

valence band multiplets, i.e., for 1S3/2 and 2S3/2 in Fig. 3.8 [45]. Because

the energy splitting between 1S3/2 and 2S3/2 is of order 0.15 eV, θF (E)

62

shows two well-defined resonances within 0.15 eV of the absorption edge. If

the crystal anisotropy is taken into account [39], both resonances split into

doublets, but the characteristic energy splitting is smaller than 25 meV.

θF (E) (displayed in Fig. 3.10 b for nCdSe = 3 and Γv = 10 meV) exhibits

only two distinct resonances, with a spectral weight that is larger for 1S3/2

than for 2S3/2 because of the larger overlap with the envelope wavefunction

of 1Se. The functional behavior shown in Fig. 3.10 b is in stark contrast to

the experimental data in Fig. 3.10 a, where at least three resonances with

comparable spectral weight can be identified close to the absorption edge.

Mixing of the 1S3/2 and 1P3/2 valence band multiplets is one possi-

ble mechanism which would explain both the additional resonances in

θF (E) and the large Stokes shift between the PL peak and the absorp-

tion edge [25]. Microscopically, mixing of 1S3/2 and 1P3/2 is effected by

broken spherical symmetry caused, e.g., by a fluctuation of the quantum

well width. For illustration, we consider a perturbation to the valence band

Hamiltonian,

δV (r) = v0 sin θ (1 + cos φ) , (3.2)

where (θ, φ) are the azimuthal and polar angle of r relative to the crys-

tal symmetry axis. For a monolayer variation of the quantum well, v0

is of order 0.1 − 0.2 eV for nCdSe = 3, 4. We diagonalize δV (r) in the

63

Figure 3.10: Calculated FR spectra of quantum shells. (a) Measured FR spectrum

for QDQWs with a 3-monolayer CdSe shell. (b) Calculated FR spectrum for a spherical

QDQW. (c) Calculated FR spectrum for a spherical QDQW with an asymmetrical

perturbation. (d) the same as c, but with a stronger perturbation. Also shown is the

measured and calculated optical absorption for each case.

subspace spanned by the 1S3/2 and 1P3/2 multiplets for v0 = 0.2 eV, ne-

glecting the perturbation δV (r) for higher valence band multiplets. All

resulting eigenstates are superpositions of 1S3/2 and 1P3/2 states and have

64

a finite transition matrix element to the conduction band ground state.

The spectral weight in Eq. (3.1) is re-distributed among all eight states

and several additional resonances emerge in θF (E) close to the absorption

edge. In Fig. 3.10 c and d, we show θF (E) calculated for QDQWs with

broken spherical symmetry for two strengths of the perturbation for ran-

domly oriented nanocrystals, v0 = 40 meV and 70 meV, with Γv = 12 meV.

Compared to spherical QDQWs (Fig. 3.10 b), the qualitative agreement

with experimental data improves when mixing of the 1S3/2 and 1P3/2 mul-

tiplets is taken into account. The anisotropy potential in Eq. (3.2) does

not provide a microscopic description for broken spherical symmetry, but

represents a simple model which allows for an analytical calculation of

θF (E). However, a perturbation of the form v0 sin θ cosφ ∝ (Y1,1 − Y1,−1)

is the lowest term in a multipole expansion for any realistic deformation

of the QDQW, such that Eq. (3.2) is expected to qualitatively capture

the essential features as long as the corresponding expansion coefficient is

finite.

These results have been further modeled using density functional the-

ory in Schreier et al.. There is some concern that the k · p description

above may be inaccurate, since it is assuming bulk crystal properties for

a structure just tens of atoms across. In this work, the crystal lattice of

65

the nanocrystal is simulated by building up the crystal atom by atom, ac-

cording to the wurtzite crystal structure. The structure is then allowed to

relax, minimizing the strain energy. This can lead to significant deviations

from the bulk crystal structure, since a relatively large proportion of atoms

in the nanocrystal are at the surface. Then, density functional theory is

used to calculate the electron and hole wavefunctions in this crystal struc-

ture. The results for the energy levels are somewhat different than the k ·p

calculation, though it is hard to say whether the DFT calculation provides

a better match to the experimental results. Regardless, it is clear that

modeling these nanocrystal QDs is not a simple task. Deviations in the

shape, large amounts of strain, and surface defects make this a complicated

system to treat theoretically.

In this chapter, we have studied the spin dynamics and quantum size

levels in QDQWs using TRFR. The variation of the energy levels and the

electron g-factor with quantum well width allows one to selectively address

quantum wells using optical techniques. This yields an additional knob to

turn in engineering nanocrystal QD structures, in that the energy and g-

factor can be controlled while maintaining the same overall nanocrystal

diameter. The observed Faraday rotation spectrum has a complex line-

shape, in contrast to the smooth absorption spectrum. This reflects the

66

spectrum of electron and hole energy levels in the nanocrystal, though the

complicated and irregular structure of these QDQWs make a quantitative

comparison to theory difficult. As we will see in the next chapter, this idea

can be taken a step further, and multiple quantum shells can be incorpo-

rated in each nanocrystal.

67

Chapter 4

Coupled Shells in Layered

Colloidal Nanocrystals

4.1 Motivation

In the field of quantum information science, semiconductor quantum dots

(QDs) are of significant interest for their ability to confine a single electron

for use as a qubit [46, 47]. However, to realize the potential offered by

quantum information processing, it is necessary to couple two or more

qubits. For nanocrystal QDs interconnected by conjugated molecules,

spin-conserving electron transfer between nanocrystals has been demon-

strated [48]. In contrast to coupling individual quantum dots, here we

demonstrate the integration of two coupled electronic states within a single

quantum dot heterostructure. These chemically-synthesized nanocrystals,

68

known as quantum dot quantum wells (QDQWs) [49, 21, 25, 50, 51], are

comprised of concentric layers of different semiconducting materials. In

the previous chapter, QDQWs were investigated with a low-bandgap shell

sandwiched between a high-bandgap core and outer shell. In contrast, now

we will look at structures with a low-bandgap core and outer shell, sepa-

rated by a high-bandgap barrier. We investigate carrier and spin dynamics

in these structures using transient absorption (TA) and time-resolved Fara-

day rotation (TRFR) measurements. By tuning the excitation and probe

energies, we find that we can selectively initialize and read out spins in

different coupled states within the QDQW. These results open a pathway

for engineering coupled qubits within a single nanostructure.

4.2 Sample structure and characterization

The samples studied in this chapter are ensembles consisting of nanocrys-

tals with a 5.5 nm diameter, low-bandgap (Eg = 1.74 eV) CdSe core, sur-

rounded by a 3 monolayer (ML), high-bandgap (Eg = 3.68 eV) ZnS barrier,

and a 4 ML outer CdSe shell [51]. The nanocrystals were prepared with

the same type of colloidal chemistry techniques as described in Chapter 3.

Transmission electron microscopy (TEM) images show that the nanopar-

ticles are fairly uniform in shape and size and are roughly spherical with

69

ZnS barrier CdSe core

CdSe

shell

V

r

e

h

Figure 4.1: Diagram of a coupled core-shell nanocrystal. In the core-shell nanocrystal

with a low band-gap core and outer shell, separated by a high band-gap barrier, electrons

and holes are confined to the core or the shell. The radial conduction and valence band

profile is shown, with schematic electron and hole wavefunctions.

some faceting at the surface [51]. A cut-away illustration of the sample

structure is shown in Fig. 4.1. Qualitatively similar results were also ob-

tained on a sample with 6.4 nm core, 2 ML barrier, and 4 ML shell. For

comparison, a control sample of 6.8 nm-diameter CdSe QDs (peak emission

at 5 K = 2.03 eV) was also prepared with nanocrystals purchased from Ev-

ident Technologies. For all of the samples, the nanocrystals were dispersed

in a solution of polyvinyl butyral (PVB) in dichloromethane. This mixture

was drop-cast into solid polymer films, following the method of Ref. [52].

The optical density of the films was around O.D. 1, giving them a slightly

translucent orange or brown appearance. The samples were placed in an

Oxford Spectromag magneto-optical cryostat. This allows measurements

at temperatures down to 4 K, and magnetic fields up to 6 T.

70

Energy (eV)

Ep

um

p =2.0

1eV

Ep

um

p =2.4

3eV

PL

Inte

nsi

ty (

a.u.)

Ab

sorp

tion (

O.D

.)

1.75 2.0 2.25 2.50.0

0.4

0.8T=295 K

Figure 4.2: Photoluminescence of coupled core-shell nanocrystals. With a high energy

pump, two PL peaks are seen – from the core and the shell. With lower pump energy,

only the core PL is present. The optical absorption is also shown. The QDQWs are in

toluene solution at room temperature.

Figure 4.1 shows the radial potential of the core-shell structure along

with the conduction- (c) and valence- (v) band wavefunctions schemati-

cally illustrated (discussed more quantitatively below). The band profile is

analogous to a pair of coupled quantum wells in which the core corresponds

to one well and the shell to the other. Indeed, under 2.43 eV excitation the

PL spectrum of these QDQWs (Fig. 4.2) shows two peaks at 2.18 eV and

1.92 eV, which have been previously attributed to radiative recombination

from an electron-hole pair in the shell and in the core, respectively [51].

When the excitation energy is tuned between the core and shell emission to

71

2.01 eV, only the lower energy (core) emission is observed. This behavior of

the PL indicates that two optically-active, metastable exciton states exist

in the QDQWs, and that by changing the pump energy either the core or

both the core and the shell can be selectively excited. We refer to these as

metastable states since they both exist for a time-scale of tens of nanosec-

onds (see below). This is typical for the lowest energy exciton state in

most nanocrystal QDs, but usually any higher energy exciton states relax

to the exciton ground state on picosecond time-scales. For some reason,

the higher energy shell state cannot rapidly relax to the core state. This

may have something to do with the spatial separation, but the specific

details of energy relaxation in these QDQWs is still an open question.

Further evidence that these two PL peaks come from states localized in

the core and the shell can be seen by varying the core and shell dimensions.

Figure 4.3 shows PL spectra from a series of samples with fixed (5.6 nm)

core diameter and varying shell thickness, and with fixed (3 monolayer)

shell thickness and varying core diameter. The dashed lines provide a

guide to the eye showing mainly a shift in the lower energy peak as the

core diameter is changed, and a shift in the higher energy peak as the shell

thickness is changed.

72

core dia. shell3.0 nm

3.5 nm

5.6 nm

6.4 nm

1 ML

2 ML

3 ML

4 ML

Figure 4.3: Coupled core-shell PL as a function of core and shell thickness. Left

column: PL from QDQWs with a 3 monolayer shell, and core diameter varied from

3.0 to 6.4 nm. Right column: PL from QDQWs with a 5.6-nm-diameter core, and

shell thickness varied from 1 to 4 monolayers. The dashed lines are guides to the eye.

Adapted from D. Battaglia et al., J. Am. Chem. Soc. 127, 10889 (2005).

4.3 Experimental setup

To perform time-resolved measurements of the spin and carrier dynamics

in these QDQWs (TRFR and TA, respectively), we use two optical para-

metric amplifiers (OPAs) seeded and pumped by an amplified Ti:Sapphire

laser, as in Chapter 3. This allows pump-probe spectroscopy using two

independently tunable, ∼ 200 fs duration pulses (∼ 30 meV linewidth).

73

Ti:Sapphire

∆t

Circ. Pol.

(PEM for FR)

lin.

pol

pump

probe

~150 fs

sample

to diode

bridge for

(KR)

(not to scale)

Regenerative

amplifier

~150 fs

OPA2

choppers

monochromator

OPA1

white light

optional

mirror

for TA

(for TA)

to spectrometer

and PMT for TAmagneto-optical

cryostat

5 K

B

Figure 4.4: Setup for TRFR and TA measurements on coupled core-shell nanocrystals.

This setup is similar to the one for single shell measurements. Here, there is additionally

a white light pulse that can be used instead of the probe for TA measurements. The

white light transmission is measured using a spectrometer and photomultiplier tube

(PMT). Also, the sample is now in a magneto-optical cryostat, allowing for measure-

ments down to 4 K and up to 6 T.

Additionally, a broad-spectrum white light pulse is produced via super-

continuum generation. The output of one OPA is focused on the sample

as a pump pulse (∼ 50µm diameter, 0.5-1.0 mW) , and the output of the

other OPA or the white light is overlapped with the pump to serve as a

probe pulse, which is delayed in time using a mechanical delay line. A

schematic of the setup is shown in Fig. 4.4.

74

For TA measurements, the white light is used as the probe. After the

probe passes through the sample it is dispersed in a 0.5 m spectrometer and

detected with a photomultiplier tube. By mechanically chopping both the

pump and probe beams at different frequencies f1 and f2, lock-in detection

at the sum frequency f1 + f2 can be used to isolate the TA signal.

To excite spin-polarized electrons into the QDQWs, the pump beam

polarization is modulated between right and left circular polarization at

a frequency fPEM = 42 kHz using a photoelastic modulator (PEM). As

the linearly polarized probe pulse is transmitted through the sample, its

polarization is rotated through an angle proportional to the spin polariza-

tion, due to the Faraday effect, as described in Section 2.4. The probe

beam is modulated by a mechanical chopper at a frequency, f2, of several

hundred Hz. The polarization rotation is measured by a balanced pho-

todiode bridge, and lock-in detection is used at both the PEM frequency

and the frequency of the mechanically chopped probe beam to isolate the

pump-induced spin polarization signal. Specifically, two lock-ins are used.

First, the diode bridge difference signal is sent to a lock-in amplifier with

reference frequency at fPEM, and a time constant of 640 µs. The “Fast X”

output of this lock-in is sent to a second lock-in, with reference frequency

f2.

75

4.4 Theoretical description

As in Chapter 3, we model the QDQWs using k ·p theory [36, 53, 38, 39].

While such a model is not rigorously applicable to structures with layers of

only a few monolayers, we find that it still provides a surprisingly accurate

description of the observed spectra. The c-band states are described in a

two-band approximation, and we assume for the v-band states a four-band

Luttinger Hamiltonian in the spherical approximation. For the c band we

use the effective masses mCdSe = 0.12m0 for CdSe, mZnS = 0.28m0 for

ZnS, and mvac = m0 for the surrounding material (which we assume to be

vacuum), where m0 is the bare electron mass. The Luttinger parameters

for the v band are given by γ1,CdSe, γCdSe = 1.67, 0.56 for CdSe and

γ1,ZnS, γZnS = 2.00, 0.75 for ZnS. For simplicity, we assume a spherical

shape of the coupled core-shell structure, with a CdSe core extending from

r = 0 to r = r1, a ZnS barrier extending from r = r1 to r = r2, and

a CdSe shell extending from r = r2 to r = r3 (see Fig. 4.1). Under the

assumption of spherical symmetry, the c-band and v-band states can be

classified by the quantum numbers nLe and nLF , respectively [38, 39, 36,

53]. Here, the total angular momentum F = |J ± L| is the sum of the

angular momentum of the Bloch function, J , and the angular momentum

of the envelope function, L. The radial quantum number is n. For both

76

the c band and the v band we assume a potential barrier of 0.9 eV for ZnS

with respect to CdSe. The barrier height to the surrounding material is

taken as 4 eV (infinite) for the c (v) band. We take the ML thickness to

be 0.31 nm for ZnS and 0.35 nm for CdSe (the interplane distances in the

[002] direction of the wurtzite crystal) [18]. We solve the radial Schrodinger

equation piecewise and take into account the boundary conditions at r = ri

as explained in Reference [36]. To calculate exciton energies we include the

Coulomb interaction to first order.

The calculated radial wavefunctions (rΨ) are shown in Fig. 4.5 for the

two c-band states, 1Se and 2Se (black) and the two v-band states, 1S3/2

and 2S3/2 (red) for r1 = 2.75 nm, r2 = 3.68 nm, and r3 = 5.08 nm. It is

clear from Fig. 4.5 that the electron and hole are localized mainly in the

core of the QDQW for the 1Se − 1S3/2 exciton, and mainly in the shell for

the 2Se − 2S3/2 exciton. This latter state is the energetically lowest state

with the electron and hole strongly localized in the shell. We therefore

assign these two states to the core and shell PL peaks.

The wave functions and energy spectra vary significantly when changing

the radii ri. We obtain fairly good agreement by comparing the calculated

energies of the lowest core and shell states to the PL data by Battaglia et

al. (Ref. [51], Fig. 4.3) for several sets of ri reported in their work. The

77

0 2 4 6-0.4

0

0.4

-0.4

0

0.4

r (nm)

rψ(n

m-2

)

core shell

2Se

2S3/2

1Se

1S3/2

Figure 4.5: Calculated core and shell wavefunctions. Top: the calculated 1Se electron

(black) and 1S3/2 hole (red) radial wavefunctions, localized in the core. Bottom: the

calculated 2Se electron (black) and 2S3/2 hole (red) radial wavefunctions, localized in

the shell. The two red curves are the L = 0 and L = 2 components of the wavefunction.

comparison of these energies is shown in Fig. 4.6. This indicates that the

k ·p model we apply here reasonably describes the size-dependent trends of

the experimentally observed low-lying exciton energies. Specifically, when

the core diameter is increased, the core state PL redshifts and the shell

state PL remains roughly constant. On the other hand, increasing the shell

thickness causes a redshift in the shell PL, leaving the core PL essentially

the same.

78

theory

experiment

core

shell

2 3 4 5

Shell thickness (ML) Core diam. (nm)

4 5 63

2.8

2.4

2.0

Ener

gy (

eV)

(a) (b)

5.6 nm core, 3 ML barrier 3 ML shell, 3 ML barrier

Figure 4.6: Measured and calculated energy levels. (a) Calculated and measured

core and shell energy levels for a 5.6-nm-diameter core as a function of shell thickness.

(b) Calculated and measured core and shell energy levels for a 3 monolayer shell as a

function of core diameter.

As in Chapter 3, in order to calculate the g-factor for the lowest c-band

core and shell states [54], we first determine the g-factor (as in Eq. 2.9) in

the two materials,

gCdSe,ZnS = 2 − 2

3

Ep∆so

(Eg + ∆so + E) (Eg + E)(4.1)

where E is the quantum confinement energy, and the band gap energy, spin-

orbit splitting, and Kane energy for bulk crystals are given by Eg,∆so, Ep =

1.75, 0.4, 20.3 eV for CdSe and Eg,∆so, Ep = 3.55, 0.09, 20.4 eV for

ZnS [40]. The g-factor is then approximated by g = (pcore + pshell)gCdSe +

pbarriergZnS + pvacgvac , where we have weighted the g-factors of CdSe, ZnS,

and the surrounding vacuum (gvac = 2) with the probability pα of the elec-

79

tron to be in the region α. Here we have neglected terms that arise due to

the interfaces [55]. For the 1Se state, which is mainly localized in the core,

we obtain gCdSe, gZnS = 0.966, 1.918 and pcore, pbarrier, pshell, pvac =

0.899, 0.076, 0.024, 0.000, yielding g = 1.04. In contrast, the param-

eters for the 2Se state, which is mainly localized in the shell, are ob-

tained as gCdSe, gZnS = 1.101, 1.922 and pcore, pbarrier, pshell, pvac =

0.043, 0.084, 0.851, 0.021, yielding g = 1.19.

4.5 Transient absorption and luminescence

measurements

We employ transient absorption (TA) measurements to probe the carrier

dynamics in these QDQWs. Here, a pump pulse excites carriers within the

QDQWs, and a spectrally broad probe pulse then measures the change in

the optical absorption spectrum induced by the pump. The left panel of

Fig. 4.7 shows the TA spectrum from the CdSe QD control sample at a fixed

pump-probe delay of 20 ps, with a pump energy of 2.30 eV (red) and 2.03

eV (black). Two effects contribute to the observed spectrum: strictly pos-

itive signal due to bleaching of interband transitions, and signal caused by

shifts in the spectrum due to multiparticle effects [56, 30]. (Here, positive

signal refers to reduced absorption.) The first effect is simply due to the

80

Pauli exclusion principle. If the pump excites an electron into a particular

conduction band state, then the probe cannot excite a second electron into

the same state, and the absorption is reduced. The second effect results

in both positive and negative contributions. This effect is due to a shift

induced in the energy level spectrum due to the pump pulse. For example,

if the pump pulse creates an exciton in a QD, the energy for the probe to

add a second exciton is not the same as to create the first one, because

of the exciton-exciton binding energy. Since we are measuring the change

in the absorption spectrum due to the pump, this pump-induced shift in

energy shows up as something like the first derivative of the absorption

spectrum.

The large peak at 2.05 eV is attributed to bleaching of the lowest opti-

cally active transition. It is apparent that the TA spectrum in the control

sample is largely independent of pump energy. The time-dependence of

the TA signal at 2.03 eV up to the maximum pump-probe separation of

2.5 ns is shown in the inset of Fig. 4.7, and is essentially the same in both

samples at all energies at the time-scales shown here. This indicates that

the lifetime of the electrons and holes is much longer than 2.5 ns, which is

usually the case in this type of QD, and which we reconfirm below.

The results of the same TA measurements performed on the QDQW

81

TA

(ar

b. unit

s)

1.75 2.25 2.75 1.75 2.25 2.75

Energy (eV)

Epump:

CdSe QDs (control) QDQWs

∆t=20 ps

T=5 K

0

TA

210

Delay (ns)

Figure 4.7: Transient absorption measurements on coupled core-shell nanocrystals.

Left: TA at a delay ∆t = 20 ps on the CdSe QD control sample, with pump energy

2.03 eV (black) and 2.30 eV (red). Right: The same measurement on the coupled core-

shell QDQWs. Inset: TA as a function of delay on the QD sample at a probe energy

2.03 eV. The black triangles indicate the calculated core and shell energies.

sample are shown in the right panel of Fig. 4.7. At the low pump en-

ergy (2.03 eV), where only the core state is excited, the TA spectrum is

very similar to the spectrum of the CdSe QDs. However, at higher pump

energy (2.30 eV) a second large positive peak appears due to bleaching

of transitions to the shell state. The black triangles in Fig. 4.7 indicate

the calculated energies of the lowest core and shell transitions, 1Se − 1S3/2

(2.010 eV) and 2Se−2S3/2 (2.254 eV), in reasonable agreement with the two

bleaching peaks in the TA spectrum. This provides further confirmation

that a second, higher-energy metastable (lifetime ≫ 3 ns) state exists in

82

0 50 100 150

Time (ns)

Norm

aliz

ed P

L (

a.u

.)

2.18 eV filter, τ = 37 ns

1.94 eV filter, τ = 20 ns

T = 295 K

Figure 4.8: Time-resolved PL from coupled core-shell QDQWs in toluene solution

taken using time-correlated photon counting. The black curve was taken with an inter-

ference filter centered at 2.18 eV (shell PL), and the red curve centered at 1.94 eV (core

PL). The band-pass FWHM of the filters was about 40 meV. The kinks at ±13 ns are

artifacts of the laser pulses.

the QDQWs. In fact, time-resolved PL measurements (shown in Fig. 4.8),

performed by time-correlated photon counting on QDQWs in toluene solu-

tion show both core and shell states to have a room-temperature radiative

lifetime of ∼ 20 ns, similar to previously measured radiative lifetimes in

CdSe QDs [57]. (The TRPL measurements were carried out on the same

sample as the PL in Fig. 4.2.) The reason why these PL measurements

were performed at room temperature and in solution is that the process

of making the PVB films substantially degrades the PL. It is likely that

additional defect states at the surface of the nanocrystals are introduced,

83

which can lead to bleaching of the PL, particularly the PL from the shell

state, as was observed. Nevertheless, even if the radiative recombination

is quenched, the TA measurements still confirm that the lifetime of both

states is much greater than 3 ns at low temperature as well.

4.6 Time-resolved Faraday rotation

spectroscopy

Using a circularly polarized pump pulse, we can excite spin-polarized car-

riers into these two metastable states, and then probe the resulting spin

dynamics through TRFR. A 2-T magnetic field is applied perpendicular to

the pump and probe direction (Voigt geometry). The left column of Fig. 4.9

shows characteristic spin precession in the QDQW sample measured by

TRFR at several pump and probe energies. The corresponding Fourier

transforms (FTs) are shown in the right column. (The non-precessing

component of the signal has been subtracted before calculating the FT.)

At this magnetic field, the spin lifetime is dominated by inhomogeneous

dephasing due to the size and shape distribution of the nanocrystals. It

has been previously observed for CdSe QDs [31], as well as QDQWs with

a single well [54], that the spin dynamics exhibit two distinct precession

frequencies. The origin of these two precession frequencies (or g-factors) is

84

FT

am

pli

tud

e (a

rb. unit

s)

0 50 100

Frequency (ns-1)

Epump Eprobe

2.07 eV 1.97 eV

2.07 eV 2.30 eV

1.97 eV

2.30 eV

2.43 eV

2.43 eV

Far

aday r

ota

tion (

arb

. un

its)

0

0

0

0

0 200 400

Time (ps)

(a)

(b)

(c)

(d)

B=2.0 T, T=5 K g=1.34g=1.48g=1.70

Data

Fit

Data

Fit

Figure 4.9: Time-resolved Faraday rotation on coupled core-shell nanocrystals. (a)

TRFR with low energy pump and low energy probe. (b) TRFR with low energy pump

and high energy probe. (c) TRFR with high energy pump and low energy probe. (d)

TRFR with high energy pump and high energy probe. Black is data, and red is a fit

(a-c are two frequency fits, d is a three frequency fit). The right column shows the

corresponding Fourier transforms (the non-precessing component is subtracted before

taking the FT). The three precession frequencies are indicated by the dotted lines.

85

still unclear, though it has been suggested that one precession frequency

may be due to lone electron spins and the other to exciton spins [31, 35].

These two behaviors might be due to different defect charge states (see the

discussion in Chapter 3). It has been shown for CdSe QDs that the relative

amplitude of the two precession components depends on the pump energy,

but is essentially independent of the probe energy [31]. In the core-shell

QDQWs, with the pump energy tuned to excite carriers only into the core

(Epump = 2.07 eV) as shown in Fig. 4.9 a and 4.9 b, two frequencies are

observed with only small changes in their relative amplitude as the probe

energy is varied. This is similar to the case in CdSe QDs. When pumping

at higher energy, however, where both the core and the shell are excited,

significant change in the ratio of the two frequency components is observed

with changing probe energy (Fig. 4.9 c and d). Furthermore, a third pre-

cession frequency appears at high probe energy. In Fig. 4.10, this is shown

in a close-up view of the FT in Fig. 4.9 d. All three of these frequencies

vary linearly with the applied field, and correspond to g-factors of 1.34,

1.48, and 1.70.

For a more detailed look at the probe energy dependence of the spin

dynamics, the probe beam was passed through a monochromator after the

sample to narrow the probe linewidth to ∼ 5 meV and TRFR measure-

86

30 40 50Frequency (GHz)

FT

am

pli

tud

e

Figure 4.10: Three precession frequencies in coupled-core shell nanocrystals. The FT

of spin precession in QDQWs with Epump = 2.43 eV and Eprobe = 2.30 eV. The red line

is a fit to the sum of three Lorentzians, shown individually in green.

ments were performed over a range of probe energies. The resulting curves

were normalized by the probe power and fit by

θF (t) = C + A0e−t/τ0 +

n∑

i=1

Aie−t/τi cos (ωit+ φi) , (4.2)

where n = 2 or 3 as needed to provide a good fit. The precession frequen-

cies, ωi, and the lifetimes, τi, were fixed for all the fits for a given probe

energy scan. That is, for one probe energy where the signal was good, all

the parameters were allowed to vary in the fit and then the resulting values

of τi and ωi were fixed for the rest of the fits in that probe energy scan. The

red curves in Fig. 4.9 show fits to the data. The origin of the non-oscillating

components C and A0 is unclear and will not be discussed further here.

The amplitudes Ai (i = 1−3) correspond to the different frequency compo-

87

nents seen in the FTs in Fig. 4.9. As was previously observed, in the CdSe

QD control sample the ratio of amplitudes of the frequency components

is largely independent of probe energy, though dependent on pump energy

(Figs. 4.11 a and 4.11 b). Furthermore, the probe energy spectrum has es-

sentially the same shape at both pump energies (though A1 is quite small

at the higher pump energy). This is expected since the carriers rapidly

relax to the lowest exciton state [56], regardless of pump energy.

As before, the QDQW sample shows similar behavior to the control

sample when only the core state is excited (Fig. 4.11 c). However, when

both the core and the shell are excited a third precession frequency ap-

pears only at high probe energy, as shown in Fig. 4.11 d. The amplitude

of Faraday rotation is a maximum for probe energies near interband tran-

sitions involving the occupied energy levels (see Eqs. 2.19, 3.1, and Ref.

[36]). This probe energy dependence implies that the spins precessing at

the third frequency are in a higher energy state than those precessing at

the other two frequencies. Since the position of the third frequency peak

corresponds to the higher energy peak in the TA spectrum in Fig. 4.7, it

is reasonable to assign this precession frequency to electrons in the shell

state.

88

ω1=118 GHz

ω2=150 GHz

ω1=99 GHz

ω2=142 GHz

ω1=101 GHz

ω2=141 GHz

ω1=119 GHz

ω2=151 GHz

ω3=131 GHz

1.9 2.1 2.3 1.9 2.1 2.3

0

0 0

0

Energy (eV) Energy (eV)

Far

aday r

ota

tion a

mpli

tude

(arb

. unit

s)

Epump

Epump

CdSe QDs

(control)

QDQWs

(a) (b)

(c) (d)

B = 1 T, T = 5 K

Figure 4.11: Faraday rotation spectra of coupled core-shell nanocrystals. Amplitude

of the two FR frequency components versus probe energy in the CdSe QD control

sample, with (a) low energy pump (Epump = 2.03 eV), and (b) high energy pump

(Epump = 2.30 eV). Amplitude of the FR frequency components versus probe energy

in the coupled core-shell QDQW sample, with (a) low energy pump (Epump = 2.03 eV,

two frequencies), and (b) high energy pump (Epump = 2.30 eV, three frequencies).

4.7 Analysis of spin dynamics and

core-shell coupling

As discussed above, based on our calculated wavefunctions we have esti-

mated the effective g-factor of the lowest core and shell c-band states by

89

g = (pcore + pshell)gCdSe + pbarriergZnS + pvacgvac, where we have weighted the

g-factors in the CdSe, ZnS and surrounding vacuum with the probability

pα of the electron to be in the region α. For the state 1Se, which is mainly

localized in the core, we obtained g = 1.04, and for the 2Se state, which

is mainly localized in the shell, we obtained g = 1.19. Comparing these

two calculated values to the observed g-factors, we see that the calculated

values are a bit smaller than any of the observed g-factors, but the differ-

ence of the two calculated g-factors is in good agreement with that of the

two lowest measured g-factors. This underestimation of the g-factors by

about 20% might not be too surprising since this k · p model used here is

not particularly rigorous. A more accurate theoretical description of these

structures could be gained through detailed first-principles calculations [58]

or by taking into account interface terms [55].

In order to obtain an estimate for the magnitude of the electron ex-

change interaction that may be expected in QDQW structures, we com-

pare our theoretical results for the conduction band states, 1Se and 2Se, of

the core-shell structure with a two-site Hubbard model to determine the

tunneling matrix element between the core and the shell.

For the conduction band, we consider the ground state Ψcore of a CdSe

core which is surrounded by a ZnS barrier and the ground state Ψshell of a

90

CdSe shell that is surrounded by vacuum and encloses a ZnS core. Using

the same ri as above, we obtain Ecore = 0.2222 eV and Eshell = 0.3655 eV

for the ground state energies of the decoupled core and shell and we define

∆ := Eshell − Ecore = 143.3 meV. In the Hubbard model, the orthogonal

states Ψcore and Ψshell are coupled by the tunneling matrix element tc. The

two lowest-lying states and energies are in lowest order in the tunneling

Ψ− ≈ N [Ψcore − (tc/∆)Ψshell] , E− ≈ Ecore − t2c/∆,

Ψ+ ≈ N [Ψshell + (tc/∆)Ψcore] , E+ ≈ Eshell + t2c/∆,

(4.3)

where N = ∆/√

∆2 + t2c . In the above states, the tunneling probabil-

ity between the core and the shell is given by pt = (tc/∆)2. Given a

calculated wavefunction in the coupled structure, we can alternatively de-

termine pt from the relative weight of the wavefunction in the core and

in the shell. From the states 1Se and 2Se discussed above we extract an

average tunneling probability pt = 0.046, and thus derive |tc| = 30.7 meV,

E− ≈ 0.2157 eV, and E+ ≈ 0.3721 eV. These energies agree well with

E1Se= 0.2188 eV and E2Se

= 0.3717 eV obtained from the coupled core-

shell model. The corresponding wavefunctions are shown in Fig. 4.12.

There is good agreement between calculations of the wavefunctions and

energies when the whole core-shell potential is used at once, and when the

core and shell are calculated separately and then coupled using this Hub-

91

0 2 4 6 0 2 4 6

0.8

0.4

0

rψ(r

)

r (nm) r (nm)

1Se

2Se

Ψ-

Ψ+

r1 r2 r3

(a) (b)

Figure 4.12: Coupled wavefunctions determined using the Hubbard model. Core (a)

and shell (b) electron wavefunctions calculated in the coupled potential (black) and the

and in the separate core and shell potentials and then coupled in the Hubbard model

(red).

bard model. This does not address the overall validity of the k · p model,

since it used in both calculations. However, the self-consistency does lend

credence to the use of the Hubbard model here.

The exchange interaction of an electron located in the core and an

electron located in the shell can be estimated in the Hubbard model as

follows. We first estimate the direct Coulomb interaction of two electrons

in the present structure, assuming a uniform dielectric constant ǫ = 9. We

obtain the value Ucore ≈ 78 meV when both electrons are in the core ground

state Ψcore. For the cases when both electrons are in the shell ground state,

Ψshell, or one is in the state Ψcore and the other one in Ψshell, we obtain

the direct Coulomb interaction energies Ushell ≈ 38 meV and Ucore,shell ≈

92

35 meV, respectively. Because the states Ψ+ and Ψ− are strongly localized

in the present structure, these Coulomb energies are a good approximation

for the corresponding interaction energies. Hence, the net on-site Coulomb

repulsions are much smaller than ∆ for the structure under study, and

the exchange interaction can be approximated by t2c/∆ ∼ 1 meV. For

comparison, we obtain a value on the same order of magnitude, 1.8 meV,

for the Coulomb exchange integral for the case when the two electrons

occupy the states 1Se and 2Se.

On the basis of these calculations, we see that QDQWs with small ∆

might provide an interesting basis to study interacting spins in coupled

nanostructures. ∆ can be tuned all the way through zero by changing

the diameter of the core and the thickness of the shell [51]. An attempt

was made to see effects of two-electron interactions in the present samples.

To do this, a second pump pulse was added to the experiment. The idea

was that the first pump would excite an electron into the core, and the

second pump would excite another electron into the core or the shell. If

one electron ended up in the core, and one in the shell, their spins would

evolve under the mutual exchange interaction which would be visible in the

TRFR measurement. Both pumps were modulated at different frequencies

using two PEMs, and the first lock-in was set to the difference or sum

93

frequency. However, it turned out that there are a number of unwanted

background signals that obscured the desired effects, if they were present.

To date, spin-spin coupling in these structures has not been observed.

In conclusion, we have fabricated and measured spin dynamics in cou-

pled core-shell nanocrystal QDQWs. We have observed evidence in the

time-resolved PL, and transient absorption measurements that these struc-

tures contain two metastable excited states. One state is localized mainly

in the core, and the other in the shell. Calculations confirm that these two

coupled states exist in the nanocrystal. The PL, TA, and TRFR measure-

ments all indicate that by varying the energy of the pump, spins can be

selectively excited into the core state alone, or into both the core and the

shell. Furthermore, as we saw in Chapter 3, the electron g-factor and the

energy dependence of the Faraday rotation depend on the confining poten-

tial in the nanocrystal. Therefore, using time-resolved Faraday rotation

spectroscopy, we can selectively read out the spin in either the core or the

shell by the differing precession frequencies and spectral dependences.

4.8 Nanocrystal QDs in an optical cavity

The experiments discussed above have all been performed on large en-

sembles (> 108) of QDs. This ensemble averaging prohibits a number of

94

interesting experiments, such as observing quantum jumps between spin

states, entangling two spins, or even measuring the transverse spin lifetime

without inhomogeneous dephasing. Additionally, virtually all proposals

for spin-based quantum information processing require the ability to work

with single electron spins.

Of course, going from a measurement of > 108 spins down to one spin

will involve a large decrease in signal. In order to address this issue, we

have explored the fabrication of integrated optical cavities to enhance the

Faraday (or Kerr) rotation measurement of spins in quantum dots [59].

A Fabry-Perot optical cavity consists of two parallel mirrors. Light

in the cavity reflects back and forth between the mirrors. The cavity

supports optical modes at wavelengths such that the round-trip path length

between the mirrors is a whole number of wavelengths. When this condition

is satisfied, then subsequent reflections interfere constructively and the

intensity of the light builds inside the cavity. This enhancement of the

light intensity has been used before to enhance the Faraday effect in other

systems [60, 61, 62, 63]. In this work, alternating dielectric layers of TiO2

and SiO2 were deposited on a glass substrate which form a distributed

Bragg reflector (DBR). Due to constructive and destructive interference,

a DBR reflects light of wavelength λ, when alternating materials with

95

different indices of refraction are layered with layer thickness equal to λ/4

(for the λ in the dielectric material). 11 layers of these materials were

deposited on the substrate using e-beam evaporation, and were capped by

a final SiO2 layer to serve as the bottom mirror of the cavity. This surface

was chemically functionalized to bind CdSe nanocrystal QDs. The sample

was then immersed in a toluene solution containing 6.6-nm-diameter CdSe

QDs [64]. The chemical functionalization and QD treatment was repeated

until the desired optical density of the QD film was reached. Then a

wedge shaped SiO2 layer is deposited on top of the QDs, to complete the

cavity. The wedge shape causes the thickness of the cavity, and therefore,

the position of the cavity resonance, to vary across the sample. Then, a

second, identical DBR was deposited on top. For comparison, half of the

sample was masked during the deposition of the top and bottom DBR

layers (though not for the SiO2 layers making up the cavity itself). Thus

half of the sample consists of CdSe QDs embedded just in SiO2, and the

other half is the same, but also embedded between two DBRs. Figure 4.13

shows a schematic of the sample, and also a cross-sectional transmission

electron micrograph (TEM) of the structure.

Two samples were fabricated, with different densities of QDs – Sample

A with higher density, and Sample B with lower density. Both samples

96

TiO2SiO2

bottom DBR

top DBR

QDs

control region

cavity

(a)

1 µm20 nm

QDs

bottom DBR

top DBR

cavity

Figure 4.13: Diagram of nanocrystals embedded in an optical cavity. Top: Schematic

of SiO2/TiO2 distributed Bragg reflectors (DBRs) forming an optical cavity around a

layer of CdSe nanocrystal QDs. Top layers are transparent to show the interior. Bottom:

cross-section electron micrograph of the sample structure, showing the alternating layers

of the DBRs. A close-up of the center of the cavity shows individual QDs.

show the same absorption spectrum, with the amplitude scaled by the QD

density. As seen by the sharp peaks in the transmission spectra shown

97

% T

ransm

issi

on 80

100

60

40

20

0

Sample A

550 600 650 700 750 800

% T

ransm

issi

on 80

60

40

20

0

550 600 650 700 750 800 λ (nm) λ (nm)

Sample B

10

20

0

30

30

0

60

630 640620 630 640

Q = 110 Q = 165100

Figure 4.14: Optical transmission spectra of the cavity/QD samples at different po-

sitions on the sample. The peak around 600-650 nm is due to the cavity resonance,

which tracks the cavity thickness. The insets show one of the resonance peaks from

each sample, with a fit that is used to determine the Q-factor.

in Fig. 4.14, the resonance of the cavity can be tuned from about 600 to

640 nm in both samples by translating across the sample. The quality

factor (Q-factor) of the cavities, which is related to the number of reflec-

tions made by the light within the cavity, was found to vary from about

80 to 180 at different positions on the sample. (The sample with lower QD

density has higher a Q-factor, since less light is lost from QD absorption.)

By comparing the TRFR signal at the peak of the cavity resonance, with

the same signal measured without the cavity, we can see that the cavity

significantly enhances the FR signal. Figure 4.15 a and b shows TRFR

traces from the two samples at room temperature, with a transverse mag-

98

FR

(m

rad.)

0 200 400 600Delay Time (ps)

Delay Time (ps)0 200 400 600 100 150 200

Q-Factor

1.5

0

1.0

0.5

0.3

-0.3

0

0.05

-0.05

0

10

15

30

25

20

Sample B

Sample A

B = 0.47 T T = 298 K λ probe = 630 nm

Control B

Control A

(a)

(b) (c)

FR

(m

rad.)

Enhan

cem

ent

Figure 4.15: Cavity-enhanced time-resolved Faraday rotation. (a) Time-resolved FR

on both samples shown on the cavity resonance (black) and on the control region with

no cavity (green and blue). (b) Zoom-in of the green and blue data in (a). (c) The

enhancement of the FR signal due to the cavity as a function of Q-factor.

netic field of 0.47 T, with and without the cavity. This enhancement is

expected, since the polarization of the light is rotated further with every

pass through the QD layer.

The enhancement factor caused by the cavity is plotted in Fig. 4.15 c as

a function of the cavity Q-factor. As expected, the enhancement is roughly

linear, with a maximum enhancement of a factor of about 28 for a Q-factor

of 180.

99

From an estimate of the QD density and the probe laser focused spot

size, we estimate that in the lower density sample, we are probing about

9×108 QDs. Given the signal-to-noise ratio, and if we were to use a smaller

spot size, we estimate a detection limit of 3×104 QDs for this sample. This

is still a ways away from a single QD, but as we will see in the next chapter,

this problem can be overcome.

100

Chapter 5

Non-destructive Measurement

of a Single Electron Spin

5.1 Motivation and Background

The prospect of quantum computation in conventional material systems

has spurred much research into the physics of carrier spins in semicon-

ductor quantum dots (QDs) [47]. An important element necessary for

spin-based quantum computing is the read-out of the qubit spin state.

Previously demonstrated schemes for single spin read-out in a quantum

dot (examples illustrated in Fig. 5.1) include optical measurements, such

as photoluminescence (PL) polarization [65, 66] or polarization-dependent

absorption [67, 68, 69]. For example, in reference [65], spins are optically

excited into GaAs interface QDs (discussed below), using above-bandgap

101

circularly

polarized PL

spin-dependent

absorption

lead

lead

dot

electrical readout

(a) (b) (c)

Figure 5.1: Previously demonstrated schemes for single spin detection. (a) Exciton

spin readout via the circular polarization of the PL. (b) Single electron spin detection us-

ing polarization-dependent absorption. (c) Electrical readout based on spin-dependent

tunneling probability out of the QD.

circularly polarized light. The PL from a single QD is collected with a

microscope objective. Due to the selection rules for interband optical tran-

sitions, the circular polarization of the resulting luminescence yields infor-

mation about the spin of the electron and/or hole in the QD. Alternatively,

in reference [69] for example, a narrow-linewidth laser is used to probe the

optical absorption of a single QD. Due to the Pauli exclusion principle, if

an electron is in the quantum dot, the optical transition to that state is

blocked, resulting in a polarization-dependent change in the optical absorp-

tion. If the light is focused tightly on the QD, this single QD absorption

can be surprisingly large.

Single spins can also be read out electrically by measuring the spin-

dependent probability for an electron to tunnel out of the dot [70]. In

102

these experiments, QDs are formed using electrical gates to deplete regions

of a two-dimensional electron gas. The QDs formed within these depleted

regions are easy to contact electrically by leaving a small gap in the depleted

area. Additional gates can be used to shift the energy levels in the QD, and

to control the tunnel coupling to the electrical leads. With a sufficiently

large magnetic field and low temperature, the spin states of the QD energy

levels can be split with an energy greater than kBT . In this case the QD

energy levels can be shifted so that one spin level is above the Fermi energy

in one of the leads, and the other spin level below the Fermi energy. In this

situation, the higher energy spin state can tunnel out of the QD, resulting

in a measurable current. This current constitutes readout of the electron

spin.

However, these previous single spin readout methods are destructive,

in that they either remove the spin from the dot, or drive transitions in the

system with a resonant optical field. In other words, these measurements

take the system out of the 2-state, spin-up/spin-down Hilbert space. To

avoid confusion, it is worth defining the term ‘destructive’. This term is

used (for example, in Ref. [71] and [72]) as a necessary, but not sufficient

condition for a quantum nondemolition (QND) measurement. A QND

measurement must satisfy two conditions: 1. The measurement must not

103

take the state out of the Hilbert space of the system being measured (non-

destructive measurement), and 2. After the measurement, the system must

be in an eigenstate with probability given by the quantum state before

the measurement (projective measurement). Another way of stating this

second condition is that the Hamiltonian of the system must commute with

the Hamiltonian of the measurement, or that “back-action” of the probe

on the system must be avoided. There has been one previous claim of

non-destructive measurement in the singlet-triplet basis in an electrically-

gated double QD system [73]. However, in this work, one of the electrons

is removed from the QD and then later replaced with an electron in the

same spin state. This seems to satisfy the second condition, not the first,

making theirs a unique definition of the term ‘non-destructive’.

In contrast, we describe measurements of a single electron spin using

Kerr rotation (KR) in which the spin state is probed non-resonantly, thus

minimally disturbing the system. This effective spin-photon interaction

has been used with great success to measure spins non-destructively (and

in fact to perform QND measurements) on ensembles of atoms in a gas.

Specifically, Faraday rotation of a probe laser has been used to observe

quantum effects such as measurement-induced decoherence, spin squeez-

ing [74, 75], and teleportation of quantum information [76]. Also, a number

104

of proposed quantum information protocols use Faraday or Kerr rotation

of single spins to generate spin-photon entanglement [77] and optically-

mediated spin-spin entanglement [78, 79].

In the last section, we saw that we were unable to measure less than

about 3× 104 QDs using Faraday rotation. So is it reasonable to hope for

single QD sensitivity? First of all, we should more carefully choose the

type of QD we are looking at. As discussed in Section 2.4, the magni-

tude of Faraday rotation depends on the momentum matrix elements con-

necting the initial and final (valence and conduction band) states. Using

the commutation relation for the position operator and the Hamiltonian,

[r, H ] ∝ p, these matrix elements can be expressed in terms of the position

operator – that is the dipole matrix elements:

〈c|p|v〉 ∝ 〈c|rH −H r|v〉 = (Ev − Ec)〈c|r|v〉. (5.1)

From these matrix elements, it can be seen that the Faraday or Kerr effect

will be larger for QDs with a larger spatial extent of the electron and hole

wavefunctions. The nanocrystal QDs in Chapters 3 and 4 are about 5 nm

in all directions. In contrast, electrons in interface fluctuation quantum

dots are confined to a region on the order of 100 × 100 × 5 nm. As a

result, transitions in nanocrystal QDs typically have dipole matrix elements

with modulus squared of several Debye [80], whereas interface QDs have

105

1 QD

(10 QDs)/10

(104 QDs)/104

Far

aday r

ota

tion (

norm

aliz

ed)

Pro

bab

ilit

y d

istr

ibuti

on

Energy (a.u.)

Figure 5.2: Single versus ensemble Kerr rotation spectrum. The calculated KR spectra

are normalized by the number of QDs in each case. The black curve shows a single odd-

Lorentzian KR spectrum, as from a single QD. The blue curve shows the sum of 10 such

KR spectra with energies distributed randomly according to the distribution shown in

gray. The red curve shows the same for 104 QDs. The normalization by the number of

QDs highlights the fact that the signal is not N times larger for N times as many QDs.

dipole strengths of 50-100 Debye [81]. Therefore, by using these interface

fluctuation QDs, we gain a significant factor in the signal.

It is also important to realize that an ensemble of N QDs will not yield

N times the Faraday rotation signal of one QD. Each QD containing a

spin contributes an odd-Lorentzian feature to the total Faraday rotation

signal centered at the optical transition energy and with the linewidth

of the transition, Γ. Since an ensemble of QDs will be spread out over

a region of energy ∆E, the maximum signal from an ensemble of QDs

106

will be reduced by a factor of roughly ∆E/Γ. Figure 5.2 illustrates this

point. (Note that in the figure, the curves are normalized by the number

of QDs. The actual, non-normalized signal does increase as the number of

QDs increases, but sub-linearly). This leads to another gain in the single

QD signal over the ensemble measurement. Finally, if we can improve the

measurement further, for example, by reducing the noise in the lasers or in

the detection electronics, we may have a hope of measuring a single spin

in a single QD.

5.2 Sample structure

A schematic of the final sample structure is shown in Fig. 5.3, and details

of the sample fabrication are given in Appendix A. The sample is grown

by molecular beam epitaxy and consists of a single 4.2-nm GaAs QW in

the center of a planar Al0.3Ga0.7As λ-cavity. A 2-min. growth interruption

at each QW interface allows large (∼ 100 nm diameter [81]) monolayer

thickness fluctuations to develop that act as QDs [82, 83]. An STM image

of this type of fluctuation, along with a cartoon of interface QDs are shown

in Fig. 5.4.

The QD layer is centered within an optical microcavity with a reso-

nance chosen to enhance the interaction of the optical field with the QD.

107

…x28

i

np

i

Ohmic

contact

80nm Ti, 1µm

aperture

QDs

Al0.3Ga0.7As AlAs GaAs

doping

top DBR

bottom DBR

focused

lasers

Figure 5.3: Schematic of the sample for single spin measurements. The sample consists

of a layer of GaAs interface fluctuation QDs surrounded by distributed Bragg reflectors

(DBRs) forming an optical cavity. The structure is doped and gated so as to control

the charging of the QDs and the QW.

See Section 4.8 for more discussion of cavity-enhanced Faraday rotation.

The front and back cavity mirrors are distributed Bragg reflectors (DBRs)

composed of five and 28 pairs of AlAs/Al0.3Ga0.7As λ/4 layers, respectively.

This asymmetrical design allows light to be injected into and emitted from

the cavity on the same side. The cavity was designed using a vertical

cavity simulator (called “Vertical”) to have a resonance aligned the QD

108

AlGaAs

AlGaAs

QDs

GaAs

(a)(b)

Figure 5.4: (a) STM image of interface fluctuations at a GaAs surface. Steps in color

indicate atomic steps in thickness. From D. Gammon et al., Phys. Rev. Lett. 76, 3005

(1996). (b) QDs form at the potential minima caused by localized islands of increased

QW thickness.

emission. However, due to some error in the growth, the cavity resonance

was somewhat lower in energy, though still with some overlap with the

QD emission. The reflectivity of the sample at 10 K (Fig. 5.5 c) shows a

cavity resonance centered at 763.6 nm (1.624 eV) with a Q-factor of 120.

The Q of the cavity could be increased by adding more layers to the front

DBR, but increasing the distance from the QDs to the surface of the sam-

ple makes isolation of single QDs more difficult, and also complicates the

sample gating (discussed below and in Appendix A). Based on previous

measurements with similar cavities [59, 62], we expect the KR at the peak

of the cavity resonance to be enhanced by a factor of ∼ 15.

109

n(1

01

6cm

-3)

0

5

10

Vb (V)0 1

(b)

En

erg

y (

eV)

0

-1

-2

1

z (µm)0 0.5 1

Vb= 1 V

Vb= -1 V

(a)

QW

750 775

R

λ (nm)

1

0.8

(c)

Top DBR

Figure 5.5: Characterization of the single spin sample. (a) Calculation of the conduc-

tion (solid) and valence (dashed) band profiles as a function of distance from the sample

surface (z = 0). As the bias voltage is changed from 1 V to -1 V (red and black lines),

the QW is lowered beneath the Fermi energy. (b) Calculated electron density in the

QW. The onset of charging occurs around Vb = 0.5 V. (c) Reflectivity of the sample at

T = 10 K showing a dip caused by the cavity resonance. The width of this dip indicates

a Q-factor of 120.

Additionally, the QDs are embedded in a diode-like structure, allowing

the charging of the QDs and the QW to be controlled with a bias voltage.

The back DBR is Si doped at n = 3 × 1017 cm−3 until a depth of 85 nm

110

beneath the QW, which is followed by a 25 nm p-type region (p = 1.5 ×

1017 cm−3), with the rest of the structure undoped. A portion of the sample

is etched down to the n-doped layer and an Ohmic Ni/GeAu contact is

deposited. On the unetched region, an 80-nm Ti layer forms a Schottky

contact with 1-µm apertures fabricated by electron-beam lithography. This

layer serves as both a front gate and a shadow mask for isolating single

QDs.

The band profile for our structure, calculated with a 1-D self-consistent

Poisson-Schrodinger solver, is shown in Fig. 5.5 a. By applying a bias

across the structure, the conduction band minimum in the QW can be

made to plunge beneath the Fermi level, charging first the QDs, then the

well itself [84, 85]. This structure is similar to that in Ref. [84] or [85],

with the addition of the thin p-doped layer. The purpose of this layer is

to provide an additional boost to the conduction band energy so that the

QW can be raised above the Fermi level at reasonable bias voltages. The

onset of this charging occurs around 0.5 V (Fig. 5.5 b) according to the

band-structure calculation.

111

5.3 Theory

As discussed several times above, the magneto-optical Kerr effect results

in a rotation of the plane of polarization of linearly polarized light with

energy E upon reflection off the sample, and is analogous to the Faraday

effect for transmitted light. For both effects, the rotation angle is deter-

mined by the difference of the dynamic dielectric response functions for σ+

and σ− circularly polarized light, which are proportional to the interband

momentum matrix elements, P±c,v = 〈ψc|px ± ipy|ψv〉 , where ψc (ψv) is a

conduction (valence) band state [86, 36]. Due to the cavity, both reflection

and transmission contribute to the measured polarization rotation. For

simplicity, we refer only to KR. For a single conduction-band energy level

in a QD containing a spin-up electron in a state ψ↑, optical transitions to

the spin-up state are Pauli-blocked, and the KR angle is then given by

θK(E) = CE∑

v

(|P+

↓,v|2 − |P−↓,v|2

) E − E0,v

(E −E0,v)2 + Γ2v

, (5.2)

where C is a constant, and E0,v and Γv are the energy and linewidth of

the transition involving the valence band state |ψv〉, respectively. We focus

on a single transition in the sum in Eq. 5.2 and drop the index v. For

Γ ≪ |∆| ≪ E, where ∆ = E − E0, we note that θK ∼ ∆−1, which

decays slower than the absorption line, (∼ ∆−2) [36, 81]. Therefore, for

112

a suitable detuning, ∆, KR can be detected while photon absorption is

strongly suppressed. This gives the KR measurement its non-destructive

property.

5.4 Experimental setup

A diagram of the experimental setup is shown in Fig. 5.6. Further details

are given in Appendix B. A cw Ti:Sapphire laser (Ti:Sa1 in the diagram)

to be used as the pump for KR or PL measurements is focused through

a mechanical chopper with frequency f1 = 4.1 kHz. The intensity of the

laser is attenuated with a neutral density (N.D.) filter. This beam is passed

through a linear polarizer and a liquid crystal variable waveplate (LCVW).

For most measurements, the LCVW is set to ±quarter-wave to convert the

light to σ+ or σ− circular polarization. A second cw Ti:Sapphire laser

(Ti:Sa2 in the diagram) used as the probe for KR measurements is focused

through a mechanical chopper with frequency f2 = 21 Hz, and is also atten-

uated with an N.D. filter. This beam is passed through a linear polarizer,

and then made collinear with the pump beam using a beamsplitter. Both

beams reflect off of a beamsplitter into a microscope objective. The ob-

jective is mounted on a motorized positioning stage for coarse positioning

in the x- and y-directions, and a piezoelectric stage for fine positioning

113

Ti:Sa 1

(cw pump)

Ti:Sa 2

(cw probe)

choppers

LCVW

polarizer

polarizer

scanning

objective

sample

He flow cryostat (10 K)

electro-

magnet

Fe yoke

longpass

filter

to spectrometer

(for PL)

to diode bridge

(for KR)

optional

mirrorf1

f2

VA-Block-in1

fref=f2

lock-in2

fref=f1

pre-amp

VAlock-in3

fref=f1

for

normalization

signal

Figure 5.6: Setup for single spin detection. Two continuous wave (cw) Ti:Sapphire

lasers provide the pump and probe for KR measurements. The lasers are focused onto

the sample, which sits in a cryostat at 10 K. An electromagnet provides a transverse

magnetic field. The outgoing light is passed though a longpass filter to remove the pump

light, and is sent to a spectrometer for PL measurements, or to the diode bridge for KR

measurements.

in all three dimensions (scanning in x and y, plus focus). The objective

focuses both beams onto the sample (spot size ∼ 1 µm), which sits inside

a liquid Helium flow cryostat. The cryostat itself is on a motorized stage

114

for coarse focus control. Additionally, the iron poles of an electromagnet

are positioned above and below the sample cold finger, to apply magnetic

fields up to ∼ 0.1 T.

For PL measurements, the probe beam is blocked, and only the pump is

focused onto the sample at T = 10 K to excite electron-hole pairs into the

continuum of states in the QW (Epump = 1.654 − 1.662 eV). The carriers

then relax into the QDs, and the subsequent PL is collected through the

same objective, and sent back through the beamsplitter. The outgoing

light from the beamsplitter is sent through a long-pass optical filter to

block the pump light. The PL is passed through the filter, and focused

through the entrance slit of a 1-m spectrometer. The PL spectrum is then

detected by a liquid-nitrogen-cooled CCD.

For Kerr rotation measurements, the pump and probe are focused on

the sample through the microscope objective onto one of the apertures.

The probe energy can be swept by means of a motor on the birefringent

filter in the laser cavity. The reflected light is collected through the objec-

tive and passed through a long-pass optical filter to block the pump beam.

The rotation of the probe polarization is then detected by a balanced pho-

todiode bridge, as described in Section 2.5. The difference channel of the

diode bridge is sent to a preamplifier with 100× gain, and a bandpass filter

115

from 3 kHz to 10 kHz. (Since the pump is modulated by the chopper at

4.1 kHz and the probe beam is modulated at 21 Hz, the signal due to spins

injected by the pump, and measured by the probe will be at 4100±20 Hz.)

After the preamplifier, the signal is sent to the first lock-in amplifier with

reference frequency f1 and time constant 640 µs. The output of this lock-in

is sent to a second lock-in amplifier with reference frequency f2 and a time

constant of one or two seconds. During this measurement time of several

seconds, the pump is repeatedly reinitializing the spin. In this sense, it is a

measurement of a single spin in a QD repeated many times, and averaged

in time. Finally, the pump polarization is switched between σ+ and σ−

with the liquid crystal variable waveplate, and a measurement of the rota-

tion angle is taken at each helicity. The difference between these two values

yields the signal modulated at both the pump and probe frequencies, and

that depends on the sign of the pump helicity. Sweeping the probe energy,

with a measurement at each energy maps out the KR spectrum. Typical

pump and probe intensities are 15 W/cm2 and 500 W/cm2, respectively.

116

5.5 Characterization: PL and Hanle

measurements

We begin by finding a good looking QD. Most of the apertures on the

sample show multiple QDs, but we look for one that has a strongly emitting

QD sufficiently separated in energy from the other QDs that it can be

individually addressed. The rest of the measurements below will be on the

same QD, unless otherwise noted. We next characterize this QD using PL

measurements, and verify that we can inject a single polarized spin into

the QD by reproducing the results of Ref. [65], in which the spin of a single

electron is measured using PL polarization.

In a typical single dot PL spectrum as a function of the applied bias

(Fig. 5.7), the sharp features (linewidth ∼ 100 µeV) are characteristic of

single-dot PL [83], demonstrating the presence of only one QD within the

laser focus and within the measured energy window. Above 0.5 V a single

line is observed at 1.6297 eV which is attributed to recombination from

the negatively-charged exciton (trion, or X−) state. Below 0.5 V this line

persists faintly, and a bright line appears 3.6 meV higher in energy due

to the neutral exciton (X0) transition. The presence of the X− line at

Vb < 0.5 V implies that occasionally a single electron is trapped in the

117

X-

0 0.5 1 1.5-0.5-1-1.5-0.2

0.2

0

1.625

1.630

1.635

PL

en

erg

y (

eV)

PL

pol.

chargeduncharged

X0 X-

XX

X-

X0

(a)

(b)

Vb (V)

3.6 meV

Figure 5.7: Photoluminescence of a single QD vs. bias voltage, and polarization

thereof. (a) Single QD PL vs. bias voltage. The neutral exciton (X0), charged exciton

(X−) and biexciton (XX) lines are identified. (b) The degree of circular polarization

of the X0 (black) and X− (red) PL lines as a function of bias voltage. The biexciton

PL is unpolarized.

dot, forming an X− when binding to an electron and a hole. In addition,

a faint line at 1.6292 eV is visible from radiative decay of the biexciton

(XX). These assignments of the observed lines are consistent with mea-

surements on similar structures [85, 65], and are further supported by the

linear dependence of the X− and X0 lines, and the quadratic dependence

of the XX line on the excitation intensity, as shown in Fig. 5.8. Figure 5.9

illustrates these three optical transitions. In this QD we see no evidence

118

X-

XXX0

slope=1.8

slope=1.2

slope=0.9

Pump power (µW)

0.1 1 10

103

104

105

PL

inte

nsi

ty (

coun

ts)

Figure 5.8: The intensity of the neutral exciton (X0), charged exciton (X−), and

biexciton (XX) PL lines as a function of pump power, on a log-log scale. The red lines

are fits to the low power, linear region of the plot. The slope of approximately 1 for

the X0 and X− curves indicates that they are mainly one-photon processes, while the

slope of approximately 2 for the XX PL indicates that it is a two-photon process.

of a positively charged exciton.

With circularly polarized excitation, spin polarized electrons and heavy

holes can be pumped into the QD due to the optical selection rules of

the GaAs QW [65, 14] (see also Chapter 2). For the purposes of this

discussion, spin polarization parallel to the optically injected electron spin

polarization will be referred to as “spin-up”, and the opposite spin as

“spin-down”. Information about the spin polarization in the QD can be

gained from the polarization of the PL [14] (see Chapter 2). The circular

polarization of the PL is determined by switching the helicity of the pump

from σ+ to σ− and measuring the intensity of the σ+-polarized PL, (I+

119

EX0

empty dot - exciton

EX- EXX

single electron - trion exciton - biexciton

Figure 5.9: Lowest energy optical transitions in a quantum dot. In all three transitions,

an electron and hole are created (destroyed). (a) In the neutral exciton (X0) transition,

the QD begins (ends) empty. (b) In the charged exciton (X−) transition, the QD begins

(ends) with a single electron. (c) In the biexciton (XX) transition, the QD begins (ends)

with an electron and a hole.

and I−, respectively). The polarization of the PL is measured using a

second liquid crystal variable waveplate set to ±1/4-wave followed by a

linear polarizer in the PL collection path. The polarization is then defined

as P = (I+ − I−)/(I+ + I−) and is shown for the X0 and X− lines in

Fig. 5.7, in agreement with earlier results [85, 65].

The X0 PL shows a small, but positive polarization over the entire

range of bias where X0 PL is present. This reflects the polarization of the

injected electrons and holes. The magnitude of the circular polarization

is most likely reduced from the polarization of the injected carriers due to

the anisotropy of the electron-hole exchange interaction in the QD. This

effect arises from elongation of the QDs along the [110] crystal axis [87],

causing electrons and holes to relax into states emitting linearly polarized

120

EX-, neg. pol.

(a)

(b)

Figure 5.10: Mechanisms for single spin initialization. (a) A single, optically injected,

spin-up electron relaxes into the QD. (b) The QD initially contains a single electron. An

optically injected electron and hole relax into the QD, forming the X− state. This state

rapidly (∼ 100 ps) decays by emitting (on average) a negatively circularly-polarized

photon, leaving behind a spin-up electron in the QD.

light.

The polarization of the X− line is determined by the hole spin, as the

two electrons in the trion form a spin-singlet state. In the uncharged regime

(Vb < 0.5 V), the negative polarization of the X− PL indicates that the

heavy hole undergoes a spin-flip before recombination in most cases. Hole

spin-flips may occur either during energy relaxation in the QW [88] or by

121

an exchange-mediated electron-hole spin-flip [89]. Regardless of the hole

spin-flip process, after the recombination of the X−, the electron left in the

QD is polarized in the spin-up direction. In this way, as shown in Fig. 5.10,

both optical injection and trion recombination serve to pump lone spin-up

electrons into the QD.

When the dot is initially charged near Vb = 0.5 V, the now dominant

X− line remains negatively polarized, resulting in continued pumping of

the spin-up state. As the electron density in the QW increases with higher

applied bias, the X− polarization becomes positive, as has been previously

observed [85, 65]. In this regime, electrically injected electrons swamp the

optically excited electrons, and the PL only depends on the spin of the

hole.

In a transverse applied magnetic field, the electron spins precess, de-

polarizing the PL. This is the Hanle effect, described in Chapter 2. The

hole spins do not precess [90] because the heavy and light hole states are

split (by ∼ 20 meV in our sample [91]), leading to an effective heavy-hole

g-factor of zero in the plane of the QW. Hanle measurements on this dot

are summarized in Fig. 5.11. In the charged regime, at Vb = 0.9 V, no

depolarization of the X− PL is observed, as expected for polarization due

to the hole spin. The case is markedly different at Vb = −0.8 V, in the

122

X0, -0.8V

X-, 0.9V

X-, -0.8V

0 1 2-1-2

0

-0.2

0.2

0.4

Magnetic Field (kG)

PL

pola

rizat

ion

T = 10 K

Figure 5.11: Single quantum dot Hanle measurements. At Vb = −0.8 V, the charged

exciton (X−) PL shows a sharp Hanle peak (red) indicating a spin lifetime T ∗

2 ∼ 10 ns.

The exciton (X0) PL shows a much broader Hanle curve (black) due to the short

recombination time of the exciton state. At Vb = 0.9 V, the X− PL polarization (blue)

shows no dependence on magnetic field. The green curves are Lorentzian fits.

uncharged regime. Here, the (negatively-polarized) X− line is depolarized

with a half-width, B1/2 = 80 G. With an estimated electron g-factor of

ge = 0.2 [65], B1/2 = 80 G corresponds to a time-averaged transverse spin

lifetime T ∗2 = h/B1/2geµB = 7 ns, where µB is the Bohr magneton, and

h is the Planck constant. This sharp Hanle peak has been previously at-

tributed to the electron spin in the QD, prior to X− formation [65]. The

X0 line shows a much broader peak (B1/2 = 4.1 kG), with a small nar-

row component at low field. The broad component is consistent with the

radiative lifetime of the exciton (∼ 50 ps) [67]. The narrow component

123

has a half-width, B1/2 = 95 G, similar to the X− width. In fact, this nar-

row peak is expected if a lone electron in the dot can bind and recombine

with a subsequently injected hole. Similar features have been observed in

ensemble Hanle measurements in GaAs QWs [92].

To summarize these PL results, in the uncharged regime spin-polarized

excitons or electrons can be pumped into the dot. Both optical injection

and trion recombination serve to pump spin-up electrons. At high bias in

the charged regime (Vb = 0.9 V) the PL polarization is due to the hole

spin, obscuring any information about the electron spin polarization. So

far, we have reproduced the results of Ref. [65]. To go further, a more

direct probe of the spin polarization is required.

5.6 Single spin Kerr rotation

The data in the top panel of Fig. 5.12 show the KR signal as a function

of probe energy for σ+ and σ− pump helicity. Here, the applied bias is

Vb = 0.2 V and the QD is in the uncharged regime. The PL at this

bias is also shown, with the X− and X0 energies indicated by the dotted

lines. These energies coincide spectrally with two sharp features observed

in the KR data, which we will refer to as Ξ− and Ξ0, respectively. In the

bottom two panels of Fig. 5.12 the sum and difference of the σ+ and σ−

124

1.62 1.63 1.64Probe energy (eV)

Ker

r R

ota

tion (

µra

d)

-100

0

100

-200

-100

0

-100

0

100

X-X0

σ+

σ-

σ++σ-

σ+-σ-

Vb=0.2 V

B=0, T=10 K

(a)

Ξ0

Ξ-

(b)

(c)

Figure 5.12: Single quantum dot Kerr rotation. (a) KR data as a function of probe

energy with σ+ (blue) and σ− (red) polarized pump at Vb = 0.2 V, in zero magnetic

field. The PL at this bias is also shown (black), and the X− and X0 energies are

indicated by the dotted lines. (b) The sum of the two curves shown in (a), representing

spin-independent effects, such as the spike (labeled Ξ0) at the X0 energy. (c) The

difference of the two curves shown in (a), representing the spin-dependent signal. The

feature Ξ− at the X− energy is attributed to single spin detection.

data is shown. The feature Ξ0 at the X0 energy clearly does not depend

on the sign of the injected spin and is similar to features seen in single

dot absorption measurements [82]. We attribute this peak to polarization-

dependent absorption in the QD. We focus here on the (σ+ − σ−) data,

125

which represents KR due to the optically oriented spin polarization. The

feature Ξ− at the X− energy only appears in the difference data, indicating

that it is due to the injected spin polarization, shown in Fig. 5.13 at four

different bias voltages. For all voltages, the Ξ− feature is centered at the

X− transition energy, indicated by the blue triangles. We can fit these

data to Eq. 5.2 including only a single transition in the sum, on top of a

broad background (red lines, Fig. 5.13). From the free parameters in these

fits we determine the transition energy E0, amplitude A (defined as half

the difference of the local maximum and minimum near E0), and width Γ

of the Ξ− KR feature.

Fig. 5.15 shows E0 compared to the energy of the X− PL line as a

function of the applied bias. Also shown for comparison is the energy of

the XX PL line. The two energies agree well and show the same quantum-

confined Stark shift. Only at the highest bias, where significant broadening

sets in, do we observe a small anti-Stokes shift between E0 and the X− PL

energy. This may be caused by interactions with electrons in the QW. (The

term “anti-Stokes shift” usually refers to a blueshift of emission, relative

to the absorption. Here we see a blueshift of emission, relative to the Kerr

rotation.) For a single electron spin in the QD ground state, the lowest

energy optical transition contributing in Eq. 5.2 is the X− transition (see

126

1.628 1.632

0

-10

-50

-100

0

0

-50

0

-50

Ker

r R

ota

tion (

µra

d)

Probe energy (eV)

fits

Vb=1.1V

Vb=0.7V

Vb=0.2V

Vb= -1V

Figure 5.13: Single spin KR feature as a function of bias voltage. The Ξ− KR

feature is present over a large range of bias voltage, though it broadens and decreases in

amplitude as the charging increases. Red lines are fits to an odd-Lorentzian plus a broad

(Gaussian) background. The blue triangles indicate the energy of the X− transition,

determined from PL measurements.

Fig. 5.9). Thus the Ξ− KR feature is due to the measurement of a single

electron spin in the QD. We have repeated this measurement on several

other QDs and observed the same Ξ− feature, also at the X− PL energy

127

-100

-50

0

50

-100

-50

0

-20

-10

10

0K

R (

µrad

)

X- PL

X- PL

X- PL

1.628 1.630 1.626 1.628 1.636 1.640

Probe energy (eV) Probe energy (eV) Probe energy (eV)

(a) (b) (c)

Figure 5.14: Single spin detection in other quantum dots. (a) KR and PL spectrum

from the QD weve been looking at so far. (b) KR and PL spectrum from the same

cavity sample as (a), but closer to the cavity resonance. (c) KR and PL spectrum from

the control sample without a cavity.

(shown in Fig. 5.14). The large, broad KR background is likely due to

transitions involving excited electron and hole states both in the QD and

in the QW, which are typically a few meV above the lowest transition [83].

If present, a KR feature due to the X0 spin should appear centered

at the XX transition energy. The signal-to-noise in our measurement is

not high enough to conclusively identify such a feature. Despite the large

amplitude of the X0 PL compared to the X− PL in the uncharged bias

regime (∼ 10 : 1), the short radiative lifetime of the X0 state results in

a low steady-state X0 population, and therefore low KR signal. To put

in some numbers, the exciton radiative lifetime in these QDs is typically

about 100 ps. On the other hand, the measured single spin lifetime (see

below) is around 10 ns. Even if a spin polarized exciton is pumped into the

128

X- PL

XX PL

Ξ- KR

1.630

1.629

Ener

gy (

eV)

-1 0 1

Vb (V)

Figure 5.15: Energy of single spin Kerr rotation feature compared to transition en-

ergies. The black points indicate the energy of the Ξ− feature as determined from the

fits as a function of bias voltage. The blue and red points show the energy of the X−

and XX transitions determined from PL measurements.

QD 10 times more than a single electron spin, the time-averaged exciton

KR signal is 10 times lower than the single electron signal. Additionally

the anisotropic electron-hole exchange interaction mentioned above, might

also play a role in reducing the spin polarization of the exciton.

As mentioned above, we might expect to see an enhancement by a

factor of about 15 of the KR signal at the cavity resonance. However, the

QD we have been measuring has its X− transition at EX− = 1.6297 eV,

whereas the cavity resonance is at Ecav = 1.624 eV. The FWHM of the

cavity resonance is about 13 meV, so there is still some overlap of the cavity

resonance with the KR feature, though the enhancement may be less than

a factor of 15. We can attempt to quantify the effect of the cavity by

129

measuring other QDs with different detuning from the cavity resonance.

Unfortunately, we were only able to find one good QD with energy lower

than the QD we’ve been looking at so far. However, it did have larger

signal. We have also fabricated a control sample in which the DBR layers

are replaced with only Al0.3Ga0.7As (same overall structure, but with no

cavity). We can still observe single electron KR in this sample, though the

signal is significantly smaller. Figure 5.14 shows single spin KR spectra

from two QDs measured in the cavity sample, and one QD in the control

sample. The cavity appears to be enhancing the signal, though many more

QDs should be measured to be sure. One open question is that, though

the signal is lower in the control sample, it appears that the noise in the

control measurement is also lower.

An effect that may contribute to this is loss in the cavity as the light

spreads out away from the focus spot. The microscope objective focuses

the light in a cone yielding an angle of incidence up to ∼ 45. As a result,

light in the cavity will spread out as it reflects multiple times, away from

the focus spot. Of course, only the portion of light that is reemitted from

the cavity from the focus spot will be re-collected by the objective and

detected. Thus the signal will still be enhanced by the cavity, but the

total amount of light making it out will be reduced. In fact, we measure

130

a 10-fold decrease in the intensity of the probe light from going into the

objective to coming out. This reduction in the detected probe light could

certainly increase the noise in the measurement. Further work is needed

to determine what is going on here. One possibility would be to etch the

planar cavity into pillar structures to create a three dimensional cavity.

By applying a transverse magnetic field B, we can monitor the depo-

larization of the single electron spin through the KR signal. In contrast to

the Hanle measurements described above, the KR probes the spin in the

QD directly and non-destructively, as opposed to being inferred from the

spin-dependent formation of the X−. The KR as a function of B is shown

for three different bias voltages (Fig. 5.16). At Vb = 0.2 V, in the uncharged

regime, a narrow peak is observed with a half-width B1/2 = 52 G, consis-

tent with the X− Hanle width measured in this regime. At Vb = 0.7 V,

where the dot has charged, but the PL remains negatively polarized, we

measure a somewhat wider KR depolarization curve, with B1/2 = 150 G.

When the QW is charged further, the spin lifetime decreases as shown at

Vb = 1.1 V, with B1/2 = 1.4 kG. Assuming an effective electron g-factor

of 0.2 [65], these half-widths correspond to transverse spin lifetimes of 11,

3.3, and 0.8 ns, respectively.

The electron spin depolarization curves measured at probe energies

131

0 1-1Magnetic field (kG)

0

1.0

0.5

0

1.0

0.5

0

1.0

0.5

Ker

r R

ota

tion (

no

rmal

ized

)

Vb=0.2 V

Vb=0.7 V

Vb=1.1 V

-0.3

∆ (meV)-2.7

5.0

T = 10 K

Figure 5.16: Single spin Kerr rotation Hanle measurements. KR measured at various

bias voltages and detunings as a function of magnetic field. The typical Hanle depo-

larization curves confirm that we are measuring spin polarization. Clearly the curve

broadens at large charging, indicating a shorter spin lifetime.

detuned from the X− transition by an energy, ∆ are shown in the top

two panels of Fig. 5.16 for ∆ = −0.3 meV (at the maximum of the Ξ−

feature), ∆ = −2.7 meV (in the low energy tail), and ∆ = +5.0 meV (on

the broad, high energy feature). The curves have been normalized by their

132

peak values, which vary with probe energy, but they show quite similar

lineshapes for a given bias. It is somewhat unclear why this is the case.

If the signal away from the Ξ− feature is due to electrons in the QW, one

might expect them to show different behavior in this Hanle measurement.

One possibility is that the same electron in the QD is being probed at

higher energy from transitions involving higher energy (QW) hole states.

It is also possible that there are differences between the curves that are too

small to be identified given the signal-to-noise. The dynamics of the single

electron spin and the background signal will be discussed in more detail in

Chapter 6.

Fig. 5.17 a shows geT∗2 = h/B1/2µB as a function of the applied bias,

measured at a probe energy, E = 1.6288 eV, near the X− transition.

The dashed line indicates the onset of QD charging. The spin lifetime

is largest in the uncharged regime. Here, geT∗2 ∼ 3 ns is consistent with

previous measurements [65] in which the spin dephasing is attributed to

the random, fluctuating hyperfine field [13, 12]. (However, in Chapter 6,

doubt will be cast on this attribution.) As the dot and well are charged,

the electron spin lifetime decreases dramatically. This can be caused by

the increasingly rapid capture of a second electron in the dot, which forms

a spin-zero singlet state. If, for example, the bias voltage is set to add

133

-1 0 1Vb (V)

1

0.1

geT

2*

(ns)

0

60

30

KR

lin

ewid

th,

Γ(m

eV)

0

0.4

0.2

charg

ed

unch

arged

KR

am

pli

tud

e,

A(µ

rad

)

(a)

(b)

(c)

Figure 5.17: Analysis of single spin measurements as a function of bias voltage. (a)

The product of the g-factor and spin lifetime on a log scale as a function of bias voltage.

The dashed line indicates the onset of QD and QW charging. The red triangle shows the

value obtained from the PL Hanle measurements. (b) and (c) Amplitude and linewidth

of the Ξ− feature a as a function of bias voltage.

two electrons to the QD in equilibrium, then a pump-injected hole can be

trapped in the QD, and lead to trion emission. After this occurs, a single

spin will be in the dot. This spin will have a very short lifetime though,

since another electron will rapidly return into the QD. Note that this spin

134

polarization is not visible in the PL Hanle measurement described above

or in Ref. [65]. Also, as discussed below, spin flips with electrons in the

QW are likely to be a relevant mechanism in this regime.

The amplitude A of the Ξ− KR signal is shown as a function of bias volt-

age (Fig. 5.17 b). The amplitude decreases in the charged regime, reflecting

the lower spin lifetime. We have argued above that spin-up electrons are

pumped into the QD in the uncharged regime. Therefore the constant sign

of the KR over the entire range of bias indicates spin-up polarization in the

charged regime as well. Contrary to this observed polarization, the posi-

tively polarized X− PL leaves a spin-down electron in the QD. However,

this electron interacts with the bath of electrons in the QW, which is, on

average, optically oriented in the spin-up direction. The predominant spin

in the QW may be transferred to the electron in the dot via a higher order

tunneling process [93] (essentially, simultaneous tunneling of an electron

out of and into the QD). The finite spin-up polarization measured up to

a large bias suggests that these electron-electron spin flips dominate over

the X−-mediated spin pumping in the charged regime.

As the bias increases above Vb = 0.5 V, the width of the Ξ− KR feature,

grows by a factor of 6, shown in Fig. 5.17 c. A similar increase in linewidth

is seen in the X− PL in the charged regime. This provides further evidence

135

for an increased coupling of the QD to other electronic states as the charg-

ing increases. There are a number of unanswered questions regarding the

observed phenomena at large bias (large charging). The broadening of the

transition, the sign of the KR signal, and the shift of the Ξ− energy from

the PL X− energy might indicate that there is some interesting physics to

be studied in this regime.

At the beginning of this chapter, the KR measurement was described

as “non-destructive” because the Kerr rotation can be measured at a de-

tuning where the optical absorption is small. Of course, there is still some

finite probability of exciting carriers in the QD. Nevertheless, the KR mea-

surement itself is inherently non-destructive in that the mechanism does

not involve the excitation of additional electrons or holes. The fact that

the measurement works is proof that any unwanted destructive effects do

not swamp the non-destructive KR measurement. The ability to do a QND

measurement is of interest in regards to probing quantum effects in a solid

state system. In principle, Kerr rotation or Faraday rotation can satisfy

both of the two requirements for a QND measurement described above.

In order to demonstrate a QND measurement, both the non-destructive

nature and the back-action-evading nature must be sufficient that the mea-

surement can be performed in a time less than the time for either of these

136

Figure 5.18: Schematic of photon-mediated coupling between two QDs via Faraday

rotation. From M. Leuenberger, Phys. Rev. B 73, 075312 (2006).

conditions to be violated. There is some hope that this limit could be

achieved. Long T1 times have been observed for spins in optically accessi-

ble QDs (> 20 ms) [94]. In our measurements, we have been able to see

signal with time constants as low as 5 ms. Additionally, estimates of the

dominant back-action mechanisms indicate that this condition may be met

as well [95].

By probing a single electron in a QD through KR non-resonantly, we

demonstrate a direct measurement of the electron spin with minimal per-

turbation to the system. As a first application, this method reveals infor-

mation about spin dynamics in single QDs, and constitutes a pathway

towards quantum non-demolition measurements and optically-mediated

entanglement of single spins in the solid state. An example of one such

scheme is shown in Fig. 5.18. The essential point of such proposals is

137

that if an electron spin is in a superposition of “up” and “down”, then

the Faraday effect results in a photon in a superposition of polarization

rotation “CW” and “CCW”. In this way, the quantum information has

been transferred from a spin to a photon. Much better optical cavities

would probably be needed to implement this scheme. (Fortunately, much

better cavities are certainly possible.) This scheme may also prove useful

for non-destructive measurements in a variety of solid-state qubits, such as

electrically-gated [70] or chemically-synthesized [59] QDs.

138

Chapter 6

Coherent Dynamics of a Single

Spin

6.1 Motivation

The ability to sequentially initialize, manipulate, and read out the state of

a qubit, such as an electron spin in a quantum dot (QD), is a requirement

in virtually any scheme for quantum information processing [96, 47, 97].

However, previous optical measurements of a single electron spin (including

those discussed in the previous chapter) have focused on time-averaged

detection, with the spin being initialized and read out continuously [98,

95, 65, 66, 69]. Here, we modify the measurement scheme of Chapter 5

to directly observe the coherent evolution of an electron spin in a single

QD, using time-resolved Kerr rotation (KR) spectroscopy. This all-optical,

139

non-destructive technique allows us to monitor the precession of the spin in

a superposition of Zeeman-split sublevels with nanosecond time resolution.

The data show an exponential decay of the spin polarization with time,

and directly reveal the g-factor and spin lifetime of the electron in the QD.

Furthermore, the observed spin dynamics provide a sensitive probe of the

local nuclear spin environment.

In Chapter 5, we describe a method for the time-averaged detection of

a single spin in a QD. By scanning the energy of a probe laser around the

lowest energy optical transition in a singly-charged QD (theX− transition),

a single electron spin produces a feature in the KR spectrum with the odd-

Lorentzian lineshape given by Eq. 5.2, centered at the energy of the X−

transition, EX− .

In the present work, as in Chapter 5, the electron is confined to a sin-

gle QD formed by monolayer fluctuations at the interfaces of a gallium

arsenide (GaAs) quantum well (QW). The QD is embedded within a diode

structure, allowing controllable charging of the dot with a bias voltage [84].

Also, the QD is centered within an integrated optical cavity to enhance the

small, single spin KR signal [98]. See Fig. 5.3 for a schematic of the sam-

ple structure. With circularly polarized excitation, spin-polarized electrons

and holes are pumped into the QW, according to the selection rules gov-

140

erning interband transitions in GaAs [14]. One or more electrons and/or

holes then relax into the QD. By measuring the subsequent single QD pho-

toluminescence (PL), we determine the equilibrium charge state of the QD

as well as the energies of various interband optical transitions as a function

of bias voltage [98, 85]. The measurements described below are performed

at a bias voltage where the QD is nominally uncharged, and the optical ex-

citation injects one or more electrons or holes. This bias voltage is chosen

to maximize the KR signal (see Chapter 5). In this regime, the QD may

contain a single spin-polarized electron through the capture of an optically

injected electron, or spin-dependent X− decay [98]. Knowing the transition

energy EX− from the PL measurements, we use the spectroscopic depen-

dence of the Kerr effect to isolate the dynamics of the single electron spin

from that of multiparticle complexes, such as charged or neutral excitons.

That is, we know that the KR feature at the X− energy is due to a single

electron spin in the QD, as opposed to, say, the exciton spin.

6.2 Experimental scheme

In previous work, only the steady-state spin polarization was measured,

concealing information about the evolution of the spin state in time. Here,

we use time delayed pump and probe pulses, shown schematically in Fig. 6.1,

141

Ti:Sa 2

(cw probe)

chopper

LCVW

polarizer

polarizer

scanning

objective

sample

He flow cryostat (10 K)

electro-

magnet

Fe yoke

longpass

filter

to spectrometer

(for PL)

to diode bridge

(for KR)

optional

mirrorf1

f2

VA-Block-in1

fref=f2

lock-in2

fref=f1

pre-amp

VAlock-in3

fref=f1

for

normalization

signal

~150 fs

monochromator

~ ps

Ti:Sa 1

(pulsed pump)

sync

EOM

pulse generator

5.1≥ ns

chopper

∆t

Figure 6.1: Setup for time-resolved single spin measurements. As compared to the

setup for continuous KR measurement, the pump laser is now mode-locked and passed

through a monochromator to narrow and select the energy of the pump pulses. Also,

the probe laser is passed though an electro-optic modulator (EOM) synchronized with

the pump laser, with electronically adjustable delay.

142

<200ps

1.5ns

Time (ns)In

tensi

ty (

a.u.) Pump

Probe

1.0

0.5

120

0 84

Figure 6.2: Temporal profile of the pump and probe pulses. The pump and probe

pulses are measured using a fast photodiode and oscilloscope. The bandwidth of the

measurement is limited to ∼ 2 GHz by the photodiode bandwidth.

to map out the coherent dynamics of the spin in the QD. The pump and

probe pulses, as detected by a fast photodiode, are shown in Fig. 6.2.

The setup used for time-resolved single spin measurements is similar to

the setup for the continuous measurement, but with a few important modi-

fications. Instead of the continuous pump laser, a mode-locked Ti:Sapphire

laser provides pump pulses with energy Epump = 1.653 eV, and duration

∼ 150 fs at a repetition period Tr = 13.1 ns. The bandwidth of the spec-

trally broad pump pulses is narrowed to ∼ 1 meV by passing the pump

beam through a monochromator. The probe pulses are derived from the

same wavelength tunable CW Ti:Sapphire laser as in Chapter 5. However,

the probe laser is now passed through an electro-optic modulator (EOM),

allowing for electrical control of the probe pulse duration from CW down

to 1.5 ns. This technique yields short pulses while maintaining the narrow

143

linewidth and wavelength tunability of the probe laser. Also, it allows us

to adjust the pulse duration so as to maintain enough average power to

achieve good signal-to-noise, while keeping the instantaneous power low

enough to avoid unwanted non-linear effects. This is an important point.

As we will see in the next chapter, high intensity pulses near the EX−

energy induce a shift of the X− energy away from the laser energy. If the

probe pulses induce an appreciable shift of this type, the KR measurement

will not be possible.

The EOM is driven by an electrical pulse generator triggered by the

pump laser, allowing for electrical control of the time delay between the

pump and the probe pulses with picosecond precision. Additionally, in

measurements with pump-probe delay t > 13 ns the pump beam has also

been passed through an electro-optic pulse picker to increase the repetition

period of the pulse train to Tr = 26.2 ns. Typical time-averaged pump and

probe intensities incident on the sample are 20 W/cm2 and 200 W/cm2,

respectively. Note that this means that the instantaneous intensities are

larger by a factor of the repetition period over the pulse duration. More

details of the setup are given in Appendix B.

The microscope objective, sample, cryostat, and KR measurement ap-

paratus is the same as in the continuous single spin measurement in Chap-

144

ter 5. As in the time-resolved measurements described in Chapters 3 and

4, though the signal is averaged for several seconds (the spin is reinitial-

ized and probed millions of times), the stroboscopic pump/probe technique

allows measurement with high time resolution.

For a fixed delay between the pump and the probe, the KR angle,

θK , is measured as a function of probe energy. At each point, the pump

excitation is switched between σ+ and σ− polarization at a rate 1/tswitch,

and the spin-dependent signal is obtained from the difference in θK at the

two helicities. The resulting KR spectrum is fit to Eq. 5.2 plus a constant

vertical offset, y0. The amplitude, θ0, of the odd-Lorentzian is proportional

to the projection of the spin in the QD along the measurement axis. The

origin of the vertical offset, y0, is unknown but might be due to the broad

KR feature from free electron spins in the QW, as discussed below (also see

the discussion in Chapter 5). By repeating this measurement at various

pump-probe delays, the evolution of the spin state can be mapped out.

6.3 Theoretical model

When a magnetic field is applied along the z-axis, transverse to the in-

jected spin (known as the Voigt geometry), the spin can be described in

the basis along the field giving eigenstates |↑〉 and |↓〉, with eigenvalues

145

Sz = ±h/2. The pump pulse initializes the spin at time t = 0 into the

superposition |ψ(t = 0)〉 = (|↑〉 ± i|↓〉)/√

2 , for σ± polarized excitation.

If isolated from its environment, the spin state then coherently evolves ac-

cording to |ψ(t)〉 = (e−iΩt/2|↑〉 ± ieiΩt/2|↓〉)/√

2 , where hΩ = gµBBz is the

Zeeman splitting. When the probe arrives at time t = ∆t, the spin state

is projected onto the y-axis, resulting in an average measured spin polar-

ization of 〈Sy(t)〉 = ±(h/2) cos(Ω ·∆t) . This picture has not included the

various environmental effects that cause spin decoherence and dephasing,

inevitably leading to a reduction of the measured spin polarization with

time.

In the simplest case, the evolution of the measured KR amplitude can

be described by an exponentially decaying cosine,

θK(∆t) = A · Θ(∆t) · exp

(−∆t

T ∗2

)

cos(Ω · ∆t), (6.1)

where A is the overall amplitude, Θ(t) is the Heaviside step function, and

T ∗2 is the effective transverse spin lifetime (though this measurement elim-

inates ensemble averaging, the observed spin lifetime may be reduced from

the transverse spin lifetime, T2, by inhomogeneities that vary in time). To

model our data, we sum over the contributions from each pump pulse sep-

arated by the repetition period Tr, and convolve (denoted ‘∗’) with the

146

σ+

σ-

Sz < 0Sz > 0

σ+

σ-

Sz = 0 Sz = 0

Bz

no misalignment misalignment

Bz

Figure 6.3: Illustration of spin misalignment leading to dynamic nuclear polarization.

Left: with the pump incident perpendicular to the magnetic field, there is no component

of the injected spin along the magnetic field. Right: A misalignment of the pump from

the perpendicular may yield a component of the spin along the magnetic field, with

opposite sign for opposite pump helicity.

measured probe pulse shape, p(t),

θ0(∆t) = p ∗[∑

n

θ(∆t− nTr)

]

. (6.2)

This measurement technique is also sensitive to small nuclear spin po-

larizations (see Section 2.1 for discussion of nuclear spin polarization). Ide-

ally there should be no induced steady-state nuclear polarization in this

experimental geometry. Since the magnetic field is applied perpendicular

to the direction of the spin, nuclear spins that are polarized by the elec-

tron spins precess around the applied field, resulting in zero steady-state

polarization. However, for any misalignment of the pump laser from the

perpendicular, there is a projection of the spin along the magnetic field, and

right (left) circularly polarized light induces a small dynamic nuclear polar-

ization (DNP) parallel (antiparallel) to the applied magnetic field [14, 99].

147

This is illustrated in Fig. 6.3. Due to the hyperfine interaction this acts on

the electron spin as an effective magnetic field, increasing (decreasing) the

total magnetic field, resulting in a different precession frequency for right

and left circularly polarized pump excitation. Since each data point is the

difference of the KR signal with σ+ and σ− polarized excitation, a small

deviation from perpendicular between the magnetic field and the electron

spin yields a measured KR signal

θ(∆t) = A ·Θ(∆t) ·exp

(−∆t

T ∗2

)

× [cos((Ω+δ)∆t)+cos((Ω−δ)∆t)], (6.3)

where δ = gµBBnuc/h is the frequency shift due to the steady-state effective

nuclear field, Bnuc.

6.4 Time-resolved single spin

measurements

The single spin KR amplitude as a function of delay, measured with a

3-ns duration probe pulse and a magnetic field B = 491 G, is shown in

Fig. 6.4 a, exhibiting the expected oscillations due to the coherent evolution

described above. Fig. 6.4 b-f show a sequence of KR spectra at several

delays, and the fits from which the data in Fig. 6.4 a are obtained. In the

inset of Fig. 6.4 a the offset y0 is shown, which oscillates with the same

148

KR

(µr

ad)

-40

-30

-35

Probe energy (eV)1.629 1.630

-25 ∆t=0.3ns

Probe energy (eV)1.629 1.630

KR

(µr

ad)

-5

5

0

∆t=2.3ns

KR

(µr

ad)

Probe energy (eV)1.629 1.630

0

15

10

5

∆t=3.3ns

Probe energy (eV)1.629 1.630

KR

(µr

ad)

-5

10

0

5

∆t=5.2 ns

Probe energy (eV)1.629 1.630

KR

(µr

ad)

-10

0

-5

∆t=6.8 ns

θ 0(µ

rad)

20

10

0

-10

-20

Pump-probe delay (ns)0 10 15 205

Delay (ns)

y0

(µra

d)

0 10

40

20

0

20

(a)

T = 10 K

B = 491 G T2* = 8.4 ± 3.5ns0θ∝

xS

zB ˆ

S

(b) (c) (d)

(e) (f)

Figure 6.4: Coherent precession of a single spin. (a) As the pump-probe delay is

varied, coherent precession is seen as oscillations in the measured projection of the spin.

The red curve is a fit to the model, and the dotted line shows the same curve but without

the probe pulse convolution. The inset shows the offset, y0 as a function of delay. The

error bars are the standard error in the fits to the KR spectra. (b-f) KR spectra at

increasing pump-probe delay showing amplitude oscillations in time. The red curves

are the fits from which the solid points in part (a) were obtained.

149

frequency as the single spin KR but decays with a shorter lifetime. This

behavior may be consistent with that of free electron spins in the QW,

previously investigated in time-averaged measurements [92]. Due to the

small confinement energy of these QDs (several meV) relative to the QW,

one does not expect a significant shift in the g-factor between the QDs and

the QW.

The solid line in Fig. 6.4 a is a fit to Eq. 6.2, yielding Ω = 0.98 ±

0.02 GHz and T ∗2 = 8.4 ± 3.5 ns. The dashed line shows Eq. 6.2 without

the probe pulse convolution, plotted with the same parameters for com-

parison. This data is taken with tswitch = 1 s, and for now, the effects of

nuclear polarization are ignored. The effects of nuclear polarization will be

addressed below.

In Fig. 6.5 a the precession of the spin is shown at three different mag-

netic fields. As expected, the precession frequency increases with increasing

field. The solid lines in Fig. 6.5 a are fits to Eq. 6.2, and the frequency Ω

obtained from such fits is shown in Fig. 6.5 b as a function of magnetic field.

A linear fit to these data yields an electron g-factor of |g| = 0.17±0.02, con-

sistent with the range of g-factors for these quantum dots found in previous

ensemble or time-averaged measurements [65, 100]. At zero magnetic field,

as shown in Fig. 6.6, the spin lifetime is found to be T ∗2 = 10.9 ± 0.5 ns.

150

θ 0(a

. u.)

T = 10 K

Pump-probe delay (ns)

B = 1195 G

B = 929 G

B = 491 G

0 2 4 6 8-2 10 12

0

0

0

(a)

10000 500

0

1.0

2.0

Ω(G

Hz)

(b)

|g| = 0.17± 0.02

Magnetic field (G)

Figure 6.5: Single spin precession as a function of magnetic field. (a) Spin precession

at three magnetic fields. The error bars are the standard error in the fits to the KR

spectra. (b) Precession frequency, Ω, as a function of magnetic field. The error bars

represent the standard deviation from repeated measurements. The linear fit (red)

yields a g-factor of ±0.17 ± 0.02.

This value agrees with previous time-averaged [65] and ensemble [92, 100]

measurements where the relevant decay mechanism is often suggested to be

dephasing due to slow fluctuations in the nuclear spin polarization. How-

ever, these polarization fluctuations are not expected to result in a single

151

exponential decay of the electron spin [13, 12]. Roughly speaking, the

random distribution of nuclear polarizations yields a Gaussian-like decay

envelope. Using these non-Markovian models to fit our data (see Fig. 6.6)

results in an increase of χ2 to 24.3 from 3.8 for the fit with a single expo-

nential decay, suggesting that other decay mechanisms than nuclear spin

fluctuations might also be relevant in this case. In these QDs, the elec-

tronic level spacing of ∼ 1 meV [87] is of the same order as kBT for this

temperature range. Therefore, thermally-activated or phonon-mediated

processes [101, 102, 103, 104] which yield an exponential decay, might be

significant in this regime.

In order to investigate the effects of nuclear polarization on the elec-

tron spin, we have varied the rate 1/tswitch at which the pump helicity is

switched. Fig. 6.7 a and b shows the single spin KR signal as a function of

time with tswitch = 1 s (as in Fig. 6.4), and tswitch = 10 s, respectively, with

otherwise identical conditions. With larger tswitch, the nuclear polarization

has time to build, as is evidenced in Fig. 6.7 b by the visible beating, as ex-

pected from Eq. 6.3. A fit of the data in Fig. 6.7 a (tswitch = 1 s) to Eq. 6.3

convolved with the probe pulse yields δ = 0 ± 0.04 GHz and T ∗2 = 8.4 ns,

whereas for the data in Fig. 6.7 b (tswitch = 10 s) δ = 0.14 ± 0.02 GHz

and T ∗2 = 8.3 ns with a χ2 of 2.3. For comparison, a single-frequency fit

152

Pump-probe delay (ns)

B = 0 G

0 10 20

40

80

0

(a)θ 0

(µra

d)

0 10 20Pump-probe delay (ns)

(b)

Figure 6.6: Single spin dynamics at zero magnetic field. (a) Decay of the spin polar-

ization with B = 0. The red line is a single exponential fit convolved with the probe

pulse, giving a reduced χ2 = 3.8. The error bars are the standard error from the fits

to the KR spectra. (b) The same data as (a) with a fit to a model of nuclear spin

dephasing (blue), yielding an obviously poorer fit, and a reduced χ2 = 24.3. The a fit

to the same model multiplied by an additional exponential decay factor, to model both

nuclear dephasing and other decoherence mechanisms, is shown in red. This fit is also

significantly worse than that in (a).

of the data in Fig. 6.7 to Eq. 6.2 yields a slightly larger χ2 of 2.8, and

a significantly shorter T ∗2 of 5.7 ns. It is unlikely that tswitch would have

such an effect on the spin lifetime, and moreover, nuclear polarization is

the only effect in this system known to act on the spin with timescales

on the order of seconds. The inset in Fig. 6.7 shows the KR amplitude

as a function of tswitch for a fixed pump-probe delay. A fit of the data to

Eq. 6.3 with δ saturating exponentially in tswitch reveals a DNP saturation

time of 1.5 s similar to what has previously been found in these QDs [105].

153

0 5 10 15 20

-20

10

0

-10

10

-20

20

0

-10

Pump-probe delay (ns)

θ 0(µ

rad)

B = 491 G, T = 10 K

(a)

(b)

tswitch = 1 s

tswitch = 10 s

θ 0(µ

rad)

Switch time (s)0 5 10

0

10

20

20

∆t = 6.7 ns

Figure 6.7: The effects of nuclear polarization on single spin precession. (a) Spin

precession with B = 491 G, and tswitch = 1 s. The red curve is a fit to the model

including DNP, yielding δ = 0 ± 0.04 GHz and T ∗

2 = 8.3 ns. (b) The same as (a), but

with tswitch = 10 s. The fit (red) now yields δ = 0.14± 0.02 GHz and T ∗

2 = 8.3 ns. The

inset shows the spin signal at a fixed delay, ∆t = 6.7 ns as a function of tswitch. The

red curve is a fit to the two-frequency model with δ saturating exponentially in tswitch.

(Specifically, δ(tswitch) = δ0[1 − exp(−tswitch/tn)], where δ0 is the maxi-

mum frequency shift, and tn is the DNP saturation time.) The nuclear

polarization seen in Fig. 6.7 corresponds to an effective magnetic field of

100 G, or an electron spin splitting of 100 neV. For comparison, when the

magnetic field and the electron spin are parallel, hyperfine spin splittings

154

∼ 1000 times larger have been observed [105]. Given the typical size of

these QDs [87], the electron interacts with ∼ 105 nuclear spins. Since the

hyperfine splitting here is smaller by a factor of 1000 than the maximum

observed splitting, the data shown in Fig. 6.7 represent the detection of at

most ∼ 105/103 = 100 polarized nuclear spins.

These measurements constitute a noninvasive optical probe of the co-

herent evolution of a single electron spin state with nanosecond temporal

resolution, which is a key ingredient for many spin/photon-based quan-

tum information proposals [77, 79]. Furthermore, this technique provides

a sensitive probe of the dynamics of the spin, revealing information about

the spin coherence time and g-factor. Future work may exploit this ability

to further explore the relevant decoherence mechanisms and the electron-

nuclear spin interactions, and to observe the coherent manipulation of sin-

gle spins in real time.

155

Chapter 7

Ultrafast Manipulation of

Single Spin Coherence

7.1 Motivation and Background

Using ultrafast optical pulses to coherently manipulate the spin state of

an electron is a key ingredient in many proposals for solid-state quantum

information processing [106, 107, 108, 109, 110, 111]. Though electrical

control of single spins has been achieved [112], the nanosecond timescales

required for such manipulation limits the number of operations that can

be performed within the spin coherence time. In that work, single elec-

trons were confined to a gate-defined 2DEG QD, and the spin control was

achieved via spin resonance induced by a stripline deposited on the sample.

The speed of such a spin rotation is limited by the maximum attainable AC

156

Sy = -1/2 Sy = +1/2

Jy = -3/2 Jy = +3/2

σ-

Effective magnetic field, BStark

EX-EX- - ∆tipping

pulse

(a) (b)

Figure 7.1: Illustration of the optical Stark effect. (a) The relevant QD transitions

illustrated in the basis along the y-axis (the growth direction). The single electron

ground state is coupled optically to the negatively charged exciton (trion) state. For

a given circular polarization, the selection rules allow a non-zero matrix element only

for one such transition. (b) When an off-resonant circularly polarized optical field is

applied, one spin state is shifted due to the OSE. This results in a spin splitting (effective

magnetic field) for the single electron.

magnetic field. In contrast, spin control via picosecond-scale optical pulses

yields an improvement of several orders of magnitude in the manipulation

time. In this chapter, we experimentally demonstrate such a scheme for a

single electron spin in a QD, monitoring the coherent evolution of the spin

state using time-resolved Kerr rotation spectroscopy. The spin is subjected

to an intense, off-resonant laser pulse, which induces a rotation of the spin

through angles up to π radians on picosecond timescales.

The optical (or ac) Stark effect (OSE) was first studied in atomic sys-

tems in the 1970s [113, 114, 115] and subsequently explored in bulk semi-

157

conductors and in quantum wells [116, 117, 118]). In recent years, the

OSE has been used to observe ensemble spin manipulation in a quantum

well [119], and to control orbital coherence in a QD [120]. Additionally,

other optical manipulation schemes have been explored on ensembles of

spins [121, 122]. Using time-dependent perturbation theory (similar to the

treatment in Section 2.4), it is found that an optical field with intensity

Itip, detuned from an electronic transition by an energy ∆, induces a shift

in the transition energy

∆E ≈ D2Itip

∆√ǫ/µ

, (7.1)

where D is the dipole moment of the transition, and ǫ and µ are the

permittivity and permeability of the material [117]. Figure 7.1 shows the

relevant energy levels for the QD system considered here. The ground

state consists of a single electron in the lowest conduction band level, spin-

split by a small magnetic field, Bz. The lowest energy interband transition

is to the trion state consisting of two electrons in a singlet state and a

heavy hole. Due to the optical selection rules (see Chapter 2), the dipole

strength of this transition in the basis along the y-direction from the spin-

up (-down) ground state is zero for σ+ (σ−) polarized light, as indicated

in the diagram. Therefore, for circularly polarized light, the OSE shifts

just one of the spin sublevels and produces a spin splitting in the ground

158

state which can be represented as an effective magnetic field, BStark, along

the light propagation direction. By using ultrafast laser pulses with high

instantaneous intensity to provide the Stark shift, large splittings can be

obtained to perform coherent manipulation of the spin within the duration

of the optical pulse (here, BStark ∼ 10 T). Note that this phenomenon can

also be described in terms of a stimulated Raman transition [108, 111], or

as an avoided crossing between excitons and photons [123].

7.2 Experimental scheme

As in Chapters 5 and 6, the sample consists of a layer of charge-tunable

GaAs interface QDs embedded in an optical cavity (see Fig. 5.3). A

schematic of the experimental setup is shown in Fig. 7.2. (Additional de-

tails of the setup are given in Appendix B.) This is similar to the setup of

the last two chapters, but again, with some significant changes. In this case,

three synchronized, independently tunable optical pulse trains are focused

onto the sample: the pump, the probe, and the “tipping pulse” (TP). The

pump and tipping pulse are both derived (by means of a beamsplitter) from

the mode-locked Ti:sapphire laser generating a train of ∼ 150-fs-duration

pulses at a repetition rate of 76 MHz. The pump is circularly polarized,

and tuned to an energy E = 1.646 eV (FWHM ∼ 1 meV), thereby injecting

159

Ti:Sa 2

(cw probe)

chopper

LCVW

polarizer

polarizer

scanning

objective

sample

He flow cryostat (10 K)

electro-

magnet

Fe yoke

bandpass

filter

to spectrometer

(for PL)

to diode bridge

(for KR)

optional

mirrorf1

f2

VA-Block-in1

fref=f2

lock-in2

fref=f1

pre-amp

VAlock-in3

fref=f1

for

normalization

signal

~150 fs

~ ps

Ti:Sa 1

(pulsed pump/TP)

sync

EOM

pulse generator

5.1≥ ns

chopper

monochromators

~ 25 ps

polarizer

λ/4 plate

del

ay

tprobe

ttip

Figure 7.2: Setup for Stark tipping measurements. As compared to the setup for time-

resolved single spin measurements, we now split the pump laser to obtain the tipping

pulse (TP). The TP is passed through a monochromator, and circularly polarized. Also

the pump is now sent through a delay line to control the pump-TP delay.

160

KR

(µr

ad)

Energy (eV)

20

-60

-140

1.624 1.628In

tensi

ty (

a.u.)

XX

X-TP θ0

Figure 7.3: Energy scales and relevant optical spectra for Stark tipping measurements.

The solid red line shows the charged exciton (X−) and biexciton (XX) PL lines from

the QD. The black circles show a single electron KR spectrum, with the odd-Lorentzian

fit (solid black line) from which the KR amplitude θ0 is obtained. The dashed red line

shows the spectrum of the tipping pulse at a detuning of ∆ = 4.4 meV.

spin-polarized electrons and holes into the continuum of states above the

QD [14]. One or more of these electrons or holes can then relax into the

QD. The circularly polarized TP (duration ∼ 30 ps, FWHM = 0.2 meV)

is tuned to an energy below the lowest QD transition (see Fig. 7.3) and is

used to induce the Stark shift. The relative time delay between the pump

pulse and the TP is controlled by a mechanical delay line in the pump

path.

As in Chapter 6, the probe pulse is generated by passing a narrow

linewidth continuous-wave laser through an electro-optic modulator syn-

chronized with the pump/TP laser. The resulting 1.5-ns-duration pulses

probe the spin in the QD through the magneto-optical Kerr effect [124].

161

This effect arises from the real part of the dielectric response function of

the QD, and results in a spin-dependent rotation of the polarization of the

linearly polarized probe upon reflection off of the sample (see Section 2.4).

As the energy of this probe light is scanned across the QD transition energy,

EX− , an odd-Lorentzian lineshape (∼ x/[1+x2]) centered at the transition

energy is seen in the Kerr rotation (KR) spectrum [98, 95]. By fitting such

a curve (as shown in Fig. 7.3) we can extract the amplitude of this feature,

θ0, which is proportional to the projection of the spin polarization in the

QD along the light propagation direction.

As before, the sample is mounted on the cold finger of a He flow cryo-

stat at the focus of a microscope objective, at a temperature T = 10 K.

The pump, probe and TP are focused and spatially overlapped on the sam-

ple with a spot size of ∼ 1 µm. The temporal profile of the probe pulse

is measured using a 2 GHz bandwidth photodiode, and the TP profile is

measured using a streak camera and found to have a FWHM of 25 ps. Po-

larized spins are initially injected into higher energy states of the QW. A

single spin can be trapped in the QD either by relaxation of a single elec-

tron into the QD, or by spin-dependent formation and recombination of a

negatively charged trion. Because of the spectral energy sensitivity of our

measurement, we only probe the QD when it is singly charged - if the dot

162

contains, for example, an exciton, there is no contribution to the signal at

the X− energy. The reflected light is collected through the same objective

and the rotation of the probe polarization is detected by a balanced photo-

diode bridge. Typical time-averaged pump and probe intensities incident

on the sample are 20 W/cm2 and 200 W/cm2, respectively. The instan-

taneous probe intensity is thus ∼ 1700 W/cm2, which is sufficiently small

to prevent the probe from inducing an appreciable (non-spin-dependent)

optical Stark shift. For example, when the probe is detuned from the X−

transition by 50 µeV (approximately equal to the linewidth of the transi-

tion), the energy shift due to the probe is expected to be ∼ 20 µeV. (This

is calculated from the measured values below.) Since the probe is linearly

polarized, this shift affects both spin states equally. This small shift may

alter the lineshape of the KR spectrum, but since the shift is less than the

linewidth of the transition, the odd-Lorentzian feature in the KR spectrum

is still visible.

In a typical measurement, the pump pulse arrives at t = 0 along the y-

axis (growth direction), and in some cases, a single spin-polarized electron

will relax into the quantum dot. For pump helicity σ±, this electron is (up

to a global phase) initially in the state |ψ(t = 0)〉 = (|↑〉 ± i|↓〉)/√

2, where

“up” and “down” are chosen as the basis along the external magnetic field

163

Bz. The spin then begins to coherently precess at the Larmor frequency

ω = gµBBz/h: |ψ(t)〉 = (exp(−iωt/2)|↑〉 ± i exp(iωt/2)|↓〉)/√

2, where g

is the effective electron g-factor, µB is the Bohr magneton, and h is the

reduced Planck constant. At time t = ttip, the TP arrives and generates

an additional spin splitting along the y-axis for the duration of the pulse.

During this time, the spin precesses about the total effective field (which

is typically dominated by BStark), and then continues to precess about the

static applied field. The probe then measures the resulting projection of

the spin in the QD, 〈Sy〉 at t = tprobe. This sequence is repeated at the

repetition frequency of the laser (76 MHz), and the signal is averaged for

several seconds for noise reduction. As described in Chapter 5, the pump

and probe are modulated using mechanical choppers, allowing for lock-

in detection to measure only spins that are injected by the pump. Also,

the pump is switched between σ+ and σ−, with a measurement made at

each helicity. The spin signal is then taken as the difference between these

values, eliminating any spurious signal from spins not generated by the

pump (e.g. phonon-assisted absorption from the TP [125]), or non-spin-

dependent rotation of the probe polarization.

To map out the coherent dynamics of the spin in the QD, KR spectra

are measured as a function of pump-probe delay. Figure 7.4 a shows a

164

Eprobe (eV)

θ K(µ

rad)

θ K(µ

rad)

0

10

-10

0

10

-10

Del

ay (

ns)

2

0

4

1.6295 1.6300

Eprobe (eV)

θ K(µ

rad)

θ K(µ

rad)

0

-20

0

-201.6295 1.6300

Del

ay (

ns)

2

0

4

(a) (b)

tprobe=0ns

tprobe=2ns

tprobe=0ns

tprobe=2ns

No TP TP at ttip = 1.3 ns

θ K(a

.u.)

1

-1

(i)

(ii)

(iv)

(iii)

(i)

(ii) (iv)

(iii)

TP

Figure 7.4: Single spin Kerr rotation vs. delay and probe energy with and without

the tipping pulse. (a) and (b) KR angle, θK as a function of probe energy and pump-

probe delay. White (black) represents positive (negative) KR. In (a) no tipping pulse is

applied. In (b) the tipping pulse is applied at ttip = 1.3 ns, with intensity and detuning

set to cause a ∼ π rotation. (i)-(iv) show linecuts from (a) and (b) illustrating the effect

of the TP. Specifically, a reversal of the sign of the KR signal after the TP. The dashed

blue line indicates the X− energy.

plot of the KR angle, θK as a function of probe energy and tprobe with

an applied field Bz = 715 G and no TP. Horizontal line-cuts display the

dispersive lineshape centered at the transition energy EX− , as shown in

Fig. 7.3. As tprobe is swept along the vertical axis, the precession of the

165

spin can be seen as the oscillations in θK . When the TP is applied at

ttip = 1.3 ns, as in Fig. 7.4 b, there is a significant change in the KR

spectra. For t < ttip, the KR signal is essentially the same as in Fig. 7.4 a,

but for t > ttip the sign of the signal is reversed. This can be clearly seen in

the line-cuts shown in Fig. 7.4 i-iv. Line-cuts (i) and (iii) are both before

the TP and show the same behavior, whereas line-cut (iv) has the opposite

sign of line-cut (ii) as a result of the TP.

As described in Chapter 5, the quality factor of the cavity (Q = 120)

implies that the FWHM of the cavity resonance is about 13 meV. There-

fore, there is significant overlap of both the probe and the tipping pulse

(TP) with the cavity resonance. As the linearly polarized probe field builds

inside the cavity, the Kerr effect is enhanced, leading to an increase in the

KR signal. From similar measurements of Kerr or Faraday rotation in a

cavity [62, 59], we expect enhancement by a factor of ∼ 15 at the peak of

the resonance. We might expect a similar enhancement of the OSE as the

TP resonates in the cavity. However, as will be discussed further below,

the magnitude of the effect actually appears to be lower than expected

from the theory, possibly due to the spreading of the light in the planar

cavity.

166

↑

↓ B

(a) (b) (c)

Figure 7.5: Sequence of rotations in the Stark tipping model. (a) Before the TP,

the spin is initially slightly misaligned from the y-axis, and then precesses about the

magnetic field, Bz. (b) At t = ttip, the spin is instantaneously rotated about the y-

axis through an angle φtip. (c) After the TP, the spin continues to precess about the

magnetic field.

7.3 Theoretical model

It is convenient to understand the observed spin dynamics in the Bloch

sphere picture, described in Section 2.1. Here, the spin state is represented

as a vector (Sx, Sy, Sz), where (0, 0,±Sz) represents the eigenstates |↑〉 and

|↓〉, and vectors with nonzero Sx and Sy represent coherent superpositions

of |↑〉 and |↓〉. In this picture, the dynamics of the spin can be calculated by

applying the appropriate sequence of rotation matrices to the initial state.

Figure 7.5 illustrates the sequence of rotations described by the model.

167

The initial spin state at t = 0 is taken to be

~S0 =

0

S0,y

S0,z

, (7.2)

where the initial component in the z-direction, S0,z is assumed to be small,

due to misalignment of the pump beam from normal incidence. Before

the tipping pulse arrives, the spin freely precesses around the applied field.

Thus for t < ttip, we apply a rotation about the z-axis at frequency ω:

~S(t) =

cosωt sinωt 0

− sinωt cosωt 0

0 0 1

0

S0,y

S0,z

=

S0,y sinωt

S0,y cosωt

S0,z

, t < ttip. (7.3)

At t = ttip we assume that the tipping pulse rotates the spin through an

angle φtip about the y-axis. Since the duration of the TP is much less than

ω−1, the tipping is assumed to occur instantaneously.

~S(ttip) =

cosφ 0 sin φ

0 1 0

− sinφ 0 cosφ

S0,y sinωt

S0,y cosωt

S0,z

, t < ttip. (7.4)

168

At t > ttip, this state ~S(ttip) then precesses freely about the z-axis at the

Larmor frequency:

~S(t > ttip) =

cosωt′ sinωt′ 0

− sinωt′ cosωt′ 0

0 0 1

~S(ttip) =

S0,y(cosφtip sinωttip cosωt′ + cosωttip sinωt′) − S0,z sinφtip cosωt′

S0,y(− cos φtip sinωttip sinωt′ + cosωttip cosωt′) − S0,z sinφtip sinωt′

S0,y sin φtip sinωttip + S0,z cosφtip

(7.5)

where t′ = t− ttip.

As can be seen from Eq. 7.5, the TP may result in a significant non-

precessing component of the spin along the z-axis. Through the hyperfine

interaction, electron spins can flip with nuclear spins. The component of

the nuclear spin along the external field Bz does not precess, and there-

fore can build over time in the process of dynamic nuclear polarization

(DNP) [14, 105] (see Section 2.1 and Chapter 6). As nuclear polarization

builds along the z-axis, it acts back on the electron spin as an effective mag-

netic field Bn. Thus in Eqs. 7.3 and 7.5, ω must be replaced by ω′ = ω+ωn.

ωn(= gµBBn/h) is proportional to the steady state nuclear polarization,

which in turn is proportional to Sz. Thus equating ωn and Sz with a

169

constant of proportionality α, we have

ωn = α(S0,y sinφtip sinωnttip + S0,z cosφtip). (7.6)

Solving this equation numerically for ωn and substituting ω′ into Eqs. 7.3

and 7.5, self-consistently yields the coherent spin dynamics of the system

as a function of time. In calculating the dynamic nuclear polarization,

we take the steady-state spin in the z-direction to be proportional to the

third component of Eq. 7.5. In principle, there is also a small component

present before the tipping pulse in Eq. 7.3, but we find that it does not

substantially affect the results so we have neglected it for simplicity. The

magnitude of the nuclear polarization seen here is consistent with that

observed in Chapter 6 (Ref. [124]). This provides a nice confirmation of

this model, since in Chapter 6 the nuclear polarization was just included

phenomenologically as a shift in the precession frequency, while here the

misalignment is included in the model.

To model the results below, we include the finite spin coherence time,

T ∗2 , and a phenomenological Itip-dependent term to account for imperfect

fidelity of the spin rotations or other background effects, with a character-

istic scale, I0. Finally, we take the difference between σ+ and σ− pump

170

helicity yielding:

θ0(t; ~S0, ω, Itip, α, T∗2 , I0) =

[Sy(t; ~S0) − Sy(t;−~S0)

]exp(−t/T ∗

2 ) exp(−Itip/I0).(7.7)

The six parameters in the model are given explicitly in Eq. 7.7: the

initial spin direction ±~S0, the precession frequency ω, the intensity of the

TP Itip, the DNP efficiency α, the time-averaged transverse spin lifetime

T ∗2 , and the phenomenological factor I0. The term with I0 is chosen to be

a decaying exponential, but other forms, such as a linear dependence work

just about equally well within the experimental uncertainty.

7.4 Ultrafast optical spin manipulation

Figure 7.6 a shows the time evolution of a single spin in a transverse mag-

netic field, with no TP applied. Each data point is determined from the fit

to a KR spectrum at a given pump-probe delay, as in Fig. 7.4. If we con-

volve Eqs. 7.3 and 7.5 with the measured profile of the probe pulse (shown

in Fig. 6.2), we can perform a least-squares fit to this data and determine

various parameters in the model: ω, T ∗2 , and the effective field from the

nuclear polarization, Bn. The red curve in Fig. 7.6 a shows the result of

this fit, and the dotted line is the corresponding plot of Eqs. 7.3 and 7.5

without the probe pulse convolution. As expected, the spin is initialized

171

Pump-probe delay (ns)

0 2 4 6-2

θ 0(a

.u.)

0

0

0

TP off

ttip = 1.3 ns

ttip = 2.6 ns

(a)

(b)

(c)

T = 10 K

Bz = 715 G

fit

model

model

x

zy

B

Sv ↓+↑

↓+↑ i

↓−↑ i

↑

Sv

TP

TP

↓

Sv

Figure 7.6: Coherent rotation of a single electron spin. (a) Coherent spin precession

with no TP. Error bars indicate the standard error in the fits to the KR spectra. The

red curve is a fit to the model convolved with the probe pulse, and the dotted line is the

same, without the probe pulse convolution. The fit yields parameters ω = 1.39 GHz,

T ∗

2 = 5.5 ns, and Bn = 68 G. (b) and (c) Same conditions as (a) but with the TP applied

at ttip = 1.3 ns and ttip = 2.6 ns, respectively, with intensity Itip = 4.7 × 105 W/cm2

and detuning ∆ = 2.65 meV, to induce a 1.05π rotation. The red curves in (b) and

(c) are from the model, using parameters obtained elsewhere. The gray dashed line

highlights the change in sign of the spin precession in (b) as compared to (a) and (c).

The diagrams at the right illustrate the effect of the TP on the spin dynamics.

172

at t = 0, and then precesses freely about the applied field.

It should be noted that in fitting the time-resolved data shown in

Fig. 7.6 a, we obtain the effective nuclear field Bn, which is not explic-

itly a parameter in the model. In fact, Bn depends on both the mis-

alignment S0,z/S0,y and the dynamic nuclear polarization (DNP) efficiency

α. However, the misalignment is determined from the data in Fig. 7.7

(S0,z/S0,y = −0.11). Using this number, α is found to be 1.2, and all of

the parameters in the model are specified.

The data in Fig. 7.6 b and c show the same coherent spin dynamics of

Fig. 7.6 a, but with the TP applied at t = ttip. The intensity of the TP

is chosen to induce a 1.05π rotation about the y-axis, which is determined

as discussed below. In Fig. 7.6 b, the TP arrives at ttip = 1.3 ns, when

the projection of the spin is mainly along the x-axis. This component of

the spin is thus rotated by the TP through ∼ π radians. The predicted

spin dynamics as given by Eqs. 7.3 and 7.5 is shown in the dotted red

line, and the same curve convolved with the probe pulse is given by the

solid red line. Note that this curve is not a fit – all of the parameters

are determined either in the fit to Fig. 7.6 a, or as discussed below. Only

the overall amplitude of the curve has been normalized. Here, the spin

is initialized at t = 0, and as before, precesses freely until the arrival of

173

the TP. After the TP, the spin has been flipped and the resulting coherent

dynamics show a reversal in sign. This can be clearly seen by comparing

the sign of the measured signal at the position indicated by the dashed line

in Fig. 7.6.

Figure 7.6 c shows the spin dynamics again with the same parameters,

but with ttip = 2.6 ns. The spin at this delay will have only a small

projection in the x-z plane and the TP-induced rotation about the y-axis

will have only a small effect on the spin state. This expectation is borne out

in the data, where the spin dynamics show essentially the same behavior as

in the absence of the TP (Fig. 7.6 a). Again, the model yields qualitatively

the same behavior.

7.5 Further exploration and control

measurements

Further details of this spin manipulation can be investigated by varying the

TP intensity, Itip, and the detuning, ∆ of the TP from the QD transition

energy for a fixed tprobe and ttip as illustrated in Fig. 7.7 b. In Fig. 7.7 a, the

KR signal, θ0, as a function of Itip is shown at tprobe = 2.5 ns with the TP

arriving at ttip = 1.3 ns, for three different values of ∆. When Itip = 0, the

spin precesses undisturbed and yields a negative signal at tprobe = 2.5 ns as

174

TP intensity (105 W/cm2)0 2 4

0

-20

-40

40

20

θ 0at

tp

rob

e=

2.5

ns

(µra

d)

∆ = 1.64 meV

∆ = 3.66 meV

∆ = 5.26 meV

T = 10 K

Bz = 715 G

π(a)

tprobe

ttip

Pump-probe delay (ns)0 4

θ 0(a

.u.)

0

Itip = 0

(b)

Figure 7.7: Dependence of Stark tipping on tipping pulse intensity and detuning. (a)

Single spin KR amplitude, θ0 as a function of TP intensity, Itip, at three detunings from

the X− transition. The tipping pulse arrives at ttip = 1.3 ns, and the probe is fixed

at tprobe = 2.5 ns, as illustrated in (b). The gray lines are fits to the model, varying

only one parameter, the strength of the OSE, β, and misalignment S0,z/S0,y = −0.11,

and I0 = 6.9 × 105 W/cm2. The tipping pulse intensity required for a π rotation at

∆ = 1.64 meV is indicated by the arrow.

in Fig. 7.6 a. As Itip is increased, the spin is coherently rotated through an

increasingly large angle, and the observed signal at tprobe = 2.5 ns changes

sign and becomes positive, as in Fig. 7.6 b. Furthermore, the strength of

the OSE is expected to decrease linearly with the detuning ∆, as seen in

Eq. 7.1. The gray lines in Fig. 7.7 a are plots of Eq. 7.5 with parameters

taken from the fit in Fig. 7.6 a, and φtip = βItip. From these curves, we

additionally obtain the phenomenological factor, I0 = 6.9 × 105 W/cm2.

175

Detuning (meV)

0 2 4 6

I π(1

05

W/c

m2)

0

5

10

Figure 7.8: Strength of the Stark effect as a function of detuning. The tipping pulse

intensity, Iπ required for a π rotation as a function of detuning from the X− transition.

The red line is a linear fit, through the origin.

This same value is used in all of the model curves shown. From this, we

can estimate the fidelity of a π-rotation to be approximately 80%. Here,

the fidelity is calculated as the actual signal after a π-rotation divided by

the difference between the ideal signal with and without a π-rotation. The

only parameter that is changed between the three curves in Fig. 7.7 a is

the strength of the OSE, β. The TP intensity required for a π-rotation,

Iπ = π/β, is shown in Fig. 7.8 as a function of detuning, displaying the

expected linear dependence.

The slope of the line in Fig. 7.8, 1.7 × 105 W/meV·cm2, is a measure

of the observed strength of the optical Stark effect. If we assume a typical

dipole moment for this QD transition, D = 50 Debye, then from Eq. 7.1,

176

we expect this slope to be 1.3× 104 W/meV·cm2. The origin of this factor

of 13 reduction of the optical Stark shift is not clear, but may be due to loss

in the planar cavity as the light spreads away from the QD to fill the cavity

volume, as discussed in Chapter 5. This effect would not significantly affect

the KR enhancement, since we only measure light that is re-emitted from

the cavity at the same point. For the OSE however, the magnitude of the

effect depends on the intensity of the light at the QD. In fact, we measure

a factor of 10 decrease in the intensity of light from going into to coming

out of the objective. Whether or not this loss is due to the spreading of

the light in the cavity, this is likely related to the smaller-than-expected

OSE. (This factor of 10 is a lower bound, since there is also some outgoing

light from back-reflections off of the objective and other surfaces.)

The data in Fig. 7.7 most clearly show the effects of DNP on the ob-

served spin dynamics. In the absence of nuclear polarization, one would

expect the curves in Fig. 7.7 a to be cosinusoidal, crossing zero at an inten-

sity half that required for a π rotation. DNP, however, which is maximal

when φtip ≈ π/2, distorts this ideal cosine form, as is well-described by

the model. Additionally, an effect of the misalignment of the initial spin

direction out of the x-y plane can be seen in the signal near Itip = 0. The

TP first rotates the spin into the x-y plane, increasing the signal slightly,

177

TP intensity (105 W/cm2)

0 1 2 3 0 1 2 3

0

-20

-40

20

40θ 0

at t

pro

be=

2.5

ns

(µra

d) TP σ-, -I0

TP σ+, I0

TP σ-, I0=∞

TP σ+, I0=∞

TP σ-

TP σ+

(a) (b)

Figure 7.9: Comparison of the Stark effect with tipping pulses of opposite helicity. (b)

Single spin KR amplitude, θ0 as a function of TP intensity for both helicities of the TP.

As above, the TP arrives at ttip = 1.3 ns and the probe is fixed at tprobe = 2.5 ns. (a)

Model curves of the data in (b), showing qualitative agreement. For the two helicities,

the sign of I0 is reversed. The dotted lines show the model with I0 = ∞ (perfect fidelity,

no background effects).

and then rotates it past the x-y plane as Itip is increased. This is confirmed

by reversing the helicity of the TP so that the spin rotation is in the same

direction as the misalignment, instead of against it (Fig. 7.9).

In order to confirm that we are manipulating a single electron spin,

we performed several additional control measurements. Even though the

modulation and lock-in detection should prevent the measurement of spin

coherence generated by the TP, we have repeated some of the measure-

ments with the helicity of the TP reversed. This should reverse the direc-

tion of the spin rotation, which, to a first approximation, should not affect

178

the measured signal. However, the symmetry of the situation is broken

by the initial misalignment S0,z/S0,y, which yields a small but measurable

difference, depending on whether the TP rotates the spin in the same or

opposite direction as the misalignment. The black points in Fig. 7.9 b show

the same type of data as in Fig. 7.7 a with TP detuning ∆ = 2.65 meV.

The red data is the same, but with the TP helicity reversed. Significantly,

the observed behavior is qualitatively the same. If the observed effect were

an artifact due to TP-generated spin polarization, one would expect to see

a sign change with the TP helicity. The fact that there is no sign change

is strong evidence for the spin manipulation model described above.

The solid curves in Fig. 7.9 a are from the model, using the same

parameters as in the text. The only change is that for the red curve,

the parameter I0 (accounting for TP-induced background effects or loss of

fidelity) was multiplied by (-1). For comparison, the dotted curves show

the prediction of the model with perfect fidelity (I0 infinite). Apparently,

the sign of I0 depends on the helicity of the TP. This provides a clue as

to the relevant mechanisms, but more study is required for a complete

understanding. Regardless of the details of the mechanisms underlying

these background effects, the model captures the essential features of the

data, including the effects of the misalignment, effectively shifting the two

179

0 2 4 6 8

0

20

40

-20

Delay (ns)

θ 0(µ

rad)

TP σ+

TP σ-

TP off

0 2 4 6

Delay (ns)

circularly pol.

pump

linearly pol.

pump

(a) (b)

Figure 7.10: Comparison of measurement with circularly and linearly polarized pump.

(a) The usual time-resolved single spin measurement, but with linearly polarized pump.

Thus non-spin-polarized electrons are injected at t = 0, and in the red and blue data,

the TP is applied at ttip = 1.3 ns. (b) For comparison, the single spin dynamics with

circularly polarized pump, and no TP.

curves horizontally, most visible at low TP intensity.

In order to further isolate any unwanted effects of the TP, we performed

another control measurement in which the pump was linearly polarized, so

that electrons were still injected into the QD, but they were not spin po-

larized, shown in Fig. 7.10 a. We then performed the usual pump-probe

delay scan measurement as described above, with the circularly polarized

TP arriving at ttip = 1.3 ns. This measurement was performed with both

helicities of the TP, and with the TP blocked. The resulting signal was

barely above the noise floor, though there appeared to be some TP-induced

signal whose sign depended on the TP helicity with a magnitude of several

180

microradians. This is an order of magnitude smaller than the signal ob-

served with the circularly polarized pump (Fig. 7.10 b), and therefore sets

an upper bound on any spurious signal due to TP-generated spins.

Much of the deviation of the data from the model can be explained by

slow drift of experimental parameters during the measurement. In par-

ticular, the observed effects are very sensitive to the focus on the sample,

since the intensities of the pump, probe and TP all vary quadratically with

the focused spot size. Additional deviations may be due to the simplistic

description of the TP-induced background effects used here. For example,

in the case of phonon-assisted transitions to the trion state, one would ex-

pect the type of spin-selective decoherence described in Ref. [122]. Further

measurements of the background effects will be needed to determine their

cause, with the aim of increasing the fidelity of these single spin rotations.

In conclusion, we have demonstrated the ability to coherently rotate a

single electron spin through angles up to π radians on picosecond timescales.

A simple model including interactions with nuclear spins reproduces the

observed electron spin dynamics with a single set of parameters for all of

the measurements. In principle, at most 200 single qubit flips could be per-

formed within the measured T ∗2 of 6 ns. However, by using shorter tipping

pulses and QDs with longer spin coherence times, this technique could be

181

extended to perform many more operations within the coherence time. A

mode-locked laser producing ∼ 100-fs-duration tipping pulses could poten-

tially exceed the threshold (∼ 104 operations) needed for proposed quan-

tum error correction schemes [47]. Additionally, the spin manipulation

demonstrated here may be used to obtain a spin echo [126], possibly ex-

tending the observed spin coherence time. These results represent progress

toward the implementation of scalable quantum information processing in

the solid state.

182

Chapter 8

Conclusion

In this dissertation, results have been presented focusing on measuring

and controlling coherent spin dynamics in semiconductor QDs. Faraday

and Kerr rotation have served as useful probes of the spin polarization.

Chapters 3 and 4 investigated novel layered nanocrystal QD structures.

First, layered nanocrystals were chemically synthesized with a low bandgap

shell, sandwiched between a high bandgap core and outer shell. These

structures were characterized using PL spectroscopy, and the spin dynam-

ics were probed using time-resolved Faraday rotation measurements. The

results revealed that these nanocrystals act analogously to a planar QW,

but in a spherical, nanoscale geometry. Specifically, the QD energy levels

and g-factors shift with the shell thickness, in agreement with calculations.

This structure can be taken further by adding two coupled regions in

183

one nanocrystal. With structures consisting of a low bandgap core and

shell, separated by a high bandgap barrier, electrons and holes are confined

to either the core or shell region. As with single shells, PL and Faraday

rotation measurements indicate that the energy levels and g-factors depend

on the core and shell dimensions. Furthermore, with appropriate selection

of the pump and probe energies, the core and shell states can be initialized

and read-out selectively.

These types of QD structures are interesting because of their simplicity

of fabrication, and tremendous flexibility in design. However, this simplic-

ity and flexibility also leads to complications such as surface defects, and

unknown factors in the environment of the QDs. In Chapters 5, 6 and

7, we switch to MBE-grown interface fluctuation QDs which have fewer

tunable properties but are more well-behaved than nanocrystal QDs.

In Chapter 5, Kerr rotation was employed as a non-destructive, optical

probe of single spins in a QD. By measuring the KR spectrum, the expected

signature was identified at the expected energy from a single electron spin

in the QD. Measurements of the spin in a magnetic field showed the ex-

pected depolarization of the spin, and yielded information about the spin

lifetime and spin decoherence mechanisms.

The single spin KR technique was then extended into the time domain

184

(Chapter 6), using pulsed pump and probe lasers. In this measurement,

we probe the coherent dynamics of the spin in the QD with nanosecond

temporal resolution. This allows us to determine the g-factor and spin

lifetime of the electron, and also gain further insight into the relevant spin

decoherence mechanisms. Additionally, we observe effects of nuclear spin

polarization on the spin dynamics, making this measurement a sensitive

probe of the local nuclear spin environment.

Finally, in Chapter 7, we demonstrate ultrafast coherent manipulation

of the spin in the QD, observing the resulting dynamics using the single

spin KR measurement. Here, a third pulse is applied to the QD, detuned

from any optical transitions to generate an effective magnetic field via the

optical Stark effect. This large, transient effective magnetic field allows the

spin to be coherently rotated through angles up to π radians, on picosecond

timescales.

These results have touched on initialization, manipulation, and detec-

tion of spin coherence in semiconductors. These are three of the key re-

quirements for applications in both spintronics and quantum information

processing. Together with the flexibility of semiconductor technology, these

results provide some hope that we might one day fulfill the promise of spin-

based technology. In the words of Martin Luther King, Jr., “He’s allowed

185

me to go to the mountain. And I’ve looked over, and I’ve seen the promised

land! I may not get there with you, but I want you to know tonight that

we as a people will get to the promised land.”

186

Appendix A

Sample structure and

processing

The cavity structure for the single spin measurements was designed using

the Vertical simulation software package. Initially, samples were fabricated

with higher-Q cavities, with more AlAs/AlGaAs layers in the top DBR.

However, this made the QD charging and the isolation of single QDs more

difficult. Both of these effects are because with more layers in the top

DBR, there is more distance from the surface of the sample to the QD

layer. Looking at Fig. 5.5 a, with a thicker top DBR, there is a larger

lever arm raising and lowering the QW with respect to the Fermi energy.

Thus large voltages would be necessary to change the charging of the QW.

Also, with a thicker top DBR, the 1 µm diameter apertures at the surface

are separated from the QDs by something like 2 µm, which doesn’t help

187

in terms of isolating single QDs. So we have compromised by making a

lower-Q cavity with a thinner top layer. However, the cavity design has

been made compatible with a three-dimensionally confined pillar design in

which Q-factors as high as 48,000 have been achieved [127]. This provides

an interesting future avenue to pursue.

The doping scheme for the sample was designed using the self-consistent

Schrodinger-Poisson solver, “1d Poisson”, made by Gregory Snider. The

structure is based on previous designs [84], but the presence of the cavity

adds an extra complication. The QW must be placed near the back contact

to allow electrons to tunnel in and out. This is fine in the structure in Ref.

[84], but with the large lever arm due to the front DBR, an unrealistic

voltage would need to be applied to pull the QW above the Fermi energy.

To solve this problem, a thin p-doped region is added, as shown in Fig.

5.5 a. This forms a p-n junction, which has the effect of pulling the bands

up in the region of the this junction, positioning the QW minimum near

the Fermi energy.

The growth of the IFQD samples was carried out by N. G. Stoltz. Care

must be taken to calibrate the growth rates accurately. Small deviations

in the thickness of cavity layer shift the resonance away from the desired

wavelength. (A 1% error in the thickness yields a 1% error in the resonance

188

wavelength, so if the target is 750 nm, instead you would get 758 nm - a shift

greater than the width of the resonance.) The samples were grown on 2 or

3 inch diameter semi-insulating GaAs substrates. Due to inhomogeneity

of the beam flux over the wafer, the cavity resonance varied across the

wafer. Using a simple reflectivity setup, we could map out the resonance

wavelength in real time to select the piece of the wafer that we wanted to

use.

A piece of the wafer with the desired cavity resonance was then cleaved

(typically 6 × 8 mm), and processed as follows:

1. Mask half the sample with AZ4110 photoresist.

2. Etch with 80:8:1 H2O:H2O2:H2SO4 down to back n-doped layer (∼

8 nm/s).

3. Remove resist, and apply AZ4251. Negative expose and develop two

square areas on the etched region, for back contacts.

4. E-beam evaporate Ni/Ge/Au/Ni/Au (5/25/65/20/200 nm) onto

the sample, and liftoff (in acetone, usually with some sonication to help it

along).

5. Anneal in forming gas, using the strip annealer at 420C for 1 minute.

Check that the two back contacts are now Ohmic (linear I-V). If not, anneal

more.

189

6. Apply maN-2403 resist for negative e-beam exposure.

7. Write the desired pattern of apertures using e-beam lithography (I

have used both a now-decommissioned JEOL system, and an FEI Sirion

system for this step). The pattern is typically 5 × 5 arrays of circular

apertures with diameters ranging from 200 to 2000 nm.

8. Develop in CD-26 for 1 minute, leaving circles of resist where the

apertures will be.

9. Evaporate 80 nm of Ti over the whole sample and lift-off the aper-

tures, resulting in the sample covered entirely in Ti, minus the apertures.

Lift-off is done using 1165 stripper at 80C, with a little sonication if it

looks like its not all coming off.

10. Define the front contact in photoresist over the apertures.

11. Etch in 100:1 H2O:HF for about 30 s, until the Ti has disappeared

everywhere but the front contact, and the underlying sample becomes vis-

ible.

12. Remove the resist, and now the sample is done. Check the I-V to

make sure it shows rectifying diode-like behavior.

The sample was mounted on the cold finger of the cryostat using con-

ductive silver paint, and wires were attached to the contacts by pressing

the wires into Indium.

190

Appendix B

Details of single spin detection

and manipulation

Figure B.1 shows again the setup for single spin measurement and manip-

ulation. Note that this diagram is not to scale, and leaves out additional

components such as lens pairs for collimating the beam and changing the

beam diameter, mirrors for additional beam steering, and filters for attenu-

ating the beams. Also, the order of some components has been rearranged

in the diagram for clarity when it doesnt affect the functionality. Here, I

will describe the setup in more detail.

191

Ti:Sa 2

(cw probe)

chopper

LCVW

polarizer

polarizer

scanning

objective

sample

He flow cryostat (10 K)

electro-

magnet

Fe yoke

bandpass

filter

to spectrometer

(for PL)

to diode bridge

(for KR)

optional

mirrorf1

f2

VA-Block-in1

fref=f2

lock-in2

fref=f1

pre-amp

VAlock-in3

fref=f1

for

normalization

signal

~150 fs

~ ps

Ti:Sa 1

(pulsed pump/TP)

sync

EOM

pulse generator

5.1≥ ns

chopper

monochromators

~ 25 ps

polarizer

λ/4 plate

del

ay

tprobe

ttip

Figure B.1: Setup for single spin measurement and control.

192

B.1 Optical path

The laser used for the pump and tipping pulses is a Coherent Mira Ti:sapphire

laser which is operated in CW for the time-averaged measurements (Chap-

ter 5) and is mode-locked for the time-resolved measurements (Chapters 6

and 7). The probe laser is a Coherent 890 Ti:sapphire ring laser, which out-

puts a tunable, narrow linewidth CW beam. The wavelength of the probe

laser is controlled by means of a stepper motor driving the birefringent

filter in the laser cavity.

For the time-resolved measurements, the probe is converted from CW to

pulses by means of an electro-optic modulator (EOM), purchased from EO-

Space, Inc. Light is coupled in and out of the EOM by means of a single-

mode, polarization-maintaining fiber. A typical extinction ratio for the

EOM was around 50:1. The EOM is driven using a pulse generator (Agilent

81110A) with minimum pulse width of 1.5 ns and electrically controllable

delay with picosecond resolution. The pulse generator is triggered by the

output of the photodiode in the pump laser cavity. As was mentioned in

Chapter 6, we cannot use a mode-locked laser for the probe pulse (though

that was our first idea). It is virtually impossible to get a pulse from a

mode-locked laser to be longer than about 100 ps. In order to maintain the

same total power to the detector, this means that the instantaneous power

193

in the pulses would be significantly higher than in the present case. As

was discussed in Chapter 7, this would lead to a significant optical Stark

shift as the probe laser approaches the transition. The effect is that the

QD resonance avoids the probe laser, making the range of small detuning

inaccessible.

The monochromator in the pump path is a Mini-chrom spectrometer

from Optometrics LLC, with manual control of the grating and path length

of 74 mm. Here, we just set the grating to the desired wavelength, and then

never touch it. The delay line in the pump path consists of a retroflector

mounted on a sliding cart, which is driven by a stepper motor. This delay

line is used to control the time delay between the pump pulse and tipping

pulse.

The monochromator in the tipping pulse path is an Acton SP2150i,

with a path length of 0.15 m. The grating here is controlled via GPIB.

The three beams are all overlapped and sent through the microscope

objective, a Nikon L-plan SLWD with a numerical aperture of 0.7 and

working distance of 6 mm. The microscope objective is mounted on a

piezo-electric-actuated stage (P-563 from Physik Instrumente) with 300 µm

range in x, y, and z, and nanometer positioning accuracy. The piezo stage

is in turn mounted on a PM500 motorized nanomover stage from Newport,

194

for coarse positioning in x, and y (where z is the focus). The cryostat is

mounted on a stepper-motor-driven translation stage which allows coarse

focus positioning, as well as allowing the cryostat to be pulled back for

putting in and taking out the sample.

The temperature of the sample is monitored by a temperature sensor

mounted at the end of the cold finger, near the sample. The bias voltage

is applied to the sample by means of a Yokogawa 7651 DC power supply.

In the outgoing light path, a filter is used to block the pump light,

and in the Stark tipping measurements, the tipping pulse. When the TP

was not present, a long-pass optical filter from Omega Filters was used. In

order to block both the pump and TP, while passing the probe, a bandpass

interference filter was used from CVI optics. The angle of the filter was

adjusted to optimize the attenuation of the pump and TP.

For PL measurements, the light was free-space coupled into an Ac-

ton AM510 1-m spectrometer, and detected using a Princeton Instruments

liquid-nitrogen-cooled CCD. For KR measurements, the probe was sent to

a diode bridge setup as shown in Fig. 2.12. In this case, the angle of

the half-wave plate was adjusted via a stepper motor to keep the diode

bridge balanced (see below for more details). The diode bridge circuit

was home-built and based on the design considerations outlined in Ref.

195

[16]. The diode bridge had outputs for each photodiode, A and B, and

the difference, A-B. The difference signal was sent to a Stanford Research

preamplifier with gain of 100 and bandpass filter from 3 kHz to 10 kHz.

The signal then went to two lock-in amplifiers (EG&G 726X models), used

as described above. Additionally, the A channel alone was sent to another

lock-in amplifier with reference frequency at the probe chopper frequency

(f1) to normalize the KR signal, and to monitor the reflectivity for posi-

tioning purposes (see below for more details).

B.2 Measurement control scheme

The measurement of the KR spectra was controlled by a Labview routine

that was set up as follows:

1. The stepper motor on the ring laser sets the initial probe energy.

2. The diode bridge is automatically balanced (see below).

3. Every nth iteration, the automatic positioning routine is run (see

below).

4. The pump helicity is set to σ+ (using the liquid crystal variable

waveplate).

5. Wait for a time tswitch.

6. A value from each of the lock-ins is stored.

196

7. The pump helicity is switched to σ−.

8. Wait for a time tswitch.

9. Take another value from each lock-in.

10. Increment the probe energy and if the end of the scan has not been

reached, return to step 2.

11. Save the results.

The above sequence is for taking a single KR spectrum. This procedure

is then incorporated into other routines that take a series of KR spectra,

incrementing some other parameter, such as pump-probe delay, or tipping

pulse intensity. In these longer scans, the routine to optimize the focus and

zero the electro-optic modulator (see below) are run in between successive

KR spectra.

B.3 Odds and ends

There are a few routines that were used to keep the system stable over long

periods of time, so that long delay scans (∼ 10 hours) could be automated.

A significant source of error in the single spin measurements is slow drift

of the sample relative to the laser spot. (For a 1 µm diameter spot, even

100 nm of drift is a significant error.) To compensate for this, the aper-

ture is automatically repositioned every so often. This is accomplished by

197

scanning the laser in a horizontal and vertical sweep over the aperture and

measuring the reflectivity using the A channel of the diode bridge. The

resulting curves are then fit to a Gaussian function to accurately determine

the position of the center of the aperture.

Additionally, drift of the focus position is also a significant issue. To

compensate for this drift, the same reflectivity measurements are performed

as for the positioning routine. Here, however, the measurement is repeated

at a number of different focus positions and the width of the fit curves

is plotted versus focus position. Having previously determined the width

when the laser is optimally focused, we use a linear fit of these values to

refocus the lasers.

For optimal operation of the diode bridge, the amount of light going

into the A and B photodiodes should be as close to equal as possible (for

“common-mode” rejection). This is accomplished by rotating the half-wave

plate to equalize the two channels. However, as the probe wavelength is

scanned the balanced position of the waveplate changes slightly, possibly

due to birefringence in the optics in the setup. For this reason, the wave-

plate is automatically rotated using a stepper motor to rebalance the diode

bridge. The balance of the bridge is detected by sending the difference out-

put (after the preamp and filter) to another lock-in amplifier at the probe

198

chopper frequency (f1). Even though this frequency is filtered out, there

is still a series of spikes in the signal at this frequency, proportional to the

unbalanced-ness of the bridge. The output of this lock-in is then zeroed by

using a simple search algorithm to move to stepper motor on the half-wave

plate.

One more source of long-term drift is in the electro-optic modulator.

The modulator is used by applying a DC bias voltage to completely null the

output of the modulator. Then the pulse generator supplies an additional

voltage to make the modulator transmissive. Over time however, the DC

bias needed to null the signal drifts (possibly due to charging effects in the

modulator). So every so often, the pulse generator is turned off, and the

DC bias voltage is scanned in a range around the present voltage, while

monitoring the A channel of the diode bridge. The resulting curve (which

has a minimum close to zero where the modulator is nulled) is fit to a

Gaussian function to determine the new optimum bias voltage.

199

Appendix C

Other theoretical views of

Faraday rotation

Though this dissertation focuses on experimental results, it is useful to

have some feeling for the theory underlying these phenomena. In Sec-

tion 2.4, the origin of the Faraday rotation effect was outlined in terms of

time-dependent perturbation theory for an electron subjected to a classical

radiation vector potential, A. In this appendix, I will describe two other

ways of thinking about Faraday rotation. There will be no rigorous calcu-

lations, just a sketch of the theoretical description to hopefully give some

additional intuition regarding these phenomena.

One way of seeing the origin of the Faraday rotation effect is to con-

sider the (classical) coefficient of absorption and index of refraction of a

material [15].

200

abso

rpti

on

ind

ex o

f re

frac

tio

n

photon energy

σ+σ−

σ− σ+

σ+ − σ−

Figure C.1: Top: Spin-dependent absorption for left and right circularly polarized

light. Bottom: Corresponding indices of refraction, and the resulting Faraday rotation

spectrum.

The real and imaginary parts of an analytic function χ(ω) = χ1(ω) +

iχ2(ω) are related by the Kramers-Kronig relations:

χ1(ω) =2

πP∫ ∞

0dω′ω

′χ2(ω′)

ω′2 − ω2

χ2(ω) = −2ω

πP∫ ∞

0dω′ χ1(ω

′)

ω′2 − ω2,

(C.1)

where P indicates the principal value.

201

The dielectric response function is such a function, whose real and imag-

inary parts yield the index of refraction and absorption spectra. The ab-

sorption spectrum near the band-edge of a semiconductor, shown schemat-

ically in the top panel of Fig. C.1, yields a peak in the index of refraction

at the energy of the band edge. In the figure, the band-edge energies are

shown to be different for oppositely polarized light, which can be caused by

Pauli blocking from a population of occupied electron states of a particu-

lar spin. The indices of refraction corresponding to the absorption spectra

(via Eq. C.1) are shown in the bottom panel of Fig. C.1. As discussed in

Section 2.4, the Faraday rotation angle is given by the difference in the

index of refraction for right and left circularly polarized light, shown in

black. Note that this curve is similar to the one in Fig. 2.10 b.

An alternative way of looking at Faraday rotation is in a picture where

both the electron and the photons are considered to be quantized [128].

Here, the effect can be thought of as forward scattering of the photons

off of the electron. In the second-quantized picture, photons are described

in terms of creation and annihilation operators: a†xkand axk, respectively,

where k labels the photon momentum and x indicates linear polarization

along x. The creation operator for a circularly polarized photon would

then be a†±,k = a†xk± ia†yk for σ± polarization.

202

The interaction between the electron and the photon is given by the

Hamiltonian Hint = −p · A/m, as in Section 2.4. The interaction Hamil-

tonian can be written down for the case of a dipole transition with energy

Ω and dipole moment D (for example, a heavy hole to conduction band

transition), and simplified by only considering the off-resonant parts. The

form of this Hamiltonian can be guessed from the symmetry considerations

discussed above. Namely, a photon of a given helicity will only interact

with a transition to one spin state. Including only non-resonant, forward

scattering the interaction Hamiltonian is given by

HS = 2Λ∑

kk′

a†+,k′a+,k| ↑〉〈↑ | + a†−,k′a−,k|↓〉〈↓|

= Λσz

∑

kk′

a†+,k′a+,k − a†−,k′a−,k,

(C.2)

where the sum is over all photon momenta, and the prefactor can be cal-

culated to be Λ = D2hΩ/2V δ. Here δ is the detuning of the photon energy

ωk away from the transition energy, δ = h(Ω − ωk). (There is a constant

offset between the two lines in Eq. C.2.)

Now we can consider the effect of this Hamiltonian as a perturbation

on an initial, coherent photon state linearly polarized along x

|ψ0〉 = eαa†

xk0 |0〉, (C.3)

where |0〉 is the vacuum, and α is a complex number related to the am-

plitude and phase of the light. The perturbation to this state from the

203

Hamiltonian above can be calculated using time-dependent perturbation

theory. However, the result can be qualitatively anticipated by rewriting

the perturbation in terms of linearly polarized photons:

HS = 2iΛσz

∑

kk′

a†xk′ayk + a†yk′axk. (C.4)

If this Hamiltonian acts on photons polarized along x, there are some

y photons mixed in, and vice versa. That is to say, the angle of the polar-

ization is rotated.

The perturbed state can be calculated quantitatively, and the Faraday

rotation angle can be calculated from the expectation value for the electric

field of the light. Assuming the rotation angle is small, the Faraday rotation

resulting from a spin-up (down) electron is given by

θF = 〈Ey〉/〈Ex〉 = ±CD2

hc

(hΩ

δ

)

, (C.5)

where C is a factor that depends on the volume in which the electron and

photon interact. This reproduces the odd-Lorentzian form of Eq. 2.19 in

the limit of no transition broadening.

204

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