Spin Engineering in Magnetic Quantum Well Structures

UNIVERSITY OF CALIFORNIA Santa Barbara Spin Engineering in Magnetic Quantum Well Structures

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

Doctor of Philosophy in Materials Science by Roberto Correa Myers

Committee in charge:

Professor David Awschalom, Co-Chairperson Professor Arthur Gossard, Co-Chairperson Professor Evelyn Hu Professor Pierre Petroff Professor James Speck

December 2006 The dissertation of Roberto Correa Myers is approved:

September 2006 Spin Engineering in Magnetic Quantum Well Structures

iii To my abuelos and grandparents, Elvira and Ricardo Correa, Betty and Vernon Myers

iv Acknowledgments

My life in graduate school has, without question, been nothing but fun. I was un- believably lucky during my studies both personally and professionally. Mostly this good fortune involves having met, worked, and played with so many talented, funny, and entertaining friends and colleagues, many of whom I hope to remain lifelong friends with. The beginning of my luck was becoming a student of my advisors, David Awschalom and Art Gossard. Both Art and David have built unusually successful research groups, which by virtue of their close proximity and collegiality, have made astounding research progress at a pace not possible anywhere other than in Santa Barbara. They are both true individ- uals, both inspiring and entertaining, with flavorful character and experience. David’s ability to maintain an incredible pace of innovative research is well known. Without a doubt, David is one of the most rewarding and enjoyable professor to work with. Art’s Zen-like aura and chivalrous style are legendary in the semiconductor community. He is always available with keen advice and infinite patience, quintessentially the perfect role model. Fortune found me in Art’s lab, the best MBE center I know of. Who taught me to grow samples? Danny Driscoll and Micah Hansen, the old-guard of famed system C, showed me how you run and maintain an MBE system, taught me the traditions of beer 30, and pub-crawling. Zeke Johnston-Halperin taught me the tricks of magnetic semiconductor growth and how to measure these samples. I’m also indebted to him for his lab example in managing collaborations and running experiments, for which I will remain his most humble and obedient servant. John English is responsible for keeping the MBE lab from exploding on any given day, for maintaining a spirit of camaraderie, and for being a lim- itless fountain of technical expertise. Both his and Art’s wisdom, consistency, and dedication have allowed the MBE lab to flourish through the years. Andrew Jackson deserves praise for countless improvements to the spintronics lab, for example using his entrepreneurial-style and MBE skills to locate, disassemble, transport, and install an MBE system in a few months...and then do it a again a couple more times for some profit; I also thank him for letting me drive his Audi

v S4 at ridiculous speeds on the 101, borrow his African drum, and his surfboard. Jason Stephens is thanked for helping co-manage the MBE system. Spintronics- growers: Roland Kawakami, Ted Kreutz, Chad Gallinat, Mark White, Francisco Bloom, and Shawn Mack are also thanked for turning bolts and risking getting arsenic poisoning with me. I’m comforted to know that the system will be left in Shawn’s capable hands. Antonio Badolato, Brian Gerardot, Jeramy Zimmer- man, Josh Zide, Christoph Kadow are thanked for keeping system C running. Dave Buell, Max Andrews, Mel Maclaurin, Christy Poblenz, Sidarth Rajan, Dan Loftgreen, Dave Buell, Chad Wang, Andrea Corrion, and Moe are thanked for setting great examples of materials research while keeping the lab lively. Whilst plodding about campus, I luckily found myself in David’s optics lab. I particularly thank Scott Crooker for writing an excellent thesis. Much of my own work followed directly from his experiments, chapter 4. Dr. Ines Meinel, now a surfer girl, provided my ﬁrst tutelage in low temperature physics and was quite patient with my initial clumsiness. Darron Young and Jay (now professor) Gupta are thanked for their dedication to gooﬁness and howling in the optics-lab. Yuichiro Kato, aside from being an astoundingly clever physi- cist and a general all-around happy guy, is a man of character. He is thanked for joining me in drinking beers (sometimes on a roof), touring me through his country, and acting in a movie. My highly distinguished friend, Nate Stern (team-Mn), is thanked for measuring samples, seemingly, before I even grew them and putting up with ridiculous arguments between the other members of team-Mn. Felix Mendoza is particularly thanked for paying the bills on time, getting me out of quite a few jams, and having a good sense of direction. Jesse Berezovsky is praised for his musical taste, hookah smoking, and dancing feats. Vanessa Sih runs in short, unpredictable bursts, and sometimes brings baked goods. I thank Hadrian Knotz for sharing his ski chalet and inspiring the “can- tenna”. I’ve had the pleasure to get to know and collaborate with many talented postdocs: Geoff Steeves, a Canadian I can call friend, Alex Holleitner, my surfing doppleganger, Min Ouyang, Yongqing Li, Dave Steuerman, Wayne Lau, Ronald Hansen, Oliver Gwyant, Florian Meier, and Sai Ghosh. Good luck to the newer students: Maiken Mikkelsen, Viva Horowitz, William Koehl, Mark

vi Nowakowski, and Brian Maertz. I’ve benefitted from collaborations with our Penn State friends: K-C Ku, Ben-Li Sheu, X. Li, Nitin Samarth, and Peter Schiffer. The CNSI staff led by Holly Wu, are praised for their consistency in keeping everything running smoothly, and for being patient with my timecard. Martino “Tino”, “Pogg” Poggio has been my comrade in capers both scientific and nautical. His scientific skills and energy were instrumental to this research, while at the same time helping me sink sailboats and bring traffic lights down. Pogg (team-Mn) was the Chair of the now defunct Smelrose Institute and a banjo player in the Shanty-town players (slanty). His knack for story telling and goofy conversation are sorely missed. The frequent and heated arguments, although draining, always lead to an enlightening perspective. Jon “Butts” Miller was the director of said institute, co-founder of SLOW, a singer in the shanty, and co-developer of various other scams, such as the Es- cher Auto-baster, and without whom the shanty is not nearly slanty enough. I thank the other members of the institute and collaborators: Kristen “the pad- dler” Madler, Chelsey and Bailey Swanson, Steeve-o Harding, R1 and M2 Ras- mussen, Vanessa Silva, Erin Muslera, Ken and Kyra, Sarita, Sara Fretwell, Ryan Aubry for festival of life, T-day extravaganza’s, BBQ’s, concerts, surfing, sing- alongs, and for keeping me aware of society outside of the crunchy-science taco shell. The Via Regina crew: Rob (slanty), Rosco and Val, Jon and Amy, Mike and Jersey are praised for living the sweet life. Jason [and Kirsten] Haaheim, Jill Heinke, Jeff Danziger, the original members of the RJ3 quartet are thanked for letting me jam with them, thereby protecting my sanity. Other musical collaborators I thank are Kristin Wiseman, April Windsor, Tommy, Rob Wallace, Ralph Lowi, Max Katz, Colter Frazer, Chris Teeples, and the gang. I still think we could make it big someday. Mi linda amor, Maria, I thank for making my last year in school the greatest one of all. All the extended families, Myers, Perreira, Griggs, Maitra, Cor- rea, Roca de Togores, Guerra, Onofryton, Carrington, Rezazadeh, are the most highly entertaining, diverse, and interesting people I know. My family has been loving and supportive through the years: My parents Albert and Mayra put up

vii with me “repairing” many things, my big sister Natasha (Jorge and Sophia), my little sister Rosie, and my little brother Timmy. Abuela, Elvira, always supported me to pursue science, and my abuelo Ricardo taught me the joy of build- ing. My grandparents Betty and Vernon, both excellent scientist and teachers, are my inspiration and role models.

viii Vitæ

Education 2001 B.S.E. Materials Science and Engineering cum laude B.A. Philosophy and Science cum laude University of Pennsylvania, Philadelphia, PA 2006 Ph.D. Materials Science University of California, Santa Barbara, CA

Publications

Magnetic quantum wells

“Conﬁnement engineering of s-d exchange interactions in GaMnAs quantum wells”, N. P. Stern, R. C. Myers, M. Poggio, A. C. Gossard, and D. D. Awschalom, cond-mat/0604576 (submitted).

“Structural, electrical, and magneto-optical characterization of paramagnetic GaMnAs quantum wells”, M. Poggio, R. C. Myers, N. P. Stern, A. C. Gossard, and D. D. Awschalom, Phys. Rev. B 72, 235313 (2005).

“Optoelectronic control of spin dynamics at near-terahertz frequencies in magnetically doped quantum wells”, R. C. Myers. K. C. Ku, X. Li, N. Samarth, and D. D. Awschalom, Phys. Rev. B 72, 041302(R) (2005).

“Antiferromagnetic s-d Exchange Coupling in GaMnAs”, R. C. Myers, M. Poggio, N. P. Stern, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 95, 017204 (2005).

“Tunable spin polarization in III-V quantum wells with a ferromagnetic barrier”, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. B 69, 161305(R) (2004).

ix Ferromagnetic semiconductors and heterostructures

“Antisite effect on ferromagnetism in (Ga,Mn)As”, R. C. Myers, B. L. Sheu, A. W. Jackson, A. C. Gossard, P. Schiffer, N. Samarth, and D. D. Awschalom, cond-mat/0606488 (submitted).

“Tunneling through MnAs particles at a GaAs p+n+ junction”, F. L. Bloom, A. C. Young, R. C. Myers, E. R. Brown, A. C. Gossard, and E. G. Gwinn, J. Vac. Sci. Technol. B 24, 1639 (2006).

“Manipulating a domain wall in (Ga,Mn)As”, A. W. Holleitner, H. Knotz, R.C. Myers, A. C. Gossard, and D. D. Awschalom, J. Appl. Phys. 97, 10D314 (2005).

“Pinning a domain wall in (Ga,Mn)As with focused ion beam lithography”, A. W. Holleitner, H. Knotz, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 85, 5622 (2004).

“Independent electronic and magnetic doping in (Ga, Mn)As based digital ferromagnetic heterostructures”, E. Johnston-Halperin, J. A. Schuller, C. S. Gallinat, T. C. Kreutz, R. C. Myers, R. K. Kawakami, H. Knotz, A. C. Gossard, and D. D. Awschalom, Phys. Rev. B 68, 165328 (2003).

“Highly enhanced Curie temperature in low-temperature annealed [Ga,Mn]As epilayers”, K. C. Ku, S. J. Potashnik, R. F. Wang, S. H. Chun, P. Schiffer, N. Samarth, M. J. Seong, A. Mascarenhas, E. Johnston-Halperin, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 82, 2302 (2003).

Non-magnetic quantum wells

“Room temperature electron spin coherence in telecom-wavelength quater- nary quantum wells”, W. H. Lau, V. Sih, N. P. Stern, R. C. Myers, D. A. Buell, A. C. Gossard, and D. D. Awschalom, cond-mat/0607702 (submitted).

x “Suppression of Spin Relaxation in Submicron InGaAs Wires”, A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 97, 036805 (2006).

“Control of electron-spin coherence using Landau level quantization in a two-dimensional electron gas”, V. Sih, W. H. Lau, R. C. Myers, A. C. Gossard, M. E. Flatté, and D. D. Awschalom, Phys. Rev. B 70, 11313(R) (2004).

“Spin transfer and coherence in coupled quantum wells", M. Poggio, G. M. Steeves, R. C. Myers, N. P. Stern, A. C. Gossard, and D. D. Awschalom, Phys. Rev. B 70, 121305(R) (2004).

“Local manipulation of nuclear spins in a semiconductor quantum well”, M. Poggio, G. M. Steeves, R. C. Myers, Y. Kato, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 91, 207602 (2003).

“Gigahertz Electron Spin Manipulation Using Voltage Controlled g-Tensor Modulation”, Y. Kato, R. C. Myers, D. C. Driscoll, A. C. Gossard, J. Levy, D. D. Awschalom, Science 299, 1201 (2003).

Spin Hall effect and current-induced spin polarization

“Generating Spin Currents in Semiconductors with the Spin Hall Effect”, V. Sih, W.H. Lau, R.C. Myers, V.R. Horowitz, A.C. Gossard, and D. D. Awschalom, cond-mat/0605672 (submitted).

“Spatial imaging of the spin Hall effect and current-induced polarization in two-dimensional electron gases”, V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and D. D. Awschalom, Nature Physics 1, 31 (2005).

“Electrical initialization and manipulation of electron spins in an L-shaped strained n-InGaAs channel”, Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 87, 022503 (2005).

“Observation of the Spin Hall Effect in Semiconductors”, Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).

xi “Current-Induced Spin Polarization in Strained Semiconductors”, Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 93, 176601 (2004).

“Coherent spin manipulation without magnetic ﬁelds in strained semiconductors”, Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Nature 427, 50 (2004).

Others

“Spatial Imaging and Mechanical Control of Spin Coherence in Strained GaAs Epilayers”, H. Knotz, A. W. Holleitner, J. Stephens, R. C. Myers, and D. D. Awschalom, Appl. Phys. Lett. 88, 241918 (2006).

“Enhancement of Spin Coherence using Q-factor Engineering in Semicon- ductor Microdisk Lasers”, S. Ghosh, W. H. Wang, F. M. Mendoza, R. C. Myers, X. Li, N. Samarth, A. C. Gossard, and D. D. Awschalom, Nature Materials, 5, 267 (2006).

“Electron spin interferometry using a semiconducting ring structure”, Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 86, 162107 (2005).

Fields of study

Major ﬁeld: Materials Science

Spin Engineering in Magnetic Quantum Well Structures

Professor David Awschalom

Professor Arthur Gossard

xii Abstract

Spin Engineering in Magnetic Quantum Well Structures

Roberto Correa Myers

Molecular beam epitaxy grown heterostructures are used to engineer spin interactions between carriers in semiconductor quantum wells and magnetic ions. In contrast to the spin-orbit and hyperfine interactions, the exchange couplings present in magnetically doped semiconductors are orders of magnitude larger in energy. The confinement afforded in quantum wells is used to enhance, localize, and control the exchange interactions both extrinsically and intrinsically. Ultra- fast optical spectroscopies monitor the spin dynamics of electrons and local magnetic moments confined to the wells revealing the effects of the exchange interaction on the spin polarization, lifetimes, and precession frequencies. In parabolically-graded (ZnCdSe) magnetic quantum wells, a gate bias shifts the confinement potential minimum, thus changing the effective exchange coupling. In these structures, both electron and magnetic ion spin dynamics are electrically controlled at close to THz frequencies with spin interactions in the meV scale. A gate bias is again used to alter the exchange coupling in (GaAs) quantum wells with a ferromagnetic barrier. The wave function overlap between spins in a ferromagnetic layer and carrier spins in a quantum well is electrically controlled, thus providing a tunable spin polarization at low magnetic fields. Lastly, quantum confinement is used to engineer the intrinsic conduction band exchange coupling in GaMnAs and InGaMnAs. The magnitude of the exchange coupling increases with confinement, which is systematically altered by adjusting the well width, barrier height, and doping density. These measurements reveal an antiferromagnetic exchange coupling in the conduction band that is

xiii increased by one-dimensional confinement. From emission spectroscopies we observe the effect of exchange coupling on the valence band and the dynamics of magnetic ion spins. Our results demonstrate carrier spin polarization and manipulation in the solid state using the exchange interaction. Electrons and holes are confined at nanometer length scales along one direction and forced to interact with magnetic ions doped within or nearby these wells. The engineering of exchange couplings in quantum wells may offer a pathway for robust spintronic devices at practical magnetic fields and temperatures.

xiv Contents

Chapter 1 Introduction 1 1.1 Heterostructures ...... 3 1.1.1 Molecular beam epitaxy (MBE) ...... 4 1.1.2 Quantum wells ...... 8 1.2 Spins in a magnetic ﬁeld ...... 11 1.2.1 Spin precession ...... 14 1.3 The exchange integral ...... 15 1.4 Overview of the thesis ...... 18

Chapter 2 Engineering spin in non-magnetic quantum wells 21 2.1 Quantum well g-factors ...... 22 2.1.1 Electron g-factor ...... 22 2.1.2 Hole g-factor ...... 25 2.1.3 Anisotropy of the g-factor in quantum wells ...... 27 2.2 Optical spin-injection ...... 29 2.2.1 Electron spin-injection ...... 30 2.2.2 Hole spin-injection ...... 32 2.3 Optical detection of spin ...... 32 2.3.1 Angular momentum of light ...... 34 2.3.2 Polarization-resolved photoluminescence ...... 36 2.3.3 Hanle effect ...... 37 2.3.4 Time-resolved Kerr (and Faraday) rotation ...... 38 2.4 Vertically biased quantum wells ...... 43

xv 2.4.1 Backgating ...... 44 2.5 Parabolic quantum wells ...... 44 2.5.1 g-factor tuning ...... 47 2.5.2 Shaping nuclear spin proﬁles ...... 48 2.5.3 Electron spin resonance by g-tensor modulation . . . . . 49 2.6 Coupled quantum wells ...... 52

Chapter 3 Magnetic semiconductors 54 3.1 s − d and p − d exchange interactions ...... 55 3.2 Paramagnetism ...... 58 3.2.1 Enhanced Zeeman splitting, a.k.a. Effective g-factors . . 60 3.3 III-V, GaMnAs ...... 62 3.3.1 Carrier-mediated ferromagnetism ...... 63 3.3.2 Growth and defect control ...... 67 3.3.3 Digital heterostructures ...... 67 3.4 Magnetic quantum wells ...... 70 3.4.1 II-VI magnetic wells ...... 70 3.4.2 III-V magnetic wells ...... 71

Chapter 4 Paramagnetic II-VI parabolic quantum wells 72 4.1 Background ...... 73 4.2 Sample design and growth ...... 77 4.3 Photoluminescence: Faraday geometry ...... 79 4.4 Time-resolved Kerr rotation: Voigt geometry ...... 80 4.4.1 Gated s − d exchange coupling ...... 81 4.4.2 Gated p − d exchange coupling ...... 85 4.4.3 Optical control of exchange coupling ...... 89 4.5 Modeling the exchange overlap ...... 93 4.5.1 PL redshift ...... 96 4.5.2 Optical control ...... 97 4.6 Conclusions ...... 98

xvi Chapter 5 Quantum wells with a ferromagnetic barrier 99 5.1 Heterostructure design ...... 100 5.2 Structure and gating ...... 102 5.3 Magnetization measurement ...... 106 5.4 PL polarization ...... 107 5.4.1 Setback spacer and magnetic layer dependence . . . . . 110 5.4.2 Anisotropic spin polarization ...... 111 5.5 Electrically tunable spin polarization ...... 112 5.5.1 Model of exchange overlap ...... 114 5.5.2 Magnetic ﬁeld dependence ...... 118 5.5.3 Optical spin-injection ...... 118 5.6 Conclusions ...... 120

Chapter 6 III,Mn-V quantum wells: band and spin-engineering 124 6.1 Growth of GaMnAs and InGaMnAs quantum wells ...... 126 6.2 Mn incorporation ...... 128 6.2.1 SIMS knock-on effect ...... 133 6.2.2 Substrate temperature dependence ...... 135 6.2.3 Effective Manganese concentration ...... 137 6.2.4 Quantum well width dependence ...... 138 6.3 Electronic properties ...... 140 6.3.1 Manganese hole-doping calibration ...... 141 6.3.2 Mn acceptor activation energy ...... 141 6.4 Substitutional Manganese Incorporation ...... 144 6.4.1 Magnetization ...... 146 6.5 Electron spin dynamics in III,Mn-V quantum wells ...... 149 6.5.1 Determining the sign and magnitude of the exchange coupling ...... 152 6.5.2 Anisotropy of s − d exchange coupling ...... 158 6.6 Conﬁnement engineering of the intrinsic exchange coupling . . 161 6.6.1 Width dependence ...... 161 6.6.2 Barrier height dependence ...... 162

xvii 6.6.3 Kinetic exchange modeling ...... 162 6.7 Electron spin lifetime versus doping ...... 173 6.8 Photoluminescence (PL): spin polarization, p−d exchange, Mn- emission ...... 175 6.8.1 Explanation of the PL spectra ...... 177 6.8.2 Polarization-resolved PL ...... 179 6.8.3 Total excitonic Zeeman splitting ...... 183 6.8.4 Hanle effect ...... 186 6.9 Absorption: the 100-quantum wells ...... 191 6.10 Summary and outlook ...... 192

Chapter 7 Conclusions 196

Appendix A Molecular Beam Epitaxy Techniques 200 A.1 A celebration of history ...... 201 A.2 Initial operation ...... 203 A.3 Knudsen cells ...... 205 A.4 Band edge thermometry ...... 206 A.5 Arsenic capping ...... 208 A.6 InGaAs on GaAs epitaxy ...... 212

Appendix B Effect of arsenic stoichiometry in bulk GaMnAs 216 B.1 Non-rotated GaMnAs: combinatorial variation of As:Ga . . . . 218 B.2 Film morphology and strain versus As:Ga ...... 221 B.3 Carrier density and Curie temperature versus As:Ga ...... 226 B.3.1 Curie temperature gradient ...... 229 B.4 Conclusions ...... 231

Appendix C UCSB samples: or how I learned to stop worrying and love MBE 233 C.1 QWs with a ferromagnetic barrier ...... 234 C.2 GaMnAs and InGaMnAs QWs ...... 236 C.3 Stoichiometric GaMnAs and non-rotated growths ...... 238

xviii References 242

xix List of ﬁgures

1.1 Band-gap map of the world ...... 5 1.2 Molecular beam epitaxy ...... 7 1.3 Quantum wells ...... 10 1.4 Regular Zeeman effect in a non-magnetic quantum well . . . . . 13

2.1 g-factor map of the world ...... 23 2.2 g-factors in quantum wells ...... 25 2.3 quantum well g-factor anisotropy ...... 28 2.4 Optical selection rules in bulk and quantum wells of zinc blende semiconductors ...... 31 2.5 Magneto-optical measurement geometries ...... 34 2.6 Hanle measurement ...... 37 2.7 Optical birefringence at the absorption edge ...... 40 2.8 Optical setup for time-resolved Kerr or Faraday rotation . . . . . 41 2.9 Parabolic quantum wells ...... 45 2.10 Wave function and g-factor and control in parabolic quantum wells ...... 46 2.11 Nuclear and electron spin manipulation in parabolic quantum wells ...... 50 2.12 Gated electron spin dynamics in coupled quantum wells . . . . . 53

3.1 Magnetization and spin polarization versus magnetic ﬁeld . . . . 59 3.2 Giant Zeeman effect in a magnetic quantum well ...... 61 3.3 Mn acceptor conﬁguration in III-V semiconductors ...... 63

xx 3.4 Curie temperature and defect structures in GaMnAs ...... 64 3.5 Carrier-mediated ferromagnetism in GaMnAs ...... 65 3.6 Digital ferromagnetic heterostructure (DFH) ...... 69

4.1 Digital magnetic quantum wells ...... 74 4.2 Electron and magnetic ion spin dynamics in II-VI magnetic quantum wells ...... 75 4.3 Schematic of magnetic parabolic quantum wells and device structure ...... 78 4.4 Photoluminescence of magnetic parabolic quantum wells . . . . 80 4.5 Electrically gated spin precession in magnetic parabolic quantum well ...... 81 4.6 Electron and magnetic ion spin precession data and ﬁts . . . . . 82 4.7 Tuning the s-d exchange overlap in magnetic parabolic quantum wells ...... 84 4.8 Mn-ion spin dynamics in parabolic quantum wells ...... 86 4.9 Optically controlled spin dynamics in magnetic parabolic quantum wells ...... 90 4.10 Data and ﬁts of optically controlled spin dynamics ...... 90 4.11 Optically tunable s − d and p − d exchange overlap ...... 91 4.12 Sublevel-dependent exchange overlap in magnetic parabolic quantum wells ...... 94

5.1 Layer scheme in quantum wells with a ferromagnetic barrier . . 101 5.2 Layer structure of quantum wells with a ferromagnetic barrier . 103 5.3 Gating structure and measurement geometry ...... 105 5.4 Magnetization and polarization versus ﬁeld and temperature . . 108 5.5 Spectrally-resolved photoluminescence and polarization versus magnetic ﬁeld ...... 111 5.6 Spectral dependence of voltage tunable spin polarization . . . . 113 5.7 Exchange overlap and polarization versus vertical bias . . . . . 115 5.8 Polarization and exchange overlap simulations ...... 117 5.9 Field dependence versus bias: control of spin-coupling . . . . . 119

xxi 5.10 The effect of optical spin-injection ...... 121

6.1 Schematic of GaMnAs and InGaMnAs QWs ...... 128 6.2 Mn SIMS profiles versus doping density and substrate temperature131 6.3 Mn SIMS profiles versus QW barrier height ...... 134 6.4 SIMS Knock-on effect for Mn in GaAs ...... 136 6.5 Well width dependence of Mn incorporation ...... 139 6.6 Mn hole-doping calibration ...... 142 6.7 Temperature dependence of hole density ...... 143 6.8 Mn acceptor activation energy ...... 145 6.9 Substitutional Mn incorporation ...... 147 6.10 Magnetization of paramagnetic GaMnAs ...... 149 6.11 Electron spin precession in III,Mn-V quantum wells ...... 151 6.12 Electron g-factor versus width in III-V quantum wells ...... 153 6.13 Temperature dependent spin splitting ...... 155 6.14 Opposing Zeeman and exchange terms ...... 157 6.15 Electron spin splitting in GaMnAs QWs ...... 159 6.16 Electron spin splitting in InGaMnAs QWs ...... 160 6.17 Angle dependence of s − d exchange in InGaMnAs QWs . . . . 161 6.18 Extrinsic exchange parameter versus Mn doping ...... 163 6.19 s − d exchange versus quantum well width ...... 164 6.20 s − d exchange versus barrier height ...... 164 6.21 Confinement engineering the s − d exchange parameter . . . . . 166 6.22 Core level transitions in the kinetic exchange mechanism . . . . 167 6.23 Effect of confinement on the conduction band exchange in II-VI magnetic quantum wells ...... 169 6.24 Electron spin lifetime versus Mn doping ...... 176 6.25 Total PL emission spectrum ...... 178 6.26 Free exciton emission in GaMnAs QWs ...... 180 6.27 PL polarization spectra of GaMnAs QWs and bulk ...... 182 6.28 PL spin splitting in GaMnAs QWs ...... 185 6.29 Mn-ion PL polarization versus pump energy and power . . . . . 187

xxii 6.30 Hanle measurement of Mn-ion spin dynamics ...... 189 6.31 Mn-ion spin lifetime versus Mn doping density ...... 191 6.32 Total excitonic spin splitting in GaMnAs QWs ...... 193

A.1 Pictures of spintronics MBE systems at UCSB ...... 204 A.2 Absorption band-edge spectroscopy for in-situ substrate temperature control ...... 209 A.3 Examples of RHEED patterns from several surfaces ...... 211 A.4 RHEED oscillations ...... 215

B.1 Non-rotated growth flux calibration ...... 220 B.2 Effect of As:Ga on film morphology ...... 222 B.3 Strain effect due to Antisites ...... 225 B.4 As-compensation gradient in non-rotated GaMnAs films . . . . 227 B.5 Curie temperature gradient in non-rotated GaMnAs films . . . . 230

xxiii List of tables

3.1 Band-gaps and exchange constants in some magnetic semiconductors ...... 57

4.1 Sublevel exchange overlap and splitting in a PQW ...... 95

5.1 Sample list and properties quantum wells with a ferromagnetic barrier ...... 106

6.1 Selected properties of GaMnAs QWs ...... 129 6.2 Selected properties of InGaMnAs QWs ...... 130

xxiv Chapter 1

Introduction

Spin angular momentum is a purely quantum mechanical property arising from Dirac’s relativistic correction to the Schrödinger equation. The first measurement of spin, performed by Stern and Gerlach in one of the most famous experiments in physics, demonstrated the existence of spin and its quantization [1]. Quantum mechanics demands that all fermions (e.g. electrons, protons, and neutrons) and bosons (e.g. photons) posses spin. Through the Pauli exclusion principle and Hund’s rules, spin is used to explain electron configurations in atoms. By extension, spin is essential for understanding chemical bonding, and thus the world around us. Magnetism in solids is a direct result of spin [2, 3]. Despite its ubiquity and importance in nature, spins are utilized only in a narrow field of electronics. While most electronics rely only on the charge of the electron, spin-based electronics (spintronics) devices obtain their functionality from spin properties and interactions [4, 5]. Metallic spintronics devices based on the giant magneto-resistive (GMR) effect [6, 7], spin-valves, are used in almost all magnetic hard disk readers. They are now being supplanted by spintronics devices based on the spin-dependent tunneling (SDT) effect [8, 9]. The increased sensitivity of these read heads enabled an exponential increase

1 Chapter 1 Introduction in the storage density in the early 90’s. Other commercially viable technolo- gies based on the spin-valve effect, such as magnetic random access memory (MRAM), offer fast and non-volatile data storage [10]. No commercial semiconductor-based spintronics devices currently exist, despite expansive research efforts over the last decade [11]. On a more positive note, every necessary criterion for utilizing spin in semiconductors has already been demonstrated [12]:

1. Spin-injection (-polarization): through polarized optical excitation [13], or from spin-polarized current ﬂow such as from a ferromagnetic layer [14–16].

2. Spin-manipulation: control of the spin-part of the Hamiltonian by g- factor control [17, 18], spin-orbit coupling [19], the hyperﬁne interaction [20, 21], or by the optical Stark effect [22].

3. Spin-detection: from polarization-resolved optical emission/absorption [13], or from spin-dependent electronic properties.

Additionally, experiments combining more than one of the above techniques have recently been implemented [23–25]. This thesis addresses the manipulation of spin in magnetic semiconductor heterostructures. We have designed such structures to control the coupling between electron and magnetic-ion spins. This exchange coupling can be orders of magnitude larger than other terms in the spin Hamiltonian, such as the g- factor, spin-orbit, or hyperﬁne terms. The strong spin-spin interactions existing in magnetic semiconductors offers a strong candidate for developing robust spintronics at practical magnetic ﬁelds and temperatures. The band engineering and electronic control possible in semiconductor heterostructures allows for the

2 Chapter 1 Introduction localization and control of free carriers over nanometer (nm)-length scales, in particular for quantum wells. Magnetic quantum wells are designed to control electrically, optically, and magnetically the wave function overlap between electrons and magnetic ions. Using time-resolved and polarization-resolved optical spectroscopies, we measure the polarization, precession, and decay of electron and magnetic-ion spins. In the time-domain, controlling the exchange overlap alters the coherent dynamics of electron and magnetic-ion spins on timescales ranging from picoseconds (ps) to nanoseconds (ns), and frequencies ranging from gigahertz (GHz) to terahertz (THz). In the energy domain, we control the sign and magnitude of electron and hole spin-splittings, thus altering the equilibrium spin polarizations. Besides providing a means to alter the extrinsic exchange coupling, quantum confinement alters the intrinsic mechanism of exchange. Confinement changes the symmetry of the electron wave function, allowing for hybridization with the magnetic ion spins. Primarily, we use optical techniques for detecting spin polarization and dynamics, thus requiring high (optical) quality samples. This is not easily achieved, in particular for III-V (GaAs-based) magnetic semiconductors, and requires the development of specialized growth techniques and procedures. The following sections briefly introduce basic techniques and concepts used throughout the thesis: the design and growth of semiconductor quantum wells, spins in a magnetic field, and the exchange integral.

1.1 Heterostructures

Solid-state optoelectronics (lasers, photo-detectors, high-speed transistors, etc.), many which contain quantum wells, are designed by engineering the electronic

3 Chapter 1 Introduction band structure. Band engineering refers to the design of the conduction and valence bands in semiconductors by altering their chemical composition, crystal structure, strain, or processing. This is accomplished by creating heterostructures; semiconductor crystals containing more than one material [26]. Joining two semiconductors with different band gaps (Eg) creates a hetero-interface.

Due to the change in Eg, these interfaces serve as barriers to both electrons and/or holes, depending on the speciﬁcs of the band alignments. The change in Eg at the interface forms regions where charge carriers will accumulate by diffusion and form local conducting channels. The band-gap discrepancy also changes the optical absorption and emission energies. Equally important as heterostructures is the formation of semiconductor alloys in which Eg is smoothly varied from one family of compounds to the next by changing the chemical composition. In Fig. 1.1,Eg is plotted as a function of lattice constant for a number of semiconductors and their alloys. According to Vegard’s law, Eg (or the lattice constant) can be smoothly varied by forming an alloy of the constituent species. Designing heterostructures and alloys requires knowledge of the lattice constant, crystal structure, band gap, and band offsets. Many additional complications exist, for example strain-induced polarization charges in GaN/AlN (wurtzite) heterostructures, surface depletion, and strain relaxation.

1.1.1 Molecular beam epitaxy (MBE)

Band engineering is usually accomplished using layer-by-layer growth techniques, such as molecular beam epitaxy (MBE), in which molecular beams of constituent atoms are directed toward the substrate in an ultra high vacuum (UHV) environment, Fig. 1.2a. The UHV environment results in a mean free path of gas molecules that is much larger than the chamber size. In this

4 Chapter 1 Introduction

Band-gap “Map of the world” 3.0 ZnSe

2.5 AlP GaP

2.0 AlAs CdSe AlSb 1.5

GaAs InP Band-gap (eV) 1.0 Si

GaSb InGaAs 0.5 Ge

InAs InSb 0.0 5.4 5.6 5.8 6.0 6.2 6.4 6.6 Lattice constant (Å)

Figure 1.1: Band gap versus lattice constant for several semiconductors. Lines connect alloy systems. Data from Ref. [27, 28] and references therein.

5 Chapter 1 Introduction molecular-flow regime, vapor atoms move from their heated Knudsen-cell sources in straight lines as molecular beams. The UHV environment allows for in-situ measurements of the growing surface by reflection high energy electron diffraction (RHEED) in which an electron beam is directed at grazing incidence to the substrate surface. The diffraction pattern is imaged on a phosphor-coated screen. RHEED provides qualitative measurements of the surface smoothness, allowing a trained eye to distinguish between smooth, rough, crystalline, amorphous, or polycrystalline films and observe changes in the surface reconstruction. RHEED also provides quantitative information, such as growth rates, by monitoring oscillations in the crystal reflectivity, see appendix A. Epitaxial layers are films in which the crystal structure maintains a well- defined relation to the crystal structure of the substrate. In heteroepitaxy of materials with identical crystal structures and similar lattice constants, the interface between the substrate and film can be perfect; the atomic periodicity between the substrate and film is not interrupted by any crystallographic defects and is only marked by a change in chemical composition and strain. Balanc- ing the strain at the interface with the strain energy of the film requires that films of dissimilar lattice constants be grown below a critical thickness beyond which defects (misfit dislocations) will nucleate at the interface and strain-relax to the natural lattice constants, see appendix A.6. Epitaxial strain alters the band-structure, especially the valence band, leading to properties not possible in strain-relaxed materials.

In the case of GaAs homoepitaxy, Ga and As2 (or As4) molecular beams condense on a heated substrate of single crystal GaAs. Once on the surface, these adatoms minimize their free energy by crystallizing into the zinc blende GaAs template provided by the substrate. By carefully adjusting the growth conditions and therefore the growth kinetics, adatoms are sufﬁciently mobile

6 Chapter 1 Introduction

(a) E l e

t r o Substrate heater and rotation n

G u a n Load/lock chamber ~10-10 torr Al 760 to 10-9 torr Transfer rod

s substrate A RHEED screen

(b) adatom

substrate

Figure 1.2: (a) Basic schematic of a molecular beam epitaxy (MBE) system. The molecular beams from the Knudsen cells are directed at the heated and rotated substrate. A ventable load/lock chamber is employed to maintain the growth chamber in a UHV environment. A RHEED system is used to monitor the surface during growth. (b) Adatom crystallization on a substrate showing layer-by-layer step-ﬂow growth. After condensing on to the substrate surface, hot adatoms will diffuse along the surface until crystallizing at step edges.

7 Chapter 1 Introduction on the crystal surface to find their lowest energy position leading to an atomic layer-by-layer cycle, such as the step-flow or island-growth mode [29], Fig. 1.2b. Typical growth rates are ∼ 1 monolayer (ML) per second. Due to these slow growth rates, the molecular beams can be shuttered to control film thicknesses with sub-monolayer precision. Diffusion usually smears out these ultra- thin films across ± 1 ML.

1.1.2 Quantum wells

Using techniques such as MBE, producing heterostructures in which the composition changes over atomic monolayer lengths scales is routine [30]. Thus it is possible to deposit a thin (∼10-100 Å) narrow band gap material within a wider band gap material, Fig. 1.3. For electrons and (or) holes confined within these wells, their motion along the growth direction (ˆz) is restricted leading to quantization of the continuous band into sub-bands. Carriers confined to these wells behave two-dimensionally since motion in thex ˆ andy ˆ directions remains approximately free assuming parabolic band dispersion in these in-plane directions. These quantum wells (QWs) are essential components in countless optoelectronic devices, such as quantum well laser diodes, avalanche photodiodes, quantum cascade lasers, and high electron mobility transistors, to name a few. In addition to modern device application, QWs have been at the forefront of condensed matter research, providing systems in which band structure, density of states, carrier density, and confinement energy are adjustable. The qualitative properties of quantum wells are easily calculable using el- ementary quantum mechanics. Solving the Schrödinger equation for a one- dimensional particle in an infinitely deep well of width d, gives energy states,

h¯ 2 nπ E = ( )2, (1.1) n 2m d

8 Chapter 1 Introduction

∗ ∗ where m = m me; m is the effective mass. In reality, the ﬁnite depth of the conﬁnement potential alters these solutions. Penetration of the wave function into the classically forbidden barrier regions is now possible by tunneling. Analytical solutions of this potential are not possible, but are found by numerical methods. Using Gregory Snider’s self- consistent numerical Poisson-Schrödinger solver [31], we plot the band edges, electron wave functions, and energy eigen values for a 7.5-nm GaAs QW within

Al0.4Ga0.6As barriers in Fig. 1.3a. The nth energy eigen values are labeled en, hhn, lhn for electron, heavy hole, and light hole states, respectively. For electrons in the conduction sub-bands, motion in the x,y plane is free giving the parabolic dispersion shown in Fig. 1.3b.

h¯ 2 E = e + (k2 + k2), (1.2) n n 2m x y where kx,y is the electron wave number (momentum) in the plane of the quantum well. The valence sub-bands are anisotropic due to the p-symmetry of the Bloch functions. In quantum wells, heavy hole states become more tightly bound along the z-axis and lighter in x,y, whereas the light hole states become weakly bound along z-axis and heavier in x,y. As seen in Fig. 1.3d, this results in a splitting between the hh1 and lh1 states with hh1 becoming the ground state for holes near kx,y = 0. Secondly, the effective masses of the light and heavy hole bands are reversed along x,y, with the effect that the bands cross near kx,y = 0. In comparison to the bulk, where the valence bands are heavily mixed at kx,y = 0, in quantum wells the valence band mixing occurs away from the zone center. This valence band anisotropy leads to g-factor anisotropy, chapter 2.1.3.

9 Chapter 1 Introduction

Al0.4Ga0.6AsGaAs Al0.4Ga0.6As (a)d = 7.5 nm (b) (c) 1.0 Ecb 0.9 E (eV) e2

0.8

e1 0.7 E 0.6 * -10 1 01 E E -1 -1 -2 g g kx,y (nm ) n (eV nm ) (d)

hh1 lh hh -0.8 1 2 E (eV) lh1 hh3 -0.9 hh1 lh2 hh4

Evb -1.0 -10 1 z -1 kx,y (nm )

Figure 1.3: (a) Conduction and valence band edges calculated for an AlGaAs-GaAs quantum well. Energy eigen values are plotted within the wells and the corresponding wave (envelope) function is overlayed. (b) Conduction sub-band dispersion and (c) density of states assuming ∗ ∗ m = 0.067. (d) Valence dispersion for hh1 and lh1 sub-bands assuming mhh = 0.082 and ∗ mlh = 0.58.

10 Chapter 1 Introduction

1.2 Spins in a magnetic ﬁeld

In a magnetic ﬁeld, the degeneracy of the energy levels for spin-up and spin- down is lifted. Recall that the observable part of electron spin is the projection 1 h¯ along the applied ﬁeld, S = ± 2 , which carries angular momentum, ± 2 . The magnetic moment µ is,

~µ = −gµB~S, (1.3) where g is the g-factor (gyromagnetic ratio), µ = eh¯ is the Bohr magneton, B 2me and the spin is written as a vector (~S). The negative sign comes from the fact that the Bohr magneton is the magnetic moment arising from a particle of charge +e h¯ with angular momentum 2 . The spin part of the electron’s Hamiltonian contains the energy associated with a magnetic dipole moment in a magnetic ﬁeld (~B),

Hµ = −~µ ·~B = HS = gµB~S ·~B. (1.4)

A magnetic moment is in its lowest energy state when it points along the applied field, but the preferred spin-orientation depends on the sign of the g-factor. For a positive g-factor an electron spin will align opposite the applied field. The g-factor for free electrons is ∼ 2.002. For consistency, the Hamiltonians in this work will always be written using the traditional definition of the g-factor, as discussed above, and spin-up (positive) refers to a spin lying along the applied field. Spin-orbit coupling changes the g-factor for electrons in atoms, and in particular for delocalized electrons in the band structure of a crystal. Depending on the symmetry of the wave functions of conduction and valence band electrons and the band gap, the g-factors for holes and electrons can vary widely from negative to positive values, see chapter 2.1. If we apply a magnetic field along the z-axis (Bz), the conduction band will split into two spin bands for either spin

11 Chapter 1 Introduction direction, 1 1 He = + g µ B , He = − g µ B , (1.5) ↑ 2 e B z ↓ 2 e B z where ge is the conduction band electron g-factor. Here we have assumed that 1 the conduction band states have S = ± 2 . To analyze the valence band we will first point out the valence band states are eigen function of total angular momentum j = L+S. This comes from the p-symmetry of wave functions, which carry orbital angular momentum L = ±1 in addition to the spin angular momentum, 1 S = ± 2 . A more complete description is given in section 2.2. The heavy hole 3 1 band is identified with j = ± 2 and the light holes with j = ± 2 . Accordingly, we substitute S with j in equation 1.4. This yields the following spin states in the valence band, 3 3 HHH = + g µ B , HHH = − g µ B , (1.6) ↑ 2 HH B z ↓ 2 HH B z for the heavy holes, and 1 1 Hlh = + g µ B , Hlh = − g µ B , (1.7) ↑ 2 lh B z ↓ 2 lh B z for the light holes, where gHH and glh are the g-factors for the heavy and light holes, respectively. Unfortunately, we must now complicate the g-factor definition for the heavy holes owing to precedent in the literature. In this thesis for purposes of comparison, we use the definition of the heavy hole g-factor 3 given by Kesteren et al. [32], where the j = ± 2 heavy hole states are given an ∗ 1 effective spin value S = ± 2 . In this case, 1 1 Hhh = + g µ B , Hhh = − g µ B , (1.8) ↑ 2 hh B z ↓ 2 hh B z where ghh = 3gHH. The regular Zeeman splitting in a non-magnetic quantum well is shown in Fig. 1.4. The conservation of angular momentum demands that the absorption

12 Chapter 1 Introduction

− + 1/ 2 e1 1/ 2 μ ge B B +1/ 2 −1/ 2

σ- ∗ Ε − + σ+ g σ σ E + 3/ 2 − 3/ 2 hh1 g μ B + hh B 3/ 2 − 3/ 2 −1/ 2 lh +1/ 2 1 g μ B +1/ 2 −1/ 2 lh B

Figure 1.4: Regular Zeeman effect in a non-magnetic quantum well. Electron energy in the conduction, heavy, and light hole bands versus magnetic ﬁeld (B). Note that the valence band spins are drawn for the electron spin scheme and not the hole paradigm. σ+ and σ− label the helicity of the emitted (or absorbed) photons from the allowed interband transitions between heavy hole and conduction bands. or emission of circularly polarized photons by band electrons (σ + and σ −) result in an equivalent change in the total angular momentum of the electron. The emission (photoluminescence) transitions are shown in Fig. 1.4 of conduction band electrons dropping into the heavy hole band, which results in a ∆ j = ±1 for σ ±. These selection rules provide an optical means of generating and detecting spin polarization in semiconductor quantum wells [13], which are reviewed in sections 2.2 and 2.3. The partial Hamiltonian in equation 1.4 is valid in the unrealistic absence of any spin-spin interactions. The exchange interaction between electron spins adds an additional ﬁeld dependent term to this equation, see sections 1.3 and

13 Chapter 1 Introduction

3.1. Also, electron and nuclear spins couple through the hyperﬁne interaction, section 2.5.2.

1.2.1 Spin precession

1 Preparing a spin- 2 particle in a superposition of basis states of the magnetic field results in spin-precession. In chapter 2.3, the time-resolved optical measurement of spin precession in quantum wells is described. Such measurements yield the g-factors, hyperfine and exchange parameters. As an illustrative example, consider a magnetic field applied along the x- axis, ~B = Bxˆ. The basis states for spin are the spin-up and spin-down states alongx ˆ, | + Sxi and | − Sxi, respectively. Applying the spin-operator (Sˆx) to 1 these basis states, trivially yields the spin-up and spin-down eigen values, ± 2 . A state at time, t = 0, with spin-up along the z-axis can thus be represented as a superposition of | + Sxi and | − Sxi, 1 |ψ(0)i = | + Szi = √ (| + Sxi + | − Sxi), (1.9) 2 and its time evolution is given by,

1 1 1 −iHˆSt/h¯ −( )igµBBt/h¯ +( )igµBBt/h¯ |ψ(t)i = e | + Szi = √ (e 2 | + Sxi + e 2 | − Sxi), 2 (1.10) where we have substituted for HS with equation 1.4 and applied the spin operator. The expectation value for spin along the z-axis is, 1 gµBB hSˆzi = hψ(t)|Sˆz|ψ(t)i = cos t , (1.11) 2 h¯ and similarly, along the y-axis, 1 gµBB hSˆyi = sin t , (1.12) 2 h¯

14 Chapter 1 Introduction

gµBB where the angular spin precession frequency ΩL = h¯ . The Larmor precession ΩL frequency is of course νL = 2π .

Note that HˆS of equation 1.4, as used in this analysis only includes the energy of a magnetic moment in a ﬁeld and neglects additional spin interactions such as the exchange or hyperﬁne couplings. These interactions modify the

ﬁeld dependence and magnitude of HS and νL.

1.3 The exchange integral

In ferromagnetic metals such as Fe and Ni, the parallel alignment of the nearest neighbor 3d electron spins is energetically favored, leading to a spontaneous spin polarization and magnetization. The orbital angular momentum (L) is quenched in these metals due to strong orbital pinning by the crystal lattice [33], thus the magnetic moment arises from spin polarization alone. Before a theory of ferromagnetism was devised, Weiss postulated the presence of an internal magnetic field (Weiss molecular field) as the origin of the spontaneous magnetization [34, 35]. The origin of this apparent magnetic field is the exchange interaction, which was demonstrated by Heisenberg in his 1928 paper [2]. When the wave functions of two fermions, such as electrons, overlap spatially, the Pauli exclusion principle puts constraints on the symmetry of the two- particle wave function. A fermion wave function must be antisymmetric with respect to particle exchange. If two electrons overlap in space, then the spin part of the wave function must be antisymmetric. On the other hand, if two electrons have the same spin, then they must be spatially antisymmetric. This Pauli exclusion principle explains, for example, Hund’s rules for the filling of atomic orbitals.

15 Chapter 1 Introduction

We illustrate the exchange interaction by considering the two-electron Hamil- tonian of the He-atom, as treated in reference [36]. The two-electron Hamil- tonian includes the kinetic energy of both electrons, their interaction with the potential of the nucleus, and the Coulombic energy between the electrons. The Hamiltonian of the entire system is,

2 2 2 2 2 h¯ 2 2e h¯ 2 2e e H = (− ∇1 − ) + (− ∇2 − ) + , (1.13) 2me 4πε0r1 2me 4πε0r2 4πε0r12

h¯ 2 2 2e2 where (− ∇ − ) = Hi is the single-electron Hamiltonian at position ~ri. 2me i 4πε0ri The solutions for the single electron Hamiltonians are known, |(n,l,m)i, where n is the principal quantum number (from solutions to the Coulomb potential) specifying the radial dependence of the wave function, and l and m are the orbital angular momentum quantum numbers specifying the angular dependence of the wave function (spherical harmonics). We consider the excited state with electrons occupying the |(1,0,0)i and

|(2,0,0)i states (1s and 2s). The two-particle wave functions are either |1s12s2i 2 or |2s 1s i. In the absence of H0 = e , the last term of equation 1.13, both 1 2 4πε0r12 wave functions are degenerate in energy with eigenvalue E1 + E2, where En is the energy of the |(n,0,0)i state. Solutions to equation 1.13 can therefore be approximated using degenerate perturbation theory. The eigen functions are,

1 |ψ12,↑↓i = √ (|1s12s2i + |(2s11s2i), (1.14) 2 which, under particle coordinate exchange, is spatially-symmetric and spin- antisymmetric, and

1 |ψ12,↑↑i = √ (|1s12s2i − |2s11s2i), (1.15) 2 which is spatially-antisymmetric, and spin-symmetric. Thus, these two-electron wave functions satisfy the Pauli exclusion principle.

16 Chapter 1 Introduction

The eigen values are,

0 0 E = E1 + E2 + h|1s12s2|H |1s12s2i ± h|1s12s2|H |1s22s1i = E1 + E2 + K ± J. (1.16)

0 K = h|1s12s2|H |1s12s2i 0 = h|2s11s2|H |2s11s2i 2 Z e ∗ ∗ 1 = 1s (r1)2s (r2) 1s(r1)2s(r2)dr1dr2, (1.17) 4πε0 r12 which is classically interpretable as the Coulombic repulsion between the 1s and 2s shells.

0 J = h|1s12s2|H |1s22s1i 0 = h|2s11s2|H |2s21s1i 2 Z e ∗ ∗ 1 = 1s (r1)2s (r2) 1s(r2)2s(r1)dr1dr2, (1.18) 4πε0 r12 which is called the direct exchange integral. Ontologically, it exists only due to the quantum mechanical requirement of Pauli exclusion. Taken at face value, this spin-dependent energy (J) is the Coulombic repulsion of identical electron orbitals (equations 1.14 and 1.15) with distinct spin conﬁgurations. This additional energy splits the degeneracy between the |ψ12,↑↓i and |ψ12,↑↑i states, where the ferromagnetic | ↑↑i state is lower in energy than the antiferromagnetic | ↑↓i state by an amount 2J. From equation 1.18, we see that the magnitude of J depends strongly on the symmetry of the wave functions on which the spins reside, as well as their spatial overlap. Also, note that J must be positive in equation 1.18, and therefore always favors ferromagnetic (parallel) spins. As we will see in chapter 3.1, hybridization between band electron and 3d electron wave functions can lead to an antiferromagnetic (antiparallel) contribution to the effective J. Such kinetic

17 Chapter 1 Introduction exchange effects typically dominate in the valence band of magnetic semiconductors leading to J < 0. Quantum-conﬁnement changes the symmetry of the conduction band, increasing hybridization with the 3d states of the magnetic impurities. In chapter 6.6, we change quantum well width and barrier height to engineer J using this conﬁnement-induced kinetic exchange effect.

1.4 Overview of the thesis

Chapter 2 is an introduction and review of spin manipulation in non-magnetic quantum wells. The optical techniques used throughout the thesis are explained. Vertically-biased quantum wells are discussed in which the electron position is electrically controlled, chapter 2.4. This is demonstrated in parabolically- graded and coupled quantum wells. The displacement of the carrier wave functions changes the g-factor and can be used to control electron and nuclear spins, see theses [37] and [38], respectively. The author’s role in these projects was primarily in the design and growth of the heterostructures. Band-grading and back-gating techniques developed in the course of these projects were later used in the magnetic QWs. Chapter 3 reviews II-VI and III-V magnetic semiconductors, discussing exchange interactions, growth aspects of GaMnAs, and digital magnetic heterostructures. The primary results of the thesis are chapters 4, 5 and 6 in which new types of magnetic quantum well structures are presented. In chapter 4, magnetically-doped quantum wells with parabolically-graded band edges (ZnCdSe-MnSe-CdSe) are studied. These structures were grown by K-C Ku in Nitin Samarth’s group at Penn State University and I performed the optical experiments and analysis in David Awschalom’s lab at UCSB. Dig- ital sub-monolayers of paramagnetic MnSe layers were incorporated into par-

18 Chapter 1 Introduction abolic quantum wells. In such II-VI magnetic quantum wells, both electron and magnetic-ion spin dynamics can be observed, see thesis [39]. By applying a vertical bias to our parabolic wells, we demonstrate the electrical control of the exchange overlap. This follows from the electrical gating techniques developed in non-magnetic quantum wells for g-factor control. The resulting changes in electron spin dynamics were orders of magnitude larger than in similar non-magnetic wells. Unfortunately, electron spin lifetimes are extremely short making it impractical to attempt spin manipulation within the coherence time. Additionally, strong magnetic interactions between the magnetic-ions are absent in these II-VI magnetic semiconductors. The paramagnetism in these systems makes it necessary to apply large magnetic ﬁelds to observe exchange effects. To overcome these limitations, the next structures developed were III-V, GaAs-based magnetic heterostructures. In chapter 5, ferromagnetic layers of sub-monolayer MnAs are incorporated into quantum well barriers. I grew these structures in Art Gossard’s MBE lab and carried out the experiments and analysis in David Awschalom’s optics and magnetics lab. Such digital ferromagnetic layers are deposited using atomic layer epitaxy to minimize low temperature crystallographic defects, see thesis [40]. Wave function overlap occurs between hole spins bound to GaAs quantum wells and ferromagnetic (MnAs) spins in the AlGaAs barriers. Again, this overlap is controlled with a vertical bias and the resulting hole spin polarization is optically detected from the quantum well emission. The ferromagnetic layer can be fully polarized at low magnetic ﬁelds. The hole spin polarization tracks the magnetization and can be electrically tuned. An unfortunate consequence of the defects introduced by the ferromagnetic layer was the loss of time-resolved optical signals necessary to observed electron spin dynamics. This obstacle is addressed in the next chapter.

19 Chapter 1 Introduction

Chapter 6 describes quantum wells of GaAs or InGaAs containing paramagnetic moments representing the ﬁrst III-V magnetic heterostructures in which electron spin dynamics are optically observable. I grew these structures in Art Gossard’s MBE lab. We (Martino Poggio, Nathaniel Stern, and the author) performed the optical measurements and analysis, in total collaboration, in David Awschalom’s lab. Growth conditions were found at which high levels of paramagnetic doping and optical quality could both be retained. Transport and structural properties were examined for a variety of dopings to determine the charge and crystallographic position of the paramagnetic dopants. Measurements of the electron spin dynamics were used to extract precise values of the exchange coupling. Conﬁnement engineering was used to modify the exchange interaction. Time-averaged measurements of magnetic-ion spin dynamics were carried out in these structures, revealing a strong dependence of the spin lifetime on the doping density. Chapter 7 summarizes the results of the thesis and suggests future experiments and structures for exchange engineering in magnetic quantum wells.

20 Chapter 2

Engineering spin in non-magnetic quantum wells

We introduce the use of quantum wells for controlling free carrier spins. As discussed in chapter 1.1.1, MBE allows the production of custom doping and band-edge profiles. The choice of quantum well and barrier material determines not only the effective band gap, but also the effective g-factor (g-factor engineering). The spin-orbit interaction and band gap are modified by chemical composition and quantum confinement. The selection rules in QWs allow for optical spin-injection and detection using the angular momentum of circularly polarized light. Polarization- and time-resolved optical measurements that monitor electron spin polarization and precession are described. Applying a vertical-bias across QWs shifts the confining potential profile and the position of carriers confined to the wells, resulting in changes in the electron effective g-factor. This g-factor tuning is particularly effective in parabolic and coupled quantum wells, where it modifies the local electron and nuclear spin Hamiltonians. These interactions are modulated at high-frequency

21 Chapter 2 Engineering spin in non-magnetic quantum wells to cause electron and nuclear spin resonance.

2.1 Quantum well g-factors

2.1.1 Electron g-factor

Band structure strongly alters the spin-orbit interaction in semiconductor QWs. The change in band structure between QW and barrier materials leads to a change in the effective g-factor. Figure 2.1 plots g-factors as a function of band gap for a number of semiconductors and their alloys [27, 28]. The spin splitting between spin-up and spin-down electrons (∆E↑↓) following from equation 1.5 is cb cb plotted, ∆E↑↓ = H↑ −H↓ = geµBB, providing a range of conduction band spin splittings possible in these systems. The sign change of ∆E↑↓ corresponds to a change in orientation of the ground state of the electron-spin in a magnetic ﬁeld.

Electron spins point opposite the ﬁeld when ge > 0 (∆E↑↓ > 0) and parallel with the ﬁeld for ge < 0 (∆E↑↓ < 0). In analogy to the band engineering map of Fig. 1.1, Fig. 2.1 provides a map to g-factor engineering using heterostructures. The Lande g-factor in zinc blende semiconductors explains the strongest effects of band structure on the splitting between spin-up and spin-down states:

2 2psp ∆ ge = g0 − ( ) , (2.1) 3m0 Eg(Eg + ∆) where g0 ∼ 2.002 is the g-factor for free electrons, psp is proportional to the conduction and valence band mixing, and ∆ is the spin-orbit splitting of the valence band (spin-split off band). Equation 2.1 explains, for example, why wider band gap materials, such as GaN and ZnO exhibit g-factors close to the free electron value (∼2) and why narrow band gap semiconductors exhibit negative g-factors, such as GaAs (∼ -0.44) and InAs (∼ -15). Further, the unusually

22 Chapter 2 Engineering spin in non-magnetic quantum wells

“g-factor map of the world” 200 GaP GaN 2 InP AlSb Si AlAs

0 ZnSe 0 ∆

CdSe E/B ( -2 GaAs µ

-200 eV/T) -4 Ge g -6 InGaAs -400 -8 GaSb -10 -600 -12 -14 -800 InSb -16 InAs 0.5 1.0 1.5 2.0 2.5

Eg (eV)

Figure 2.1: Summary of electron g-factors versus band gap in several semiconductors. Lines connect alloy systems. The corresponding spin splitting in the conduction band is plotted in the right vertical axis per unit of magnetic ﬁeld. Data from Ref. [27, 28] and references therein.

23 Chapter 2 Engineering spin in non-magnetic quantum wells large and negative value of the conduction band electron g-factors in InSb (∼ -51) and InAs are due to the large ∆ in these materials. Alloying semiconductors provides one route for engineering g-factors. For example in GaAs, by alloying with Al the conduction-band g-factor can be increased [41, 42] from -0.44 in pure GaAs to +0.6 in Al0.4Ga0.6As. It was also shown that alloying GaAs with In decreases the g-factor from -0.44 to -0.8 in

In0.1Ga0.9As [41, 42] and reaches -4.07 in In0.53Ga0.47As lattice-matched to InP [43]. An additional modification of g-factors exists in QWs [44]. The spatial confinement of carriers along the growth direction increases the effective band gap ∗ ∗ (Eg ) as seen in Fig. 1.3. The simple substitution Eg → Eg in equation 2.1 demonstrates the change in g-factor in QWs. Optical measurements of electron g-factors in a number of QW systems experimentally show the effect of confinement on the conduction band g-factors. Snelling et al. used measurements of polarization-resolved photoluminescence (PL) in the Voigt geometry – the Hanle effect – to extract effective electron g-factors and spin lifetimes in GaAs-AlxGa1−xAs QWs [45]. These measurements demonstrated that for Al concentrations of 30%, QW width modifies the g-factor from negative values in wide wells, to positive values in narrow wells, and an effective ge = 0 for widths near 5.5 nm. These time-averaged measurements were later con- firmed by optical measurement of electron spin dynamics in the time-domain of similar structures using the technique of time-resolved Kerr or Faraday rotation [46–48]. Note that such measurements only provide information about the magnitude of the g-factor and not its sign. Figure 2.2 plots the effective electron g-factor and ∆E↑↓ from references [45, 47, 48] showing the wide range in sign and magnitude available in GaAs-AlGaAs and InGaAs-GaAs QWs.

24 Chapter 2 Engineering spin in non-magnetic quantum wells

40 0.6 30 GaAs-Al Ga As 0.4 0.4 0.6 20 ∆

GaAs-Al Ga As E/B ( 0.2 0.33 0.67 In Ga As-GaAs 10 0.11 0.89 µ

g 0.0 0 eV/T) -0.2 -10 -20 -0.4 -30 -0.6 -40 246810121416182022 d (nm)

Figure 2.2: Effective in-plane electron g-factors in some III-V quantum wells plotted as a function of well width (w). Data reproduced from Ref. [45, 47, 48].

2.1.2 Hole g-factor

The energy landscape for electron spins in the valence band is more complex than for the conduction band due to the strong spin-orbit interaction and band mixing of the heavy, light, and spin-split-off bands. In QWs, the degeneracy of the heavy and light hole bands is lifted due to symmetry breaking from conﬁne- ment and/or strain, whereby the heavy hole band becomes the ground state for holes, chapter 1.1.2. The label of these valence bands becomes somewhat of a misnomer in QWs since the heavy hole band in bulk becomes lighter in QWs d2E d2E (larger dk2 ), and the light hole band becomes heavier (smaller dk2 ). Away from kx,y = 0, the bands therefore cross and result in strong mixing. This results in an 3 anisotropy of the angular momentum that pins the heavy hole spin ( j = 2 ) par- 1 allel to the growth axis, and light hole spin ( j = 2 ) perpendicular to the growth axis [49]. Polarization resolved PL was used to measure the Zeeman splittings in GaAs- AlGaAs and InGaAs-GaAs QWs [50, 51]. From these excitonic spin splittings,

25 Chapter 2 Engineering spin in non-magnetic quantum wells the g-factor for heavy holes and heavy hole excitons were extracted. In the GaAs-AlGaAs system, heavy hole g-factors undergo a similar sign change as does the electron g-factor near 5.5 nm well width. In the GaAs-AlGaAs system however, the complexity of the valence band results in non-linear Zeeman splittings at high magnetic fields, making precise determination of the hole g-factors difficult. Usually the short spin lifetimes ( ∼110 fs in GaAs) [52] of hole spins in bulk semiconductors renders it inaccessible in the time-domain, resulting from the four-fold degeneracy of the valence band at k ∼ 0. However, the degeneracy lifting of the heavy and light hole bands in QWs, discussed above, increases the hole spin lifetime [53]. Measurements of polarization-resolved PL at zero field have shown hole spin lifetimes of up to 1 ns in 7.5 nm GaAs-Al0.33Ga0.67As QWs [54]. In n-doped QWs of GaAs-AlGaAs , time- and polarization-resolved PL was used to measure the spin precession of optically injected heavy holes in a transverse field allowing for the observation of heavy hole spin precession with lifetimes up to 700 ps [55]. Surprisingly, hole-spin procession has not been observed in almost identical samples using the technique of time-resolved Kerr rotation, where only electron spin precession has been observed [46–48]. The difference in Fermi-level is a possible explanation since the sensitivity of these techniques changes depending on the density of states. PL is sensitive only to non-equilibrium spin polarization that is optically-injected, whereas Kerr and Faraday rotation are sensitive to any spin polarization. It should also be considered that sample to sample variation could alter the relaxation processes, especially in samples where both electron and hole g-factors change in sign and magnitude with small changes in well width.

26 Chapter 2 Engineering spin in non-magnetic quantum wells

2.1.3 Anisotropy of the g-factor in quantum wells

Quantum confinement alters the g-factor (g// = gz) parallel and perpendicular to the growth axis (g⊥ = gx,y) leading to anisotropic spin splittings [44]. This results from the anisotropy inherent to the valence band, where the heavy and light hole bands are split in QWs. Such a splitting affects the anisotropy of both holes and electrons. Heavy hole spins are pinned along the growth axis, while light holes align perpendicular to it, and consequently the g-factors have large anisotropy. The anisotropy of the hole g-factors in QWs has been measured in GaAs-AlGaAs QWs. The g-factors parallel to the growth axis for heavy holes vary from about -2 to +1 with QW width as measured from the Zeeman splitting in the Faraday geometry [50]. The in-plane heavy hole g-factor is more than ten times smaller, ∼ 0.03 as measured from hole spin precession in a transverse field [55] (Voigt geometry), reflecting the strong pinning of heavy hole spins along the growth axis. The effective band gap has an anisotropy depending on whether the band gap from the light hole or heavy hole is considered. Following from equation (1), the electron g-factor is therefore anisotropic in QWs depending on the splitting between the heavy and light hole bands. Optically detected magnetic resonance (ODMR) was used to measure the anisotropy of the electron g-factor in single-sided modulation p-doped InGaAs-InP QWs [56]. In the bulk limit of In0.53Ga0.47As the g-factor was measured to be isotropic with ge ∼ −4. In QWs the anisotropy increases as the well width narrows, plotted in Fig. 2.3. In g the narrowest wells the measured ratio was, // = 4. Strain was also modified g⊥ by changing the concentration of Ga while maintaining the same quantum well width. Both g// and g⊥ were modified by the change in strain, however the ratio of the two remained roughly constant, suggesting that the g-factor anisotropy

27 Chapter 2 Engineering spin in non-magnetic quantum wells

g⊥ (in-plane) g g

g// (out-of-plane)

quantum well width d (nm)

Figure 2.3: Anisotropic electron g-factor in In0.53Ga0.47As-InP quantum wells measured at a temperature of 1.6 K. The g-factor parallel to the growth axis, g// (solid circles), and perpendicular to the growth axis, g⊥ (open circles), are plotted as a function of well width (w). The line is a self-consistent subband calculation. Reproduced from Ref. [56]. was strain independent. Similar measurements of g-factor anisotropy by angle- dependent time-resolved Kerr rotation in strained InGaAs-GaAs wells and in unstrained GaAs-AlGaAs wells confirm the strong g-factor anisotropies due to quantum confinement and weaker dependence on strain [57, 58]. Electron spin precession in a transverse magnetic field is altered by g-factor ↔ anisotropy. Since the g-factor is a tensor (g), the Larmor frequency (νL) gains ↔ a vector-dependence. It is necessary to define the spin precession vector, ν~L =g

µB~B/h, which deﬁnes the frequency and axis about which electron spins precess. The g-factor anisotropy is responsible for an enhancement of dynamic nuclear spin polarization [58]. Additionally, in a later section we discuss how these anisotropies can be controlled with a vertical bias leading to a new form of electron spin resonance based on manipulation of ν~L through the g-tensor.

28 Chapter 2 Engineering spin in non-magnetic quantum wells

2.2 Optical spin-injection

The electronic band structure of bulk zinc blende semiconductors can be com- puted using ~k · ~p theory. An introductory treatment is given in chapter 10 of Davies, [59]. An accessible calculation of the Bloch functions in the vicinity of the zone centers,~k = 0, is given by Bastard [60]. Descriptions of this calculation have also been given from different perspectives in these theses [37, 38, 61]. Qualitatively, the bottom of the conduction band is s-symmetric with projec- 1 tions of total angular momentum j = S = ± 2 . In the valence band in addition 1 to having S = ± 2 , the p-symmetric Bloch functions carry an additional orbital angular momentum L = ±1. Thus the valence band breaks into six degenerate states of angular momentum: 1 2 states with L = 0 and S = ± 2 , 1 1 | 2 ,± 2 i (split-off) 1 and 4 states with L = ±1 and S = ± 2 , 3 3 | 2 ,± 2 i (heavy), 3 1 | 2 ,± 2 i (light). The spin orbit coupling, ∆, pushes the spin split-off valence band low enough in energy to not be involved in optical interband transitions for photon energies hν ∼Eg, allowing only transitions from the heavy and light valence bands to the conduction band, as shown in Fig. 2.4a. The optical selection rules in zinc blende semiconductors provides a convenient means of polarizing carrier spins through the absorption of circularly polarized light. Qualitatively, the optical selection rules obey a conservation of total angular momentum. When a circularly-polarized photon (σ ±, j = ±1) is absorbed or emitted by an electron, the change in j of the electron, ∆ j = ±1. Quantitatively, optical transition rates in zinc blende semiconductors have been treated using Fermi’s golden rule in

29 Chapter 2 Engineering spin in non-magnetic quantum wells the electric dipole approximation [13]. The results allow ∆ j = ±1 transitions of electrons from heavy or light bands to the conduction band with well-deﬁned spin-states as illustrated in Fig. 2.4b.

2.2.1 Electron spin-injection

Optical spin-injection into the conduction band is accomplished, for example, + 3 using a σ polarized optical excitation. In this case, j = − 2 heavy hole elec- 1 1 trons are excited into j = S = − 2 conduction band states. Additionally, j = − 2 1 light hole electrons are excited into j = S = + 2 conduction band states, which cancel out the net spin of the conduction band. Fortunately, the transitions of the heavy hole band are 3-times more favorable than light hole transitions, thus 1 3 out of 4 conduction band electrons will have j = S = − 2 and 1 out of 4 will 1 have j = S = + 2 giving a net spin polarization of 2 out of 4. Thus optical spin- injection in bulk gives a maximum spin polarization of 50% in the conduction band. In quantum wells, as we have already seen in sections 1.1.2 and 2.1.2, quantization along the growth axis splits the degeneracy of the heavy and light hole bands, and reverses the effective masses. One may ask, if “heavy-holes” are actually light (small m∗) and “light-holes” are actually heavy (large m∗), why do we not change the label of the bands? Here we must remember that the valence band labels refer to Bloch functions that are eigenfunctions of total angular momentum, thus the labels “heavy” and “light” refer to states with eigen values of 3 1 total angular momentum ± 2 and ± 2 , respectively. In imposing the boundary conditions in quantum wells, states with well-deﬁned total angular momentum are preserved, but their effective masses are altered. The selection rules in quantum wells are roughly identical to bulk, with the

30 Chapter 2 Engineering spin in non-magnetic quantum wells

Bulk GaAs Quantum well (7.5-nm) (a)(b) (c) (d) 0.8 − 1 + 1 − 1 + 1 e 2 2 e 2 2 1 0.6

σ+ σ+ E σ+ σ+ E * 0.4 g g ××

E (eV) - - σ σ σ- σ-

hh 3 1 1 3 E -0.8 − − + + hl − 3 + 3 2 2 2 2 lh1 1 1 = = = = 2 − + 2 lh hh1 2 2 + 3 + 1 − 1 − 3 2 2 2 2 Hole picture Δ -1.0 Δ

so − 1 + 1 so − 1 + 1 2 2 2 2 -1.2 -10 1 -1 kx,y,z kx,y (nm )

Figure 2.4: Optical selection rules in bulk and quantum wells of zinc blende semiconductors. (a) The conduction (e), heavy (hh), light (lh), and spin split-off (so) bands are shown near the zone center. (b) Optical absorption (or emission) transitions for σ ± photons are shown for hν ∼Eg. Heavy hole transitions are 3-times more favorable than light hole transitions, as indicated by line weight. Angular momentum states are labeled for electrons (holes in the boxed region) in the valence bands. (c) Subbands (e1, hh1, lh1) in a GaAs-Al0.4Ga0.6As quantum well ∗ ∗ from simulations of Fig. 1.3. (d) Optical transitions for photon energy Eg < hν

31 Chapter 2 Engineering spin in non-magnetic quantum wells exception that heavy hole and light hole degeneracy is broken by an energy Ehl at kx,y = 0, Fig. 2.4c. With a properly narrow line width optical excitation, it is ∗ ∗ possible to select only heavy hole transitions (Eg < hν

2.2.2 Hole spin-injection

Up to this point, we have discussed transitions from the point of view of electrons in valence and conduction band states, the electron picture. A physi- cally equivalent interpretation considers the creation of an electron and hole pair during photon absorption, the hole picture. The absorption of a σ + pho- 3 ton with hν ∼Eg excites an electron from the j = − 2 heavy hole band into the 1 3 j = S = − 2 conduction band. This process leaves behind “the lack of a j = − 2 3 electron”, which is equivalent to a j = + 2 hole. As seen in Fig. 2.4b, switching to the “hole picture” simply requires a sign change of the total angular momentum. From this paradigm, conservation of angular momentum requires that j of the electron and hole pair (exciton) is equal to the j of the photon absorbed. Thus absorption of circularly-polarized photons injects angular momentum into both the conduction and valence bands. We note an important distinction that the angular momentum injected into the conduction band states is purely spin, j = S, while in the valence band both orbital and spin angular momentum are injected.

2.3 Optical detection of spin

Spin, or rather, the total angular momentum of carriers is detected using polarization- resolved optical measurements. This follows directly from the selection rules

32 Chapter 2 Engineering spin in non-magnetic quantum wells we described in the previous section, where the angular momentum of circularly- polarized photons couples to the total angular momentum of band carriers. Fol- lowing trivially from the polarization-selective absorption shown in Fig. 2.4 b and d, a polarization-resolved measurement of absorption will be sensitive to the spin polarization. Application of a magnetic field breaks the degeneracy of the angular momentum states along the field direction, the Zeeman effect (Fig. 1.4) giving a spectral dependence for the polarization-resolved optical transitions. The reverse of absorption, emission, also conserves angular momentum producing circularly-polarized photons depending on the angular momentum of the carriers. In this thesis, polarization-resolved absorption and emission are measured to monitor spin polarization, spin-splitting, and dynamics. In the following sections we describe the details of these optical techniques. Two fundamental measurement geometries (Faraday and Voigt) specify the position of the optical axis relative to the magnetic field axis as shown in Fig. 2.5. Note that these geometries are defined without respect to the sample orientation, which must also be specified:

1. In the Faraday geometry the optical axis is parallel to the magnetic field (longitudinal). The magnetic field induced spin-splitting couples to σ ± photons with angular momentum along the field direction, allowing for spectral measurement of the Zeeman splitting. Spins that are optically injected in this geometry are static basis states of the magnetic field, whose only time-dependence is spin-relaxation, with a characteristic longitudi-

nal spin lifetime (T1).

2. In the Voigt geometry the optical axis is perpendicular to the magnetic ﬁeld (transverse). The magnetic-ﬁeld induced spin-splittings and angular momentum states are orthogonal to the optical axis, thus Zeeman split-

33 Chapter 2 Engineering spin in non-magnetic quantum wells

σ+ Faraday (longitudinal) , j = 1 Voigt (transverse) B σ+ x-basis z j = 1 + 1 B x 2 x-basis g μ B 1 e B + 1 + 1 + 1 2 2 2 − μ 2 ge BB

z (y)-basis Sx Sz(y) − 1 t t 2 1 1 + 1 − 1 − − z (y)-basis 2 2 2 2 + 1 − 1 2 2

Figure 2.5: Magneto-optical measurement geometries. Spins are injected into the conduction band under σ + excitation and in a magnetic ﬁeld along the x-axis. The spin-basis states along the optical axis and magnetic ﬁeld are shown for the Faraday and Voigt geometries. The time evolution of the injected spins in the basis state along the optical axis is shown.

tings cannot be measured spectrally in this geometry. Spins that are optically injected are perpendicular to the ﬁeld direction and can be represented as superpositions of the magnetic ﬁeld basis states. This induces

gµBB a spin-precession at the Larmor frequency (νL = h ) as described in section 1.2.1, which decays exponentially with a characteristic transverse ∗ spin lifetime (T2 ).

2.3.1 Angular momentum of light

In this thesis, the accurate determination of the sign of the angular momentum of light is necessary. This is complicated by the fact that the angular momentum of light is measured, using wave retarders, with respect to its linear momentum vector, i.e. as right circularly polarized (RCP), or left circularly polarized (LCP)

34 Chapter 2 Engineering spin in non-magnetic quantum wells light. However, in our case the photon’s angular momentum is coupled to a spin, whose direction is defined with respect to the magnetic field direction. The photon’s angular momentum is thus measured with respect to a laboratory framework (defined by the field direction), i.e. σ + and σ −. RCP light is defined as the angular momentum vector being parallel to the linear momentum vector, jˆ = kˆ. That is, the angular momentum vector points toward the observer of the incoming photon. LCP is defined as the opposite, with jˆ = −kˆ. An optics textbook or web page suffices to accurately adjust the linear polarizer and quarter wave retarder with respect to each other to understand if you are measuring RCP or LCP, also discussed in Poggio’s thesis [38]. Now consider a photon being emitted from a heavy hole exciton in a quan- 3 tum well. Due to the selection rules, we know that a + 2 heavy hole (equal to 3 1 a − 2 electron in the heavy hole band) combines with a − 2 conduction band electron to emit a j = +1 photon, i.e. σ +. This photon has angular momentum j, defined with respect to the spin of the recombining carriers, which is in turn defined by the magnetic field axis. If the photon is emitted along the field direction, then jˆ = kˆ. That is, a σ + photon moving along the field direction is equal to an RCP photon. If this photon is instead emitted against the field direction, then jˆ = −kˆ. Thus, a σ + photon moving against the field direction is equal to an LCP photon. From the above discussion it is clear that the experimenter needs to carefully measure the direction of the magnetic field with respect to the optical path in order to properly ascribe σ ± polarizations.

1. Determine the retardance of the wave plate.

2. Determine the direction of the fast-slow axis with respect to the linear polarization.

35 Chapter 2 Engineering spin in non-magnetic quantum wells

3. Reference an optics handbook to determine which case is RCP and which is LCP based on 1 and 2.

4. Determine the absolute direction of the magnetic ﬁeld with respect to the optical axis.

5. Using 3 and 4, determine the relation between RCP/LCP and σ ±, which is speciﬁc to your experiment.

2.3.2 Polarization-resolved photoluminescence

In the Faraday geometry, polarization-resolved photoluminescence (PL) yields direct measurement of the Zeeman splitting in the conduction and valence band.

In quantum wells with sufﬁciently large Elh, we can ignore the light hole transitions. The PL emission energy (E) for σ + and σ − polarizations follow from Fig. 1.4 and equations 1.5 and 1.8,

∗ e hh ∗ 1 E + = E + H − H = E − µ B(g − g ), (2.2) σ g ↓ ↓ g 2 B e hh and, ∗ e hh ∗ 1 E − = E + H − H = E + µ B(g − g ). (2.3) σ g ↑ ↑ g 2 B e hh The total excitonic Zeeman splitting is,

∆Eex = Eσ + − Eσ − = −(ge − ghh)µBB = −gexµBB, (2.4) where we deﬁne the exciton g-factor gex = ge − ghh. The polarization of photoluminescence is deﬁned as,

I + − I − P = σ σ , (2.5) Iσ + + Iσ − +,− where Iσ +,− is the intensity of σ polarized PL. In the absence of spin- injection, polarization arises from the Zeeman effect, as just discussed, and can

36 Chapter 2 Engineering spin in non-magnetic quantum wells

S (0) t z

sample y

Bx σ± Laser

Figure 2.6: Hanle effect measurement of time-averaged spin-precession. be measured only in the presence of a magnetic ﬁeld in the Faraday geometry. In a quantum well, for example, the heavy hole exciton will split into j = ±1 states with energy splitting given by equation 2.4. The thermal occupation of the two states corresponding to σ + and σ − emission dictates the relative intensities. Thus the P in equation 2.5 tracks a Brillouin function. In the presence of spin-injection (either optical or electrical) the PL polarization is proportional to the average spin polarization of the valence and conduction bands. Such measurements are convoluted with the recombination lifetimes and spin lifetimes of electrons, heavy, and light holes.

2.3.3 Hanle effect

The Hanle effect refers to time-averaged measurements of spin precession [62]. Optical spin-injection in the Voigt geometry sets up spin polarization along the z-axis at time zero, Sz(0), as shown in Fig. 2.6. The transverse magnetic field causes a spin precession. At zero magnetic field, the spin polarization remains fixed along the z-axis thus the PL polarization is maximized. At higher magnetic fields the spin polarization rotates away from the z-axis, decreasing the PL polarization. This time-averaged measurement of spin precession can be

37 Chapter 2 Engineering spin in non-magnetic quantum wells modeled simply by integrating the spin precession of equation 1.11. Thus,

Z ∞ Sz(Bx) = Sz(Bx,t)dt 0 ∞ Z t gµBBx = exp ∗ cos t dt 0 T2 h¯ S (0) = z , (2.6) ∗ 2 gµBBxT2 h¯ + 1

2h¯ which is a Lorentzian with a full width at half maximum FWHM = ∗ . gµBT2 Noting that the spin polarization is proportional to the PL polarization, P(0) P(Bx) = + P0, (2.7) ∗ 2 gµBBxT2 h¯ + 1

∗ where P0 is an experimental offset. If g is known, then T2 can be extracted.

Here we have assumed that the recombination time (Tr) is much greater than ∗ T2 .

2.3.4 Time-resolved Kerr (and Faraday) rotation

The technique of time-resolved Kerr or Faraday rotation has been used exten- sively to monitor the time dependence of electron spin precession in a number of semiconductors and their heterostructures. More comprehensive discussions of the history and implementation of this technique are provided in a number of theses [37–40, 61, 63, 64]. Here we provide a brief description of the measurement scheme and technique sufﬁcient to explain the results presented in the thesis. Faraday rotation (FR) refers to the rotation of the polarization plane of linearly polarized light as it passes through a magnetically polarized material, ﬁrst observed by Faraday in 1845 [65]. The effect is due, classically, to a difference in the index of refraction for right and left circularly polarized light (nσ + and

38 Chapter 2 Engineering spin in non-magnetic quantum wells

+ − nσ − ) in the medium, i.e. σ and σ move at difference velocities through the material. Linearly polarized light can be decomposed into a superposition of σ + and σ − components. During the ﬁnite time required to transit the material, the velocity difference leads to a phase lag in the σ + and σ − components, effectively rotating the polarization vector of the light after passing through the medium.

The origin of the difference in nσ + and nσ − , called birefringence, is a magnetization component oriented along the optical axis. FR is observed not only due to magnetic ﬁeld induced birefringence, but by any source of magnetization, such as spin polarization and remnant magnetizations. Kerr rotation (KR) is similar to FR but is observed in reﬂection. Though the details of the physical origin are different, the phenomena are used interchangeably to optically monitor magnetization in a material. In semiconductors, the optical absorption edge occurs at energies of ∼ Eg. Through the Kramers-Kronig relation, this corresponds to a peak in the index of refraction. As illustrated in Fig. 2.7, the difference in absorption edges (due to spin-splitting at the band edge) results in an optical birefringence, which is the origin of FR or KR signals in semiconductors. The FR is proportional to the difference in index of refraction (nσ + −nσ − ). This results in a characteristic energy dependence of the FR, with a maximum slightly above the absorption edge, a sign change and minimum for energies slightly below the absorption edge. The FR (and KR) effect combined with optical spin-injection form the basis of a powerful technique for measuring the time dynamics of carrier spins (and localized moments) in bulk zinc blende semiconductors and their heterostructures [66–68]. In the this technique, known as time-resolved FR (or KR), an ultrashort laser pulse (∼ 100 fs) is beam split into a 2 pulses (pump and probe) each with distinct optical paths. By passing through a linear polar-

39 Chapter 2 Engineering spin in non-magnetic quantum wells

* (a) Eg

σ− σ+

absorption

(b) σ− σ+ n σ+−σ− ∝θ K

Energy

Figure 2.7: Optical birefringence at the absorption edge in semiconductors. (a) The difference in absorption edge energies for the two optical helicities due to spin-splitting. (b) The corresponding optical birefringence due to the absorption edge splitting in (a). izer and quarter-waveplate the pump pulse is prepared in σ ± polarization state and is incident upon the sample. The absorption of the pump pulse results in a non-equilibrium spin population through optical spin-injection, section 2.2. Meanwhile, the probe pulse passes through a linear polarizer and down a mechanical delay line. The computer controlled delay line alters the length of the optical path of the probe pulse, thus providing control of the relative arrival time of the probe pulse with respect to the pump pulse. A schematic of the pulsed laser and delay line geometry is shown in Fig. 2.8a. After transmitting through (reﬂecting off of) the sample, the FR (KR) of the probe pulse is measured using a balanced photodiode bridge. Detailed description of the photodiode bridge

40 Chapter 2 Engineering spin in non-magnetic quantum wells

L/2 (a) 13 ns probe Ti:sapphire

~ 100 fs Δt = L/c pump

Bx (b)

Δt Sz σ± π probe θ ∝ pump F Sz

θ ∝ K Sz

Figure 2.8: Optical setup for time-resolved Kerr or Faraday rotation. (a) The pulsed laser and delay line schematic. (b) Time-resolved Kerr or Faraday measurements in the Voigt geometry. circuit design is given in Crooker’s thesis [39] as well as a new, lower noise design in Kato’s thesis [37]. In practice, data are not acquired by single shot (single laser pulse) measurements described above, but by averaging ∼ 100,000,000 repeated measurements. The laser system continuously emits laser pulses at a repetition rate near ∼ 100 MHz. Optical choppers and/or electro-optical modulators alter the polarization state or intensity of the laser pulses. Usually a cascaded lock-in amplification technique is employed, whereby the pump beam is modulated at high frequency (3-100 kHz), and the probe beam is modulated at low frequency (1 kHz). The first lock-in amplifier isolates the FR or KR signal due to the

41 Chapter 2 Engineering spin in non-magnetic quantum wells pump pulse, thus only signals generated by spin-injection are amplified and input into a second amplifier. The second lock-in amplifier further isolates the signal due to the probe pulses. A detailed discussion of lock-in techniques and signal processing is provided in Kato’s thesis [37]. Electron spin precession is measured by time-resolved Kerr rotation (KR) in the Voigt geometry as shown in Fig. 2.8b [69]. Typical laser intensities are 1 mW for the pump beam and 0.2 mW for the probe beam. The helicity of the pump beam polarization is modulated at ∼50 kHz by a photo-elastic modulator and the intensity of the linearly polarized probe beam is modulated by optical chopper at 1 kHz. The pump and probe beams are focused to overlapping ∼ 50 µm diameter spots on the sample, which sits in a magneto-optical cryostat with a split-coil superconducting magnet.

We measure the rotation of the axis of polarization (θK) of the reﬂected probe beam as a function of time (∆t). As described above, the mechanical delay line is used to electronically control the arrival time of the probe pulse with respect to the pump pulse.

The dependence of the Kerr or Faraday rotation angle (θK,F ) with delay time (∆t) is, −∆t θK,F (∆t) = Aeexp ∗ cos(2πνe∆t + φe), (2.8) T2e ∗ where Ae is the amplitude of electron spin precession, T2e is the transverse elec- geµBB tron spin lifetime, νe = h is the electron spin precession frequency, and φe is a phase offset. Ae is the amplitude of Kerr rotation at ∆t = 0 and is therefore proportional to the total electron spin-injected by absorption of the pump pulse.

42 Chapter 2 Engineering spin in non-magnetic quantum wells

2.4 Vertically biased quantum wells

QWs may have tilted bands due to internal electric fields from electronic doping, surface depletion, polarization charge, and band gap gradients. Applying a vertical bias to QWs provides a convenient means for controlling the carrier density and band bending within these structures. Thus, wave function position and shape can be controlled with vertical bias. In this section we describe how vertically biased QW structures can be used to manipulate the spin of electrons and local moments. The g-factor gradient and band profiles are tailored by heterostructure design. Applying a vertical-bias shifts the electron wave function into regions of varying g-factor allowing control over the sign and magnitude of the electron spin-splitting using an electric field. Spatial control of the electron wave function is also used to selectively polarize and depolarize local nuclear spins via the hyperfine coupling. Finally, by applying a high-frequency vertical- bias, electron and nuclear spin can be resonantly modulated and depolarized. The g-factor will shift with band bending due to wave function penetration into the barriers. This g-factor gradient in semiconductor QWs provides a means for controlling the effective g-factors by changing the position of carrier wave functions confined to these structures. This effect was observed in a two- dimensional electron gas (2DEG) of GaAs-AlGaAs in which the electron spin resonance (ESR) frequency was measured as a function of gate bias [17]. A shift in the ESR frequency corresponding to a change in effective electron g-factor of ∼0.003 (change in ∆E↑↓ ∼0.17 µeV/T) was observed due to gate bias. In the square well and 2DEGs, the application of an electric field along the growth axis (gate bias) results in a triangular distortion of the QW, in which case wave function distortion and energy shifting due to the quantum confined Stark effect are the main results. Although the wave function minimum is displaced,

43 Chapter 2 Engineering spin in non-magnetic quantum wells penetration into the energetically forbidden barriers remains a small effect.

2.4.1 Backgating

To attain significant displacements of the electron wave function in these structures, large electric fields need to be applied across the front and backgates. This requires the application of DC voltages at which typical backgated structures break down [70]. The development of low-temperature (LT) grown GaAs conduction barriers by Maranowski et al. allows for the application of un- typically large electric fields to GaAs-based heterostructures [71]. LT GaAs resistive layers are grown between the active layer and the conducting back- gate, usually n+GaAs. At a substrate temperature of 250˚C and in an As overpressure, the excess As incorporates into GaAs at concentrations of ∼1-2% (∼1020 cm−3). After annealing such layers at high temperature, metallic As precipitates form creating a highly resistive metal/semiconductor/metal com- posite [72], massively reducing leakage currents. Further modification of these layers can be performed to reduce photocurrent leakage. Simply increasing the Al content of the LT layer shifts the absorption edge to higher energy removing the photocurrent while maintaining the highly resistive character of LT GaAs [21].

2.5 Parabolic quantum wells

A convenient band structure for spatial wave function control exists in parabolic QWs, in which the ﬁrst-order effect of linear band bending (constant electric ﬁeld) is to shift the parabola’s minimum [73]. Since the potential maintains the same curvature under a linear distortion, the electronic wave functions remain undistorted and shift their center position to the new potential minimum, Fig.

44 Chapter 2 Engineering spin in non-magnetic quantum wells

Parabolic Quantum Well (a) (b) ECB d = 51 nm Photoluminescence (arb. units) (arb. Photoluminescence z Photon Energy (eV)

Figure 2.9: Parabolic quantum wells of GaAs-AlGaAs. (a) Schematic conduction band edge diagram in a parabolically graded quantum well. The ground state electron wave function dis- places along the growth axis (z) with vertical bias. (b) Photoluminescence as a function of excitation energy (PLE) in a 51-nm parabolic quantum well graded from 30%Al in the barriers to 0% in the center of the well. Reproduced from Ref. [74].

2.9a. GaAs-AlGaAs type parabolic QWs can be grown using MBE digital-alloying techniques. The Al concentration is digitally-graded to achieve the desired harmonic oscillator potential along the growth axis [30, 74, 75]. This is accomplished by shuttering the Al beam to grow a digital alloy (bilayer) in which the thickness of AlGaAs and GaAs are varied to attain the desired average Al concentration throughout the QW region. Each bilayer period is typically ∼20 Å, in order for enough GaAs to be grown in each bilayer to maintain purity and smoothness throughout the structure. Concentration grading is possible by analog alloying, in which the ﬂux of each effusion cell is adjusted by changing its temperature, but has the disadvantage that the effusion cell must be stabilized at each temperature setting during which time the growth is paused for several minutes. Such pauses are undesirable during QW growths since the probability

45 Chapter 2 Engineering spin in non-magnetic quantum wells

20 20 (a) 0.05 (c) 0.05 0.2 70 13% Δ E ( 15 10% μ 15 eV) g 0 0 0.00 0.00

7% g

-0.2 -70 10 10

z (nm) -0.05 -0.05 0% 5 5 1.63 (b) 100 1.60 30 1.58 10 -0.10 -0.10 V0 3 7% PL (a.u.) 1.55 0 0 -10123 Energy (eV) -4 -2 0 2 -1 0 1 2 3 V (V) g Vg (V)

Figure 2.10: (a) g-factor tuning in vertically-biased GaAs-Al0.4Ga0.6As parabolic quantum wells from optical measurements of electron spin dynamics via time-resolved Kerr rotation at a temperature of 5 K in a 6 T magnetic field. The wells are 100-nm wide and the % value represents the concentration of Al in the center of the well. The corresponding spin splitting in the conduction band is plotted in the right vertical axis. (b) Photoluminescence spectrum showing the quantum confined Stark effect in a parabolic quantum well at 5 K. Reproduced from Ref. [18]. (c) The g-factor versus vertical bias for parabolic quantum well with the corresponding wave function position (z). Reproduced from Ref. [21]. of impurity incorporation increases with wait time. Digital alloys are controlled through shutter timing and therefore require no growth interrupts to grade composition. As seen in Fig. 2.9b, Energy-resolved PL measurements confirm the harmonic shape of the potential since the rich excitonic spectrum shows a large family of equally spaced energy levels. The analytically soluble nature of the harmonic potential allows for the determination of conduction and valence band offsets from these spectra.

46 Chapter 2 Engineering spin in non-magnetic quantum wells

2.5.1 g-factor tuning

Parabolic QWs of GaAs-AlGaAs with LT GaAs (or LT AlGaAs) backgates were used to demonstrate electric field control of the electron g-factor [18, 76]. By measuring the electron spin dynamics optically using time-resolved Kerr rotation, the effective electron g-factor was extracted for various bias voltages. Figure 2.10a plots the effective electron g-factor as a function of gate bias in a number of these structures along with the ∆E↑↓. In particular, for samples with 7% Al at the center of the well, g-factor tuning through zero is possible. The g-factor remains constant over a large bias range between 0 V and -2 V due to exciton binding as evidenced by the loss of PL at biases outside this range, Fig. 2.10b. PL of these structures display the quantum confined Stark effect, appearing as a red shift in PL with gate bias, a typical spectroscopic feature of gated QWs [77]. Once the exciton binding between electrons and holes is overcome, PL disappears, electron wave function shifting occurs with less impedance, and the g-factor response to gate bias increases. The effective electron g-factor in parabolic QWs changes as a function of position along the growth axis due to its dependence on Al concentration [42]. It can be calculated quantitatively as a weighted average between the electron 2 probability density (|Ψs(z)| ) and the g-factor at any given position along the QW axis, ∗ 2 g (z) = ∫ g(z)|Ψs(z)| dz + ∆gc, (2.9) where g∗(z) is the effective electron g-factor, g(z) is the g-factor of electrons at each position in the QW due to the local Al concentration, and ∆gc is a phe- nomenological constant that accounts for confinement effects not included in this simple model. Based on the measured values of g∗ as a function of gate bias, shown in Fig. 2.10a, the center position of the wave function (z) can be

47 Chapter 2 Engineering spin in non-magnetic quantum wells calculated using equation 2.9. Figure 2.10c plots the results of this calculation for a 100-nm wide parabolic QW with 7% Al at the center. The wave function can be displaced ∼20 nm at 5 nm/V. Recent models of these structures using ﬁve-level k·p theory show close agreement with the experiment indicating that the simple picture of g-factor tuning, described above, captures the essential physics [78].

2.5.2 Shaping nuclear spin proﬁles

The spatial wave function control in parabolic QWs has also been used to shape nuclear spin profiles [21]. This experiment exploits the contact hyperfine interaction between electron and nuclear spins to selectively polarize and depolarize nuclear spin profiles at specific positions within a QW [79]. Optically injected spin-polarized electrons in the conduction band of a semiconductor polarize nuclear spins through dynamic nuclear polarization (DNP) [80]. This effect only occurs however, if a longitudinal component of electron spin polarization exists [58]. This is easily accomplished in the Faraday geometry, where the field is parallel to the optical axis, and therefore injects spins parallel to the field. In the Voigt geometry, which is necessary to observe spin precession, spins are, in principle, injected perpendicular to the field and DNP should not be observed since the transverse spin polarization averages to zero. In large magnetic field, the average longitudinal electron spin polarization will not be zero and DNP may occur through the equilibration of electron spins along the field, which were originally polarized transverse to the field. Additionally, a direct longitudinal component of spin-injection can be created by simply tilting the sample since the excitation beam refracts slightly along the field. In QWs, the g-factor anisotropy leads to a spin precession axis (ΩL) which is not parallel with the

48 Chapter 2 Engineering spin in non-magnetic quantum wells optical axis. This provides a component of spin injected along the quantization axis and enhances DNP [58]. If electrons overlap with nuclear spin, the contact hyperfine interaction will modify their spin Hamiltonians, such that the spin splitting, ∆E↑↓ = hΩL = gzµBBz + AH < Iz >, where AH is the hyperfine constant and < Iz > is the component of nuclear spin along the magnetic field. In analogy to the exchange interaction described later in equation (3.1), the hyperfine constant is weighted by the overlap between electrons and nuclear spin. Thus control over the electron probability density, as shown in Fig. 2.9a, will modify the effective strength of the hyperfine term. Nuclear spin profiles were created at arbitrary positions within parabolic QWs by fixing the gate bias, and therefore the electron wave function position, and waiting twenty minutes for DNP to saturate.

Time-resolved Kerr rotation was used to measure ∆E↑↓ as a function of gate bias once the nuclei were polarized. Subtracting the signal in the unpolarized case allows for the extraction of the hyperfine term plotted in Fig. 2.11a, where the vertical lines plot the center position of the electron wave function during the polarization process [21]. These nuclear spin profiles can be resonantly depolarized by applying RF to the vertical gates at isotope specific frequencies.

2.5.3 Electron spin resonance by g-tensor modulation

Thus far we have limited the discussion of electric ﬁeld effects to the case of DC bias. It is also possible to apply AC biases to parabolic QWs at frequencies close to the spin precession frequency and resonantly control electron spins [81]. This spin resonance technique, called g-tensor modulation resonance (gTMR), does not require a transverse magnetic ﬁeld, but makes use of the inherent g-factor anisotropy of QWs. As described in a previous section, the g-factor in a QW

49 Chapter 2 Engineering spin in non-magnetic quantum wells

(b) g (a) 0.8 (a.u.)

0 0.6 KR (a.u.) KR 0.4 0.0 0.5 1.0 1.5 Time delay (ns) 0.2 8 0.0 1 6 0 0.8 A (c)

H -1 0.6 ( 0.4 µ

eV) 2 0.2 Magnetic field (T) Magnetic field 0 0.0 -2 -1 0 2.0 Vg (V) 1.5 (d) 1 V = -1.5 V 1.0 0 0 0.5 -1

0.0 (a.u.) KR 0 2 468 10 12 -2 z (nm) Microwave induced 02468 Magnetic field (T)

Figure 2.11: Nuclear and electron spin manipulation in parabolic quantum wells of GaAs- AlGaAs. (a) Spatial proﬁling of nuclear spin polarization in parabolic QWs at 6 K and a ∼4

T field. (AH < Iz >) due to nuclear spin polarization is plotted along the z, Gaussian fits are shown as lines. Vertical lines mark the center of the electron wave function when dynamic nuclear polarization was performed. Reproduced from Ref. [21]. (b) Frequency-modulated electron spin precession due to microwave-modulation of the electron g-factor at 5 K and a 6 T field. A signal of at 2.356 GHz applied across the front and back gates is plotted as a sinusoid. (c) The change in KR due to the microwave signal as a function of both magnetic field and DC vertical bias. Black squares plot the expected position of g-TMR resonance calculated from the measured anisotropy of the g-factor. (d) Linecut of data in (c) at fixed DC bias showing the resonance structure. Reproduced from Ref. [81].

50 Chapter 2 Engineering spin in non-magnetic quantum wells

is anisotropic due to lateral confinement. Since g⊥ and g// have different gate bias dependences, electron spins precess about different axes depending on the vertical bias. In gated parabolic QWs, the g-factor anisotropy is modulated by applying AC bias between the front and back gates, effectively modulating the spin precession vector. Microwave-frequency is applied across the front and backgates of a parabolic QW while measuring the electron spin precession using time-resolved Kerr rotation. The repetition rate of the laser (∼ 76 MHz) is used as a reference clock for a frequency tunable microwave amplifier. Thus, ∼ GHz frequency voltages are applied to the QWs in phase with the laser. Additionally, the laser is operated in a phase-locked loop to maintain a stable repetition rate. Figure 2.11b plots the Kerr rotation as a function of time showing electron spin precession in the presence of ∼ 2 GHz gate bias. Since the effective g-factor is modulating at ∼ 2 GHz, electron spin precession is frequency modulated. When the modulation frequency is brought into resonance with the spin- splitting along the quantization axis (g-TMR) electron spins can be tipped either toward or away from the magnetic field. The requirements to observe g-TMR are that the g-tensor must be modulated at the spin-splitting frequency and that the g-factor must be anisotropic. Both criteria are satisfied in these parabolic QW structures. Lock-in amplification was used to measure the microwave- induced Kerr rotation as a function of magnetic field and an additional DC vertical bias, Fig. 2.11c. The measurements show a clear resonance line that shifts with magnetic field and vertical bias. The position of the resonance matches calculated values (black squares) based on angle dependent measurements of the g-factor anisotropy. Line cuts at fixed bias voltage, Fig. 2.11d, reveal the shape of the resonance lines and match numerical simulations (blue line).

51 Chapter 2 Engineering spin in non-magnetic quantum wells

2.6 Coupled quantum wells

In parabolic QW structures the electron wave function shifts in a continuous manner, but it is also possible to achieve discrete changes in electron position in coupled QWs. These structures consist of adjacent QWs connected by barriers that are thin enough for wave function tunneling to occur [82]. Vertically-biased coupled QWs have been used to transition from the spatially-direct exciton state, where the exciton resides in one well, to the spatially-indirect exciton state, where the exciton is split between adjacent wells [83]. The resulting change in spatial overlap between the electron and hole drastically alters the transition rates and can be used to modulate absorption [84]. If the widths of the coupled wells are different, then shifting the electron wave function between wells changes the electron g-factor due to its strong width dependence, Fig. 2.2. The electron g-factor was measured in coupled QWs of GaAs-AlGaAs as a function of vertical bias for samples with different tunnel barriers widths [47]. Figure 2.12 plots the g-factors in these structure measured by time-resolved Kerr rotation together with the schematic band edge diagrams. For uncoupled QWs with a large 20-nm tunnel barrier, Fig. 2.12a, two g-factors are visible over the full bias range corresponding to the different g-factors in the two wells. Tunneling between the wells does not occur within the carrier recombination time. A sharp g-factor transition with vertical bias occurs for a sample with a 6-nm tunnel barrier, Fig. 2.12b, where electron tunneling is incoherent. Finally, strong coupling of the wells is observed in narrow 2-nm tunnel barriers, Fig. 2.12c, where the g-factor is smoothly varied between its value in the two wells.

52 Chapter 2 Engineering spin in non-magnetic quantum wells

Barrier = 20 nm a 0 (a)

-2

-4 (b) Barrier = 6 nm 0 b (V) g

V -2

-4 (c) Barrier = 2 nm 0 c -1 -2 -3 0.1 0.2 0.3 |g|

Figure 2.12: Tuning the electron g-factor in coupled quantum wells of GaAs-Al0.3Ga0.7As. The two wells are 7-nm and 10-nm wide. Grayscale intensity plots the fourier transform of electron spin precession as a function of vertical bias and g-factor at a temperature of 5 K in a 6 T magnetic ﬁeld. Schematic band edge diagrams show the effect of vertical bias for coupled wells with different tunnel barrier widths. Vertical arrows indicate the strong (solid line) and weak (dashed line) photoluminescence transitions and curved arrows show the inter-well tunneling. (a) Two distinct g-factors in uncoupled wells with 20-nm wide tunnel barrier. (b) Sharp g-factor switching for incoherently coupled wells with 6-nm tunnel barriers. (c) Smooth g-factor tuning in coherently coupled wells with 2-nm tunnel barriers. Reproduced from Ref. [47].

53 Chapter 3

Magnetic semiconductors

The doping of transition metal ions into semiconductors has been the focus of several decades of study. The initial work in dilute magnetic semiconductors (DMS) began and quickly ﬂourished in the 70’s and 80’s with the discovery of giant Zeeman splitting induced in II-VI DMS due to the interaction between local magnetic ion spin and the delocalized carriers in the host semiconductor’s band structure. Aided by the rapid progress in the growth of extremely high purity semiconductor heterostructures by molecular beam epitaxy (MBE), the II-VI DMS provided a wealth of physics and an ideal playground to test and reﬁne band theory for this novel class of materials. Advances in the past decade have spawned a renewed interest in magnetic semiconductors, which are seen as prime candidates as systems for spin-based electronics, or spintronics, in which both electron charge and electron spin are utilized for increased device functionality. In particular, the discovery of ferromagnetism in Mn-doped GaAs fueled high paced research efforts to increase the Curie temperature of this material to room temperature and/or achieve ferromagnetism in other transition metal doped semiconductors. The physics of II- VI DMS has been adapted for the new class of ferromagnetic semiconductors

54 Chapter 3 Magnetic semiconductors and explains the long-range ferromagnetic coupling between Mn ion spins as due to the secondary spin coupling between delocalized valence band holes and the two Mn ions, which are typically separated by tens to hundreds of atomic distances. In this chapter we discuss the magnetic interactions present in II-VI and III- V DMS systems. The exchange interactions between electrons or holes in the semiconductor band structure and magnetic ions are explained. The exchange interactions are responsible for an enhancement of the Zeeman splitting of the spin split electronic bands in a magnetic ﬁeld. The growth of the ferromagnetic semiconductor GaMnAs by MBE is described, in which highly non-equilibrium conditions are necessary to overcome the solubility limit of Mn in GaAs. Shut- tered MBE techniques for the growth of digital ferromagnetic heterostructures are described. A brief discussion of carrier-mediated ferromagnetism is included to illustrate the importance of defect control in GaMnAs.

3.1 s − d and p − d exchange interactions

The spin interactions between conduction band electrons and the local magnetic- ion 3d electrons is called the s−d exchange interaction, while the valence band electrons also interact with 3d electrons through the p−d exchange interaction. These two effects are the fundamental Hamiltonian terms usually dominating the spin physics of DMS. The origin of these exchange interactions is discussed in chapter 1.3 within a single He atom. In DMS the same exchange interactions occur but within a continuous semiconductor crystal. The exchange interactions can be approximated by summing all the interactions (J) between an electron

55 Chapter 3 Magnetic semiconductors and each magnetic ion,

~ ~ Hex = −∑J(~r − Ri)~σ · Si, (3.1) i where ~σ is the electron spin operator at position ~r, ~Si is the spin operator of 2+ the local magnetic moment at position ~Ri (3d, e.g. Mn ), and J(~r −~Ri) is the exchange constant. Here the exchange operator, J, is assumed to be highly localized such that it only takes on constant values at δ-functions localized on the magnetic impurity. This is equivalent to the Heisenberg exchange interaction implemented in a dilute magnetic system, rather than a ferromagnetic metal [2, 85]. J(~r −~Ri) is a difficult quantity to calculate by first principles. Its definition involves two particle exchange, in this case between band electrons and localized 3d electrons, which is written as the convolution between the two particles’ wave functions and the electrostatic interaction between them, similarly to the exchange in the He-atom as in equation 1.18. In semiconductors this calculation can be approximated using the fact that the conduction band and valence band electrons have s-like and p-like symmetry at the bottom of the band. A momentum independent J can then be calculated using k · p theory. Non-vanishing diagonal terms in the k · p matrix appear in the form, 1 (± )N αxhS i,α = hΨ |J|Ψ i, (3.2) 2 0 z cb cb for the conduction band (s − d exchange), and, 1 (± )N βxhS i,β = hΨ |J|Ψ i, (3.3) 2 0 z vb vb for the valence band (p − d exchange), where N0 is the cation density, x is the fraction of magnetic impurities occupying cation sites (e.g. Zn1−xMnxSe), hSzi is the projection of the total spin of the magnetic impurity along the applied magnetic field, Ψcb is the s-like wave function at the conduction band edge, and

56 Chapter 3 Magnetic semiconductors

Ψvb is the p-like wave function at the valence band edge. N0α and N0β are frequently referred to as the s − d and p − d exchange constants, respectively. While any semiconductor doped with magnetic ions can be deﬁned as a DMS, research has focused mainly on the doping of Mn2+ ions in either II- VI and more recently, III-V semiconductors in which case the Mn2+ ion lies on the III or II site. In II-VI systems, Mn2+ is both electronically neutral and 5 2+ contains a half-ﬁlled 3d shell of spin 2 . Additionally, the solubility of Mn in II-VI is generally quite large, usually more than 10%. Compared with ZeSe and CdSe, MnSe is wide band gap semiconductor. This allows for the simul- taneous engineering of the band gap and exchange overlap. Typical values of the exchange parameters and the band gaps are listed in Table 3.1. As we dis-

Table 3.1: Band-gaps and exchange constants in some magnetic semiconductors. The range of x and temperature at which the band gaps (Eg) were measured are listed. References in “[]”.

Material Eg (eV) x range Temp (K) N0α (eV) N0β (eV)

Cd1−xMnxTe 1.585 + 1.51x 0 < x < 0.5 77 [86] +0.22 -0.88 [87]

Zn1−xMnxTe 2.381 + 0.68x x > 0.02 2.2 +0.19 -1.09 [88]

Zn1−xMnxSe 2.80 0 < x < 0.2 6.5 [89] +0.29 -1.4 [90]

Cd1−xMnxSe 1.821 + 1.54x 0 ≥ x ≤ 0.277 2 [91] +0.23 -1.27 [92]

Ga1−xMnxAs - - - −0.025 [48, 93] −1.2 [94] +0.023 [95] cussed in chapter 1.3, one would expect that the exchange interaction would be positive favoring the parallel alignment of magnetic-ion and band electron spins, i.e. ferromagnetic exchange coupling. Equations 3.2 and 3.3 should also be positive assuming J > 0. As an example consider the case of parallel spins ~ 1 (~σ · S = + 2 hSzi), in which case, following from equation 3.1,

~ ~ 1 Hex = −∑J(~r − Ri)~σ · Si = −(+ )N0αxhSzi. (3.4) i 2 For II-VI DMS, β is always found to be negative (antiferromagnetic), whereas

57 Chapter 3 Magnetic semiconductors

α is positive (ferromagnetic). The antiferromagnetic p − d exchange indicates that the approximation of a purely Heisenberg-like, direct, exchange coupling is not accurate for the electrons in the valence band. This has been explained in terms of hybridization effects between the p-like and 3d orbitals, discussed in section 6.6.3.

3.2 Paramagnetism

The magnetic properties of II-VI DMS are generally quite simple since long- range Mn-Mn coupling does not occur. Nearest neighbor Mn2+ ions couple antiferromagnetically and can coalesce into clusters under certain crystal growth conditions. Barring signiﬁcant clustering, the uncoupled Mn2+ ion spins act as 5 j = S = 2 paramagnets, with magnetization given by,

M = −gMnµBN0xhSzi, (3.5)

2+ where gMn is the g-factor for Mn , µB is the Bohr magneton, and hSzi is the average spin polarization. To calculate hSzi we must ﬁrst know the thermal occupancy of the spin states split along the magnetic ﬁeld. The Brillouin function

B j is such a function dictating the equilibrium thermal occupancy given that j = L + S is quantized and split by a magnetic ﬁeld into levels separated by energy ∼ gµBB,

(2 j + 1) 2 j + 1 1 η jgµBB B j(η) = coth( η) − coth( ) , η = , (3.6) 2 j 2 j 2 j 2 j kB(T − θP) where kB is Boltzmann’s constant, and θP is the paramagnetic Curie temperature. The ﬁeld dependence of the magnetization following from equations 3.5 and 3.6 is plotted in Fig. 3.1.

58 Chapter 3 Magnetic semiconductors

2 j = 3/2 -2

x) j = 4/2 0 1 -1 N j = 5/2 〈 S B z μ

0 0 〉 g = 2 -1 1

M / (g T = 5 K -2 2 -8 -6 -4 -2 0 2 4 6 8 B (T)

Figure 3.1: Magnetization versus magnetic ﬁeld for various total angular momentum states ( j) at constant magnetic ﬁeld. The corresponding average spin polarization is plotted on the right vertical axis hSzi.

The equilibrium spin polarization is just, 5 5gMnµBB hSzi = − jB j(η) = − B5/2 . (3.7) 2 2kB(T − θP) 2+ Note that since the g-factor for Mn (gMn = 2) is positive, for B > 0, then hSzi < 0 and M > 0. The spin polarization is plotted on the right vertical axis in Fig. 3.1. These signs can usually be figured out, regardless of the nomenclature used, by simply remembering the two fundamental points that: 1) magnetic moments always point along the applied magnetic field, and 2) for a positive g-factor the electron spin points opposite to the field (unless the Hamiltonians define the g-factor in a nontraditional way). One interesting point is that the g-factor defines, equation 3.5, the direction of the electron spin with respect to its magnetic moment. In the case of delocalized band electrons, by tuning the spin-orbit coupling and/or the exchange interactions the sign of the g-factor can be changed such that the electron spin is either parallel or antiparallel to its own magnetic moment, and, therefore, to the applied field. In the case of the 3d electrons localized on the magnetic impurity (Mn2+) described here, the highly localized character of the orbitals maintains the g-factor at close to its free space

59 Chapter 3 Magnetic semiconductors value of ∼ 2.002. We can derive the Curie-Weiss law by writing the Taylor expansion of equation 3.6, J + 1 [(J + 1)2 + J2](J + 1) B (η) = η − η3 .... (3.8) j 3J 90J3 Taking the ﬁrst term and using equation 3.7 gives,

gµB j( j + 1)B hSzi = − . (3.9) 3kB(T − θP) Substituting into equation 3.5 gives,

(N x)g2µ2 j( j + 1)B C M = 0 B = χB = B, (3.10) 3kB(T − θP) T − θP where χ is the susceptibility, C is the Curie constant, (N0x) is the density of moments, and we have expressed the function generically for any value of j. This first term approximation is valid at sufficiently small field or high temperature yielding a magnetization that is linear with field.

3.2.1 Enhanced Zeeman splitting, a.k.a. Effective g-factors

The exchange interaction, equation 3.1, adds another magnetic ﬁeld term to the spin splitting in the conduction and valence bands. The spin Hamiltonians given by equations 1.5-1.8 now include the exchange terms in equations 3.2 and 3.3. Thus for the conduction band,

1 1 1 1 He = + g µ B − N αxhS i , He = − g µ B + N αxhS i, (3.11) ↑ 2 e B z 2 0 z ↓ 2 e B z 2 0 z where, as discussed in the previous section, hSzi follows the magnetization of the magnetic dopant, equation 3.5. Similarly, for the valence bands,

1 1 1 1 Hhh = + g µ B − N βxhS i , Hhh = − g µ B + N βxhS i, (3.12) ↑ 2 hh B z 2 0 z ↓ 2 hh B z 2 0 z

60 Chapter 3 Magnetic semiconductors

− +1/ 2 1/ 2 e1 − geμBB xN0α Sz + 1/ 2 −1/ 2 σ− σ+ σ− σ+ +3/ 2 −3/ 2 E

hh1 − gHHμBB xN0β Sz

+3/ 2 −3/ 2

Figure 3.2: Giant Zeeman effect in a magnetic quantum well. The splitting of the electron spin states in the conduction, heavy, and light hole bands in a magnetic ﬁeld. As deﬁned in the text, the energy splittings are shown here for positive g-factors and positive exchange constants (α and β). Note that the valence band spins are drawn for the electron scheme and not the hole paradigm. σ+ and σ− label the helicity of the emitted photons from the allowed transitions.

Also, note that for a 3d magnetic dopant with g = 2, then hSzi < 0 for B > 0. Thus the exchange and Zeeman terms add together. and,

1 1 1 1 Hlh = + g µ B − N βxhS i , Hlh = − g µ B + N βxhS i. (3.13) ↑ 2 lh B z 2 0 z ↓ 2 lh B z 2 0 z

In Fig. 3.2 the spin split bands are shown for a magnetic semiconductor in which the exchange interaction adds a non-linear (Brillouin function) ﬁeld dependence compared to the purely linear regular Zeeman effect shown in Fig. 1.4. The giant Zeeman splitting is easily observed in Faraday geometry measurements of the polarization-resolved PL, section 2.3.2. Using equations 3.11 and 3.12, we redeﬁne the PL emission energies given by equations 2.2 and 2.3 to

61 Chapter 3 Magnetic semiconductors include the exchange terms,

∗ e hh ∗ 1 1 E + = E + H − H = E − µ B g + (N x)hS i(α − β), (3.14) σ g ↓ ↓ g 2 B z ex 2 0 z and,

∗ e hh ∗ 1 1 E − = E + H − H = E + µ B g − (N x)hS i(α − β). (3.15) σ g ↑ ↑ g 2 B z ex 2 0 z

The total excitonic Zeeman splitting, equation 2.4 becomes,

∆Eex = Eσ + − Eσ − = −gexµBB + (N0x)hSzi(α − β). (3.16)

3.3 III-V, GaMnAs

Magnetic doping in the case of III-V GaAs-based DMS is quite different than II- VI materials. Transition metal dopants, such as Mn2+ are no longer electrically neutral. In GaAs, AlAs, and InAs, an isolated substitutional Mn2+ ion is a deep acceptor lying ∼ 110 meV above the top of the valence band [96]. Initial work on Czokralski grown Mn:GaAs crystals found a low solubility of Mn in GaAs. Electron paramagnetic resonance and magnetometry conﬁrmed two different acceptor conﬁgurations:

0 − + 1. (AMn = AMn+h ) The neutral Mn acceptor in GaAs. The hole is bound close to the Mn-impurity, but does not reside in the valence shell of the 5 defect. The antiferromagnetic p − d exchange coupling maintains the 2 5 3 spin of the 3d Mn acceptor opposite to the hole spin ( 2 ). The acceptor complex therefore has j = 1 as veriﬁed by ESR and magnetometry [97, 98].

− 2. (AMn) The charged Mn acceptor. The hole resides in the valence band of GaAs, or at sufﬁciently high Mn-doping, resides in the Mn impurity

62 Chapter 3 Magnetic semiconductors

(a)3d5 + h (b) 3d4 (c) 3d5

+5/2 -3/2 +4/2 +5/2 + h 0 - Mn- Mn Mn -3/2 h+ VB

Figure 3.3: Mn acceptor conﬁguration in III-V semiconductors. (a) Neutral Mn acceptor in GaAs and (b) GaP. (c) Ionized Mn acceptor in GaAs.

5 5 band. The 3d Mn acceptor has spin 2 as veriﬁed by ESR and x-ray photoemission [94].

These two Mn-acceptor configurations are illustrated in Fig. 3.3. Depending on the concentrations of Mn and the Fermi level, the configuration of the Mn acceptor will change between the two cases. The charged acceptor is favored when the Mn concentration is high and the carrier compensation is low since the Fermi level is near or in the valence band. The neutral Mn acceptor is favored at low doping levels and/or high compensation. In GaP, the neutral Mn acceptor fully binds the hole within the valence shell, such that the Mn core configuration is no longer 3d5 but is 3d4 [99], also illustrated in Fig. 3.3b.

3.3.1 Carrier-mediated ferromagnetism

Orders of magnitude larger doping densities of Mn can be incorporated into GaAs by using highly non-equilibrium growth techniques. This was pioneered by Munekata et al. who developed a low temperature MBE method for growing In1−xMnxAs with x < 18% [100]. Low substrate temperatures overcome the thermodynamic solubility limit creating a supersaturated solid solution of

63 Chapter 3 Magnetic semiconductors

200 (a) (b)

100 80 (K) C T 60 50 40

30 0.1 1 p (1021 cm-3)

Figure 3.4: Curie temperature and defect structures in GaMnAs. (a) The Curie temperature as a function of the hole density in annealed GaMnAs grown by MBE. (b) The atomic structure of GaMnAs and the most common crystalline defects. Figures are adapted from reference [102].

III,Mn-V. Ohno et al. used this same low temperature MBE method to prepare

Ga1−xMnxAs crystals in which they observed the ﬁrst spontaneous ferromagnetism in a semiconductor in 1996 [101]. It was found experimentally that the ferromagnetic ordering temperature, the Curie temperature (TC) in GaMnAs is proportional to the hole density independently of the Mn doping density. Figure 3.4a plots this dependence explicitly for a number of ﬁlms with varying Mn doping density. Depleting the hole density (p) by doping or electronic gating reduces TC or removes ferromagnetic ordering entirely. The experimental observation of a carrier-mediated ferromagnetism drove theoretical models in the early years. It was already known from work in II-VI DMS, that the p − d exchange coupling between holes in the top of the valence band and magnetic ions in magnetic semiconductors is large and negative, section 3.1. An obvious theoretical explanation to the hole-mediated ferromagnetism in GaMnAs therefore considers a long-range and indirect cou-

64 Chapter 3 Magnetic semiconductors

+5/2 p-d p-d +5/2 Mn- Mn- -3/2 h+ VB

Figure 3.5: Schematic of carrier-mediated ferromagnetic coupling in GaMnAs. pling between two Mn-ion spins. Qualitatively, the Mn-ions spin couples to the hole spin through p−d exchange, and the hole spin similarly couples to another Mn-ion spin, shown in Fig. 3.5. Such a theory was first quantitatively described by Dietl et al., who also predicted a number of additional magnetic semiconductor systems where ferromagnetism could occur due to the same mechanism [103–105]. In particular, above room-temperature ferromagnetism was predicted in GaN and ZnO. This theoretical prediction stimulated a large, world-wide materials effort to find ferromagnetism in other systems than GaMnAs, and, in particular, for above room- temperature ferromagnetic semiconductors. Countless papers have reported observing ferromagnetism in semiconductors at room temperature, however much controversy surrounds these findings. For example, many of conclusions of the papers are not widely accepted in the magnetic semiconductor community since nanoscopic ferromagnetic precipitates could account for the ferromagnetic signals, usually observed by SQUID alone. These precipitates and other ferromagnetic compounds that are unintentionally formed often have Curie temperatures that are well above room temperature, e.g. GaMn or MnAs. Magnetometry of such films have only yielded S-shaped hysteresis loops indicative of polycrystalline samples. Perhaps the main drawback of these papers is the lack of clear proof of the coexistence of magnetism and semiconducting properties as has

65 Chapter 3 Magnetic semiconductors been thoroughly demonstrated in GaMnAs. For example, band edge observations of exchange splitting that tracks the magnetization, hole density dependent Curie temperatures, or optically-induced magnetization changes. In recent years, the controversy of the theoretical understanding of ferromagnetism in semiconductors has even encompassed the founding system of GaMnAs [106]. The difficulty lies in the increasing level of complexity in the experimental understanding of the microstructure and magnetic properties [102, 107]. As we will discuss briefly, improvements in the magnetic properties of GaMnAs have occurred due to reductions in electronic and magnetic defects by progress in crystal growth and post-growth treatments. In a clear-cut example of Kuhnian ”scientific progress”, GaMnAs research has entered the ”refinement” mode, where theoretical paradigms (such as carrier-mediated ferromagnetism) are already widely accepted, but new experimental findings cast doubt on many of the theoretical details. Of particular difficulty is the understanding of the actual valence band structure and position of the Fermi level. Large dopant and defect densities lead to huge variations in hole density for samples of nominally identical composition. Additionally, dopants and defects create impurity bands that alter the valence band structure. Despite the theoretical nightmare briefly outline above, materials and experimental breakthroughs have abounded. The Curie temperature of GaMnAs and related heterostructures has increased, mainly by post-growth annealing, from 110 K to 230 K [102]. Magnetic domain walls can be driven and reversed using electrical currents allowing for the measurement of the domain wall speed [108–110].

66 Chapter 3 Magnetic semiconductors

3.3.2 Growth and defect control

The MBE growth of GaMnAs requires low substrate temperatures to incorporate sufﬁcient Mn for ferromagnetism to occur. This requires ∼ 5% Mn and a substrate temperature of ∼ 250◦ C. At these conditions, large concentrations of crystalline defects are formed that reduce the hole density by carrier compensation and therefore reduce the stability of the ferromagnetic phase. One such defect, is the Mn interstitial (Mni), which is considered to be a double donor compensating two holes each [111], shown schematically in Fig. 3.4b. These defects have shown a surprisingly high diffusivity allowing for their removal by using post-growth annealing at relatively low temperatures (∼ 200◦ C) that drive them to the surface [112]. Consequently, the hole density and TC after annealing can be drastically increased [113]. Another well-studied defect in low temperature (LT) grown GaAs is excess arsenic. For substrate temperatures less than 300˚C, excess arsenic is incorporated by occupying a Ga-site, called an antisite defect (AsGa), shown in Fig. 3.4b. Like the Mni defect, it is also a double donor [114]. The density of these defects can reach 1020 cm−3, ∼1% of Ga-sites, which is the same order of magnitude as typical Mn doping levels [115]. By post-growth annealing at temperatures above 500˚C, AsGa can be removed by the formation of metallic As precipitates,[116] however at these temperatures Mn also precipitates out of the (Ga,Mn)As solid solution as MnAs nano-particles [117]. Appendix B describes the effect of arsenic antisites on the magnetic and electronic properties of GaMnAs.

3.3.3 Digital heterostructures

In addition to GaMnAs random alloy, in which Mn-atoms are randomly posi- tioned throughout the GaAs matrix, the layer-by-layer MBE method allows for

67 Chapter 3 Magnetic semiconductors digital Mn-doping at selective and thin positions within a heterostructure. The substrate temperatures and arsenic overpressures are similar to those required for GaMnAs growth, the only difference being the shutter sequence. GaAs is deposited with the Mn-shutter closed. GaAs growth is paused while MnAs is deposited. It was found by Kawakami et al. that sub-monolayers (ML) of MnAs deposited within a GaAs matrix in this form are ferromagnetic, which they coined digital ferromagnetic heterostructures (DFH) [118]. Typical thickness of the MnAs layers are 0.5 ML and typical GaAs spacer layers are 20 ML. Fig- ure 3.6 plots a schematic layer structure showing the partially completed MnAs ferromagnetic regions contained within a GaAs matrix. Varying the number of periods of MnAs layers within this matrix and the spacing between them, Kawakami et al. found that although individual MnAs sub-monolayers are themselves ferromagnetic, bringing the layers closer increased the overall TC, thus indicating a possible interlayer ferromagnetic coupling through the spacer layer. With the aim of improving and expanding the electronic and magnetic properties of such structures, Johnston-Halperin et al. adapted an advanced MBE technique to reduce the defect densities [119]. The technique, referred to as atomic layer epitaxy (ALE) or migration enhance epitaxy (MEE), deposits each constituent beam species at a time. Each beam species is shuttered within each monolayer, e.g. 1 ML of GaAs requires one 0.5 ML deposition of Ga and one 0.5 ML of As. A detailed description of the growth method is provided in Johnson-Halperin’s thesis [40]. Each adatom is given additional time to mi- grate along the substrate, effectively smoothing the ﬁlms. Additionally, dosing the arsenic within each monolayer reduces the density of As-antisites which increases the total hole density. In the case of GaMnAs random alloy, such an increase in hole density increases the TC. Counterintuitively, although the ﬁlm

68 Chapter 3 Magnetic semiconductors

0.5 ML MnAs GaAs spacer

Figure 3.6: Idealized schematic of a digital ferromagnetic heterostructure (DFH). Open squares represent adatoms in the GaAs layers and grey squares represent adatoms in the partially completed MnAs layers. In reality, diffusion would smear out the Mn concentration proﬁles.

quality was improved, Johnston-Halperin et al. found a decrease in the TC of ALE grown DFH’s compared with regular DFH’s. Even more strangely, the ALE grown DFH’s indicated no interlayer coupling, which indicates that the coupling observed by Kawakami et al. was probably structural in origin, i.e. strain mediated. The beneﬁt of the ALE grown DFH structures is the ability to independently dope the spacer layers n or p-type [119]. As described above, carrier compensation in these ﬁlms is reduced probably since As-antisites are more carefully controlled. A semiconductor-compatible ferromagnetic heterostructure with the possibility of electron or hole based conduction opens the possibility of more advanced semiconductor spintronic devices, such as the spin- transistor [40, 120].

69 Chapter 3 Magnetic semiconductors

3.4 Magnetic quantum wells

In QWs, the spatial confinement for carrier wave functions along the growth axis leads to a modification for the effective exchange splitting. An important distinction in the definition of x is made between the case for bulk or QW DMS. This can be understood, from equations 3.2 and 3.3, as a modification of the effective TM concentration experienced by carriers confined to the QWs. The effective x is equal to the overlap integral of the carrier probability density 2 |Ψs,p| with the TM concentration profile along the growth axis (z).

3.4.1 II-VI magnetic wells

In II-VI DMS, magnetic impurities are electrically inert, and the solubility of these impurities is relatively high; for example 10% Mn in ZnCdSe is easily at- tainable while maintaining high optical quality. These systems enjoy a long and successful history in the literature including the engineering of spin interactions via quantum conﬁnement in II-VI DMS QWs [121]. Giant Faraday rotations, the high solubility of magnetic impurities, and the large energy scale of the resulting exchange splittings enabled these structures to be ideal test systems for conﬁnement and exchange interactions. Since TM doping usually increases the band gap in II-VI materials, magnetic QWs can be constructed by simply doping the barrier [122]. In this case, exchange overlap between carriers in the well and magnetic impurities in the barrier occurs through the penetration of the wave function into the barrier, leading to relatively small x values as compared to bulk DMS alloys. Alternatively, magnetic ions can be doped directly in the well. This can be accomplished, for example, by digital doping submonolayers of MnSe layers into ZnSe-ZnCdSe square QWs [68, 123]. By changing the thickness and position of the digi-

70 Chapter 3 Magnetic semiconductors tal MnSe layers within a QW the effective x values are significantly increased leading to giant Faraday rotations as large as ∼107 deg/cm T. Finally, uniform magnetic doping in both QWs and barriers is possible in CdMnTe-CdMnMgTe, where the exchange constants in both the QW and Mg containing barriers are identical [124]. The uniform magnetic doping profiles in these structures allow for reliable measurements of the effects of quantum confinement on the exchange parameters [125, 126], discussed in later sections.

3.4.2 III-V magnetic wells

Although most work in magnetic QWs, to date, has been limited to II-VI paramagnetic DMS systems, III-V magnetic semiconductors offer the prospect of ferromagnetic interactions, i.e. GaMnAs. Unlike for the case of II-VI, Mn- doping in GaAs is electronically active, such that a Mn-ion occupying a Ga-site acts as an acceptor providing a free hole that takes part in long-range ferromagnetic interaction between Mn-ion spins. Unfortunately, the large density of growth defects severely reduce the electronic and optical performance compared with II-VI paramagnetic structures. As we will discuss in chapters 5 and 6, unique layer structures and growth conditions have been employed to im- prove the electronic and optical properties of magnetic III-V quantum wells.

71 Chapter 4

Paramagnetic II-VI parabolic quantum wells

In chapter 2, we saw how g-factor engineering in quantum wells could be used to control electron spin dynamics. Vertically-biased quantum wells allow one to electrically change the position of carriers conﬁned to these structures, particularly in coupled and parabolic quantum wells, section 2.4. These experiments exploit variations in the spin-orbit interaction due to compositional gradients. The energy of these effects in conventional semiconductors, however, is on the µeV scale and is manifested as variations in the spin precession frequency in the GHz range. In this chapter we discuss how magnetic dopants are incorporated into such structures to increase the energy splitting into the meV scale and precession frequency for electron spins into the THz range. Vertically biasing these structures provides electrical control of these energy splittings and precession frequencies through the exchange overlap between electrons, holes, and magnetic ions. The exchange coupling is controlled optically and electronically in magnetically-doped II-VI parabolic quantum wells (PQWs).

72 Chapter 4 Paramagnetic II-VI parabolic quantum wells

4.1 Background

In magnetically doped quantum wells, section 3.4, the s−d and p−d exchange interactions increase the effective g-factors for electrons and heavy holes, 3.1. The additional magnetic field dependent terms in the spin Hamiltonians mani- fests in the giant Zeeman effect, section 3.2.1 and Fig. 3.2. In II-VI magnetic quantum wells, the effective magnetic-ion concentrations, x, experienced by electrons or holes confined to the wells can be engineered by changing the magnetic doping profile. Magnetic doping in II-VI semiconductors increases the band gap, Table 3.1, thus requiring selective placement of the magnetic impurities to retain quantum confinement. In II-VI digital magnetic heterostructures, submonolayers of magnetic semiconductors can be grown within quantum wells while retaining high optical quality. The magneto-optical study of such structures by time- and polarization-resolved spin spectroscopies was the subject of Scott Crooker’s thesis [39], and the associated publications [68, 69, 123, 127]. In the experiments described in this chapter we use quantum wells composed of ZnCdSe with ZnSe barriers and doped with MnSe magnetic submonolayers. The effective g-factors can reach ∼ 100 in such structures as evidenced by the giant Zeeman splitting. Figure 4.1a shows the electron wave function and conduction band edge in such structures along the growth axis. The well is 12-nm wide and composed of Zn0.8Cd0.2Se with ZnSe barriers; 3 equally spaced MnSe single monolayers are deposited within the quantum well. The polarization-resolved absorption spectrum of this well is plotted in Figure 4.1b. Application of a magnetic field leads to a net magnetization of the paramagnetic 2+ 5 Mn spin- 2 ions. As described in section 3.2.1, this results in the giant Zeeman splitting between σ ± absorption edges. At 2 T the splitting is ∼ 10−15 meV,far exceeding the normal Zeeman splitting from the exciton g-factor, gexµBB ∼ 0.1

73 Chapter 4 Paramagnetic II-VI parabolic quantum wells

(a)1 ML MnSe (b) B=0T B=2T T=5K σ+ T=5K σ+ σ- σ- Ψ e

Ecb

z Absorption (arb. units) 2.63 2.66 2.63 2.66 12 nm E (eV) E (eV)

ZnSe Zn0.8Cd0.2Se ZnSe

Figure 4.1: (a) The conduction band edge (Ecb) and the electron wave function (Ψe) in a magnetic quantum well of ZnSe-ZnCdSe. Three layers of 1ML MnSe are evenly spaced in the well. (b) Polarization-resolved absorption spectra in such wells. Giant Zeeman splitting due to the exchange interaction is evident with the application of a magnetic ﬁeld measured in the Faraday geometry. Data from [68, 127]. meV. Using the technique of time-resolved KR (or FR) in the Voigt geometry, section 2.3.4, the spin precession of electrons conﬁned to these structures can be observed, Fig. 4.2a. In the presence of the giant s − d exchange coupling, the electron precession frequency increases into the THz range. Just as the exchange interaction increases the spin splitting, it decreases the electron spin ∗ lifetime (T2e), which is ∼ 2 ps in Fig. 4.2a. Following from equations 3.11, the total electron spin splitting in the conduction band is,

e e e ∆E = H↑ − H↓ = geµBBx − N0αxhSxi, (4.1) where hSxi is deﬁned in equation 3.7 with the magnetic ﬁeld (Bx) applied along the x-axis rather than the z-axis. Thus the electron spin precession frequency is

74 Chapter 4 Paramagnetic II-VI parabolic quantum wells

(a) 4 (b) 10 electron spin

0 0

(arb. units) 4.6 K 2+ 2+ Mn spin

FR -10 Mn spin 2 T θ -4 0 20 40 60 80 0 200 400 600 Δt (ps) Δt (ps)

(c) z Δt < 0 z Δt = 0 z Δt > 0

Mn2+ p-d Mn2+ Mn2+ hh Bx Bx Bx pump (jz = 1)

Figure 4.2: (a) Electron and magnetic ion spin precession in a 12-nm wide Zn0.77Cd0.23Se quantum wells with a 4ML Zn0.9Mn0.1Se barrier. (b) Data in (a) are plotted on a longer timescale to show the long-lived Mn2+ spin precession. (c) Origin of the Mn2+ spin precession is schematically illustrated at times before (∆t < 0), during (∆t < 0), and after (∆t > 0) the pump pulse. The p − d exchange between optically-injected holes and the Mn2+ spins torques the magnetic moments away from the field direction. Data from [68, 127]. now, e e H↑ − H↓ g µ B − N αxhS i ν = = e B x 0 x . (4.2) e h h ∗ We can define an effective g-factor (ge) by noting that hSxi will have a linear magnetic field dependence at sufficiently small magnetic field. Using equation 5 3.9 with j = 2 , e ∗ 35(N0αx)gMn ∆E = ge µBBx = ge + µBBx. (4.3) 12kB(T − θP) Thus, ∗ 35(N0αx)gMn ge = ge + . (4.4) 12kB(T − θP) Although hole spin precession is not observed in our experiments using time-resolved KR, it is illustrative to note that the heavy hole band spin split-

75 Chapter 4 Paramagnetic II-VI parabolic quantum wells ting is modiﬁed by the p−d exchange coupling. Just as discussed above for the conduction band, following from equations 3.12,

hh hh hh ∆E = H↑ − H↓ = ghhµBBx − N0βxhSxi. (4.5)

Intriguingly, an additional coherent spin precession is observed at long delay times well after the electrons spins have dephased and recombined, Fig. 4.2b. This precession frequency is linear with magnetic field yielding a g-factor of ∼ 2. Ascribing this precession as due to the coherent precession of the Mn2+ paramagnetic moments explains both the g-factor value and the fact that it is observed well after all the excitons have decayed. To explain this behavior, Crooker et al. proposed that a transient exchange field tips the Mn2+ spins away from the field direction resulting in a precession, [127]. Only the s − d or p − d exchange interaction between optically injected electron or heavy hole spins could explain the pump pulse dependence of this effect. Since the Mn2+ spins are uncoupled (paramagnetic), then the fact that Mn2+ spins are precessing in-phase requires that the transient exchange field be short in time compared with the Mn2+ spin precession period. This allows the ∗ s − d exchange couping to be ruled out as a possible mechanism since T2e is on par with the Mn2+ precession period. The heavy hole spin coherence time is expected to be much shorter than the electron spin lifetime < 0.5 ps, thus the p − d exchange coupling is the probable mechanism. As schematically shown in Fig. 4.2c, before the pump pulse (∆t < 0) Mn2+ spins are oriented along the magnetic field. During the pump pulse (∆t =∼ 0) spin-polarized holes injected along the z-axis producing a p − d exchange field that rotates the Mn2+ spins along z. The hole spins decay before the Mn2+ spins precess more than half a period. After the pump pulse (∆t > 0) the component of Mn2+ spin along x, precesses in the yz plane.

76 Chapter 4 Paramagnetic II-VI parabolic quantum wells

The two components of precession can be ﬁt to two exponentially decaying cosines, slightly modifying equation 2.8 yields,

−∆t θK(∆t) = Aeexp ∗ cos(2πνe∆t + φe) + T2e −∆t AMnexp ∗ cos(2πνMn∆t + φMn), (4.6) T2Mn

2+ where νe is given by equation 4.2, AMn is the amplitude of Mn spin preces- ∗ sion, T2Mn is the transverse electron spin lifetime, and φMn is a phase offset. The Mn2+ precession frequency is given by,

g µ B ν = Mn B x . (4.7) Mn h

4.2 Sample design and growth

We design II-VI parabolic quantum wells (PQWs) in order to control the position of electrons and holes with a vertical bias. Digital magnetic heterostructures are used to locally engineer exchange interactions, as described in the previous section. A schematic of the structures is shown in Fig. 4.3a, where a vertical bias is used to decrease the spatial overlap between electrons conﬁned to the PQWs and a thin magnetic layer at the center of the wells. The samples are produced by MBE, see section 1.2, and consist of 100 nm ZnSe /50 or 100 nm PQW/500 nm ZnSe/ (001) n+GaAs wafer. These samples were grown by Keh-Chiang Ku in Professor Nitin Samarth’s group at Penn State University.

The PQWs consist of ZnSe-Zn0.85Cd0.15Se produced by grading the Cd concentration using digital shuttering to produce a parabolic band edge proﬁle as we discussed in section 2.5. The digital alloy period is 2.5 nm for a 50-nm wide sample (Sample 1) and 5 nm for a 100-nm sample. Mn doping is performed 1 using digital shuttering such that four 8 ML of MnSe are deposited within a

77 Chapter 4 Paramagnetic II-VI parabolic quantum wells

(a) Vg (b) 50-nm PQW

100 Large overlap 95

Zn (%)Zn 90 85 -20 -10 01020 z (nm) ZnSe 4 x 1/8 ML MnSe ZnCdSe (15% ZnCdSe (15% Cd) Cd)

(c) ZnSe

Zn1-0.85-1Cd0-0.15-0Se (PQW)

Small overlap ZnSe

n+GaAs (wafer) Vg

Figure 4.3: (a) Schematic of electric ﬁeld tunable exchange overlap in a magnetic parabolic quantum well (PQW). (b) Layer structure within a 50-nm thick magnetic PQW of ZnCdSe. (c) Layer structure and vertical bias across a transparent Ti/Au front gate and an Ohmic contact to the n+GaAs wafer.

78 Chapter 4 Paramagnetic II-VI parabolic quantum wells

5-nm wide region at the center of each well. The PQW layer structure is shown in Fig. 4.3b. We also prepared control samples of both 50-nm (Sample 2) and 100-nm wide PQWs that contain no Mn doping.

The vertical bias (Vg) is applied between a transparent front gate of evapo- rated Ti (5 nm) / Au (5 nm) and the n+GaAs substrate as shown in Fig. 4.3c.

4.3 Photoluminescence: Faraday geometry

The PL is measured normal to the sample surface and as a function of magnetic field (B) in the Faraday geometry, section 2.3. The excitation consists of linearly polarized light from a pulsed-laser (femto-mode MIRA) with a 76-MHz repetition rate and ∼ 200 fs pulses. The laser energy is 2.88 eV and the power density is 56 W/cm2 which is focussed to a ∼ 50 µm spot on the sample surface. PL spectra at two different magnetic fields and with a constant vertical bias are shown in Fig. 4.4a. The PL emission spectra of the QWs can be fitted with a Gaussian. These fits yield intensity, emission energy (E), and line width. We plot E as function of magnetic field in Fig. 4.4b for a number of vertical biases. The PL emission energy of the magnetically doped wells shows a large red shift as the magnetic field is increased from 0 to 6 T. For the magnetic 50-nm PQW (sample 1) the redshift is ∼ 20 meV. In the corresponding non-magnetic sample (sample 2) the shift is < 0.3 meV.

The PL emission energy shifts by up to 2 meV with the application of Vg. This quantum conﬁned Stark effect (QCSE) shift occurs in both the magnetic and non-magnetic PQW, which can be seen in Fig. 4.4b comparing closed and open symbols. It is difﬁcult to measure the QCSE shift since Vg rapidly decreases the intensity of PL and broadens the line width.

79 Chapter 4 Paramagnetic II-VI parabolic quantum wells

3 2.635 (a) 0T 2 Non-magnetic 2.630

(b) 1

2.625 0

5 E (eV) Magnetic 4 2.620 6T Magnetic 3 Vg < 0 2.615 Photoluminescence (arb. Units) Photoluminescence 2 Non-magnetic 1 0 2.610 2.61 2.63 2.65 0123456 E (eV) B (T)

Figure 4.4: (a) PL of a 50-nm Mn-doped parabolic quantum well (Sample 1) and a 50-nm control (sample 2) at 5 K and B = 0 T (top) and 6 T (bottom) under various Vg. (b) PL emission energy (E) as a function of magnetic ﬁeld, extracted from Gaussian ﬁts to spectra shown in (a).

Closed symbols indicate Vg = 0 V and open symbols represent Vg = −1,−2 V.

4.4 Time-resolved Kerr rotation: Voigt geometry

Spin dynamics are measured by time-resolved Kerr rotation (KR) in the Voigt geometry with the optical path parallel to the growth axis, section 2.3.4. The pump beam power is 1 mW and probe beam power is 0.2 mW, which are both focused to overlapping ∼ 50µm diameter spots.

In Fig. 4.5a the Kerr rotation is plotted as a function of both Vg and ∆t for the 50-nm wide magnetic PQW. As discussed in section 4.1, two spin precession frequencies are observable due to short-lived electron spin precession and long-lived Mn2+ precession. The frequency of the electron spin precession de-

80 Chapter 4 Paramagnetic II-VI parabolic quantum wells

Magnetic Non-magnetic 0 (a) (b)

-1

(V) 5K, 3T g V -2 KR (a.u.) KR (a.u.) -0.02 +0.02 -0.15 +0.15 -3 0204060 80 020406080100 ∆t (ps) ∆t (ps)

Figure 4.5: Kerr rotation (KR) as function of Vg and ∆t at 5 K and B = 3 T with a pump-probe energy (Ep) of 2.638 eV (a) the 50-nm wide magnetically doped parabolic quantum well, and (b) for the 50-nm non-magnetic control sample.

2+ creases with Vg, while the Mn precession changes in amplitude. In the control sample, Fig 4.5b, only electron spin precession is observed. The Vg dependence is clearly weaker than in the magnetically doped samples. ∗ The data are fit to equation 4.6 in order to extract Ae,Mn, νe,Mn, and T2e,Mn as a function of Vg. Line cuts of data from Fig 4.5 at constant Vg are plotted in Fig. 4.6. The line cuts reveal the fast precession and decay of electron spins close after 10 ps with the Mn2+ spin precession persisting for ∆t > 400 ps. Data for the control sample fit well to a single decaying cosine, i.e. assuming AMn = 0 or fit to equation 2.8.

4.4.1 Gated s − d exchange coupling

The parabolic band edge proﬁles allow for a spatial translation of the electron wave function, see section 2.5. The effective electron g-factor is plotted in

81 Chapter 4 Paramagnetic II-VI parabolic quantum wells

Magnetic (e-)

01020304050

Magnetic (Mn2+)

(arb.units) K θ

0 100 200 300 400

Non-magnetic (e-)

020406080100 ∆t (ps)

Figure 4.6: KR as a function of ∆t at ﬁxed Vg. Raw data (squares) are plotted in three panels: (top) sample 1 at short ∆t, (middle) sample 1 at long ∆t, and (bottom) sample 2. Fits to equation 4.6 are plotted as lines

82 Chapter 4 Paramagnetic II-VI parabolic quantum wells

Fig. 4.7 as a function of Vg. We note that the non-magnetic sample (sample 2) reveals the expected g-factor tuning due to the Cd concentration gradient. ∗ Also, the spin lifetime T2e increases with negative bias. As described in section 2.5.1, the g-factor tuning is attributed to the spatial translation of the electron wave function which has a z dependence due to the concentration gradient in the barrier, equation 2.9. The increase in lifetime is due to the spatial separation of electron and holes with bias [18].

In the magnetic sample, Vg = 0 V corresponds to the maximum overlap ∗ ∗ with the MnSe layers. Consequently, the maximum ge and minimum T2e are ∗ observed. The spin splitting is ∼ 1.4 meV and ge ∼ 8, which are about 10-times larger than in the non-magnetic sample. Applying a negative bias decreases the ∗ ∗ s − d exchange overlap leading to a decrease in ge and an increase in T2e. The ∗ minimum spin splitting ∼ 0.3 meV and ge ∼ 2, which are both still larger than in the control sample. The effective Mn concentration (x∗) experienced by electrons in the PQWs is proportional to the wave function overlap, which is modiﬁed by Vg. As is evident in equation 4.2, changing x∗ strongly alters the electron spin splitting and precession frequencies. In this structure, the conduction band spin splitting tunes over 1 meV with Vg, corresponding to changes in the spin precession frequencies of ∆νe = 270 GHz. These results have been reproduced in several additional samples, including a Mn-doped 50-nm wide PQW with a 1-nm period thickness, and in a 100-nm wide Mn-doped PQW. The spin-splitting and g-factors tune over 60-times more than the non-magnetic PQWs, section 2.5.1. Having completed our original goal, we now point out additional features of the magnetic PQWs not present in non-magnetic QWs.

83 Chapter 4 Paramagnetic II-VI parabolic quantum wells

5K, 3T 8 1.4

Magnetic ∆

6 1.0 E (meV) eff g

1.21. 2 4 0.7 1.11. 1

-2 -1 0 -2 -1 0

2 0.3 Non-magnetic

0 0.0 50 40 (ps) *

2 30 T 20 10 0 -3 -2 -1 0

Vg (V)

Figure 4.7: (a) Conduction band spin splitting (right vertical axis) and effective electron g- factor (left axis) as a function of gate bias, Vg. Inset: control sample data enlarged to show non-magnetic g-factor tuning. (b) Transverse electron spin lifetime as a function of Vg.

84 Chapter 4 Paramagnetic II-VI parabolic quantum wells

4.4.2 Gated p − d exchange coupling

So far we have only discussed the electron spin dynamics. As discussed in section 4.1, the p − d exchange interaction between optically-injected holes spins and Mn2+ spins results in Mn-ion spin precession. The Mn-ion spin dynamics parameters extracted from fits to data in Fig. 4.5 are plotted as a function of gate bias in Fig. 4.8. 2+ For Vg = 0 V, the Mn spin precession amplitude (AMn) and the spin life- ∗ 2+ time (T2Mn) are at a maximum. Here the Mn spin precession amplitude is large since the hole exchange (p − d) overlap and, therefore, the effective tipping field is large. Application of Vg < 0 V leads to a decrease in both AMn ∗ and T2Mn. The decrease in p − d overlap explains the decrease in Mn-ion spin precession amplitude. The weaker the tipping field, section 4.1, the smaller is the precessing component of Mn-ion spin. The decrease in Mn2+ spin lifetime with vertical bias may be due to an increase in the exciton recombination time once the electron and hole are separated by the vertical bias. As the recombination time increases, so does the temporal cross-section for carrier-Mn spin scattering, which would reduce the Mn-ion spin lifetime. Mn spin heating is expected to be negligible since we bias the devices in the regime of negligible current flow [128]. Mn-spin heating could also result from laser pulses. For example Baumberg et al. observed spin heating due to subsequent laser pulses. This occurred since the decay constants were large, ∼ 4 µs. Spin heating was reduced in their experiments by using a pulse picker to reduce the laser repetition rate such that all Mn-ion spin precession had decayed before the arrival of the subsequent pulse [67]. For the structures studied here, laser pulse heating of Mn-spin is not expected since Mn-ion spin decay constants are no longer than ∼ 300 ps, much shorter than the time between successive laser

85 Chapter 4 Paramagnetic II-VI parabolic quantum wells

2.04 354 ∆ E ( µ 2.03 352 eV), ∆ Mn g 2.02 351 1 m =

2.01 349

0.010

0.005 (a.u.) Mn

A 0.000

320

300 (ps) *

2Mn 280 T

260 -3 -2 -1 0 1

Vg (V)

Figure 4.8: The effective Mn g-factor (left axis) and the spin splitting (right axis) for the ∆ j = 1 transition, Mn-spin precession amplitude, and lifetime are plotted as a function of gate bias, Vg.

86 Chapter 4 Paramagnetic II-VI parabolic quantum wells pulses, ∼ 13 ns. Our attention now turns to the Mn-ion spin precession frequency, which does not vary with Vg, Fig. 4.8. The g-factor remains constant within a high precision, gMn = 2.0240 ± 0.0006, corresponding to a constant Mn-ion spin splitting for the ∆ j = 1 transitions of ∆EMn = 351.3 ± 0.5 µeV. The question arises: under what circumstances ought the Mn-ion spin splitting change? Sim- ply following from the Newtonian principle of action-reaction, the fact that s−d and p − d exchange coupling alters the energies for electron and hole spins, respectively, then they should correspondingly alter the Mn-ion spin energy. Starting from the basic exchange equations 3.2 and 3.3, the Hamiltonian for a 5 3 1 1 3 5 Mn-ion spin with angular momentum states j = − 2 ,− 2 ,− 2 ,+ 2 ,+ 2 ,+ 2 along the magnetic ﬁeld Bx is,

Mn 3D e 3D h Hj = gMnµB jBx − ne α jhSxi − nh β jhSxi, (4.8) where the density of Mn-ions (N0x) has been replaced with the density of elec- 3D 3D trons (ne ) for the s − d term and the density of holes (nh ) for the p − d term, which are all in units of number·cm−3. In this mirror image, the number of electrons or holes that a Mn-ion spin experiences is the relevant weighting parameter for these terms rather than the Mn-ion density. The equilibrium Mn-spin polarization hSxi has been replaced by the equilibrium electron spin polarization for the s − d term and by the hole spin polarization for the p − d term, which are given by, e 1 geµBBx hSxi = − B1/2( ), (4.9) 2 2kBT and, h 1 ghhµBBx hSxi = − B( ), (4.10) 2 2kBT where the Brillouin function (B j(η)) determines the thermally averaged spin polarization, equation 3.6. The spin splitting for the Mn-ion spin in a ∆ j = 1

87 Chapter 4 Paramagnetic II-VI parabolic quantum wells transition,

Mn Mn Mn 3D e 3D h ∆E∆ j=1 = hνMn = Hj − Hj−1 = gMnµBBx − ne αhSxi − nh βhSxi, (4.11) which corresponds to the energy splitting that causes Mn-ion spin precession at frequency νMn. Thus equation 4.7 is strictly incorrect; it give an effective precession frequency not taking into account the effect of exchange coupling on the Mn-ion spin dynamics. Mn We can make a rough estimate for ∆E∆ j=1 in our experiment. For the optical 3D 3D 16 −3 pump power used here we expect ne = nh ∼ 10 cm . Values of N0α =

0.28 eV and N0β = −1.38 eV are taken from Table 3.1 and linearly interpolating for Zn0.85Cd0.15MnSe. The lattice constant for zinc blende ZnSe, a = 0.5667 nm (reference [27]), and for hypothetical zinc blende CdSe, a = 0.6077 nm (reference [129]). Linear interpolating according to Vegard’s law gives a = 0.5729 22 nm for Zn0.85Cd0.15Se, which gives a cation site density of N0 = 2.14 × 10 cm−3 and thus α = 1.32 × 10−23 eV·cm3 and β = −6.46 × 10−23 eV·cm3. We e h ﬁnd hSxi = −0.46 and hSxi = 0.50 using T = 5 K, Bx = 3 T, and assuming ∗ ge ∼ 8 for zero bias as observed in Fig. 4.7. The effective hole g-factor is cal- ∗ ∗ culated assuming the gh is proportional to and dominated by β, just as ge ∝ α ∗ ∗ geβ in equation 4.4. Thus, gh = α , which is ∼ −39.3 for zero bias. The Zeeman splitting is, gMnµBBx = 351.5 µeV assuming that gMn = 2.024. Mn Using the above estimates and inputting into equation 4.11 gives ∆E∆ j=1 = (351.5+0.4) = 351.9 µeV, where the change in Mn-ion spin splitting due to the s−d and p−d exchange is 0.4 µeV. This small effect is within the noise of our measurements, explaining the lack of bias dependence to the Mn-ion precession frequency, Fig. 4.8. The above analysis shows that in order to increase the Mn-ion spin exchange coupling, we need to increase the electron or hole densities. Quantum well car-

88 Chapter 4 Paramagnetic II-VI parabolic quantum wells rier densities can easily reach 1018 cm−3. We also note, that our analysis con- ∗ tains a recursive problem; the value of gMn depends on ge,h and vice versa. If the Mn-ion spin exchange splitting is sufﬁciently increased, this forward-back action leads to an avoided level crossing. Such behavior has been observed in n-doped CdMnTe quantum wells. Electron paramagnetic resonance measurements show an avoided crossing and Knight shift between the electron and Mn-ion spin resonances [130].

4.4.3 Optical control of exchange coupling

Time-resolved KR is measured in these samples at Vg = 0 as a function of the degenerate pump-probe energy of the laser (Ep). As seen in Fig. 4.9 for small

∆t, the electron spin dynamics show a strong sensitivity to Ep evident by the change in period length precession and its amplitude. The Mn2+ spin dynamics change in amplitude but the period length does not change, similar to what occurs for the Vg dependence in section 4.4.2. Figure 4.10 displays line cuts of the ∆t dependence of the Kerr rotation for three different values of Ep. The effective electron g-factor and spin-splitting, extracted from these ﬁts, are plotted as a function of Ep in Fig. 4.11. At the ∗ lowest Ep = 2.611 eV we observe the largest ge ∼ 18 corresponding to a spin e ∗ splitting ∆E ∼ 3.2 meV. Increasing Ep ∼ 2.627 to 2.650 eV saturates ge ∼ 6 corresponding to ∆Ee ∼ 1.1 meV. The conduction band spin splitting tunes over 2 meV simply by decreasing the laser energy. This corresponds to a shift in the spin precession frequency from 252 to 750 GHz, ∆νe = 0.5THz.

The large changes in electron spin dynamics due to Ep are twice as large as the Vg effects shown in Fig. 4.7 and must also be explained by changes in the s − d exchange coupling. We interpret the Ep effect in the context of sublevel

89 Chapter 4 Paramagnetic II-VI parabolic quantum wells

KR (a.u.) 2.64 2 0 -2

2.63 (eV) p E 2.62

2.61

0204060 80 100 ∆t (ps)

Figure 4.9: KR at 5 K and B = 3 T of a magnetically doped parabolic quantum well (sample

1) as a function of the laser energy, Ep.

Ep = 2.635 eV

Ep = 2.619 eV (arb. units) (arb. K

Ep = 2.611 eV

01020304050 ∆t (ps)

Figure 4.10: Line cuts at constant Ep of data from Fig. 4.9, marked by colored arrows, are plotted as a function of ∆t. Both raw data (squares) and ﬁts (lines) to equation 4.6 are presented. Colors correspond to arrows in Fig. 4.9 indicating the position of the line cuts.

90 Chapter 4 Paramagnetic II-VI parabolic quantum wells

20 3.5 (a) 16 2.8 Δ E (meV) 12 2.1 eff g 8 1.4

4 0.7 ~15 meV 0 0.0 (b) 2 n = 1

0 (a.u.)

e n = 2 A -2

-4

(c) 3

(ps) 2 * 2

T 1 0 2.602.61 2.62 2.63 2.64 2.65

Ep (eV)

Figure 4.11: Optically tunable s − d and p − d exchange overlap. (a) Effective g-factor and conduction band spin splitting, (b) amplitude, and (c) lifetime are plotted as a function of Ep. Vertical colored lines label features corresponding to heavy hole exciton resonance lines for the n = 1 (red line) and n = 2 state (blue line).

91 Chapter 4 Paramagnetic II-VI parabolic quantum wells

ﬁlling in the PQWs. The ground state wave function for n = 1 is populated and probed at small Ep. The n = 1 wave function has a maximum in the center of the PQW thereby giving a maximum overlap with the MnSe layers. The ﬁrst excited state wave function, n = 2, has a node exactly in the center of the PQW that minimizes the s − d overlap. This interpretation is qualitatively consistent ∗ with Fig. 4.11 in which ge decreases as Ep increases. We also observe an increase in electron spin lifetime with increased Ep, which is indicative of a ∗ reduction in the s − d exchange coupling that tends to reduce T2e.

Another indication for sublevel resonance arises from the Ep dependence of KR. The electron spin precession amplitude (KR signal) has sharp features when Ep is resonant with an absorption edge, section 2.3.4. This has been observed by Crooker et al. in quantum wells for Ep resonant with the interband transitions between successive QW sublevels [69]. Similar features are shown in Fig. 4.11 in the Ae spectral dependence indicative of the exciton resonances.

The resonances for the n = 1 (e1-hh1) and n = 2 (e2-hh2) heavy hole exciton states are labeled in Fig. 4.11. Such features follow our qualitative interpretation of the Ep effect on the s − d exchange coupling. As Ep is increased, ∗ photo-excited electrons occupy both n = 1 and n = 2 states decreasing ge. The laser pump-probe energy dramatically changes the exchange coupling.

This raises the possibility that the Vg effects shown in Figs. 4.5-4.8 are not due to spatial wave function translation, but are due to a QCSE redshift with gate bias [77]. In this interpretation, Vg red shifts the absorption edge. Since

Ep is ﬁxed for these measurements, the redshift causes higher sublevels to be probed. As we have seen above, this would reduce the s − d exchange effect. This hypothesis can be rejected since large QCSE shifts are not observed in the PL data shown in Fig. 4.4. Under applied bias the PL energy is redshifted by 2 meV at most. For this value of Vg, the change in s−d exchange leads to a more

92 Chapter 4 Paramagnetic II-VI parabolic quantum wells

∗ ∗ than 4-times drop in ge. The corresponding change in Ep required to modify ge by this amount is ∼ 20 meV, much larger than the ≤ 2 meV of QCSE observed by PL.

4.5 Modeling the exchange overlap

We ﬁrst calculate the band edge, wave functions, and sublevel energies in the PQW using a numerical one-dimensional Poisson-Schrödinger solver [31]. The ∗ ∗ ∗ materials parameters (Eg, me, mhh, mlh) of zinc blende Zn1−xCdxSe are quadratic functions of x and are taken from references [131, 132]. The materials parameters of zinc blende MnSe are not well known from the literature. For this analysis we assume Eg = 2.85 eV, larger than the barriers of ZnSe, and a conduction band offset from ZnSe of 0.08 eV. The effective masses for MnSe are assumed: ∗ ∗ ∗ me = 0.16, mhh = 0.45, mlh = 0.145. The structure used to simulate the 50-nm magnetic PQW parabolically grades the average Cd concentration along the growth axis from 0% Cd in the barriers to 15% Cd at the center of the well. The steps in concentration are digital 2.5-nm wide regions as in the real structure. The MnSe layers are each 0.07- nm wide and are equally spaced at the center 5-nm region of the well; they are 1 larger than 8 ML = 0.04 nm since 0.07 nm is the minimum step size for the solver. Figure 4.12 plots the conduction and valence band edge near the center of the PQW showing the position of the MnSe layers, and the sublevel energies for n = 1,2 with the corresponding probability density functions along the growth axis (|Ψ(z)|2). We can calculate the effective Mn2+ concentration (x∗) for the n = 1,2 elec-

93 Chapter 4 Paramagnetic II-VI parabolic quantum wells

0.48 n = 2

0.47 n = 1 (eV)

F 0.46 E – E

n = 1

-2.05 n = 2

-15 -10 -5 0 5 10 15 z (nm)

Figure 4.12: One-dimensional solutions to the Poisson and Schrödinger equations for a 50-nm wide magnetic parabolic quantum well. The probability density for the n = 1 (red line) and n = 2 states are plotted.

94 Chapter 4 Paramagnetic II-VI parabolic quantum wells tron (e1,2) and heavy hole (hh1,2) states, Z x∗ = |Ψ(z)|2Mn(z)dz, (4.12) which is the weighted average of the probability density function with the Mn concentration proﬁle Mn(z). This is calculated by numerical integration assuming that Mn(z) is simply a series of step functions of height 0.125 and width 0.3 nm, the average Mn concentration within 1 ML. The values of the sublevel energy and x∗ are listed in Table 4.1.

Table 4.1: Calculations of the exchange overlap in a 50-nm wide Mn-doped parabolic quantum. The energy of each sublevel is listed (relative to Fermi level as in Fig. 4.12 simulations). Using s,p−d x∗, the s − d or p − d exchange splitting at 3 T and 5 K is calculated, ∆E .

∗ e,h Ψ Energy (eV) ∆Eg (meV) x ∆Es,p−d (meV) ∆E (meV)

e1 0.4678 0.17 0.0087 4.25 4.43

e2 0.4757 0.17 0.0017 0.82 0.99

hh1 -2.047 0.51 0.0058 -14.02 -13.52

hh2 -2.048 0.51 0.0014 -3.46 -2.95

The conduction band Zeeman splitting,

e ∆Eg = geµBBx, (4.13)

e can be estimated by observing ge = 1.2 from Fig. 4.7, thus g µBBx = 0.21 meV. The heavy hole Zeeman splitting,

hh ∆Eg = ghhµBBx, (4.14) is estimated by noting that the Faraday geometry PL measurement observed a 1 red shift in the control sample of 2 gexµBBx < 0.3 meV, Fig. 4.4. Since gex = ge − ghh, section 2.3.2, then |ghh| < 2.9. Thus ghhµBBx < 0.5 meV.

We estimate hSxi = −1.75 assuming that g = gMn = 2.024, Bx = 3 T, T = 5 5 K, θP = 0 K, and plugging into the j = 2 Brillouin function, B5/2(η), equation

95 Chapter 4 Paramagnetic II-VI parabolic quantum wells

3.6. Recall that for Zn0.85Cd0.15MnSe, we estimated N0α = 0.28 eV and N0β = −1.38 eV, section 4.4.2. Using these values we calculate the conduction band exchange splitting, ∗ ∆Es−d = −N0αx hSxi, (4.15) the heavy hole exchange splitting,

∗ ∆Ep−d = −N0βx hSxi. (4.16)

The spin splitting between spin-up and spin-down states (H↑ − H↓) as given by equations 4.1 and 4.5 is,

e e ∆E = ∆Eg + ∆Es−d, (4.17) for the conduction band and,

hh hh ∆E = ∆Eg + ∆Ep−d, (4.18) for the heavy hole band. Estimates are listed in Table 4.1. Clearly, the exchange splitting dominates over the Zeeman splitting for both the conduction and valence bands. The sublevel effect, qualitatively described in section 4.4.3 is quantitatively substantiated since for electrons or holes, the exchange splitting in moving from the n = 1 to the n = 2 states.

4.5.1 PL redshift

The shift in PL emission energy for the magnetic sample is due to the giant Zee- man splitting from the s−d and p−d splitting in the conduction and heavy hole bands, respectively. The ground state emission is either Eσ + or Eσ − . Looking at equations 3.14 and 3.15 we notice that since (α − β) > 0 and hSxi < 0, then the lowest energy emission is from Eσ + . The PL shift is given by,

∗ 1 1 E + − E = − µ B g + (N x)hS i(α − β), (4.19) σ g 2 B z ex 2 0 z

96 Chapter 4 Paramagnetic II-VI parabolic quantum wells which in terms of equations 4.17 and 4.18 is,

hh e ∗ ∆E − ∆E E + − E = , (4.20) σ g 2 which ∼ −9 meV using values in Table 4.1 valid for Bx = 3T and at a temperature of 5 K. Under the same magnetic ﬁeld and temperature, the PL shift is measured to be ∼ 10 meV, from Fig. 4.4 at 3 T. The model accurately predicts this PL redshift.

4.5.2 Optical control

The results of the model are now compared to the exciton resonance features observed in Fig. 4.11, which are separated by ∼ 15 meV. Similarly to the PL redshift calculations, above, the absorption edge is in the spin-ground state of

Eσ + for each sublevel,

hh e ∗ ∆En − ∆En E + = E + , (4.21) σ ,n g,n 2

∗ where Eg,n is the effective band gap (Ee − Eh) for the nth sublevel. The energy spacing between the n = 2 and n = 1 absorption edges is Eσ +,2 − Eσ +,1 = 15.9 meV, using values from Table 4.1. This compares well with the measured value of ∼ 15 meV. Finally we look at the change in conduction band spin splitting from the n = 1 to n = 2 states, which is ∼ 3.4 meV from our model. In Fig. 4.11, the electron spin splitting is observed to change by ∼ 2.2 meV. This is smaller than what is expected from our model. Error in our estimate of N0α could arise due to Kinetic exchange effect in the quantum well, section 6.6.3, or error in xe could arise due to nonideal Cd or Mn concentration proﬁles in the PQWs.

97 Chapter 4 Paramagnetic II-VI parabolic quantum wells

4.6 Conclusions

In this chapter, paramagnetic dopants were incorporated into parabolic quantum wells. A vertical bias was used to move the spatial part of the electron and hole wave functions. This resulted in strong changes in their spin states due to the s − d and p − d exchange coupling to the paramagnetic spins. As a result electron precession frequencies could be electrically tuned over hundreds of GHz compared with tens of GHz in similar non-magnetic parabolic wells. The exchange coupling could also be controlled by selectively populating and prob- ing different sublevels within the quantum wells. This laser energy effect took advantage of the wave function symmetry and magnetic doping proﬁle within the quantum wells. Simple models of the effective magnetic-ion concentration account for these effects. Additionally the p−d exchange interaction altered the Mn-ion spin dynamics. Although lifetime and amplitude of the Mn-ion spin precession was varied with a vertical bias, the precession frequency remained constant. This was explained by the orders of magnitude lower electron and hole concentrations in these un-doped wells compared with the density of magnetic-ions. By electronically doping carriers into the quantum wells, the magnetic-ion spin splittings should also be affected by the exchange interactions providing a route for Mn- ion spin resonance and coherent control, as has been demonstrated for electron spins.

98 Chapter 5

Quantum wells with a ferromagnetic barrier

In chapter 4, we examined how heterostructure band and spin engineering in II- VI magnetic semiconductors can be used to control the strength of the exchange interactions. In these wells the magnetic impurities are strictly paramagnetic. This arises from the lack of hole-mediated ferromagnetism in II-VI DMS since the magnetic dopants, e.g. Mn2+ are isoelectronic, chapter 3. As a result, the magnetic moments (and their spins) can only be aligned using large magnetic fields of ∼ 1 − 10 T and at liquid Helium temperatures. In this chapter we discuss III-V based quantum wells in which ferromagnetic layers are incorporated. As a result, the moments can be fully polarized at low magnetic fields ∼ 0.1 T. A vertical-bias is again used to control the spatial overlap of carriers confined to the quantum wells and the ferromagnetic layer. This exchange overlap leads to a spin polarization of the confined carriers that tracks the magnetization of the ferromagnetic layer, which can be electrically tuned in sign and magnitude.

99 Chapter 5 Quantum wells with a ferromagnetic barrier

5.1 Heterostructure design

In III-V GaAs-based semiconductors, Mn2+ contributes a hole to the valence band, and at sufficiently high dopings a long-range ferromagnetic coupling arises mediated by the hole-spin, section 3.3. Recall that special growth conditions are required to incorporate sufficient Mn into GaAs for ferromagnetism. Substrate temperatures are typically ∼250˚C, which results in the incorporation of excess As as antisites. These are double donors and affect the ferromagnetism of GaMnAs by compensating holes, which reduces the Curie temperature, see appendix B. These defects degrade the optical properties of GaMnAs. Low temperature MBE growth also leads to oxygen incorporation, especially in Al containing barriers. Oxygen impurities act as non-radiative centers, efficiently killing optical emission and absorption even at concentrations as low as ∼ 1016 cm−3. Our goal is to incorporate these ferromagnetic layers into quantum well structures. Since optical characterization methods are the primary method we use for examining spin polarization and exchange interactions, then the electronic and optical defects associated with the ferromagnetic layers pose the greatest obstacle. One possible structure is to directly dope Mn into AlGaAs-GaAs or GaAs- InGaAs quantum wells. Such structures are discussed in chapter 6 in which both PL and time-resolved KR measurements are possible. Unfortunately, these optical signals disappear for Mn concentrations necessary for ferromagnetism. Thus only optical-quality paramagnetic GaMnAs quantum wells are, as yet, possible. A second doping scheme is to incorporate a set-back layer (d) between the ferromagnetic layer and the quantum well, illustrated in Fig. 5.1. The advantage

100 Chapter 5 Quantum wells with a ferromagnetic barrier

Mn-containing layer set-back = d

GaAs AlGaAs GaAs Surface Wafer Barrier QW

High Tsub Low Tsub

Growth direction (z)

Figure 5.1: Layer scheme for isolating crystalline defects in a ferromagnetic barrier from the active quantum well region. is two-fold:

1. Defects in the ferromagnetic region are spatially separated from the quantum well, reducing their effect on the optical characteristics.

2. The quantum well region and set-back barrier can be grown at high substrate temperatures to obtain optically clean QWs.

In ferromagnetic GaMnAs and its heterostructures, heating above ∼300˚C leads to the precipitation of a second crystal phase, MnAs, which is a ferromagnetic metal. To maintain a precipitate-free epitaxial structure the substrate temperature must, therefore, not be reheated after deposition of the magnetic layers. Such a requirement demands that the ferromagnetic layer be included only in the top-most QW barrier. The structure of the ferromagnetic layer will also affect the optical performance and characterization. Ferromagnetic GaMnAs in the random-alloy structure is a degenerate p-type semiconductor. It has strong absorption throughout the infrared spectrum. More importantly, GaMnAs exhibits a strong magneto- circular dichroism (MCD) effect, whereby linearly-polarized light transmitted

101 Chapter 5 Quantum wells with a ferromagnetic barrier through the sample becomes circularly-polarized, similar in origin to the KR or FR effect, section 2.3.4. The MCD spectrum of ferromagnetic GaMnAs exhibits strong polarization features throughout the infrared spectrum, [133, 134]. Incorporating GaMnAs random-alloy layers into the QW barriers would therefore spectrally convolute with the polarization-resolved optical signals from the QWs. In the samples studied here, we have incorporated digital ferromagnetic layers of sub-monolayer MnAs into the AlGaAs QW barriers. As described in section 3.3.3, these digital ferromagnetic heterostructures (DFHs) are composed of alternating layers of sub-monolayer MnAs with GaAs spacers [118]. Ferromag- netism is retained even in the case of a single 0.5 ML of MnAs. Since MCD is an absorption effect, it scales with layer thickness. By changing the number of ferromagnetic layers included in the DFH barrier, we can account for MCD artifacts. One ﬁnal complication is our use of the low temperature atomic layer epitaxy (ALE) technique to grow the DFHs and top-most QW barrier. This growth method reduces the density of compensating As-defects (antisites) in the GaAs spacers and barrier by shuttering the As for each monolayer of growth [119].

5.2 Structure and gating

Details of the sample growth and structures are described in appendix C.1. The samples are produced using two MBE systems. Chamber 1 can produce high mobility III-V heterostructures and clean QWs. In this chamber the gated QW structure is grown by MBE including the set-back layer (d). The detailed layer structure is shown in Fig. 5.2. The LT AlGaAs layer serves as a resistive layer for backgating, 2.4.1. After As-capping the surface, the samples are transferred in air to the second MBE system for overgrowth of the magnetic layers.

102 Chapter 5 Quantum wells with a ferromagnetic barrier

7.5 nm GaAs Chamber 2: 50 nm Al0.4Ga0.6As Atomic Layer Epitaxy ° Magnetic Layer Tsub = 240 C 85 ML = 24 nm First Mn layer

Al0.4Ga0.6As d = 5 or 9 nm 7.5nm GaAs QW

350 nm Al0.4Ga0.6As

500 nm LT AlGaAs Chamber 1: resistive structure MBE ° Tsub = 585 C (except for LT layer) 200 nm GaAs 200 nm n-GaAs

SI-GaAs+ buffer and SL

Magnetic Layers (0.5 or 0.3) ML MnAs / 0.5 ML GaAs Sample B or C

20 ML Al0.4Ga0.6As Sample A

20 ML Al0.4Ga0.6As

20 ML Al0.4Ga0.6As 84 ML Al0.4Ga0.6As 20 ML Al0.4Ga0.6As

0.5 ML MnAs 0.5 ML GaAs

Figure 5.2: Layer structure of quantum wells with a ferromagnetic barrier (not to scale). The magnetic layers for three different gated samples are shown.

103 Chapter 5 Quantum wells with a ferromagnetic barrier

The magnetic layers consist of DFHs grown by ALE. The composition of each monolayer is controlled by sequentially depositing each constituent ele- ment with sub-monolayer precision allowing for digital alloying within a single monolayer (ML) and the formation of thin ferromagnetic layers of MnAs [40]. Three distinct magnetic structures are prepared, as shown in Fig. 5.2. For the magnetic samples, the first ML deposited contains MnAs. The magnetic layer is capped by a top QW barrier of 177 ML Al0.4Ga0.6As / 27 ML GaAs. A control sample without MnAs is also prepared using the ALE procedure. Standard photolithography and wet etching are used to define ∼ 1 µm tall mesas of the QW structures. The 400-nm back contact of n-GaAs is contacted in the mesa-free region using In soldering for Ohmic contact. Photolithography and e-beam evaporation are used to define transparent front gate of 50 Å Ti / 40

Å Au on the top of the mesas. We apply a bias (Vg = front - back) from -15 V to +2 V, which leads to a current of < 10 µA and current density < 0.25 mAcm−2. These large voltages can be applied due to the 500-nm LT AlGaAs resistive layer. In Fig. 5.3a, a schematic of the mesa and gating scheme is shown. the device Table 5.1 summarizes the structure and preparation of the samples. Sample

H represents our initial attempt at structures incorporating In0.2Ga0.8As quantum wells. The advantage is that the barriers are GaAs and thus the magnetic layers display sharp and square hysteresis with in-plane magnetic easy axes. Room temperature Hall measurements in the Van der Pauw geometry are carried out to determine the resistivity, carrier density, and mobility, Table 5.1. Several samples display Ohmic behavior, which is indicative of modulation hole doping from the adjacent MnAs layer into the QW forming a two-dimensional hole gas. The magnetic samples that do not show Ohmic behavior, display a non-linear I-V, which has been ascribed to hopping conductivity through the DFH layers [119].

104 Chapter 5 Quantum wells with a ferromagnetic barrier

(b) Spectrometer (1.33 m) (a) CCD

PL Pump B

PL pump Ti/Au LP LP Magnetic Layer y Vg ±λ/4 l/4 ±

d-spacer x QW

LT AlGaAs In nGaAs

(001) GaAs wafer, buffer, SL B

Figure 5.3: (a) Device structure and wiring for application of a vertical bias (not to scale). (b) Polarization-resolved photoluminescence measurement geometry (Faraday). The cone indicates the direction of PL surface emission. Arrow “PL pump” shows the path of the pump beam and the direction of the applied magnetic ﬁeld (B).

105 Chapter 5 Quantum wells with a ferromagnetic barrier

Table 5.1: Sample list, selected electronic properties at room temperature, and results of optical spin polarization are listed. Samples A-G contain GaAs quantum wells with AlGaAs DFHs. Sample H contains an InGaAs quantum well with GaAs DFHs.

−2 Sample d (nm) Magnetic layer Gated? P? Ohmic? p2D (cm ) A (030508C) 5 1 × (0.5/84) yes yes no - B (030508B) 5 5 × (0.5/20) yes yes no - C (030508D) 5 5 × (0.3/20) yes yes no - control (030508A) 5 - yes no no - D (030211C) 9 1 × (0.5/84) no no yes 8.58 × 1011 E (030211A) 9 5 × (0.5/20) no no yes 2.08 × 1012 F (030130B) 5 1 × (0.5/84) no yes yes 1.56 × 1012 G (030130A) 5 5 × (0.5/20) no yes yes 9.67 × 1011 H (030326A) 5 5 × (0.5/20) no no yes 3.70 × 1013

5.3 Magnetization measurement

The magnetic properties of the samples are characterized using superconducting quantum interference device (SQUID) magnetometry. The data for a single magnetic layer of 0.5-ML MnAs, sample A, are plotted in Fig. 5.4a and in Fig. 5.4b (black data) showing the magnetic ﬁeld and temperature dependences, respectively. These results are representative of the magnetic properties of the other structures; samples containing additional DFH periods have qualitatively identical magnetic properties with larger total magnetic moments. The magnetization at B = 2 kGs reaches 4µB/Mn past where the hysteresis loop is closed, 5 2+ approaching the theoretical limit of 5µB/Mn given spin- 2 Mn moments. The fact that the magnetization continues to increase past the hysteresis region indicates a paramagnetic component. This is consistent with the discrepancy between the observed and theoretical total magnetization.

In general, the properties of these DFH layers with Al0.4Ga0.6As spacers are quite different than for DFHs that contain GaAs spacer layers, see section

106 Chapter 5 Quantum wells with a ferromagnetic barrier

3.3.3 and references [118, 119]. For our samples, a magnetic easy axis is oriented out-of-plane, while this direction is a hard axis for normal GaAs DFH. The in-plane direction shows a weaker hysteresis, not a purely hard-axis. These observations indicate that the anisotropic easy axis likely lies along a non-trivial crystal direction. The Curie temperature (TC) of these structures is ∼15 K compared with up to ∼40 K for GaAs DFH. Takamura et al. reported a similar rotation of the easy axis and low values of TC for bulk layers of (Al,Ga,Mn)As random alloy.[135].

5.4 PL polarization

Polarization-resolved photoluminescence (PL) is measured in the Faraday geometry, section 2.3.2. The linearly polarized excitation laser is pulsed at 76 MHz and tuned to an energy of 1.731 eV with a fluence of ∼56 W/cm2. Polariza- I −I tion is defined as in equation 2.5, as P = σ− σ+ . PL spectra are obtained Iσ− +Iσ+ using a fiber-coupled 1.33-m spectrometer with a liquid-Nitrogen cooled CCD array. Polarization-resolution is obtained using a variable wave plate (VWP) and Glan-Thompson linear polarizer. Figure 5.5 plots the PL and polarization spectrum of the quantum well emission as a function of magnetic field. The data are shown for sample A at a fixed

Vg = −3.8 V. In Fig. 5.5a the PL emission of the QW is observed, which is spectrally isolated from GaAs or AlGaAs bulk emission energies. The intensity, line width, and emission energy show a negligible magnetic ﬁeld dependence up to 2 T. On the other hand, the corresponding polarization, Fig. 5.5b shows a sharp magnetic ﬁeld dependence, where P switches sign at below 1 T. Line cuts of the data in Fig. 5.5a and b are replotted as a function of emission energy E in Fig. 5.5c and d. PL polarization switches sign and magnitude with the application of

107 Chapter 5 Quantum wells with a ferromagnetic barrier

4 (a) 4 2 5 K

2 P (%) /Mn) B

µ 0 0

M ( -2 -2 A control -4 -4 -2 -1 0 12

B (kG) 4 (b) +1 kG 3 3 /Mn) P (%) B

µ 2 2 M ( 1 1

0 0 5101520 T (K)

Figure 5.4: (a) Magnetization and PL polarization versus magnetic field for a quantum well with a ferromagnetic barrier with d = 5 nm (Sample A). Open and closed symbols indicate the direction of field sweep as up and down, respectively. Control sample at the bias value of maximum polarization is included for comparison (red line). (b) Temperature dependence of magnetization and polarization for sample A with a fixed magnetic field. The magnetization data are taken without bias in a magnetometer and the polarization data are taken under a fixed

Vg = −3.8 V.

108 Chapter 5 Quantum wells with a ferromagnetic barrier

fields as low as ±2 kGs. Note that this large change in P is unaccompanied by a spectral shift or intensity change in the total PL emission. Also, note that PL polarization is spectrally associated with the quantum well exciton emission. This indicates that either/both electron and heavy hole spins in the quantum wells are spin-polarized, section 2.3.2. Since the excitation is linearly-polarized it produces no net spin polarization in the QWs, thus photoexcited carriers in the quantum well become spin-polarized before the recombine through interaction with the ferromagnetic layer in the top barrier. To illustrate the magnetic field and temperature dependence of the polarization, the PL spectra area integrated over the full range of the QW emission, indicated by dashed lines in Fig. 5.5c and d. This produces a single data point of polarization or PL for each value of applied field and temperature. Such data are plotted in Fig. 5.4 a and b (red circles) overlayed with the magnetization data. Clearly, the polarization tracks the magnetization of the adjacent ferromagnetic layer, demonstrating that the spin polarization in the QWs is associated with the magnetization of the ferromagnetic barrier. Note that the polarization field dependence shows no hysteresis as compared to the magnetization data, which could be due to photo-induced changes in the magnetization of the ferromagnetic layer [136] or a change in the magnetic properties from device processing. The control sample data (red line) shows a weak linear field dependence in the PL polarization, which is consistent with the weak spin polarization expected due to only Zeeman effect. At low magnetic fields the Brillouin function appears linear, see section 2.3.2 and equation 3.8.

109 Chapter 5 Quantum wells with a ferromagnetic barrier

5.4.1 Setback spacer and magnetic layer dependence

The setback spacer is varied to test the spatial length scale of the coupling between the magnetic layer and carriers in the QWs. In Samples D and E, which have d = 9 nm, no spin polarization is observed in the QWs. Spin polarization is only observed with samples that have d = 5 nm. Initial samples with d ≤ 2 nm did not exhibit PL emission from the QW. The spatial extent of the spin-polarizing interaction is also examined by changing the number of periods in the DFH layers. In sample B, four additional 0.5 ML MnAs layers are inserted at 20 ML (5.7 nm) spacings compared with sample A. There is no qualitative difference in the ﬁeld dependence of the PL polarization between samples A and B. The insensitivity of the spin polarization in the QW to the additional MnAs layers occurs since only the ferromagnetic layer closest to the QW causes the spin polarization. Since the second nearest magnetic layer has an effective d = 10.7 nm, this distance gives an upper limit on the spatial extent of the polarizing interaction. Spin-coupling between the QW bound carriers and the ferromagnetic layer occurs within 10.7 nm. The insensitivity of the polarization to the total magnetic layer thickness also helps us rule out the possibility of optical polarization artifacts, such as magnetic circular dichroism (MCD). MCD is an absorption effect, and is therefore proportional to the total thickness of magnetic material, exhibiting MCD. Since the polarization is roughly equal between samples A and B, MCD cannot explain this signal. The location of the polarization in the QW emission spectrum and the low polarization background away from the QW emission line also strongly exclude the possibility of other polarization artifacts.

110 Chapter 5 Quantum wells with a ferromagnetic barrier

1.562 PL (arb. units) (c) 2 kG 1.566 5 K 1.570 0 kG

1.574 -2 kG 1.578

(a) units) (arb. PL 1.582

E (eV) Polarization P (%) 10 1.566 8 (d) 1.570 4 5 0 1.574 0 1.578 (b) -4 P (%) -5 1.582 -8 1.586 -10 -2 -1 0 1 2 1.565 1.575 1.585 B (T) E (eV)

Figure 5.5: The magnetic ﬁeld dependence of the emission spectrum of an unbiased quantum well with a ferromagnetic barrier with d = 5 nm (sample A), (a) PL and (b) polarization-resolved PL. Line cuts at constant ﬁeld of data from (a) and (b), as indicated by the dashed lines, are plotted in (c) and (d). Dashed lines indicate the bounds of integration used to calculate the values of polarization presented.

5.4.2 Anisotropic spin polarization

As we have seen in sections 2.1.2, the anisotropy of the valence band due to quantum confinement and strain pin heavy hole spins along the growth direction and light hole spins in-plane. Thus hole spin polarization will result in a PL polarization that strongly depends on collection geometry. To examine this effect, we measure PL polarization from the edge-emission in the Faraday geometry (field in-plane). For all values of Vg and magnetic field, no polarization is observed. This suggests that the spin polarization observed from the surface

111 Chapter 5 Quantum wells with a ferromagnetic barrier emission (ﬁeld out-of-plane) is pinned along the growth direction. Since the conduction band is fairly isotropic, electron spin polarization is ruled out. From the known anisotropy of the valence band we expect that only light hole spin polarization results in PL polarization for edge-emission, whereas only heavy hole spins results in spin polarization in the surface emission. The above “selection rule” considerations indicate that the PL polarization results from recombination between unpolarized electron spins in the n = 1 conduction subband and spin-polarized heavy holes in the n = 1 valence subband.

5.5 Electrically tunable spin polarization

The dependence of the PL and polarization spectra as a function of bias are plotted in Fig. 5.6, where the magnetic field is held constant. Data are shown for the control sample and for sample A. As expected, the PL of both samples shows the quantum confined Stark effect (QCSE) redshift of the emission energy as the negative bias is applied across the gates [77]. In the magnetic sample, large and electrically tunable spin polarization is evident since the PL polarization increases with negative bias. Qualitatively identical behavior is observed in the other magnetic samples under bias. On the other hand, the control sample shows no significant polarization (<0.5%) at any bias. The PL intensity of the magnetic sample drops at the same bias voltages where the polarization increases, but the PL intensity of the control sample remains bright throughout this bias range. Since the PL quenching only occurs in the magnetic samples, it must be due to interaction with the MnAs layer and not because of generic LT growth related defects (the top-most barrier in the control sample is also grown by low temperature ALE). The laser pump (1.73 eV) is tuned below the band gap of the Al0.4Ga0.6As barriers (∼1.92 eV), such

112 Chapter 5 Quantum wells with a ferromagnetic barrier

Photoluminescence (PL)

1.585 (a) Control (b) Sample A 1.580

1.575 4 3 2 Energy (eV) Energy 1.570 1 1.565 0 PL (arb. Units)

Polarization Control 8 Sample A 1.585 6 4 1.580 2 0 1.575 P (%) Energy (eV) Energy 1.570 1.565 -12 -8 -4 0 -12 -8 -4 0 Vg (V) Vg (V)

Figure 5.6: Spectral dependence of voltage tunable spin polarization at 5 K and +1 kG for (a) non-magnetic control sample and (b) sample A with d = 5 nm. Top panels show the PL intensity, while corresponding PL polarization is plotted in the bottom panels.

113 Chapter 5 Quantum wells with a ferromagnetic barrier that all photoexcited carriers are confined to the QW. Also, the QWs are biased in a regime of negligible current flow. Thus any loss of PL intensity is due to non-radiative losses. The above observations lead us to conclude that carriers tunnel through the barrier resulting in non-radiative recombination with defect states in the first MnAs layer. This indicates that the heavy hole wave function is overlapping considerably with the magnetic layer. The gate voltage dependence of the PL polarization is plotted in Fig. 5.7(b) for three different magnetic samples. PL polarization increases with decreasing bias in each sample. We also note that there is a voltage at which the sign of the polarization changes. Below this cross-over bias voltage (Vcb) the polarization increases to its maximum value accompanied by a large QCSE red shift of the

PL. Above Vcb, the polarization decreases below zero without any QCSE shift in the PL emission. The control sample polarization shows no sensitivity to bias, supporting the validity of the cross-over effect observed in the magnetic samples. The negative PL polarization is attributed to the heavy hole spin polarization. Negative bias pulls holes toward the magnetic layer, which would increase the exchange coupling. The change in sign maybe due to the effect of electron spin polarization at positive bias. At these conditions electrons are pulled toward the magnetic layer.

5.5.1 Model of exchange overlap

The effect of gating on the wave function overlap is modeled using a one- dimensional Poisson-Schrödinger simulation of the structure [31]. Calculations were performed assuming Mn acts as a deep acceptor in Al0.4Ga0.6As with 110 meV binding [137]. We also assume that a ﬁctitious 2 V Schottky barrier forms at the surface in order to functionally simulate the backgating structure. Results

114 Chapter 5 Quantum wells with a ferromagnetic barrier

4 0V (a) control (b)

(a.u.) -0.5V 3 A 5 K -1.2V B HH1 +1 kG

Ψ 2 C

0.0 P (%) 1 (eV) v

-0.4E 0 0.5 ML MnAs

-1 100 95 90 85 80 75 -15 -10 -5 0 5 z (nm) Vg (V)

Figure 5.7: (a) Valence band edge diagram at three values of Vg (bottom panel) and corresponding heavy hole (HH) n = 1 wave functions (top panel). (b) PL polarization as a function of Vg for biased samples. of these simulations are plotted in Fig. 5.7a, showing the valence band edge along the growth axis of Sample A at several biases. The corresponding ground state heavy hole wave functions are shown in the top panel. For Vg = 0 V, large band bending due to the Mn2+ acceptors results in a triangular distortion of the square well potential, shifting the center of the wave function toward the magnetic layer. This explains the zero-bias spin polarization observed in several of the non-gated samples, Table 5.1. For Vg < 0 V, the heavy hole wave functions increases their overlap with the Mn2+ layer. This overlap is responsible for PL quenching through tunneling and non-radiative recombination and for hole spin polarization, Fig. 5.6. Using the wave function gating dependence we model the exchange splitting and expected heavy hole spin polarization due to the p − d exchange interaction (section 3.1). Figure 5.8a plots the heavy hole probability density function |Ψ(z)|2 at the Mn-layer (5-nm from the top edge of the QW). Negative bias voltage increases the heavy hole density over the Mn. The effective Mn-

115 Chapter 5 Quantum wells with a ferromagnetic barrier concentration experienced by heavy holes is given by equation 4.12, where we use our calculated values of Ψ(z) and assume that Mn(z) is equal to 50% in a 1-ML region 5-nm from the edge of the QW. These values of x∗ allow us to cal- ∗ culate the total p − d exchange splitting x N0βhSzi assuming that the exchange 5 coupling N0β = −1.2 eV, as shown in Table 3.1. We also assume that hSzi = 2 , i.e. fully polarized Mn2+ spins, which should be the case given that the moment is almost fully saturated for Bz = 1 kG. The p − d exchange splitting of the heavy hole subband is plotted in Fig. 5.8b. The polarization of the heavy hole subband is given by the thermal occu- 1 pancy of the two-level system, i.e. the spin- 2 Brillouin function, equation 3.6. Thus the spin polarization is given by,

1 P = B (η), (5.1) 2 1/2 where the argument of the Brillouin function is given by,

x∗N βhS i η = 0 z . (5.2) 2kBT Note that the argument of the Brillouin function is the energy splitting between the spin-levels divided by the thermal energy, which in this case is due to the p − d overlap between QW heavy holes and ferromagnetic Mn2+ spins. Also, 1 note that we have assumed the heavy hole band as an effective spin- 2 system, rather than the more accurate treatment that includes the orbital angular momentum L = ±1. The spin polarization is plotted in Fig.5.8c as a function of gate bias. The simulation qualitatively follows the trend of the data shown in Fig. 5.7b. The quantitatively predicted spin polarization is fairly accurate given the simplicity of our model. Sources of error include the assumption of perfect top-hat Mn-proﬁles, and error in the Poisson solutions yielding the band bending in the QWs. Overlap increases exponentially with bias thus results of our

116 Chapter 5 Quantum wells with a ferromagnetic barrier

1.0 (c) 0.5 spin-P (%) 0.0 0

eV) (b) μ ( -4 p-d E Δ -8 0.0010

(a)

(%) 0.0005 at the MnAs layer HH 2 | Ψ | 0.0000 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

Vg (V)

Figure 5.8: (a) The heavy hole n = 1 probability density function at a position 5-nm from 2 the top QW edge (|Ψ(z)|HH ) in the MnAs-doped region following from the one-dimensional Poisson-Schrödinger simulations. (b) The total p − d exchange splitting for the heavy hole

(∆p−d) due to wave function overlap. (c) The spin polarization (spin-P) of the heavy hole band, equation 5.1.

117 Chapter 5 Quantum wells with a ferromagnetic barrier simulation are highly sensitive to error in the calculation of the built-in band bending due to the Mn-acceptors.

5.5.2 Magnetic ﬁeld dependence

The magnetic field dependence of the PL polarization is measured for many values of Vg (sample A). Representative results are plotted in Fig. 5.9. The polarization increases with the magnetic field for Vg ≤ −1 V. The magnetic field dependence of the polarization changes sign once Vg > −1 V. Here the polarization decreases with magnetic field. Following from the previous discussion, this indicates Vcb ∼ −1 V. We determined the angular momentum of the PL using the procedure described in section 2.3.1. From the QW selection rules, Fig. 2.4, a positive − 1 polarization (net σ , j = −1) indicates + 2 electron spins occupying the heavy 1 hole band (or − 2 heavy holes in the hole picture). Also recall that the spin of polarization of the Mn-layer aligns opposite the field (parallel to the field in the hole picture). Since we observe positive polarization at positive field in the bias range of Vg ≤ Vcb, the heavy hole electron spin is oppositely aligned to the Mn-ion spin under these conditions. This is exactly what we expected from our simple model (above) due to the negative p − d exchange coupling. In the bias range, Vg > Vcb, the opposite spin coupling is observed. This may indicate, as we discussed above that the spin polarization is no longer due to heavy hole, but rather, electron spin polarization in the conduction band.

5.5.3 Optical spin-injection

As a ﬁnal test, the PL intensity and polarization are studied while optically injecting spin using a circularly polarized excitation. Fig. 5.10a plots the PL

118 Chapter 5 Quantum wells with a ferromagnetic barrier

6 5 K 0.50.5 4

0.00.0

-0.5-0.5 2 -2 0 2

0 P (%) -3.8-3.8V V -2 -2-2V V -1-1V V -4 +1+1V V Control,control, +2.5 +2.5V V -6 -2 -1 0 12 B (kG)

Figure 5.9: Field dependence of sample A at various values of Vg. Open and closed symbols indicate the direction of ﬁeld sweep as up and down, respectively. Control sample at the bias value of maximum polarization is included for comparison (blue line). Inset shows an enlargement of data for Vg = −1 and +1.

119 Chapter 5 Quantum wells with a ferromagnetic barrier intensity for sample A as a function of gate bias for σ ± polarized excitation. The quenching of the PL is unaffected by the polarization state of the pump beam. This indicates that the non-radiative losses due to the MnAs layer are not spin-sensitive, i.e. spin dependent tunneling is not observed. Fig. 5.10b plots the PL polarization for all three polarization states of the excitation. The σ − and σ + excitation result in increased and decreased PL polarization, respectively. This conﬁrms that optical spin-injection is possible in these structures with the correct sign as expected from the PL polarization measurements. The polarization for the two opposite cases of spin-injection are averaged (magenta) and perfectly match the bias dependence in the case of no spin-injection (black line). We interpret this to mean that the spin polarization mechanism shows no sensitivity to the initial spin state of the interacting carriers; optical spin-injection simply shifts the net PL polarization magnitude without changing the strength of the heavy hole and Mn-ion spin interaction. This makes sense if the coupling is due to p − d exchange coupling by wave function overlap since the equilibrium spin polarization would be unaffected. The time-averaged measurements presented here are insensitive to any transient effects of optical spin-injection.

5.6 Conclusions

Ferromagnetic layers were incorporated into the QW barriers of quantum wells. Polarization-resolved measurements of the optical emission of these QWs was used to examine the spin polarization of carriers conﬁned to the wells. Heavy holes in these structures become spin-polarized by p−d exchange overlap with the ferromagnetic layer in the barrier. In contrast to the exchange effects observed in paramagnetic structures, see chapter 4, the spin polarization of the

120 Chapter 5 Quantum wells with a ferromagnetic barrier

15 (a) (b) σ- +1 kG + σ 10 5 K

0 P (%) PL (arb. units)

-5 Linear σ- + -10 σ (σ- + σ+)/2

-15 -10 -5 0 -10 -5 0 Vg (V) Vg (V)

Figure 5.10: PL as a function of Vg for sample A under optical excitation with different helicities in a constant magnetic ﬁeld. Both (a) unpolarized PL emission and (b) polarization are plotted showing the effect of optical spin-injection. In (b) the polarization values for σ − and σ + excitation are averaged (Pσ − +Pσ + )/2 and compared with the case of zero optical spin-injection (linear pump).

121 Chapter 5 Quantum wells with a ferromagnetic barrier ferromagnetic layer can be fully saturated with application of small magnetic ﬁelds. Thus the heavy hole spin polarization tracks a hysteresis loop as opposed to a Brillouin function. A vertical gate was used to control the spatial overlap between the heavy holes and ferromagnetic layer. This allows for an electrically tunable heavy hole spin polarization in quantum wells in the absence of optical or electrical spin- injection and at modest magnetic ﬁelds. A simple model of the exchange overlap captures the qualitative effect of biasing. The expected antiferromagnetic coupling of heavy hole and magnetic ion spins was observed increasing with negative bias as the exchange overlap increases. Remarkably the spin polarization changes sign when electrons are pulled toward the ferromagnetic layer, positive bias. The heterostructures studied here were grown using a two chamber MBE process. Although we were able to measure PL in these structures, time-resolved KR signals of the electron spin precession were not observed. Using the current spintronics MBE system it would be possible to grow this entire structure in a single chamber. Newer structures can be produced more rapidly, making it possible to search for sample structures in which time-resolved KR and electron spin precession could be observed. For example, InGaAs QWs are favorable since the DFH layers contain GaAs-spacers. The magnetization behavior is much more ideal (sharp and square hysteresis loops and higher TC) than in the GaAs quantum wells where DFHs contain defective AlGaAs spacers. An InGaAs ferromagnetic quantum well was prepared (sample H). This sample showed much larger hole density indicating a lower density of compensating defects. No spin polarization was observed, but a vertical bias was never applied to these samples. Recommended future work on these structures would be to produce gated InGaAs-GaAs QWs with ferromagnetic layers. Time-resolved

122 Chapter 5 Quantum wells with a ferromagnetic barrier

FR could be measured in transmission without needing to remove the substrate.

123 Chapter 6

III,Mn-V quantum wells: band and spin-engineering

In chapter 5, we discussed III-V quantum wells with a ferromagnetic barrier. The advantage of these structures over their II-VI paramagnetic counterparts, such as discussed in chapter 4, is the ability to incorporate ferromagnetism allowing for low field switching and magnetization saturation. Thus large exchange splittings are possible at low fields. Unfortunately, the time-resolved KR signal of electron spin dynamics in these structures was lost due to defects in the low temperature ALE grown top barrier. In this chapter, we describe a new class of III-V magnetic quantum structures in which most of the optical signals present in the non-magnetic QWs are retained. In Ga1−xMnxAs and (InGa)1−xMnxAs quantum wells with x ≤ 0.1% grown under specific MBE conditions, the time-resolved KR and FR measurement of electron spin dynamics is possible [138]. Although these wells are too lightly doped to allow for ferromagnetic coupling between the Mn-ions, the paramagnetism of these structures leads to easily observable exchange splitting of the conduction and valence bands.

124 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Time-resolved KR and FR measurements of the electron spin precession in the presence of the conduction band s − d exchange coupling are performed. Such measurements represent the first observation of time-resolved spin dynamics in III-V magnetic semiconductors. Careful comparison between the control samples and the temperature dependence of the exchange splitting allow for the determination that the s−d exchange parameter is negative [48]. By varying the magnetic doping, barrier height, and well width systematically in several series of samples, we determine the effect of confinement on the exchange coupling. Quantum confinement is observed to increase the magnitude of the exchange splitting. This effect arises from the change in electron wave function symmetry leading to hybridization with the Mn-ion localized states [93], see section 6.6.3. The valence band exchange splitting is measurable through the polarization- resolved PL and absorption. Such measurements of the QW exciton are used to estimate a negative p − d exchange parameter, which agrees with previous measurements in compensated GaMnAs. The emission arising from the neutral Mn2+ acceptor bound exciton in the QW is also observed in the PL spectra. The polarization of this emission tracks the spin of the Mn2+. Time-averaged measurements of the Mn-ion spin precession are measured using the Hanle technique. We observe an exponential increase in the Mn2+ spin lifetime as the doping concentration is reduced. The results presented in this chapter are published in references [48, 93, 138]. The first three authors contributed equally to every aspect of the published research. As such, many of the results presented in this chapter are also shown in Poggio’s thesis, [38], albeit from a different point of view. Unpublished results described in this chapter include the InGaMnAs s − d measurements, magneto- optical absorption determination of the p − d exchange parameter, and Hanle

125 Chapter 6 III,Mn-V quantum wells: band and spin-engineering measurements of Mn2+ spin lifetime.

6.1 Growth of GaMnAs and InGaMnAs quantum wells

The current methodology of GaMnAs and III-V magnetic semiconductor research is to maximize the TC by varying the sample design and growth conditions. We abandon this strategy with the goal of retaining the time-resolved KR or FR signals. Recall that for GaMnAs MBE (section 3.3) two interdependent materials problems occur:

◦ 1. Mn is weakly soluble in GaAs. Low temperature growth (Tsub ∼ 250 C)

is required to incorporate enough Mn substitutionally (MnGa)to achieve

ferromagnetism. Too much Mn doping leads to interstitial Mn (Mni).

◦ 2. At Tsub < 300 C, excess As in the form of antisites (AsGa) are incorporated.

Both Mni and AsGa are double donors that compensate the holes from the MnGa acceptors. The varying level of these compensating defects in ferromagnetic

GaMnAs explains the broad range of TC’s that are reported for samples with nominally the same Mn doping level, see section 3.3.1. These defects are present at densities of 1019 cm−3 or greater, which is more than sufﬁcient to kill the band-edge optical properties. One strategy for reducing AsGa defects is to shutter the As by ALE as was used in the previous chapter. This tech- 18 −3 nique has been shown to reduce the AsGa compensation threshold to 10 cm [119]. Another technique is to precisely adjust the As-ﬂux to minimize the

AsGa compensation. We used this method to study the effect of AsGa on the ferromagnetism and electronic properties of GaMnAs, see appendix B.

126 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

◦ Samples of n-GaAs (Si-doped) were grown at Tsub = 250 C, using various As-ﬂux values, see appendix C.3. The Si-doping level was calibrated for high temperature growth in which each Si impurity contributes one electron. At low temperature any reduction in the carrier density can be ascribed to AsGa compensation. In order to maintain band edge optical signals, the As-ﬂux must be controlled within 1% resulting in a compensation threshold of less than 1016 cm−3 [139]. Unfortunately, Mn-doping densities required to achieve ferromagnetism quench the optical signals even in this case of nearly stoichiometric

GaAs. Examining (2), an easier solution to avoid AsGa is to increase Tsub to > 300◦ C, at which an As-overpressure can be maintained without any excess As incorporation. Following from (1), these substrate temperatures reduce the solubility limit of Mn. A series of non-magnetic AlGaAs-GaAs quantum wells were grown with ◦ Tsub = 300 to 580 C. Strong time-resolved KR signals are maintained in non- ◦ magnetic AlGaAs-GaAs QWs grown with Tsub ≥ 400 , but are lost for Tsub ≤ ◦ 350 . This effect is not related to AsGa, but due to oxygen impurities incorporated into the Al containing QW barriers, well-known for its impurity gettering of oxygen defects during MBE growth [140, 141]. This follows since InGaAs- GaAs QWs can be grown at 350◦C while retaining optical signals. The lower substrate temperatures possible for the InGaMnAs QWs allow for higher Mn doping levels than in the GaMnAs QWs since Mn solubility increases at lower temperatures.

In the following sections we discuss Ga1−xMnxAs / AlyGa1−yAs and

(In0.2Ga0.8)1−xMnxAs / GaAs quantum wells grown in an As overpressure and ◦ with Tsub ≥ 350 . At these conditions excess As does not incorporate, but Mn solubility is decreased below the ferromagnetic limit. Although only paramagnetism occurs in these QWs, exchange couplings are easily observable by op-

127 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

InGaMnAs QWs GaMnAs QWs (b) As (a) As As As x x 1-y 1-y Mn Mn GaAs GaAs Ga Ga 1-x 1-x y y ) Al Al Ga 0.8 GaAs GaAs Ga 0.2 (In AlyGa1-yAs d (nm) n=1 ΔECB∝ y Ee GaAs d (nm) Ga1-xMnxAs n=1 Ee

(In0.2Ga0.8)1-xMnxAs

Figure 6.1: Conduction band edge proﬁles and layer structure along the growth direction for (a) InGaMnAs QWs, and for (b) GaMnAs QWs. tical spin spectroscopies. The layer structure and band edge along the growth axis are shown schematically in Fig. 6.1. A detailed discussion of the sample growth and structure is given in appendix C.2. Tables 6.1 and 6.2 list selected properties of the samples used to determine the s − d exchange parameter.

6.2 Mn incorporation

The Mn concentration proﬁle along the growth axis (Mn(z)) is measured using secondary ion mass spectroscopy (SIMS). Representative proﬁles are plotted in Fig. 6.2a for the series A samples of 7.5-nm QWs grown on the same day with ◦ Tsub = 400 C and with different Mn-doping levels. These scans were performed by Charles Evans and Associates (CEA). The primary ion beam consists of Cs+ at 2 keV impact energy. These conditions provide a depth resolution of 3.25

128 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Table 6.1: Selected sample data from GaMnAs QWs used for s − d exchange analysis. The QW width (d), Al-barrier concentration (y), effective Mn concentration (x), conﬁnement energy

(Ee), electron g-factor (ge), and the intrinsic s − d exchange parameter (N0α) are listed.

Series Sample d (nm) y (%) x (%) ge Ee (meV) N0α (meV) A 040714A 7.5 40 0.0065 -0.03 47 -82 040714B 0.0537 040714C 0.1266 B 040830A 3 40 0.0025 0.46 136 -208 040830B 0.0117 040830C 0.0274 C 041122A 10 40 0.0134 -0.13 31 -70 041122B 0.0310 041122C 0.0804 041122D 0.1274 D 041122G 5 40 0.0044 0.22 80 -140 041123A 0.0110 041123B 0.0318 041123C 0.0574 E 050829A 10 10 0.0046 -0.265 22 -43 050829B 0.0108 050829C 0.0326 050829D 0.0694 F 050919A 10 20 0.0029 -0.218 27 -62 050919B 0.0100 050919C 0.0328 050919D 0.0311 G 050920A 5 20 0.0021 0.028 62 -106 050920B 0.0052 050920C 0.0140 050920D 0.0267

129 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Table 6.2: Similar data as in Table 6.1 for InGaMnAs QWs.

Series Sample d (nm) x (%) ge Ee (meV) N0α (meV) H 050216A 5 0.0240 -0.449 61 -59 050216B 0.0501 050216C 0.0964 I 051103A 3 0.0092 -0.394 91 -77 051103B 0.0212 051103C 0.0303 051103D 0.0413 J 050118A 7.5 0.0183 -0.519 41 -36 050118B 0.0382 050118C 0.0829 nm/e, calibrated using the atomically abrupt interfaced between AlGaAs and GaAs. Mn is detected by collecting the CsMn+ complex, which is insensitive to the AlGaAs/GaAs interface. The SIMS concentration is quantitatively calibrated by comparing the Mn signal from Mn-ion implanted GaAs with a known dose as a reference. These calibration scans are performed within the same SIMS runs. We also checked for changes in ionization yield of CsMn+ between AlGaAs and GaAs layers by scanning Mn-ion implanted AlGaAs calibration samples. The z-axis calibration for the SIMS scans is performed using the Al signal as a reference for the QW region as well as the lower edge of the barrier, both layers grown at low temperature. Here we assume that the growth rate calibrations of the MBE growth are accurate. The depth of the crater should not be used to calibrate the z-axis since low temperature grown layers sputter faster than the high temperature grown buffer layers. Thus if the SIMS run penetrates past the LT layer, the sputter rate is not constant.

130 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

d surface Al0.4Ga0.6As Al0.4Ga0.6As (a) Al 19 10 Mn 660° C ) +

-3 Cs beam 2 keV 1018 Mn 642° C Mn 604° C

(z) (cm 1017 Mn 1016 control 1015

(b) 104 Mn 660° C Al

2+ 3 O beam 2 keV 10 Tsub 400° C

2 10 Tsub 350° C

Tsub 325° C SIMS yield (arb. units) SIMS 1 + 10 Mn

50 100 150 200 z (nm)

Figure 6.2: Mn concentration profile measured by SIMS, (a) for 4 samples with 7.5-nm QWs and varying Mn doping (series A). The Mn effusion cell temperature is labeled in each figure. (b) Mn+ SIMS profile for 3 sample with 7.5-nm QWs and varying the substrate temperature

(Tsub). The un-calibrated Al SIMS signal (plotted as black lines) serves as a marker for the QW region.

131 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Surface segregation

As expected, the maximum in Mn concentration occurs near the center of the QW region, where the Mn shutter is opened. Unfortunately, on the surface side of the QW a large concentration of residual Mn incorporates into the structure even after the Mn shutter is closed. Since the growing surface is not at equilibrium, this surface segregation effect can occur even if the Mn-doping density is below the equilibrium solubility limit. Mn surface segregation was reported for Mn δ-doped GaAs two-dimensional electron gases (2DEGs) grown at 400◦C [142]. Our results are not in full quantitative agreement with the report by Naz- mul et al. This numerical discrepancy is attributed to the MBE chamber to chamber irreproducibility of growth conditions. In particular, different methods were used for substrate temperature measurement comparing our methods to those of Nazmul et al. Mn incorporation is known to be highly sensitive to substrate temperature. In our MBE system, the substrate temperature is measured directly by ABES (appendix A.4). Nazmul et al used a radiatively coupled thermocouple, which in our MBE system can be more than 50◦C from the actual substrate temperature. The surface segregation may lead to Mn clustering, which allows second- phase MnAs precipitates to form. Although these impurities are likely present in our samples, the Schottky barrier around such precipitates prevents their detection in the electrical or optical signal of free carriers in the quantum wells. They are, however observable in the magnetization of the samples. In the next section, the hole conductivity and carrier densities measured in these samples indicate, in comparison with the SIMS data, that most of the Mn impurities present in the sample are substitutionally incorporated. Thus Mn surface segregation and related growth defects have a negligible effect on our optical studies

132 Chapter 6 III,Mn-V quantum wells: band and spin-engineering of the exchange splittings.

Unintentional doping

Below the QW ∼ 200 nm from the surface, the Mn proﬁle becomes constant at a value near the detection limit of SIMS (1015 cm−3). The background Mn density scales with the effusion cell temperature, Fig. 6.2a, suggesting that Mn-ﬂux escapes around the closed shutter and incorporates into the growing epilayer even with the shutter closed. The control samples are grown with the Mn effusion cell at the cold idling temperature (200◦ C) resulting in Mn background that is well below the SIMS detection limit. The PL emission for Mn in bulk GaAs due to the neutral Mn-acceptor bound exciton emission (1.4 eV emission at 5 K [143]) is observed in all samples grown with a hot Mn cell. This observation supports our hypothesis regarding unintentional Mn doping.

Incorporation in AlGaAs

Figure 6.3 plots the Mn concentration profile between samples of varying Al barrier height (y), but with identical Mn-flux and well width. The Mn profiles show no appreciable change as the Al concentration is varied, suggesting that Mn incorporation is independent of y.

6.2.1 SIMS knock-on effect

Immediately below the QW region, the Mn concentration decreases toward the substrate reaching a minimum point after ∼ 100 nm below the QW. This Mn-tail is broader than the Al depth proﬁle. Thus it is possible that it is a real effect and Mn diffuses toward the substrate. We eliminate the possibility of Mn downward diffusion since the Mn-tail shows no substrate temperature or Mn-doping level

133 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

surface AlyGa1-yAs AlyGa1-yAs 1019 y = 10 % )

-3 18 y = 20 % 10

y = 40 %

(z) (cm 1017 x ~ 0.01% d Mn d = 10 nm 1016 0 50 100150 200 250 z (nm)

Figure 6.3: Mn concentration proﬁle measured by SIMS for three 10-nm QWs with varying Al concentration in the QW barriers. dependence, which ought to change the diffusion length. A second possibility is that the Mn-tail is due to a measurement artifact from SIMS. During the sputtering process energetic ions bombard the surface. The measurement assumes that atoms in the sample are uniformly sputtered. If one species is more mobile, such as charged impurities or defects, they may be pref- erentially sputtered and knocked into the sample. This Knock-on effect results in broader proﬁles pushing one species below the sample surface during the sputtering process. A knock-on effect for Mn is expected since it is a charged impurity with a high atomic mobility in GaAs. Such behavior was reported by Nazmul et al in their Mn δ-doped 2DEG structures. To test the knock-on effect, we perform two separate SIMS scans on a sample (series A) at two different impact beam energies, 2 keV and 8 keV. Atomic force microscopy (AFM) is used to measure the roughness of the SIMS craters sputtered using the two beam energies, Fig. 6.4a-c. The 2 keV crater has 0.32 nm RMS roughness and the 8 keV crater has 0.30 nm RMS, respectively. The

134 Chapter 6 III,Mn-V quantum wells: band and spin-engineering roughness of the starting surface is 0.14 nm RMS. The lack of a clear difference in roughness in the two craters indicates that the sample is uniformly sputtered for either beam energy, thus any changes in the profile are due to preferential sputtering of Mn. The Mn profiles from the knock-on test are plotted in Fig. 6.4d. On the surface side of the QW, the Mn surface segregation tail shows no dependence on the beam. On the other hand, the Mn-tail on the substrate side of the QW changes shape with beam energy; the decay length us 17 nm/e at 8 keV and 10 nm/e at 2 keV. An increased beam energy pushes Mn deeper into the sample during the SIMS runs indicating preferential sputtering. We conclude that the Mn profiles on the substrate side of the QW are sharper than the SIMS data indicate.

6.2.2 Substrate temperature dependence

SIMS profiles are taken for three 7.5-nm wide QWs grown with the same Mn cell temperature (same Mn beam flux), but different Tsub, Fig. 6.2b. The impact beam was O2+ at 2 keV and the detected species was Mn+ (UCSB by T. Mates). Due to oxygen impurities incorporated during growth in the AlGaAs layers the SIMS profiles taken with the O2+ beam show a large sensitivity to AlGaAs/GaAs interfaces. Thus, an absolute calibration of the Mn density for these scans is difficult. We restrict ourselves to qualitative comparisons in the interface free regions within the top AlGaAs layer.

The Mn proﬁles become sharper as Tsub is reduced; the surface segregation tail is eliminated and the peak in Mn density within the QW region increases. As expected, Mn solubility increases at lower Tsub . The sharp Mn doping proﬁles at lower Tsub allow for more complex magnetic quantum structures to be grown,

135 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

(a) 2 keV Cs+ crater (b) 8 keV Cs+ crater (c) sample surface 4 1 3 nm m

μ 2 0 1 -1 0 0123401234 12345 μm μm μm

1033 (d) Al 8 keV

2 16 nm/e 102

SIMS Mn Knock-on test 1 10 Cs+ 2-8 keV 2 keV

SIMS yield (arb. units) 10 nm/e + 1000 CsMn 20 40 60 80 100 120 140 160 180 200 Depth (nm)

Figure 6.4: Beam energy test for preferential Mn sputtering (knock-on). (a) and (b) show AFM images of the sample morphology in the center of SIMS craters formed by a 2 keV and 8 keV Cs+ beam, respectively. (c) AFM image of the bare sample surface. (d) The Mn profiles in sample 040714C measured at two different beam energies as labeled. Solid lines are exponential fits to the Mn profile tail below the QW.

136 Chapter 6 III,Mn-V quantum wells: band and spin-engineering such as coupled QWs. As discussed previously, however, optical signals quench ◦ ◦ in the GaAs QWs for Tsub = 325 C and 350 C. A secondary Mn peak occurs 50 nm below the QW region but only for ◦ ◦ Tsub = 325 C and 350 C. This position corresponds to the interface between the high temperature (HT) and low temperature (LT) grown QW barrier. After the HT layer is completed, Tsub is cooled to LT requiring a long pause at this interface. The Mn peak occurs due to Mn ﬂux escaping around the closed shutter, which scales with Mn cell temperature as discussed previously, and accumulat- ◦ ing into the sample. This effect does not occur for the sample with Tsub = 400 C, which illustrates the temperature dependent solubility of Mn. Alternatively, the second Mn peak could be due to changes in the SIMS ionization yield due to oxygen impurities at this interface; the Al signal also shows a change in intensity at this same region though the Al concentration is constant.

6.2.3 Effective Manganese concentration

The presence of Mn in the AlGaAs barriers (surface segregation) does not interfere with our measurement of the carrier spin splittings at the band edge within the QWs. The effective Mn concentration (Mn) experienced by electrons in the QWs is defined, as in equation 4.12, as the SIMS concentration profile Mn(z) 2 weighted by the electron probability density |Ψ(z)e| , which is calculated using our favorite one-dimensional Poisson-Schrödinger solver [31]. Values of x calculated in this fashion are listed in Tables 6.1 and 6.2. One additional complication exists since the decay of the Mn profiles on the substrate side of the QW is due to the knock-on effect, an artifact of the SIMS measurement. The uncertainty of x is estimated by recalculating the x assuming that the Mn profile on the substrate side of the QW is atomically sharp, i.e.

137 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Mn(z) = 0 below the QW. A standard deviation is determined from these two x calculations. This error is generally < 3%, except in QWs with signiﬁcant wave function penetration in the barrier. The maximum uncertainty occurs for the d = 3 nm QWs (series B) where the error reaches 15%. These errors are factored into the error bars in our determination of N0α.

6.2.4 Quantum well width dependence

The Mn concentration (x) is plotted in Fig. 6.5a for 23 samples (series A-D and H-J) as a function of the inverse Mn effusion cell temperature. For each sample series, the Mn concentration increases exponentially with Mn cell temperature indicating a linear relation between the Mn beam flux and doping density. Com- paring samples with the same Mn cell temperature, however, the Mn density drops as the with QW width narrows. The data in Fig. 6.5a are replotted in Fig. 6.5b to show the explicit QW width dependence. The effective Mn density in the QW is a strong function of well width for the growth conditions used here. In particular, at high Mn beam fluxes the difference in x between samples of different d is more apparent. The d dependence of Mn incorporation maybe due to the kinetics of Mn incorporation. Many dopants do not immediately incorporate during MBE growth. For example, In is well-known to only incorporate into GaAs once a critical density of In is built-up on the surface [144]. For this reason sharp InGaAs/GaAs QWs are grown via predeposition of one monolayer of InAs followed by the alloy growth and subsequent desorption of the “In floating layer” [145]. Narrow wells have lower total In densities than wider wells, which is what we observe in the case of the GaMnAs QWs. Similarly, we observe Mn surface segregation indicating that Mn does not fully incorporate immediately.

138 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

InGaMnAs 10 nm (a) 0.1 GaMnAs 7.5 nm 5 nm 3 nm x (%)

0.01

0.00108 0.00112 0.00114 Inverse Mn effusion cell temperature (K-1)

° (b) 0.12 660 C 654°C 0.10 642°C 0.08 623°C

x (%) ° 0.06 604 C

0.04 InGaMnAs 0.02 GaMnAs 0 234567891011 d (nm)

Figure 6.5: QW width dependence of Mn incorporation in GaAs (black) and InGaAs (red) quantum wells from sample series A-D and H-J. (a) The percentage of x is plotted as a function of the inverse Mn effusion cell temperature. (b) Data are replotted to show x as a function of quantum well width. Lines connect samples grown with the same Mn effusion cell temperature as labeled.

139 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Thus, wider wells have larger Mn densities for the same Mn-ﬂux since a larger total amount of Mn is deposited. The Mn incorporation behavior in InGaMnAs and GaMnAs QWs is qualitatively identical. Comparing data with the same symbol but different color in Fig 6.5a, we note that InGaAs QWs have higher Mn doping densities for the same Mn effusion cell temperature as compared to the GaAs QWs, which is due to the lower Tsub used to grow the InGaAs QWs.

6.3 Electronic properties

Compensation threshold

The carrier concentration of the samples is obtained by Hall effect measurements at T = 300 K in the Van der Pauw geometry. To investigate possible As defect-related carrier compensation two n-GaAs samples where prepared but grown at two different temperatures. The samples are 1 µm of n-GaAs (Si- 17 −3 ◦ doped ∼ 1×10 cm ) with one grown at HT (Tsub = 580 C) and the other at ◦ ◦ ◦ Tsub = 400 C. The carrier density in the 400 sample is lower than the 580 C sample by 1.1 × 1016 cm−3, providing an estimate for the compensation due excess As. Si doping is amphoteric, meaning that it occupies both Ga or As sites. This leads to both n and p doping. For HT growth with Si doping level < 5 × 1018 cm−3, however, only n-type doping occurs [146]. For LT growth Si also may act as an acceptor [147], thus part of the compensation in the LT layer maybe due to Si self-compensation. Thus the carrier compensation we observe in the 400◦ C sample is an upper limit on the density of excess As, < 1 × 1016 cm−3. This compensation threshold is drastically smaller than for GaAs grown ◦ 20 −3 at Tsub = 250 C, where ∼ 1 × 10 cm excess As is typical [115]. The compensation threshold for our samples is also much lower than for ALE grown

140 Chapter 6 III,Mn-V quantum wells: band and spin-engineering n-GaAs, 2 × 1018 cm−3 [119].

6.3.1 Manganese hole-doping calibration

As we just discussed, non-Mn related carrier compensation is negligible for ◦ Tsub = 400 C. Thus the hole density (p) in Mn-doped samples reflects the incorporation behavior of Mn. We investigate the hole density dependence on the Mn-flux by growing a series of bulk GaMnAs samples using the same condi- ◦ tions as for the QWs (Tsub = 400 C). The Mn-doping calibration samples consist of 1µm thick GaAs:Mn with Mn at various effusion cell temperatures. The sheet concentration of holes (p2D) per hour of Mn shutter time is plotted as a function of the inverse temperature of the Mn effusion cell, Fig. 6.6. The doping rate demonstrates exponential thermal activation (Arrhenius) due to a linear relationship between the Mn-flux and the hole density. This indicates a lack of Mn-self compensation, and therefore substitutional Mn incorporation is occurring. Data for two QW sample series with d = 15 nm and 30 nm are also plotted in Fig. 6.6. The activation energy from exponential fits matches the fit for the bulk Mn-doping calibration series indicating that MnGa incorporation is sustained at these higher doping levels regardless of QW structure. Mn penetration into the QW barriers and surface depletion could modify electrical properties, particularly for the narrower wells.

6.3.2 Mn acceptor activation energy

Figure 6.7 plots the temperature dependence of the hole density for a sample series of bulk GaMnAs (500-nm thick) epilayers and for GaMnAs QWs (series A). At high temperature the Mn-doped holes show clear exponential (Arrhe- nius) thermal activation from which we extract the effective acceptor activation

141 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

QW, d = 30 nm 1015 /hr)

-2 fit QW, d = 15 nm fit 1014 Bulk fit

1013

p2D per hour of Mnof p2D per hour (cm shuttertime 1012 0.00105 0.00112 0.00119 0.00126 Inverse Mn effusion cell temperature (K-1)

Figure 6.6: Mn hole-doping calibration. The room temperature two-dimensional hole density (p2D) per hour (hr) of Mn shutter time as a function the inverse Mn source temperature. Three sample sets grown at 400◦C are plotted and the data are ﬁt to an exponential thermal activation (Arrhenius) equation, plotted as a line, which gives a reasonable activation energy of ∼ 1 eV for Mn sublimation.

energy (Ea). Bulk samples show Arrhenius type behavior over a larger temperature range than the QWs. At low temperature, holes freeze out in all Mn doped samples. As the Mn concentration is increased, the temperature dependence deviates from pure this simple model; the exponential ﬁts are carried out over smaller temperature ranges as x increases. The activation energies extracted from ﬁts as in Fig. 6.7 are plotted as a function of Mn concentration in Fig. 6.8. We observe Ea that is lower than 110 meV, the known position of the MnGa acceptor in the band gap of GaAs from photoemission [143]. Similarly, Blakemore et al observed a thermal activation

142 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

14 10 (a) 3×1011 (d) Bulk QW ) )

2 13 - 10 x ~ 0.06% -2 2×1011 x ~ 0.007% m

1012

p2D (c p2D (cm 1011 1011 4812 71421

3×1012

) (b) ) (e) 2 Bulk 12 QW - 14 -2 2×10 m 10 x ~ 0.28% x ~ 0.05%

1012 p2D (c p2D (cm 1013

4812 71421

(c) 3×1012 (f) ) 14 Bulk ) 12 QW 2 - 10 -2 2×10

m x ~ 0.63% x ~ 0.13% 12 10 p2D (c p2D (cm

1013 4812 3812 T-1 (K-1 × 1000) T-1 (K-1 × 1000)

Figure 6.7: The two-dimensional hole density (p2D) as a function of temperature for 500-nm thick bulk GaMnAs (a-c) and for GaMnAs QWs from series A (d-f). Lines plot exponential ﬁts of the data over a speciﬁc temperature range.

143 Chapter 6 III,Mn-V quantum wells: band and spin-engineering energy of acceptors in Mn doped GaAs below the optical ionization energy of 110 meV [96, 148]. This was explained as due to impurity band broadening. Qualitatively, the activation energy of acceptors should decrease continuously as the doping level of Mn is increased. The Mn acceptor level broadens into an impurity band eventually merging with the GaAs valence band at the insulator to metal transition. In the bulk GaMnAs samples, the activation energy decreases with increased Mn doping as expected. For the GaMnAs QWs, this effect is not observed perhaps since additional temperature dependent compensation effects mask the acceptors’ thermal activation. In 3-nm wide QWs (series B), Ea can not be extracted since exponential thermal activation is not clearly observed over a large enough temperature range. Similarly for the 5-nm wide QWs (series D),

Ea is found for the samples with the highest doping levels, but not in the other

2 samples. Ea is measured in the bulk 500-nm GaMnAs and in the sample series A and C. The well width effect is likely as result of carrier compensation from impurities gettered by Al during the growth of the AlGaAs QW barriers, discussed previously, see section 6.1. As the QW width is decreased, the wave function barrier penetration increases, making the effects of defective barriers more signiﬁcant.

6.4 Substitutional Manganese Incorporation

We estimate the fraction of substitutional Mn by assuming that each MnGa do- nates one free hole and each Mni compensates two holes [111]. A similar calculation was recently used in ferromagnetic GaMnAs (x ∼ 1%) to estimate the concentrations of Mni and MnGa [149]. The effect of As defect compensation can be ignored since it was shown to be negligible for our chosen growth con-

144 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

120

5 nm 100 7.5 nm 10 nm 80 500 nm

(meV) 60 a E

0 0.20.4 0.6 x (%)

Figure 6.8: The activation energy (Ea) for Mn-doped holes as a function of x in bulk GaMnAs and in GaMnAs QWs extracted from ﬁts shown in Fig. 6.7. For reference, a dashed line marks the 110-meV activation energy for an isolated Mn2+ ion in GaAs. ditions (< 1 × 1016 cm−3) as discussed in section 6.3. Thus,

p = MnGa − 2Mni, (6.1) where MnGa is the concentration of substitutional Mn and Mni is the concentration of interstitial Mn. Note that since Mn is the sum of MnGa and Mni, then

MnGa we may rewrite equation 6.1 in terms of Mn , Mn 1 p 2 Ga = + . (6.2) Mn 3 Mn 3 Values of this ratio are plotted for sample series A-D in Fig. 6.9. The fact that all the Mn-doped QWs, including InGaMnAs wells, demonstrate hole con-

145 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

MnGa 2 duction indicates that at a minimum, Mn = 3 . This model assumes that Mn impurities are static in the crystal and assumes that MnGa and Mni are fully thermally activated at room temperature. The ﬁrst assumption is likely valid; we can ignore thermodynamic equilibrium since our crystal is grown very far from equilibrium as a supersaturated solid solution of Mn in GaAs. The assumption regarding thermal activation of MnGa and Mni is not valid, but it is expected that the deep MnGa acceptors have a larger activation energy than the shallow Mni

MnGa donors. Thus the actual value of Mn should be larger than we modeled since at room temperature the shallow donors are more ionized than the deep acceptors.

MnGa As the Mn density (x) increases, Mn decreases. This indicates that high Mn doping levels lead to the incorporation of hole compensating defects, such as

Mni or Mn-containing second phases such as MnAs. Precipitates remove Mn from the lattice and form Schottky barriers leading to a reduction in hole concentration. Hole compensation may also occur due to interstitial-substitutional pairs and dimers of two nearest neighbor substitutional Mn [150, 151]. At the doping densities in our samples, these defects are unlikely. Optical signals, predominantly PL, degrade with increased Mn, see section 6.8). Particularly in the narrow QWs with d = 3 and 5 nm, compensation of holes due to barrier defects

MnGa also deﬂates our estimate of Mn . Despite these effects, MnGa is about 70-90% in all the GaMnAs QWs, Fig. 6.9.

6.4.1 Magnetization

The magnetization of the samples are measured using superconducting quantum interference device (SQUID) magnetometry. Mn2+ ions are expected to behave 5 as paramagnetic S = 2 moments. Thus their magnetization is given by equations

146 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0.8

0.6 /Mn Ga 3 nm

Mn 0.4 5 nm 7.5 nm 10 nm 0.2

0 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 x (%)

MnGa Figure 6.9: The fraction of substitutional Mn ( Mn ) as a function of x, estimated assuming no surface depletion and full thermal activation of both MnGa doped holes and Mni doped 2 electrons. The minimum fraction of substitutional Mn (dashed line, 3 ) for p-type GaMnAs is plotted.

147 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

3.5-3.7 as, 5 5gMnµBB M = gMnN0x µBB5/2 , (6.3) 2 2kB(T − θP)

5 µB where the saturation magnetization is M0 = ( 2 gMnN0x)µB in units of cm3 . Thus 5 the magnetization tracks the j = 2 Brillouin function as a function of temperature and field. To demonstrate the paramagnetism of the samples, a 1 µm thick GaMnAs layer is grown using the same conditions as for the GaMnAs QWs. The magnetization is measured as a function of magnetic field. Since the doping level of Mn and the thickness of the layer are small in comparison to the wafer, the diamagnetic background of the GaAs wafer dominates the raw magnetization signal. We assume that at 2 K and at 8 T the magnetization of the paramagnetic layer is saturated allowing the diamagnetic background to be subtracted. The magnetization is plotted for 3 different temperatures in Fig. 6.10 after background subtraction. Fits to equation 6.3 are performed assuming that gMn = 2 and are plotted as lines over the data. The two fit parameters, M0 and θP are 5 listed. The values of M0 are equal to 2 gMnN0x, which is 5 times the Mn concentration N0x. These fits demonstrate that paramagnetism is the dominant behavior for Mn dopants at the chosen growth conditions. The negative paramagnetic

Curie temperatures (θP) indicate a weak antiferromagnetic interaction between Mn-moments. Similar scans were attempted in the single QW samples, however the magnetization of the Mn-doped QW layers is a factor of 100× smaller than the bulk sample discussed above, below the detection limit of the SQUID.

148 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

2 × 1020 θ , M P 0 2 K -1.5 K , 1.7 × 1020 cm-3 20 5 K ) 1 × 10 20 -3 3 -2.4 K, 1.3 × 10 cm 20 -3 10 K /cm -0.5 K, 0.9 × 10 cm B

μ 0

M ( 041006G 20 μ ° -1 × 10 1 m GaMnAs, Tsub = 400 C

-2 × 1020 -8-6 -4 -2 0 2 4 6 8 B (T)

Figure 6.10: The magnetization of paramagnetic GaMnAs. Magnetization versus magnetic ﬁeld for a bulk GaMnAs epilayer grown at the same conditions as the GaMnAs quantum wells. Fits to equation 6.3 are plotted as lines over the data.

6.5 Electron spin dynamics in III,Mn-V quantum wells

Electron spin dynamics are measured by time-resolved Kerr (KR) and Faraday (FR) rotation in the Voigt geometry. The optical path is parallel to the growth axis, section 2.3.4. The pump beam is 1-5 mW and probe beam is 0.1-0.5 mW which are focused to overlapping ∼ 50µm diameter spots. GaMnAs samples are measured with KR since transmission is not possible without additional processing. However, the band edge of InGaMnAs is below the absorption edge of GaAs allowing for transmission of the probe beam through the wafer. We therefore measure FR for the InGaMnAs QWs.

Measurements of θK,F (∆t) reveal electron spin precession in Mn-doped III- V QWs, Fig. 6.11a and b. The data ﬁt to a single exponentially decaying cosine, equation 2.8. In the GaMnAs samples, the electron spin precession signal dis-

149 Chapter 6 III,Mn-V quantum wells: band and spin-engineering appears for x > 0.13%. As we discussed in section 4.1 and illustrated in Fig. 4.2c, Mn2+ spin precession is observable in the KR signal of II-VI magnetically doped QWs leading to an additional frequency component in the delay scans. In principle this precession should occur and be observable in the III,Mn-V QWs. Unfortunately, the orders of magnitude smaller x in the III-V QWs compared to the II-VI QWs reduces the Mn2+ spin precession signal to well below the noise ﬂoor of the KR and FR measurement. As we will discuss later in later section 6.8.4, the time-averaged spin dynamics of Mn2+ are observable in the PL emission associated with the Mn-acceptors within the QWs. Another possibility to observe this signal would be to use transmission spectroscopy on multiple QWs of InGaMnAs to amplify the signal. Such samples would need to be designed properly to avoid strain relaxation, appendix A.6. As we discussed in chapter 4, the total spin splitting in the conduction band e of the magnetic QWs is the sum of the Zeeman (∆Eg) and s − d exchange term

(∆Es−d), which is proportional to the electron spin precession frequency. Re- calling equations 4.1, 4.2, and 4.17,

e e ∆E = hνe = ∆Eg + ∆Es−d = geµBBx − N0αxhSxi. (6.4)

Figure 6.11c and d plot νe as a function of B for InGaMnAs and GaMnAs QWs grown in the same day but with different Mn-doping densities, sample series A and H. For the non-magnetic control samples with x = 0, only the Zeeman term is present leading to a linear field dependence of νe. Such fits provide |ge| values, listed for the complete sample sets in Tables 6.1 and 6.2. Increasing the value of x turns on the s−d exchange term adding a non-linear field dependence to νe. At high x values the s − d term begins to dominate the spin splitting resulting in a curved Brillouin function field dependence clearly visible in Fig. 6.11c and d.

150 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

(a) (c) 1 InGaMnAs 050216A 75 x ~ 0.02%

d = 5 nm 5 K 50 ν e

0 5 K (GHz) (arb. units) F θ 25 5K, 8 T InGaMnAs -1 Series H

0 75 (b) GaMnAs (d) 1 040714B x ~ 0.05%

d = 7.5 nm 5 K ν GaMnAs 50 e (GHz) Series A 0

(arb. units) K θ 25

-1 5K, 8 T

0 0 200 400 0 2468 Δ t (ps) Bx (T)

Figure 6.11: Electron spin precession in III,Mn-V QWs. Examples of time-resolved KR data (points) together with ﬁts (lines) for an InGaMnAs (a) and a GaMnAs QW (b). The precession frequency extracted from such ﬁts are plotted as points in (c) and (d) for a series of InGaMnAs and GaMnAs QWs, respectively. Fits to equation 6.4 are plotted as lines. Increasing symbol size indicates larger x, and the control samples (x ∼ 0) are plotted with open symbols.

151 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

6.5.1 Determining the sign and magnitude of the exchange coupling

We emphasize that a measurement of νe alone, because of phase ambiguity, does not determine the sign of ∆E. In other words, measurement of νe can only yield a relative sign between the Zeeman and exchange terms in equation 6.4. In this section we describe how an absolute measurement of the sign of the s−d exchange parameter is possible. The strategy is as follows:

1. Determine the absolute sign of the Zeeman term.

2. Gradually switch-off the s − d term by increasing the sample temperature. Thus, the relative sign of the s − d term with respect to the Zeeman splitting is determined.

3. Since from (1) the sign of Zeeman splitting is known, equation 6.4 yields

the absolute sign of N0α.

First, the absolute sign of the Zeeman term (ge) is determined. As we described in section 2.1, the dependence of the electron g-factor’s sign and magnitude in both GaAs-AlGaAs and InGaAs-GaAs quantum wells is well known from the literature. In GaAs-AlGaAs QWs, the g-factor changes from negative to positive near d = 6 to 8 nm for y = 40%. In InGaAs-GaAs QWs, the g-factor is always negative. Figure 6.12 plots ge as a function of d in all of the non- magnetic (x = 0) QWs used for the s − d analysis, where the sign is ascribed by comparison to literature [45, 47]. The thickness dependence reproduces the known behavior. Increasing the value of y shifts the g-factor to more positive values as in bulk AlGaAs [42]. Next we examine changes in the total electron spin splitting as the temperature is increased and the s − d exchange term is diminished. Recall that hSxi

152 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0.6 From Lit. T = 5 K 0.4 This work 0.2 e

g 0.0 GaAs-AlGaAs -0.2 -0.4 InGaAs-GaAs -0.6 5101520 d (nm)

Figure 6.12: ge as a function of d in GaAs-AlyGa1−yAs and In0.2Ga0.8As-GaAs quantum wells. Black circles are from GaAs QWs with y = 40% of this study. Red circles and squares are from references,[45, 47] respectively, with y = 33%. Stars are for y = 20% and triangles indicate y = 10%. Black squares indicate data for InGaAs QWs. Lines guide the eye. has the opposite sign as the magnetic ﬁeld, equation 3.7. Thus, from equation 6.4 three cases are possible:

1. ge and N0α have the same sign. As T is increased, νe will decrease con- e tinuously without going lower than ∆Eg/h.

e 2. ge and N0α have the opposite sign and |∆Eg| > |∆Es−d|. As T is increased, e νe will increase continuously to ∆Eg/h.

e 3. ge and N0α have the opposite sign and |∆Eg| < |∆Es−d|. As T is increased, e νe will decrease continuously to zero then rise up to ∆Eg/h.

Case 1: Constructive Zeeman and exchange terms

Figure 6.13a and b plot the temperature and ﬁeld dependence for sample 040714A from series A. As T is increased and the s − d exchange term is reduced, νe decreases asymptotically toward the value of νe for the control sample without

153 Chapter 6 III,Mn-V quantum wells: band and spin-engineering dropping below geµBB/h. This indicates (1), that the g-factor and s − d exchange parameter have equal sign. For this sample d = 7.5 nm ge < 0, thus

N0α < 0. Similar temperature dependences are carried out on all samples used for the s − d analysis, Tables 6.1 and 6.2. A negative N0α is consistently measured in all GaMnAs and InGaMnAs QWs. Having calibrated the absolute sign of the Zeeman and exchange terms, it is possible to ﬁt the magnetic ﬁeld and temperature dependent νe data to equation 6.4, shown as red lines in Fig. 6.11c and d, as well as Fig. 6.13. We assume that gMn = 2 and that hSxi obeys equation 3.7, as indicated by the paramagnetic behavior of the magnetization shown in Fig. 6.10. The only free parameters are

(xN0α) the extrinsic s−d exchange parameter and (θP) the paramagnetic Curie temperature.

Large and negative values of θP (-24 K) are observed in the lowest doped

GaMnAs QW, shown in Fig. 6.13. A negative θP is consistently observed in all InGaMnAs QWs as well as GaMnAs QWs for various x and y values. This indicates that either long range Mn-Mn coupling is antiferromagnetic or the Mn2+ spin temperature is larger than the lattice temperature. The ﬁts to the magnetization data indicate a smaller but negative value of θP in bulk paramagnetic GaMnAs. We have also attempted to introduce ferromagnetic interactions by modulation hole doping the III,Mn-V QWs. Preliminary studies show no effect of increasing the hole density on the ﬁeld dependence of the exchange splitting indicating that Mn-Mn are probably too far apart to cooperate ferromagnetically, which requires order of magnitude higher x. The large negative values of 2+ θP may be explained by an increased spin temperature of Mn due to photoexcitation. Mn2+ spin heating has also been observed in II-VI DMS at low doping densities [152]. Smaller values of |θP| (< 7 K) are observed as x is increased.

154 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

5 K 9 K (a) 10 17 K 30 K 54 K 5 160 K (GHz) e ν Control (x ~ 0) 0 0 246810 B (T)

10 (b)

(GHz) 8 T e ν 6 T 4 T

0 0 100 200 300 T (K)

Figure 6.13: νe as a function of temperature for a GaMnAs QW (sample 040714A) with d = 7.5 nm. (a) The field dependence of νe for various temperatures. Open points show the data for the control sample of the same series A with x = 0. (b) The temperature dependence at constant field. Lines are fits to equation 6.4.

155 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Case 2 (or 3): Opposing Zeeman and exchange terms

As shown in Fig. 6.12, the g-factor becomes positive in samples with y = 40% at d < 7 nm. When ge > 0, the antiferromagnetic s−d exchange coupling opposes the Zeeman splitting, case (2). This is well illustrated for the sample with d = 5 nm (041123B) in which at a certain temperature and magnetic field the total spin splitting is canceled. Figure 6.14a plots the magnetic field dependence of the ∆Ee for this sample at various temperatures. Despite their highly non-linear nature, the data are well fit to equation 6.4. e Consider the high temperature case, when ∆Es−d ∼ 0. When B > 0, ∆E > 0 e since ∆Eg > 0 for this sample. At low temperature ∆Es−d < 0 due to the increased spin polarization of Mn. At T = 10 K and B = 7 T, ∆Ee = 0 since the s − d exchange splitting is equal and opposite to the Zeeman splitting. Be- e e low this temperature, ∆E < 0 since |∆Es−d| > |∆Eg|. The exchange enhanced Zeeman splitting of the conduction band edges is plotted if Fig. 6.14b for this sample, illustrating the unusual field dependence of the electron spin splitting.

Calibrated spin splitting

The total spin splitting in the conduction band is measured as a function of magnetic ﬁeld for all the GaMnAs samples (Table 6.1) in which the QW width, barrier height, and Mn-density are varied, Fig. 6.15. The same data for InGaMnAs e samples (Tables 6.2) is plotted in Fig. 6.16. Note that the sign of |∆E | = hνe is e e e calibrated as discussed above, where ∆E = H↑ −H↓. The absolute signs follow the convention we established in chapter 1.2. Each sample series with varying values of x are grown on the same day, which reduces QW thickness variations between samples to < 1%. Variation in the sample thicknesses would change the ge value and results in errors in the determination of xN0α. We estimate

156 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

(a) 10 d = 5.0 nm x ~ 0.032% T = 13 K eV) μ 0 ) ( ↓ -H ↑

e -10 E Δ

-20 T = 2.0 K

0 2468 Bx (T)

g > 0 and N α < 0 (b) Ee e 0 +1/2 -1/2 μ α ge BBx -xN0

-1/2 +1/2

Figure 6.14: Opposing Zeeman and exchange terms in a 5-nm GaMnAs QW. (a) ∆Ee as a function of magnetic ﬁeld for sample 041123B at various temperatures. Fits to equation 6.4 are shown as lines. (b) Schematic of conduction band edges for spin-up and spin-down electrons in this QW. Red arrows and circles indicate the position where the exchange and Zeeman terms exactly cancel.

157 Chapter 6 III,Mn-V quantum wells: band and spin-engineering an error in the determination of xN0α from this variation of 3%. In all samples, ∆Ee decreases as x increases due to the antiferromagnetic (negative) s − d exchange coupling (N0α). As discussed in chapter 3.1, ferromagnetic s − d exchange is always observed in II-VI DMS, Table 3.1. This makes our observation of antiferromagnetic s−d exchange in III,Mn-V quantum wells very surprising. In II-VI, theory argues that conduction band symmetry forbids hybridization of s and d orbital [153]. Only direct Heisenberg exchange is possible, which as we discussed in chapter 1.3 is always ferromagnetic. In the next sections we discuss how quantum conﬁnement makes the s − d exchange increasingly more negative, which occurs due to a changes in the symmetry of the conduction band as the kinetic energy of the electrons is increased.

6.5.2 Anisotropy of s − d exchange coupling

InGaMnAs QWs are measured in transmission via time-resolved FR. This is possible without additional processing since the GaAs barrier and wafer are optically transparent at the absorption edge of the InGaMnAs QWs. Transmission spectroscopy allows for an investigation of the angle dependence of the g-factor and s − d exchange coupling. The anisotropy of the valence band in InGaAs QWs occurs not only from the effect of quantum conﬁnement but due to the epitaxial strain. The anisotropy of the g-factor in InGaAs QWs is observable in the angle dependence of the total conduction band spin splitting for the control sample (open symbols) Fig. 6.17a. Subtracting the g-factor angle dependence from that of the magnetically doped samples yields the angle dependence of the s − d exchange coupling plotted in Fig. 6.17b. From this limited data set it appears that the exchange parameter is isotropic, comparing 2 different in-plane

158 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

200 300 B, 3-nm, 40% D, 5-nm, y = 40% 100 200 eV) eV) μ μ ( ( 0

e e E 100 E Δ Δ -100 0 -200 0 0

-100 eV) -100 μ eV) ( μ e (

E e

-200 Δ

E -200 Δ

-300 A, 7.5 nm, 40% -300 C, 10-nm, 40% 0 246810 0 246810

Bx (T) Bx (T) 0 eV)

μ -100

( e E Δ

G, 5-nm, 20% F, 10-nm, 20% E, 10-nm, 10% -200 04 8 04 8 04 8 Bx (T) Bx (T) Bx (T)

Figure 6.15: The total electron spin splitting (∆Ee) as a function of B at T = 5 K for GaMnAs QWs. Solid circles of increasing size indicate larger x. Open circles are for the control samples. Fits to Eq. 6.4 are shown as lines. The sample series letter is marked in bold and listed together with d and y values.

159 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0 0 -100 H, 5-nm I, 3-nm eV) eV)

μ -100 μ ( (

e -200 e E E Δ

-300 Δ -200 -400 -300 0 246810 0 246810 Bx (T) Bx (T) 0 J, 7.5-nm -100 eV) μ (

e -200 E Δ -300

0 246810 Bx (T)

Figure 6.16: The total electron spin splitting (∆Ee) as a function of B at T = 5 K for InGaMnAs QWs. Solid circles of increasing size indicate larger x. Open circles are for the control samples. Fits to Eq. 6.4 are shown as lines. The sample series letter is marked in bold and listed together with values of d. directions with the out of plane angle dependence (θ). It appears that while the strong anisotropy of the valence band changes the g-factor, it does not affect the exchange coupling.

160 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

InGaMnAs QW, 5-nm -0.2 0

(a) (b) Δ E

-0.25 s-d (meV)

B = 8 T (meV) e

-0.5 E Δ T = 5 K

-0.3

-0.1 -25 025 -25 025 θ (°) θ (°)

Figure 6.17: Angle dependence of s − d exchange in InGaMnAs QWs (series H) as measured by time-resolved FR measurements of electron spin precession. (a) The total electron spin splitting (∆Ee) as a function of the angle (θ) between the sample normal and optical axis. Control sample data are plotted with open symbols. (b) The s − d exchange splitting as a function θ calculated by subtracting the Zeeman splitting in the control sample. Larger symbol size indicates increasing x. Red and black data correspond to a 90 degree in-plane sample rotation, e.g. < 110 > to < 110 >.

6.6 Conﬁnement engineering of the intrinsic exchange coupling

6.6.1 Width dependence

Figure 6.18a and b plot xN0α from fits to equation 6.4 as a function of x for GaMnAs and InGaMnAs QWs. Linear fits for each sample set of constant d are also plotted. The linear behavior of the extrinsic exchange parameter with Mn doping demonstrates that the intrinsic parameter N0α is constant over the measured doping range for III,Mn-V QWs with the same width. Finite values of xN0α at x = 0, extrapolated from the linear fits, are due to either the

161 Chapter 6 III,Mn-V quantum wells: band and spin-engineering experimental error in ge or error in x. The latter dominates the error of these ﬁts. The slope varies with d, which is explicitly plotted in Fig. 6.19 for both GaMnAs QWs (series A-D) and InGaMnAs QWs (series H-J). Reducing the

QW width makes N0α increasingly more negative. In GaMnAs QWs, N0α appears to saturate for the widest QWs. QW width has the same effect on InGaMnAs QWs, though the effect is weaker. As we will see, this difference occurs since InGaAs QWs are more strongly confined, thus larger changes in the well width are needed to alter the confinement energy compared with the more weakly confined GaAs QWs.

6.6.2 Barrier height dependence

Figure 6.20 plots the variation of N0α as a function of the concentration of Al in the AlGaAs QW barriers. By varying the QW barrier height, the confinement energy (Ee) is varied independently from the QW width. This addresses the possibility that the d-dependence of N0α shown in Fig. 6.19 occurs due to other parameters besides Ee. This specifically tests for the possibility that N0α variation occurs due to different Mn incorporation behavior or Mn-profile measurement artifacts related to QW width, as we observed in Fig. 6.5. As seen in

Fig. 6.20, N0α becomes more negative as y is increased. Increasing y or decreasing d both lead to larger conﬁnement energy (Ee) causing N0α to become more negative.

6.6.3 Kinetic exchange modeling

In II-VI DMS QWs, a negative change in N0α as large as −170 meV was previously reported for increasing conﬁnement and was attributed to a kinetic ex-

162 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

(a) 0 GaMnAs QWs y = 40%

eV) -50 10-nm

μ 3-nm (

α 0 xN -100 5-nm 7.5-nm

0.00 0.05 0.10 0.15 x (%)

(b) 0 InGaMnAs QWs 7.5-nm -25 3-nm eV) μ (

α 0 -50 5-nm xN

0.00 0.05 0.10 x (%)

Figure 6.18: The extrinsic exchange parameter versus Mn doping. xN0α from ﬁts to equation 6.4 is plotted as a function of x; error bars are the size of the points. Linear ﬁts are shown for each sample set of constant d. (a) plots data for GaMnAs QWs in series A-D, and (b) plots data for InGaMnAs QWs of series H-J.

163 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0.00

In Ga As QWs -0.05 0.2 0.8

-0.10 (eV)

0 GaAs QWs

N -0.15 Al0.4Ga0.6As barriers -0.20

-0.25 0 210468 d (nm)

Figure 6.19: s − d exchange versus quantum well width. Values of N0α extracted from ﬁts in Fig. 6.18 are plotted as a function of quantum well width (d). Data from GaMnAs QWs with y = 40% and for InGaMnAs QWs are shown.

-0.02 AlyGa1-yAs barriers -0.04 -0.06 d = 10 nm

-0.08 (eV)

α 0 -0.10 N -0.12 d = 5 nm -0.14

-0.16 10 15 20 25 30 35 40 y (%)

Figure 6.20: s − d exchange versus barrier height (y) for Ga1−xMnxAs-AlyGa1−yAs QWs.

164 Chapter 6 III,Mn-V quantum wells: band and spin-engineering change coupling due to the admixture of valence and conduction band wave functions [126].

The confinement energy (Ee) as defined in Fig. 6.1 is calculated for each QW using our favorite one-dimensional Poisson-Schrödinger solver [31] with values listed in Tables 6.1 and 6.2. We replot N0α as a function of Ee in Fig. 6.21, and the data are roughly linear. This effect is modeled taking into account the kinetic exchange (chapter 6.6.3) contribution to the conduction band exchange coupling. Kinetic exchange may occur in the conduction band due to changes in the symmetry of the electron wave function as the confinement energy is increased. p − d Kinetic exchange

As we discussed in chapter 3.1, the interactions between dilute magnetic ion and carrier spins are modeled using the Hamiltonian in equation 3.1 with the effective exchange integral J. As we discussed in chapter 1.3, direct Coulomb exchange is always a positive (ferromagnetic) interaction (J > 0). But experimental work has shown, Table 3.1, that the effective coupling between holes and localized transition metal states in most magnetic semiconductors is antiferromagnetic, i.e. Jp−d < 0, or N0β < 0. To explain this behavior, DMS theorists looked to the well established models for localized exchange interactions of dilute magnetic impurities in metals constructed in the 60’s [154, 155]. These models were developed to explain the Kondo effect [156]. Schrieffer and Wolff, very simply and elegantly, found that electrons in the Fermi sea that hop onto the localized d-states produce an effectively negative exchange interaction, i.e. opposite spin orientations are preferred [155]. Their model requires that the Fermi electrons are higher in energy than the localized d-states, and the mechanism is usually referred to as kinetic exchange.

165 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0.00

In Ga As QWs (eV) 0.2 0.8

α -0.05 0

Linear approx. GaAs QWs -0.10

-0.15 Linear approx. Kinetic exchange Intrinsic s-d exchange, N model -0.20

0.00 0.05 0.10 0.15

Confinement energy, Ee (eV)

Figure 6.21: Confinement engineering the s − d exchange parameter. Intrinsic exchange coupling of the conduction band (N0α) as a function of the electron confinement energy (Ee) for GaMnAs (black) and InGaMnAs QWs (red). Linear fits to the data are shown as solid lines. The dashed line is a fit to the kinetic exchange model.

Such a theory was adapted to the case of magnetic semiconductors, where the Fermi level resides in the band gap between the top of the valence band and bottom of the conduction band. These theories explain the antiferromagnetic p − d exchange observed in all known DMS [153, 157]. Hopping transitions responsible for the kinetic exchange, depicted in Fig. 6.22, dominate in the valence band since d levels strongly hybridize with the valence band (p-like) states. In the conduction band (s-like symmetry), kinetic exchange vanishes at k = 0 by symmetry [85], leaving ferromagnetic direct exchange dominant.

166 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

(a) d-level transitions (b) p-d exchange (c) s-d exchange

4 d 6 2 CB d L0 * ε- -Eg -Ee U - δ * eff eff E ε-= δeff E d6L 2 g 0 VB d5L δ 0 eff * ε+= δeff – Ueff ε++E g + Ee d5L 0 d4

Figure 6.22: Core level transitions in the kinetic exchange mechanism. (a) Schematic of the conﬁguration-interaction model for core-level transitions from a d5 Mn core with a valence 5 6 6 2 4 band hole (d L0) to a d core with two valence holes (d L0) or a d core. The effective parameters Ueff −δeff and δeff measure energies from the bottom of the core multiplet to the valence band edge[94]. (b) The same core-level scheme shifted into a one-particle picture. The transitions between Mn core levels correspond to the electron and hole capture energies, ε− and

ε+, responsible for valence band kinetic exchange. (c) The corresponding scheme adjusted for kinetic exchange involving conduction band electrons; γ(Ee) in equation 6.6 corrects for the energy difference between the carriers in each scheme.

Conﬁnement-induced s − d kinetic exchange

In the case of quantum conﬁned structures, however, changes in band structure alter the exchange mechanism. Assumptions made for bulk of a pure s-like conduction band Bloch state and purely p-like valence band state is no longer valid. Namely, the effect of conﬁnement is to add a kinetic energy (zero point energy) above the bottom of the conduction band for electrons. The additional kinetic energy pushes the ground state electrons away from k = 0 allowing for a kinetic exchange mechanism to occur not only for the valence band states, but also for the conduction band states. In effect, the conduction band states are no longer purely s-like and attain some p-character as they rise above the bottom of the band. As usual, advances in the DMS theory in this area have been driven by pioneering experimental work.

167 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Mackh et al. reported an effective reduction in the Zeeman splitting for DMS quantum wells as their width was reduced [125]. Polarization-resolved PL measurements in CdMnTe-CdMnMgTe wells showed a reduction in the exciton exchange splitting of ∼ 15% compared with bulk. This effect was modeled taking into account the confinement dependence of the valence band exchange parameter, i.e. the effect of reduced dimensionality on the p − d coupling due to kinetic exchange [158]. The exchange constant is treated as k-dependent within k · p theory, which leads to a reduction in the magnitude of valence band kinetic exchange with increasing quantum confinement energy. Although this model replicated the qualitative features of the data in II-VI materials, the calculated reduction in the exchange splitting was a factor 5 smaller than what was observed experimentally indicating an inconsistency in either the experimental work or the theory. In addition, this model amounts to a reduction of |N0α| with increasing k, which is inconsistent with what we observed in III,Mn-V quantum wells, Fig. 6.21. The situation seemed to be resolved by the report by Merkulov et al. who measured the s − d exchange splitting independently of the p − d exchange [126]. The samples used were identical to those in the report of Mackh et al.. By honing in on the conduction band exchange splitting only, Merkulov et al., observed that the s − d parameter (N0α) is reduced by up to 25% as the well width is narrowed, whereas in the earlier report measuring the total excitonic exchange splitting (N0[α −β]) only a 15% reduction was observed. Figure 6.23 plots the conduction band exchange splitting as a function of electron kinetic energy (quantum confinement) in these II-VI magnetic quantum wells. These data indicate that the conduction band exchange mechanism is more strongly altered than the valence band, which explains the quantitative failure of the theory of Reference [158] that did not take into consideration kinetic exchange of

168 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

1.0

(0) 0.9 α )/ e (E

α 0.8

01020 30 40 50 Ee (meV)

Figure 6.23: Effect of confinement on the conduction band exchange in II-VI magnetic quantum wells. The conduction band exchange parameter is plotted as a function of electron kinetic energy. Adapted from Ref. [126]. the conduction band. Merkulov modeled the effect of kinetic exchange on the effective conduction band exchange parameter simply by considering the effect of quantum confinement on the band mixing. Increasing the quantum confinement by decreasing the well width (or increasing the barrier height) increases electron kinetic energy (and thus k), the electron wave function takes on more p- symmetry character, increasing the antiferromagnetic contribution to N0α from kinetic exchange. Depending on the degree of the admixture, a positive conduction band exchange constant will decrease and even become negative for sufficiently large kinetic energy [126]. We quantitatively apply the envelope function model of Merkulov et al. to the data of Fig. 6.21. To lowest order in Ee, the dimensionless slope of N0α vs.

Ee is given by:

2 2 d(N0α(Ee)) 2(Eg + ∆) + Eg = − × {N0αpot d|Ee| Eg(Eg + ∆)(3Eg + 2∆) 4∆2 −[N0βpot + N0βkinγ] × [1 − 2 2 ]} (6.5) 3[2(Eg + ∆) + Eg ]

169 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

, where ∆ is the spin-orbit coupling, Eg is the band gap, N0αpot (N0βpot) is the direct conduction (valence) band edge exchange integral, and N0βkin is the kinetic exchange integral of the valence band. The conduction band kinetic exchange integral is assumed to be zero in the vicinity of k = 0. γ is a parameter that corrects the βkin given by Schrieffer-Wolff kinetic exchange in the valence band for electrons in the conduction band,

(Ev − ε+)(ε− − Ev) γ(Ee(k)) = , (6.6) [(Eg + Ee(k)) + (Ev − ε+)][(ε− − Ev) − (Eg + Ee(k))] where Ev is the energy of the valence band edge, ε+ and ε− are the energies for virtual hole and electron capture in the Mn d levels, respectively, as deﬁned in Fig. 6.22. These energies are estimated from the conﬁguration-interaction analysis of core-level photoemission spectroscopy where the core ground state is assumed to be the Mn2+ ion (Fig. 6.22a) [94]. The d5 core and valence 6 2 4 band hole L0 hybridize with the d L0 state and the d state [159], leading to virtual transition energies in a one-particle picture of ε+ = −5.2 ± 0.5 eV and

ε− = 2.7±0.5 eV relative to the valence band edge (Fig. 6.22b). γ(Ee) accounts for the energy shift to the conduction band shown in Fig. 6.22c.

Valence band potential exchange is assumed insignificant (N0βpot = 0), but the model is sensitive to both the sign and magnitude of N0βkin, which are in- fluenced by the average electronic configuration of the Mn acceptors in the material. While N0βkin has been found to be positive in uncompensated paramagnetic GaMnAs with neutral Mn acceptors [160, 161], the MBE-grown samples studied here are compensated by Mn interstitials [138] which act to ionize the

Mn acceptors and cause a negative N0βkin[162, 163]. Therefore, we estimate valence band kinetic exchange as N0βkin = −1.2 eV for MBE-grown GaM- nAs based on photoemission [94] and transport measurements[164] and from calculations[165]. This value is also supported by direct measurements of a

170 Chapter 6 III,Mn-V quantum wells: band and spin-engineering negative N0βkin in multiple GaMnAs QWs grown in the same conditions as the samples discussed here, see Fig. 6.28. N0αpot is the only free parameter in the model, appearing both in equation 6.5 and as the y-intercept value of N0α(Ee) at Ee = 0. Equation 6.5 is generally insensitive to N0αpot since N0βkin is typically an order of magnitude larger, so the extrapolation to Ee = 0 determines the ﬁt value for N0αpot. A linear approximation of equation 6.5 and a more detailed calculation of the envelope function theory of Merkulov et al. are plotted in Fig. 6.21. The slope of the calculation is −1.3 ± 0.3, agreeing well with a linear ﬁt of

−1.42 ± 0.12. The model extrapolates a value of N0α = −25 ± 1 meV for bulk

GaMnAs. The envelope function theory calculation accounts for the weak Ee dependence in γ(Ee) (equation 6.6), yielding curvature at high Ee, but since

Ee is small compared to ε− and ε+, the curvature is too small to be easily observed. The good agreement between the model and the data support the con- clusion that band mixing dominates the behavior s − d exchange as a function of 1D confinement energy in GaMnAs. While other treatments of kinetic exchange in QWs, such as the sp3 tight binding model,[166] are not excluded by this experiment, these other models in their current form do not provide such numerical agreement in both II-VI and III-V DMS experiments as the model of Merkulov et al.. Recently, independent calculations confirm the results of our model [167, 168]. The authors also suggested that negative s − d exchange coupling, such as observed in our quantum wells due to confinement, could be used to produce n-type ferromagnetic semiconductors. To summarize, we observe that the kinetic exchange mechanism dominates in the conduction band in GaMnAs quantum wells. This effect is identical to what was earlier observed in II-VI magnetic quantum wells. In the case of II-VI, however the s − d exchange parameter begins at a large and positive value, thus

171 Chapter 6 III,Mn-V quantum wells: band and spin-engineering the kinetic exchange acts simply as a reduction in the absolute value of N0α. In III,Mn-V quantum wells we have observed the very same decrease in exchange coupling as the conﬁnement is increase, however the bulk value is much smaller and perhaps even negative. Thus the reduction in the exchange parameter increase the absolute value of the exchange coupling. Our model extrapolates a value of N0α = −25±1 meV for bulk GaMnAs. This value is an order of magnitude smaller than that typically measured in II-VI magnetic semiconductors, and of the opposite sign. Its magnitude is similar to the +23 meV measured by Raman spin-ﬂip scattering in GaMnAs [95], but the sign is inconsistent with both previous measurements and band-edge s-d exchange theories. One recent proposal to explain the anomalously small value of N0α in bulk GaMnAs argues that the charged Mn2+ acceptors add a repulsive Coulomb potential that screens conduction band electrons and reduces the exchange overlap [169]. This model may explain the small value but not the sign. Since density of charged defects is highly sensitive to growth conditions, we expect a large variation in the charge screening and hence the measured values of the exchange constants for varying deposition conditions.

Neutral versus ionized Mn-acceptor

Thus far we have not discussed the exact character of the Mn-acceptor in our 5 III,Mn-V quantum wells and have assumed that they behave as perfect S = 2 moments. In reality the Mn-acceptors take on two different conﬁgurations, as discussed in chapter 3.3 and plotted in Fig. 3.3. The neutral and charged Mn- acceptors differ in their conﬁgurations with the location of the hole [97, 98]. 3 Since the heavy hole has total S = 2 and couples antiferromagnetically the Mn, then the spin state of the neutral Mn acceptors is vastly different than for the 5 charged acceptor, with distinct g-factors (2 versus 2.77) and total j ( 2 versus

172 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

1). Regardless of the spin orientation of the loosely or unbound hole, the Mn 5 5 core remains 3d with S = 2 ). Since s − d exchange involves the interaction between s electrons in the conduction band and core d electrons of the Mn-ion, the value of N0α should remain unchanged regardless of the presence of the loosely bound hole [162]. In contrast, p − d exchange is strongly modiﬁed by this hole. The presence of a loosely bound hole in the neutral complex opens a ferromagnetic exchange path whereas the ionized acceptor offers only antiferromagnetic channels. In the literature, such a dependence on the nature of the Mn acceptor core is offered as an explanation for the apparent sign ﬂip of the p−d term as the Mn concentration was increased from the very dilute limit (paramagnetic) to the high doping regime (ferromagnetic). Our exchange splitting model takes gMn = 2 and thus neglects any effect of the loosely bound hole on the core state g-factor since the exchange interaction between the hole and the core is expected to be small [170]. Measurements in III-V DMS support this assumption by consistently showing gMn = 2.0. Therefore, the relative concentration of neutral and charged Mn-acceptors in our samples should have a negligible effect on both N0α and gMn, allowing us to ignore this detail in our extraction of N0α from the data.

6.7 Electron spin lifetime versus doping

∗ The transverse electron spin lifetime (T2 ) is extracted from ﬁts to the electron spin precession measured by time-resolved KR and FR. As shown in Fig. 6.24a and b, the lifetime shows a large sensitivity to the Mn doping level in either ∗ GaMnAs and InGaMnAs quantum wells. Figure 6.24c and d plot T2 as a function of the percentage of Mn (x) for GaMnAs and InGaMnAs QWs, respectively, revealing qualitatively identical behavior. For the lowest Mn doping,

173 Chapter 6 III,Mn-V quantum wells: band and spin-engineering the lifetime increases with Mn doping as compared to the un-doped control samples. This increase is consistent with the D’Yakonov-Perel (DP) spin relaxation mechanism since increasing impurity concentration makes the process of motional narrowing more efficient by providing additional momentum scatters [171, 172]. Simply put, the addition of mobility reducing impurities smear out the k-dependent spin-orbit splitting that dephase electron spins. After reaching ∗ a maximum at very low Mn-doping (x ∼ 0.01%), T2 drops off as a function of x. In pursuit of a maximal electron spin lifetime, we varied the Mn density in InGaMnAs quantum wells with an otherwise fixed structure. Data from 14 samples are plotted in Fig. 6.24d revealing a maximum in lifetime at the same Mn concentration as observed in the GaMnAs samples. The decrease in lifetime occurs for x > 0.01%, indicating that the DP mechanism is no longer dominant in this regime. For these high doping levels either the Elliot-Yafet (EY) or the Bir-Aronov-Pikus (BAP) relaxation mechanisms may limit conduction electron spin lifetimes, since both should increase in strength with increasing x [13]. EY relaxation, due to the spin-orbit interaction, grows stronger with larger impurity concentration while, the BAP process, based on the electron-hole exchange interaction, increases with increasing hole doping. A study of the temperature dependence of the lifetime is underway to more conclusively identify the deco- herence mechanisms in III,Mn-V quantum wells [173, 174]. In II-VI magnetic quantum wells, magnetic doping leads to shorter electron spin lifetimes due to the strong extrinsic s − d exchange coupling [127]. Mag- netic impurity doping in these materials results in relaxation through spin-flip scattering. In contrast, the III,Mn-V quantum wells have extrinsic s − d exchange energies which are several orders of magnitude smaller than in typical II-VI DMS due to the large difference in Mn doping levels. In this case the effect of the exchange interaction on the spin lifetimes is expected to be negli-

174 Chapter 6 III,Mn-V quantum wells: band and spin-engineering gible.

6.8 Photoluminescence (PL): spin polarization, p − d exchange, Mn-emission

In this section we discuss magneto-optical characterization of the optical emission spectra of our III,Mn-V quantum wells. While the time-resolved KR and FR measurements provide data on the dynamics of electrons in the conduction band of the magnetic quantum wells, PL (optical emission) provide access to the total excitonic exchange splitting since the band-edge optical emission involves recombination of carriers between the conduction and valence band. As we discussed in chapters 3.2.1 and 4.5.1, polarization-resolved PL is sensitive to the spin splitting of holes, which are otherwise invisible to KR and FR studies. First we explain the optical spectra of the III,Mn-V quantum wells. The rich spectra show characteristic emission lines from free excitons and Mn-bound excitons within the quantum wells. Polarization-resolved PL is used as an optical magnetometer to monitor the spin polarization of Mn within the quantum wells. Band-edge emission is also used to measure the exchange enhanced (giant) Zeeman effect. From these data the total excitonic spin splitting in the quantum wells can be used to extract the p − d exchange parameter. Lastly, we discuss time-averaged measurements of Mn-spin dynamics via the Hanle effect. Polarization-resolved PL from Mn-acceptors within the quantum wells provides a means for measuring the Mn-spin lifetime revealing a strong doping dependence.

175 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

GaMnAs QWs GaMnAs QWs

(a) 400 (c) x = 0 10-nm 300 Bx = 4 T T = 5 K * (ps)

x = 0.01 2 200 T (arb. units)

K 7.5-nm θ 100 x = 0.03% 5-nm 0 3-nm 0 1000 2000 Δt (ps) 0.00 0.04 0.08 0.12 x (%)

InGaMnAs QWs 150 (d) (b) T2* ~ 33 ps 100 Bx = 8 T d = 5 nm T2* ~ 142 ps * (ps) 2 (arb. units) T 50 T2* ~ 76 ps K

θ InGaMnAs QWs

T2* ~ 23 ps 0 0 100200 300 0.0 0.1 0.2 0.3 Δt (ps) x (%)

Figure 6.24: Electron spin lifetime versus Mn doping. (a) and (b) Plot the time-resolved KR and FR signals of electron spin precession in GaMnAs (series G) and InGaMnAs (series H), respectively. The transverse electron spin lifetime extracted from such ﬁts is plotted as a function of x in (c) for GaMnAs QWs of series A-D, and in (d) for InGaMnAs QWs of series H- ∗ J. The plotted values of T2 in (c) are the mean values from 0 to 8 T and the error bars represent the standard deviations.

176 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

6.8.1 Explanation of the PL spectra

Figure 6.25 plots the full PL spectrum of a GaMnAs quantum well at low temperature. The spectrum reveals a large family of emission lines. Each peak is identified and the emission mechanism is schematically shown in the bottom of the figure. Three main large peaks are labeled due to emission from the GaAs bulk. This originates from the buffer layers of the sample below the quantum well barriers close to the starting substrate. The two highest energy peaks are the well known free exciton and donor bound exciton emission lines that dominate at low temperature in high purity GaAs [175]. Approximately 110 meV redshifted from the bulk free exciton emission the Mn-emission line is observed in the bulk. This is the well-known 1.4 eV emission in Mn:GaAs occurring due to a neutral Mn-acceptor bound exciton [143]. Two peaks are identified with quantum well emission. At higher energy than the bulk emission, the exciton emission between conduction and valence subbands in the wells occurs due to the increased effective band gap in the wells. It is this peak from which the total excitonic exchange splitting is measured. Approximately 110 meV redshifted from this peak we observe emission similar to what is observed for neutral Mn-acceptor bound exciton emission in the bulk. Due to the spectral location of this peak it is clearly due to the Mn-acceptor bound exciton emission in the well, where the position of the emission line is shifted by the increased effective band gap in the well. Polarization-resolved as a function of magnetic field confirms the identification of this line. The Mn emission line in the quantum wells is used to measure the Mn-ion spin lifetime via the Hanle effect, section 6.8.4.

177 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Emission Energy (eV) 1.381.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 106 GaMnAs QW, d = 10 nm, y = 10% (050829A) Bulk QW Bulk Bulk QW 0 0 0 (A Mn, X) (A Mn, X) (D , X) (X) (X) 105

104

PL (counts/s) 103

107 meV 102 116 meV

n=1 n=1

* Eg –EMn Eg -EMn E * Eg –Ed Eg g (Mn2+ + h) (Mn2+ + h) 116 meV 107 meV n=1 n=1 VB

Figure 6.25: The full PL emission spectrum for a GaMnAs QW at 5 K and 0 T. The free exciton emission lines in the QW and bulk are indicated by X. Neutral Mn acceptor bound 0 exciton emission is labeled (AMn, X) observed in bulk and QWs. Neutral donor bound exciton emission in bulk GaAs is labeled (D0, X). Schematic band edge diagrams show the origin of these emission lines.

178 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

6.8.2 Polarization-resolved PL

The polarization-resolved PL is measured in the Faraday geometry (chapter 2.3) with PL collected normal to the sample surface. The linearly-polarized excitation laser is focused to a spot 100 µm in diameter with an energy set above the QW free exciton absorption edge. As seen in Fig. 6.26, increasing the Mn concentration leads to a PL that broadens, red shifts, and decreases in intensity, eventually quenching. The decreasing intensity of the PL is similar to the degradation in KR and FR signal with Mn-doping. The loss of these two optical signals with distinct physical origins, reflects the increasing density of crystalline defects with Mn-doping. QWs with no Mn or very small Mn doping levels have emission spectra well fit by two Gaussians. Since the narrower and higher-energy Gaussian peak tracks the magnetic field dependence expected for the Zeeman splitting in QWs, it is identified with the free exciton emission due to heavy hole and electron recombination (Fig. 6.25). The wider and lower-energy emission line does not occur in the InGaMnAs QWs, but only in GaMnAs wells. We have preliminarily identified this peak as due to donor-bound exciton emission from shallow 0 donors in the QWs, DMn. Mn interstitials (Mni) are expected in our samples, which act as double donors (section 6.4). The emission line width of this peak increases as the calculated Mni concentration increases, and it even appears in some non-magnetic GaMnAs QW samples grown with a cold Mn cell (Fig. 15 −3 6.26), perhaps due to an impurity level of Mni (≤ 10 cm ). The PL polarization (P), defined in equation 2.5, is measured as a function of emission energy at various magnetic fields for a GaMnAs quantum well with d = 7.5 nm. Near the QW band edge, two polarization peaks from the free 0 exciton emission (X) and Mn interstitials bound exciton emission (DMn,X) are

179 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Emission energy (eV) 1.55 1.56 1.57 1.58 1.59 A, 7.5-nm 1k x ~ 0.007% B = 0 T x = 0% T = 5 K x ~ 0.05% PL (counts) 100

σ+, σ- 2k A, 7.5-nm C, 10-nm 1k 1k x = 0 x = 0

0 0 3k

PL (counts) 1k 2k x ~ 0.007% x ~ 0.013% ×10 1k 0 0 1.561.57 1.58 1.531.54 1.55 1.56 B = 8, W/cm2 10k B, 3-nm D, 5-nm 1k 5k x = 0 x = 0 0 0

PL (counts) x ~ 0.0025% 500 x ~ 0.004% 1k

0 0 1.66 1.681.70 1.72 1.60 1.62 1.64 Emission energy (eV) Emission energy (eV)

Figure 6.26: Emission spectra for GaMnAs QWs (series A). Pump power is 1 W/cm2 at 1.722 eV. Polarization-resolved PL for GaMnAs QWs (series A-D) at two different values of x (T = 5) K and at a magnetic ﬁeld of 8 T in the Faraday geometry. 2-Gaussian ﬁts to the spectra are shown as plotted as lines.

180 Chapter 6 III,Mn-V quantum wells: band and spin-engineering observed, Fig. 6.27a. The polarization has opposite sign for the two peaks. In the same sample we observe the neutral Mn acceptor bound exciton emission in bulk GaAs [143], which arises in this sample due to unintentional Mn doping in the GaAs buffer layers grown above the starting substrate. The polarization of this bulk Mn emission line shows a paramagnetic (Brillouin function) magnetic

field dependence, Fig. 6.27d, which tracks the magnetization of the MnGa acceptors in bulk GaAs. The polarization of the PL for the low-energy peak in the QW PL polarization coincides with the low-energy PL peak which we assigned to emission from Mni donor bound excitons in Fig. 6.26. Its polarization is plotted in Fig. 6.27c (solid points) and shows a similar paramagnetic (Brillouin function) behavior with field and temperature to that of the bulk Mn-acceptor line. 0 The polarization of the bulk (AMn,X) line is proportional to the spin polarization of local Mn2+ moments since the PL from this line results from conduction band electrons recombining with holes trapped on MnGa acceptors. The spin of 3 2+ 5 these j = 2 holes is coupled to the local Mn S = 2 antiferromagnetically via 0 the p − d exchange interaction [176]. Similarly, the (DMn, X) emission in the QW PL should occur due to recombination between holes in the valence subband of the QW and electrons bound (and exchange coupled) to Mni donors. The clear Brillouin-like field dependence indicates that the recombinant polarization originates around isolated paramagnetic Mn impurities in the lattice, i.e. either MnGa acceptors or Mni donors. The Brillouin-like behavior is not consistent with coupled Mn centers such as interstitial-substitutional pairs, which couple antiferromagnetically and would therefore exhibit a hard axis (linear magnetic field dependence) up to very high fields. These centers are unlikely to be present in samples due to their low Mn density [150]. The ∼ 20 meV redshift of the polarization peak from the main QW peak does not match the 110 meV

181 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

40 40 0 (a) (DMn ,X) (A 0,X) (X) (b) Mn +B 30 +B 30 QW Bulk -B 20 20

10 10 (%)

0 0 P -10 -10

-20 -20 +B -30 -B -30 -B

1.521.53 1.54 1.55 1.56 1.57 1.381.39 1.40 1.41 1.42 1.43

E (eV) E (eV)

20 30 (c) (d) 20 10 10

(%) 0 0 1.85 K P -10 5 K -10 -20 10 K -30 20 K -20 -8 -6 -4 -2 02468 -8 -6 -4 -2 0246 8 B (T) B (T)

Figure 6.27: PL polarization spectra of GaMnAs QWs and bulk (pump at 1.72 eV with 1.0 0 mW). (a) The QW exciton (X) and neutral Mn-donor bound exciton (DMn, X) emission from Mn interstitials. Data are from sample 040714A (series A). (b) The neutral Mn-acceptor bound 0 exciton emission line (AMn, X) of bulk GaAs.(c) PL polarization as a function of B for the 0 higher energy (X) emission line (black line) and for the lower energy (DMn, X) emission line 0 (closed symbols), and (d) for the bulk (AMn, X) emission peak.

182 Chapter 6 III,Mn-V quantum wells: band and spin-engineering binding energy of the substitutional acceptor. While there is an excited state of the MnGa acceptor with a binding energy of 26 meV [177], recombination from this excited state is unlikely when each acceptor is ﬁlled with, at most, one hole and given that PL usually originates from the lowest available energy levels.

Rather, we assume that the recombination originates from the Mni donor and the valence band in the QW. Comparison to the experimentally measured Mni donor binding energy is not possible since none are reported. The postulated coupling of the electron spin to Mni spin results in polarized emission which follows the magnetization of Mni within the QW. These measurements open the possibility of indirectly measuring the magnetization of the Mn impurities in the QWs using polarization-resolved magneto-PL. As observed in Fig. 6.25, emission from the Mn acceptors is observable in GaMnAs quantum wells. The polarization of this center also reﬂects the spin polarization of the Mn-centers within the wells. As previously described, the high-energy feature in the PL is due to excitons bound within the wells, but delocalized relative to the Mn states. Due to the exchange interactions in both the valence and conduction band, the exchange enhanced Zeeman effect should result in a polarization with a Brillouin functional dependence, equations 3.6 and 3.16. For the small values of x studied here, however, these effects are not resolvable and the polarization shows a weak, linear ﬁeld dependence Fig. 6.27c (line) with an opposite sign compared to the polarization of the low-energy peak (points).

6.8.3 Total excitonic Zeeman splitting

Figure 6.26 plots the polarization-resolved PL spectra at two different helicities for a number of GaMnAs quantum wells in a ﬁxed magnetic ﬁeld. The split-

183 Chapter 6 III,Mn-V quantum wells: band and spin-engineering ting of the QW exciton emission line with magnetic field, shown in this figure, constitutes a measurement of the (giant) Zeeman shift, defined in Fig. 3.2 and equation 2.4. A full magnetic field dependence of this exciton spin splitting, therefore allows us to extract the exchange parameter N0(α − β). Since we already have measured N0α by time-resolved KR and FR, then we now have a method to determine N0β.

The magnetic ﬁeld dependence of ∆Eex attained by polarization-resolved PL is plotted in Fig. 6.28 for two different GaMnAs quantum wells. In the control samples with x = 0 (black circles), ∆Eex depends linearly on ﬁeld for

B < 2T. The slope gives the out-of-plane heavy hole exciton g-factor (gex). The extracted values of gex agree within the experimental error with previously published values [50]. At higher fields, ∆Eex deviates from linearity. For the magnetically doped samples (red circles), fits to equation 2.4 are performed with only one fit parameter, N0β. Here we assumed gex from the control samples measurements, while hSzi and N0α are assumed from the time- resolved KR and FR data analyses, section 6.5. Fits become unreliable at high magnetic fields in all samples. Strangely we find widely differing values of

N0β = −0.85 ± 0.38, −2.9 ± 1.5, +24.5 ± 1.8, and +4.3 ± 0.4 eV for QWs with d =10, 7.5, 5.0, and 3.0 nm, respectively. This huge disagreement between samples indicates an incompleteness of our model for the valence band; the mixing of valence band states may be contributing to the problematic extraction of the p−d exchange coupling especially for small d [50]. Simply put, our assumption that ∆EPL = ∆Eex is probably invalid. Previous measurements in bulk GaMnAs provide little guidance with one report suggesting positive p − d exchange for low x (paramagnetic) [160], and others ﬁnding negative p − d exchange for much larger x (ferromagnetic) [94, 178, 179]. In the last section of this chapter (section 6.9) we discuss polarization-resolved absorption measure-

184 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0.5 5-nm (041122G)

(meV) 0

PL E

Δ -0.5

-1

0.25 3-nm (040830A)

(meV) 0

PL E Δ -0.25

-10-5 0 5 10

Bz (T)

Figure 6.28: Spin splitting in GaMnAs QWs measured from polarization-resolved PL. ∆EPL as function of Bz at T = 5 K is plotted for GaMnAs QWs with y = 40%. Data from the control samples (black symbols) and Mn-doped samples (red symbols) are plotted. Fits to equation

3.16 are plotted as solid lines, where we assumed that ∆EPL = ∆Eex.

185 Chapter 6 III,Mn-V quantum wells: band and spin-engineering ments that directly provide ∆Eex. From these experiments we ﬁnd a reliable value of N0β.

6.8.4 Hanle effect

0 The QW emission line arising from (AMn,X) provides optical access to the MnGa 0 spin state. The polarization state of the (AMn,X) line reflects the spin orientation of the electron in the conduction subband of the QW as well as the hole bound to the Mn-defect. In the neutral Mn acceptor configuration (Fig. 3.3) the Mn bound hole is strongly antiferromagnetically coupled to Mn. In this section we 0 describe Hanle effect measurements of the (AMn,X) emission line and argue that they reflect the time dynamics of Mn-ion spins within the quantum wells. PL-polarization is measured in the Voigt geometry with the σ + polarized excitation laser set at an energy above the QW absorption edge and the spec- 0 trometer detector fixed at the (AMn,X) emission line. Figure 6.29 plots the pump energy dependence of the PL (a) and polarization (b) at zero magnetic field. The 0 intensity of (AMn,X) emission drops precipitously at the absorption edge of the 0 QW. This reflects the localized nature of the (AMn,X) centers. The polarization 0 of the (AMn,X) increases continuously as the energy is brought close to the QW 0 absorption edge. We interpret this as evidence that spin is injected into the AMn centers from the optically injected hole spins. We hypothesize the following process to explain optical spin-injection into 0 + the conduction subband and the AMn centers. With a (σ ) laser energy above the absorption edge of the QW, spin-down electrons are injected into the conduction band and spin-up heavy holes are injected into the valence band. The 0 heavy hole spins relax into the stable AMn configuration, where they couple antiferromagnetically to the Mn-ion spins, thus imparting some of their angular

186 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

Detect (A 0,X) QW (X) QW (X) Mn 12 4 (a) (c) 8 x ~ 0.010% 2 4 x ~ 0.003% x ~ 0.003% 0 0

(b) (d) 12 8 10 (%) PL (arb. units)

P 8 6 6 4 2 1.540 1.545 1.550 1.555 1.560 4 024681012 Pump energy (eV) Pump power (mW)

Figure 6.29: Mn-ion PL polarization versus pump energy and power in zero magnetic ﬁeld. The PL and polarization of the neutral Mn acceptor bound exciton emission is detected in GaM- nAs QWs with d = 10 nm and y = 20% using a σ + polarized excitation. (a) PL intensity and (b) polarization of the Mn-emission as a function of pump energy for two QWs of different Mn densities, as labeled. Dashed line indicate the position of the QW free exciton emission serving as a marker for the exciton absorption edge of the QW. (c) PL intensity and (d) polarization of the Mn-emission as a function of pump power.

187 Chapter 6 III,Mn-V quantum wells: band and spin-engineering momentum to the Mn spins. The spin-up electrons and spin-down holes (cou- + 0 pled to Mn spin) recombine emitting a σ photon with an energy at the (AMn,X) emission line. Figure 6.29c and d plot the laser power dependence of the PL and polar- 0 ization. While the polarization of the (AMn,X) emission remains constant at all laser powers, the PL increases linearly with power. This indicates that over the power range studied, saturation of the defects states does not occur. 0 Since spin-injection of the (AMn,X) center is possible at zero magnetic field, we can employ the Hanle technique to measure time-average spin dynamics of this complex (chapter 2.3.3). Figure 6.30 plots the magnetic field dependence of the PL polarization for a GaMnAs quantum well measured in the Voigt geom- 0 etry at the quantum well exciton emission line (a) and at the (AMn,X) emission line (b). In both cases a characteristic Lorentzian field dependence indicates time-average spin precession. In Fig. 6.30a, the polarization of the quantum well (X) emission decreases over a large ∼ 1 T magnetic field range. Time-resolved KR measurements of this QW at this energy indicate that hole spins decohere rapidly. We fit the data to equation 2.7 using ge as extracted from the time-resolved KR measurements ∗ and we find a spin lifetime of ∼ 140 ps which closely agrees with T2 extracted from KR data of electron spin precession. 0 Figure 6.30b plots the Hanle curve of the (AMn,X) center, which is well fit by a Lorentzian that is much narrower in field than in Fig. 6.30a. Since the 1 width of the Lorentzian is proportional to ∗ , we must first know the g-factor gT2 ∗ before we can extract T2 . The g-factor of the electron should not be used since we know already that the electron spin lifetime is too short (by a factor of > 10) to account for this narrow a Hanle curve. The g-factor of the a valence band hole should not be used since we also know this lifetime to be even shorter than

188 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

15 Pump (σ+) 1.653 eV, (a) g = 0.2651 Detect 1.540 eV, e (TRKR) 10 QW (X)

(%) e-spin P 5

* ± 0 e, T2 = 143 6ps -1.0-0.5 0.00.5 1.0

Bx (T)

5 Pump (σ+) 1.554 eV, (b) Detect 1.440 eV, gMn = 2.0 4 0 QW (AMn , X)

(%) 3 Mn-spin P

* ± 1 Mn, T2 = 2,700 200 ps -0.01 0.00 0.01

Bx (T)

Figure 6.30: Hanle measurement of Mn-ion spin dynamics in GaMnAs QWs with y = 20%. (a) Transverse magnetic field dependence of the polarization of the QW free exciton emission. ∗ Red line is a fit to equation 2.7. T2 is calculated from this fit assuming ge as determined by time- resolved KR measurements of electron spin precession. (b) Polarization of the QW neutral Mn acceptor bound exciton emission as a function of transverse magnetic field. Red line is a fit to ∗ equation 2.7. T2 of Mn-ion spin is calculated from this fit assuming gMn = 2.

189 Chapter 6 III,Mn-V quantum wells: band and spin-engineering the electron spin lifetime. The spin that directly accounts for the observed PL polarization is the loosely bound hole that is coupled to the Mn-acceptor within 0 the AMn defect. However the spin lifetime of this hole is also expected to be short since it is strongly exchange coupled. Finally we can consider that the Hanle curve in Fig. 6.30b reflects the spin dynamics of the Mn-ion spin itself. In this case, the spin state of the hole, loosely bound to the Mn, simply acts as a compass needle indicating the direction which the Mn-ion spin is pointing. Thus the Hanle curve reflects the dynamics of the Mn-ion spin of which the PL polarization is an indirect measure. Recall our hypothesis regarding spin- 0 injection in which valence band hole spins relax into the AMn state where they exchange couple (p − d) to the Mn. In this process the Mn-ion spins become temporarily polarized along the optical axis, which in the Voigt geometry sets up a Mn-ion spin precession. Based on the above argument, we assume that g = gMn ∼ 2.0 yielding a spin lifetime of 2.7 ns. A series of GaMnAs quantum wells are prepared with various Mn doping levels. The spin lifetime of electrons and Mn-ion spins are measured as in Fig. 6.30, and plotted as a function of x in Fig. 6.31. Over the entire doping range the electron spin lifetime remains constant, and values extracted from time-resolved KR and Hanle measurements are in close agreement. For 0 < x < 0.01%, the Mn-ion spin lifetime demonstrates a dramatic dependence on Mn concentration; the spin lifetime decreases exponentially with Mn doping. At the lowest Mn doping densities, the PL intensity and polarization of the 0 (AMn,X) center becomes weak, making Hanle measurement difficult to perform. Despite this problem, Mn spin lifetimes approaching 10 ns have been observed in the lowest Mn-doped GaMnAs quantum wells. To our knowledge these data represent the first measurements of Mn-ion spin lifetimes in GaMnAs.

190 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

0 10 (AMn , X) (X) KR (ns)

* 1 2 T

0.1

0.00 0.01 0.02 0.03 Mn concentration, x (%)

Figure 6.31: Mn-ion and electron spin lifetime versus Mn doping density in GaMnAs QWs with y = 20%. Hanle measurements of Mn-ion (black) and electron spin (red) lifetimes are extracted from ﬁts as shown in Fig. 6.30. Electron spin lifetimes measured by time-resolved KR measurements are also plotted (blue).

6.9 Absorption: the 100-quantum wells

On a ﬁnal experimental note, we describe initial efforts to measure the p−d exchange constant via magneto-optical absorption. Recall that measurements of the spin splitting of the QW exciton peak yielded strange and widely varying estimates of N0β. This error was attributed to the sensitivity of PL to complexities in the valence band, making our assumption that ∆EPL = ∆Eex incorrect. With absorption the situation becomes easy to interpret since the exciton absorption peak in the quantum well is easily identiﬁable, and from which the exciton spin splitting (∆Eex) is directly observable. Unfortunately, measuring absorption in a single GaMnAs quantum well poses an incredible challenge. Since absorption is proportional to thickness, a 10-nm quantum well poses an remarkably thin cross section. Our sledgeham- mer solution (far from elegant) is to simply include multiple quantum wells,

191 Chapter 6 III,Mn-V quantum wells: band and spin-engineering in this case 100 GaAs quantum wells with d = 8.5 nm are grown with 20-nm

Al0.4Ga0.6As spacers using our usual MBE methods. The only difficulty is to maintain stable growth conditions over a much longer deposition time. Magneto-optical absorption measurements of such samples were performed. Figure 6.32a plots the absorption spectra as a function of longitudinal field for the Mn-doped sample (x = 0.00104 as measured by SIMS). Comparing data from the two helicities a clear spin splitting is observed. The ground state heavy hole exciton (hh1) is easily identifiable. Such data are fit to a Gaussian to extract the absorption energy as a function of magnetic field, plotted in Fig. 6.32b.

The difference of these two curves corresponds directly to ∆Eex and can be ﬁt to equation 3.16. The heavy hole exciton Zeeman splitting ∆Eex = −gexµBB, where gex is the exciton g-factor, is found from the same measurements carried out on the undoped control sample. Fits to the exciton exchange splitting give xN0(α − β) = 1.48 meV. Using our previous time-resolved KR measurements of N0α, allows us to estimate N0β = −1.5 ± 0.5 eV, agreeing in sign and magnitude with the accepted literature values for MBE-grown GaMnAs [94, 180].

6.10 Summary and outlook

The MBE growth of III,Mn-V based heterostructures in which coherent electron spin dynamics and PL can be observed is demonstrated. The main innovation is the use of intermediate temperature near 400◦C, where As-antisites are not included, but enough Mn can be incorporated to observed exchange effects. At these growth conditions the fraction of substitutional Mn incorporated is estimated at 70-90%. Time-resolved measurements of the electron spin precession allow a high precision method to determine the spin splitting in the conduction band of

192 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

1.68 σ+ (a) σ-

5 Abs. (arb. units) 1.64

1.60 hh1

Energy (eV) 1.56 0 1.52

-50055 -5 Bz (T) Bz (T)

1.558 (b) (c) hh1 4 hh1 σ+ σ- 1.556 2

(meV) 0 ex

1.554 E Δ -2

Abs. energy (eV) 4 1.552 -50 5 -50 5 Bz (T) Bz (T)

Figure 6.32: Total excitonic spin splitting in GaMnAs QWs measured from polarization- resolved absorption at T = 5 K. (a) The longitudinal magnetic field dependence of the absorption spectrum of 100 QWs of GaMnAs with d = 8.5 nm and y = 40%. (b) Gaussian fits of the absorption lines in (a) are plotted for the ground state heavy hole exciton transition (hh1) identified by a grey box. (c) The difference in energy of the data in (b), ∆Eex = Eσ + − Eσ − , is plotted as a function of longitudinal field (black symbols). Fits to equation 3.16 are plotted as a solid line.

193 Chapter 6 III,Mn-V quantum wells: band and spin-engineering

III,Mn-V quantum wells. The magnitude and sign of the s − d exchange parameter is accurately determined by systematic variation of the Mn doping density, magnetic field, and sample temperature. For all III,Mn-V quantum wells measured thus far, the s − d exchange coupling is observed to be antiferromagnetic, the first such observation in a magnetic semiconductor. This effect is due to quantum confinement induced changes in the electron wave function symmetry. By varying the well width and barrier height we found a roughly linear dependence of the exchange coupling on the confinement energy. A model of confinement-induced kinetic exchange fits the slope with no adjustable fitting parameters. The accuracy of the fit is remarkable, attesting to the universality of the model put forth by Merkulov et al. for confinement induced kinetic exchange in II-VI magnetic quantum wells [126]. Electron spin lifetimes in the QWs increase for the lowest Mn dopings compared with undoped samples indicating the dominance of the D’Yakonov-Perel mechanism over spin-flip scattering in this regime. Variation in the quantum well parameters, low Mn-doping and barrier height, reveal anomalous regions of very long electron spin lifetimes (up to 1 ns) in GaMnAs and InGaMnAs quantum wells, which could be used in future experiments for coherent spin control within the electron spin lifetime. Polarization-resolved PL reveals the complexity of the Mn defect structure in the quantum well structures. We observe the spin polarization of electrons, holes, and Mn spin through the emission spectra. These techniques allowed an estimate of the p − d exchange parameter, but are hampered by the complexity of the valence subband in the quantum wells. Future PL measurements should be carried out on InGaMnAs quantum wells. The pre-existing sample sets are already well characterized by SIMS and time-resolved FR. It is expected that the additional strain modification of the valence band would make the PL data more easily interpretable. The observation of time-averaged Mn-ion spin precession

194 Chapter 6 III,Mn-V quantum wells: band and spin-engineering is perhaps the most interesting of our PL experiments. Hanle measurements of the Mn-related PL emission within the quantum well reveal extremely long spin lifetimes. More dramatically, the Mn spin lifetimes decrease exponentially with Mn doping density. Polarization-resolved absorption spectroscopy of multiple GaMnAs quantum wells yield an accurate measure of the p − d exchange parameter. System- atic measurements of such wells is lacking and should be the focus of future work. Multiple InGaMnAs quantum wells offer the advantage that they can be measured in transmission, which reduces the difﬁculty in acquiring an absorption spectrum, however such structures would require careful heterostructure design to avoid unintentional strain relaxation at the quantum well interfaces. We attempted to induce ferromagnetic interactions between the Mn-ions by modulation doping holes or electrons in the barriers of GaMnAs and InGaM- nAs quantum wells. Preliminarily, it appears that it is not possible (using our current techniques) to add sufﬁcient Mn-doping or modulation doping to create ferromagnetic interactions while retaining a clear KR or FR signal. One totally unexplored possibility is to abandon the requirement of good optical signals. Samples could instead be characterized by SQUID or transport measurements as are most III-V magnetic semiconductors. Without the requirement of high optical quality, samples can be grown at lower substrate temperatures at which high Mn-doping levels are possible. Using techniques such as ALE [40] or non-rotated growth (appendix B.1), the low temperature defects (As-antisites) can be minimized.

195 Chapter 7

Conclusions

We have developed magnetic quantum well structures using advanced semiconductor growth techniques. These structures were developed to demonstrate the carrier spin polarization and manipulation in the solid state using the exchange interaction occurring between free carrier and magnetic ion spins. Compared with weaker spin-orbit effects occurring in non-magnetic semiconductors, the strong exchange interactions offer one pathway for robust spintronics at practical magnetic fields and temperatures. In all our experiments, polarization- resolved optical spectroscopies (absorption and emission processes) were used to monitor the time-resolved and time-averaged spin dynamics of electrons, holes, and magnetic ions. The time-resolved spin dynamics offers a particularly rich look into the time and energy landscapes of quantum confined spins and their interactions with local magnetic moments. In magnetic quantum well structures, electrons and holes are confined at nanometer length scales along one direction and forced to interact with magnetic ions doped within or nearby these wells. In all structures examined, the strong exchange interactions existing within magnetic semiconductors couples the spin-state of the carriers confined within the quantum wells to the spin-state

196 Chapter 7 Conclusions of the magnetic dopants. In addition, II-VI and III-V magnetic semiconductor systems allow for both paramagnetic and ferromagnetic long-range interactions between the magnetic ions themselves. Electrons or holes confined within these quantum wells can be spatially shifted along one direction using vertical gates. By parabolically-grading quantum wells using band engineering, the effect of vertical gating is optimized. The controllable interaction between carrier spins and paramagnetic moments deposited within these II-VI wells allows for a tunable electron spin precession frequency orders of magnitude larger than is possible in analogous non-magnetic structures. In III-V semiconductors, magnetic dopants can interact ferromagnetically allowing for a fully saturated magnetization at low magnetic fields. Unlike in II- VI magnetic semiconductors, transition metal doping in III-V’s is more difficult to accomplish. Quantum wells containing digital submonolayer ferromagnets within the barriers were developed in which the optical emission signal of the quantum wells is maintained. The spin polarization of holes within the wells track the magnetization of the ferromagnet. A vertical bias is used to adjust the overlap integral, and thereby tune the extrinsic exchange coupling between hole and magnetic ion spins. Finally, III,Mn-V magnetic quantum wells were developed in which optical emission and time-resolved absorption signals are maintained. In such structures electron spin dynamics in the technologically important class of III- V magnetic semiconductors were observed for the first time. These structures represent a compromise between magnetic and optical performance in which optical defects are eliminated at the cost of a decreased solubility limit for magnetic dopants. Despite the technologically uninteresting paramagnetism of these structures, the high precision observations of electron spin precession allowed a new method to more directly observe exchange interactions in these materials.

197 Chapter 7 Conclusions

A full and systematic study of the s − d (conduction band) exchange parameter between electrons and magnetic ions was carried out in which quantum well width, barrier height, doping density, and band gap were varied. In all samples a surprisingly antiferromagnetic s − d exchange interaction was observed, unlike all other known magnetic semiconductors to date. Quantum confinement was used to smoothly increase the absolute value of the exchange coupling, and this dependence is well fit by a model of kinetic exchange due to confinement- induced band mixing. In this case, confinement energy allows for an intrinsic control of the exchange mechanism. Initial absorption spectroscopy yields precise data on the p − d exchange coupling between holes in the valence subband and magnetic ions. Lastly, optical emission measurements in these structures reveal a rich defect structure of the magnetic ions. These experiments reveal optical spin-injection, not only of electron spin, but also of hole and magnetic- ion spin, probably mediated by the p − d coupling. Time-average measurements of the spin precession of magnetic defect centers allows a first look into the dynamics of magnetic ions within these III,Mn-V quantum wells already displaying long lifetimes and a high sensitivity to doping. The above list of conclusions may lead the reader to wonder if magnetic quantum wells could be practicably implemented in any spintronics devices, given that the spin polarization and manipulations demonstrated above were performed at low temperature and, usually, high magnetic fields. One reply is to note that while these structures may not be very practical, they do represent excellent laboratories in which to study the exchange couplings of free carrier spins and magnetic moments. Improvements in our understanding of the origin and mechanism of the exchange interaction may guide future materials devel- opments. Additionally, control over the time dynamics of spins may lead to fundamentally new spintronics based devices. A second reply to this pessimism

198 Chapter 7 Conclusions is to point out that we have compromised on magnetic (and therefore temperature) performance to maintain very good optical performance. It is possible to simply abandon the requirement of good band-edge optical properties and explore spintronic effects through magnetometry and/or transport experiments. From this perspective, III,Mn-V quantum wells offer great promise; it was in quantum conﬁned GaMnAs, not bulk, that the current record Curie temperature has been observed [181].

199 Appendix A

Molecular Beam Epitaxy Techniques

This appendix will hopefully offer a technical details to current and future crystal growers in the spintronics MBE system. Here the reader is cast into an on- slaught of acronyms. UHV, MBE, RHEED, XRD, LT, HT, BET, and etc... But once past this barrage of mumbo-jumbo we will ﬁnd a calm even in the cacoph- onous, occasionally hot, and always sweaty world of the MBE lab. Out of the din of cryopump compressors that creak, high-power chillers that leak, electrical fans, and humming power supplies, a rhythmic pattern of electro-pneumatic clicks emerges. Somewhere shutters are opening and closing, somewhere a crystal is growing. We pass from lab to lab, and notice, ahh, the quick trills of a short-period superlattice, the arrhythmic pattern of a parabolic well, the steady drum roll of a DBR. This is the joyful Spring time. Far from our memories is the Winter of system openings: the sweaty work in a Tyvec suits, and a respirator digging into your face, and the sound of metal tools scraping toxic ﬂakes from a chamber wall, and a man named English swearing as a shutter blade falls into the sump.

200 Appendix A Molecular Beam Epitaxy Techniques

This section discusses technical details on the MBE system known as the “spintronics”, or magnetic semiconductor chamber. First we review some of the history of the ﬁrst-age of spintronics MBE at UCSB, then discuss the acquisition and installation of a new MBE system. Some details on system components and operation in the new chamber are discussed, such as the in-situ temperature monitoring, As-ﬂux gradient techniques, growth rate calibration by RHEED oscillations on different materials (GaAs, AlAs, InGaAs, and MnAs), x-ray diffraction calibration of growth rates, defect and mobility calibration samples. Additional details are available in the MBE notebooks and system logs (both paper and computerized).

A.1 A celebration of history

I will give a brief discussion of the history of the spintronics systems on campus, however I offer the disclaimer that some of this information may be inaccurate, but subjectively precise. Despite a contamination problem, an amazing amount of useful samples were cranked out by then post-doc Roland Kawakami (now Prof. at UC River- side), grad student Ezekiel “Zeke” Johnston-Halperin (now Prof. Ohio State), who initially got the system up and running and learned how to grow GaMnAs. They developed digital ferromagnetic heterostructures (DFH), discussed in previous chapters, some of the ﬁrst spin-LED structures, and samples in which ferromagnetic proximity polarization was ﬁrst measured, along with the more traditional random alloy of GaMnAs, to name a few of the more successful projects. The limitation of the old spintronics system was a problem of contamination. It was acquired from Phillips research and was used to grow II-VI semi-

201 Appendix A Molecular Beam Epitaxy Techniques conductors, and therefore contained elements such as Zn, Cd, Se, and S. Un- fortunately sulfur has an unusually impractical vapor pressure (which also resulted in an unhealthy stink whenever the entry exit chamber was vented), and therefore the Arsenide samples grown in this system suffered from background sulfur contamination on the order of 1016 cm−3 (this figure was measured from unintentionally doped samples (UID) which we sent to Charles Evans and As- sociates (CEA) for secondary ion mass spectroscopy (SIMS) analysis. Electri- cally, this led a n-type conduction in UID samples (sulfur is a shallow donor in GaAs). Optical performance in the old system was poor: AlGaAs-GaAs quantum wells did not show PL, although low temperature PL and TRFR signal could be observed in InGaAs-GaAs quantum wells. A decision was made to purchase a newer MBE system that did not have any background contamination, when research interests moved toward direct doping of magnetic impurities in high optical quality heterostructures, such as the structures discussed in this thesis. Andrew Jackson located an opportunity at a telecom start-up company in Fremont, CA ( east bay, silicon valley) called Bandwidth 9. This company was selling several Veeco (formerly Applied-EPI) Gen-II MBE systems. The system we acquired had been used only for Arsenide growths: GaAs, AlAs, Be, and Si. The system was manufactured in 2000 and had only been used for three growth campaigns. Bandwidth 9 had recently opened the chamber, removing the entire source flange and paying Veeco to clean the chamber as well as replace all the shutters. We packed the system in October of 2003 and had it installed and growing at UCSB by November 2003.The fast turn around was facilitated by the near perfect condition of the system. A series of system modifications had to be made, redesign of the quartz rod (light pipe) which feeds through the CAR substrate heater assembly allowing for white light

202 Appendix A Molecular Beam Epitaxy Techniques transmission through the wafer. Modifications to the CAR took several quick openings. The final bake was performed over winter break and in early 2004 the first samples began to emerge from the system. Figure A.1 gives a pictorial history of the removal of the old system and acquisition and installation of the new system.

A.2 Initial operation

After the uncontaminated system was installed, we proceeded to contaminate it ourselves by installing both a Mn and Cr cell. The original (and current at time of writing) UCSB source conﬁguration is as follows: Si (10cc high temp doping cell), Al (400g downward SUMO), Ga (400g SUMO), In (400g SUMO), Mn (60cc Dual heater Cone), Cr (10cc high temp doping cell), Be (10cc high temp doping cell), As (valved/cracker sublimator). A 2DEG calibration sample was grown before heating Mn or Cr (µ ∼ 500,000 cm2/Vs). After heating Mn, we observed no decrease in mobility, in fact the mobility increased (probably due to the usual reduction of background impurities as the system remains under vacuum). In this initial period we also discovered a problem of Cr and crucible material. Heating the Cr cell resulted in a sudden increase in the background pressure, which appeared to be the usual outgassing of new source material. The RGA showed that this was mainly nitrogen gas (peaks at 28 and 14). Unlike normal outgassing, the pressure did not decrease exponentially with time and instead increased with cell temperature. The Cr cell was kept cold until the next opening. Using the MBE lab baking chamber, we determined that the problem was due to a catalytic reaction between Cr and PBN-type crucible resulting in the decomposition of PBN which evolves nitrogen into the chamber. Using a tung-

203 Appendix A Molecular Beam Epitaxy Techniques

(a) (b)

(c) (d)

(e)

Figure A.1: Pictures of spintronics MBE system moves. (a) Final take-down of the old spintronics MBE system originally from Phillips research. (b) Packing up the new system from Bandwidth 9 in Silicon Valley. (c) Installing and servicing the system at UCSB. (d) Spintronics system under bake, and (e) fully operational.

204 Appendix A Molecular Beam Epitaxy Techniques sten crucible removes the problem. 2DEG cals confirm that the unintentional doping of Cr into non-magnetic samples does not decrease the mobility of the samples. These results corroborate the findings of Wagenhuber et al. [182], since transition metal dopants, such as Mn and Cr, do not restrict our ability to produce high mobility samples. In our system, the record low-temperature mobility is greater than 1.7 × 106 cm2/Vs, which was measured in a typical modulation Si-doped AlGaAs/GaAs two-dimensional electron gas grown at 630◦C (sample 040713A). To avoid contamination of Mn or Cr via the substrate blocks into the non- magnetically doped samples, several blocks are reserved for non-magnetic growths only. In contrast to previous procedures used on the old spintronics system, we do not clean the magnetic blocks after each growth. It is possible that this results in a Mn or Cr contamination in non-magnetically doped parts of samples grown on these blocks via surface diffusion. However, in our studies of magnetic QWs (chapter 6) we observed that the unintentional Mn density is ∼ 1015 cm−3, too low to pose a significant problem for most experiments.

A.3 Knudsen cells

The spintronics chamber has 3 effusion cells incorporating the SUMO style crucible with dual heating zones: the In, Ga, and Al (downward looking) cells. The Mn cell is a cone-type crucible with dual heating zones. The Cr cell is a 10cc high temperature doping cell, identical to the Si and Be doping cells. The In cell is operated with the tip at 100◦ C above the base temperature. The Ga tip is operated at 200◦ C above the base temperature. This minimizes splattering defects. The Al cell is only heated using the base; heating the Al tip

205 Appendix A Molecular Beam Epitaxy Techniques above the base temperature would wick the Al source material out of the cell and short out the effusion cell. The Mn tip is operated 100◦ C above the base temperature. No systematic tests of Mn or In tip temperatures versus defect densities have been carried out yet. The As valved/cracker sublimator has had a couple problems. The ﬁrst problem was the failure of the cracking zone thermocouple T/C; it no longer reads accurate temperatures. Before it failed we recorded two constant power values at which the temperature is known. A second problem is a leak that developed at the nipple connecting the cracking zone to the effusion cell port. This is due to the large torque on this joint. This leak disappeared after a jury-rigged seal was attached to the port. Both problems should be addressed in the next system opening.

A.4 Band edge thermometry

The new spintronics system uses absorption band edge spectroscopy (ABES) to monitor the substrate temperature. The transmission spectrum of the wafer is measured by white light spectroscopy. A 400W halogen bulb is fiber coupled and directed through a mini viewport on the CAR On the chamber side the viewport couples to a fiber bundle that couples to a quartz rod (light pipe) carefully installed through the substrate heater. Due to modifications made to the CAR, the VEECO designed quartz rod did not fit. A custom rod was made using the UCSB glass shop (Chemistry Dept.). Additionally, a small window was cut into the ion gauge shield, exposing the glass rod to As deposition that absorbs IR. The ion gauge shield was fitted with a foil shield, and the quartz rod was coated with gold. Coating with gold reduces the transmission efficiency since IR is reflected by the gold layer and not by total internal reflection at the

206 Appendix A Molecular Beam Epitaxy Techniques glass interface. Backing wafer free wafer mounts provide the best ABES signal, although quartz backing wafers may also be used. Double polished wafers are preferred since light scattering at an unpolished surface reduces transmission. A rough schematic of the optical setup is shown in Fig. A.2a. The transmitted light passes through a BOMCO type heated viewport (a.k.a. the pyroport) and is coupled into an optical fiber attached to the detection spectrometer. The detection system uses the Ocean Optics model NIR512 spectrometer connected by USB. This spectrometer results in much faster and reliable data than the older DRS system, which used a monochromator with a rotatable grat- ing. The new spectrometer uses an InGaAs diode array as the detector, thus full spectra are collected as quickly as 25-ms. Labview VI’s are employed to collect and fit the absorption spectra. An example of the band edge data and fit is shown in Fig. A.2b. Since the detector cannot be lock-in detected, background subtraction is critical to attain good fits. VI’s can be corrupted by user error, which has resulted in the past in the temperature readings being incorrect. Always check using an independent method if the temperatures are correct. For example, 2 × 4 to 4 × 4 reconstruction shift on GaAs upon cooling past ∼ 520◦ C, oxide desorption at ∼ 620◦ C. The temperature reading calculated by the VI can then be output through a thermocouple (T/C) cable to the substrate temperature controller. This passes from an analog output card, through a voltage pre-amp, and into a T/C switch box (More details are available from Andy Jackson). This allows real-time feedback on the ABES temperature, providing a typical substrate temperature stability of ±2◦C . The switch box allows one to switch from ABES to regular T/C feedback, Fig. A.2a. An example of a growth temperature history employ- ing ABES feedback is shown in Fig. A.2c. Note that during large temperature swings the PID settings for the Eurotherm temperature controller must be al-

207 Appendix A Molecular Beam Epitaxy Techniques tered as shown in the ﬁgure. The LT PID settings should be implemented as soon as the wafer reaches the desired temperature. If you input the LT PID values too soon, then the substrate temperature will take forever to stabilize. The HT PID values should be input as soon as you begin to heat the substrate away from LT. If you don’t change the PID until it is hot, you risk overshooting the temperature. Do not feedback on the ABES signal unless you have a robust band edge. If the signal is lost for even a few seconds, the substrate heater may uncontrollably heat the wafer and fry your sample; the wafer will turn into liquid Ga as all the arsenic sublimates away. Know your wafer. If the wafer is doped, the ABES signal will be dramatically changed. Electronic doping lowers the band edge, and the temperature calibration must be changed. Since the doping level dramatically changes the band edge temperature dependence, do not use ABES on doped wafers unless you are relatively insensitive to wafer temperature. Also, the wafer will absorb more of the heat from the substrate heater. MnAs containing layers are metallic and heavily absorb throughout the IR spectrum where the GaAs band edge appears. Growing more than 25-nm of MnAs will totally ﬁlter the band edge signal. Also, just as in doped wafers the MnAs layer absorbs the IR from the substrate heater as well. Expect the output power to drop as the thickness of the layer increases in order to maintain a constant substrate temperature.

A.5 Arsenic capping

The arsenic cap is an efﬁcient method for protecting the surface of a sample from oxidation [183]. Oxide layers form immediately on all Arsenides. This is not a problem for many experiments, but can corrupt surface sensitive activities,

208 Appendix A Molecular Beam Epitaxy Techniques

(a) Substrate NIR 512 r PC e USB wafer b i Analog f output

l a c i t p

Heater O Eurotherm Preamp cable T/C HV power

MBE growth ne li T/C chamber er ow switchbox p V T/C sensor H 800 (c) White light input T/C 700 C) (b) 959 nm ° ° 600 ABES Tsub = 214 C 500 LT Temperature ( 400 PID

300 HT PID HT PID Transmission (arb. units) 900 1000 1100 Wavelength (nm) 01234 Time (hours)

Figure A.2: Absorption band-edge spectroscopy (ABES) for in-situ substrate temperature control. (a) Schematic of ABES implementation in the spintronics Gen-II MBE system. (b) Example of the band edge signal from a semi-insulating GaAs wafer (double-side polished). The absorption edge ﬁtted by computer simulation is marked by a red line and the calculated temperature is plotted. (c) Temperature history for a growth with temperature control by ABES feedback (sample 050210A). Red line plots the T/C temperature and black line plots the actual substrate temperature measured by ABES. Dashed lines mark the time when the PID integration value was switched from HT (I = 28.4) to LT (I = 600).

209 Appendix A Molecular Beam Epitaxy Techniques such as STM, or MBE regrowth. As-capping is an easy procedure to perform in an arsenide system. All that is required is that the substrate be cooled down sufficiently for As-dimers or tetramers to stick to the sample surface. Arsenic con- densation will not occur efficiently until the substrate is at below room temperature. The substrate heater should be completely turned off, the cooling from the water lines to the CAR as well as the radiative cooling from the LN2 cryopanel will eventually reduce the sample temperature below room temperature. In our chamber, robust As-capping has been performed by increasing the As-BEP to −5 ∼ 10 torr, slightly above the normal As2 flux necessary for smooth high temperature 1 ML/s GaAs growth. The As deposition for 5 hours should proceed once the substrate T/C reads < 60◦ C. The film is an amorphous elemental arsenic layer and can be observed in the RHEED pattern. The two-dimensional reconstruction pattern will disappear giving way to smooth rings indicating an amorphous surface. After an in-air transfer to another UHV system, the As-cap is removed by heating the substrate to around 300-350◦C at which temperature the As layer desorbs from the surface leaving behind a smooth two-dimensional surface usually displaying 4 × 4 reconstruction pattern in the RHEED. Figure A.3 shows examples of typical RHEED images for several growth conditions and surface qualities. Before As-capping and after desorbing the As-cap a 4 × 4 reconstruction pattern would be observed as in Fig. A.3e and f. After the cap is formed an amorphous pattern would emerge, similar to the polycrystalline pattern in Fig. A.3b except that the rings would be blurred together, i.e. evenly spread in k-space.

210 Appendix A Molecular Beam Epitaxy Techniques

(a)° (b) ° Tsub = 220 C Rough growth, Tsub = 220 C non-2D

3D040607C 040607B Polycrystalline

(c)° (d) ° Tsub = 580 CT2D growth (HT), sub = 580 C As overpressure

90° rotation 2× streaky4× streaky 040901C (e) ° (f) ° Tsub < 400 CTsub < 400 C 90° rotation

Growth complete, cooling 4× streaky4× streaky

(g) ° (h) ° Tsub = 270 CT2D growth (LT) sub = 270 C stoichiometric

90° rotation × × 1 spotty050419D 1 spotty

Figure A.3: Examples of RHEED patterns from several surfaces. In (a) a rough surface produced by low temperature strained growth (sample 040607C, a 0.1ML CrAs/ 20ML GaAs digital superlattice). (b) shows a sample grown with identical conditions but with more strain (sample 040607B 0.5ML CrAs/ 20ML GaAs) leading to a polycrystalline surface. (c) and (d) show the common 2 × 4 reconstruction pattern for GaAs at high temperature in an As-overpressure (sample 040901C, GaAs-AlGaAs QW). Upon cooling the same sample changes to the c4 × 4 pattern as shown in (e) and (f). (g) and (h) show the RHEED patterns typically observed for low temperature grown GaAs based samples in which the As is reduced to stoichiometric level to avoid As-defects. The pattern is spotty yet remains two-dimensional.

211 Appendix A Molecular Beam Epitaxy Techniques

A.6 InGaAs on GaAs epitaxy

We briefly discuss some details of InGaAs on GaAs epitaxy. The samples discussed in this thesis were only thin ∼ 10 nm InGaAs quantum wells. In the case of InxGa1−xAs it is necessary to understand that the lattice constant depends on x: aInAs = 6.0583 Åand aGaAs = 5.6533 Å. From epitaxy 101, we know that at some thickness of strained film, the strain energy will overcome the energy to nucleate misfit dislocations at the interface, the so-called Matthews- Blakeslee critical thickness (h). For epitaxial films this can be roughly estimated 2 as, h = a f /∆a = a f /ε, where a f is the lattice constant of the film and ε is the film strain. For InGaAs with x = 20%, h ∼40 nm. Since strain is a critical part of InGaAs-GaAs epitaxy it is necessary to first design the strain state and thickness of the layer. If you go past the critical thickness you will have misfit disloca- tion at the interface and probably an anisotropic strain state (strain relaxation is usually never isotropic). An excellent introduction is given in Max Andrews’ thesis [184]. As an example, if x=5% and your film is thicker than h ∼160 nm, the film should be partly relaxed (orthorhombic distortion). If you remain below the critical thickness the strain will be perfectly epitaxial (tetrahedrally distorted film). Indium is well-known to only incorporate into GaAs once a critical density of In is deposited on the surface [144]. For this reason sharp InGaAs/GaAs QWs are grown via predeposition of one monolayer of InAs followed by the alloy growth and subsequent re-evaporation of the “In floating layer” [145]. The In incorporation effect will change the rate of the oscillations early on, i.e. the rate will appear to get faster as you grow. To avoid In incorporation effect, you should grow at as low a temperature as possible and still get oscillations. This will give you greatest accuracy in determining the In composition by RHEED

212 Appendix A Molecular Beam Epitaxy Techniques oscillations. For bulk InGaAs films where the strain and composition are very ◦ important I use Tsub = 490 − 500 C. The best quality InGaAs QWs are grown ◦ ◦ at as high a temperature possible Tsub ≤ 520 C. Above 520 C you risk re- evaporation of the In floating layer which decreases the In density and blurs the interfaces. The As overpressure typically used for InGaAs is about 10% smaller than for HT GaAs, i.e. 7E-6 instead of 8E-6 torr. We used RHEED oscillations of InGaAs on GaAs to calibrate the growth rate. As for the case of AlAs or GaAs oscillations, several tricks of the trade are involved, namely (finding the specular spot, fiddling with the RHEED GUN controls, and moving the crystal position). I assume that the reader is somewhat familiar with these tricks. Additional details on RHEED oscillations are given in the appendices of Zeke Johnston-Halperin’s thesis [40]. Since this is performed on a GaAs RHEED sample, the oscillations correspond to strained MLs. We must remember to never exceed the critical thickness with the RHEED oscillations so you must have some idea of the growth rate before you begin. Always deposit plenty of GaAs (3× as much as InGaAs) to restabilize the surface and avoid strain relaxation. Choose your substrate temperature between 520 and 480 C (lower temperature gives better growth rate accuracy) and make sure you are in T/C feedback. Do not use the ABES for temperature control during RHEED oscillations as the program will freeze and change the substrate heater suddenly. Make sure you have the As shutter MANUALLY opened. I do this since if the computer crashes the As shutter will close. Measure the specular spot using a crystal orientation near the reconstruction (should be 4×). Sometimes the specular spot will not be clear but will emerge once you open the shutters. You can test this by first opening the Ga shutter and you should observe GaAs oscillations. Open the Ga shutter first, then the In shutter. The RHEED intensity will first drop well below the original strength and should

213 Appendix A Molecular Beam Epitaxy Techniques begin to oscillate. InGaAs oscillations for x = 20% will decay and become un- observable after only 6-9 periods. Close the In shutter and deposit 3× as much GaAs as InGaAs. Figure A.4 plots several examples of RHEED oscillations including InGaAs on GaAs (part d).

214 Appendix A Molecular Beam Epitaxy Techniques

(a) (b) 140 GaAs AlAs 580 °C 580 °C 1266

120 1200

4 RHEED intensity (arb. units) 02040 2040 Time (s) Time (s) (c) (d)

MnAs In0.2Ga0.8As ° ° 240 C 106 500 C 60

100

RHEED intensity (arb. units) 150 200 300 0612 18 Time (s) Time (s)

Figure A.4: RHEED oscillations of (a) GaAs on GaAs, (b) AlAs on GaAs, (c) MnAs on MnAs, and (d) InGaAs on GaAs.

215 Appendix B

Effect of arsenic stoichiometry in bulk GaMnAs

In this appendix we review investigations of the effect of arsenic non-stoichiometry on the structural, electronic, and magnetic properties of bulk GaMnAs. As this work does not involve magnetic quantum wells, it is not included in the main part of the thesis. However, the molecular beam epitaxy (MBE) techniques discussed here could be used for quantum well structures. This work is moti- vated by the problem of excess arsenic defects that occur in low temperature (LT) grown GaAs and its alloys, such as GaMnAs. As we discussed in chapter 3.3.2, excess As accumulates into GaAs grown with a substrate temperature ◦ below ∼ 300 C and incorporates predominantly as an As-antisite (AsGa) as illustrated in Fig. 3.4b. These antisites are double donors and therefore compensate the holes due to Mn-doping in GaMnAs. Since the Curie temperature

(TC) is proportional to the hole density (p), then the incorporation of excess As reduces TC. Near the ferromagnetic limit of ∼ 1% Mn, compensation due to

AsGa could dramatically alter the TC’s since here the defect and dopant densities are roughly equal. Note that theoretical work predicts a change in the

216 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs mechanism of ferromagnetism in the low doping limit, near the insulator-metal transition [111, 185]. The effect of arsenic non-stoichiometry on the ferromagnetism of GaM- nAs has been theoretical discussed, but has surprisingly not been systematically treated experimentally. Here we discuss the first systematic investigation into the effect of AsGa incorporation on the structural, electronic, and magnetic properties of GaMnAs in the low Mn-doping regime [186]. Previous work on this subject in the literature only varied the As content up to 3 different values. The effect of excess As on the lattice constant of GaMnAs was treated by Schott et al.[187], who used one different As-flux than in the first work on ferromagnetic GaMnAs [101]. They noted a change in the extrapolated lattice constant of zinc blende MnAs due to AsGa incorporation. They followed-up their study by using 2 different As:Ga ratios, differing by 500% (2 different As contents), noting large changes in the lattice constant of GaMnAs over a broad range of Mn-doping and substrate temperatures [188]. Sadowski and Domagala then measured lattice constant variation in Ga0.96Mn0.04As grown using 3 different

As:Ga ratios and they noted an interplay between Mni and AsGa incorporation for this Mn doping regime [189]. Campion et al. reported an improvement in structural and magnetic properties of GaMnAs films grown using As2 instead of As4 [190], unfortunately they do not describe the fluxes used for each As- species nor how the effective III-V ratio should vary between these cases. They have also reported high TC’s over a broad range of Mn-doping levels stressing the importance of minimizing the As:Ga ratio although no data on the variation of this parameter were provided [191]. Avrutin et al. carried out the first study that measured both the structural, electrical, and magnetic properties of GaM- nAs using 2 different As:Ga ratios [192]. They noted an optimization of the hole density, conductivity, and TC for the low As:Ga ratio, where fewer AsGa should

217 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs be incorporated. Thus it would seem that the experimental literature has treated the effect of arsenic defects only roughly, even though it is widely accepted that this parameter can strongly affect the ferromagnetism. By using a pseudo-combinatorial LT MBE technique (non-rotated growth), the As:Ga ratio is continuously varied within individual growth runs. The geometry of the MBE chamber inherently provides a gradient in the As:Ga ﬂux ratio across the substrate of ±50%, while maintaining a roughly uniform Mn ﬂux. The Mn doping concentration was chosen to be between 1 to 2%, in the region of the onset of ferromagnetism. This avoids the formation of Mn interstitials

(also double donors) since at these concentrations Mni has a larger formation energy than MnGa [106], such that all hole compensation should originate from

AsGa. We observe the conductivity and hole density spanning over two orders of magnitude due to small variations in As:Ga. This leads to a transition from paramagnetism to ferromagnetism for constant Mn densities, see section B.3. Also, the As:Ga ratio signiﬁcantly alters ﬁlm morphology, see sections B.2.

B.1 Non-rotated GaMnAs: combinatorial variation of As:Ga

The details of LT MBE growth used for these samples is discussed in appendix C.3. Here we describe the method of non-rotated growth in which the geometry of the MBE system provides a continuous variation in the As:Ga ratio across the wafer [139]. This provides the advantage of a combinatorial approach in which a single non-rotated growth provides the equivalent of twenty or more rotated growths of various As:Ga ratios. The effect of As:Ga and Mn concentration can be mapped-out in high-resolution while performing a much smaller number of growths than otherwise required.

218 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

Figure B.1a shows the position of the Ga Knudsen cell and the As sublimator relative to the substrate position. The angle between the Ga and As2 molecular beams provides a continuous variation in the As:Ga ratio across the y-axis of the substrate. The variation of the As flux is measured by depositing a thick As film on a quartz wafer, covered by a shadow mask along the y-axis, with the substrate stabilized below room temperature. We have assumed that the deposition rate of As under these conditions is proportional to the As flux. A profilometer is used to measure the thickness of the As film along the y-axis. The normalized flux variation of As, plotted in Fig. B.1b, is calculated by normalizing the thickness along y by the thickness at y = 0. The variation of the Ga-flux is measured using an optical measurement of a distributed Bragg reflector (DBR) and cavity structure grown in 2 separate steps. First, a 10 period GaAs/AlGaAs DBR is grown on a rotated 2-inch substrate in a larger, more uniform system (Veeco Gen III) with a source to substrate distance of 11-inches. This results in an extremely uniform DBR which is As-capped and transferred into the Gen II system. The As-cap is desorbed and a GaAs optical cavity layer is grown on the sample without substrate rotation. The reflectance spectrum of the sample is measured ex-situ at different points across the wafer and is fit to an optical model of the reflectance. Because the 10 period DBR was almost completely uniform, the thickness of the non-uniform GaAs layer grown in the Gen II could be accurately and unambiguously determined by fitting the measured optical spectrum to a computer model of the structure. Since the deposition is performed at 585˚C, the growth rate is independent of the As overpressure, being limited by the group-III flux only. The normalized Ga rate, proportional to the Ga-flux, along the y-axis is plotted in Fig. B.1b. Both the As and Ga flux variation are well fit by a second order polynomial (lines), which are used to convert y-axis position for non-rotated growths to the

219 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

As (a) As +y y Mn Mn -x +x -y Ga substrate Ga

1.4 (b) As:Ga variation 1.2 Ga As 1.0 0.8 0.6 Normalized Flux Normalized -40 -30 -20 -10 0 10 2030 40 y (mm)

× 17 (c) (d) 1.10 4.7 10 Normalized Flux

17 ) 4.5×10 1.05 -3 Mn:Ga variation 4.3×1017 1.00

p (cm 4.1×1017 0.95 3.8×1017 0.90 3.6×1017 0.85 -20-10() 0 10 20 -20-10 0() 10 20 x (mm) y (mm)

Figure B.1: Non-rotated growth flux calibration. (a) The relative positions of the As, Ga, and Mn Knudsen cells with respect to the substrate.(b) The flux gradient of Ga and As molecular beams along the y-axis of the wafer normalized to the center flux value. (c)-(d) The Mn:Ga flux variation along the x-axis (c) and y-axis (d) of the wafer measured from hole density (p) variation in non-rotated samples grown at 400 C, where As-flux does not affect carrier density.

220 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

As:Ga ratio. We also consider the Mn ﬂux variation across the substrate. Since the Mn Knudsen cell is roughly perpendicular to the y-axis, variation along this direction is expected to be small. In order to measure the Mn variation across the substrate, a 1-µm thick GaAs:Mn layer with Mn ∼ 4×1017 cm−3 is deposited at 400˚C with in an As2 overpressure. At these conditions Mn incorporation is fully substitutional and excess As incorporation is negligible (< 1 × 1016 cm−3), thus the hole density is proportional to the Mn:Ga ﬂux ratio [138]. The hole density variation across the x and y axes due to Mn:Ga variation is plotted in Fig. B.1 c and d, respectively. Along either axis, the Mn:Ga ratio varies by 5-10%, which corresponds to variation in the Mn concentrations of 0.05 to 0.1%. As will be clear, see section B.3), this variation in Mn concentration is negligible with respect to the As compensation effect under investigation.

B.2 Film morphology and strain versus As:Ga

The roughness of non-rotated samples is measured across the y-axis by atomic force microscopy (AFM). We describe results from two non-rotated films, without Mn doping and with 1.5% Mn. Both 5×5-µm2 and 1×1-µm2 area scans are measured every 3 to 4 mm along the y-axis of both wafers. The RMS roughness from these scans is plotted as a function of As:Ga (converted from y-position from Fig. B.1b calibration) and representative AFM images of each region are shown in Fig. B.2. The As-rich region from 11 < As:Ga < 14 results in fully coalesced films with RMS roughness of 0.2 nm or less indicative of a 2D growth mode. RHEED observations of such films, from rotated growths at the same As:Ga condition, show sharp and streaky 2D reconstruction. In the boarder region between

221 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

1.0 Droplets (3D) Clovers (3D) 0.8 15 10 0.6 5 nm

m 0 μ 0.4 -5 0.2 -10 -15 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 μm μm 0.0% Mn 10 1.5% Mn

Ga-rich As-rich

1 Stoichiometric RMS roughness (nm) 0.1 6 8 10 12 14 As:Ga 1.0 Grains (2D) Coalesced (2D) 0.8 1.5 1.0

0.6 0.5 nm

m 0 μ 0.4 -0.5 0.2 -1.0 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 μm μm

Figure B.2: Morphology of GaAs and GaMnAs ﬁlms as a function of the As:Ga ratio (non- rotated growths). AFM micrographs (1×1 µm2 scans) are plotted for a GaAs growth (sample 051101A). The RMS roughness measured from such AFM scans is plotted as a function of the As:Ga ratio (along y-axis of wafer) for both a GaAs and GaMnAs sample (sample 051011B). The stoichiometric region is shaded grey, where excess As is minimized, two-dimensional growth (2D) is maintained, however, some granular structure is observed.

222 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs the As-rich to stoichiometric regions (from 9 < As:Ga < 11), films show a nanoscale-granular structure. For decreasing As:Ga, this granular structure becomes more distinct and the film roughness increases, however it remains below 1 nm RMS. RHEED observations for films grown in this As:Ga range show a spotty reconstruction pattern indicative of a rough, but 2D surface, which we have found to be characteristic of films near the stoichiometric condition. In this region, the hole density of Mn-doped films reaches a maximum indicating the minimum of compensating AsGa defects, see section B.3. Immediately below As:Ga ∼ 9, the RMS film roughness increases by a factor of 10 as the As:Ga ratio transitions from the stoichiometric condition to Ga- rich. AFM images of this region show the formation of 10-nm tall clover-shaped features 60-nm in diameter with a crater in the center. The RHEED pattern of these films shows additional spots indicative of 3D features on the surface. For As:Ga < 9, the surface roughness continues to increase with the appearance of larger droplet-like features. At the lowest As:Ga ratio, these hemispherical droplets are 50-nm tall and 240-nm in diameter. The large change in roughness across the non-rotated wafers is observable by eye under intense illumination, where the surface transitions from mirror finish at the As-rich region to a hazy surface in the Ga-rich region. It appears that Mn-doping suppresses the onset of roughening toward smaller As:Ga. Since Mn-doping increases the effective group-III flux, one would expect a shift in the onset of roughness to higher As:Ga. This smoothing may be due to a surfactant effect of Mn on the surface. As discussed in section B.3, the highest hole densities are observed in GaMnAs grown with As:Ga just before the transition from 2D to 3D surface, indicating that the growth kinetics at this stoichiometric condition do not significantly increase the density of hole compensating Mni donors.

223 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

HRXRD scans are obtained using a triple-axis (Phillips X’Pert MRD Pro) thin film diffractometer. We measure ω-2θ scans near the (004) Bragg peak of the (001) GaAs substrate. The diffraction patterns obtained for the non-rotated LT GaAs film at the position of highest As-compensation (As:Ga = 13:1) and at the stoichiometric condition (As:Ga = 9:1) are plotted in Fig. B.3 a and b, respectively. In the As-rich position, the epilayer peak appears as a shoulder at lower angle, reflecting the increase in the out-of-plane lattice constant, due to

AsGa incorporation. We also note the thickness fringes indicative of the high quality of the film surface and interface. A dynamical HRXRD program is used to simulate the data [193] assuming that the lattice constant of (Ga1−z,Asz)As, a = 5.654 + 1.356z [114]. Best fit simulations in Fig. B.3a give a film thickness 19 −3 19 −3 90 nm and AsGa = 4×10 cm . The additional fit with AsGa = 5×10 cm 19 −3 shows the error in the AsGaestimate is about 1×10 cm . The thickness of the epilayer is 10-15% thinner at the top edge than at the center of the wafer due to the smaller Ga-flux at this position, as shown in Fig. B.1b. At the stoichiometric position the epilayer shoulder becomes small, only visible as a slight asymmetry on the left side of the substrate diffraction peak in Fig. B.3b. Simulations using 18 −3 a film thickness of 100 nm yield AsGa = 4×10 cm , which is smaller than the uncertainty of the fits. A more precise estimate of the AsGa density is found from the hole density variation, discussed in section B.3. Thickness fringes disappear since the lattice constant of the epilayer is almost indistinguishable from that of the substrate. Figure B.3 c and d plot the diffraction patterns from a non-rotated GaMnAs film with 1.5% Mn from the As-rich and stoichiometric positions, respectively. For these layers, strain due to Mn incorporation dominates the epilayer lattice constant obscuring the AsGa effect. Simulations are performed assuming that the lattice constant of (Ga1−x,Mnx)As, a = 5.654 + 0.476x [187]. The fits in-

224 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

data (a) × 19 -3 As:Ga = 13:1 AsGa = 4 10 cm × 19 -3 AsGa = 5 10 cm

data (b) Intensity (cps) × 18 -3 As:Ga = 9:1 AsGa = 4 10 cm × 19 -3 AsGa = 1 10 cm

-0.1 0.0 0.1 Δω 004 (deg) data Mn = 1.5% As:Ga = 13:1 (c)

data As:Ga = 10:1 (d) Intensity (cps) Mn = 1.3%

-0.3-0.2 -0.1 0.0 0.1 0.2 0.3 Δω 004 (deg)

Figure B.3: Strain effect due to Antisites. (a)-(b) HRXRD data from a GaAs ﬁlm (sample 051101A) at the As-rich wafer position (a) and stoichiometric position (b). HRXRD (points) and dynamical simulations (lines) are plotted for various AsGa densities. (c)-(d) Data from a GaMnAs ﬁlm (sample 051011B) at the As-rich wafer position (c) and stoichiometric position (d). Data (points) and dynamical simulations (lines) are plotted using different Mn densities.

225 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs dicate a ﬁlm thickness of 85 nm and 95 nm, and Mn concentration of 1.5% and 1.3% at the As-rich and stoichiometric positions, respectively. The variation in Mn concentration is not explained by the Mn:Ga ratio variation along y, Fig. B.1d, which would predict a decrease in Mn concentration of up to 0.1%.

The additional strain reduction may be due to the minimization of AsGa defects at the stoichiometric position, but the uncertainty of the HRXRD simulations prevents a more quantitative analysis.

B.3 Carrier density and Curie temperature versus As:Ga

Figure B.4 a and b show the variation in hole density for a series of non-rotated GaMnAs films of various Mn concentrations (a) along the y-axis, and (b) with y converted to As:Ga, using the Fig. B.1b calibration. All the hole density data in the figures are from room temperature measurements of samples prepared in the Van der Pauw geometry (3×3 mm2) in a magnetic field up to 0.2 T. Selective measurements of the hole density of patterned Hall-bars in magnetic fields up to 2 T agree with the Van der Pauw measurements to within 50%. Additional high field measurements at 7-9 T deviate from the low field data by up to 50 %. The deviation is believed to come from anomalous hall effect in GaMnAs. We did not perform complete measurements of the hall bars across the non- rotated wafers since it is a time consuming process. On the other hand, VDP samples are rapidly measurable. The large uncertainty is actually not significant in comparison to the orders of magnitude variations in hole density that we observe due to changes in As : Ga. The hole density varies by over two orders of magnitude across each wafer showing a maximum value near y ∼ 6 mm corresponding to As:Ga ∼ 9. The

226 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

Mn% As:Ga y=0 = 8:1 1.5 1019 (a) (b) 1.25

) 1.0 -3 1018 0.75

p (cm

1017

01020 81410 12 y (mm) As:Ga (c)1.5% Mn (d) 1019 As:Ga y=0 8:1

) 9:1 -3 18

10 p (cm

1017

-20 -10 01020 61681410 12 y (mm) As:Ga

Figure B.4: As-compensation gradient in non-rotated GaMnAs ﬁlms. The room temperature hole density (p) variation along ﬁlms grown with different Mn concentrations (samples 051011B,C and 051018A,B), (a) as a function of y-position, and (b) as a function of As:Ga ratio. (c)-(d) Similar data for samples with the same Mn concentrations but different As:Ga ratios at y = 0 (samples 051011A,B).

227 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs results are explained by a decreasing number of AsGa donors that compensate

MnGa acceptors as the As:Ga ratio is reduced from the As-rich condition. The maximum of p along y corresponds to the position at which AsGa are minimized, the stoichiometric condition. A further reduction in As:Ga leads to a Ga-rich surface that limits Mn incorporation, and leads to a ten-fold increase in surface roughness marking a transition from 2D to 3D ﬁlm morphology, see section B.2). The y-position of the maximum in p matches in four separate non-rotated growths demonstrating the reproducibility of this technique. When the Mn density is reduced, the hole-density peak narrows on the As-rich side, indicating that samples with lower Mn-doping density are more sensitive to the

AsGa compensation effect. This follows if the As:Ga ratio controls the density of hole compensating defects independently from the doping density of Mn, as expected for this doping regime. We note that for the 0.75% Mn doped wafer, when As:Ga ∼ 10 the hole compensation is nearly complete, p ∼ 1016 cm−3. For As:Ga ∼ 9, the compensation is minimized, p ∼ 1019 cm−3. In this case, a 10% reduction in the

As:Ga ratio reduces the AsGa compensation by three orders of magnitude. This demonstrates the strong sensitivity of the AsGa incorporation to As:Ga, and the high degree of As-ﬂux control, within 10%, necessary to properly minimize this defect. Previous studies of the effect of As:Ga ratio reported only two or three values of this parameter in increments of 200% or more [188, 189, 192]. In the Ga-rich position (As:Ga < 9), the hole density decreases along with As:Ga, but remains constant as the Mn doping density is varied, Fig. B.4. This behavior is explained by a solubility limit for Mn, which is below any of the Mn doping levels used here. Decreasing the As:Ga ratio below the stoichiometric value leads to a Ga-rich surface that serves as a kinetic barrier for the incorporation of group-III substitutional dopants, such as Mn. One might consider the

228 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs alternative possibility that interstitial incorporation of Mn might occur in these circumstances and could explain the reduction in hole density in the Ga-rich region. However, if Mni donors were incorporating and compensating the MnGa acceptors, then the carrier density should change as the Mn-doping density is altered, which is not the case here. The position of the hole density peak can be controlled by adjusting As:Ga at y = 0, which shifts the position of the stoichiometric condition along y. To demonstrate this, two wafers with identical Mn concentration of 1.5 % are grown but with different As:Ga at y = 0. The variation of p along y is plotted in Fig. B.4c. By increasing As:Ga at y = 0 from 9:1 to 10:1, the peak position (stoichiometric condition) is shifted by ∼ 1 cm toward the center of the wafer in a direction away from the As source. This shift is observable by eye as a change in the position of the transition region, where the surface changes from mirror ﬁnish to hazy. Converting the y-position to As:Ga should fully overlap the hole density peaks, Fig. B.4d. The peaks do not completely overlap indicating that error may be present in the y-axis conversion.

B.3.1 Curie temperature gradient

The Curie temperature variation with As:Ga for two non-rotated GaMnAs films is plotted in Fig. B.5 a and b, together with the hole density variation. TC is determined from magnetization versus temperature scans by selecting the point of maximum second derivative in M(T). The scans are performed during a field warm (5 mT) after a field cool (1 T). TC tracks the p variation with As:Ga, displaying a maximum in Curie temperature in the same region where the hole density peaks. In the As-rich portion of the wafer, AsGa donors limit the hole density and therefore TC. This effect is very clear, for example, with As:Ga >

229 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

(a) 1019 1.25% Mn 20

Ga-rich 15 T C ) (K) -3 18 m 10 As-rich 10 p (c 5 Stoichiometric

17 0 10

(b) 1.50% Mn 1019 20

15 T ) C -3 (K) 1018 10 p (cm

1017 0 81410 12 As:Ga

Figure B.5: Curie temperature gradient in non-rotated GaMnAs ﬁlms. The hole density (p) and

TC are plotted as a function of the As:Ga ratio for two wafers with different Mn concentrations, (a) 1.25% (sample 051011C), and (b) 1.50% (sample 051011B). The stoichiometric region is shaded grey, where excess As is minimized.

230 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs

12, where ferromagnetism is completely suppressed in both 1.5% and 1.25% samples due to hole compensation from As non-stoichiometry in the form of

AsGadonors.

B.4 Conclusions

The electronic and magnetic properties of GaMnAs reveal a strong sensitivity to AsGa donors that compensate MnGa acceptors. By varying the As:Ga ratio, the hole densities of GaMnAs films are varied by more than two orders of magnitude, and, as a result, ferromagnetism can be completely suppressed or optimized. For As-rich growth, films show the maximum smoothness, but lowest p and TC. If As:Ga is reduced, the film roughness increases up to the stoichiometric condition, where p and TC are maximized. For Ga-rich growth, the film roughness increases by an order of magnitude as excess Ga accumulates on the surface. Our results emphasize that precise control of hole compensation due to AsGa is critical for attaining optimal TC’s in the low Mn-doping regime. The combinatorial MBE growth approach used here could be applied to more complex LT grown GaAs-based heterostructures in which As:Ga is a strong control parameter. The rapid and high-resolution mapping of growth parameter phase-space by this technique allows for systematic investigations not practica- ble using more traditional methods. An immediately obvious next step to this project would be to attempt higher Mn doping levels. Since increasing Mn-doping now introduces the Mn-interstitial defect, it is probably necessary to grow at lower substrate temperatures. This would make the excess arsenic effect even stronger, but it can be optimized as usual using the non-rotated growth method. This technique could be used to optimize very low temperature growth (≤ 100◦ C), where Mn > 10% may

231 Appendix B Effect of arsenic stoichiometry in bulk GaMnAs be possible. This presents a new and unexplored method for increasing TC of GaMnAs.

232 Appendix C

UCSB samples: or how I learned to stop worrying and love MBE

This chapter describes additional details of the MBE growth of samples I made at UCSB for the experiments discussed in this thesis. Awschalom group members searching for samples grown for other projects should look in the directory: “\\Titan\group\Shared Pro jects\Roberto Sample Log”, which includes sample descriptions and growth sheets. The detailed layer structure can be determined by examining the growth recipe of each sample, and asking an MBE grower to translate it for you. Kato and Poggio also included appendices in their theses describing some of these details [37, 38]. There is no reason to avoid old samples for new experiments. The advantage is that old samples are already well characterized. Also, Arsenide heterostructures are pretty much immortal; getting “fresh” samples is not relevant unless you’re talking about fast oxidizers like AlAs or MnAs. I have a couple words of caution regarding old samples:

• Smoothing superlattices are frequently used to grow the buffer layers. These are usually composed of AlAs/GaAs superlattices. Such layers

233 Appendix C UCSB samples: or how I learned to stop worrying and love MBE

may interfere with wet etching, for example in spray etching to remove the substrate. They will also appear in the optical spectrum as AlGaAs depending on their average Al concentration.

• Some recipes were altered on the ﬂy so the layer structures may not perfectly correspond to the sample. Consult the growth sheets (computer) to be sure, and as a ﬁnal recourse the growth notebooks (paper).

The II-VI samples used in chapter 4 were grown by Keh-Chiang Ku at Penn. State University in Professor Nitin Samarth’s group.

C.1 QWs with a ferromagnetic barrier

These structures were used in experiments described in chapter 5, see Fig. 5.2 and Table 5.1. The structures were grown in both system C and the “old” spintronics system in late 2002 and early 2003. At this time the spintronics MBE system at UCSB was capable of growing Mn-doped GaAs-based heterostructures, but could not produce good quantum wells owing to contamination, see appendix A.1. To grow clean, optical-quality QWs with ferromagnetic barriers, we used a two-chamber growth method with As-capping. The As-cap allows for in-air transfer between systems without oxidation, see appendix A.5. All samples are grown on (001) oriented semi-insulating (SI) GaAs wafers with a 2-inch diameter. The As2:Ga beam equivalent pressure ratio (BEP) is

30 : 1. At a substrate temperature (Tsub) of 585˚C, the 300-nm GaAs buffer layer and a 180-nm wide superlattice (30×[3-nm AlAs / 3-nm GaAs]) is grown, which serves as a smoothing layer and diffusion barrier for carriers. A 400- nm region of GaAs [200-nm n-GaAs (Si-doped: n = 1 × 1018 cm−3) / 200-

234 Appendix C UCSB samples: or how I learned to stop worrying and love MBE nm i-GaAs] comprises the back contact. An additional 9-nm wide superlattice (6×[0.9-nm AlAs / 0.6-nm GaAs]) keeps electrons in the back contact from diffusing into and being compensated by the LT AlGaAs resistive layer.

Tsub is cooled to 300˚C for the deposition of a 461-nm wide LT Al0.4Ga0.6As layer. Note that for LT AlGaAs, surface roughness increases for Tsub <300˚C as evidenced by a change in the RHEED pattern to 3D, see Fig. A.3. For LT

GaAs the Tsub =250˚C is possible.

Upon completion, Tsub is heated to 630˚C for 10 minutes, which precipitates out As-particles and is responsible for the resistive properties of LT GaAs and

AlGaAs. Tsub is cooled to 585˚C, followed by deposition of a ﬁnal 30-nm superlattice diffusion barrier (20×[0.9-nm AlAs / 0.6-nm GaAs]). The two diffusion barriers plus the annealed LT layer comprise what we refer to as the 500-nm LT structure. The following layers complete the QW and set-back layer structure: 350 nm Al0.4Ga0.6As / 7.5 nm GaAs QW / d Al0.4Ga0.6As (where d is either 5 nm or 9 nm). After these layers the samples are As capped, appendix A.5, and transferred in air to the “old spintronics” system. After desorbing the As cap at 630˚C the samples are cooled to 230˚C for LT magnetic overgrowth. The magnetic layers are grown by ALE [40]. Sam- ples with magnetic structure 5×(0.5/20) and 5×(0.3/20) consist of 5 periods of DFH superlattice of Mn-rich layers (0.5 ML and 0.3 ML of MnAs, respectively) spaced by 20 ML Al0.4Ga0.6As. Samples with magnetic structure 1×(0.5/84) consist of a single Mn-rich layer, 0.5 ML MnAs / 84 ML Al0.4Ga0.6As. Finally, a Mn-free control sample consisting of 85 ML of ALE grown Al0.4Ga0.6As is prepared. The magnetic layer is capped by 177 ML Al0.4Ga0.6As / 27 ML GaAs also grown by ALE. The ALE growth of AlGaAs digital alloys proceeds, just as in the case of

235 Appendix C UCSB samples: or how I learned to stop worrying and love MBE

MnAs/GaAs DFH, by doping each constituent species within each ML. Thus 1ML of AlGaAs consists of 0.5ML (Al + Ga, deposited separately) and 0.5ML As (deposited after AlGa).

C.2 GaMnAs and InGaMnAs QWs

These structures were used in experiments described in chapter 6, see Tables 6.1 and 6.2. The structures were grown in the “new” spintronics system in 2004- 2005, using absorption band edge spectroscopy (ABES) to monitor and control the substrate temperature, A.4. The growth rate of GaAs is ∼ 0.7 ML/sec and of Al0.4Ga0.6As is ∼ 1 ML/sec as calibrated by RHEED intensity oscillations of the specular spot, see Fig. A.4. The As2:Ga beam flux ratio for all samples is 19:1 as measured by the beam equivalent pressure of each species using a bare ion gauge in the substrate position. Mn cell temperatures for doping were extrapolated from growth rate calibrations of MnAs measured at much higher growth rates using RHEED oscillations. Refined estimates of Mn-doping densities were found by growing a p-doping series of bulk GaAs:Mn at 400◦C, such that most Mn incorporated substitutionally (Mn doping cals). Finally, the Mn doping densities were measured directly for each sample used in the quantitative analysis by SIMS, see section 6.2. The QWs, shown schematically in Fig. 6.1, are grown on (001) semi- insulating GaAs wafers using the following procedure. The substrate is heated to 635◦C under an As overpressure for oxide desorption and then cooled to 585◦C . With the substrate rotating at 10 RPM throughout the growth, a 300- nm GaAs buffer layer is first grown using 5-s growth interrupts every 15 nm for smoothing, which results in a streaky 2 × 4 surface reconstruction pattern as observed by RHEED, see Fig. A.3. A 500-nm layer of Al0.4Ga0.6As is

236 Appendix C UCSB samples: or how I learned to stop worrying and love MBE grown followed by a 20-period digital superlattice of 1-nm AlAs and 1.5-nm GaAs. The sample is then cooled to the growth temperature (usually 400◦C, but also 350◦C and 325◦C) during which the RHEED pattern changes to a 4×4 reconstruction, see Fig. A.3. All the GaMnAs QWs used for the s − d ex- ◦ change analysis were grown with Tsub = 400 C and all the InGaMnAs QWs ◦ were grown with Tsub = 350 C.

For GaMnAs QWs, the ﬁrst QW barrier consists of a 50-nm Al0.4Ga0.6As layer; during its growth the 4 × 4 pattern becomes faint and changes to 1 × 1. Before the QW layer, a 10-s growth interrupt is performed to smooth the interface; during this wait the RHEED partially recovers a 4 × 4 reconstruction pattern. The GaMnAs QW layer deposition causes the 4 × 4 to again become faint leading to 1 × 1, but during the next 10-s wait on the top side of the QW, a 4 × 4 partially recovers. Increased Mn-doping leads to surface roughening as evidenced by the development of a spotty RHEED pattern during and after the QW growth. In contrast, lower-doped samples, which emit PL and show time-resolved KR signal, display a streaky two-dimensional RHEED pattern throughout their growth. The structure is completed by a top QW barrier of

100-nm Al0.4Ga0.6As and a 7.5-nm GaAs cap after which a streaky 4×4 reconstruction pattern is observed. For InGaMnAs QWs, the ﬁrst barrier consists of 50-nm GaAs grown at ◦ Tsub = 350 C. The RHEED pattern maintains a streaky 4×4 pattern throughout this growth. A 10-s growth interrupt is performed before and after the QW to smooth the interfaces. During the InGaMnAs QW deposition the RHEED changes to a 1 × 2 pattern, but recovers a 4 × 4 during the last 10-s interrupt. A 107.5-nm GaAs layer completes the top of the structure. We have also investigated the limit to which Mn can be incorporated into GaMnAs and InGaMnAs QWs while maintaining a strong optical signal, see

237 Appendix C UCSB samples: or how I learned to stop worrying and love MBE section 6.1. GaAs QWs cannot be grown below 400◦ C without the loss of optical signal, which is independent of Mn-doping and related to defects in the Al-containing barriers. InGaAs QWs can be grown down to 300◦ C. The Mn concentration can be increased to 0.35% before the optical signal are no longer present. Additional growth were performed that investigated the effect of modulation doping of hole or electrons into the QWs. It is possible to dope electrons or hole into the QWs and change the carrier type. The doping levels were so high that it is likely that no carriers were doped into the QWs. Simulations of these structures show that the degenerately doped barriers trap carriers instead of allowing them to diffuse into the QW region. Future samples should either have directly doped carriers in the QW region by Si or Be doping in the QW region, or use different modulation doping schemes such as lower doping densities or delta doping.

C.3 Stoichiometric GaMnAs and non-rotated growths

These samples were used to study the effect of arsenic non-stoichiometry on the electronic and magnetic properties of GaMnAs, see appendix B. With non- rotated growths, the As:Ga ratio and thus the excess As content can be smoothly varied in a single growth. Since electronic and optical properties are extremely sensitive to As defects, a systematic and rapid optimization of this parameter is very useful. We also discuss here a method to grow (with rotation) stoichiometric LT GaAs based samples. The As level can be quickly calibrated using the non-rotated method. Optimized samples can then be grown using the calibrated As:Ga value.

238 Appendix C UCSB samples: or how I learned to stop worrying and love MBE

All samples are grown on 2-inch diameter GaAs (001) substrates. The substrate temperature is monitored by absorption band edge spectroscopy (ABES). The growth rate of GaAs at the center of the wafer is 0.7 ML/sec as calibrated by RHEED intensity oscillations of the specular spot at 580˚C. The Mn doping density is inferred from RHEED intensity oscillation measurements [194] of the MnAs growth rate performed at 240˚C. The arsenic cracker temperature was set to approximately 850˚C which should lead to predominantly only As2 molecules. The T/C of the cracker was broken during these growths so we grew in constant power mode. The beam equivalent pressure (BEP) of As2 and Ga are measured with a nude ion gauge at the center position of the substrate just prior to growth. The ratio of As2 and Ga BEPs, defined as As:Ga, is proportional to the ratio of the atomic fluxes of As and Ga delivered to the growing surface. Unfortunately, a reliable quantitative conversion from BEP to atomic flux is difficult due to the uncertainties in the ionization efficiencies of each beam species [29]. A rough estimate gives that the atomic flux ratio is 2.5 times smaller than the BEP ratios, where we assume that only As2 are present with a temperature equal to that of the cracking zone. The following procedure is used for both non-rotated and rotated growths prior to deposition of the low temperature (LT) films. The wafer is first heated to 635˚C with As:Ga ∼ 35 for oxide desorption, monitored by RHEED, and sub- sequently cooled to 585˚C. While rotating at 10 RPM, a 300-nm GaAs buffer layer is grown using 5-s growth interrupts for every 15 nm of deposition resulting in a streaky 2 × 4 surface reconstruction RHEED pattern. While cooling the substrate to 250˚C, the As-valve setting is reduced at 400˚C for the desired As:Ga value for LT growth. Prior to LT growth, a streaky 4 × 4 surface reconstruction is observed in the RHEED pattern. For rotated growths, once the substrate stabilizes at 250˚C the LT growth begins. For non-rotated growths,

239 Appendix C UCSB samples: or how I learned to stop worrying and love MBE the substrate rotation is stopped at 400˚C and the wafer [110] crystal axis is aligned along the As:Ga gradient direction (y-axis). This can be reproducibly accomplished noting the radial angle between the RHEED e-beam and substrate (the RHEED axis is 30 degrees from the y-axis), and finding a high symmetry crystal axis from the surface reconstruction pattern. An easy was to do this is to rotate the crystal until you think it is approximately aligned along the y-axis, then rotate it 30 degrees in either direction, you will see a high symmetry point in one of the two directions. Calibrate this position as home, then rotate the crystal backwards 30 degrees and re-home the center. Since the crystal can slip during the buffer growth, it is best to set the wafer position after the buffer is grown and during the cool down to LT. All LT films are 100-nm thick at the center wafer position. After completion of the LT layer, they are cooled in a consistent manner to room temperature. Since the Ga rate is fixed for our study, then the As-flux is the only parameter we need to vary to find the stoichiometric point. It may also be necessary to optimize the Ga-rate for future (even lower temperature) studies of Mn incorporation in GaMnAs. In the non-rotated growths, see appendix B.1, the Ga-rich region of the wafer can be easily seen as a hazy region (by eye) since it is 10× rougher than the As-rich region, which has a mirror finish. We found that the stoichiometric value for As:Ga is ∼ 9 : 1, which for our designated Ga-flux, −6 requires an As2 BEP of ∼ 2.00 × 10 torr. Calibrating the As-flux for LT stoichiometric growth is the most important step since variation in this parameter of 1% can significantly alter the electronic and optical properties of the film. After first opening the As-valve for the first time, stabilize for 10 minutes before measuring the flux and adjusting. Close the As-valve to 0mil before each change in valve position. Wait an additional 10 minutes after each change before recording the flux value. These long waits are

240 Appendix C UCSB samples: or how I learned to stop worrying and love MBE necessary since the flux drifts slowly in time. After 30 minutes total of having the As-valve open, the flux will be stable to within ∼1% comparing flux values before and after the growth. During the long wait time necessary to calibrate the As-flux you can bake your wafer in the growth chamber (T/C to 400˚C). If you change the sublimator temperature you will need to wait for several hours until the flux is restabilized. For stoichiometric growths you should wait a day after you change the sublimator temperature before the next growth.

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266 Select materials parameters

GaAs band gap (0 K) direct, Γ8v − Γ6c 1.51914 eV lattice constant Zinc Blende 5.65359 Å

AlAs band gap (4 K) indirect, Γ15v − X1c 2.229(1) eV

direct, Γ15v − Γ1c 3.13(1) eV lattice constant Zinc Blende 5.66139(5) Å

InAs band gap (4.2 K) direct, Γ8v − Γ6c 0.4180(5) eV lattice constant Zinc Blende 6.0583 Å MnAs lattice constant Zinc Blende, ext. 5.90 − 5.98 Å NiAs, α (a) 3.724 Å (c) 5.706 Å Useful Constants

Planck constant h 6.62606876(52) × 10−34 J s 4.13566727(16) × 10−15 eV s 4.13566727(16) meV ps h¯ = h/(2π) 1.054571596(82) × 10−34 J s 6.58211889(26) × 10−16 eV s Electronic charge e 1.602176462(63) × 10−19 C −31 Free electron mass me 9.10938188(72) × 10 kg −1 Bohr magneton µB = eh¯/2me 57.88381749(43) µeV T −1 Boltzman constant kB 86.17342(15) µeV K h/e2 25.812807572(95) kΩ

h¯/µB 11.37125914(94) ps T hc/e 1239.84185(68) eV nm e/h 0.241798948(88) GHz µeV−1