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Spin dynamics and electronic structure of muonium and its charged states in silicon and gallium arsenide Chow, K. H. 1995

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SPIN DYNAMICS A N D ELECTRONIC STRUCTURE OF M U O N I U M A N D ITS CHARGED STATES IN SILICON A N D GALLIUM ARSENIDE By K.H. Chow Honors B.Sc. in Physics and Chemistry, UBC, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © K.H. Chow, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of The University of British Columbia Vancouver, Canada Date D - e c \%l<Wr DE-6 (2/88) Abstrac t This thesis describes recent fiSH measurements on muonium (Mu=yu+e~) centers in crystalline Si and GaAs. Spin-exchange scattering and charge-exchange of Mu with conduction electrons are found to be important dynamical processes in these systems. Longitudinal muon spin relaxation (1/Ti) measurements in intrinsic Si and p-type Si:B from 350 K to 900 K can be explained within a model where Mu cycles rapidly between its positive and neutral charge states via interaction with conduction electrons. The average muon-electron hyperfine parameter in the neutral s tate is consistent with Mu at the tetrahedral (T) interstitial site. This indicates that at the highest temperatures measured neutral Mu spends significant time away from the bond center site, the calculated minimum in the adiabatic potential energy surface. Measurements of the 1/Ti rates were made in intermediate doped n-type Si and heavily doped n-type GaAs under conditions where coherent spin precession of Mu is unobservable. A peak in 1/Ti as a function of magnetic field is observed and shown to be characteristic of neutral bond-centered Mu ( M u ^ c ) undergoing spin-exchange scattering with free carriers. These results establish that neutral M u # c does not convert to M u - , the expected stable state in n-type Si, at temperatures below approximately 200 K. Furthermore, we conclude M.UgC is present in heavily doped n-type GaAs below « 30 K. At very high spin-exchange rates, such as in heavily doped Si:P, and under the application of a large magnetic field, M u ^ c is "spin-polarized" and a frequency shift from the Larmor precession frequency of a free muon is observed. n Muon level-crossing resonance and muon spin rotation measurements on heavi ly-doped n-type GaAs:Si and GaAs:Te show that the majority of positive muons im-planted at room temperature form an isolated diamagnetic Mu center located at a high-symmetry site with Ga neighbors along the (111) direction(s). The experimental results, together with simple theoretical considerations, imply that the center is M u " located at or near a Tca site. These studies on Mu have bearing on research on hydrogen in semiconductors. In particular, the spin dynamics taking place for Mu should also be occuring with hydrogen. The importance of such processes is usually not taken into account by researchers of hydrogen diffusion and related dynamics. The characterization of M u " is the first experimental determination of the detailed local structure of a charged isolated hydrogen or Mu center in any semiconductor. This is especially interesting since the existence of isolated H - in semiconductors is still being debated. iii Table of Contents Abstract ii Table of Contents iv List of Tables v List of Figures vi List of Acronyms vii Acknowledgements ix 1 INTRODUCTION and BACKGROUND 1 1.1 Muonium Centers in Si and GaAs 7 1.2 Charge and Spin Dynamics of Muonium 11 2 nSK 14 2.1 Muons 15 2.2 The Various fxSR Techniques 18 2.3 Electronics 22 2.4 / iSRData 24 3 THEORETICAL CONSIDERATIONS 28 3.1 A List of Hamiltonians 29 3.2 Time-Dependent Muon Polarization of Muonium Centers 36 iv 3.3 Relaxation of the M u ° Transverse Field Signal 40 3.4 Quadrupole Level Crossing Resonance of Mu D 44 3.5 SPIN DYNAMICS - INTRODUCTION 49 3.6 Qualitative Discussion of Muonium Spin Dynamics in a Longitudinal Field 50 3.7 Isotropic Mu undergoing Charge Exchange in a Longitudinal Field-Strong Collision Approach 56 3.8 Celio/Odermatt Method and Spin Exchange of M u ^ c in LF 64 3.9 TF-/ /SR in the Fast Spin Exchange Limit 72 4 A P P A R A T U S 74 4.1 T R I U M F Spectrometers 74 4.2 Cryostats and the Oven 78 4.3 L A M P F Spectrometer 81 5 M E A S U R E M E N T S and D I S C U S S I O N 84 5.1 Useful Information 84 5.2 Analysis of //SR Data 85 5.3 Samples 87 5.4 Intrinsic S i — High Temperatures 88 5.5 p-type Si:B - High Temperatures 98 5.6 n-type Si:P with Intermediate Doping Levels 101 5.7 Spin Polarized Muonium-Heavily Doped Si:P 109 5.8 Heavily-Doped GaAs — Low Temperature 116 5.9 Diamagnetic Centers in Heavily-Doped GaAs 120 6 S U M M A R Y and C O N C L U D I N G R E M A R K S 134 6.1 Muonium Spin Dynamics in Other Materials 135 V 6.2 Hydrogen 136 6.3 Extensions of Current Studies 137 Bibliography 139 A PROBABILITIES in ALCR and TF-^SR 145 A.l Probabilities in ALCR 145 A.2 Probabilities in TF-//SR 147 VI List of Tables 1.1 Physical Properties of Electron, Muon and Proton 7 1.2 Hyperfine Parameters for muonium in Si and GaAs 9 5.1 Nearest and Next-Nearest Neighbors for muon in a T site 85 5.2 Muon in a BC site 85 5.3 Summary of Si samples studied 88 5.4 Summary of GaAs samples 89 5.5 Physical properties of Ga and As 121 5.6 Fit results for n-type GaAs 132 A. l Probabilities for Ga when N = 4 146 A.2 Second moment calculations when 2?0|| (HO) a x i s for muon in a Toa site 148 A.3 Second moment calculations when i?o|| ( H I ) axis for muon in a Tea site. 148 vii List of Figures 1.1 Zincblende structure of Si and GaAs 8 2.1 Elements of Muon Decay 17 2.2 Counter arrangement 20 2.3 Time-differential Electronics 23 2.4 T F Raw Spectrum and Corrected Asymmetry Spectrum 25 2.5 LF Corrected Asymmetry Spectrum 27 3.1 Breit-Rabi diagram for Mu^ in Si 34 3.2 Breit-Rabi diagram for Mug C in Si 36 3.3 Har tmann curves 43 3.4 LCR for a muon and spin 3/2 nucleus at 9 = 0° 46 3.5 L C R for a muon and spin 3/2 nucleus at 8 = 54.74° 47 3.6 LCR of up to four equivalent spin 3/2 nuclei 48 3.7 Qualitative description of charge exchange 52 3.8 Isotropic Muonium undergoing charge exchange - Qualitative 55 3.9 Isotropic muonium undergoing charge exchange- Quantitative 62 3.10 Comparison between exact and approximate expressions for isotropic muonium undergoing charge exchange 63 3.11 Simulations of the field dependence of l /T i for Mug C in Si as a function of spin exchange rate 68 3.12 Field dependence of l /T i and amplitudes for the two component relax-ation due to MUJBC in Si undergoing spin exchange 71 viii 4.1 OMNI' 76 4.2 HELIOS 77 4.3 Oven 80 4.4 LAMPF spectrometer 82 5.1 Intrinsic Si - Raw data 90 5.2 Intrinsic Si- Field dependence of 1/Ti 91 5.3 Mu level and charge exchange 92 5.4 Temperature dependence of 1/Ti and A^ in intrinsic Si 94 5.5 Temperature dependence of 1/Ti in Si:B 99 5.6 Temperature dependence of 1/Ti in SiP-14 103 5.7 M u ^ peak in Si:P 104 5.8 Donor Ionization in Si:P 106 5.9 Field Dependence of 1/Ti in Si:P 108 5.10 Asymmetry and frequency spectra for heavily doped Si:P, Soil (100) . . I l l 5.11 Temperature dependence of Knight shift in heavily doped n-type Si:P . 113 5.12 Asymmetry and frequency spectra for heavily doped Si:P, BQ\\ (110) . . 115 5.13 Data in n-type GaAsrSi 117 5.14 Mn°BC peak in GaAsrSi 118 5.15 Temperature dependence of the MuD amplitude in GaAs 122 5.16 n-type GaAsrSi - LCR data 124 5.17 Typical TF spectrum for GaAsrSi 127 5.18 n-type GaAs:Si,Te - TF data 128 IX List of Acronyms Acronym AB Anti-bonding site ALCR Avoided level crossing resonance BC Bond-center site CE Charge exchange CFD Constant fraction discriminators C-V Capacitance-voltage ESR Electron Spin Resonance HELIOS and OMNI Magnetic field spectrometers IR Infra-red LF Longitudinal field Mu Muonium Muy Neutral muonium at the tetrahedral site MugC Neutral muonium at the bond-center site Mu c Diamagnetic muonium NIM Nuclear Instrumentation Modules SE Spin exchange SIMS Secondary ion mass spectrometry T Tetrahedral intersitial site Toa Tetrahedral site with four nearest neighbor Ga nuclei x Acronym TD TDTF TF TM /iSR VAX WTF ZF Time differential Time differential transverse field Transverse field Thin muon detector Muon spin rotation, relaxation and resonance Virtual Access extension Weak transverse field Zero field XI A c k n o w l e d g e m e n t s I wish to express my sincerest grati tude to the numerous number of collaborators I have had the fortune to work with in a scientific capacity during my tenure as a graduate student. Foremost, I would like to thank my advisor Robert Kiefi for his patience, interest and support in all my research undertakings, including financial support for at tending conferences and performing experiments abroad. Without his help, a significant par t of this thesis would not have been possible. I would also like to especially thank Jiirg Schneider and Bassam Hitti , who have made major contributions both to the research in connection with this thesis and to my education. Interaction with them has been helpful, stimulating and fun. In addition, I wish to express my grati tude to Thomas Estle and Roger Lichti for their help and tutelage during the last several years. Thanks also go out to Curtis Ballard, Keith Hoyle, Shaun Johnston, Andrew MacFarlane, Mee Shelley, Dirk Schumman, Sarah Dunsiger and Jeff Sonier for experimental assistance at TRIUMF and Masa Senba for stimulating discussions. In addition, I would like to acknowledge our European collaborators Steve Cox, Ted Davis and Claude Schwab for invaluable assistance at PSI and ISIS and for the privilege of working on experiments with them. Special thanks goes again to Steve Cox, Thomas Estle, Bassam Hitti and Roger Lichti for making the time-integral measurments on the n-type GaAsrTe sample at ISIS. I am also very grateful to our collaborators at LAMPF, especially Wayne Cooke, Mel Leon, Mike Paciotti , A. Morrobel-Sosa and L. Zavieh for making the time-integral measurements of the n-type GaAs:Si sample. xii Chapter 1 I N T R O D U C T I O N and B A C K G R O U N D The study of muonium (Mu = fJ,+ e~) via the technique of fj,SK (muon spin resonance, relaxation and rotation) in semiconductors is interesting from two perspectives. Muo-nium is an example of a relatively simple isolated impurity which can provide a test for theoretical methods used to investigate the electronic structure and dynamics of other, more complicated defects. In addition, it provides information about the behavior of its more massive cousin, atomic hydrogen. The properties of muonium are closely re-lated to those of atomic hydrogen since their electronic structures are virtually identical except for small zero point motion effects. The advantage of muonium is that it is far more easily studied in isolated form than atomic hydrogen. Since the discovery that atomic hydrogen easily enters most semiconductors and is chemically active within these materials, a great deal of research (starting in the early 1980s) has gone into studying both hydrogen and its muonium analogue. The fact tha t hydrogen can passivate (deactivate or neutralize) the dangling bonds in amorphous Si (a-Si) is primarily responsible for the semiconducting properties in these materials.[1] This surprising result led to much research into H in Si until the amount of work in this material currently exceeds that in other semiconductors. It was soon demon-strated through combined spreading-resistance measurements and secondary ion mass spectrometry (SIMS) profiling that substitutional B near the Si surface in crystalline Si (c-Si) could be passivated by exposure to an H plasma. [2] Microscopic information on the B-H complex was obtained, using structure sensitive techniques such as infrared I Chapter 1. INTRODUCTION and BACKGROUND 2 (IR) vibrational spectroscopy and ion-channeling. [2] These measurements support a bond-centered structural model where H is positioned along the (111) axis between the B and an adjacent Si. Since the initial experiments on acceptor passivation in Si:B, the array of defects and dopants in crystalline semiconductors that are known to be electrically neutralized by H is growing rapidly in elemental and compound semicon-ductors. Hydrogen forms stable bound states with many defects and impurities; it ties up dangling bonds, passivates shallow donors or acceptors and some deep-level impuri-ties, activates a few originally inactive impurities, and forms various complexes within extended defects. IR studies indicate that when H-acceptor (shallow) complexes are formed, H is positioned in a bond-center site. When H-donor complexes are formed, H is a t tached to one of the donor's Si nearest neighbors rather than to the donor it-self (anti-bonding position). The reactivity of hydrogen in semiconductors can either be beneficial or detrimental. Technological applications involve use of passivation to improve materials properties, isolate devices and fabricate device structures. Perhaps equally important is the unintentional introduction of H during semiconductor growth and processing since such H might migrate during device operation and hence alter device characteristics. Although the passivation complexes have been extensively studied, information on the structure and dynamics of isolated hydrogen in any of its charge states is almost non-existent. The charge state of isolated hydrogen within semiconductors is an important issue since it may influence the interstitial location of hydrogen, its electronic structure and its diffusivity. Furthermore, the charge state will almost certainly influence the reaction rates with other impurities. Thus far, there are only two spectroscopic observations of isolated H in a semicon-ductor, bo th electron spin resonance (ESR) experiments of the paramagnetic "AA9" center in Si. This center was first reported by Gorelkinskii and N.N. Nevinnyi [3] and Chapter 1. INTRODUCTION and BACKGROUND 3 recently confirmed by Bech Nielsen et. al. [4] These experiments show that H° is sit-uated at the center of a Si-Si bond. Although neither the H + nor the H~ centers have been experimentally characterised from a structural perspective, their existence ( H + in p-type materials and H~ in n-type materials) in both Si and GaAs are inferred from electric field-induced migration experiments. [5,6,7,8,9] More recently, Johnson et. al. [10] studied the rates for charge state changes of atomic hydrogen (H + +2e~ <-» H - ) in Si:P by analyzing capacitance transients under reverse bias. These results showed that the hydrogen acceptor level is lower in energy than the donor level, indicating tha t hydrogen has a large negative "effective Coulomb energy", i.e. hydrogen is a negative U impurity in Si 1 . Nevertheless, some researchers still debate the existence of H -[11,12] since the assumed "debonding" of the H-donor complex (in n-type Si) into H~ and the charged donor is found to be at least partially reversible [13]; i.e. on removal of the electric field, the complex is observed again. These reversibility experiments were performed under conditions where all H~ should have been swept out of the region and hence repassivation should not take place without an additional source of hydrogen. Alternative explanations for the da ta in n-type GaAs have also been proposed. [12] The mobility of the various charge states can be inferred from studies of the degree of passivation as a function of depth. These measurements are performed by introducing H into the samples, such as by exposure to an hydrogen plasma at high temperatures (> 150°C), and then studying the "depth profiles" of passivation with methods such as SIMS, spreading resistance and capacitance-voltage (C-V) profiling.[14] In general, the data are usually difficult to interpret and often not reproducible from one lab to the next. The difficulty stems from the fact that H may exist in multiple charge states, some of which have not been convincingly established, and because H appears to be present in a number of different configurations including atomic, diatomic and as a complex. 1In a negative U system, two H°s can lower their energy by changing to H+ and H - . Chapter 1. INTRODUCTION and BACKGROUND 4 In general, in both Si and GaAs, the time dependences of the H depth profiles depart greatly from tha t expected for the diffusion of a single species with a constant diffusion coefficient.[2] Nevertheless, a number of the at tempts to explain the current H data are implicitly based on an assumption that the diffusing species is in a single charge s tate although it has been recognized that monatomic hydrogen may fluctuate between its neutral and positive or negative charge states. Neutral muonium (Mu) is formed when an implanted positive muon (ju+) captures an electron. The muonium atom is thus a very light pseudo-isotope of H, with a mass only l / 9 t h tha t of atomic hydrogen (p+e~). Since /J,+ and p+ have the same charge and spin, muonium and hydrogen in vacuum have nearly identical electronic structures. The muonium and hydrogen hyperfine parameters are proportional to the magnetic moments of the muon and proton, respectively, which are in the ratio of 3.183:1. The muon is relatively short-lived with a mean lifetime of only 2.2 fxs (see Table 1.1). When comparing results on hydrogen and muonium, it is useful to keep in mind some of the important differences and similarities which are as follows: • Since only a very small number (sometimes only one) of muons are in the sample at a time, muon-muon interactions are negligible. This allows experiments on isolated atomic muonium centers to be carried out with ease. In fact, it is these experiments which have provided most of the knowledge of states formed by atomic hydrogen as an isolated impurity. • The muon is short-lived (lifetime of 2.2 fis) and one observes muonium centers which are formed within several ns after implantation. The short t ime scale of muonium studies also implies that the muonium centers observed may not be the equilibrium states, especially at low temperatures where the equilibration t ime is long. This should be contrasted with studies on hydrogen where the experiments Chapter 1. INTRODUCTION and BACKGROUND 5 are performed a significant amount of time (minutes, hours or even longer) after the initial hydrogen implantation or incorporation has occured. Furthermore, if the reaction time to form the Mu-dopant complex is much longer than the muon lifetime, muonium passivation will not be observed. • In the absence of any zero point motion effects due to the mass difference between Mu and H, the potential energy surfaces of both centers are identical. Hence, one expects that under such conditions, the structures of hydrogen and muonium will tu rn out to be the same. If this is true, the energy level positions within the semiconducting gap are expected to be essentially the same for hydrogen and muonium. 2 However, the difference between the zero point energies of the muon and proton, in addition to modifying the potential energy surface slightly, implies that the energy level of muonium in a given potential well lies higher than the analogous proton state. This reduces Mu site change barriers compared to those of hydrogen and may even influence the relative stability of different sites under certain circumstances [17]. Furthermore, the lighter mass of muonium can lead to significant differences in rates for processes which involve diffusion. It will be evident from the discussions in Sec. 1.1 and Sec. 1.2 below that there have been significant advances in the past regarding the study of muonium in semicon-ductors, in particular regarding characterization of the isolated paramagnetic centers. However, many open questions still remain regarding their behavior at high temper-atures where introduction of atomic hydrogen is usually done and in doped materials 2In Si, both hydrogen and muonium are thought to have deep donor levels in the semiconductor band gap. It is interesting to contrast with positronium (Ps = e+e~). Ps is believed to be weakly bound and is a shallow impurity. [15] No clear experimental results have demonstrated the existence of Ps in bulk Si between 10 - 300 K.[16] Chapter 1. INTRODUCTION and BACKGROUND 6 which are the systems of interest for researchers on hydrogen in semiconductors. Fur-thermore, experiments characterizing the charged states of muonium have been virtu-ally non existent. The measurements described in this thesis represent recent a t tempts to investigate the behavior of muonium centers in doped Si and GaAs and include high temperature work in intrinsic Si, studies in p-type and n-type Si as well as experiments in heavily-doped GaAs. The various //SR techniques are powerful tools for studying the spin dynamics of muonium in semiconductors. It is found that at low temperatures, two processes in-volving conduction electrons are important: (1) electron capture to form M u - and (2) repeated spin exchange scattering. In n-type Si and GaAs, the muonium center lo-cated in the bond-center position is found to have a much larger cross-section for spin exchange scattering than for electron capture to form M u - . At high temperatures, sig-nificant ionization of the neutral Mu centers (to M u + ) takes place but retrapping of a conduction electron occurs rapidly, resulting in cyclic charge state changes of muonium (i.e. muonium cycles repeatedly between its positive and neutral charge states). At temperatures above 450 K, Mu spends a significant amount of time near the tetrahedral interstitial site rather than the bond-center position, the theoretical global minimum in the potential energy (see Sec. 1.2). The first experimental determination of the local structure of a charged muonium (or hydrogen) center in a semiconductor is described, in this case the M u - center in heavily doped n-type GaAs at room temperature. Although there is no magnetic hyperfine interaction, M u - is characterised by studying the muon-nuclear dipole interaction with the neighboring nuclear spins and the induced quadrupole interaction on these spin 3/2 nuclei. These studies show that in n-type GaAs, M u " exists as an isolated center with the muon and nearest neighbor(s) Ga on the same (111) axis. The M u - center is likely to be located at or near a Tca site (tetrahedral interstitial site with Ga nearest Chapter 1. INTRODUCTION and BACKGROUND 7 Physical Properties Mass (MeV) Spin Gyromagnetic ratio 7 (rad s_ 1T_ 1) 7 = 7 /2TT (MHz/T) Lifetime r (//s) e 0.51100 1/2 1.75882 xlO11 27992.48 Stable »+ 105.66 1/2 8.516154 xlO8 135.54 2.19703 P+ 938.28 1/2 2.675221 xlO8 42.58 Stable Table 1.1: Some physical properties of Electron, Muon and Proton. [18] neighbors). 1.1 M u o n i u m Centers in Si and G a A s Prior to the initiation of the work described in this thesis, a great deal of experimental and theoretical effort was expended to identify the muonium centers responsible for the various fj,SK signals and to determine their electronic and structural properties. Excellent reviews of the history of Mu in semiconductors are available, such as the comprehensive article by Patterson [19], which details works in the field up to 1988, and the more recent paper by Kiefl and Estle.[20] Conventional transverse-field (TF) , time-differential (TD) muon-spin-rotation (//SR) measurements have demonstrated that when //+ is implanted into covalent semiconduc-tors (Si, GaAs, GaP, Ge and diamond) at low temperatures, three distinct centers are formed — two neutral paramagnetic centers known historically in the /J.SK l i terature as normal (Mu) and anomalous (Mu*) muonium, and a third diamagnetic center referred to as Mu£>.[19] Recently, these labels have been changed to Mu^ for normal muonium and M\igC for anomalous muonium to reflect their charge state and location in the lattice (see below). Chapter 1. INTRODUCTION and BA CKGROUND 8 Figure 1.1: Interstitial sites in the zincblende structure of a III-V semiconductor such as GaAs. The tetrahedral (T) and bond-center (BC) sites that are associated with the paramagnetic Mu centers are shown, as well as the anti-bonding (AB) site. The shaded circles indicate one type of host nuclei while the open circles indicate the other type. This figure will also be valid for a Group IV semiconductor such as Si where both the shaded and open circles represent Si nuclei. Chapter 1. INTRODUCTION and BACKGROUND 9 Semiconductor Si GaAs Center Mug. Mu°c Mug. Hyperfine Parameters A„ = 2006.3 MHz muon: A\\ = -16.82 MHz, Ax = -92.59 MHz 29Si: A\\ = -137.5 MHz, A± = -73.96 MHz AM = 2883.6 MHz muon: A\\ = 218.54 MHz, A± = 87.87 MHz 75As: A\\ = 563.1 MHz, Ax = 128.4 MHz 69Ga: A|| = 1052 MHz, Ax = 867.9 MHz Table 1.2: Muon and nearest neighbor nuclear hyperfine parameters for muonium in Si and GaAs at low temperatures. [20] Further discussion will concentrate on Si and GaAs. Figure 1.1 shows the interstitial sites associated with the paramagnetic centers. The Mug. center has an isotropic muon-electron hyperfine (hf) parameter A^ and is established in both Si and GaAs to be rapidly diffusing between interstitial T sites [20,21,22]. In contrast, Mug C is immobile on the timescale of the muon lifetime and is located between a stretched (Si-Si or Ga-As) bond - i.e. the bond-center (BC) site. As a consequence of the muon location, the hyperfine interaction of M u ^ c is axially symmetric about a (111) crystalline axis and thus described by two parameters A\\ and A± which are approximately an order of magnitude smaller than AM. The electronic equivalence of M u ^ c and the hydrogen AA9 center has been experimentally established by the similarity of their hf parameters after scaling by the magnetic moments, establishing that the larger zero-point motion of the Mu centers relative to that of the analogous hydrogen centers has only a small effect on the hyperfine parameters. The hf parameters for the two Mu centers in Si and GaAs are given in Table 1.2. (For completeness, the nuclear hyperfine parameters are also included). The hf parameters of Mug. are certainly positive, in analogy with Chapter 1. INTRODUCTION and BACKGROUND 10 vacuum Mu, and correspond to a positive spin density at the muon. In the case of the Mujj^ centers, the relative signs of A\\ and Aj_ have been experimentally measured, see Ref. [20]. Also, the absolute signs of the muon hyperfine parameters relative to those of a nuclei are measured quantities. The absolute signs have not been directly measured but can be inferred, as discussed in detail in Ref. [20]. In Si, most theories [2] predict tha t the global minimum in the adiabatic potential energy of neutral muonium or hydrogen is at or very near the bond center (BC) site, if the surrounding lattice atoms are allowed to relax fully. These electronic structure investigations are equally applied to both hydrogen and muonium since both are treated as classical point particles. The conclusion that Mug C is the stable state and hence Muy the metastable state in Si is supported by a study of electron-irradiated Si at 15 K which indicates tha t MuJ- converts to Mu#c .[23] Note, however, that a recent theoretical calculation which takes into account the "quantum nature of the proton and the muon" [17] claims that it is M u ^ c rather than MuJ which is metastable (and only exists till « 100 K); furthermore, the authors find that in contrast to the situation for muonium, H g C is stable. Although the two paramagnetic Mu centers have been spectroscopically well char-acterised, the nature of the diamagnetic center M u D remains a mystery. The M u D centers are frequently observed, but little is known experimentally about their struc-ture (charge state or site) due to the absence of an unpaired electron spin and the accompanying magnetic hyperfine interactions. Either charged state, Mu + or M u - , or any diamagnetic complex containing Mu contributes to the diamagnetic yuSR signal. In the past few years, Mu D has been assigned in the /iSR literature to (1) M u + at the BC site in p-type semiconductors and (2) M u - at the T site in n-type semicon-ductors. This assignment is based primarily on theoretical expectations rather than experimental results. Chapter 1. INTRODUCTION and BACKGROUND 11 The most stable equilibrium charge state of isolated muonium or hydrogen depends on the location of the electronic chemical potential or Fermi level with respect to the muonium level (which is assumed to be in the gap). For example, in heavily doped n-type semiconductors where the Fermi level is close to the conduction band edge, the negative charge state (i.e. H - or Mu~) is favoured while the positive center (i.e. H + or M u + ) is likely to be more stable in heavily doped p-type semiconductors where the Fermi level is close the the valence band edge. Current theoretical results regarding the location of charged hydrogen (or muonium) centers in semiconductors are sparse. In Si, supercell based adiabatic density functional calculations [2,24] predict tha t H + (or M u + ) located at the BC site should be the stable center in p-type materials while H~ (or Mu~) at the T site is the stable center in n-type materials 3 . A similar situation also exists in GaAs where H~ is predicted to be located near the To a site. [2,25] To date, there is no direct experimental information on the structure of isolated H + or H~ in any semiconductor. 1.2 Charge and Sp in D y n a m i c s of M u o n i u m In the previous section, we have given an overview of muonium centers at low tem-peratures and in the absence of any interactions with charge carriers or ionization of muonium. The question naturally arises: "What happens at high temperatures?" In the past , measurements at tempting to answer this question have been performed mostly in Si. 3In supercell approaches, an unterminated cluster of host nuclei containing the defect is periodically reproduced so that translational symmetry can be invoked. The Born-Oppenheimer approximation is used to decouple the nuclear and electronic motions. The basic premise in the density functional treat-ments [2] is that the many electron system is replaced by an effective- particle model. Electron interac-tions beyond the Hartree potential are described with an exchange and correlation potential expressed as a functional of the charge density. The exchange and correlation potential must be approximated and the charge density and effective potential calculated self-consistently. Chapter 1. INTRODUCTION and BACKGROUND 12 In high resistivity Si, measurements of the temperature dependence of the 1/T2 linewidth of the muon spin precession signals show that Mu^ and M u ^ c ionize with activation energies of a few tenths of an eV. The M u ^ c and M u j centers undergo sig-nificant ionization to form Mu f l above « 130 K and sa 230 K respectively. [19] It is unclear whether the final state is Mu~ or Mu + although muon-decay positron channel-ing results [26] at room temperature imply that the ionized species resides near the BC site, leading to the assignment of Mu + (see Sec. 1.1). Above 400 K, the 1/T2 linewidth of the diamagnetic signal rises rapidly, indicating that the ionization is reversible and the inverse reaction Mu* —+ Mu also occurs.[27] In n-type Si, muonium interacts with conduction electrons and as a consequence, muon spin precession signals for both MugC and Mu^ are only observed at low tem-peratures where there is no significant donor ionization. With increasing temperatures, these signals are rapidly damped and become increasingly difficult to observe. [28] The 1/T2 rates of Muj, below room temperature in n-type Si are sample-dependent and also not convincingly explained. [19] Longitudinal field (LF) ^ J S R measurements in n-type Si below room temperature show exponential relaxation of the muon polarization; how-ever, the center responsible is not clearly identified and the physical processes respon-sible are not well understood. Nevertheless, the existence of fast 1/Ti spin relaxation indicates the presence of a paramagnetic species undergoing repeated spin or charge exchange with the conduction electrons. The experimental situation regarding the charge and spin dynamics of Mu centers in GaAs is even less understood than in Si. These studies have been primarily focussed on high-resistivity GaAs. Results of TF-//SR experiments (i.e. 1/T2 linewidth mea-surements) show that significant ionization of Mug C and Mu^ occurs above « 30 K and « 200 K respectively. [19] The diamagnetic center is not observed at low temperatures in high-resistivity GaAs. Chapter 1. INTRODUCTION and BACKGROUND 13 In summary, previous work on muonium in semiconductors has led to character-ization of the structure of isolated paramagnetic centers at low temperatures in the absence of free charge carriers. However, measurements are clearly lacking in several areas: (1) the spin dynamics of muonium at elevated temperatures in intrinsic Si and at all temperatures in doped Si and GaAs and (2) structural information, such as that available for the paramagnetic centers, on the charged diamagnetic centers in semi-conductors. The measurements described in this thesis were undertaken to increase our knowledge in these two areas. The experiments also have relevance for studies of hydrogen in semiconductors. The remaining chapters in the thesis are organized as follows. Chapter 2 describes the //SR technique. Chapter 3 is devoted to a discussion of the theoretical methods used to model the da ta in the thesis. Descriptions of the experimental apparatus with emphasis on the oven follow in Chapter 4. The measurements in Si and GaAs are presented and discussed in Chapter 5 and the concluding remarks in Chapter 6 end the main text of the thesis by providing a summary of the thesis and comments on the past and possible future directions. Appendix A details the complications that arise in the calculation of the yuSR spectra when two isotopes are considered. Chapter 2 / /SR This chapter provides a short description of the versatile //SR technique used to obtain the da ta in this thesis. The acronym //SR stands for muon spin Rotation, Relaxation or Resonance. Detailed expositions of this technique have been presented in a number of previous works, for example Ref. [29]. Sec. 2.1 describes the fundamental aspects of muon decay. This is followed by a general description, in Sec. 2.2, of the time-differential and integral //SR techniques. This section also discusses the longitudinal-field (LF) and transverse-field (TF) //SR techniques. In this thesis, the time-differential LF-^tSR technique ("Relaxation" in the acronym ^SR) was used to study the spin dy-namics of muonium in intrinsic Si, p-type Si:B and intermediate doped n-type Si:P. The time-integral LF technique ("Resonance" in the acronym / J S R ) was used to pro-vide structural information on the charged center, i.e. Mu~, in heavily-doped n-type GaAs:Si. The time-differential TF-y«SR technique ("Rotation" in the acronym /uSR) was also used to study the spin dynamics of muonium in heavily-doped n-type Si:P and to provide complementary information on the structure of Mu~ in heavily doped n-type GaAs:Si. In Sec. 2.3, the s tandard time-differential electronics at T R I U M F is described. The form of the da ta obtained via time-differential TF-/ /SR and LF-^/SR is discussed in Sec. 2.4. A description of how to extract the all-important "muon spin polarization " function from the raw data is given and examples of their appearance are shown. 14 Chapter 2. fiSR 15 2.1 M u o n s All //SR experiments are possible because muons are naturally spin-polarized. In meson factories such as TRIUMF, collisions of protons of energy > 500 MeV with suitable production targets such as carbon or beryllium produce reactions such as those listed below: p + p-*iv++p + n (2.1) p + n->Tr++n + n (2.2) p + n—nr~+p + p (2.3) The work in this thesis regards /x+ and hence only the positive pion will be further discussed. Pions have zero spin and decay with a mean lifetime of 26.03 ns. The dominant decay mode is: 7T+ -> fi+ + v„ (2.4) Low energy pions which stop near or at the production target can be used to produce spin-polarized muons. The muon neutrino v^ has its spin antiparallel to its momentum (i.e. negative helicity) and hence the muon is also emitted with its spin and momentum antiparallel to conserve angular momentum. Such "surface muons", so-called because they originate from pions at rest at the surface of the target, are thus 100% polarized and in the pion rest frame are emitted with momentum 29.8 MeV/c. A beam of such muons has a range of « 120 mg/cm 2 in carbon. These muons stop in the target and decay with a lifetime of rM = 2.197 fis (see Table 1.1) into a positron, an electron neutrino and a muon anti-neutrino: H+ —• e+ + ue + v^ (2.5) Chapter 2. fiSR 16 In a conventional (time-differential) //SR experiment, one measures the time interval between detection of the incoming muon and its subsequent decay positron. Since the muon decays into three particles, the emitted positron energy varies continuously from zero (if the two neutrinos are emitted in opposite directions and carry away all the kinetic energy) to Emax = 52.8 MeV (if the two neutrinos travel together and antiparallel to the positron). The decay probability of the muon can be calculated from the Electroweak theory and involves parity violating interactions. [29] After integrating over the neutrino momenta, the probability per unit time for positron emission at an angle 9 to the fi+ spin is: dW(e, 9) = — [1 + a(e) cos(0)]n(e) de dcos(9), (2.6) where e = E/Emax is the reduced positron energy, a(e) = (2e — l ) / ( 3 — 2e) and n(e) = e2(3 — 2e). The term "1 + o(e) cos# ", which is plotted for various values of e in Fig. 2.1(a), leads to spatial asymmetry particularly for large e where the positron is emitted preferentially in the direction of the muon. The average of dW(e, 9) over all positron energies (e = 0 t o e = l ) yields dW(9) = W(9)dcos(9) (2.7) where (2.8) W(9) is also plotted in Fig. 2.1. It should be emphasized that the feasibility of //SR rests on the fact that the decay positron is not emitted isotropically; instead, it is correlated with the muon spin direction at the time of the decay. W(6) = 2TIL 1 + cos(0) Chapter 2. fiSR 17 € = 1 6 = 0.8 a 6=0.6 muon spin e=0.4 e=0.2 6 = 0 b muon spin Figure 2.1: (a) Polar diagram of the term " l + a ( e ) c o s # " for various reduced energies e and (b) the energy averaged distribution W(8). The distance from the origin to any point on the curve is proportional to the decay rate at angle 9 with respect to the muon polarization. The arrows to the right labelled the muon spin. 'muon spin" indicate the direction of Chapter 2. fiSR 18 2.2 T h e Various /xSR Techniques A fxSR experiment involves implanting highly polarized positive muons into the sample of interest. The decay positrons are monitored to give information on how the muon spin polarization evolved prior to decaying. T R I U M F and the Paul Scherrer Institut (PSI) provide continuous fxSK beams where muons arrive essentially at random times while facilities such as ISIS [30] provide pulsed muon beams where a number of muons enter the sample within a small time range which defines t = 0. The beam structure of LAMPF will be discussed in more detail in Sec. 4.3. Further discussion in this chapter will concentrate primarily on the ^SR facility at TRIUMF. The polarized muons are manipulated by a combination of bending (dipole) mag-nets, focussing (quadrupole) magnets and momentum selection slits as they travel in the beam pipe toward the target sample. Positron contamination of the beam can be effectively eliminated by a Wien (velocity) filter, available on certain beam lines, located prior to the last set of quadrupole focussing magnets before the target. The Wien filter consists of crossed electric and magnetic fields which are both perpendicular to the muon beam, effectively eliminating contamination in the beam from positrons which have the same momentum but different velocity. At the same time, the magnetic field also precesses the muon spin and can therefore be used to align the muon spin in directions from 0° to 90° relative to the muon's momentum - i.e. the Wien filter also acts as a spin rotator. Collimators are usually used to reduce the muon beam spot size since muons tha t miss the sample can produce a background signal. The //SR techniques used to obtain the results discussed in this thesis can be di-vided into two general classifications: (1) time-differential (TD) and (2) time-integral. In a time-differential experiment, the time dependence of the polarization of a muon Chapter 2. fiSR 19 ensemble is measured after its implantation in a sample. Therefore, at continuous b e a m facilities such as T R I U M F or PSI, where muons arrive essentially at random, only a single muon can be allowed in the sample at a time. This requirement is necessary to ensure tha t a detected decay positron can be unambigiously associated with its parent muon. Hence, the primary task of the fast electronics and da ta acquisition system is to select events in which only one muon and one positron are detected within a given t ime interval, which is typically « 20 (is (see Sec. 2.3). Recording many such correlated fi+ — e+ events provides an ensemble average of the behavior of the single muon. At pulsed beam facilities such as RAL, each beam pulse contains a large number of muons which arrive at approximately the same time. Hence, there is no need to associate a given positron with an individual muon. One disadvantage of pulsed facilities is tha t they have a timing resolution limited by the width of the beam pulse. There are also experiments where determination of the time-dependence of the muon polarization function is unnecessary. Sometimes, it is sufficient to measure its integrated value over some length of t ime which is long relative to the muon lifetime — this is the basis of time-integral //SR experiments. Since one is no longer restricted to having only one muon in the sample at a time, there is no theoretical upper limit on the muon rate. The ability to use greatly increased count rates provides increased sensitivity in some types of experiments. In general, the muons and positrons are detected by "counters" in the vincinity of the sample. These counters incorporate plastic scintillators that emit flashes of light whenever they are traversed by an ionizing particle. These light flashes can be manipulated by lucite light guides into photomultipliers. The photomultipliers give rise to voltage pulses which are carried by coaxial cables to the electronics in the "counting room" for processing. Chapter 2. fiSR 20 muo momentum X I y —z —z —z X 0 TF - spin and field perpendicular wTF - spin and field perpendicular LF — spin and field parallel ZF — no field Figure 2.2: Schematic of the typical arrangements for the counters, muon spin and magnetic field in transverse-field (TF) , weak transverse field (wTF), longitudinal field (LF) and zero field (ZF) ^SR. TM labels the Thin Muon counter. B, L, F , R are labels for the Back, Left, Forward and Right positron counters. The arrows under the column labelled "muon spin" indicate the muon spin direction while the arrows under the column "field" show the direction of the applied magnetic field. This diagram is modified from one originally obtained from J. Brewer. Chapter 2. /J,SR 21 Figure 2.2 shows a schematic of a ^/SR experiment (at TRIUMF) . The backward polarized muons are detected by a thin muon (TM) counter and then stop in the sam-ple. In order to allow incoming muons to pass through and also to minimize multiple scattering which will cause the muons to miss the sample and add to the background signal, the muon counter must be thin (?a 250 fim). Depending on the type of experi-ment, either two or four positron counters are used: the Left (L), Right (R), Forward (F) and Backward (B). These are shown in Fig. 2.2. In addition to these, occasions arise where Up (U) and Down (D) counters are also used. The labelling is from the viewpoint of an imaginary person "riding along" the muon beam just before it enters the sample. The B counter has a hole in the middle in order to avoid the path of the muon beam and the F counters are often "split" to accommodate axial cryostats. Typical arrangements of muon spin and magnetic field are also shown in Fig. 2.2. In the ideal (time-differential) transverse field (TF) measurement, the initial muon spin is perpendicular to the applied field. In this configuration, the magnetic field is usually applied parallel (or antiparallel) to the muon momentum while the muon spin is rotated (by the Wien filter) until it is perpendicular to the magnetic field. All four positron counters contain precession information. A weak transverse field (WTF) experiment is similar except that the muon spin is not rotated and a weak field is applied (usually with small coils) perpendicular to the muon spin. This allows for transverse field studies on beamlines where no Wien Fi l ter / spin rotator is available. In a TD longitudinal field (LF) experiment, the muon spin and magnetic field are parallel (antiparallel). In a zero field (ZF) experiment, tr im coils are usually used to cancel out stray magnetic fields and ensure zero field at the sample. In both ZF and LF experiments, only counters along the initial muon spin direction are used. A typical integral experiment has the same orientations of field and initial muon spin as in the LF arrangement. Chapter 2. fiSR 22 2.3 E lec tron ics The s tandard time-differential electronics at TRIUMF is shown in Fig. 2.3. The voltage signals from the muon or positron counters are passed into constant fraction discriminators (CFD) through variable delays. These delays can be used to synchronize the various counters. The output from the CFD is a well-defined NIM (Nuclear Instrumentation Module) pulse (—0.7 V) of width « 50 ns. The muon signal generates the pile-up "gate" (EG & G GP100/NL) of duration » 10 /*s. The length of gate is usually chosen to be at least several muon lifetimes. If another muon or positron pulse is detected during the gate interval, the event will be discarded; hence, very long gates are impractical. A valid start of the clock (LeCroy 4204 time to digital converter) occurs if the muon has arrived when the pileup gate was not triggered. This muon signal also triggers the da ta gate (LeCroy 222 gate generator), generally 500 ns shorter t han the pile-up gate, which determines the time period when a valid clock stop can be detected. If a positron signal (from any one of the positron counters) is detected when there is no pileup and during the data gate, its logic pulse is sent to the NIM/ECL (emitter coupled logic) converter (Creative Electronic Instruments Level Converter LC 5170) which in turn routes the signal to the stop input of the clock. The clock then digitizes the time interval and increments the appropriate "bin" in a CAMAC histogramming memory module (Creative Electronic Instruments HM-2161). The pileup signal is also fed to one of the unused inputs of the converter. Double hit events (one of which could indicate pileup) are rejected by the clock. In summary, a good event is recorded when a single positron is detected after there has been a muon counted (data gate, G) but only one muon (no pileup, P) and no subsequent muon (bit 0) or positron (bits 1,2) is detected within the gate. If the event is valid, then the corresponding time bin in the appropriate decay positron counter Chapter 2. /J.SR 23 TDC start Figure 2.3: Only the L and R counters are shown. Chapter 2. fiSR 24 histogram is incremented by one. Typically about 107 decay events are recorded and the resulting time histograms may be regarded as the ensemble average of such decays. Note that a 250 ns delay is added to the positron counter electronics in order to allow for accumulation of positron events which occur before a muon enters the sample, providing an estimate of the uncorrelated random background. Virtual Address extension (VAX) computers which are interfaced with the CAMAC histogramming modules periodically read, store and handle the display of these data . 2.4 ^ S R D a t a The number of events Ni(t) recorded in a single histogram labelled i has the form: Nt(t) = Ni(0)e-*'T» [1 + AiPi(t)] + bt (2.9) where Ai is a constant whose value depends on factors such as the probability distribu-tion of the emitted positron and the solid angle of the counter, Pi(t) is the projection of the muon spin polarization P(t) on the symmetry axis of the ith detector and bi is the random background which is assumed to be time-independent. Examples of such single histograms are shown in Fig. 2.4a for a T F experiment. Superimposed on the muon decay is the signal of interest, in this case a precession at the Larmor frequency of the muon. Rather than single histograms, the da ta are usually displayed and analysed as "asymmetries" of two opposing matched counters (labelled +i and —i) calculated as follows: ^ ; [Ni(t) - bi] + [N.i(t) - &_,-] y J where the only remnants of the muon's exponential decay (e~//T'1) are the error bars, which are proportional to the square root of the number of events in the time bin and which grow exponentially with time. The 180° phase condition due to the fact Chapter 2. /J.SR 25 Run 5 4 2 : G a R s : S i TF=46 G T=296 K O w 2000 c CD Q_ j2 1000 h c o O 0 IP Q] QI • D O G " 1 m m I 1 1 1 1 i • ID 1 1 1 1 1 1 (a) ' -i^^'Jf "'T*" 0.2 -X CL 0 X < - 0 . 2 -1 I 1 I I 1 I I T " J I I I L J L 0 2 4 6 8 TIME (ILLS) 10 Figure 2.4: Time spectra for fiSR in heavily doped n-type GaAs:Si (concentration of Si is w 5 x 1018 cm" 3 ) at T = 296 K and T F = 46 G: (a) raw da ta and (b) time-dependent corrected asymmetry. Chapter 2. fiSR 26 that the counters are "opposite" implies that P-i(t) = — Pi(t). Further defining a = N-i(0)/Ni(0) and assuming f3 = A_i/Ai = 1 we obtain (1 - a) + Ajl\(t)(l + a) A{t) ~ (1 + a) + AiPlWl - a) {2A1) A(t) in this form is known as the "raw asymmetry" and a is usually fitted to obtain the "corrected asymmetry" A,-P,-(<), the form most common in analysing and presenting //SR data . The corresponding corrected asymmetry plots for the T F example in Fig. 2.4a is shown in Fig. 2.4b. An example of LF data is shown in Fig. 2.5. Note that in an integral experiment, one counts the total number of events i.e. f£ N(t)dt. Ultimately, as discussed in Ref. [31], we are interested in J/2 e_ '/T^P,(i)(it where in this thesis, we are primarily concerned with the LF geometry. It should be emphasized that Pi(t) (or its time-integral) is the quantity of interest in any /xSR experiment, from which one hopes to infer the underlying physics. The form of P{(t) under various physical circumstances relevant to this thesis will be discussed further in Chapter 3. Chapter 2. fxSR 27 Run 978: Si:B, LF=200G, T=546K 2 3 4 5 TIME Gas) ^favvAV^ 8 Figure 2.5: Time-dependent corrected asymmetry spectrum for Si:B (boron concentra-tion « 1016 cm"3) at T = 546 K and LF = 200 G. C h a p t e r 3 T H E O R E T I C A L C O N S I D E R A T I O N S This chapter describes the theoretical methods that form the framework for analysing and interpreting the da ta described in Chapter 5. More specifically, the influence of various physical processes on the muon spin polarization, the quantity of interest in a //SR experiment, will be discussed. We begin by listing and discussing various spin Hamiltonians relevant to the the-sis (Sec. 3.1). In particular, the Hamiltonians describing (i) a diamagnetic muonium center interacting with the neighboring nuclei via the muon-nuclear dipole and nuclear quadrupole interactions and (ii) muonium centers with isotropic and axially anisotropic hyperfine interactions are discussed. Then, a description of a general method for calcu-lating the muon spin polarization for Mu centers is presented (Sec. 3.2). As mentioned in Chapter 2, all physical information is contained in this function. Calculation of the TF-//SR and the time-integral signals for Mu D in the presence of nuclei with spin (in particular spin 3/2) is then discussed (Sec. 3.3 and Sec. 3.4). In Chapter 5, the meth-ods discussed in these sections will be used to obtain structural information on M u - in heavily-doped n-type GaAs:Si. It is found theoretically that , in contrast to a free muon in a magnetic field, a Gaussian damping of the diamagnetic precession signal occurs in a TF-yuSR experiment. The field dependence of this relaxation can help establish site symmetry, the type of nearest neighbor nuclei and the distances to those nuclei. Complementary structural information can be obtained via an integrated LF-/uSR ex-periment. At certain values of the magnetic field, the muon polarization is transferred 28 Chapter 3. THEORETICAL CONSIDERATIONS 29 resonantly to the nuclei. In this situatioin, dips show up in the field dependence of the time-integrated muon spin polarization and can provide valuable information on the location of the muon in the lattice as well as the strength of the quadrupole and dipole interactions. In Sec. 3.5 to Sec. 3.9, spin dynamics is considered where we are interested in the muon spin polarization when muonium is undergoing spin-exchange scattering or cyclic charge s tate changes. Various quantitative approaches are discussed after a qualitative exposition. Theoretically, it is found that both spin exchange scattering and cyclic charge exchange lead to an exponential decay of the LF muon spin polarization. The field dependence of the decay rate provides a clear signature of the muonium center, i.e. whether it has an isotropic or anisotropic hyperfine parameter. In a high field TF-yuSR experiment, muonium undergoing extremely fast spin exchange precesses at a frequency different from the Larmor frequency. This shift is a consequence of the unequal populations in the Zeeman split levels of the muonium electron when the Zeeman interaction becomes comparable to or much greater than the the thermal energy kgT. The methods described here will be used to analyze and interpret (i) the LF-//SR da ta for intrinsic Si, p-type Si:B and intermediate doped n-type Si:P and (ii) the TF-yuSR da t a for heavily doped n-type Si:P (Chapter 5). 3.1 A List of H a m i l t o n i a n s The t ime evolution of the muon spin polarization in semiconductors can be calculated by using quantum mechanics. Hence, in order to describe the work in this thesis, several spin Hamiltonians must be considered. These Hamiltonians will be listed and discussed below. Chapter 3. THEORETICAL CONSIDERATIONS 30 The relevant spin Hamiltonian for the muonium center in a magnetic field B (mag-nitude B) may be written as [20] H = / * 7 e B - S - / r j v B - I + S - A ' i - I + J2[l D ' T + S A8' • T - hfnB • T + J ' • Q4 • J ' ] (3.1) where h is Planck's constant and %, y^ (see Table 1.1) and 7^ are the electron, muon and nuclear gyromagnetic (magnetogyric) ratios respectively; S, I and J' are the spin operators for the electron, muon and ith nucleus respectively; AM and A* are the hyperfine tensors for the muon and the ith neighboring nucleus respectively; D* is the dipolar tensor between the muon and nucleus i and Q* is the nuclear quadrupole tensor for nucleus i (it can be shown that only nuclei with J > 1/2 have a quadrupole moment [32]). The g tensors are all assumed to be isotropic, hence the gyromagnetic ratios are scalars (7 oc g). The very weak dipolar interactions between different nuclei are invariably neglected. Depending on the nature of the Mu center and the neighboring nuclei, Eq. (3.1) can be simplified. If one is interested in the interaction of the diamagnetic center M u ° , then S = 0 and Eq. (3.1) has the form H/h = nz/h + HD/h + HQ/h (3.2) where the spin Hamiltonian consists of the muon and nuclear Zeeman interactions CHZ)i the muon-nuclear dipole interactions (HD) and the nuclear quadrupole interac-tion arising from the electric field gradient at the nuclei (HP): Hz/h = -%BI, + '£-KBJi (3.3) i HD/h = £^ ( - 2 J ^' + w:' + ^ 4 ) i HQ/h = 2 W " - ^ V ' ' + l)/3) Chapter 3. THEORETICAL CONSIDERATIONS 31 where D* = frhwUticrf (3.4) Qi = 3V\zleqi/4Ji(2Ji-l) The magnetic field B is assumed to be applied along the z direction. The nuclear electric quadrupole parameters, Q\ depend on the electric field gradient, Vz>j (which in this case is assumed to be induced mainly by the muon) and the nuclear electric quadrupolar moment (e<?4). The magnetic dipole parameters Dl vary inversely with the cube of the distance from the muon to the nucleus ( r t ) . Since an adequate description of the da ta in this thesis does not require more generality, the magnetic dipole interaction and the nuclear electric quadrupole tensors are assumed to be axially symmetric about a common axis z which is given by the muon-nucleus direction. Since z is tilted by an angle 8 from z, Ix, = Ixcos6-Izsin0 (3.5) h' = Jv Iz' = Ix sin 9 + Iz cos 8 and similarly for S and J . If M u D is located in a system where the dipole and quadrupole interactions with the surrounding nuclei are negligible, then the expres-sion for a free muon in a magnetic field is recovered: H/h = -%BIZ (3.6) The Hamiltonian described by Eq. (3.6) will be a good description for the diamagnetic center M u D in Si. This is due to the fact that in Si, the majority of the host nuclei have zero spin (28Si occurs with 95.3% isotopic abundance) and only a small number (29Si occurs with 4.7 % isotopic abundance) have spin 1/2. Hence, there are no quadrupole Chapter 3. THEORETICAL CONSIDERATIONS 32 interactions and there will be dipole interactions with only a small number of the neighboring nuclei. The basis set labelled by the muon magnetic quantum number m j are eigenstates of Eq. 3.6. Hence, |m/ = +1 /2 ) and |m/ = —1/2) correspond to energy eigenvalues —j^B/2 and j^B/2 respectively. When discussing the interactions of the neutral Mu centers Mu° and M u # c in this thesis, the following simplified spin Hamiltonian will be used: H/h = %BSZ - %BI, + AnSjIt. + A±(SX>IX, + Sy,Iy.) (3.7) The magnetic field is assumed to be directed along z. The last two terms represent an anisotropic, axially symmetric hyperfine (hf) interaction with symmetry axis z (tilted at an angle 6 from z). The parallel and perpendicular hf constants are labelled A\\ and A± respectively. It can be seen by comparing Eq. (3.7) with Eq. (3.1) that all interactions between the muon or the muonium electron with the nuclei have been neglected. This approximation is quite good for Mu in Si for the same reasons as discussed above for Mu f l [Eq. (3.6)] but is less appropriate for Mu in GaAs where all host nuclei have spin 3/2. For completeness, it should be noted that the s and p electron spin densities can be calculated by [20,33] where the free atom values A{ and A* are given by , t 87r n0h2 . . . A{ = y ^ i w . , / ( 0 ) (3.9) where fi0 = 4TT X 1 0 - 7 V s A - 1 m _ 1 is the permeability of vacuum, psj is the s spin density on the the muon, ppj(r) is the p spin density at position r*with respect to the muon and a is the angle between z and f. Note that the s(p) spin density is equal to the square of the normalized valence s (p) wave function of the muon. Chapter 3. THEORETICAL CONSIDERATIONS 33 When A\\ ^ A±, Eq. (3.7) describes M\igC (or similar anisotropic centers with axial symmetry). When A\\ — A± = A^, Eq. (3.7) describes Mu^ (or similar isotropic centers) and can be rewritten as: H/h = %BSz-%BIz + AliS-I (3.10) It can be seen from Eq. (3.8) that A^ is proportional to the unpaired electron spin density at the muon, ^ (O)! 2 [for isotropic Mu in vacuum, 1^(^01 is the Is electronic wave function]. The eigenvectors and energy eigenvalues of Eq. (3.7) are orientation dependent and their determination generally requires the numerical solution of a fourth-order algebraic equation. There are only a handful of cases where they can be solved analytically for arbitary B: (a) isotropic muonium [given by Eq. (3.10)]; (b) anisotropic muonium with 9 = 0° and (c) anisotropic muonium with 6 = 90°. We use the basis set \mjms) consisting of the muon (m/ = ±1/2) and the electron (ms = ±1/2) magnetic quantum numbers in the z direction. The analytic solutions are listed below. The solutions to isotropic muonium [Eq. (3.10)] are as follows: Ei/h = Aj4 + (%-%)B/2 (3.11) E2/h = -AJA + y/Al + a + 7„)2£2 /2 E3/h = Aj4-(%-%)B/2 E4/h = -AJA - y/Al + (Te + 7 , ) 2 # 7 2 and the corresponding eigenvectors are: ki) = I + +) (3-12) |e2) = coso;| - +) + s i n a | + - ) h) = 1--) |e4) = — s ina |—h) + cosa|H—) Chapter 3. THEORETICAL CONSIDERATIONS 34 N ^ ouuu 4000 2000 0 2000 4000 --1 1 1 Mu T in Si i i i i i 1 ^ ^ ^ ^ 2 ^ ^ ^ ^ 3 i i -z ---— 0.00 0.05 0.10 0.15 0.20 0.25 Field (T) 0.30 Figure 3.1: Breit-Rabi diagram for Mu§. in Si (A,, = 2006 MHz). The labels 1,2,3,4 correspond to the value of n in Eq. (3.12). where the symbol "+" indicates mi,ms = +1/2 and "—" indicates mi,ms = —1/2 and cos 2a = x VTT x< (3.13) where x = B/B0 is a dimensionless quantity with B0 — A^j{^e + 7^). The field depen-dence of the energy levels constitutes a so-called "Breit-Rabi diagram", an example of which is shown in Fig. 3.1. The energy eigenvalues for Mu^ c where 0 = 0° are E,/h = A||/4 + ( 7 e - 7 , ) - B / 2 E2/h = -An/4 + y/Al + (% + %)*By2 E3/h = A | | / 4 - ( 7 e - 7 „ ) J 3 / 2 (3.14) Chapter 3. THEORETICAL CONSIDERATIONS 35 with the corresponding eigenvectors |ex) = | + +) (3.15) |e2) = c o s / ? | - + ) + s i n / ? | + - ) N = 1--) |e4) = - s i n / ? | - + ) + cos/?| + - ) where cos2/?=^=JL= (3.16) v i + y2 andy = B/B where B = Ax/(7e + 77i) (i-e. the eigenvectors are the same as isotropic Mu with A^ -+ A±). The energy eigenvalues for MugC where 8 = 90° are Ek/h = \ (A± + v/4(7e - 7 , ) 2# 2 + (A,, - Ax)2) (3.17) J5fc/* = I ( - A x + ^/4(7e + %YB* + (A,, + A± ) 2 ) Es/h = \ (AL - ^4(% - %YB* + (A,, - A 0 » ) E4/fc = i ( - A x - ^/4(7e + 7^252 + (A,, + A± ) 2 ) with the corresponding eigenvectors |ci> = COS77I + +) + sin^j ) (3.18) |e2) = c o s £ | - + ) + s i n £ | + - ) |e3) = -sinryl ++)+COS77I ) |e4) = - s i n £ | - +) + cos£| + - ) where 2(% - %)B cos 2rj — -[ 4 ( T e - 7 ^ i ? 2 + (A | | -Ax) 2 ]* Chapter 3. THEORETICAL CONSIDERATIONS 36 N -400 0.000 0.005 0.010 Field (T) 0.015 0.020 Figure 3.2: Breit-Rabi diagram for Mu^ c in Si at 6 = 70.5° with A\\ = -16.819 MHz and A± = -92.59 MHz. cos2<!; 2(% + %)B [4(7e+7, ) 2 5 2 + (A | |+^ ± ) 2 ]5 (3.19) The field-dependence of the energy levels of Mu# c in Si for 6 = 70.5° is calculated numerically and shown in Fig. 3.2. 3.2 Time-Dependent Muon Polarization of Muonium Centers The density matrix approach is useful for calculating the time dependence of the muon spin polarization in systems consisting of a small number of particles. The problem is the following: given a spin Hamiltonian H, such as those in Sec. —* 3.1, what is the muon polarization P(t) with components Ps(t) (s = x,y,z)7 In the Chapter 3. THEORETICAL CONSIDERATIONS 37 density matr ix notation [32,34], P(t) = (<?*(*)) = Tr[p(t)*»] (3.20) Here, the magnetic field is assumed to be applied along the z direction and crtf are the Pauli matrices corresponding to the r t h component of the muon spin (r = x, y, z): ( 0 1 1 0 ; ^ 0 I ~l \ oj ;< = (i 1° o\ -1 / (3.21) If \s{) are energy eigenstates of H with energy eigenvalues E{, then P(i) = E^IP(0^|e.-> i where in the Heisenberg representation Pit) jnt/n p(0)t -iHt/h (3.22) (3.23) If the initial muon spin is 100 % polarized in the r t h direction and the electron and nuclear spins are unpolarized, then a p(0) = - ( 1 + < ) (3.24) where 1 is the unity operator and K is the dimension of the matrix representation of H.. In a system consisting of the muon and N nuclei of spin J , K = 2(2 J + 1 )^ while K = 4(2 J + 1)N for a system consisting of Mu (fi+ + e~) and N nuclei. By defining ^ ^ (3.25) UJij = n we obtain from Eq. (3.22): P(t) E e-lUJijt K (e.-|(l + 0|ei>(eJ-ph> p-iuiiji E-ir-(e.-krl£i>(eil^k.-> hj K (3.26) 1Ks in most fiSR literature, we do not explicitly use different symbols to distinguish between math-ematical operations between scalars, operators or matrices. These differences should be clear from the context. Chapter 3. THEORETICAL CONSIDERATIONS 38 keeping in mind that Trier*1] = 0. Furthermore, since (v?)2 = 1, it follows that E Ke.K|e.-)|2 = K - £ l<e.-Kki)|a (3-27) and since E = E (3-28) (3.29) the sth component of P(t) is: + i E ( e _ ^ t ( ^ l < k , ) ^ l < K ) + eiw°'t(eik1e.->(e.-k.1eJ-)) (3.30) which can be further simplified in the case where s — r to give: P' ( t) = 1 ~ I E K^Kki) |2 x [1 - oosK-01 (3-31) The r"1 component of the integrated polarization, Pr, the observable quantity in a time-integral experiment (see Chapter 2), is given by Pr = &— / y (3.32) Eq. (3.32) gives Pr = i - fEIN<l e ;> l 2 *(!-*); (3-33) ~ _ e~<l/T" cosfajti) - e-t2/T» cos^-fr) - e"*1^"^-^ sinfu^) + e"* 2^^-^ sinfu;^) ~~ (1 + ufjTlXe-^l^ - e-Wn.) In the special case of <i = 0 and <2 = oo, ^ = 1 - I E I^K%-)I2 x , " * , , (3.34) Chapter 3. THEORETICAL CONSIDERATIONS 39 which is a sum of Lorentzians. In time-integral LF experiments, such as those described in this thesis, r = z. (The choice of i i ^ 0 and t? ^ oo enables one to enhance the signal, as discussed in Ref. [35]). There are only a handful of cases where "approximation-free" analytical solutions can be obtained. These deal primarily with free Mu centers, i.e. the spin Hamiltonians for these centers do not contain interactions with neighboring nuclei. Several examples shall be discussed below to illustrate the use of the above equations. The evaluation of P(t) and P for Mu D interacting with neighboring nuclei shall be deferred till Sec. 3.3 and Sec. 3.4. Consider Mu + or M u " in a magnetic field directed along z. The Hamiltonian is given by Eq. (3.6). In a LF experiment where the initial muon spin is parallel to the applied magnetic field, Eq. (3.31) with s = r = z gives a time-independent muon spin polarization along z of Pz(t) = 1. [Note that Px(t) = Py(t) = 0.] This implies tha t the initial muon spin direction is preserved, a fact that is obviously expected since its initial s tate (spin either parallel or antiparallel to the field) is an eigenstate of the spin Hamiltonian. The muon polarization for a T F experiment can also be calculated in a straightforward manner. Assuming that the initial muon spin is in the x direction and that we are also interested in the muon spin polarization along the same direction, one obtains from Eq. (3.31), with s = r = x, Px{t) = 1 — [1 — cos{u^t)] = cos^^t) where Up = 2ir;yfl is the muon Larmor or Zeeman frequency of the spin 1/2 particle, i.e. muon [36]. A precession signal is expected, as illustrated in Fig. 2.4. The next example is that of isolated isotropic muonium. In a LF experiment, Eq. (3.31) with Eq. (3.12) and Eq. (3.13) yield: P*W = 2 ( I T S + 2(TT^ros^<> <3-35> where UJ2A = 2-KA^y/l + x2. [Note that Px(t) = Py(t) = 0.] Similarly, a straightforward Chapter 3. THEORETICAL CONSIDERATIONS 40 calculation shows that the time dependence of the muon spin polarization in a T F experiment is Px(t) = - (cos 2 a coscvi2t + sin2 a cosu^tf + cos2 a cosu^t + sin2 a cosco^t) (3.36) At high fields, cos2 a —* 1 while sin2 a —> 0 and only the c o s u ; ^ and c o s u ^ t terms have significant amplitude. In the more general case of the (isolated) anisotropic muonium center [Eq. (3.7)], analytical expressions for the time-dependence of the muon polarization are available for 9 = 0° and 0 = 90°. For future reference, we point out that the LF and T F P(t) for 0 = 0° is identical to that of isotropic muonium discussed above with A^ = A±. The interested reader is referred to Ref. [19] for the 8 = 90° situation. Although approximate solutions are available for other values of 9, such as with the effective field approach described in Sec. 3.8, the exact calculation must be done numerically. 3.3 R e l a x a t i o n of t h e Mir 0 Transverse Fie ld Signal As discussed in Sec. 3.2, in a TF-/iSR experiment on a Mu center in Si, the measured muon polarization is well-described as a signal precessing at the Larmor frequency of the free muon since the majority of the neighboring nuclei have zero spin. However, in many other systems, a significant fraction of the neighboring nuclei around a MnD center have non-zero spin. In this section, we will be concerned with the effect of many static, randomly oriented spins on the muon polarization. These random dipolar fields give rise to a spread in the internal field at the muon, resulting in a distribution of precession frequencies (centered about u^ in frequency space) and hence a loss of coherence of the muon precession. This distribution of internal fields (and frequencies) can be shown to approach a Gaussian, and is therefore completely characterised by the second moment, when the number of nuclear spins is large. Therefore, in "time space", Chapter 3. THEORETICAL CONSIDERATIONS 41 the depolarization of the precession signal can be described by a Gaussian damping function exp(—a2t2). Note that in this section as well as in Chapter 5, we use the symbol a to characterize the relaxation while in the previous section, it was used to denote the Pauli matrices. However, no confusion should arise since the meaning of a should be clear from the context. (Furthermore, the symbols for the Pauli matrices contained subscripts and superscripts.) We shall assume that the appropriate spin Hamiltonian for the system is given by Eq. (3.2). In principle, one can calculate P{t) and P exactly using the techniques described in Sec. 3.2. However, this approach can be quite cumbersome for a system containing many spins. Fortunately, a, the parameter of interest, can be calculated from the secular contributions (terms in the Hamiltonian which change the muon Zee-man energy in first order) of Eq. (3.2). The influence of both the dipole-dipole and the quadrupole interactions on a was first calculated by Har tmann [37]. An excellent detailed discussion of the Har tmann formalism developed from a second moment ap-proach is presented by T.L. Estle [38]. Before proceeding with outlining the actual calculation, it is useful to point out that there are two extreme field regions. At low magnetic fields, the nuclear Zeeman interaction is smaller than the quadrupole inter-action. The latter controls a and the nuclei are quantized along the muon-nuclear direction. In the opposite extreme, the quantization axis is along the applied field B and a reduces to the more familiar Van Vleck expression for the dipolar width [39]. Har tmann and Estle show that in a T F experiment the dipole and quadrupole in-teractions cause a distribution of frequencies about the muon Larmor frequency and a2 is proportional to the second moment of this distribution. The second moments are additive and hence one only needs to calculate the second moment contribution for one nucleus (if there are "shells" of equivalent nuclei, then one calculates the contribution due to one representative nucleus and multiplies by the number of equivalent nuclei). Chapter 3. THEORETICAL CONSIDERATIONS 42 The strength of the applied field affects o by determining the axis of nuclear quantiza-tion and hence the eigenfunctions \uk) of the single nucleus Hamiltonian [a special case of Eq. (3.2)]: H/h = -jnBJz + Q[J$ - J(J + l ) /3] (3.37) At high fields where the nuclear Zeeman interaction is much larger than the quadrupole interaction, the applied field direction z is a good quantization axis and the basis set \mj) are eigenfunctions of Eq. (3.37). Clearly, the eigenfunctions will be different when the Zeeman interaction becomes comparable to (or less than) the quadrupole interaction and in general must be determined numerically. The frequency shifts from the muon Zeeman frequency can then be found by calculating the expectation values of the secular dipolar Hamiltonian (the explicit form is given in Ref. [38]) and leads to the following expression for the second moment of the field distribution: M 2 = 2 T ? T ^ ( w ^ ) [(u^\u^^S29-l) + (uk\Jx\uk)3coseSm0}2 (3.38) The sum extends over all 2 J + 1 eigenstates of Eq. (3.37). A sum over all nuclei is then made to obtain the total contribution of M2 and finally a2 = 7^M 2 /2 . At high fields, or in the absence of the quadrupole interaction, |u^) = \mj) and hence [uk\ Jx\uk) = 0. Therefore, in this situation, Eq. (3.38) reproduces the field-independent Van Vleck expression for dipolar coupling between the muon and the nuclei in the system: M 2 = ( f i g s ) 2 ( W r f - 1 ) 2 J ( J 3 + 1 ) (3.39) In contrast, inclusion of the quadrupole interaction produces a strong field dependence of a, particularly when 3 cos2 6 - 1 = 0 (or 6 = 54.74°). This is easy to understand since, in the limit of very high fields, a approaches the Van Vleck value which vanishes for 6 = 54.74°. Fig. 3.3 shows examples of the field dependence of a for several values 6 in the situation of the muon interacting with one spin 3/2 nucleus. (The particular Chapter 3. THEORETICAL CONSIDERATIONS 43 w 3 O .12 .10 .00 0 0 = 0° 5 10 15 MAGNETIC FIELD (orb. units) 20 Figure 3.3: Field dependence of the transverse field linewidth parameter cr as a function of 6 for a system consisting of the muon and one spin 3/2 nucleus interacting via the dipole-dipole interaction and quadrupole interaction. choice of a nucleus with spin 3/2 as an illustrative example is made since this is the spin of the host nuclei in GaAs, see Chapter 5.) Measurements of a as a function of 6 and B can therefore be used to help establish the site symmetry, the type of nearest neighbor nuclei and the distances to those nuclei. Appendix A discusses the complications which arise when the neighboring nuclei exist as two isotopes and when there are two values of 0, such as for Ga (6 9Ga and 71Ga) when B || (110) or (111) crystalline directions. Chapter 3. THEORETICAL CONSIDERATIONS 44 3.4 Quadrupo le Level Cross ing R e s o n a n c e of M u D In this section, quadrupole level-crossing resonance in a system consisting of the dia-magnetic muonium center and N spin 3/2 neighbors will be discussed. We shall label such phenomenon as avoided level-crossing resonance (ALCR) in this thesis. The reason for this name will be discussed in more detail below. Since ALCR has been discussed in much detail elsewhere, such as in Ref. [40], only a brief summary of the main features will be presented. The spin Hamiltonian for this system is given by Eq. (3.2) with J = 3/2 and describes the situation of Mu D in GaAs (see Chapter 5). The experimen-tal investigation of ALCR in this thesis occurs in the time-integral mode and in the geometry of a LF experiment (see Chapter 2) where one is interested in Pz as defined by Eq.(3.32) with r = z. In a muon ALCR experiment, one seeks magnetic field values for which the po-larization Pz(t) of the implanted / i + , which is originally parallel to B , is transferred resonantly to the neighboring nuclei. Recall from Sec. 3.1 that in the absence of inter-actions between the muon and nuclei the initial muon spin state is a good eigenstate of Ti and hence Pz(t) = 1 and Pz = 1 [Eq. (3.32)]. When nuclear interactions exist, this is still correct at most fields and far away from a resonance. A resonance occurs when the muon Zeeman splitting is matched to the appropriate energy splitting of neighboring atoms determined primarily by the quadrupole and nuclear Zeeman interactions. In a plot of the energy levels of the spin Hamiltonian versus field, this would correspond to a crossing of two energy levels if there were no muon-nuclear dipole interaction [HD = 0 in Eq. (3.2)]. An avoidance of the levels will occur if the dipole interaction 7iD, which is usually small compared to the muon Zeeman interaction and the nuclear Zeeman and quadrupole interactions, "mixes" the approximate (i.e. HP = 0) eigenstates cor-responding to these two energy levels. In this situation, a resonant transfer of muon Chapter 3. THEORETICAL CONSIDERATIONS 45 polarization to the nuclei (and vice versa) can occur. At the same time, there is a slight shift in the location of the avoided crossing as compared to the "crossing". The behavior of the energy levels is illustrated in Fig. 3.4(a) and Fig. 3.5(a) for a system consisting of the muon and one spin 3/2 neighbor for 6 = 0° and 8 = 54.74° respec-tively. The size of the energy gap Egap due to the avoidance of the two levels is on the order of the dipole interaction and as can be seen from Eq. (3.31), a fraction of the muon polarization will oscillate at Egap/li. This (slow) oscillation(s) in Pz(t) leads to Pz < 1. (Note that although the phenomenon is frequently referred to in the literature as LCR or level-crossing resonance, this is a misnomer since the resonances occur when there is an avoided crossing. Hence, the name ALC or avoided-level-crossing, which is also used in the literature, is more appropriate. We compromise by using the name ALCR.) As with most experiments of this type, the ALCR spectra in this thesis are obtained by mapping the field dependence of Pz. Such spectra can provide valuable information on the location of the muon in the lattice as well as the strength of the quadrupole [i.e. Q') and dipole (i.e. Dl) interactions. It should be mentioned that (1) groups ("shells") of equivalent nuclei contribute to the same resonance and (2) this resonance is hardly influenced by nuclei in other shells. This ability to selectively study shells of nuclei is one of the most powerful features of ALCR. The fields at which these resonances appear are determined primarily by Q% and the orientation 8 between the applied field B and z while their widths and relative intensities are governed by Dl, 8 and the number of equivalent nuclei. Fig. 3.4(b) and Fig. 3.5(b) show examples for 8 = 0° and 8 = 54.74° respectively in a system consisting of the muon and one spin 3/2 neighbor. Note that the resonances are plotted as 1 — Pz; hence, they appear as peaks rather than dips. The approach we adopt for calculating Pz in a system consisting of a muon and N Chapter 3. THEORETICAL CONSIDERATIONS 46 0.16 0.20 0.24 FIELD (kG) 0.28 Figure 3.4: (a) The field dependence of the energy levels of a system consisting of a muon and one spin 3/2 nucleus (71Ga) with 8 = 0°. Q = —1.472 MHz and D = 2.1 x 10 - 2 MHz. Only one of the four "crossings" shown is an avoided level crossing. This ALC is indicated by an arrow, (b) The corresponding ALCR spectrum. Chapter 3. THEORETICAL CONSIDERATIONS 47 0.16 0.20 0.24 0.28 FIELD (kG) Figure 3.5: (a) The field dependence of the energy levels of a system consisting of a muon and one spin 3/2 nucleus (71Ga) with 8 = 54.74°. Q = -1.472 MHz and D = 2.1 x 10 - 2 MHz. All four of the "crossings" shown are in reality avoided level crossings, (b) The corresponding ALCR spectrum. Chapter 3. THEORETICAL CONSIDERATIONS 48 spin J = 3/2 nuclei involves "brute force" numerical diagonalization of a matrix of size 2(2J + 1 )^ to obtain the eigenvalues and eigenvectors of 7i [Eq. (3.2)]. The Hamil-tonian 7i is represented in matrix form using the product states Im/^mjj) \mjN) = \mimj1....mjN) as the basis set. For very large N, the brute force diagonalization of 7i will become very CPU intensive. (As an example, calculation of the spectrum in Fig. 3.6 which consists of 148 field points requires a total of « 12 hours of CPU time on a VAX 4000-lOOA.) The eigenvalues and eigenvectors are then used with Eq. (3.34) to calculate Pz. In the situation where D <C 1/TM, additional equivalent nuclei produce more intense and slightly broader resonances, as shown in Fig. 3.6. The intensity is roughly proportional to \/Neq where Neq is the number of equivalent nuclei. The situation of neighbouring nuclei with two possible isotopes, such as with Ga (69Ga and 7 1Ga), is discussed in Appendix A. 3.5 S P I N D Y N A M I C S - I N T R O D U C T I O N The remaining sections in this chapter will be devoted to discussion of the spin dynamics of the muonium centers. By spin dynamics, we mean cyclic charge exchange and spin-exchange scattering, where in this thesis, the primary interest will be in the t ime evolution of the muon spin polarization in a LF-//SR experiment. The organization of the later sections will be as follows: Sec. 3.6 contains a qualita-tive description of spin and charge exchange in LF. Quantitative treatments for charge exchange will be discussed via two approaches: (a) the so-called "one-component strong collision approach" (Sec. 3.7) and (b) the more general (Liouville) approach developed by Celio and Odermatt (Sec. 3.8). Also in Sec. 3.8, the quantitative theory of spin-exchange scattering in a LF will be developed from an approach analogous to (b), developed initially by Nosov and Yakovleva [41]. Finally, in Sec. 3.9, the limit of fast Chapter 3. THEORETICAL CONSIDERATIONS 49 0.025 0.020 0.015 0.010 0.005 0.000 0.15 0.20 0.25 0.30 0.35 0.40 0.45 B (kG) Figure 3.6: The ALCR of up to four spin 3/2 atoms at 8 = 54.74°. This simulation was done for 71Ga with Q = -1.472 MHz and D = 1.0 x 10 '2 MHz. 4 atoms Chapter 3. THEORETICAL CONSIDERATIONS 50 spin exchange of spin-polarized muonium in a T F experiment will be developed within the framework of effective magnetic fields. We also note that an alternative approach based on the "stochastic time-ordered method" has been presented by Masayoshi Senba in various papers on cyclic charge exchange and spin-exchange scattering [42,43,44,45]. We shall not discuss these cal-culations in great detail except to point out that those results are (i) currently only applicable to muonium with an isotropic hyperfine parameter and (ii) are in good agreement (where available) with the calculations described below. 3.6 Qual i ta t ive D i scuss ion of M u o n i u m Spin D y n a m i c s in a Longi tudinal F ie ld Although some of the subtler points are missed in a qualitative presentation of spin and charge exchange, it is included in order to give the reader an intuitive understanding of the parameters which are important in such dynamics and how they influence the muon spin polarization. This will be especially helpful when the behavior of M u ^ c undergoing such dynamics is considered in Sec. 3.8. As Chapter 1 pointed out, coherent precession signals from two long-lived paramag-netic muonium states Mu^ and M u ^ c are observed at low temperatures in semiconduc-tors such as intrinsic Si. However, at high temperatures the signals are rapidly damped, indicating tha t dynamical changes occur, the most obvious being ionization of muo-nium. Now, suppose there are also a significant number of free carriers in the system. The ionized muonium can then recapture an electron and cyclic "charge exchange" (CE) processes can occur, e.g. Mu <-> Mu + . In general, the ionization and capture rates will not be the same. Another scenario is the following: suppose the tempera-tures are low such that Mu ionization does not occur but there are many free carriers Chapter 3. THEORETICAL CONSIDERATIONS 51 in the system. This would represent the situation in n-type Si at low temperatures. Mu can also interact with these charge carriers through spin-exchange (SE) scattering whereby the Mu electron and the conduction electron "flip-flop" repeatedly - i.e. the muonium electron cycles rapidly between its ms = + 1 / 2 and ms = —1/2 states. As will be discussed below, both SE and CE lead to similar field and temperature dependences of the muon spin polarization in LF and hence can be interchanged for purposes of qual-itative discussions. The electron dynamics influence the muon polarization through the hyperfine interactions coupling the muon and the electron. We begin by discussing muonium with an isotropic hyperfine parameter undergoing charge exchange in LF. The assumption of an isotropic hyperfine parameter implies that the discussion is strictly valid for a center such as Muj, in Si but not for Mug C . Although the isotropic center is denoted by Mu° in the following discussion, the treatment is also valid for any isotropic Mu center. As was shown earlier in Sec. 3.1, the muon spin polarization in LF for Mu with an isotropic hf parameter A^ is described by the sum of a constant and a term oscillating at the field-dependent o»24 frequency: 1 + 2a;2 1 Pz{t) = 27iT^) + 2aT^)cos(u;24') Px(t) = 0 Py(t) = 0 (3.40) x = B/B0, B0 = A^/ (7 e + 7/i) and u>24 = 2irAti\/l + x2. In contrast, the spin of the diamagnetic center M u + is "locked" along the field direction and the muon polarization does not evolve with time: P,W = 1 Px(t) = 0 Py(t) = 0 (3.41) Chapter 3. THEORETICAL CONSIDERATIONS 52 c o D N _D O CL (a) Slow Time (b) Fast Time to t, t2 Mu Mu + Figure 3.7: Isotropic muonium undergoing charge exchange in the (a) slow and (b) fast limits. Chapter 3. THEORETICAL CONSIDERATIONS 53 Suppose tha t Mu^ is formed at t = 0. The time dependence of the muon polar-ization is shown in Fig. 3.7. At <i, Mu^ is ionized and becomes M u + . The ionized electron will carry off some spin polarization, the amount depending on the ionization rate as described below. The polarization remains unchanged until t2 at which time Mu + recaptures an electron to form Muj- again. Note that the average initial muon polarization at t-i after the first charge exchange cycle is less than at t = 0. This ionization/recapture process continues until the muon decays. The observed muon polarization is simply an average of many such Mu <-> M u + processes. Several qualitative conclusions can be drawn from the discussion above: 1. The muon spin polarization decreases as a function of time - i.e. LF relax-ation occurs or, in standard ^SR or NMR terminology, the muon experiences 1/Ti spin relaxation. 2. There are two extreme regions — fast CE and slow CE. The terms "fast" and "slow" are defined by comparing the Muy ionization rate with the u;24 frequency. In the slow limit, the mean lifetime r of Mu^ (i.e. the inverse of the ionization rate) is much greater than the period of a hyperfine oscilla-tion (27r/u>24). In other words, the muon polarization undergoes numerous oscillations before MuJ ionizes. In this situation, the fractional loss in po-larization per CE cycle is the amplitude of the oscillating component, i.e. A24 = 1/2(1 + x2) , and is independent of r . In contrast, in the "fast" CE limit, r is much less than the period of a hyperfine oscillation; i.e. Mu^ ionizes before the muon polarization can undergo one oscillation. In this sit-uation, UJ24T <C 1 and the fractional loss in polarization per charge exchange cycle is equal to A 2 4^ | 4 r 2 / 2 = TT2A^T2 which is (i) much less than the am-plitude A24 of the oscillating component and (ii) independent of magnetic Chapter 3. THEORETICAL CONSIDERATIONS 54 field. The two limits are schematically illustrated in Fig. 3.7(a) and Fig. 3.7(b). One can now qualitatively predict the behavior of the muon polarization as a function of temperature and magnetic field B , two of the typical independent variables in a /uSR experiment. As the temperature is raised, the ionization and retrapping rates for the Mu <-» M u + process are expected to increase. Hence, we will go from a regime of slow CE to fast CE. In view of the discussion (item 2) above, one expects 1/Ti to rise with temperature initially and then to decrease again when the charge exchange is very rapid. (The ionization rates have been assumed to increase with temperature much more rapidly than the retrapping rate.) This maximum in 1/Ti should occur when XMU ~ w24, the "cross-over" between the slow and fast regimes. A cross-over from the fast to slow spin exchange limits can also be obtained by increasing the magnetic field (this is due to the nature of the CJ24 frequency which increases with increasing field). As discussed above, in the fast charge exchange limit (low magnetic fields), the amount of polarization lost is independent of magnetic field; hence, so is the 1/Ti rate. However, in the slow CE regime, one expects 1/Ti to be proportional to the amplitude of the oscillating u;24 component, i.e. 1/(1 + x2). Hence, at very high fields, 1/Ti oc B~2. The expected temperature and field dependences of 1/Ti for isotropic muonium undergoing CE are summarized pictorially in Fig. 3.8(a) and Fig. 3.8(b) respectively. The case of SE is very similar. Roughly speaking, in the case of spin-exchange scattering, repeated cycling occurs between the following two states: (1) muon and electron spin parallel and (2) muon and electron spin antiparallel. State (1) is an eigenstate of Eq. (3.10) and hence the muon polarization behaves similarly to M u ° in the case of charge exchange, i.e. independent of time. The muon polarization of state Chapter 3. THEORETICAL CONSIDERATIONS 55 o o 1 Slow CE - (a) i ' i i ' Fast \ C E i TEMPERATURE Fast CE -: (b) i i 1 i Slow . CE B" 2 \ • i Log (MAGNETIC FIELD) Figure 3.8: Qualitative (a) temperature dependence and (b) field dependence of \/T\. A B~2 field dependence is reached at high fields. Chapter 3. THEORETICAL CONSIDERATIONS 56 (2) is qualitatively similar to Mu^ (oscillating and non-oscillating components). Hence, the qualitative conclusions drawn from the discussion on charge exchange also applies to spin-exchange scattering. It turns out that these predictions are in excellent agreement with the results of more quantitative t reatments such as those described below. The qualitative discussion of M u g C (or similar anisotropic centers with an axially symmetric hyperfine interaction) undergoing spin exchange scattering will be delayed until Sec. 3.8. 3.7 Isotropic M u undergoing Charge Exchange in a Longi tudinal F i e l d -S trong Col l i s ion Approach The so-called "one-component" strong collision approach used for modelling isotropic muonium undergoing cyclic charge exchange follows quite intuitively from the discus-sion in Sec. 3.6. Consider the process Mu ^ fi (3.42) studied in LF where the applied magnetic field is chosen without loss of generality to be directed along z. Mu denotes the neutral center with an isotropic hf parameter A^ and H the diamagnetic center such a Mu + or M u - . We assume that the time evolution of the muon polarization of each state is completely described by its z-component (i.e. the x and y components are zero). Mu and /J, are assumed to change states at random times with average rates \MU and AM respectively or equivalently, the mean lifetimes of Mu and fj. are 1/XMU and 1/AM respectively. This means that the probability that the muon is still in a certain state after some period of time is simply given by an exponential decay (e_AMu* or e~Xf,t). Furthermore, the jump is assumed to occur instantaneously, Chapter 3. THEORETICAL CONSIDERATIONS 57 or at least much faster than the "fastest time-scale" in the problem. (For example, in the case of charge cycling between the neutral and positive charge states of isotropic muonium, this is usually l/u>2^.) If the time evolutions of the muon polarization while in the // and Mu states are denoted by Q^it) a n d QMu(t) respectively, the process Eq. (3.42) can be described by the coupled integral equations: Gfu(t) = P M „ ( 0 ) Q f " ( * ) e - W + A M £ d r G £ ( T ) Q f " ( t - T ) e - W * - T ) (3.43) G>(t) = P„(0)Q»z(t)e-^ +\MuftdTGM*(T)Q%(t-T)e-x^) where -PM(0) and P M « ( 0 ) denote the initial /i and Mu fractions respectively and G%(t) and G^u(t) are the LF muon polarizations of the appropriate state. In this section, it is assumed that Q\{t) is calculated from the spin Hamiltonian describing the state i. Note, however, that this may not be the case; for example one can use an Ansatz expression for Qlz(t) and hence, the approach described above can be used in situations where the Liouvillian approach in Sec. 3.8 is not clearly applicable. The above equations are similar to those first discussed by Ivanter and Smilga [46] and are very intuitive. Consider the first equation in Eq. (3.43). The first term corresponds to the decay of the muon polarization of the prompt (t = 0) Mu state. The second term is interpreted as follows: X^G^^dr is the "amount" of fi polarization entering the Mu state at time r . Consequently, this polarization evolves as Mu from r to t, subject to the decay law 6-AMU(<-T) Obviously, r must be integrated from 0 to t to account for all possible times. Exactly the same interpretation applies to the second equation except p and Mu are interchanged. The total muon polarization Gzot{t) = G»(t) + G™u(t) is the observed quantity in a time-differential experiment. Note again that in Eq. (3.43), only the time evolution of the ^-components of the polarizations is followed. This approach is valid for isotropic muonium undergoing charge exchange in a longitudinal field since QM«(t) = £}*«(*) = Q£(<) = Q*(t) = 0 (see Sec. 3.2) but fails for M < c since Q f u(t) Chapter 3. THEORETICAL CONSIDERATIONS 58 and Q™u(t) ^ 0 in general. By defining Q and Q as the Laplace transforms of Q and G respectively, g™\s) = PMu(0)Q^u(s + \Mu) +AM<2f*(s + \Mu)g^s) or upon rearrangement, oris) -KQ*(* + *MU)GM = PMu(0)Qr(s + \Mu) (3.44) (3.45) -\MUQ»(S + K)G™\s) +gft(s) = PM(0)Cjf(« + A,) The "solution" of the above equations is straightforward. For example, a possible way to proceed is to use Kramer's rule to express the solution as a ratio of determinants. •>Mu en*) PMU(0)Q?U(S + \Mu) -A„er(s + Aw„) A (3.46) where GM = 1 PMU(0)Q™ U(S + \Mu) (3.47) -XMUQ^S + A J 1 (3.48) Recall tha t we are interested in calculating (7*ot(£) which therefore has the Laplace transform Qi°\s) = Q™u(s) + 0£(a): QT{s) = PMU(0)Q7U(S + XMU) Chapter 3. THEORETICAL CONSIDERATIONS 59 +P„(0)eS(* + A|l) +Q?(s + K)Q™u(s + A M „ ) [ A ^ ( 0 ) + XMUPMU(0)} / l - XMUKQ^S + K)Q™u{s + \Mu) (3.49) @lot(s) with s = l/rM is the quantity of interest in an integral experiment where the integration limits are from 0 to oo [see Eq. (3.32)]. The relevant Laplace transforms of the appropriate expressions in Sec. 3.2 are O ^ , , * 1 + 2x2 1 1 s + \Mu J,z {S + AMu) - 2 ( l + x2)s + x^ + 2 ( 1 + x2) {s + ^ ^ + w a ( 1 + x 2 ) ^ . W J j where the same notation is used and LO0 = 2TTAII and Substituting into Eq. (3.49) yields a equation which is the ratio of a third and fourth order polynomial in s, implying that a partial-fractions expansion is possible: <3l0\s) = - 2 L - + —^2— + - 2 2 - + _ ^ i _ (3.52) s + a i s + a2 5 + 0:3 s + a4 where a; and a t are complex numbers. The above equation in turn implies that Gt?(t) = J2aie-a>t (3.53) t = i which is simply a sum of exponentials. It is straightforward to show from comparing Eq. (3.49) with Eq. (3.52) that at- are roots of the following fourth degree polynomial: a4 -Aa3 + Ba2-Ca + D = 0 (3.54) where A = 3AMu + AM (3.55) B = 3A2,„ + 2AM„AM+u;2(l + :z2) C = (AMU + A ^ A ^ + U ^ I + Z 2 ) ) D = \MUWO/2 Chapter 3. THEORETICAL CONSIDERATIONS 60 and the amplitudes a8 can be obtained by solving the linear equations conveniently written in matr ix form as Ma = Y (3.56) where / M = a2 + Oi3 + «4 Oil + «3 + «4 oti +a2 + a4 Oi\ + a2 + a3 and a2a3 + a2a4 + 030:4 a^a3 + a1a4 + a3a4 axa2 + a1a4 + a2a4 ara2 + axa3 + a2a3 a2a3a4 a\ot3a4 oc\oc2a4 <X\<x2oiz (3.57) ' ^ a = 0.2 0.3 V ° 4 I (3.58) and / Y = \ (3.59) 3AJV/« + AM 3X2Mu + 2XMuXfi + u,2(l + x2) - 0.5u,2(l - P^O)) \ (AM„ + AM)(A^U +o;02(l + x2)) - 0 . 5 O ; 2 ( ( A M U ( 1 - PM(0)) + A,) / In general, Eq. (3.54) and Eq. (3.56) must be solved numerically. It turns out that two of the a ' s are complex and are approximately equal to XMU ± iw24. For most practical purposes, these rapidly damped oscillations are not experimentally observable. The remaining two roots are real: one of them has a small amplitude and a value equal to approximately AM« + AM while the remaining real root (of significant amplitude) is the 1/Ti that is experimentally observable in a LF experiment. These results imply that for isotropic Mu undergoing CE, the experimentally measurable muon polarization is Chapter 3. THEORETICAL CONSIDERATIONS 61 an exponentially decaying function ae~ ' / T l . Examples of the field dependences of 1/Ti for fixed A^ and variable \MU a r e shown in Fig. 3.9a. Similarly, the field dependence of 1/Ti f ° r fixed \MU and variable AM is shown in Fig. 3.9b. Note the agreement with Fig. 3.8 obtained from intuitive arguments. For example, the field dependence of 1/Ti is relatively flat at low fields and approaches a B~2 behavior at high fields. An approximate analytical equation for 1/Ti is more difficult to obtain. From Eq. (3.49), Eq. (3.50) and Eq. (3.51) Ql°\s) = *g (3.60) hot where and top = 1 + 1 - ,A+ A1*AP*<°> w? 1 (s + \Mu)2 + "IJ 2 (3.61) bot = *+ { YMu x ) (?—r4i—r) I (3-62) Now we make the approximation that the "timescale" of observation is much longer than \/\MU- In terms of the inverse Laplace transform, this implies that we are inter-ested in the range where s <C Ajv/«- This yields 1 + <T(-) - ^ - ^ ^ K - i f a ) * (3.63) This equation implies tha t Cr'oi(i) has the form aexp(—t/Ti) with *Mu -r -V \ A M U -f- u>24 ) z This simple expression is a very good approximation to the exact solution and also contains all the essential intuitive features relevant to charge exchange. A typical quantitative comparison of Eq. (3.64) and the exact solution is illustrated in Fig. 3.10. The solution to the problem of M u ^ c undergoing charge exchange is significantly more difficult and shall be discussed in the section below. Chapter 3. THEORETICAL CONSIDERATIONS 62 00 3. IU 101 10° 10'1 1 0 " 10'3 10"4 10"5 10"6 10"7 i n " 8 -------I (a) \ / I 1 1 ' „~2 - 1 10 ps 104 /zs"1 I I ' ---1 _ 1 -\ /LLS . ^ 6 - 1 " 10 /US -N^/ / K . ^S>«N^ ^\T~ -^ -I 1 10 10 10" 10' 10' MAGNETIC FIELD (kG) Figure 3.9: Isotropic Muonium with A^ = 100 MHz undergoing cyclic charge exchange. (a) The rate AM is fixed at 100 / / s - 1 and the field dependence of 1/Ti on XMU is shown. (b) The rate \MU is fixed at 100 ps'1 and the field dependence of 1/Ti °n A^  is shown. Chapter 3. THEORETICAL CONSIDERATIONS 63 10' 10' -10" -10 -10"' -10 10 10 10" 10 10 10 10 MAGNETIC FIELD (kG) 10 Figure 3.10: Isotropic Muonium with A^ — 100 MHz undergoing cyclic charge exchange at a rate of A^ = XMU = 100 pis~l. The approximate expression for 1/Ti given by Eq. (3.64) is plotted as the dashed line while the exact solution as the solid line. Both curves are indistinguishable. Chapter 3. THEORETICAL CONSIDERATIONS 64 3.8 C e l i o / O d e r m a t t M e t h o d and Spin Exchange of M u ^ c in LF As pointed out above, the approach in Sec. 3.7 fails for M\igC undergoing charge exchange since the time evolution of its polarization cannot be described completely by the ^-component alone. Below, we outline the approach described by Celio and Odermatt [27] which adequately describes the transitions among muon states which are of interest in this thesis. The formalism enables one to calculate the full muon polarization for each center provided tha t it can be described by a spin Hamiltonian, and just as importantly, easily extend it to treat the situation where transitions between centers occur. This approach is appropriate for discussion of Mug C undergoing charge exchange and spin-exchange scattering. (Obviously, it can also be used to describe Muy undergoing cyclic charge exchange.) Consider the isolated Mu center. The starting point is the general density matrix p for this two-particle muon-electron system: p = -(l+pli.a+pe.r + pjkajTk) (3.65) where a and r are the Pauli matrices for muon and electron respectively and PM = Tr(pa) pe = Tr(pr) p>k = Tr(pajTk) (3.66) are the muon, electron and mixed polarizations. The commutation relations for the Pauli matrices can be used [27] along with the equation of motion for the density matrix i*% = [H,p] (3-67) to give the following 15 coupled first order differential equations for the polarizations Chapter 3. THEORETICAL CONSIDERATIONS 65 in Eq. (3.66): Pej = ejkl['^pk, + Ynknmpml + u!;ple] U\( x ,U) Pjk = enlm[SjmSkn — (p1^ - p[) - 8jn(—nmnkplll - u™plk) + ,w* hniY^PT+u'eP3"1)} (3.68) where the Einstein summation convention is used and j , k = 1,2,3. This spin Hamilto-—* —* nian H is assumed to be given by Eq. (3.7) and LJ^ = 2^^ilB, ue = 27r%B, UJ\ — 2TTA±, u* = 27r(^4|| — Ax)\ n is a unit vector along the hf symmetry axis of Mu^ c . The above equation can be written compactly in matrix form as: dP dt = MP (3.69) where ' a^ P = Pe )k (3.70) \p"J and M is a 15 x 15 matrix obtained from Eq. (3.68). Obviously, P = P(t) is time-dependent. The solution to Eq. (3.69) is well-known and has the form 15 P = P(t) = J2 c^ exp(-a,-<) (3.71) i where c,- and «; can be complex numbers and at- is a (complex) 15 x 1 column matrix. The eigenproblem M -di = aidi (3.72) is solved to obtain a, (eigenvalues) and a,- (normalized eigenvectors). Then ct- can be obtained from the following system of linear equations: P(0) = f > a t - (3.73) 4 = 1 Chapter 3. THEORETICAL CONSIDERATIONS 66 The initial muon spin direction determines P(0) . For example, in a LF experiment, pM(0) = (0 ,0 ,1) . Similarly, in a T F experiment pM(0) = (1, 0,0) where the muon spin is initially in the ^-direction. In both geometries, we assume pe(0) = 0 and all p^k(0) = 0. Note that since M is anti-symmetric, the eigenvalues <*i are pure imaginary and P consists of undamped precession components. Although Mug C undergoing charge exchange is not treated quantitatively in this thesis and hence the approach discussed in Sec. 3.7 is completely adequate, we include for completeness a discussion of charge exchange within the Celio/Odermatt framework. Suppose there are two states 1 and 2 whose polarizations P1 and P2 evolve with time as described by Eq. (3.69) with Mi and M2 respectively. As a specific example, these two states might be the Mu and // centers of Sec. 3.7. Furthermore, we assume that the transition 1 —• 2 occurs at a rate A12 and the reverse process 2 —• 1 occurs at a rate A21. By a generalization of the one-component coupled integral equations Eq. (3.43), Celio and Odermatt [27] show that : dt / 3 \ / \ I 6 (3.74) P i \ P 2 , Mi - A12 A21.4 \ A12A M2 - A2i J \ Each "element" in the above matrix is a 15 x 15 block matrix of the form Eq. (3.68). The "off-diagonal" terms Ai2-4 and A2i«4 describe the polarization transfer. Loss of transfer of electron polarization during the transition is accounted for by the 15 x 15 matrix A Aij = &ij if i , i = 1,2,3 (3.75) Aij — 0 if i,j = 4....15 If electron polarization is conserved during the transition, A is the identity matrix. Solution of Eq. (3.74) is a obviously a straightforward generalization of the solution Chapter 3. THEORETICAL CONSIDERATIONS 67 of Eq. (3.69). The total observed muon polarization is P = P\ + P2. Note that since the matrix of interest in Eq. (3.74) is no longer anti-symmetric, the eigenvalues can in general be complex. Hence, P can in general consist of precessions (a t imaginary), relaxations (a ; real) or damped precessions (a, complex). As expected, in the case of charge exchange of the isotropic Mu center, numerical calculations show that the muon polarization is a single-component exponential decay with a relaxation rate 1/Ti identical to those obtained as described in Sec. 3.7. The quantitative model used for spin exchange (in an unpolarized medium in LF) of M u ^ c (and Mu^) follows the treatment originally developed by Nosov and Yakovl-eva [41] where spin exchange between the Mu electron and the surrounding leads to "flipping" of the Mu electron. This can be accounted for by adding the rate param-eter V$E in a purely phenomenological fashion to the electron and mixed polarization components in Eq. (3.68): Pi = ....-2p>euSE ^ = ....-2pjkuSE (3.76) The numerical solution is similar to that described above. Approximate analytic ex-pressions derived from this formalism and for isotropic Mu in the limits of very slow and very fast spin exchange are available and the reader is referred to Ref. [19] and references therein. It is interesting to compare the phenomenological rate USE with the so-called "spin-flip rate" \$F defined in the stochastic time-ordered method of Ref. [43]. The XSF is interpreted as the probability that the electron spin of muonium is flipped after a collision with a medium electron. It was shown in Ref. [43] that USE = ^SF/2. Nevertheless, in the following discussions, we shall still refer to USE a s the "spin-flip ra te" . Fig. 3.11(a) shows a simulation of the field dependence of the muon 1/Ti a s a Chapter 3. THEORETICAL CONSIDERATIONS 68 100 0.01 -I 0.0001 -if) =1 -• ^ ---,. 1 /XS I J 10 /XS"1 / (a) 10° 1 -1 /IS I -10000 /xs"1 - ^ / 1000 JUS'\ ^ ~ \ \ ^ " ^ ^ i ^ - -100 0.01 -^ \ ^ (b) 1 /IS 10000 /us~'V^\. f \f 1000 /xs-1 / 100 ^s~1 10 /xs 1 --" ^ s -10 100 FIELD (mT) 1000 Figure 3.11: Theoretical simulations of the field dependence of 1/T\ for M.\x°BC in Si as a function of various USE with (a) 8 = 0° which is equivalent to isotropic muonium with Ap = Ax. and (b) with 6 = 70.53°. Chapter 3. THEORETICAL CONSIDERATIONS 69 function of the Nosov-Yakovleva spin-exchange rate USE specifically for the case of M u £ c in Si (A|| = -16 .82 MHz, A± = -92 .59 MHz) where the angle 9 = 0°. As indicated in Sec. 3.1, Mug C at 9 = 0° is exactly equivalent to isotropic muonium (with A^ — Ax)- Therefore, this situation is equivalent to spin exchange of isotropic muonium with Ap — A±. As expected, the depolarization of the muon spin is well described by a single relaxing component of the form exp(—t/T\) (see Sec. 3.7). Regardless of the spin-exchange rate USE-, the field dependence of 1/Ti for isotropic muonium is relatively flat at low fields (fast spin-exchange regime) and approaches a B~2 dependence at high fields where USE is less than the CU24 frequency (slow spin-exchange regime). In contrast, in the case of Mug C for 9 ^ 0°, the LF muon spin polarization is described by two relaxing components for slow spin-exchange rates. The two 1/Ti rates and their amplitudes are shown in Fig. 3.12 for M u ^ c in Si, 9 « 70.53° and VSE = 5 i±s~x. The two rate constants approach USE a n d the two amplitudes are roughly equal at B w Bp (Bp is defined below). In the field ranges far away from Bp, one component has a very small amplitude and hence will not be observable. For fields near Bp, the two rates approach each other and the polarization will appear to have only one relaxing component. The dashed line in Fig. 3.12 is the "experimentally observable" field-dependence of 1/Ti. This "average" curve is obtained by least-squares fitting of the two-component theoretical relaxation assuming a single relaxing component using roughly the same binning and time range as the experimental da ta (in this example, 100 points in a range of 8 fis). As expected, this curve closely mimics the field-dependence of 1/Ti for the component with larger amplitude. The average 1/Ti rates are plotted in Fig. 3.11(b) for 9 = 70.53° for various USE- The most startling feature is the peaked resonant-like feature in 1/Ti for slow spin flip rates. Note the maximum value of 1/Ti at the peak is approximately USE- A similar peak is observed for any 9 away from 0° and 90°, with the peak field depending on 9 as demonstrated below. As one reaches Chapter 3. THEORETICAL CONSIDERATIONS 70 the fast spin-exchange regime, the peak disappears although there are still significant differences in the detailed field dependences compared with the isotropic or 9 = 0° case. Such resonant-like spin relaxation provides an unmistakable signature for M u ^ c . In principle, it can also be used to estimate hyperfine parameters and spin-exchange rates under conditions where muonium spin-precession signals are difficult, if not impossible, to observe. A qualitative understanding of the origin and characteristics of the peak in 1/Ti is easily understood from the "effective field" picture. At "high" fields such that the electron Zeeman interaction greatly exceeds the hf interaction, an approximate spin Hamiltonian describing the muon subsystem can be obtained [39] from Eq. (3.2). For each value of the electron magnetic quantum number m s , the approximate Hamiltonian gives rise to an effective magnetic field [19,39] acting on the muon spin which in general has components both parallel and perpendicular to B 0 . If the effective field is labelled as B e / / then {„ (A±-An)sin29\ .„ 1 Beff = \B0 - Bp, -V X "; J if ms = +-(n (AL-A»)sm29\ .„ 1 , „„, Beff = \B0 + Bp, +y J*> J if ms = -- (3.77) where the first (second) term in parantheses is the parallel (perpendicular) component of the effective field. Furthermore, _ ( A ± s i n 2 0 + ,4||cos2fl) P ~ 2% • ( } The total muon polarization is the average over the precession of the muon spins about these two effective fields. For example, in a T F experiment involving isotropic Mu (.A|| = A± = A,,,), the two (ms = + 1 / 2 and ms = —1/2) frequencies (-j^Befj) obtained approximate the high field u>12 and u>34 frequencies (see Sec. 3.1). Since we are consid-ering LF, the initial muon spin is parallel to B 0 and the muon precesses in a cone of Chapter 3. THEORETICAL CONSIDERATIONS 71 CO 3. \— 10 8 -4 -I -: A / N-i i (a) -B (in kG) Figure 3.12: Longitudinal-field (a) relaxation rates and (b) amplitudes for M.\i°BC in Si with orientation 8 = 70.53° and VSE = 5/J.S'1. The dashed curve in (a) is the "experimentally observable" curve (see text). The field-dependent amplitude in (b) traced by the solid line is associated with the l/Ti rate traced by the dashed curve in (a). Chapter 3. THEORETICAL CONSIDERATIONS 72 fixed angle about each effective field. Hence, the projection of its polarization onto an axis parallel to Bo consists of a constant and an oscillating component. As shown in Eq. (3.77), when Bo = Bp, the muon polarization for states with ma = + 1 / 2 precesses about an effective field which is perpendicular to the applied field (and the initial muon polarization) since the parallel component of the effective field is zero. At this field, the ampli tude of the oscillating component is the largest. When B0 is far from Bp, this ampli tude rapidly goes to zero. As discussed in Sec. 3.6, a useful approximate picture of LF depolarization is that for small USE (slow spin-exchange scattering), the amount of muon polarization lost per electron spin-exchange cycle is proportional to the ampli tude of these oscillations. Hence, in the slow spin-exchange limit, l/T^ (1) scales linearly with VSE and (2) is expected to reach a maximum near Bp. For M u ^ c in Si, Bp = 0.32 T when 0=70.53°, as shown in Fig. 3.11(b) and Fig. 3.12. 3.9 TF-^uSR in the Fast Spin Exchange Limit Consider a TF-//SR experiment on M u ^ c in the fast spin exchange limit. As the muonium electron flips rapidly between the ms = + 1 / 2 and ms = —1/2 states, there will be a rapid averaging of the two effective fields described in Eq. (3.77) such that Bav = Beff (m. = - ) + B e / / ( m s = - - ) = (B 0 ,0 ) (3.79) In a T F experiment, this implies that muonium undergoing very fast SE is indistin-guishable from the diamagnetic center undergoing Larmor precession. However, now suppose tha t the m s = + 1 / 2 and ms = —1/2 states are not equally populated; in other words, some small spin polarization has been achieved. Such a situation might occur at low temperatures and high magnetic fields such that the electron Zeeman energy (%B0) is comparable to or much greater than the thermal energy (kgT). If fi/2 and /_i/2 denote the populations in the ms = + 1 / 2 and ms = —1/2 states respectively and Chapter 3. THEORETICAL CONSIDERATIONS 73 the spin polarization is measured by A P — f-1/2 — f+i/2, Eq. (3.79) will be modified to Bav = h,2 x B e / / (m, = + 1 / 2 ) + /_ 1 / 2 x B e / / (m , = - 1 / 2 ) (3.80) This leads to B „ = (B. + A P > S i ° 2 ' t * » " " ' ' , AJ» • < * • ~ > > " - 2 " ) . (3.81) (The dynamics for Muj. will be obtained if one sets A\\ = A± = A^. In the special case of isotropic Mu in the extremely fast spin exchange limit, Eq. (3.81) is identical to the result in Ref. [45] obtained via the stochastic time-ordered method.) The muon precesses about Hav at frequency jfj.Bav. Hence, a frequency shift will occur when A P 7^  0. In the limit where A P <C 1 and neglecting terms with ( A P ) r where r > 1, BlKBJ1 + Ap.A^^colO\ ( 3 8 2 ) Consider a simple example of an isolated electron spin (e.g. isolated muonium) in a magnetic field Bo- The energy levels corresponding to the ms = —1/2 and ms — + 1 / 2 are separated by the Zeeman interaction by an energy 7e-Bo. Assuming that the population of each level is governed by Boltzmann statistics, _ 1 _ e-h%B0/kBT hXfeBo A P ~ 1 + e-h%B0/kBT ~ ^ T ( 3 ' 8 3 ) where ks is Boltzmann's constant and the last approximation in Eq. (3.83) is valid when A P <C 1 (e.g. at high temperatures). The 1/T dependence of the polarization leads to the so-called Curie law prevalent throughout discussions of the magnetization or magnetic susceptibility [47]. In this limit, Eq. (3.82) and Eq.(3.83) give = Pp - Bav = -h%(Ax sin2 6 + A\\ cos2 6) Bo 4%kBT [ ' ' where we call the normalized frequency shift K, the Knight Shift. We also point out that as in LF, if the spin exchange is extremely rapid, then there will be negligible relaxation in the precession signals. Chapter 4 A P P A R A T U S This chapter describes the peripheral apparatus used to obtain the //SR data in this thesis. This includes a brief description of various spectrometers at TRIUMF as well as cryostats and the oven. The fxhCK spectrometer used at LAMPF will be also described. 4.1 T R I U M F S p e c t r o m e t e r s Three magnetic field spectrometers, all part of the TRIUMF fiSH facility, were used to obtain the data. These instruments have also been discussed in detail elsewhere, such as in Refs. [31], [48] and [49]. OMNI and OMNI' , shown in Fig. 4.1, are nearly identical spectrometers which provide magnetic fields up to « 4 kG along the direction of beam travel and smaller fields of < 100 G in perpendicular directions. OMNF is also equipped with x — y — z t r im coils which allow compensation of stray fields in order to obtain zero field. Magnetic fields are generated by coils positioned in a Helmholtz arrangement. Incoming muons are detected with thin (typically < 0.4 mm) plastic scintillators which enable the muons to pass through. The TM counter also detects contamination positrons and the decay positrons from muons, but with a far lower efficiency. Decay positrons are detected with thick ( « 1 cm) plastic scintillators coupled to photomultiplier tubes through Lucite light guides. The arrangement of counters discussed in Sec. 2.2 is a reasonable representation of the actual situation for both OMNI and OMNF. In a LF experiment, the F and B counters (see Fig. 2.2) form matched pairs from which one can obtain the 74 Chapter 4. APPARATUS 75 corrected asymmetry spectra. In a T F experiment, the L and R counters contain the relevant precession information, while in a weak T F experiment all four counters are used. HELIOS, shown in Fig. 4.2, is a warm-bore superconducting magnet capable of gen-erating magnetic fields up to 7 T. The thin muon (TM) counter is a circular disk about 0.4 m m thick attached to cylindrical light guides and eventually to photomultipliers (not shown in the diagram) outside of the bore of the magnet. The forward positron counters are made up of four segments « 0.6 cm thick; these counters are arranged cylindrically because of geometrical restrictions imposed by the magnet 's cylindrical bore. The counters are usually labelled Forward Top Right (FTR) , Forward Top Left (FTL) , Forward Bottom Right (FBR), and Forward Bottom Left (FBL). In a LF-juSR experiment (see Sec. 2.2), the forward counters are positioned so that there is no over-lap with the sample, as shown in Fig. 4.2. These counters are logically connected ("or"ed) using coincidence units to form a single telescope. HELIOS is also equipped with a set of Backward Counters, not shown in Fig. 4.2, positioned upstream of the sample. These were originally similar to the Forward positron counters (consisting of four segments) but were changed eventually to the style of the TM counter (with a hole to allow passage of muons). This was done to reduce background events from decay positrons of muons stopping in a collimator upstream of the sample. In a LF-//SR experiment, all Backward counter segments are also logically ORed together to form a single telescope. Asymmetry spectra can be formed from the Forward and Backward telescopes. In a T F experiment, the forward counters are repositioned so there is some overlap over the sample. These four counter segments function as U,D,L and R de-scribed in Sec. 2.2 during the discussion of the TF-^SR geometry. (Opposing segments are used to form the asymmetry spectra.) The Backward counters are not used in a TF-/uSR experiment. Chapters APPARATUS 76 Approximate I I Scale 0 20 Inches Figure 4.1: Schematic of the Top View of the OMNI' spectrometers. Also shown is the beam pipe with some collimation and a cryostat or an oven. This figure was modified from one originally provided by T.M. Riseman.[48] Chapter 4. APPARATUS 77 Forward e Counter W/////////W/M Sample Rod He Space Vacuum Cryostat TM Counter Forward e Counter XpJ /x+beam Sample Beam Vacuum HELIOS Magnet Bore 0150mm Approx . Scale: B, 0 75 150 m m Figure 4.2: A cutaway showing the central portion of a high-field experimental arrange-ment with HELIOS. Also shown is a schematic of the He gas flow cryostat. Chapters APPARATUS 78 4.2 Cryos ta t s and the O v e n Several cryostats and an oven were used to obtain the data discussed in this thesis. Data obtained for temperatures below 300 K were obtained using a horizontal Cu cold-finger cryostat or horizontal or vertical He flow cryostats. Thin (about 0.002 inch) Kapton windows allow beam access. These apparatus are part of the TRIUMF yuSR-facility and have been discussed in previous works such as Ref. [48], [31] and [49]. The samples studied were all significantly larger than the beam spot size (usually w 10 — 20 mm) after collimation. They were greased with Apezion N onto a thin (0.25 mm) silver plate which was in tu rn greased onto the Cu sample holder. In the case of the cold-finger cryostat, thin aluminum foil was wrapped around both the sample (and silver backing) and the cold-finger to improve temperature homogeneity. The temperatures were measured with a Pt resistor and/or a Si diode from Lakeshore Inc. located within « 15 m m from the sample. In the gas flow cryostat, the samples (and holders) were held in place with a combination of thin transparent X-ray Mylar film (of thickness 3.5 /mi) from Chemplex Industries, Inc. and teflon tape. Temperatures were measured with a P t resistor and /or a Carbon Glass resistor. Temperature control was achieved by use of either a commercial temperature con-troller from Lakeshore (e.g. DRC82C or DRC93C) or a homemade PID controller (designed by Dr. Sydney Kreitzman) connected to the thermometers and bifilar wound resistance heater wires on the sample holders. In order to reach temperatures from 300 K to 1000 K, a horizontal access hot-finger oven was constructed. Since this apparatus has not been described in previous works, it is described in slightly more detail below. The outside of the oven must be kept at room temperature since it is always in proximity to positron or muon counters. As with all /iSR apparatus , muons must be able to reach the sample and positrons must Chapter 4. APPARATUS 79 be allowed to leave. The schematic of the oven is shown in Fig. 4.3. The oven consists of a cylindrical jacket constructed from SS316L stainless steel and an insertable sample rod. The jacket is terminated by a groove for a Viton O-ring which, in conjunction with a mask containing an 8 hole screw pat tern, holds a 0.002 Kapton window in place. Thin-walled Cu tubes soft soldered onto the end of the shell allows for water cooling. During high temperature operation, the inside of the jacket is evacuated. At low temperatures ( « 400 K and below), about 1/4 a tm. of an inert gas was sometimes let into the oven to improve thermal conduction with the surroundings. The sample rod is a thin-walled SS316L stainless steel tube with a copper sample holder screwed in place at the end. The rod and holder are held concentric in the tube by two triangular SS316L concentricity spacers. The triangle shape minimizes thermal conduction from the rod to the outside. Bifilar wound resistance wires with an Inconel sheath from Philips Electronic Instruments Company served as heater wires. These wires consist of a high resistance ("hot end", « 13 0,) and a low resistance ("cold end", w 0.2 ft) section. The hot end was wound in a bifilar fashion around the sample holder and held in place by ceramic epoxy. Winding the heater wires on the outside allowed the sample holder to be designed in such a way that there is only minimal mass behind the sample which would impede positrons leaving in that direction. The sample sits on a thin copper plate (thickness « 0.8 mm) located in the middle of the sample holder. This plate is held snugly by copper posts and specially constructed copper nuts. The sample is "wrapped" in thin tantalum foil (the thickness of one layer of Ta foil is 0.0127 mm) which acts as a diffusion barrier between copper and Si [50]. Samples (and Ta foil) are held in place by copper nuts. A P t thermometer behind the plate is used to monitor the temperature. Copper foil of thickness « 0.01 m m was used to tightly enclose the Cu sample holder to improve temperature homogeneity and Chapter 4. APPARATUS 80 Shel Sample Rod Vacuum Space Spacers Holder Post Water Cooling Window Heater. 4 cm Sample^ Plate Figure 4.3: Schematic of the relevant features of the oven, shown in cross-section. The approximate scale applies to the enlargement of the sample holder. Chapter 4. APPARATUS 81 reduce heat loss through the Kapton window. About 100 W was required to reach 1000 K. 4.3 L A M P F S p e c t r o m e t e r The /iSR facility LAMPF in New Mexico, U.S.A, and the //LCR spectrometer located there, are discussed in this section since the ALCR data in this thesis were obtained there. Recall from Sec. 2.4 that in an ALCR experiment, one is interested in measuring the integrated polarization P as a function of applied magnetic field. Unfortunately, during the time of writing this thesis, the future of LAMPF, which has the world's best /xSR facility for this type of measurements, (i.e. integrated LF experiments) looks bleak. A more detailed discussion of the facility and the spectrometer is given in Ref. [51]. The peak fx+ rate is « 3 x 10 8 ^ + / s whereas the average /J,+ rate is « 2 x 107//+/s> which is approximately 10 x higher than the M20B p+ channel at TRIUMF. The proton beam has a 675 fj,s pulse length, 120 Hz (hence a duty factor of 8.1%) and an average current of 650 fiA. The muon beam is collimated to w 4 cm FWHM. The spectrometer and counter arrangements are shown in Fig. 4.4. There is no muon counter and the positrons are detected by six Forward (F1-F6) counters cut and fitted together in a roughly spherical shape. Five Side (S1-S5) counters are also available for monitoring the beam. Because of the high fi+ peak rates, aluminum degraders (D) were used to help avoid saturation of the e+ counters. This also has the advantage of increasing asymmetry since the lower energy positrons are discarded. Long-time instabilities in the muon beam are "averaged out" by applying a ramped supplementary field (via M l in Fig. 4.4) at a rate of 400 G/s (instead of using a "flip coil"). Many such scans are performed and added together to form an ALCR spectrum. Flux coils (FC) inside a cold finger cryostat containing the sample sense the applied Chapters APPARATUS 82 Side View Beam's View of Counters -10 c m -SCALE Figure 4.4: ;/LCR spectrometer. S: Sample; H: Sample Holder; C: Cryostat; D: Al degrader; Sl-5: side positron counters for monitoring beam; Fl-6: downstream positron counters; Ml: longitudinal field scan coils; FC: flux coils; BP: beam pipe. The 5 kG main coils are not shown. Figure modified from that provided by M. Paciotti; see also Ref. [51]. Chapter 4. APPARATUS 83 ramp field. In order to obtain magnetic fields greater than 400 G, a static longitudinal field of up to 5 kG can also be applied. The da ta can either be presented as NF/NS or Np/Np, where Np is the total count rate in the F counters, Ns is the total count rate in the S counters and Np is the total proton count rate. Sometimes, one choice is bet ter than another. Chapter 5 M E A S U R E M E N T S and D I S C U S S I O N This chapter describes and discusses the results obtained. After some preliminary comments that will be useful for the reader who is at tempting to follow the ensuing discussion, the da ta in intrinsic, p-type, intermediate n-type and heavily doped n-type Si will be presented where the emphasis is on the charge and spin exchange dynamics. This will be followed by discussions of the data in heavily doped GaAs which include spin exchange dynamics and characterization of the negatively charged muonium center. 5.1 Useful Informat ion In the sections below, the units Gauss (G) and Tesla (T) will both be used for describing the strength of magnetic fields. Note that 1 T = 10 kG = 104 G. Also, there is often mention of applying a magnetic field Bo parallel to a certain high symmetry axis, i.e. (100), (110) or (111), of the semiconductor sample. The corresponding values of 6 that must be used in the theoretical equations in Chapter 3 as well as the number and the nature of the neighboring nuclei depend on the muon site. Suppose the muon or muonium is located in a tetrahedral (T) interstitial site in a zincblende or diamond semiconductor such as GaAs or Si. The angles 9 between the muon-nucleus axis and Bo and the number of nuclei with a certain value of 6 are summarized in Table 5.1. Now, consider the situation where the muon or muonium is located at a bond-center (BC) site, as would be the case for Mug C in Si or GaAs. All bond directions in these semiconductors are along one of the (111) directions. If one makes the reasonable 84 Chapter 5. MEASUREMENTS and DISCUSSION 85 Orientation (100)|| Bo (111)11 Bo (110)|| B 0 Nearest Neighbors Equiv Nuclei 9 (deg.) 4 54.74 1 0 3 70.53 2 90 2 35.26 Next Nearest Neighbors Equiv Nuclei 9 (deg.) 2 0 4 90 6 54.74 2 90 4 45 Table 5.1: The angles 9 between the muon-nucleus axis and the applied magnetic field for a muonium center at the T site. (100)||Bo (1H)I|B0 <110)||Bo Number Ratio 9 (deg.) 4 54.74 Equivalent 1 0 3 70.53 2 90 2 35.26 Table 5.2: The angles 9 between the applied magnetic field and the M u ^ c hyperfine symmetry axis (also a bond axis). assumption that Mug C thermalizes at random in one of these bonds, the ratio of the number of centers at 9 between Bo and the hyperfine symmetry axis (also bond axis) is as summarized in Table 5.2. 5.2 Ana lys i s of / iSR D a t a The analysis of /fSR data consists of two general steps. The first consists of extracting parameters from the "raw" /xSR data. For example, in a T F experiment, one is usually Chapter 5. MEASUREMENTS and DISCUSSION 86 interested in parameters such as amplitudes, frequencies and relaxation rates. Then, the dependence of these parameters on physical variables such as temperature, magnetic field, doping levels, etc. are examined and at tempts are made to extract the important physics. Invariably, this means fitting to equations which are used to represent the physical system (such as those in Chapter 3). In general, the TF-//SR signal consists of a sum of oscillations, possibly damped. In the time-differential //SR experiments, one or more pairs of opposing counters are used to form the asymmetry spectrum (spectra). These signals are usually analyzed online with fast Fourier transforms in order to obtain an initial estimate of the precession frequencies. Accurate estimates of the experimental parameters (such as asymmetry, phase, relaxation rate, and precession frequency), are then obtained by fitting the "corrected asymmetries" (see Sec. 2.4). If the muon precession frequency is very high the da ta are often analysed by first transforming to a reference frame rotating at a frequency slightly less than u^. In this way the da ta can be packed heavily thereby increasing the speed of the fitting procedure. The observable LF-//SR signal usually consists of a sum of several relaxing components. These data are also fitted as corrected asymmetries to obtain the experimental parameters. Estimates of all model parameters and their errors, whether in the "raw data" or the off-line "physical interpretation", were obtained by using the minimization program MINUIT [52] (for on-line /iSR data, MSRFIT, a program written by Jess Brewer which interfaces MINUIT, is used) to minimize the weighted least squares function. Unless specifically mentioned, all statistical errors quoted from fits are full MINOS errors, which give statistical 'lcr' error, i.e. change in fit parameter such that the value of \ 2 increases by one, taking correlations between the various parameters into account. Chapter 5. MEASUREMENTS and DISCUSSION 87 5.3 S a m p l e s This thesis describes fiSR measurements in Si and GaAs. The Si samples studied are summarized in Table 5.3. The concentration of dopants as obtained from photolumi-nescence (measured by M. Thewalt) and /or resistivity values are summarized where available. The "diameter" of the samples varies between 15 m m and 25 mm and ex-cept for SiP-19 and SiP-20, the samples are about 2 mm thick. SiP-19 and SiP-20 have thicknesses between 200-250 fim. Except for SiP-18, SiP-19 and SiP-20, all the samples were grown by various commercial companies using the float zone refining technique and doped during the growth process. The TOPSIL sample was grown by the com-mercial company TOPSIL, the SiB-16 and SiP-14 and SiP-16 samples were grown by General Diode Corporation. The SiB-14 and SiP-15 samples were provided by T.L. Estle from Rice University, the SiP-18 sample was provided by M. Thewalt from Simon Fraser University and the SiP-19 and SiP-20 samples were obtained from the University of Paris at Orsay. The TOPSIL sample, which has a very low net carrier concentration, will be referred to as the "intrinsic" or "very pure" Si sample in later discussion. The GaAs wafers studied, all of which are heavily doped, are summarized in Table. 5.4. The GaSi-18-100, GaSi-18-110, GaSi-18-111, GaTe-19-100 and GaZn-19-100 sam-ples were obtained from Laser Diode Inc. and are grown by the Liquid Encapsulated Czochralski method. Secondary ion mass spectroscopy (SIMS) measurements on the GaAs:Si samples confirm the concentration to be « 3 x 1018 c m - 3 . The GaTe-18-100 sample is obtained from Hewlett-Packard and the GaSi-17-100 sample from Sumitomo Electric Industries Ltd. Both were grown by the horizontal Bridgman method. The di-ameter of all GaAs samples is between approximately 1.5 to 2 inches and the thickness is > 400 //m. The mnemonic G&AA — ij — nlm is used for the GaAs samples where A A denotes the dopant (Si, Te or Zn), ij the approximate concentration (10 tJ c m - 3 Chapter 5. MEASUREMENTS and DISCUSSION 88 Name Face Type TOPSIL 110 p SiB-14 100 p SiB-16 111 p SiB-18 111 p SiP-14 111 n SiP-15 100 n SiP-16 111 n SiP-19 100 n SiP-20 111 n Photoluminescence CF(cm-3) CB(cm-3) 1.4 x 1012 1.1 x 1012 0.3 x 1014 4.4 x 1014 4.9 x 1014 2.4 x 1012 1.5 x 1015 5.0 x 1015 Resistivity p(Q,.cm) Concentration (cm -3) 30,000 « 5 x 1011 cm - 3 « 1 0 1 6 7.5 x 1018 1 x 1014 0.5 1.2 x 1016 0.007 9 x 1018 0.0008 88 x 1018 Table 5.3: List of Si samples studied. The entries under the column labelled "Name" are mnemonics for the samples. The direction of the face is given by the column labelled "Face". Cp and CB denote the P and B concentrations respectively, as determined by photoluminescence studies (where available). dopants) and nlm the direction of the flat face. 5.4 Intrinsic Si H i g h Temperatures In this section, extensive measurements of the muon spin 1/Ti relaxation in intrinsic Si as a function of longitudinal magnetic field (0.015 — 6 T) and temperature (350 K-850 K) are described. This sample is labelled TOPSIL in Table 5.3. These measurements were performed on the M15 and M20B beam lines at T R I U M F with the high-field spectrometer HELIOS (Sec. 4.1) and the oven (Sec. 4.2). The majority of the mea-surements were obtained with the magnetic field B 0 parallel to a (110) crystallographic direction. Recall from the discussion in Sec. 1.1 that both the Mu^ and M u ^ c centers are undergoing rapid ionization above 230 K and that an inverse reaction Mu + -> Mu may also be occurring at higher temperatures. The current LF-//SR measurements Chapter 5. MEASUREMENTS and DISCUSSION 89 Name Sample Face Type GaSi-17-100 GaAs:Si 100 n GaSi-18-100 GaAs:Si 100 n GaSi-18-110 GaAs:Si 110 n GaSi-18-111 GaAs:Si 111 n GaTe-18-100 GaAs:Te 100 n GaZn-19-100 GaAs:Zn 100 p Resistivity p(£ Concentration (cm - 3) 9.0 xlO16 0.0014-0.0010 2.5-5 xlO18 0.0010 4.5 x 1018 0.0013 3.2 x 1018 4.5 x 1018 0.0034 2.80 x 1019 Table 5.4: List of GaAs samples studied. The entries under the column labelled "Name" are mnemonics for the samples. The direction of the face is given by the column labelled "Face". establish that cyclic charge state changes of Mu are indeed taking place. Furthermore, estimates of the average hyperfine parameter of Mu can be obtained. As shown in Fig. 5.1, excellent fits to the fj.SK time spectra were obtained by assuming that the LF muon spin polarization has the form exp[—t/T\]. Fig. 5.2 shows the fitted values of 1/Ti at four of the six temperatures where detailed field scans were carried out. Note that at low fields, l / 2 \ is relatively flat and a BQ2 dependence is approached at high fields. As discussed in Sec. 3.6, this is a strong signature that the neutral center involved in the charge dynamics has an isotropic hf parameter. This is also supported by several measurements made with Bo along a (100) direction (by tilting the sample) which showed the same 1/Ti as the (110) data. The relaxation da ta is assumed to be described by a two-state model where a single muonium center with isotropic hyperfine parameter A^ undergoes repeated cycles of ionization followed by capture of an electron from the conduction band, i.e. Mu <-* M u + + e - . Since the rates for the charge exchange are very rapid, it is reasonable to Chapter 5. MEASUREMENTS and DISCUSSION 90 Q_ N < 0.15 0.10 0.05-, 0.00-TIME fas) Figure 5.1: The LF muon spin polarization in intrinsic Si (TOPSIL) at 500 K for 0.1 T (circles), 0.5 T (squares), and 1.5 T (triangles). The curves are fits to a single exponential relaxation function. Chapter 5. MEASUREMENTS and DISCUSSION 91 i en =1 10 10 FIELD (T) Figure 5.2: The field dependence of l/Ti at various temperatures in the intrinsic Si sample (TOPSIL). The curves are the best global fit to the charge-exchange model described in the text. Chapter 5. MEASUREMENTS and DISCUSSION 92 <W/////////////A \ Mi l V 0 n + "-M E. V Figure 5.3: Charge exchange process where muonium undergoes repeated cycles of ionization followed by capture of an electron from the conduction band. assume that the muon is in thermal equilibrium with the thermally generated conduc-tion electrons. Furthermore, the 0 / + level of Mu (E^) is assumed to be in the upper part of the Si band gap (Eg) and to have a two-fold spin degeneracy (g = 2). In this case, the probability p0 that Mu is neutral may be written as: A, Po AjV/u "+• A. i + i-nW) leading to 1 (E,-EF A M „ = 2 ^ e x P ( - ^ -(5.1) (5.2) where A^ and \MU a r e the rates for electron capture and ionization respectively and Ep is the Fermi level. The charge exchange process is illustrated in Fig. 5.3. x The most direct approach to the analysis would be to fit the field dependence of 1/Ti at each temperature for the free parameters A^, AM and \MU using the theoretical methods described in Sec. 3.7. Unfortunately, this is not possible since the data at each temperature cannot be fitted with a single set of independent parameters A^, A^ 1The reaction Mu° <->• Mu - is also possible. However, we argue that the rate for Mu" —• Mu+ is much more rapid than the rate for Mu° —• Mu - since the latter involves electron capture and hence has the functional form <rnv (see below). In our near intrinsic sample, the term with the most significant temperature dependence is n which has the form e~Egl2kBT where Eg is the band gap (see Eq. (5.3) for example). Since the ionization rate of Mu° —* Mu+involves an activation energy significantly less than Eg/2 (e.g. the 0/+ level is in the upper part of the band gap), the charge exchange cycle involving the neutral and positive charge states will dominate. Chapter 5. MEASUREMENTS and DISCUSSION 93 and \MU- Instead, the entire set of data was fit by assuming the capture rate has the form AM = avn, where a is an average cross section for an electron in the conduction band to be captured by M u + , n is the electron concentration in the conduction band and v is the average thermal velocity of the electron. There is uncertainty regarding the values of n in the literature. The two expressions used in this thesis, where we are interested in the temperature range from 350 K to 850 K, are from Refs. [53] and [54]: n = 3.87 x 1016 T§ exp(-0.605eV/fcBT) cm" 3 n = 5.71 x 1019 (T /300 ) 2 3 6 5 exp ( -6733 /T) cm- 3 . (5.3) The electron thermal velocity is assumed to have the form [53]: v = (8kBT/*m;h)± (5.4) where the thermal velocity effective mass m*h = 0.28me for the temperature range of interest (m e is the mass of the electron). The ionization ra te XMU is given by Eq. (5.2.) Three parameters, a, E^ and A^ were then extracted from a global fit of the data at all temperatures; a and E^ — Ep were assumed to be temperature independent with Ep located at midgap while A^ was allowed to vary with temperature. (The fitted parameters change only slightly if one assumes E^/Eg is constant and the temperature dependence of Eg is as given by [55]. 2) The results of the global fit are shown as curves in Fig. 5.2. It is interesting to plot the temperature dependence of l / T i a t low fields, as shown in Fig. 5.4a. Such behavior is expected for charge exchange, as discussed in Sec. 3.6, providing further confirmation that l /T i is d u e to rapid cycling of the muonium charge state. Above 600 K, the mean time spent as the neutral center, i.e. \/\MU, is less than the period of a hyperfine oscillation (27r/u>24) and hence the amount of polarization lost 2Eg (eV) = 1.170 - 4 • 7 j x + 1 6 0 3~6 T2; units of T are Kelvin. Chapter 5. MEASUREMENTS and DISCUSSION 94 300 500 700 900 TEMPERATURE (K) Figure 5.4: (a) 1/71 in TOPSIL (circles) in low fields between 15 m T and 40 m T . These da ta closely approximate the temperature dependence for constant low field since at these low fields l / 7 \ is only weakly field-dependent. The curve indicates the smooth interpolation of the best global fit results (described in the text) , (b) The average muon-electron hyperfine parameter A^ and the fraction of time in the neutral state p0 in nominally pure Si. The points are the fitted hf parameters obtained. The solid line is an extrapolation of /iSR results for the hf parameter of Mu^ below 300 K. The dashed line indicates the temperature dependence of p0 calculated from Eq. (5.1). The approximate hf value expected if the neutral center were M u ^ is also indicated on the figure. Chapter 5. MEASUREMENTS and DISCUSSION 95 per cycle diminishes and consequently l /T i falls off ("motional narrowing"). As will be discussed in Sec. 5.5, the temperature dependence of this l /T i curve on the type and concentration of dopants can be used to help determine whether the charge-exchange interaction occurs with electrons or holes. This was used to confirm that E^ is in the upper part of the band gap since reduced l /T i rates were observed for the p-type material, indicating that the charge exchange process involves a positive ionized state and conduction electrons, i.e. Mu <-> M u + + e~ rather than Mu + h+ <-> Mu + . Fig. 5.4b shows the fitted values for A^ and the fraction of time the muonium is in the neutral s tate (i.e. p0 determined via Eq. [5.1)] as a function of temperature. The error bars in AM are primarily systematic originating from the uncertainty in the literature values for n (see Eq. [5.3)]. The solid curve is an extrapolation of ^uSR measurements of A^ for Mu° in Si [56]. These data, which were taken below 300 K, were well described the following formula based on an empirical Debye model: m 4 /e/T x3dx A„(T) = AM 1-C a: (5.5) .0J Jo ex - 1 where A(0) = 2006(2) MHz, 0 = 655(25) K, C = 0.68(5). The agreement between the experimental data and the extrapolated curve is remarkable considering that the present da ta were obtained indirectly by fitting high temperature measurements of l /T i to a dynamical charge exchange model. Two other parameters which govern the charge exchange dynamics of muonium, and presumably hydrogen, were obtained from the fit to the two state model. The thermal average cross section for an electron in the conduction band to be captured by Mu+ is a - 2.8(3) x 10 - 1 5 cm2 and the Mu 0 / + level is 0.34(1) eV below the conduction band edge, where the errors are purely statistical. As indicated in Fig. 5.4(b), the fitted values for A^ are more than an order of magnitude larger than the isotropic hf parameter of M u ^ c and are similar to that for Chapter 5. MEASUREMENTS and DISCUSSION 96 Muy, implying that when Mu is in the neutral charge state, it spends a significant fraction of its t ime near T sites. This result is surprising since theoretical calculations [24,57] (see Sec. 1.1) of the adiabatic potential energy surface for H or Mu place the total energy of the muon/muonium at a BC site to be at least « 0.3 eV below that of the T site, implying that the BC site should be occupied much more frequently. One possible explanation for the above result comes from the recent theoretical work by Ramirez and Herrero [17]. They suggest that quantum effects can signifi-cantly modify the conclusions of previous "standard" electronic structure calculations. In particular, they find that Mug C is metastable with a lifetime that decreases with increasing temperature and eventually converts to Mu^. This result, they claim, is a consequence of the larger zero-point kinetic energy of muonium located at the BC site as compared to muonium near the T site since Muy has a "larger dereal izat ion" than Mu%c; i.e. the uncertainty principle states that confinement of a quantum particle in a small spatial region leads to a larger kinetic energy. Within this picture, our result is not surprising. However, it should be kept in mind that the majority of other theoreti-cal calculations predict that M\XgC is the stable state. If this result is correct, then the above experiemental result may be a consequence of the nature of the lattice response to the motion of muonium. The LF-//SR data in intrinsic Si were obtained at high temperatures where it is likely that theoretical calculations based on the assumption that the lattice can relax fully to the presence of the muon are no longer valid. This is supported by molecular dynamics calculations of the interstitial proton in Si (T ^> 1200 K) which show that the surrounding host atoms, which are much more massive, do not have time to respond to the fast motion of the proton [58] and sites such as the BC site which were originally low in energy are no longer so. Recall tha t the BC site is only the global minimum in the potential energy if the surrounding lattice atoms are allowed to relax fully. Within this model, we argue that the muon spends a significant Chapter 5. MEASUREMENTS and DISCUSSION 97 fraction of its t ime near the T site since a much smaller lattice relaxation is required there than for Mu occupying the BC site. Since the muon has only l / 9 t h the proton mass, it is expected to move even more rapidly than the proton at any (high) tem-perature. Therefore, the inability of the lattice to respond to the muon will probably occur at lower temperatures than for the proton. The postulate that the host nuclei cannot respond quickly enough to a rapidly moving muonium center has been invoked previously to explain the existence of metastable Mu^ at low temperatures in Si [24], the disagreement between the theoretical and experimentally determined isotropic hf parameter of normal muonium in GaAs [59], and the local tunneling motion of the Mu 7 center in CuCl [60]. Although the experimental data above suggest that the muon is experiencing a large isotropic hf parameter for a significant fraction of its lifetime (p0 in Fig. 5.4) and can be completely explained by a two state model, the actual dynamics may be more complicated since a diffusing muon is likely to be sampling many different locations with many different hf parameters. An example of such a multi-site model has been recently proposed to explain RF-//SR data [61,62], all taken below 450 K. 3 These da ta suggest that the ionization of M u j inferred from TF-^SR experiments [19] at « 230 K (see Chapter 1) involves the intermediate state Mug C . If this more complicated model involving two sites is valid at high temperatures, a T <-» BC site change may be the limiting process in electron ionization. In this case, the fitted activation energy of 0.34 eV could be interpreted as an activation energy for the reaction Muy —*• M u J c + e~, involving bo th ionization and a site change, and would not be directly associated with 3In a RF experiment, the experimental geometry is the same as in an (integrated) LF-^SR experiment where the main applied field Bo and the initial muon spin are parallel except that a small ( « 10G) field of magnitude B\ oscillating at angular frequency uiosc is applied perpendicular to the main field. A resonance condition occurs when w0,c matches the appropriate energy splittings (for the diamagnetic muon, this is just y^Bo) in which case the muon spin precesses at frequency j^Bi [39]. Clearly, this corresponds to a reduction in the integrated LF asymmetry (as in ALCR). This allows one to selectively study one center at a time provided transitions between various centers do not occur. Chapter 5. MEASUREMENTS and DISCUSSION 98 the 0 / + level for either site. 5.5 p - type Si:B - H i g h T e m p e r a t u r e s In the previous section, it was implicitly assumed that electrons in the conduction band were involved in the charge exchange. However, it is not clear a priori that holes do not play a direct role in the charge exchange cycle, i.e. Mu + h+ <-> M u + . The importance of negatively charged carriers in governing the spin dynamics at high temperatures is confirmed by studying p-type Si, in particular Si:B. These experiments were performed with the OMNI' spectrometer and the oven at the M20B and M15 beamlines at TRIUMF. The Si:B samples studied are SiB-14, SiB-16 and SiB-18 in Table 5.3. In the slow charge exchange limit where the rate XMU for the reaction Mu —> M u + (i.e. conversion of Mu to M u + ) , is much less than the the u>24 frequency, it can easily be seen from Eq.(3.64) that 1/ri« *£" ( i + i )"' * (6.6) where tc is the mean time required to complete one cyclic charge exchange cycle (i.e. Mu -> M u + —» Mu). 1. Suppose the charge exchange process involves holes, i.e. Mu + h + <-> Mu + . One expects the rate AM for the thermal excitation of a hole to the valence band, i.e. M u + —* Mu + h+, to be essentially independent of the hole concentration. It is also reasonable to assume that the rate \MU for hole capture by Mu is proportional to the concentration of holes. Hence, at fixed temperature, as the hole concentration is increased, tc should decrease and 1/Ti increase. Chapter 5. MEASUREMENTS and DISCUSSION 99 30 25 -i ,_ 20 15 10 5 -0 (l-B 300 400 500 600 700 800 900 TEMPERATURE (K) Figure 5.5: The low field temperature dependence of 1/Ti for the samples TOPSIL (open squares), SiB-14 (open circles), SiB-16 (closed circles) and SiB-18 (open dia-monds). LF ranges from 150 G to 400 G in TOPSIL and is 200 G in the SiB-14, SiB-16 and SiB-18 samples. Chapter 5. MEASUREMENTS and DISCUSSION 100 2. On the other hand, suppose the the charge exchange process involves elec-trons, i.e. Mu «-• Mu + + e~. In this case, one expects the thermal ionization process Mu —> M u + + e~ to be independent of the number of free electrons and the capture process M u + + e~ —• Mu to be proportional to the number of free electrons. Hence, if the free electron concentration decreases, such as by increasing the boron doping in Si, tc increases and 1/Ti decreases. In summary, at fixed temperature, 1/Ti wiU increase with increasing boron concentra-tion if the charge exchange cycle involves holes. Conversely, 1/Ti will decrease with increasing boron concentration if the charge exchange cycle involves electrons. The high temperature experimental situation in Si:B is similar to intrinsic Si de-scribed in Sec. 5.4. The LF muon spin polarization function Pz(t) is well-described by a one-component exponential relaxation. The temperature dependence of 1/Ti in the Si:B samples at low fields is shown in Fig. 5.5. The observation that the onset of 1/Ti occurs at higher temperatures with increasing boron concentration (most evident by comparing pure Si with SiB-16 and SiB-18) indicates that electrons and not holes are important in the charge exchange process. The curves, which are in reasonable agreement with the experimental data, are obtained by using Eq. (3.64) and assuming that the retrapping rate AM is proportional to the electron carrier concentration n (with the same values of a and v as for the TOPSIL Si sample described in Sec. 5.4) and the Mu ionization rate \MU is given by Eq. (5.2). 4 In Eq. (5.2), the Fermi level EF is of course dependent on n and the hole concentration p and has the form [63] EF = EF,i + -^-ln- (5.7) 2 p 4It should be pointed out that the shapes of the theoretical curves are quite sensitive to the actual parameters used. Although the agreement between theory and experiment is actually quite reasonable, one should not lose sight of the fact that the qualitative behavior alone can be used to verify the importance of the negatively charged carriers. Chapter 5. MEASUREMENTS and DISCUSSION 101 where Epi is the intrinsic Fermi level, assumed to be pinned at mid-gap. The carrier concentrations for a semiconductor containing ND donors and NA acceptors which are fully ionized (high temperatures) are given by [47]: n = I[(JVD - NA)2 + 4n?]i + \(ND - NA) P = \[(ND - NA)2 - 4 r # + ^(ND - NA) (5.8) where n,- is the intrinsic carrier (either e~ or h+) concentration [e.g. Eq. (5.3)]. Several points are noteworthy: (a) 1/Ti decreases at high temperatures, the result of the "motional narrowing" effect discussed in Sec. 3.6 and Sec. 5.4. (b) The curves eventually merge when n is the same in all samples, i.e. in the intrinsic regime, (c) The observation that holes play a minor role in the charge exchange cycle confirms that the Mu level is in the upper part of the gap. In this situation, thermal excitation of a hole into the valence band is highly unlikely due to the large ionization energy required, hence the reaction Mu + -+ Mu + h+ of the charge exchange cycle involving holes is suppressed. (There is evidence that the reverse reaction Mu + h + —> M u + occurs in p-type Si below « 150 K. This is illustrated in Ref. [19] and also recently confirmed by our measurements of the relaxation rates of the precession signals of the M u ^ c center in SiB-14.) 5.6 n- type Si:P w i t h In termedia te D o p i n g Levels As demonstrated in the previous two sections, Mu undergoes significant ionization at elevated temperatures. At high temperatures, there are numerous electrons in the conduction band and the dominant observable physical process involving muonium is cyclic charge exchange. However, at low temperatures in n-type Si, although there are still many conduction electrons, there is insufficient thermal energy available to ion-ize Mu. Nevertheless, there is little doubt that Mu is interacting strongly with these Chapter 5. MEASUREMENTS and DISCUSSION 102 electrons since, as discussed in Sec. 1.2, muonium precession signals (see Sec. 3.2) are not observed above the phosphorus donor ionization temperature in n-type Si:P. Two distinct processes involving the neutral Mu center and the conduction electron are ex-pected to be relevant: (1) electron capture to form M u - and (2) repeated rapid electron spin-exchange scattering between the muonium electron and the conduction electrons. One expects the relative importance of process (1) and process (2) to be dependent on the relative stabilities of Mu" at various sites in the lattice. For Si, supercell-based adiabatic calculations predict that Mu^p ( M u - at the T site) is the overall ground state in n-type materials and that Mug C is very high in energy.[2,24] Consequently, one ex-pects that M u ^ c is more likely to undergo spin-exchange scattering than to form Mu^ since the latter would involve electron capture as well as a simultaneous change to a more energetically favorable site. On the other hand, the electron capture process Muy —• Muy can take place without a change in muon site and thus the corresponding cross-section should be much larger. In order to test this hypothesis, measurements of the LF muon spin 1/Ti relax-ation in n-type Si:P (samples SiP-14, SiP-15 and SiP-16 in Table 5.3) were performed. The experiments were carried out on the M13 and M15 beamlines at TRIUMF with the OMNI' and HELIOS spectrometers using the horizontal gas-flow and horizontal cold-finger cryostats respectively. It was shown in Sec. 3.8 that the field dependence of the relaxation rate provides a distinctive signature of the nature of the muonium center undergoing spin exchange; more specifically, whether it is M u j or M\igC. Our experimental results, in conjunction with the theoretical predictions, clearly establish the presence of long-lived M u ^ c in heavily doped n-type semiconductors at tempera-tures where the precession signals are not observable. These measurements also help establish the identity of the Mu center in n-type Si at low temperatures, which was unclear from the sparse LF-/^SR measurements on n-type Si:P summarized in Ref. [19]. Chapter 5. MEASUREMENTS and DISCUSSION 103 CO =1 3 2 0 0 100 200 TEMPERATURE (K) 300 1014 cm"3 Figure 5.6: Temperature dependence of muon 1/Ti relaxation rates in Si:P for B0 = 0.2 T (closed circles) and 20 mT (open circles). The solid curves through the points serve as guides to the eye. The three regions indicated in the figure are discussed in more detail in the text. Chapter 5. MEASUREMENTS and DISCUSSION 104 I 00 FIELD (T) Figure 5.7: Field dependence of 1/Ti at 60 K (open squares) and 73 K (closed circles) in SiP-14. The dashed and solid curves are best fits to the da ta using the Nosov-Yakovleva theory (Sec. 3.8) for 73 K and 60 K respectively and assuming 1/Ti is due to Mug C in Si with 6 = 70.53°. The fitted Nosov-Yakovleva spin exchange rates are VSE= 11-2(5) / i s - 1 and 15.4(5) / / s _ 1 at 60 K and 73 K respectively. Chapter 5. MEASUREMENTS and DISCUSSION 105 In the SiP-14 sample, the applied field B 0 was parallel to one of the [111] crystalline axes where for MugC, 6=70.53° and 0=0° orientations occur in a 3 to 1 ratio (see Table 5.2). At temperatures greater than w 45 K, the LF muon spin relaxation is well described by two components. One component decays exponentially [ex exp(—t/Ti)] while the second is non-relaxing. Here, we will be concerned with the 1/Ti rates of the relaxing component. The constant component is at t r ibuted to a charged and non-paramagnetic species; most likely M u j . The temperature dependence of 1/Ti for B 0 = 200 m T (2 kG) and B0 = 20 m T (200 G) is shown in Fig. 5.6. There are three distinct temperature regions labelled in Fig. 5.6. In region 1, 1/Ti increases rapidly with temperature. In region 2, 1/Ti is essentially temperature-independent while in region 3, there is a sharp initial increase of 1/Ti for both fields at « 150 K followed by a crossover occurring at « 230 K. One should note that the relaxation at 20 m T is smaller than at 200 m T for T < 230 K. Field scans of 1/Ti at selected temperatures in Regions 1 and 2 are shown in Fig. 5.7. The resonant-like feature in 1/Ti occurring at ?s 0.32 T is predicted for M\igC (0 = 70.53°) in Si undergoing spin-exchange scattering. The displayed quantitative fits, using the theoretical approach described in Sec. 3.8 with 0=70.53° fixed and the Nosov-Yakovleva spin-exchange rate USE a s the only adjustable parameter, are in good agreement with the data. Note that in addition to being of smaller amplitude due to the smaller population in that orientation, the 1/Ti rates for 0 = 0° around the peak region (from « 150 to 600 mT) are predicted to be a factor of 100 or more than for the 0=70.53° centers (see Fig. 3.11). Hence, the main contribution to the observed 1/Ti is from the 0=70.53° centers. A further assumption is made that VSE = anv where <r is the cross-section for spin-exchange scattering, n is the free electron concentration and v is the average thermal velocity of the electron. [53] If the only relevant impurities in the semiconductor are due to phosphorus at a concentration ND (in this case, No = 5 x 1014 c m - 3 ) and the intrinsic carrier concentration is assumed Chapter 5. MEASUREMENTS and DISCUSSION 106 1.0 0.8 0.6 0.4 0.2 0.0 0 50 100 150 200 250 300 Temperature (K) Figure 5.8: Temperature dependence of n/Np in the case of Si:P for various concen-trations of donors (No in units of cm - 3) . to be negligible, then n is given by Ref. [63] as Nr. I \(N^.\2 I oNz_ „ v r . EH ]j n/Nj D ND 4exPfcfr (5.9) where Nc = 2.86 x 1019 cm_3(T/300K)15 is the effective density of states in the conduc-tion band and E^ (=45.5 meV for the P level in Si) is the energy separation between the impurity level and the bottom of the conduction band. The fraction n/Nu is plotted in Fig. 5.8. The fitted values of USE correspond to an average a = 4.4(5) x 10~15 cm2. The unmistakable identification of Mu#c leads to the following interpretation of the results in Fig. 5.6 in the three regions: The rapid increase of l/Ti in Region 1 tracks Chapter 5. MEASUREMENTS and DISCUSSION 107 an increasing n (and hence V$E) due to ionization of the phosphorus donors. In Region 2, n remains essentially constant, typical of extrinsic behavior (see Fig. 5.8); thus, USE a n d l / ^ i are relatively flat.5 In Region 3, it is clear that Mug C still exists since the 1/Ti rates at the higher field is larger than the relaxation rates at the lower field. Previous work has shown that there is significant ionization of M u ^ c in lightly-doped Si starting at w 130 K (see Sec. 1.2). There is sufficient extrinsic electron density in this sample that a M u + ion can quickly retrap an electron; therefore, in addition to spin-exchange scattering, the cyclic charge-exchange process M u ^ c <-• Mu + +e~ also becomes active, occurring at much lower temperatures than for intrinsic Si. As discussed in Sec. 3.6, spin-exchange scattering and charge exchange have similar field dependences, as confirmed by calculations for M u # c using the theoretical formalism discussed in Sec. 3.8. In particular, there is a peak in the field dependence of 1/Ti for slow ionization rates which disappears when the ionization is fast. The onset of a second process, i.e. charge exchange, would cause additional loss of muon polarization and hence explains the initial sharp increase of 1/Ti in region 3. A possible explanation for the da ta above 200 K is that as the effective spin-exchange ra te increases, the fast spin-exchange regime is eventually reached where 1/Ti decreases with increasing field (see Fig. 3.11)-hence the crossover at 230 K. However, as discussed in Sec. 5.4, the actual dynamics above 200 K, which clearly involve M u ^ c , may be more complicated and could involve a BC—»T site change. The general features shown in Fig. 5.6, in particular the existence of the three distinct temperature regions, is also evident in all Si:P samples studied with phosphorus doping levels ranging from « 2 x 1013 c m - 3 to « 1016 c m - 3 (i.e. SiP-16). However, in SiP-16 a peak in 1/Ti due to M u ^ c is not observed at temperatures where there is significant impurity ionization since the 5Actually, since v oc y/T and m*th does not change significantly over this temperature range, VSE should change by ta 50%. The flatness of VSE hence suggests that <r is inversely proportional to v. Chapter 5. MEASUREMENTS and DISCUSSION 108 10' CO 3, 10 10v 10 - 1 0 1 2 FIELD (kG) Figure 5.9: The field dependence of 1/Ti at 73 K in various Si:P samples. The lines serve as a guide to the eye. increased electron concentration leads to very large USE values and behavior expected for the fast spin exchange limit. This is illustrated in Fig. 5.9. The spin exchange rate is "slow" in SiP-14 and SiP-15 and hence there is still evidence for a peak. However, USE is "fast" in SiP-16 and the peak is washed out. The results show that in Si:P, M u ^ c is resistant to forming a negative ion for T < 200 K although it interacts repeatedly with conduction electrons. Such behavior implies a substantial energy barrier for the electron capture/si te change reaction M u ^ c +e~ —> Muy, as discussed above. 6 An additional comment is in order: Since the 1/Ti rates can be at t r ibuted entirely to M.UgC, it is likely tha t Mu^ exists only as a precursor state which is rapidly converted to a diamagnetic center such as Mu". 7 A rough estimate of the lower limit for such a barrier can be obtained by assuming that the rate A0_ for the reaction Mu^c +e~ —»• Mu^ has the Arrhenius form A0_ = Ae~ElkBT. If one assumes that A = 10 -13 s_1 (optical phonon-type frequencies), claims that Mu^c can only make at most one transition to Mu" in 10 fis (A0_ = 0.01 / JS - 1 ) , and assumes that MVL°BC persists till 200 K, then E > 0.3 eV. 7This is consistent with the RF-//SR and microwave measurements in samples of similar or lower concentration which either fail to detect Mu^ or if Mu .^ is observed, it appears as a very broad signal.[61] Chapter 5. MEASUREMENTS and DISCUSSION 109 The detection of M u ^ c in these n-type materials in the same temperature range as in intrinsic Si (see Chapter 1) implies that H ^ may also exist as a metastable species in the former. Recall from the discussion in Chapter 1 that there have only been two spectroscopic observations of isolated H in a semiconductor, both via ESR [3,4]. These experiments, which show that the hydrogen center detected is H ^ c , were both performed in high-resistivity Si. If H%c exists in n-type Si, it will undoubtedly be undergoing rapid spin-exchange scattering with conduction electrons. 5.7 Spin Polarized Muonium—Heavily D o p e d Si:P It has been established in Sec. 5.6 that long-lived neutral paramagnetic muonium is still observed in Si with intermediate n-doping. (A paramagnetic center is also believed to have been detected recently in Sb (metal) [49] where, as expected from the discus-sion above, it undergoes rapid spin-exchange scattering.) It is interesting to study a very heavily doped n-type semiconductor: does paramagnetic muonium exist and if so, what is the nature of its interaction with free carriers in the system? In a semi-conductor, one has the ability to change the electron concentration by changing the dopant concentration such that the material goes from insulating/semi-conducting to "metallic", in the sense that the conductivity remains finite even when extrapolated to zero temperatures. [64] Roughly speaking, an insulator to metal transition takes place when the concentration of impurities (e.g. phosphorus) is so high that the ground-state wavefunctions of electrons on neighboring impurity atoms overlap significantly, creating band-like states. This transition occurs when the impurity concentration No exceeds a critical concentration Nc, which in Si:P is « 3.5 x 1018 c m - 3 . There is increasing ex-perimental evidence in n-type Si:P that conduction initially occurs in a donor impurity band well separated from the conduction band. When No }t 3iVc, this impurity band Chapter 5. MEASUREMENTS and DISCUSSION 110 merges completely with the conduction band. [65] From the studies in the previous section, one would expect that since the free elec-tron concentration is very high in a heavily-doped semiconductor, any paramagnetic muonium that exists in the system will be undergoing very rapid spin-exchange scatter-ing. As discussed in Sec. 3.8 and Sec. 3.9, if the Mu electron is unpolarized and if Mu is in the extremely fast spin-exchange limit, it is indistinguishable from the diamagnetic center M u ° (e.g. M u + or Mu") . Namely, in a TF-//SR experiment, it appears as a signal precessing at the Larmor frequency of the free muon while in a LF-//SR experi-ment, it appears as a non-relaxing signal. However, if a large magnetic field is applied at low temperatures, the muonium electron can be spin-polarized. Consequently, in the frequency spectrum of a TF-//SR experiment, the precession signal appears as a line (or lines) shifted from the diamagnetic Larmor frequency [see Sec. 3.9 and Eq. (3.81)]. This should be contrasted with the case of intermediate rates of spin exchange, such as that discussed in Sec. 5.6, where the precession signals from the paramagnetic centers are rapidly damped. In this latter case, one must resort to LF-//SR measurements to characterize the center. This section presents preliminary high TF-//SR measurements in heavily doped n-type SiP-19 (see Table 5.3 ). In this sample, ND ~ 2.6iVc. Since this sample is relatively thin, it was either "double-stacked" and /or a narrow range of less-energetic muons were selected. Naturally, the lower energy implies that such muons have a smaller stopping range. The TF-//SR experiments were performed at the M15 and M20B beamlines at TRIUMF with HELIOS and the horizontal gas flow cryostat (see Chapter 4). These measurements show the existence of a paramagnetic center with hf parameters similar to that of Mug C undergoing rapid spin exchange scattering. The temperature depen-dence of the frequency shift is measured and agrees with that expected for an isolated paramagnetic impurity in contact with a heat ba th at temperature T. Chapter 5. MEASUREMENTS and DISCUSSION 111 SiP-19 T=100K TF=1.5T, B || [100] 0.2 0.1 -< -0.1 -0.2 0.15 0.1 "5. E < 0.05 ~o J I I I I L 196 196.5 197 197.5 198 198.5 199 199.5 200 Frequency (MHz) Figure 5.10: (a) TF-//SR asymmetry spectrum for SiP-19 at T=100 K with B 0 parallel to a (100) axis. The field was « 1.5 T. The data are shown and fitted (solid line) in a rotating reference frame, (b) Real amplitude of the Fourier Transform of the TF signal. Chapter 5. MEASUREMENTS and DISCUSSION 112 At high magnetic fields, the TF-//SR signal consists of more than one oscillating component. This is illustrated by Fig. 5.10(a), which shows the corrected asymmetry spectrum of a typical TF-//SR run in SiP-19 with the applied magnetic field Bo parallel to a (100) crystallographic axis. The corresponding Fourier transform shown in Fig. 5.10(b). The fit shown in Fig. 5.10(a) was made to a sum of two oscillating signals in a reference frame rotating at « 2.2 MHz below the Larmor precession frequency of the diamagnetic center. One signal oscillates at frequency fsc while the other precesses at frequency fjy. 8 The component oscillating at / D , which is essentially temperature independent, is at tr ibuted to diamagnetic centers formed in the sample (probably Mu~) and to some muons which miss the sample and hence stop in the holder. (Mu~~ is diamagnetic since its two muonium electrons are "paired up" ; hence, the local spin susceptibility of Mu~ is much less than for paramagnetic muonium.) The component oscillating at fsc is at t r ibuted to Mug C undergoing rapid spin-exchange scattering. As in Sec. 3.9, we define the term "Knight shift", denoted by the symbol £ , to be K = f° ~ fBC (5.10) JD Fig. 5.11 shows the temperature dependence of K, when B 0 is parallel to a (100) crystallographic axis in SiP-19. The magnitude of the magnetic field is « 1.5 T. In the notation described above, K is the positive quantity ( /o — / B C ) / / D - The solid line is a fit to the function K = | (5.11) If we equate Eq. (5.11) with Eq. (3.84), which assumes that the polarization of the muonium electron is simply that of an isolated electron spin in a magnetic field B 0 , 8Both signals are damped where the relaxation of the fee signal is much greater than that of the fo signal. At some temperatures, neither a single Gaussian or an exponential fuction adequately describes the relaxation (in either signal). Slight misorientation of the sample was experimentally verified to be responsible for a significant part of the damping of the /BC signal. The fitted frequency shifts turn out to be quite insensitive to the actual relaxation function used during the fitting procedure. Chapter 5. MEASUREMENTS and DISCUSSION 113 Knight Shift in SiP-19, TF=15 kG 0.016 0.014 0.012 I— u. X 0.01 CO x 0.008 O ^ 0.006 0.004 0.002 0 0 20 40 60 80 100 120 140 160 180 200 TEMPERATURE (K) Figure 5.11: Temperature dependence of Knight shift in heavily doped n-type SiP-19 (9 xlO18 cm - 3) with B 0 applied parallel to a (100) crystallographic axis. The solid line is a fit to the data as described in the text. Chapter 5. MEASUREMENTS and DISCUSSION 114 then -h%(A±sm29 + Allcos29) mth = —— u . (5.12) 47M^S Using the measured hf parameters for M u ^ c in Si with much lower donor concentation (see Table 1.2) and 9 = 54.74° ( since B 0 is parallel to a (100) crystallographic axis), mth = 0.167 K. Our fit yields m = 0.147(3) K; the deviation of « 10% may indicate a slight reduction in the hyperfine parameters of the M u ^ c center in our heavily doped Si:P sample. (It is interesting to compare K. in Si:P to the metal Sb [49] where the Knight shift in the latter levels off at low temperatures. An isolated Kondo impurity is proposed to explain the data in Sb.) The assignment of the Knight Shift to a M u ^ - l i k e center is further established by studying the orientation dependence of the TF-//SR signal. The TF-//SR signal clearly depends on the angle 9 between Bo and the bond axis, which is also the hf symmetry axis of Mu%c. This is illustrated by comparing Fig. 5.10 with Fig. 5.12. In the former all 9 = 54.74° while in the latter, the sample rotated such that B 0 is parallel to a (110) crystallographic axis and hence centers with 9 = 35.26° and 9 = 90° occur with equal probability (see Table 5.2). The Fourier Transform in Fig. 5.12(b) shows indications of three frequencies and the fit shown in Fig. 5.12(a) is to a sum of three oscillating components. As in Fig. 5.10, one component is a t t r ibuted to the Mu D center. The other two components are at tr ibuted to the other two inequivalent M u ^ c centers (i.e. two values of 9) and were assumed to have the same amplitude. The ratio of the three fitted values of K, obtained from Fig. 5.10(a) and Fig. 5.12(a) are £ i : £ 2 : fC3 = 2.13(1) : 1.57(1) : 1.00 (5.13) where K\ and £ 3 are the Knight shifts obtained from the da ta where B 0 is parallel to a (110) axis and 1C2 is obtained from the da ta where B 0 parallel to a (100) axis. The error limits quoted are entirely statistical. One can compare the experimental ratio Chapter 5. MEASUREMENTS and DISCUSSION 115 SiP-19 T=100K TF=1.5T, B || [110] 0.2 0.1 -x 0 - ' X < X J Z3 Q. E < (U 0d -0.1 -0 .2 0.15 0.125 0.1 -0.075 -0.05 0.025 -0.02 2 3 4 TIME (jis) 196 196.5 197 197.5 198 198.5 199 199.5 200 Frequency (MHz) Figure 5.12: (a) Raw asymmetry spectrum for SiP-19 T=100 K with B 0 parallel to a (110) axis. The field is w 1.5 T. The data are shown and fitted (solid line) in a rotating reference frame, (b) Real amplitude of the Fourier Transform of the TF signal. Chapter 5. MEASUREMENTS and DISCUSSION 116 with the theoretical Knight shift Kik = B°~Bav (5.14) -DO where Bav is defined by Eq. (3.81). By assuming that the spin polarization A P of the muonium electron is given by Eq. (3.83) at T = 100 K and B0 = 1.5 T, and A\\ and A± are the same as for MugC in Si, one obtains fCth(90°) : /C,fc(54.7°) : JCth(35.3°) = 2.18 : 1.60 : 1.00 (5.15) in good agreement with the experimental results. (Further discussion of A P follows below.) The appearance of the Mug C line(s) at lower frequencies compared with the dia-magnetic line in the frequency spectrum is noteworthy since it implies that the absolute signs of A\\ and A± are negative, as can be verified by Eq. (3.81). Recall from Sec. 1.1 that prior to the present measurement, only the relative signs of A\\ and A± in Si were experimentally determined. 5.8 H e a v i l y - D o p e d G a A s — Low Temperature Analogous to the situation in Si, one expects that paramagnetic muonium centers which are formed in n-type GaAs at low temperatures will either undergo spin exchange scattering or capture an electron to form Mu" . This section describes measurements on GaSi-17-100 and GaSi-18-100 (see Table. 5.4) with B 0 | | (100) crystallographic axes. As shown in Table 5.1, in this orientation of B 0 and crystallographic axes, all 9 = 54.74°. Both samples have dopant concentrations on the metallic side of the metal-insulator transition (Nc « 1.5 x 1016 c m - 3 ) in GaAs:Si. These experiments were performed on the M15 and M20 beamlines at TRIUMF with HELIOS and the horizontal gas flow Chapter 5. MEASUREMENTS and DISCUSSION 117 D.Q2 Q_ N (X 0 200 400 FREQUENCY (MHz) 2 4 6 TIME (jus) Figure 5.13: (a) / J S R frequency spectrum in GaSi-18-100 at 5 K with T F = 1.5 T. The frequencies labelled by v\2 and v^4 are due to M u # c and Vd labels the strong diamagnetic line, (b) Corrected asymmetry AzPz(t) in GaAs:Si with LF of 740 G along a (100) axis at 4 K. Only the relaxing component is shown. There is a large non-relaxing component also present. cryostat described in Chapter 4. The results described clearly establish the presence of long-lived M u ^ c undergoing spin exchange scattering at low temperatures. In a T F experiment at 5.5 K and 1.5 T in GaSi-18-100, a strong diamagnetic signal is seen as well as two (high-field) broad frequencies corresponding to MVL%C, as indicated by the frequency spectrum shown in Fig. 5.13(a). The notation v*- labels the transitions between the appropriate energy levels in the Breit-Rabi diagram for Mug C in Si. In a LF experiment near the same temperature, a two component muon polarization is observed, one component relaxing and the other time-independent [see Fig. 5.13(b) — only the relaxing component is shown in the figure]. The magnitude of the non-relaxing component could not be determined directly from these LF measurements. This component is a t t r ibuted to the diamagnetic center which appears as a precession signal close to the Larmor frequency of a free muon in T F (see Fig. 5.13a) but a non-relaxing signal in LF. The relaxation rates rapidly increase with temperature and the signals in both LF and T F become unobservable above R J 3 0 K . This is consistent with Chapter 5. MEASUREMENTS and DISCUSSION 118 T 1 1 1 1 1 r 0 0.4 0.8 1.0 1.4 FIELD (T) Figure 5.14: The field dependence of \jTx in GaSi-17-100 at 5.5 K. The line is a best fit to the da ta assuming the 1/Ti is due to M u ^ c undergoing spin-exchange scattering in GaAs with 9 = 54.7°. The fitted uSE = 2.09(6) /is"1 . the previous TF-//SR measurements on M u # c in high-resistivity GaAs which showed significant ionization at this temperature (see Sec. 1.1). Field dependences of l / 2 \ in both GaSi-17-100 and GaSi-18-100 (at « 5 K ) show the characteristic signature of M\igC (i.e. peak at 4.8 kG) undergoing spin exchange scattering with electrons in the conduction band. The LF dependence of the 1/Ti rates in GaSi-17-100 is shown in Fig. 5.14 along with the corresponding best fit to the Nosov-Yakovleva spin flip theory-described in Sec. 3.8. The poor agreement between the theory and the da ta at low fields is a t t r ibuted to the fact that the theoretical calculations do not take into account the nuclear hyperfine interaction of M u ^ c with the Ga and As host atoms. Nevertheless, the existence of a peak in 1/Ti near Bp = 0.48 T (calculated for Mug C in GaAs using A\\= 218.54 MHz and A±= 87.87 MHz) demonstrates that Mug C is present in heavily doped n-type GaAs and is responsible for most of the 1/Ti relaxation. It is noteworthy to point out the differences between the /iSR signals in heavily Chapter 5. MEASUREMENTS and DISCUSSION 119 doped and high-resistivity GaAs. The first difference is the large diamagnetic amplitude in the heavily doped samples. Recall from Sec. 1.1 that the diamagnetic center is not observed at low temperatures in high-resistivity GaAs. The nature of the diamagnetic center will be discussed in more detail in Sec. 5.9. Furthermore, in high-resistivity GaAs, the 1/Ti relaxation observed in LF is due to Mu^ undergoing quantum diffusion [21,22]. In contrast, although a small fraction of the low-field relaxation may be due to Muj. in the heavily doped samples studied, most of the observed spin relaxation is due to M u ^ c . With these large concentrations of free electrons, Muj. appears to be rapidly converted to Mu~ (see Sec. 5.9). As in Si:P (Sec. 5.6), the stability of M u ^ c to a transition to Mu" implies a significant energy barrier for the reaction M u ^ c +e~ —+ Muy. Note that although the sample is on the metallic side of the metal-insulator tran-sition, the value of USE in GaSi-17-100 is relatively small, indicating that ionized elec-trons, i.e. those in the conduction band rather than the impurity bands, contribute to the spin-exchange process 9. An impurity band has been detected in GaAs:Si up to concentrations Np f=a 5iVc.[66] It is only when NE> ^> Nc that the impurity band merges with the conduction band. As a rough approximation, we assume that a narrow im-purity band does indeed exist and is at Ed — 5.81 eV below the conduction band (as in lightly doped GaAs:Si). If we assume that USE = °nv where n is calculated from Eq. (5.9) with ND = 9 x 1016 cm" 3 and v is given by Eq. (5.4) with m*th = 0.067me, the fitted uSE corresponds to av(5.5K) = 1.3(1) x 1 0 - 7 s _ 1 cm3 and a = 2.3(1) x 10 - 1 4 cm2, roughly comparable to the values in intermediate doped Si:P (Sec. 5.6). (On the other hand, if it is assumed that n ss No, ie. all dopant electrons contribute to the spin exchange process, then a = 1.3 x 10 - 1 0 c m - 2 , which seems too large.) 9The carriers in a narrow impurity band have a much lower v than electrons in the conduction band and so would make a much smaller contribution to USE • Chapter 5. MEASUREMENTS and DISCUSSION 120 No evidence for a Knight shift is observed in the heavily doped GaAs samples at T F = 1.5 T up to room temperature. Since previous studies of M u ^ c in high resistivity GaAs have shown that this center undergoes significant ionization above approximately 30 K, it may be difficult to observe a Knight shift above this temperature, provided that a site change to another state, e.g. M u - , is also taking place. 5.9 D i a m a g n e t i c Centers in H e a v i l y - D o p e d G a A s As discussed in Chapter 1, the charged isolated muonium and hydrogen centers are poorly characterised. The position of the Fermi level determines the stable equilibrium charge state of the hydrogen or muonium center. This can be illustrated as follows: suppose that the muonium level (located in the band gap) which is singly occupied by one electron, either spin up or spin down (hence the spin degeneracy is two), is at energy E& above the top of the valence band and tha t the extra Coulomb energy for double electron occupancy of the level is U. In this simplified system, the relative equilibrium populations of Mu + :Mu°: M u " can be shown to be [47] -, o &F — E4 (2Ep — 2Ed — U 1 : 2 exp —;——— : exp —— p kBT P V kBT In a heavily doped n-type semiconductor, the Fermi level Ep is close to the conduction band Ec, while in a p-type semiconductor it is close to the valence band. Hence, in heavily doped n-type semiconductors, the negative charge state will be favored (pro-vided the quantity Ec — Ed is more than a few kBT greater than U), while if Ep is close to the valence band, such as in heavily doped p-type semiconductors, the positive charge state will be preferred (provided U is a few kBT more than —2Ej). The primary aim of this section is to report and discuss measurements regarding the diamagnetic center Mu f l in GaAs. In particular, we discuss measurements on (5.16) Chapter 5. MEASUREMENTS and DISCUSSION 121 Nat. Abundance j n eq Nucleus (%) (MHz/T) (e x 10"2 8 m2) 7 1Ga 39.8 12.984 0.112 6 9 Ga 60.2 10.219 0.178 75As 100.0 7.292 0.3 Table 5.5: Natural abundance, gyromagnetic ratio j n and quadrupole moment eq for the three isotopes of GaAs. All nuclei have a spin of 3/2. GaAs heavily doped with Zn (p-type) and Si (n-type) using time-integral LF and TF-//SR. The GaAs samples studied are part of the list tabulated in Table 5.4. TF-^tSR experiments at TRIUMF were performed in a He gas flow cryostat (TF measurements) at the M15 and M20 beamlines. As described below, measurements were also made at the 7rM3 channel in the PSI //SR facility, the ISIS facility at the Rutherford Appleton Laboratory and in the LAMPF /«SR facility (see Sec. 4.3). The most relevant aspect of these measurements is the characterization of the local electronic structure of Mu~ in heavily doped n-type GaAs. Our ability to characterize the diamagnetic centers in GaAs rests on the fact that the host atoms have a spin of 3/2. The natural abundances, gyromagnetic ratios and quadrupole moments of these nuclei are summarized in Table 5.5. Since there is no unpaired electron spin density at the muon, Mu D does not have a hyperfine interaction. However, it is still possible to observe resonances in an integrated LF experiment (see Sec. 3.4) and depolarization of the muon spin rotation signal (see Sec. 3.3) because of the existence of the muon induced quadrupole and the magnetic dipole-dipole interactions between the muon and these neighboring nuclei. Such measurements provide information on the muon site as will be detailed below. Chapter 5. MEASUREMENTS and DISCUSSION 122 & A A A A A A -;**W>0° i i i i o i $ • • i i 0 50 100 150 200 250 Temperature (K) Figure 5.15: Temperature dependence of the T F Mu D amplitude (Ad) for n-type GaSi-18-100 (circles) and p-type GaZn-19-100 (triangles) at T F = 1 . 5 T. Although the formation of Mu D from the paramagnetic centers is not well-understood, we would be remiss not to briefly mention some qualitative features of the data and current speculations. Fig. 5.15 displays the temperature dependence of the T F diamag-netic amplitude Ad in GaSi-18-100 (n-type) and GaZn-19-100 (p-type) with the applied magnetic field B 0 | | (100) axis in both samples. The absolute fractions of Mu D are not well-determined since there were no comparisons made under the same experimental conditions with a sample which has a 100 % diamagnetic signal (such as high purity Ag). Note, however, tha t the low field value of Ad in Ag is « 0.25 and this value can only decrease at high fields due to the finite t ime resolution of the particle detectors [56], indicating that the Mu D fraction in these samples is large. The significant dia-magnetic amplitude at all temperatures should be contrasted with the much smaller Mu f l fraction in high resistivity GaAs. Note that Ad in the p-type sample appears to be temperature independent whereas this is clearly not the case in n-type GaAs. The sum of the Mu D and M u ^ c amplitudes in GaSi-18-100 at low temperatures (e.g. 5 K) is significantly less than the full precession amplitude, implying a so-called "missing fraction" of the polarization. Since M u ^ c is directly observable in T F and LF (see Fig. 5.13 and Sec. 5.8), the missing fraction is a t t r ibuted to Mu^. It has been suggested Chapter 5. MEASUREMENTS and DISCUSSION 123 from studies of the field dependence of the non-relaxing component of the LF polar-ization in n-type GaSi-17-100 that below 150 K, the cross-sections for spin-exchange scattering and electron capture by Muy to form M u - are similar in magnitude. [67] The most prominent feature in Fig. 5.15, i.e. the minimum at 50 K for Aj, in n-type GaAs, is not fully understood at present 10. We now emphasize measurements on the diamagnetic center at room temperature in n-type GaAs. Based on the simple theoretical arguments outlined above, M u - is ex-pected to be responsible for the Mu D signal in n-type GaAs while M u + the diamagnetic signal in p-type GaAs. This is experimentally supported by the different temperature dependence of A^, as shown in Fig. 5.15. Further confirmation comes from the differ-ences between the field dependences of the 1/T2 linewidth and the ALCR spectra in the n- and p-type samples, which are discussed below. Henceforth, we will refer to Mu D as M u - in the n-type samples and Mu + in the p-type samples. However, the ensuing discussion is valid regardless of the charge state of the diamagnetic center. Fig. 5.16 shows the integrated LF (ALCR) spectrum on GaAs:Si, obtained by our collaborators at LAMPF, with Bo parallel to a (100) axis in n-type GaSi-18-100 at room temperature. The y-axis is 1 — Pz where the same notation as in Sec. 3.4 is used where t\ = 0 and t2 = oo in Eq. (3.34). Except for the low field zero-crossing, the five resonances shown were the only ones clearly seen up to « 120 m T . The absolute intensities of the resonances in the GaAsrSi spectrum are estimated by comparing with integrated T F measurements on copper using the methods described in Ref. [68] and by taking into account that M u - does not give rise to the full T F precession amplitude in this field range (the M u - amplitude is « 87% of the diamagnetic 10Note that if Mu" forms from Mu§>, then Ad will always increase with increasing M u - formation rate. As a matter of fact, from using a strong collision approach similar to that in Sec. 3.7, one can show that in the high field limit, Ad = yt+wtu w n e r e ^ is the Mu - formation rate and u0 — 2irAfi where A^ is the muon hyperfine parameter of Mu^. Chapter 5. MEASUREMENTS and DISCUSSION 124 0 10 20 30 MAGNETIC FIELD (mT) Figure 5.16: The field dependence of the integrated longitudinal polarization at room temperature in GaSi-18-100 with B 0 parallel to a (100) axis. The experimental data were obtained by the group at the LAMPF facility. The curves are fits to the data as described in the text. The fits obtained by assuming Mu" is in a Toa site are plotted as the solid curves. Fits assuming Mu - is in the BC site (dashed curves) are virtually indistinguishable. 0.008 I CLN 0.004 0.000 Chapter 5. MEASUREMENTS and DISCUSSION 125 amplitude in high purity 99.99 % Ag). ALCR measurements were also made on the n-type GaAs:Te sample, i.e. GaTe-19-100, by our collaborators at the ISIS pulsed fxSR facility. Although the statistics is much lower, the spectrum is consistent with that shown in Fig. 5.16. In contrast, no resonances were observed up to 0.2 T in p-type GaZn-19-100 with B 0 parallel to a (100) axis. The four high field (?» 20 — 45 mT) resonances show unambiguously that the muon and nearest neighbor(s) Ga lie on the same (111) axis. This clear identification is possible because the shapes and positions of the resonances are determined by just two independent parameters, Q' and Dl, associated with one of the Ga isotopes. (The same notation as in Chapter 3 is used.) The dipole and quadrupole parameters of the other isotope satisfy the conditions D(69Ga)/D(71Ga.) = 7 n ( 6 9 Ga)/7„( 7 1 Ga) and <2(6 9Ga)/Q(7 1Ga) = q(69Ga)/q(nGa). The strength of Q{ determines the positions of the resonances whereas the widths and intensities are determined by D\ In the situation where there are N equivalent nuclei, the intensity of a resonance is essentially determined by VNDi although its position remains virtually unchanged (see Sec. 3.4). Hence, N and £), cannot be obtained independently. Note that the small number of resonances is due to the fact that , as noted earlier, with B 0 along a (100) direction, the four (111) axes are equivalent with 6 = 54.74°. Any site on a (111) axis satisfies this. The highest symmetry site is a tetrahedral interstitial site. Other possibilities include the BC and AB^a (anti-bonding to a Ga nucleus) site (see Fig. 1.1). Quantitative fits through the four high field resonances are obtained with the following assumptions: 1. T G 0 site: Pz is calculated from Eq. (3.34) and Eq. (A.6) by assuming that only the four nearest neighbor Ga atoms contribute to the quadrupole and dipole interaction and that 6 = 54.74°. 2. BC site: only the nearest neighbor Ga on the (111) axis is assumed to Chapter 5. MEASUREMENTS and DISCUSSION 126 contribute. 3. ABGO site: the muon lies close to one of the four Ga nuclei in the T^a c&ge and only this nucleus contributes (mathematically, this is equivalent to the situation of the muon at the BC site). As discussed in Sec. 3.4 and Appendix A, one must also consider the appropriate probability distributions due to fact that Ga exists as two isotopes. In the model where Mu~ is at a Toa or a BC site, the most sensible assignment for the remaining lower field resonance at PS 9 m T is that it is due to the As nuclei. If the Mu" was located at the AB site, the lower field line is due to several of the As nuclei in the next shell. This resonance cannot be due to Ga since the isotopic signature of two lines (one due to 6 9Ga and the other to 71Ga) is not observed in the magnetic field region examined. A quantitative fit of this resonance is obtained with the following assumptions: 1. Toa site: from Table 5.1, it can be seen that when Bo is parallel to a (100) direction, there are six As next nearest neighbors, four at 8 = 90° and two at 6 = 0°). 2. BC site: the resonance is due to the single neighboring As atom on the (111) axis. 3. AB<3a site: the distance from the muon to the Ga can be estimated from the Ga resonances (see below). As a rough approximation, the Toa cage is assumed to be undistorted and the values of 6 for the three As nuclei can be obtained. Fig. 5.16 displays examples of the fits to the five resonances observed in GaAs:Si. The lower field region from « 50 — 100 m T containing the low field resonance and Chapter 5. MEASUREMENTS and DISCUSSION 127 i r i i 0 2 4 6 8 TIME (/xs) Figure 5.17: TF-^SR da ta at room temperature in GaSi-18-100 with B 0 | | (100), B0 » 0.19 T, which has been displayed for convenience in a reference frame rotating about w 0.46 MHz below the Larmor frequency of the free muon. The solid curve is a fit to a function proportional to e~° t cosiuj^t + <f>). The dashed line is the fitted damping envelope ±e_<T f . the region from « 180 — 430 m T containing the four high field resonances are fitted separately. There is little difference in the quality of the fits for any of the three site assumptions. Hence, the strengths of the quadrupole interactions (Q ! 's) extracted from these fits are essentially independent of the site assumption. These parameters, as well as those describing the GaAs:Te ALCR spectrum, are listed in Table 5.6. Although the relative strengths of the muon-Ga and muon-As dipolar interactions (for the TGO and BC sites) can be accurately estimated from the ALCR spectrum, the 0 . 2 -0 . 1 -^ 0 . 0 -Q_ X < - 0 . 1 -- 0 . 2 -Chapter 5. MEASUREMENTS and DISCUSSION 128 =1 U. 12 0.08 0.04 n 1-h * -1 • GaAs :Si i i • • l — i <111> <110>_ <100> 0.12 0.08 0.04 -0 4 = f * (b) GaAs:Te <111> <110> <100> 1 10 100 1000 MAGNETIC FIELD (mT) Figure 5.18: Plots of the room temperature T F depolarization rate <r as a function of magnetic field in (a) GaAs:Si and (b) GaAs:Te. The filled squares, triangles and circles are results of measurements where B 0 is parallel to a (100), (110) and (111) axis respectively. The solid curves in (a) and (b) are obtained by assuming Mu~ is in a T G 0 site. For comparison, the long-dashed curve in (a) shows the expected field dependence of o when B 0 || (100) direction for M u " in an undistorted Toa site in GaAs. Chapter 5. MEASUREMENTS and DISCUSSION 129 absolute strengths are only known approximately since the normalization of the ALCR intensities may be subject to considerable systematic error. More accurate values of the absolute dipolar parameters and hence the muon-nuclear distances are obtained from TF-//SR. Recall from the theoretical discussion in Sec. 3.3 that the dipole-dipole and muon-induced quadrupole interaction causes damping of the Larmor precession signal. A typical example is shown in Fig. 5.17 which shows a TF-//SR run in GaSi-18-100 where the relaxation of the precession signal is fitted to a Gaussian damping function exp[—cr2t2]. Fig. 5.18a shows the room temperature field dependence of a for the three major crystallographic directions in n-type GaAs:Si. Three samples, GaSi-18-100, GaSi-18-110 and GaSi-18-111 were studied with B 0 parallel to the (100), (110) and (111) faces respectively. In Fig. 5.18, the dramatic decrease of a with increasing magnetic field when B 0 is parallel to a (100) direction is a signature that the nearest neighbor nuclei are at 6 = 54.74° (see Sec. 3.3), confirming that the dominant nuclear dipole interaction comes from a nucleus on the same axis. Assuming the Taa site and the quadrupole parameters obtained from the ALCR data, the muon nuclear magnetic dipole parameters and the corresponding muon nuclear distances were obtained from a global fit of the TF-^SR linewidth data for the three orientations u . The contributions from the nearest neighbor Ga nuclei and next nearest neighbor As nuclei are considered. An estimate of the systematic errors involved is obtained by assuming (a) the ratio of dipole parameters is also fixed to that obtained from the ALCR and (b) the ratio is not constrained. The fits and Ga and As distances obtained by assuming Mu~ in a TGO site are shown in Fig. 5.18 and Table 5.6 respectively. At this point, it should be emphasized that none of the T(j a , BC or AB^a sites can be ruled out experimentally. An analysis based on the assumption tha t M u " is at a BC site estimates that the Ga and As nuclei on the (111) axis are distorted outwards u T h e theoretical approach used is described in Sec. 3.3 and Sec. A.2. Chapter 5. MEASUREMENTS and DISCUSSION 130 by « 42% [rGa = 1.75(2) A] and « 60% [rAs = 1.95(3) A] respectively. In the case of the ABGO site, rca « 1.75(2) A and TA* = 2.42(3) A where, as indicate above, the To a cage is assumed to be undistorted 12. The quoted error bars are statistical. However, the BC site remains highly unlikely for Mu" on theoretical grounds since this site is found to be the highest energy (least stable) of all the sites tested. [25] This supports the qualitative viewpoint that placement of another electron into an already electron-rich state is energetically unfavorable due to the strong electron-electron repulsive energy. All existing theoretical results predict that M u - is lowest in energy when occupying regions of low valence charge, such as the tetrahedral interstitial site. Adiabatic po-tential calculations [25,69] place the overall minimum off-center from Tca but at only slightly lower energy. If this is correct, the zero point energy for the muon would likely establish a state centered at the T(ja site. The magnitude and width of the ALCR resonances and the TF-/^SR line broad-ening imply that M u - at room temperature does not undergo long-range diffusion on the timescale of the muon lifetime. This result should be contrasted with the rapid diffusivity of Mu§. at all temperatures in high resistivity GaAs ( « 1010 hops/s at room temperature) . Several factors may lead to a small diffusion ra te in the case of Mu" . In particular, the large lattice distortion implied by the results discussed above would reduce the tunneling matr ix element and diffusivity of M u - . Furthermore, the ionicity of GaAs serves to lower the energy of the T o a region for M u " due to Coulomb interac-tions, while at the same time raising the energy of the TAs site, compared to that for M u ° . Consequently, the energy barrier for M u - to hop to the next equivalent site may be much larger. Our M u - results should closely model the structure of isolated H~ in GaAs. On the other hand, the diffusivity of H~ is expected to be even smaller since its 12In this situation, since rGa = 1-75 A, there will be two As nuclei at 6 = 80.5° and one at 9 = 13.5° contributing to the resonance at « 9 mT. Chapter 5. MEASUREMENTS and DISCUSSION 131 heavier mass lowers the zero-point energy and reduces the tunneling matr ix element. In heavily-doped GaAs, there is a possibility of muonium-impurity formation. Mea-surements on n-type GaAs:Si were compared to the room temperature ALCR and TF-fj.SK measurements on n-type GaAs:Te, i.e. GaTe-18-100. The TF-^SR measurements were obtained by rotat ing the wafer to obtain Bo parallel to the three major crys-tallographic axes. These da ta are shown in Fig. 5.18b and are very similar to that in GaAs:Si. The near identical values of the fitted parameters between GaAs:Si and GaAs:Te shown in Table 5.6 is strong evidence that the center is not closely associated with a dopant atom. Note tha t Si substitutes at a Ga site while Te substitutes at an As site. Consequently, the Mu-Si and Mu-Te complexes should have very differ-ent signatures, particularly in the ALCR spectrum at room temperature. The lack of muon-impurity formation is not surprising. The only mobile center is Mu^ where the hop rate in high-resistivity GaAs is found to be « 1010 s _ 1 [21] at room temperature. The M u D (Mu~) center is formed within « 10 - 1° s or sooner, see Fig. 5.15. 13 Hence, if Muy is "thermalized" at random locations in the sample, it would only make ap-proximately one hop at most before converting to Mu D , implying that the Mu-complex formation probability is negligibly small. Currently, no similar information regarding the M u + center in the p-type GaAs sample is available. The TF-//SR linewidth a « 0.12 /us - 1 and remains essentially field-independent up to 2 T when B 0 is applied parallel to a (100) axis. The lack of "drop-off" in a implies that the positions of any ALCR resonances cannot be estimated, making a search very time-consuming. Above 150 K, the linewidth decreases dramatically, implying that M u + is undergoing rapid diffusive motion, hence "motionally averaging" 13At high fields, the transition rate A for Mu§, —• MuD is given by A = \J J°"j' •, as can be shown by using a strong collision approach similar to that in Sec. 3.7 where fo is the strength of the diamagnetic signal normalized to that in silver. Chapter 5. MEASUREMENTS and DISCUSSION 132 GaAs:Si GaAs:Te |£(71Ga)|//i (MHz)a 1.472(4) 1.532 (4) \QC5As)\/h (MHz)° 0.621(6) 0.695 (8) Ratio of Dipole Parameters Unrestricted Fixed Unrestricted Fixed r(Ga) A 2.192 [-10.5%] 2.206 [-10.0%] 2.176 [-11.2%] 2.200 [-10.2%] r(As) A 2.77 [-2.1%] 2.68 [-5.3%] 2.92 [+3.2%] 2.68 [-5.3%] °<9(69Ga)/Q(71Ga) = ? ( 6 9 Ga) / 9 ( 7 1 Ga) = 0.178/0.112. The quoted error estimates are statistical. Table 5.6: The quadrupole parameters Q and the Ga and As distances obtained from the fits to the GaAs:Si and GaAs:Te da ta as described in the text for the assumption of TGO site. The percentage numbers in square parenthesis below the values for r (Ga) and r(As) indicate the deviation from the unrelaxed nearest neighbor and next nearest neighbor distances of 2.45 A and 2.83 A respectively. The same Q values will also be valid if M u " is assumed to be in the BC site. The meanings of "unrestricted" and "fixed" are described in the text. Chapter 5. MEASUREMENTS and DISCUSSION 133 the random dipolar fields giving rise to the damping of the precession signal. Rapid diffusion of Mu" occurs above « 450 K in the GaAs:Si and GaAs:Te samples. These aspects are still being actively investigated at TRIUMF. The absence of Mu" diffusion implies that H~ will probably not be mobile at temper-atures significantly below 450 K. Considering the stability of M u - , it may be surprising that passivation of donors in n-type GaAs involves diffusion of a H - center, as proposed by some researchers. [70] However, it should be noted that some aspect of hydrogen pas-sivation and "diffusion" experiments are usually performed at high temperature and on much longer timescales. For example, intentional hydrogenation of the GaAs samples are usually done at temperatures in excess of 150 °C and on a timescale of hours. Sim-ilarly, measurements of diffusion are performed at temperatures of typically 100° C or greater, and again on timescales of minutes to hours. Under these conditions, it is still conceivable tha t motion of H~ is important . Furthermore, it was shown earlier (Sec. 5.4) that Mu can cycle between its charged and neutral states at high temperatures and in the presence of free charged carriers. A similar situation is expected to hold for M u " and H~. The diffusivity of Mu^ is much higher than that of Mu", a situation which is probably also valid for hydrogen. Hence, one intriguing possibility is tha t under certain conditions, the diffusion rate of hydrogen may be governed by the fraction of time it spends as H^. C h a p t e r 6 S U M M A R Y and C O N C L U D I N G R E M A R K S In this thesis, recent ^SR measurements on muonium centers in Si and GaAs have been described. Spin-exchange scattering and charge-exchange with conduction electrons are found to be important dynamical processes in these semiconductors. Studies of the magnetic field and temperature dependences of the 1/Ti spin relaxation rates via the longitudinal field ^ S R technique allow one to characterize the muonium center in situations where the precession signals are not directly observable. Furthermore, they allow one to extract dynamical parameters which characterize the spin and charge exchange processes. Extensive measurements of muon spin 1/Ti relaxation in intrinsic Si and p-type Si:B above room temperature confirm that muonium is rapidly cycling between its neutral and positive charge states by interaction with conduction electrons, i.e. Mu <-> Mu + +e~. The neutral s tate is similar to the rapidly diffusing Mu° center, as indicated by the magnitude of its average hyperfine parameter, indicating that muonium spends a significant amount of t ime away from the predicted adiabatic potential energy mini-mum, the bond-center site. The field dependences of the 1/Ti rates in intermediate doped n-type Si and heavily doped n-type GaAs:Si provide clear signatures of MugC centers undergoing relatively slow spin-exchange scattering with conduction electrons. These results significantly expand the temperature and concentration regions for which M u ^ has been shown to exist in n-type Si and firmly establish its presence in heavily-doped n-type GaAs, implying that in these semiconductors the cross-section for electron 134 Chapter 6. SUMMARY and CONCLUDING REMARKS 135 capture by Mug C to form Mu" is significantly smaller than that for spin-exchange scat-tering. At very high spin-exchange rates, such as in heavily doped Si:P, and under the application of a large magnetic field, M u ^ c is "spin polarized" and a frequency shift (Knight shift) from the Larmor frequency of a free muon is observed. The frequency shift approximates a Curie-law dependence above 30 K but decreases dramatically at low temperatures. This indicates the local spin susceptibility is strongly affected by donors below this temperature. A structural study of the charged muonium center in a semiconductor is also pre-sented, in particular the M u " center in n-type GaAs. At room temperature, Mu" exists as an isolated center with the muon and nearest neighbor Ga on the same (111) axis. The M u - center appears to be located at or near a T^a site. 6.1 M u o n i u m Spin D y n a m i c s in Other Mater ia ls The spin dynamics of muonium described in this thesis are not expected to be restricted to Si and GaAs alone; clearly, they can occur in other semiconductors and, for that matter , in any metallic or conductive systems where muonium is stable. Such dynamics have indeed been observed recently in various materials such as Ge [19,71], the fullerene superconductor A3Ceo and Sb (antimony) [49]. The importance of such processes in Si has also prompted researchers to reevaluate and extend previous /xSR studies in other semiconductors. For example, the rapid loss of polarization due to spin and charge exchange has been recently postulated to account for experimenters' inability to observe paramagnetic muonium in InP and has revitalized research on this material. Chapter 6. SUMMARY and CONCLUDING REMARKS 136 6.2 H y d r o g e n The similarity between muonium and hydrogen implies that spin and charge exchange processes that are observed in muonium are also occurring for hydrogen. However, the studies of hydrogen are made on a timescale much longer than for muonium. This means that direct quantitative comparisons between the muonium dynamics described in this thesis and hydrogen experimental work are difficult because of the inability of experimental techniques used to study hydrogen to probe the same dynamics. Never-theless, one of the important consequences of the current studies on muonium dynamics should be a caution to researchers working on hydrogen diffusion and other dynamic properties tha t a complicated set of state and possibly site changes are active on short time scales within the temperature and doping range usually investigated x. Results which interpret final state hydrogen products in terms of a single diffusing hydrogen state at high temperatures are clearly not valid and should probably be reexamined within this more realistic dynamical framework. One of the most important aspects of hydrogen in semiconductors is the formation of hydrogen complexes and passivation of electrical activity. All a t tempts to observe direct muonium-dopant interaction in semiconductors have been unsuccessful.[72,73] Several reasons have been proposed for Si, and may also be valid for other semiconductors. In general, the dopant concentration must be high enough so that the muon or muonium, which probably thermalizes at random sites in the lattice, can reach an impurity and form a complex within the ^s timescale or quicker. However, in heavily doped p-type Si, hole capture is likely to occur and hence the majority of the prompt centers in these samples are Mu + , which may not diffuse rapidly until very high temperatures. Unfortunately, at these elevated temperatures, rapid reorientation of the M u ^ - d o p a n t 1It is interesting to point out that a recent publication by Johnson e l al [10] utilize the charge state changes of hydrogen (H+ +2e~ <-*• H~) to establish the "negative U" character of this defect in Si. Chapter 6. SUMMARY and CONCLUDING REMARKS 137 complex (or even breakup) can occur, analogous to the effect observed for the Si(B,H) passivation complex [74], rendering direct spectroscopic observation via ^SR unlikely. In n-type Si, direct muonium-donor interactions may be difficult to observe even if the complex is formed since the analogous hydrogen-donor complex is located in an anti-bonding position adjacent to a Si host atom. It remains to be shown that muonium complexes can be observed. So far, the most important contribution from /xSR to this field has been the information that it provides on electronic structure and dynamics of isolated muonium/hydrogen centers. 6.3 E x t e n s i o n s of Current S tudies Although a considerable amount of work has been done regarding the dynamics of muonium centers in semiconductors, in particular in Si, a large amount of research is still in progress. Several examples pertaining directly to the discussions in this thesis are mentioned briefly below. In addition to extending the experiments of spin and charge dynamics to other semiconductors, as mentioned above, there are also ongoing experiments in Si aimed at building a more complete quantitative picture of the transitions among the various Mu states by using the RF-^tSR and LF-yuSR techniques. The potential of being able to obtain site information on the charged muonium centers, as demonstrated by the work on heavily doped n-type GaAs described in this thesis, is very exciting. Researchers in this field are now in a position to obtain the same sort of structural information on the charged centers as for the neutral states, which was considered one of the major triumphs in muonium semiconductor studies. The ability to characterize charged centers in GaAs has also led to a flurry of experiments to investigate the formation and diffusion of these muonium centers. These measurements will undoubtedly have direct Chapter 6. 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Depolarization of M u + mesons in solids. Sov. Phys. JETP, 16:1236, (1963). M. Senba. Spin dynamics of positive muons during cyclic charge exchange and muon slowing down time. J. Phys. B, At. Mol. Opt. Phys., 23:1545, (1990). M. Senba. Repeated electron spin exchange of muonium with spin-1/2 species. J. Phys. B, At. Mol. Opt. Phys., 23:4051, (1990). M. Senba. Spin dynamics of the positive muon radicals in the presence of rapid electron spin exchange: frequency shift and relaxation. J. Phys. B, At. Mol. Opt. Phys., 24:3531, (1991). M. Senba. Muonium spin exchange in spin-polarized medium: Spin-flip and -nonflip collisions. Phys. Rev. A, 50:214, (1994). I.G. Ivanter and V.P. Smilga. The theory of Mu+ -meson depolarization with al-lowance for the process of charge exchange or formation of unstable chemical com-pounds. Sov. Phys. -JETP, 33:1070, (1971). N.W. Ashcroft and N.D. Mermin. Solid State Physics. Holt, Rinehart and Win-ston, 1976. T.M. Riseman. fiSR Measurement of the Magnetic Penetration Depth and Co-herence Length in the High-Tc Superconductor YBa2Cu30e.95- PhD thesis, The University of British Columbia, 1993. T.M.S. Johnston. Anomalous electronic structure of the muon in Antimony: evi-dence for an isolated Kondo impurity. Master 's thesis, The University of British Columbia, 1993. K. Holloway. Tantalum as a diffusion barrier between copper and silicon. Appl. Phys. Lett, 57:1736, (1990). Bibliography 143 [51] M.A. Paciotti, D.W. Cooke, M. Leon, B.L. Bennet, C. Pillai, O.M. Rivera, B. Hitti, T.L. Estle, S.F.J. Cox, R.L. Lichti, T.R. Adams, C D . Lamp, A. Morrobel-Sosa, 0 . Richter, C. Boekema, J. Lam, S. Alves, J. Oostens, and E.A. Davis. Develop-ment of a uLCR Facility at LAMPF. Hyp. Int, 87:1111, 1994. [52] Application Software Group. Minuit: Function Minimization and Error Analysis. Technical report, CERN, 1992. [53] M.A. Green. Intrinsic concentration, effective densities of states, and effective mass in silicon. Physica B, 67:2944, (1990). [54] F.J. Morin and J.P. Maita. Electrical Properties of Silicon Containing Arsenic and Boron. Phys. Rev., 96:28, (1954). [55] C D . Thurmond. . J. Electrochem. Soc, 122:1133, (1975). [56] E. Holzschuh. Direct measurement of muonium hyperfine frequencies in Si and Ge. Phys. Rev. B, 27:102, (1983). [57] C.H. Chu and S.K. Estreicher. Similarities, differences, and trends in the proper-ties of interstitial H in cubic C, Si, BN, BP, A1P, and SiC. Phys. Rev. B, 42:9486, (1990). [58] F. Buda, G.L. Chiarotti, R. Car, and M. Parinello. Proton Diffusion in Crystalline Silicon. Phys. Rev. Lett, 63:294, (1989). [59] C G . Van de Walle and L. Pavesi. Spin-polarized calculations and hyperfine pa-rameters for hydrogen or muonium in GaAs. Phys. Rev. B, 47:4256, (1993). [60] J.W. Schneider, R.F. Kiefl, K. Chow, S.F.J. Cox, S.A. Dodds, R .C DuVarney, T.L. Estle, R. Kadono, S.R. Kreitzman, R.L. Lichti, and C. Schwab. Local Tunneling and Metastability of Muonium in CuCl. Phys. Rev. Lett, 68:3196, (1992). [61] B. Hitti, S.R. Kreitzman, T.L. Estle, R.L. Lichti, K.H. Chow, J.W. Schneider, C D . Lamp, and P. Mendels. RF-^SR Study of Muonium Charge States and Dynamics in Si. Hyp. Int., 86:673, (1994). [62] S.R. Kreitzman, B. Hitti, R.L. Lichti, T.L. Estle, and K.H. Chow. Muon Spin Resonance of Muonium Dynamics in Si and its Relevance to Hydrogen, submitted to Phys. Rev. B. [63] Adir Bar-Lev. Semiconductors and Electronic Devices. Prentice Hall, 1984. [64] Nevill Mott. Metal-Insulator Transitions. Taylor and Francis, 1990. Bibliography 144 [65] A. Gaymann, H.P. Geserich, and H.V. Lohneysen. Far-Infrared Reflectance Spec-t ra of Si:P near the Metal-Insulator Transition. Phys. Rev. Lett, 71:3681, (1993). [66] H. D. Drew D. Romero, S. Liu and K. Ploog. Observation of a metallic impurity band in n-type GaAs. Phys. Rev. B, 42:3179, (1990). [67] R. Kadono, A. Matsushita, K. Nagamine, K. Nishiyama, K.H. Chow, R.F. Kiefl, A. MacFarlane, D. Schumann, S. Fujii, and S. Tanigawa. Charge State and Diffu-sivity of Muonium in n-type GaAs. Phys. Rev. B, 50:1999, (1994). [68] M.A. Paciotti . Muon Beam Polarization at the L A M P F Biomedical Channel. IEEE Transactions on Nuclear Science, NS-32:3338, (1985). [69] R.L. Lichti, private communication. [70] J . Chevallier, B. Clerjaud, and B. Pajot. Neutralization of Defects and Dopants in III-V Semiconductors. In J.I. Pankove and N.M. Johnson, editors, Hydrogen in Semiconductors. Academic Press in New York, (1991). [71] R.L. Lichti, K.H. Chow, D.W. Cooke, S.F.J. Cox, E.A. Davis, R.C. DuVarney, T.L. Estle, B. Hitti, S.R. Kreitzman, R. Macrae, C. Schwab, and A. Singh. Longitudinal Relaxation of Muonium in Ge and GaAs. Hyp. Int., 86:789, (1994). [72] Dj M Marie, P.F. Meier, S. Vogel, S.F.J. Cox, E.A. Davis, and J .W. Schneider. Passivation of boron in silicon by hydrogen and muonium: calculation of electric field gradients, quadrupole resonance frequencies and cross relaxation functions. J. Phys. Condens. Matter, 3:9675-9686, (1991). [73] D.W. Cooke, M. Leon, M.A. Paciotti, B.L. Bennett, O.M. Rivera, S.F.J. Cox, C. Boekema, J. Lam, A. Morrobel-Sosa, P.F. Meier, T.L. Estle, B. Hitti, R.L. Lichti, E.A. Davis, J. Oostens, and E.E. Haller. Muon Level-Crossing Resonance in Si:Al. Hyp. Int., 86:639, (1994). [74] M. Stavola, K. Bergman, S.J. Pearton, and J. Lopata. Hydrogen Motion in Defect Complexes: Reorientation Kinetics of the B-H Complex in Si. Phys. Rev. Lett., 61:2786, (1988). Appendix A P R O B A B I L I T I E S in A L C R and TF-^uSR A . l Probabi l i t i e s in A L C R The calculation of the LCR spectrum, e.g. 1 - Pz, from the Hamiltonian described by Eq. (3.2) is straightforward when the muon is surrounded by N equivalent nuclei which are 100 % abundant. However, this is not the case in GaAs where the spin 3/2 nucleus Ga has two isotopes 6 9Ga and 7 1Ga with natural abundances fa = 0.602 and / r i = 1 — /69 = 0.398 respectively. Clearly, complications arise. Suppose that the muon sits in a site with TV nearest neighbor Ga atoms and all 9 are equivalent, a situation that would be true if the muon was located in a Tcra or BC site and the applied field B 0 is parallel to a (100) axis. The probabilities of finding n 7 1Ga atoms (and hence N — n 6 9Ga atoms) is given by the binomial distribution: Pn(n) = (N - n)lnl ( A - 1 } and can be derived from lJV = ( / 7 1 + / 6 9 ) 7 V = E p 7 l ( n ) (A.2) For a muon in a Toa site, N = 4 and the quantities P7i(n) [and p&9(n)] are tabulated in Table A.l which follow from Eq. (A.2). l = P 7 i ( 4 ) + . . . . + p n ( 0 ) = (A.3) fn + 4/73i/69 + 6 / ^ / 1 , + 4/71/639 + /649 (A.4) 145 Appendix A. PROBABILITIES in ALCR and TF-fiSR 146 n 0 1 2 3 4 P7i(n) 0.1313 0.3473 0.3444 0.1518 0.0251 Peo(n) 0.0251 0.1518 0.3444 0.3473 0.1313 Table A. l : Probabilites for Ga when JV = 4 The RHS of Eq. (A.2) also shows the method for calculating Pz, the integrated po-larization (the relevant quantity in LCR spectra). One should calculate the integrated polarization from Eq. (3.2) for the following five cases: 1. 4 7 1Ga neighbors, 2. 3 n G a and 1 6 9Ga neighbor 3. 2 7 1Ga and 2 6 9Ga neighbors 4. 1 71Ga and 3 6 9Ga neighbors 5. 4 6 9Ga neighbors Then, the weighted sum is obtained: P, = I > i ( n ) P , ( n 71Ga) (A.5) n=0 where Pz(n 71Ga) is the total polarization obtained from the Hamiltonian consisting of n 7 1 G a and N — n 6 9 Ga nuclei. It should be noted that all the integrated polarizations take on the value of unity far away from a resonance. The numerical calculation involving five Hamiltonians, each of which has dimension 2 x 44 = 512 is numerically very time consuming. Fortunately, in the case of the ALCR Appendix A. PROBABILITIES in ALCR and TF-fiSR 147 data in this thesis (concerning the Ga nuclei), the resonances due to each isotope are well-separated. This implies that the resonances, or more correctly, the deviations from unity, satisfy Rz = f^PriHlRnin) + R^9(N - n)} (A.6) n=0 where ^ ( n ) = 1 - Pz(n 6 9 Ga), Rn(n) - 1 - Pz(n 7 1Ga) and Rz(n) = 1 - Pz, (the .Rs will be zero far away from a resonance). A total of 10 Hamiltonians must now be considered and the integrated polarizations calculated for each. However, only two of these consist of four-nuclei Hamiltonians. As a mat ter of interest, since we will be interested in the regime of parameters where Re^n) = TIRQ9(1) [and R71(n) = nRn(l)}, i.e. dipolar parameter much less than 1/TM, a further approximation can be made: Rtot = (n)Rn(l) + (N-n)Re9(l) = Nf71Rn(l) + Nf69Re9(l) (A.7) where the expectation value of n, i.e. (n), is given by Nf. In the case of TV = 4, Nf71 = 1.592 and iV/69 = 2.408. The above form Eq. (A.7) is useful since it allows one to obtain excellent estimates of the n-atom resonances in a trivial manner from the much easier-to-calculate one atom contribution. A . 2 Probabi l i t i e s in T F - ^ S R Similar complications arise when calculating the "Har tmann contribution" to a due to two isotopes of Ga and also in the case where several values of 6 are possible. This condition on 8 will occur if the the muon was sitting in a T<ya or BC site and the applied field Bo is parallel to the (110) or (111) axes, for example. The number of equivalent nuclei with a certain value of 9 is summarized in Sec. 5.1. Appendix A. PROBABILITIES in ALCR and TF-fiSR 148 M(n 71Ga,iV - n 69Ga) M(4 71Ga,0 69Ga) M(3 71Ga,l 69Ga) M(2 71Ga,2 69Ga) M ( l 71Ga,3 69Ga) M(0 71Ga,4 69Ga) Second Moment Contribution 2mGa7i(90°) + 2mG a7i (35.26°) |[2mGa7i(90°) + mG a7i (35.26°) + mGa69(35.26°)] + 5[™Ga71(90°) + 2 m G a n (35.26°) + mGa69(90°)] |[2mGa7i(90°) + 2mGa69(35.26°)] + |[mGa7i(90°) + mGa69(90°) + mG a7i (35.26°) + mGa69(35.26°)] + |[2mGa69(90°) + 2mGa7x (35.26°)] |[2mG a 6 9(90°) + mGa69(35.26°) + mG a7i (35.26°)] + M™Ga69(90°) + 2mGa69(35.26°) + mG an(90°)] 2mGa69(90°) + 2mGa69(35.26°) Table A.2: Second moment contributions for the muon in a T©a site when Bo is applied parallel to a (110) crystallographic axis. M(n 71Ga,iV - n 69Ga) M(4 71Ga,0 69Ga) M(3 71Ga,l 69Ga) M(2 71Ga,2 69Ga) M ( l 71Ga,3 69Ga) M(0 71Ga,4 69Ga) Second Moment Contribution mGa71(°O) + 3 m Ga 7 1 ( 7 0 - 5 3 ° ) f [mG an(0°) + 2mG a7i (70.53°) + mGa69(70.53°)] + l[mGa69(0°) + 3mGa7i(70.53°)] | [mGa7 1(°°) + mGa7 1(7 0-5 3°)] + 2mGa69(70.53°)] + |[mGa69(0°) + mGa69(70.53°) + 2mGa7i (70.53°)] |[mGa69(0°) + 2mGa69 (70.53°) + mG a7i (70.53°)] + i[mGa7 1(0 O) + 3 mGa6 9(7 0-5 3°)J mGa6 9(°°) + 3mGa69(70.53°) Table A.3: Second moment contributions for muon in a Toa site when B 0 is applied parallel to a (111) crystallographic axis. Appendix A. PROBABILITIES in ALCR and TF-fiSR 149 Recall that the second moments are additive. If the muon is in a T<?a site, then 4 MNN = £ > 7 1 ( n ) M ( n n G a , N - n69Ga) (A.8) n=0 where MJVJV is the second moment contribution due to the four nearest neighbor Ga and the quantity M(n 71Ga, iV — n 69Ga) is summarized in Table A.2 for Bo parallel to a (110) direction and in Table A.3 for B 0 parallel to a (111) direction. For example, in Table A.2, M(n71Ga, N — n69Ga) is calculated by considering all the possibilities of distributing N 71Ga and hence N — n 69Ga nuclei among the four nearest neighbors, two of which at at 35.26° and the remaining two at 90°. The notation rrii(x°) denotes the second moment contribution due to isotope of type i located at 0 = x°. Note that when Bo is parallel to a (100) direction, all four Ga nuclei are equivalent with 6 = 54.74° and MNN = [iV/71mGa7i(54.74°) + iV/69mGa69(54.74°)] (A.9) As an addendum, the next nearest neighbor contributions M/VAW are (recall that 75As is 100 % abundant): MNNN = 2mAs75(0°) + 4mAs75(90°) (A. 10) for B 0 parallel to a (100) direction; MNNN = 2mAs75(90°) + 4mAs75(45°) (A.ll) for the case of B 0 parallel to a (110) direction and MNNN = 6mAs7s(54.74°) (A. 12) in the case of B 0 parallel to a (111) direction. In the case of the muon in a BC site, there are two "nearest-neighbor" atoms on the (111) axis, one Ga and one As. In this case, the contributions due to these two Appendix A. PROBABILITIES in ALCR and TF-fiSR 150 nuclei to the second moment are as follows: MNN = /69mGa69(54.74°) + / 7 1 m G a 7 1 (54.74°) + mAs75(54.74°) (A.13) when Bo is parallel to a (100) direction; MNN = - [/69mQa69(90°) + / 7 i m G a 7 i (90°) + mAs75(90°)] + i [/69mGa69(35.26°) + /7 1mG an(35.26°) + mAs75(35.26°)] (A.14) when B 0 is parallel to a (110) direction and MNN = j [/69mGa69(0°) + /7 1mG a7i(0°) + mAs75(0°) 4 4 + 7 [/69mGa69(70.53°) + /7 imG an(70.53°) + mAs75(70.53°)j (A.15) when B 0 is parallel to a (110) direction. We shall not discuss quantitatively the "next nearest neighbors" for the situation of the muon in a BC site. 


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