Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-ubc_1995-982705.pdf [ 5.99MB ]
JSON: 831-1.0085683.json
JSON-LD: 831-1.0085683-ld.json
RDF/XML (Pretty): 831-1.0085683-rdf.xml
RDF/JSON: 831-1.0085683-rdf.json
Turtle: 831-1.0085683-turtle.txt
N-Triples: 831-1.0085683-rdf-ntriples.txt
Original Record: 831-1.0085683-source.json
Full Text

Full Text

S P I N D Y N A M I C S A N D ELECTRONIC S T R U C T U R E OF M U O N I U M A N D ITS C H A R G E D STATES I N 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 O F THE REQUIREMENTS FOR THE DEGREE OF D O C T O R 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 degree at the University of  of  the requirements  for  an advanced  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 department  or  by  his  or  her  representatives.  It  is  understood  that  copying  my 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  DE-6 (2/88)  D-ec  \%l<Wr  Abstract  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. T h e average muon-electron hyperfine parameter in the neutral state is consistent with M u 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 t h a t 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 heavilydoped n-type GaAs:Si and GaAs:Te show that the majority of positive muons implanted 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.  T h e importance of such processes is usually not taken into account by  researchers of hydrogen diffusion and related dynamics. T h e 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 I N T R O D U C T I O N and B A C K G R O U N D 1.1  Muonium Centers in Si and GaAs  1.2  Charge and Spin Dynamics of Muonium  2 nSK  3  1 7 11 14  2.1  Muons  15  2.2  The Various fxSR Techniques  18  2.3  Electronics  22  2.4  /iSRData  24  T H E O R E T I C A L CONSIDERATIONS  28  3.1  A List of Hamiltonians  29  3.2  Time-Dependent Muon Polarization of Muonium Centers  36  iv  4  5  6  3.3  Relaxation of the M u ° Transverse Field Signal  40  3.4  Quadrupole Level Crossing Resonance of M u 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 FieldStrong 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  APPARATUS  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  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 D a t a  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  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  135  M u o n i u m Spin Dynamics in Other Materials  V  6.2  Hydrogen  136  6.3  Extensions of Current Studies  137  Bibliography  139  A PROBABILITIES in ALCR and TF-^SR A.l Probabilities in ALCR  145 145  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 G a 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)  ax  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 F i g u r e s  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  L F Corrected Asymmetry Spectrum  27  3.1  Breit-Rabi diagram for Mu^ in Si  34  3.2  Breit-Rabi diagram for M u g C in Si  36  3.3  H a r t m a n n curves  43  3.4  L C R 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  L C R of u p 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 m u o n i u m undergoing charge exchange  63  3.11 Simulations of the field dependence of l / T i for M u g 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 relaxation due to MUJB C in Si undergoing spin exchange  viii  71  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 Mu D amplitude in GaAs  122  5.16 n-type GaAsrSi - LCR data  124  5.17 Typical T F spectrum for GaAsrSi  127  5.18 n-type GaAs:Si,Te - T F 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  Mug C  Neutral muonium at the bond-center site  Muc  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  Time differential  TDTF  Time differential transverse field  TF  Transverse field  TM  Thin muon detector  /iSR  Muon spin rotation, relaxation and resonance  VAX  Virtual Access extension  WTF  Weak transverse field  ZF  Zero field  XI  Acknowledgements  I wish to express my sincerest gratitude 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 attending conferences and performing experiments abroad. Without his help, a significant p a r 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 t h e m has been helpful, stimulating a n d fun. In addition, I wish to express my gratitude to T h o m a s Estle and Roger Lichti for their help and tutelage during the last several years. T h a n k s 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 T R I U M F 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, T h o m a s 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 L A M P F , 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  T h e 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. Muonium 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 related to those of atomic hydrogen since their electronic structures are virtually identical except for small zero point motion effects. T h e advantage of muonium is t h a t it is far more easily studied in isolated form than atomic hydrogen. Since t h e discovery t h a t 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 b o t h hydrogen and its muonium analogue. T h e fact t h a 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 demonstrated t h r o u g h 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 t h e 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 semiconductors. Hydrogen forms stable bound states with many defects and impurities; it ties up dangling bonds, passivates shallow donors or acceptors and some deep-level impurities, activates a few originally inactive impurities, and forms various complexes within extended defects. IR studies indicate t h a t when H-acceptor (shallow) complexes are formed, H is positioned in a bond-center site. When H-donor complexes are formed, H is attached to one of the donor's Si nearest neighbors rather t h a n to the donor itself (anti-bonding position). T h e 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 nonexistent. T h e 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 t h e  reaction rates with other impurities. Thus far, there are only two spectroscopic observations of isolated H in a semiconductor, b o t h 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 t h a t H° is situated 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 b o t h 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 t h a 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 t h e 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 d a t a in n-type GaAs have also been proposed. [12] T h e 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 d a t a are usually difficult to interpret and often not reproducible from one lab to the next. T h e 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 n u m b e r of different configurations including atomic, diatomic and as a complex. 1  In 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 t h a t expected for t h e diffusion of a single species with a constant diffusion coefficient.[2] Nevertheless, a number of the attempts to explain the current H d a t a are implicitly based on an assumption t h a t the diffusing species is in a single charge state 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. T h e muonium atom is thus a very light pseudo-isotope of H, with a mass only l / 9 t h t h a 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. T h e 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. T h e muon is relatively short-lived with a mean lifetime of only 2.2 fxs (see Table 1.1). W h e n 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. • T h e muon is short-lived (lifetime of 2.2 fis) and one observes muonium centers which are formed within several ns after implantation. T h e short time scale of muonium studies also implies that the muonium centers observed may not be the equilibrium states, especially at low temperatures where t h e equilibration time 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 m u o n 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 t u r n 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 t h a t the energy level of muonium in a given potential well lies higher t h a n 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 semiconductors, in particular regarding characterization of the isolated paramagnetic centers. However, many open questions still remain regarding their behavior at high temperatures where introduction of atomic hydrogen is usually done and in doped materials 2  In 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. Furthermore, experiments characterizing the charged states of muonium have been virtually non existent. T h e measurements described in this thesis represent recent a t t e m p t s 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. T h e various //SR techniques are powerful tools for studying the spin dynamics of muonium in semiconductors. It is found that at low temperatures, two processes involving 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 located in the bond-center position is found to have a much larger cross-section for spin exchange scattering t h a n for electron capture to form M u - . At high temperatures, significant 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 t h a n the bond-center position, the theoretical global minimum in the potential energy (see Sec. 1.2). T h e 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  e  »+  P+  Mass (MeV) Spin Gyromagnetic ratio 7 (rad s _ 1 T _ 1 ) 7 = 7 /2TT (MHz/T) Lifetime r (//s)  0.51100 1/2 1.75882 xlO 11 27992.48 Stable  105.66 1/2 8.516154 xlO 8 135.54 2.19703  938.28 1/2 2.675221 xlO 8 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 C e n t e r s in Si a n d 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 u p to 1988, and the more recent paper by Kiefl and Estle.[20] Conventional transverse-field ( T F ) , time-differential (TD) muon-spin-rotation (//SR) measurements have demonstrated that when // + is implanted into covalent semiconductors (Si, GaAs, GaP, Ge and diamond) at low temperatures, three distinct centers are formed — two neutral paramagnetic centers known historically in the /J.SK literature 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  Semiconductor  Center  Hyperfine Parameters  Si  Mug. Mu° c  A„ = 2006.3 MHz muon: A\\ = -16.82 MHz, Ax = -92.59 MHz 29 Si: A\\ = -137.5 MHz, A± = -73.96 MHz  GaAs  Mug.  AM = 2883.6 MHz muon: A\\ = 218.54 MHz, A± = 87.87 MHz 75 As: A\\ = 563.1 MHz, Ax = 128.4 MHz 69 Ga: A|| = 1052 MHz, Ax = 867.9 MHz  9  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 t h e interstitial sites associated with the paramagnetic centers. The Mug. center has an isotropic muonelectron 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, M u g C is immobile on the timescale of the muon lifetime and is located between a stretched (Si-Si or GaAs) 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 t h a n AM. T h e 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 t h a t the larger zero-point motion of the M u centers relative to t h a t of the analogous hydrogen centers has only a small effect on the hyperfine parameters. T h e 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). T h e hf parameters of Mug. are certainly positive, in analogy with  Chapter  1. INTRODUCTION  and  BACKGROUND  10  vacuum Mu, a n d correspond to a positive spin density at the muon. In the case of t h e 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. T h e absolute signs have not been directly measured but can be inferred, as discussed in detail in Ref. [20]. In Si, most theories [2] predict t h a 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 t h e surrounding lattice atoms are allowed to relax fully. These electronic structure investigations are equally applied to both hydrogen and muonium since b o t h are treated as classical point particles. T h e conclusion that M u g 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 t h a 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 t h e muon" [17] claims that it is M u ^ c rather t h a n M u J which is metastable (and only exists till « 100 K); furthermore, the authors find t h a t in contrast to the situation for muonium, H g C is stable. Although the two paramagnetic Mu centers have been spectroscopically well characterised, t h e n a t u r e of the diamagnetic center M u D remains a mystery. T h e M u D centers are frequently observed, b u t little is known experimentally about their structure (charge state or site) due to the absence of an unpaired electron spin and the accompanying magnetic hyperfine interactions. Either charged state, M u + or M u - , or any diamagnetic complex containing Mu contributes to the diamagnetic yuSR signal. In the past few years, M u 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 semiconductors. This assignment is based primarily on theoretical expectations rather t h a n experimental results.  Chapter  1. INTRODUCTION  and  BACKGROUND  11  T h e most stable equilibrium charge state of isolated muonium or hydrogen depends on t h e location of t h e 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 b a n d 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 t h e Fermi level is close the the valence b a n d 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 t h a t H + (or M u + ) located at the B C site should b e 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 T o 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  C h a r g e a n d S p i n 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 temperatures and in the absence of any interactions with charge carriers or ionization of muonium. T h e question naturally arises: "What happens at high temperatures?" In the past, measurements attempting to answer this question have been performed mostly  in Si. 3  In 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 treatments [2] is that the many electron system is replaced by an effective- particle model. Electron interactions 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/T 2 linewidth of t h e muon spin precession signals show t h a t 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 significant 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 M u + although muon-decay positron channeling results [26] at room temperature imply that the ionized species resides near the BC site, leading to the assignment of M u + (see Sec. 1.1). Above 400 K, the 1/T 2 linewidth of the diamagnetic signal rises rapidly, indicating that the ionization is reversible and the inverse reaction M u * —+ Mu also occurs.[27] In n-type Si, muonium interacts with conduction electrons and as a consequence, muon spin precession signals for both M u g C and Mu^ are only observed at low temperatures where there is no significant donor ionization. W i t h 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; however, the center responsible is not clearly identified and the physical processes responsible are not well understood. Nevertheless, the existence of fast 1/Ti spin relaxation indicates t h e presence of a paramagnetic species undergoing repeated spin or charge exchange with the conduction electrons. T h e experimental situation regarding the charge and spin dynamics of Mu centers in GaAs is even less understood t h a n in Si. These studies have been primarily focussed on high-resistivity GaAs. Results of TF-//SR experiments (i.e.  1/T 2 linewidth mea-  surements) show that significant ionization of M u g C and Mu^ occurs above « 30 K and « 200 K respectively. [19] T h e 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 characterization 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) t h e 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 t h a t available for the paramagnetic centers, on the charged diamagnetic centers in semiconductors. T h e measurements described in this thesis were undertaken to increase our knowledge in these two areas. T h e experiments also have relevance for studies of hydrogen in semiconductors. T h e 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 d a t a in the thesis. Descriptions of the experimental apparatus with emphasis on the oven follow in Chapter 4. T h e measurements in Si and GaAs are presented a n d discussed in Chapter 5 and the concluding remarks in Chapter 6 end the m a i n 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 d a t a in this thesis. T h e 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 timedifferential and integral //SR techniques. This section also discusses the longitudinalfield (LF) and transverse-field ( T F ) //SR techniques. In this thesis, the time-differential LF-^tSR technique ("Relaxation" in the acronym ^ S R ) was used to study the spin dynamics of muonium in intrinsic Si, p-type Si:B and intermediate doped n-type Si:P. The time-integral L F technique ("Resonance" in the acronym / J S R ) was used to provide structural information on the charged center, i.e. Mu~, in heavily-doped n-type GaAs:Si. T h e 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 standard time-differential electronics at T R I U M F is described. T h e form of the d a t a 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 d a t a is given and examples of their appearance are shown.  14  Chapter 2.  2.1  fiSR  15  Muons  All //SR experiments are possible because muons are naturally spin-polarized. In meson factories such as T R I U M F , 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. T h e muon neutrino v^ has its spin antiparallel to its m o m e n t u m (i.e. negative helicity) and hence the muon is also emitted with its spin and m o m e n t u m antiparallel t o 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 m g / c m 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 t o the positron). T h e 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 t h e fi+ spin is: dW(e, 9) = — [1 + a(e) cos(0)]n(e) de dcos(9), where e = E/Emax  (2.6)  is the reduced positron energy, a(e) = (2e — l ) / ( 3 — 2e) and n(e) =  e 2 (3 — 2e). T h e 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 W(6) =  2TIL  1+  cos(0)  (2.8)  W(9) is also plotted in Fig. 2.1. It should be emphasized that the feasibility of //SR rests on the fact t h a t the decay positron is not emitted isotropically; instead, it is correlated with the muon spin direction at the time of the decay.  Chapter 2.  fiSR  17  €= 1  6 = 0.8  a  6=0.6  muon spin e=0.4  6= 0  e=0.2  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) t h e energy averaged distribution W(8). T h e distance from the origin to any point on t h e curve is proportional to the decay rate at angle 9 with respect to the muon polarization. T h e arrows to the right labelled 'muon spin" indicate the direction of the muon spin.  Chapter 2.  2.2  fiSR  18  T h e V a r i o u s /xSR T e c h n i q u e s  A fxSR experiment involves implanting highly polarized positive muons into the sample of interest. T h e 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. T h e beam structure of L A M P F will be discussed in more detail in Sec. 4.3. Further discussion in this chapter will concentrate primarily on the ^ S R facility at T R I U M F . T h e polarized muons are manipulated by a combination of bending (dipole) magnets, 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. T h e Wien filter consists of crossed electric and magnetic fields which are both perpendicular to the muon beam, effectively eliminating contamination in the b e a m from positrons which have t h e 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 t h a t miss the sample can produce a background signal. T h e //SR techniques used to obtain the results discussed in this thesis can be divided 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 t h a t a detected decay positron can be unambigiously associated with its parent muon. Hence, the primary task of the fast electronics and d a t a acquisition system is to select events in which only one muon and one positron are detected within a given t i m e 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 t h e same time. Hence, there is no need to associate a given positron with an individual muon. One disadvantage of pulsed facilities is t h a 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 t h e muon polarization function is unnecessary. Sometimes, it is sufficient to measure its integrated value over some length of time which is long relative to the muon lifetime — this is t h e 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. T h e ability to use greatly increased count rates provides increased sensitivity in some types of experiments. In general, t h e 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. T h e photomultipliers give rise to voltage pulses which are carried by coaxial cables to t h e electronics in the "counting room" for processing.  Chapter 2.  fiSR  20  X  I muo momentum  y  TF -  —z  X  wTF -  —z —z  spin and field perpendicular spin and field perpendicular  LF — spin and field parallel 0  ZF — no field  Figure 2.2: Schematic of the typical arrangements for the counters, muon spin and magnetic field in transverse-field ( T F ) , weak transverse field (wTF), longitudinal field (LF) and zero field (ZF) ^SR. T M labels the Thin Muon counter. B, L, F , R are labels for t h e Back, Left, Forward and Right positron counters. T h e arrows under t h e 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 T R I U M F ) . The backward polarized muons are detected by a thin muon (TM) counter and then stop in the sample. 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 experiment, 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. T h e labelling is from the viewpoint of an imaginary person "riding along" the muon beam just before it enters the sample. T h e B counter has a hole in the middle in order to avoid the p a t h 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 ( T F ) 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 t h e Wien filter) until it is perpendicular to the magnetic field. All four positron counters contain precession information. A weak transverse field ( W T F ) experiment is similar except t h a t 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 Filter/ spin rotator is available. In a T D longitudinal field (LF) experiment, the muon spin and magnetic field are parallel (antiparallel). In a zero field (ZF) experiment, trim coils are usually used to cancel out stray magnetic fields and ensure zero field at the sample. In b o t h ZF and L F 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 L F arrangement.  Chapter 2.  2.3  22  fiSR  Electronics  T h e s t a n d a r d time-differential electronics at T R I U M F is shown in Fig. 2.3. T h e voltage signals from the muon or positron counters are passed into constant fraction discriminators ( C F D ) through variable delays. These delays can be used to synchronize the various counters. T h e output from the C F D 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. T h e 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 d a t a gate (LeCroy 222 gate generator), generally 500 ns shorter t h a n 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 d a t a gate, its logic pulse is sent to t h e N I M / E C L (emitter coupled logic) converter (Creative Electronic Instruments Level Converter LC 5170) which in t u r n 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 ( d a t a 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.  24  fiSR  histogram is incremented by one. Typically about 10 7 decay events are recorded and the resulting time histograms may be regarded as the ensemble average of such decays. Note t h a t 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 r a n d o m background. Virtual Address extension (VAX) computers which are interfaced with the CAMAC histogramming modules periodically read, store and handle the display of these d a t a .  2.4  ^SR Data  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 distribution of the emitted positron and the solid angle of the counter, Pi(t) is the projection of t h e 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 t h e muon. Rather t h a n single histograms, the d a t a 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  25  Chapter 2. /J.SR  Run  542:  GaRs:Si  1  IP  1  1  TF=46 G T=296 K  1  1  1  1  Q] QI  O  • D  w 2000 c  G  CD Q_  '  (a)  O  m  j2 1000 h" c o O  -  m  Ii  • ID  0 1  i^^'Jf  1  1  1  I  1  I  I  1  J  I  I  I  L  1  I  "'T*"  I  T"  J  L  0.2 -  X  CL  0  X  <  -0.2 -  0  2  4  6  8  10  TIME (ILLS) Figure 2.4: Time spectra for fiSR in heavily doped n-type GaAs:Si (concentration of Si is w 5 x 10 18 c m " 3 ) at T = 296 K and T F = 46 G: (a) raw d a t a and (b) time-dependent corrected asymmetry.  Chapter 2.  26  fiSR  that the counters are "opposite" implies that P-i(t) N-i(0)/Ni(0)  and assuming f3 = A_i/Ai  A{t)  = — Pi(t).  Further defining a =  = 1 we obtain  (1 - a) + Ajl\(t)(l ~ (1 + a) + AiPlWl  + a) - 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. T h e corresponding corrected asymmetry plots for the T F example in Fig. 2.4a is shown in Fig. 2.4b. An example of L F d a t a 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 L F 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 t o infer the underlying physics. T h e form of P{(t) under various physical circumstances relevant to this thesis will be discussed further in C h a p t e r 3.  27  Chapter 2. fxSR  Run 978:  Si:B, LF=200G, T=546K  ^favvAV^ 2  3  4  5  8  TIME Gas)  Figure 2.5: Time-dependent corrected asymmetry spectrum for Si:B (boron concentration « 1016 cm" 3 ) at T = 546 K and LF = 200 G.  Chapter 3  THEORETICAL  CONSIDERATIONS  This chapter describes the theoretical methods that form the framework for analysing and interpreting the d a t a 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 thesis (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 calculating t h e muon spin polarization for M u 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 M u 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 methods 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 experiment. At certain values of the magnetic field, the muon polarization is transferred  28  Chapter 3.  THEORETICAL  29  CONSIDERATIONS  resonantly t o the nuclei. In this situatioin, dips show up in the field dependence of t h e 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 t h e muon spin polarization when muonium is undergoing spin-exchange scattering or cyclic charge state 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 L F muon spin polarization. T h e 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 t h e Zeeman interaction becomes comparable to or much greater t h a n the the thermal energy kgT.  T h e methods described here will be used to analyze and interpret (i) the LF-  //SR d a t a for intrinsic Si, p-type Si:B and intermediate doped n-type Si:P and (ii) the TF-yuSR d a 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 time evolution of the muon spin polarization in semiconductors can be calculated by using q u a n t u m 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  30  CONSIDERATIONS  T h e relevant spin Hamiltonian for the muonium center in a magnetic field B (magnitude B) may be written as [20] H  =  /*7eB-S-/rjvB-I + S-A'i-I + D ' T + S A8' • T - hfnB  J2[l  • T + J ' • Q 4 • 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 t h e electron, muon and ith nucleus respectively; A M 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 t h a t only nuclei with J > 1/2 have a quadrupole moment [32]). T h e g tensors are all assumed to be isotropic, hence the gyromagnetic ratios are scalars (7 oc g). T h e 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 interaction arising from the electric field gradient at the nuclei (HP): Hz/h  =  -%BI,  +  '£-KBJi i  HD/h = £ ^ ( - 2 J ^ ' + w:' + ^ 4 ) i  Q  H /h = 2 W " - ^ V ' ' + l)/3)  (3.3)  Chapter 3.  THEORETICAL  31  CONSIDERATIONS  where D*  =  Qi  =  frhwUticrf  (3.4)  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). T h e magnetic dipole parameters Dl vary inversely with t h e cube of t h e distance from the muon to the nucleus ( r t ) . Since an adequate description of the d a t a 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' and similarly for S and J .  =  Ix sin 9 + Iz cos 8  If M u D is located in a system where the dipole and  quadrupole interactions with the surrounding nuclei are negligible, then the expression for a free muon in a magnetic field is recovered: H/h = -%BIZ  (3.6)  T h e Hamiltonian described by Eq. (3.6) will be a good description for the diamagnetic center M u D in Si. This is due to t h e fact that in Si, the majority of the host nuclei have zero spin ( 28 Si occurs with 95.3% isotopic abundance) and only a small number ( 29 Si occurs with 4.7 % isotopic abundance) have spin 1/2. Hence, there are no quadrupole  Chapter 3.  THEORETICAL  32  CONSIDERATIONS  interactions and there will be dipole interactions with only a small number of the neighboring nuclei. T h e basis set labelled by the muon magnetic q u a n t u m 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.  W h e n discussing t h e 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).  T h e 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 A{  =  87r n0h2 . . 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 t h e muon.  Chapter 3. THEORETICAL  33  CONSIDERATIONS  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 fourthorder 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  E3/h  =  Aj4-(%-%)B/2  E4/h  = -AJA  - y/Al + (Te + 7 , ) 2 # 7 2  + 7„) 2 £ 2 /2  and the corresponding eigenvectors are: ki)  =  |e2) =  I + +) coso;| - +) + s i n a | + - )  h) = 1--) |e4) =  — s i n a | — h ) + cosa|H—)  (3-12)  Chapter 3. THEORETICAL  ouuu  34  CONSIDERATIONS  1  1  i  1  i  1 4000 N  M u T in Si  -z  ^ ^ ^ ^ 2  2000 -  -  0 ^  2000  -  ^ ^ ^ ^ 3  —  4000 i  0.00  i  0.05  0.10  i  i  i  0.15  0.20  0.25  0.30  Field (T) 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  (3.13)  VTTx<  where x = B/B0 is a dimensionless quantity with B0 — A^j{^e + 7^). The field dependence 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  E3/h  =  + y/Al + (% +  A||/4-(7e-7„)J3/2  (3.14) %)*By2  Chapter 3. THEORETICAL  CONSIDERATIONS  35  with the corresponding eigenvectors |ex) =  | + +)  (3.15)  |e2) =  cos/?|-+)+sin/?| + - )  N = 1--) |e4) =  - s i n / ? | - + ) + cos/?| + - )  where  cos2/?=^=JL= v i + y2 andy = B/B  (3.16)  where B = Ax/(7 e + 77i) (i-e. the eigenvectors are the same as isotropic  Mu with A^ -+ A±). The energy eigenvalues for Mug C where 8 = 90° are Ek/h  = \ (A± + v/4(7e - 7 , ) 2 # 2 + (A,, - Ax) 2 )  J5fc/* =  (3.17)  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) =  cos£|-+)+sin£| + -)  |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  0.010  0.005  0.015  0.020  Field (T)  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,)252 + (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  37  CONSIDERATIONS  density m a t r i x 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  0  1  ~ l \ ;<  ;^  1  0  oj  I  o\  (i =  1°  -  1  (3.21)  /  If \s{) are energy eigenstates of H with energy eigenvalues E{, then  P(i) = E^IP(0^|e.->  (3.22)  i  where in the Heisenberg representation jnt/n p(0)t -iHt/h  Pit)  (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 UJij  =  ^  ^  n  (3.25)  we obtain from Eq. (3.22): P(t)  E  e-lUJijt  K  (e.-|(l + 0|ei>(eJ-ph>  p-iuiiji  E-ir-(e.-k rl£i>(eil^k.-> K  (3.26)  hj  1  Ks in most fiSR literature, we do not explicitly use different symbols to distinguish between mathematical operations between scalars, operators or matrices. These differences should be clear from the context.  38  Chapter 3. THEORETICAL CONSIDERATIONS  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)  E =E  (3-28)  and since  (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 ^ K k i ) | 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 - f E I N < l e ; > l 2 *(!-*);  (3-33)  ~ _ e~<l/T" cosfajti) - e-t2/T» cos^-fr) - e"* 1 ^"^-^ s i n f u ^ ) + e " * 2 ^ ^ - ^ sinfu;^) ~~ (1 + ufjTlXe-^l^ - e-Wn.) In the special case of <i = 0 and <2 = oo,  ^ = 1 - I E I^K%-)I 2 x , " * , ,  (3.34)  Chapter 3.  THEORETICAL  39  CONSIDERATIONS  which is a sum of Lorentzians. In time-integral L F experiments, such as those described in this thesis, r = z. (The choice of ii ^ 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. T h e evaluation of P(t) and P for M u D interacting with neighboring nuclei shall be deferred till Sec. 3.3 and Sec. 3.4. Consider M u + or M u " in a magnetic field directed along z.  The Hamiltonian is  given by Eq. (3.6). In a L F 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 t h a t Px(t)  = Py(t)  = 0.] This implies t h a t  the initial muon spin direction is preserved, a fact that is obviously expected since its initial state (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. T h e next example is t h a t 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  40  CONSIDERATIONS  calculation shows t h a t the time dependence of the muon spin polarization in a T F experiment is Px(t) = - ( c o s 2 a coscvi2t + sin 2 a cosu^tf + cos 2 a cosu^t  + sin 2 a cosco^t)  (3.36)  At high fields, cos 2 a —* 1 while sin 2 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 L F 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 M i r 0 T r a n s v e r s e F i e l d 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 r a n d o m 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(—a 2 t 2 ). Note that in this section as well as in Chapter 5, we use the symbol a t o 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 t h a t t h e 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 Zeem a n 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 H a r t m a n n [37]. An excellent detailed discussion of the H a r t m a n n formalism developed from a second moment approach 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 t h a n the quadrupole interaction.  T h e latter controls a and t h e 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]. H a r t m a n n and Estle show t h a t in a T F experiment the dipole and quadrupole interactions cause a distribution of frequencies about the muon Larmor frequency and a2 is proportional to the second moment of this distribution. T h e 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  42  CONSIDERATIONS  T h e strength of the applied field affects o by determining the axis of nuclear quantization 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. T h e 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: M2 =  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 M 2 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: M2 = ( 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 cos 2 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  43  CONSIDERATIONS  .12  .10  0 = 0°  w 3  O  .00 0  5  10  15  20  MAGNETIC FIELD (orb. units)  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 m a d e since this is t h e 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 t h e neighboring nuclei exist as two isotopes and when there are two values of 0, such as for Ga ( 6 9 Ga and 71  Ga) when B || (110) or (111) crystalline directions.  Chapter 3.  3.4  THEORETICAL  CONSIDERATIONS  44  Q u a d r u p o l e L e v e l C r o s s i n g 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. T h e reason for this n a m e 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. T h e spin Hamiltonian for this system is given by Eq. (3.2) with J = 3/2 and describes the situation of M u D in GaAs (see Chapter 5). T h e experimental investigation of ALCR in this thesis occurs in the time-integral mode and in the geometry of a L F 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 polarization 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 interactions 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 t h e 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 corresponding 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". T h e 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° respectively. T h e size of the energy gap Egap due to the avoidance of the two levels is on the order of t h e 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 L C R 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 t h a t the resonances are plotted as 1 — Pz; hence, they appear as peaks rather t h a n dips. T h e approach we adopt for calculating Pz in a system consisting of a muon and N  Chapter 3. THEORETICAL  0.16  CONSIDERATIONS  0.20  0.24  46  0.28  FIELD (kG)  Figure 3.4: (a) The field dependence of the energy levels of a system consisting of a muon and one spin 3/2 nucleus ( 71 Ga) 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  0.16  CONSIDERATIONS  0.20 0.24 FIELD (kG)  47  0.28  Figure 3.5: (a) The field dependence of the energy levels of a system consisting of a muon and one spin 3/2 nucleus ( 71 Ga) 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  48  CONSIDERATIONS  spin J = 3/2 nuclei involves "brute force" numerical diagonalization of a matrix of size 2 ( 2 J + 1 ) ^ to obtain the eigenvalues and eigenvectors of 7i [Eq. (3.2)]. T h e Hamiltonian 7i is represented in matrix form using the product states I m / ^ m j j ) \mimj1....mjN)  \mjN)  =  as the basis set. For very large N, the brute force diagonalization of 7i  will become very C P U 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 C P U time on a VAX 4000-lOOA.) T h e eigenvalues and eigenvectors are then used with Eq. (3.34) to calculate Pz. In the situation where D <C 1/T M , 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.  T h e situation of neighbouring nuclei with two possible isotopes, such as with Ga ( 6 9 Ga and  3.5  71  Ga), is discussed in Appendix A.  SPIN DYNAMICS - INTRODUCTION  The remaining sections in this chapter will be devoted to discussion of the spin dynamics of the muonium centers. spin-exchange  scattering,  By spin dynamics, we mean cyclic charge exchange  and  where in this thesis, the primary interest will be in the time  evolution of the muon spin polarization in a LF-//SR experiment. T h e organization of the later sections will be as follows: Sec. 3.6 contains a qualitative 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 spinexchange scattering in a L F 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  4 atoms  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 71 Ga with Q = -1.472 MHz and D = 1.0 x 10' 2 MHz.  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 t h a t 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 calculations 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  Q u a l i t a t i v e D i s c u s s i o n o f M u o n i u m S p i n D y n a m i c s in a L o n g i t u d i n a l Field  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 paramagnetic muonium states Mu^ and M u ^ c are observed at low temperatures in semiconductors such as intrinsic Si. However, at high temperatures the signals are rapidly damped, indicating t h a t dynamical changes occur, the most obvious being ionization of muonium. 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 <-> M u + . In general, the ionization and capture rates will not be the same. Another scenario is the following: suppose the temperatures 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 t h e M u 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, b o t h SE and CE lead to similar field and t e m p e r a t u r e dependences of the muon spin polarization in L F and hence can be interchanged for purposes of qualitative discussions. T h e 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 M u g 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 L F 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:  x = B/B0,  1 + 2a;2  +  1  Pz{t)  = 27iT^) 2aT^) cos(u;24 ' )  Px(t)  =  0  Py(t)  =  0  (3.40)  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 t h e muon polarization does not evolve with time:  P,W = 1 Px(t) = 0 Py(t) = 0  (3.41)  Chapter 3. THEORETICAL  CONSIDERATIONS  52  (a) Slow  c o  Time  D N _D O CL  (b) Fast  Time  to  t, Mu  t2 Mu +  Figure 3.7: Isotropic muonium undergoing charge exchange in the (a) slow and (b) fast limits.  Chapter 3.  THEORETICAL  CONSIDERATIONS  53  Suppose t h a t Mu^ is formed at t = 0. T h e time dependence of the muon polarization is shown in Fig. 3.7. At <i, Mu^ is ionized and becomes M u + . T h e ionized electron will carry off some spin polarization, the amount depending on the ionization rate as described below. T h e polarization remains unchanged until t2 at which time M u + recaptures an electron to form Muj- again. Note t h a t the average initial muon polarization at t-i after the first charge exchange cycle is less t h a n at t = 0. ionization/recapture process continues until the muon decays.  This  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. T h e muon spin polarization decreases as a function of time - i.e. LF relaxation occurs or, in standard ^ S R or NMR terminology, the muon experiences 1/Ti spin relaxation. 2. There are two extreme regions — fast C E and slow CE. T h e 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 t h a n the period of a hyperfine oscillation (27r/u>24). In other words, the muon polarization undergoes numerous oscillations before M u J ionizes. In this situation, the fractional loss in polarization per C E cycle is the amplitude of the oscillating component, i.e. A24 = 1/2(1 + x 2 ), and is independent of r . In contrast, in the "fast" C E limit, r is much less t h a n the period of a hyperfine oscillation; i.e.  Mu^  ionizes before the muon polarization can undergo one oscillation. In this situation, 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 t h a n the am-  plitude A 24 of the oscillating component and (ii) independent of magnetic  Chapter 3.  THEORETICAL  field.  CONSIDERATIONS  54  T h e 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 t e m p e r a t u r e 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 t h a n 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 C E 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. T h e 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  1  i  '  i  '  Fast  Slow CE  -  55  C E  \  (a)  i  i  TEMPERATURE 1  i  i  Fast CE  Slow CE  . -  o o  :  (b) B"2  i  \  •  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 t h a t these predictions are in excellent agreement with the results of more quantitative treatments such as those described below. T h e 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  I s o t r o p i c M u u n d e r g o i n g C h a r g e E x c h a n g e in a L o n g i t u d i n a l F i e l d Strong Collision Approach  The so-called "one-component" strong collision approach used for modelling isotropic muonium undergoing cyclic charge exchange follows quite intuitively from the discussion in Sec. 3.6. Consider the process  Mu  ^  fi  (3.42)  studied in L F 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 M u + 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 M u and fj. are 1/XMU and 1/AM respectively. This means t h a t the probability that the muon is still in a certain state after some period of time is simply given by an exponential decay (e _ A M u * or e~Xf,t).  Furthermore, the j u m p is assumed to occur instantaneously,  Chapter 3.  THEORETICAL  57  CONSIDERATIONS  or at least much faster t h a n 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)  an  d QMu(t)  respectively, the process Eq. (3.42) can be described by  the coupled integral equations: Gfu(t)  =  PM„(0)Qf"(*)e-W  G>(t)  =  P„(0)Q»z(t)e-^  +AM£drG£(T)Qf"(t-T)e-W*-T) (3.43) x  +\MuftdTGM*(T)Q%(t-T)e-  ^)  where -PM(0) and P M « ( 0 ) denote the initial /i and Mu fractions respectively and G%(t) and G^u(t)  are the L F 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. T h e 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). T h e first term corresponds to the decay of the muon polarization of the prompt (t = 0) Mu state. T h e 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. T h e 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)  +A M <2f*(s +  \Mu)g^s)  (3.44)  or upon rearrangement,  oris) -\MUQ»(S  -KQ*(* + K)G™\s)  + *MU)GM  +gft(s)  = PMu(0)Qr(s + \Mu)  (3.45)  = PM(0)Cjf(« + A,)  T h e "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.  PMU(0)Q?U(S  +  \Mu)  en*)  -A„er(s + Aw„)  •>Mu  (3.46)  A  1  PMU(0)Q™  U  (S +  \Mu)  (3.47)  GM = where  -XMUQ^S  + AJ  1  (3.48) Recall t h a t we are interested in calculating (7*ot(£) which therefore has the Laplace transform Qi°\s)  QT{s)  = Q™u(s) + 0£(a):  =  PMU(0)Q7U(S  +  XMU)  Chapter 3. THEORETICAL  CONSIDERATIONS  59  +P„(0)eS(* + A|l) +Q?(s + K)Q™u(s + /l -  AM„)[A^(0)  +  XMUPMU(0)}  + K)Q™u{s + \Mu)  XMUKQ^S  (3.49)  @lot(s) with s = l/r M 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 ^ , , * J,z  {S + AMu) -  1 + 2x2 2 ( l + x2)s  1 +  1  x^  where the same notation is used and  + LO0  s + \Mu  2 ( 1 + x2)  =  2TTAII  {s +  ^ ^  + wa(1 + x2)  ^.WJj  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 + ai 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 B  = 3A2,„ + 2A M „A M +u; 2 (l + :z2)  C  =  D  =  (AMU + A ^ A ^ + U ^ I + Z 2 ) ) \MUWO/2  (3.55)  Chapter 3. THEORETICAL  60  CONSIDERATIONS  and t h e amplitudes a8 can be obtained by solving the linear equations conveniently written in m a t r i x form as Ma = Y  (3.56)  where / a2 + Oi3 + « 4  M =  Oil + « 3 + « 4  a2a3 + a2a4 + 030:4 a^a3 + a1a4 + a3a4 a2a3a4  oti +a2 + a4  Oi\ + a2 + a3  axa2 + a1a4 + a2a4  ara2 + axa3 + a2a3  a\ot3a4  oc\oc2a4  <X\<x2oiz (3.57)  and '  ^ 0.2  (3.58)  a = 0.3  V°4 I and /  \ 3AJV/« + AM  Y = 2  3X  2  Mu  2  (3.59) 2  + 2XMuXfi + u, (l + x ) - 0.5u, (l - P ^ O ) )  \ (A M „ + A M )(A^ U +o; 0 2 (l + x 2 )) - 0 . 5 O ; 2 ( ( A M U ( 1 - P M (0)) + A,) / In general, Eq. (3.54) and Eq. (3.56) must be solved numerically. It turns out that two of t h e 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 t h e 1/Ti t h a t is experimentally observable in a L F experiment. These results imply t h a t for isotropic Mu undergoing CE, the experimentally measurable muon polarization is  Chapter 3.  THEORETICAL  61  CONSIDERATIONS  an exponentially decaying function a e ~ ' / T l . Examples of the field dependences of 1/Ti for fixed A^ and variable \MU  are  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  top = 1 + 1 -  +A  P  ,A 1*A *<°>  w?  1  2  (s + \Mu) + "IJ 2  (3.61)  and  bot  = *+ { YMu x ) (?—r4i—r)  I  ( 3 - 62 )  Now we make the approximation that the "timescale" of observation is much longer t h a n \/\MU-  In terms of the inverse Laplace transform, this implies that we are inter-  ested in the range where s <C Ajv/«- This yields  <T(-) - 1 + ^ - ^ ^ K - i f a ) *  (3.63)  This equation implies t h a 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  IU  I 1  10  1  -  I  '  '  -1  „~2  -  10 ps 104 /zs"1  10° -  10' 1  1  62  -  -  _ 1  -1  -  \ /LLS 10"  10'  -  10  10"4 10"5  -  10"6  -  7  -  10"  in"8  00  3.  .^6  3  N^/  (a) \ /  ^ S >« N ^  /K.  ^\T~ ^ I  10  -1 "  /US -  10  I  I  10"  10'  1  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  63  CONSIDERATIONS  10' 10'  -  10"  -  10  -  10"'  -  10 10 10  10" 10  10  10  10  10  MAGNETIC FIELD (kG)  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.  3.8  THEORETICAL  64  CONSIDERATIONS  C e l i o / O d e r m a t t M e t h o d a n d S p i n E x c h a n g e o f M u ^ c in L F  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. T h e formalism enables one to calculate the full muon polarization for each center provided t h a 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 M u g 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 = Pjk  ejkl['^pk,  + Ynknmpml U\(  + u!;ple] ,U)  = enlm[SjmSkn — (p1^ - p[) - 8jn(—nmnkplll x  - 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  (3.70)  )k  \p"J and M is a 15 x 15 matrix obtained from Eq. (3.68). Obviously, P = P(t) is timedependent. 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 4= 1  (3.73)  Chapter 3.  THEORETICAL  66  CONSIDERATIONS  The initial muon spin direction determines P ( 0 ) . For example, in a LF experiment, p M (0) = ( 0 , 0 , 1 ) . Similarly, in a T F experiment p M (0) = (1, 0,0) where the muon spin is initially in the ^-direction. In b o t h geometries, we assume pe(0) = 0 and all p^k(0) = 0. Note t h a t since M is anti-symmetric, the eigenvalues <*i are pure imaginary and P consists of u n d a m p e d precession components. Although M u g 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 t h a t the transition 1 —• 2 occurs at a rate A 12 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 t h a t :  dt  / P3i \  / Mi - A12  \ P 2 ,  \  A12A  A21.4  \  M2 - A2i J  I 6 (3.74)  \  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 Aij — 0  if i , i = 1,2,3  (3.75)  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  67  CONSIDERATIONS  of Eq. (3.69). T h e 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 t h a t 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. T h e 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 Yakovleva [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 parameter 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 expressions derived from this formalism and for isotropic M u in t h e 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 rate". Fig. 3.11(a) shows a simulation of the field dependence of the muon 1/Ti  as  a  Chapter 3. THEORETICAL  CONSIDERATIONS  100 -  68  1  ,.  -  10000 /xs"1 •^  - ^ /  -  ^~\\  0.01 -1 /XS  I  if)  0.0001 --  =1  ^"^^  J 1  I  1000 JUS'\  10 /XS"  /  10  (a)  °  -1 /IS  I  i  ^ - -  100 -  -  (b)  10000  /us~'V^\. -  ^ "^s  \ ^  f 0.01 1  10  /IS  / 10 /xs  1000 /xs -1  \f 1  100 ^s~  1  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  69  CONSIDERATIONS  function of the Nosov-Yakovleva spin-exchange rate USE specifically for the case of M u £ c in Si (A|| = - 1 6 . 8 2 MHz, A± = - 9 2 . 5 9 MHz) where the angle 9 = 0°. As indicated in Sec. 3.1, M u g 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, t h e 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 t h a n the CU24 frequency (slow spin-exchange regime). In contrast, in the case of M u g C for 9 ^ 0°, the L F 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 V  SE  = 5 i±s~x.  T h e 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. T h e 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 d a t a (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  70  CONSIDERATIONS  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  {„  V  Beff = \B0 - Bp, (n  (A±-An)sin29\ X  "  ;  .„  J  (AL-A»)sm29\  y  Beff = \B0 + Bp, +  J*>  if ms = +.„  J  1 1  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||co s 2 fl) 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 considering L F , the initial muon spin is parallel to B 0 and the muon precesses in a cone of  Chapter 3. THEORETICAL  10  i  -  (a)  -  CO  i  I  8  71  CONSIDERATIONS  3. \—  4  : A -  / N-  -  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  72  CONSIDERATIONS  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 amplitude of the oscillating component is the largest. W h e n B0 is far from  Bp,  this amplitude rapidly goes to zero. As discussed in Sec. 3.6, a useful approximate picture of L F depolarization is t h a t for small USE (slow spin-exchange scattering), the amount of muon polarization lost per electron spin-exchange cycle is proportional to the amplitude 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 t h e Fast S p i n E x c h a n g e 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 t h a t Bav = Beff  (m. = - ) + B e / / ( m s = - - ) = ( B 0 , 0 )  (3.79)  In a T F experiment, this implies t h a t muonium undergoing very fast SE is indistinguishable from the diamagnetic center undergoing Larmor precession. However, now suppose t h a 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 t h a t 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 a n d  Chapter 3.  THEORETICAL  73  CONSIDERATIONS  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 t o B „ = (B. + A P >  Si  ° 2 ' t * » " " ' ' , AJ» • < * • ~ > > " - 2 " ) .  (The dynamics for Muj. will be obtained if one sets A\\ = A± = A^.  (3.81)  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\  (382)  Consider a simple example of an isolated electron spin (e.g. isolated muonium) in a magnetic field Bo- T h e 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 A P  ~ 1 + e-h%B0/kBT  hXfeBo  ~ ^ T  (3  '83)  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 Bo  =  -h%(Ax  sin 2 6 + A\\ cos 2 6) 4%kBT  [  ' '  where we call the normalized frequency shift K, the Knight Shift. We also point out that as in L F , if the spin exchange is extremely rapid, then there will be negligible relaxation in the precession signals.  Chapter 4  APPARATUS  This chapter describes the peripheral apparatus used to obtain the //SR d a t a in this thesis. This includes a brief description of various spectrometers at T R I U M F as well as cryostats and the oven. The fxhCK spectrometer used at L A M P F will be also described.  4.1  T R I U M F Spectrometers  Three magnetic field spectrometers, all part of the T R I U M F 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 u p 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 trim 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 m m ) plastic scintillators which enable the muons to pass through. The T M 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. T h e arrangement of counters discussed in Sec. 2.2 is a reasonable representation of the actual situation for both OMNI and O M N F . 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 generating magnetic fields u p to 7 T. T h e 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. T h e forward positron counters are m a d e u p of four segments « 0.6 cm thick; these counters are arranged cylindrically because of geometrical restrictions imposed by the magnet's cylindrical bore. T h e counters are usually labelled Forward Top Right ( F T R ) , Forward Top Left ( F T L ) , Forward Bottom Right ( F B R ) , and Forward Bottom Left (FBL). In a LF-juSR experiment (see Sec. 2.2), the forward counters are positioned so that there is no overlap 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 t h e sample. These were originally similar to the Forward positron counters (consisting of four segments) but were changed eventually to the style of the T M 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 described in Sec. 2.2 during the discussion of the T F - ^ S R geometry. (Opposing segments are used to form the asymmetry spectra.) The Backward counters are not used in a TF-/uSR experiment.  Chapters  76  APPARATUS  Approximate Scale  I 0  I 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  Sample Rod He Space Vacuum Cryostat TM Counter  Forward e Counter  W/////////W/M  XpJ  B,  /x + beam Sample Beam Vacuum HELIOS Magnet Bore 0150mm  Forward e Counter A p p r o x . Scale: 0  75  150  mm  Figure 4.2: A cutaway showing the central portion of a high-field experimental arrangement with HELIOS. Also shown is a schematic of the He gas flow cryostat.  Chapters  4.2  78  APPARATUS  Cryostats and the Oven  Several cryostats and an oven were used to obtain the d a t a discussed in this thesis. D a t a 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) K a p t o n windows allow beam access. These apparatus are part of the T R I U M F yuSR-facility and have been discussed in previous works such as Ref. [48], [31] and [49]. T h e samples studied were all significantly larger t h a n the beam spot size (usually w 10 — 20 m m ) after collimation. They were greased with Apezion N onto a thin (0.25 m m ) silver plate which was in t u r n 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 P t resistor a n d / o r 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 a n d / o r a Carbon Glass resistor. Temperature control was achieved by use of either a commercial temperature controller 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.  79  APPARATUS  be allowed to leave. T h e schematic of the oven is shown in Fig. 4.3. T h e 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 pattern, 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 atm. of an inert gas was sometimes let into the oven to improve thermal conduction with the surroundings. T h e sample rod is a thin-walled SS316L stainless steel tube with a copper sample holder screwed in place at the end. T h e rod and holder are held concentric in the tube by two triangular SS316L concentricity spacers. T h e 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. T h e 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 t h a t there is only minimal mass behind the sample which would impede positrons leaving in that direction. T h e sample sits on a thin copper plate (thickness « 0.8 m m ) located in the middle of the sample holder. This plate is held snugly by copper posts and specially constructed copper nuts. T h e sample is "wrapped" in thin tantalum foil (the thickness of one layer of Ta foil is 0.0127 m m ) 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.  80  APPARATUS  Water Cooling  Shel  Window  Spacers Sample Rod Vacuum Space  Heater. Holder Sample^  Plate Post 4 cm  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.  81  APPARATUS  reduce heat loss through the Kapton window. About 100 W was required to reach 1000 K.  4.3  L A M P F Spectrometer  T h e /iSR facility L A M P F in New Mexico, U.S.A, and the //LCR spectrometer located there, are discussed in this section since the ALCR d a t a 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 L A M P F , which has the world's best /xSR facility for this type of measurements, (i.e. integrated L F experiments) looks bleak. A more detailed discussion of the facility and the spectrometer is given in Ref. [51]. T h e peak fx+ rate is « 3 x 1 0 8 ^ + / s whereas the average /J,+ rate is « 2 x 10 7 // + / s > which is approximately 10 x higher t h a n the M20B p+ channel at T R I U M F . T h e 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. T h e muon beam is collimated to w 4 cm F W H M . T h e 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  82  APPARATUS  Beam's View of Counters Side View -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  r a m p field. In order to obtain magnetic fields greater t h a n 400 G, a static longitudinal field of u p to 5 kG can also be applied. The d a t a can either be presented as or Np/Np,  NF/NS  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 better t h a n 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 t h a t will be useful for the reader who is attempting to follow the ensuing discussion, the d a t a 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 d a t a in heavily doped GaAs which include spin exchange dynamics and characterization of the negatively charged muonium center.  5.1  Useful Information  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 = 10 4 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 a n d 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 M u g C in Si or GaAs. All bond directions in these semiconductors are along one of the (111) directions. 84  If one makes the reasonable  Chapter 5. MEASUREMENTS  and DISCUSSION  85  Nearest Neighbors  Next Nearest Neighbors  Equiv Nuclei  9 (deg.)  Equiv Nuclei  9 (deg.)  (100)|| Bo  4  54.74  2 4  0 90  (111)11 Bo  1 3  0 70.53  6  54.74  (110)|| B 0  2 2  90 35.26  2 4  90 45  Orientation  Table 5.1: T h e angles 9 between the muon-nucleus axis and t h e applied magnetic field for a muonium center at t h e T site. Number Ratio  9 (deg.)  (100)||Bo  4 Equivalent  54.74  (1H)I|B 0  1 3  0 70.53  <110)||Bo  2 2  90 35.26  Table 5.2: T h e angles 9 between the applied magnetic field and the M u ^ c hyperfine symmetry axis (also a bond axis). assumption that M u g C thermalizes at random in one of these bonds, the ratio of t h e number of centers at 9 between Bo and the hyperfine symmetry axis (also bond axis) is as summarized in Table 5.2.  5.2  A n a l y s i s of / i S R D a t a  The analysis of /fSR d a t a consists of two general steps. T h e 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 a t t e m p t s 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 d a t a are often analysed by first transforming to a reference frame rotating at a frequency slightly less t h a n u^.  In this way the d a t a can be packed heavily thereby  increasing the speed of the fitting procedure. T h e observable LF-//SR signal usually consists of a sum of several relaxing components. These d a t a 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, M S R F I T , 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  5.3  and  DISCUSSION  87  Samples  This thesis describes fiSR measurements in Si and GaAs. T h e Si samples studied are summarized in Table 5.3. The concentration of dopants as obtained from photoluminescence (measured by M. Thewalt) a n d / o r resistivity values are summarized where available. T h e "diameter" of the samples varies between 15 m m and 25 m m and except for SiP-19 and SiP-20, the samples are about 2 m m 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. T h e T O P S I L sample was grown by the commercial company T O P S I L , 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. T h e GaAs wafers studied, all of which are heavily doped, are summarized in Table. 5.4. T h e GaSi-18-100, GaSi-18-110, GaSi-18-111, GaTe-19-100 and GaZn-19-100 samples 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 10 18 c m - 3 . T h e 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 diameter 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  Photoluminescence C F (cm- 3 ) CB(cm-3)  Name  Face  Type  TOPSIL SiB-14 SiB-16 SiB-18  110 100 111 111  p p p p  1.4 x 1012 0.3 x 1014  SiP-14 SiP-15 SiP-16 SiP-19 SiP-20  111 100 111 100 111  n n n n n  4.9 x 1014 1.5 x 1015 5.0 x 1015  1.1 x 1012 4.4 x 1014  88  p(Q,.cm) 30,000  Resistivity Concentration (cm - 3 ) « 5 x 1011 c m - 3 «1016 7.5 x 1018  2.4 x 1012  1 x 1014 0.5 0.007 0.0008  1.2 x 1016 9 x 1018 88 x 1018  Table 5.3: List of Si samples studied. T h e entries under the column labelled "Name" are mnemonics for the samples. T h e 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 t h e direction of the flat face.  5.4  I n t r i n s i c Si  High 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). T h e majority of the measurements were obtained with the magnetic field B 0 parallel to a (110) crystallographic direction. Recall from t h e discussion in Sec. 1.1 that b o t h t h e M u ^ and M u ^ c centers are undergoing rapid ionization above 230 K and that an inverse reaction M u + - > Mu may also be occurring at higher temperatures.  T h e current LF-//SR measurements  Chapter 5. MEASUREMENTS  and  DISCUSSION  Name  Sample  Face  Type  GaSi-17-100 GaSi-18-100 GaSi-18-110 GaSi-18-111  GaAs:Si GaAs:Si GaAs:Si GaAs:Si  100 100 110 111  n n n n  GaTe-18-100  GaAs:Te  100  n  GaZn-19-100  GaAs:Zn  100  p  p(£ 0.0014-0.0010 0.0010 0.0013  89  Resistivity Concentration (cm - 3 ) 9.0 xlO 16 2.5-5 xlO 1 8 4.5 x 1018 3.2 x 1018 4.5 x 1018  0.0034  2.80 x 1019  Table 5.4: List of GaAs samples studied. T h e entries under the column labelled "Name" are mnemonics for the samples. T h e direction of the face is given by the column labelled "Face".  establish t h a t 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 t h a t t h e 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 t h e 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 t h e sample) which showed the same 1/Ti as the (110) data. T h e relaxation d a t a 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 t o  Chapter 5. MEASUREMENTS  and DISCUSSION  90  0.15  0.10 Q_ N  <  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  i  and DISCUSSION  91  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  92  DISCUSSION  <W/////////////A \  Mil  V  0  +  E.  n  "-M  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 t h a t the muon is in thermal equilibrium with the thermally generated conduction 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.  (5.1)  i + i-nW)  leading to 1 (E,-EF A M „ = 2 ^ e x P ( - ^ where A^ and \MU  are  (5.2)  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  T h e 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 d a t a at each temperature cannot be fitted with a single set of independent parameters A^, A^ 1  The reaction Mu° <->• M u - is also possible. However, we argue that the rate for Mu" —• Mu + is much more rapid than the rate for Mu° —• M u - 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  93  DISCUSSION  and \MU- Instead, the entire set of d a t a 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 b a n d and v is the average thermal velocity of the electron. There is uncertainty regarding the values of n in the literature. T h e 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 10 16 T§ exp(-0.605eV/fc B T)  n  =  5.71 x 10 19 ( T / 3 0 0 ) 2 3 6 5 e x p ( - 6 7 3 3 / T )  cm"3 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 = 0.28m e for the temperature range of interest (m e is t h e mass of the electron). T h e ionization r a t e XMU is given by Eq. (5.2.) Three parameters, a, E^ and A^ were then extracted from a global fit of the d a t a 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 t h e 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 2  Eg (eV) = 1.170 - 4 • 7 j x + 1 6 0 3~6 T2; units of T are Kelvin.  Chapter 5. MEASUREMENTS  300  and  DISCUSSION  94  500 700 900 TEMPERATURE (K)  Figure 5.4: (a) 1/71 in T O P S I L (circles) in low fields between 15 m T and 40 m T . These d a t a closely approximate the temperature dependence for constant low field since at these low fields l / 7 \ is only weakly field-dependent. T h e curve indicates the smooth interpolation of the best global fit results (described in the text), (b) T h e 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. T h e 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  95  DISCUSSION  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 b a n d 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+ <-> M u + . Fig. 5.4b shows the fitted values for A^ and the fraction of time the muonium is in the neutral state (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)]. T h e 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: A„(T) =  AM  1-C  m  a: 4  /e/T  . 0 J Jo  x3dx ex - 1  (5.5)  where A(0) = 2006(2) MHz, 0 = 655(25) K, C = 0.68(5). T h e agreement between the experimental d a t a and the extrapolated curve is remarkable considering t h a t t h e present d a t a 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 b a n d to be captured by Mu+ is a - 2.8(3) x 1 0 - 1 5 cm 2 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 t h a n 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 t h a t when Mu is in the neutral charge state, it spends a significant fraction of its time 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 m u o n / m u o n i u m at a BC site to be at least « 0.3 eV below that of the T site, implying t h a t 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 significantly modify the conclusions of previous "standard" electronic structure calculations. In particular, they find that M u g C is metastable  with a lifetime t h a t 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 t h e T site since Muy has a "larger derealization" t h a n 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 theoretical 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 n a t u r e of the lattice response to the motion of muonium. T h e LF-//SR d a t a 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 t h a 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  97  DISCUSSION  fraction of its time near t h e T site since a much smaller lattice relaxation is required there t h a n 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) temperature. Therefore, the inability of the lattice to respond to the muon will probably occur at lower temperatures t h a n for the proton. T h e postulate t h a t 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 M u 7 center in CuCl [60]. Although the experimental d a t a 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 d a t a [61,62], all taken below 450 K.  3  These d a t a  suggest t h a t t h e ionization of M u j inferred from T F - ^ S R experiments [19] at « 230 K (see Chapter 1) involves the intermediate state M u g 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 b o t h ionization and a site change, and would not be directly associated with 3  In 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 - High Temperatures  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 + . T h e importance of negatively charged carriers in governing t h e 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 T R I U M F . T h e 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 t h a n the the u>24 frequency, it can easily be seen from Eq.(3.64) t h a t  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 + <-> M u + . 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 t h a t 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 20 15 i ,_  10 5 0 (l-B 300  400  500 600 700 TEMPERATURE (K)  800  900  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 diamonds). 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  100  DISCUSSION  2. On the other hand, suppose the the charge exchange process involves electrons, i.e. Mu «-• M u + + 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  w  iU 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. T h e high temperature experimental situation in Si:B is similar to intrinsic Si described in Sec. 5.4. The L F muon spin polarization function Pz(t)  is well-described  by a one-component exponential relaxation. T h e temperature dependence of 1/Ti in the Si:B samples at low fields is shown in Fig. 5.5. The observation t h a t 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. T h e 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 T O P S I L 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 + -^-ln2 4  (5.7) p  It 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  101  DISCUSSION  where Epi is the intrinsic Fermi level, assumed to be pinned at mid-gap. T h e carrier concentrations for a semiconductor containing ND donors and NA acceptors which are fully ionized (high temperatures) are given by [47]: n = I[(JV D - NA)2 + 4n?]i + \(ND P = \[(ND  - NA)2 - 4 r # + ^(ND  -  NA)  - NA)  (5.8)  where n,- is t h e 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) T h e 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 M u + - + Mu + h+ of the charge exchange cycle involving holes is suppressed. (There is evidence t h a t 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 t h e M u ^ c center in SiB-14.)  5.6  n - t y p e Si:P w i t h I n t e r m e d i a t e D o p i n g L e v e l s  As demonstrated in t h e previous two sections, Mu undergoes significant ionization at elevated temperatures.  At high temperatures, there are numerous electrons in the  conduction b a n d 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 ionize 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 t h e neutral Mu center and the conduction electron are expected 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 M u " 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 M u g C is very high in energy.[2,24] Consequently, one expects t h a t M u ^ c is more likely to undergo spin-exchange scattering t h a n 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 L F muon spin 1/Ti relaxation 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 T R I U M F with the OMNI' and HELIOS spectrometers using the horizontal gas-flow and horizontal cold-finger cryostats respectively. It was shown in Sec. 3.8 t h a t 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 temperatures 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  3  CO  2  =1  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  I  and  DISCUSSION  104  00  FIELD (T) Figure 5.7: Field dependence of 1/Ti at 60 K (open squares) and 73 K (closed circles) in SiP-14. T h e dashed and solid curves are best fits to the d a t a using the Nosov-Yakovleva theory (Sec. 3.8) for 73 K and 60 K respectively and assuming 1/Ti is due to M u g C in Si with 6 = 70.53°. T h e 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 t h a n w 45 K, t h e L F 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. T h e constant component is attributed to a charged and non-paramagnetic species; most likely M u j . T h e 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 t h a n 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. T h e 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 m T ) are predicted to be a factor of 100 or more t h a n 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 m a d e t h a t V  SE  =  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 10 14 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 concentrations of donors (No in units of cm - 3 ). to be negligible, then n is given by Ref. [63] as Nr. I \(N^.\2  n/Nj D  I oNz_ „ v r .  ND 4ex  EH ]j  (5.9)  Pfcfr  where Nc = 2.86 x 1019 cm _ 3 (T/300K) 1 5 is the effective density of states in the conduction 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  an  d l / ^ i are relatively flat.5 In Region 3, it is clear that M u g C still exists since  the 1/Ti rates at the higher field is larger t h a n 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 <-• M u + + e ~ also becomes active, occurring at much lower temperatures t h a n 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 d a t a above 200 K is that as the effective spin-exchange r a t e 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 10 13 c m - 3 to « 10 16 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 5  Actually, 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  108  DISCUSSION  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. T h e 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. T h e 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/site change reaction M u ^ c +e~ —> Muy, as discussed above.  6  An additional comment is in order: Since the 1/Ti rates can be attributed entirely to M.UgC, it is likely t h a 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 - 1 3 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 / J S - 1 ) , and assumes that MVL°BC persists till 200 K, then E > 0.3 eV. 7 This 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  109  DISCUSSION  T h e 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  S p i n P o l a r i z e d Muonium—Heavily D o p e d S i : 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 discussion 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 semiconductor, 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 t h a t the conductivity remains finite even when extrapolated t o 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 10 18 c m - 3 . There is increasing experimental 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 electron concentration is very high in a heavily-doped semiconductor, any paramagnetic muonium that exists in the system will be undergoing very rapid spin-exchange scattering. 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 M u " ) . 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 experiment, 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 t o LF-//SR measurements to characterize the center. This section presents preliminary high TF-//SR measurements in heavily doped ntype SiP-19 (see Table 5.3 ). In this sample, ND ~ 2.6iVc. Since this sample is relatively thin, it was either "double-stacked" a n d / o r a narrow range of less-energetic muons were selected. Naturally, the lower energy implies that such muons have a smaller stopping range. T h e TF-//SR experiments were performed at the M15 and M20B beamlines at T R I U M F 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 M u g C undergoing rapid spin exchange scattering. T h e temperature dependence of the frequency shift is measured and agrees with t h a t expected for an isolated paramagnetic impurity in contact with a heat b a t h 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  196  J  I  I  I  I  L  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 t h a n 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. T h e corresponding Fourier transform shown in Fig. 5.10(b). T h e 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 at frequency fjy.  8  while the other precesses  The component oscillating at / D , which is essentially temperature  independent, is attributed 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 u p " ; hence, the local spin susceptibility of Mu~ is much less than for paramagnetic muonium.) T h e component oscillating at fsc  is attributed to M u g 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 t e m p e r a t u r e 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 t h a t the polarization of the muonium electron is simply that of an isolated electron spin in a magnetic field B 0 , 8  Both 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  ^  0.006  O  0.004 0.002 0 0  20  40  60  80  100  TEMPERATURE  120  140  160  180  200  (K)  Figure 5.11: Temperature dependence of Knight shift in heavily doped n-type SiP-19 (9 xlO 18 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  114  DISCUSSION  then mth =  -h%(A±sm29  ——  +  2  uAllcos  9)  .  (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 t o 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 d a t a in Sb.) T h e 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. T h e 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 t h a t 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 t h e fit shown in Fig. 5.12(a) is to a sum of three oscillating components. As in Fig. 5.10, one component is attributed to the M u D center. T h e other two components are attributed to the other two inequivalent M u ^ c centers (i.e. two values of 9) and were assumed to have the same amplitude. T h e 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 d a t a where B 0 is parallel to a (110) axis and 1C2 is obtained from the d a t a where B 0 parallel to a (100) axis. T h e 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  -  0  -'  x X  < -0.1  -0.2 2  3  4  TIME  (jis)  0.15 0.125 XJ Z3  0.1  -  0.075  -  Q.  E < (U  0d  0.05 0.025  -0.02 196  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.) T h e appearance of the M u g C line(s) at lower frequencies compared with the diamagnetic 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 t h a t prior to t h e 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 T e m p e r a t u r e  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 M u " . 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 10 16 c m - 3 ) in GaAs:Si. These experiments were performed on the M15 and M20 beamlines at T R I U M F with HELIOS and the horizontal gas flow  Chapter 5. MEASUREMENTS  and  117  DISCUSSION  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. T h e 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 L F 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. T h e 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). T h e notation v*- labels the transitions between the appropriate energy levels in the Breit-Rabi diagram for M u g C in Si. In a L F 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]. T h e magnitude of the nonrelaxing component could not be determined directly from these L F measurements. This component is attributed 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 nonrelaxing signal in L F . T h e relaxation rates rapidly increase with temperature and the signals in both L F and T F become unobservable above R J 3 0 K . This is consistent with  Chapter 5. MEASUREMENTS  T  0  and DISCUSSION  1  0.4  1  1  0.8 FIELD (T)  1  118  1  1.0  r  1.4  Figure 5.14: T h e field dependence of \jTx in GaSi-17-100 at 5.5 K. The line is a best fit to the d a t a 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 b a n d . T h e L F dependence of t h e 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 theorydescribed in Sec. 3.8. The poor agreement between the theory and the d a t a at low fields is attributed to the fact t h a t the theoretical calculations do not take into account t h e 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 M u g C in GaAs using A\\= 218.54 MHz and A±= 87.87 MHz) demonstrates t h a t M u g 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 L F is due to Mu^ undergoing q u a n t u m 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 . W i t h 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 M u " 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 transition, the value of USE in GaSi-17-100 is relatively small, indicating that ionized electrons, i.e. those in the conduction band rather than the impurity bands, contribute to the spin-exchange process 9 . An impurity b a n d 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 t h a t a narrow impurity band does indeed exist and is at Ed — 5.81 eV below the conduction band (as in lightly doped GaAs:Si). If we assume t h a t USE = °nv where n is calculated from Eq. (5.9) with ND = 9 x 10 16 c m " 3 and v is given by Eq. (5.4) with m*th = 0.067m e , the fitted uSE corresponds to av(5.5K)  = 1.3(1) x 1 0 - 7 s _ 1 cm 3 and a = 2.3(1) x 1 0 - 1 4  cm 2 , 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 1 0 - 1 0 c m - 2 , which seems too large.) 9  The 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  120  DISCUSSION  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 t h a t this center undergoes significant ionization above approximately 30 K, it may be difficult to observe a Knight shift above this temperature, provided t h a t 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 C e n t e r s 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 u p or spin down (hence the spin degeneracy is two), is at energy E& above the top of the valence band and t h a t the extra Coulomb energy for double electron occupancy of the level is U. In this simplified system, the relative equilibrium populations of M u + : M u ° : M u " can be shown to be [47] -, o &F — E4 (2Ep — 2Ed — U 1 : 2 exp —;——— : exp —— p  kBT  P  V  (5.16)  kBT  In a heavily doped n-type semiconductor, the Fermi level Ep is close to the conduction b a n d 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 (provided the quantity Ec — Ed is more t h a n a few kBT  greater t h a n 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 t h a n  —2Ej).  T h e primary aim of this section is to report and discuss measurements regarding the diamagnetic center M u f l in GaAs.  In particular, we discuss measurements on  Chapter 5. MEASUREMENTS  Nucleus  and DISCUSSION  121  Nat. Abundance (%)  j n (MHz/T)  eq (e x 10" 2 8 m 2 )  39.8 60.2 100.0  12.984 10.219 7.292  0.112 0.178 0.3  71  Ga Ga 75 As  69  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 L F and T F //SR. T h e GaAs samples studied are part of the list tabulated in Table 5.4. TF-^tSR experiments at T R I U M F were performed in a He gas flow cryostat ( T F 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 L A M P F /«SR facility (see Sec. 4.3). T h e 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 t h a t 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, M u D does not have a hyperfine interaction.  However, it is still possible to observe resonances in an integrated L F  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 dipoledipole 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  122  DISCUSSION  &AAA A A A  $  o  -  •  ;**W>0° i  0  i  50  i  i  100  •  i  i  150  200  i  250  Temperature (K)  Figure 5.15: Temperature dependence of the T F M u 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 M u 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 diamagnetic amplitude Ad in GaSi-18-100 (n-type) and GaZn-19-100 (p-type) with the applied magnetic field B 0 | | (100) axis in b o t h samples. T h e absolute fractions of M u D are not well-determined since there were no comparisons m a d e under the same experimental conditions with a sample which has a 100 % diamagnetic signal (such as high purity Ag). Note, however, t h a 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 time resolution of the particle detectors [56], indicating that the M u D fraction in these samples is large. T h e significant diamagnetic 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 t e m p e r a t u r e independent whereas this is clearly not the case in n-type GaAs. The sum of the M u D and M u ^ c amplitudes in GaSi-18-100 at low temperatures (e.g. 5 K) is significantly less t h a n the full precession amplitude, implying a so-called "missing fraction" of the polarization. Since M u ^ c is directly observable in T F and L F (see Fig. 5.13 and Sec. 5.8), the missing fraction is attributed to Mu^. It has been suggested  Chapter 5. MEASUREMENTS  and  123  DISCUSSION  from studies of t h e field dependence of the non-relaxing component of the L F polarization 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. t h e 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 expected to be responsible for the M u 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 differences between the field dependences of the 1/T 2 linewidth and the ALCR spectra in the n- and p-type samples, which are discussed below. Henceforth, we will refer to M u D as M u - in the n-type samples and M u + 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 L F (ALCR) spectrum on GaAs:Si, obtained by  our collaborators at L A M P F , with Bo parallel to a (100) axis in n-type GaSi-18-100 at room temperature. T h e 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 u p to « 120 m T . T h e 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 10  Note 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 ^ i s the M u - formation rate and u0 — 2irAfi where A^ is the muon hyperfine parameter of Mu^.  Chapter 5. MEASUREMENTS  and DISCUSSION  124  0.008  I CLN  0.004  0.000  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 M u - is in the BC site (dashed curves) are virtually indistinguishable.  Chapter 5. MEASUREMENTS  and  125  DISCUSSION  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 u p to 0.2 T in p-type GaZn-19-100 with B 0 parallel to a (100) axis. T h e four high field (?» 20 — 45 m T ) 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.) <2( 6 9 Ga)/Q( 7 1 Ga) = q(69Ga)/q(nGa).  = 7 n ( 6 9 G a ) / 7 „ ( 7 1 G a ) and  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. T h e 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 t o 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 t h a t it is due to the As nuclei. If the M u " 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  69  Ga and the other to  71  Ga) 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. T h e lower field region from « 50 — 100 m T containing the low field resonance and  Chapter 5. MEASUREMENTS  and  DISCUSSION  127  0.2-  0.1-  ^  0.0-  Q_ X  <  -0.1-0.2i  0  2  r  4 TIME (/xs)  i  i  6  8  Figure 5.17: T F - ^ S R d a t a 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. T h e solid curve is a fit to a function proportional to e~° t cosiuj^t + <f>). T h e 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 A L C R spectrum, the  Chapter 5. MEASUREMENTS  and  DISCUSSION  1-  *-  —i  i  1  U. 12 h  128  <111>  •  •  •  <110>_  0.08  0.04 <100>  GaAs :Si =1  n  i  0.12  l  <111>  4=f*  <110>  0.08  0.04 -  (b) GaAs:Te  <100>  0 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. T h e 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 t o 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 r u n in GaSi18-100 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, GaSi18-100, 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 t h a t 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 T F - ^ S R linewidth d a t a 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 t h a 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 T o a cage is assumed to be undistorted  12  . The quoted error bars are statistical. However,  the BC site remains highly unlikely for M u " 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 t h a t 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 potential 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(j a site. T h e magnitude and width of the ALCR resonances and the TF-/^SR line broadening 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 ( « 10 10 hops/s at room temperature). Several factors may lead to a small diffusion r a t e in the case of M u " . In particular, the large lattice distortion implied by the results discussed above would reduce the tunneling matrix 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 interactions, 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 12  In 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  131  DISCUSSION  heavier mass lowers the zero-point energy and reduces t h e tunneling matrix element. In heavily-doped GaAs, there is a possibility of muonium-impurity formation. Measurements on n-type GaAs:Si were compared to the room temperature ALCR and T F fj.SK measurements on n-type GaAs:Te, i.e. GaTe-18-100. T h e T F - ^ S R measurements were obtained by rotating the wafer t o obtain Bo parallel to the three major crystallographic axes. These d a t a are shown in Fig. 5.18b and are very similar to that in GaAs:Si. T h e 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 t h a 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 different signatures, particularly in the ALCR spectrum at room temperature. T h e 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 « 10 10 s _ 1 [21] at room temperature. T h e 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 approximately one hop at most before converting to M u 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 fieldindependent up to 2 T when B 0 is applied parallel to a (100) axis. T h e lack of "drop-off" in a implies t h a t the positions of any ALCR resonances cannot be estimated, making a search very time-consuming.  Above 150 K, the linewidth decreases dramatically,  implying t h a t M u + is undergoing rapid diffusive motion, hence "motionally averaging" 13  At high fields, the transition rate A for Mu§, —• Mu D 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  GaAs:Si  and  |£( 7 1 Ga)|//i (MHz) a  \QC5As)\/h  Ratio of Dipole Parameters  r(Ga)  r(As)  (MHz)°  A  A  1.472(4)  0.621(6)  Unrestricted  2.192 [-10.5%] 2.206 [-10.0%]  2.77 [-2.1%] 2.68 [-5.3%]  2.176 [-11.2%] 2.200 [-10.2%]  2.92 [+3.2%] 2.68 [-5.3%]  Fixed GaAs:Te  132  DISCUSSION  1.532 (4)  0.695 (8)  Unrestricted Fixed  °<9( 6 9 Ga)/Q( 7 1 Ga) = ? ( 6 9 G a ) / 9 ( 7 1 G a ) = 0.178/0.112. T h e quoted error estimates are statistical. Table 5.6: T h e quadrupole parameters Q a n d t h e Ga a n d As distances obtained from the fits t o t h e GaAs:Si and GaAs:Te d a t a as described in t h e text for t h e assumption of TGO site. T h e percentage numbers in square parenthesis below t h e values for r ( G a ) and r(As) indicate t h e deviation from t h e unrelaxed nearest neighbor a n d next nearest neighbor distances of 2.45 A and 2.83 A respectively. T h e same Q values will also be valid if M u " is assumed to be in t h e BC site. T h e meanings of "unrestricted" and "fixed" are described in t h e text.  Chapter 5. MEASUREMENTS  and  DISCUSSION  133  the random dipolar fields giving rise to the damping of the precession signal. Rapid diffusion of M u " occurs above « 450 K in the GaAs:Si and GaAs:Te samples. These aspects are still being actively investigated at T R I U M F . T h e absence of M u " diffusion implies that H~ will probably not be mobile at temperatures significantly below 450 K. Considering the stability of M u - , it may be surprising t h a t 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 passivation 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. Similarly, 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 t h a 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~. T h e diffusivity of Mu^ is much higher than that of M u " , a situation which is probably also valid for hydrogen. Hence, one intriguing possibility is t h a t under certain conditions, the diffusion rate of hydrogen may be governed by the fraction of time it spends as H^.  Chapter 6  S U M M A R Y and C O N C L U D I N G  REMARKS  In this thesis, recent ^ S R 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 t h e 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 t e m p e r a t u r e confirm that muonium is rapidly cycling between its neutral and positive charge states by interaction with conduction electrons, i.e. Mu <-> M u + +e~. T h e neutral state 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 time away from the predicted adiabatic potential energy minimum, 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 heavilydoped n-type GaAs, implying that in these semiconductors the cross-section for electron  134  Chapter 6. SUMMARY  and CONCLUDING  REMARKS  135  capture by M u g C to form M u " is significantly smaller t h a n t h a t for spin-exchange scattering. 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. T h e 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 presented, in particular the M u " center in n-type GaAs. At room temperature, M u " exists as an isolated center with the muon and nearest neighbor G a on the same (111) axis. T h e M u - center appears to be located at or near a T^a site.  6.1  M u o n i u m S p i n D y n a m i c s in O t h e r M a t e r i a l s  T h e 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  6.2  and CONCLUDING  REMARKS  136  Hydrogen  T h e similarity between muonium and hydrogen implies that spin and charge exchange processes t h a t are observed in muonium are also occurring for hydrogen. However, the studies of hydrogen are made on a timescale much longer t h a n 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. Nevertheless, 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 t h a 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 t e m p t s 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 t h a t 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 M u + , 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 1  It 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 ^ S R unlikely. In n-type Si, direct muonium-donor interactions may be difficult t o observe even if the complex is formed since the analogous hydrogen-donor complex is located in an antibonding 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 C u r r e n t S t u d i e s  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. T h e 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. SUMMARY  and CONCLUDING  REMARKS  138  consequences for the hydrogen analogues. This important class of materials will continue to be actively studied via //SR in the forseeable future and t h e resultant area of research promises to be an exciting field.  Bibliography  [1] J.I. Pankove. Hydrogen Neutralization of Defects. In D. Adler and B. Schwartz, editors, Tetrahedrally-Bonded Amorphous Semiconductors. Academic Press in New York, (1985). [2] S.M. Myers, M.I. Baskes, H.K. Birnbaum, J.W. Corbett, G.G. DeLeo, S.K. Estriecher, E.E. Haller, P. Jena, N.M. Johnson, R. Kirchheim, S.J. Pearton, and M.J. Stavola. Hydrogen interactions with defects in crystalline solids. Review of Modern Physics, 64:559, (1992). [3] Yu. V. Gorelkinskii and N.N. Nevinnyi. Electron paramagnetic resonance of hydrogen in silicon. Physica B, 170:155, 1991. [4] B. Bech Nielsen, K. Bonde Nielsen, and J.R. Byberg. Charge and site-change dynamics of muonium (hydrogen) in Si. Materials Science Forum, 143-147:909, (1994). [5] Hoon Young Cho, Suk-Ki Min, K.J. Chang, and C. Lee. Negatively charged state of atomic hydrogen in n-type GaAs. Phys. Rev. B, 44:13779, (1991). [6] C.H. Seager and R.A. Anderson. Real-time observations of hydrogen drift and diffusion in silicon. Appl. Phys. Lett., 53:1181, (1988). [7] A.J. Tavendale, S.J. Pearton, A.A. Williams, and D. Alexiev. Injection and drift of a positively charged hydrogen species in p-type GaAs. Appl. Phys. Lett., 56:1457, (1990). [8] M.H. Yuan, L.P. Wang, S.X. Jin, J.J. Chen, and G.G. Qin. Negative-charge state of hydrogen species in n-type GaAs. Appl. Phys. Lett., 58:925, (1991). [9] J. Zhu, N.M. Johnson, and C. Herring. Negative-charge state of hydrogen in Si. Phys. Rev. B, 41:12354, (1990). [10] N.M. Johnson, C. Herring, and Chris G. Van de Walle. Inverted Order of Acceptor and Donor Levels of Monoatomic Hydrogen in Silicon. Phys. Rev. Lett., 73:130, (1994). [11] S.K. Estreicher. Theoretical and //SR studies related to hydrogen in compound semiconductors. In S.J. Pearton, editor, Hydrogen in Compound Semiconductors. Trans Tech Publications, (1993). 139  140  Bibliography  [12] Richard A. Morrow. Silicon donor-hydrogen complex in GaAs: A deep donor? J. Appl. Phys, 74:6174, (1993). [13] C.H. Seager and R.A. Anderson. Irreversible changes of the charge state of donor/hydrogen complees initiated by hole capture in silicon. Appl. Phys. Lett., 63:1531, (1993). [14] C. Herring and N.M. Johnson. Hydrogen Migration and Solubility in Silicon. In Jacques I. Pankove and N.M. Johnson, editors, Hydrogen in Semiconductors, chapter 10. Academic Press in New York, (1991). [15] A.Z. Varisov. Influence of spatial dispersion on hydrogen-like atoms (Mu,Ps) in nonpolar semiconductors. Sov. Phys. Semicond., 16:330, (1982). [16] Y.C. Chen, K.G. Lynn, and Bent Nielsen. Monitoring the surface oxidation process with an energy-tunable monoenergetic positron beam. Phys. Rev. B, 37:3105, (1988). [17] R. Ramirez and C.P. Herrero. Distinct Quantum Behavior of Hydrogen and Muonium in Crystalline Silicon. Phys. Rev. Lett., 73:126, (1994). [18] J u r g W. Schneider. Avoided-Level-Crossing: A New Technique in Muon Spin Rotation to Study the Nuclear Hyperfine Structure of Muonium Centers in Semiconductors. P h D thesis, Universitat Zurich, 1989. [19] B.D. Patterson. (1988).  Muonium States in Semiconductors.  Rev. Mod. Phys.,  60:69,  [20] R . F . Kiefl and T.L. Estle. Muonium in Semiconductors. In Jacques I. Pankove and N.M. Johnson, editors, Hydrogen in Semiconductors, chapter 15. Academic Press in New York, (1991). [21] J.W. Schneider, R.F. Kiefl, E.J. Ansaldo, J.H. Brewer, K. Chow, S.F.J. Cox, S.A. Dodds, R.C. DuVarney, T.L. Estle, E.E. Haller, R. Kadono, S.R. Kreitzman, R.L. Lichti, Ch. Neidermeier, T. Pfiz, T.M. Riseman, and C. Schwab. Q u a n t u m motion of muonium in GaAs and CuCl. Mat. Sci. For., 83-87:569, (1992). [22] R. Kadono, R.F. Kiefl, J.H. Brewer, G.M. Luke, T. Pfiz, T.M. Riseman, and B.J. Sternlieb. Quantum Diffusion of Muonium in GaAs. Hyp. Int., 64:635, (1990). [23] E. Westhauser, E. Albert, M. Hamma, E. Recknagel, A. Weidinger, and P. Moser. Mu to Mu* Transition in Electron Irradiated Silicon. Hyp. Int., 32:589, (1986).  141  Bibliography  [24] C.G. Van de Walle. Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors. In Jacques I. Pankove and N.M. Johnson, editors, Hydrogen in Semiconductors, chapter 16. Academic Press in New York, (1991). [25] L. Pavesi and P. Giannozzi. Atomic and molecular hydrogen in gallium arsenide: a theoretical sutdy. Phys. Rev. B, 46:4621, (1992). [26] H. Simmler, P. Eschle, H. Keller, , W. Kundig, W. Odermatt, B.D. Patterson, I.M. Savic, J.W. Schneider, B. Stauble-Pumpin, U. Straumann, and P. Truol. Muon stopping sites in semiconductors from decay-positron channeling. Materials Science Forum, 83-87:1121, (1992). [27] W. Odermatt. Spin Dynamics of transitions among muon states in semiconductors. Helv. Phys. Acta, 61:1087, (1988). [28] E. Albert, A. Moslang, E. Recknagel, and A. Weidinger. Relaxation of Anomalous Muonium in Silicon. Hyp. Int., 17-19:611, (1984). [29] A. Schenck. Muon Spin Rotation  Spectroscopy.  Adam Hilger Ltd., 1985.  [30] G.H. Eaton, A. C a m e , S.F.J. Cox, J.D. Davies, R. De Renzi, 0 . H a r t m a n n , A. Kratzer, C. Ristori, C.A. Scott, G.C. Stirling, and T. Sundqvist. Commisioning of the Rutherford Appleton Laboratory Pulsed Muon Facility. Nucl. Instrum. and Methods Phys. Res. A, 269:483, (1988). [31] Grame Luke. Quantum Diffusion and Spin Dynamics thesis, T h e University of British Columbia, 1988. [32] Mitchel Weissbluth. Atoms and Molecues.  of Muons in Copper. P h D  Academic Press, Inc., 1978.  [33] J.W. Schneider, K. Chow, R . F . Kiefl, S.R. Kreitzman, A. MacFarlane, R.C. DuVarney, T.L. Estle, R.L. Lichti, and C. Schwab. Electronic Structure of anomalous muonium in G a P and GaAs. Phys. Rev. B, 47:10193, (1993). [34] E. Roduner and H. Fischer. Muonium substituted organic free radicals in liquids: theory and analysis of /iSR spectra. Chemical Physics, 54:261, (1981). [35] M. Leon. Use of time information in muon level-crossing resonance spectroscopy. Phys. Rev. B, 46:6603, (1992). [36] C. Cohen-Tannoudji, B. Diu, and F. Laloe. Quantum J o h n Wiley & Sons, Inc., 1977.  Mechanics.  Herman and  [37] 0 . H a r t m a n n . Quadrupole Influence on t h e Dipolar-Field W i d t h for a Single Intersitial in a Metal Crystal. Phys. Rev. Lett., 39:832, 1977.  142  Bibliography  T.L. Estle. Hamiltonians, Line Shapes, Summer School Notes, 1993. C.P. Slichter. Principles berg New York, (1980).  of Magnetic  and Speciras.  Resonance.  Unpublished /J.SK Maui  Springer-Verlag Berlin Heidel-  R . F . Kiefl and S.R. Kreitzman. Muon Level-Crossing Resonance. In T. Yamazaki, K. Nakai, and K. Nagamine, editors, Perspectives of Meson Science, chapter 9. North-Holland, 1992. V.G. Nosov and I.V. Yakovleva. Depolarization of M u + mesons in solids. Phys. JETP, 16:1236, (1963).  Sov.  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. T h e theory of M u + - m e s o n depolarization with allowance for the process of charge exchange or formation of unstable chemical compounds. Sov. Phys. -JETP, 33:1070, (1971). N.W. Ashcroft and N.D. Mermin. Solid State Physics. ston, 1976.  Holt, Rinehart and Win-  T.M. Riseman. fiSR Measurement of the Magnetic Penetration Depth and Coherence Length in the High-Tc Superconductor YBa2Cu30e.95- P h D thesis, T h e University of British Columbia, 1993. T.M.S. Johnston. Anomalous electronic structure of the muon in Antimony: evidence for an isolated Kondo impurity. Master's thesis, T h e University of British Columbia, 1993. K. Holloway. Tantalum as a diffusion barrier between copper and silicon. Phys. Lett, 57:1736, (1990).  Appl.  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. Development 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 properties 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 parameters 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 Spect r a 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 b a n d 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 Diffusivity 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 a n d TF-^uSR  A.l  P r o b a b i l i t i e s in A L C R  T h e calculation of the L C R 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  69  G a and  71  Ga 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. T h e probabilities of finding n N —n  69  71  Ga atoms (and hence  Ga atoms) is given by the binomial distribution: Pn(n)  =  (N - n)lnl  (A  -1}  and can be derived from l JV = ( / 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=P7i(4) + ....+pn(0)=  (A.3)  fn + 4/ 7 3 i/ 6 9 + 6 / ^ / 1 , + 4/ 7 1 / 6 3 9 + /649 (A.4) 145  Appendix  A.  PROBABILITIES  in ALCR  n  P7i(n)  and TF-fiSR  146  Peo(n)  0 0.1313 0.0251 1 0.3473 0.1518 2 0.3444 0.3444 3 0.1518 0.3473 4 0.0251 0.1313 Table A . l : Probabilites for Ga when JV = 4 T h e RHS of Eq. (A.2) also shows the method for calculating Pz, the integrated polarization (the relevant quantity in L C R spectra). One should calculate the integrated polarization from Eq. (3.2) for the following five cases: 1. 4  71  Ga neighbors,  2. 3 n G a and 1 6 9 Ga neighbor 3. 2  71  Ga and 2  69  Ga neighbors  4. 1  71  Ga and 3  69  Ga neighbors  5. 4  69  Ga neighbors  Then, the weighted sum is obtained:  P, = I > i ( n ) P , ( n  71  Ga)  (A.5)  n=0  where Pz(n  71  Ga) is the total polarization obtained from the Hamiltonian consisting of  n 7 1 G a and N — n  69  G a nuclei. It should be noted that all t h e integrated polarizations  take on the value of unity far away from a resonance. T h e numerical calculation involving five Hamiltonians, each of which has dimension 2 x 4 4 = 512 is numerically very time consuming. Fortunately, in the case of the ALCR  Appendix  A.  PROBABILITIES  in ALCR  and  147  TF-fiSR  d a t a in this thesis (concerning the Ga nuclei), the resonances due to each isotope are well-separated. This implies t h a t t h e resonances, or more correctly, t h e deviations from unity, satisfy Rz = f^PriHlRnin)  + R^9(N  - n)}  (A.6)  n=0  where ^ ( n ) = 1 - Pz(n  69  G a ) , Rn(n)  - 1 - Pz(n  71  Ga) 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 m a t t e r 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/T M , a further approximation can be made: Rtot  = (n)Rn(l) = Nf71Rn(l)  where the expectation value of n, i.e. Nf71  +  (N-n)Re9(l) + Nf69Re9(l)  (n), is given by Nf.  (A.7) In the case of TV = 4,  = 1.592 and iV/ 6 9 = 2.408. T h e 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  P r o b a b i l i t i e s in T F - ^ S R  Similar complications arise when calculating the " H a r t m a n n 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. T h e number of equivalent nuclei with a certain value of 9 is summarized in Sec. 5.1.  Appendix A. PROBABILITIES  69  in ALCR and TF-fiSR  148  M(n  71  M(4  71  Ga,0  69  Ga)  2m Ga 7i(90°) + 2m G a 7i (35.26°)  M(3  71  Ga,l  69  Ga)  |[2m G a 7i(90°) + m G a 7i (35.26°) + m Ga69 (35.26°)] + 5[™Ga 71 ( 90 °) + 2 m G a n (35.26°) + m G a 6 9 (90°)]  M(2  71  Ga,2  69  Ga)  |[2m G a 7i(90°) + 2m Ga 6 9 (35.26°)] + |[m G a 7i(90°) + m Ga 6 9 (90°) + m G a 7i (35.26°) + m Ga 6 9 (35.26°)] + |[2m Ga 69(90°) + 2m Ga 7x (35.26°)]  M(l  71  Ga,3  69  Ga)  |[2m G a 6 9 (90°) + m Ga 6 9 (35.26°) + m G a 7i (35.26°)] + M™Ga 69 ( 90 °) + 2m Ga 6 9 (35.26°) + m G a n(90°)]  M(0  71  Ga,4  69  Ga)  2m Ga 6 9 (90°) + 2m Ga 6 9 (35.26°)  Ga,iV - n  Ga)  Second Moment Contribution  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  71  M(4  71  Ga,0  69  Ga)  M(3  71  Ga,l  69  Ga)  M(2  71  Ga,2  69  Ga)  |[ m Ga 7 1 (°°) + m Ga 7 1 ( 7 0 - 5 3 °)] + 2m Ga 6 9 (70.53°)] + |[m G a 6 9 (0°) + m G a 6 9 (70.53°) + 2m G a 7i (70.53°)]  M(l  71  Ga,3  69  Ga)  |[m Ga 69(0°) + 2m Ga 69 (70.53°) + m G a 7i (70.53°)]  Ga,iV - n  69  Ga)  Second Moment Contribution m  Ga 7 1 (° O ) + 3 m G a 7 1 ( 7 0 - 5 3 ° ) f [m G a n(0°) + 2m G a 7i (70.53°) + m Ga 6 9 (70.53°)] + l[m Ga 69(0°) + 3m Ga 7i(70.53°)]  + i[ m Ga 7 1 ( 0 O ) + M(0  71  Ga,4  69  Ga)  m  3m  Ga 6 9 ( 7 0 - 5 3 °)J  Ga 6 9 (°°) + 3m Ga 6 9 (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 - n 69 Ga)  (A.8)  n=0  where  MJVJV  is the second moment contribution due to the four nearest neighbor Ga  and the quantity M(n  71  Ga, iV — n 69 Ga) 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(n 7 1 Ga, N — n 69 Ga) is calculated by considering all the possibilities of distributing N  71  Ga and hence N — n 69 Ga 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/ 71 m Ga 7i(54.74°) + iV/ 69 m Ga 6 9 (54.74°)] As an addendum, the next nearest neighbor contributions  M/VAW  (A.9)  are (recall that  75  As  is 100 % abundant): MNNN = 2m As 7 5 (0°) + 4m As 7 5 (90°)  (A. 10)  for B 0 parallel to a (100) direction; MNNN = 2m As 7 5 (90°) + 4m As 7 5 (45°)  (A.ll)  for the case of B 0 parallel to a (110) direction and MNNN = 6m As 7s(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 = / 6 9 m G a 6 9 (54.74°) + / 7 1 m G a 7 1 (54.74°) + m As75 (54.74°)  (A.13)  when Bo is parallel to a (100) direction; MNN = - [/ 6 9mQ a 69(90°) + / 7 i m G a 7 i (90°) + m As 7 5 (90°)]  + i [/69mGa69(35.26°) + / 7 1 m G a n(35.26°) + m As 7 5 (35.26°)]  (A.14)  when B 0 is parallel to a (110) direction and MNN = j [/ 6 9m Ga69 (0°) + / 7 1 m G a 7i(0°) + m A s 7 5 (0°) 4 + 7 [/69mGa69(70.53°) + / 7 im G a n(70.53°) + m As75 (70.53°)j 4  (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.  


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items