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Electron paramagnetic resonance of phosphorus doped silicon in the intermediate impurity concentration… Cullis, Pieter Rutter 1972

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ELECTRON PARAMAGNETIC RESONANCE OF PHOSPHORUS DOPED SILICON IN THE INTERMEDIATE IMPURITY CONCENTRATION RANGE by PIETER RUTTER CULLIS B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1967 M . S c . , U n i v e r s i t y of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of Physics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada Date O c r o W v 2bs /972. ABSTRACT The EPR proper t ies of phosphorus-doped, s i l i c o n samples were i n v e s t i g a t e d through the temperature range 1 „ 1 0 K ^ T ^ 35°K and the 16 3 18 impuri ty concentrat ion range 5 x 10 donors/cm ^ ^ 2 x 10 donors/cm <, Rapid passage spectra obtained are i n t e r p r e t e d d i r e c t l y as u n d i f f e r e n t i a t e d absorption envelope s p e c t r a . Slow passage absorption d e r i v a t i v e s i g n a l s were integrated to obtain s i m i l a r absorption envelope These absorption envelope spectra were decomposed into the three major components observed--a broad centre l i n e (BCL), two hyperf ine l i n e s , and a d i s c r e t e c e n t r a l p a i r l i n e . Each of these components i s found to be charac ter ized by a symmetric " V o i g t p r o f i l e " l i n e s h a p e — i . e . by a convolut ion of a Gaussian envelope f u n c t i o n with Lorentzian " s p i n pac-kets" o Spin l a t t i c e r e l a x a t i o n times and spin packet widths were measured f o r each of these components as funct ions of temperature and concentrationo These data appear to be consis tent with the proposal that the hyperf ine l i n e spins c r o s s - r e l a x with f a s t e r r e l a x i n g BCL . s p i n s . The e x t r i n s i c hyperf ine s p i n - l a t t i c e r e l a x a t i o n times are noted H — 3 /2 to be temperature dependent according to T^ cx T~ . The " g " values of the hyperf ine l i n e s and BCL were determined at var ious temperatures and impurity c o n c e n t r a t i o n s „ Both l i n e s are observed to e x h i b i t ef -f e c t i v e " g " s h i f t s as the impurity concentrat ion i s i n c r e a s e d . These " g " s h i f t s are shown to be i n c o n s i s t e n t with the ferromagnetic exchange model of Morigaki and Maekawa. The r e l a t i v e s u s c e p t i b i l i t y of the BCL as a f u n c t i o n of impurity concentrat ion i s shown to be reasonably con-s i s t e n t wi th the conjecture that the BCL ar i ses p r i m a r i l y from c l u s t e r s of three or more i m p u r i t i e s . A l l observed e f f e c t s would i n d i c a t e that the BCL i s the precursor of the s i n g l e EPR l i n e observed f o r more h e a v i l y doped samples. i i i TABLE OF CONTENTS Page Abstl?ciCt • • o o o o o o * o o * * o o o o » « » « « * * « o * o o o o * o « o o * * « o * o o » * o * o o • o o IX 1? 3. ID 1.6 Of COn*t6ntS 0 0 0 0 0 O * 0 0 « » « » 0 * 0 « 0 0 O 0 0 0 « » 0 0 « * 0 O 0 « « * « « * * 0 * 0 0 XXX L i s t of Tables ». v i i L i s t of Figures ..................... v i i i Acknowledgements x i Chapter 1 INTRODUCTION 1.1 General Introduction and Motivation 1 1.2 Outline of This Thesis 4 Chapter 2 BASIC THEORY 2.1 Introduction 5 2.2 C h a r a c t e r i s t i c Spectra at Low Impurity Con-centrations 5 2.2.1 Impurity El e c t r o n Wavefunction .... 5 2.2.2 Iso l a t e d Donor Spectrum ........... 8 2.3 C h a r a c t e r i s t i c Behaviour of Highly Concentra-tGd ScUTip3.es o * « o o o o o o o o o * o o o * o o o o o o o » * o * o o o o * 1.0 2.4 C h a r a c t e r i s t i c Spectra of the Donor P a i r and Higher Order Clusters 13 2.4.1 Introduction 13 2.4.2 C h a r a c t e r i s t i c Spectra of the Donor Pair 14 2.4.3 Clusters of Three or More Donor AtOITlS o o « « « o o o o o o o « * « o « « « o « o o o « o o « o 17 i v Page Chapter 3 EXPERIMENTAL METHODS 3 ol: Intl?0(l\J,CtlOTl •ooooeoooooo**ooooo****oe*ooooee 3.2 Apparatus „ 19 3 .2 .1 Microwave Spectrometer 19 3.2.2 Temperature Measurement and C o n t r o l ••o»oofro***ooo»»o»oo*oo**oo 21 3.2.3 Samples and Sample Preparat ion . . . . 23 3.3 Homogeneous and Inhomogeneous Broadening; The Spin Packet 25 3.4 Slow Passage Absorpt ion Spectra 27 3 .4 .1 Homogeneously Broadened Systems . . . 27 3.4.2 Slow Passage Response of Inhomo-geneously Broadened Systems . . . . . . . 31 3.5 Rapid Passage Signals C h a r a c t e r i s t i c of Inhomogeneously Broadened Systems 33 3 .5 .1 S u s c e p t i b i l i t y of an I s o l a t e d Spin 33 3.5.2 Rapid Passage S u s c e p t i b i l i t y of the T o t a l D i s t r i b u t i o n of Spins . . . 41 3.5.3 Comments on Inhomogeneously Broad-ened Rapid Passage Spectra 45 3.6 Relaxat ion Time Measurement Techniques . . . . . . 49 3 .6 .1 In t roduct ion 49 3.6.2 Recovery Method f o r Measuring T^ (Rapid Passage) 49 3.6.3 Phase Method of Obtaining (Rapid Passage) „ 52 V Page 3.6.4 "Increase of Modulation" Tech-nique f o r Measuring AHp (or Tg) (Rapid Passage) 53 3.6.5 Saturat ion Technique f o r Measuring and T 2 (Slow Passage) 55 Chapter 4 EXPERIMENTAL RESULTS 4.1 I n t r o d u c t i o n 62 4.2 The Hyperfine Lines 63 4 .2 .1 In t roduct ion 63 4.2.2 Hyperfine Line Lineshape and Spin Packet Width 67 4 .2 .3 Hyperfine Line Spin L a t t i c e Relaxa-t i o n Time Measurements 71 4 .2 .4 Hyperfine Line " g " Value and Hyperfine Constant Observations . . . 75 4 .2 .5 I n t e r a c t i o n of the Hyperfine Spins and the Broad Centre Line S p i n s . . . . 78 4.3 The C e n t r a l P a i r Line 79 4.4 The Broad Centre Line 80 4 .4 .1 In t roduct ion 80 4.4.2 Broad Centre Line Detect ion and Ex-perimental L i m i t a t i o n s . . . . . . . . . . . . 81 4 .4 .3 Lineshape and Linewidth Observa-t ions f o r the Broad Centre Line . . 88 4 .4 .4 Broad Centre Line " g " Value D a t a . . 92 4 .4 .5 Broad Centre Line Spin L a t t i c e Relaxat ion Time Measurements . . . . . . 95 v i Page 4.5 Relative I n t e n s i t i e s of the Spectral Components .... ......... ..... 100 4.5.1 Introduction 100 4.5.2 Temperature and Concentration Depen-dence of the Relative I n t e n s i t i e s of the Spectral Components ........ 101 Chapter 5 DISCUSSION OF RESULTS 5.1 Introduction v 106 5.2 O r i g i n of the Broad Centre Line 107 5.3 "g" Value Results 109 5.4 Relaxation Time and Linewidth Measurements .. 115 Chapter 6 SUMMARY AND CONCLUSIONS o o 120 Appendix A Relations Between the Signals Observed and the Complex Magnetic S u s c e p t i b i l i t y of the Sample 122 Appendix B Absorption Envelope 124 Appendix C Possible "g" S h i f t s Using an " E f f e c t i v e F i e l d Approximation" 127 Bibliography 129 v i i LIST OF TABLES Table Page 3 . 1 Room Temperature R e s i s t i v i t i e s and Sample Concentra-tions Determined Using the I r v i n Curve . . . . . . . . . . . . . . . . . . 2 4 3 . 2 Table of Rapid Passage "Passage Conditions" and Corres-ponding Signals Observed 3 . 3 Table of Rapid Passage "Passage Conditions" and Corres-ponding Signals Observed f o r H^< A Hp . . . . . . . . . . o . . . . . . . . 4 4 5 . 1 Allowed Transitions f o r a Highly Coupled Donor Pair Which Contribute at or near the Hyperfine Line Frequen-c i e s , and the Corresponding Second Order T r a n s i t i o n Frequency 1 1 4 v i i i LIST OF FIGURES 2 „ 1 ( a ) Energy Levels and Allowed T r a n s i t i o n s f o r a Highly COU.pJ.GCl ( J ^ ^ A ) Pel 11? • o * o o * o o * * * o * o o « * * * o « « o « « o o o * « o o 16 2 o 1 ' PclU?" SpCCtrUm • » 0 » * * 0 0 0 0 » 0 * * 0 0 « 0 » 0 0 0 0 0 0 « 0 * 0 0 * 0 0 16 3 . 1 "Bucking Arm" Arrangement f o r a R e f l e c t i o n Spectro-ITIG tCX* • • o o o e o o o * * o * « o * o o o o o * o * o » e » o o o « * o * » o * o o o « * o e o o 22 3 0 2 Temperature C o n t r o l and Temperature Measurement Apparatus . . . . . . . . . . . . . . . . . . . 22 3 . 4 Magnetizat ion i n the Rota t ing Frame . . 35 3 . 5 D i s t o r t i o n s of the u Out of Phase Rapid Passage S i g n a l due to V i o l a t i o n s of E i t h e r the Saturat ion or the A d i a b a t i c Passage C o n d i t i o n ( B u g a i ) . . . . . . . . . . . 47 3 . 6 Experimental Device f o r Measuring T^ by a "Recovery" Technique 50 3 . 7 Phase D i f f e r e n c e 6 as a Funct ion of Modulation F r e -quency f o r Various Values of T^ 54 H l 3 .8 ' " S a t u r a t i o n Curves" H j f i ' * (HQ) v s . f o r Various 2 Values of " a " (Castner) 57 3 . 9 The Rat io (H,) / ( H - , ) i (defined i n the text) v V upper v V lower v J v s . the Inhomogeneity Parameter " a " 58 i x F igure 3.10 The H a l f w i d t h Correc t ion Factor "w" as a Funct ion Of ci (POSenei?^) • • • o o * o o * « « « Q O « o o o o o o « « « * « o « o a o « < 3.11 Rat io of the Height of the Absorption Curve when H, = Hj^ to the Maximum Height of the Absorpt ion 1 p CUrVe o o o o o o . o o o « o o . * . o . o o o o o o o o 0 0 0 0 0 0 . . . 0 0 . 0 0 0 . . 0 4.1(a) Rapid Passage Absorpt ion Envelope Spectra f o r Samples 5.0E16 and 1.2E17 O 0 0 » 0 » » 0 O O 0 O * » « O O O O O 0 O O i 4.1(b) Slow Passage Absorpt ion D e r i v a t i v e and Absorpt ion Envelope Spectra f o r Samples 2.2E17 and 3.0E17 . . . 4.1(c) Slow Passage Absorpt ion D e r i v a t i v e and Absorpt ion Envelope Spectra f o r Samples 6.0E17 and 1.0E18 . . . 4.2 Hyperfine Line Spin Packet Halfwidth (AHp1) v s . Impurity Concentration o o o . o . o o . . . o . . . . . o . o o . . . o < 4.3 Hyperfine Spins S p i n - L a t t i c e Relaxat ion Times CT^) v s . Temperature f o r Samples with Intermediate Con-4.4 The " g " Values of the Hyperfine Lines (g^) and the BCL (gg) v s . Impurity Concentration at 1 . 2 ° K . 4.5 " P a i r " Spectra Obtained f o r Large " P r e - S a t u r a t i o n " E f f e c t s o o o o o » « 0 * o 0 0 0 0 0 « O Q O O » O Q O O O « 0 0 0 0 « 0 « 0 * 0 0 0 * 0 0 d A ' Asymmetry Introduced Into the ^ Spectra f o r Sample 6.0E17 due to the " M i x i n g In" of Some Absorpt ion Component . 0 . 0 . . . . . . 0 . 0 0 0 . . . . 0 0 . . . . . . . . . o Observed Decrease of the BCL I n t e n s i t y R e l a t i v e to Hyperf ine Line I n t e n s i t y wi th Increase of Tempera-ture f o r Sample 1.2E17 . . . . . . . . . . . . . . o o o o o « * * o * o * « * o o o o o Slow Passage Absorpt ion D e r i v a t i v e and Absorpt ion Envelope Spectra f o r Sample 1.2E17 at 13°K „ . . . . o « o o o o o The H a l f w i d t h of the BCL at 1 . 2 ° K v s . Impurity Con-c e n t r a t i o n . . o o o o o o o o o o * « o o o o o o o o o o o o o o o o o o o « o « o o o o o * o o H a l f w i d t h of the BCL v s . Temperature f o r Samples w i t h Intermediate Concentrations Spin Packet Halfwidths of the BCL (AHp) v s . Tempera-ture f o r Samples 2.2E17 and 3.0E17 . . . . . . . . 0 0 0 0 0 9 0 0 0 0 0 0 The "g" Values of the BCL v s . Temperature f o r Sample 6.5E17 . . ooooooooooooooooooaoooooooooooooooooooo BCL S p i n - L a t t i c e Relaxat ion Times v s . Temperature f o r Samples 2.2E17 and 3.0E17 . , oooooooooooovooooooooooo R e l a t i v e S u s c e p t i b i l i t y of the BCL v s . Impurity Con-C e i " l t X * 3 tlOn. O * O O O O Q * O O * O O « O O O O Q * O O O O O O O O O O O O O O * * O « * O O O « I Energy L e v e l Structure of a H i g h l y Coupled Donor P a i r C a l c u l a t e d to Second Order (Jerome and Winter) . . . xi ACKNOWLEDGEMENTS I am g r e a t l y i n d e b t e d t o my s u p e r v i s o r Dr. J.R. Marko f o r h i s u n f a i l i n g e n t h u s i a s m and encouragement t h r o u g h a l l phases o f t h i s work. H i s p a t i e n c e and u n d e r s t a n d i n g d u r i n g times o f p e r s o n a l d i s c o u r a g e m e n t a r e much a p p r e c i a t e d . S e c o n d l y , I w i s h t o thank D r . R. B a r r i e who took over the s u p e r v i s o r y r o l e and e x e c u t e d t h i s r o l e w i t h t a c t and d i s p a t c h . H i s sound a d v i c e d u r i n g t h e s i s p r e p a r a t i o n and p r i o r t o the Ph.D. e x a m i n a t i o n s was i n v a l u b l e . There a r e , o f c o u r s e , many p e o p l e who c o n t r i b u t e d b o t h d i r e c t l y and i n d i r e c t l y t o the s u c c e s s f u l c o m p l e t i o n o f t h i s s t u d y . My f a m i l y , e s p e c i a l l y my mother and f a t h e r , Mr. and M r s . H.E. C u l l i s , were a n e v e r - f a i l i n g s o u r c e o f s t r e n g t h . I cannot e x p r e s s a d q u a t e l y how much I a p p r e c i a t e them. My s i s t e r , T a r a , t y p e d t h i s t h e s i s and made o r d e r o f the chaos o f my o r i g i n a l t e x t . I t was a j o y t o work w i t h h e r . A n o t h e r whose g e n e r o s i t y and u n d e r s t a n d i n g I can never r e p a y i s my a u n t , M rs. H.M. C u l l i s . F i n a l l y , t o my w i f e Penny go s p e c i a l thanks f o r c o n t i n u i n g m o r a l s u p p o r t . I n p a r t i c u l a r h e r p a t i e n c e ( and o c c a s i o n a l i m p a t i e n c e I ) w i t h t h i s work h a s t e n e d i t s c o m p l e t i o n . S u p p o r t f o r t h i s work came from the N a t i o n a l R e s e a r c h C o u n c i l , g r a n t number 67-4624. I a l s o w i s h t o acknowledge p e r s o n a l s u p p o r t from NRC i n the form o f a g r a d u a t e s c h o l a r s h i p . 1 CHAPTER 1 INTRODUCTION 1.1 GENERAL INTRODUCTION AND MOTIVATION There have been many studies of the helium temperature electron paramagnetic resonance (EPR) properties of n-type silicone'' - The observed c h a r a c t e r i s t i c s of samples with r e l a t i v e l y low impurity concentrations (N^) are reasonably well understood i n terms of an i s o l a t e d donor or pseudo-hydrogen atom model. In this model the electrons are s t r i c t l y l o c a l i z e d to a given impurity s i t e . On the other hand, the properties of very heavily doped samples can be interpreted i n terms of the formalisms usually applied to high den-2-5 s i t y d e l o c a l i z e d electron ("metallic") systems. This work i s con-cerned with the more complicated EPR properties of samples with impurity concentrations intermediate to these extremes--the " i n t e r -mediate concentration range". The intermediate range i n the case of phosphorus-doped s i l i c o n ( Si(P) ) i s used i n t h i s work to describe impurity concen-16 3 18 t r a t i o n s over the i n t e r v a l 1 x 10 donors/cm ^ 2 x 10 ^ d donors/cm , Some quantitative understanding of the EPR spectra p e c u l i a r to the lower end of the intermediate range has been obtained 6 13 through the "donor p a i r " approach. ~ This pair model treats the impurities as a c o l l e c t i o n of pseudo-hydrogen molecules, each of which i s composed of nearest neighbour impurity atoms. However, 16 3 even at concentrations as low as N = 1 x 10 donors/cm , several d 2 aspects of EPR and o p t i c a l data have not been s a t i s f a c t o r i l y explained i n terms of the donor p a i r . In p a r t i c u l a r , a s o - c a l l e d "broad background" EPR l i n e (which, f o r the sake of c l a r i t y , i s r e f e r r e d to i n t h i s work as the "broad c e n t r a l l i n e " (BCL) ) has been d i r e c t l y 14-17 and i n d i r e c t l y observed over the e n t i r e intermediate range. A t the h i g h concentrat ion end of t h i s range t h i s l i n e i s dominant and appears to be the precursor of the d e l o c a l i z e d resonance c h a r a c t e r i s t i c of more h e a v i l y doped samples. This BCL cannot be understood i n terms of p a i r s . A primary objec t ive of t h i s study i s therefore to del ineate the EPR proper t ies and p o s s i b l e o r i g i n s of the BCL. Samples w i t h intermediate concentrations have not been e x t e n s i v e l y i n v e s t i g a t e d p r i o r to t h i s work. Morigaki and Maekawa}7 however, have s tudied the BCL at impurity concentrations 17 3 1.7 x 10 donors/cm . They report c h a r a c t e r i s t i c asymmetric l ineshapes as w e l l as s t rong concentrat ion and temperature dependences 18 of i t s e f f e c t i v e " g " values and i n t e n s i t i e s . They i n t e r p r e t e d t h e i r observations i n terms of a ferromagnetic coupling between " c l u s t e r s " of donor atoms. These c l u s t e r s are assumed to contr ibute to the BCL. The donor-donor i n t e r a c t i o n w i t h i n each of these c l u s t e r s g i s s t i l l assumed tn be of the expected antiferromagnetic form. The above data and i n t e r p r e t a t i o n s are complicated by the very s trong concentrat ion dependence of the s p i n - l a t t i c e r e l a x a -t i o n times i n Si(P) f o r samples i n the intermediate r a n g e . A t low temperatures (T ^ 10°K) these r e l a x a t i o n times e n t a i l that " r a p i d 3 passage" EPR spectra are observed f o r samples with 2 x 10 donors/cm , whereas "slow passage" signals are monitored at higher temperatures and/or impurity concentrations. The d i f f e r e n t c h a r a c t e r i s t i c s of these two types of spectra have hithe r t o prevented observation of signals that vary i n a smooth and interpretable manner over the " whole of the intermediate range,, The anomalous lineshapes reported by Maekawa and Morigaki, and, concomitantly, the lack of d i s t i n c t i o n between rap i d and slow passage signals, introduce s u f f i c i e n t ambi-guity into t h e i r experimental data to prevent i t s complete acceptance. I t i s shown that some of the reported observations can, i n f a c t , be reproduced by (or interpreted as) appropriate v i o l a t i o n s of the r a p i d passage conditions, or by varying basic experimental detection modes. In t h i s work, a technique i s used whereby both r a p i d passage and slow passage spectra are unambiguously interpreted i n terms of a s o - c a l l e d "absorption envelope". (This absorption envelope i s d i r e c t l y r e l a t e d to the imaginary part of the magnetic suscepti-b i l i t y Q f T ' ) of the sample, f o r slow passage s i t u a t i o n s ) . The absorption envelope spectra are observed to vary smoothly with concentration and d i r e c t comparison of spectra over the whole of the intermediate range i s allowed. In t h i s study, relevant EPR parameters pertaining to both the "hyperfine" l i n e s ( c h a r a c t e r i s t i c of l i g h t l y doped samples) and the BCL were measured as a function of concentration and tempera-ture (1.1°K^ T ^ 35°K) . I t i s found that the BCL can be interpreted i n terms of c l u s t e r s of donor atoms. Several of the anomalous 4 r e s u l t s previously reported were not observed, or were found to be manifestations of passage e f f e c t s . In general, a clearer and more comprehensive picture of the EPR behaviour of Si(P) samples with intermediate concentrations i s reported. ' 1.2 OUTLINE OF THIS THESIS The main body of t h i s thesis i s divided into f i v e chapters. Chapter 2 discusses the t h e o r e t i c a l EPR response c h a r a c t e r i s t i c of very l i g h t l y doped and very heavily doped samples. Subsequently, the intermediate spectra due to c l u s t e r s of two donor atoms are analysed, and, thus armed, q u a l i t a t i v e observations are made con-cerning the spectra c h a r a c t e r i s t i c of higher order c l u s t e r s . Chapter 3 presents the experimental apparatus, materials, and methods em-ployed In t h i s i n v e s t i g a t i o n . A f a i r l y comprehensive examination of r a p i d and slow passage phenomena i n r e l a t i o n to inhomogeneously broadened paramagnetic systems i s included. In Chapter 4 the experimen-t a l r e s u l t s obtained i n the work are presented. Chapter 5 consists of a discussion of these r e s u l t s and the conclusions that can be drawn from them. These r e s u l t s are summarized i n Chapter 6 , which closes the t h e s i s . 5 CHAPTER 2  BASIC THEORY 2.1 INTRODUCTION Phosphorus "impurity" atoms enter the s i l i c o n l a t t i c e s u b s t i t u t i o n a l l y . Four of t h e i r f i v e valence electrons e s t a b l i s h covalent bonds with nearest neighbour s i l i c o n s i t e s . The f i f t h unpaired or paramagnetic electron i s responsible f o r the magnetic resonance phenomena observed. The following sections discuss the t h e o r e t i c a l low temperature EPR response of t h i s system at the low concentration (N^ ^ 1 x 10^ donors/cm 3) and high concentration C N ^ ^ 2 x lO''"8 donors/cm ) extremes of the intermediate range. A discussion of the c h a r a c t e r i s t i c spectra of an i n t e r a c t i n g p a i r of impurities f o l l o w s . This leads to a q u a l i t a t i v e d e s c r i p t i o n of the possible e f f e c t s of " c l u s t e r i n g " on the observed EPR spectrum, which .closes the chapter. 2.2 CHARACTERISTIC SPECTRA AT LOW IMPURITY CONCENTRATIONS 2.2.1 The Impurity E l e c t r o n Wavefunction At low impurity concentrations and helium temperatures, the unpaired donor electron i s l o c a l i z e d to an impurity s i t e and experiences l i t t l e i n t e r a c t i o n with other donor electrons. This system can be described i n terms of a hydrogen atom, with 19 c e r t a i n modifications, which we now discuss. F i r s t l y , the impurity electron i s loosely bound, and i t s correspondingly 6 Large "Bohr" o r b i t includes many s i l i c o n s i t e s that surround the parent nucleus. The p o l a r i z a b i l i t y of these s i l i c o n atoms produces a s h i e l d i n g e f f e c t that reduces the Coulomb a t t r a c t i o n between the electron and the impurity s i t e by the d i e l e c t r i c constant K of the medium (K=12 f o r s i l i c o n ) . Secondly, the electron experiences a p e r i o d i c p o t e n t i a l due to the regular arrays of s i l i c o n s i t e s i t encounters i n i t s movement through the c r y s t a l . This e f f e c t i s conveniently treated by replacing 19 the free electron mass m by an e f f e c t i v e mass m*. This e 20 quantity has been measured, and i s anisotropic (m* depends on the d i r e c t i o n of propagation of the electron through the c r y s t a l ) . We s i m p l i f y the problem by replacing these aniso-t r o p i c e f f e c t i v e masses by a mean e f f e c t i v e mass m*. (This mean m* i s chosen to be consistent with the e l e c -t r o n i c "Bohr o r b i t " a* = 17.2 & obtained i n a previous work.^^) 19 I t can then be shown that the wavefunctions of the bound impurity electron are products of the form ¥(r) = F(r) * k (V) Eq. 2.1 where F(r) i s a r e l a t i v e l y slowly varying hydrogenic "envelope" function that r e f l e c t s the binding e f f e c t of the impurity, and ^^Cr) i s a "Bloch wave" that i s r a p i d l y o s c i l l a t i n g and r e f l e c t s the p e r i o d i c i t y of the lattice„ We think of the electron as being i n i t i a l l y i n a state at the bottom of the conduction band ( i . e . i n a pure Bloch state) and subsequently 7 being bound by an ionized impurity. The ^ ^ ( r j j ' s are therefore chosen from Bloch states at l o c a l conduction band minima. S i l i c o n has an i n d i r e c t band gap with six conduction band minima at [ k o > 0 , 0 3 and equivalent points i n k space. ^ i s therefore written as a l i n e a r combination of the Bloch waves at each of these minima. We obtain .j . Eq. 2.2 *v(v) must r e f l e c t the tetrahedral symmetry of the l a t t i c e , which places demands on the allowed The ground state wavefunction i s distinguished from other states consistent with the l a t t i c e symmetry as i t has 19 equal contributions from each of the conduction band minima ( a ^ = a j f ° r a -^"- ^ T s a n ( ^ J t s ) # X I~ i e r e s u l t i n g symmetric wavefunction gives a non-zero probability of the electron being situated at the (nuclear spin j) donor nucleus. A hyperfine i n t e r a c t i o n between the electron and nucleus i s therefore observed. This state l i e s somewhat lower i n energy (~14 meV) than other nominally degenerate (under the e f f e c t i v e mass approximation) w that have a ^ consistent with other i r r e d u c i b l e representations of the tetrahedral point group of the c r y s t a l . This i s because the electron described by the symmetric wavefunction spends more time i n the (unshielded) 8 region near the donor nucleus , where the approximations made break down. The normalized "Is' ' ground state wavefunction can be w r i t t e n as - r 6 l k ^ • r v(*> = "TT 1 \ / ? e a * E e u (r) E q . 2.3 J 6 IT ^a* n=l k n V —o where the ^ ( r ) have the p e r i o d i c i t y of the l a t t i c e . 2.2.2 The I s o l a t e d Donor Spectrum The s p i n Hamiltonian ffl of the i s o l a t e d donor e l e c t r o n i n the presence of a magnetic f i e l d H (which defines the " z " d i r e c t i o n ) can be w r i t t e n as # = fl E q . 2 . 4 9 1 s z hyp ^ *ffz - §^HS z i s the Zeeman i n t e r a c t i o n term, where " g " i s the e l e c t r o n i c g f a c t o r , P i s the e l e c t r o n Bohr magneton, and S i s the p r o j e c t i o n of the e l e c t r o n s p i n S_ on the " z " a x i s . Nuclear Zeeman terms have been discounted as being s m a l l . foyp = A (!.*.§) ^ s t n e hyperf ine i n t e r a c t i o n term where I i s the nuclear s p i n and A i s the hyperf ine s p l i t t i n g constant . We wri te the basis wavevectors f o r as the d i r e c t products | m s » m ] ; > where mg and ITL- are the allowed p r o j e c t i o n s of S^  and I_ r e s p e c t i v e l y on the " z " a x i s , ^ h y p 1 S n o t d iagonal ^ n such a representat ion ( i . e . $ ' h y p does not commute with $ ^ « As gPH » A we t r e a t ^ j jyp a s a p e r t u r b a t i o n . The f i r s t order c o r r e c t i o n to the energy l e v e l s i s then given b y : 9 E ^ = < m , rruj. | A (I • S) | m , nij > E q . 2.5 = A m^rrLj. The energy l e v e l s of the ground state are therefore E = gPHm + A m mT E q . 2.6 m m_ ° s s I s I A microwave f i e l d of angular frequency co = 8 H ( # i s the gyromagnetic r a t i o f o r f ree electrons) i s switched o n . The corresponding Hamiltonian can be w r i t t e n as H l [ s + e - 1 " t + S_ e + i u t ] E , . 2.7 where H ^ i s the amplitude of the microwave magnetic f i e l d , and S + , S are e l e c t r o n s p i n r a i s i n g and lowering opera tors . T r a n s i t i o n s are therefore observed according to the s e l e c t i o n r u l e s A m = +1, A m = 0 „ W r i t i n g 7 = gPH and A i n s JL e frequency u n i t s , t r a n s i t i o n s are observed at the microwave frequencies - 2 " ^ e s e spectra are termed the "hyperf ine l i n e s " , and are c h a r a c t e r i s t i c of bound electrons that experience 31 a hyperf ine i n t e r a c t i o n w i t h t h e i r parent P n u c l e u s . In c l o s i n g t h i s s e c t i o n we note that the i n d i v i d u a l h y p e r f i n e l i n e s f o r samples with low impurity concentrations 16 3 23 (N^ ^ 1 x 10 donors/cm ) are very inhomogeneously broadened. T h i s broadening i s a t t r i b u t e d to unresolved hyperf ine i n t e r -a c t i o n of the l o c a l i z e d e l e c t r o n with l o c a l (nuclear s p i n |) 29 S i s i t e s , that have a 4.8% n a t u r a l abundance i n the c r y s t a l . 10 2.3 CHARACTERISTIC BEHAVIOUR OF HIGHLY CONCENTRATED SAMPLES A t the h i g h concentration end of the intermediate range 18 3 (N^ ^ 2 x 10 donors/cm ) the impurity e lectrons i n t e r a c t s t rongly and are e f f e c t i v e l y d e l o c a l i z e d at a l l experimental temperatures. The t h e o r e t i c a l behaviour of such a system n e c e s s a r i l y involves a many body problem, which has not been solved s a t i s f a c t o r i l y f o r the r e a l i s t i c case where the donor i m p u r i t i e s are randomly d i s t r i -b u t e d . A q u a l i t a t i v e understanding of the observed phenomena can, however, be achieved . We consider the l i m i t i n g case of a high densi ty d e l o c a l i z e d conduction e l e c t r o n system such as may be found i n meta ls . An i o n i z e d impurity i s introduced i n t o such a m e t a l l i c system. The Coulomb p o t e n t i a l due to t h i s impuri ty i s then s a i d to be "screened" by the c o l l e c t i v e e f f e c t s of the e lec t rons i n 24 the conduction band. T h i s screening i s most e a s i l y understood i n terms of the "Thomas-Fermi" approximation, which assumes that the Coulomb p o t e n t i a l of the s i t e i s s lowly v a r y i n g over a volume of the c r y s t a l c o n t a i n i n g s e v e r a l e l e c t r o n s . These (conduction) e lec t rons a l l f e e l some a t t r a c t i o n toward the s i t e , but , i f the densi ty of conduction e lec t rons i s large enough, the a t t r a c t i v e p o t e n t i a l may not be enough to b i n d any p a r t i c u l a r e l e c t r o n to that s i t e . I t i s found that the modif ied Coulomb p o t e n t i a l i s given by 2 V(r) = f - exp [ - £ ] E q . 2.8 kr ^ L r where r Q i s the " s c r e e n i n g l e n g t h " . The screening l e n g t h , under 1 1 the Thomas-Fermi approximation, i s given by^~V -o - ( ? [ w ] 1 / 3 ) 1 where a^ i s the e f f e c t i v e Bohr radius of the bound impurity e l e c -tron, and N i s the density of electrons i n the conduction band. I t i s i n t e r e s t i n g to note that a p o t e n t i a l of the form of Eq, 2 . 8 would not have any bound states f o r r < a T r. Therefore, i f we o -^ H i n i t i a l l y assume a l l impurity electrons to be i n the conduction band (as we d i d f o r the case of i s o l a t e d impurities) we would not expect to f i n d bound electrons f o r concentrations greater than some c c c r i t i c a l concentration N^. may be determined from Eq. 2 . 9 f o r ° 26 r = In the case of Si(P) , where a^ = 1 7 . 2 A , N^ j = 3.2 x lO''"8 donors/cm 3 o This agrees remarkably w e l l with the experimental value = 3.5 x 10"*"8 donors/cm3,^ even though such p o t e n t i a l l y important e f f e c t s as impurity l e v e l broadening have not been considered. The main objection to t h i s mechanism i s that f o r concentrations s l i g h t l y l ess than N^, and T=0 K, the i o n i z a -t i o n energy would be the same as that observed at very low impurity concentrations. This i s because a l l electrons would be bound, and no screening could take place. The i o n i z a t i o n energy i s observed 2 7 28 to decrease smoothly as i s increased ' however, which indicates another conduction mechanism. "Impurity banding" processes due to overlap of the donor electron wavefunctions could give r i s e to 29 ~ such e f f e c t s . A l t e r n a t i v e l y , a "hopping" mechanism whereby elec-trons progress from n e u t r a l impurity s i t e to neu t r a l impurity s i t e 30 has been proposed. The extra electron giving r i s e to an H ion 12 complex at each impuri ty s i t e i t encounters has a very smal l b i n d i n g 31 energy and therefore the corresponding wavefunction i s s p a t i a l l y extended. Appreciable overlap between such wavefunctions may be expected at impuri ty concentrations appreciably less than , l e a d i n g to formation of a (conducting) "H b a n d " . The d e r e a l i z a t i o n of the e x t r i n s i c e lectrons has the e f f e c t of s i m p l i f y i n g the observed EPR s p e c t r a . The h i g h l y mobile e lec t rons i n t e r a c t w i t h many donor n u c l e i (with nuclear s p i n "up" or "down") and the hyperf ine term i n the one-elec t ron Hamiltonian i s therefore averaged to z e r o . S i m i l a r l y , the e f f e c t s of l o c a l f i e L d s due to 29 "inhomogeneous" i n t e r a c t i o n s with S i s i t e s are a lso averaged out . The only non-zero term l e f t i n the one-electron Hamiltonian i s the Zeeman term. Under the inf luence of a microwave magnetic f i e l d as given by E q . 2 .7 , we may expect that a s i n g l e resonance l i n e at frequency Y w i l l be observed f o r such a system. Further considera-t i o n of t h i s "motional narrowing" type of mechanism suggests that the resonance l ineshape should be L o r e n t z i a n . 7 ^ The p r i n c i p a l r e s u l t of d e l o c a l i z a t i o n of the e x t r i n s i c e lec t rons i s , therefore , a s i n g l e EPR l i n e . We note, however, that the d e l o c a l i z a t i o n c r i t e r i o n p r e v i o u s l y discussed describes t o t a l d e l o c a l i z a t i o n to the extent that the e l e c t r o n i s conducting. An e l e c t r o n system does not have to be conducting before we may expect to monitor a s i n g l e EPR l i n e . The EPR r e s u l t implies only that e l e c -trons are d e l o c a l i z e d over p a r t i c u l a r c l u s t e r s of donors that are 31 29 not n e c e s s a r i l y connected. Hyperfine i n t e r a c t i o n s with P and S i s i t e s may e a s i l y be averaged out over such a c l u s t e r . 13 In summary, we may expect to monitor a sin g l e EPR l i n e f o r samples with high impurity concentrations. Further, as the im-p u r i t y concentration i s increased to approach this l i m i t i n g case, more and more clu s t e r s of donors should be formed. We may expect to monitor EPR manifestations of such clusters as the concentration i s increased through the intermediate range , culminating i n a single EPR l i n e at high impurity concentrations. The c h a r a c t e r i s t i c spectra of these c l u s t e r s i s discussed i n the following sections. 2.4 CHARACTERISTIC SPECTRA OF THE DONOR PAIR AND HIGHER ORDER  CLUSTERS 2.4.1 Introduction At the low concentration end of the intermediate range 16 3 (N^ ^ 1 x 10 donors/cm ) the low temperature ESR response can be interpreted i n terms of the l o c a l i z e d electron, pseudo hydrogen atom model presented i n Section 2.2.2. The spectra observed at the high concentration extreme of the ' intermediate range can be discussed i n terms of the d e l o c a l i z e d (screened) electron model of Section 2.3. As the concentration i s increased 1hto.iJgithe intermediate range, we may expect to monitor a t r a n s i t i o n from l o c a l i z e d to d e l o c a l i z e d behaviour as electron-electron i n t e r a c t i o n becomes l a r g e r . We discuss the- form and e f f e c t of these interactions i n terms of a low concentration version of the "inhomogeneity model" which appears to agree with experimental r e s u l t s at very high impurity 5 32 concentrations. ' A random d i s t r i b u t i o n of impurity s i t e s 14 i n the host l a t t i c e i s assumed» This assumption i s con-33 s i s t e n t with various experimental data. Given t h i s random impurity d i s t r i b u t i o n } > r e g i o n s of "low" and "high" donor concentration may be expected to coexist i n a sample of nominal impurity concentration N^. For the purposes of t h i s work, we r e f e r to the high concentration regions as " c l u s t e r s " of two or more impurities„ A donor may be con-sidered to be part of a c l u s t e r i f the electron - electron (exchange) i n t e r a c t i o n i s large enough so that a p a r t i c u l a r donor electron cannot be considered to be indigenous to a p a r t i c u l a r s i t e , but may be found with approximately equal p r o b a b i l i t y on any other impurity s i t e of the c l u s t e r . As the donor concentration i s increased the f i r s t manifestations of d e l o c a l i z a t i o n are due to clusters of two 26 donor i m p u r i t i e s — t h e donor p a i r . At s t i l l higher concen-t r a t i o n s , c l u s t e r s of three should contribute. We may increase the concentration f u r t h e r — o b t a i n i n g higher order c l u s t e r s — u n t i l a l l the impurities i n the sample may be considered to constitute one large cluster., In this s i t u a t i o n a l l the impu-r i t y electrons are delocalized and the d e l o c a l i z e d resonance phenomena outlined i n Section 2 . 3 i s monitored«, 2«,4o2 C h a r a c t e r i s t i c Spectra of the Donor Pair We consider the s i t u a t i o n where the exchange i n t e r a c t i o n J i s considerably greater than the hyperfine i n t e r a c t i o n A. The spin Hamiltonian of the two electron system can then be written as: 15 if = gPH ( S L Z + S 2 Z ) + J (S • S_2) + A ( I L • S L + 1 2 • S 2 ) Eq» 2.10 For the s i t u a t i o n J2J>A , S_ = + i s nearly a good quantum numbero The energy l e v e l s of the system (applying the hyperfine i n t e r a c t i o n as a perturbation) are then given by E 0 / * 1 = gPHms + | nig (mj^ + mj ) + £ Eq. 2.11 E° = - 1 J E^ j, and Eg are the t r i p l e t and s i n g l e t energies (using the usual notation f o r the ground state of the hydrogen molecule); mg represents the allowed projections of the t o t a l spin S on the "z" axis; and mT , mT are the corresponding projections x l J-2 of the i n d i v i d u a l nuclear spins. The introduction of micro-wave r a d i a t i o n as given by Eq. 2.7 induces t r a n s i t i o n s accor-ding to A m = ±1 , A (mT + I T L . • ) = 0. Figure 2.1 depicts the energy l e v e l s and allowed t r a n s i t i o n s f o r the highly coupled donor p a i r system. 26 34 In a previous work ' the p a i r spectra were calculated f o r J ~ A . In t h i s s i t u a t i o n , S i s no longer a good quantum number and the hyperfine i n t e r a c t i o n couples t r a n s i t i o n s between the s i n g l e t and t r i p l e t s t a t e s . These t r a n s i t i o n s are 34 allowed over a wide band of frequencies according to y = T e + | (J 2 + A 2 ) ^ + ^ Eq. 2.12 and y = T e + i ( J 2 + A 2 ) 7 + 1 Eq. 2.13 16 7T ~7W t ~7K—W T F i g . 2 01 (a). The Energy Levels and Allowed Transitions f o r a Highly Coupled ( J » A) P a i r . gpH-A/2 gpH gpH +A/2 F i g . 2.1 (b). The "Pair" Spectrum. 17 Eq. 2.12 reveals that t r a n s i t i o n s are allowed at frequencies intermediate to the highly coupled p a i r l i n e s at 7 , 7 g + <> Eq. 2.13 shows t r a n s i t i o n s occur at frequencies e x t e r i o r to A A the i n t e r v a l 7 g - ^ , 7 + « In general, we may conclude that exchange int e r a c t i o n s on the order of A produce t r a n s i t i o n s that tend to " b l u r " the e n t i r e spectrum. 2.4.3 Clusters of Three or More Donor Atoms Shimuzu"'"''" has calculated the spectra due to c l u s t e r s of three donor atoms, but there appears to be some inconsistency i n h i s r e s u l t s . In p a r t i c u l a r , the calculated t r a n s i t i o n frequencies are not i n v a r i a n t under a permutation of the donor spin indices (interchange of donor atoms) and s i g n i f i c a n t ESR l i n e s at frequencies Y g + -|- A are predicted. These l i n e s are not observed. This c a l c u l a t i o n does ind i c a t e , however, that any discrete t r a n s i t i o n s w i l l be more blurred or smeared out than c l u s t e r s of two were observed to be. This i s to be expected when one examines the most l i k e l y conformation of c l u s t e r s of three. We may expect that two of the i n t e r a c t i n g donors w i l l be closer to each other than to the t h i r d . This t h i r d donor may be expected to produce perturbations i n the " p a i r " t r a n s i t i o n frequencies. Of course, any p a r t i c u l a r c l u s t e r of three w i l l have discrete t r a n s i t i o n frequencies. The random d i s t r i b u t i o n of impurities, however, indicates that the t h i r d donor may assume a wide v a r i e t y of positions r e l a t i v e to the other two. The net response of a l l such c l u s t e r s w i l l consist therefore of a smeared out version of the p a i r spectra. Similar remarks can be made for higher order c l u s t e r s of n donors: the "more i s o l a t e d " donor has the net e f f e c t of smearing out the discrete t r a n s i t i o n s p e c u l i a r to the n-1 c l u s t e r . 18 CHAPTER 3  EXPERIMENTAL METHODS 3ol INTRODUCTION An ESR microwave spectrometer i s generally operated i n e i t h e r of two modes. In the "absorption" mode the s i g n a l de-tected i s due to the resonant absorption of microwave power by the paramagnetic spins i n the sample. This absorption causes a reduc-t i o n i n the Q f a c t o r of the sample cavity, which can be experimentally observed. On the other hand, when tuned to the "dispersion" mode the s i g n a l observed i s due to changes i n the s u s c e p t i b i l i t y of the sample when the resonant conditions are satisfied„ These changes i n the s u s c e p t i b i l i t y cause s h i f t s i n the resonant frequency of the c a v i t y , which are again experimentally detectable. Two d i s t i n c t types of response which resonant spins may e x h i b i t are encountered i n t h i s work. These are usually des-cribed as e i t h e r "slow passage" or " f a s t passage", and have the following d i s t i n g u i s h i n g features. We consider the two l e v e l system describing unpaired paramagnetic electrons whose spin degeneracy i s removed by Zeeman i n t e r a c t i o n with an external magnetic f i e l d . When microwave r a d i a t i o n of appropriate frequency i s applied, t r a n s i t i o n s are induced that tend to equalize the populations of both s t a t e s . I f the spin relaxation times (the spins "relax" back to the ground state v i a interactions with each other and with the l a t t i c e ) are short enough so that there i s always a net excess of spins i n the ground state, "slow passage" phenomena are observed. 19 Energy i s absorbed by the sample, and s i g n a l s are observed i n both the absorpt ion and d i s p e r s i o n modes of the spectrometer. Rapid passage behaviour i s observed when the s p i n r e l a x a t i o n times are so long that the s p i n system i s " sa tura ted" (the two states obtain equal p o p u l a t i o n s ) . Because l i t t l e net 'power i s absorbed by the sample, no s i g n a l s can be observed i n the absorpt ion mode of the spectrometer. Signals can, however, be detected i n the d i s p e r s i o n mode, and are found to be w e l l 36 descr ibed by the treatment of P o r t i s „ In Sect ion 3.2 the apparatus used, the spectrometer d e t e c t i o n modes employed, and the samples i n v e s t i g a t e d are sum-marized . Slow passage spectra are discussed f o r both homogeneously and inhomogeneously broadened systems i n Sections 3.3 and 3 .4 . A d i s c u s s i o n of inhomogeneously broadened f a s t passage spectra i n the l i g h t of P o r t i s ' a n a l y s i s f o l l o w s , which suggests a method f o r d i r e c t comparison of slow passage and r a p i d passage s i g n a l s . A treatment of the var ious methods employed to determine the spin r e l a x a t i o n times (observed under f a s t and slow passage) c loses the chapter . 3.2 APPARATUS 3 .2 .1 The Microwave Spectrometer A standard X band ESR r e f l e c t i o n spectrometer was 37 employed. The k l y s t r o n i s a V a r i a n Associa tes r e f l e x type , d e l i v e r i n g approximately 55 mw over the frequency range 9.2 GHz to 9.7 GHz. The TE.. „ sample c a v i t y was constructed 20 of brass and had a loaded Q value of approximately 4000 at helium temperatures. This c a v i t y was kept c r i t i c a l l y coupled ("matched") by a v a r i a b l e c o u p l e r . Fol lowing standard "narrow band" ESR p r a c t i c e , v a r i a b l e frequency (50 Hz < ^ ^ 14 5 x 10 Hz) magnetic f i e l d modulation was employed and f i r s t harmonic s i g n a l s detected by a phase-sensi t ive l o c k i n a m p l i -f i e r . The spectra observed were d i g i t a l l y and g r a p h i c a l l y recorded . The k l y s t r o n frequency was s t a b i l i z e d by an automatic frequency c o n t r o l (AFC) that was referenced to the sample c a v i t y f o r detec t ion i n the absorption mode. The sample c a v i t y was therefore kept at resonance i n a l l s i t u a t i o n s , which e f f e c t i v e l y prevents the observation of d i s p e r s i o n e f f e c t s . When observing the d i s p e r s i v e response however, the k l y s t r o n frequency was referenced to a h igh Q ex ternal c a v i t y of v a r i a b l e f requency. This wavemeter was s i t u a t e d o n e - t h i r d to one-half way up the " d i p " caused by the sample c a v i t y i n the k l y s t r o n mode. I t was noted that the slow passage d i s p e r s i o n s i g n a l obtained was often d i s t o r t e d due to the "mixing i n " of absorption components. In the case of f a s t passage, however, there i s a n e g l i g i b l e absorption com-ponent (as noted i n Sect ion 3.1) and hence no d i s t o r t i o n was observed. The c r y s t a l detector was biased by u s i n g a "bucking arm" independent of the c a v i t y arm, ( i s o l a t i o n = 30 dB) as 21 suggested by Wilmshurst and depicted i n Figure 3 . 1 . The same c r y s t a l b ias current (and therefore the same s e n s i t i v i t y to r e f l e c t i o n s from the cavity) could then be maintained independently of the power fed i n t o the c a v i t y . T h i s c o n d i t i o n was e s s e n t i a l f o r accurate sa tura t ion s t u d i e s . 3.2.2 Temperature Measurement and C o n t r o l The temperature of the samples was v a r i e d from 1.2 to 3 5 ° K . This was f a c i l i t a t e d by a s t a i n l e s s s t e e l jacket immersed i n a helium bath , which surrounded the whole of the c a v i t y arm as shown i n Figure 3 .2 . This jacket was u s u a l l y evacuated us ing a small auxi 1 i a r y pump because the conductive "heat leak" alone between the cavi ty and the helium bath af forded e x c e l l e n t temperature s t a b i l i t y . The brass block that formed the bottom of the sample c a v i t y was heated by applying up to 10 v o l t s across a standard 80 ohm Ohmite r e s i s t o r which was i n good thermal contact with- the b l o c k . j The sample was attached to the bottom of the c a v i t y by Apezion vacuum grease. T h i s grease allowed good thermal contact between the sample and the brass b l o c k . A spurious r a p i d passage EPR s i g n a l with a g value of approximately 2.000 and a h a l f w i d t h of 120 gauss due to the grease was observed. E r r o r introduced by t h i s s i g n a l could be e f f e c t i v e l y e l iminated by j u d i c i o u s v a r i a t i o n of experimental 50 parameters. The temperature was measured by a res is tance thermometer (a 470 ohm Speer r e s i s t o r ) that was again i n good thermal contact with the block and s i t u a t e d .05 inches below the base of the c a v i t y . The 22 Directional Attenuator Phase shifter coupler^ \ \ „ Circulotor J Klystron Sample Detector f Output <0 Magnet F i g . 3 . 1 . The "Bucking Arm" Arrangement f o r a R e f l e c t i o n Spectrometer. (Wilmshurst) waveguide •stainless steel jacket -liquid He 4 reservoir •sample cavity sample thermometer heater brass block F i g . 3 .2 . The Temperature C o n t r o l and Temperature Measurement Apparatus . 23 r e s i s t a n c e of the thermometer was measured using a n u l l c i r -cui to T h i s res is tance thermometer was c a l i b r a t e d against a S o l i t r o n germanium thermometer at the beginning and end of t h i s i n v e s t i g a t i o n . L i t t l e d e v i a t i o n between the two c a l i b r a t i o n s was observed. The accuracy of the temperatures measured i s e s t i -mated to be ± 5 % . Temperature c o n t r o l , as p r e v i o u s l y i n d i c a t e d , was achieved by heat ing the brass block forming the bottom of the c a v i t y . A f t e r switching on the heat , an e q u i l i b r i u m temperature was reached i n approximately ten seconds. Temperatures up to 35°K were r e a l i z a b l e . Temperatures below 4 = 2°K were achieved by pumping the helium bath us ing a Stokes pump with a pumping rate of 150 f t . / m i n . on a U i n c h l i n e . The minimum temperature thus achieved was 1 . 1 ° K . 3.2.3 Samples and Sample Preparat ion Table 3.1 summarizes the samples used . The ingots from which these samples were c u t were f l o a t zone r e f i n e d w i t h a (111) ingot a x i s . Sample s i z e s were adjusted (de-pending on concentration) so that at l e a s t 5 x 1 0 i m p u r i t y 39 spins were p r e s e n t . Surface e f f e c t s were e f f e c t i v e l y removed by e tching the samples i n CP-4 and/or heat ing them ( i n a i r ) to 500°C f o r about h a l f an hour . The concentrat ion of donor i m p u r i t i e s was determined from room temperature d . c . r e s i s t i v i t y measurements (using the standard four p o i n t probe t e c h n i q u e ) . These r e s i s t i v i t y measurements were converted to concentrations (donors/cm ) by r e f e r r i n g to the 40 I r v i n c h a r t . T h i s conversion agreed w e l l w i t h that obtained u s i n g the Dow-Corning S i l i c o n S l i d e R u l e . Concentrations thus obtained have a p o s s i b l e e r r o r of at most 10%. 24 RESISTIVITY |l-cmo) DONOR CONCENTRATION (donors/cc) SAMPLE DESIGNATION .155 ± 3% 5.0 X 1 0 1 6 5.0E16 .081 1.2 x 1 0 1 7 1.2E17 .058 2.2 x 1 0 L 7 2.2E17 .048 3.0 X 1 0 " 3.0E17 .0330 6.0 x 1 0 1 7 6.0E17 .0245 1.0 x 1 0 1 8 1.0E18 .0180 1.6 X 1 0 1 8 1.6E18 Table 3 . 1 . Room Temperature R e s i s t i v i t i e s and Sample Concentrations Determined Using the I r v i n Curve. 25 3.3 HOMOGENEOUS AND INHOMOGENEOUS BROADENING. THE SPIN PACKET We f i r s t d i s t i n g u i s h between homogeneous and inhomo-geneous broadening phenomena. The comments apply s p e c i f i c a l l y to the Si(P) system, but may be genera l ized to include other paramag-n e t i c m a t e r i a l s . Homogeneous broadening of ESR l i n e s i s u s u a l l y observed i n systems that have r e l a t i v e l y high d e n s i t i e s of paramagnetic s p i n s . As p r e v i o u s l y noted, these spins i n t e r a c t s t rongly with one another v i a exchange processes , wi th the r e s u l t that the e lec t rons become " d e l o c a l i z e d " — t h a t i s , they have equal p r o b a b i l i t y of being at any one of a r e l a t i v e l y large number of impurity s i t e s at a p a r t i c u l a r t ime . C e r t a i n e f f e c t s can be noted. F i r s t l y , the e f f e c t s of any " l o c a l " f i e l d ( i . e . due to hyperf ine i n t e r a c t i o n 31 29 with a P or S i nucleus) w i l l tend to be averaged out as the e l e c t r o n samples a l l p o s s i b l e l o c a l f i e l d s above and below the mean r e s o n a n t . f i e l d . Secondly, microwave power introduced at any par t of the ESR l i n e d i f f u s e s throughout the whole of the l i n e , due to the good communication between s p i n s . F i n a l l y , the experimental absorpt ion lineshape observed i s L o r e n t z i a n . Inhomogeneously broadened ESR l i n e s are u s u a l l y observed i n r e l a t i v e l y d i l u t e systems where the s p i n - s p i n i n t e r a c t i o n i s not a p p r e c i a b l e . I f the temperature i s low enough so that the impuri ty e l e c t r o n i s i n a bound state and not thermally exc i ted to the conduction band, i t w i l l be l o c a l i z e d to a p a r t i c u l a r donor s i t e . The e l e c t r o n w i l l therefore experience a l o c a l magnetic 26 f i e l d due to nearby nuclear spins i n the host l a t t i c e , or due to other e f f e c t s that are independent of impurity concentration. The ESR spectra observed f o r such inhomogeneously broadened systems w i l l then consist of many small, r e l a t i v e l y independent contributions from each of these l o c a l i z e d spins. The absorption "envelope" of a l l these resonances i s found to have Gaussian shape. For future reference we write the halfwidth of t h i s Gaussian l i n e as A H G (in f i e l d u n i t s ) . P o r t i s has described these inhomogeneously broadened 41 l i n e s by introducing the concept of a "spin packet". This spin packet consists of i n t e r a c t i n g spins that act c o l l e c t i v e l y . I t i s found that i n order to obtain the correct behaviour of the absorption (slow passage) s i g n a l at high microwave powers, t h i s 47 packet must have Lorentzian shape. We write i t s halfwidth, i n f i e l d u n i t s , as A H p . The "absorption envelope" of these non-i n t e r a c t i n g components can then be described as a convolution of Lorentzian spin packets with a Gaussian envelope function ( A H p < < A H G ) . Perhaps the c l e a r e s t discussion of the spin packet concept and such parameters as the spin packet width i s contained i n the following remarks of Anderson. As the in t e r a c t i o n s (between electrons) become stronger, the in d i v i d u a l — s p i n s which we t r e a t take on much more of a " q u a s i - p a r t i c l e " status, as being a kind of e f f e c t i v e spin, which i s not, i n f a c t , the spin of a single electron, but an elementary e x c i t a t i o n of energy hwj obtained by some perturba-t i o n procedure. The l i f e t i m e of such an e f f e c t i v e spin i s presumably what i s meant by T2» and thus , 27 i t s frequency can only be determined to w i t h i n I/T2 . We can then meaningfully ask only about motions of the frequency W j (= rS/tiHj) over distances greater than I / T 2.00 . P e r h a p s a second way to define I / T 2 , or the width of the spin packet, i s to say that i t i s i n f a c t impossible to meaningfully e x c i t e , by any experimental arrangement, a group of frequen-c i e s w i t h s t ructure narrower than I / T 2 . In the f o l l o w i n g s e c t i o n the slow passage s i g n a l s c h a r a c t e r i s t i c of homogeneously broadened systems are descr ibed, 43 Subsequently Castners treatment of inhomogeneously broadened absorption spectra i s d i s c u s s e d , and a general p r o f i l e corres -ponding to the " a b s o r p t i o n envelope" d e r i v e d . Thus armed, the f a s t passage response of very inhomogeneously broadened systems 3 6 as o u t l i n e d by P o r t i s i s presented, and t h e i r r e l a t i o n to the absorpt ion envelope emphasized. A phenomenological method of d e a l i n g with such systems where s p i n - s p i n i n t e r a c t i o n s are appreciable i s suggested. 3.4 SLOW PASSAGE ABSORPTION SPECTRA 3 .4 .1 Homogeneously Broadened Systems Slow passage homogeneously broadened ESR spectra are commonly observed, and many discuss ions of the corresponding 37 l ineshapes are given i n the l i t e r a t u r e . The s i g n a l s detected are d i r e c t l y r e l a t e d to the r e a l ( f t ' ) and imaginary ( f t " ) components of the complex magnetic s u s c e p t i b i l i t y of the sample, as shown i n Appendix A - 1 . In the absorption mode of the spectrometer s i g n a l s p r o p o r t i o n a l to ft"H^ are seen, whereas i n the d i s p e r s i o n mode we observe ' H ^ . 28 (H^ i s the amplitude of the microwave magnetic f i e l d .») These components are der ived below. Consider a sample containing paramagnetic spins to be placed i n a s t a t i c magnetic f i e l d = H ^ , thereby d e f i n i n g the " z " d i r e c t i o n . The magnetization M of the sample (M = ] C » where the yx . are the i n d i v i d u a l u n i t volume 1 s p i n magnetic moments) , i f i n i t i a l l y perturbed from i t s e q u i l i b r i u m value M 0 = / ^ 0 H _ o ( f t Q i s the " s t a t i c suscep-t i b i l i t y " of the sample), w i l l precess about the d i r e c t i o n of H according to —o ° dM ^ = (M x i y E q . 3. where o* i s the gyromagnetic r a t i o of the spins ( 8 = -1.76 x 10' Hz/G auss f o r f ree e l e c t r o n s ) . A s o l u -t i o n of E q . 3.1 i s given by M(t) = M s in ( co t) i + M cos (co t) ^ + M ^  E q . 3 — y j x v o J v v o y j z ^ where COQ= O H q i s the (Larmor) precession frequency. The magnetization M(t) w i l l re lax back to v i a i n t e r a c t i o n s among the spins and between the spins and the l a t t i c e . These r e l a x a t i o n processes are descr ibed by the 44 Bloch phenomenological r e l a x a t i o n times T^ and T^0 The z component of M grows as i n d i v i d u a l m o m e n t s " f l i p " to b r i n g M c l o s e r to M , which i s descr ibed v i a : z o ' 29 dM M -M z _ o z „ „ _ dt T L b q ° T h i s process requires the moments to give up energy to the l a t t i c e , and thus T^ i s c a l l e d the " s p i n - l a t t i c e " r e l a x a t i o n time o The components M^ and M^ are destroyed as the i n d i v i -dual spins experience s l i g h t l y d i f f e r e n t magnetic f i e l d s ( i . e . due to d i p o l e f i e l d s of neighbours) which causes them to become dephased. T h i s process i s . descr ibed phenomenolo-g i c a l l y by dM M dM - M dt Tg , dt T b q ° T 2 i s the " s p i n - s p i n " r e l a x a t i o n t ime. Tg i s of ten less than T^ , as t h i s process does not require t r a n s f e r of energy to the l a t t i c e . Tg can never be greater than T ^ , however. I f T 2 = T l ' T 2 i s r e f e r r e d t o a s " T i l i m i t e d " . A c i r c u l a r l y p o l a r i z e d microwave magnetic f i e l d of amplitude ( H ^ « H Q ) r o t a t i n g i n the x-y plane with angular frequency u> i s now i n t r o d u c e d . The net magnetic f i e l d seen by the sample i s H ( t ) = H cos(wt)x - HjSinCwt)^ + H j k E q . 3.5 The equation of motion of the magnetization M ( t ) , i n c l u d i n g r e l a x a t i o n e f f e c t s , i s , therefore : 30 — v 7 = " 0 [ i l ( t ) x H ( t ) 1 - x i - y 3 + o z k E q . 3 . 6 d t \ / T 2 T 2 T L As mentioned i n the i n t r o d u c t i o n to t h i s c h a p t e r , slow passage phenomena r e q u i r e t h a t an e q u i l i b r i u m e x i s t s between the n e t excess o f s p i n s t h a t are e x c i t e d to the h i g h e r energy s t a t e , and those t h a t r e l a x from i t . T h i s i s expr e s s e d dM by the slow passage demand t h a t z = 0 . I f E q . 3.6 i s d t s o l v e d under t h i s assumption, we o b t a i n the m a g n e t i z a t i o n 4 5 i n d u c e d i n the "x" d i r e c t i o n as M = fi coT, H cos (cot) T (co-co) + H s i n (cot) x o 2 M 1 + T 2(co -co) 2 + tf2H,2TnT„ 2 v o ' 1 1 2 Comparison o f E q . 3 . 7 w i t h Eq.A.3 o f Appendix A r e v e a l s t h a t the component o f the magnetic s u s c e p t i b i l i t y observed i n the a b s o r p t i o n mode o f the spe c t r o m e t e r i s T\ * T where ft" = floHo* T2 1 E q . 3.8 1 + £ 2 T 2 2 ( H Q - H ) 2 + tf^^TjTg and the component observed i n the d i s p e r s i o n mode i s 7^' A S g i v e n by rt< = * o H o * T 2 * V H o - H ) E q . 3.9 1 + 2 T 2 2 ( H Q - H ) 2 ' + ft 2 H 1 2 T 1 T 2 I n E q . 3.8 and 3.9 the r e l a t i o n co= ftH has been used t o express 31 fiT and ft" as functions of the magnetic f i e l d H rather than the microwave frequency co. This r e f l e c t s the experi-mental device whereby the magnetic f i e l d (not the microwave frequency) i s swept through the resonant condition. Y 2 2 The f a c t o r 0 T^Tg i n the denominator of these slow passage s u s c e p t i b i l i t i e s i s termed the "saturation parameter". For large values of t h i s parameter, slow passage signals w i l l be (power) broadened and have reduced amplitude. The slow passage "passage condition" f o r o b s e r v a b i l i t y i s therefore given by X H^-v/T^Tg < 1 . In the absence of satu-r a t i o n e f f e c t s (& H ^ V T ^ T 2 « 1) the absorption ( f t " ) s i g n a l i s Lorentzian with a halfwidth AH= =V~ « 6 -Lg As magnetic f i e l d modulation and subsequent f i r s t harmonic detection was employed i n t h i s i n v e s t i g a t i o n , d e r i -*. d f l ' dfl " , . . ,52 vative spectra ,TT or — — were detected under slow cui dH passage conditions. 3.4.2 The Slow Passage Response of Inhomogeneously Broadened  Systems As mentioned previously, absorption ( f t ' * ) lineshapes f o r very inhomogeneously broadened systems are Gaussian. I f we use the P o r t i s spin packet approach, t h i s inhomogeneously broadened l i n e i s considered to consist of numerous non-i n t e r a c t i n g Lorentzian components of width A Hp where 43 A H p « A Hfi, Castner has extended t h i s concept i n the assumed absence of s p e c t r a l d i f f u s i o n — i . e . i n t e r a c t i o n 32 between s p i n p a c k e t s „ He suggests that the absorption en-velope can be described as a convolution of Lorentz ian s p i n packets w i t h a Gaussian envelope, even when A Hp i s on the order o f , or l a r g e r than, A H ^ 0 As shown i n Appendix A - 2 , the general absorption envelope lineshape i s a V o i g t p r o f i l e that i s Gaussian f o r A H p « A H G (inhomo-geneously broadened systems) and Lorentzian f o r A H p ^ A H ^ Two comments should be made about the const i tuent s p i n packets . We consider them to be ( i n d i v i d u a l l y ) homo-geneously broadened. We are concerned with the width of these s p i n packets when 2fH^ i s greater than both T~^~ and -1 M7 Tg . Then, according to the remarks of Hyde, the micro-wave f i e l d i n t e r a c t s with spins precessing i n the frequency ef f 1 i n t e r v a l H , - - i . e . T_ = c^r~ « This e f f e c t i v e packet width H^ i s i n t e r p r e t e d as a r i s i n g from the uncertainty p r i n c i p l e . The l i f e t i m e of a s p i n state i s l i m i t e d by the a p p l i e d microwave f i e l d . As Hyde s ta tes , the change i n populations of the l e v e l s occurs i n a time t = jrrr~ > accompanied by an uncer ta inty i n energy V H ^ h . As H^ be-comes s m a l l , the l i f e t i m e w i l l be l i m i t e d by Tg or T^ pro-e f f 1 cesses . The s a t u r a t i o n parameter, when T„ = C T T — , may be w r i t t e n as ^ H ^ T g = tfH^. A second p o i n t should be made about the s p i n packet concept f o r systems i n which " s p i n d i f f u s i o n " phenomena are observed,, T h i s behaviour i s i n t e r p r e t e d as i n d i c a t i n g that 33 the s p i n packets are no longer n o n - i n t e r a c t i n g . However, 46 other authors who have inc luded spin d i f f u s i o n e f f e c t s i n t h e i r d e r i v a t i o n s of absorption envelope spectra have a r r i v e d at l ineshapes that can be descr ibed by the V o i g t p r o f i l e , and which have the same sa tura t ion behaviour as 43 obtained by the Castner f o r m u l a t i o n . I t i s assumed, t h e r e f o r e , that the s p i n packet concept s t i l l holds when s p i n d i f f u s i o n i s present . I t should be noted, however, that the i n t e r p r e t a t i o n of the s p i n packet width i n terms of 46 T,, may not be c o r r e c t . T h i s point i s f u r t h e r explored (with regard to some of the experimental r e s u l t s of t h i s i n v e s t i g a t i o n ) i n Chapter 4. F i n a l l y , we mention that the absorption d e r i v a t i v e s i g n a l s obtained experimentally are q u a l i t a t i v e l y s i m i l a r to those observed f o r homogeneously broadened systems. I n -t e g r a t i o n of these spectra w i l l then r e v e a l the "absorpt ion envelope" V o i g t p r o f i l e . 3.5 RAPID PASSAGE SIGNALS CHARACTERISTIC OF INHOMOGENEOUSLY  BROADENED SYSTEMS 3.5.1 The S u s c e p t i b i l i t y of an I s o l a t e d Spin I n d i v i d u a l spins i n very inhomogeneously broadened systems have very l i t t l e i n t e r a c t i o n with one another, and T 2 — t h e s p i n - s p i n r e l a x a t i o n t ime—is correspondingly l o n g . We describe the magnetization M of a s i n g l e s p i n . F i r s t we consider the r a p i d passage case where no r e l a x a t i o n 34 e f f e c t s are a l l o w e d — i . e . i s a lso extremely l o n g . The equation of motion of the magnetization i n the presence of a large s t a t i c magnetic f i e l d H q and microwave r a d i a t i o n of frequency co can then be w r i t t e n (from E q . 3.6) as —tfj^MCt) xH(t ) j E<I« 3 " L 0 We move i n t o the r o t a t i n g frame ( i . e . the frame r o -t a t i n g at frequency co) and assume to l i e along the "x" a x i s . T h i s s i t u a t i o n i s depicted i n F igure 3 .4 . The equa-t i o n of motion of M i n t h i s frame i s given by dt " X H eff] Eq. 3.11 where 5 e f f = H l * + ( H o " f ) ^  E<5- 3.12 In t h i s frame M precesses about the d i r e c t i o n of the " e f f e c -t i v e " magnetic f i e l d Qleff) with angular frequency co* = X j igff • W e d i v i d e M i n t o two components, M|| p a r a l l e l to the d i r e c t i o n of the e f f e c t i v e magnetic f i e l d and Mj_ r o t a t i n g with frequency coT i n a plane perpendicular to i t . The components of the magnetization i n the r o t a t i n g frame are then given by M x = M^cosCco ' t ) cos 9 + Mn s i n 6 E q . 3.13a M y = Mj_ sin(co't) E q . 3.13b M z = M ( 1 cos e - Mj_cos(co't) s i n 6 E q . 3.13c 35 36 where 5 = ( H Q - J) M± and 9 = s i n f ^ L / C l + 6 ) 2 ] . The super-s c r i p t (~) s i g n i f i e s that these are components i n the r o t a t i n g frame. We assume, as s t a t e d , that no r e l a x a t i o n of M toward i t s e q u i l i b r i u m value w i t h the e f f e c t i v e magnetic f i e l d i s a l l o w e d . and M|j therefore have the same values as they had f a r o f f resonance, where the magnetization and the magnetic f i e l d were e q u i l i b r i a t e d . As H ^ « H Q , M | j » M ^ i n a l l cases . MJJ can then be approximated by the t o t a l i n i -t i a l magnetization of the s p i n as given by M = ftQHQ . The components of the induced magnetization are w r i t t e n (to f i r s t order) as ~ M M = E q . 3.14a ( l + a * ) 2 M Y = 0 E q . 3.14b ~ M 5 M = — E q . 3.14c z ~ l (1 + 5 2 Thus, from E q . A . 5 of Appendix A, the observed s u s c e p t i -b i l i t i e s are M R' - 2HT " " " P i E q ° 3 ° 1 5 a 1 2HL(1 +6 ) 2 M ft" = 2H~ ~ ° E q ' 3 o l 5 b ft H  1 1 o o We now examine the more p h y s i c a l s i t u a t i o n i n which 3 7 the magnetization M i s allowed to re lax s l i g h t l y to the e f f e c t i v e magnetic f i e l d that i t experiences as , the r e -sonant c o n d i t i o n i s t raversed ( i . e . as 5 i s v a r i e d from 6 < 0 to 6 > 0 o This may be accomplished by v a r y i n g the microwave frequency co o r , as i s u s u a l l y the case i n ESR, by v a r y i n g H and keeping co fixed) „ The assumption made i s that the components of the induced magnetization M , M and M a x y z have the same form as i n Equations 3 . 1 4 , but M and 5 are e x p l i c i t funct ions of t ime. We r e f e r to Figure 3 . 5 and denote the r o t a t i n g frame whose " z " axis l i e s along the d i r e c t i o n of the e f f e c t i v e magnetic f i e l d by the s u p e r s c r i p t (') . The equation of motion of the magnetization i n t h i s r o t a t i n g frame, i n c l u d i n g r e l a x a t i o n e f f e c t s , i s then given by — _ (Mr x H ' j, f) - x I - _y 3 + o z k E q . 3 .16a dt f£ f j T' L / / MM Where T^ and Tg are the phenomenological Bloch r e l a x a t i o n times i n the r o t a t i n g frame. In t h i s frame H i s less than the a p p l i e d magnetic f i e l d H that the s p i n sees i n the laboratory frame. We o note , however, that according to t h e i r phenomenological d e f i n i t i o n s , T^ and Tg are independent of magnetic f i e l d . T^ and Tg are there-fore equal to the usual and Tg defined i n the laboratory frame. The z component of E q . 3 .16a can then be w r i t t e n as dM' M ' - M ' •jj-jr- = rji E q . 3 .16b Using the r e l a t i o n M' = (M +6M ) / ( l + 6 ) 2 and s u b s t i t u t i n g Z X. Z 38 f o r M and M from E q . 3.14 we obtain the (obvious) r e -l a t i o n = M. i s simply the component of the i n i t i a l magnetization (M ^ ) i n the d i r e c t i o n of as given by M' = M cos 8 = M 6/ (1 + S ) 2 . S u b s t i t u t i n g f o r M f and o o o J z i n E q . 3.16b we obtain an equation d e s c r i b i n g the time e v o l u t i o n of M to be 3 6 T h i s equation has the same form as that der ived by P o r t i s . In h i s f o r m u l a t i o n , however, T^ i s replaced by some mean r e l a x a t i o n time r . The i n t e r p r e t a t i o n of t h i s T i n terms of T^ or T^ i s not c l e a r . Our d e r i v a t i o n , on the other hand, gives the r e s u l t that i n extremely inhomogeneously broadened systems the s p i n r e l a x a t i o n processes depend only on T^ ( s p i n - l a t t i c e ) mechanisms. This i s reasonable, f o r i n such systems the s p i n - s p i n i n t e r a c t i o n s should be n e g l i g i b l e by d e f i n i t i o n . 36 P o r t i s obtains an approximate s o l u t i o n to E q . 3.17 which corresponds to the experimental s i t u a t i o n where magnetic f i e l d modulation (of angular frequency a>m and amplitude H ) i s employed. The corresponding r a p i d pas-sage magnetic s u s c e p t i b i l i t y (using E q . 3.14) observed when H < H, and co T ^ » l i s given as", m l m l H m _ tr - TT- COS (w t) -u 1 ftoH H l m ^ ft' " 2 " H ^ ( 1 + £2^2 E q ' 3 ' 1 8 39 where/" = — / H^. The response corresponding to the /*• term i s a non-steady state phenomenon, and can always be made n e g l i g i b l e by reducing the sweep rate — . I t dt i s not considered i n the following discussion,, At this point we consider the main defect i n a single spin rapid passage s u s c e p t i b i l i t y such as that given by Eq. 3ol8„ The obvious d i f f i c u l t y i s that any spin-spin i n t e r a c t i o n has been ignored i n formulating t h i s s u s c e p t i b i l i t y . We rethink t h i s problem i n terms of spin packets. We may expect that i n the case where spin-spin i n t e r a c t i o n s are appreciable that a group of spins (whose i n d i v i d u a l spin s u s c e p t i b i l i t i e s are otherwise given by equations such as Eq. 3.18) w i l l act c o l l e c t i v e l y and have approximately the same resonant frequency. We consider the s i t u a t i o n when the frequency width of this packet G^ - = u A H„) T 2 P i s greater than the width of the single spin s u s c e p t i b i l i t y (=.414$ H^ f o r Eq. 3.18). R e c a l l i n g the remarks of Ander-42 son, the s u s c e p t i b i l i t y as given by Eq. 3.18 i s then unphysical—we cannot excite a group of frequencies whose width i s less than tfAHp. The most obvious phenomenological approach to take is° to propose an envelope function f o r the i n d i v i d u a l spin s u s c e p t i b i l i t i e s corresponding to spins that have strong interactions with one another. This envelope function i s assumed to be a Lorentzian (of width A H p i n f i e l d units) 40 a l t h o u g h we s h o u l d n o t e t h a t t h e r e a p p e a r s t o be no a p r i o r i j u s t i f i c a t i o n f o r t h i s i n the r a p i d passage c a s e . T h i s l i n e s h a p e has s t r o n g a p p e a l h o w e v e r , as i t i s n e c e s s a r i l y the l i n e s h a p e t h a t the s p i n p a c k e t s u s c e p t i b i l i t y o b t a i n s u n d e r s low p a s s a g e , as p o i n t e d o u t i n S e c t i o n 3 „ 3 „ F u r t h e r , i t i s assumed t h a t the i n d i v i d u a l s p i n s u s c e p t i b i l i t y i s g i v e n (when m a g n e t i c f i e l d m o d u l a t i o n i s used) by s u c h s i n g l e s p i n s u s c e p t i b i l i t i e s as E q , 3.18 even i n the p r e s e n c e o f s p i n - s p i n i n t e r a c t i o n , as l o n g as the p a c k e t i s s a t u r a t e d . T h i s i s r e a s o n a b l e , f o r i f a l l the s p i n s c o n s t i t u t i n g the p a c k e t are s a t u r a t e d t h e y c a n n o t r e l a x one a n o t h e r ( v i a s p i n - f l i p s ) . T h e r e f o r e , the r e l a x a t i o n e f f e c t s o f the i n -t e r a c t i o n s a r e n o t o b s e r v a b l e . The n e t r a p i d passage s u s -c e p t i b i l i t y o f a l l the s p i n s I n a p a r t i c u l a r p a c k e t i s then g i v e n by the f o l l o w i n g c o n v o l u t i o n ( f o r i n d i v i d u a l s p i n s u s c e p t i b i l i t i e s o f the f o r m g i v e n by Eq, 3 , 1 8 ) : Eq, 3 .19 X H H cos (co t) r , i • _ - Q o o m v m J I 1 . 1 . c 2 H L H L i r A H p J 1 +/ H,_H \ 2 / l + / H ' - H \ 2 \ 2 IAV) V \ H w ) I n the case where r L ^ < A H p , t h i s r e d u c e s t o the L o r e n t z i a n 7) H H cos(co t) _ , _ - 1 ' o o m v m f „ o o n ft' = ~WT , ... n , 2 Eq. 3 .20 1 + / H - H \ V A H p / whereas i f H L » A H p , Eq, 3 .19 r e d u c e s to Eq. 3 . 1 8 . We t h e r e f o r e assume P o r t i s ' a p p r o a c h to be c o r r e c t f o r s i t u a -t i o n s i n w h i c h H ] L > . A H p , b u t p r e f e r our v e r s i o n when H j < A H p c •41 3o5»2 The Rapid Passage S u s c e p t i b i l i t y of the T o t a l  D i s t r i b u t i o n of Spins In Sect ion 3.5.1 the s u s c e p t i b i l i t y of an i s o l a t e d , n o n - i n t e r a c t i n g s p i n was considered . Subsequently, a method of taking i n t o account s p i n - s p i n i n t e r a c t i o n s was suggested, which used the f a m i l i a r spin-packet concept. These s p i n packets are again not considered to i n t e r a c t wi th one another. These independent responses, be they from spins or s p i n packets , contribute to an inhomogeneously broadened resonance l i n e that i s a convolution of a l l these indepen-3 6 dent cont r ibut ions with some envelope f u n c t i o n . P o r t i s has evaluated such convolutions i n the s i t u a t i o n where s p i n -s p i n i n t e r a c t i o n s are n e g l i g i b l e (H, > A H ) and H , « A H r , 1 P u. where A H i s the width of the envelope. He used an i n -u t e g r a l s o l u t i o n to E q . 3.17, and convoluted the r e s u l t i n g s p i n s u s c e p t i b i l i t y over the whole d i s t r i b u t i o n of s p i n s . The r e s u l t s he obta ins , which are considered "accurate when H ^ > A H p , are given i n Table 3.2. A s l i g h t g e n e r a l i z a t i o n has been made to P o r t i s * o r i g i n a l r e s u l t s , as i s explained i n the f o l l o w i n g paragraph. P o r t i s gives a lower l i m i t f o r co T, f o r both cases & m l 2a and 2b (co T, < 1) of Table 3.2. This l i m i t nominally m l ensures that r a p i d passage condit ions w i l l be maintained as the i n d i v i d u a l resonances comprising the l i n e are t r a v e r s e d . TT We note that the corresponding spectra are observed ^ out of phase with the modulat ion. When systems with very small 42 T ^ ' s are observed (co^Tj^Cl) the bas ic r a p i d passage s a t u r a t i o n c o n d i t i o n $ H^T^>1 may be v i o l a t e d . This v i o l a t i o n gives r i s e to the commonly detected "slow passage" s i g n a l s . However, as noted by P o r t i s f o r case 1 48 of Table 3,2, and by other authors , the slow passage (derivat ive) response i s i n v a r i a b l y i n phase with the modula-t i o n . Subsequently the out of phase s i g n a l s as seen on a phase-sensi t ive l o c k i n detec tor , are not a f f e c t e d by v i o l a -t i o n of the sa tura t ion c o n d i t i o n . Case 2b was extended to include t h i s s i t u a t i o n where co T.<$C1, and the expression m l ^ corresponding to the case 2a obtained. The amplitude of these s i g n a l s i s observed to be l i n e a r i n T ^ , i n d i c a t i n g that f o r very short T ^ T s these spectra become unobservable, IT I t i s a lso i n t e r e s t i n g to note that these out of phase s i g n a l s are not d i s t o r t e d by v i o l a t i o n of the adiabat ic c o n d i t i o n ft H, 2 ~> co H by exact ly the same reasoning as 1 ^ m m 3 3 • & 49 above. Bugai has pointed out that v i o l a t i o n of t h i s con-d i t i o n i s equivalent to v i o l a t i o n of the s a t u r a t i o n c o n d i -t i o n , g i v i n g r i s e to slow passage i n phase e f f e c t s that w i l l TT not a f f e c t out of phase components. Table 3.3 reveals the r a p i d passage s u s c e p t i b i l i t i e s obtained f o r the case when A H > H n , when the Lorentz ian P 1 s p i n packet i s assumed. T h i s table i s obtained d i r e c t l y from Table 3.2, by assuming that A H p « A H G , and by making the f o l l o w i n g a s s o c i a t i o n . The s p i n packet must be sa tura ted , as p r e v i o u s l y noted . The sa tura t ion c o n d i t i o n General C o n d i t i o n s : dH dH „ HL & H < A H » —2. < co A H , "XH T > l (saturation c o n d i t i o n ) , - 5 — & co H <SH; (adiabat ic c o n d i t i o n ) , i m b d t m l l d t m m l Case Passage Conditions Amplitude of F i r s t Harmonic ft* Lineshape of F i r s t Harmonic fi* Phase, R e l a t i v e to Modulation 1 m AH H < H l co T . m 1 * & < \ cit L fto-"— H m AHpp AHpp D i s p e r s i o n D e r i v a t i v e 0 2a m 1 H <H, 1 dt i ft H co T, Hm A H , H L Absorpt ion Envelope TT 2 2b H K " T , < 1 H ~ m 1 m Hm> H l m JL dH „ o<:co H T T m m dt ~ „ , 2H co T, A H I n , m m L . ^ C «, > Absorption Envelope IT 2 co T , < H 1 m 1 ^ m H > H , m 1 dH „ o<--co H ~r~r m m dt ft H 2 co T , H m 1 oA T T — m l r r— A H , IT Absorpt ion Envelope IT 2 3a co T , > 1 m 1 H < H , m 1 m dH „ Tn o<H, 1 dt • ft H Hm °AHG H L Absorpt ion Envelope TT 3b m 1 H >H_ m 1 T ^ H 1 " d t < m TifH l n f ^ ^ 1 H l Broadened Absorption Envelope tr 4 u T , > l m 1 dH 1 - d t 1 m ± f t H Hm l n f ^ \ AHpp A H , <• H L J Absorpt ion D e r i v a t i v e Sign Changes with Reversal Of. T r a v e l Table 3 .2 . Table of Rapid Passage "Passage Condi t ions" and Corresponding Signals Observed f o r H. > A H . General C o n d i t i o n s : dH , / T i H i \ H 1 < A H p , H L & H m & A H p < A H G , -rr9- < comAH° , H^ I J > 1 ( sa turat ion condit ion) , dH " — - & co H < $H A H (adiabatic condition) . dt m m 1 P Case Passage Conditions Amplitude of F i r s t Harmonic rtT Lineshape of F i r s t Harmonic 7^' Phase, R e l a t i v e to Modulation 2a' co T- < A H P m L < H L H m< A H P T , d H o < A H P clt r - TT -P H . H ^ M 1 C A H p W p Absorpt ion Envelope IT 2 2b* ^ P < c o T l < A H P H m 1 H . m 1 H m > A H P dH ^ u o < co H —TT mm dt H ln,-co T, m.. A H ^ ( m fe^ Absorpt ion Envelope TT 2 co T n < A H P m l H m H > A H m P d H o < c o H dt m m 2 H co T, Hm T T A H G M L A H P Absorpt ion Envelope TT 2 3aT » T . > m l m H m< A H P T . d H o < A H P cit rt H H m ° A H G AHp Absorpt ion Envelope TT 3b» co T. > A H P m l H m H m > A H P T , d H o < A H •""dt r Broadened Absorpt ion Envelope TT Table 3 .3 . Table of Rapid Passage "Passage Condit ions" and Corresponding Signals Observed f o r H, < A H P . 45 thus observed i s # T.jT 2 > 1, where T 2 = # A H P • Subs-v / H l \ t i t u t i n g f o r enables us to wri te 0 y y> 1 . We compare t h i s expression with the normal r a p i d passage s a t u r a t i o n c o n d i t i o n $ ^ T ^ > 1 , which leads us to i d e n t i f y an e f f e c t i v e r e l a x a t i o n time of the spins when AH p > H, T LH L as T = • „ o Table 3.3 f o l l o w s . A H P 3.5.3 Comments on Inhomogeneously Broadened Rapid Passage  Spectra 36 The main r e s u l t of P o r t i s * a n a l y s i s of r a p i d passage s i g n a l s i s the c l e a r i d e n t i f i c a t i o n of the f i r s t harmonic d i s p e r s i v e response with the u n d i f f e r e n t i a t e d " a b s o r p t i o n envelope" . As p r e v i o u s l y mentioned, the f i r s t harmonic slow passage absorption response i s the f i r s t d e r i v a t i v e of t h i s absorption envelope. Therefore , i f these slow passage absorption s i g n a l s are i n t e g r a t e d , we again obtain t h i s envelope. A l l spectra can therefore be presented i n t h i s unambiguous absorption envelope form. This f a c t proved e s p e c i a l l y u s e f u l f o r the study of the Si(P) s y s -tem over the intermediate concentrat ion range. A t lower 17 3 concentrations (N^ ^ 1.2 x 10 donors/cm ) and tempera-tures (T ^ 10°K) the long ( T > 10 sec) r e l a x a t i o n times e n t a i l r a p i d passage behaviour . More h i g h l y concentrated samples have shorter s p i n r e l a x a t i o n times and give slow passage response. Absorpt ion envelope spectra can be ob-tained f o r a l l concent ra t ions . These spectra are d i r e c t l y comparable and vary smoothly as a f u n c t i o n of c o n c e n t r a t i o n . 46 Previous s tudies d i d not compare the slow passage and f a s t passage spectra i n t h i s manner, leading to some d i f f i c u l t i e s i n i n t e r p r e t a t i o n . ^ ' ' Three aspects of the r a p i d passage spectra as pre -sented i n Tables 3.2 and 3.3 deserve f u r t h e r d i s c u s s i o n » F i r s t l y , as noted i n Sect ion 3 0 5 0 2 , v i o l a t i o n of the a d i a -2 b a t i c c o n d i t i o n (co H < 2 H.. ) i s equivalent to v i o l a t i o n of v m m 1 J ^ the s a t u r a t i o n c o n d i t i o n (6 H 1 T 1>1) . V i o l a t i o n of e i t h e r c o n d i t i o n therefore produces s i m i l a r d i s t o r t i o n s of the observed s p e c t r a . I f the s a t u r a t i o n c o n d i t i o n i s not obeyed, t y p i c a l ( in phase) slow passage e f f e c t s w i l l be seen. As i s a lso mentioned i n Sect ion 3 .5 .2 , only the TT out of phase (comT^> 1) r a p i d passage s i g n a l w i l l be a f f e c t e d . As r a p i d passage s i g n a l s are monitored i n the d i s p e r s i o n mode of the spectrometer, the slow passage component d i s t o r t i n g the d X T r a p i d passage spectra w i l l be a • ^ term. This term dominates as e i t h e r the s a t u r a t i o n c o n d i t i o n or a d i a b a t i c c o n d i t i o n i s f u r t h e r v i o l a t e d , as shown i n Figure 3 .5 . T h i s i s the only reproducible steady state d i s t o r t i o n of r a p i d passage s i g n a l s that i s observed. Secondly, f o r case 3b of Table 3.2 (H > H 1 > A H , m 1 p comT^>l) the absorption envelope spectra observed are broad-23 ened by e f f e c t s . T h i s has a lso been noted by Feher . The broadening i s due to the large h a l f w i d t h of the i n d i v i -dual s p i n s u s c e p t i b i l i t y l ineshape , which i s given as ~— i „ S i m i l a r broadening i s not observed f o r case 2b 47 F i g . 3 . 5 . D i s t o r t i o n s of the tr Out of Phase Rapid Passage Signal due to V i o l a t i o n s of E i t h e r the Saturation or the Adiabatic Passage Condition. 48 of Table 3.2 (H > H, > A H D ; co T, < 1) as the i n d i v i d u a l m L r m l s p i n s u s c e p t i b i l i t y i s "narrowed" by an exponential term. The corresponding lineshape i s of the form e" ' 3 ^ / (1 + 8 2) 2 where P 0 * ^ « We observed t h i s e f f e c t experimentally f o r the Si(P) system. The TT out; of phase w m T ^ > 1 absorption envelope r a p i d passage signals were observed to be appre-c i a b l y broader than the ^ out of phase comT^< 1 s i g n a l s , f o r a given H^„ F i n a l l y , we discuss the rather r e s t r i c t i v e passage dH c o n d i t i o n T, -r-— < H which should nominally be observed 1 dt m J f o r case 3b of Table 3 .2 . I f t h i s c o n d i t i o n i s v i o l a t e d , an absorption d e r i v a t i v e type of s i g n a l corresponding to case 4 of Table 3.2 should be seen. Exper imental ly , how-ever , t h i s c o n d i t i o n was v i o l a t e d by as much as two orders of magnitude before any d e r i v a t i v e components could be ob-s e r v e d . The i n s e n s i t i v i t y to t h i s passage c o n d i t i o n i s p o s s i b l y due to the same e f f e c t that produces "the loss of magnetization" phenomena (some magnetization i s "destroyed" on every r a p i d passage, through resonance) observed i n 2 . dH Si(P) by Feher . The c o n d i t i o n T^ ^ 2 . < H m e s s e n t i a l l y requires that each par t (of width 2 H ) of the inhomogeneous broadened l i n e should be sampled f o r a time on the order of T ^ , presumably i n order that a steady state response may be o b s e r v e d . ^ Given the loss of magnetization phenomena, t h i s steady s tate s i t u a t i o n w i l l be achieved much more q u i c k l y . In genera l , we consider t h i s passage c o n d i t i o n to 49 be f a r too r e s t r i c t i v e f o r the Si(P) system. Tables 3.2 and 3.3 and some of the o r i g i n a l r e s u l t s of P o r t i s i n d i c a t e var ious methods that may be employed to measure T^ and the s p i n packet width A H p , The methods employed i n t h i s i n v e s t i g a t i o n are explored i n d e t a i l i n the next s e c t i o n . A method not used, but u s e f u l i n some s i t u a t i o n s can be a p p l i e d when a> T . . « l and case 2b of m 1 Table 3,2 i s monitored. The amplitude of t h i s s i g n a l depends l i n e a r l y on T ^ . R e l a t i v e T ^ ' s could therefore be e a s i l y estimated as the s p i n - l a t t i c e r e l a x a t i o n time i s changed (by, f o r example, heat ing the sample). 3.6 RELAXATION TIME MEASUREMENT TECHNIQUES 3.6.1 In t roduct ion Four methods of measuring the phenomenological s p i n r e l a x a -t i o n times T^ and T^ were employed. Three of these are a p p l i c a b l e to spectra observed under r a p i d passage c o n d i t i o n s . The f o u r t h , 43 due to Castner , appl ies to slow passage s i t u a t i o n s when the absorption i s monitored. 3.6 o2 Recovery Method f o r Measuring T^ (Rapid Passage) In our a p p l i c a t i o n of t h i s standard technique, the usual l i n e a r sweep of the " s t a t i c " f i e l d was replaced by a low frequency (.01 Hz to 1 Hz) t r i a n g u l a r or sawtooth modulation of per iod T . The experimental s i t u a t i o n i s depicted i n Figure 3 . 6 together with sketches of the observed s i g n a l s as a func t ion of t ime. 50 51 I f we assume an exponential r e l a x a t i o n time T ^ , and that a c e r t a i n f r a c t i o n e of the magnetization i s not saturated upon each pass through the l i n e , the e q u i l i b r i u m r a t i o of the s i g n a l amplitudes can be w r i t t e n as - A t / T - A t / T ( T - A t ) / T M A ^ T ) = i - ^ S - + JLS Ce - 1) E q > 3 o 2 1 - ( T - A t L ) / T - T / T A t . / T 1 - e 1 + e e 1 (e -1) For reasonable modulation frequencies and microwave magnetic f i e l d s , e i s observed to be qui te smal l because of F e h e r ' s l o s s - o f -magnetization phenomenon. In our case we note that Equation 3 . 2 1 i n d i c a t e s that appreciable values of e (incomplete sa tura t ion of the spins) lead to reduced r a t i o s R. Therefore , f o r a given At-^ we increased and/or the s i n u s o i d a l modulation frequency a>m u n t i l some maximum r a t i o R(At^,T) was a t t a i n e d . Ratios R(At^>T) f o r var ious A t^ were then obtained by simply adjus t ing the s t a t i c f i e l d H q . The r a t i o s obtained as a f u n c t i o n of A t ^ could then be f i t t e d (for a p a r t i c u l a r Tj) to the equation - A t A R ( A t r T ) = l ^ e E q e 3 o 2 2 - (T- At ) / T 1 - e Experimental ly i t was observed that the most accurate estimates of T^ could be made when T ~ 4T^. Due to response time l i m i t a t i o n s of the chart recorder employed, t h i s method was con-f i n e d to s i t u a t i o n s where T^ > .5 s e c . 52 3 . 6 o 3 Phase Method of Obtaining (Rapid Passage) I f the rapid passage saturation and adiabatic condi-tions together with the a n c i l l ary conditions H < H, and & m l H^> are obeyed, the observed f i r s t harmonic rapid pas-sage s i g n a l has a phase dependence on the magnetic f i e l d 36 modulation frequency which can be written as K T a s i n (co t - 0 ) where 0 = tan _ 1(oj T.) ' v m m l or E q . 3 . 2 3 a cos (co mt-0-iT) Given that the f i e l d modulation o s c i l l a t e s according to H cos (co t) , "In phase" or "IT out of phase" f i r s t harmonic signals m v m 7 have maximum amplitudes at some lockin phase s e t t i n g 0 O + nir, n=0, In t h i s study determination of 0 O at various modulation frequen-cies was f a c i l i tated by introducing an a u x i l l i a r y sample with T^ ^ 10" sec. into the sample cavity, adjacent to the sample whose we wished to measure. This f a s t relaxing sample (for 18 3 our s i t u a t i o n Si(P).with >^ 2 x 10 /cm ) gives a slow passage (in phase) response and allowed a r e l a t i v e l y accurate determina-ti o n of 0 O C± 2 ° ) to within a factor of TT. The response from the sample undergoing rapid passage can then be analyzed and found to have a maximum at some lo c k i n phase s e t t i n g 0£p« Then: 0 + mr - 0 D T J = - 0 - TT E q . 3 . 2 4 Choosing n=-l, 0 - 0 R P = IT - 0 = 8 E q . 3 . 2 5 53 where 0 ^ 8 ^ T T , and 8 i s unique . This merely means that we deter-2 mine 0 o and 0 R p , subtract one from the other , and add or subtract IT • • m u l t i p l e s of TT to obtain some angle between 0 and > which i s 9. Using the r e l a t i o n 0 = t a n - 1 C ^T^) we can e a s i l y o b t a i n : 6 = t a n " 1 ^ 1 -j E q . 3.26 co T, m 1 We show how the phase d i f f e r e n c e 8,as measured on the l o c k i n , v a r i e s wi th the modulation frequency f o r var ious values of T^ i n Figure 3 .7 . i As tanQ = comT^ , T^ may be determined by p l o t t i n g cot 8 v s . co . The slope of the l i n e obtained gives T n d i r e c t l y , m ^ a 1 J In c l o s i n g t h i s s e c t i o n , we note that experimentally i t was found to be more convenient to detect the phase where the s i g n a l was n u l l e d out , rather the the phase at which the maximum amplitude of the resonance concerned occurred . - The phase of the s i g n a l at maximum amplitude i s simply obtained by adding TT + 2 to the n u l l phase value obta ined . 3.6.4 "Increase of Modulation" Technique f o r Measuring A H p (or T^) (Rapid Passage) This technique a r i s e s i n connection with the t r a n s i t i o n from case 3a' to case 3b 1 i n Table 3 .3 . The breakdown of the l i n e a r r e l a t i o n between H and the s i g n a l amplitude as H m t= e m i s increased through the range where H =; A H „ can be used to m P obtain an approximate ( ± 20%) measure of A H p and therefore Modulation Frequency (Hz) F i g . 3.7. The Phase Difference 6 as a Function of Modulation Frequency for Various Values of T,. 5 5 Tg. Experimentally, we reduce H u n t i l the values of A Hp obtained by t h i s technique are independent of any further decrease i n H^. This i s to ensure that we are not at such high microwave power that Hj>AHp, i n which case we would be measuring by th i s technique. 3 * 6 . 5 The Saturation Technique f o r Measuring T^ and Tg (Slow Passage) We use the method which i s b a s i c a l l y that o r i g i n a l l y 43 proposed by Castner, to which several minor additions have been made. As stated i n Section 3.4.2, he derives an i n t e -g r a l form of the inhomogeneously broadened absorption l i n e -shape under the following assumptions. F i r s t , the "spin packets" behave independently of one another (spectral d i f f u -sion e f f e c t s s mall). Secondly, the i n d i v i d u a l spin packet lineshape i s Lorentzian, with the form / A H , x 2 g ( H - H , ) = ^ * 1 + / H ' - H X 2 + S 2 E E q - 3 - 2 7 2 ( A H / A H where Tg = l / g (-g—E-) , A H p i s the f u l l spin packet width at h a l f maximum, and H T i s the resonant f i e l d at the centre of the packet. F i n a l l y , Castner assumes that the "envelope" of the spin packets i s Gaussian which we write as h(H-H) = K e*p -(1"!° \ E<*«3-28 oJ >,2 AH n AH f 1 1.665 where AH„ i s the f u l l Gaussian halfwidth at h a l f maximum, u and H i s the resonant f i e l d at the centre of the envelope. The o absorption s u s c e p t i b i l i t y ft'' (H) obtained using the above func-56 t ions i s c a l c u l a t e d i n Appendix A . I t i s found that the semi-normalized maximum amplitude (at H=H q ) of the experimen-t a l l y detected s i g n a l i s given by 2 2 Y (H ) = x e a x ( 1 - e r f a ( l + x 2 ) 2 ) E t 3- 3 * 2 9 max v o' ir—r (1 + x^) 2 ( 1 - e r f (a)) H l . H , = 1 a n r i a = A H r where x = _ 1 , i = 1 , and a = .832 T h i s i s i d e n t i c a l to the corresponding expression o r i g i n a l l y 43 der ived by Castner . Figure 3,8 i s a reproduction of C a s t n e r ' s " s a t u r a t i o n curves" p l o t obtained by p l o t t i n g Y (H ) against x, f o r var ious values of the parameter a . maxv oJ & ' ^ We see that as the l i n e becomes more inhomogeneous ( i . e . , a gets smaller) the s a t u r a t i o n curve becomes broader . There-fore a measure of the h a l f w i d t h of the s a t u r a t i o n curve can be used to evaluate a . In order to expedite t h i s c a l c u -l a t i o n , we define the q u a n t i t i e s ( x i ) and ( x i ) , v f J u p p e r v 2 lower ( x i ) i s the value of x such that the corresponding I upper maximum s i g n a l amplitude V r ( ( x i ) = jrV^ , on the high x K 2 upper K-max (saturating) side of the s a t u r a t i o n curve . (xi) ] _ o w e r i s s i m i l a r l y def ined on the low x (non-saturating) s ide of the s a t u r a t i o n curve . We can p l o t the r a t i o ^x-p upper = upper lower ^ Slower against a as obtained from E q . 3.29, as seen i n Figure 3 .9 . Thus, f o r a given s a t u r a t i o n curve of an inhomogeneously broadened l i n e we can determine a (assuming we have s u f f i -c i e n t range of microwave power) simply by determining the microwave magnetic f i e l d s where the s i g n a l i s one-half i t s F i g . 3.10. The Halfwidth Correction Factor "w" as a Function of "a" (Posener). 59 maximum v a l u e — i . e . , on the non-sa tura t ing side of the s a t u r a t i o n curve, and on the s a t u r a t i n g s i d e . T h i s gives (MO u s m K U ^ E R and we can then read o f f the value of a f o r the 1 lower inhomogeneously broadened l i n e from Figure 3.10. This deter -mination of a i s r e l a t i v e l y accurate i n that we have only to determine r e l a t i v e , rather than absolute , values of the quanti ty Hj_. Given now that we have determined a, what does t h i s A H t e l l us? From our i n i t i a l d e r i v a t i o n s , a = .832 _ P , A H G where A H „ i s the Gaussian h a l f w i d t h of the envelope of the s p i n packets . We note that t h i s A H , i s not n e c e s s a r i l y the experimentally observed inhomogeneously broadened l i n e h a l f w i d t h , as was assumed by Castner . I f we consider a range of power so that we are not s a t u r a t i n g , the s u s c e p t i b i l i t y i s given by v- ( _ v 2 C-y*) H \ j e y ( c f . Appendix A) ftTTCH) = -4 a / - 2 2 T T 2 A H ' G / J a + (v-y) 1 E q . 3.30 The term i n | | i s commonly c a l l e d a V o i g t p r o f i l e . Po-sener"'''' has determined the h a l f w i d t h of h a l f amplitude of t h i s p r o f i l e as a f u n c t i o n of a by a numerical technique. He obs f i n d s that the observed envelope h a l f w i d t h A H , i s r e l a t e d to A H v i a A H G = ~MT Eq° 3 o 3 L where w i s p l o t t e d as a f u n c t i o n of a i n F igure 3.10. Thus, i f we know what a i s , we determine A H , from A H ^ S by f i n d i n g w i n Figure 3.10. A H P — t h e s p i n packet h a l f - w i d t h — i s then given as : 60 . wobs A Hp = a G Eq. 3.32 w Of course, i f A H p i s determined, i s also given v i a 2 X r A \ We now determine T^. This i s equivalent to deter-1 H l mining H^  = (x = rr-) . From Eq. 3.29 we can determine the r a t i o Y(x=l) f o r various values of a. (When Y max x=l, H =H^ .) The r a t i o of the height of the absorption curve 1 2 when H = Hi^  to the maximum height of the absorption curve 1 2 can be p l o t t e d . This i s given i n Figure 3.11. We obtain Y f H =H ") H^ by simply c a l c u l a t i n g the r a t i o V 1 Y and reading o f f 2 Y max the corresponding value of H, (- Hi) 62 CHAPTER 4  EXPERIMENTAL RESULTS 4.1 INTRODUCTION T h i s chapter d e t a i l s the EPR response of Si(P) sam-16 3 p i e s w i t h intermediate donor concentrat ion (1 x 10 donors/cm 18 3 ^ 2 x 10 donors/cm ) over the temperature range 1 . 1 ° K ^ T ^ 3 5 ° K . The impurity s p i n r e l a x a t i o n times of such samples depend s t r o n g l y on both temperature and concentra t ion . '^ ' ' ' "~ , , As pointed out i n Chapter 3, the type of EPR response observed ( rapid or slow passage) depends on the r e l a x a t i o n time of the s p i n . Rapid passage spectra were therefore observed f o r con-17 3 centra t ions N^ ^ 2.2 x 10 donors/cm and temperatures T <^  1 0 ° K , whereas slow passage s i g n a l s were seen at higher temperatures and/or concent ra t ions . The r a p i d passage s i g n a l s were i n t e r p r e t e d ( fo l lowing Por t i s ) as the (undifferent ia ted) "absorpt ion envelope" . Slow passage absorption d e r i v a t i v e s i g n a l s were integra ted to again obtain t h i s absorption envelope. I t was found that a l l absorp-t i o n envelopes could be descr ibed i n terms of Cas tner ' s extension of P o r t i s * s p i n packet hypothesis ( c . f . Sect ion 3 . 4 . 2 ) . The gen-e r a l absorption envelope lineshape observed i s a V o i g t p r o f i l e . T h i s p r o f i l e i s Gaussian f o r very inhomogeneously broadened systems ( A H r ) < < A H „ ) . A l t e r n a t i v e l y , f o r homogeneous broadening phenomena ( A H >^iH ) a Lorentz ian lineshape i s observed. Jr b Low temperature (T = 1„2°K) absorption envelope 63 s p e c t r a , obtained from Si(P) samples with impurity concentra-t i o n s spanning the intermediate range, are shown i n Figure 4 . 1 . As the concentrat ion i s increased , the appearance and eventual dominance of a broad c e n t r a l l i n e (BCL) i s observed. Con-c o m i t a n t l y , the r e l a t i v e i n t e n s i t y of the hyperf ine spectra ( c h a r a c t e r i s t i c of l i g h t l y doped samples) decreases . These hyperf ine l i n e s become experimentally undetectable i n the 18 / 3 18 3 region 1 x 10 donors/cm < N^ < 1.6 x 10 donors/cm . For the purposes of t h i s d i s c u s s i o n we d i v i d e these spectra i n t o t h e i r three major components—the hyperf ine l i n e s , the d i s c r e t e c e n t r a l " p a i r " l i n e , and the BCL. T h i s decomposition ignores 54 the d i s c r e t e spectra due to c l u s t e r s of three or four donors . These features are r e l a t i v e l y s m a l l , however, and can be taken i n t o account i n a simple manner when necessary . In the next three s e c t i o n s , parameters r e l a t e d to each of the three dominant l i n e s are d e f i n e d , a synopsis of re levant p r e v i o u s l y reported r e s u l t s presented, and the r e s u l t s of t h i s i n v e s t i g a -t i o n d e s c r i b e d . The r e l a t i v e i n t e n s i t i e s of these components as func t ions of concentrat ion and temperature are discussed i n Sec t ion 4 .5 , which closes the chapter . 4.2 THE HYPERFINE LINES 4.2.1 In t roduct ion The hyperf ine l i n e s are the only spectra d i r e c t l y 16 observable at low impuri ty concentrations (N^ <C 1 x 10 donors/cm ) and helium temperatures. As shown i n Sect ion 2.2.2, such l i n e s are c h a r a c t e r i s t i c of l i g h t l y doped 64 L A J 5 0 E 1 6 1.2 E17 F i g . 4.1 (a ) . Rapid Passage Absorpt ion Envelope Spectra f o r Samples 5.0E16 and 1.2E17. F i g . 4.1 (b) . Slow Passage Absorption D e r i v a t i v e and Absorp-t i o n Envelope Spectra f o r Samples 2.2E17 and 3.0E17. 67 samples, where each donor e l e c t r o n i s l o c a l i z e d to a p a r t i c u l a r impuri ty s i t e . The hyperf ine i n t e r a c t i o n of 31 the ground state e l e c t r o n with the s p i n \ P nucleus produces two w e l l resolved spectra separated by the hyperf ine s p l i t t i n g A = 42 gauss ( in f i e l d u n i t s ) . These spectra are centred at the resonant magnetic f i e l d s 23 given by where P i s the Bohr magneton, g the impurity e l e c t r o n g v a l u e , and co the angular microwave f r e q u e n c y » In the f o l l o w i n g subsections the concentrat ion and temperature dependent c h a r a c t e r i s t i c s of these hyperf ine l i n e s are presented and b r i e f l y d i s c u s s e d , 4 » 2 , 2 The Hyperfine Line Lineshape and Spin Packet Width The hyperf ine spectra of samples with low impurity 16 3 concentrations (N^ ^ 1 x 10 donors/cm ) are extremely inhomogeneously broadened ( A H p « ^ H ^ ) , T h i s broadening has been a t t r i b u t e d to unresolved hyperf ine i n t e r a c t i o n of the impuri ty e l e c t r o n with neighbouring (nuclear) s p i n 29 23 \ S i s i t e s . The h y p e r f i n e l i n e absorption envelope then has a Gaussian shape. A t higher concentrations 17 3 (N^ ^ 6 x 10 donors/cm ) i t was observed that the h y p e r f i n e l i n e assumed Lorentz ian c h a r a c t e r i s t i c s , i n d i c a -t i n g a homogeneous broadening mechanism ( A H > A H ) . 68 The spectra of more h i g h l y concentrated samples 17 3 ( N ^ > 2,2 x 10 donors/cm ) were u s u a l l y observed under slow passage. A t a p a r t i c u l a r concentra t ion , no appre-c i a b l e change i n the hyperf ine absorption envelope l ineshape was observed as the temperature was increased from l o l ° K 0 I t was noted that the hyperf ine l i n e s of these more concentrated samples disappeared at tempera-tures somewhat lower than 30°K„ The hyperf ine spectra 18 3 of the 1 x 10 donors/cm sample, f o r example, could not be detected at temperatures greater than 1 0 ° K o H The lineshape parameter A H p (the f u l l spin packet h a l f w i d t h at h a l f maximum) was measured at 1 0 2 ° K f o r a l l 16 3 samples i n the concentrat ion range 5 x 10 donors/cm <C 1SL 17 3 ^ 6 x 10 donors/cm » The r e s u l t s obtained are p l o t t e d i n Figure 4 ,2 , The experimental techniques employed were the increase of modulation method and the Castner s a t u r a t i o n method f o r r a p i d passage and slow passage s i t u a t i o n s r e s p e c t i v e l y . The s p i n packet width A Hp i s seen to be p r o p o r t i o n a l to the second or t h i r d power of 17 3 the concentrat ion f o r 6 x 10 donors/cm , A t higher impuri ty concentra t ions , A H p i s l i m i t e d by the observed absorption envelope l i n e w i d t h . These measured p i n packet w dths are s i g n i f i c a n t l y 5 6 l a r g e r than those der ived v i a spin-echo measurements of Tg ( A H n = —rrr -) which give A U ~ 5 x 10""^ gauss. The very f o±2 69 CONCENTRATION (donors/cc) F i g . 4.2. The Hyperfine Line Spin Packet Halfwidth (AH^) vs. Impurity Concentration. 70 weak concentrat ion dependence of T 2 reported i n t h i s work i s a l s o not consis tent wi th the data of Figure 4.2, assuming, of course, that i t i s correc t to make the i d e n t i f i c a t i o n 1 T 2 ii A H p -2 There i s , however, another p o s s i b l e i n t e r p r e t a t i o n of T 2 as measured i n t h i s work that i s consis tent wi th a pre -v i o u s l y proposed model of the hyperf ine l i n e s p i n r e l a x a t i o n mechanismo I t has long been thought that the i s o l a t e d hyper-57 f i n e l i n e spins r e l a x v i a an intermediate " c r o s s - r e l a x a t i o n " process with f a s t r e l a x i n g centres comprised of c l u s t e r s of 59 i m p u r i t i e so I f we think of the BCL as being due to such cen-t res ( this hypothesis i s strengthened by the r e s u l t s of Sect ion 4 .5 . 2 as discussed i n Sect ion 5.2) we may expect the i s o l a t e d h y p e r f i n e l i n e spins to re lax v i a c r o s s - r e l a x a t i o n with BCL f a s t centres that have t r a n s i t i o n frequencies at or near the i s o l a t e d s p i n t r a n s i t i o n f r e q u e n c i e s . As noted by Mims and 63 McGee , energy t r a n s f e r or cross r e l a x a t i o n between these two sets of spins w i l l take place i n some e f f e c t i v e s p i n - s p i n r e l a x a t i o n t ime. I t would, t h e r e f o r e , seem reasonable to i d e n -t i f y the hyperf ine l i n e s p i n - s p i n r e l a x a t i o n times measured i n t h i s work with the c r o s s - r e l a x a t i o n time T„_ between the rli) hyperf ine l i n e spins and the BCL s p i n s . This proposal gains c r e d i b i l i t y i n the l i g h t of the r e l a t i o n A H p tv ( N d ) n (where n = 2.5 + .5) e x h i b i t e d i n Figure 4.2, and also by the observed temperature independence of A H p . The concentrat ion dependence would i n d i c a t e that T H B ( N d ) " 2 ° 5 . Mims and McGee 6 3 noted that 71 the c r o s s - r e l a x a t i o n time between two sets of spins—A and B — -2 4 v a r i e d according to T^g tX (N) , where N i s the r e l a t i v e concen-t r a t i o n of A spins to B s p i n s , and f u r t h e r , that T ^ B was temperature independent. The very s i m i l a r concentration dependence of T^g and T ^ B i s not p a r t i c u l a r l y s i g n i f i c a n t , however, as N measures the r e l a t i v e concentrations of the two spin s p e c i e s , whereas i s the con-c e n t r a t i o n of donorso The main a t t r a c t i o n s of t h i s c r o s s - r e l a x a t i o n hypothesis are that i t appears to be consis tent with a strong concen-t r a t i o n dependence of A Hp such as that observed, and that i t p r e d i c t s r e l a t i v e l y good communication between the i s o l a t e d hyperf ine spins and the BCL s p i n s . 4 .2 .3 Hyperfine Line Spin L a t t i c e Relaxat ion Time  Measurements Spins c o n t r i b u t i n g to the hyperfine l i n e s i n Si(P) 16 3 samples with low impurity concentration 1 x 10 donors/cm ) have s t rongly temperature dependent s p i n l a t t i c e r e l a x a t i o n times ( T I J ) over the range 1 „ 2 ° K ^ T ^ 2 0 ° K o This behaviour i s i l l u s t r a t e d i n Figure 4.3 by the r e s u l t s of Feher 1.S S3 -L6 3 and Gere and Castner f o r a 1 x 10 donors/cm sample. The var ious r e l a x a t i o n mechanisms involved have been 15 57 ex tensively i n v e s t i g a t e d ' and quite s a t i s f a c t o r y agree-ment has been obtained between theory and experiment. These r e l a x a t i o n mechanisms are u s u a l l y r e f e r r e d to as " i n t r i n s i c " which i n d i c a t e s that they have n e g l i g i b l e concentrat ion dependence, and that they are not due to i n t e r a c t i o n s be-tween donor e l e c t r o n s . Other experiments conducted at 1 . 2 ° K 72 r e v e a l a very strong concentration dependence of the hyper-f i n e spins s p i n - l a t t i c e r e l a x a t i o n time over the intermediate range The re levant e x t r i n s i c (concentration dependent) 58 s p i n r e l a x a t i o n mechanism i s not understood. Figure 4.3 presents the T^ data that were obtained i n t h i s s e r i e s of experiments. For those samples observed 17 3 under r a p i d passage ( N ^ 1.2 x 10 donors/cm , T < C 1 0 ° K ) , the recovery or phase methods p r e v i o u s l y described were used to determine T ^ . A t higher temperatures and/or con-c e n t r a t i o n s , Castner ' s (slow passage) sa tura t ion tech-1 nique was employed, and the r e l a t i o n T^ = ft A H p was as-sumedo At temperatures T ^ 1 0 ° K , sample 2.2E17 was found to have s p i n r e l a x a t i o n times such that spectra could be observed under both r a p i d and slow passage, and thus both H the phase and s a t u r a t i o n methods of determining T^ could H be used . The r a p i d passage phase technique gave T ^ T s that were two or three times longer than those obtained by the slow passage s a t u r a t i o n method. We note , however, that other experimenters have observed a d i s t r i b u t i o n of r e l a x a -t i o n rates among the spins of a given sample, depending 59 on the l o c a l environment of a p a r t i c u l a r s p i n . We i n -t e r p r e t the d i f f e r e n c e between the r a p i d passage and slow passage T ^ T s to i n d i c a t e that the slow passage technique emphasizes the f a s t e r r e l a x i n g (and thus more d i f f i c u l t to saturate) s p i n s . A l l data presented i n Figure 4.3 f o r 73 icf4 10 &105 104 \ \ Symbol Sample 1.0E16 5.0E16 1.2E17 J3 0 X + 2.2E17 3.0E17 6.0E17 \ \ 0 \ v V 10 2 4 T E M P E R A T U R E (°K) F i g . 4 . 3 . The Hyperfine Spins Spin-Lattice Relaxation Times (T^) vs. Temperature f o r Samples with Intermediate Concen-tr a t i o n s . 74 17 3 samples with N ^ ^ 2 . 2 x 10 donors/cm were obtained using the slow passage technique. ' H I t should be noted that although the r e s u l t s f o r sample 2„2E17 were d i f f e r e n t depending upon the experimental method (fast passage or slow passage) employed, both methods gave the same dependence of on the temperature. We conclude that the absolute values of given f o r samples 2„2E17 and 3.0E17 are somewhat ambiguous, but that the r e l a t i v e values of as the temperature i s increased 'are meaningful . The more h i g h l y concentrated sample 6.0E17 d i d not d i s p l a y a d i s t r i b u t i o n H of r e l a x a t i o n times, and the absolute values of T-^ are p o s s i b l y more meaningful . There i s , however, a prima f a c i e d i f f i c u l t y i n that to determine T^ an estimate of i s necessary. This est imate, obtained i n t h i s work from a measurement of the c a v i t y Q and the power i n c i d e n t on the c a v i t y as o u t l i n e d by 37 Poole , i s subject to e r r o r . In summary, the slow passage H r e s u l t s give accurate r e l a t i v e values of T^ as the temperature or concentrat ion i s v a r i e d , but the absolute values of T^ are somewhat i n doubt. The temperature dependencies of T^ were found to be governed by two b a s i c processes . The rtintrinsic" concen-16 t r a t i o n independent mechanisms, as given f o r the 1 x 10 donors/cm sample, become dominant at higher temperatures 75 17 3 f o r samples with N^ <C 1.2 x 10 donors/cm . A t higher 17 3 concentrations ( N ^ ^ 2 „ 2 x 10 donors/cm ) , e x t r i n s i c pro-cesses c o n t r o l the r e l a x a t i o n rate at almost a l l experimen-t a l l y observed temperatures. F u r t h e r , i t was found that the r e l a t i o n T " T " N E q . 4.2 where n - 1.5 +.2, was obeyed by a l l hyperf ine l i n e spins i n samples with concentrations i n the range 2.2 x 1 0 ^ 3 17 3 donors/cm <^  6 x 10 donors/cm . This power law was p r e v i o u s l y found to be d e s c r i p t i v e of the e x t r i n s i c mechan-i s m ' s temperature dependence f o r samples i n the 3„9 x 1 0 ^ d o n o r s / c m ^ N^^C 6 .4 x 10"^ donors/cm^ concentrat ion r a n g e . ^ 4 .2 .4 Hyperfine Line " g " Value and Hyperfine Constant  Observations The value of g appropriate to E q . 4 .1 , as given by g g = 1.9988, has been reported f o r sample concentrations 1 x 1 0 ^ donors /cn? A p r e v i o u s l y unreported but d i s t i n c t concentrat ion dependence of t h i s hyperf ine l i n e .g value (gH) was observed i n t h i s work. This dependence i s i l l u s t r a t e d by the 1 . 2 ° K r e s u l t s given i n Figure 4 .4 . 17 3 For concentrations N^<C1.2 x 10 donors/cm , t h i s g^ remains r e l a t i v e l y constant and i s equal to g g (within experimental e r r o r ) . A smooth decrease i n g^ i s then observed as the concentrat ion i s f u r t h e r increased to 18 , 3 = 1 x 10 donors/cm . (At higher concentrations the 2.0002 1 .9998" 19994 " g 19990 + "5 > 19986 -1.9982 -1.9978 • BCL x Hyperfine Lines — x T N N H r- 1 I I I -I i 1 I—«-J7 id 10" , 10 CONCENTRATION (donors/cc) F i g . 4.4. The "g" Values of the Hyperfine Lines (g^) and the Rf.T. fcr_ "i v s . T m n i i T ' i t v Tonnentration at 1.2°K. ai 77 hyperf ine l i n e s are no longer o b s e r v a b l e . ) These hyperf ine l i n e g values were not observed to be temperature dependent, although the accuracy of the measurements decreases somewhat wi th temperature. These measurements were taken by a standard technique 61 that e n t a i l e d i n t r o d u c i n g an a u x i l i a r y L i : L i F (g = 2.0023) sample i n t o the sample c a v i t y . A l l g values quoted were c a l c u l a t e d by observing the f i e l d d i f f e r e n c e A H between the centre of the ESR resonance whose g value was d e s i r e d , and the centre of the narrow ( ~ . 3 gauss wide) resonance of the L i : L i F sample. The g value i s given by g = g L i : L i F - A H g L . : L i F E q . 4.3 to w i t h i n two or three parts i n the s i x t h decimal p l a c e . F i e l d d i f f e r e n c e s were measured using an NMR proton probe. In conjunct ion w i t h t h i s s e r i e s of experiments, i t was observed that the hyperf ine s p l i t t i n g (A) was inde-pendent of concentra t ion . For the more h i g h l y concentrated 17 3 samples (N^^-2.2 x 10 donors/cm ) the q u a l i t a t i v e obser-v a t i o n was made that the hyperf ine s p l i t t i n g s ta r ted to decrease s l i g h t l y at temperatures somewhat lower than 3 0 ° K , as contrasted to the r e s u l t s of Lepine"'"' f o r samples with lower concentra t ions . The experimental accuracy of these measurements do not allow a q u a n t i t a t i v e d e s c r i p t i o n , however, 78 4.2.5 I n t e r a c t i o n of the Hyperfine Spins and the Broad  Centre Line Feher has demonstrated the existence of a background l i n e that i s coupled to the hyperf ine l i n e s f o r a sample 16 / 3 w i t h (the low) concentrat ion = 1 x 10 donors/cm . T h i s i n d i r e c t " o f f - l i n e sa tura t ion" experiment reveals that the magnetization of the hyperf ine l i n e s can be destroyed (saturated) by applying microwave power at magnetic f i e l d s up to 200 gauss away from the resonant f i e l d s of the hyper-f i n e l i n e s . T h i s suggested the existence of a "broad background l i n e " that can t r a n s f e r power across the spec-trum v i a a " c ross r e l a x a t i o n " or " s p i n d i f f u s i o n " process.''""' S i m i l a r types of o f f - l i n e sa tura t ion processes were observed i n t h i s ser ies of experiments. These e f -f e c t s were e s p e c i a l l y not iceable f o r samples with concen-17 3 t r a t i o n s <_ 1.2 x 10 donors/cm and at temperatures T ^ 4 . 2 ° K . In these experimental s i t u a t i o n s the c h a r a c t e r i s -t i c s p i n l a t t i c e r e l a x a t i o n times of the hyperf ine spins are quite long (on the order of seconds or l a r g e r ) „ At reasonable microwave power l e v e l s the hyperf ine spectra could be appreciably reduced by " p r e - s a t u r a t i o n " e f f e c t s whereby the l i n e s were saturated by the very act of approach-i n g toward them (by v a r y i n g the " s t a t i c f i e l d " ) . Power i s presumably t r a n s f e r r e d to the i s o l a t e d hyperf ine spins through the broad centre l i n e , by a s i m i l a r s p e c t r a l d i f f u -s i o n process as that observed by Feher . T h i s e f f e c t was 79 very pronounced at higher microwave powers and magnetic f i e l d modulation f r e q u e n c i e s . Figure 4.5 shows the l i m i t i n g (large H^) spectra f o r sample 5.0E16 to be the c h a r a c t e r i s -26 t i c spectra of the h i g h l y coupled donor p a i r . These spins have r e l a x a t i o n times that are appreciably f a s t e r than 59 the i s o l a t e d hyperf ine s p i n s , Honig has suggested that these exchange coupled " f a s t centres" provide a mechanism whereby the slow r e l a x i n g hyperf ine spins relax to the l a t t i c e v i a a cross r e l a x a t i o n process with the p a i r s p i n s . T h i s conjecture i s not i n c o n s i s t e n t wi th the above obser-v a t i o n s , although i t may not account f o r the whole e f f e c t , as i s f u r t h e r discussed i n Chapter 5. 4.3 THE CENTRAL PAIR LINE T h i s r e l a t i v e l y narrow and d i s c r e t e l i n e l y i n g approximately midway between the hyperf ine spectra becomes a prominent part of the EPR r a p i d passage spectra at concentra-16 3 t i o n s greater than N^ = 1 x 10 donors/cm . I t has been suc-c e s s f u l l y i n t e r p r e t e d as due to h i g h l y exchange coupled donor p a i r s , where the exchange energy J i s greater than the hyperf ine 26 s p l i t t i n g A . In a previous i n v e s t i g a t i o n the experimental i n t e n s i t y of t h i s l i n e r e l a t i v e to that of the hyperf ine l i n e s , as a f u n c t i o n of concentrat ion (1 x 10 donors/cm <^  N^ ^ 16 3 4 x 10 donors/cm ) , has been used to obtain an estimate of J as a f u n c t i o n of the interdonor separat ion r . 80 The c e n t r a l p a i r l i n e i s nominally centred about a resonant f i e l d IT given by Eq. 4.4 c The value of g appropriate to t h i s expression has been observed to be s l i g h t l y larger than the value of g be due to the same e f f e c t that produces the s h i f t of the broad centre l i n e to lower f i e l d at low concentration?, which i s discussed l a t e r i n Section 5.2. p a r t i c u l a r l y germane to t h i s i n v e s t i g a t i o n . Except f o r the p a t h o l o g i c a l s i t u a t i o n described i n Section 4.2.5, the inten-s i t y of t h i s l i n e was i n v a r i a b l y much smaller than that of the hyperfine spectra. Indeed, the i n t e n s i t y of t h i s l i n e r e l a t i v e to that of the hyperfine l i n e s appeared to be roughly independent of concentration f o r sample concentrations N,>5 x 10^ donors/cm 3, d 4.4 THE BROAD CENTRE LINE 4.4.1 Introduction The absorption envelope spectra presented i n Figure 4.1 reveal the gross concentration dependence of the broad centre l i n e (BCL) absorption envelope over the intermediate range. This l i n e i s the dominant s p e c t r a l component for obtained f o r the hyperfine l i n e s . 62 This s h i f t i s thought to Other properties of the c e n t r a l p a i r l i n e are not 81 17 3 a l l concentrations N, > 2.2 x 10 donors/cm . As previous-a ^ l y stated, 'we consider the background l i n e observed by Feher at the lower extreme of the intermediate range to be of the same o r i g i n as the l i n e d i r e c t l y observed at higher concentrations. At the upper extreme of concentra-t i o n , the behaviour of th i s BCL would strongly indicate that i t i s the precursor of the single (delocalized elec-18 tron) ESR l i n e observed f o r concentrations ~D> 2 x 10 donors/cm . In the following subsections we present and discuss relevant concentration and temperature dependent features of the BCL over the intermediate range. •+.4.2 Broad Central Line Detection and Experimental Limitations Other experimenters have investigated the behaviour of the BCL over the intermediate concentration r a n g e . A t temperatures T < 4.2°K and concentrations ^ 2 x 1 0 ^ 3 donors/cm , the rapid passage response they observe i n the dispersive mode of the spectrometer revealed absorption envelope spectra that are comparable to those we obtained i n t h i s i n v e s t i g a t i o n . These authors continue to monitor the dispersive mode at higher concentrations, however, even though the samples have shorter relaxation times and ex h i b i t slow passage behaviour. In such a s i t u a t i o n , the d X ' spectra observed are nominally proportional to ^ , assuming magnetic f i e l d modulation i s employed. The BCL spectra thus observed were found to be asymmetric.^ In t h i s study the absorptive (ft"' T) component of the suscep-82 F i g . 4 . 5 . The " P a i r " Spectra Obtained f o r Large " P r e - S a t u r a t i o n " E f f e c t s . H H : K l y s t r o n " l o c k e d " t o low f r e g u e n c y s i d e o f c a v i t y . : K l y s t r o n " l o c k e d " t o h i g h f r e g u e n c y s i d e o f c a v i t y . dT» F i g . 4 . 6 . Asymmetry Introduced Into the T J J J - Spectra f o r Sample 6.0E17 Due to the " M i x i n g In" of Some Absorpt ion Component. 83 t i b i l i t y i s monitored f o r slow passage s i t u a t i o n s . The d i s p e r s i v e component i s r e j e c t e d by " l o c k i n g " the k l y s t r o n frequency to the c a v i t y frequency v i a an A F C . The f i r s t harmonic absorption d e r i v a t i v e slow passage s i g n a l ob-ta ined i s the d e r i v a t i v e of the absorption which, when i n t e g r a t e d , gives the absorption envelope. No reproducible asymmetry of the BCL absorption envelope was observed f o r such s p e c t r a . I t i s suggested i n t h i s work that the reported asymmet-ry of the BCL — s i g n a l s can be i n t e r p r e t e d i n terms of an admixed absorptive ( f t T t ) component of the suscep-t i b i l i t y . In order to demonstrate the p o s s i b l e e f f e c t s , the d i s p e r s i v e response f o r sample 6.0E17 was monitored i n the u s u a l fashion at 1 „ 2 ° K . A somewhat d i s t o r t e d BCL lineshape was observed. However, i t was noted that as the k l y s t r o n was AFC " l o c k e d " to the other s ide of the resonant c a v i t y " d i p " ( c . f . Sect ion 3.2) the apparent asymmetry of the background l i n e was r e v e r s e d . This behaviour i s i l l u s -t r a t e d i n Figure 4.6 , and can be i n t e r p r e t e d i n terms of mixing i n some of the absorptive component. The s h i f t of the k l y s t r o n reference frequency from one side of the d "X"' c a v i t y dip to the other reverses the s ign of ^ , but d X ' 1 does not reverse the s ign of any mixed-in — - T J T — component. an d A~ r Thus absorptive components that would be added to the — — -C u i s i g n a l f o r the f i r s t s i t u a t i o n w i l l be subtracted i n the second case, g i v i n g r i s e to the observed e f f e c t . 84 The BCL spectra are extremely broad, r e l a t i v e to the d i s c r e t e hyperf ine l i n e spec t ra , i n a l l samples with 17 3 concentrations N ^ ^ 6 x 10 donors/cm „ The absorption d e r i v a t i v e slow passage spectra obtained give l i t t l e d i r e c t information about the BCL. Indeed i t i s often d i f f i c u l t to a s c e r t a i n i f i t i s there at a l l . ( c . f . Figure 4.1.) The integrated absorption envelope spec t ra , on the other hand, can be e a s i l y decomposed i n t o the p a i r , BCL, and hyperf ine l i n e s wi th reasonable accuracy. A l l BCL s a t u r a t i o n data f o r samples with concentrations i n the 17 3 17 3 range 2.2 x 10 donors/cm < N , < 6 x 1 0 donors/cm ^ d were therefore obtained from integrated absorption envelope d a t a . We discuss the BCL i n samples that were observed under r a p i d passage. The BCL absorption envelope appeared to decrease i n i n t e n s i t y ( r e l a t i v e to the hyperf ine l ines) as the temperature was r a i s e d over the range 5°K < T < 1 5 ° K . T h i s e f f e c t i s e x h i b i t e d i n Figure 4.7 f o r sample 1.2E17. T h i s behaviour i s not " r e a l " however, as i s demonstrated by the integra ted (slow passage) absorption envelope spectrum taken at 15°K f o r sample 1.2E17 shown i n Figure 4 .8 . This f i g u r e reveals the r e l a t i v e s i z e of the BCL to the hyperf ine l i n e s to be roughly the same as i n the 4 . 2 ° K r a p i d passage spectrum of Figure 4 .7 . The observed decrease i n the background l i n e i n t e n s i t y wi th increase of temperature (as seen under r a p i d passage) 8 5 F i g . 4 . 7 . Observed Decrease of the BCL I n t e n s i t y R e l a t i v e to The Hyperfine Line I n t e n s i t y w i t h Increase of Temp-erature f o r Sample 1.2E17. 86 h 1.2 E17 F i g . M-.8. Slow Passage Absorption Derivative and Absorption Envelope Spectra f o r Sample 1.2E17 at 13°K„ 87 could be due to e i t h e r or both of the following e f f e c t s . F i r s t l y , the rapid passage signals obtained f o r T ^ 5°K correspond to the ( w m T j > 1) passage case 3b of P o r t i s , as discussed i n Section 4.2.1. The absorption envelope spectra obtained can be appreciably broadened i f H l -3 23 . „ > 10 , where AH~ i s the Gaussian envelope halfwidth. A H G G This broadening w i l l obviously a f f e c t the narrow hyper-f i n e l i n e s more than the broad background l i n e . A t higher temperatures (T ^  S°K) signals corresponding to the (co T, < 1) passage case 2b of Po r t i s are monitored. v m 1 In t h i s s i t u a t i o n , produces broadening only i f i t i s of the same order as the envelope halfwidth. Thus as the temperature i s r a i s e d the hyperfine l i n e s become narrower and concomitantly assume greater amplitude r e l a t i v e to the background l i n e . However, t h i s e f f e c t would not account f o r the smaller r e l a t i v e area under.the B C L . A more s a t i s f a c t o r y explanation can be formulated as follows. I f the rapid passage signals described by Table 3.3 (H^ < AHp) are c o r r e c t l y formulated, we can make the f o l -lowing observation. I f co T. < 1 and H n< AH^, the ampli-m l 1 r IT tude of the TJ- out of phase s i g n a l (case 2a) i s inversely proportional to AHp. The hyperfine l i n e spin packet width was not observed to vary appreciably with temperature over the range 1.2°K ^  T ^ 10°K„ We would therefore not expect any temperature dependence of the amplitude of the hyperfine absorption envelope due to changes i n A H . 8 8 As shown i n Section 4.4.3 following, however, the BCL spin packet widths (for more highly concentrated samples observed under slow passage) do increase with increase of temperature. I f t h i s phenomenon occurs at lower (rapid passage) impurity concentrations, we may expect the ampli-tude of the BCL absorption envelope to be reduced (hy A ^ ) a s the temperature i s r a i s e d . This e f f e c t could A Hp e a s i l y account f o r r e l a t i v e BCL decrease noted i n Figure 4.7. 4.4.3 Lineshape and Linewidth Observations f o r the Broad  Central Line At helium temperatures and concentrations ^ 1.2 x 17 3 10 donors/cm the BCL absorption envelope has Gaussian c h a r a c t e r i s t i c s i n the "wings". Conversely, at high con-17 3 centration ( N ^ ^ 6 x 10 donors/cm ), t h i s envelope assumes Lorentzian shape. The behaviour of the observed lineshape indicates inhomogeneously broadened ( A H p « A H G ) structure f o r the low concentration BCL, and homogeneously broadened (AH_> AhVO structure i n the high concentration JV"> b region. We also note, as shown i n Figure 4.9, that the BCL absorption envelope narrows appreciably over the i n t e r -mediate range. This e f f e c t i s a t t r i b u t e d to a "motional narrowing" mechanism as discussed i n Section 5.3. The BCL linewidth i s also observed to decrease as the tempera-ture i s raised' and can be explained by a s i m i l a r mechanism. This dependence i s i l l u s t r a t e d i n Figure 4.10. 6 0 501 40+ a c n O o 30+ < 20+ 10-\ \ \ \ •I N 0 -1 h 37 -J r-10" 10 Concentration (donors/cc) .18 F i g . 4 . 9 . The Halfwidth of the BCL at 1 . 2 ° K . v s . Impurity Concentrat ion. 60 5 0 -# 4 0 D D c n m °n:O30 20 10+ 0 Symbol Sample 2.2E17 + - 3.0E17 6CE17 • 1.0E18 0 10 15 2 0 TEMPERATURE (°K) 25 3 0 35 F i g . M-.10. The H a l f w i d t h of the BCL v s . Temperature f o r Samples with Intermediate Concentrat ions . 91 A g a i n , as for the hyperf ine l i n e s , an important experimental parameter i s the width of the Lorentzian s p i n packets , A»Hp. Unfor tunate ly , due to the small amp-17 3 l i t u d e of the BCL f o r concentrations ^ 1.2 x 10 donors/cm , i t was not experimentally possible to measure the s p i n packet width by the r a p i d passage " increase of modulation" tech-nique „ However, quite reproducible measures of A Hp could be obtained f o r the more concentrated samples observed under slow passage, us ing the sa tura t ion technique. The values are p l o t t e d as a f u n c t i o n of temperature i n Figure 4 .11 . I t i s i n t e r e s t i n g to note that as the BCL narrows with increase of temperature, the s p i n packets broaden. F u r t h e r , i f the s p i n packet width i s not l i m i t e d by the width of the absorption envelope, A H p v a r i e s according to A Hp ex T ° 5 E q . 4.5 As the concentrat ion i s increased to 6 x 10 donors/cm , the s p i n packet widths are l i m i t e d by the l i n e w i d t h of the BCL at a l l experimental temperatures. 92 We are again faced with the question of what these BCL s p i n packet widths and corresponding T ^ ' s correspond to p h y s i c a l l y . The observed temperature dependence of t h i s parameter would not be expected of a purely s p i n - s p i n process . As p r e -46 v i o u s l y s t a t e d , Clough and Scott have included the e f f e c t s of 5 7 s p i n d i f f u s i o n phenomena i n a c a l c u l a t i o n of the form of the s a t u r a t i o n p l o t s that are used i n t h i s work to c a l c u l a t e T^ and Tg• Results q u a l i t a t i v e l y s i m i l a r to those of Castner are obtained, but the form of the various parameters (such as the inhomogeneity parameter "a") are somewhat d i f f e r e n t . In par-t i c u l a r , they note , f o r very inhomogeneously broadened systems at l e a s t , that the s p i n d i f f u s i o n time " T ^ " corresponds to the widths of P o r t i s ' s p i n packets . Fur ther , i t i s noted by K l a u -42 der and Anderson that s p i n d i f f u s i o n times may be temperature dependent i n some cases . I t i s tempting, therefore , to i n t e r p r e t the observed BCL T ^ ' s as measures of the s p i n d i f f u s i o n time T . This a s s o c i a t i o n i s i n no sense r igorous however, and i s suggested only i n the absence of a bet ter e x p l a n a t i o n . 4 .4 .4 Broad Centre Line " g " Value Data I t i s rather questionable to assign a d e f i n i t e g value to the BCL, as i t i s the envelope of many r e l a t i v e l y indepen-dent c o n t r i b u t i o n s . I t i s , however, a convenient nota t ion to describe the observed s h i f t s i n the resonant f i e l d at the l i n e centre as the concentrat ion and/or temperature i s v a r i e d . The r e s u l t s obtained should be viewed i n t h i s sense. We define t h i s g value of the BCL (g_) convention-93 16-14-a 12+ o U) ^ 1 0 8 " 6 -0 ' 4 f / / / • 2.2E17 • 3.0E17 2 4 6 8 Temperature ( °K ) 10 F i g . 4 .11 . The Spin Packet Halfwidths of the BCL (AHp) v s . Temperature f o r Samples 2.2E17 and 3.0E17. 1.9996 19994 OQ1,9992 1.9990-1.9988 0 2 4 6 Temperature (°K ) F i g . 4 .12. The " g " Values of the BCL v s . TemDeraiure^for 94 a l l y , as i n the equation „B _ J L _ (nto) E q . 4.6 where H i s the resonant f i e l d at the centre of the BCL. Our experimental values f o r gg at 1 . 2 ° K as a f u n c t i o n of concentrat ion (obtained using the same technique as f o r the hyperf ine l ines ) are i n d i c a t e d i n Figure 4 .4 . We note that gg tends smoothly with i n c r e a s i n g concentrat ion toward the value g g = 1.9988. This i s the g value a s s o c i a -ted with the s i n g l e ( d e l o c a l i z e d electron) resonance l i n e that i s observed f o r samples i n the semiconductor-metal t r a n s i t i o n concentration range.^ F u r t h e r , the steady dec l ine i n g wi th i n c r e a s i n g concentration mirrors the D r e s u l t already noted f o r the hyperf ine l i n e s , and g f i - g^ = .0012 ± . 0 0 0 2 f o r a l l samples s tudied i n the 1.2 x 1 0 1 7 3 18 3 donors/cm ^ N^ $C 1.0 x 10 donors/cm concentrat ion range. T h i s value of gg - g^ i s consis tent with the s h i f t of the BCL r e l a t i v e to the centre of the hyperf ine l i n e 17 pat tern p r e v i o u s l y reported by Morigaki and Maekawa. In p a s s i n g , we note that they report t h i s s h i f t to drop 17 3 sharply when the concentrat ion exceeds 5 x 10 donors/cm . No s i m i l a r reduct ion was observed i n t h i s work. This c o n t r a d i c t i o n i s due mainly to the d i f f e r e n t methods used i n t h i s work to obtain the impurity concentrat ion N ^ . As s ta ted by Q u i r t and Marko^ the s tated concentrations of the 17 3 Morigaki and Maekawa samples i n the N^ — 5 x 10 donors/cm 9 5 concentration region are roughly a f a c t o r of two smaller than the correct values. Such an adjustment brings the r e s u l t s of Morigaki and Maekawa into reasonable consistency with the r e s u l t s of t h i s i n v e s t i g a t i o n . Figure 4 . 1 2 exhibits the g_ temperature depen-D dence f o r sample 6 . 0 E 1 7 . We see that g^ again decreases smoothly toward the value g g = 1 . 9 9 8 8 associated with the sin g l e s p e c t r a l l i n e obtained f o r more highly concentrated samples. Similar behaviour was observed in a l l samples where the BCL was detectable and had an i n i t i a l low tempera-ture g value greater than 1 . 9 9 8 8 . As the g value of the hyperfine l i n e s (g^) was found to be r e l a t i v e l y independent of temperature, there i s a deviation "g,, - g„" at a l l rj n experimental temperatures f o r the more highly concentrated samples. This i s due to the s h i f t of g^ away from the con-duction electron value at higher concentrations (N^ ^ 2 . 2 1 7 3 x 1 0 donors/cm ) , which was discussed i n Section 4 . 2 . 4 . 4 . 4 . 5 Broad Central Line Spin L a t t i c e Relaxation Time  Measurements Quantitative measures of the BCL spin l a t t i c e relaxa-t i o n time T^ were d i f f i c u l t to obtain f o r samples with con-centrations i n the lower h a l f of the intermediate concen-1 7 3 t r a t i o n range (N^ ^ 1 . 2 x 1 0 donors/cm ) . As can be ob-served, the BCL spectra f o r such "rapid passage" samples have small ( ~ 2 0 % ) amplitude r e l a t i v e to the hyperfine l i n e . 9 6 I t was found that nei ther the recovery or phase method a p p l i c a b l e to such a r a p i d passage s i g n a l gave meaningful r e s u l t s . This was due mainly to the r e l a t i v e l y large s i z e of the hyperf ine l i n e s , which obscured such d e t a i l s as the amplitude of the BCL. There were, however, strong i n d i c a t i o n s as to the r e l a t i v e r e l a x a t i o n times of the H BCL and hyperf ine l i n e s . In the s i t u a t i o n where the T^ of the hyperf ine l i n e s was being measured by the recovery technique, i t was noted that the BCL i n v a r i a b l y recovered to i t s f u l l amplitude no matter how short the time between "sweeps" through the s p e c t r a . This i n d i c a t e s that the BCL has a much f a s t e r T^ than the hyperf ine l i n e s . F u r t h e r , as the temperature was r a i s e d through the range 4 . 2 ° K ^ T ^ 7°K (monitoring the ^ out of phase "jT-^ < 1 response) , the BCL s i g n a l was observed before the hyperf ine l i n e s again became dominant at higher temperatures. T h i s a lso i n d i c a t e s that the BCL has an appreciably f a s t e r r e l a x a -t i o n time than the hyperf ine l i n e s , even at higher tempera-t u r e s . 17 3 A t higher concentrations ( N ^ ^ 2.2 x 10 donors/cm ) the slow passage r e s u l t s are analyzed from the integrated absorp-t i o n envelope spectra. I t was found f e a s i b l e to measure the B BGL s p i n l a t t i c e r e l a x a t i o n time (T^) from these spectra v i a the s a t u r a t i o n technique. The r e s u l t s obtained are p l o t t e d against temperature i n Figure 4 .13. Included i n t h i s f i g u r e f o r comparison are the s p i n l a t t i c e r e l a x a t i o n 97 1C? O -5 10 \ I \ X 2.2E17 X ^ + 3.0E17 BCL Hyperfine Lines S. \ . \ <\ \ X. \ \ \ X. \ N X \ V X \ \ \ \ x x X \ X \ X \ \ \ X \ \ N \ \ x ^ X> \ \ \ \ \ \X X. \ \X x \ v \ X. \ \ \ \ V \ \ \ _ • \ \ T \ ^ N \ \ A\ \ \ \ \ \ \ V \ 1 i 1 \ \ \ i i i i i n 2 4 6 8 10 TEMPERATURE (°K) F i g . 4.13. The Spin-Lattice Relaxation Times of the BCL Spins vs. Temperature for Samples 2.2E17 and 3.0E17. 98 times of the spins c o n t r i b u t i n g to the hyperf ine l i n e s . The T - ^ s of the BCL spins i n samples 2.2E17 and 3.0E17 appear to obey the r e l a t i o n The hyperf ine s p i n l a t t i c e r e l a x a t i o n times depend more H — 1 5 s t rongly on the temperature (T^ <x T~ ) as noted i n Sec-t i o n 4 . 2 . 3 . I t i s a lso observed from Figure 4.13 that , 17 3 f o r ^ . 2.2 x 10 donors/cm at l e a s t , the BCL spins may have longer s p i n l a t t i c e r e l a x a t i o n times than the hyperf ine s p i n s . F u r t h e r , i t was observed f o r sample 6.0E17 at 1 . 2 ° K H and 4 . 2 ° K that T^ was approximately an order of magnitude •g smaller than T^„ These r e s u l t s would appear to c o n t r a d i c t our e a r l i e r suggestion that the more i s o l a t e d spins c o n t r i b u -t i n g to the hyperf ine l i n e s re lax v i a a c r o s s - r e l a x a t i o n process with the f a s t r e l a x i n g BCL s p i n s . A l s o , i f the hyperf ine l i n e T ^ ' s are i n t e r p r e t e d as c r o s s - r e l a x a t i o n times we note that they are appreciably shorter (by at l e a s t an order of magnitude) than the measured s p i n - l a t t i c e r e l a x a t i o n t imes . T h i s would suggest that the bott leneck f o r r e l a x a t i o n of the hyperf ine spins i s not the c r o s s - r e l a x a t i o n rate but the r e l a x a t i o n times of the BCL spins themselves. We mig therefore expect T^ ? to equal T ^ , and f u r t h e r , that both s p i n l a t t i c e r e l a x a t i o n times w i l l have the same temperature dependences. I f we i n t e r p r e t the T2*s measured f o r the BCL spins as s p i n d i f f u s i o n times, however, the e f f e c t i v e r e l a x a -t i o n times of the BCL spins may be appreciably shorter than -1 E q . 4.7 99 B T^, p o s s i b l y removing the a m b i g u i t y c o n c e r n i n g the r e l a -H B t i v e s i z e s o f and T^. The q u e s t i o n o f the t e m p e r a t u r e dependences o f t h e s e r e l a x a t i o n t i m e s i s n o t r e s o l v e d , however, and i s d i s c u s s e d i n the f o l l o w i n g p a r a g r a p h . We examine the p r o p o s e d r e l a x a t i o n p r o c e s s f o r an i s o l a t e d s p i n c o n t r i b u t i n g t o the h y p e r f i n e l i n e s . G i v e n t h a t c r o s s - r e l a x a t i o n i t s e l f i s n o t a b o t t l e n e c k , i t i s s u g g e s t e d H t h a t t h e r e l a x a t i o n time T^ o f such s p i n s w o u l d be d i r e c t l y p r o p o r t i o n a l t o the r e l a x a t i o n time o f the BCL s p i n s w i t h w h i c h t h e y c r o s s - r e l a x , and i n v e r s e l y p r o p o r t i o n a l t o the number o f BCL s p i n s t h a t a r e a v a i l a b l e . I t i s p l a u s i b l e t h a t the- number o f a v a i l a b l e s p i n s w o u l d be d i r e c t l y p r o p o r -B —1 B —1 t i o n a l t o the s p i n d i f f u s i o n r a t e ( T 2 ) ~ . G i v e n t h a t ( T 2 ) ~ 5 B —1 i s p r o p o r t i o n a l t o T° and T^ ol T~ , t h i s mechanism would g i v e T? <X T~^"°^; as was o b s e r v e d . 1 0 0 4 . 5 RELATIVE INTENSITIES OF THE SPECTRAL COMPONENTS 4 . 5 . 1 Introduction In t h i s section, we discuss the measured r e l a t i v e i n t e n s i -t i e s of the hyperfine l i n e s , the BCL, and the discrete c e n t r a l p a i r l i n e that are obtained from the absorption envelope data. These r e l a t i v e i n t e n s i t i e s can be used to determine the f r a c t i o n of the spins i n the sample that contribute to each s p e c t r a l com-ponent. In the following paragraphs, some of the experimental d i f f i c u l t i e s and possible errors involved i n such measurements are discussed. Samples with impurity concentrations N^ ^ 2 . 2 x 1 0 ^ " 7 donors/ 3 cm were observed under slow passage. The hyperfine spins were observed to "saturate" at lower microwave powers than the BCL spins (due to the longer relaxation times of the hyperfine spins) . Saturation e f f e c t s would therefore tend to unduly emphasize the s i z e of the BCL r e l a t i v e to the hyperfine l i n e s . Care was taken to work at low enough microwave powers so that such saturation e f f e c t s were n e g l i g i b l e . A comparison of the areas under each of the s p e c t r a l components should then give a measure of the r e l a -5 2 t i v e s u s c e p t i b i l i t y of each component. Some d i f f i c u l t i e s were experienced f o r s i m i l a r analyses of the•rapid passage absorption envelope. For signals obeying the H B passage conditions co T , » l , H , > A H N , A H , the observed areas ^ & m 1 ' 1 P' P can be d i r e c t l y compared to give r e l a t i v e i n t e n s i t i e s . However, i n order to be sure of s a t i s f y i n g H > A H f o r the BCL, r e l a t i v e -101 l y large H ^ ' s had to be used . This e n t a i l e d " p r e - s a t u r a t i o n " effects—power was transmitted (via s p e c t r a l d i f f u s i o n ) through the BCL "ahead" of the sweep f i e l d H, as noted i n Sect ion 4 . 2 . 5 . This power i s coupled to the hyperf ine l i n e s by a c r o s s - r e l a x a t i o n process , thus d i m i n i s h i n g the ob-served i n t e n s i t y of the hyperf ine l i n e s . This e f f e c t would i n d i c a t e that the 1 „ 2 0 K absorption envelope data f o r samples S 0 O E I 6 and 1.2E17 give BCL's that are p o s s i b l y too large r e l a t i v e to the hyperf ine l i n e s . In the case of the 1.2E17 sample, i t was p o s s i b l e to obtain more unambiguous measures of the r e l a t i v e i n t e n s i t y of the s p e c t r a l components by u t i l i z i n g integra ted absorption envelope data taken at T = 13°K as i s explained i n the f o l l o w i n g s e c t i o n . The 5.0E16 sam-p l e , on the .o ther hand, could be observed f o r quite small values of ( A H p was very small) at 1 , 2 ° K and p r e - s a t u r a t i o n e f f e c t s were not n o t i c e a b l e . 4.5.2 The Temperature and Concentration Dependence of the  R e l a t i v e I n t e n s i t i e s of the S p e c t r a l Components The area under a p a r t i c u l a r absorption envelope s p e c t r a l l i n e ( in the absence of s a t u r a t i o n or other passage e f f e c t s ) i s d i r e c t l y p r o p o r t i o n a l to the number of c o n t r i -6 3 but ing s p i n s , and, concomitantly, to the s u s c e p t i b i l i t y ( ^ of the sample. We denote the areas under the BCL, hyperf ine l i n e s , and d i s c r e t e c e n t r a l p a i r l i n e s as ?^B> ftitt and ftp r e s p e c t i v e l y . The parameter of i n t e r e s t i s 102 the experimental r a t i o P R as def ined by R B * B + ftB Eq. 4 . 8 We discuss the temperature dependence of f o r the more h i g h l y concentrated "slow passage" samples i n v e s t i g a t e d 17 3 (N > 2.2 x 10 /cm ) . P was not observed to change 15°K f o r samples 2.2E17 and 3.0E17. A t higher tempera-tures (T > 2 0 ° K ) , P was, however, observed to be an i n c r e a s i n g f u n c t i o n of temperature. More h i g h l y concen-t ra ted samples (6.0E17 and 1.0E18) showed P g to be an i n c r e a s i n g f u n c t i o n of temperature even over the range 1„1°K ^ T ^ 1 5 ° K . A l s o , P was observed to depend more s t rongly on the temperature f o r the 1.0E18 sample than the 6.0E17 sample. We i n t e r p r e t these r e s u l t s as evidence f o r l o c a l i z e d and d e l o c a l i z e d behaviour of the var ious s p e c t r a l components. 7S JJ has been p r e v i o u s l y r e p o r t e d ^ to have a " C u r i e Law" (ft H ^ 7j7 where C = constant) dependence on the temperature that i s c h a r a c t e r i s t i c of l o c a l i z e d systems. The r e l a t i v e invar iance of P^ with i n c r e a s i n g temperature over the range 1 . 1 ° K ^ T < 15°K f o r samples 2.2E17 and 3.0E17 would i n d i c a t e that 7\ a lso has a Curie Law temperature dependence. We would therefore consider that the BCL i s due to r e l a t i v e l y l o c a l i z e d spins f o r concentrations as h igh as 3 x WL^ appreciably (,< 6%) over the temperature range 1 . 1 ° K ^ T 103 donors/cm .. The r e s u l t s f o r more h i g h l y concentrated sam-ples on the other hand show to be more and more depen-dent on temperature. T h i s i s i n agreement with our e a r l i e r s p e c u l a t i o n that the BCL i s the precursor of the t o t a l l y d e l o c a l i z e d resonance obtained f o r very h i g h l y doped samples. The s u s c e p t i b i l i t y f o r such systems i s temperature independent. In Figure 4.14 a graph of Pg versus impurity concen-t r a t i o n i s presented. The r e s u l t s given f o r samples with 17 3 N ^ ^ . 2 . 2 x 10 donors/cm were taken from (unsaturated) slow passage absorption envelope data at 1 . 2 ° K „ The r e s u l t shown f o r sample 1.2E17 was obtained from s i m i l a r slow passage data taken at 1 3 ° K . The Curie behaviour of both the BCL and hyperf ine l i n e s at such concentrations and the r e s u l t i n g invar iance of P f o r temperatures at l e a s t up B to 15°K would i n d i c a t e that t h i s P^ i s a f a i r measure of the r a t i o P at 1 „ 2 ° K . The a c t u a l P obtained from r a p i d B B passage absorption envelope data at 1 . 2 ° K was discarded due to the d i f f i c u l t i e s noted i n the i n t r o d u c t i o n to t h i s s e c t i o n . Some comments on the spins c o n t r i b u t i n g to the hyper-f i n e spectra are i n order at t h i s p o i n t . C l u s t e r s of two, three and four donor atoms have d i s c r e t e t r a n s i t i o n s at the 54 hyperf ine l i n e f r e q u e n c i e s . The hyperf ine l i n e s observed i n the more h i g h l y concentrated samples have appreciable components due to such c l u s t e r s as i s i n d i c a t e d by some of the data already presented i n t h i s chapter . The large 104 Concentration (donors/ cc) F i g . 4.14. The Relative S u s c e p t i b i l i t y of the BCL vs. Impurity Concentration. 105 increase i n the hyperf ine l i n e s p i n packet width and the dramatic decrease i n the hyperf ine spins s p i n - r e l a x a t i o n times as the concentrat ion i s increased would suggest appre-c i a b l e i n t e r a c t i o n among the c o n t r i b u t i n g s p i n s . F u r t h e r , the e f f e c t i v e g s h i f t s of the hyperf ine l i n e s a lso points to c l u s t e r e f f e c t s . ^ We cannot s t a t e , t h e r e f o r e , that t r a n -s i t i o n s at the hyperf ine l i n e frequencies are due only to i s o l a t e d s p i n s , e s p e c i a l l y f o r impuri ty concentrations 17 3 >^ 2.2 x 10 donors/cm , where the above mentioned e f f e c t s become very n o t i c e a b l e . In summary we see from Figure 4.14 that the r e l a t i v e i n t e n s i t y of the BCL as given by P i s a monotonically i n -creas ing f u n c t i o n of c o n c e n t r a t i o n . The observed tempera-ture dependence of P^ f o r samples with ^ 3.0 x 1 0 ^ donors/cm would i n d i c a t e that the spins c o n t r i b u t i n g to the BCL are b a s i c a l l y s t i l l l o c a l i z e d , at l e a s t f o r tempera-tures T <. 1 5 ° K , The behaviour of P„ with temperature f o r more h i g h l y concentrated samples shows that the BCL assumes i n c r e a s i n g l y d e l o c a l i z e d c h a r a c t e r i s t i c s as the concentrat ion i s i n c r e a s e d . F i n a l l y , the hyperf ine l i n e s observed i n samples with concentrations i n the upper h a l f of the i n -17 3 termediate range ( N ^ ^ 2.2 x 10 donors/cm ) are not n e c e s s a r i l y due only to i n d i v i d u a l l o c a l i z e d s p i n s . 106 CHAPTER 5  DISCUSSION OF RESULTS 5.1 INTRODUCTION In t h i s chapter, an attempt i s made to correlate the data presented i n Chapter 4 i n terms of some reasonable model of the BCL. In the f i r s t section, i t i s shown that the r e l a t i v e s u s c e p t i b i l i t y r e s u l t s obtained i n Section 4 . 5.2 can be s a t i s f a c t o r i l y explained i f the BCL i s due to clu s t e r s of three or more donor atoms. Subsequently, the g value r e s u l t s of Sections 4.2 . 4 and 4 . 4 . 4 are revealed to be incon-s i s t e n t with an e f f e c t i v e f i e l d model proposed by Morigaki and Maekawa.''"7 I t i s proposed that clusters of three impurities could give r i s e to the observed e f f e c t s . This proposal gains some c r e d i b i l i t y when i t i s shown that clusters of two (pairs) can give r i s e to s i m i l a r , though smaller, e f f e c t s as those we report. In the l a s t section of this chapter the linewidth and r e l a x a t i o n time data obtained are discussed. The relaxation time data obtained f o r the hyperfine spins are seen to suggest that the relevant e x t r i n s i c (concentration dependent) spin l a t t i c e r e l a x a t i o n mechanism i s also a t t r i b u t a b l e to clusters of three or more donor atoms. Qu a l i t a t i v e observations are then made concerning the observed spin packet width and l i n e -width data. 107 5.2 THE ORIGIN OF THE BROAD CENTRAL LINE There have been s e v e r a l suggestions as to the o r i g i n of the BCL, and we discuss two of these i n some d e t a i l . Some ex-perimenters suggest that the BCL may be i n t e r p r e t e d i n terms of c l u s t e r s of two donor atoms with exchange coupling J on the order 23 of the hyperf ine i n t e r a c t i o n A . Such pai rs give t r a n s i t i o n s over a broad range of f requencies , as discussed i n Sect ion 2 .4 .2 , and must account f o r at l e a s t a small par t of the observed B C L . ^ There are , however, two quite ser ious object ions to such a model. F i r s t l y , the number of such p a i r s i n samples of intermediate concen-t r a t i o n s i s f a r too smal l to account f o r the observed i n t e n s i t y of 6 5 the BCL. Secondly, at higher intermediate concentrations we may expect the d i s c r e t e c e n t r a l l i n e to become l a r g e r than the hyper-f i n e l i n e s when near ly a l l donors are found i n c l u s t e r s of at l e a s t two impurity atoms. T h i s e f f e c t was ; not observed, as i s noted i n Sect ion 2 . 4 . 3 . We therefore d i s -card the p a i r model, and proceed to the next highest order c l u s t e r c o n s i s t i n g of three i n t e r a c t i n g donors. Other experimenters have suggested that c l u s t e r s of 11 66 three may make a c o n t r i b u t i o n to the BCL. ' The main d i f f i c u l t y w i t h t h i s model i s the lack of a r e l i a b l e c a l c u l a t i o n of the ESR spectrum a r i s i n g from such a c l u s t e r . S h i m u z u ^ has attempted such a c a l c u l a t i o n ; but there are some i n c o n s i s t e n c i e s i n h i s r e s u l t s , as pointed out i n Sect ion 2 .4 .3 , and i t i s not f e l t that they can be completely t r u s t e d . 108 As an a l t e r n a t i v e approach to the problem, we attempt to i n t e r p r e t the observed r e l a t i v e s u s c e p t i b i l i t y of the BCL as a f u n c t i o n of impurity concentrat ion (Figure 4.14) i n terms of c l u s -ters of three donors . S p e c i f i c a l l y , we i d e n t i f y the BCL as a r i s i n g from c l u s t e r s containing three or more donor i m p u r i t i e s . Holcomb 67 and. Rehr present a numerical Monte Carlo c a l c u l a t i o n d e s c r i p t i v e of c l u s t e r i n g phenomena i n samples with random d i s t r i b u t i o n of im-p u r i t i e s . T h e i r s i m p l i s t i c model defines two donor i m p u r i t i e s to be members of a c l u s t e r of two i f they l i e w i t h i n a distance 2r o of each other , and to be " i s o l a t e d " i f they are f u r t h e r a p a r t . A c l u s t e r of three i s formed i f another donor i s w i t h i n 2r of e i t h e r o member of the p a i r , and so on f o r higher order c l u s t e r s . The s o l i d l i n e i n F igure 4.14 gives the f r a c t i o n of the i m p u r i t i e s that are members of c l u s t e r s of three or more i m p u r i t i e s f o r the best f i t o value 2r Q = 132 A , us ing the r e s u l t s of Holcomb and Rehr . A d i s -crepancy i s observed between t h i s curve and the r e l a t i v e s u s c e p t i -b i l i t y of the BCL (P .^) f o r the more h i g h l y concentrated samples. JD T h i s can be explained i f c l u s t e r s of three give n o n - n e g l i g i b l e cont-r i b u t i o n s at the hyperf ine l i n e f requencies , as would be expected from the d i s c u s s i o n i n Sect ion 4 . 5 . 2 . The reasonable agreement thus obtained i n Figure 4.14 between the r e l a t i v e s u s c e p t i b i l i t y of (or r e l a t i v e number of spins c o n t r i b u t i n g to) the BCL and the f r a c t i o n of spins that are i n o c l u s t e r s of three or more (where 2r Q = 132 A) provides considerable support f o r the conjecture that the major EPR c o n t r i b u t i o n to the BCL comes from c l u s t e r s of three or more donors . Furthermore, the 109 interdonor separat ion c r i t e r i o n of 132 A i n order that a donor may be considered par t of a c l u s t e r i s consis tent wi th p r e v i o u s -l y noted p a i r e f f e c t s . For such a p a i r separat ion , the exchange 26 energy J i s equal to.38 A , which i n t e r a c t i o n leads to such "smearing out" e f f e c t s as were discussed i n Sect ion 2 . 4 . 2 . 5.3 THE " g " VALUE RESULTS I n t h i s s e c t i o n , the g value r e s u l t s f o r the BCL and hyperf ine l i n e s presented i n Sections 4 .4 .4 and 4.2.4 of Chapter 4 are d i s c u s s e d . We f i r s t comment on the concentrat ion dependence of the 1 . 2 ° K gg (where gg i s the e f f e c t i v e g value of the BCL) data shown i n Figure 4 .4 . g n i s observed to be B 17 3 approximately 2.0000 f o r samples with ~ 1.0 x 10 donors/cm , and subsequently decreases w i t h increase of impurity concentra t ion . The l i m i t i n g h i g h concentrat ion value f o r g- i s the charac-JL5 t e r i s t i c d e l o c a l i z e d e l e c t r o n g value f o r s i l i c o n (as g i -ven by g g = 1.9988). As noted i n Sect ion 4 .4 .4 , t h i s concen-t r a t i o n dependence has been p r e v i o u s l y reported by Morigaki 17 and Maekawa. They i n t e r p r e t the s h i f t of gg away from the free e l e c t r o n value g g at lower impuri ty concentrations i n 35 terms of an e f f e c t i v e f i e l d approximation. The model used assumes the BCL to a r i s e from c l u s t e r s of donor atoms, and considers that a p a r t i c u l a r c l u s t e r " f e e l s " an e f f e c t i v e f i e l d due to i n t e r - c l u s t e r exchange i n t e r a c t i o n s . In Appendix C> t h i s e f f e c t i v e f i e l d approximation i s a p p l i e d to the simplest such system which c o n s i s t s of a c l u s t e r of two donors experien-c i n g an e f f e c t i v e f i e l d due to a t h i r d (more i so la ted) i m p u r i t y . LIO We f i n d that the e f f e c t i v e g value difference can be written (in the "high temperature" limit) as J 2 7 T P gB - g e = - " E q ' S ' 1 N dg e P where Jg i s the average exchange between the c l u s t e r of two donors and the t h i r d impurity, and ft p i s the magnetic suscep-t i b i l i t y of the sample. For a system obeying the Curie Law ( f t p 0 * ^ ) we obtain ge^2 gB " g e = " "kT~ E q ° 5 " 2 Similar c a l c u l a t i o n s can be made f o r the i n t e r a c t i o n s of larger c l u s t e r s . The only correction involves a f a c t o r f (that depends on the size of the clusters) that m u l t i p l i e s the r i g h t hand 17 3 5 side of Eq. 5.1 and 5.2. ' We note that i n order f o r these equations to give the observed p o s i t i v e value of Sg-ge> J must be negative or "ferromagnetic". The observed-temperature dependence of g.,,-g would seem to be r e l a t i v e l y w e l l explained n e by Eq. 5.2. There are two d i f f i c u l t i e s with t h i s i n t e r p r e t a t i o n . F i r s t , the observed decrease of g„-g as the concentration i s B e increased would require a corresponding decrease of the factor />w» 3 S rs*> fj<. As f increases f o r larger c l u s t e r s , J must decrease with concentration increase. One might expect i n t u i t i v e l y that, as impurity concentration, i s in/nr-eased, more i n t e r a c t i o n between progressively larger clusters would lead to the reverse I l l behaviour of J . Secondly, r e l a t i v e l y l o c a l i z e d spins have been 4 5 observed to e x i s t ' i n samples with very h igh impurity concen-18 3 t r a t i o n s (N d ^ 3 x 10 donors/cm ) „ A Curie-Weiss s u s c e p t i b i -v 1 l i t y ( (]°<Y~Q ' w ^ e r e 9 ^ s t n e "paramagnetic Curie temperature") can be f i t t e d to the observed s u s c e p t i b i l i t y of such " l o c a l i z e d 35 moments". As noted by Smart, the negative values of 8 ob-tained from such f i ts "* i n d i c a t e antiferromagnetic coupl ing phen-omena. A decreasing importance of i n t e r c l u s t e r i n t e r a c t i o n s as the concentrat ion i s increased may remove the ambiguity i n t -roduced by both of the above e f f e c t s , but the reason f o r such a decrease i s not obvious . We now examine the p r e v i o u s l y unreported s h i f t s of the hyperf ine l i n e g value (g^) over the intermediate concentrat ion range. As seen i n Figure 4 „ 4 , g^ i s observed to approximately equal g f o r sample concentrations as h igh as 1.2 x 10^ donors/cm , and subsequently decreases as the impurity concentra-t i o n i s f u r t h e r i n c r e a s e d . I n t e r p r e t a t i o n of t h i s s h i f t i n terms of the Morigaki-Maekawa e f f e c t i v e f i e l d model would i n d i -cate the f J i s an i n c r e a s i n g f u n c t i o n of impurity concentrat ion , and, furthermore, that J i s p o s i t i v e or ant i fer romagnet ic . Furthermore, the observed temperature independence of g^ could not be descr ibed by a dependence such as that given by E q . 5 .2 . We therefore conclude that the observed behaviour of g^ can-not be descr ibed i n terms of an e f f e c t i v e f i e l d model i n a manner that i s consis tent with the observed c h a r a c t e r i s t i c s of the BCL. 112 In view of the association that has been made between the BCL and clusters of three or more donor atoms, i t i s sug-gested that an a l t e r n a t i v e explanation of the BCL and hyper-f i n e l i n e s g value data could be obtained from a d e t a i l e d c a l -c u l a t i o n of the spectra due to clusters of three. Such a c a l -c u l a t i o n has not been attempted i n t h i s work, but i t can be shown that highly coupled clusters of two donors (pairs) do show s i m i l a r , though smaller, g s h i f t s and temperature depen-dences to those experimentally observed. We f i r s t consider the possible s h i f t s of the hyperfine l i n e s with Increase of impuri-ty concentration, assuming that an appreciable component of the l i n e i s due to such p a i r s . We r e f e r s p e c i f i c a l l y to Figure 5.1, which shows the energy l e v e l structure of a pair system calculated to second order, under the assumption that J , Y e » A 68 and J < y . Under the influence of microwave r a d i a t i o n , the e t r a n s i t i o n s that contribute at or near the hyperfine l i n e f r e -quency are given i n Table 5.1, together with the (second order) t r a n s i t i o n frequency. I t i s observed that both hyperfine l i n e s due to pairs are s h i f t e d to higher frequency (or higher f i e l d ) A 2 by an amount on the order of vp^— ^ .14 gauss (in f i e l d u n i t s ) . Further, one of the components contributing to the high f i e l d hyperfine l i n e can be s h i f t e d to yet higher f i e l d by the term A 2 • q - ^ p — j j as J approaches Y g . This p a r t i c u l a r second order ef-f e c t would be consistent with the observed e f f e c t i v e s h i f t of gj_j to lower g values as the concentration i s raised, except that i t i s too small to account f o r the observed e f f e c t which 113 e n t a i l s a s h i f t on the order of a gauss or more. Also, of course, both hyperfine l i n e s were observed to s h i f t together. Secondly, as noted by Shimuzu,^ the observed tempera-ture dependent s h i f t s to lower magnetic f i e l d of the c e n t r a l p a i r l i n e can be explained by the following mechanism. We make reference to Figure 5,1. Highly exchange coupled pairs ( J » A) have a non-magnetic s i n g l e t state (S) . At low tempera-tures, the t r i p l e t state may therefore become appreciably depopulated. The t r a n s i t i o n s that contribute to the low f i e l d side of the discrete c e n t r a l pair l i n e (JT ^, t Q > , |T ^, s > ^ | T q , t Q > » | T Q , S > ) w i l l therefore be more intense than the corresponding t r a n s i t i o n s (T *"T^) that contribute to the high f i e l d side of t h i s l i n e . This would give r i s e to an apparently larger e f f e c t i v e g g f o r the c e n t r a l p a i r l i n e . As the temperature i s r a i s e d , the spins w i l l populate the states more evenly, and g c w i l l assume the value g g . This behaviour i s again q u a l i t a t i v e l y s i m i l a r to that observed fo r the BCL. I t i s not inconceivable that clusters, of three could give r i s e to the observed behaviour of the g values of the BCL and hyperfine l i n e s . I f q u a l i t a t i v e l y s i m i l a r e f f e c t s are exhibited by c l u s t e r s of two, there would appear toheroreason why such e f f e c t s should not be exhibited by clusters of three. I t should be noted, however, that a second order c a l c u l a t i o n of the energy l e v e l structure seems indicated. /-r\TDt.,> /r-*-\hto>,\Tgs> Ji tJ "ETC 5 r.,t.,> ° \ ir-'5> ~WrJ) Second-crdcr shift of energy nuclear sublevels in the dif-ferent triplet and singlet electronic states (not to scale). The good wave functions in zero-order ap-proximation are noted beside each level. We assume an antiferromag-netic coupling lower than vt. F i g . 5.1. Energy Level Structure of a Highly Coupled Donor Pair Calculated to Second Order (Jerome and Winter). T r a n s i t i o n T r a n s i t i o n Frequency Low F i e l d Hyperfine Line I v ^ -Hv 1^ ». A , A Ye ~ 2 W e Iv^i) -Hvu) Y e " 2 " 2\yi 1 + 4 ( Y +J) e High F i e l d Hyperfine Line 0 Ki^i) -Hv*]) y + A + A * e 2 4 ( Y e - J ) A A 2 y + n + e 2 4Y e Table 5.1. Allowed Transitions f o r a Highly Coupled Donor Pair Which Contribute At or Near the Hyperfine Line Fre-quencies , and the Corresponding Second Order Trans-i t i o n Frequency. 115 5.4 RELAXATION TIME AND LINEWIDTH MEASUREMENTS We f i r s t discuss the temperature dependence of the s p i n l a t t i c e r e l a x a t i o n times of the hyperf ine l i n e spins (T^) . The H observed behaviour of T^ i s I l l u s t r a t e d i n Figure 4 .3 , which shows the i n c r e a s i n g dominance of e x t r i n s i c (concentration dependent) pro-cesses as the impurity concentrat ion i s r a i s e d . The temperature de-H — 3 /2 pendence of the e x t r i n s i c mechanism (T^ T~ f o r sample concen-17 3 t r a t i o n s N ^ ^ 2.2 x 10 donors/cm ) i s the i n t e r e s t i n g f e a t u r e . A s i m i l a r T^ temperature dependence f o r e x t r i n s i c processes has been observed f o r r e l a t i v e l y d i l u t e samples with concentrations i n the i n t e r v a l 3.9 x lO"*"6 donors/cm^ < N , < 6 x 10^ donors/cm^ We d note , t h e r e f o r e , that t h i s e x t r i n s i c mechanism appears to e x h i b i t the same temperature dependence throughout the intermediate range, which would suggest that the same e x t r i n s i c concentration dependent r e l a x a t i o n mechanism i s dominant throughout the intermediate range. We discuss t h i s observation i n terms of var ious models of hyperf ine s p i n - l a t t i c e r e l a x a t i o n . Other experimenters have suggested that the concentrat ion de-H 59 pendence of T^ i s due to " f a s t centres" or h i g h l y coupled p a i r s . The i s o l a t e d hyperf ine l i n e spins are proposed to re lax to the l a t t i c e by a process that involves s p i n d i f f u s i o n (mutual s p i n f l i p s between r e l a t i v e l y " i s o l a t e d " donor impuri t ies ) and subsequent cross r e l a x a t i o n with the " f a s t centres" that relax to the l a t t i c e more q u i c k l y than i s o l a t e d s p i n s . The Hamiltonian of such h i g h l y coupled p a i r s i s diagonal i n the exchange energy (the exchange term commutes with 116 the r e s t of the H a m i l t o n i a n ) . The r e l a x a t i o n mechanisms i n p a i r s does not , therefore , proceed through a simple lat t ice-exchange modu-l a t i o n (electron-phonon interac t ion) process , but requires a mixing of s tates of d i f f e r e n t e l e c t r o n i c spins by the hyperf ine i n t e r a c t i o n . There are some d i f f i c u l t i e s involved when t h i s model i s extended to higher concentra t ions . We would then expect (from the d i s c u s s i o n i n Sec t ion 5.1) that an appreciable number of the spins are i n " c l u s -t e r s " of three or more donor atoms. As pointed out by H a r r i s and 69 Yngvesson, i n connection with t h e i r work on I r C l c complexes, a b simple modulation of the exchange energy by the l a t t i c e v i b r a t i o n s (electron-phonon interac t ion) can d i r e c t l y induce s p i n r e l a x a t i o n i n c l u s t e r s conta ining three s p i n s . We would therefore expect that c l u s t e r s of n ^ 3 donors would re lax to the l a t t i c e at a f a s t e r rate than would a p a i r wi th s i m i l a r interdonor s e p a r a t i o n . A l s o , we might not expect that the p a i r and c l u s t e r of three e x t r i n s i c r e l a x a t i o n processes would have the same temperature dependences. The ob-H served concentrat ion dependent e x t r i n s i c T^ r e s u l t s would there-fore i n d i c a t e that the i s o l a t e d hyperf ine s p i n r e l a x a t i o n mecha-nism v i a intermediary c l u s t e r s of three or more donors i s the dominant concentrat ion dependent mechanism over the whole of the intermediate range. Previous s tudies of the hyperf ine spins s p i n - r e l a x a t i o n 16 3 times f o r samples w i t h N ^ ^ 5 x 10 donors/cm have neglected the p o s s i b i l i t y of s i g n i f i c a n t r e l a x a t i o n e f f e c t s due to c l u s t e r s of threes , as the densi ty of such c l u s t e r s i s assumed to be very s m a l l . On the bas is of the c l u s t e r c r i t e r i a discussed i n Sect ion 5.1 how-117 1.6 3 ever , even a sample with = 3 x 10 donors/cm has approximately 5% of donors belonging to c l u s t e r s of three impurity atoms. An e v a l u -a t i o n of the a b i l i t y of these c l u s t e r s to re lax more i s o l a t e d spins obviously requires a d e t a i l e d theory of the energy t r a n s f e r (cross-relaxat ion) between the f a s t and slow r e l a x i n g components of the s p i n system. Such a theory would have to be consis tent wi th the s p i n packet widths and observed l inewidths of the BCL and hyperf ine l i n e s presented i n Sections 4.2.2 and 4 . 4 . 3 . These packet and en-velope hal fwidths are discussed i n the f o l l o w i n g paragraphs. We should note, however, that no s a t i s f a c t o r y theory of cross r e l a x a t i o n phenomena i s present ly a v a i l a b l e . I n t e r p r e t a t i o n of t h i s data i s not based on any f i r m model, therefore , and only q u a l i t a t i v e i n f e r -ences are made. The measured s p i n packet widths of the hyperf ine l i n e s are observed to be temperature independent below 2 0 ° K , and to be a s t rongly i n c r e a s i n g f u n c t i o n of impurity concentrat ion, as shown i n Figure 4 .2 . The h a l f w i d t h of the hyperf ine s p i n packets A H p i s a lso 2.5 observed to be roughly p r o p o r t i o n a l to N^* . As discussed i n Sect ion 4 .2 .2 , these data are reasonably consis tent wi th a c r o s s - r e l a x a t i o n mechanism between the BCL and hyperf ine s p i n s . These spin packet widths A H p do not agree with e a r l i e r measurements of > assuming the r e l a t i o n T o = —Vrr- h o l d s . 2 & A H p •a Measurements of the BCL spin-packet widths (AHp) seem 118 to i n d i c a t e a s i m i l a r increase of A H ^ with i n c r e a s i n g concentra t ion . The A j i p ' s measured f o r samples 2.2E17 and 3.0E17, however, a l so e x h i b i t e d a temperature dependence that appears to be descr ibed by A H p <x T ° 5 E q . 5.3 A s i m i l a r BCL s p i n packet broadening with increase of temperature was i n d i r e c t l y observed f o r sample 1.2E17, as i s discussed i n Sect ion 4 .11 . In conjunction with t h i s broadening of the spin packets , a narrowing of the BCL absorption envelope (Voigt) l ineshape was a lso observed. We discuss such' an e f f e c t i n terms of Anderson's argu-42 ments concerning i n t e r a c t i n g exchange coupled s p i n s . The o f f -diagonal components of the exchange i n t e r a c t i o n JCS-^S^) may be w r i t t e n i n terms of the s p i n r a i s i n g and lowering operators as exchange^ o f f diagonal = 2 ^ S l+ S 2- + S l - S 2+^ E q . 5.4 A s p i n may therefore jump backwards and forwards between two d i f f e r -j ent s p a t i a l p o s i t i o n s at a frequency c o g = • p ^ . I f the ESR t r a n s i -t i o n frequencies at these two d i f f e r e n t points i n the c r y s t a l d i f f e r by an amount 2tico , Anderson notes that f o r to < co , (1) the i n t r i n s i c J o e o v J ESR width of the t r a n s i t i o n corresponding to e i t h e r of the two spa-t i a l l y separated Impurit ies broadens, and (2) these t r a n s i t i o n s are moved c l o s e r together i n energy ( i . e . 2h"coQis reduced) . I t would appear that i n our case the s p i n packet broadening Cfcwe) induced by r a i s i n g T and/or i s more than compensated f o r by a corresponding " p u l l i n g together" of the a l t e r n a t i v e t r a n s i t i o n energies , tending to smaller envelope h a l f w i d t h s . 119 In c l o s i n g t h i s s e c t i o n , we b r i e f l y discuss the rather tenuous model suggested, i n which Tg i s a. measure of the BCL-hyperfine l i n e c r o s s - r e l a x a t i o n time and Tg i s some s p i n - d i f f u s i o n time p e c u l i a r to the BCL s p i n s . This model appears to give s e l f - c o n s i s t e n t agreement with the observed temperature dependences of T ^ , T ^ , B B T^ and T g . F u r t h e r , i t i s consis tent with the r e l a t i v e l y good communication observed between the i s o l a t e d hyperf ine l i n e spins and the BCL s p i n s . T h i s good communication i s witnessed by the 23 o f f - l i n e s a t u r a t i o n e f f e c t s observed by Feher , the p r e - s a t u r a t i o n e f f e c t s discussed i n Sect ion 4 .2 .5 , and the comparable values of T^ and T^ noted f o r samples 2.2E17 and 3.0E17. A l s o , t h i s model does o f f e r a reasonable i n t e r p r e t a t i o n of the concentrat ion depen-H dence of T^ through the intermediate range i n terms of a c ross -r e l a x a t i o n process with f a s t e r r e l a x i n g BCL s p i n s . U n t i l the i d e n t i -f i c a t i o n s of T^ and Tg as c r o s s - r e l a x a t i o n and s p i n - d i f f u s i o n times r e s p e c t i v e l y are put on a somewhat f i rmer t h e o r e t i c a l bas is however, t h i s model must be presumed to be somewhat circumspect . 120 CHAPTER 6  SUMMARY AND CONCLUSIONS In t h i s work the EPR proper t ies of Si(P) have been e x t e n s i v e l y i n v e s t i g a t e d over the temperature range 1 0 1 ° K ^ T 16 ^ 35°K and the intermediate concentrat ion range 5 x 10 donors/cm 18 3 < N , < 1»6 x 10 donors/cm . Signals obtained under both slow passage and r a p i d passage are Interpreted i n terms of the absorp-t i o n envelope formal ism. These s i g n a l s are decomposed i n t o the three major s p e c t r a l components observed—the broad c e n t r a l l i n e (BCL), the hyperf ine l i n e s , and the d i s c r e t e c e n t r a l p a i r l i n e . Each of these components i s charac ter ized by a symmetric l ineshape that can be descr ibed by a V o i g t p r o f i l e , o r , i n other words, as a convolut ion of a Gaussian envelope f u n c t i o n with Lorentz ian s p i n packets . The r e l a t i v e s u s c e p t i b i l i t y of the BCL as a f u n c t i o n of impuri ty concentrat ion i s shown to be reasonably consis tent w i t h the proposal that the BCL a r i s e s p r i m a r i l y from c l u s t e r s of three or more donor atoms. T h i s conjecture i s a lso supported by the uniform temperature dependence of the e x t r i n s i c . h y p e r -f i n e s p i n - l a t t i c e r e l a x a t i o n rate f o r a l l samples with intermediate concentra t ions . The observed e f f e c t i v e g s h i f t s of the hyperf ine l i n e s and the BCL are shown to be i n c o n s i s t e n t wi th 17 the ferromagnetic exchange model of Morigaki and Maekawa. Some j u s t i f i c a t i o n i s presented f o r the proposal that c l u s t e r s of three i m p u r i t i e s give r i s e to the observed g s h i f t s . The strong concentrat ion dependence of the hyperf ine s p i n - l a t t i c e r e l a x a t i o n times i s i n t e r p r e t e d as due to c r o s s - r e l a x a t i o n processes 121 with the BCL spins. A l l observed e f f e c t s are consistent with the proposal that the BCL i s the precursor of the single EPR l i n e observed f o r heavily doped S i ( P ) . In conclusion, the obvious need f o r a (second order) c a l c u l a t i o n of the c h a r a c t e r i s t i c spectra due to cl u s t e r s of three donor impurities has been demonstrated. Some of the data presented has a somewhat ambiguous i n t e r p r e t a t i o n due to the presence of s p e c t r a l d i f f u s i o n e f f e c t s , f o r which there i s no adequate theory at present. In general, however, the e f f e c t s observed are s e l f -consistent and behave i n a l o g i c a l manner. 122 APPENDIX A Relations Between the Signals Observed  and the Complex Magnetic S u s c e p t i b i l i t y of the Sample The microwave cavity can be thought of as a tuned RLC c i r c u i t . ^ Nuclear magnetic resonance (NMR) studies use such RLC c i r c u i t s and the r e l a t i o n s between the signals and the complex mag-n e t i c s u s c e p t i b i l i t y of the sample are d i r e c t l y analogous to the higher frequency ESR situation., In NMR experiments the sample i s placed inside a c o i l of inductance L, which i s driven by a s i n u s o i d a l voltage Vcos (cot) = R ^ ^ ) (where ^  - Ve1"^" and the s c r i p t l e t t e r s denote complex quantities) . This d r i v i n g voltage produces a l i n e a r l y polarized o s c i l l a t i n g magnetic f i e l d 2H 1cos (cot) = R (#) (where 2H 1e l a ) t) along the axis of the c o i l , which i s assumed to l i e along the "x" a x i s . The magnetization induced i n the sample i s then given by M x = Re(fW) where fo-=fi# andft ( ft ="ftT - i f t t T ) i s the "complex magnetic s u s c e p t i b i l i t y " o f the sample. According to Lenz's Law t h i s Induced magnetization w i l l i n turn induce a voltage i n the c o i l that w i l l oppose the d r i v i n g voltage producing the induced magneti-z a t i o n . The magnetic f i e l d vector ^ corresponding to the induced magnetization i s then given by (8. = M-fr'to}'. The voltage induced i n the c o i l i s then seen to be SY= L | | = L n A ^ = 8TrLnAco(ift' +ft " ) H L e i w t Eq. A . l where 0 i s the magnetic f l u x , and n and A r e f e r to the number of turns and the c r o s s - s e c t i o n a l area of the c o i l r e s p e c t i v e l y . The 123 p h y s i c a l l y observable voltage change S V i s then given by SV = Re (.rf/) = c H L (ft 1 1 s i n cot - ft'cos cot) Eq. A . 2 where c i s a constant f o r a p a r t i c u l a r experimental s i t u a t i o n . For the case of ESR , a vol tage e n t i r e l y analogous to that given by Eq. A . 2 i s developed across a c r y s t a l detector due to r e f l e c t i o n s (or transmissions) from the c a v i t y when the resonant condit ions are s a t i s f i e d . We r e l a t e the components of the magnetization to the r e a l and imaginary parts of the magnetic s u s c e p t i b i l i t y . Obviously In order to see the connection with the components of the magneti-z a t i o n i n the r o t a t i n g frame ( ro ta t ing with angular frequency co) 72 we use the r e l a t i o n where the s u p e r s c r i p t (~) s i g n i f i e s components i n the r o t a t i n g frame. Comparison of Eq. A . 3 to Eq. A . 4 leads immediately to the r e l a t i o n s M x = M cos cot - M s i n cot x y Eq. A . 4 and ft 1 1 M 2H. 1 Eq. A . 5 1 2 4 APPENDIX B The Absorpt ion Envelope The absorption s u s c e p t i b i l i t y i s w r i t t e n as the f o l l o w i n g convolut ion of the Gaussian envelope and Lorentzian s p i n packets def ined by Equations 3 . 2 „ 7 and 3 . 2 . 8 . X / -I A Hp j v , , , m " o I H'e \ b / dHT _ _ . X " ( H ) = T ff 7 ro E q . B . L A H p A H _ 2 2 „ where A H p = and AH£ = ^ . We Let s = i + 0 H ^ T L T 2 * The integrand i n E q . B . L i s smaLL except when H — H T , and H* i s siowLy v a r y i n g over t h i s i n t e r v a l compared to the other terms i n the i n t e g r a n d . We therefore assume H 1 = H and remove i t from the i n t e -g r a L . As the integrand i s smaLL except where H* — H , we extend the Lower L i m i t of the i n t e g r a l to -oo . Changing v a r i a b l e s accor-ding to H " = H ' - H we obtain * o H / e " H " 2 d H " X , " ( H ) = __^2 § S OH_ B ^ 2 2 t T 2 A H p A H P ( H - H Q ) Rearranging terms i n E q . B . 2 and w r i t i n g a = • „ t , v = • „ , • , y = H T T G G H f + °* 2 X " ( H ) = — r 2 ° i a S / - ? e " E q . B . 3 2 T T 2 A H ^ s /-oo a s + (v-y) 125 and s u b s t i t u t i n g b = as: 7V'(H) = ft H ' 1 o T 2 T T 2 A H £ e- y dy (v-y) Eq. B.4 We evaluate the term i n the curly brackets: I = | | . Using some of the r e s u l t s of M i t c h e l l and Zemansky 73 I = 2TT 2 Re Eq. Bo5 where z = b + i v . This a form of the complex error function. We evaluate Eq. B„5 f o r the p a r t i c u l a r case where v = 0 ( i . e . H=Ho) and obtain I = 2ir 2e i 2 2 z_a s [ l - e r f (as) J Eq. B.6 S u b s t i t u t i n g from Eq. B.6 back into Eq. B.4 we have £l - e r f (as)J Eq. B.7 ft I V A H n 1 s 2 2 a s Experimentally the quantity V R = f t T T H ^ i s observed. Therefore the s i g n a l amplitude at H = H Q i s given by V R K Q H O H * AHA xe 2 2 ( l + O 1 - e r f [ a ( l - b c 2 ) 2 1} Eq. B.8 H 1 jl where the i d e n t i t i e s \ = / f t -y T^Tg and x = l / H i have been su b s t i -„ 2 ? tuted f o r s = (1 + 6 HjTjTg) 2 . Eq. B.7 i s i d e n t i c a l to that obtained 43 z by Castner, except our a = .832 AHp/AH^ whereas Castner obtained 126 a = A H / A H o Figure 3 „8 i s a p l o t of a semi-normalized v e r s i o n Jr b of E q . B„7 as given by V. R 2 2 a x xe 2~ 1 (1+x ) 2 { 1 - e r f [^n4-v 2-> z a(l+x ) 1 - e r f (a) 01 E q . B . 9 Castner assumes that A H ^ i s given by the (unsaturated) h a l f w i d t h of the observed envelope f u n c t i o n . This i s not n e c e s s a r i l y so i f AHp^> A H ^ . We consider a range of power such that the s a t u -2 2 r a t i o n parameter K H^T-^T^ « 1, and therefore s = 1 and b = a . S u b s t i t u t i n g these r e l a t i o n s i n t o E q . B.4 the absorption p r o f i l e i s given by TVT (H) = * o H 2ir 2 A H £ 1 - y dy 2 2 oo a + (v-y) E q . B.10 The term i n c u r l y brackets i s a s o c a l l e d V o i g t p r o f i l e . As noted i n the main text , the h a l f w i d t h of t h i s p r o f i l e (AH2^S) can be r e l a t e d to the h a l f w i d t h of the Gaussian envelope f u n c t i o n v i a the r e l a t i o n A H obs w A H G .832 E q . B . l l where w i s a f u n c t i o n of a , and has been tabulated by Posener 51 1 2 7 APPENDIX C Poss ible " g " S h i f t s Using an  " E f f e c t i v e F i e l d Approximation" We consider a h i g h l y coupled p a i r of impurity atoms to experience an e f f e c t i v e f i e l d due to a t h i r d , more i s o l a t e d , donor. The Hamiltonian f o r t h i s system i n an external f i e l d H q may be w r i t -74 ten i n the Heisenberg model as H = g P S z H o • Sms3z - f c s 2 - f ) - J r f f S • S 3 z E q . C . l where S_ = + S^ C^he impurity atoms 1 and 2 are h i g h l y coupled) and impurity atom 3 produces the e f f e c t i v e magnetic f i e l d " f e l t " by the p a i r . ^ e f f ^ s a measure of the coupling between the more i s o l a t e d impurity e l e c t r o n and the p a i r . We neglect hyperf ine coup-l i n g as i t i s not germane to the present d i s c u s s i o n . In the e f f e c -t i v e f i e l d approximation we write J r . r - S • S_ J „r. S S„ E q . C . 2 e f f — — 3 e f f z 3 z ^ and can therefore rewrite E q . C . l as ^ = g P [ H o " lF S 3 z ] S z + gPHS 3 z - \ ( S 2 - | ) E q . C . 3 The e f f e c t i v e f i e l d produced by the more i s o l a t e d impurity i s therefore J e f f S 3 z H = E q . C.4 rff gP We average over the e n t i r e c r y s t a l and obtain the mean such e f f e c t i v e . 128 f i e l d to be J < S 3 z > H e f f ~ " E q ' C ° 5 where J = <J r?r> . eff We use the relations M = fi^H . NgP<S > Eq. C.6 P o ° z where M is the magnetization of the sample, "ftp is the susceptibility, and N i s the concentration of spins. H r. — — — n — K — Eq. C o 7 e f f NgP 2 This can be expressed as an effective g shift via W o = SCH G+H e £ f) Eq. C.8 and we obtain *eff " g = " 7 7 2 C ' 9 NgP 129 BIBLIOGRAPHY 1. LoD. Bogomolova, V.N. L a z u k i n , and I.V. S h e p e l e v a , Usp. F i z . Nauk (83), 433 (1965) o 2. J.D. Q u i r t and J.R. Marko, P h y s . Rev. L e t t e r s 2j5, 318 (1971). 3. H. Ue and S. 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