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Effects of donor-donor interaction on the absorption spectra of shallow donors in silicon Kuwahara, Ronald Hirokazu 1971

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THE EFFECTS OF DONOR-DONOR INTERACTION ON THE ABSORPTION SPECTRA OF SHALLOW DONORS IN SILICON by RONALD HIROKAZU KUWAHARA B.Sc. (Eng.)5 Queen's University, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF rtifc KjiUUi.rx.rui-itJ.sia run in£. L > & \ i i \ i i , h i Or DOCTOR OF PHILOSOPHY In the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In present ing th is thes is in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 for reference and Study. I fu r ther agree that permission for extensive copying of th is thes is fo r s c h o l a r l y purposes may be granted by the Head of my Department or by his r e p r e s e n t a t i v e s . It is understood that copying or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT In uncompensated phosphorus or arsenic-doped s i l i c o n , observation of the shallow, donor absorption l i n e spectrum for donor con-centrations from 10^ to 2 x 1 0 ^ / c c , y i e l d s information about the bound states of the donor electrons and the behavior of the conduction band edge. With increasing concentration, the photon induced IS(A) -> 2 P q and IS(A) -y 2 P + t r a n s i t i o n l i n e s are observed to broaden i n energy, and s h i f t to s l i g h t l y lower energies; the lineshapes, Lorentzian at low donor concen-t r a t i o n s , become asymmetric with the formation of a t a i l to the low energy side. A superposed absorption background i s observed for donor concentra-16 17 tions greater than 2 x 10 /cc. Above 2 x 10 /cc an absorption edge i n the 10-to-30 meV region with an exponential energy dependence i s observed. xne change i n the absorption ixne spectrum xs due primarxxy to the f i n a l s t a t e s . The halfwidths of the t r a n s i t i o n s are well explained by a donor-pair model (or e f f e c t i v e hydrogen molecule (Macek, 1 9 7 1 ) ) , with the donor d i s t r i b u t i o n assumed random. The di f f e r e n c e i n broadening of the 2 P + l e v e l compared to the 2 P q l e v e l i s due to the anisotropic e f f e c t i v e mass of the donor e l e c t r o n . The t r a n s i t i o n lineshapes are q u a n t i t a t i v e l y explained i n terms of the convoluted Fano function (Bhatia, 1970; Fano, 1 9 6 2 ) . The Fano parameters Q and T, are interpreted i n terms of the evidence f o r the conduction band t a i l i n g and the possible configuration i n t e r a c t i o n . i i i The integrated absorption coefficient of the line spectrum and the decrease of the absorption cross section above the conduction band threshold with increasing concentration are also accounted fox in.terms of band tailing. The experimental restilts are in agreement with the infrared absorption data for antimony-doped germanium (Nisida and Horii, 1969). iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv List of Tables v i i List of Figures v i i i Acknowledgements xi Chapter 1 INTRODUCTION 1.1 General Introduction 1 1.2 The Purpose and Outline of This Thesis 3 Chapter 2 THEORY OF DONOR STATES 2 . 1 I n t r o d n r t l o n . - . , , , , , , , , » . , , , , , » . , , » . ; „ , . , . . 6 2.2 Optically Induced Transitions in Donor Impurities .. 7 2.2.1 Absorption Coefficient 7 2.2.2 Low Impurity Concentrations - The Isolated Impurity Approximation 10 A. Shallow Donor States in Silicon 10 B. The Quantum Defect Method 16 2.2.3 Intermediate Concentrations -Overlap Effects 18 A. Baltensperger's Wigner Seitz Approximation 18 B. The H2 Molecule-like Donor Pair Approximation 22 2.2.4 Higb Impurity Concentrations -Conduction Band Tailing 28 V page 2.3 Configuration Interaction between Excitation Channels < 34 Chapter 3 GENERAL EXPERIMENTAL PROCEDURES 3.1 Introduction 41 3.2 Optical Instruments 44 3.2.1 The Grating Monochromator 44 3.2.2 The Fourier Interferometer 45 3.3 Low Temperature Techniques 50 3.4 Samples 51 3.5 Calculation of the Absorption Coefficient .......... 53 Chapter 4 EXPERIMENTAL RESULTS AND INTERPRETATION 4.1 Introduction 58 4.2 Phosphorus-Doped Silicon 59 4.2.1 Low Concentrations 60 4.2.2 Intermediate Concentrations 65 A. Halfwidth of IS (A) -> 2PQ and 2P± Transitions 65 B. Peak Positions of the Transitions .... 75 C. Absorption Coefficients - Evidence for a Conduction Band Tail 78 D. Lineshapes 91 E. Disappearance of Higher Excited Levels 105 4.2.3 Higher Concentrations .....107 v i page 4.3 Arsenic-Doped Silicon 117 4.4 Comparison with Experimental Studies in Similar Materials 120 4.4.1 Ge(Sb) .120 4.4.2 Si(B) 123 Chapter 5 ' CONCLUSIONS AND SUGGESTIONS FOR FURTHER ' EXPERIMENTAL STUDY 5.1 Conclusions 125 5.2 Suggestions for Further Study 128 Appendix A The Absorption Coefficient of a Thin Absorbing Appendix B Properties of the Fano Function 137 Appendix C Instrumental Corrections to Absorption Spectra 142 Appendix D Occupation Probabilities for a Multiple Level System of Donors 152 Appendix E Evaluation of the Energy Level Splitting of the 2P States Using the Method of Baltensperger 157 Bibliography 160 v i i LIST OF TABLES Table Page 2.1 Calculated and Observed Donor Energy Levels in Silicon .... 13 2.2 2P Level Broadening for tbe Baltensperger Formalism ....... 20 3.1 Properties of the Host and Donor Elements 43 3.2 Operating Conditions for a Fourier Spectrophotometer 49 3.3 Donor Concentrations of the Silicon Samples 54 4.1 Effective Bohr Radii for the IS, 2P and 2P+ Levels 71 /; ? KoriTnat-oA B«rVf,T-oimd Absorotion Coefficient at Various Donor Concentrat ions 87 4.3 Estimated Contributions of Various Excited Levels to the Total Integrated Absorption Coefficient 89 4.4 Parameters for the Convoluted Fano Function 98 4.5 Values for the Principal Quantum Number of tbe Highest Discrete Level . 106 A.l Reflection and Transmission Amplitudes for a Parallel-Sided Sample 132 C l Spectral Slit Widths of the Monochromator 145 D.l Population Distribution among the Donor Levels at Various Temperatures .' 156 v i i i LIST OF FIGURES Figure Page 2.1 Electric Dipole Transitions between Donor Levels 8 2.2 Energy Band Structure of Silicon in the (1,0,0) Direction .. 11 2.3 The Baltensperger Broadening Estimate vs. Effective Wigner-Seitz Radius 21 2.4 Probability Distribution of Nearest Neighbor Separations ... 23 2.5 Schematic of a Two-Donor "Molecule" 24 2.6 Conduction Band Density of States in a Heavily Doped SemxcoauucLOi- Jo 2.7 The Asymmetric Fano Function for Different Negative Q Values 37 3.1 Photon-Induced Transitions in Phosphorus and Arsenic-Doped Silicon 42 3.2 Michelson Interferometer 46 3.3 Plan View of Beckman Fourier Spectrophotometer 47 4.1 Parameters Describing a Peak Lineshape 59 4.2 Comparison of a 2P+ Lineshape to a Lorentzian Profile 62 4.3 Absorption Coefficient vs. Photon Energy (31-46 meV) for Several Phosphorus Concentrations 66 ix Figure Page 4.4 Halfwidth of the 2P and 2P^  Transitions vs. o ± 1/3 Concentration 67 4.5 Energy Shifts of the 2P , 2?' and 3P_, Peak Positions vs. o' ± ± Phosphorus Concentration 76 4.6 Integrated Absorption Coefficient vs. Phosphorus Concentration for IS (A) -> 2PQ, IS (A) -> 2P+ Transitions 79 4.7 Absorption Cross Section vs. Photon Energy above the Conduction Band Edge for Several Concentrations 82 4.8 Integrated Absorption Cross Section vs. Phosphorus 4.9 Shift of Peak Position and the Asymmetry o r 1^/4^  °^ 2P and 2PJ Lines vs. Halfwidth 92 o ± 4.10 Comparison of the Convoluted Fano Function with Experimental 2PQ Lineshapes 101 4.11 Comparison of the Convoluted Fano Function with Experimental 2P+ Lineshapes 102 4.12 Comparison of the Convolution Fano Function with Other Experimental 2P+ Lineshapes « 103 4.13 Absorption Coefficient vs. Photon Energy in the Far-Infrared Region for Phosphorus-, Arsenic and Boron-doped Silicon 109 X Figure 'J^M. 4.14 Decrease in Donor Activation Energies vs. 1/3 (Phosphorus Concentration) 114 4.15 Change in the Band Gap and Ground State of Doped Silicon with Increasing Concentration 116 4.16 Halfwidth and Peak Position T S . Arsenic Concentration H8 4.17 Absorption Spectra for Antimony-doped Germanium (from Nisidaand Horii, 1969) '. .. 122 A.l Transmission and Reflection of a Ray from a Parallel Sided Absorbing Sample 132 A. 2 Optical Path Difference 133 B. l Contour of Integration for the Fano Function 138 B. 2 Relation of Fano Parameters to the Lineshape Measurements .. 140 C. l Comparison of a Spectral Slit Profile to a Gaussian Lineshape 143 C.2 Deconvolution Procedure for Absorption Lines 143 C.3 Computer Correction of Instrumental Broadening for Several Slit Settings 146 C.4 Least Squares Fit of a Cubic Polynomial 148 C. 5 Effect of Least Squares "Smoothing" on Lineshapes 149 D. l Relative Population of the IS(A), (T) and (E) Levels vs. ACKNOWLEDGMENTS I wish to thank my supervisor, Dr. J.W. Bichard, for his help, ideas, interest and his friendship during these studies and for his advice during the thesis preparation. To Dr. R. Barrie, I am grateful for his honest criticism and advice regarding the interpretation of these investigations, and also to Mr. V. Macek for allowing the presentation of part of his work prior to publication. I would like to thank Dr. J. Marko for many valuable discussions related to this work. The research described in this thesis was supported by the National Research Council in grants to Dr. J.W. Bichard. I also wish to thank the National Research Council for tbe award of an NR.C 1967 Science Scholarship. CHAPTER 1 1. INTRODUCTION 1.1 General Introduction When an atom from Group V of the periodic table replaces an atom in the crystalline structure of silicon or germanium (Group IV elements) the excess electron is loosely bound to tbe impurity site (Kobn, 1957). The energy states which this electron can occupy are called the donor states. The Group V elements - phosphorus, arsenic, and antimony -produce shallow donor levels with binding energies in the 40-55 meV region. At low temperatures, far infrared absorption due to photon induced transi-tions from the donor ground state to the donor excited states are observed. It has been found that these levels are associated with the 6-fold degenerate conduction band minima of the silicon lattice (Kohn, 1957). Previous optical studies of the shallow donors were concerned with identifying the levels and determining their energies (Kohn, 1957; Bichard and Giles, 1962), determination of the symmetry (group theoretical representations) and level degeneracy using uniaxial strain measurements (Aggarwal et. al., 1965). The difference in binding energy of each'chemical species of donor (the chemical shift) was also investigated, and explained in terms of differences in their core potentials. In the isolated donor limit the EPR signal of the donor electron was studied to investigate properties of the donor ground state wavefunctions (see for example, Feher, 1959). 2. The basic theoretical treatment in the low concentration limit is the Kohn-Ltittinger formalism (1955). Other variational treatments of the effective mass equations have been used to obtain better agreement with experiment (Tefft, Bell, and Romero, 1969; Faulkner, 1969). Other interactions have also been included: the effect of higher lying conduction band minima on the ground state (Castner, 1971); the valley-orbit splitting of the IS levels (Baldereschi, 1969). The Quantum Defect Method of atomic physics has been adapted to interpret, phenomenologically, the chemical shifts and changes in electron-phonon coupling (Bebb, 1969; Bebb and Chapman, 1969). Recent interest has developed, both experimentally and theoretically, in heavily-doped and amorphoxis semiconductors; the effect cf random spatial potential variations on the band structure is similar in both types of materials. Work has been done at low temperatures on the infrared absorption of the indirect gap (Balkanski et. al, 1969) to measure doping effects on the band structure, and on the EPR signals of the donor electron in silicon heavily doped with phosphorus to measure clianges in the localiza-tion of the ground states (see for example Maekawa, 1966; Quirt and Marko, 1971). Findings of other experimental studies in higlaly-doped n type group IV semiconductors (for example, electrical conductivity, Hall coefficient, magnetoresistance, NMR) have been summarized by Alexander and Holcomb (1968) and interpreted in terms of the non localization of donor states and the semiconductor-metal transiton proposed by Mott (1961). Conductivity data has also been compared with percolation studies of a random donor distribu-tion model (Holcomb and Rehr, 1969). Other theoretical treatments of the random impurity potential give rise to a density of states t a i l on the 3. valence and/or conduction bands. Little work has been done in the concentration region between the low and heavily doped limits. The concentration dependence of the acceptor spectra has been measured in this region in boron-doped silicon (White, 1967; Newman, 1956), and in antimony-doped germanium (Nisida and Horii, 1969). Only a qualitative explanation of the concentration dependence of these spectra have been presented. Cullis and Marko (1970) evaluated the effect of the donor-donor exchange interaction on the EPR spectra of 16 phosphorus-doped silicon of concentrations below 5 x 10 /cc. They found the interaction could be considered as due to a collection of donor pairs having random separations. A well known phenomenon in manv svstems is the presence of configuration interaction between excitation channels. Phillips (1966) interpreted the excitation spectra above and below the continuum threshold in semiconductors and insulators in terms of this interaction, using the Fano formalism. Shibatani and Toyozawa (1968) have reformulated the theory for particular application to optical absorption in semiconductors. Phillips (1964) has suggested that configuration interaction should be present wherever two excitation channels compete. 1.2 The Purpose and Outline of This Thesis The main purpose of this thesis is to investigate, in detail, the behavior of the shallow donor levels in the low, intermediate and high impurity concentration regions by observing their absorption spectra, and to compare the results with the older Baltensperger, and the recent donor-pair "*"See section 2.2.4 for references. theoretical treatments of the donor-donor interaction. Complementing this study is the investigation of possible effects due to the formation of a density of states t a i l at each conduction band minima at high concentrations. The possibility of configuration interaction between the shallow donor levels and the degenerate conduction band t a i l states is also considered in the interpretation of the data. The experiments performed f a l l into three main groups. A) The detailed structure of the IS (A) -»- 2 P q and IS (A) -»- 2 P + transition lineshapes were studied in phosphorus-doped silicon in the concentration 14 17 range 1 0 to 6 x 1 0 /cc at liquid helium temperatures. To ascertain any differences in the behavior of different chemical species, the IS(A) -> 2 P Q t''*?T^7-tLOTI ?_n ZT"cr.z-C—dcpzi c x i x c c n _.rcc 1r.7icLi5c.Lcu f._>r cuiiCcut-ioiXoiiS xu 1 6 to 6 x 1 0 /cc at the same temperature. B) Absorption coefficients for transitions from the phosphorus IS(A) ground state to continuum states above" the conduction band edge, as well as to the 3 P , 4P and 5 P levels ( 4 0 - 5 5 meV region) at low temperature were measured. C) An investigation of the absorption t a i l in the far infrared region (<30 meV) at low temperatures was made for concentrations greater than 1 0 1 7 / c c . The thesis is divided into four sections. Chapter 2 describes the theoretical analysis of previous experimental work in shallow donors in silicon and the theoretical models used to interpret the present data. Chapter 3 is a description of the experimental techniques. Chapter 4 is a 5. presentation of the results of the experiments. The interpretation of the data, also contained in this chapter, is consistent with the theoretical models and compatible with other experimental resnlts, both in silicon and in similar semiconductors. In chapter 5 the findings of this work are summarized and suggestions for further related experimental work are presented. CHAPTER 2 '•THEORY OF DONOR STATES 2.1 Introduction In t h i s chapter i s presented the t h e o r e t i c a l background and the development of models used to describe the observed behavior of phos-phorus doped-silicon. In section 2.2, the i n t e r a c t i o n of electromagnetic r a d i a t i o n with matter, applied to the system of doped semiconductors i s discussed. An i n v e s t i g a t i o n of the donor l e v e l s follows. The Kohn-LuttInger formalism f o r shallow donor states i s reviewed, as i s the semi-empirical Quantum Defect Method. In t r e a t i n g the i n t e r a c t i o n s of the donors 5 several concen-t r a t i o n regions are d i s t i n g u i s h a b l e , depending on the i n t e r a c t i o n s of primary importance. In the low concentration region (<5 x 10^^/cc) the donors are e s s e n t i a l l y i s o l a t e d from each other. In the intermediate concentration region (5 x 10''*"' to 2 x lO^^/cc) the s p a t i a l l y extended excited-state wave-functions overlap, and the photon-induced t r a n s i t i o n s are a f f e c t e d p r i m a r i l y by tbe f i n a l s t ate i n t e r a c t i o n s . Two d i f f e r e n t treatments of the donor-donor i n t e r a c t i o n s are developed, which are suited to a quantitative, comparison with experiment. In the high concentration region (>2 x 10^/cc) the overlap e f f e c t s of the ground states may be significant."'" In the h e a v i l y doped 19 region (>2 x 10 /cc) the conduction band and donor ground states are merged with each other. The f i r s t three regions are of i n t e r e s t in. t h i s study. "'"Evidence suggests (Alexander and Holcomb, 1968) that the formation of a s p a t i a l l y continuous IS band (the Mott t r a n s i t i o n ) occurs at 3 x lO-^/cc. A general survey of theoretical treatments, which show the possible effects of high donor concentrations on the conduction band, follows. The effects of configuration interaction between two excitation channels i s discussed, and the Fano formalism i s developed. The t h e o r e t i c a l developments of t h i s chapter are the bases for the interpretation of the experimental results given i n chapter 4. 2.2 Op t i c a l l y Induced Transitions i i i Donor Impurities 2 .2.1 Absorption Coefficient The theory of the in t e r a c t i o n of electromagnetic radiation with matter, w e l l known for atomic spectra (Condon and Shortley, 1964) has been applied to shallow donor spectra. In the semi-classical treatment, the atom or impurity interacts with an external radiation f i e l d described i n terms of i t s vector potential A. The vector potential per photon can be written as a superposition of normalized plane waves of frequency u) i n the form, 1/2 k 2 a cos (k*r - cot) (2.1) where a i s the p o l a r i z a t i o n vector, and i s the wavevector (in MKS u n i t s ) . The Hamiltonian operator describing a p a r t i c l e of mass m and charge e i n a rad i a t i o n f i e l d i s given by H = _1 [p - eA(?,t)] 2 + V(r) (2.2) 2m where V(r) i s the p o t e n t i a l energy of the p a r t i c l e . Using the gauge condition V*A = 0, the expression i s s i m p l i f i e d to give 8. H = H° - e p- A ( r , t ) + A 2 ( r , t ) (2.3) m m 2 where H = ^— + V ( r ) i s the unperturbed Hamiltonian. The t h i r d term of 2,3 2m i s much l e s s than the second term and s h a l l be neglected. The i n t e r a c t i o n H amiltonian, thus becomes T * - J p - A ( r , t ) . ( 2.4) m ' T r a n s i t i o n p r o b a b i l i t i e s between s t a t e s i and ,f w i t h wavefunctions iD^ and ij; , r e s p e c t i v e l y i n v o l v e m a t r i x elements of the form <i|T|f> = "| / **(p • A ) i p f d? ( 2 . 5 ) For a b s o r p t i o n of a photon of propagation v e c t o r k w i t h a simultaneous t r a n -s i t i o n of the e l e c t r o n from i n i t i a l s t a t e i to f i n a l s t a t e f , the ma t r i x element i s w r i t t e n '<±|T| f> = -e 2HTICJ <i|p«a|f> (2.6) ra 2 eiii i k . r where ( 2 . 1 ) and the standard d i p o l e approximation, e & 1, were s u b s t i t -uted i n t o ( 2 . 5 ) . photon T)CJ e x c i t a t i o n | r e l a x a t i o n - * — E ; Figure 2.1 E l e c t r i c D i p o l e T r a n s i t i o n s Between Donor Levels The formula for the absorption cross section, a, per donor involves tbe above-mentioned electric dipole transition rate, the number of fil l e d i n i t i a l states n_^ , and the number of empty final states njj. The number of filled states,n, or empty states,n*j is given by the total number of states g, times the occupational probability f, as found from Fermi-Dirac statistics. Allowing for photon emission processes, Bebb and Chapman (1969) have shown the absorption coefficient can be written a (no) = 2TT N, '{|<i|Tl f >| (n.n' - n_n?) 6 (E . . --tiu) } (2.7) —— a 1 i t i i i t where N . = donor concentration d n^,n^ = number of occupied i n i t i a l , final states per donor n_^ ,n. = number of unoccupied i n i t i a l , final states per donor E.. = E_ - E. if f i The bar over the transition probability indicates an average over a l l final states. For discrete levels, the occupation probabilities are either 1 or 0. At low temperatures n^ = n^ = 0 and n^n^ = g^g^. For excitations to non stationary states, the delta function of equation 2.7 is replaced by a normalized shape function which describes the lifetime of the excited state. If relaxation processes result in a mean lifetime T for the excita-tion, a Lorentzian lineshape obtains. At low temperatures, for transitions to non stationary states, the absorption coefficient is given by atfta) = 2TT |<i|T]f>|2 g ± g f (T. 1 } (2.8) "ft. 17 1 + ( W T ) ^ The integrated absorption coefficient, a n^t.> I s defined as a . = / a(nio) d(nto) (2.9) xnt T N allto 10. 2.2.2 Low Impurity Concentrations - The Isolated Impurity Approximation In the limit of low donor concentrations (less than 5 x 10^ ~* donors/cc), each donor is essentially isolated from the others. States are spatially localized, and the problem is treated similar to that of the hydrogen atom. A. Shallow Donor States in Silicon The basic theory of shallow donor states in silicon was developed by Kohn and Luttinger (1955) and later expanded by Kohn (1957). More recent work, based on the Kohn-Luttinger formalism, has been done to better account for the chemical species-dependent ground state energies (Castner, 1971; Bebb and Chapman, 1967) and the mass anisotropy (Faulkner, 196S). Studies have shown conclusively that there are six equivalent con-duction band minima at k^^ = {ko,0,0} in the {1,0,0} directions (see figure 2,2). The energy at one minima is given by E = E + h 2 (k - k' ) 2 + ft2 (k 2 + k 2) (2.10) ° 2^ * ° 2m" * •* i t where m. = .98 ia , ro = .19 m , m being the free electron mass. I O t C O The donor wave functions satisfy a Schroedinger equation of the form f V2 + V(r)'+ U(5?) ] iK2) =' E i K * ) . (2.11) 2m where V(r) is the periodic potential of the Si lattice and U(r) is the potential due to the isolated donor ion at the coordinate origin. In our 2 -* 2 case i t can be shown that for large r, U(r) = -e /KT and solutions to equation 2.11 have the form 2 Kohn (1957) gives a complete treatment of this problem. 11. Valence Bands Donor levels (0,0,0) {1,0,0} Figure 2.2 Energy Band Structure of Silicon in the (1,0,0) Direction „, ( i )(r) = F<i>(*) K £ ( i ) , ? ) , i = 1, (2.12) where \pflc^"""^ ,r) is the Blocb. function of the k ^ ^ minimum. The envelope functions, F^"""^(r), satisfy the effective mass equations, -t i 2 V - f i 2 ( b2 + b2 2m,, -» 2 2ni \ . 2 s , 2 & Oz t ox oy e2.- E F ( i ) ( r ) = 0, (2.13) i = 1, ... 6 where z lies along the {1,0,0} direction. The donor electron is described as an electron of anisotropic, effective mass moving in a coulomb field in a medium of dielectric constant K . 12. If m^  - m^ , or i f these masses are replaced by a suitable average m , the solution of equation (2.13) is identical to that for the hydrogen atom. The energy levels form a Rydberg series with quantum numbers n, i , m and the hydrogenic wayefunctions are well known. If the mass aniso-tropy is not neglected, a solution is found using a variational calculation. Using the fact that the constant energy surfaces at the conduction band minima are ellispoids, axial symmetry reduces the problem to two dimensions. For ! n t / r o j = «19 the parameters, a and b, for the variational trial function. F(r) e - f ^ - ^ + 4 1 J~T: L & \>L vira b (2.14) c o are a = 25.OA , b = 14.2 A . The variational envelope functions for a simDlI.e valley minima for the IS, 2S, 2P _ and 2P .., levels are written J ' m = 0 i n — ±x. *ls<*> " J— *~P (2'15) / 2 rra b F o c(r) = 1 e~ p / 2 (2-p) (2.16) 2S' F 9 p (r) = 1 e p / 2 (z. (2.17) 2 Po rrx (b } v32ira b F 2 P (?) - 1 e- p / 2 x + iy, (2.18) ^± J 2, a^ - a1 v32ira b 2 2 2 2 where p = x + y + z_ 2 I2 a b The energy levels are labelled by the familiar atomic quantum numbers n, JL, m for convenience. These are only rigorous in the 1 3 , in the hydrogenic limit. Since the solutions to equation 2.13 are invariant under rotation about the z axis and under inversion through the origin, they are labelled by the magnetic quantum number m and a parity P = ± 1. The degeneracies of the ± m states remain, but the accidental degeneracies due to 3 different JL and m are lifted. The calculated and observed energy levels for phosphorus and arsenic donors in silicon are listed in table 2.1 along with the group theoretical representations and degeneracies of each level. En ergy of Level* State Representations Degen- Eff. Mass P As n,l,m of T, d eracy Theory Experiment (meV) IS, m = 0 A 1 1 -45.31 -53.51 T "1 3 1 + i -33,6C» — 31 - 03 E 2 J -32.36 -32.42 2P, 'm = 0 A± + Xj+ E. 6 -11.3 ± .6 -11.1 -11.1 2P, m = + 1 2TX + 2T? 12 -5.9 ± .1 -6.1 -6.1 2S, m = 0 A l " 1 -8.8 ± .6 2S, m = 0 E + T± 5 -8.8 ± .6 3P, m = 0 A l + T i + E 6 -5.7 ± .6 -5.2 -5.2 3P, m = + 1 2TX + 2T2 12 -2.9 ± .05 -2.9 4P, m = 0 6 -3.6 4P, m = + 1 12 -1.9 5P, m = + 1 12 -1.2 *Measured with respect to the conduction band minima Table 2.1 Calculated and Observed Shallow Donor Energy Levels in Silicon Spin degeneracies are not important in this problem and are not explicitly included. As noted above, there exist six equivalent solutions to equation (2.13), one for each of the minima, and these must reflect the tetrahedral symmetry, T^ , of the system. The total wavefunctions are written 1 j - 1 1 i - 1, . (2.19) using a group theoretical analysis of T^ , the irreducible representation of the states are found. The representations of the six {IS} states are A, + E + T_. The coefficients, a . ^ , for the A, E and T, representations 1 1 l 1 (Kohn and Luttinger, 1955) are given by a l ( j ) = V /6 (1,1,1, 1 ,1,1) „. (j) = 1/2 (1,1,-1,-1,0 ,0) ^  a3 = 1/2 (1,1,0,0,-1, - J = 1//2 (1,-1,0,0,0 ,0) -j „ 5«> = 1//2 (0,0,1,-1,0 ,0) ^ = 1//2 (0,0,0,0,1, -1) J (2.20) In the effective mass approximation (EMA) these six states remain degenerate Selection rules for electric dipole transitons strongly forbid S + S or P -> P transitions, but true donor states are not strictly S-like or P-like. Each will have a small mixture of the other. As only S -» P transitions have been observed, this effect is neglected. The matrix elements for these radiative transitions can be calculated. Using the commutation relation, 15. 2 [x ,H j = [x,p /2in] = ift/iu p and equation 2.5 one obtains <i|p*a|f> = ra_ <tjji [x,H]iJjf> ±h = m_ (E f - E1) <F(?)|x|F^}> (2.21) The ratio of transition probabilities has been calculated (Kohn, 1957) to be (IS(A) -> 2P±)/(1S(A) -»• 2P q) = 10.6/4.0 The accuracy of the ratio depends on the validity of the wavefunctions used in the calculation. In the isolated donor limit, comparison between theory and experiment (Bichard and Giles, 1962) shows that the energies of the P-like levels are in excellent agreement with the theoretical estimates. This is to be expected as their envelope functions, F v~^(r), vanish at the donor nucleus where the EMA is least valid. The P state orbitals are also larger in spatial extent then the S orbitals. Faulkner (1969), using a more sophisticated treatment of the shallow donor problem, has affirmed the accuracy of the Kohn Luttinger theory for these states;. Contrary to the EMA, the IS states are split, and the energies of these states vary with the type of donor impurity. While the 1S(E) and 1S(T) lewels (31 to 33 meV) l i e near the effective mass estimate of 29 meV, the .IS(A) ground state lies much lower in energy (45.3 and 54.3 meV for P and As donors, respectively). From the a^-^ coefficients (2.16) i t is noted that the IS (.A) state wavef unction is non zero at r = 0; disagreement is therefore expected due to the core potential. Explanation of the chemical splitting of the IS levels has been attempted. Baldereschi (1969) explains the splitting of A^ , E and levels in terms of valley orbit interaction - the scattering of electrons between equivalent conduction band minima. Castner (1971) estimates the influence of higher lying conduction band minima. Also there exist several central cell corrections which attempt to explain the deviation of the IS(A) state in terms of the core potential. B. The Quantum Defect Method The EMA cannot properly account for the different observed binding energies, E » of the various donors-in silicon. Kohn and Luttinger proposed a correction in which the hydrogenic envelope x^avefunction of each -r/b* level has the asymptotic behaviour F(r) oCe . The effective Bohr radius * 2 * a = "h "c is replaced by b , where * 2 me E , = - e 2 / ( 2 K b * ) ( 2 . 2 2 ) obs Essentially this procedure does not change the form of the wavefunctions, only their spatial extent. Strictly speaking a or b is governed by K and m , * ft parameters of the host crystal. Therefore a and b should be independent of the chemical species of the dopant. Recently the quantum defect method (QDM) of atomic physics has been adopted to study deep impurities in semiconductors (Bebb and Chapman, 1969) and successfully applied to study phonon-assisted transitions of acceptors in silicon (Bhatia, 1970). Fundamental to the QDM is the fact that tbe impurity potential, while possibly complicated near the core itself, must become 2 coulombic (-e /KIT) for large r. In this exterior region the solutions to the impurity problem are well described by the Kohn-Luttinger wavefunctions as shown for the P states. The wavefunctions. valid outside the core, must be continuous with the core functions (which are s t i l l unspecified). Through the continuity requirement, the coulombic potentials reflect the core behaviour. The observed binding energies are parameters sensitive to the core 4 potential, and thus the QDM wavefunctions are solutions of [ * ! ? _ - e l - E o b s ] F v ( r ) - 0 (2.23) 2m For IS states where the core correction is required, the envelope functions become F (r) = P (r) Y°(e,&), with P (r) = N r v _ 1 e - r / v a . (2.24) V M O V N is a normalization factor and v is the effective principal quantum number} determined from the observed binding energy: \ E , = -R*/v2 v = n - 6 (2.25) obs * Here, R is the effective Rydberg, 6 is the quantum defect, and n is an integer. The wavefunctions are scaled from the binding energy in terms of v * * rather than a or m . The hydrogenic approximation is a limiting case of the QDM. The advantage of this treatment over more sophisticated ones is that no detailed information of the core potential is required to estimate the effect of the core in the exterior region. The QDM has been useful in evaluating transition probabilities and electron-phonon interaction. ^The solutions, divergent at r = 0, are valid for r > 0. 18. 2.2.3 Intermediate Concentrations - Overlap Effects Modifications to the isolated donor model are required when impurity concentrations are increased since the donor electrons now must interact with each other, perturbing the levels. This "overlap" of wave-functions is treated theoretically in two ways: an early calculation by Baltensperger ( 1 9 5 4 ) using a Wigner Seitz approximation, and a recent calcu-lation by Macek (to be published) using a donor-pair approximation. A. The Baltensperger Wigner Seitz Approximation In this model, the donor electron is assumed to be character-ized by an effective mass m and the medium by tbe dielectric constant K ; the single donor wavefunctions are described by the simple hydrogenic model mentioned in section (2.2.3.A). An important approximation is now made: the donors are arranged in a regular close-packed "lattice", the unit cells of which are found by the Wigner-Seitz method. These cells are then approximated by spheres of equal volume, whose radius r g is related to tbe donor concentra-tion, by . 4 i r r 3 = 1/N, (2.26) 3 S d The interdonor separation R = 2r g. Using the EMA approximation, the effec-tive mass equation (2.13) simplifies to ^ _ V 2F(f) + (ef + E)F(r) = 0 (2.27) 2m* where E is an energy eigenvalue. This equation has the familiar solution F(r) = R n > S L(r) Y l Q ( e , * ) ' (2.28) 19. where R , and Y„ are the radial and angular parts of the wavefunction. Due n,l Am to the surrounding impurities, the wavefunctions are considered finite in extent and new boundary conditions are found from the requirement that the wavefunctions satisfy Bloch's equation for the donor "lattice". These wave-functions have the form F(r) = e 2 T r i l* r u(r) , u(f + %) = u(r) , (2.29) where c~ is the translation vector of the superlattice. Considering the center, (k = 0), and the boundary, (2k-^  = 1), of the reduced Brillouin zone, i t can be shown (Baltensperger,1953) that i remains a good quantum number in the spherical approximation, and that the desired boundary conditions are d_!(r) = 0 lT= 0, SL even , 2k£ = 1, fl. odd (2.30) F(r ) = 0 2k-£ = 1, St even , k = 0, i odd (2.31) The eigenvalues of (2.27) are given by E = -eAm* . _1 (2.32) -.2 2 2 2n K n Due to the requirements (2.30) and (2.31) the quantum number, n, is no longer an integer; n = Y, - e where K = 1,2 ... and e denotes the deviation of n from integer values and is found by the solution of (2.30) and (2.31). The values of e substituted into equation (2.32) determine the lower and upper edges of the band for a given value, of r . The solution, first carried out by Baltensperger, is given in detail in appendix E. The results for the 2P band, determined by a computer calculation, are listed in table 2.2 and plotted in 20. Wigner Seitz Radius Upper Limit Lower Limit Bandwidth Shift of Mean Value r /a meV meV meV meV s 5 .44 -2.50 7.94 1.47 4 3.47 -4.01 7.48 -.27 5 2.29 -4.63 6.92 -1.17 6 1.53 -3.99 5.51 -1.23 7 1.00 -2.81 3.81 -.90 8 .645 -1.77 2.41 -.56 9 . .401 -1.04 1.44 -.32 10 .240 -.577 .817 -.17 11 .138 -.304 .442 -.083 - 07^ 5 - T1 S4 .-. 2 30 —. 003-5 i i 13 .0408 -.0755 .116 -.017 14 .0211 -.0363 .0574 -.0076 15 .0106 -.0171 .0278 -.0033 16 .0052 -.0080 .0132 -.0014 17 .0025 -.0037 .0062 -.0006 18 .00118 -.00168 .0029 19 .0005 -.00076 .0013 20 .00025 -.00034 .0006 30 6.4 x 10~8 -1.7 x 10~7 Table 2.2 2P Broadening for the Baltensperger' s Formalism. The upper and lowe energy of the mid equivalent Wigner-r band edges, the point of the band Seitz radius. bandwidth, and are listed as the shift in functions of the 21. no i a h 6 E 0 ' *-> > 1 0 5 BROADENING OF THE 2P LEVEL (BALTENSPERGER) BANDWIDTH \ \ \UPPER  \ I IMIT \ \ \ SHIFT \ A |__|_ \ \ LOWER LIMIT -4 0 r s / a J L 1 ! 1_JL 10 12 14 Figure 2.3 The Baltensperger Broadening Estimate of the Upper and Lower Band Edges, Bandwidth, and Shift: as" a Function of Wigner Seitz Radius r /a". 22. figure 2.3. The upper and lower band edges, as well as the total bandwidth and energy shift of the mean value of the band, are listed as a function of the Wigner-Seitz cell equivalent radius. For the 2P levels, broadening is significant at a Wigner Seitz radius (r ) of 12a . Similar calculations for the IS level behaviour show no shift in energy of the IS level and broaden-ing becomes significant for r g = 5a . The Hydrogen Molecule-like Donor Pair Approximation Perhaps a more reasonable treatment of substitutional donors, diffused into a silicon lattice, is to assume their spatial distribution is of a random nature. The donor concentrations of interest are sufficiently 4 dilute (less than one donor per 10 silicon) that Poisson statistics are valid. The problem of evaluating the nearest neighbor distribution was first formulated by Chrandrasekhar (1943). Letting w(r)dr denote the probability that the nearest neighbor donor is situated at a distance between r and r + dr from the donor in question, then w(r) = exp(-_4iTr3Nd) 4irr2Nd (2.33) where is the donor concentration. This distribution is shown, for several 15 17 concentrations, in figure 2.4. In the concentration range 10 to 10 /cc, the nearest-neighbor donor is assumed to have the largest influence on the donor at the origin. In other words, the crystal is treated as an assembly of N^ /2 two-donor molecules."' Photon absorption occurs with the excitation of one of the ground (ISIS) state electrons to an excited level, (1S2P , 1S2P+ ^ etc.). The transition exhibits a distribution of energies due to the random pair separation giving rise to a broadened absorption line spectrum. "'The donor pair model was used successfully to explain exchange energies in the EPR spectra of phosphorus-doped silicon up to 5 x lO-^/cc. (Cullis and Marko, 1970). •24. E Schematic diagram of the energy variation Figure 2.5 Two-donor Molecule The two-donor molecule is shown in figure 2.5 where A and B are the donor ions and 1 and 2 are the two localized electrons. The Hamiltonian of the four particle system, considering only the coulomb inter-actions, is written H(l,2) = H(l) + H(2) + V(l,2) (2.34) where H(l) and H(2) are the isolated donor Hamiltonians; V(l,2) represents the interaction between the two donors; ie., V(l,2) = (2.35) It is assumed that the interdonor separation is much larger than the effective Bohr r a d i u s so that the two e l e c t r o n s are co n s t r a i n e d to move about t h e i r own donor cores. The Heltler-London method i s used to c o n s t r u c t molecular wavefunctions from the s i n g l e donor e l e c t r o n wavefunctions. One obtains the energy eigenvalues of the system by m i n i m i z i n g the energy of the system w i t h respect to a l i n e a r combination of s u i t a b l e t r i a l b a s i s f u n c t i o n s . Using the Kohn-Luttinger wavefunctions, the ground s t a t e IS(A) s i n g l e e l e c t r o n v a v e f u n c t i o n i s given by V s = V/S E F. (5 )(r*)*(lt ( j ),r) (2.36) i = 1 J S as d e f i n e d i n s e c t i o n 2.2.2.A. (The higher l y i n g 1S(E) and 1S(T) s t a t e s are not considered as they are not occupied at low temperatures.) S i m i l a r l y , the 2 P / m _ n S s t a t e s are described by ; (.)2P°(?) = Z a < j ) (r)TJ/(k ( j ), r) (2.37) and the 2P, , , N s t a t e s by (m = ± 1) ^ ( i ) ( ? ) = 1 ^ 4 1 1 ^ ^ ^ ^ <2-38) = E B i j ) F f f l 2 P y ^ ^ k ( J ) ' f ) ( 2 - 3 9 ) Here, the a f ^ and 8.^ d e s c r i b e the c o n t r i b u t i o n s from the j * " * 1 v a l l e y to the x x i 1 " * 1 s t a t e and the F ( r ) ' s are given by expressions (2.15) to (2.18). The Heitler-London t r i a l f u n c t i o n s are i c i c + U/ , X O J - 1 OA 1 Cfl Y (1,2) = ( 1 ± P 1 2 ) ( 1 ± D <I>(1) / b (2) (2.40) 26. U j l S J 2 F > 1 c A O P R . (1,2) = (1 ± P l 2 ) ( l ± I) ij; (1) ^ (2) (2.41) where 2P = 2P , , , , or 2P , and P.. „ and I denote the space co-ordinate (x ± iy) o' 12 1 permutations and the inversion operators, respectively. Using a variational calculation with (2.40) and (2.41) and the Hamiltonian (2.34), Macek has shown the (lS2Po) and<(1S2P+ 1) states form two bands for |R| > 50 A ° . The upper and lower bounds are determined by the totally symmetric states with a =? 3^  - ^~I(1,1,1,1,1,1). The other states l i e in between. The Bloch functions i K k ^ ,r) give rise to interference coefficients that vary from -1 to +1 depending on the angular coordinates of the interdonor separation R. These coordinates depend on the relative lattice sites of the two impurities. Upon averaging over a l l angular coordinates, a band of levels results.^ These upper and lower limits are calculated for |R| >50 A° using the symmetric Heitler-London wavefunctions. In the concentration region of interest (10^/cc < < 10^^/cc) the average interdonor separa-tions vary from 150 to 600 A ° . Thus, the following approximation is made: | r 1 A h | r 2 B | « |R| (2.42) V(l,2) is expanded in a Taylor series and terms of O^/R"*) and smaller are neglected. In view of approximation (2.42), only the direct matrix elements are calculated. (The exchange energy matrix elements are found to The effect of discrete lattice sites, as opposed toa continuous range of sites, on the pair distribution has been treated (Cullis and Marko, 1970). This effect disappears with the angular averaging. 27, be much smaller than these).7 The equation H(l,2) ¥(1,2) E Kl,2) (2.43) is solved using the expanded V(l,2) terms in 11(1,2). After averaging over - i . R/R, i.e. the angular variables, the upper and lower energy bounds for the "molecular" levels are found to be E 1 S 1 S ( R ) = -e2, + (HV5) (2.44) K R E l S 2 P o ( R ) = -e^ ± e 2 c- 2b 2 + O ^ / R 5 ) ; (2.45) KR K D3 K lS2P + /_. . 2 . 2 c2 2 . _ "* ^ V f H-^O(V) , (2.46) R where the E(R)'s are measured with respect to their energies in the isolated donor limit, and the parameters a and b are the variational parameters of the anisotropic Bohr radius (see equation 2.14) and _ _ _1S, p_2P _,1SI i„2P „1S, ,^ 2P ,„ . _ N 5 = <F |z|F o> = <F Ix|F x> = <F |y|F y> (2.47) The bandwidths of the 2P and 2Pin levels given by 2.45 and 2.46 are o ±1 . quadraticaily proportional to a and b respectively. The transition energy bandwidths, at interdonor separation R, are given by A (R) = |E^ S 2 P°(R) - E 1 S 1 S ( R ) | = e 2 g 2b 2 ± O^/R 5) (2.48) K R 3 A±1(R)'= |E^ S 2 P±1(R) - E 1 S 1 S ( R ) | = e 2 J^ a*_ ± O ^ / R 5 ) (2.49) K R3 Cullis (1970) has summarized the detailed calculations of the I S - I S exchange integral and has calculated the exchange energy as a function of the donor pair separation. These results justify the above assumption. * ' 28. For a given donor concentration, Macek estimated the bandwidths of the IS ->. 2P q and IS -»- 2P+^ transitions by summing the bandwidths A.(R) over the pair separation distribution given by (2.33) and averaging appropriately.. In Section 4.2.2 the numerical results of this corn-outer calculation are compared with the experimentally observed, concentration-dependent broad-ening of these transitions. 2.2.4 Higher Impurity Concentrations As the impurity concentration is increased, the behavior of the impurity-crystal system becomes more complicated. The following section is a discussion of the theoretical work describing qualitatively the possible behavior of highly doped semiconductors. Apart from possible chemical effects (such as the formation of aggregates or the precipitation of impurities), two effects are important: the effect of impurity concen-tration on the impurity levels, and the effect of impurity concentration on the band structure of the host lattice. As noted in the previous section, the discrete impurity levels broaden in energy with increasing concentration. This is the result of a spatial "overlap" of the impurity wavefunctions. At sufficiently high concentrations, these impurity levels become non localized. The activation energy of these impurity electrons to conducting states is expected to decrease due to the screening effect of nearby electrons in the nonlocalized (or impurity) band.(Ando and Uemura, 1971). Holcomb and Rehr, (1969) in their percolation calculation for a random impurity distribution assume that the impurity electrons are completely localized i f no impurity is within 2r^ of that impurity, but can freely move to the other site i f the separation is less than this distance (2r^ = 4.4 d.* for agreement with con-ductivity data in Si(P)). Mott (1961) suggests with the formation of the impurity band and the consequent electron screening, the semiconductor should change to a m e t a l l i c or conducting state, and he estimates the onset * 3 18 of t h i s behavior to occur at ^ (25/a ) ( N ^ 2 x 10 /cc) . In the h e a v i l y doped l i m i t the impurity ground state and the host l a t t i c e band become degenerate and the Fermi l e v e l of the electron system passes into the conduction band ( F i s t u l , 1968). Matsubara and Toyozawa (1961) using the hydrogenic impurity model have estimated t h i s t r a n s i t i o n to occur at *3 19 N,o.l/(4Tra ) ( N , ^ l x 10 / c c ) . a a I t i s expected that the presence of a high impurity concen-t r a t i o n can produce changes i n the conduction or valence bands. In recent years, heavily-doped and amorphous semiconducting materials have become objects of experimental study, and the behavior of t h e i r band structure has become a popular t h e o r e t i c a l subject. Tbe problem of c a l c u l a t i n g tbe structure at the band edges for the case of heavily-doped c r y s t a l s i s s i m i l a r to that f o r amorphous semiconductors, doped or undoped. Amorphous systems lack long range s t r u c t u r a l order. Consequently, the s p a t i a l period-i c i t y of l a t t i c e s i t e s i s absent. For heavily-doped c r y s t a l l i n e semiconduc-tors, c e l l u l a r disorder p r e v a i l s ; i n an otherwise p e r i o d i c l a t t i c e , there i s a lack of p e r i o d i c i t y i n the impurity p o t e n t i a l s from unit c e l l to unit c e l l . The l a t t e r case i s more simple to analyze and i s a p p l i c a b l e to phosphorus-doped s i l i c o n . In the t h e o r e t i c a l treatments of band properties, there are several commom assumptions regarding the properties of the disordered systems. At high impurity concentrations, the random s p a t i a l d i s t r i b u t i o n of i m p u r i t i e s , which gives r i s e to a s p a t i a l l y f l u c t u a t i n g and slowly vary-ing impurity p o t e n t i a l , i s treated as a perturbation to the p e r i o d i c poten-t i a l of the i n t r i n s i c l a t t i c e . The perturbation i s usually assumed to be 30. some type of screened coulomb potential. The calculations regard the screen-ing radius to be less than the most probable inter-impurity separation. The fluctuations are found to cause a "smearing" of the band edges, forming a so-called density of states " t a i l " . The quantitative description of the " t a i l " is uncertain. The mathematical treatments are as varied as the number of authors. A common result is a density of states which has the form of an Urbach t a i l (first observed in the inter-barid transitions of the alkali halides); the absorption coefficient has an exponential energy dependence of the form ct^ exp (y (E-E ) /kT), where k is the Boltzmann constant and y is an adjustable parameter. (The temperature dependence is not valid at low tem-peratures). A Gaussian density of states ta i l with an absorption coefficient 2 of the form a^exp(E-EQ) also has been investigated. The states in these "tails" are formed by the fluctuating potential produced by the impurities. Thus, such "tailing" can appear in systems of identical atoms if potential fluctuations are present. Various theoretical treatments of the band tailing problem are outlined below. g Early investigations considered a random chain of impurities in one dimension; the interaction, through screened impurity potentials, was treated by perturbation methods. The solutions revealed that the presence of disorder tended to smear out the transition region between allowed and forbidden bands. Takano (1962) used a Wannier function representation in the investigation of the energy spectrum of a lattice containing a large number of impurities, whose potentials were assumed to be highly localized. Parti-cular attention was paid to the transition region separating the allowed and forbidden bands. The deviations from the average lattice potential, due to g See Takano (1962) for a complete l i s t of these investigations. 31. the assumed random impurity d i s t r i b u t i o n , were handled by a Green's function method. The density of states at the edge of the. main continuum, as w e l l as i n the impurity band., was c a l c u l a t e d . I t was. found that, i f the i n t e r a c t i o n between states of. d i f f e r e n t energy bands was considered, a repulsive force between, neighboring bands was present; t h i s i s s i m i l a r to the non-crossing ru.le of atomic spectra. For small perturbations a. density of states t a i l from the continuum enters the forbidden region. The band t a i l i n g produced by the f l u c t u a t i n g impurity p o t e n t i a l r e s u l t s from a s h i f t and broadening of the density of states of the i n t r i n s i c l a t t i c e . I t was found q u a l i t a t i v e l y that a s i n g l e impurity band should broaden towards the low energy side and become asymmetric with increased doping. More recent development of the theory has been summarized by Halperin and Lax J (1966). Their theory, an extension of t h i s development, assumes that the c r y s t a l l i n e p o t e n t i a l i s modified by the f l u c t u a t i n g impuri screened coulomb p o t e n t i a l producing bound states. The d i s t r i b u t i o n of thes lowest l e v e l s , Gaussian i n the high concentration l i m i t , gives r i s e to a density of states t a i l on the low energy side of the band. In a one dimen-s i o n a l model t h i s density of states i n the Gaussian approximation has the f o r m 1 0 p(E)^|E|exp(-4/3|2E| 3 /' 2). Stern (1971) has used the Halperin and Lax model to i n t e r p r e t o p t i c a l interband absorption r e s u l t s i n t h i n f i l m s of amorphous s i l i c o n . The many assumptions made were j u s t i f i e d only by q u a l i t a t i v e reasoning, the. knowledge of the behavior at energies higher or 9 Halperin and Lax also give reference to relevant experimental evidence i n tunneling, o p t i c a l absorption, and luminescence i n semiconductors. 1 0 E n e r g i e s i n these " t a i l s " are measured with respect to the i n t r i n s i c conduction band minimum. lower i n the bands, and i n some cases, s o l e l y to permit numerical compai:-isons to be made. Shklovski and Efros (1970) treated the screened p o t e n t i a l f l u c t u a t i o n s i n the. form of homogeneous spheres whose charge and radius vary. In the heavily-doped l i m i t the resultant density of states t a i l has the form p^exp(-E ' ~) l n ( E ) . Poor agreement was found with the data on interband t r a n s i t i o n s i n GaAs (Pankove, 1965). T s i t s i s h v i l i (1970) ca l c u -l a t e d a t h e o r e t i c a l density of states t a i l on the valence and conduction bands of heavily-doped semiconductors but found no t a i l on the interband absorption c o e f f i c i e n t . Bonch-Bruevich (1962) has treated the three dimensional case of a heavily-doped semiconductor- The i n t e r a c t i o n between impurity electrons by screened or unscreened coulomb p o t e n t i a l s was treated using a Green's function method. The calculated density of states d i f f e r s every-where from zero i n the forbidden region; at the edge of the i n t r i n s i c conduction band, the density of states changes s u b s t a n t i a l l y , decreasing f i r s t l i n e a r l y i n energy for E < 0, and then approaching an exponential energy dependence. The exact form of the t a i l i s uncertain because the c a l c u l a t i o n i s s e n s i t i v e to the type of screening p o t e n t i a l . A more recent paper on disordered semiconductors has been published (Bonch-Breuvich, 1970). The interband absorption c o e f f i c i e n t was c a l c u l a t e d for two types of random force f i e l d s : a screened p o t e n t i a l having a continuous, d i f f e r e n t i -able c o r r e l a t i o n function, and a coulorabic p o t e n t i a l . Phonon i n t e r a c t i o n s were not considered. The EMA formalism was assumed v a l i d , 'and a Green's fun c t i o n method was used. At low temperatures, where the mean p o t e n t i a l 33. fluctuation is greater than kT, and for the nearly intrinsic case where the Fermi level s t i l l lies below the intrinsic conduction band edge, an E — absorption t a i l was calculated having the form ofi-exp ( o - -frio), where w is -w" temperature independent and may be a function of concentration. This op-tical t a i l is. correlated with the density of states t a i l but does not reproduce i t . p(E) ?0 / / 0 E Figure 2.6 Conduction Band Density of States in a Pure ( p Q ) and Heavily Doped (p) Semiconductor. The zero level represents the bottom of the conduction band of a pure semiconductor (from Fistul, 1968). Pankove (1965) has suggested another possible mechanism for producing tails on the continuum bands. Localized strains due to the impurity cores or dislocations cause localized areas of compression or tension. The deformation potential decreases or increases the energy gap locally. While ample theoretical support for conduction band tailing exists in the heavily-doped limit, the experimental verification is meager. Little is reported in the way of possible effects at lower concentrations. Experimental data appears necessary before a quantitative analysis can be attempted. The interpretation of the optical absorption results at high donor concentrations (section 4.2.«3 ) elaborates on the concepts of this section. 2 .3 Configuration Interaction between Excitation Channels When the electronic states of two different configurations are degenerate in energy, the actual states may have to be represented by a superposition of the states of the two configurations. The configuration interaction, m m to terms of the Hamiltonian neglected in an independent channel approximation, can be significant. For the case of discrete levels degenerate with a continuum of levels, the lineshape for electronic transi-tions to these states takes on a characteristic form. Configuration interaction of this type has been observed in a large variety of systems: for example, the exciton spectra in semicon-ductors and insulators (Phillips, 1966); ultraviolet spectra of rare gas solids (Jain, 1965); the internal acceptor levels of silicon (Bhatia, 1970). The interaction effects are particularly conspicuous for energy levels above the lowest ionization threshold, but are not necessarily restricted to this 11 region. "^'"Jain (1965) points out that below the threshold at least a small amount of scattering is required to transfer photon energy to the lattice; such scattering would produce a weak background, and configuration interaction would result. Phillips suggests that the phonon fie 3d may do the same. For s i m p l i c i t y the development of the formalism for c o n f i -guration i n t e r a c t i o n i s l i m i t e d to a system of a single d i s c r e t e e l e c t r o n i c l e v e l of wavef unction <j> l y i n g within a continuum of energy l e v e l s of wave-function ty„. In the zeroth order approximation (no configuration i n t e r -im action) the wavef unctions cj> and if) are independent of each other; thus «j>|<f» = i , <-J>E|>ly> = <5(E-E') <*|*e> = 0 (2.50) The i s o l a t e d channel solutions are assumed known and the energy eigenvalues are obtained from «f>|Ho|<i» =E<J) <^E.|Ho|^E> = E'5(E-E') (2.51) The Hamiltonian of the two channel system i s given by H q + H^, where H^ represents the i n t e r a c t i o n between the channels, and <^\n±\^> = V E' (2.52) The t o t a l f i n a l state wavefunction describing t h i s system, $_, can be written as • $ E = acb + / dE-b E^ E" (2.53) where a and b ' may be functions of energy. The Schroedinger equation for E the t o t a l system can be written (H + H ) § = E$ (2.54) o 1 E E The procedure of Fano (1962) i s used to evaluate t h i s problem. The c o e f f i c i e n t s a and b " are determined below. Multi-E * r pl y i n g 2.54 by <p , and s u b s t i t u t i n g 2.53 for f , 36. a<j> H <j> + /dE'b ' ty H ty' + d/lLa* + ty H. /dE'b ' i ^ ' " = Ea<f>*<|> + tf^E/dE'b^iJ; (2.55) Ji li Integrating 2.55 over a l l space, and substituting in the expressions 2.50, 2.51 and 2.52, one obtains a<ty\n \ty> + JdE'b , <cp |H \ty > = aE«j> | <j» (2.56) O Hi X ill or, aEA + /dE-X- V* = aE ty tj Jl Similarly, by multiplying 2.54 by and repeating the above sequence it is found that, b^E' + a V^ = b„.E (2.57) The solution of 2.56 and 2.57, plus the normalization requirement for i„(<$J$' >' = 1) yields for a and b„ (Fano, 1962), Ji L hi & a = sin(A) ; b , = VE' sin(A) - cos(A) 6(E-E') (2.58) E E E-E' where the phase shift A is given by A = - arctan ^ E ^ -, (2.59) E-E -F_ ty (E) and the principal value integral, F(E), resulting from the degeneracy of ty and ty , is F(E) = J r 9 dE'|V..J ( 2 - 6 0 > 1 E 1 E-E' Figure 2.7 Plot of the Asymmetric Fano Function Versus Dimensionless energy f o r . D i f f e r e n t Negative Values of Q, from Fano (1961). 38. Thus, the t o t a l f u n c t i o n ¥ can be w r i t t e n as a s u i t a b l e combination of E s i n g l e channel s t a t e s $ = 1 vE TfV, E-E' sinA — j|> cosA = 1 sinA cosA E (2.61) 7fV, E where £ can be i n t e r p r e t e d as a new d i s c r e t e s t a t e wavefunction c o n t a i n i n g an admixture of continuum s t a t e s . To c a l c u l a t e photon-imduced t r a n s i t i o n s of t h i s system, the t r a n s i t i o n p r o b a b i l i t y between the d i s c r e t e ground s t a t e |i> and the f i n a l ""*—-» " r ' i ~ E > • -••--•—•-•-•> - ' - ' of the form |<$ |T|1>| , where T i s the e l e c t r i c d i p o l e moment t r a n s i t i o n operator. S u b s t i t u t i o n of expression (2.61) f o r 5 i n t o t h i s m a t r i x element y i e l d s <& |T| ±> = 1 <$|T|i> sinA - <ip' ,|T.|i> cosA (2.62) Se v e r a l new parameters are in t r o d u c e d f o r convenience. Let e =-cotA = E-E -F(E) (2.63) r = 2-rr | V | , and Q = <$1T|J> irV E * < i | » E | f|i> (2.64) 3 9 . The ratio of transition rates can now be. written, with the new parameters. as |<£ |f|i>|2 2 1 E1 1 1 = (Q+e) |<*jT|i>|2 l + £ 2 1 + Q 2-l + 2Qs (2.65) 2 1+e 1+e The terms of expression 2.65 represent, the background absorption due to the continuum of energy states, the Lorentzian lineshape associated with the discrete transition and an interference term due to the configuration interaction, respectively. The lineshape is referred to as a Fano line-shape and is illustrated in figure 2.7 for several values of Q . It is important to note the asymmetric lineshape with a low energy t a i l for negative values of Q . r , a measure of the strength of the interaction (V ) , is also a measure of the lifetime of the discrete state before decay via the continuum channel. As V , the strength of the configuration interaction, approaches infinity (Q 0), an antiresonance dominates; as V becomes weak E (Q ->• »), a Lorentzian profile obtains. The integrated cross section of the discrete lineshape, a. , divided by the background cross section at the r int energy corresponding to the peak position, a , can fee evaluated i f the parameters Q , T and F are assumed slowly varying in energy; i.e., r{|<$"jT|i>|2 - |4_|T|i>f} dE = rE<^jT|i>l 2{Q 2-l+2Qe}d^ - C W U i l X-J ry 1+e = |<^p|T|i>|2 TT(Q 2-1)F 2 o r o. / a = T r ( Q 2 - l ) r (2.66) int o • ^ 4 0 , The other properties of the Fano function are given i n appendix B. It i s noted that configuration i n t e r a c t i o n can be expected to occur where two channel e x c i t a t i o n s are p o s s i b l e . The i n t e r p r e t a t i o n of experimental data i n terms of two channel e x c i t a t i o n s i s presented i n se c t i o n 4.2.2.E. 41. CHAPTER 3  GENERAL EXPERIMENTAL PROCEDURES 3.1 Introduction The motivation f o r the experimental techniques used i n the i n v e s t i g a t i o n i s explained i n t h i s s e c t i o n . Subsequent sections describe the apparatus and the procedures used. The i n v e s t i g a t i o n of concentration e f f e c t s i n shallow donors was c a r r i e d out p r i m a r i l y on phosphorus-doped s i l i c o n f o r several reasons. S i l i c o n i s an extensively studied material whose shallow donor impurities are adequately described i n the low concentration l i m i t by the Kohn-Luttinger formalism. S i l i c o n i s e a s i l y obtained commercially i n large, high p u r i t y s i n g l e c r y s t a l s . Phosphorus was chosen as the most s u i t a b l e donor. As shown i n table 3.1, phosphorus most nearly exhibits the e l e c t r o n i c structure of s i l i c o n , and therefore, spurious e f f e c t s due to the core electrons ought to be minimized. Experiments were also performed on ars e n i c -doped s i l i c o n ; changes i n the electron-phonon coupling due to core electron e f f e c t s are p o s s i b l e . For phosphorus and arsenic, e f f e c t s due to the d i f f e r -ent t e t r a h e d r a l r a d i i of the host and donor impurity are expected to be small. The energy regions of i n t e r e s t i s determined by the IS(A) to P l e v e l t r a n s i t i o n s of the phosphorus and arsenic donors. The t r a n s i t i o n s are l a b e l l e d i n f i g u r e 3.1. > 4 5 . 3 4 3 . 5 7: 4 2 . 5 3 D CONDUCTION 5p±, 3p± 2p± 39.17 I T 2pc 34.13 15(T) BAND I T 4 2 . 4 Figure 3.1 Observed Photon-Induced Transitions in Phosphorus and Arsenic-Doped Silicon. The energies (in meV) are valid at low donor concentrations. Silicon. Phosphorus Arsenic Atomic Number 14 15 33 Atomic Weight 28.09 30.95 74.92 E l e c t r o n i c Structure 2-8-4 2-8-5 2-8-18-5 Tetrahedral Radius'1" 1.17 A 0 1.10 A° 1.18 A° Ionic Radius (A°) .41 S i 4 + .34 P 5 + 5+ .46 AS Table 3.1 Properties of the Host and Donor Elements In the i n v e s t i g a t i o n , two e f f e c t s were minimized. i ) The thermal population of the higher l y i n g 1S(E) and 1 C /''r\ 1 ,^--^1 the IS(A) l e v e l to be observed. The thermal propuiation d i s t r i b u t i o n f o r noninteracting donors i s c a l c u l a t e d i n appendix F. If the c a l c u l a t i o n i s v a l i d f o r the concentrations of i n t e r e s t , the f r a c t i o n a l population d i s t r i -bution at 20°K i s c a l c u l a t e d to be 0.995, .0038 and .001 f o r the IS(A), 2 1S(T) and 1S(E) l e v e l s r e s p e c t i v e l y . i i ) . The e f f e c t of the donor electron-phonon i n t e r a c t i o n on absorption linewidths i s minimized. The olectron-phonon i n t e r a c t i o n at lox<r donor concentrations has been investigated both experimentally and theoret-3 i c a l l y . Results show that acoustic phonons couple neighboring states and L. Pauling, 1960. The Nature of the Chemical Bond, Cornell Univ. Press. "Lepine (1970) has used a s i m i l a r c a l c u l a t i o n f o r concentrations up to 8 x 10^ -6 phosphorus per cm3 i n s i l i c o n , 3For example Colbow (1962), Parsons (1968), Kane (1960), Barrie and Nishikawa (1963). 4 4 . enhance the r e l a x a t i o n of electrons to the ground state and thus, a f f e c t the absorption linewidth. Calculations by Barrie and Nishikawa (1963) show that the coupling i s strong above a c r i t i c a l temperature Tc; for example, in phosphorus doped s i l i c o n Tc i s greater than 50°K for most states coupled to the 2P and 2P l e v e l s . o ± For samples at l i q u i d helium temperatures, these two e f f e c t s were small and could be accounted f o r . 3.2 O p t i c a l Instruments Two d i f f e r e n t spectrometers were used to cover the desired i n f r a r e d regions. 3.2.1 The Grating Monochromator A Perkin Elmer model 83 monochromator, employing a Littrow mounted Bausch and Lomb grating with 30 grooves per mm and blazed at 30 microns, was used i n the f i r s t order to cover the s p e c t r a l region 32 to 45 m i l l i - e l e c t r o n v o l t s (meV) (263 - 370 cm "*"). To minimize unwanted r a d i a t i o n , NaF r e s t r a l e n plates were used i n the entrance and e x i t o p t i c s i n conjunction with sooted mirrors. Less than 1% of the r a d i a t i o n trans-mitted by the s i l i c o n samples was second order d i f f r a c t i o n r a d i a t i o n . The t h e o r e t i c a l r e s o l v i n g power, a measure of the ultimate a t t a i n a b l e r e s o l u -t i o n at zero s l i t width, i s given by X_ = mM, where m i s the d i f f r a c t i o n AX order and M i s the t o t a l number of grooves on the g r a t i n g . For t h i s apparatus X/AX = 1920. Water vapor linewidths, p l o t t e d as a function of s l i t width y i e l d e d , upon extrapolation to zero s l i t width, X/AX equal to 45. 65% of the theoretical limit, in agreement with the performance rating listed by Bausch and Lorab. The source of infrared radiation was a globar operated at 250 watts. The radiation was chopped at 13 Hertz for phase sensitive detection. Assuming black body radiation, the maximum useable photon flux in a typical spectral s l i t width was estimated to be ^10photons/second. After monochromation, the beam was focused on the sample which was maintained at low temperatures, and the transmitted radiation was detected by a Reeder vacuum thermocouple equipped with a Csl window. The amplified signal was displayed on a Brown strip chart recorder. The s l i t settings used were a compromise between high resolution and good signal-to-noise ratios.- The signal-to-noise ratio was limited by the detector sensitivity and the photon flux. Tbe grating drive was calibrated using the accurately known water vapor absorption lines (Blaine et. al., 1962). The calibration curve in the region of interest was linear in wavelength. During experiments the monochromator was purged with dry nitrogen gas; nevertheless, several strong water vapor lines remained a minor nuisance. 3.2.2 Fourier Interferometer The second spectrometer used to investigate the 4-44 meV region (50 to 350 cm """), was a Beckman RIIC FS-720 Fourier Spectrophotometer modified for use with samples at low temperatures. The detailed description of the theory of this spectrometer has been published by Gebbie and Twiss (1966). In essence, the device is a Michelson interferometer. The i n f r a r e d source i s a P h i l l i p s HPK 125 watt high-pressure mercury arc lamp which produces a continuous spectrum. The p a r a l l e l hetrochromatic beam i s divided by a mylar beam s p l i t t e r i n t o two paths. The two mirrors (M^ and of f i g u r e 3.2). one fi x e d , the other movable, recombined the beams which subsequently i n t e r f e r e with each other. For an inc i d e n t beam of the form cos (wt-2irkx), where tt> and k are the frequency and wavenumber of the r a d i a t i o n , the e x i t beam i n t e n s i t y at an o p t i c a l path d i f f e r e n c e x i s given by I(x) = / I ( k ) (1 + cos(2ffkx)) dk o oo = + /I(k) cos(2-,Tkx) dk (3.1) o When the mirrors and M.^ are equidistant from the beam s p l i t t e r , the o p t i c a l path d i f f e r e n c e i s zero and K o ) = 2.1 («>"), With the appropriate J^~ZZrZ=ZI3 M, f i x e d E x i t beam Figure 3.2 Michelson Interferometer choice of beam s p l i t t e r thickness and f i l t e r s , the maximum detectable wave-number, K, i s v a r i e d . Thus the detected i n t e n s i t y , f o r an o p t i c a l path 48. difference x, is given by K I(x) = I(») + /I(k) cos(2iTkx)dk (3.2) o The exit beam from the interferometer is focused by the paraboloid mirror (5 of figure 3.3) onto the sample at 7. The transmitted fraction, refocus-ed into the light pipe, 9, by a black polyethylene lens-filter, 8, is then collected and quantified by the golay detector, 10, equipped with a diamond window. The light beam, mechanically chopped at 15 Hertz is phase-sensitive detected and amplified. The movable mirror, H^, is driven by a stepping motor, allowing the signal to be averaged during each step inter-val. This integrated average is digitized and punched onto paper tape in binary code. The intensity I(x) is recorded as a function of the optical path difference x. The interferogram is also recorded on chart paper as a check for spurious signals. The intensity of interest, I(k) is obtained 4 when a fourier transformation is performed on the data. One advantage of this system is that a wide spectral region, unavailable with the other monochromators, became accessible. Spectral regions could be made to overlap for comparison purposes. The necessity of fourier transforming the signal I(x) into I(k) before any interpretation could be made, was a minor drawback. The resolution of the device depends primarily on the maxi-mum optical path difference used . In practice, tbe resolution was deter-mined by other factors given below. The beam splitter and filters were ^The basic computer program for optimum fourier transformation was devel-oped by Dr. J.E. Bertie of the University of Alberta. 49. chosen to maximize the i n t e n s i t y i n the s p e c t r a l region of i n t e r e s t (see table 3.2). The stepping drive increment, A, was selected to be l e s s than one h a l f the shortest wavelength to be detected. The strength of the i n t e r -ferogram s i g n a l diminished as the path d i f f e r e n c e increased, so that the maximum u s e f u l path d i f f e r e n c e , X, was ul t i m a t e l y determined by the s i g n a l -to-noise r a t i o which i s proportional to the s i g n a l averaging time, Z."* Since the t o t a l scanning time i s given by T = XZ/A, i t became a matter of adjusting X and Z to give the best r e s u l t s during the a v a i l a b l e scanning time T. The optimum conditions are indicated i n table 3.2. Increment Stepping Scanning Size Time Resolution F i l t e r s A(y) T(minutes) (meV) 3-27 50G 12u Beckman f'/4 20 58 .46 18-50 25G 6u CsI 1.68mm 10 115 .46 3-50 25G 6y 10 115 .46 Path d i f f e r e n c e 8.0 mm Signal averaging i n t e r v a l -8.53 sec. Table 3.2 Operating Conditions for Fourier Spectrophotometer. Spectral Region Beam S p l i t t e r (meV) (Gauge) (Thickness) With the a v a i l a b l e gain, l a r g e r path d i f f e r e n c e s produced s i g n a l v a r i a t i o n s l e s s than the noise l e v e l . This may i n d i c a t e that, for the samples examined, there were no s p e c t r a l components r e q u i r i n g large path d i f f e r e n c e s for t h e i r r e s o l u t i o n . 50, The sensitivity of the Golay detector is presently the limiting factor in the resolution. Consequently, the accurate analysis of the shape, of the absorption peaks was restricted to the monochromator data, the interfer-ometer being used to observe broad features in the far infrared. The use of a cooled germanium detector should allow higher resolution to be attained. A wavenumber calibration was unnecessary as this obtains directly from the path difference, which is measured accurately to 2 parts in 5000. Previous comparison with water vapor lines has shown good agree-ment. Atmospheric*absorption was not a problem as the entire spectrophoto-meter was evacuated to a pressure of less than 15 torr during experiments. 3.3 Low Temperature Techniques Liquid helium at 4.2°K was used to cool the silicon samples to the required low temperatures, and standard metal double dewars were employed. Two thin rectangular samples were mounted on a copper block in thermal contact with the liquid helium container. Care was taken in the design and use of the mounting to eliminate strains in the samples which may distort the absorption spectrum. To achieve a strain-free mount, samples were attached to the block at one end only; good thermal contact was assured by the application of silver impregnated grease between the sample and the block. The mounting block used with the monochromator was soldered to the helium can and positioned at a beam focus of area approximately 1 mm x 10 mm. A copper radiation shield with narrow entrance and exit slits for the beam surrounded the samples, and the shield was maintained at 77°K to minimize the influx of room temperature radiation from the surrounding apparatus vacuum jacket. Csl windows on the vacuum jacket allowed for infrared radiation less than 45 microns to pass. With this apparatus the sample, temperature, measured by previous workers, (White, 1967) was less than 6°K. For low temperature work on the interferometer a Beckman purge kit with vacuum "tight polyethylene windows replaced the standard sample cell box (see figure 3.2). The copper block used with the inter-ferometer was screwed onto the t a i l of the helium pot with an indium pad press-fitted between for maximum thermal contact. The interferometer had a large focal area of approximately 3/4 inch diameter, and the beam cross section was limited by the copper sample holders. To minimize radiation heating from the heterochromatic beam, a double layer of black polyethylene film was placed in the entrance port of the copper radiation shield. In spite of these precautions a germanium resistance thermometer showed oper-ating temperatures of 12 ± 2°K. Three samples could be accommodated within this dewar. 3.4 Samples The samples used in the investigation were cut from, commer-cially obtained single crystal ingots of silicon of approximately 1 inch diameter. The phosphorus-doped silicon was float zone refined with a (111) ingot axis. The arsenic-doped silicon ingots had the same properties but were pulled by the Czochralskii method. Most ingots were of uniform concen-tration, a few having concentration gradients along the ingot axis. 52. The desired sample s l i c e s were cut perpendicular to the ingot axis by a rotary diamond saw. The s l i c e s were ground to the desired thickness using #600 alumina abrasive on window gla s s . The samples, t y p i c a l l y 1/2 by 3/4 inches i n area, were wax mounted onto a s p e c i a l p o l i s h i n g j i g and polished to a mirror surface using a s u i t a b l e mixture of #600 g r i t , water and suspendex s o l u t i o n on Astromet c l o t h . A microsccpi look at the surface showed p i t t i n g to be n e g l i g i b l e . I d e a l l y , the sample thickness was chosen to optimize the absorption s i g n a l and yet to prevent saturation of the absorption peak. In pr a c t i c e at high donor concentrations t h i s became a secondary concern. White (.1967) has shown for boron-doped s i l i c o n that s t r a i n could s e r i o u s l y a f f e c t the absorption linewidths as may have been the case i n previous work, u It was the l i m i t a t i o n of producing and using s t r a i n f r e e , s e l f supporting samples that determined the high concentration sample thickness. An attempt was made to use very t h i n samples (^10 microns) by mounting them onto a supporting i n t r i n s i c s i l i c o n frame with epoxy r e s i n . Large s t r a i n produced by the d i f f e r e n t thermal contraction of the epoxy f i l m d i s t o r t e d the spectrum and made the technique unfeasible. Another problem with t h i n samples was the elim i n a t i o n of interference f r i n g e s (section 3.5). Various methods were t r i e d ( p o l ishing one surface only, making lens-shaped samples, e t c . ) . The method used most 6Colbow, 1964. Goruk, 1964. 7 T h i s also l i m i t e d the highest concentration for which absorption spectra could be analysed. 53. s u c c e s s f u l l y was to make the sample cross section i n the beam vary i n thickness by 10%. This v a r i a t i o n was s u f f i c i e n t for phase changes across the beam to remove the interference fringes yet the sample was p a r a l l e l sided enough to insure m u l t i p l e r e f l e c t i o n s would not d e f l e c t away from the detector. (The wedge angle was approximately to mounting, the samples were u l t r a s o n i c a l l y cleaned i n a bath of toluene and then i n a bath of e t h y l alcohol to remove wax and grease from the surface. room temperature dc r e s i s t i v i t y measurements obtained using a standard tour point probe. The r e s i s t i v i t y measurements were converted to concentrations 3 (donors/cm ) using, the I r v i n chart (1962). This conversion agreed well with that of the Dow-Corning S l i d e Rule (1967). Concentrations were known g to 5 - 10% accuracy. Table 3.3 l i s t s properties of the c r y s t a l s used. 3.5 C a l c u l a t i o n of the Absorption C o e f f i c i e n t As shown i n section 2.2.1 the absorption of r a d i a t i o n by donor electrons i s characterized by a frequency dependent absorption co-e f f i c i e n t , cx(co), which can be evaluated from the measurement of the i n t e n -s i t y , I^(w), transmitted through a p a r a l l e l - s i d e d sample of doped s i l i c o n 8 The f r i n g e problem i n tbe interferometer scans remained a nuisance f o r t h i n samples i n the f a r i n f r a r e d (X >100 microns). 9 In several samples, the absorption spectrum revealed mixtures of As, P and possibly Sb impurities; these samples were discarded. Sample thicknesses were measured with a micrometer. P r i o r The concentration of donor impurities was determined from 54. TABLE 3.3 DONOR CONCENTRATIONS OF SILICON SAMPLES Sample Number md Manufacturer' 44 - A 140151 6 - B 10694 140153 11981 7 - B 10259 10181 A 10181 B 10227 10217 10239 B 10141 14 - C 19 - D 17 - D 18B- D 18A- D Dopant P P P P P P P P P P P P P P P A Intrinsic Resistivity Q cm 300°K 100 + 10 48 ± 5 8.5 ±...4 4.5 ± .3 1.1 ± .05 .75 ± .04 .67 ± .06 .38 ± .02 .28 ± .02 .215 ± .01 .20 ± .01 .165 ± .01 .080 ± .008 .034 ± .03 3.5 ± .2. .56 ± .05 .20 ± .01 .14 ± .02 >3500 Concentration 3 /cm" x 10 13 1.05 x 10 1* 5.7 x 10 14 1.1 x 10 15 4.8 x 10 15 7.5 x 10 15 8.6 x 10 15 ,16 1.7 x 10 2.4 x 10 3.3 x 10J 3.6 x 10 4.6 x 10 1.2 x 10 5.7 x 10 1.4 x 10 1.1 x 10 3.6 x 10 5.5 x 10 <io12 16 16 .16 16 17 17 15 16 16 16 *Unless specified otherwise, a l l samples were manufactured by General Diode Ltd. A ~ Semi Elements B - Merck C ~ Knapic D - Northern Electric of thickness d. This r e l a t i o n s h i p , developed i n appendix A, i s given by I,rO) B (1-R.)2 ( 3 . 3 ) " ' ' i ^ ' 1 r a ( < - o ) d / o „ -a(u))d/-_,2 , , . 2 / n N [e '2 - Re 2] + 4Rsin (,3-g) where g - . 2Trnd . cos6 , K o R(w) = R = r e f l e c t i v i t y , n = r e a l part of the index of r e f r a c t i o n = 3 . 5 for S i , 6p = angle of r e f r a c t i o n , X q = wavelength of r a d i a t i o n i n free space, g = tan~'"| -2k | ^ 0 , and "-I-n -k J k = e x t i n c t i o n c o e f f i c i e n t ; imaginary part of the index of r e f r a c t i o n A s expressed i n appendix A, the source of interference fringes i n the trans-mitted i n t e n s i t y measurements i s the sin"((3-g) term. The fringes were eliminated by using a s u i t a b l e choice of the var i a b l e s determin ing g. The v a r i a t i o n s i n g are given by A g = 2rad cosG (-AX + Ad + tan0A6) ( 3 . 4 ) —. . „-, — o - — x d o X o The f i r s t term of 3 . 4 i s due to the v a r i a t i o n of wavelength across the s l i t p r o f i l e ; the second term, the v a r i a t i o n of sample thickness across the beam cross section; the t h i r d term, the v a r i a t i o n i n the angle of r e f r a c t i o n for the impinging l i g h t beam. I t i s evident that f o r thick samples, the f i r s t term (cO dAX ) i s dominant i n the v a r i a t i o n . For t h i n samples, the second term 56. must dominate; typically, for a thin sample, d = 100 microns AX_ - AE 1.02 meV = .0005, Ad = 10%, tan6A.e ^  0, X^ 30 microns, X E ' 40 meV d and Ag = 2tix3.5 x 100 (.0005 + . 10 + 0) 2^.3TT (3.5) Upon averaging, the expression 3.3 becomes I (to) = I (u>) (1-R)2 e ~ a ( a ) ) d .. (3.6) ° 1-R2 e " 2 o ( ( a ) ) d The incident beam intensity, Io(io),was not measured, but rather the intensity, I^Oo), transmitted through an intrinsic silicon sample at low temperatures was measured. The incident intensity I0(w) was much larger than I^ ,(co) and thus gain correction factors would have had to be introduced because the coarse amplifier gain was uncalibrated. The intrinsic sample did not absorb in the region of interest and therefore the intensity was solely a function of reflectivity I t(u) =(l-R((o)7 I (3.7) 1 il+R(u))l 0 The reflectivity in the 25 to 50 meV region, measured by Giles and Bichard (1962) was constant at .31 ± .02 in agreement with the value obtained from the known index of refraction (n = 3.5). As noted in section 3.4, the samples were prepared slightly wedge shaped to remove interference fringes. However, they v/ere sufficiently parallel to insure that the multiple reflection terms were not dispersed. In tbe high concentration samples where the loss of some of these terms may have been p o s s i b l e , the absorption was large j for ad =^  1, the i n c l u s i o n 2 *~2ctd of the mult i p l e r e f l e c t i o n term (the R e i n the denominator of equation 3.6) created a maximum possible error of 1.3%. P a r a l l e l - s i d e d samples, e x h i b i t i n g large, widely spaced interference fringes were studied with the hope of determining a. (to) from these patterns. Unfortunately, the r e s u l t s did not coincide with any e a s i l y analysed expression. Contraction of the sample with temperature also changed the fringe pattern. The data from the monochromator(grating p o s i t i o n , I T and I j ) were punched onto IBM cards together with a photon energy c a l i b r a t i o n . Absorption c o e f f i c i e n t s were calculated on an IBM 360 model 67 computer. The spectra were corrected for instrumental broadening and the estimated true spectra of absorption c o e f f i c i e n t as a function of photon energy were p l o t t e d . See appendix B for various instrumental c o r r e c t i o n s . 58. C H A P T E R 4 E X P E R I M E N T A L R E S U L T S A N D I N T E R P R E T A T I O N 4.1 Introduction The absorption spectra of photon-induced transitions from the IS ( A ) level to the excited donor levels and to the ionization continuum 14 17 were studied for phosphorus concentrations from 10 /cc to 2 x 10 /cc. Detailed lineshape measurements were made on the IS(A) -» 2P q and IS (A) -> 2P+ transitions in phosphorous-doped silicon and on the IS ( A ) -> 2P Q transition in arsenic-doped silicon. In this chapter the experimental data are presented together with an analysis of the results in terms of the existing theory. The various experimentally determined parameters for a typical absorption line, shown in figure 4.1, are defined below: Peak_pjDs^tjion _0^)_: the energy at which the slope of the lineshape is zero (i.e. the energy of the peak maximum); Haifwidth _(W^: the f u l l width of the absorption line at one half the peak maximum; Asymmetry (6 . ): a measure of the skewness of the lineshape -N 1 th 6 . = ( H - H ) , at /N the peak maximum; 'N 'N . Integrated absorption coefficient ( a ^ n t ) : t ne area under the lineshape including the line "tails". 59. a 12 o <da/dE>E = ° o 6l/2 ~ H2~ H1 W = H 2 +H 1 Figure 4.1 Parameters Describing a Single Absorption Peak Lineshape. 4.2 Phosphorus-Doped Silicon After instrumental corrections were applied to the data. the above mentioned parameters (peak position, half width, etc.) for the 1 13 2P and 2P, lines were measured in the concentration region of 5 x 10 o + -The absorption spectral lines are labelled in subsequent discussions by the final state only; the IS(A) level is the ground state unless specified otherwise. 60. to 1.2 x 10"""^  phosphorus donors/cc. Observations \rere also made of the higher energy absorption l i n e s and the i o n i z a t i o n continuum at these donor concentrations. The sample temperature was l e s s than 6°K. The r e s u l t s of the i n v e s t i g a t i o n s are interpreted i n terms of three concentration regions. 4.2.1 Low CQrtcentrations; The Isolated DonOr Limlt For donor concentrations l e s s than i O ^ / c c the i s o l a t e d impurity approximation should be v a l i d . This corresponds to a most probable donor separation (see f i g u r e 2.4) of greater than 550 A 0 (27 a ) and an ft equivalent Wigner-Seitz radius of greater than 620 A (31 a ), for an e f f e c -t i v e Bohr radius of 20 A°. The absorption spectra of phosphorus - doped s i l i c o n i n the i s o l a t e d impurity l i m i t has been studied i n d e t a i l by various workers (Kohn, 1957; Bichard and G i l e s , 1962), and experiments on strained samples have confirmed the t h e o r e t i c a l p r e d i c t i o n s regarding the symmetry and degeneracies of the various donor l e v e l s . The peak p o s i t i o n s of the 2 P q and 2P + l i n e s are found to be 34.13 ± .01 and 39.17 ± .01 meV, r e s p e c t i v e l y , i n agreement with previous work. The halfwidth, i n the l i m i t of zero donor concentration, was extra-polated from measured values for concentrations l e s s than 5 x l O ^ / c c . This concentration independent halfwidth was found to be 0.05 ± .02 and 0.07 ± .02 mev for the 2P Q and 2P + l i n e s , r e s p e c t i v e l y . (Previously, Bichard and G i l e s (1962) placed an upper l i m i t on the halfwidth at 0.08 meV for both t r a n s i t i o n s , for a concentration of 5 x l O ^ / c c . ) The lineshapes, a f t e r instrumental c o r r e c t i o n s , were'found to be Lorentzian for both the 2P and 2P + t r a n s i t i o n s . Figure 4.2 compares the corrected 2P + lineshape 61. 14 with a Lorentzian f i t at a concentration of 5.7 x 10' /cc. L i n e a r i t y of /aji versus |E-E | " indicates a Gaussian or a Lorentzian lineshape, r e s p e c t i v e l y . From these p l o t s (figure 4.2) the uncorrected lineshape suggests the former lineshape while the instrumentally corrected l i n e suggests the l a t t e r . The zero concentration halfwidth may be due to several e f f e c t s . It i s possible that the techniques used f o r instrumental c o r r e c t i o n (see appendix C) of the narrow low-concentration l i n e s i s inad-equate to f u l l y deconvolute the instrumental s l i t p r o f i l e from the observed lineshapes. Extrapolation of the computer corrected halfwidths to zero concentration thus y i e l d s only an upper l i m i t to the true r e s i d u a l h a l f -width. However., i f the r e s i d u a l broadening 'were p r i m a r i l y instrumental, the corrected lineshapes should be nearly Gaussian. This i s not the case. The true r e s i d u a l halfwidths of the 2P and 2P^ absorption o ± l i n e s i n the i s o l a t e d donor approximation can be a t t r i b u t e d to two e f f e c t s : i n t e r n a l s t r a i n produced by d i s l o c a t i o n s and donor-electron acoustic phonon i n t e r a c t i o n . The e f f e c t of d i s l o c a t i o n s on donor spectra has been discussed by Kohn (1957). D i s l o c a t i o n s produce i n t e r n a l s t r a i n s which s h i f t i n energy the r e l a t i v e p o s i t i o n s of the s i x conduction band minima. The s h i f t s i n the minima produce a s p l i t t i n g of the donor l e v e l s . A broadening of the spectrum obtains upon averaging the e f f e c t s of the random s t r a i n . Kohn estimated the s p l i t t i n g of the T^ representation t r i p l e t of an excited l e v e l -8 r~ 2 to be ^ 7 x 10 v£ eV cm, where JL i s the d i s l o c a t i o n density (lines/cm ). Since "good" s i l i c o n s i n g l e c r y s t a l s have d i s l o c a t i o n d e n s i t i e s l e s s than 62. ENERGY (mev) Figure 4.2 Comparison of a 2'P-t Lineshape to a Lorentzian Profile. True peak has been corrected for instrumental broadening. 63. 5 2 .10 lines/cm , a d i s l o c a t i o n broadening of l e s s than 0.025 meV i s expected. Kohn's c a l c u l a t i o n gives the same value of d i s l o c a t i o n broadening i n acceptor l e v e l s . This estimate i s i n reasonable agreement with the experi-A 2 mental value of 0.02 meV f o r 4 x 10 lines/cm i n boron-doped s i l i c o n (White, 1967). The observed absorption l i n e s are zero-phonon t r a n s i t i o n s , thus, the broadening due to the electron-phonon i n t e r a c t i o n i s a r e l a x a t i o n or l i f e t i m e e f f e c t . Various t h e o r e t i c a l i n v e s t i g a t i o n s of the e l e c t r o n -phonon i n t e r a c t i o n i n shallow impurities have been c a r r i e d out (Kane, 1960; Nishikawa and Ba r r i e , 1963; Barrie and Nishikawa, 1963). The formalism of Barrie and Nishikawa y i e l d s values which are i n reasonable agreement with the observed temperature dependent halfwidths of t r a n s i t i o n s i n boron-doped r 4 i < ~rin V -al .".TT < - f ! " T f TV- - £-ITI r T.-.TT .rjf i-h'-h^rt— uJ f-1; T j r » T 7 i , ~ u e ,o — -! rp ~ 7-. *• on donors i n s i l i c o n at 4.2°K suggests only that there i s no disagreement. In the Barrie-Nishikawa theory applied to s i l i c o n , the dom-inant phonon broadening mechanism was shown to be a phonon l i f e t i m e e f f e c t of the excited l e v e l due to the decay of electrons to neighbouring eigen-states through emission or absorption of long wavelength accoustic phonons. For an absorption peak associated with the t r a n s i t i o n i -»- f (where i i s the i n i t i a l state and f i s the f i n a l state) the phonon co n t r i b u t i o n to the h a l f width of the peak AE^(T), due to coupling of the f i n a l state f to a nearby state j which i s k T ^ energy units from f, has the following values: AE.(T) = h 3 o f j r 1 + • 1 j T f l / T ) C c - 1 -1 , j below f (4.1) 64. I T ^ / T K c - 1 - 1 , j above f (4.2) is called the characteristic temperature and h ^ is a constant. In c o the problem of interest, the sample temperature, T, is much less than f for states lying near the 2 P q and 2 P + levels, and AE^  (T) is only weakly temperature dependent. For a single conduction band minima, Barrie and Nishikawa have calculated the approximate halfwidth contributions of levels neighboring the 2 P q and 2 P + donor levels at temperatures of 0 ° and 75°K. Using their results and expression ( 4 . 1 ) and ( 4 . 2 ) the phonon broadening at 1 0 ° K was estimated; the results are essentially the same as the 0 ° K half-width calculation. The phonon broadening estimates yield a probable upper limit of 0.02 meV for the 2P level and 0.03 meV for the 2P, level. The main con-o ± tribution to the 2 P q level broadening comes from the 2S level which is assumed to l i e at its EMA value.below 2P . The contributions to the 2P, o ± level halfwidth come from the 3 S , 3 P q and 2S levels. Changes in the tran-sition linewidths due to the phonon broadening of the ground state were found to be negligible at low temperatures. If i t can be assumed that the dislocation broadening and phonon broadening halfwidths simply can be summed, the theoretical estimates of the upper bounds, 0.045 meV and 0.055 meV, compare well to the experi-mental zero concentration halfwidth upper bounds of 0.05 ± .02 meV and 0.07 ± .02 meV for 2 P q and 2 P + transitions, respectively. Theoretically, the dislocation interaction should split the 2 P q and 2P+ level degeneracies s l i g h t l y and the electron-phonon i n t e r a c t i o n should broaden each of the t r a n s i t i o n s due to the l i f e t i m e e f f e c t . It has been shown (Barrie and Nishikawa, 1 9 6 3 ) that only 3 of the 6 2P and 3 of the 1 2 2 P + states are o p t i c a l l y a c t i v e from the IS (A) ground state. Thxis, the expected l i n e -shape i s a superposition of three Lorentzian l i n e s s i t u a t e d at s l i g h t l y d i f f e r e n t energies. Experimentally, the two broadening mechanisms cannot be distinguished at l i q u i d helium temperatures. The observed Lorentzian lineshape i s i n general agreement with the predicted lineshape. 4.2.2 Intermediate Concentration - Donor-Donor Interactions As the concentration of phosphorus donors i s increased from 1 . 2 x i O ^ / c c to 1.2 x l O ^ / c c the absorption spectrum changes appreciably. Q u a l i t a t i v e l y , the absorption l i n e s are seen to increase i n halfwidth and s h i f t to lower photon energies with increased doping. The higher energy peaks ( 4 P + , 5 P ) broaden and, at s u f f i c i e n t l y high concentrations, are replaced by a featureless absorption continuum. The lower energy 2 P q and 2 P + t r a n s i t i o n s are seen to take on an asymmetric lineshape with a t a i l on the low energy sid e . A sloping background absorption, superimposed on the• d i s c r e t e spectrum, i s observed at concentrations greater than 2 x l O ^ / c c . The general features of the spectra are evident i n f i g u r e 4.3. The quanti-t a t i v e a nalysis and i n t e r p r e t a t i o n of t h i s behaviour follow. A. Halfwidths of the 2P and 2 P ^ T r a n s i t i o n s - Overlap of Donor Wavefunctions o ± _ The halfwidths of the 2 P and 2 P ^ t r a n s i t i o n s were measured o ± from the peak p r o f i l e s corrected for instrumental e f f e c t s for each doped sample. Figure 4.4 shows the. halfwidths of the 2P and 2 P + t r a n s i t i o n s at a 66. 5 '14 10/bc 10 .15 5x10 T—r-r 10° ~ ~ i — n ~ 2x1 C 1 8 5x10 o A 1S(A)~>2P0 1S(A) ->2P± H2 MOLECULE APPROX. WIGNER SEITZ APPROX. a a=26/J b a=16Ac a / / / / / / / / 10 ,17 ~T~_ 20 30 yf, /• . 40 50 ( DONOR CONCENTRATION ) •> (10 CM ) 1/3 F i g u r e 4.4 H a l f w i d t h of the 2 P and 2 P . T r a n s i t i o n vs. Concentration o ± The zero c o n c e n t r a t i o n h a l f w i d t h has been subtracted. ..The s o l i d l i n e i n d i c a t e s r e s u l t s o the H 9 molecule model (Macek); the dashed l i n e the Wigner-Seitz Sphere model (Baltensperg 68. 2 temperature less than 6°K as a function of phosphorus concentration. The zero concentration halfwidths of 0.05 and 0.07 meV have been subtracted from the 2P q and 2P+ halfwidths, respectively. The error bars represent the maximum range of halfwidths of several scans of the peak, and the scale of the concentration axis was chosen arbitrarily. Examination of figure 4.4 reveals that for a given concentration, the 2P+ transition is broader than the 2P q, and with increasing concentration the 2P+ transition broadens faster than the 2P q line. Thus i t is concluded that the broadening is mainly due to changes in the excited state rather than the ground state. In the intermediate concentration region, the concentration 3 dependent broadening due to electron-donor interaction is an extra pertur-bation. Before this concentration-dependent broadening of. the transition halfwidths can be evaluated, the other line broadening mechanisms have to be considered. In the limit of zero concentration, the Hamiltonian of the donor electron consisted of the unperturbed, isolated impurity Hamiltonian plus an electron-phonon perturbation and a dislocation produced strain perturbation. Dislocations directly affect the properties of the host lattice and are independent of the donor concentrations in the crystal. Our assumption is that in the intermediate concentration region, the electron-phonon and electron-donor interactions are independent, and donor-donor-phonon inter-actions may be neglected. In other words, the dislocation and phonon 2 For donor concentrations where background absorption was appreciable, the halfwidth was measured with the background removed. 3 This refers to the interaction of a donor electron with nearby donors. b r o a d e n i n g a r e assumed to be independent o f d o p i n g . T h i s assumption was used i n th e a n a l y s i s of boron-doped s i l i c o n s p e c t r a (White, 1967), S i n c e the data, was t a k e n on -uncompensated samples a t l i q u i d h e l i u m t e m p e r a t u r e s , c o n c e n t r a t i o n b r o a d e n i n g due to the p r e s e n c e of i o n i z e d 4 donors i s n e g l i g i b l e . Two t h e o r e t i c a l t r e a t m e n t s of c o n c e n t r a t i o n depen-dent b r o a d e n i n g ; d i s c u s s e d i n section-2.2.3, c o n s i d e r the e l e c t r o n - d o n o r i n t e r a c t i o n : the B a l t e n s p e r g e r e s t i m a t e of c o n c e n t r a t i o n b r o a d e n i n g , and t h e d o n o r - p a i r model f o r m u l a t e d by Macek. B a l t e n s p e r g e r 1 s c a l c u l a t i o n o f the bandwidths of the 2P l e v e l i s g i v e n i n s e c t i o n 2.2.3.A where the l e v e l b r o a d e n i n g i s c a l c u l a t e d as a f u n c t i o n of the e q u i v a l e n t W i g n e r - S e i t z r a d i u s , r . S i m i l a r c a l c u l a t i o n s o f t^ViQ K o n H _Hn>-j f o r » r th? i s "Level show s ^ i f f - or f-tr^ i^er!",*T?<? f o r a w i n n e r — v z r a d i u s g r e a t e r t h a n 5a . S i n c e e x p e r i m e n t a l e v i d e n c e s u g g e s t s t h a t the b r o a d e n i n g o f the IS —>• 2P t r a n s i t i o n s i s due p r i m a r i l y to the e x c i t e d l e v e l s , the 2P l e v e l i s c o n s i d e r e d as the o n l y s o u r c e of the b r o a d e n i n g . To o b t a i n bandwidth as a f u n c t i o n of donor c o n c e n t r a t i o n , the v a l u e of t h e e f f e c t i v e Bohr r a d i u s a has to be e v a l u a t e d . The e q u i v a l e n t W i g n e r - S e i t z r a d i u s i s used to r e l a t e donor s e p a r a t i o n s to donor c o n c e n t r a t i o n , t h r o u g h the e x p r e s s i o n N" 1 = 4 3 W d ( r s ) J (4.3) A l l donors e x c e p t t h e photon a b s o r b i n g donor i n q u e s t i o n a r e assumed n e u t r a l and i n t h e i r ground s t a t e . C a l c u l a t i o n s e s t i m a t i n g the i n c i d e n t photon f l u x and e x c i t a t i o n l i f e t i m e s show t h a t the f r a c t i o n o f donors i n e x c i t e d c o n f i g -u r a t i o n s due t o r a d i a t i o n e x c i t a t i o n i s n e g l i g i b l e . 7 0 . As discussed i n s e c t i o n 2.2.2.B, a should be a. non-adjustable parameter. However, because of the mass anisotropy of the donor e l e c t r o n , a i s some s u i t a b l e average of the e l l i p s o i d a l parameters a and b defined by expression 2.14. A v a r i e t y of methods- have been used to o b t a i n a s u i t a b l e v a l u e f o r a T e f f t , B e l l and Romero (1969) suggest that to p r o p e r l y account f o r the s p l i t t i n g of the 2S, 2P A and 2P l e v e l s , a d i f f e r e n t e f f e c t i v e mass r ° ' m = 0 m = ±1 5 should be assigned to each l e v e l to approximate the a n i s o t r o p i c e f f e c t i v e mass by an i s o t r o p i c , one. The expression f o r m developed by T e f f t e t . a l . i s given by m L m nu J m t t j l A- (4.4) 9 9 where A. = 2 (i*~-m'~) + 2JL -1 , and and m are the o r b i t a l and magnetic l m ( 2 1 - 1 ) (21+3) quantum numbers, r e s p e c t i v e l y . The expression leads to the Kohn-Luttinger values of the l e v e l energies given i n t a b l e 2 . 1 when used w i t h the hydrogen! * approximation. To r e l a t e the Kohn-Luttinger values to observed energies, a 1 / 2 i s r e p l a c e d by b where b = a \ L obs Using the c y c l o t r o n resonance values of Hensel, e t . a l (1965) f o r the e f f e c t i v e mass parameters, m^/m^ = * 9 .1905 ± .0001 and m /m = .9163 ± .0004, and the expression a = i i K , the * 2 m e the e f f e c t i v e Bohr r a d i i may be c a l c u l a t e d . These are l i s t e d i n t a b l e 4.1. IS 2'P o 2 P ± nlm 1 0 0 2 1 0 2 1 1 A5-ra V3 * / 5 3/5 ft m / n i o . 2 5 9 . 2 2 6 . 3 6 3 a 2 3 . 2 A ° 1 6 . 6 A ° 2 6 . 6 A ° ft b 1 8 . 5 A 0 1 5 . 3 A ° 2 6 . 1 A 0 T a b l e - 4 . 1 E f f e c t i v e Bohr R a d i i for the IS, 2 P and 2 P , Levels o ± ft Using the Baltensperger r e s u l t s of table 2.2 and the values b^p — 1 6 A° o ft and b^p ^ 2 6 A" i n expression 4 , 3 the broadening of the 2 P q and 2 P + t r a n s i t i o n s as a function of concentration was determined. The r e s u l t s , shown by the dashed l i n e s of f i g u r e 4.4,are i n poor agreement with the experimental data.. The Baltensperger values show a broadening that increases ft too r a p i d l y with concentration. Adjusting the parameter b would improve the agreement with experiment, but there i s l i t t l e j u s t i f i c a t i o n on other ft grounds for changing b . The most serious d i f f i c u l t y with Baltensperger s treatment l i e s i n the basic assumption of a non-random impurity d i s t r i b u t i o n . The Wigner-Seitz sphere approximation requires that the separations between the donor impurities are maximized. The repulsion of the donor ion cores during the process of t h e i r d i f f u s i o n into the s i l i c o n l a t t i c e would a f f e c t the donor d i s t r i b u t i o n i n t h i s way. But surely, the e f f e c t i s small for the concentrations of i n t e r e s t . I f the d i s t r i b u t i o n i s random, and an average Wigner-Seitz radius i s defined, t h i s e f f e c t i v e radius would have to have a value le s s than r . because the donor separations would be le s s than s maximal. (The d i s t r i b u t i o n of nearest neighbor separations for a random donor d i s t r i b u t i o n and the equivalent Wigner-Seitz inter-donor separation (indicated by arrows) are shown i n f i g u r e 2.4). If the remainder of the c a l c u l a t i o n remains the same, i n c l u s i o n of a factor allowing for a random donor d i s t r i b u t i o n makes the Baltensperger values for broadening even l a r g e r than before. rUsing the estimate of r /b = 5 to be the onset of broaden-ing of the IS l e v e l , the value b..„ = 18.5 A° suggests broadening of the ground state does not commence u n t i l a donor concentration of 3 x 1 0 X //cc i s reached. The second treatment of the concentration dependent broaden-ing problem used a. random donor d i s t r i b u t i o n as a basic assumption. The donor energy l e v e l s are s h i f t e d due to -interactions with the nearest neigh-bor donor whose separation v a r i e s for each donor-pair. The donor-pair treatment was evaluated by Macek for both IS (A) -> 2P q and IS (A) -> 2P + t r a n s i t i o n s . The following values v/ere used i n the c a l c u l a t i o n s which are discussed i n section 2.2.3.B. a = 25.0 A° = 1.23a* (4.5) b = 14.2 A° = .70a* a* = 20.4 A° Using these values, the dipole matrix elements are found to be <F 1 S iz|F 2 Po> - <F 1 S | z|F 2 Px>^<F 1 S|y|F 2 Py> - . 7 8 2 and the t r a n s i t i o n bandwidths for an .inter-donor separation R are found to be A q(R) =17.4 meVft*3 for 2?o (4.6) R 3 A (R) = 54.0 meV "ct*3 for 2P L, (4.7^ The average bandwidths are evaluated by a computer computation from the expression S0,_ (E) - / [6(E-A (R)) + 6(E+A (R)) ] W(R) dR (4 .8) Zr O O O O S.)V (E) = / [<5(E~A+(R)) + (E+A +(R))] W(R)dR (4.9) '"*' ±1 o where W(R) i s the Chandrasekhar expression 2 . 2 3 for the pro b a b i l i t y of a nearest neighbor at a separation R. While the expressions 4.8 and 4.9 y i e l d halfwidth estimates, the transition, lineshape p r o f i l e s are not obtained since the density of states between the band edges, A^(R) and A +(R), are unknown. Values of halfwidth for the 2 P q and 2 P + transitions were calculated using the donor pair formalism i n the concentration range 3 5 17 10" 4 ^ 10 /cc and they are shown i n figure 4.4 by the two s o l i d l i n e s . Considering that no parameters i n the calculation were adjustable, the agreement between theory and experiment i s good. Several conclusions can be made i n the l i g h t of this com-parison. The difference i n halfwidth of the 2 P and 2 P ^ , l e v e l s , found at r o ±1 74. a l l concentrations investigated, i s due to the Bohr radius parameters, a and b, which r e s u l t from the anisotropy of the electron e f f e c t i v e mass. The donor-pair formalism suggests that i n the hydrogenic l i m i t of an i s o t r o p i c e f f e c t i v e mass, the broadening of the 21' and 2P l e v e l s becomes the same (they also become degenerate i n energy). I t i s also, expected that i f the shallow donor approximations are v a l i d , semiconductors having a greater mass anisotropy than s i l i c o n should have a proportionately l a r g e r d i f f e r e n c e i n the broadening of the 2P and 2 P q l e v e l s . Germanium i s such a semiconductor, having e f f e c t i v e mass constants m. = 1.60 m and I o m - .081 m and v a r i a t i o n a l Bohr radius parameters a = 64.5 A 0 and t o b = 22.7 A°. The r e s u l t s of Nisida and H c r i i (1969) discussed i n d e t a i l i n section 4.4 e x h i b i t t h i s behaviour. A comparison ox L n e higher enet'^y x e v e i w o . v s . t u u c i i o i i i w ^ > 4P ...) with the 2P l e v e l wavefunctions suggest that for a given n sthe nP ., (n = 2, 3, 4, 5) t r a n s i t i o n s should be broader than the nP t r a n s i t i o n s i n approximately the same r a t i o . Unfortunately, d e t a i l e d halfwidth measure-ments could not be made for the 3P and 4P l e v e l s . The 3P and.4P l i n e s are o o much weaker than the other lines,. The high energy t r a n s i t i o n s are c l o s e l y spaced so that at high concentrations, the overlap of the lineshapes makes an a l y s i s d i f f i c u l t . Observations of the 3P Q and 3P + l i n e s confirm that the 3P l i n e i s narrower than the 3P,. o ± The donor-pair t h e o r e t i c a l analysis shows that the nearest neighbor donor i n t e r a c t i o n s h i f t s the energy l e v e l s s l i g h t l y so that the re s u l t a n t lineshape i s inhomogeneous, a superposition of t r a n s i t i o n s at d i f f e r e n t energies. 75. At s u f f i c i e n t l y high concentrations the pair approximation i s expected to break down as donor t r i p l e t and second nearest neighbor interactions become more s i g n i f i c a n t . Macek's calculations were assumed v a l i d for inter-donor separations greater than 50A°. The pair separation 17 pro b a b i l i t y function suggests that for a donor concentration of 2 x 10" /cc, ten percent of the donors are separated by less than 50A°. Calculations for the (IS,IS) ground state give a broadening of t h i s l e v e l for interdonor. ft separations less than 3.5a or 70A°. Thus, other models should perhaps be used to describe the concentration region above 2 x lO^^/cc. B. Peak Po s i t i o n of the Transitions The peak positions of the 2P o, 2P(_, and 3P( t r a n s i t i o n s , measured for various donor concentrations, are shown, i n figures 4.5a and 4,5b as functions of concentration. The peak positions are plotted r e l a t i v e to the low concentration values of 34.13, 39.17, and 42.50meV for the 2P , 2P + , 3P + , respectively. The larger possible errors on the 3P + t r a n s i t i o n data are due to the lower resolution of the scans made of this l i n e , the • influence of neighboring peaks and the presence of a sloping absorption back-ground. The s h i f t of peak po s i t i o n with concentration shows a behaviour s i m i l a r to that of the halfwidth: for a given concentration, the 2P y, 2P +, and 3P + l i n e s s h i f t d i f f e r i n g amounts (the 3P + l i n e having a larger peak s h i f t by almost a factor of 5). Thus, the s h i f t s are mainly due to changes i n the f i n a l states. Baltensperger's calculations suggest the p o s s i b i l i t y of a peak s h i f t to lower energies with concentration. The s o l i d l i n e i n figure 76. •.04 0 -.04 > £-08 > o % -.12 LL! O f— o X -.04 .08 1014 1015 5x1015 ip 1 6 5x10 16 1S(AH 2P+ T T : i T T T 1 1S(A)*2Po 1 A r i A, 17 10"/cc 39.17 meV 34.13 meV -Qr-0 10 20 30 1/5 4 0 4 PHOSPHORUS CONCENTRATION^ (10/cm) 50 Figure 4.5a Energy Shifts of Peak Positions vs. (Phosphorus Concentration) 1^ 3 for IS (A) -*• 2P Q and IS (A) + 2P + Transitions. Error bars represent the r e p r o d u c i b i l i t y of the experiments. 77. 14 15 i t r i o I J >2 0 cu o £ ~ l >~ CD LU ~ 4 2Z LU 2Z £ - 6 o X C L -8 0 7"~— 10 16 1S(A)-»3P+ o n 1fi 1*7 5x10 10 /cc 42.50 meV PHOS. CONC? / 3(104/cm) - o -O U Figure 4.5b Energy S h i f t of Peak Positions vs. (Phosphorus Concentration) 1'' 3 for IS (A) -v 3P + T r a n s i t i o n s . The s o l i d curve i s the Baltensperger estimate of peak s h i f t for a* = 26 A°. 4«5b indicates the Baltensperger value of peak s h i f t , from table 2.2, using an e f f e c t i v e Bohr radius of 26 A°. The f i t of the theory to the observed peak p o s i t i o n i s s i m i l a r to i t s f i t of the observed halfwidths -poor q u a n t i t a t i v e agreement. With the approximations used, Macek's c a l -c u l a t i o n of the donor-pair i n t e r a c t i o n gives no s h i f t of the peak p o s i t i o n The i n c l u s i o n of higher order terms may give a net s h i f t to the t r a n s i t i o n lineshapes. As the s h i f t of the peak, p o s i t i o n i s c l o s e l y r e l a t e d to the behaviour of the peak asymmetry and the lineshape, a c l o s e r examination of the s h i f t s w i l l be made i n a l a t e r s e c t i o n . C. Absorption C o e f f i c i e n t s - Evidence for a Conduction Band T a i l From the p r o f i l e s of the absorption peaks the integrated J-Ii i_ O j l tions i n the phosphorus concentration region 10"^^ N:^  ^1.2 x lO^^/cc. The area under each absorption c o e f f i c i e n t p r o f i l e was c a l c u l a t e d over an energy i n t e r v a l of several halfwidths. The omitted area outside the i n t e r v a l was estimated using the method of Bhatia (1970). For t h i s c o r r e c t i o n the area was assumed to be that of a Fano or Lorentzian p r o f i l e . This assumption i s j u s t i f i e d l a t e r . A f t e r the c o r r e c t i o n s , described i n appendix C, were applied, the logarithm of the integrated absorption coeff ci e n t was p l o t t e d versus the logarithms of the donor concentration; t h i s p l o t i s i n d i c a t e d by the t r i a n g u l a r points of f i g u r e 4.6. For donor concentrations below 10"^/cc, the a n a l y s i s i s straightforward. In the s p e c t r a l region of i n t e r e s t ao other e x c i t a t i o n channels are present. Since the t r a n s i t i o n peaks are well separated, the 7 9 . . ] (—r~i—i 1 1—I~T"I 1 1—r~i' J U L_i IMPURITY CONCENTRATION ( CM~3) Figure 4.6 Integrated Absorption C o e f f i c i e n t vs. Phosphorus. Concentration for IS (A) -> 2P and ->- ?.P^  T r a n s i t i o n s , o ± ^ in d i c a t e the integrated area i n c l u d i n g background 0 i n d i c a t e the cal c u l a t e d value of [TT peak ax halfwidth] t y p i c a l error bars are shown. 2 80. area i n the wings of the Lorentzian p r o f i l e i s e a s i l y estimated. The main u n c e r t a i n t i e s are i n the determination of the impurity concentration and the instrumental c o r r e c t i o n . For donor concentrations greater than 16 10 /cc the 2 P + peak overlaps the 3 P q peak. Allowance for the integrated absorption of the 3 P q t r a n s i t i o n was made by assuming the r e l a t i v e t r a n s i -t i o n p r o b a b i l i t i e s of the 2 P , and 3 P l e v e l s are i n the same r a t i o f o r a l l ± o concentrations. An absorption background i n the 3 2 to 45 meV region, decreasing slowly for energies below the conduction band edge, i s included i n the estimate of integrated absorption, c o e f f i c i e n t s . The examination of fig u r e 4.6 allows several comments to be made. The s t r a i g h t l i n e f i t of slope 1.0 through both the 2 P and 2¥A data ~ o ± points shows that within experimental error, the integrated absorption c o e f f i c i e n t i« n r o n o r t i o n a l to the donor concentration f o r IO"*""' -14 .1 vx 1 6 2 x 10"1" /cc. Above t h i s upper l i m i t , the data points l i e above the l i n e a r extrapolation; f o r N, = 1.2 x 10^/cc the values of a. _ for both peaks r d i n t are more than 50% greater than those obtained from tbe extrapolated curve. 1.0 For a given concentration, the values of a. determined from the l i n e a r N, & ' xnt d dependence for the 2P and 2P, t r a n s i t i o n s are i n the r a t i o o ± a. (2P)/a. \2P.J) = .38 ± .02. The value i s i n s u r p r i s i n g agreement with xnt o xnt ~ Kohn Ts t h e o r e t i c a l estimate of 4.0/10.6 = .378, and agrees with the e a r l i e r 14 experimental value of .43 ± 25% for a concentration of 5 x 10 /cc (Bichard and G i l e s , 1962). From the expression (2.8) for the absorption c o e f f i c i e n t and i t s integrated value (2.9), the N^*° dependence of a i n t below 16 2 2 x 10 /cc implies that the dipole matrix elements J<i|T|f>| are constant 81. and r e l a t i v e l y unaffected by the i n t e r a c t i o n between neighboring donors. The changes i n O j n j . a t higher concentrations can be a t t r i b u t e d to changes ei t h e r i n the matrix elements of the t r a n s i t i o n p r o b a b i l i t i e s or to changes i n the density of a v a i l a b l e f i n a l s tates. These p o s s i b i l i t i e s are examined i n the next se c t i o n . The absorption c o e f f i c i e n t v/as measured for photon energies greater than the IS(A) ground state to conduction band threshold energy (45 meV) for several donor concentrations. For these t r a n s i t i o n s a broad featureless absorption i s expected. The absorption cross section G"(E) ( i . e . absorption c o e f f i c i e n t a(E) divided by the donor concentration N.,) i s p l o t t e d i n f i g u r e 4.7 as a function of photon energy. Data from other work i s included, where a v a i l a b l e . The -.large error bars, mainly the r e s u l t of u n c e r t a i n t i e s i n donor concentration and sample, thickness, give the maximum allowed v a r i a t i o n s of the r e l a t i v e positions of the traces. The uneven v a r i a t i o n s i n the spectrum are due to incomplete c a n c e l l a t i o n of atmospheric water vapor spectra. The graph shows that for energies greater than the low concentration i o n i z a t i o n absorption edge, the absorption cross s e c t i o n i s decreased with increasing donor concentration. For the d i f f e r e n t concentrations i n v e s t i g a t e d , the cross sections approach a common value at an energy approximately 10 meV above the threshold. From the comparison of 2? o and 2P+ t r a n s i t i o n broadening, i t was concluded that the IS(A) l e v e l i s r e l a t i v e l y unbroadened for concen-t r a t i o n s below 10"^/cc. Therefore, for a d i s c r e t e ground state, the measured absorption c o e f f i c i e n t should monitor the density of states of the conduction band. Two i n t e r p r e t a t i o n s of the data seem p o s s i b l e . The 3.0 28 2.6 to°22 ~Z0 z 218 \— Lul 1-6 CO co1.4 00 §1.2 o z « o P . 8 CL o .6 co CD < .4 T T T 14 0 0 5.6x10 /cc A A 1.7x10 .16 * 3.5x10 .16 4.8x10 ,16 * * 1.2x10 ,17 .2 1 3P0 3P+ 4R. 5B ; a 0 e * ? ° o o o o O c 4, ° ° o o 0 00, 'AA, A*A T A A ^ I + t t- •i- 4. x»-x-n -*C.B. . A P O / 34 36 38 AO 42 44 46 48 •PHOTON ENERGY (mev.) 5 0 52 54 55 Figure 4.7 Absorption Cross-section above the Conduction Band Edge vs. Photon Energy f o r Several Phosphorus Concentrations.. a (5 x llP-Vcc), b (5 x 10 1 5/c.c) from data of other workers-Error bars show r e l a t i v e positions of the curves. 83. expression for absorption coefficient (2.8) divided by the donor concen-tration suggests that either the matrix elements for the transition probability decrease with increasing concentration, or that the density of final states at the conduction band minima is altered. The integrated absorption cross sections, (5". , for the int 2P q and 2P+ transition regions have been evaluated from the data of figure 4.6 and are plotted in figure 4.8 as a function of concentration. It is convenient to consider the cross sections at energies both above and below the conduction band edge. If the data are interpreted in terms of changes in the IS (A) -*- 2P and IS (A) -* (conduction band) transition proba-bil i t i e s , these changes are opposite to each other: the matrix elements connecting the ground state to the discrete levels increase with concen-tration and those connecting the ground state to the continuum Bloch states decrease. This must be due to changes in the final states. Little work has been done theoretically to estimate the influence of donor-donor interaction on the transition probabilities of donor levels. A similar problem is the difference in the transition probabilities for different donor species. Bebb and Chapman (1967) have considered this problem and, using both the QDM and the EMA, have calculated 2 the dipole matrix elements j <vs|T| 2P>| as a function of the quantum defect parameter, v, (i.e. as a function of the ground state energy). Baltensperger*s broadening calculation is similar to the QDM in that the donor-donor "overlap" is treated in terms of a change in the principal quantum number from n to vn where v is the defect. If this model were 8 4 , 0 7 6 5 4 2 1 0 9 TTPXHV 2 0 , u m t i — i r~rTT~r • j — i — T T T T T T " a j T u La n n 1SA>*2P0 IJ r r Cl n n 6 I l 1 I I 1 i I i i M I 10 16 10 17 IMPURITY CONCENTRATION ( C K 3 ) Figure 4.8 Integrated Absorption C r o s s - s e c t i o n vs. Phosphorus Concentratio f o r 2P and 2P T r a n s i t i o n s , o ± t o t a l c r o s s - s e c t i o n i n c l u d i n g background o - rr/2 (peak x h a l f w i d t h ) a - cone, of 1,05 x 10^^/cc b - cone, of 5.7 x 1 0 1 4 / c c H o r i z o n t a l l i n e s i n d i c a t e constant c r o s s - s e c t i o n s . 8 5 . valid, the effect of donor-donor interactions on the transition probabil-ity could be estimated from the matrix elements |<1S|TJV2P>| where v is the defect parameter due to broadening of the 2.P level. An estimate of 2 2 this calculation was made assuming that | <1S | T | \>2P> | ^ j <v>s | T | 2P> | . For a mean broadening of 1 meV of the 2 P + ^ level, the Baltensperger value of v = .96, and the Bebb and Chapman calculation yields a possible change in the transition probabilities of ^±20% for the QDM5 and <10% for the EMA. The estimate is crude, and may not be meaningful. The Baltensperger model was found to overestimate the concentration broadening, and the QDM over-estimates the differences in the cross-sections of the I S ->• 2P transitions o in A and P donors due to the energy differences of the ground state.^ The donor-pair model suggests that the donor-donor inter-actions produce a shifting of the 2P states from their low concentration energy values. White (1967) has investigated the cross sections of tran-sition lines in boron-doped silicon, compensated with phosphorus which were split by a Stark electric field. The changes in the observed cross section with the Stark splitting were attributed to changes in the neutral boron concentration, and the transition probabilities were assumed to be unchanged. If a change in the transition probability is the proper explan-ation for the observed behaviour below the ionization edge, the behaviour of the cross section above the ionization energy is s t i l l unexplained. If the conduction band states are Bloch states of the intrinsic lattice i t is ~*The actual result gives a decrease of 20% for a v decreased from 1.0 to .96. ^ QDM predicts the ratio of As to p[lS 2 P ^ transition probabilities to be 1 to 1.5. The experimental result of section 4.3 is a ratio of 1.0. improbable that the t r a n s i t i o n p r o b a b i l i t i e s to these states should have a concentration dependence. The second a l t e r n a t i v e i s to assume that the density of f i n a l states for t r a n s i t i o n s above and below the i n t r i n s i c conduction band edge has a l t e r e d . This appears to be a more consistent hypothesis. If the change of density of states were due to a p a r t i a l "smearing" of the conduc-t i o n band states into the region below the i n t r i n s i c edge, the absorption cross s e c t i o n below the edge would increase, with a corresponding decrease. i n cross section above the threshold. Experimentally, at energies below 1 6 45 meV and for concentrations greater than 2 x 10 ' /cc, the excess cross s e c t i o n can be associated with the presence of a slowly varying, featureless background, whose absorption increases with photon energy; i t s features To pursue the hypothesis further, the estimated height of the background was subtracted from the a versus E plots f o r concentrations above 2 x 10^^/cc. The remaining absorption was a t t r i b u t e d to the IS(A) •+ 2P l e v e l t r a n s i t i o n s . Their peak height and halfwidth, with the background removed, were ca l c u l a t e d and the integrated absorption c o e f f i c i e n t , taken to be ( 12 x peak height x halfwidth) was evaluated. These values for the integrated absorption c o e f f i c i e n t , and for the integrated cross section are shown by the open c i r c l e s i n figures 4.6 and 4.8. Within the accuracy of the estimate, the cross sections without background, l i e along the h o r i -zontal extrapolation of a concentration independent cross section. From the d i f f e r e n c e between the t o t a l integrated cross section (a. ) and the concen-o v m t t r a t i o n independent cross section, the excess absorption c o e f f i c i e n t at the 87. 2P and 2P, transition energies was evaluated. These values were taken to o ± be the height of the background at 34 and 39 meV and are listed in table 4.2. Donor Background Absorption Coefficient (cm "*") Concentration - N, d 34 meV 39 meV <2 x 1 0 1 6 / c c 0 0 2.4 i n 1 6 x 10 0 - 4 2 - 1 0 3.3 x 1 0 1 6 2 - 1 0 8 - 20 3.5 x 1 0 1 6 ' 3 - 1 0 1 0 - 30 4.8 i n 1 6 x 1 0 8 - 2 0 20 - 50 1.2 x 1 0 3 2 - 5 5 5 0 - 90 Table 4.2 Estimated Background Absorption Coefficient for Various Phosphorus Concentrations. It is seen that the background absorption decreases by a factor of about 1/2 from 39 to 34 meV for * 10^^/cc; this decrease is slowly varying compared to the. discrete donor transitions. The background absorption coefficient appears to increase with concentration by a factor greater than N,^ "^  but its actual concentration dependence was not measurable. The values Q of background cross section are included in figure 4.7. The area of the background t a i l , suggested by the values at 34 and 39 meV, is comparable with the cross section decrease above the ionization threshold. It is 88. assumed that the s h i f t i n g of the density of states of the conduction band i s the dominating e f f e c t and that possible changes i n the matrix elements of the t r a n s i t i o n p r o b a b i l i t i e s are of l e s s e r importance. The p o s s i b i l i t y that the excess integrated absorption c o e f f i c i e n t i n the 2 P and 2 P , t r a n s i t i o n energy regions i s due to a o ± "smearing" of higher l y i n g l e v e l s ( 3 P Q , 3 P + , 4 P q . . . ) was investigated. The integrated absorption c o e f f i c i e n t was measured i n the energy i n t e r v a l from 3 2 to 4 3 . 5 meV, the photon energy range corresponding to t r a n s i t i o n s to the 2 P , 2 P , , 3 P , 3 P . and 4P l e v e l s . At low donor concentrations the o ± o ± o r a t i o of the integrated cross sections of the sum of the 3P 3 3P and 4P b o" ± o l e v e l s to the cross section of the 2 P + l e v e l i s 0.90 ± 10%. I t was assumed that the absorption i n t e n s i t i e s of the higher l e v e l s ( 3 P q etc.) r e l a t i v e *-v-,.~ ?P 1 .----•} v - — ~~ J T . J f-y. - ~— ~.- — - t- A N-, r p u - - i - ^ p . - i - j n„ v ~ ground was removed from the estimates of integrated absorption c o e f f i c i e n t of the 2 P and 2 P . t r a n s i t i o n s . L i s t e d i n table 4 . 3 are the t o t a l observed o ± integrated absorption c o e f f i c i e n t s (a. ), the estimated contributions of to r int the various l e v e l s (o^p > A 2 P + ^ ' a n d t * i e discrepancy between the observed o and estimated t o t a l absorption c o e f f i c i e n t ^ ~ £«)• Comparison of l e v e l s i n the same sample eliminates errors due to sample thickness and impurity concentration. For the highest impurity concentrations, the r e s u l t s are l e s s accurate.^ Assuming no concentration dependence, of the t r a n s i t i o n The contributions of each l e v e l may be i n error as they are much broader, and the 3P + l e v e l may extend beyond the 43.6 meV cut o f f . Conversely the 4P, l e v e l may overlap into the region below 43.6 meV. 89. p r o b a b i l i t i e s , the r e s u l t s suggest that the. higher excited l e v e l s (3P , 3P +, 4P q) do not account f or the t o t a l background absorption. Concentration 5.5xlO" U 1 . 7 x l 0 1 6 3 . 5 x l 0 1 6 4 . 8 x l 0 1 6 1 . 2 x l 0 1 7 / cc Energy In t e r v a l (meV) 36-43,6 36-43.6 32.0-43.6 36.0-43.7 32.0-43.6 a. , 2,5±5% 135±5% 454±5% 560±5% 1400+10% a 2 p 60 180 1.30 60 150 200 500 a,„ . * 1.21 60* 150* 200* 500* 3P 3P, 4P o + o 1 Ea 2.5±5% 120±5% 360+10% 400±10% 1180±10% np n = 2, 3, 4 a. - Ea 0 15 90±40 160±70 220±200. mt np % discrepancy 0 M.2% ^25% V33% <50% Integrated absorption c o e f f i c i e n t s of sum of 3P Q, 3P +, 4P Q t r a n s i t i o n s are assumed equal to that of the 2P + t r a n s i t i o n . The upper l i m i t s of these values are noted. The units for the a's are cm "'"meV. Table 4.3 Estimated Contributions of Various Excited Levels to the To t a l Integrated Absorption C o e f f i c i e n t . For donor concentrations greater than 2 x 10 /cc the experimental data i s consistent with a model having a set of excited d i s c r e t e donor l e v e l s broadened by nearest donor i n t e r a c t i o n s , degenerate i n energy with states continuous i n energy, which are formed from the conduction band. Electrons i n the ground states are photon activated through the two p o s s i b l e e x c i t a t i o n channels - the P l e v e l s or the c o n t i n -uum levels= Above the i o n i z a t i o n threshold the change i n the conduction band density of states i s consistent with the observed decrease i n absorp-t i o n cross s e c t i o n . As noted i n section 2.3 a number of t h e o r e t i c a l treatments of h i g h l y doped systems have suggested a density of states t a i l forming at the conduction band minima. Our model i s consistent with work, done by Bonch Bruevich (cf. F i s t u l , 1969) and also by Takeno (1962) who suggested that for weak perturbations, a t a i l on the conduction band edge should form because of a s h i f t i n g and rearranging of the. continuum l e v e l s . The perturbations to the conduction band edge are thought to be due to f l u c t u a t i o n s i n the c r y s t a l p o t e n t i a l a r i s i n g from the random donor d i s t r i bution. The energy associated with these f l u c t u a t i o n s i s small compared t the gap energy: a t a i l extending 10 meV below the conduction band minima i s l e s s than a one percent change i n the i n d i r e c t gap of s i l i c o n . A background absorption has been observed i n the external spectrum of boron-doped s i l i c o n (White, unpublished). The background, 16 observed at higher dopings (^1.2 x 10 /cc) was a t t r i b u t e d to multiphonon t r a n s i t i o n s (absorption of one photon with the emission or absorption of one or ip.ore phonons and the e x c i t a t i o n of a h o l e ) . This process i s weak for impurities i n group IV materials, and u n l i k e l y to produce a sloping, concentration dependent background., Multi-phonon processes also cannot explain the decrease i n cross section above the. i o n i z a t i o n energy. In the following s e c t i o n , the lineshapes w i l l be evaluated i n the l i g h t of the possible two channel e x c i t a t i o n s . D. Lineshapes The lineshapes of the IS (A) -> 2P q and IS (A) -> 2P + t r a n s i t i o n s have been examined i n d e t a i l . As noted e a r l i e r , for donor concentrations greater than 2 x 1 0 ^ / c c the lineshapes, superposed on a background absorption, are asymmetric with a low energy t a i l . Several lineshape parameters have been evaluated: the halfwidth VJ, the peak p o s i t i o n E , the background absorption, the integrated absorption coe-f f i c i e n t , a. » and a measure, of the asvmmetrv. 6. ,, and 6.. defined i n t " ' 1/4' i n s e c t i o n 4.1 In f i g u r e 4.9b the values of and <5^y2 versus h a l f -width are p l o t t e d for the 2P + t r a n s i t i o n . Figure 4,9c shows versus halfwidth for the 2P t r a n s i t i o n . I t i s noted that the S's are l i n e a r o i n halfwidth and the graphs of 5__/4 ^ o r an<^ * m v e t* i e s a m e slope and i n t e r c e p t . For the 2P + l e v e l the s t r a i g h t l i n e of the P^ o t ^ a s 1/3 the slope of the S-^/^ P^-ot ( s e e appendix B), For peaks narrower than 0.15 mev (or concentrations l e s s than l O ^ / c c ) the asymmetry i s n e g l i g i b l e . P l o t t e d i n f i g u r e 4.9a are the s h i f t s of the 2P q and 2P + peak p o s i t i o n s versus halfwidth. Again, the s h i f t s are n e g l i g i b l e f or halfwidths l e s s than 0.15 meV; a l i n e a r dependence of s h i f t versus halfwidth i s evident for the broader peaks. > cu E L i — i x CO < UJ CL .6 .8 1.0 HALFWIDTH (meV) 1.2 Figure 4.9a S h i f t of the 2P and 2P + Peak Positions vs. Halfwidth. Difference between data points and s t r a i g h t l i n e i s E^. Figure 4.9b and c Asymmetry (5., and 6, vs. Halfwidth for 2P, and 2P „ . -i' 1/4 1/2 ± o Lines Respectively. Slope of curve i s 1/3 the slope of. 6^^; slope of the 6n ,,. curves for 2P and 2P, i s the same. 1/2 o ± X = 1.2 x 1 0 1 7 / A = 4,8 x 1 0 1 6 / c c 0 = 2.4 x 10 1 6/cc cc HALFWIDTH (meV) Figure 4 . 9 c 9 4 . As noted e a r l i e r , a two channel e x c i t a t i o n of donor electrons i s consistent with experimental observations. A two channel e x c i t a t i o n from a common ground state cannot be regarded as a simple sum of two independent processes. As discussed i n s e c t i o n 2.3 a configuration i n t e r a c t i o n between channels i s p o s s i b l e ; the i n c l u s i o n of an i n t e r a c t i o n between the channels produces a c h a r a c t e r i s t i c asymmetric lineshape described by the so-called Fano function. To obtain a comparison of the t h e o r e t i c a l lineshape with the experimental lineshape, several approximations are necessary. It i s assumed that configuration i n t e r a c t i o n occurs between the d i s c r e t e channel (the excited 2P , 2F donor l e v e l s ) and the continuum channel (the postulated o i density of states t a i l ) . The photon-induced t r a n s i t i o n s to each state can be expected to possess a Fano lineshape. characterized by the parameters T. Q and (see section 2.3) As noted i n the donor-pair approximation, the concentration broadening i s inhomogeneous. The resultant lineshape i s a superposition of l i n e s at s l i g h t l y d i f f e r e n t energies; i . e . the broadened l i n e i s considered to be a band of c l o s e - l y i n g sharp states. Each sharp state possesses the Fano lineshape and the t o t a l lineshape i s treated as a convolution of a broadened envelope function and the Fano lineshape of the sharp s t a t e . The observed lineshape i s a s p a t i a l superposition of trans-i t i o n s from each donor. The background absorption may be composed of t r a n s i t i o n s to q u a s i - l o c a l i z e d l e v e l s , discontinuous i n energy, when c o n s i -dered over a small region of the c r y s t a l . The s p a t i a l summation of .these t r a n s i t i o n s over the whole c r y s t a l would then produce the observed back-ground absorption, which i s continuous i n energy, It i s assumed f or the sake of s i m p l i f y i n g c a l c u l a t i o n s , that the s p a t i a l l y averaged parameters for the continuous background can be used. Shibatani and Toyozawa (1968) reformulated Fano 1s theory f o r semiconductors and considered the energy dependence of V, Q and and redefined them as s p a t i a l l y and energy averaged values* These assumptions have been used by others to compare t h e o r e t i c a l curves with experimental data (Fano, .1961; J a i n , 1965; Bhatia, 1970). The experimental lineshape i s assumed to be the convolution of a Fano lineshape and a Lorentzian envelope function, plus a slowly varying background. The Lorentzian envelope i s used because at concentra-1 ft 8 *-h"~ "* C1- 'c". rh^ « ! > c ! p y : , ' ? ^ l i , r , . » o K " . i>fs r-y z> 1pv.r - ; T - A, o i n appendix B.2 the convoluted Fano function parameters F, Q, H and E ^ can be r e l a t e d to the experimental parameters (the r a t i o of the integrated absorption c o e f f i c i e n t of the d i s c r e t e t r a n s i t i o n to the background, a ^ n t / a Q ; the halfwidth, W; the peak height, P, the asymmetry f a c t o r , 6n and the peak s h i f t , A E q ) by the following r e l a t i o n s h i p s : ' __int = fjLnt = i ^ 2 - 1 ^ ' ^-10) a a 2 o o A E = r+K - E ' ( E ) , (4.11) • ° 2Q The convolution'of Lorentzian and Fano functions can also be cal c u l a t e d r i g o r o u s l y . 96. 6l/4 = 3 6 l / 2 " 3 ( n ' H ) > ( 4- 1 2> a. — TT PxW, (4.13) i n t W = T+H for o/»l (4.14) Here, a Q i s the average background a b s o r p t i o n c o e f f i c i e n t at the energy of the oeak maximum. I t was noted that the c a l c u l a t i o n of the i n t e g r a t e d ab s o r p t i o n c o e f f i c i e n t f o r the 2P and 2P, t r a n s i t i o n s from expression f o ± (4.13) y i e l d e d values which l a y on the l i n e a r N^ "*"^  e x t r a p o l a t i o n of the a. values at lower con c e n t r a t i o n s (points i n d i c a t e d by open c i r c l e s i n i n t r J f i g u r e s 4.6 and 4.8). This i s c o n s i s t e n t w i t h our model, as i s the obser-ved r e s u l t that the slope of 8. ,. versus W i s 3 times the slope f o r 6, / n 1 / 4 • 1/2 versus V.7 ( f i g u r e 4.10b). The parameter r+H was determined from the h a l f w i d t h , Q from the asymmetry measurements, E ^ ( E ) from the s h i f t of the peak p o s i t i o n and F from the r a t i o of a. /a . The s i g n of the parameter Q was det e r -m t o mined from the s i g n of the peak s h i f t and the asymmetry; the t a i l on the low energy s i d e of the l i n e i n d i c a t e s a negative Q. I t i s noted that i f the parameter E ^ = 0, the s h i f t of the peak p o s i t i o n should equal (T+H)/2Q which i s i n d i c a t e d by the dashed l i n e i n f i g u r e 4.9a, E ^ = E - F ( E ) represents a s h i f t of the resonance p o s i t i o n w i t h respect to the s i n g l e channel peak p o s i t i o n E , and i s only zero i f the i n t e r a c t i o n between the 9 d i s c r e t e l e v e l and the continuum i s independent of energy. As the continuum background i s s l o p i n g , F ( E ) i s expected to be non zero, as suggested by the s h i f t data. The estimate of a. /a (a. ^  from f i g u r e 4.6, a . from i n t o i n t o 97. table 4 . 2 ) are the least accurate measurements, and yield only an order of magnitude estimate of F, The value of Q is more accurate. For values of | Q| less than 1 2 the estimate was accurate to ^10%. For JoJ greater than this value, the accuracy was less. For |Q|^>20 the differences between the Fano line and the Lorentzian line ClQl"^) were not detectable. The expression for a convoluted Fano function assuming a 9" linearly varying background of slope X per meV , is given by F Q , r + H ( E ) * r~2 TT(H+D Q - 1 + 4 Q r+H 1 +(_?_ (E-Ef)l (r+H • > TTT (1+X(E-Ef)) ( 4 . 1 5 ) -To obtain a more accurate value of T , the values of the other parameters, r+H, Q and E^ were inserted into expression ( 4 . 1 5 ) , and values of V consistent with the a. /a constraint ( 4 . 1 0 ) were substituted. Comparison xnt o of the computed and observed lineshapes, with peak heights normalized to 1 . 0 , yielded a better estimate of Y. The values of the parameters used in the comparison are listed in table 4 . 4 for lineshapes at several concentra-tions. Figures 4 . 1 0 , 4 . 1 1 and 4 . 1 Z show the lineshapes; the solid lines represent the convoluted Fano function and the points are experimental. The dashed line in figure 4 . 1 2 represents a Lorentzian line for comparison. As the background is slowly varying across the discrete transition peaks the linear slope should be a good approximation. Concentration H + r Q r E f / cc (meV) (meV) (meV) 1.2 x 1 0 1 7 1.05 -6 .12 -.06 4.8 x 1 0 1 5 .80 -7.5 .09 -.04 3.5 x 1 0 1 6 .47 „g .09 -.01 2,4 x 1 0 1 6 .30 -10 .09 0 1.7 x 1 0 1 6 .27 -11 .09* 5 x 1 0 1 5 .12 <-30 <.09* 14 5 x 10 .06 <-30 2? u <.09* i | 1.2 x I O 1 7 .26 • -8 .02 -.04 4.8 x 1 0 1 6 .21 -10 .03 -.02 2.4 x I O 1 6 .15 -15 • .03 -.01 5 x 1 0 1 A .06 <-30* ^Ele c t r o n phonon i n t e r a c t i o n contributes s u b s t a n t i a l l y to th i s value. Table 4.4 Parameters for the Convoluted Fano Function - Best F i t With Experimental P r o f i l e s . 99. The theory of configuration interaction suggests that an antiresonance should occur at the high energy side of the observed absorption peaks. For |Q| greater than 5 and with the Lorentzian envelope halfwidth, H» T, the antiresonance is weak and broad. Strong water vapor absorption in tbe 34.4 to 34.6 meV region"^ make experimental observation of the weak antiresonance unlikely for the 2P_ transition. Similarly, the 3 P q transition makes the analysis of the high energy side of the 2 ? + transition difficult. Several computer corrected scans did suggest the possibility of an antiresonance, but the results were not sufficiently consistent for further comment. The comparison of the configuration interaction calculation with experimental results shows them to be consistent. The presence of a asymmetric lineshape due to configuration interaction. The parameter Q, which is the ratio of <$[T|i> , (where the numerator is the dipole uV*<^E|f|i> matrix element connecting the discrete final and init i a l levels, and the denominator is a product of the discrete final state-continuum coupling energy and the dipole matrix element connecting the continuum and i n i t i a l levels) is determined from the peak asymmetry. |Q] decreases with increas-ing concentration. This is expected from the definition of Q\ <4>E|T|i> * and irV , the coupling between the excited level and the continuum, are E expected to increase as the background absorption increases. For "^Effects of incomplete cancellation of water vapor absorption are evident in figure C.3. I O C . concentrations less than 2 x 1 0 ^ / c c , |Q| is large and the lineshape becomes Lorentzian indicating that the configuration coupling is weak, I * i 2 The coupling interaction JTTV^  | is a measure of the lifetime of the discrete level excitation before decay via the continuum levels. In our f i t of the convoluted Fano function to the experimental lines, the effect of phonon broadening was not explicitly considered. In the large |Q| limit the lineshape becomes Lorentzian. Hence, at low concentrations, the phonon broadening is included in the T+H halfwidth estimates and T is largely a measure of phonon broadening. At high concentrations where the asymmetry and concentration broadening are dominant, the parameter T is 2 mainly a measure of the configuration coupling, (IVJ irV^ , j ). This suggests 2 that the {trV 1 part ofT increases with donor concentration which is 1 6 For concentrations greater than 2 x 1 0 /cc, where 2 — 1 1 n|Vg| = T, the configuration coupling limits the lifetime to ^  1 0 seconds for the 2P state and a slightly longer lifetime for the state 2 P q . This is comparable with; phonon relaxation lifetimesof 2 x 10 ^ seconds obtained from the zero concentration halfwidth measurements. At higher temperatures, the phonon relaxation lifetime decreases; with phonon processes dominating the relaxation, the configuration interaction should be less evident. A transition line asymmetric at 4 . 2 ° K (concentration 1.7 x 10^/cc) was found to be Lorentzian at 5 5 ° K with a phonon relaxation - 1 2 lifetime ^ 1 0 seconds. The sizes of the Fano parameters compare, well with the values found by Bhatia ( 1 9 7 0 ) for configuration interaction between the internal acceptor states of boron-doped silicon and the P3/2 valence band. r EXPERIMENTAL DMA O 0 EX 298 5X10XX14 *6 A fl EX38 2.4X10XX16 *101813 + E EX41 5.0X10XX16 *102399 V C EX 43 1.2X10XX17 10141 FANO PARAMETERS 1.2X10KK17 Q = -8 n= .02 rvH= .26 O 4.6X10SX15 a= - io . r= .03 T+H= .21 2P0 2.4X10SK16 Q= -15. r= .03 T+H= •15. 2PJ 5 X10BX14 Q= -30. r= .03 f>H= .06 2P0 2Po X X X * K • X „ X X V » X 32.6 - 1 32.8 —1 33.C -1 33.2 I 33.4 I 33.6 33.8 X.Q ENERGY rr.jV —T S<1.2 34.4 34.6 34.E I 35.0 1 35.2 35. Figure 4.10 Comparison of the Convoluted Fano Fui c t i o n with the Normalized Experimental P r o f i l e s of the 2P Tran s i t i o n s at. 6°K. Figure 4.11 Comparison of the Convoluted Fano Function with the Normalized Experimental P r o f i l e s of the 2P Transitions' at 6 K. CONC. 3 . 5 X 1 0 X X 1 6 1 . 7 X 1 0 X X 1 6 FANO PARAMETERS Q= - 9 . o r = . 0 9 T+H= Q = -11. r= . 0 9 r + H = LORENTZIAN LINESHAPE o CO 37.8 38.0 38.2 38.4 38.6 38.2 39.0 3:3.2 ENERGY lvn=iV] 39.4 3 9 . e 39. a 4C.0 40.2 40.4 40. S Figure 4.12 Comparison of the Convoluted Fano Function with the Normalized Experimental P r o f i l e s of the 2P + Transitions at 6°K„ 104. The experimental lineshape is well described by a convolu-ted Fano lineshape and the asymmetry and shift are explained in terms of configuration interaction. Although this explanation is consistent with all observations thus far noted, i t is not necessarily the only explanation. Indeed, Baltensperger's calculation produced a shift of the mid value of the 2P band to lower energies with increasing concentration, which may be interpreted as a shift of the lineshape to lower energies. Takeno (1962), as noted in section 2.2.4, suggests that at high concentrations, the impurity bands become asymmetric with a t a i l on the low energy side. Majlis (1967) has also calculated banding effects of shallow impurity levels. His result suggests a highly asymmetric peak with a tail to the low energy side, a sharp cut off on the high energy side, and a significant shift of peak position to higher energies, contrary to the direction of the observed shift. The donor-pair calculations were not carried out to sufficient order to ascertain asymmetries. Fistul (1969) suggests that the levels broaden symmetrically about the isolated impurity value. Unfortunately calculable lineshape functions are unavailable for a compar-ison with experimental data, to support these possibilities. It should also be noted that the assumption of the presence of a conduction band density of states t a i l is not a necessary condition for configuration interaction effects to occur. Jain (1965) suggests that some background intensity must always be present to allow transfer of energy from the photon-electron field to the lattice. Thus, configuration interaction is also expected to occur i f the observed background were due to multi-phonon processes. 105. E. Disappearance of Higher Excited-'Levels From the examination of the absorption spectra(figure 4.3) i t is seen that with increasing phosphorus concentration, the discrete levels nearest the ionization continuum (4P+, 5P+) broaden and transform into a featureless absorption continuum. Because the wavefunctions of the higher excited levels are more extended spatially than those for lower levels, they should become spatially non localized at lower concentrations. This is boras out by the calculations of Baltensperger and Macek. From ESR studies on phosphorus-doped silicon, i t is known that at sufficiently high 19 donor concentrations (~2 x 10 /cc) the IS band merges with the conduction band. The excited levels should merge with the conduction band at lower concentrations. Fano (1961) has considered the effect of spatial overlap on a kydberg series level scheme iii.'rate gas-solids. The QDM was utilised * ft with the energy of a particular level defined by E^ = -_R where R is (n-An)2 the effective Rydberg and An is the quantum defect. For large distances r away from the core, the bound electron moves in a coulomb field proportional 2 1 to e__. If the /r dependence of the potential is modified due to spatial Kr overlap with other potentials, the relationship is changed. If the poten-t i a l beyond a critical radius, r , remains at a value e^r^, the discrete • 2 # * * localized levels with n values (n-An) > r^/a (where a is the effective Bohr radius) should be replaced by a non localized continuum of levels. Using this criterion as an estimate of the disappearance of excited levels, a comparison is made with the observed spectrum. Using the QDM correction for the ground state energy, the defect parameter for phosphorus is 106. An = 0.2; the average of the v a r i a t i o n a l parameters, a and b, (equation 2,14) gives a = 20.4 A°. Using the s i m p l e s t approximation that i s 3 given by r == 3 , the v a l u e of the p r i n c i p a l quantum number 0 4TTN, d r — + ,2) above which d i s c r e t e l e v e l s should no longer be. seen, "> a" (n . v c r x t i s c a l c u l a t e d f o r s e v e r a l donor c o n c e n t r a t i o n s . The values are l i s t e d i n t a b l e 4.5 as w e l l as the highest e x p e r i m e n t a l l y observed d i s c r e t e l e v e l . N J (donors/cc) r ( A ° ) r /a n c r x t Highest Observed 101'* 1340 66 8 6 P + , I 10"'^ 620 30 6 5P + I 1 0 1 6 290 14 4 4P + 3.5 x 1 0 1 6 190 9.3 3 3 P + S 3P 0 5 x 1 0 1 6 . 1 7 0 8.2 3 3P 1 0 1 7 135 6.6 <3 ^3P 5 x 1 0 1 7 80 3.8 2 + + Table 4,5 C a l c u l a t e d and Observed Values f o r the P r i n c i p a l Quantum Number ofth e Highest Observable D i s c r e t e L e v e l The r e s u l t s are i n reasonable agreement. The c r i t e r i o n used to estimate r ° o assumes a maximum se p a r a t i o n of donors and should g i v e an upper l i m i t f o r n, The estimate suggests that l e v e l s as high as 8P be observable where i n f a c t 107. levels only up to 6P have been confirmed. For concentrations greater 17 than 5 x 10 /cc the. 2 P q and 2 P + levels should merge with the conduction band. The calculation raises the question whether the disappearing discrete levels may contribute to the hypothesized ta i l of continuum levels. 4.2.3 Higher Concentrations Optical absorption measurements using the fourier transform spectrometer were carried out on silicon samples with phosphorus dopings higher than 1 0 ^ / c c in the far infrared region from 5 to 35 meV. The results for samples at liquid helium temperatures are shown in figure 4.13, as well as data obtained from intensity transmission profiles for doped 9 uncompensated silicon at 2.5°K. The large absorption coefficients which saturated the transmission signals.prevented analysis in the higher photon energy regions. It is evident from the data- that a definite exponential absorption t a i l is present, which shifts to lower photon energies with increasing donor concentration. The slopes of the exponential edges for phosphorus and arsenic donors have no observable concentration dependence. The relative positions of the boron, phosphorus and arsenic absorption 18 edges for concentrations ^2 x 10 /cc are in the same order as their respective single-impurity ionization energies of 43.8, 45.3 and 53.5 meV.. Neuringer, L.J. and Milward, R.C. ( 1 9 6 7 ) . Appl. Opt. J5 978: The elec-tronic transitions giving rise to the absorption could not be identified. The profiles were shown for the application of doped silicon as a far infrared transmission f i l t e r . 108. The data cf f i g u r e 4.13 has a s i m i l a r q u a l i t a t i v e appear-ance to the behavior of the o p t i c a l a b s o r p t i o n edge of the d i r e c t gap of h e a v i l y doped I I I - V compound semiconductors at low temperatures (Pankove, 1965). The presence of an exponential a b s o r p t i o n edge i s con-s i s t e n t - w i t h the t h e o r e t i c a l models f o r d e n s i t y of s t a t e s t a i l i n g of the conduction band, which i s expected to increase w i t h c o n c e n t r a t i o n . As the i m p u r i t y c o n c e n t r a t i o n i s i n c r e a s e d , the donor wave-f u n c t i o n s i n c r e a s e t h e i r o v e r l a p , and f o r a s u f f i c i e n t l y c l o s e spacing, the ground s t a t e wavefunctions become s p a t i a l l y non l o c a l i z e d . I t i s expected that- the greater the o v e r l a p , the lower the a c t i v a t i o n energy (the energy necessary to e x c i t e a donor e l e c t r o n i n i t s ground s t a t e to q u a s i - l o c a l i z e d s t a t e s ) . An estimate of the a c t i v a t i o n energy may be found as f o l l o w s . In the donor p a i r approximation each e l e c t r o n , i n a ( I S , I S ) ground s t a t e , i s l o c a l i z e d on a separate donor s i t e . However., the a c t i v a t i o n energy may be a s s o c i a t e d w i t h photon-induced t r a n s i t i o n s to q u a s i - l o c a l i z e d s t a t e s such as the p o l a r or i o n i c s t a t e s of the donor p a i r , which have an e f f e c t i v e H-H*" e l e c t r o n i c c o n f i g u r a t i o n . These s t a t e s are neglected i n the donor p a i r c a l c u l a t i o n s of Macek ( s e c t i o n 2.2.3.B). I t has been suggested (Mott, 1961) that the "weight" of the i o n i z e d s t a t e s i n the donor wavefunction should i n c r e a s e as the. mean i n t e r a t o m i c d i s t a n c e i s decreased. We propose that these p o l a r (but not n e c e s s a r i l y conducting) s t a t e s are a s s o c i a t e d w i t h a d e n s i t y of s t a t e s t a i l and are " p u l l e d " down from the conduction band. The normal s t a t e of the e f f e c t i v e hydrogen molecule may be given by the Heitler-London model where the donor e l e c t r o n s are constrained 109. FAR INFRARED SPECTRAL REGION P-1.6x10 f l 17 / P-1.2X10 / 0 10 20 PHOTON ENERGY (meV) 30 Figure 4.13 Absorption C o e f f i c i e n t vs. Photon Energy i n the Far-Infrared Region for P, A , and B Doped S i l i c o n , s a - indicates data from transmission curves i n uncompensated s i l i c o n f o r Neuringer e t . a l , 1967, at 2.5°K. The s t r a i g h t l i n e s through the data points marked 1, 2, 3 are best, fi'.fcg 110. to t h e i r o r i g i n a l i o n cores. The e l e c t r o n i c energy of the normal s t a t e , E^, n e g l e c t i n g the exchange i n t e r a c t i o n s i s given b y 1 ^ E = -2E + 2J-+-J 1 ' (4.16) N O r 1+A where E q i s the i o n i z a t i o n energy of the i s o l a t e d hydrogenic donor; J =/ ' f ' l^(i) / - e ^ Vs ! '^(D dr, , the coulomb- i n t e r a c t i o n of a IS e l e c t r o n A V ~7^~l A x  k j " l B on donor A with donor core B (see s e c t i o n 2.2.3.B, f i g u r e 2.5, f o r the d e f i n i t i o n of v a r i a b l e s ) : J 1 = e/ J7 I S (1) ^ ^ S ( 2 ) ! 2 dr..dr„, the A B ' l / i K r i 2 ' coulomb i n t e r a c t i o n of a IS e l e c t r o n on donor A with a IS e l e c t r o n on donor i ~\ c 1S 1S 1 ^  l i ; and A = ffty, (1) ^  (2) < ( 1 ) < " ( 2 ) d r . d r 0 the overlap i n t e g r a l of A b D A .L Z the. IS hydrogenic wavef u n c t i o n s . S i m i l a r l y , the. e l e c t r o n i c energy of the 4- • e f f e c t i v e hydrogenic molecule i o n , R^, i s found to be E I = E o + ^ - (4.17)" i s ^ 1 s where J i s defined above and A = fip^ (1) <JJ^  ( l ) d r ^ . The e l e c t r o n i c energy of the p o l a r or i o n i z e d H H l e v e l , E + _ , i s the sum of the e l e c t r o n i c energy of the l i t i o n , E . minus the a f f i n i t y of the i o n i z e d e l e c t r o n f o r the 2 ' i ' J n e u t r a l donor neighbor, I £ e > plus the coulomb energy between the H i o n and + 2 the H i o n (=-e / K R ) , t h a t i s E ^ = E + J - I - e 2 • (4.18) +- o ——• - ee —„ 1+A KR • ^ P a u l i n g and Wilson (1935) give an ext e n s i v e study of the hydrogen molecule and r e l a t e d problems; the i n t e g r a l s mentioned above are evaluated i n d e t a i l . 111. t -2R/a" * N e g l e c t i n g terms i n J and J smaller than e ' ' ( v a l i d f o r R> a ), the d i f f e r e n c e i n energy, E^, i s found to be approximately = + E - I - e2/k-R (4.19) o ee The I term, the i o n i z a t i o n u o t e n t i a l of the H i o n i s 0.754 eV, which ee * i s much l e s s than the i o n i z a t i o n p o t e n t i a l f o r atomic hydrogen, E - 13.53 As I <<E , the e f f e c t i v e I term f o r the donor p a i r i s ne g l e c t e d , ee o ee As shown f o r intermediate c o n c e n t r a t i o n s , the assumption of a random donor d i s t r i b u t i o n gives b e t t e r agreement w i t h experiment than the Wigner-Seitz approximation of e q u a l l y spaced donors. R e c a l l i n g the Chandrasekhar expression (2.33) f o r W(R), the p r o b a b i l i t y of the nearest neighbor s e p a r a t i o n R, a. mean value tor the. decrease i n a c t i v a t i o n energy f o r a random donor d i s t r i b u t i o n i s found to be oo E = E - . /Ve 2\4irR 2 N, exp( -4TTR 3 N,\ dR o l icR = E - <a> N . 1 / 3 (4.20) o d From &_ = 30 meV and a* = 20.4 A°, e^ = 1,23 x 10~ 5 meV cm. Thus, * K ica E = E - 2.7 x 10" 5 N,1''3 meV (4.21) A o d I t i s noted that the a c t i v a t i o n energy f a l l s to zero at a c o n c e n t r a t i o n 18 of 4.8 x 10 /cc. .Other experimental measurements i n t h i s c o n c e n t r a t i o n r e g i o n (summarized by Alexander and Holcomb, 1968) are c o n s i s t e n t w i t h a t r a n s i t i o n from s p a t i a l l y l o c a l i z e d to non l o c a l i z e d ground s t a t e s f o r 18 phosphorus co n c e n t r a t i o n s i n s i l i c o n around 3 x 10 /cc. 112. A straightforward comparison of the optical data to theoretical models is difficult; changes in the donor ground states should be considered as well as changes in the final states. To explain the experimental data, the theoretical expression derived by Bonch Bruevich (1970) is first considered. The interband optical absorption coefficient at low temperatures was derived for a highly doped semiconductor with .the impurities producing a random force field perturba-tion to the regular periodic lattice. The energy spectrum of a disordered semiconductor treated this way consists of allowed and forbidden bands with a density of states extending somewhat below the conduction band edge. The transitions to the continuous spectrum states within the forbidden band were calculated for a "smooth" (non singular) random field. For the concentra-tion region where the Fermi level remains below the conduction band, the absorption coefficient was found, using a Green's function technique, to be a = a Q exp^A-h^l where A is the band gap, and w is a characteristic energy which is independent of temperature and is related to the gradient of the random potential. A semi-empirical expression, consistent with the experi-mental data is obtained for the absorption coefficient, a, by setting A equal to the donor activation energy, E ^ . Thus a = N,a exp f E - <a>NJ1/'3 - fiw} (4.22) d o . . J o d 1 -w is used to f i t the data of figure 4.13. Good agreement was found for the —18 2 phosphorus data using the values = 12.4 x 10 cm , <a> = 3.27 x 10" 1 5 meV cm, w = 2.75 meV"1, (and E q = 45.3 meV). Bebb and Chapman (1967) using the QDM have estimated the effect of the ground state energy on S ->- ? type t r a n s i t i o n p r o b a b i l i t i e s and suggest that the t r a n s i t i o n p r o b a b i l i t y f o r a r s e n i c should be smaller than the phosphorus value by 2/3. I f O " q i s a c c o r d i n g l y decreased by 2/3 and E q i s set to 2 8 53.5 meV, the a r s e n i c data at 2 x 10 ' /cc i s a l s o w e l l described by expres s i o n 4.19. The c a l c u l a t e d values of Os are i n d i c a t e d i n f i g u r e 4,.13 by the s t r a i g h t l i n e s f o r donor con c e n t r a t i o n s greater than 1.2 x 1 0 1 7 / c c . The f i t of the 1.2 x 10 1 7/cc data i s obscured by the presence of the IS (A) -> 2P ab s o r p t i o n peak. o The f i t t e d value of <a> = 3.27 x 10 ^ meV cm i s i n reason-~5 able agreement w i t h the c a l c u l a t e d value of 2.7 x 10 ^ mev cm. The c a l -c u l a t e d value of <a> i s l a s s than the f i t t e d v a l u e . The e f f e c t s of donor t r i p l e t and l a r g e r c l u s t e r s on the a c t i v a t i o n energy have not been'included i n t h i s estimate of <a> and should have the e f f e c t of i n c r e a s i n g the c a l c u -l a t e d <a> value. . I t i s of i n t e r e s t to make a comparison of the o p t i c a l a b s o r p t i o n data w i t h other experimental work. Since the a b s o r p t i o n edge i s w e l l described by the se m i - e m p i r i c a l expression 4.22, and the f i t t e d v alue of <a> i s c l o s e to the t h e o r e t i c a l v a l u e , i t i s assumed that the mean value of the a c t i v a t i o n energy d e s c r i b i n g the experimental data i s given by _5 expression 4.17 w i t h <a> = 3.27 x 10 meV cm. This curve i n f i g u r e 4.14 3 '3 i s compared w i t h other data. (The curve of E q - < a > N c j ' w i t h <a> = 2.7 x 1 0 ~ 1 5 i s a l s o i n c l u d e d i n f i g u r e 4.17). Pearson and Bardeen (1949) measured the e l e c t r i c a l r e s i s t i v i t y and H a l l c o e f f i c i e n t s of boron-doped and phosphorus-doped s i l i c o n as f u n c t i o n s of temperature. They determined the energy r e q u i r e d to f r e e the e l e c t r o n (or hole) from the im p u r i t y i o n (Phosphorus Concentration) /cc Figure 4.14 Decrease i n Donor A c t i v a t i o n Energy vs. (Phosphorus Concentration)" 1 Solid l i n e i s an empirical f i t to the experimental absorption data <a> = 3.27 x 10 meV cm Dashed l i n e represents the calculated value of <a> = 2.7 x 10T5 meV x Indirect Gap Shrinkage - Balkanski et. a l . (1969) M . A H a l l Coefficient. Data - Pearson aid Barde.en (1949) £ * AlexanderJHolcomb (1968) * so that i t can c o n t r i b u t e to the c o n d u c t i v i t y . Their r e s u l t s f o r phos-phorus are i n d i c a t e d by the t r i a n g l e s i n f i g u r e 4.14. With the formation of a n o n . l o c a l i z e d ground s t a t e , the e l e c t r o n s can. c o n t r i b u t e to conduction even at low temperatures, which'explains the r e s u l t of Pearson and Bardeen at 1 0 1 8 / c c . The r e s u l t s of work on the EPR s i g n a l of phosphorus-doped s i l i c o n at low temperatures (Quirt and Marko, 1971) i n d i c a t e that at donor conc e n t r a t i o n s greater than 3 x 1 0 ^ / c c the IS (A) l e v e l s have overlapped s u f f i c i e n t l y to form a n o n - l o c a l i z e d band and the resonance s i g n a l i s that of non l o c a l i z e d e l e c t r o n s . B a l k a n s k i e t . a l . (1969), using t r a n s m i s s i o n measurements on h i g h l y doped s i l i c o n at 35°K i n the phosphorus c o n c e n t r a t i o n r e g i o n 6 x 10 to 4.9 x 10" /cc, measured the ab s o r p t i o n below the fundamental gap. A f t e r f r e e c a r r i e r a b s o r p t i o n was sub t r a c t e d , an estimate of the i n d i r e c t gap between valence and conduction bands was made, using a two parameter f i t of the d a t a . 1 " A f t e r c o r r e c t i n g f o r the B u r s t e i n s h i f t (the r i s e of the Eermi l e v e l i n t o the conduction band) a shrinkage of tha i n d i r e c t gap w i t h i n c r e a s i n g donor c o n c e n t r a t i o n was found, p r o p o r t i o n a l 1/2 to . The X's i n f i g u r e 4.14 i n d i c a t e . t h e .energy decrease of the 17 18 i n d i r e c t gap r e l a t i v e to the i n t r i n s i c gap; the. p o i n t s at 10 and 10 /cc • ]/2 • were- e x t r a p o l a t e d from the dependence of the gap decrease. Their r e s u l t i n d i c a t e s that the conduction band edge decreases i n energy (with i n c r e a s i n g concentration) which decreases both the donor a c t i v a t i o n energy and the i n d i r e c t gap. The B a l k a n s k i r e s u l t thus adds support to the 11 P a r a b o l i c band edges were assumed. 116. contention of the presence of an optical t a i l on the conduction band at high concentrations. To match the Balkanski data with the estimated activation energies for greater than lO^/cc, the IS (A) levels must rise in energy towards the conduction band. Thus, the probable explana-tion of the high concentration infrared absorption is that the transitions occur between ground states which rise, and quasi-localized or conduction band states which decrease in energy with increasing donor density. This is sketched in figure 4.15. It is evident that more experimental data are required before a clear interpretation of behavior in the high concentra-tion region is found. Low donor concentration High donor concentration Figure 4.15 Change in the Band Gap and Ground State of Doped Silicon with Concentration. 117. 4"3 Absorption'Spectra - Arsenic-Doped S i l i c o n The lineshape parameters l i s t e d i n s e c t i o n 4.1 were measured f o r the IS(A) -> 2P t r a n s i t i o n s i n arsenic-doped s i l i c o n . The o r e s u l t s are s i m i l a r to those f o r phosphorus-doped s i l i c o n . The computer c o r r e c t e d values of peak p o s i t i o n and h a l f w i d t h are p l o t t e d as f u n c t i o n s of As c o n c e n t r a t i o n i n f i g u r e 4.16. Goruk (1964) had p r e v i o u s l y examined the c o n c e n t r a t i o n broadening of As donors f o r 5 x 10"^ 4 N,4.1.2 x l O ^ / c c , d Some samples f o r t h i s previous data may have been a c c i d e n t a l l y s t r a i n e d i n ] 2 mounting, producing a s t r a i n broadened t r a n s i t i o n . " A d i f f i c u l t y w i t h samples i n v e s t i g a t e d w i t h low As co n c e n t r a t i o n was the presence of phos-phorus contamination. The 2P l i n e of As l i e s at 42,24 meV and the 3P, o ± line, of P l i e s at 42.5 meV. The overlap of the two t r a n s i t i o n s produce a IS . i 'J broadened . L i n e . T n e 10"" aonor/cc sample' " was found to have. 10% P i m p u r i t i e s and 90% A.s. At higher a r s e n i c concentrations the phosphorus contamination was n e g l i g i b l e . E x t r a p o l a t i o n of the h a l f w i d t h to zero a r s e n i c c o n c e n t r a t i o n gives a p h o n o n - p l u s - d i s l o c a t i o n broadening of 0.10 meV. D i s l o c a t i o n broad-ening should be independent of the chemical species of donor and i s assumed to be <0.03 meV (as f o r phosphorus). For As the phonon broadening estimate of the 2P t r a n s i t i o n i s <0.07 meV compared w i t h <0.03 meV f o r P donors, o This can be explained i n terms of d i f f e r e n c e s i n the electron-phonon c o u p l i n 12 p o i n t b of f i g u r e 4.16 13 p o i n t a of f i g u r e 4.16 118. > 4 e .3 a 9 Lu .1 Oh-I i, S f ( A s ) 1S(A)->2Po ' \* ' T/>, / I fli" ir 'r ' s ^ v i i T 1 l i ' i " ' IF. 1 1C As CONC. (/cc.) • is D A i i i A / + .04 h > 0 o i—« I— 1—-! g-JO A C L £ - . 0 8 -.12 0 Goruk (1964) 10 4 -b 4-20 „ + —<|>— As CONCENTRATION' ° ^1/S doVcm) 30 1/3 40 Figure 4.16 'Halfwidth and Peak P o s i t i o n vs. As Concentration The dashed l i n e i n d i c a t e s the. phonon and d i s l o c a t i o n broadening. The s o l i d l i n e i s the. c o n c e n t r a t i o n broadening f o r the donor paxr approx. a - phosphorus contaminated sample, b - s t r a i n e d sample. 119, An obvious d i f f e r e n c e between P and As donors are the ground s t a t e energies of 45.3 (P) and 53,5 meV (As). The Quantum. Defect Method allows f o r t h i s d i f f e r e n c e by a s s i g n i n g a quantum defect 6 = n - v f o r the IS(A) l e v e l s i I" where v =fE_,. , For phosphorous v = ,80'j;,l: f o r a r s e n i c K b s i v = .73 i .1. Bebb and Chapman (1967) have considered the e f f e c t cf the v a l u e v on. the c o u p l i n g of bound c a r r i e r s to a c o u s t i c phonons using a deformation p o t e n t i a l description-. I t can be shown, using the method of B a r r i e and Nishikawa (1963) that the phonon broadening h a l f w i d t h c o n t r i -b u t i o n , AW, i s p r o p o r t i o n a l to the c o u p l i n g strength parameter Y.. Bebb and Chapman have evaluated the r a t i o Y^/Y^ ^ f o r a c o u s t i c phonons as a f u n c t i o n of v, A strong v dependence f o r Y was found, I t was found f o r P donors broadening f o r As donors should be 1.6 times stronger than f o r P donors, or ^ .05 meV. The c a l c u l a t e d phonon + d i s l o c a t i o n broadening i s thus <.08 meV compared to <,10 meV determined from experiment. With the zero c o n c e n t r a t i o n h a l f w i d t h added to the t h e o r e t -i c a l c o n c e n t r a t i o n broadening of the donor-pair c a l c u l a t i o n of Macek, the s o l i d l i n e i n f i g u r e 4.16 i s obtained. The c o n c e n t r a t i o n dependence i s s i m i l a r to that f o r phosphorus. •The i n t e g r a t e d a b s o r p t i o n c o e f f i c i e n t f o r the c o n c e n t r a t i o n -j 1(3 1 0 i n t e r v a l 10 " <N^ <6 x 10 /cc shows a s t r i c t N^ * dependence. I n s u f f i c i e n t data was a v a i l a b l e to note any d e v i a t i o n from t h i s at the highest concentra-t i o n . The r a t i o of the i n t e g r a t e d a b s o r p t i o n c o e f f i c i e n t s of the 2P 1.20. t r a n s i t i o n of P to the i n t e g r a t e d c o e f f i c i e n t of As was found to be 1.0 ±0.1 f o r a l l concentrations measured,, as expected from the EMA. Although the a r s e n i c donor data i s i n s u f f i c i e n t to i n v e s -t i g a t e i n d e t a i l c o n f i g u r a t i o n i n t e r a c t i o n or conduction band t a i l i n g , the 2P q t r a n s i t i o n peak does show an asymmetry to the low energy s i d e ( l e s s n o t i c e a b l e than f o r phosphorus because of the greater phonon broad-ening) w i t h a peak s h i f t to lower energies. In f i g u r e 4.13 i t was noted t h a t arsenic-doped s i l i c o n has a s i m i l a r e x p o n e n t i a l absorption edge at high c o n c e n t r a t i o n s . The r e s u l t s f o r a r s e n i c donors are c o n s i s t e n t with the behavior examined i n greater d e t a i l f o r phosphorus donors. 4.4 Coapariduu wj_i.ii liixperxuifenLai Studies m s i m i l a r r i a t e r i a i s Experimental work, s i m i l a r to that of the present study, has been c a r r i e d out i n two other m a t e r i a l s ; antimony-doped germanium and boron-doped s i l i c o n . 4.4.1 Ge(Sb) The shallow donor l e v e l scheme of germanium i s s i m i l a r to that found i n s i l i c o n except that the IS l e v e l s are a nearly-degenerate doublet. The l e v e l s , l a b e l l e d i n the same bydrogenic n o t a t i o n , are shown i n f i g u r e 4.17b. N i s i d a and H o r i i (1969) have i n v e s t i g a t e d the a b s o r p t i o n s p e c t r a of Sb-doped Ge i n the concentration, r e g i o n 6 x 1 0 ^ / c c to 7 x 1 0 ^ / c c at 2.3°K. Because of the l a r g e r e f f e c t i v e Bohr r a d i i of 150 A° 121. and 90A° f o r 2P, and 2P l e v e l s , r e s p e c t i v e l y , the donor-donor e f f e c t s ± o r J begin, at lower concentrations i n germanium than i n s i l i c o n . The absorp-t i o n s p e c t r a of f i g u r e 4.17a i s taken from t h e i r work. The h a l f w i d t h s of the 2P and 2P t r a n s i t i o n s from the doublet ground s t a t e s show a o ± beha\'iour s i m i l a r to that i n phosphorus-doped s i l i c o n - the 2P + l e v e l s broadening w i t h donor c o n c e n t r a t i o n f a s t e r than the 2P l e v e l s . N i s i d a o and H o r i i r e p o r t , a l s o , -that w i t h i n c r e a s i n g c o n c e n t r a t i o n , the peak p o s i -t i o n s s h i f t s M i g h t l y ( l e s s than 0.1 meV) to lower energy. For concentra-t i o n s g r e a t e r than 5 x l O ^ / c c , the lineshapes become asymmetric w i t h a t a i l on the low energy s i d e . A background a b s o r p t i o n , appearing f i r s t at higher photon ene r g i e s , i s evident. The asymmetric lineshapes and back-ground are observable i n t h e i r data i n f i g u r e 4.17a. At a constant 2 6 en e x y y o £ 7 m eV i.l i e 1 > ackg r o u n d ab s o t n t i o n c o e f f i c i e n t increases as N, \... • ^ as opposed to the a b s o r p t i o n c o e f f i c i e n t s of the d i s c r e t e l e v e l s which v -l<° i n c r e a s e as N. d An e x p l a n a t i o n of c o n c e n t r a t i o n broadening i n Ge(Sb) was attempted by Pomerantz (1.970) i n terms of an i n t e r n a l s t a i n e f f e c t . This model was r e f u t e d and the data r e i n t e r p r e t e d q u a l i t a t i v e l y by Stoneham (1971) i n terms of an overlap of donor wavefunctions. Macek suggests that concen-t r a t i o n broadening estimates using the donor-pair formalism q u a n t i t a t i v e l y agree w i t h the experimental data f o r Ge(Sb)."^ I t was suggested by N i s i d a and H o r i i that the background i s due to formation of a conduction band P r i v a t e communication. .122. v r i 1 r : T 1 1 ' r>'-H H photon energy irtieV) • ^  Concentration dependence of she absorption spear.- of antimony in germanium at 2.3°K. 9 -8 7 6 5 J 3 2 A • \20 •I50 I GO •200 -500 -1000 250 4-f-300 A,A3 T 7T2\ • 3 p . i l • 2p.±l -3p.O B, I?, I C, C 3 xp.O E , E 3 Is IT) ts (S) 0 -J -E(meV) M/<) g V. Energy levels of antimony clonor ia gci'mia-munt. Optical transitions are also indicated. ! 0 1 5 IO" Sb concentiati'oa C ..; Line width o f the various excitation lines as a function of aniiinonv concentration. Figure 4.17 Absorption" Spectra i n Antimony-Doped Germanium at 2.3°K from Nisida and H o r i i (1969), Journ. Phys. Soc. (Japan) 26, 388. 123. d e n s i t y of s t a t e s t a i l . No other q u a n t i t a t i v e a n a l y s i s i s c i t e d to support t h i s h y pothesis, other than reference to luminescence s t u d i e s which show evidence of t h i s t a i l i n g . Considering the c l o s e s i m i l a r i t i e s of the shallow donor p r o p e r t i e s of Ge and S i (Koh:i, 1957) i t i s h i g h l y probable that the two systems can be explained i n terms of the same theor-e t i c a l model. B. S i ( B ) The t r a n s i t i o n s between the e x t e r n a l boron acceptor l e v e l s i n s i l i c o n l i e i n the same photon energy region as the phosphorus t r a n s i -t i o n s i n s i l i c o n , and have roughly the same, e f f e c t i v e Bohr r a d i u s . The behavior of boron i m p u r i t i e s at v a r y i n g concentrations has been s t u d i e d . 1 6 The s t u d i e s up to 1.2 x 10" boron/cc by White (1966) are d e t a i l e d , but the higher c o n c e n t r a t i o n s t u d i e s up to 2 x 10 /cc are l e s s so (Newman, 1956). I t was found that as the co n c e n t r a t i o n was i n c r e a s e d , the d i s c r e t e t r a n s i t i o n s were broadened and were e v e n t u a l l y r e p l a c e d by a f e a t u r e l e s s background continuum. For example, w h i l e the background increases i n prominence and the ab s o r p t i o n edge s h i f t s to lower e n e r g i e s , boron peak #4, 18 v i s i b l e f o r N^ ^ 10 /cc, does not appear to s h i f t , suggesting l i t t l e change i n the ground s t a t e energy w i t h doping. An e x p o n e n t i a l edge"'""' (see 18 f i g u r e 4.14) has been found f o r a high doping of 2.8 x 10 /cc i n the f a r i n f r a r e d which complements the data of Newman. This suggests that w h i l e the ground s t a t e energy remains independent of c o n c e n t r a t i o n , a valence band t a i l may be formed f o r l a r g e acceptor c o n c e n t r a t i o n s . White observed an an a b s o r p t i o n background at a boron c o n c e n t r a t i o n of 1.2 x l O ^ / c c which "^From the data of Neuringer, e t . a l . (1967), 124. was a t t r i b u t e d to multi-phonon processes. His data r e v e a l s no asymmetry of l i n e s h a p e s , and the data of Newman i s i n s u f f i c i e n t l y r e s o l v e d to detec such behaviour. CHAPTER 5 INCLUSIONS AND SUGGESTIONS FOR FURTHER ' EXPERIMENTAL'STUDY 5.1 Conclusions The f a r i n f r a r e d a b s o r p t i o n of photon-induced t r a n s i t i o n s i n the shallow donor l e v e l s have been s t u d i e d at l i q u i d helium temperatures i n the s p e c t r a l r e g i o n of 10 - 55 meV f o r phosphorus c o n c e n t r a t i o n s from 10"^ to 6 x 1 0 ^ / c c and a r s e n i c concentrations from 10''""' to 6 x l O ^ / c c . These s t u d i e s have y i e l d e d i n f o r m a t i o n on the concentration-dependent behavior of the e x c i t e d and ground s t a t e donor l e v e l s and the conduction bar edge. The conclusions drawn from the i n v e s t i g a t i o n a r e summarized below. With i n c r e a s i n g donor c o n c e n t r a t i o n (N^) up to 2 x lO'^/cc 1) the IS (TO ->• 2 P and IS (A) -> 2 P , t r a n s i t i o n s broaden: the d i f f e r e n c e i n o ± t h e i r t r a n s i t i o n h a l f w i d t h s i n d i c a t e that the f i n a l s t a t e s of the e x c i t a t i o n s are the dominant source of broadening; 2) the lineshapes of these t r a n s i t i o n s change from symmetric L o r e n t z i a n , to asymmetric Fane lineshapes having a low energy t a i l ; 3 ) the peak p o s i t i o n s of the 2 P Q , 2 P + and 3 P _ , . t r a n s i t i o n s s h i f t to lower energies - the 3 P + t r a n s i t i o n s h i f t i n g by a f a c t o r of ^5 more than the other measured t r a n s i t i o n s . 4) an ab s o r p t i o n background superposed on the d i s c r e t e donor t r a n s i t i o n s appears f o r conc e n t r a t i o n s g r e a t e r than 1 0 ^ atoras/cc; 126. 5) the i n t e g r a t e d a b s o r p t i o n c r o s s - s e c t i o n , i n c l u d i n g the background, i n the r e g i o n of the 2F and 2P, t r a n s i t i o n s i n c r e a s e s f o r N.. > l O ^ / c c , o + d 7 w h i l e the c r o s s - s e c t i o n f o r t r a n s i t i o n s from the I S ( A ) s t a t e to s t a t e s above the low c o n c e n t r a t i o n conduction band minisum decreases; 6) the observed i n t e g r a t e d a b s o r p t i o n cross s e c t i o n s of the I S ( A ) -> 2 P Q t r a n s i t i o n s f o r phosphorus and a r s e n i c donors are the same w i t h i n .experimental e r r o r ; 7) the t r a n s i t i o n to the highest phosphorus donor l e v e l (5P +) broadens and i s r e p l a c e d by a f e a t u r e l e s s continuum; the t r a n s i t i o n to the next l e v e l 4P^) f o l l o w s s u i t , 8) For phosphorus and a r s e n i c concentrations g r e a t e r than lO^'/cc an a b s o r p t i o n edge at low temperatures i s present below 30 mev, having an e x p o n e n t i a l energy dependence. The slope of the e x p o n e n t i a l edge has no observable c o n c e n t r a t i o n dependence, but the edge s h i f t s to lower energies w i t h i n c r e a s i n g donor d e n s i t y . The r e s i d u a l t r a n s i t i o n h a l f w i d t h , found by e x t r a p o l a t i n g to zero donor c o n c e n t r a t i o n i s explained i n terms of electron-phonon and d i s l o c a t i o n broadening. The d i f f e r e n c e i n t h i s zero- c o n c e n t r a t i o n h a l f w i d t h between phosphorus and a r s e n i c donors i s explained iiu terms of d i f f e r e n c e s i n the e l e c t r o n - a c o u s t i c phonon co u p l i n g a r i s i n g froffi t h e i r core p o t e n t i a l d i f f e r e n c e s . The Quantum Defect Method of t r e a t i n g the core e f f e c t s gives a reasonable estimate of t h i s phonon c o u p l i n g , but f a i l s to account f o r the i d e n t i c a l observed i n t e g r a t e d c r o s s - s e c t i o n s f o r the 2P t r a n s i t i o n of phosphorus and a r s e n i c donors. 127. The broadening of the 2P and 2P + t r a n s i t i o n s at intermed-i a t e P and As concentrations i s best explained by t r e a t i n g the problem i n terms of nearest-neighbor i n t e r a c t i o n s , w i t h the donor-pair s e p a r a t i o n determined by a p r o b a b i l i t y d i s t r i b u t i o n f o r random donor s i t e s . The broadening i s inhomogeneous; the donor-donor i n t e r a c t i o n produces a s h i f t i n g of the l e v e l s , and the random donor d i s t r i b u t i o n produces the broadened l i n e . The assumption of a random d i s t r i b u t i o n appears j u s t i f i e d on the b a s i s of the good f i t of t h i s model to the data. The d i f f e r e n c e i n broadening of the 2P and 2P l e v e l s i s due to the anis o t r o p y of the e f f e c t i v e mass of the o ± 1 J donor e l e c t r o n . For semiconductors w i t h a l a r g e r mass a n i s o t r o p y , such as germanium, the d i f f e r e n c e s between these l e v e l s should be g r e a t e r ; and t h i s i s so (data of N i s i d a and H o r i i , 1969). Although perhaps v a l i d f o r the acceptor l e v e l s i n s i l i c o n (Colbow, 1963; White, 1967; B h a t i a , 1970), the r e s u l t s of Baltensperger's broadening model g i v e poor agreement with the experimental data f o r donors. The changes i n the ab s o r p t i o n c r o s s - s e c t i o n s above and below the conduction band t h r e s h o l d and the appearance of a s l o p i n g c o n t i n -uum background at higher concentrations are c o n s i s t e n t w i t h the hypothesis of a d e n s i t y of s t a t e s t a i l forming on the conduction band. The observed peak s h i f t s and asymmetries are explained i n terms of a c o n f i g u r a t i o n i n t e r -a c t i o n between the d i s c r e t e donor and continuum e x c i t a t i o n channels as a r e s u l t of t h i s t a i l . The convoluted Fano lineshapes are found to be good d e s c r i p t i o n s of the experimental l i n e s h a p e s . The decrease i n the Fano para-meter Q and t h e r e f o r e the in c r e a s e i n | TTV j w i t h c o n c e n t r a t i o n , r e f l e c t the Ground s t a t e broadening i s found to be l e s s important. i n c r e a s e i n c o u p l i n g to the continuum channel (regardless of the channel's o r i g i n ) . The o b s e r v a t i o n of an ex p o n e n t i a l a b s o r p t i o n edge s h i f t i n g to lower energies i s c o n s i s t e n t w i t h a conduction band " t a i l i n g " e f f e c t . The i n t e r p r e t a t i o n of the d e n s i t y of s t a t e s t a i l i n terms of the p o l a r or i o n i c s t a t e s of the donor p a i r s allowed an estimate of the a c t i v a t i o n energy to be made. The decrease i n the a c t i v a t i o n energy, estimated, from, the a b s o r p t i o n edge data i s close, to the c a l c u l a t e d estimate. These r e s u l t s are compatible w i t h the observed decrease i n the i n d i r e c t gap -of phosphorus doped s i l i c o n ( B a l k a n s k i e t . a l , , 1969). The experimental r e s u l t s are i n general agreement w i t h the behavior of antimony donors i n germanium i n the eq u i v a l e n t c o n c e n t r a t i o n r e g i o n s , observed by N i s i d a and H o r i i , 1969, 5.2 Suggestions f o r F u r t h e r Experimental Work Further i n f r a r e d a b s o r p t i o n s t u d i e s i n the c o n c e n t r a t i o n i 7 r e g i o n above 2 x 10"" /cc would complement present work on doped s i l i c o n i n t h i s r e g i o n . The exp o n e n t i a l a b s o r p t i o n edge of both phosphorus and a r s e n i c donors should be examined i n d e t a i l up to concentrations where the o p t i c a l 18 "gap" between IS and conduction band s t a t e s approaches zero (3 x 10 < 19 < 2 x 10 / c c ) . The broadening of the IS (A) ->• 2P t r a n s i t i o n s and t h e i r expected t r a n s f o r m a t i o n i n t o a f e a t u r e l e s s a b s o r p t i o n continuum at these h i g h c o n c e n t r a t i o n s would be of i n t e r e s t . However, the development of t h i n sample (or t h i n f i l m ) p r e p a r a t i o n and mounting i s re q u i r e d before t h i s i n v e s t i g a t i o n , i s p o s s i b l e . For ab s o r p t i o n c o e f f i c i e n t s greater than 1000 cm i t may be p o s s i b l e to determine the a b s o r p t i o n c o e f f i c i e n t from the expres-s i o n f o r r e f l e c t a n c e (A.3 of appendix A) by measuring the r e f l e c t a n c e of a t h i n h i g h l y doped sample. 129. As the donor c o n c e n t r a t i o n i s i n c r e a s e d , i t i s expected that the higher donor l e v e l s become s p a t i a l l y non l o c a l i z e d , and even-t u a l l y form a band capable of conduction. I t would be of i n t e r e s t to i n v e s t i g a t e the n o n - l o c a l i z a t i o n c f the f i n a l s t a t e s of the photon-induced t r a n s i t i o n s at high c o n c e n t r a t i o n s . P h o t o - c o n d u c t i v i t y measurements of phosphorus-and arsenic-doped s i l i c o n i n the 5 to _35 meV r e g i o n f o r concen-17 t r a t i o n s >10" //cc should r e v e a l much about the s p a t i a l extend of the donor s t a t e s , and the s t a t e s producing the exponential t a i l . I t has been suggested that the random i m p u r i t y d i s t r i b u t i o n produces f l u c t u a t i o n s i n the p e r i o d i c p o t e n t i a l of the c r y s t a l . These f l u c t u a t i o n s are thought to cause the formation of o p t i c a l and/or d e n s i t y of s t a t e s t a i l s on the valence and conduction band edges (e.g. Bench-Bruavich, 1969: F i s t u l , 1969), I t may be. of i n t e r e s t to perform experiments on high c o n c e n t r a t i o n , doubly doped semiconductors", i n p a r t i c u l a r , on s i l i c o n doped w i t h two donors of d i f f e r e n t i o n i z a t i o n energy. A high con-c e n t r a t i o n of the "deeper" donor (% 10^/cc) would produce the p o t e n t i a l f l u c t u a t i o n s that a f f e c t the conduction band edge. The " s h a l l o w e r " donor t r a n s i t i o n s would be monitored to observe the changes i n the a b s o r p t i o n spectrum due to the conduction band t a i l i n g e f f e c t s . A lower c o n c e n t r a t i o n 16 of these shallower i m p u r i t i e s (<10 /cc) minimizes the lineshape broadening due to the " o v e r l a p " of the "shallower", donor wavefunctions, At s u f f i c i e n t l y low temperatures the "deeper" donors should remain n e u t r a l . The proper choice of chemical species of the donors would prevent the s u p e r p o s i t i o n of the a b s o r p t i o n s p e c t r a . The donor-pair treatment of Macek may a l s o be be extended to t r e a t the i n t e r a c t i o n between the double-doped donors. Rather than e v a l u a t i n g the molecule problem, the eq u i v a l e n t H-X molecule problem where X i s a hydrogenic atom w i t h d i f f e r e n t mass and core p o t e n t i a l may be evaluated to estimate the i n t e r a c t i o n between the two donor s p e c i e s . B h a t i a (1970) observed the i n f r a r e d a b s o r p t i o n s p e c t r a of s i l i c o n , doubly-doped with indium and boron f o r t o t a l c o ncentrations below 17 2 x 10 /cc. The samples were doped w i t h a c o n c e n t r a t i o n of indium (a deeper acceptor) higher than boron. I t was found that the highest energy t r a n s i t o n s of the e x t e r n a l l e v e l s of boron, broaden and are. replaced by an ab s o r p t i o n continuum w i t h i n c r e a s i n g indium doping. T r a n s i t i o n s to the i n t e r n a l l e v e l s of boron were s h i f t e d to lower energy, and the stre n g t h of the c o n f i g u r a t i o n i n t e r a c t i o n w i t h the P-^^ v a i e n c e band was changed. These r e s u l t s may be i n d i c a t i v e of the experimental e f f e c t s i n double-doped donors. APPENDIX A ' ' T H E'ABSORPTION COEFFICIENT OF A THIN *ABSORBING"SAMPLE The i n t e r a c t i o n of -ra d i a t i o n w i t h matter has been discussed i n chapter 2, T h e . r e l a t i o n s h i p between the r a d i a t i o n i n t e n s i t y t r a n s m i t t e d through, or r e f l e c t e d from a t h i n absorbing f i l m immersed i n a i r , and the ab s o r p t i o n c o e f f i c i e n t i s evaluated i n t h i s s e c t i o n . Consider a ray of l i g h t , c h a r a c t e r i z e d by i t s e l e c t r i c v e c t o r E , i n c i d e n t at an angle 6^ to a p a r a l l e l - f a c e d sample of thickness. •k d and complex index of r e f r a c t i o n n = n - i k as shown i n f i g u r e A . l . For ob l i q u e i n c i d e n c e , e l e c t r i c v e c t o r s p o l a r i z e d p e r p e n d i c u l a r and p a r a l l e l to the plane of i n c i d e n c e have to be considered. The r e f l e c t i v i t y r ^ i s defined as the f r a c t i o n of the i n c i d e n t amplitude r e f l e c t e d at i n t e r f a c e 1, Using Maxwell's equations and S n e l l ' s law of r e f r a c t i o n to evaluate the boundary c o n d i t i o n s at the air-sample i n t e r f a c e , i t can be shown"*" f " a s cos © \ - r t * cos OR, <cii= co Se r - n * c o s 6 i ( A ' 1 ) cos &i r\* cosOv c o 5 <£1 r +V \ v cesQl f o r p e r p e n d i c u l a r and p a r a l l e l p o l a r i z a t i o n s , where© i s the angle of r e f r a c t i o n . At i n t e r f a c e 2 r ^ = - r - j j _ and r ? ^ - "~rlj(S, * The change i n phase of the ray due to t r a v e r s i n g the sample i s now considered, See any standard t e x t on electromagnetic theory, e.g. R e i t z , J.R,, and M i l f o r d , F.J., (I960). Foundations of Electromagnetic Theory, CAddison Wesley) P. 321. Figure A . l R e f l e c t i o n and Transmission of a L i g h t Ray from a Thin, Paralle Sided, Absorbing Sample. -t , . ..7.'»S tg= Wv-r)e Ev -3«S & r ^ ? r (we Ei Table A . l M u l t i p l e R e f l e c t i o n and Transmission Amplitudes at the Air Sample I n t e r f a c e s . r = r e f l e c t i v i t y at air-sample i n t e r f a c e 1. 133. Figure A. 2 Optical Path Differences The construction of point G allows A to be written 2L\ = n (GB) - AD. Since AC and DB are successive wavefronts reflected from the lower surface, The difference in optical path due to traversing the sample once, A s - £ AD = n"(CB). The path difference A is thus, A " j£ ~ 3 VV? <SC = rfd cos 0Y z z and the phase difference S= ZTT A ^ 2TT n*d cos &r > where is the free space wavelength. The infinite number of reflected and transmitted amplitudes can be characterized by r and 8, and are listed in table A.l. Consider the sum of these amplitudes; for samples of high reflectance, the entire series or transmitted rays are summed. The relative refelcted amplitude r~r(\-r * ; e \ 1 +• (re ) + (.re J • . . . } 1 3 4 . - r 6 , e = r V \ -- e ; (A. 2; l - r e S i m i l a r l y ' } summation of the s e r i e s y i e l d s f o r the transmitted amplitude —» _ O-r'-je i I +[re j ( r e J + ... ( E-,(Go 1 . ' J (A. 3 ) I - r e To f i n d the r e l a t i v e i n t e n s i t i e s , the complex values f o r r and 8 are introduced f o r normal i n c i d e n c e ; the d i s t i n c t i o n between perpen-2 d i c u l a r and p a r a l l e l p o l a r i z a t i o n s disappears. Irl 2"- (l-nAW1 and -iS = -<xcl * 2Tr'md g ~ c d x i a ( A . 5 ) ____ e A o T 3 The a b s o r p t i o n c o e f f i c i e n t ot = 4 lTk . S u b s t i t u t i o n of A.4 and A.5 i n t o A.3 y i e l d s f o r the r e l a t i v e t r a n s m i t t e d amplitude £r = U-vx)e e J ( A . 6 ) The r e l a t i v e , t r a n s m i t t e d i n t e n s i t y i s given by 2 For the monochromator, the maximum angle of inc i d e n c e due to f o c u s s i n g was < 8 ° ; thus 9 R ~ 2 . 5 ° and cos 8 ° ~ cos 2 ° & 1 i n expressions A . l . 135. it I H i ) \-r%\ e MV"* where we have introduced the observable quantity, reflectance, (A. 7) r""A nr^foA o l on t.Vigt - l » - l r f ^ ) ( r - l r r e ^ ) 2,2 For most semiconducting materials, k « n and the k /n term is neglected. A similar evaluation for the relative reflected intensity gives Z I*. 3This is inherent in the assumption that the reflectances of the doped and intrinsic samples are the same. ] + 4 s w i 1 ( zirnd) j X<? (A. 8) 1 X., Consider now the transmitted f r a c t i o n of the in t e n s i t y . The 2 argument of the sin'C(J>~ ( 3 ) term i n equation A./ i s averaged over the sair.pl area i n the l i g h t beam; the averaging i s due to the va r i a t i o n of the thick-ness' d. the suecirral s l i t width of the beam A X and the s o l i d angle of o incidence dug to the focussing of the beam onto the sample. Removing the interference term by averaging the r e l a t i v e transmitted in t e n s i t y becomes 2T[ - jk Z-TT i 2.TT "fe [V +41? |_ -g, CA.9) where AS (1 - R ) 2 and B = e 0 ^ 2 - Re_* c d / 2. Expression A.. 9 i s the standard relationship between absorption c o e f f i c i e n t and transmitted intensity for p a r a l l e l sided absorbing samples. 137. APPENDIX B • PROPERTIES OF THE'FANO'FUNCTION The Fano f u n c t i o n was derived as the lineshape f o r a two channel e x c i t a t i o n i n . s e c t i o n 2.3. Several p r o p e r t i e s of the f u n c t i o n are evaluated below, B.l Convolution of a Fano'Function w i t h a L o r e n t z i a n 'Function The Fano f u n c t i o n F (E) can be w r i t t e n can i r r . L p ^ J 1 CB.l) where Q = | T [ i > (B.2) (see s e c t i o n 2.3) The L o r e n t z i a n f u n c t i o n L^(E) can be w r i t t e n L.,(E) = fL ( * ? (B.3) where H i s the peak width at h a l f maximum. The. c o n v o l u t i o n of the two f u n c t i o n s 5(e) J F Q R L U ( E - E i ) a£ - F;5P ( E 1 ) * • L U ( E - E O can be solved by the method of f o u r i e r transforms ( B h a t i a , 1970) or by the method, of r e s i d u e s . The l a t t e r method i s chosen to 'evaluate S(E). 138, Introducin- the v a r i a b l e s h = 11/2, t = P/2, x - E* - E^, y = E - E^ f o r s i m p l i c i t y , we consider the c o n v o l u t i o n of the f i r s t term A(E) of the Fano f u n c t i o n , Ate')* Lrtfe-e')- I f ! wt J C B . 4 ) - 1^-0 hi C9 1 Consider the i n t e g r a t i o n along the closed contour of f i g u r e B . l 4* I c it . i o \ F i g u r e B . l Contour of I n t e g r a t i o n . I t i s noted that O O 1 <V> d X  The four poles of the f u n c t i o n occur at x = 1 ih+y, as shown i n f i g u r e B . l . Only the two p o s i t i v e poles w i t h i n the contour are considered; the value- of the i n t e g r a l i s found to be dx 7 2 2 2 ( t / + x Z ) ( h Z + ( y - x ) Z ) = 2TT i = 2TV i sum of the. residues 1 - 2 i t ( y + i ( h -i C t ) ) ( y - i ( h + t ) ) 139, + -2ih Cy+i Ct+h)) Cy+i (h-t)) = IT U"-c ) ht Cv2+(t+h)2) (13. 5) In the l i m i t as the. r a d i u s , R, of the s e m i - c i r c l e extends to i n f i n i t y the i n t e g r a t i o n around the dashed curve C -*0, and equations B.4 and B.5 y i e l d A(E')*L w (E-E')« C M ( B . 6 ) L(t+hf. In a s i m i l a r maimer the co n v o l u t i o n of the second term, B(E), c f the Fano (B.7) The t h i r d term of the convol u t i o n i s obvious, and the convoluted Fano f u n c t i o n can be w r i t t e n F Q R J . E ' ) * L H (E -E 1 ) •=. & T(H*D -f Q*-l + 2 Q [ £ _ (E-E-)] ] Lr+t4 i i e 14- r JL- ( E - E ^ - 1 * L r-4-H •irr1 (B.8) B,2 P r o p e r t i e s Of the Fano Function The u s u a l lineshape parameter's, i n d i c a t e d i n f i g u r e B2 are r e l a t e d to the f , H and Q parameters of the Fano f u n c t i o n f o r the d i s c r e t e S o l u t i o n of ' _r|_ f ^ (<E)\^Q y i e l d s expressions f o r the peak p o s i t i o n i n g , E , and the antiresonance p o s i t i o n E . O K Eo = JZi I i + E? E R = - ( T H H ) Q + E . ? (B.9) The peak h e i g h t , P - F ( M + E ^ ^ ' l Q Z (B. 10) - + The energies E, ^  a n < l E ^ / 2 a r e r o u n Q w i t h the d i s c r e t e Fano f u n c t i o n at h a l f i t s maximum value: + P+H r i ±ua^z)\ i — I t V r 1 (B-U) The h a l f w i d t h W = | - El, | « J!+H_ \ 0^ +2 I f Q 2 » 2 , then W * P + M CB.12) 1 4 1 . A measure of the asymmetry of the lineshape at h a l f maximum i s the parameter , . (B.13) = (r+H)/Q S i m i l a r 1}', the asymmetry at one quarter the peak maximum i s given by \ = * s ^ r r t / Q CB..14) The r a t i o of the i n t e g r a t e d c r o s s - s e c t i o n of the d i s c r e t e t r a n s i t i o n , , to the background cross s e c t i o n at E , C e , was found i n s e c t i o n 2.3 to be ' The. i n t e g r a t e d area under the d i s c r e t e p o r t i o n ( i . e . n e g l e c t i n g the 2/jrp term) of the Fano l i n e , 0(.. , i s given by i n t p o r oi,. . = ( F (E) <i£ - G"M (B.I6) •it j Q,r+M -co Using the values B.10 and B.12 f o r the peak height and h a l f w i d t h OL. ™ X i f Q 2 » 2 (B.17) x n t £ • 142, APPENDIX C INSTRUMENTAL CORRECTIONS TO'ABSORPTION SPECTRA A q u a n t i t a t i v e a n a l y s i s of a b s o r p t i o n l i n e s p e c t r a r e q u i r e s the c o n s i d e r a t i o n of the e f f e c t the instrument has on the measurements. The major e f f e c t i s that due to the f i n i t e s p e c t r a l s l i t w i d t h of the spec-trometer; f o r t u n a t e l y , the e f f e c t can be minimized. For low donor concen-t r a t i o n samples, where the a b s o r p t i o n l i n e s are narrow, the c o r r e c t i o n i s most s i g n i f i c a n t . One attempts to minimize such problems by u s i n g the narrowest s p e c t r a l s l i t widths p o s s i b l e , m a i n t a i n i n g reasonable s i g n a l - t o -noise, r a t i o s . The a v a i l a b i l i t y of l a r g e memory, high speed, d i g i t a l computer i n recent years has allowed new and more accurate c o r r e c t i o n techniques to be used. These techniques are described below. C.I C o r r e c t i o n s f o r Instrumental Broadening of Absorption'Lines A w e l l e s t a b l i s h e d r e l a t i o n s h i p e x i s t s between the observed, p r o f i l e of an a b s o r p t i o n peak f (E), i t s true p r o f i l e f^CE), and the i n s t r u -mental p r o f i l e f (E). This i s The s p e c t r a l p r o f i l e of a monochromator i s defined as the a c t u a l energy d i s -t r i b u t i o n of the i n t e n s i t y from the e x i t s l i t f o r r a d i a t i o n from a q u a s i -monochromatic source. I t i s apparent that a knowledge of f g ( E ) should allow f T ( E ) to be estimated from the measured values of f^CE). Parsons (1968) has shown that the i n s t r u m e n t a l p r o f i l e f o r a L i t t r o w mounted g r a t i n g i s a Gaussian d i s t r i b u t i o n of the form ( C l ) o o MEASURED GAUSSIAN FIT • / / / 6 / / / / H 0 *7i B 2 0 4-\ \ ZZ 2.3 24 2.5 2.6 ENERGY (ARB.) 2.7 2.8 2.S Figure C . J . Convorxson of the S p e c t r a l SI.it P r o f i l e w x t h a Gau s s i a n ijinesna'pe. ENERGY Figure C.2 Deconvolution. Procedure f o r Absorption L i n e s . 144. (C.2) where 2 J ln2 i s the f u l l l i n e w i d t h at h a l f maximum. The above expression was confirmed by experimental measurements of f ^ ( E ) , The method used to s determine the s p e c t r a l p r o f i l e i s described below. A He - Ne laser.beam of wavelength 0.6328 microns was detected i n the 4 7 t h and 4 8 t h orders at 29.74 and 30.37 microns (-347 and 329 cm" 1). The l i n e s were measured at s e v e r a l s l i t w i d t h s . Also scanned at v a r i o u s s l i t s e t t i n g s were the a b s o r p t i o n p r o f i l e s of the water vapor l i n e s at 278,32 and 315.03 cm \ whose true l i n e w i d t h s were expected to much l e s s than the ex p e r i m e n t a l l y observed widths. A l l of these p r o f i l e s were found to be Gaussian as a f u n c t i o n of g r a t i n g angle. (See f i g u r e C l f o r a t y p i c a l water vapor p r o f i l e . ; Valu.es OJ. the. s p e c t r a l width S were p l o t t e d as a f u n c t i o n of mechanical s l i t w i d t h . I t i s noted that <$ i s not independent of the photon energy. For a L i t t r o w mount A L = Constant where A L i s the l i n e a r d i s p e r s i o n . Hence, f o r the same mechanical s l i t s e t t i n g A L , the s p e c t r a l s l i t width 6" , v a r i e s s l i g h t l y w i t h energy, E , ( i . e . = A E = = c o n s t a n t ) . A E E For l i n e s l e s s than 0.5 meV wide, the assumption of a constant Cf across t h e i r observed p r o f i l e s i ntroduced l e s s than 1% e r r o r . For broad a b s o r p t i o n l i n e s , the i n s t r u m e n t a l c o r r e c t i o n i s n e g l i g i b l e s i n c e the s p e c t r a l s l i t width i s much l e s s than the true l i n e w i d t h . The s p e c t r a l s l i t widths f o r v a r i o u s s l i t s e t t i n g s are given i n t a b l e C l . The l i n e a r d i s p e r s i o n r e l a t i o n s were used to s c a l e these r e s u l t s to 34 and 39 meV. The l a s e r obtained values of <y were used i n the c o r r e c t i o n technique; the greater width of the water vapor l i n e estimates of ©- i n d i c a t e s that the true w i d t h of the water vapor l i n e s are not n e g l i g i b l y s m a l l compared to the s p e c t r a l s l i t w i d t h . 145. S l i t S e t t i n g O" at 34 meV G at 39 meV ram R^O Peaks He Ne Laser . HO.Peaks He Ne Laser 1.0 .077 meV .106 meV 1.2 .103 ,099 . 118 .106 1.4 .115 .107 . 131 .124 .1.6 .129 .123 .145 .140 7 n .155 .152 . 180 .175 Table C l C a l c u l a t e d S p e c t r a l P r o f i l e s of the Monochromator An adaptation of a simple method of deconvolution (equation C.I) suggested by S a v i t s k y (1966) was used to determine these p r o f i l e s . The method i s i l l u s t r a t e d s c h e m a t i c a l l y i n f i g u r e C.2. The observed curve (A c f f i g u r e C.2) was convoluted w i t h the s l i t f u n c t i o n f , This r e s u l t e d i n curve E which i s f u r t h e r broadened and reduced i n peak i n t e n s i t y . The a l g e b r a i c d i f f e r e n c e s at each point: between the observed curve A and the convoluted curve B were added to A to produce curve C, the f i r s t approxima-t i o n to the true l i n e s h a p e . C vras then convoluted w i t h f , the s l i t s f u n c t i o n , to y i e l d curve D which should be equal to A. I f they f a i l e d to match, the a l g e b r a i c d i f f e r e n c e s between A and D were added to C and the second i t e r a t i o n proceeded. The method has the advantage that i t i s s e l f c o n s i s t e n t and can produce not only the l i n e w i d t h but a l s o the e n t i r e true a b s o r p t i o n p r o f i l e . The use of a d i g i t a l computer makes t h i s an e f f i c i e n t d e c onvolution procedure. Se v e r a l m o d i f i c a t i o n s were made to the deconvolu-t i o n procedure to decrease the number of i t e r a t i o n s r e q u i r e d . In general three i t e r a t i o n s were s u f f i c i e n t to reduce d i f f e r e n c e s between curves D and SLIT -CT UJ o C J O o . «—aa a. o to ( X • a a a ' a a CORRECTED 5 If CD -J - • P 1.2 MH EXP 20 SI (F) 140153 B 1.2 MM ND= 5.H10XX15/CC C 1..4 MM TEMP=4.2 K PEAK 2P0 D 1.6 MM MEASURED Figure C,3 Computer C o r r e c t i o n of Instrumental Broadening f o r Sev e r a l S l i t S e t t i n g s Computer Output) Note the incomplete c a n c e l a t i o n of wa-:er vapor s p e c t r a at 34.5 msv. 33.7 33.9 34.1 ENERGY MEV 3 4 . 3 3 4 , : .9 3 4 , 1 ENERGY MEV 147. A of f i g u r e C . 2 to w i t h i n the noise l e y e l of trie apparatus. R e s u l t s i n d i c a t e that bands of width l e s s than twice the s p e c t r a l w i d t h G" can be a c c u r a t e l y r e s t o r e d . Figure C . 3 i n d i c a t e s the r e p r o d u c i b i l i t y of lineshapes 1 f o r v a r i o u s s l i t s e t t i n g s of a narrow l i n e , C .2 Reduction of F l u c t u a t i o n s by Least Squares "Smoothing" In the experiments performed, the spectrometer was operated such t h s t the i n s t r u m e n t a l broadening c o r r e c t i o n s were minimized by using the narrowest p r a c t i c a l s l i t w i d t h , at the expense of optimum s i g n a l - t o -n o i s e r a t i o s . For high i m p u r i t y c o n c e n t r a t i o n samples, t h i s s l i t w idth requirement was not as r e s t r i c t i v e because the absorption l i n e s were broader and the i n s t r u m e n t a l c o r r e c t i o n s m a l l e r . The l e s s than optimum s i g n a l - t o -l l ois-s r " t i o c-'ir. ho improved iv s e v e r a l W I Y P, I t can he iutovoved i n the spectrometer e l e c t r o n i c s i t s e l f by e l e c t r o - m e c h a n i c a l f i l t e r i n g : u sing longer time constant f i l t e r s and longer scanning times. However, the e l e c -t r o n i c improvement i s l i m i t e d by the accuracy that the c h a r t recorded data can be d i g i t i z e d f o r subsequent a n a l y s i s . The d i g i t a l conversion was done manually. I t was found that the s l i g h t i r r e g u l a r i t i e s i n t h i s d i g i t i z e d i nput data could cause l a r g e r f l u c t u a t i o n s i n the computer c o r r e c t e d p r o f i l e . A d i g i t a l smoothing technique has been u t i l i z e d to minimize the i r r e g u l a r -i t i e s i n the d i g i t i z e d data. •*"As an a d d i t i o n a l check of the deconvolution procedure, a peak measured at s p e c t r a l s l i t w idth 8- was convoluted v/ith the p r o f i l e of width Cf by computer to give a new p r o f i l e o b t a i n a b l e w i t h a s p e c t r a l s l i t s e t t i n g of J^ TCF . The computed curve was compared to the p r o f i l e measured at s l i t s e t t i n g .nt8*; the two p r o f i l e s agreed, as they should. 1 4 8 . Ir. i s assumed that each datum point i s the sum o i a random noise component and the true a b s o r p t i o n s i g n a l , and that the true a b s o r p t i o n l i n e i s r e l a t i v e l y "smoothly" v a r y i n g over the i n t e r v a l of i n t e r e s t . 2 (This i s t r u e i f the data p o i n t s are taken s u f f i c i e n t l y c l o s e together.) Y CORRECTED 901NT F i g u r e CU 4 Least Squares F i t of a Cubed Polynomial f o r 9 P o i n t s , To estimate the tr u e v a l u e of each of the data p o i n t s a l e a s t squares f i t of a cubic polynomial through the eig h t neighboring p o i n t s is- used. Least squares f i t t i n g using a computer, i s a simple and r a p i d procedure. I t can be shown ( S a v i t s k y , 1966) that a l e a s t squares f i t of a polynomial can be accomplished by m u l t i p l y i n g each p o i n t by the proper weight f a c t o r , summing the r e s u l t i n g terms, and n o r m a l i z i n g by the proper weight f a c t o r . For a nine p o i n t l e a s t squares f i t of the poi n t X , the p o i n t s X -5, X -4 ... This should not c o n t r a d i c t Shannon's theory of i n f o r m a t i o n t r a n s m i s s i o n where only two samples per s p e c t r a l width are r e q u i r e d to gather a l l of the a v a i l a b l e i n f o r m a t i o n . This i s true only i f the tra n s m i t t e d pulses-are of known shape; where noise i s present and the lineshape i s unknown, two p o i n t s per time constant i s a b e t t e r c r i t e r i o n . 1 4 9 . L E A S T S Q U A R E S F I T S K P > * 8 8x10 Y c c T E S T 2?± . 8 ! . _| ! _J P H O T O N E N E R G Y Figure C .5 The E f f e c t of Least Squares "Smoothing 1 1 on Lineshapes. The change of the observed lineshape due to the "smoothing" i s w i t h i n the experimental e r r o r . The lineshapes c o r r e c t e d f o r i n s t r u m e n t a l broadening are s h i f t e d f o r c l a r i t y . 150. X , ...,. X +5 are m u l t i p l i e d by -21, 14, 39, 54, 59, 54, 39, 14, -21 respec-o o t i v e l y and the sum of these terms i s d i v i d e d by 231. The d i g i t a l " f i l t e r " has the advantage over an RC time-constant f i l t e r i n that the " c u r r e n t " r e a d i n g i s weighted most and p o i n t s "ahead" as w e l l as "behind" are sampled; the RC f i l t e r .samples only the previous events, weighting them w i t h an ex p o n e n t i a l f a l l o f f . A nine p o i n t cubic polynomial f i l t e r should reduce the f l u c t u a t i o n s by a f a c t o r of three. A f i v e p o i n t cubic polynomial " f i l t e r " w i t h a noise r e d u c t i o n f a c t o r of two, was a l s o used i n some cases. F i g u r e C.5 i l l u s t r a t e s the e f f e c t "smoothing" has on the computer-corrected p r o f i l e , when i t i s a p p l i e d to the observed peak p r o f i l e . The p r o f i l e s are s h i f t e d upward f o r c l a r i t y . C.3 Integrated Absorption C o e f f i c i e n t The i n t e g r a t e d a b s o r p t i o n c o e f f i c i e n t , QL. j S i s the c a l c u l a t e d . xnt' area under the measured absorption c o e f f i c i e n t curve. I t has been shown by Kostkoxv'ski and Bass (1956) that t h i s i s true only i n the l i m i t of zero spec-t r a l s l i t w i dth. In general . J o C ( E ) d E > ] c < M e a s u r e d (E)dE They have a l s o shown that t h i s e r r o r i s small i f the instrument s p e c t r a l s l i t width i s l e s s than, the true h a l f w i d t h . For most narrow l i n e s , the true i n t e g r a t e d a b s o r p t i o n c o e f f i c i e n t was obtained by measuring Oi. at v a r i o u s s l i t s e t t i n g s . E x t r a p o l a t i o n of the s l i t s e t t i n g to zero y i e l d e d an estimate of the true i n t e g r a t e d a b s o r p t i o n . The m a j o r i t y of c o r r e c t i o n s were l e s s than ten percent of the measured v a l u e . For l i n e w i d t h s greater than .3 meV the c o r r e c t i o n was n e g l i g i b l e . 151. A second c o r r e c t i o n to.the i n t e g r a t e d absorption considers the a b s o r p t i o n i n the extended wings of the p r o f i l e . For a L o r e n t z i a n l i n e , the e r r o r can be a p p r e c i a b l e . For narrow l i n e s , the a b s o r p t i o n c o e f f i c i e n t was integrated, over ten h a l f w i d t h s and the omitted f r a c t i o n was l e s s than 5 percent. For broader l i n e s (greater than .3 meV) the method suggested by B h a t i a (1970) was employed to i n c l u d e the a b s o r p t i o n in the t a i l s . 152. APPENDIX D ' OCCUPATION PROBABILITIES FOR' A MULTIPLE'LEVEL"SYSTEM OF DONORS . The detailed study of the statistics of semiconductors dominated by impurity levels has been done by Blakemore (1963), and the formulae discussed below are derived in this reference. The specific problem of interest is the evaluation of the population distribution of the donor electrons among the donor levels and the conduction band in silicon at a temperature T. Only the temperature region below 70°K is investigated. At T = 0°K the problem is simple; a l l electrons are in the IS(A) ground state, at energy E measured with respect to the energy of the cordvetTon hand edge (E = 0). For temperatures above absolute zero, the electrons are statistically distributed among the donor levels. It can be shown (see for example, Blakemore, 1963) that the number of free electrons n^ in the conduction band at a temperature T is n = N eVRT (D.l) f c where E^ is the Fermi energy measured with respect to the conduction band minima. (D.2) and M = number of conduction band minima and m„ and in are the longitudinal c it t and transverse effective masses of the donor electron. The expression, D.2, for N is strictly true only for parabolic conduction band minima. 153. Using the known values of m. , ja and M fox s i l i c o n , the yal.ue of N i s • 2 . t c ' c found to be N = 5,43 x l O ^ T ^ ^ c r a ^. The. condition of complete non-c degeneracy, JE^|^>^T, i s also assumed, Consider now the excited l e v e l s l y i n g between the conduction band and the ground s t a t e Csee f i g u r e D . l ) . I f we denote the density of donors with electrons trapped at the r^1 l e v e l by Nd , then ftU kvels I 4- o c p ^ e r ] where we have introduced the notation r] = E^/"RT, €/*E /l?T; i s the concen--t i l t r a t i o n of donor impurities and g^ i s the t o t a l degeneracy of the r l e v e l . S£.c.T .udin£ st>ir>, The numerator of D.3 can be written (D.4) -where N , . i s the ion i z e d donor density. Substituting D.4 i n t o D.3 i t i s d i found that *W - « * ^ .5) Hi', The t o t a l density of neutral donors N, i s found by summing N, over a l l r ; J dn . dr levels Since [N, + N,. = N , l , expression D.6 leads to dn d i d " ^ 9r e x p ^ - . e r ] - ( D . 7 ) 14-. Z. gr exp^e^ th Hence, the d e n s i t y of donor e l e c t r o n s i n the r " s t a t e i s given by (D.8) The p o p u l a t i o n ratio of two l e v e l s , 1 and j i s 3.\ The t o t a l d e n s i t y of donor e l e c t r o n s i s given by N. = 2! N, + n,. (D.9) d ^ dr r W r i t t e n e x p l i c i t l y u s i n g expressions D.l and D.8 1= M, e.^fE,/kT] + ^  f a r eyp|(E.f-Ev~)/i?T? ] »I ' ' \ — • — / ^ . ( I j 4- Z. 5 s { E f - E s > / t e T j J The a p p l i c a b i l i t y of the above expressions to the a c t u a l donor l e v e l s under i n v e s t i g a t i o n depends on the assumptions that 1) the de n s i t y of e l e c t r o n s e x c i t e d to the conduction band, n^, i s s m a l l compared to N^; 2) the number of e l e c t r o n s l o s t to compensating i m p u r i t i e s (of den s i t y N^) i s very s m a l l compared to n^, i . e . n, « N^; 3) the s t a t i s t i c s governing the po p u l a t i o n d i s t r i b u t i o n are unaffe c t e d by the s p a t i a l overlap of the l e v e l s ; 4) the system i s i n thermal e q u i l i b r i u m . 1 5 5 . 156. The p o p u l a t i o n d e n s i t i e s of the l e v e l s were, determined w i t h the help of a d i g i t a l computer. The values of the g 's, E's and N are known. For a given temperature T and c o n c e n t r a t i o n N , the expressions D.l and D.8 were s o l v e d f o r and the 's, r e s p e c t i v e l y , f o r a range of 2 ' reasonable values of the Fermi l e v e l E,. The c o r r e c t v a l u e f o r E_, t r s a t i s f i e s the r e s t r i c t i o n D,9 that N, - /• N, -!- nr' the f r a c t i o n a l , popu-d or f r l a t i o n s are then determined u s i n g t h i s Fermi l e v e l . The parameters used i n the a n a l y s i s and the c a l c u l a t e d populations are shown.in D . l ; and the Fermi l e v e l and p o p u l a t i o n d i s t r i b u t i o n as f u n c t i o n s of temperature are p l o t t e d i n f i g u r e D . l . L e v e l E.r me v 8 r Frac sS 1U t i o n a l P o p u l a t i o n at Temperature IS (A) -45.31 2 1.0 .995 .952 .735 .49 1S(T) -33.6 6 .0034 .031 .146 .210 1S(E) -32.21 4 .001 .012 .070 .112 2P o -11.1 12 .0016 .01 2P -6.1 24 .0023 .023 .07 3P o -5.2 12 .0008 .01 .03 2S -8.8 12 .003 .022 .06 3P -2.9 24 .0007 .01 .047 AP o -3.6 12 C.B. 0 .004 .035 Table D.l P o p u l a t i o n D i s t r i b u t i o n of Donor E l e c t r o n s . 2 The f u l l form of expression D.8 was used to describe the p o p u l a t i o n of the IS ( A ) , 1S(T) and 1S(E) l e v e l s ; f o r the much higher energy p l e v e l s the Boltzmann form of the expression was assumed; i . e . , Ndp l e V £ l = N d S p expJ ( E f - EpVfcT} 157. 'APPENDIX E •' •' EVALUATION OF ' THE ' ENERGY LEVEL' SPLITTING ' OF THE 2P ' STATES USING .'THE METHOD OF BALTENSPERGER Schroedinger's equation in the hydrogenic approximation for * a donor electron of mass m and charge e moving in a coulomb field of dielectric constant K is The solution is of the form (1/ - R ft , . Y (9,Q>), (E.2) T n , i(r) l,m ^  where n, 1 and m are integers, K M - ^9{ZSL.\ Ur_ \* F ({+i-n , 2£_ ) - ( E . 3 ) F l ' X j , x)= | + Z ocUUl) ••• (o<4v>_0 a » (E.4) F(o(, IT, x) is the associated Laquerre Polynomial for integer values of n. The following definitions are used, (Xc _JQ_ and e is the me 7- K-cx unit of energy. For s type wavefunctions 2. = 0 and = eyp ^~(-fYH,'5U 'x) (E.5) for p type wavefunctions 1 = 1 and vb = eyp^^ - X Ir (-\1V2>4,(E.6) The upper.and lower limits of the 2P band are calculated below using tbe wavefunction (E.6). Applying the boundary condition .158, one obtains the expression r-r* 1+ZL (\^ -n)6^  5" ~i.f.Z-n\ f ' ^ (3-ti)(4-nv,...U -^n)x ] ( E > 7 ) ? / L \ M : 7 7773" J or LL J L l F (z-n,4,^) - J_ fJLiA Fb - n , s , ^ ) = o L n xv-. J n i u I E , 8) To s a t i s f y the new boundary c o n d i t i o n s , i t i s assumed t h a t n v a r i e s s l i g h t l y from an i n t e g e r v a l u e . The v a r i a b l e n = 2 i s s u b s t i t u t e d i n t o (E.8) to [ 1- %~\\ I +• Z e ( G 4 i ^ ... (e + *-0 £ a:* * * ( . > r » * ~ ; r r — 3 v" l 0 + 3 ) 1 (E.9) Erginsoy (1952) has shown t h a t , +- S"1 ( I 4-1 4 - . . - - 1 - )(>-0i ( E . I O ) 2 ^-1 Using t h i s approximation i n equation (E.9) terms i n & are c o l l e c t e d and one obt a i n s , A c * z B = 1 O I-+ u £ 4 C = 0 where 1-2 2_S> ? >+4 ' vM-4 * 3 ( E . l l ) (E.12) (E.1.3) CE.14) X The q u a d r a t i c (E.I1) can be solved f o r £ . This y i e l d s the low l i m i t of the 2F band. The upper l i m i t of the band obtains form the boundary c o n d i t i o n (E.15) Using the non-integer s u b s t i t u t i o n f o r n, (. n - 2 + b) , equation (E.15) can be e a s i l y s o l v e d . For n = 2, 4 => 1, the r e s u l t i s 0 « to Z.. ^ 1 1 * 0+3)1 j (EJ.6) The i n f i n i t e s e r i e s i n expressions (E.12), (E.13) and (E.16) were summed on the computer using a s u f f i c i e n t number of terms to reach convergence of the "1 s e r i e s . To evaluate the •5 y.' (*ts) 1J o 1 ___x _ i terms i n (E.12), (E.13) and Where i s an i n t e g e r and j =^ + 3 is greater than 10. The upper and lower band edges at the Wigner-Seitz r a d i u s r r are given, r e s p e c t i v e l y , by an For 6 « b <?s fl the bandwidth of the 2P l e v e l i s given by (E.18) K a n (E.19) 160. BIBLIOGRAPHY Aggarwal, R.L., et. al.?(1965), Phys. Rev., 138, A882. Aggarwal, R.L. and Ramdas, A.K., (1965), Phys. Rev.,140, A1246. Alexander, M.N. and Holcomb, D.F., (1968), Rev. Mod. 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