UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Infrared absorption lines in boron-doped silicon. Colbow, Konrad 1963

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

Item Metadata

Download

Media
831-UBC_1963_A1 C55 I5.pdf [ 3.24MB ]
Metadata
JSON: 831-1.0085855.json
JSON-LD: 831-1.0085855-ld.json
RDF/XML (Pretty): 831-1.0085855-rdf.xml
RDF/JSON: 831-1.0085855-rdf.json
Turtle: 831-1.0085855-turtle.txt
N-Triples: 831-1.0085855-rdf-ntriples.txt
Original Record: 831-1.0085855-source.json
Full Text
831-1.0085855-fulltext.txt
Citation
831-1.0085855.ris

Full Text

INFRARED ABSORPTION LINES IN BORON-DOPED SILICON by KONRAD COLBOW B . SCo , McMaster Uni v e r s i t y , Hamilton, Ont., .1959 M.Sc, McMaster Un i v e r s i t y , Hamilton, Ont., 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1963 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r 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 tha t 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 f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. May 31 19fi3 The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL•EXAMINATION FOR THE.DEGREE OF DOCTOR OF PHILOSOPHY B . S c , McMaster U n i v e r s i t y , 1959 M. Sc., McMaster U n i v e r s i t y , 1960 WEDNESDAY, MAY 29, 1963, AT 2:00 P.M. IN ROOM 452, BUCHANAN BUILDING of KONRAD COLBOW COMMITTEE IN CHARGE Chairman: F.H. Soward R. Ba r r i e A.M. Crooker F.W. Dalby J.C. G i l e s K.B. Harvey G.M. Volkoff G.B. Walker External Examiner: H.J. Hrostowski Department of Physics, U n i v e r s i t y of Oregon INFRARED ABSORPTION LINES IN BORON-DOPED SILICON ABSTRACT In boron-doped s i l i c o n , o p t i c a l absorption takes place through the e x c i t a t i o n of bound holes from the ground ixate. to excited states. This leads to a l i n e spectrum. Due to a lack of s u f f i c i e n t r e s o l u t i o n and a f a i l u r e to make proper allowance for l i n e d i s t o r t i o n by the f i n i t e spectrometer s l i t width, previous au-thors gave a misleading p i c t u r e of the low temperature half-width, the temperature dependence of t h i s h a l f -width, and the onset of concentration broadening at low temperatures. New experimental data are presented and explained by introducing the mechanism of s t a t i s t i c a l Stark broadening due to ionized impurities, and by modifying Baltensperger' s (1953) theory for concentration broad-ening. At low impurity concentration the width i s att r i b u t e d to phonon broadening (Barrie and Nishikawa 1962) and i n t e r n a l s t r a i n s (Kohn 1957). GRADUATE STUDIES F i e l d of Study: Physics Physics of the s o l i d state R. Ba r r i e D i e l e c t r i c s and Magnetism M. Bloom Noise i n Physical Systems R.E. Burgess Ele c t r o n Dynamics R.E. Burgess Related Studies: Applied Electromagnetic Theory G.B. Walker D i g i t a l Computer Programming J.R.H. Dempster PUBLICATIONS 1. Temperature Dependence of Absorption Line Width i n Boron-Doped S i l i c o n . Konrad Colbow, J„W. Bichard, and J.C. G i l e s . Can. J . Phys. 40 s 1436 (1962). 2. Absorption Line Width i n Boron-Doped S i l i c o n . Konrad Colb ow. B u l l . Am. Phys. Soc. Series I I S I, 485 (1962). ABSTRACT In boron-doped s i l i c o n , o p t i c a l a bsorption takes place through-the e x c i t a t i o n of bound.holes from the ground s t a t e to e x c i t e d s t a t e s . This leads to a l i n e spectrum,, Due to a lack of s u f f i c i e n t r e s o l u t i o n and a f a i l u r e to make proper allowance f o r l i n e d i s t o r t i o n by the f i n i t e spectrometer s l i . t width,, previous authors gave a misleading p i c t u r e of the low temperature h a l f - w i d t h , the temperature dependence of t h i s h a l f - w i d t h , and the onset of co n c e n t r a t i o n broadening at low temperatures. NewJ#% experimental data are explained by i n t r o d u c i n g the r^ ew" mechanism of s t a t i s t i c a l Stark broadening due to i o n i s e d i m p u r i t i e s , and by modifying Baltensperger's (1953) theory f o r conc e n t r a t i o n broadening. At low impurity c oncentration the width i s a t t r i b u t e d to phonon broadening ( B a r r i e and Nishikawa 1962) and i n t e r n a l s t r a i n s (Kohn 1957). ACKNOWLEDGEMENTS My s i n c e r e thanks are due to Dr. R. B a r r i e f o r h e l p f u l advice and c o n s t r u c t i v e c r i t i c i s m . d u r i n g the pr e p a r a t i o n of t h i s t h e s i s . I a l s o wish to thank Drs. J . W. Bicha r d , J . C. G i l e s , and A. M. Crooker f o r v a l u a b l e d i s c u s s i o n s . The research f o r t h i s t h e s i s was supported by the Defence Research Board of Canada, Grant No. 9512-26. I a l s o l i k e to express my g r a t i t u d e to the N a t i o n a l Research C o u n c i l of Canada f o r the award of a studentship. i i i TABLE OF CONTENTS Page Abstract i i Table of Contents . i i i L i s t of F i g u r e s v Acknowledgements , . . v i i I Chapter I - I n t r o d u c t i o n 1 Chapter I I - Experimental 1. Apparatus and Experimental Procedure . 4 2. Spectrometer Broadening . . 7 3. R e s u l t s and D i s c u s s i o n of E r r o r s 12 Chapter I I I - Theory and I n t e r p r e t a t i o n of Data 1. Impurity States and the Hydrogenic Approximation . 15 2. General I n t e r p r e t a t i o n of L i n e Broadening . . . . 20 3. S t a t i s t i c a l Theory of Stark Broadening 21 4. The E f f e c t of Screening 26 5. Temperature Dependence of the H a l f - w i d t h due to the Linear and the Quadratic Stark E f f e c t 27 6. E v a l u a t i o n of the Half-width due to the Quadratic Stark E f f e c t „ . . 29 7. S t a t i s t i c a l Broadening due to van der Waals Forces. 31 8. Broadening due to Overlap Forces 32 9. Broadening due to I n t e r n a l S t r a i n s 35 10. Phonon Broadening 36 - iv -Page Chapter IV - Conclusions. 40 Table 1: Standard Voigt profiles 43 Table 2; Integrated absorption, A . . . . . . . 44 Table 3: Integrated absorption cross-section, 44 Appendix I; Distribution of holes between the ground state, excited states, and the valence band as a function temperature 45 Appendix II: Second-order Stark shift for hydrogenic Is and 2p states 48 Bibliography 49 V LIST OF FIGURES To f o l l o w page: FIG. 1. Absorption constant vs. wave number f o r boron-doped s i l i c o n at v a r i o u s temperatures and three boron concentrations 6 FIG. 2. Water vapor l i n e - w i d t h vs. wave number 8 FIG. 3. T y p i c a l V o i g t a n a l y s i s (water vapor l i n e at 298.6 cm"'')' 10 FIG. 4. E x t r a p o l a t i o n of the t r u e h a l f - w i d t h to zero c o n c e n t r a t i o n of i m p u r i t i e s 12 FIG. 5. True h a l f - w i d t h vs. temperature f o r three boron concentrations 12 FIG. 6. H a l f - w i d t h minus zero-concentration h a l f - w i d t h vs. temperature f o r l i n e 4 12 FIG. 7. Absorption constant vs. wave number f o r 11 ohm cm boron-doped s i l i c o n at four temperatures 12 FIG. 8. Energy l e v e l diagram f o r boron-doped s i l i c o n . . . . 14 FIG. 9. D i s t r i b u t i o n , of holes between the ground s t a t e , e x c i t e d s t a t e s , and the valence band vs. temperature. 14 FIG. 10. Screened e l e c t r i c f i e l d , and Coulomb f i e l d w i t h cut-o f f , as a f u n c t i o n . o f separation 26 FIG. 11. Screening parameter vs. temperature f o r three concentrations of boron 27 FIG. 12. F i e l d d i s t r i b u t i o n f u n c t i o n f o r various values of the screening parameter 27 v i To f o l l o w page: FIG. 13. H a l f - w i d t h of the f i e l d d i s t r i b u t i o n as a f u n c t i o n of the screening parameter 27 FIG. 14. H a l f - w i d t h i n u n i t s of the s t r e n g t h parameter v s . temperature f o r the l i n e a r Stark e f f e c t 28 FIG. 15. H a l f - w i d t h i n u n i t s of the st r e n g t h parameter vs. temperature f o r the quadratic Stark e f f e c t 28 FIG. 16. Broadening of hydrogenic l e v e l s v s . distance between i m p u r i t i e s i n u n i t s of the e f f e c t i v e Bohr r a d i u s . . 32 1 CHAPTER I  INTRODUCTION The introduction of a group III impurity,, l i k e boron, into a perfect s i l i c o n l a t t i c e produces a hole (electron deficiency) loosely bound to the impurity ion. In boron-doped s i l i c o n , i n f r a r e d absorption takes place through the e x c i t a t i o n of bound holes from the ground state to excited states. This leads to a l i n e spectrum. If the bound hole states were unaffected by, each other and by l a t t i c e v i b r a t i o n s , these s p e c t r a l lines would have no width other than th e i r n atural width, which i s about 10"^ electron v o l t s (ev), (e.g Herzberg 1944). However, the lines that have been observed are considerably broader. This gives r i s e to two i n t e r e s t i n g f i e l d s of study,- namely l i n e broadening due to i n t e r a c t i o n with the l a t t i c e v i b r a t i o n s , and so-called concentration broadening, that i s broadening due to the i n t e r a c t i o n of bound holes with each other. Most of the discussions in t h i s t hesis apply i n a s i m i l a r way to group V donor impurities (bound electrons). The reason f o r choosing a more complicated acceptor impurity l i k e boron i s purely h i s t o r i c a l . Neglecting instrumental broadening, Burstein et a l (1953) found in boron-doped s i l i c o n half-widths of about 1 mev (10" ev) at 4 02°K and an increase of the half-width ( f u l l width at half-maximum) of about 4d percent at ,77 °K. This was l a t e r concluded to be in reasonable agreement with the c a l c u l a t i o n s of Lax and Burstein (1955) which gave a half-width of 3.6 mev at 4.2 °K and predicted roughly the observed increase in half-width at 77 °K. Lax and Burstein (1955) suggested that the width of the impurity levels i s due to the simultaneous emission or absorption of one or more phonons accompanying the change of state of the bound hole. Sampson and Margenau (1956) further improved on the agreement between the c a l c u l a t e d and the 2 observed widths. Using the simple Lorentz broadening approach they c a l c u l a t e d a low temperature h a l f - w i d t h of 1.6 mev. Kane (1960) pointed out that the observed broadening at 4.2 °K may be e n t i r e l y i n s t r u m e n t a l . Kane suggested that due to the weakness of the electron-phonon i n t e r a c t i o n i n s i l i c o n and germanium the dominant o p t i c a l l y induced t r a n s i t i o n i s pur e l y e l e c t r o n i c without any change of phonon occupation number. He suggested that the width arose from a f i n i t e l i f e -time of the e x c i t e d s t a t e due to the electron-phonon i n t e r a c t i o n , . Kane -3 estimated a width of about 3 x 10 mev. In t h i s l i f e - t i m e broadening, the widths of the absorption l i n e s are mostly determined by those of the exc i t e d l e v e l s . Thus the width of the l i n e s w i l l c r i t i c a l l y depend on the energy l e v e l s t r u c t u r e , and one expects d i f f e r e n t l i n e s to have d i f f e r e n t widths. This was i n disagreement w i t h the experimental r e s u l t s of B u r s t e i n et a l (1953), and the theory by Lax and B u r s t e i n (1955), which p r e d i c t e d that the widths were mostly determined by the ground s t a t e and thus should be the same f o r a l l l i n e s . Colbow et a l (1962) showed that the observed broadening at 4.2 °K i s not e n t i r e l y instrumental as suggested by Kane (I960)a However, t a k i n g proper care of instrumental broadening they found a h a l f - w i d t h of 0.2 mev at 4.2 °K and an increase of the h a l f - w i d t h between 4.2 °K and 90 °K by a f a c t o r of 6, r a t h e r than the 40 percent p r e d i c t e d by Lax and-Burstein (1955). Their data a l s o showed that d i f f e r e n t l i n e s have d i f f e r e n t h a l f - w i d t h s , and that the l i n e shape at 4.2 °K i s predominantly l o r e n t z i a n . The theory of phonon broadening has been c l a r i f i e d by B a r r i e and Nishikawa (1962). T h e i r r e s u l t s are i n f u l l agreement w i t h Kane's (1960). B a r r i e and Nishikawa obtained e x p l i c i t l y the l i n e shape f u n c t i o n of the zero-phonon process, which turns out to be approximately l o r e n t z i a n near the peak. 3 They pointed out that the multiphonon processes which Lax and Burstein (1955) considered contribute to the continuous background and become important only at higher temperatures (about 90°K). However, the theory of Barrie and Nishikawa did not account for the rather steep temperature dependence observed by Colbow et al (1962). Up to this point it had always been assumed that for the rather low impurity concentrations used by Colbow et al (1962) and Burstein et al (1953), concentration broadening was negligible. This appeared to follow from the experimental data of Newman (1956), which showed that for boron-doped silicon concentration broadening starts above 10^ boron impurities 3 per cm . In addition Baltensperger•s (1953) calculations suggested that concentration broadening in boron-doped silicon should start at approximately 6 x lO-^ boron impurities per cm"*. In contrast to this, the data presented 15 -3 here show that at an impurity concentration of 1.2 x 10 cm concentration broadening is already important. It is believed that the main shortcoming of Baltensperger1s (1953) theory lies in the assumption of a regular lattice qf impurities. The implications of an alternative, and more likely assumption of a random distribution of impurities are discussed in this thesis. A new mechanism is suggested to account for the observed rapid increase of the half-width with temperature, namely statistical Stark broadening due to ionized impurities. 4 CHAPTER II lo Apparatus and Experimental Procedure Radiation from a Globar source was dispersed i n a Model 83 Perkin-Elmer spectrometer modified to house a Bausch and Lomb grating with 30 grooves per millimeter blazed at 333 cnf * i n the f i r s t order. The t h e o r e t i c a l r e s o l v i n g power was X/AX = 315, for a s l i t width of 0.8 mm. The i n t e n s i t y of shorter wavelength r a d i a t i o n was reduced by sooting the mirror i n the entrance optics and by using two r e f l e c t i o n s from sodium f l u o r i d e r e s i d u a l ray plates. The remaining short wavelength r a d i a t i o n was measured and corrected f o r by passing the r a d i a t i o n through a Rocksalt window, which w i l l transmit only r a d i a t i o n shorter than about 500 cm°^, The e f f e c t of the short wavelength photons on the occupation number of the impurity states i s n e g l i g i b l e . The short wavelength r a d i a t i o n merely contributes to the continuous background, and was within experimental error the same through a pure and the doped sample. Any possible heating of the specimen due to this r a d i a t i o n i s n e g l i g i b l e as well- Near the entrance s l i t the r a d i a t i o n i s chopped by a semi-circular d i s c at a frequency of 13 cycles sec°^, After detection by a thermocouple with a cesium iodide window, the amplified s i g n a l was displayed on a Brown s t r i p - c h a r t recorder. In each burst of l i g h t the number of photons i s very small compared to the t o t a l number of boron impurities i n the sample. During the time i n t e r v a l when the r a d i a t i o n i s stopped by the chopper blade, the sample i s believed to return to equilibrium. Thus no saturation effects are expected to occur. The spectrometer was c a l i b r a t e d by means of the atmospheric water vapor absorption spectrum, using the data of P l y l e r and Acquista (1956), and Randall et a l (1937). In the course of a measurement the spectrometer was flushed continuously, with dry nitrogen gas to reduce the absorption due to .. a tmo s ph er i'cf1 wa't'e'r^ Vapo r s; . 5 The experiment c o n s i s t s of measuring the transmission r a t i o s of boron-doped s i l i c o n and i n t r i n s i c s i l i c o n as a f u n c t i o n of the wave number of the r a d i a t i o n f o r v a r i o u s temperatures and r e s i s t i v i t i e s of boron-doped s i l i c o n . The measurements were made at four temperatures corresponding to the b o i l i n g p o i n t s of helium (4.2°K), n i t r o g e n (77.4°K), oxygen (90.1°K), and pumped n i t r o g e n (60 + 3°K). The r e f r i g e r a n t s were contained i n a metal dewar v e s s e l , the base of which was loc a t e d near a focus of the r a d i a t i o n l e a v i n g the monochromator. The r a d i a t i o n ports were covered wit h cesium iodided' windows. P a r a l l e l -sided specimens were attached to a f l a t copper block at the base of the coolant c o n t a i n e r , w i t h f a c i l i t i e s f o r r o t a t i n g a l t e r n a t i v e l y the doped or an i n t r i n s i c sample i n t o the path of the r a d i a t i o n . Thermal contact between the copper block and the samples was achieved by means of vacuum grease c o n t a i n i n g a suspension of s i l v e r powder. The samples were h e l d i n place by a f l a t copper s t r i p ( w i t h a hole i n i t s center f o r the passage of the r a d i a t i o n ) , Carewas paid to avoid s t r a i n i n g the samples. The samples were cut from ingots grown by Merck and Company by the f l o a t i n g - z o n e technique. They had thicknesses between 0.0361 cm (most h i g h l y doped m a t e r i a l ) and 0.5085 cm ( l i g h t l y doped m a t e r i a l ) , the l a t t e r being the l a r g e s t thickness that could be r e a d i l y accommodated i n the metal dewar a v a i l a b l e . These thicknesses were chosen to g i v e a compromise between the s i g n a l strength ( h i g h transmission) and the observable absorption (low t r a n s m i s s i o n ) . The specimen surfaces were ground w i t h p r o g r e s s i v e l y f i n e r grades of carborundum and f i n a l l y p o l i s h e d w i t h l e v i g a t e d alumina.. Before mounting i n the d,ewar v e s s e l , the,.surfaces ,were degreased i n an u l t r a s o n i c c l e a n i n g bath. Impurity concentrations were determined from the room-temperature r e s i s t i v i t y of each sample. The r e s i s t i v i t i e s were 1.3 ohm cm, 11 ohm cm, 130 ohm cm, and 3600 ohm cm ( i n t r i n s i c ) . A l l samples were 6 e s s e n t i a l l y uncompensated. Using the data of I r v i n (1962) and Morin and Maita (1954), the corresponding concentrations of boron are re s p e c t i v e l y 1.2 x 1 0 1 6 cm"3, 1.2 x 1 0 1 5 cm"3, 1.0 x 1 0 1 4 cm"3, and less than 5 x 1 0 1 2 cm"3 ( i n t r i n s i c ) . With one exception, the specimens used in the present i n v e s t i g a t i o n were thick enough that interference fringes could not be observed with the r e s o l u t i o n a v a i l a b l e . For the one specimen (0.0361 cm thick,, 1.3 ohm cm) for which interference fringes did appear, they were narrow and shallow enough to be r e a d i l y averaged out in. reading the data of the recorder chart. Under these conditions, the monochromatic absorption c o e f f i c i e n t c< i s obtained from the transmission at a given wave number by (Moss 1959) T _ (1 - R ) 2 exp. (- d) . 1 - R 2 exp. ( - 2 c< d) The transmission of the i n t r i n s i c material i s then given by T Q = (1 - R ) 2 / (1 - R 2 ) . (1-2) Hence the transmission r a t i o of doped and i n t r i n s i c s i l i c o n becomes (1 - R 2 ) exp. ( - * d) T/T 0 = ~ Y-2 2—* ~ . (1-3) ° 1 - R exp. ( - 2 o(d) This i s the r a t i o that was a c t u a l l y measured in.a t y p i c a l experiment. Here d i s the specimen thickness. The surface r e f l e c t i v i t y R was assumed to be the same for the i n t r i n s i c and the boron-eloped s i l i c o n . Using Eq. ( 1 - 2 ) a va lue of 0.31 + 0.03 was determined for R, which i s the same as that used by Bichard and G i l e s (1962). Within experimental error the r e f l e c t i v i t y remained constant over the frequency region under study (360 cm°^ to 240 cm"*-), and the temperature range (4.2°K to 90°K) investigated. With t h i s value for the r e f l e c t i v i t y Eq.(l-3) leads to the graphs of absorption FIGURE 1. Absorption constant vs. wave number for boron=doped s i l i c o n at various temperatures (T) and three boron concentrations (N A). To follow page 6. 7 constant as a f u n c t i o n of wave number shown in, F i g u r e 1. Instead of measuring the transmission f o r a doped sample r e l a t i v e to that f o r an i n t r i n s i c sample and using Eq.(l-3), one could have determined the t r a n s m i s s i o n f o r a doped sample r e l a t i v e to a i r , using Eq„(l~l)o However, there are s e v e r a l advantages to the f i r s t method. The exact v a l u e f o r the r e f l e c t i v i t y i s c o n s i d e r a b l y l e s s important f o r c a l c u l a t i n g the absorption c o e f f i c i e n t i f Eq.(l-3), r a t h e r than Eq,(l-=2), i s used. In a d d i t i o n , s c a t t e r e d short wavelength r a d i a t i o n can be expected to be more n e a r l y the same and thus cancel i n the r a t i o , when the doped sample i s compared w i t h an i n t r i n s i c one, r a t h e r than w i t h no sample i n the l i g h t path. The same ap p l i e s f o r the c a n c e l l a t i o n of atmospheric water vapor abs o r p t i o n . In the preceeding o u t l i n e f o r the 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 i t was i m p l i c i t l y assumed that a p a r a l l e l beam of l i g h t i s at normal incidence on a p a r a l l e l - s i d e d specimen. Any angles between the specimen surfaces were l e s s than 0.01°. A c t u a l l y , the l i g h t beam was not p a r a l l e l . However, s i n c e the maximum angle of incidence i n the converging beam was only 8°, no c o r r e c t i o n f o r convergence needed to be a p p l i e d . 2. Spectrometer Broadening Using the curves of Figure 1 and " p e e l i n g o f f " the c o n t r i b u t i o n s of neighboring l i n e s , one can. determine an."observed l i n e shape" and an "observed h a l f - w i d t h " . The p e e l i n g o f f i s achieved by assuming a symmetric l i n e p r o f i l e . The r e s u l t i n g absorption l i n e s (dotted l i n e s ) may be seen i n F i g u r e 6, which gives a magnified p i c t u r e of the l i n e s 3 and 4 of F i g u r e 1(a). This " p e e l i n g o f f " becomes d i f f i c u l t at higher temperatures. However, i f one assumes that the areas under the weaker l i n e s i n F i g u r e 6 stay n e a r l y constant and that t h e i r heights decrease w i t h temperature i n the same way as the strong a b s o r p t i o n l i n e , one can. decompose the experimental curves i n t o 8 the dotted lines, as shown in Figure 60 Errors in these assumptions give rise to considerably smaller errors' in the observed half-width of the strong line. These observed absorption lines are not only broadened by processes in the specimen but also by the finite resolution of the spectrometer. It is the purpose of the present section to outline the method used for obtaining from the observed line the "true line shape" and the "true half-width", that is the line profile obtained by using a spectrometer with infinite resolution. The intensity distribution in a spectral line broadened by two independent effects is expressed by a convolution integral of the form (Unsold 1955) -t-00 f (x) = / f ( x - y) f " (y) d y . (2-1) -00 Here f '(x) or f (x) is the profile the line would assume if only one broadening effect were present; x is the distance from the centre of the line in terms of either wavelength or frequency units. If f (x) is the observed line shape, Eq.(2-1), determines the true line,profile f "(y), provided we know the s l i t function f'(y). This s l i t function is the profile a monochromatic signal assumes after passing through the spectrometer. It may be obtained experimentally from the observed profiles of single atmospheric water vapor absorption lines, which have a true width considerably smaller than the observed width and thus approximate well enough to monochromatic lines. The small true width of single water vapor lines was inferred from a plot,of their observed half-width as a function of spectrometer sl i t width. The observed half-widths of suitable wafer vapor absorption lines have been plotted in Figure 2. The corresponding spectrometer s l i t width is 0.8 mm in the region from 370 cm"*- to 260 cm"*-, and 1.2 mm in the region from 260 cm"*" to 240 cm"*-. The same s l i t widths were used to obtain the spectra in Figure 1. FIGURE 2. Water vapor line-width vs. wave number. To follow page 8. FIG. 2 Numerous methods f o r s o l v i n g the i n t e g r a l equation (2-1) have been a p p l i e d (Unsbld 1955, van de Hulst 1946). However, a l l the general methods are very l a b o r i o u s . The problem becomes q u i t e simple i f the s l i t f u n c t i o n and the observed l i n e p r o f i l e may be f i t t e d e i t h e r both by l o r e n t z i a n or both by gaussian curves. In t h i s case the true l i n e p r o f i l e comes out to be r e s p e c t i v e l y l o r e n t z i a n or gaussian as w e l l , and one has the simple r e l a t i o n between the h a l f - w i d t h s l o r e n t z i a n % h" = h - h' ir 2 2 ' 2 (2-2) gaussian : h = h - h Here h" i s the true h a l f - w i d t h , h the observed h a l f - w i d t h , and h' the h a l f -width of the s l i t f u n c t i o n . However, no s i n g l e f u n c t i o n of e i t h e r type gives a good f i t to both the observed l i n e shape and the s l i t f u n c t i o n . I t has been found that the present experimental curves f o r the observed l i n e p r o f i l e and the s l i t f u n c t i o n can be q u i t e adequately f i t t e d by Voigt f u n c t i o n s (Voigt 1912). These f u n c t i o n s are defined as the co n v o l u t i o n i n t e g r a l between a gaussian and a l o r e n t z i a n f u n c t i o n . They may be used to f i t any p r o f i l e which l i e s between a gaussian and a l o r e n t z i a n curve. I f f ( x ) and f'(x) are both taken to be Voigt f u n c t i o n s , then the true l i n e shape i s also a V o i g t f u n c t i o n by Eq. (2-1). One may thus w r i t e observed line shape, +oo r, ^ w "f e x P ' [ " (y / P 2 ) 2 ] , , 0 ,v f ( x ) = M J b. ±~ dy ; (2-3) -oo 1 + t ( x - y) / P L ] 2 s l i t f u n c t i o n , +oo - » ft 1 \ 2 1 £ , w _ M. j **P.[- ( y / P 2 > ] D Y . ( J . 4 ) 1 + C ( x - J * / P [ ] 2 t r u e l i n e shape, f • • ( . ) . - M " 7 °xp-[~  i j ' r i > 2 i * > «-» -00 10 I II where M, M , and M are constants. I t i s a property of these f u n c t i o n s and Eq. ( 2 - 1 ) that - ?i - p; (2-6) and F i g u r e 3 i l l u s t r a t e s a t y p i c a l V o i g t a n a l y s i s . In a d d i t i o n to the width h at h a l f maximum hei g h t , the width at one-tenth the height, b2 at two-tenth the maximum h e i g h t , e t c . , are measured. A f t e r c a l c u l a t i n g the r a t i o s b^/h, where i goes from one to e i g h t , Table 1 (van de Hulst and Reesinck 1947) was used to o b t a i n the corresponding V o i g t parameters p]_/h and 2^ ^ / n ^ > which were a p p r o p r i a t e l y averaged over d i f f e r e n t bf f o r a g i v absorption l i n e . A l o r e n t z i a n l i n e shape gives P^/h = 0.500, ?2 2 = 0> w h i l e a g a u s s i a n . p r o f i l e leads to B^/h =0, P2 /" = 0-36 . The atmospheric water vapor l i n e s i n v e s t i g a t e d to determine the s l i t f u n c t i o n ( F i g u r e 2) were a l l found to lead to P[/h' = 0.28 t 0.08 ; Pj 2 / h ' 2 = ° ' 1 6 - ° ' 0 6 • (2"8> Using the water vapor absorption l i n e h a l f - w i d t h s from Figure 2, Eq. (2-8) enables us to f i n d the s l i t f u n c t i o n V o i g t parameters B[ and P2 2 • Performing the a n a l y s i s i n the same f a s h i o n f o r the observed absorption l i n e s i n boron-doped s i l i c o n , one f i n d s the Voigt parameters J3^ and p2« 0 n e m a v then use Eq. (2-6) and (2-7) to o b t a i n the V o i g t D" D" 2 parameters , P2 f° r t* i e t r u e absorption l i n e s . W i t h i n experimental IT O e r r o r P2 w a s found to be equal to zero, which means the V o i g t f u n c t i o n s f o r the t r u e absorption l i n e s s i m p l i f y to l o r e n t z i a n p r o f i l e s . That i s , Eq. (2-5) becomes FIGURE 3. Typical Voigt analysis (water vapor line at 298.6 cm" ). The numbers b^/h (where i goes from 1 to 8) give the line width at i/10 of the maximum height, relative to the line width at half maximum height, (h). The values Bi/h where obtained from Table 1. To follow page — I b 8 / h = 0.53 0.300 299 298 Wave number (cm'h FIG. 3 f"(x) = c" (1 + x 2/f^ 2) ~l , (2-9) where c" is the maximum height of the curve, occurring at x = 0; and one has I! V |, £ i/h = 0.50o It was found that for the observed absorption lines in boron-doped silicon P^/h varied between 0.45 for narrow lines to about 0.5 for broad lines. Under these conditions, with h = 2 p , Eq. (2-6) leads to h":= gh - g'h1 , (2-10) where g = 2 P^ /h varies between 0.9 and 1.0 for different absorption lines, and by, Eq. (2-8) one has g' = 2 pj/h' = 0.56 + 0.16. The values listed in Table 1 under p may be used to evaluate the integrated absorption A. Both A and p are defined by +oo A = ( f(x) dx = phc (2-11) -oo for the observed line, or similarly for the true line by A" = f"(x) dx = p" h" c" , ' (2-12) -oo where c and c" are respectively the observed and the true central ordinate (maximum height). Since the integrated absorption is not changed by spectrometer broadening provided one scans over a reasonably wide wave number range, one has p h c = p h c Knowing the observed integrated absorption, h", and p^ '/h", Eq, (2-13) and Table 1 enables one to find the true central ordinate c", if desired. With the present experimental data it is a good enough approximation to take in Eqs. (2-11), (2-12), and (2.13) from Table 1 p ^ p" = 1.54 ± 0.03, (2-14) for a l l absorption lines investigated. 12 3o Results and Discussion of Errors Correcting the data in Figure 1 for spectrometer d i s t o r t i o n by the methods outlined i n the previous section, the true half-widths h" were obtained. These have been plotted in Figure 5 as a function of temperature for three concentrations of boron impurities.. In Figure 4 these data are extrapolated to zero impurity concentration. The r e s u l t i n g curves are shown i n Figure 5 as dotted l i n e s . For the purpose of extrapolation a II W 3 p l o t of In h vs. was found convenient and r e s u l t s i n reasonably s t r a i g h t l i n e s . The main con t r i b u t i o n to the uncertainty in h" (Figure 5) has varying o r i g i n s f or d i f f e r e n t temperatures and l i n e s under study. This w i l l p a r t l y be i l l u s t r a t e d by reference to Figure 7, which shows i n magnified form the absorption constant of lines 3 and 4 in 11 ohm cm boron~doped s i l i c o n at four d i f f e r e n t temperatures. For rather sharp l i n e s (low temperature and impurity concentration) the main source of error comes from the uncertainty of g and g' in Eq. (2-10), together with the f a c t that one subtracts two sim i l a r numbers. The extreme case for t h i s i s l i n e 1 in 130 ohm cm at 4o2°K. Here h = 0.153 mev, h' = 0.095 mev, and using Eq. (2-10) one obtains h" = 0.08 f 0.04 mev. On the opposite extreme, for wide l i n e s (high temperature and impurity concentration) the uncertainty in the half-width r e s u l t s p r i m a r i l y from the d i f f i c u l t y i n peeling-off the absorption due to neighboring l i n e s . Two other sources of uncertainty are important only for s p e c i a l cases. The f i r s t i s the rather large absorption in the 1,3 ohm cm material, which gives r i s e to an uncertainty of about f i v e percent in the peak-heights of l i n e s 2 and 4 at 4 ,2°K and 60°K, The second source of error i s the presence of some background absorption l i n e s due to atmospheric water vapor. FIGURE 4. Extrapolation of the true half-width (logarithmic scale) to zero concentration of boron impurities (N A). To follow page 12. 0 2 4 6 8 10 12 14 16 18 20 N A X/ 3 (104 cm"1) FIG. 4(a) FIGURE 5. True half-width (corrected f or spectrometer broadening) vs. temperature for boron concentrations of i) 1.0 x 1 0 ^ cm"3, i i ) 1.2 x 1 0 1 5 cm"3, i i i ) 1.2 x 1 0 1 6 cm"3, and iv) very low concentrations (extrapolated). i To follow page T (°K) FIG. 5(a) FIGURE 6. Half-width minus zero-concentration half-width (extrapolated) vs. temperature for line 4. To follow page 12. 0 1 0 2 0 . 3 0 4 0 5 0 6 0 7 0 8 0 9 0 T (°K) F I G . 6 FIGURE 7. Absorption constant vs. wave number for 11 ohm cm 15 ^ (1.2 x 10 cm ) boron-doped s i l i c o n at four temperatures. This f i g u r e i l l u s t r a t e s some of the d i f f i c u l t i e s and un c e r t a i n t i e s involved i n "peeling" o f f the influence of neighboring l i n e s . To follow page 12. 13 Incomplete c a n c e l l a t i o n of water vapor l i n e s may be r e s p o n s i b l e f o r the small bump on the high energy s i d e of l i n e 4 (see F i g . 7 ) . S i m i l a r l y , the dip i n l i n e 2 reported e a r l i e r (Colbow et a l . 1962) may be due to water vapor. The f a c t that a r a t h e r strong water vapor absorption l i n e occurs r i g h t at the p o s i t i o n of l i n e 2 gives the data f o r t h i s l i n e ( F i g u r e 5b) l e s s r e l i a b i l i t y than those f o r l i n e 4 and 1 (Figures 5a, c ) . Since a more complete set of h a l f - w i d t h measurements was a c c e s s i b l e f o r l i n e 4, Figure 5a w i l l be used e x c l u s i v e l y i n the q u a n t i t a t i v e i n t e r p r e t a t i o n s i n Chapter I I I , i Other pieces of information e x t r a c t e d from the data i n Figure 1 are the p o s i t i o n s of the absorption l i n e s ( F i g u r e 8) and the i n t e g r a t e d a b s o r p t i o n (Table 2 ) . The energy l e v e l s i n Figure 8 have been p l o t t e d by assuming an i o n i z a t i o n energy of 46 mev ( B u r s t e i n et a l . 1956, Kohn 1957), The present author b e l i e v e s that an i o n i z a t i o n energy of p o s s i b l y as low as 44 mev cannot be r u l e d out by the present experimental data. This value would g i v e b e t t e r agreement between the experimental and the c a l c u l a t e d (Schechter 1962) p o s i t i o n s of e x c i t e d s t a t e s (Figure 8 ) . The experimentally determined p o s i t i o n s of the absorption l i n e s are e s s e n t i a l l y i n agreement w i t h those obtained e a r l i e r by Hrostowski and Ka i s e r (1958), At 4.2°K an a d d i t i o n a l l i n e was resolved near the i o n i z a t i o n edge i n the 130 ohm cm m a t e r i a l . The u n c e r t a i n t y of the i n t e g r a t e d absorption i n Table 2 i s about 10 percent f o r a l l l i n e s from the u n c e r t a i n t y i n the product p h c, i , e . Eq. (2-11), In a d d i t i o n an u n c e r t a i n t y of approximately 5 percent e x i s t s due to the f a c t that the wings of the absorption l i n e s may not be w e l l f i t t e d by V o i g t p r o f i l e s . However, t h i s i s d i f f i c u l t to estimate due to an overlap of neighboring l i n e s . 14 The i n t e g r a t e d absorption c r o s s - s e c t i o n s (]E) i n Table 3 were obtained by d i v i d i n g the i n t e g r a t e d absorption (Table 2) by the number of boron i m p u r i t i e s i n the ground s t a t e (Figure 9). The u n c e r t a i n t y of i s estimated to be approximately 25 percent. FIGURE 8. Energy level diagram for holes in boron-doped silicon. The "experimental" energy level diagram (left side) was obtained by taking the ground state at -46 mev. Along the vertical arrows the positions of the observed absorption lines are given in mev and cm"*-. The numbers in the center of the diagram (1 to 9) serve to identify the levels for discussion in this thesis. The relative intensities quoted in the diagram refer to a boron-concentration of 1.0 x 10 1 4 cm"3 at 4.2 °K. For comparison, the right side of Figure 8 shows the calculated positions (Schechter 1962) for the ground state and the lowest four excited states. The numbers in bracket give the theoretical degeneracies. To follow page W/MIIIIIIIIIIIIIIIIIIIIIIIIIIIIII,, VALENCE BAND W//M. 12 -16 -20 > -24 B -28 -32 • 36 -40 -44 — — y r e l a t i v e i n t e n s i t i e s ..0 4.7 .73 4.7 0.3 0.4 0.6 0.3 0.1 ' B 'a 'fl 'a 'e 'a 'a 'a 'a C J o o o o , o o O o s t m 00 to o C O • • • • • • • • • 00 O N O N St 0> " 0 bo C O s t r ~ o I—1 C O C O St St LO C O C O C O C O C O C O C O > > > > > > > > > <D CD <D B a a 6 B a a a a LO s t St C M to LO o o o o o O o o o • • • « • • • « • o o o o o o o o o +1 + + 1 +1 + 1 +1 +1 +1 +1 CNI O <t O N N O LO O N St iO C O N O <t © i—i • • • . • e o s t 00 O N C M C M C O C O C O co co C O St St St St St •4 •3 T h e o r e t i c a l (Schechter ,1962) B<0 ; - — - B>0 7 ~ (2) (2) (4) (4) (4) FIG. 8 FIGURE 9. i D i s t r i b u t i o n of holes between the ground s t a t e ( a ) , e x c i t e d s t a t e s ( b ) , and the valence band (c) vs. temperature. The boron concentrations are: 1) 1.0 x 10 cm" , 2) 1.2 x 1 0 1 5 cm"3, and 3) 1.2 x 1 0 1 6 cm"3. The o r d i n a t e giv e s the number of holes i n u n i t s of 1 0 1 4 , 1 0 1 5 , and 1 0 1 6 -3 cm f o r cases 1), 2 ) , and 3) r e s p e c t i v e l y . To f o l l o w page 15 CHAPTER III lo Impurity States and the Hydrogenic Approximation S i l i c o n i s a group IV semiconductor, and i t s electr o n i c structure i s such that at zero temperature the electrons f i l l the valence band, leaving the conduction band empty. When a small quantity of a group III or a group V element i s added, a set of energy l e v e l s , l o c a l i z e d at the impurity atom i s formed. For group V impurities these are referred to as donor l e v e l s . They correspond to an electron loosely bound to the impurity ion. This electron can be "donated" to the conduction band with a small expenditure of energy. In the case of group III elements the le v e l s are referred to as acceptor l e v e l s , since they can accept an electron from the valence band. A l t e r n a t i v e l y one may p i c t u r e acceptor le v e l s as containing holes loosely bound to the impurity ion. Boron i s a group III element and gives r i s e to acceptor l e v e l s i n s i l i c o n . At room temperature the electrons and holes are removed from the impurity s i t e s by thermal excitations, and e x i s t as free charge c a r r i e r s i n the c r y s t a l , while at l i q u i d helium temperature nearly a l l these c a r r i e r s are bound to s p e c i f i c impurity s i t e s and occupy the lowest energy state. These bound states have been extensively studied (Kohn 1957), and i t has I I been shown that they can be approximately, described by the same Schrodinger equation as the states of an electron i n a hydrogen atom, but with the Coulomb p o t e n t i a l modified by the s t a t i c d i e l e c t r i c constant H , and the electron mass replaced by an e f f e c t i v e mass m*, ( K i t t e l and M i t c h e l l 1954, Luttinger and Kohn 1955): ( 2ni* V 2 . e 2 / K r ) F ( ? ) = E F ( r ). (1-1) In analogy with the solutions of the hydrogen atom, one expects an energy l e v e l scheme with the energies reduced by m*/mX2 and the Bohr o r b i t s 16 enlarged by Xm/m . Taking for boron-doped s i l i c o n X =12 and m /m = 0,5 one obtains roughly the observed value for the i o n i z a t i o n energy of 46 mev„ o The radius of the f i r s t Bohr o r b i t turns out to be 13 A for acceptors, o rather than the 0o53 A i n hydrogen. These large Bohr o r b i t s give an explanation of why such a simple d e s c r i p t i o n of impurity states i s possible. The mean distance between the bound c a r r i e r and the impurity i s s u f f i c i e n t l y large f o r a macroscopic quantity l i k e the d i e l e c t r i c constant to be u s e f u l . Also, the o r b i t s of the bound c a r r i e r s are s u f f i c i e n t l y large that we need not be concerned with the d e t a i l e d nature of the wave function rig h t at the impurity s i t e . This i n any case i s true of excited states, such as 2 p states, whose wavefunctions vanish at the impurity s i t e , 'The s-states are l i k e l y to be much more complicated and depressed in energy, since the eigenfunctions are large r i g h t at the impurity s i t e where the p o t e n t i a l i s no longer modified by the d i e l e c t r i c constant. This d e s c r i p t i o n of impurity states would be quite good, i f the energy vs, wave number vector diagram were of parabolic shape and had a simple extremum at k = 0. The wave functions of the acceptor states are l i n e a r combinations of Bloch waves chosen from near the top of the valence I ! band, and the e f f e c t i v e mass Schrodinger equation (1-1) corresponds simply to an expansion of the energy around k = 0, keeping terms to order k . The f a c t that we are concerned only with small k follows from the uncertainty p r i n c i p l e . The large s p a t i a l extension of the Bohr o r b i t implies a small extension for the c r y s t a l momentum k. Unfortunately, the valence band i n s i l i c o n i s not simple. Information about the structure of t h i s band *has been obtained from cyclotron resonance experiments by Lax, Zeiger, and Dexter (1954) and Dresselhaus, Kip, and K i t t e l (1955). If i t were not for s p i n - o r b i t coupling the valence band maximum at k = 0 would be s i x f o l d degenerate, 17 including the double degeneracy a r i s i n g from spin. The simples way to understand t h i s degeneracy i s to consider the t i g h t binding l i m i t , i n which the wave functions corresponding to the maximum go over into 3p atomic wave functions for s i l i c o n . The s p i n - o r b i t coupling l i f t s the degeneracy p a r t i a l l y . According to E l l i o t t (1954) the top of the valence band remains f o u r f o l d degenerate, corresponding to atomic J. = 3/2 states, s p l i t t i n g away from k — 0 in the (100) d i r e c t i o n into twofold degenerate bands. Another twofold degenerate state l i e s an amount X below the top of the valence band corresponding to J = 1/2. A t h e o r e t i c a l estimate (Kohn 1957) gives X = 35 mev. Since in boron-doped s i l i c o n the highest acceptor states (lowest energy states for bound holes) l i e above the valence band by about 40 mev, one expects that a l l six valence bands w i l l enter with appreciable amplitude in the acceptor states. Instead of a single p a r t i a l d i f f e r e n t i a l equation, Eq. (1-1), one finds a set of s i x coupled p a r t i a l d i f f e r e n t i a l equations. These have been derived by K i t t e l and M i t c h e l l (1954) and Luttinger and Kohn (1955). They are err ^  [ 0 W •$ ( 1_ 2 ) Here £ i s an acceptor state energy r e l a t i v e to the valence band, maximum. The ^, are numbers re l a t e d to three e f f e c t i v e mass constants A, B, C (which are determined from cyclotron resonance data£, and £b»wthe spin-or b i t s p l i t t i n g X). The s p e c i f i c form of the j^ , may be found i n the a r t i c l e s by K i t t e l and M i t c h e l l (1954) and Luttinger and Kohn (1955). In terms of the e f f e c t i v e mass constants, the energies of the three valence bands are given by 18 E 1 > 2 (k) = Ak2 + [B 2 k 4 + C 2 (k x 2 k y 2 + k y 2 k/ + k z 2 k x 2 ) ] 1 / 2 , (1=3) E 3(I) = -X + Ak2, each eigenvalue being doubly, degenerate. The total wave functions of these acceptor states have the form V(?) = IZ An (k) V n,l? n,k 6 , -> F. (r) 0. (r). (1-4) ~ 1 j j Here the (j^j (?) are the Bloch functions at the top of the valence bands in the unperturbed crystal. The Fj (r) are slowly varying functions which modulate the Bloch functions. For this reason they are often referred to as envelope functions. In the limit where the mass constants B and C become zero (corresponding to spherical constant energy surfaces), the envelope functions reduce to simple hydrogenic functions. Schechter (1962) attempted to solve Eq. (1-2). In order to simplify the calculations he first calculated the energy levels in the extreme limits of X = 0 and X =0*. First-order perturbation calculations were then made, in \ and X~^ in order to obtain estimates for the energy levels corresponding to the actual, finite value of X. Expanding the envelope functions F.. ("r) in spherical harmonics, Schechter»s acceptor state trial functions had the form V ^ Z I r 1 exp. <-r/r ) X I C j Y < 9 , <J> ) 0 <r). (1-5) 1 1 j,m lm lm ' j The complete acceptor Hamiltonian is invariant under the operations of the full tetrahedral group T^ (Margenau and Murphy 1943) if spin is ignored, or, the tetrahedral double group (Dresselhaus 1955) if spin is included from the beginning. The acceptor state wave functions 19 f o r each- l e v e l form bases for i r r e d u c i b l e representations of the appropriate symmetry group. Since the spherical harmonics for each 1 go into spherical harmonics of the same 1 under the symmetry operations, each term i n the sum over 1 in Eq.(l-5) must transform i n the same way as the complete wave function. A f t e r using group t h e o r e t i c a l arguments to reduce the number of s p h e r i c a l harmonics appearing in the t r i a l functions, and r e s t r i c t i n g 1 to values less than some IQ , the best approximations to the wave function and the energy were obtained by maximizing the expectation value of the energy f o r an electron state. (This corresponds to minimizing the energy for a hole s t a t e ) . The sign of the e f f e c t i v e mass constant B i s experimentally not determined, since the valence band energies depend only on B 2 by Eq. (1-3). However, B occurs l i n e a r l y i n the D j j > o f E f l * (l- 2)» Schechter (1962) cal c u l a t e d the ground state energy and those of the highest four excited electron acceptor states for both p o s i t i v e and negative B. The positions of these c a l c u l a t e d l e v e l s , together with t h e i r degeneracies ( i n brackets) are shown i n F i g . 8. The agreement with experiment i s seen to be only f a i r . The l e v e l s are either f o u r f o l d degenerate corresponding to the representation P , or twofold corresponding to V, and T% of the 8 o ' tetrahedral double group. The ground state i s predominantly I s - l i k e and the lowest four excited states 2p - l i k e i n the sense that they reduce to hydrogenic Is and 2p states r e s p e c t i v e l y , in the l i m i t when a l l three valence bands c o l l a p s e into one. In the following sections these rather complicated wave functions w i l l not be used. Instead simple hydrogenic wave functions w i l l be a r b i t r a r i l y employed in which the f i r s t Bohr o r b i t i s replaced by an e f f e c t i v e Bohr o r b i t a*. It w i l l be required that t h i s e f f e c t i v e 20 Bohr orbit satisfy the hydrogenic energy equation E n = - e2/ 2 X a* n 2 , (1-6) where E n is the experimental binding energy. 2. General Interpretation of Line Broadening Consider a semiconductor with a random distribution.of ionized and - neutral acceptor or donor impurities. At an absorbing neutral impurity there will be a certain electric field, the magnitude of which depends on the specific spatial configuration of the surrounding impurities. This electric field may give rise to a first order Stark splitting, or second order Stark shift, of different magnitudes for different impurity sites. Thus one obtains a broadening of the resultant absorption line. Let us suppose the distribution of intensity of absorption for one neutral impurity which is in an electric field F, is given by I (F,W); then in order to obtain the total intensity distribution in the Stark broadened line one must integrate over a l l such distributions weighted by the probability of the distribution occurrences oo I (fiW),diO = dOO j" I (F,(0) W (F) dF. (2-1) o Here W (F) is the field strength probability function, which has been studied by Holtsmark (1919, 1924) and others in connection with gravitational problems and pressure broadening in gases. This probability function will be discussed in the following section. Each component of the statistically- broadened absorption line will be treated as a sharp line, that is to say the I (F,C0) in Eq.(2-1) will be replaced by a delta function. Actually the true width of I(F,W) may be obtained by extrapolating the experimental true width in Figure;; 5, to zero concentration of impurities (dotted lines). These widths are seen to be 21 already quite considerable, and are believed to result primarily from phonon broadening, and broadening by internal strains due to dislocations. The latter two broadening mechanisms will be discussed in sections 10 and 9 respectively. After the width resulting from statistical Stark broadening of two sharp energy levels has been obtained it has to be combined with the width at zero concentration by a convolution integral of the form fc (x) = (x - y) I(y) dy , (2-2) -oo where f^ (x) is the true line shape at concentration c, f Q (x - y) is the true line shape extrapolated to zero concentration of impurities, and I(y) is the line shape obtained from, statistical Stark broadening under the assumption that the I(F,Cu>) in Eq. (2-1) are sharp lines. If one makes the " it simplifying assumption that f c , f 0 , and I(y) have lorentzian profiles, " tt then the half-width of f c is simply the sum of the half-widths of f 0 and I(y). This assumption is at least approximately justified (Colbow et al. 1962). 3. Statistical Theory of Stark Broadening The basic question is, "What is the probability that the frequency of a sharp absorption line is displaced by an amount between A CO and ACO + dACO"? Let us assume that this frequency displacement results from the displacement of only one of the two levels involved in the absorption, by means of either the linear or the quadratic Stark effect. This will be justified later. The probability of a certain frequency displacement due the surrounding impurities is then simply related to the probability of a certain electric field at the absorbing impurity. This in turn depends on 22 the probability of finding another impurity at a distance r from. i t . Since the I(F,to) in Eq. (2-1) is taken to be a delta function, the probability function for a frequency displacement between ACO and ACJ+ dACO will give directly the desired line shape of the absorption line I(AtO). The problem may be divided into several parts according to the dependence of the field on the distance from the field producer. In the present section it will be assumed that the field at the absorbing neutral impurity results mainly from the surrounding ionized impurities, and has the form of a Coulomb field: F = (e/4H"€) r . This case becomes important above 40°K, when an increasing number of impurities will be ionized (Figure 9). (In the following section the effect of screening by free holes will be discussed in an approximate fashion, and in sections 7 and 8 line broadening resulting from the interaction with neutral impurities will be considered). For the purpose of illustration it is useful to consider first the effect of only the nearest neighbor. This approximation may be referred to as the binary form of the Holtsmark theory. Consider a neutral impurity atom and ask what is the probability W(r) dr that there is no ionized impurity closer than r and one ionized impurity in the spherical shell volume element 4TCr dr at r. Let there be N.j_ ionized impurities per cm . The probability of having one ionized impurity in the shell between r and r + dr is assumed to be given by p(r, dr) = (4TTr 2 dr) Nt. The probability of no ionized impurity closer than r, P(r) is obtained from the relation P(u + du) = P(u) (1 - 4rTTu2 du Nt). Thus dP / P = - 4TTu 2 N£ du , 23 which gives after integration P(r) = exp.[ - (4TT/3) r 3 Ni] . Thus one has W(r) dr = P(r) p(r, dr) = 4 l N t r 2 dr exp.[- (4TC/3) Nt r 3 ] . After introducing the mhan s^ae*«g r Q , defined by (4TT/3) r 0 3 = 1/ % , (3-L) one may finally write W(r) dr = exp.(-r/r 0) 3 d(r/r 0) 3 . (3-2) Since a given r defines an electric field F at a neural impurity site, and an electric field defines a frequency displacement via the Stark effect, Eq. (3-2) also represents the distribution of these last two quantities. Thus if the field is given, by F = (e/4T6) r - 2 , substitution into Eq. (3-2) gives W(F) dF = C exp. [- ( F 0 / F ) 3 / 2 ] d ( F 0 / F ) 3 / 2 = (3/2F) (F 0/F) 3/ 2 exp. [- (F 0/F) 3 / 2'] dF , (3-3) 2 where F D — (e/4TT£) r Q . C was taken as minus one, in order that the field probability be normalized to T W(F) dF = 1 . (3-4) o For the linear Stark, effect AW = s F and A C0Q = s F Q, where s is a constant. Using a delta function for l(F,W) , one obtains from Eq. (2-1) oo KAOO) = S <f(AlO - s F) W(F) dF o oo = / I s ! " 1 £(F - iW/s) W(F) dF = I s|" 1 W(kW/s) . Substituting for W(AW/s) from Eq. (3-3), this becomes 24 I(AU>) ,= (s/|s|) (3/2 LU) (AW 0/AU3) 3 / 2 e x p . [ - ( L » 0 / * 0 ) 3 / 2 ] . (3-5) One obtains two peaks corresponding to p o s i t i v e and negative s, each having the i n t e n s i t y d i s t r i b u t i o n function I(AU) as given by Eq. (3-5). 9 2 For the quadratic Stark e f f e c t Au?= t F and A bJQ - t F Q , where t i s another constant. Again using Eqs. (2-1) and (3-3) t h i s gives oo I(ACO) = / ^(AW - t F 2 ) W(F) dF o oo = / I t l " 1 «f(F 2 - A(0/t) W(F) dF = (1/2 | t | ) ( A ( 0 / t ) " 1 / 2 J [^[F - ( A W / t ) I / 2 ] + 6~[F + ( A c O / t ) 1 / 2 ] j W(F) dF = ( l / 2 | t | ) ( A U ) / t ) - 1 / 2 w [ ( A W / t ) 1 / 2 3 , since F i s p o s i t i v e . Or f i n a l l y , using Eq. (3-3) I(AU» = (t/|t|)(3/4A(0)(ALJ o/AW) 3 / 4 exp.[ -(At0o/Au;) 3 / 4] . (3-6) For any given energy l e v e l t can be either p o s i t i v e or negative, depending on the s p e c i f i c energy, l e v e l structure. The foregoing elementary considerations, which y i e l d the binary approximations to the i n t e n s i t y d i s t r i b u t i o n Eq. (3-5) and (3-6), have been superseded by the work of Holtsmark (1919, 1924) and others, who included the cooperative e f f e c t of many ions. Review a r t i c l e s on t h i s f i e l d where written by Chandrasekhar (1943), Breene J r . (1957), and Margenau and Lewis (1959). The f i e l d F, which appears i n Eq. (3-3), must be written as a vector sum when i t i s compounded from the f i e l d s of many impurities in d i f f e r e n t places; 25 F =|2__Fn| . (3-7) The analysis leading to Eq. (3-3) must be carried out in a configuration space of 3N dimensions. In place of Eq. (3-3) it yields the function W(B) dp plotted in Figure 12 (curve labelled Holtsmark), where F/FQ . For small and large values of B one obtains respectively the two alternate expressions: oo W(B ) = (2/TtB) / v sin v exp. [ -(v/ B ) 3 / ] dv o f(4/3TT) B 2 (1 - 0.463 B 2 + 0. 1227 B A + ... ), p small 496 B " J / i (1 + 5.107 B" J /' + 14.93 p " 3 + ..),B large. L 1. 5 / 2 B~ 3 / 2 (3-8) In the binary theory F Q was defined by F Q = (e/4TC6) r 0 " 2 = (e/4TC6) [ ( 4 T / 3 ) 2/ 3 = 2.60 (e/4TU6) N i 2 / 3 . The many-ion calculation gives nearly the same parameter: F D = 2.61 (e/4TTe) N t 2 / 3 . (3-9) QJne then passes to the frequency distribution in the same way as before: In the linear Stark effect ku?/Au50 = F/Fc = B , and since in general I(kiJ) d(&u3) = W(£) dB, (3-10) one finds K A O ) = i/LW0 w(Au;/t^u;0) . (3-11) 1 /o For the quadratic Stark effect (&U?/tW0) ' = F/FQ = |3 , and Eq. (3-3) gives KA00) ,= 1/2 (£\0Jo&vOr1/2 W t ( W / f c u ) Q ) 1 / 2 ] . (3-12) 26 4. The E f f e c t of Screening When.the c r y s t a l contains ionized impurities per cm , there w i l l also be an equal number p = of free charge c a r r i e r s , the influence of which has been neglected so f a r . The e f f e c t of these free charge c a r r i e r s may be approximated by using a screened Coulomb p o t e n t i a l of the form U ,= (e/4Tre) 1/r e " r / X , (4-1) . IT Where A i s the Debye-Huckel (1923) screening length. In c l a s s i c a l s t a t i s t i c s A = [ kT/4TT p e 2 ] 1 / 2 , (4-2) k being the Boltzmann constant. D i f f e r e n t i a t i n g the expression for U in Eq. (4-1) one obtains the screened Coulomb f i e l d F> = (e r/4Tl*r 2) [(1/r) + ( 1 / A ) ] e " r / X , (4-3) which i s to replace the simple Coulomb f i e l d i n Eq. (3-3). However, analysis of the line-broadening problem even with t h i s s i m p l i f i e d f i e l d remains formidable. Using the same approach as Ecker (1957), F has been approximated by '4TC6 r 3 i f r < X (4-4) 0 if r U , Figure 10 shows a comparison between the screened f i e l d at 90°K and the Coulomb f i e l d with i t s cut-off at r = A for three concentrations of impurities, as a function of r in units of r s . Where the raeaw apae-iog r s i s defined by (4TU/3) r s 3 = 1/NA. (4-5) Using Eq. (4-4), the method of section 3 leads to a dependence of the l i n e shape on the screening parameter FIGURE 10. Screened electric field (dotted lines), and Coulomb field with cut-off (full lines), as a function of separation. The separation between field point and field producer is expressed in units of the ntonni uporing "hr.twr.r.n . impnr "I ti r.n 14 -3 (r s ) . The boron concentrations are: i) 1.0 x 10 cm , ii ) 1.2 x 10 1 5 cm"3, and i i i ) 1.2 x 10 1 6 cm-3. The screened electric fields were calculated from the equation F = (e/4Tie) r - 1 ( r " f +X" 1) e ° r / X , using the calculated screening length X at 90°K. To follow page 26. 27 £ = (4Tt\ 3 /3) % = (1/6)TU° 1 / 2 (kT/e 2) 3 / 2 p~ 1 / 2 . (4-6) The physical meaning of the screening parameter is the number of ionized impurities within the Debye radius A . Clearly as X tends to infinity the results must agree with those of Holtsmark. Figure 12 shows the results of Ecker's (1957) computations for several values of £ . From these curves one can obtain the half-widths h(B), which have been plotted in Figure 13 as functions of the screening parameter. The dependence of the screening parameter on temperature has been shown in Figure 11 for three different concentrations of impurities. Figure 11 was obtained by using Eq. (4-6) and the values for p from- Figure 9. 5. Temperature Dependence of the Half-width due to the Linear and  the Quadratic Stark Effect For the linear Stark effect &(0 = s F = s F 0 B , and hence in units of frequency, the half-width of I(A.(0) is given, by h^AtO) = s F D h(B ) . (5-1) For the quadratic Stark effect = t F 2 = t F Q 2 p 2 , and hence in units of frquency, the half-width of I(&CO) is given by h2(Lo!>) - t F 0 2 h 2(B ) . (5-2) Taking for silicon the dielectric constant 6/ 6 Q = 12 , one finds, using Eq. (3-8) F Q = 2.61 (e/4Tue) N i 2 / 3 =3.13 x 10'8 N i 2 / 3 volt cm"1, (5-3) -3 where Ni is in units of cm . Using Eqs. (5-1), (5-2), (5-3), and Figures 11 and 13, the curves for h^(AcJ) / s and h2(AoO) / t as a function of temperature for three FIGURE 11. Screening parameter, £,,vs. temperature f o r boron concentrations of i ) 1.0 x 1 0 1 4 cm" 3, i i ) 1.2 x 1 0 1 5 cm"3, and i i i ) 1.2 x 1 0 1 6 cm"3 . To f o l l o w page 27. FIGURE 12. F i e l d d i s t r i b u t i o n function, W(B), for various values of the screening parameter, <T. (Ecker 1957). To follow page 27. 0 0.5 1 1.5 2 2.5 3 3.5 4 FIG. 12 FIGURE 13. Half-width of the f i e l d d i s t r i b u t i o n , h ( B ) , as a function of the screening parameter, 6*; (logarithmic s c a l e ) . To follow page 27. 28 impurity concentrations have been calculated. These graphs are shown in Figures 14 and 15 respectively, and their temperature dependence must be compared with Figure 6, which was obtained by subtracting the half-width extrapolated to zero concentration of impurities (Figure 4) from the true half-width in Figure 5. Let us assume that the electric field at the impurity center has its direction along the z-axis. The perturbation Hamiltonian is then «fV = - F e z . (5-4) The wave functions of the bound carrier have the form yr (!)(•?) = ZIoCj F. ( i )(r) 0.(r) (5-5) in the effective mass theory of donor or acceptor states. Here the are numerical coefficients, the 0j(r) Bloch waves, and the F^^(r) are hydrogen-like envelope functions. The matrix element of <fV between two such states is given by ( y ( i ) , y ( i ' ) ) . F j ( O f £ v F j(i»> ) . ( 5 . 6 ) Since the effective mass Hamiltonians for both types of states are invariant under inversion and there are no accidental degeneracies, such as the 2s, 2p degeneracy in hydrogen, the first-order Stark effect vanishes. However, the full Hamiltonian of the impurity, problem has only tetrahedral symmetry and is not invariant under inversion. As a result, if the effective mass theory is seriously in error, states belonging to the representation T^ , a n <* Vg can have an appreciable first-order Stark effect. This might possibly be of significance for the acceptor ground state (Yq) in silicon (Kohn 1957), but the resulting effect on the half-width is difficult to estimate. By Eq. (3-5) one would presumably obtain two peaks, corresponding to positive and negative strength parameter, whose unresolved resultant may give rise to FIGURE 14. Half~width in units of the strength parameter s vs. temperature for the linear Stark effect. The boron concentrations are: i) 1.0 x 10 1 4 cm"3, ii) 1.2 x 10 1 5 cm"3, and i i i ) 1.2 x 10 1 6 cm"3 . To follow page 28. 0 10 20 30 40 50 60 70 80 90 T (°K) FIG. 14 FIGURE 15. Half-width in units of the strenght parameter t vs. temperature for the quadratic Stark effect. The 14 3 boron concentrations are: i) 1.0 x 10 cm , ii) 1.2 x 10 1 5 cm"3, and i i i ) 1.2 x 10 1 6 cm"3 . To follow page 28. 29 an appreciable broadening of the absorption line. Fortunately, however, Figures 6, 14, and 15 seem to suggest by their temperature dependence that the second-order Stark effect might be the dominant effect and, as will be seen in the next section, sufficient to explain the experimental data. If the broadening in Figure 6 were caused predominantly by the first-order Stark effect a steeper rise of the half-width at low temperature would have to be expected, as will be seen from Figure 15, especially for the 1.3 ohm cm material. 6. Evaluation of the Half-width due to the Quadratic Stark Effect The second-order Stark shift for a nondegenerate state y ^ is given by *2E<D = ' | ( y ( i ) , X v V i , ) ) | 2 / ( E < ^ - E^')) . (6-1) Replacing the denominators E^^ - E^*'^ by an average value AE and assuming that the linear Stark effect ( y ( i ) , &V vanishes, Eq. (6-1) becomes * 2 E ( 1 ) = 1/AE £ ! ' | £v y ( i , ) ) l 2 = 1 / A E 5Z* ( > ( i ) , «sv y ( i , ) ) ( > ( i , ) , iv y ( i ) ) i 1 = 1 / A E ( > ( i ) , (cfV) 2 ) = 1/AE ( F ( i ) , (cJV) 2 F ( 1 ) ) . (6-2) The shifts will be of a similar order of magnitude for degenerate states. Using Eq. (5-4) for £ V and simple hydrogenic wave functions for F^^, Eq. (6-2) has been evaluated in Appendix II. The result is £2E = t F 2 = c (e F a*) 2 / AE . (6-3) Here a* is the effective Bohr radius, and c is a constant which turns out to be 1 for the Is state and 18.for the 2p0 state (Appendix II). Thus, the 30 quadratic Stark e f f e c t may be neglected f o r the ground s t a t e , when compared w i t h that f o r the e x c i t e d s t a t e . What i s the value of t r e q u i r e d , such that h 2 ( A ^ ) / t i n Figure 15 gives the temperature dependence of the h a l f - w i d t h in> F i g u r e 6? The curve f o r the 1.3 ohm cm boron-doped s i l i c o n seems best s u i t e d f o r numerical comparison, s i n c e i t shows the l a r g e s t change w i t h temperature. From F i g u r e 6 the change i n h a l f - w i d t h between 0°K and 80°K i s M h J ! - h^) .= 1.06 mev. (6-4) Over the same temperature range one obtains from F i g u r e 15: A [ h 2 ( A O ) / t ] = 0.95 x 10 6 v o l t 2 cm"2 . (6-5) Equating k ^ t ^ A v O ) ] and A (h£ - h^) one obtains t = 1.1 x 10" 6 mev v o l t " 2 cm 2 . (6-6) Let us assume that energy l e v e l 4 (Fig u r e 8) may be approximated by a hydrogenic 2p s t a t e . The experimental value f o r i t s binding energy i s E 2 = (46 - 39.7) mev = 6.3 mev . (6-7) Using t h i s v a l u e and the hydrogenic energy equation E n = - m* e 4 / 2 Ag2 Y2 n 2 = - e 2 / 2 X a* n 2 , (6-8) wi t h n = 2 , one obtains f o r the e f f e c t i v e Bohr ra d i u s of s t a t e 4: a* = 24 1 . (6-9) Using Eqs. (6-6), (6-9), and t a k i n g c = 18 f o r a 2p 0 s t a t e , Eq. (6-3) leads to A E = c e 2 a* 2 / t = 1 mev , (6-10) which has the order of magnitude one would expect from the general nature of the energy, l e v e l diagram ( F i g u r e 4 ). In equating A[h2(AW)J and A ( h c - h©) i t was assumed i m p l i c i t l y , that the i n t e r a c t i o n between n e u t r a l i m p u r i t i e s , which i s r e s p o n s i b l e f o r the low temperature broadening, changes l i t t l e w i t h temperature below 80°K. 31 Screening and a decrease of NA - will tend to reduce this contribution to the half-width with increasing temperature. However, since these forces will be appreciable only for nearest neighbors, for which the screening is small below 80°K, the temperature dependence of their contribution to the half-width should be relatively small below 80°K. 7. Statistical Broadening due to van der Waals Forces In this and the following section the interaction between neutral impurities will be discussed. The present sectionideals with van der Waals forces, which arise when the overlap between the wave functions is negligible. All other forces between neutral atoms may be classed as "overlap forces" and arise only when the wave function of one atom overlaps that of the other. These forces will be considered in the following section. Margenau (1933) developed the statistical theory for broadening by van der Waals forces in order to explain pressure broadening in gases. The line shape of this distribution is I(AV>) = 2TUb1/2 N A t f " 3 / 2 exp.^Ti: 3 N2 / 9 A*) . (7-1) The half-width of this distribution is h w(AV) = 0.82TC3 b N2 . (7-2) Where b is the strength constant of the van der Waals forces. That is Vi = - b r f 6 (7-3) is the van der Waals interaction between the absorbing impurity and the ith neutral impurity, and r i is the distance between them. When applied to the present problem N = - Ni is the number of neutral impurities per cm3. Thus at very low temperatures one has N = N^ . Figure 6 shows that at low temperatures when the number of ionized impurities is negligible, the half-widths are 0.03, 0.12, and 0.60 mev, for impurity c o n c e n t r a t i o n s of ( 1 , 12, and 120) x 10 cm r e s p e c t i v e l y . I f the van der Waals f o r c e s were to e x p l a i n t h i s c o n c e n t r a t i o n broadening at low temperatures, then according to Eq. (7-2) the h a l f - w i d t h should increase w i t h the square of the c o n c e n t r a t i o n . That i s , they should be i n the r a t i o s 1 : 144 : 14400. However, they are observed to go l i k e 1.: 4 : 20 . I f the expression by London and E i s e n s c h i t z (1930) f o r the s t r e n g t h parameter b i s modified to s u i t the present problem, one obtains b = 6.48 ( e 2 / K a * ) a*° . (7-4) S u b s t i t u t i n g t h i s i n t o Eq. (7-2), and t a k i n g X = 12 f o r s i l i c o n , leads to h w(A\^) = 2.0 x 10" 3 a * 5 N 2 mev, (7-5) where a* i s i n cm and N i n cm". Taking f o r the e f f e c t i v e Bohr ra d i u s of the ground s t a t e a value of a* = 13 X, Eq. (7-5) g i v e s f o r N = 1.2 x 1 0 1 6 cm"3 h w(Av>) = 1.1 x 10" 5 mev. (7-6) This has to be compared w i t h an experimental h a l f - w i d t h (he - h^) of 0.60 mev. Thus i t would seem that the van der Waals i n t e r a c t i o n i s too weak to e x p l a i n the v a r i a t i o n of the h a l f - w i d t h w i t h c o n c e n t r a t i o n of i m p u r i t i e s at low temperatures ( F i g u r e 6 ) . 8. Broadening due to Overlap Forces In t h i s s e c t i o n i t w i l l be shown i n a q u a l i t a t i v e way how overlap forces may account f o r the observed e a r l y onset of c o n c e n t r a t i o n broadening at low temperatures ( F i g u r e 6 ) . B a l t e n s p e r g e r 1 s (1953) work, the r e l e v a n t part of which i s shown i n F i g u r e 16, suggests that c o n c e n t r a t i o n broadening should s t a r t at r s = 12 a* . (8-1) Here a* i s the e f f e c t i v e Bohr radius of a bound c a r r i e r of e f f e c t i v e mass m* FIGURE 16. Broadening of hydrogenic levels vs. distance between impurities in units of the effective Bohr radius, a*. (Baltensperger 1953). To follow page 32. 33 in a crystal with dielectric constant X, and r g is defined by ( 4 T/3)r s 3 = 1/NA. (8-2) If one takes (Conwell 1956) for boron-doped silicon "K ~ 12, m /m = 0.5, and hence a* = X a m/m* = 13A , (8-3) then Eqs. (8-1) and (8-2) would suggest that concentration broadening should set in at about N = 6 x 10 1 6 cm"3 . (8-4) This, as was mentioned earlier, seemed to agree with Newman's (1956) data. In contrast to this, however, the present data (Figure 6) shows that concentration broadening starts below NA = 1.2 x 10 1 5 cm"3 . (8-5) How can one bring Baltensperger's calculated values in agreement with this earlier onset of broadening? Two ways suggest themselves. The first involves replacing r s in Eq. (8-1) by Xr s, where X<1. This takes care of the fact that the impurities do not form a regular lattice as assumed by Baltensperger (1953). The second modifiction, that of a*, takes into consideration that the simple hydrogenic model used to obtain Figure 16 gives rather inadequate values for the binding energy. Instead of a regular spacing of impurities, let us assume a random distribution. Since for overlap forces we are only concerned with small separations, we may use the binary approximation of Holtsmark's theory. By Eq. (3-2) the probability of a nearest neighbor at a distance r = Xr s is given by W (X) dX = exp. (-X3)dX3 . (8-6) Now let us ask the question: "What is the value of X = Xjy2> such that half the impurities find themselves closer than X]y2 r s t o t n e neighboring one, 3 4 and half of them are further away?" Eq. (8-6) gives if3 _ i _ a v n (_ v 3 1^/2 1/2 = f exp.(-X3) dXJ = 1 - exp. (- X J ) , o or, Xl/2 = 0.885. The mean spacing of the impurities closer than rs t o t n e i r nearest neighbor is 0.648 r s , which suggests that the effective mean spacing in Eq. (8-1) should be about r g * = 0.7 r s . (8-7) For computing the edges of the Is and the 2p.bands (Figure 16) Baltensperger (1953) used a simple hydrogenic model in conjunction with the cellular method. This involves assuming the validity of the effective mass Schrodinger equation *2/2m* V 2 V + (e 2/Xr + E n ) V = 0 (8-8) within a sphere of radius rs,,defined by Eq. (8-2). The general solution of Eq. (8-8) has the form Yn,l,m = R n , l ( r ) Yl,m <e,4>) . (8-9) and is well known from the hydrogen problem. The energy is given by E n = - m* e4/2«fi2 K 2 n 2 = - e2/2Xa*n2. (8-11) where, in the cellular method, n is to be determined by appropriate boundary * o conditions. A value of a = 13A for boron doped silicon gives one the observed ionization energy E^ — 46 mev for the bound hole. In estimating the concentration where the overlap forces become appreciable, one is concerned with the 2p states, since these broaden long before the Is state (Figure 16). It is now realized that acceptor states are much more complicated than simple hydrogenic wave functions. However, if one wants to approximate the particular line under study by a simple 35 hydrogenic ls-*2p transition, the most reasonable approach might be to require that a*, in the 2p wave function V2IO ' E q* (8-1-0), satisfy Eq. (8-11) with n = 2. Thus, using the experimental value for the binding energy of E 2 = 6.3 mev for energy level 4 (Figure 8), one obtains from.Eq. (8-11) a* = 24 %' . (8-12) With the modifications given by Eqs. (8-7) and (8-12), one obtains from Eqs. (8-1) and (8-2), for the onset of concentration broadening at low temperatures, a value of NA = 3 x 10 1 5 cm"3 , (8-13) which compares rather well with Eq. (8-5), considering the crudeness of the assumptions involved. Not only is the hydrogenic model a rather crude approximation, but the usefulness of the band scheme itself becomes doubtful, when one is dealing with a random distribution of impurities. 9. Broadening due to Internal Strains In t h i s and the following section an attempt i s made to account for the half-width, extrapolated to zero impurity concentration (Figure 5). It i s believed that an appreciable part of t h i s width r e s u l t s from i n t e r n a l s t r a i n s due to d i s l o c a t i o n s . This contribution to the half-width i s expected to be e s s e n t i a l l y temperature independent over the temperature range investigated. The remaining contribution to the half-width w i l l be a t t r i b u t e d to phonon broadening in the following section. The samples used in t h i s experiment have a quoted (Merck and Co. 4 2 1962) d i s l o c a t i o n density of about n = 5 x 10 d i s l o c a t i o n lines per cm . These d i s l o c a t i o n l i n e s are expected to give r i s e to i n t e r n a l s t r a i n s whose magnitude i s given approximately by (Kohn 1957) s .= ( n 1 / 2 ) x 10" 8 cm ^ 2.2 x 10" 6 . (9-1) 36 As discussed i n section I I I - l , i n the absence of strains a l l acceptor le v e l s are according to theory either twofold or f o u r f o l d degenerate. Under a shear s t r a i n s^ the f o u r f o l d degenerate levels w i l l s p l i t into two twofold degenerate l e v e l s , L I V I pron n__t"Vi <r_twr> f rt 1 rl nnrr. rmiin-iw^pl I . < - . - J - Q f i r s t arde»=±ja=s. The magnitude of the s p l i t t i n g i s ft 1/2 A E = s £ ^ (10 _ o cm) n € ^ 0.03 mev . (9-2) In t h i s and the following section £ i s an e f f e c t i v e deformation p o t e n t i a l constant, which depends on the geometry of the s t r a i n and i s of the order of 15 ev (Lax and Burstein 1955). Thus f or kT > s £ ( i . e . T > 1/2 °K), tr a n s i t i o n s from the ground state to a twofold degenerate l e v e l w i l l be broadened by about 0.03 mev due to i n t e r n a l s t r a i n s ; while t r a n s i t i o n s from the ground state to a fo u r f o l d degenerate l e v e l w i l l be broadened by about 0.06 mev. For T<Tl/2 °K only the twofold degenerate lower energy l e v e l w i l l be occupied. Hence no broadening of absorption l i n e s due to the ground state s p l i t t i n g would be expected, and only a t r a n s i t i o n to a f o u r f o l d excited state would be broadened by about 0.03 mev . 10. Phonon Broadening Let us consider absorption l i n e 4, corresponding to the t r a n s i t i o n from the ground state to the energy l e v e l Tp ( l e v e l 4 i n Figure 8). The electron-phonon i n t e r a c t i o n w i l l cause both t h i s l e v e l , and the ground state to have a f i n i t e l i f e - t i m e , r e s u l t i n g i n a broadening of these l e v e l s , and thus also i n a broadening of the corresponding absorption l i n e . The magnitude of t h i s broadening w i l l depend on the exact p r o b a b i l i t i e s for the d i f f e r e n t modes of decay and e x c i t a t i o n a v a i l a b l e to these states. The l i f e - t i m e of the ground state w i l l be much longer than that of the excited state, and hence we s h a l l neglect i t s contribution to the half-width of the 37 absorption line. Let us denote by X the state that has the most influence on the life-time of the state jj . According to Nishikawa (1962), the contribution of this state X to the half-width of the zero-phonon line is AU) K = (62/v2 a* 3 ?2TT) y ^ D e J t P ] , x P X ^ Jq=y/a* ^ v T < T (10-1) where •y p x ,= I T p - Tx| a* / *f v , (10-2) v> = ( e T c / T - 1 y l , (10-3) and T c = -K v / a* k (10-4) is the characteristic temperature, above which the half-width starts increasing. e p x ( t ) =|dr FpCft* F x(r) e 1 t ' ? , (10-5) F(r) being the eigenfunctions of the unperturbed electronic Hamiltonian, which will be approximated- by simple hydrogenic functions. The other parameters have the following meanings "q* = wave number vector of the phonon, v = sound velocity in silicon= 8.3 x 10^  cm sec."*", € = deformation potential constant fa 15 ev , ^ = density of silicon = 2.33 g cm , -16 -1 k = Boltzmann's constant = 1.38 x 10" erg deg~ Unfortunately, the nature of the states ^ , and X> the value of T^, and the proper choice for the effective Bohr radius a*, are rather uncertain. Let us start with the experimental value T c 47 °K (Figure 5) and the assumption that £ has 2p character (Kohn 1957). If one takes a* = 13.5 A* (Nishikawa 1962), Eq. (10-4) gives y = 1 , and thus by Eq. (10-2) one obtains | T^ - T^| = 4.0 mev. This would suggest identifying the state X 38 with the state 2 in Figure 8. Assuming this state is also 2p-like, one obtains by using simple hydrogenic wave functions With this value Eq. (10-1) gives (0.106 mev for T = T AU)A = 1 (10-6) (0.170 mev for T = 2 T c . If on the other hand one takes a* = 24 A* (the value one obtains from Eq. (8-11) for the energy T D , with n = 2), Eq. (10-4) gives P y = 1.8 . By Eq. (10-2) this implies |T^ - T^ | =4.1 mev, and thus suggests again identifying the state X with the state 2 in Figure 8. However, for — L.8 one obtains y Px [ l e ^ l 2 ] ^ v / a * - o- 7 0 * 1 0 - 3 > which after substitution into Eq. (10-1) leads to ( 0.0029 mev for T = T c Au)x= 1 (10.7) (.0.0046 mev for T = 2 T c . Another possibility is that the state X , responsible for the shortened life-time of B, is not the 2p-like state 2 in Figure 8, but rather a 2s-like state, which one may expect in this general region, and which one would presumably not see in optical absorption from the Is-like ground state. Making this assumption one obtains for a* = 13.5 8, corresponding to y ^ = 1 , that At*^ = 0. For a* = 24 8, corresponding to y ^ = 1.8 one obtains V 0 ^ 1 ) | ^ q = y p x / a , - 2,75 x 10- . Substituting this value into Eq. (10-1) one obtains '0.0074 mev for T = T atJ1= < ° (10-8) ' 0.0117 mev for T = 2 T c . 39 If one considers the crudeness of the wave functions used, and the nature of the approximations involved,, the agreement between the above estimates, especially, Eq. (10-6), and the width extrapolated to zero concentration in Figure 5, seems to be amazingly good. One may observe, that the increase in width of the extrapolated line in Figure 5 is some-what steeper than that predicted by Eq. (10-6) between T = T c and T = 2 Tc. However, this would be expected, if one remembers that while the multi-phonon processes are of l i t t l e importance to the half-width below T = Tc, their relative importance when compared with the zero-phonon process increases with temperature. They may be expected to make an appreciable contribution to the experimentally observed half-width above 40 CHAPTER IV  Conclusions The data presented in this thesis concerning a study of absorption line width in boron-doped silicon, differs considerably from that obtained by previous authors. The reason for this disagreement is a previous lack of sufficient resolution and a failure to make proper allowance for line distortion by the finite spectrometer s l i t width. The low tempjerature half-width is considerably smaller, and its temperature dependence above 50°K much steeper than found previously. The onset of concentration broadening is seen to occur at a considerably smaller impurity concentration than seemed to follow from previous experiments. The positions of the absorption lines are essentially in agreement with those obtained by Hrostowski and Kaiser (1958). It is believed that four effects make significant contributions to the true half-widths. These are a) statistical Stark broadening, b) phonon broadening, c) broadening due to the overlap of impurity wave functions, and d) broadening by internal strains. To the approximation that the absorption lines (corrected for spectrometer distortion) have lorentzian profiles, their half-widths are just the sums of the widths for the four broadening mechanisms when each is considered by itself. This follows, since the absorption lines may be thought of as resulting from a combination of the separate broadening effects by three consecutive convolution integrals. In a semiconductor containing a random distribution of neutral and ionized impurities, different absorbing impurities will be in different electric fields due to the surrounding ionized impurities. . These fields (F) should be expected to give rise to appreciable second-order Stark shifts 41 ((f2E = tF ) of the excited states, resulting in a broadening of the total absorption lines. The contribution to the half-width from this effect was obtained from a knowledge of the field strength probability function W(F), which had been extensively studied previously in connection with gravitational problems, gases, and plasmas. Ecker (1957) computed W(F), including the effect of screening by mobile charge carriers. Using these results, and the calculated dependence of the ionized impurity concentration on the temperature, this statistical Stark broadening was found to account satisfactorily for the rapid increase of the half-width above 50°K. Agreement with the data was achieved by using a strength parameter of -9 -2 2 t.= 1.1 x 10" ev volts cm' , which has the magnitude one would expect from an approximate calculation. At very low impurity concentrations an essential contribution to the half-width is expected to result from the finite life-time of the excited state due to the electron-phonon interactions. Theoretical calculations show that the half-width for this process is given by h.= h c/ (1 - e - V T ) . The calculated value of h Q depends on the choice of the characteristic temperature T c. The best agreement between this theory and the data was obtained for T c = 47°K and h = 1.0 x 10"^ ev. For these values the theory suggests that the life-time of the state responsible for the absorption line under study is mostly influenced by a state about 4.0 x 10-3 e v below i t . The multi-phonon processes are of l i t t l e importance to the half-width below T = Tc, but their relative importance when compared to the zero-phonon processes increases with temperature. They may be expected to make an appreciable contribution to the experimentally 42 observed half-width above T = Tc. In addition, a nearly temperature independent contribution to the half-width of about 3 x 10"^  ev can be expected from internal strains due to dislocations (corresponding to a quoted dislocation density of 4 9 about 5 x 10 dislocation lines per cnr). A cellular calculation, when modified to f i t the assumption of a random distribution of impurities, gives an order of magnitude estimate for the onset of concentration broadening at low temperatures. Broadening of a 2 p state is predicted to start at about 3 x 10 i J impurities per cm , if one replaces the mean spacing (r s) between impurities by an effective mean o -r/2a* spacing of 0.7 r g , and takes a* equal to 24A in the expression e ' occurring in the "2p" wave function under consideration. (This value of -3 a* gives the experimental binding energy of 6.3 x 10 ev for this state). Considering the crudeness of some of the approximations involved, in particular the use of hydrogenic wave functions, the agreement between theory and experiments seems to be rather good. An important question regarding the theoretical approach chosen in this problem, concerns the validity of the assumption of a random distribution of impurities. This assumption is believed to be more realistic than that of a regular array of impurities, and it is the only possible one in the absence of any concrete evidence for a clustering of boron impurities in silicon. Table 1 : Standard Voigt P r o f i l e s Parameters Ordinates i n Terms of Central Ordinate p x/h ^ / ^ P2/h p- 2 2/h 2 p 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 Widths i n Terms of Half-width (b^h) 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.00 0.04 0.09 0.14 !0.19 0.24 0.30 0.36 0.43 0.51 0.59 0.69 0.79 0.92 1.07 1.26 1.50 1.83 2.38 3.54 oo 0.60 0.59 0.57 0.55 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.35 0.33 0.30 0.27 0.23 0.19 0.13 0.00 0.36 0.34 0.32 0.31 0.29 0.27 0,25 0.23 0.21 0.20 0.18 0.16 0.14 0.12 0.11 0.09 0.07 0.05 0.04 0.02 0.00 1.06 1.08 1.11 1.13 1.16 1.18 1.20 1.23 1.25 1.28 1.30 1.33 1.35 1.38 1.40 1.43 1.45 1.48 1.51 1.54 1.57 0.57 0.56 0.56 0.56 0.56 0.56 0.55 0.55 0.55 0.54 0.54 0.53 0.53 0.53 0.52 0.52 0.52 0.51 0.51 0.51 0.50 0.72 0.72 0.71 0.71 0.71 • 0.71 0.71 0.70 0.70 0.70 0.70 0.69 0.69 0.68 0.68 0.68 0.67 0.67 0.66 0.66 0.66 0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.83 0.83 0.83 0.82 0.82 0.82 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.15 1.15 1.15 1.16 1.16 1.17 1.17 1.17 1.18 1.18 1.18 1.19 1.19 1.19 1.20 1.20 1.21 1.21 1.22 1.22 1.22 1.32 1.33 1.33 1.33 1.34 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.44 1.45 1.47 1.48 1.50 1.52 1.53 1.52 1.53 1.54 1.56 1.57 1.59 1.60 1.62 1.64 1.66 1.68 1.71 1.74 1.77 1.81 1.85 1.88 1.92 1.96 1.98 2.00 1.82 1.84 1.87 1.90 1.94 1.98 2.02 2.06 2.10 2.15 2.19 2.24 2.29 2.34 2.40 2.46 2.54 2.64 2.74 2.87 3.00 2.08 2.12 2.19 2.25 2.34 2.42 2.54 2.64 2.75 2.87 2.98 3.12 3.26 3.39 3.54 3.70 3.85 4.00 4.13 4.25 4.36 2.38 2.49 2.63 2.79 3.00 3.24 3.52 3.80 4.14 4.44 4.73 5.03 5.32 5.57 5.83 6.07 6.30 6.55 6.76 6.92 7.00 2.58 2.82 3.13 3.56 4.08 4.58 5.05 5.50 5.96 6.40 6.78 7.15 7.52 7.86 8.21 8.55 8.86 9.18 9.50 9.77 9.95 44 2 Table 2. Integrated absorption, A (cm" ) Resistivity Line No. Temperature (°K) (ohm cm) frig. 1) 4.2 60 77 90 130 1 1.4 1.8 - -130 2 6.7 4.4 - -130 4 6.6 5.3 1.8 -11 1 18 20 23 -11 2 54 109 50 53 11 3 6.6 - - -11 4 44 73 60 45 1.3 1 160 - 220 250 470 1.3 2 630 920 600 650 1.3 4 620 800 810 950 Table 3. Integrated absorption cross-section, cm). Resistivity Line No. Temperature (°K) (ohm cm) (Fig.l) 4.2 60 77 90 130 1 1.4 4.1 - -130 2 6.7 10 - -130 4 6.6 12 17 -11 1 1.5 2.2 4.3 -11 2 4.5 12 9.4 ,19 11 3 0.55 - - -11 4 3.7 7.9 11 16 1.3 1 1.3 2.0 2.9 7.2 1.3 2 5.2 8.4 6.9 10 1.3 4 5.2 7.3 9.3 15 45 Appendix I: Distribution of holes between the ground state, excited states, and the valence band as a function of temperature. The present appendix outlines the formalism used to calculate the curves shown in Figure 9. Let us begin by assuming that the number of donor impurities and free electrons is negligible. There are NA acceptor impurities with energy levels E ,= - £ a above the valence band, and p free holes. The probability P(E) of an acceptor level having four electrons with paired spins when an extra electron is attached (ionized condition) is given by P(E) = {l + 2 exp.[(E - EF)/kT]} - 1 , (l-l) where Ep is the Fermi energy. The number of ionized acceptors is thus given by p = NA ( l + 2 exp.[( ea + E p)/kT]| _ 1 . (1-2) Writing p = Nv exp. (Ep/kT) , (1-3) one obtains a quadratic equation for p, whose appropriate solution is p = -Nv/2 + (1/2)(NV 2 + 4 < N A ) 1 / 2 . (1-4) Here Ny = (Nv/2) exp.(- €a/kT) , and the effective density of states Nv is given by Nv = 2 (2TTkT/h2)3/2 (m h l 3/ 2 + mh23/2) . (1-5) Eq. (1-5) has been derived under the assumption that the actual energy surfaces in silicon may be approximated by two spherical constant-energy surfaces with effective masses mni and mn2 . It was assumed that (E F - E)/kT»l. This approximation is good as long as p<0.4 Ny (Smith 1958), which applies for a l l cases of interest in this thesis. The valence band may be regarded to this approximation as a single level with 46 degeneracy NV,, placed at the top of the band. In deriving Eq. (1-2) from the expression for the probability of occupation P(E), it was assumed that one is only concerned with a single level. However, the impurity levels have excited states, and these should be included in Eq. (1-2) with the appropriate probabilities. One then has for the number of un-ionized acceptors N A P = N A 1 + S i " 1 exp, [ ( E F - E.)/kT] , (1-6) where E^ is the energy of the i th excited state, and g^ is a number to take account of degeneracy and spin (E Q = - 6 a, g 0 = 2). Since below 90°K one has (E^ - E 0)/kT»l , the first term in the sum predominates, and hence Eq. (1-2) is a good approximation. Taking for the effective masses the values (Bube 1960) = 0.5 me ; mh2 = 0.16 me , (1-7) Eq. (I-t5) leads to N V .= 2.02 x 10 1 5 T 37 2 cm-3 . (1-8) Substitution into Eq. (1-4), taking the ionization energy of = 46 mev for boron-doped silicon, leads to the curves of Figure 9 for the number of ionized impurities as a function of the absolute temperature T. The number of bound carriers in excited states N E X is then computed from Nex = (% " P) ^ Gi exp.(- fi/kT) , (1-9) where the assumption was made that N 6 X<^(NA - p). The energy differences €^ between the i th excited state and the ground state are obtained from Figure 8. The degeneracies of the excited states relative to the ground state, G^ , were taken as 1, 1, 1/2, 1/2, for the lowest four excited states 47 (Kohn 1957), which make the main c o n t r i b u t i o n to N e x . A value of equal to one was taken f o r the remaining f i v e experimentally observed energy l e v e l s ( F i g u r e 8 ) . The number of i m p u r i t i e s i n the ground s t a t e , N , was then obtained from N = N A - p - N e x . ( I -48 Appendix I I ; Second-order Stark s h i f t f o r hydrogenic Is and 2p states. In the present appendix the matrix elements occurring i n Chapter I I I , Eq. (6-2), w i l l be evaluated: < F < I > , a v ) 2 F<*>> , where £v = - F e z , and for F ^ 1 ) hydrogenic wave functions ^ m > w i l l be taken. In p a r t i c u l a r one has f o r the Is and the 2p states: Y i 0 0 ( r , © , c £ ) = T T 1 / 2 a * " 3 / 2 exp.(-r/a*) > 2 1 0 ( r , ©,tf» = ( 1 / 4 ) ( 2 T T ) - 1 / 2 a * _ 5 / 2 r cos9 exp.(-r/2a*) "^21±l ( r' 9 » 0> " (1/8) T r " 1 / 2 a * " 5 / 2 r sin9 exp.[(-r/2a*) ± i £J . Hence one obtains 2TC i t 0 0 di00> (^ V> 2 >100> - / d(J) | d e / dr r 2 s i n 6 T 1 0 0 r c o s ^ Q Q 0 0 0 2 2 -3 9 ^ 4 = 2 F " e* a* J J w^  dw J r exp.(-2r/a*) dr -1 a* 2 . S i m i l a r l y , .2 2 -5 L '• °° <V$10» <* V> >210> " ( 1 / 1 6 ) F 6 a j w dw J r exp.(-r/a*) dr = 18- a * 2 . 1 oo <>21+1' (<f V ) 2 >2H-1>= ( 1 / 3 2 ) F 2 e 2 a * " 5 / w 2 ( l - w 2 ) dw J r 6exp.(-r/a*) dr - - - 1 0 = 6 a * 2 . Here w = cosG . Thus one obtained Eq. (6-3) with c given by 1, 18, and 6 for the Is state, 2p 0 state, and 2p + states r e s p e c t i v e l y . 49 B I B L I O G R A P H Y Baltensperger, W. 1953. Phil. Mag. 44, 1355. Bichard, J.W. and Giles, J.C. 1962. Can. J. Phys. 40, 1480. Breene, Jr., R.G. 1957. Revs. Modern Phys. 29, 94. Bube, R.H. 1960. Photoconductivity of Solids (John Wiley & Sons, Inc., New York, London), p.209. Burstein, E., Bell, E.E., Davisson, J.W., and Lax, M. 1953. J. Phys. Chem. 57, 849. Burstein, E., Picus, G.S., Henvis, B., and Wallis, R. 1956. J. Phys. Chem. Solids I, 65. Chandrasekhar, S. 1943. Revs. Modern Phys. _15, 1. Colbow, K., Bichard, J.W., and Giles, J.C. 1962. Can. J. Phys. 40, 1436. Colbow, K., 1962. Bull. Am. Phys. Soc, Series II, ]_, 485. Conwell, E.M. 1956. Phys. Rev. 103, 51. Debye, P. and Huckel, E. 1923. Physik. Z. 24, 185. Dexter, D.L., Zeiger, H.J., and Lax, B. 1956. Phys. Rev. 104, 637. Dresselhaus, G., Kip, A.F., and Kittel, C. 1955. Phys. Rev. 98, 368. Dresselhaus, G. 1955. Phys. Rev. 100, 580. Ecker, G. 1957. Z. Physik JL48, 593; _149, 254. Elliott, R.J. 1954. Phys. Rev. 96, 266. Herzberg, G. 1944. Atomic Spectra and Atomic Structure (Dover Publications, New York), 2nd ed., p.51. Holtsmark, J. 1919. Am. Physik, 58, 577; Physik. Z. 20, 162. Holtsmark, J. 1924. Physik. Z. 25, 73. Hrostowski, H.J., and Kaiser, R.H. 1958. J. Phys. Chem. Solids, 4, 148. Irvin, I. C. 1962. Bell System Tech. Journ. 41, 387. Kane, E.O. 1960. Phys. Rev. _119, 40. Kittel, C. and Mitchell, A.H. 1954. Phys. Rev. 96, 1488. 50 Kohn, W. and Luttinger, J.M. 1955. Phys. Rev. 97, 869. Kohn, W. 1957. S o l i d State Physics, edited by F. S e i t z and D. Turnbull, V o l . 5 (Acadmeic Press, Inc., New York),p.257. Lax, M. and Burstein, E. 1955. Phys.Rev. 100, 592. London, F. and E i s e n s c h i t z , R. 1930. Z. Physik 60, 491. Margenau, H. 1933. Phys. Rev. 43, 129. Margenau, H. and Murphy, G.M. 1943. The Mathematics of Physics and Chemistry (Van Nostrand, New York), p.559. Margenau, H. and Lewis, M. 1959. Revs. Modern Phys. _3_1, 569. Merck and Co., 1962. Private communication. Morin, F.J. and Maita, J.P. 1954. Phys. Rev. 96, 28. Moss, T.S. 1959. Op t i c a l properties of semiconductors (Butterworths, London), p.14. Newman, R. 1956. Phys. Rev. 103, 103. Nishikawa, K. 1962. Phys. L e t t e r s , I, 140. Nishikawa, K. and Barrie, R. 1962. B u l l . Am. Phys. S o c , Series I I , 7_,U85. Nishikawa, K. 1962. Ph.D. Thesis, University of B r i t i s h Columbia. P l y l e r , E.K. and Acquista, N. 1956. J.Res. N.B.S., ^6, 149. Randall, H.M., Dennison, D.M., Ginsburg, N. and Weber, L.R. 1937. Phys. Rev. 52, 160. Sampson, D. and Margenau, H. 1956. Phys. Rev. 103, 879. Schechter, D. 1962. J. Phys. Chem. Soli d s , 23, 237. Smith, R.A. 1958. Semiconductors (Cambridge Uni v e r s i t y Press), p.80. Unsold, A. 1955. Physik der Sternatmospharen (Springer-Verlag, B e r l i n , Gottingen,. Heidelberg), Chap. IX. Van de Hulst,, H.C. 1946. B u l l . Astron. Inst. Neth. _10, 75. Van de Hulst, H.C. and Reesinck, I.I.M. 1947. Astrophys. J . 106, 121. Voigt, W. 1912. Munch. Ber. 603. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085855/manifest

Comment

Related Items