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UBC Theses and Dissertations

Effects of strain on the exciton spectrum of gallium antimonide Rickards, Bradley Alfred 1972

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THE EFFECTS OF STRAIN ON THE EXCITON SPECTRUM OF GALLIUM ANTIMONIDE by BRADLEY ALFRED RICKARDS B.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 5 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1 9 7 2 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s for an a d v a n c e d d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e 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 r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s fo r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date ABSTRACT The o p t i c a l a b s o r p t i o n s p e c t r a of undoped GaSb, having a c a r r i e r c o n c e n t r a t i o n of 8 . 6 x l O 1 6 cm-3, has been s t u d i e d u s i n g an i s o t r o p i c and an a n i s o t r o p i c b i a x i a l s t r a i n a p p l i e d i n the fc)Ol] and f i l l ] c r y s t a l l o g r a p h i c p l a n e s . T h i n (2 t o 5 ^m) s i n g l e c r y s t a l s of GaSb were mounted on g l a s s sub-s t r a t e s which, when co o l e d to 2°K, would produce an i s o t r o p i c b i a x i a l s t r a i n i n the plane of the sample. U s i n g t h i s t e c h -nique s t r a i n s r a n g i n g from +10 ( t e n s i l e ) t o - 4 . 2 x 10~ (compressive) were o b t a i n e d . The a p p l i c a t i o n of a u n i a x i a l or b i a x i a l s t r a i n causes a s p l i t t i n g of the degenerate valence band edge at k=0 i n the z i n c blende type semiconductors, with a c o r r e s p o n d i n g s p l i t t i n g of the f r e e e x c i t o n a b s o r p t i o n peak ( o<- peak). From the s p l i t t i n g of the e x c i t o n a b s o r p t i o n peak, the v a l e n c e band shear deformation p o t e n t i a l s have been determined t o be T)u = + 3 . 5 ± 0 . 3 eV (b = - 2 . 4 ± 0 . 2 eV) and D ^ = + 4 . 4 + 0 . 3 eV ( d = - 5 . 1 ± 0 . 3 eV). The s t r a i n s p l i t e x c i t o n a b s o r p t i o n peaks e n e r g i e s are found to s h i f t l i n e a r l y over the range of s t r a i n c o v ered. From the mean s h i f t of the e x c i t o n a b s o r p t i o n peaks w i t h s t r a i n , the h y d r o s t a t i c deformation p o t e n t i a l ( D d - ) has been found to be - 1 0 . 4 ± 0 . 9 eV per u n i t s t r a i n a p p l i e d i n the {00l} plane and -8 . 9 ± 0 . 5 eV per u n i t s t r a i n a p p l i e d i n the f i l l ] p l ane. The s i g n s of the deformation p o t e n t i a l s have been determined by o b s e r v i n g t h a t , f o r a t e n s i l e s t r a i n , the " c e n t r e of g r a v i t y " of the s t r a i n s p l i t v alence band moves towards the c o n d u c t i o n band with the m - r = ± 3 / 2 valence band moving "up" and the m =+1/2 valence band moving "down" wit h r e s p e c t t o the "centre of g r a v i t y " of the s t r a i n s p l i t valence bands. Under a compressive s t r a i n the "centre of g r a v i t y " of the s t r a i n s p l i t v a l e nce bands moves away from the con-d u c t i o n band wi t h the mj=+3/2 valence band moving "down" and the mj=+l/2 valence band moving "up" with r e s p e c t t o the "c e n t r e of g r a v i t y " of the s t r a i n s p l i t valence bands. The m =-+1/2 and m = +3/2 valence bands have been i d e n t i f i e d from the a n i s o t r o p i c b i a x i a l s t r a i n measurements u s i n g p o l a r i z e d l i g h t . From the b i a x i a l s t r a i n measurements the c* a b s o r p t i o n peak has been i n t e r p o l a t e d t o have an energy value of 0.8107 +0.0002 eV i n the u n s t r a i n e d l a t t i c e . The /3 and Y a b s o r p t i o n peaks have been observed i n t h i c k u n s t r a i n e d samples a t photon e n e r g i e s of 0.8055 +0.0005 eV and 0.7962 ±0.0002 eV, r e s p e c t i v e l y . These two a b s o r p t i o n peaks are a l s o observed i n the t h i n b i a x i a l l y s t r a i n e d samples but are not observed t o s p l i t as a r e s u l t of the s t r a i n . These two peaks are a l s o observed t o have almost an i d e n t i c a l energy s h i f t w i t h s t r a i n which appears t o be independent of the plane of a p p l i c a t i o n of the s t r a i n . There i s a l s o no p o l a r i z a t i o n dependence of the o p t i c a l a b s o r p t i o n . These r e s u l t s r e g a r d i n g the /3 and t a b s o r p t i o n peaks c o n t r a d i c t s the p r e v i o u s i d e n t i f i c a t i o n of these l i n e s as b e i n g due t o a bound e x c i t o n complex. No a l t e r n a t e e x p l a n a t i o n as t o the ^ a b s o r p t i o n peaks can be g i v e n a t pr e s e n t . XV TABLE OF CONTENTS Pajge A b s t r a c t i i Table of Contents i v L i s t of T a b l e s . . „ v i i i L i s t of F i g u r e s . . . . . x Acknowledgements «... x i i i Chapter 1 - INTRODUCTION 1 Chapter 2 - THEORETICAL BACKGROUND 2 . 1 I n t r o d u c t i o n 7 2 . 1 . 1 U n s t r a i n e d Z i n c Blende L a t t i c e ( P h y s i c a l D e s c r i p t i o n ) 9 2 . 1 . 2 U n s t r a i n e d Energy Band S t r u c t u r e . . . 11 2 . 1 . 3 E f f e c t of S t r a i n on Band S t r u c t u r e . 13 S e l e c t i o n R ules f o r E l e c t r o n i c T r a n s i t i o n s 17 2.1.4 Deformation P o t e n t i a l s 20 B i a x i a l S t r a i n i n the {00 l ] Plane. 23 B i a x i a l S t r a i n i n the [ i l l ] Plane. 25 B i a x i a l S t r a i n i n the [lio] Plane. 27 S i g n of the Deformation P o t e n t i a l s . 30 2 . 2 Review of E x c i t o n Theory 32 2 . 2 . 1 Review of E x c i t o n Theory 32 2 . 2 . 2 E f f e c t o f S t r a i n on E x c i t o n A b s o r p t i o n 35 V P a g e 2 . 2 . 3 o(-> ^> and JT A b s o r p t i o n Peaks 4 2 The E f f e c t of S t r a i n on Acceptor L e v e l s 4 3 2 . 3 Review of A b s o r p t i o n Due t o D i r e c t T r a n s i t i o n s 4 5 2 . 4 Temperature Dependence of E l a s t i c C o n s t a n t s . c 4 8 Chapter 3 - EXPERIMENTAL DETAILS 3 . 1 I n t r o d u c t i o n 5 3 3 . 2 O p t i c a l System 5 5 3 . 2 . 1 Source and Chopper Assembly........ 5 5 3 . 2 . 2 F o r e - O p t i c s 5 7 3 . 2 . 3 E b e r t Monochromator 5 7 3 . 2 . 4 Image D i s t o r t i n g System 5 8 3 . 2 . 5 P o l a r i z a t i o n 5 9 3 . 2 . 6 E l e c t r o n i c D e t e c t i o n System 6 0 3 . 3 C a l i b r a t i o n . . 6 2 3 » 4 C r y o g e n i c s 6 3 3 . 4 . 1 Helium Immersion Dewar 6 3 3 . 4 . 2 U n i a x i a l S t r e s s Dewar 6 5 3 . 5 Samples and Sample P r e p a r a t i o n , 6 6 3 . 5 . 1 Samples „ o . . . . . o . . 6 6 3 . 5 . 2 C r y s t a l O r i e n t a t i o n „ 6 7 3 . 5 . 3 G l a s s S u b s t r a t e s and T h e i r Thermal Expansion.. . 6 7 v i Page 3 . 5 . 4 Sample Mounting 7 3 3 . 5 . 5 G r i n d i n g and P o l i s h i n g of Samples.. 7 6 A: Sample Thickn e s s M o n i t o r i n g . . . . 7 8 B: G r i n d i n g . .. 7 8 C: Rough P o l i s h i n g 7 9 D: F i n a l P o l i s h 8 0 E: S u r f a c e Damage Due t o Sample P r e p a r a t i o n 8 1 3 . 5 • 6 Mounting of Prepared Sample i n U n i a x i a l S t r e s s Dewar 8 5 3 . 6 C a l c u l a t i o n of A b s o r p t i o n C o e f f i c i e n t from Measured T r a n s m i s s i o n I n t e n s i t y 8 6 Chapter 4 - PRESENTATION AND DISCUSSION OF EXPERIMENTAL  RESULTS 4 . 1 I n t r o d u c t i o n 9 1 4 . 2 E x c i t o n A b s o r p t i o n S p e c t r a 9 2 4 . 2 . 1 B i a x i a l S t r a i n Measurements 9 3 4 . 2 . 2 U n i a x i a l S t r a i n Measurements 1 0 3 4 . 3 C a l c u l a t i o n of Deformation P o t e n t i a l s I l l 4 . 4 The f3 and ^ A b s o r p t i o n Peaks. 1 1 8 4 . 4 . 1 Dependence of the f3 A b s o r p t i o n Peak on S t r a i n 1 1 8 4 . 4 . 2 Dependence of the f A b s o r p t i o n Peak on S t r a i n 1 2 3 Chapter 5 - CONCLUSIONS 1 2 7 v i i Page Appendix A - WAVELENGTH CALIBRATION OF THE EBERT MONOCHROMATOR 1 3 1 Appendix B - WOOD'S METAL SEAL ON HELIUM DEWAR 1 4 3 Appendix C - FORMULAE FOR THE ABSORPTION COEFFICIENT FOR AN ABSORBING MEDIUM ON A SUBSTRATE. 1 4 5 Appendix D - REVIEW OF STRESS AND STRAIN COMPONENTS AND THEIR TRANSFORMATIONS 1 5 0 BIBLIOGRAPHY 1 5 6 v i i i LIST OF TABLES Table Page 2 . 1 The s t r a i n , G;j , v a l e n c e band s p l i t t i n g , 2A, mean s h i f t of the v a l e n c e bands, 6E H, and the s t r e s s , , f o r GaSb 29 2 . 2 Parameters used i n c a l c u l a t i n g t h e e x c i t o n b i n d i n g e n e r g i e s f o r the d i r e c t e l e c t r o n i c t r a n s i t i o n s i n GaSb 39 2 . 3 The e x c i t o n peak energy s p l i t t i n g and the c a l c u l a t e d b i n d i n g e n e r g i e s f o r the v a r i o u s s t r a i n s a p p l i e d i n the <001> and < l l l ) d i r e c t i o n s 41 2 . 4 The e l a s t i c c o n s t a n t s , and the reduced e l a s t i c c o n s t a n t s f o r GaAs, InSb, and GaSb f o r v a r i o u s temperatures. 50 3 . 1 The thermal e x p a n s i v i t y of the g l a s s s u b s t r a t e s used and the s t r a i n on the GaSb sample 72 3 . 2 The g r i n d i n g compounds and l a p used i n the sample p r e p a r a t i o n i n c l u d i n g an estimate of the s u r f a c e damage produced by each compound.. 77 i x T able Page 4 . 1 The o<y2and <?<>/*. e x c i t o n peak energy p o s i t i o n f o r the v a r i o u s o r i e n t a t i o n s and g l a s s s u b s t r a t e s used 9 7 4 . 2 The defor m a t i o n p o t e n t i a l parameters obtained i n t h i s work and i n p r e v i o u s work 1 1 4 4 . 3 The energy of the j3 and t a b s o r p t i o n peaks f o r t h e v a r i o u s b i a x i a l s t r a i n s a p p l i e d i n the { 0 0 l ] and ( l l l \ c r y s t a l l o g r a p h i c p l a n e s 1 2 0 X LIST OF FIGURES F i g u r e Page 2.1 (a) Z i n c blende l a t t i c e s t r u c t u r e . (b) Model of the z i n c blende s t r u c t u r e i n d i c a t i n g the p o l a r Till] d i r e c t i o n s 10 2.2 C a l c u l a t e d energy bands of GaSb a l o n g important symmetry axes 12 2.3 The valence and lowest c o n d u c t i o n bands of diamond and z i n c blende type m a t e r i a l s near k=0 f o r the u n s t r a i n e d and the s t r a i n e d l a t t i c e 15 2.4 The temperature dependence of the reduced e l a s t i c c o n s t a n t (a) c^ , (b) c,£, and (c) cf,/, f o r GaAs, InSb and the estimated dependence f o r GaSb 51 3.1 Schematic diagram of the E b e r t monochromator o p t i c s 56 3.2 Block diagram of the d e t e c t o r e l e c t r o n i c s . . . . 61 3.3 (a) Helium immersion dewar. (b) The sample h o l d e r f o r the u n i a x i a l s t r e s s dewar.... 64 3.4 The thermal e x p a n s i v i t y between 294°K and 0°K f o r GaSb and the v a r i o u s g l a s s s u b s t r a t e s used 69 3.5 (a) D e t a i l e d drawing of the a d j u s t a b l e sample h o l d e r . (b) Device used f o r m o n i t o r i n g the sample t h i c k n e s s d u r i n g p r e p a r a t i o n . . . 74 x i F i g u r e Page 3 . 6 Sample mounting d e t a i l f o r u n i a x i a l s t r e s s dewar 85 4 . 1 A b s o r p t i o n c o e f f i c i e n t at 2°K as a f u n c t i o n of photon energy f o r the v a r i o u s g l a s s s u b s t r a t e s used 94 4 . 2 The e x c i t o n peak energy p o s i t i o n as a f u n c t i o n of the b i a x i a l s t r a i n a p p l i e d i n the {00l} plane 98 4 . 3 The e x c i t o n peak energy p o s i t i o n as a f u n c t i o n of the b i a x i a l s t r a i n a p p l i e d i n the { i l l ] plane 99 4 . 4 The e x c i t o n peak energy s e p e r a t i o n as a f u n c t i o n o f the b i a x i a l s t r a i n a p p l i e d i n the { l l l ^ plane 101 4 . 5 The mean s h i f t o f the f o r b i d d e n energy gap, SEW, f o r the s t r a i n a p p l i e d i n the [ 0 0 l ] and { i l l ] c r y s t a l l o g r a p h i c p l a n e s . . . . 102 4 . 6 The change i n the s p e c t r a of a sample, w i t h an i n i t i a l i s o t r o p i c b i a x i a l t e n s i l e s t r a i n i n the { 0 0 l | plane, due to a u n i a x i a l s t r e s s a p p l i e d a l o n g a <C00l) d i r e c t i o n 104 4 . 7 The change i n the s p e c t r a of a sample, w i t h an i n i t i a l i s o t r o p i c compressive b i a x i a l s t r a i n i n the { m l plane, due t o a u n i a x i a l s t r e s s a p p l i e d a l o n g a <^22l)> d i r e c t i o n . . . . 107 x i i F i g u r e Page 4 . 8 The ^k, ^ 3 / 2 and /3 a b s o r p t i o n peak energy-p o s i t i o n as a f u n c t i o n of the s t r e s s a p p l i e d t o the g l a s s s u b s t r a t e 1 0 9 4 . 9 The y# and If a b s o r p t i o n peaks i n u n s t r a i n e d samples of GaSb 1 1 9 4 . 1 0 The p and If a b s o r p t i o n peak e n e r g i e s as a f u n c t i o n of the a p p l i e d b i a x i a l s t r a i n 1 2 1 4 . 1 1 The 2f a b s o r p t i o n peak f o r an u n s t r a i n e d and a b i a x i a l l y s t r a i n e d GaSb sample 1 2 4 A l Schematic diagram of the Ebert monochromator wave d r i v e mechanism.. 1 3 2 BI D e t a i l e d diagram of the Wood's Metal S e a l . . . 1 4 3 C l Schematic diagram of a two l a y e r m a t e r i a l c o n s i s t i n g of an a b s o r b i n g media and g l a s s s u b s t r a t e bound on e i t h e r s i d e by a vacuum 1 4 6 ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r , Dr. J . W. B i c h a r d , f o r h i s i n t e r e s t and a d v i c e g i v e n d u r i n g the course of t h i s work and f o r h i s sug g e s t i o n s and c o n s t r u c t i v e c r i t i c i s m r e l a t i n g t o the p r e p e r a t i o n of t h i s t h e s i s . I would l i k e t o thank my w i f e , Pat, f o r her for b e a r a n c e , p a t i e n c e and encouragement g i v e n throughout the course of my s t u d i e s . I would a l s o l i k e t o thank her f o r t y p i n g t h i s manuscript. The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s was supported by the Defence Research Board of Canada, Grant No. 9512-29 and Grant No. 9510-35. The N a t i o n a l Research C o u n c i l of Canada p r o v i d e d t h r e e s t u d e n t s h i p s f o r the author. 1 CHAPTER 1 INTRODUCTION The e f f e c t on the band s t r u c t u r e of a semiconductor, due t o a deformation of the l a t t i c e , can be c o n v e n i e n t l y t r e a t e d by the use of the deformation p o t e n t i a l t h e o r y . The concept of a deformation p o t e n t i a l was f i r s t i n t r o d u c e d by Bardeen and Shockley (1950) f ° r the study of s c a t t e r i n g of e l e c t r o n s and h o l e s by l o n g wavelength a c o u s t i c phonons. An accurate knowledge of the v a l u e s of the deformation p o t e n t i a l parameters i s t h e r e f o r e d e s i r a b l e t o permit the c a l c u l a t i o n of the l a t t i c e m o b i l i t y f o r h o l e s i n u n s t r a i n e d m a t e r i a l s . The d e f o r m a t i o n p o t e n t i a l parameters can a l s o be used t o d e s c r i b e the e f f e c t s of i n t e r n a l and e x t e r n a l l y a p p l i e d s t r a i n s on the band s t r u c t u r e of a semiconductor. K l e i n e r and Roth (1959) developed the th e o r y , i n terms of the deformation p o t e n t i a l parameters, f o r the shear s t r a i n s p l i t t i n g , and hydro-s t a t i c s h i f t , of the va l e n c e bands under a b i a x i a l s t r a i n . P i c u s and B i r (1959) a l s o c o n s i d e r e d the e f f e c t o f a u n i a x i a l s t r a i n on the hole energy spectrum of Germanium u s i n g deforma-t i o n p o t e n t i a l s . In g e n e r a l , the a p p l i c a t i o n of a u n i a x i a l or b i a x i a l s t r e s s , t o a s o l i d , produces a s t r a i n which reduces the symmetry of the m a t e r i a l and r e s u l t s i n s i g n i f i c a n t changes i n the e l e c t r o n i c energy l e v e l s . In p a r t i c u l a r , an a n i s o t r o p i c s t r e s s r e s u l t s i n the s p l i t t i n g of the degenerate v a l e n c e band edge at k=0 i n the diamond and z i n c blende type semiconductors. 2 The removal of t h i s degeneracy i s observable i n the s t r e s s dependence of the o p t i c a l p r o p e r t i e s a s s o c i a t e d w i t h the band edge t r a n s i t i o n s . One of the band edge t r a n s i t i o n s , on which an a p p l i e d s t r e s s has a ve r y n o t i c a b l e e f f e c t , i s the fo r m a t i o n of the e l e c t r o n - h o l e p a i r complexes c a l l e d e x c i t o n s . When an e l e c t r o n i s e x c i t e d from the val e n c e band t o the co n d u c t i o n band, the coulomb i n t e r a c t i o n between the e l e c t r o n i n the conduction band and the ho l e i n the va l e n c e band g i v e s r i s e t o a bound s t a t e of the two p a r t i c l e s . The energy necessary t o c r e a t e an e x c i t o n i s l e s s than th e energy s e p a r a t i o n of the two bands, by an amount equal t o the b i n d i n g energy of the e l e c t r o n - h o l e p a i r . For d i r e c t t r a n s i t i o n , i n v o l v i n g the i n t e r a c t i o n of e l e c t r o n s w i t h photons o n l y , the t r a n s i t i o n s t o the bound e x c i t o n s t a t e g i v e s r i s e t o a l i n e spectrum. T h i s i s observable as a sharp peak i n the a b s o r p t i o n spectrum. When an a n i s o t r o p i c s t r e s s i s a p p l i e d t o a diamond or a z i n c blendetype semiconductor the e x c i t o n a b s o r p t i o n peak i s observed t o s p l i t i n t o two peaks. These two a b s o r p t i o n peaks are due t o t h e t r a n s i t i o n of e l e c t r o n s from the two s t r a i n s p l i t v a l e n c e bands t o the e x c i t o n l e v e l . T h i s e f f e c t was f i r s t observed by Zwerd l i n g e t a l . ( 1 9 5 9 ) f o r t h i n Germanium samples glued t o a g l a s s s u b s t r a t e . K l e i n e r and Roth ( 1 9 5 9 ) i n t e r p r e t e d the r e s u l t s of Zw e r d l i n g e t a l . and were able to determine the deformation p o t e n t i a l s of Germanium. T h i s technique was l a t e r used by G l a s s ( 1 9 6 4 ) t o determine the deformation p o t e n t i a l s of 3 Germanium u s i n g both t e n s i l e and compressive s t r a i n s . R e c e n t l y , a v a r i e t y of t e c h n i q u e s have been used t o de-termine the d e f o r m a t i o n p o t e n t i a l s of GaSb. However, t h e r e i s a c o n s i d e r a b l e v a r i a t i o n among the v a l u e s r e p o r t e d (see Table 4 * 2 ) i n d i c a t i n g a p o s s i b l e dependence on the experimental t e c h n i q u e s used. Benoit a l a Guillaumeet a l . ( 1 9 7 0 ) determined the de-f o r m a t i o n p o t e n t i a l s by s t u d y i n g the e f f e c t of a u n i a x i a l compressive s t r e s s on the photoluminescence of GaSb ( P i e z o e m i s s i o n ) . T h e i r i n v e s t i g a t i o n was done at 35°K w i t h an energy r e s o l u t i o n of 0.5meV. No e r r o r s were quoted f o r t h e i r r e s u l t s and no temperature adjustment was performed on the e l a s t i c c o n s t a n t s used i n c a l c u l a t i n g the deformation p o t e n t i a l s . G a v i n i and Cardona ( 1 9 7 0 ) measured the deformation p o t e n t i a l s at 77°K by measuring the r e f l e c t i v i t y o f n-type GaSb u s i n g a s m a l l ( 1 0 6 dyne cm"2 ) u n i a x i a l s t r e s s modulation ( P i e z o r e f l e c t a n c e ) . The e r r o r i n t h e i r v a l u e s was estimated t o be ±2Q%. There i s a l a r g e d i s c r e p a n c y between the de-f o r m a t i o n p o t e n t i a l s determined by t h i s technique and v a l u e s determined by o t h e r t e c h n i q u e s . The d i f f e r e n c e has q u a l i t a t i v e l y been e x p l a i n e d ( G a v i n i and Timosk ( 1 9 7 0 ) ) as due to the f a c t , t h a t i n modulated p i e z o r e f l e c t a n c e , one measures the s m a l l -s t r e s s deformation p o t e n t i a l s of the e x c i t o n s , i n s t e a d of those of the o n e - e l e c t r o n band edges. I t has been shown i n the present work ( s e c t i o n 2 . 2 . 2 ) t h a t , f o r s t r e s s e s l e s s than + 1 0 * dyne cm"2 ( s t r a i n - l x l O 4 ) , the e x c i t o n b i n d i n g energy i s v e r y s t r e s s dependent and cannot be n e g l e c t e d when c a l c u l a t i n g the deformation p o t e n t i a l s . A l s o , i n the p i e z o r e f l e c t a n c e method one measures the r a t i o of the shear deformation p o t e n t i a l s t o the h y d r o s t a t i c deformation p o t e n t i a l , and not the i n d i v i d u a l d e f o r m a t i o n p o t e n t i a l s as i n the present work. P o l l a k e t a l . ( 1 9 7 1 ) dete rmined the deformation p o t e n t i a l s of GaSb by s t u d y i n g the e f f e c t of a compressive u n i a x i a l s t r e s s on the f r e e e x c i t o n i n GaSb, at 1 . 7°K, u s i n g wavelength mbdulated r e f l e c t i v i t y and s t r e s s e s of up t o 5 x l 0 9 dyne cm'2. Yu et a l . ( 1 9 7 1 ) used the s t r e s s - i n d u c e d b i r e f r i n g e n c e due t o i n t e r - b a n d t r a n s i t i o n s ( i n t r i n s i c p i e z o b i r e f r i n g e n c e ) t o determine the deformation p o t e n t i a l s of GaSb at room temper-a t u r e . Yu s t a t e s t h a t one.of the two deformation p o t e n t i a l s o b t a i n e d by t h i s method always agrees q u i t e w e l l w i t h the v a l u e o b t a i n e d by other methods. There i s a good agreement between one of Yu's v a l u e s and the v a l u e o b t a i n e d i n the p r e s e n t work. I t was t h e r e f o r e f e l t t h a t i n view of the l a r g e d i s c r e p a n c y i n the r e p o r t e d deformation p o t e n t i a l s of GaSb, an independent method should be used t o determine t h e i r v a l u e s . A d i r e c t way f o r d e t e r m i n i n g the deformation p o t e n t i a l c o n s t a n t s of the v a l e n c e bands, i s by o b s e r v i n g the changes i n the a b s o r p t i o n edge s p e c t r a , induced by a s t a t i c s t r a i n ( p i e z o a b s o r p t i o n (PA)). T h i s method i s u s e f u l i n the case o f i n d i r e c t energy gap m a t e r i a l s s i n c e the a b s o r p t i o n i s s m a l l e r than f o r d i r e c t gap m a t e r i a l s , and hence t h i c k samples can be used. In the case of d i r e c t gap m a t e r i a l s , s i n c e the a b s o r p t i o n i s v e r y s t r o n g i n the r e g i o n of i n t e r e s t , t h i n 5 samples (2 t o 10 jj.m) must be used i n order t o perform PA measurements. One way of p e r f o r m i n g PA measurements, on t h i n samples, i s the method of Z w e r d l i n g e t a l . ( 1 9 5 9), i n which the sample i s r i g i d l y cemented t o a t r a n s p a r e n t s u b s t r a t e . The sample w i l l then be s t r a i n e d , upon c o o l i n g , due t o the d i f f e r e n t i a l thermal e x p a n s i v i t y between the sample and the s u b s t r a t e . T h i s i s the method used i n the p r e s e n t work. The advantage of PA measurements, i n which the sample i s r i g i d l y cemented t o a s u b s t r a t e , i s t h a t both p o s i t i v e and n e g a t i v e s t r a i n s can be o b t a i n e d . In the p r e s e n t work, p o s i t i v e ( t e n s i l e ) s t r a i n s of up t o + 1 0 x l 0 " ¥ and n e g a t i v e (compressive) s t r a i n s of - 4 . 2 x 1 0 " ' ' were o b t a i n e d . By showing t h a t , f o r the s t r a i n s used,the e x c i t o n b i n d i n g energy was r e l a t i v e l y s t r a i n independent, the v a l e n c e band de-f o r m a t i o n p o t e n t i a l s c o u l d then be determined by assuming t h a t t h e separation of the e x c i t o n peaks and t h e i r energy, r e p r e s e n t e d t h e behaviour of the s t r a i n s p l i t v a l e n c e bands. A disadvantage of t h i s method, ot h e r than the requirement o f t h i n samples, i s t h a t the s t r a i n i s homogeneous i n the plane of the sample and t h e r e f o r e the e x c i t o n a b s o r p t i o n s p e c t r a , f o r r a d i a t i o n i n c i d e n t normal t o the s u r f a c e , w i l l be p o l a r i z a t i o n independent and the s i g n s of the deformation p o t e n t i a l s can not be determined. In the p r e s e n t work, a u n i a x i a l s t r a i n was superimposed on the b i a x i a l l y s t r a i n e d sample thereby e l i m i n a t i n g t h e b i a x i a l homogeneity of the s t r a i n . The s i g n s of the deformation p o t e n t i a l s c o u l d then be determined by u s i n g p o l a r i z e d l i g h t t o i d e n t i f y the v a l e n c e 6 bands r e s p o n s i b l e f o r the e x c i t o n a b s o r p t i o n peaks. In the p r e s e n t work, the i n f r a - r e d t r a n s m i s s i o n was ob-served, u s i n g a h i g h r e s o l u t i o n monochromator, i n the r e g i o n of the fundamental a b s o r p t i o n edge of GaSb. The a b s o r p t i o n s p e c t r a was observed f o r t h i c k (100 t o 200 jura.) unmounted samples and f o r t h i n (2 t o 5 /<m) samples mounted on g l a s s s u b s t r a t e s c o o l e d t o 2°K. The t h i n samples, upon c o o l i n g t o 2°K, were homogeneously b i a x i a l l y s t r a i n e d i n the plane of the sample. The f i n e s t r u c t u r e , near the fundamental absorp-t i o n edge, was i n v e s t i g a t e d as a f u n c t i o n of both compressive and t e n s i l e b i a x i a l s t r a i n s . The f3 and t a b s o r p t i o n peaks, p r e v i o u s l y observed by Johnson (1964) near the fundamental a b s o r p t i o n edge, were a l s o observed and s t u d i e d f o r both s t r a i n e d and u n s t r a i n e d samples. 7 CHAPTER 2 THEORETICAL BACKGROUND 2.1 INTRODUCTION In "this c h a p t e r we wish t o b r i e f l y summarize the t h e o r e t i c a l background necessary f o r a n a l y z i n g the e x p e r i -mental r e s u l t s . An e x c e l l e n t review and d i s c u s s i o n of the t h e o r y and e x p e r i m e n t a l work done on the I I I - V compounds i s g i v e n i n t h e s e r i e s "Semiconductors and Semimetals" (Ed. by R. K. W i l l a r d s o n & A l b e r t C. Beer, Academic P r e s s ) . In s e c t i o n 2.1.1 and 2.1.2 the u n s t r a i n e d z i n c blende l a t t i c e and band s t r u c t u r e are d i s c u s s e d . The e f f e c t of s t r a i n on the band s t r u c t u r e i s then c o n s i d e r e d i n s e c t i o n 2.1.3. The d e f o r m a t i o n p o t e n t i a l t h e o r y of K l e i n e r and Roth (1959), which i s used to d e s c r i b e the e f f e c t of a b i -a x i a l s t r a i n on the band s t r u c t u r e of a semiconductor, i s reviewed i n s e c t i o n 2.I.4. The t h e o r y of K l e i n e r and Roth does not c o n s i d e r the s t r a i n induced c o u p l i n g between the 3=3/2 m T=il/2 and the 3=1/2 m^=±\/2 valence bands. The s t r e s s induced c o u p l i n g was f i r s t c o n s i d e r e d by Hasegawa (I963) and a d i s c u s s i o n g i v e n by P o l l a k and Cardona (1968) f o r the s t r e s s induced c o u p l i n g , due t o a u n i a x i a l s t r e s s , i s reviewed and adapted t o the b i a x i a l s t r a i n case. The e x c i t o n t h e o r y , i n the e f f e c t i v e mass approximation, i s then reviewed i n s e c t i o n 2.2.1 and i n s e c t i o n 2.2.2 and the e f f e c t of s t r a i n s of d i f f e r i n g magnitudes and d i r e c t i o n s 8 on the e x c i t o n b i n d i n g e n e r g i e s are c o n s i d e r e d u s i n g the method developed by Osipov ( 1 9 6 7 ) . As a r e s u l t of t h i s d i s c u s s i o n we show t h a t , f o r the s t r a i n s used i n t h i s work, the e f f e c t of s t r a i n , on the b i n d i n g energy of the e x c i t o n , can be n e g l e c t e d . Johnson ( 1 9 6 4 ) i n GaSb, are then d i s c u s s e d i n s e c t i o n 2 . 2 . 3 . The y peak has p r e v i o u s l y been a t t r i b u t e d t o an e x c i t o n -a c c e p t o r complex and t h e r e f o r e the e f f e c t of s t r a i n on the a c c e p t o r l e v e l s i s d i s c u s s e d . F i n a l l y , s i n c e the e l a s t i c constants'at 2°K are necessary f o r the c a l c u l a t i o n of the deformation p o t e n t i a l parameters, th e y are g i v e n i n the f i n a l s e c t i o n . The X a b s o r p t i o n peaks, f i r s t observed by 9 2.1.1 UNSTRAINED ZINC BLENDE LATTICE (PHYSICAL DESCRIPTION) The group I I I - V i n t e r m e t a l l i c compounds c r y s t a l l i z e i n a z i n c blende s t r u c t u r e . T h i s s t r u c t u r e can be c o n s i d e r e d as b e i n g made up of two i n t e r p e n e t r a t i n g f a c e - c e n t r e d c u b i c sub-l a t t i c e s o r i e n t e d p a r a l l e l to each other and d i s p l a c e d from one another by the v e c t o r T=(a/4, a/4, a/4) where a i s the edge l e n g t h of the elementary cube of the f a c e - c e n t r e d l a t t i c e , (see F i g u r e 2.1). With t h i s arrangement each atom i s bonded t e t r a h e d r a l l y t o f o u r atoms of the o p p o s i t e group. The z i n c blende s t r u c t u r e belongs to the space group T^ (F43m), which i s c o n t a i n e d i n the c u b i c system, w i t h i t s f a c e - c e n t r e d c u b i c B r a v a i s l a t t i c e , but i t s p o i n t group i s o n l y T^ i n accordance w i t h the t e t r a h e d r a l environment of each atom. The z i n c blende l a t t i c e d i f f e r s from the diamond l a t t i c e i n t h a t i t l a c k s a c e n t r e of i n v e r s i o n symmetry s i n c e each s u b l a t t i c e c o n s i s t s of atoms of d i f f e r e n t k i n d s . As a r e s u l t o f the l a c k of i n v e r s i o n symmetry and the p a r t i a l i o n i c bonding of the atoms, the z i n c blende c r y s t a l has a p o l a r i t y a l o n g the ( l l ] ) ^ axes which g i v e s r i s e t o a d i f f e r e n c e i n a * The n o t a t i o n used throughout t h i s t e x t t o d e s i g n a t e c r y s t a l p l a n e s and d i r e c t i o n s , i s t h a t g i v e n by K i t t e l (1963) page 34, which i s fhkl] = planes e q u i v a l e n t by symmetry (hkl) = s i n g l e c r y s t a l l o g r a p h i c plane <hkl^ = a f u l l s e t of e q u i v a l e n t d i r e c t i o n s [hkl] = s p e c i f i c c r y s t a l d i r e c t i o n where h, k, 1, are the M i l l e r i n d i c e s . 10 F i g u r e 2 .1 (a) Z i n c blende l a t t i c e s t r u c t u r e ( from Banerjee - et a l . ( 1 9 6 9 ) ) . (b) Model of the z i n c blende s t r u c t u r e i n d i c a t i n g the p o l a r 111 d i r e c t i o n s . The white spheres r e p r e s e n t A atoms (Ga) and the dark spheres B atoms (Sb) (from MacRae ( 1 9 6 6 ) ) . 1 1 number of the p r o p e r t i e s of [ i l l ] or { 1 1 1 } surfaces""" (see F i g u r e 2 . 1 b ) . Due t o the s m a l l e r bond d e n s i t y and hence s m a l l e r b i n d -i n g energy between c e r t a i n p l a n e s , any s e p a r a t i o n of the c r y s t a l (due t o e i t h e r cleavage or a b r a s i o n ) would tend t o occur between the p l a n e s w i t h the s m a l l e r b i n d i n g energy l e a v i n g , i n the i d e a l case, the ( i l l ) plane c o n s i s t i n g of A(Ga) atoms on the one f a c e and the f i l l ) plane with B(Sb) atoms on the o t h e r f a c e . The e f f e c t of t h i s p o l a r a x i s i s observed i n v a r i o u s p r o c e s s e s such as cle a v a g e , o x i d a t i o n , c r y s t a l growth, e t c h i n g and g r i n d i n g or p o l i s h i n g . 2.1.2 UNSTRAINED ENERGY BAND STRUCTURE In the u n s t r a i n e d z i n c blendetype m a t e r i a l , i f s p i n - o r b i t i n t e r a c t i o n i s n e g l e c t e d , the v a l e n c e band extrema i n GaSb and o t h e r I I I - V type semiconductors, l i e s at k=0, at which p o i n t the band i s t r i p l y degenerate ( n e g l e c t i n g s p i n ) . With s p i n - o r b i t i n t e r a c t i o n , the degeneracy i s p a r t i a l l y removed, b e i n g s p l i t i n t o a doubly degenerate p - ^ ^ - m u l t i p l e t with double group symmetry (see F i g u r e 2.2) and a f o u r -f o l d degenerate p2/2~mxx^ ^-P^- e^  w i t h double group symmetry f^^. The f o u r - f o l d degenerate f~~ band ( l a b e l l e d J = 3/2 mj=±3/2, ±l/2 i n angular momentum n o t a t i o n ) and the t w o - f o l d degener-ate fjv band (J=l/2 mj=±l/2) are separated by a s p i n o r b i t The {ill} or A s u r f a c e s are d e f i n e d as those t e r m i n a t i n g i n group I I I atoms t r i p l y bonded t o the matrix and the { i l l } or B s u r f a c e s are those t e r m i n a t i n g i n t r i p l y bonded group V atoms. See f o r i n s t a n c e Barber e t a l . ( 1 9 6 5 ) . 1 2 > (0,0,0) (1,0,0) (f,-|,0) (0,0,0) k VECTOR (~p) F i g u r e 2 . 2 C a l c u l a t e d energy bands of GaSb al o n g important symmetry axes. The energy l e v e l s are desig n a t e d a c c o r d i n g t o the c o n v e n t i o n a l double-group n o t a t i o n ( from Zhang and Callaway ( 1 9 6 9 ) ) . s p l i t t i n g energy of Aa(=0.749 ± 0 . 0 0 2 eV. f o r GaSb (Reine e t a l ( 1 9 7 0 ) ) ) . A l s o , because o f the s p i n - o r b i t i n t e r a c t i o n , s m a l l terms l i n e a r i n k appear i n the energy e q u a t i o n s f o r the va l e n c e bands which cause the extrema t o be d i s p l a c e d from the p o i n t k=0 i n the < ( l l i ^ d i r e c t i o n . C o u p l i n g between the de-generate bands d i s t o r t s the energy s u r f a c e s i n t o q u a r t i c s u r f a c e s , o f t e n c a l l e d "warped" or " f l u t e d " s u r f a c e s . These warped energy s u r f a c e s , which cannot be re p r e s e n t e d by a mass t e n s o r , are u s u a l l y d e s c r i b e d i n terms o f the i n v e r s e mass band parameters (Dresselhaus e t a l . ( 1 9 5 5 ) ) A, B and N. The c o n d u c t i o n band of GaSb possesses a minimum at k=0 and i s doubly degenerate ( i n c l u d i n g s p i n ) w i t h symmetry J£ The c o n d u c t i o n band has s u b s i d i a r y minima i n the ^ 1 1 1 ^ and <(100)> d i r e c t i o n s l y i n g about 0 . 0 9 eV and 0 . 7 0 eV i n energy r e s p e c t i v e l y above the [000] minima (Sagar ( 1 9 6 1 ) ) . The con-d u c t i o n band i s p a r a b o l i c near the minimum and,with i n c r e a s i n g k, the c u r v a t u r e decreases w i t h a consequent i n c r e a s e i n the e f f e c t i v e mass of the e l e c t r o n s . The d i r e c t (or fundamental) a b s o r p t i o n edge, o f energy E Q i s due t o t r a n s i t i o n s from the [7 v a l e n c e band t o the two-ov f o l d degenerate s - l i k e 17 c o n d u c t i o n band at k=0 . oc 2 . 1 . 3 EFFECT OF STRAIN ON BAND STRUCTURE The nature of the energy band extrema i n semiconductors depends on the symmetry of the c r y s t a l l a t t i c e and, i f the c r y s t a l i s d i s t o r t e d by a n o n - i s o t r o p i c s t r a i n , the form of these extrema w i l l be a l t e r e d . When an i s o t r o p i c ( h y d r o s t a t i c ) s t r e s s i s a p p l i e d t o a semiconductor, such t h a t the symmetry of the c r y s t a l i s not 14 a l t e r e d , the net e f f e c t i s t o s h i f t , r e l a t i v e t o one another, the band e n e r g i e s not r e q u i r e d by symmetry t o be degenerate. T h i s s h i f t i n g of the energy l e v e l s can remove a c c i d e n t a l degeneracies but not symmetry induced degeneracies such as those t h a t occur at the band extrema. When a u n i a x i a l or b i a x i a l s t r e s s i s a p p l i e d t o a z i n c -blende semiconductor , the shear components of the s t r e s s t e n s o r remove the c u b i c symmetry and p a r t i a l l y remove the degeneracy of the f o u r - f o l d degenerate J=3/2 valence band at k=0 s p l i t t i n g i t i n t o two t w o - f o l d degenerate bands (see F i g u r e 2 . 3 ) . To f a c i l i t a t e d i s c u s s i o n , the th r e e valence bands shown i n F i g u r e 2.3 have been l a b e l l e d v l , v2 and v3 f o l l o w i n g the convention of P o l l a k et a l . (1968). The t r a n s i t i o n s between these valence bands and the 17 conduction band have been oc l a b e l l e d E Q ( 1 ) , E Q ( 2 ) , and E Q ( 3 ) where the s u b s c r i p t o i n -d i c a t e s t h a t these t r a n s i t i o n s occur at the c e n t r e (k=0) of the B r i l l o u i n zone. When a b i a x i a l s t r e s s i s a p p l i e d i n the {lOO} or {ill} c r y s t a l l o g r a p h i c planes (or the u n i a x i a l s t r e s s along the <^L00> or <^Lll) d i r e c t i o n s ) the s p l i t bands v l and v2 can be i d e n t i f i e d (when the bands are decoupled) by the magnetic quantum numbers ±mj. A l s o , i n these two cases, the energy s u r f a c e s are e l l i p s o i d s of r e v o l u t i o n w i t h the p r i n c i p l e symmetry a x i s along the s t r e s s d i r e c t i o n f o r the u n i a x i a l case or alo n g the normal t o the s t r e s s plane f o r the b i a x i a l c ase. 15 Eo(l) era ir-Eo+A 0 7 A 0 E„(l) r6c E G(2) —-cr v, J=3/2 m, =±i/2 v J=3/2 2 rrtj =±3/2 J= 1/2 mj = ± 1/2 ( a ) (b) (c) TENSILE STRAIN UNSTRAINED COMPRESSIVE STRAIN F i g u r e 2.3 The v a l e n c e and lowest c o n d u c t i o n bands of diamond and z i n c b l e n d e t y p e m a t e r i a l s near k=0 f o r the u n s t r a i n e d l a t t i c e (b), and f o r a t e n -s i l e ( a ) , and compressive s t r a i n ( c ) , a p p l i e d i n the <00l) or CL11> d i r e c t i o n . The l a b e l s o* and TT i n d i c a t e t h a t the e l e c t r o n i c t r a n s i t i o n i s allowed f o r l i g h t p o l a r i z e d p e r p e n d i c u l a r or p a r a l l e l , r e s p e c t i v l y , t o the a p p l i e d u n i -a x i a l s t r a i n . 1 6 The c o n d u c t i o n band, b e i n g o n l y s p i n degenerate, i s not s p l i t by the a p p l i e d s t r e s s and the deformation can cause changes i n the e f f e c t i v e mass on l y . The h y d r o s t a t i c component of the u n i a x i a l or b i a x i a l s t r e s s , causes a net s h i f t of the c o n d u c t i o n band energy minimum r e l a t i v e t o the " c e n t r e of g r a v i t y " of the s t r a i n s p l i t v a l e n c e band d o u b l e t . K l e i n e r and Roth ( 1 9 5 9 ) have d e r i v e d equations r e l a t i n g the s h i f t of the "c e n t r e of g r a v i t y " and s p l i t t i n g of the v a l e n c e bands i n terms o f deformation p o t e n t i a l s and a p p l i e d s t r e s s which w i l l be d i s c u s s e d i n s e c t i o n 2 . 1 . 4 * The " c e n t r e of g r a v i t y " of the s t r a i n s p l i t v a l e nce band dou b l e t i s i n t e r p r e t e d t o mean a p o i n t h a l f way, i n energy, between the v l and v 2 v a l e n c e bands. The mean energy gap between these v a l e n c e bands and the c o n d u c t i o n band i s d e f i n e d as E = l / 2 ( E 0 ( l ) + E Q ( 2 ) ) where E Q ( l ) and E Q ( 2 ) are the e n e r g i e s a s s o c i a t e d w i t h the t r a n s i t i o n s between the s t r a i n s p l i t v a l e n c e bands v l and v 2 r e s p e c t i v e l y and the condu c t i o n band. 17 SELECTION RULES FOR ELECTRONIC TRANSITIONS For a u n i a x i a l s t r e s s a l o n g the ^lOO^ or <(111^ d i r e c t i o n s t h e band Hamilton!ans are d i a g o n a l ( i . e . the elements of the s t r a i n H a m i l t o n i a n m a t r i x which connect d i f f e r e n t bands v a n i s h i n the a n g u l a r momentum r e p r e s e n t a t i o n q u a n t i z e d along the s t r e s s d i r e c t i o n ; see s e c t i o n 2 . 1 . 4 ) and we may speak of m j = ± 1 / 2 ( v l ) and n i j = ± 3 / 2 ( v 2) bands. O p t i c a l t r a n s i t i o n s are t h u s made from an m T = ± 1 / 2 or an m T = ± 3 / 2 v a l e n c e band v J t o an nij = ±l / 2 s t a t e of the J = l / 2 c o n d u c t i o n band. A t r a n s i t i o n from the mj = ±l/ 2 v a l e n c e band t o the cond u c t i o n band can s a t i s f y t h e s e l e c t i o n r u l e ^m=0 and Am=±\ and so can o c c u r f o r l i g h t p o l a r i z e d both p a r a l l e l and p e r p e n d i c u l a r t o t h e s t r e s s d i r e c t i o n ( If and C p o l a r i z a t i o n , r e s p e c t i v e l y ) . T r a n s i t i o n s from the mj = ± 3 / 2 v a l e n c e band t o the c o n d u c t i o n band can o n l y s a t i s f y the s e l e c t i o n r u l e Anij = ± 1 and so can o n l y o c cur f o r l i g h t p o l a r i z e d p e r p e n d i c u l a r t o the s t r e s s d i r e c t i o n ( c T p o l a r i z a t i o n ) . Thus, t r a n s i t i o n s from both the v l and v 2 v a l e n c e bands t o the c o n d u c t i o n band are allowed f o r <f p o l a r i z a t i o n whereas o n l y the v 2 v a l e n c e band to the c o n d u c t i o n band t r a n s i t i o n o c c u r s f o r If p o l a r i z a t i o n (see F i g u r e 2 . 3 ) . F o r a u n i a x i a l s t r e s s i n o t h e r d i r e c t i o n s , ( i . e . <(llO)> ), m i s i n g e n e r a l no l o n g e r a good quantum number and we w i l l KJ no l o n g e r have simple s e l e c t i o n r u l e s . However, Hensel and Feher ( i 9 6 0 ) have shown t h a t i f a g i v e n s t r e s s a l o n g the ^lOO} and ^111^ d i r e c t i o n s produces v a l e n c e band s p l i t t i n g s of the * T h i s i s not s t r i c t l y t r u e s i n c e the s t r e s s c o u ples bands w i t h the same v a l v e of m,. However i f the valence band energy s p l i t t i n g i s much l e s s than the s p i n o r b i t s p l i t t i n g (as i s the case i n the p r e s e n t work) the Ha m i l t o n i a n ( 2 . 6 ) i s , to a good a p p r o x i m a t i o n , s t i l l d i a g o n a l . 18 same magnitude and s i g n per u n i t s t r e s s , then the s t r a i n H a m i l t o n i a n becomes d i a g o n a l f o r s t r e s s i n an a r b i t r a r y d i r e c t i o n and then the simple p o l a r i z a t i o n p r o p e r t i e s are p r e s e r v e d f o r a u n i a x i a l s t r e s s i n any d i r e c t i o n . When a sample i s cemented to a s u b s t r a t e , an i s o t r o p i c b i a x i a l s t r a i n i s produced i n the plane of the sample and, s i n c e ah i s o t r o p i c s t r a i n i n the plane of the sample i s e q u i v a l e n t (as f a r as o p t i c a l s e l e c t i o n r u l e s are concerned) t o the presence of a u n i a x i a l s t r e s s p e r p e n d i c u l a r t o the plane of the sample, i t i s o n l y p o s s i b l e to observe t r a n s i t i o n s w i t h l i g h t p o l a r i z e d p e r p e n d i c u l a r t o the s t r e s s d i r e c t i o n ( c T p o l a r i z a t i o n ) . T h e r e f o r e , f o r an i s o t r o p i c s t r a i n i n the p l a n e bf the sample and f o r p o l a r i z e d l i g h t i n c i d e n t normal t o the sample s u r f a c e , both t r a n s i t i o n s w i l l be p e r m i t t e d and w i l l be independent of the angle of p o l a r i z a t i o n . Hence, i n orde-r t o determine the o p t i c a l s e l e c t i o n r u l e s and t h e r e -f o r e the s i g n s of the v a l ence band deformation p o t e n t i a l s , a u n i a x i a l s t r a i n p e r p e n d i c u l a r t o the i n c i d e n t l i g h t i s r e q u i r e d . The e l e c t r i c - d i p o l e s e l e c t i o n r u l e s have been obtained by B a i l e y (1970) f o r the a b s o r p t i o n l i n e s a r i s i n g from weakly bound two-or t h r e e - c a r r i e r complexes i n a u n i a x i a l l y s t r a i n e d d i r e c t gap z i n c blende semiconductor. He a l s o determines e x a c t l y the r e l a t i v e s t r e n g t h s of each l i n e f o r both i f and p o l a r i z a t i o n . These e l e c t r i c - d i p o l e s e l e c t i o n r u l e s can T h i s p r o p e r t y was used t o v e r i f y t h a t the b i a x i a l s t r a i n was indeed i s o t r o p i c i n the plane of the sample. be used to determine whether the ^ or if a b s o r p t i o n peaks, observed i n t h i s work ( s e c t i o n 2 . 2 „ 3 and 4 . 4 ) , are due t o any of these complexes. 2,1.4 DEFORMATION POTENTIALS The e f f e c t on the band s t r u c t u r e o f a semiconductor o r a metal, due t o a deformation o f the l a t t i c e , can be conveniently-t r e a t e d by the use o f the deformation p o t e n t i a l t h e o r y . The concept o f a deform a t i o n p o t e n t i a l was f i r s t i n t r o d u c e d by Bardeen and S h o c k l e y ( 1 9 5 0 ) f o r the study o f s c a t t e r i n g o f e l e c t r o n s and h o l e s by l o n g wavelength a c o u s t i c phonons and was l a t e r used by K l e i n e r and Roth ( 1 9 5 9 ) f o r t h e i r t h e o r y of the shear s t r a i n s p l i t t i n g o f the v a l e n c e bands under a b i a x i a l s t r a i n . K l e i n e r and Roth, i n t h e i r treatment o f the s t r e s s i nduced s p l i t t i n g o f the v a l e n c e bands, n e g l e c t e d the s t r e s s -i nduced c o u p l i n g between the J = 3 / 2 mj =• i l / 2 v a l e n c e band and the s p i n - o r b i t s p l i t o f f J = 1 / 2 mj = i l / 2 v a l e n c e band. Hasegawa ( I 9 6 3 ) and P o l l a k and Cardona ( I 9 6 8 ) l a t e r c o n s i d e r e d t h e strless induced c o u p l i n g between the s e bands and found t h a t i t c o u l d have a n o t i c a b l e e f f e c t upon the s t r a i n dependence o f t h e energy bands. P o l l a k and Cardona t r e a t e d the va l e n c e band s p l i t t i n g and c o u p l i n g f o r a u n i a x i a l s t r e s s . T h e i r r e s u l t s w i l l be b r i e f l y reviewed here and g e n e r a l i z e d t o the b i a x i a l s t r a i n case u s i n g the de f o r m a t i o n p o t e n t i a l n o t a t i o n o f K l e i n e r and Roth. In t he deformation p o t e n t i a l t h e o r y , the e f f e c t o f the s t r a i n i s assumed t o be a f i r s t o r d e r p e r t u r b a t i o n so t h a t a m a t r i x form f o r the o r b i t a l - s t r a i n H a m i l t o n i a n H£ can be w r i t t e n i n the r e p r e s e n t a t i o n o f the unperturbed wave f u n c t i o n s . 2 1 Symmetry arguments are then used t o reduce t h e form o f t h e m a t r i x elements t o a minimum number o f undetermined p a r a -meters c a l l e d the deform a t i o n p o t e n t i a l s which are then e v a l u a t e d e m p i r i c a l l y , . When a b i a x i a l s t r e s s i s a p p l i e d t o a z i n c blende c r y s t a l , t h e c u b i c symmetry i s removed and the 3=3/2. v a l e n c e band s p l i t s i n t o a p a i r of Kramer's d o u b l e t s . Hasegawa ( 1 9 6 3 ) has shown t h a t t h e o r b i t a l s t r a i n H a m i l t o n i a n , which d e s c r i b e s the e f f e c t o f a homogeneous s t r a i n on the co n d u c t i o n and v a l e n c e bands, can be w r i t t e n as = Dj ( 2 . 1 ) + 2D„ [(J J f-jy3 ) e M + ( J / - J 2 / 3 ) € W +(Jj - J V 3 ) € a , ] where ,—£Xj , — are the components o f the s t r a i n t e n s o r (see Aplpendix D), t h e D's are the deformation p o t e n t i a l c o e f f i c i e n t s , J i s the ang u l a r momentum o p e r a t o r and x,y and z r e f e r t o the c r y s t a l l o g r a p h i c a x i s . The parameters D d and D j " are t h e h y d r o s t a t i c p r e s s u r e d e f o r m a t i o n * P i c u s and B i r ( i 9 6 0 ) d e f i n e d deformation p o t e n t i a l s a,b and d which r e f e r t o h o l e e n e r g i e s . The K l e i n e r and Roth deform a t i o n p o t e n t i a l s r e f e r t o e l e c t r o n e n e r g i e s and are r e l a t e d by the f o l l o w i n g a = ~ ( 2 / 3 ) D f b = -(2/3)D« d = ~ ( 2 / V T ) t i p o t e n t i a l s f o r the c o n d u c t i o n and v a l e n c e bands r e s p e c t i v e l y . The q u a n t i t i e s and are u n i a x i a l deformation p o t e n t i a l s a p p r o p r i a t e t o s t r a i n s of t e t r a g o n a l (<00l) ) and rhombohedral ( ( i l l ^ ) symmetries r e s p e c t i v e l y . The f a c t o r s | j J e t c . r e p r e s e n t the symmetrized product ( j v J„]=1/2(J J + J J ) I x yj / v x y y x' The H a m i l t o n i a n f o r a g i v e n hand, i n c l u d i n g s p i n - o r b i t i n t e r a c t i o n , i s g i v e n by H 1 - H S Q + He 1 (2.2) where H g o i s the s p i n o r b i t H a m i l t o n i a n i n the absence of s t r e s s and i i s e i t h e r c or v r e p r e s e n t i n g the c o n d u c t i o n or v a l e n c e bands, r e s p e c t i v e l y . The s t r a i n H a m i l t o n i a n f o r the b i a x i a l s t r a i n i n a p a r t i c u l a r c r y s t a l l o g r a p h i c plane w i l l now be determined from the o r b i t a l s t r a i n H a m i l t o n i a n g i v e n by e q u a t i o n 2.1. C o n s i d e r a C a r t e s i a n c o o r d i n a t e system ( x T , y" , z f ) such t h a t the z T a x i s i s a l o n g the normal t o the c r y s t a l f a c e and the i s o t r o p i c b i a x i a l s t r a i n i s a p p l i e d i n the x f , y T plane. For a t h i n s i n g l e - c r y s t a l i n a s t a t e of uniform and i s o t r o p i c s t r a i n ( e i t h e r t e n s i l e or compressive) i n the plane of the sample, the components of the s t r a i n t e n s o r i n terms of t h i s c o o r d i n a t e system are (2.3) V M 1 £<jj = T -*T where T i s the s t r a i n and A i s a parameter (see Table 2.1 and Appendix D) determined by the c o n d i t i o n t h a t the s t r e s s normal t o the c r y s t a l f a c e v a n i s h ( i . e . <7jti= 0). The parameter }\ 23 i s c a l c u l a t e d in, terms of t h e e l a s t i c c o n s t a n t s c - d i s c u s s e d i n s e c t i o n 2 . 4 . For the s t r a i n a p p l i e d i n any plane, o t h e r than the {ooi] plane, the s t r a i n t e n s o r \^±j] w i l l have t o be transformed (see Appendix D) from the primed system i n t o the c r y s t a l co-o r d i n a t e system (x, y, z) t o o b t a i n the s t r a i n t e n s o r [£••] ( i , j = x, y, z) i n terms o f the p r i n c i p a l c r y s t a l l o g r a p h i c axes. We w i l l now b r i e f l y c o n s i d e r the s p e c i a l cases of an i s o t r o p i c b i a x i a l s t r a i n a p p l i e d t o t h e p r i n c i p a l c r y s t a l -l o g r a p h i c p l a n e s . BIAXIAL STRAIN IN THE fooij PLANE For an i s o t r o p i c b i a x i a l s t r a i n a p p l i e d i n the {001} plane, the s t r a i n components r e f e r r e d t o the p r i n c i p a l c r y s t a l l o g r a p h i c c o o r d i n a t e s are: e,= ez= T e3= ~?\T £ V= 6 R = efc = 0 The s i n g l e s u f f i x n o t a t i o n i s d e f i n e d i n Appendix D and i s as f o l l o w s : x x = l , yy = 2, zz = 3 , yz = zy = 4 * zx = xz = 5 , xy = yx = 6 . The s t r a i n H a m i l t o n i a n s ( 2 . 1 ) then become Kl = Da ( 2 - A ) - T C - D - R ( 2 - A ) ' T - 2 D l X ( l + A ) ( j ? i - J ^ 3 ) T ( 2 * 4 ) and the s t r a i n H a m i l t o n i a n f o r the energy d i f f e r e n c e between t h e and the |gv bands becomes 24 H 6= H N H ; = (D d c-Dr)(2-A)T + 2D„ (l+A) ( J * - f/3)T ( 2 5 ) - S E H + (3/2)SE..,(Ji*- J/3) where 8E H = (Dj-Dj) (2-A)T = (dE0/bP)P i s the s h i f t of the energy gap, E Q , due to the h y d r o s t a t i c component of the s t r a i n , and SE,,,,,^  (4/3) (1+ TOD^T, i s the l i n e a r s p l i t t i n g of the P3 /2 m u l t i p l e t . I f the wave f u n c t i o n s f o r the v a l e n c e band s t a t e s are taken i n the ( J , nij) r e p r e s e n t a t i o n i n which the s p i n - o r b i t H a m i l t o n i a n H i s d i a g o n a l then the H a m i l t o n i a n matrix f o r t h e valence bands can be w r i t t e n as (see P o l l a k ( 1 9 6 8 ) ) |3/2,3/2> |3/2,l/2> | l / 2 , l / 2 > H A/3-8EM-6E0.,/2 0 0 0 A/3 - SE H +£E„,/2 8 EL / V 2 0 6E M )/V2 -2A/3-3E H ( 2 . 6 ) where A>is the s p i n - o r b i t s p l i t t i n g of the v a l e n c e bands at k=0. Only s t a t e s of p o s i t i v e mj have been c o n s i d e r e d s i n c e the s t r e s s does not remove the Kramers degeneracy of each s t a t e . I t should be noted t h a t s i n c e the s t a t e |3/2,3/2^> i s not coupled by the s t r e s s t o the other two v a l e n c e bands i t s energy w i l l have a l i n e a r s t r e s s dependence on s t r a i n w h i l e the s t a t e s w i t h mj = l / 2 w i l l have a n o n - l i n e a r s t r e s s de-pendence caused by the o f f - d i a g o n a l terms of the m a t r i x ( 2 . 6 ) . D i a g o n a l i z i n g " the above H a m i l t o n i a n , P o l l a k ( 1 9 6 8 ) f i n d s , f o r the change i n the energy d i f f e r e n c e between the c o n d u c t i o n S i n c e the H a m i l t o n i a n i s d i a g o n a l f o r the s t r a i n a p p l i e d i n the {00l} plane, the magnetic quantum numbers mj are good quantum numbers. 25 and valence bands at k= 0 , the f o l l o w i n g e x p r e s s i o n s E Q (2 )= (E e - E 2 )=-V3+ 5EM +SEM,/2 E Q (1 )= (E c -E, )= A/6+SE M - SE M ( / 4 - \&Af>E„,+9 (SE.J/V}/^ ( 2 . 7 ) E Q (3 )= (E t -E, )= A/ 6+6E H - SEeo/4+{Ai.+^EM,+9 (SE0J/a}'% where E,,E 2, and E 3 are the e n e r g i e s of the s t r a i n s p l i t v a l e n c e bands v l , v2 and v 3 r e s p e c t i v e l y . For S E 0 O L ^ A 0 e q u a t i o n ( 2 . 7 ) can be expanded i n powers of S E „ , / A o t o g i v e " E Q (2 )= ( E C - E A )=-A/3+8E H + 8E ( ) 0/2 E n ( l ) = ( E c - E , ) = - - y 3 + 6 E H - S E~ , / 2 -(SE „ , ) / / ( 2 ^ ) + ° 2 ( 2 . 8 ) E Q (3 )= (E t -E, ) = + 2 A / 3 + S E M + ( SE..,)/ (2 A,)+ The s p l i t t i n g of the v a l e n c e band at k=0 i s g i v e n ( E 0 ( 2 ) - E 0 ( l ) ) = SE^,+(SE e „ ) / ( 2 A . ) ( 2 . 9 ) where 8 E „ = 4 ( 1 + ^)D«T/3 i s the l i n e a r s p l i t t i n g of the m u l t i p l e t f o r an i s o t r o p i c b i a x i a l s t r a i n a p p l i e d i n the {001} p l a n e . The s h i f t of the energy gap, E Q , due t o the hydro-s t a t i c component of the s t r a i n i s giv e n by SE„= (Dj -D^) • ( 2 -A)T ( 2 . 1 0 ) BIAXIAL STRAIN IN THE f i l l ] PLANE For a b i a x i a l s t r a i n a p p l i e d i n the ( i l l } plane, the s t r a i n component, 6^ - , g i v e n i n terms of the c r y s t a l plane c o o r d i n a t e system (equation 2 . 3 ) have t o be r o t a t e d so as t o be expressed i n terms of the p r i n c i p a l c r y s t a l l o g r a p h i c axes. -"-Note t h a t K l e i n e r and Roth ( 1 9 5 9 ) d i d not c o n s i d e r the s t r e s s induced i n t e r a c t i o n between the m T = l / 2 bands. 26 The strain components in terms of the principal crystallographic axes are 6,= £z= 63= (2-A)T/3 ( 2 . 1 1 ) The strain Hamiltonian ( 2 . 1 ) then becomes ( 2 . 1 2 ) He= p£(2-A)T H^ = D/(2-A)T -(2/3)D;(l+A)T [fjxJy]+ Je]+ & J»] and the strain Hamiltonian for the energy difference between the [ftc and the lg v bands becomes H6= (DJ'-Dj) ( 2 - A)T+(2/3 )K (1+A)T [{j, Jjjf {j, J4]+ f j e J X J ( 2 1 3 ) = SEH+8EHI(l/2) [ f j x fj^J^+fJa J„J where <SE,„= ( 4 / 3 )D^ (l+^)T i s the linear s p l i t t i n g of the P-/_ 3 / 2 multiplet for an isotropic biaxial strain applied in the [ i l l ! plane. At this point the problem i s simplified i f we rotate J so that (J* J £ )—**(J, Jx J3 ) with J 3 directed along the [ i l l ] direction. (The choice of the perpendicular axes J, and J 2 i s immaterial.) Making such a transformation we obtain He= 6E H + (3/2)5E H /[(j/-J^3)] ( 2 . 1 4 ) which i s similar to the result ( 2 . 5 ) . The equations relating the energy change between the conduction and valence bands w i l l be the same as equation ( 2 . 8 ) in the approximation $Em<£&<>, with the quantity SEeo, being replaced by $Em giving E Q (2 )= (E C-E A 3+8EH + $Elu/2 ( 2 . 1 5 ) E Q (1 )= (E c -E , )=-A/3+8EH - SEM I/2- (6E m) 2/ (2&)+— E Q (3 )= (E t -E 3 )=+2A/ 3 +SE w + (SE„, f/ ( 2 A o ) + — 27 The s p l i t t i n g o f the V^j^ valence band f o r the b i a x i a l s t r a i n i n the [ i l l ] plane i s t h e r e f o r e g i v e n by ( E 0 ( 2 ) - E 0 ( D ) = 8 E w + ( 5 E „ / f / ( 2 ^ + — ( 2 . 1 6 ) BIAXIAL STRAIN IN THE f l l O ] PLANE For the case of a b i a x i a l s t r a i n i n the [OOl] and [ i l l ] p l a n e s the c h o i c e of the q u a n t i z a t i o n axes p e r p e n d i c u l a r to the s t r a i n plane l e d t o a simple form f o r the H a m i l t o n i a n and the wave f u n c t i o n . T h i s c h o i c e p r e s e r v e s m as a good quantum number and hence l e a d s t o w e l l d e f i n e d s e l e c t i o n r u l e s and an i d e n t i f i c a t i o n of s t a t e s i n terms of t h e i r quantum numbers. However, when s t r e s s i s a p p l i e d t o a plane of lower symmetry, such as the [ l l O ] plane, the s i t u a t i o n i s c o n s i d e r -a b l y more c o m p l i c a t e d w i t h mj no l o n g e r b e i n g a good quantum number and cannot be used t o d e s i g n a t e the v a r i o u s v a l e n c e bands. However, Hensel and Feher ( i 9 6 0 ) show t h a t i f the v a l e n c e band s p l i t t i n g e n e r g i e s per u n i t s t r e s s are the same ( SE„01 = SE,„) i n the [lOO] and [ i l l ] p l a n e s then the H a m i l t o n i a n becomes d i a g o n a l f o r any o r i e n t a t i o n and we have the c o n d i t i o n f o r " i s o t r o p i c q u a n t i z a t i o n " . The s t r a i n components i n terms of the c r y s t a l l o g r a p h i c c o o r d i n a t e system f o r a b i a x i a l s t r a i n , T, a p p l i e d i n the [HO] p i ane are £,= Bl= ( l - A ) T / 2 €3= T ( 2 . 1 7 ) 6^=65-= 0 e*= -(l+A)T / 2 The v a l u e s f o r the energy d i f f e r e n c e s between the con-d u c t i o n and v a l e n c e bands f o r a b i a x i a l s t r e s s i n he [ l i o ] 2 8 plane have been d e r i v e d by P o l l a k e t a l . ( 1 9 6 8 ) and w i l l not be quoted here s i n c e experiments r e l a t i n g t o the f l l O ] plane were not performed i n t h i s study. A summary of the r e s u l t s f o r the t h r e e main c r y s t a l l o g r a p h i c p l a n e s of GaSb i s g i v e n i n Table 2 . 1 . Row 1 of Table 2 . 1 g i v e the components of the s t r a i n t e n s o r , l^ij], i n terms of the c r y s t a l c o o r d i n a t e system (x,y,z) f o r a uniform and i s o t r o p i c b i a x i a l s t r a i n a p p l i e d i n the plane of the sample. The s t r a i n components a s s o c i a t e d w i t h the sample c o o r d i n a t e system (x^y'jz') are equal to ^ x x = £ a y = T and £«=-AT where the parameter, g i v e n i n row 2 i s determined from the c o n d i t i o n t h a t the normal component o f the s t r e s s t e n s o r , Oi\, e q u a l s z e r o . The v a l u e s f o r c^-( 0 ° K ) used i n c a l c u l a t i n g A were taken from Table 2.4, s e c t i o n 2 . 4 - The s t r e s s induced valence band s p l i t t i n g , 2 A , i s t a b u l a t e d i n row 3, and the s h i f t , S E ^ , i n the fundamental energy gap, E Q, due t o the h y d r o s t a t i c component of the s t r a i n i s g i v e n i n row 4 « The v a r i a t i o n of the mean width o f the f o r b i d d e n energy gap, SEH= (Dj ~Dj )(AV/V), i s i n v a r i a n t f o r any d i r e c t i o n of s t r a i n i n the c r y s t a l f o r a g i v e n d i l a t a -t i o n , (AV/V). The l a s t row of Table 2 . 1 g i v e s the form of the s t r e s s t e n s o r , <f^ , i n terms of the sample c o o r d i n a t e s (x^y^z') f o r a g i v e n s t r a i n , T, which p e r m i t s the s t r e s s e s developed i n the plane of the sample d u r i n g c o o l i n g t o be e s t i m a t e d . I t can a l s o be observed from the Table 2 . 1 (row 3 ) t h a t i t i s s u f f i c i e n t t o analyze the cases of s t r a i n f o r the sample Table 2 . 1 The s t r a i n , e;y,valence band s p l i t t i n g , 2 A , mean s h i f t of the valence bands, SE,/, and the stress, atf, f o r GaSb b i a x i a l l y strained i n the [001] , f i l l ] and fllOj c r y s t a l planes Sample ^Normal Parameter [001] r m ] [no] or ( 2 ) o„ \a ^ ( 4 . 2 U K ) ( 3 ) 2 A = ( E 0 ( 2 ) - E 0 ( l ) ) ( 4 ) (dE 0/dP )=SE„ ( 5 ) T £ j = -AT £ = 6 f = 6t= o 2 c r t = c„ 0 . 9 1 4 8 E M , + i ( S E e J Z / A . 6E M (= 4(l+»D uT/3 ( D ; - D y ) ( 2 - " \ ) T CT.'O 0 o <r,! o _o o o^  [c„ +c a-Ac^)T e,= e,= e3= ( 2 - A ) T / 3 6V= T/3 + 2 c / x - 2 c y y +2c«+4ci / v ; = 0 . 4 8 5 S E ^ t S E , , , ) 2 / ^ , SE,„= 4(l+A)D^T/3 ( D S - D d v ) ( 2 - A ) T Vo 0 ] 0 <o Lo 0 0 J 6cfy(c,, + 2 c „ . ) > T c,; + 2 c / z + 4 c # i V f ( = € 2 = ( l - A ) T / 2 e 3= T € ^ £ r = 0 6t= -(l+A)T/2 (c „ + 3 c , ^ 2 c„v) (c„ + c« + 2 c v vJ = 0 . 6 1 8 i [ ( S E M , ) 2 + 3 ( S E , „ ) 2 > SE W= 4 ( l + A ) D M T / 3 SE„= 4(1+A )D,:T/3 ( D r f c - D ^ ) ( 2 - A ) T tfl 0 0 0 fli'o -0 0 0 ; ^ ' = 4 c ^ ( c „ + 2 c , J t T c„ +c a + 2 c ^ <"2c/^ (crf-c/t) +c,-,lT c, i 1 (a) Lin et a l . ( 1 9 7 2 ) (b) Osipov ( 1 9 6 7 ) 30 normals i n the <£)01) and <CL11^ d i r e c t i o n s o n l y , i n order t o determine the v a l u e s of the deformation c o n s t a n t s D^ and . SIGN OF THE DEFORMATION POTENTIALS The s i g n s of the deformation p o t e n t i a l s are determined by the " d i r e c t i o n of s h i f t " o f the va l e n c e bands, r e l a t i v e t o the c o n d u c t i o n band, under an a p p l i e d s t r e s s . I f , under a t e n s i l e ( p o s i t i v e ) s t r e s s , the v 2 ( J = 3/2 mj = ±3/2) v a l e n c e band moves "up"" ( i . e . towards the condu c t i o n band) and the v l ( J = 3/2 I » J = l / 2 ) v a l e n c e band moves "down" ( i . e . away from the c o n d u c t i o n band), then the va l e n c e band deformation p o t e n t i a l s D u and D^ w i l l be p o s i t i v e . I f however, the v2 v a l e n c e band moves "down" wh i l e the v l v a l e n c e band moves "up" (under a t e n s i l e s t r a i n ) , then the deformation p o t e n t i a l s w i l l be n e g a t i v e . Note t h a t i f under a t e n s i l e s t r a i n the v l band moves "up", then under a compressive (negative) s t r a i n the v l band w i l l move "down" w i t h a s i m i l a r behaviour f o r the v2 v a l e n c e band. S i m i l a r l y , i f S E H i s p o s i t i v e f o r a t e n s i l e s t r a i n then t h e q u a n t i t y (D^ - D j ) i s p o s i t i v e , and i f SE^ i s n e g a t i v e then (Dj - T)j) i s n e g a t i v e . There are s e v e r a l methods of d e t e r m i n i n g the d i r e c t i o n o f s h i f t of the v a l e n c e bands under an a p p l i e d s t r e s s . One method of i d e n t i f y i n g the bands i s by o b s e r v i n g the s t r e s s The terms "up" and "down" d e s c r i b e the d i r e c t i o n of motion o f the s t r a i n s p l i t v a l e n c e bands, r e l a t i v e t o the " c e n t r e of g r a v i t y " of the bands (see f o o t n o t e page 16) and the con-d u c t i o n band. With r e f e r e n c e t o F i g u r e 2 . 3 , a valence band moves "up", w i t h r e s p e c t t o i t s " c e n t r e of g r a v i t y " , i f i t moves towards the co n d u c t i o n band. A v a l e n c e band moves "down", w i t h r e s p e c t t o i t s " c e n t r e of g r a v i t y " , i f i t moves away from the c o n d u c t i o n band. 31 induced i n t e r a c t i o n between the v l ( J = 3/2 m = ± l / 2 ) and th e v 3 ( J = 1/2 m T = ± l / 2 ) v a l e n c e bands. I t can be noted from Equations ( 2.8) and ( 2 . 1 5 ) t h a t the v l v a l e n c e band has a q u a d r a t i c dependence on the s t r e s s whereas the v 2 band does not. I f t h i s q u a d r a t i c dependence can be observed, then the v a l e n c e bands can be i d e n t i f i e d . However, i n the work r e p o r t e d here, the s t r e s s e s were not l a r g e enough t o observe a q u a d r a t i c dependence of the energy on s t r e s s . " A second method f o r i d e n t i f y i n g the v a l e n c e bands i s by means of the s e l e c t i o n r u l e s f o r p o l a r i z e d l i g h t a l r e a d y d i s c u s s e d i n s e c t i o n 2 . 1 . 3 . The t r a n s i t i o n from the v l v a l e n c e band t o the c o n d u c t i o n band occurs f o r both Tfand <f p o l a r i z a t i o n ( l i g h t p o l a r i z e d p a r a l l e l and p e r p e n d i c u l a r t o u n i a x i a l s t r e s s d i r e c t i o n ) f o r u n i a x i a l s t r e s s a l o n g e i t h e t h e ^POl)' or ^ l l l ) d i r e c t i o n s , whereas the t r a n s i t i o n from t h e v 2 v a l e n c e band t o the c o n d u c t i o n band o c c u r s o n l y f o r <f p o l a r i z a t i o n ( p e r p e n d i c u l a r t o the s t r e s s d i r e c t i o n ) . T h i s second method was used here t o i d e n t i f y the v a l e n c e band and thereby determine the s i g n s of the deformation p o t e n t i a l s -xPollak and Aggarwal ( 1 9 7 1 ) observed a q u a d r a t i c dependence f o r v l but o n l y f o r u n i a x i a l s t r e s s i n excess of 2x10-9 dynes cm~2 . 2 . 2 o l REVIEW OF EXCITON THEORY When an e l e c t r o n , i n a c r y s t a l , i s e x c i t e d from the v a l e n c e band t o the c o n d u c t i o n band i t l e a v e s a p o s i t i v e l y charged h o l e i n the v a l e n c e band which i s f r e e t o move through the c r y s t a l . The e l e c t r o n and hole w i l l a t t r a c t each other, through t h e i r coulomb i n t e r a c t i o n , and can form a s e r i e s o f bound s t a t e s , l i k e a hydrogen atom. In t h i s case they w i l l move, through the c r y s t a l , as a bound e l e c t r o n -h o l e p a i r c a l l e d an e x c i t o n (or f r e e - e x c i t o n ) . I f the c r y s t a l c o n t a i n s i m p e r f e c t i o n s , ( i m p u r i t i e s o r d i s l o c a t i o n s ) the e x c i t o n may be l o c a l i z e d near the imper-f e c t i o n i n the c r y s t a l and we then have what i s c a l l e d a bound (or tr a p p e d ) e x c i t o n . A f u r t h e r d i s c u s s i o n of e x c i t o n -i m p e r f e c t i o n complexes i s g i v e n i n s e c t i o n 2 . 2 . 3 . The t h e o r y of e x c i t o n s has been e x t e n s i v e l y surveyed by Dexter e t a l . (1965) and Knox (1963) and w i l l o n l y be summarized here. For the treatment of e x c i t o n s i n semiconductors, the e f f e c t i v e mass approximation i s most a p p l i c a b l e (Dresselhaus ( 1 9 5 6 ) ) . The e f f e c t i v e mass t h e o r y f o r e x c i t o n s was f i r s t f o r m u l a t e d by Wannier (1937) f o r a simple band model. He p i c t u r e d the e x c i t o n as an e l e c t r o n i n a c o n d u c t i o n band b e i n g bound t o a h o l e i n a v a l e n c e band. The v a l i d i t y o f the e f f e c t i v e mass approximation f o r e x c i t o n s depends on h a v i n g the i n t e r a c t i o n p o t e n t i a l s l o w l y v a r y i n g over the dimensions of a u n i t c e l l . T h i s i s g e n e r a l l y a p p l i c a b l e f o r semiconductors s i n c e the e l e c t r o n and hole 33 have a s e p a r a t i o n much l a r g e r than the l a t t i c e s p a c i n g o f th e c r y s t a l . In the e f f e c t i v e mass approximation, the H a m i l t o n i a n f o r t he case o f simple (non-degenerate) energy bands, i s 2me 2m, \ Kjre - rhj <2-19) where me and m 4 are the e l e c t r o n and h o l e e f f e c t i v e masses, r e and rK are t h e i r r e s p e c t i v e p o s i t i o n coordinates^ and -K ds the d i e l e c t r i c c o n s t a n t o f the c r y s t a l . Making t h e c o o r d i n a t e t r a n s f o r m a t i o n , R = (m er e+ m, r\ )/(m E4- mA ) and r = r c - r^ (Dresselhaus ( 1 9 5 6 ) ) the e x c i t o n e i g e n f u n c t i o n s can be w r i t t e n i n the form ¥ = e ' * % ? ) (2.20) where K i s the wave v e c t o r f o r the e x c i t o n and 'f'(r) s a t i s f i e s t h e e q u a t i o n *L V7"- £l + tL K l ] V(r) = E f (?) (22 21) 2JK Kv If where jj. ~ (mem^ )/(me+m^ ) i s t h e reduced e x c i t o n mass. The s o l u t i o n s o f equation (2.21) are s i m i l a r t o those o f a Hydrogen atom w i t h the coulomb i n t e r a c t i o n b e i n g reduced from -e2/? t o -e2/Kr„ From t h i s we can conclude t h a t f o r each v a l u e o f the wave v e c t o r , K, t h e r e e x i s t s a s e t of bound s t a t e s at e n e r g i e s g i v e n by E (K) = E 0 - /x(ez/K f -f h Z K* / 2 h r n r ~ 2M* (2.22) where M = me+m^ i s t h e e f f e c t i v e e x c i t o n mass and E,, i s the energy r e q u i r e d t o c r e a t e one unbound (n=o°) electron_-hole p a i r of zero k i n e t i c energy (K=0). 34 The e x c i t o n b i n d i n g energy i s g i v e n by (n=l) E X = = (M/m, ) R 0 ( 2 . 2 3 ) where R 0= 1 3 o 6 eV and m0 i s the f r e e e l e c t r o n mass. For d i r e c t t r a n s i t i o n s , e l e c t r o n - h o l e p a i r s can o n l y be c r e a t e d i f t h e e x c i t o n wave v e c t o r , K, i s approximately e q u a l to z e r o . The e n e r g i e s o f t h e e x c i t o n s t a t e s are then g i v e n by E(K=0) = E 0 - E*_ ( n = l , 2 , 3 — ) ( 2 . 2 4 ) n* The r a d i u s o f the e x c i t o n ground s t a t e o r b i t , i s g i v e n by a x = K^%£.) a ° ( 2 . 2 5 ) where a„ = 0 . 5 3 A" i s the Bohr r a d i u s o f the ground s t a t e o f t h e Hydrogen atom. The q u a n t i t y K„is the s t a t i c d i e l e c t r i c constant""", ( = 1 5 . 7 , W i l l a r d s o n & Beer V o l . I l l ( 1 9 6 7 ) page 1 4 ) a n d y U . i s t h e reduced e x c i t o n e f f e c t i v e mass as p r e v i o u s l y d e f i n e d . The allowed e x c i t o n t r a n s i t i o n s appear as sharp l i n e s i n a H y d r o g e n - l i k e spectrum. However, s i n c e t h e i r i n t e n s i t y f a l l s o f f as l / r i 3 , u s u a l l y o n l y the f i r s t l i n e i s observed o n the edge o f the fundamental a b s o r p t i o n . I n the energy band p i c t u r e , the e x c i t o n l e v e l o c c u r s a t an energy E x (equation 2 . 2 3 ) below the c o n d u c t i o n band. The c r e a t i o n o f an e x c i t o n t h e n i n v o l v e s the e x c i t a t i o n o f an e l e c t r o n from the v a l e n c e band to the e x c i t o n l e v e l . -* Knox ( I 9 6 3 ) shows t h a t f o r an e x c i t o n r a d i i g r e a t e r than 35a 0 , the s t a t i c d i e l e c t r i c c o n s t a n t K„is a p p r o p r i a t e . 35 2.2.2 EFFECT OF STRAIN ON EXCITON ABSORPTION When an a n i s o t r o p i c s t r a i n i s a p p l i e d t o a c r y s t a l the e x c i t o n a b s o r p t i o n peak i s observed t o s p l i t i n t o two peaks. The appearance of the two peaks i s due t o e l e c t r o n i c t r a n s i -t i o n s , from the two s t r a i n s p l i t v a l e n ce bands v l and v2 to the e x c i t o n l e v e l . The s p l i t t i n g and s h i f t i n g of the e x c i t o n w i t h s t r a i n can be used t o o b t a i n i n f o r m a t i o n about the de-fo r m a t i o n p o t e n t i a l s of the band edges. K l e i n e r and Roth (1959) i n t h e i r a n a l y s i s of the s t r a i n s p l i t e x c i t o n l e v e l , assumed t h a t the e x c i t o n b i n d i n g energy was independent of the magnitude and d i r e c t i o n of the s t r a i n and was equal to the b i n d i n g energy i n the absence of s t r a i n . They argued t h a t t h i s was reason-a b l e , s i n c e the b i n d i n g e n e r g i e s of the e x c i t o n s , due t o t r a n s i -t i o n s from the two s t r a i n s p l i t v a l e n c e bands, i s determined l a r g e l y by the s m a l l e l e c t r o n mass. They then used the p o s i t i o n s of the two e x c i t o n peaks t o determine the s p l i t t i n g 2 A (see Table 2.1) of the valence band edge and the mean s h i f t E-E„ of the band gap. Osipov (1967) has co n s i d e r e d the e f f e c t of s t r a i n on the e x c i t o n b i n d i n g energy. H i s r e s u l t s w i l l be reviewed here and a p p l i e d t o GaSb. Osipov c a l c u l a t e d the ground s t a t e energy l e v e l s , of the d i r e c t e x c i t o n s , i n s t r a i n e d Germanium i n the e f f e c t i v e mass approximation. He a p p l i e s the t h e o r y , developed by Kohn & L u t t i n g e r (1955), K i t t e l & Mitchell(1955) and Keyes (I96I), c o n c e r n i n g the shallow i m p u r i t y s t a t e s i n the Germanium and S i l i c o n l a t t i c e . Osipov argues t h a t f o r a weakly bound d i r e c t e x c i t o n i n Germanium, co n s i d e r e d as a very shallow i m p u r i t y c e n t r e , the e f f e c t i v e mass method should be more a p p l i c a b l e , than i n the shallow a c c e p t o r case, s i n c e the mean e x c i t o n r a d i u s i s approximately f o u r times (three times f o r GaSb) the mean Bohr r a d i u s of the a c c e p t o r ground s t a t e . He f i r s t c a l c u l a t e s the ground s t a t e b i n d i n g energy of e x c i t o n s b e l o n g i n g t o each e l l i p s o i d of the s t r a i n e d v a l e n c e band s e p e r a t e l y ( i . e . s t r a i n T-*«>)" and then a p p l i e s the t h e o r y of P r i c e (l°6l) t o determine the b i n d i n g energy f o r an i n t e r -mediate s t r a i n T. In the i n t e r m e d i a t e s t r a i n case, where the b i n d i n g energy of the e x c i t o n i s comparable t o the band s p l i t t i n g , e x c i t o n energy, E ( T ) , f o r a s t r a i n T, has the form E (T ) = E ( o o ) + %/i (2.26) A c c o r d i n g t o P r i c e (1961) equation (2.26) i s v a l i d when E(T) - E(e*>) i s s m a l l i n comparison w i t h E(<?°) with, i n the i n t e r m e d i a t e s t r a i n case, the f i r s t o r d e r c o r r e c t i o n term, W,/^ , b e i n g g i v e n by *L = Zo_ (2.27) T 2A Here 2A i s the v a l e n c e band s p l i t t i n g at k=0 (see Table 2.1) and Z, i s a parameter c a l c u l a b l e f o r the i n f i n i t e s t r a i n ground s t a t e i n the e f f e c t i v e mass approximation ( P r i c e ( l 9 6 l ) The c r i t e r i o n of an i n f i n i t e s t r a i n i s the c o n d i t i o n t h a t t h e valence band s p l i t t i n g , 2A, be much g r e a t e r than the b i n d i n g energy of the e x c i t o n groundstate i n the u n s t r a i n e d l a t t i c e (T=0). and H a l l ( I 9 6 2 ) ) . E x p r e s s i o n s f o r the q u a n t i t y Za , i n terms of a t r a n s v e r s e and l o n g i t u d i n a l e x c i t o n r a d i i , a x ( c o ) and a , , ! 0 0 ) , and the i n v e r s e c y c l o t r o n mass c o n s t a n t s B and D have been d e r i v e d f o r a u n i a x i a l s t r a i n i n the d i r e c t i o n <00l)> by P r i c e ( 1 9 6 1 ) and i n the d i r e c t i o n <111> by H a l l ( 1 9 6 2 ) , and are T ||<00l> Z. =P/ + D1/! + 2 a x 3 \a* a,* a i ; .. ( 2 . 2 8 ) T |<111) Z0= 2 B V l + 2 \ + 2DXI2 + 1 \ 3 U l at at) 9 (a* a* a* / In e q u a t i o n ( 2 . 2 6 ) , E(oo) i s the b i n d i n g energy of the e x c i t o n ground s t a t e f o r s t r a i n , T-+oo} and i s c a l c u l a t e d u s i n g e q u a t i o n 2 . 2 3 i n the form E ( o o ) = (Mo°))lR. ( 2 . 2 9 ) \ m 0 / «Jn* where R,,^  1 3 . 6 eV, mQ i s the f r e e e l e c t r o n mass and i s the s t a t i c d i e l e c t r i c c o n s t a n t , as p r e v i o u s l y d i s c u s s e d . The q u a n t i t y jUx(oo) ±s the averaged reduced e f f e c t i v e masses of e x c i t o n s f o r both the v l and v 2 bands, and i s giv e n by the r e l a t i o n (Osipov ( 1 9 6 7 ) ) M^t mX)( *o) m^(<-) ( 2 . 3 0 ) The q u a n t i t i e s mKII(°°) and mxx(°°) are the l o n g i t u d i n a l and t r a n s v e r s e e f f e c t i v e e x c i t o n masses, r e s p e c t i v e l y , d e f i n e d by 1 + 1 mxl,(">) me m,„(°°) 1 = _1- 1 me ( 2 . 3 1 ) where me i s the i s o t r o p i c e f f e c t i v e e l e c t r o n mass f o r the co n d u c t i o n band at k=0 and i s equal t o 0 . 0 4 7 mD ( B r a u n s t e i n e t a l . ( 1 9 6 2 ) ) . 38 The q u a n t i t i e s mAX(o°) and m,,,,^ ) are the t r a n s v e r s e and l o n g i t u d i n a l e f f e c t i v e masses f o r the v a l e n c e bands i n the l i m i t T-^oo. These q u a n t i t i e s can be c a l c u l a t e d from the i n v e r s e c y c l o t r o n masses A,B,C and D f o r the v a l e n c e bands v l and v 2, as i s i n d i c a t e d i n rows 1 and 2 of Table 2 . 2 (In the c a l c u l a t i o n s of Table 2 . 2 , when two q u a n t i t i e s have been e i t h e r added or s u b t r a c t e d , the e r r o r quoted i s the most probable e r r o r . Topping (1955 ) ) • The i n v e r s e c y c l o t r o n mass parameters f o r GaSb, have been taken from S t r a d l i n g ( 1 9 6 6 ) and are, A = - 1 1 . 0 ± 0 . 6 , B = - 6 . 0 ± 1 . 5 and |C| = 11 ±4 i n u n i t s of tf/2m, (= 3 - 7 8 eV A* ). Using the r e l a t i o n D 2 = C 2+ 3BZ we get |D| = 15 ± 5 . In the c a l c u l a t i o n s f o r Table 2 . 2 , i t i s assumed t h a t D i s n e g a t i v e s i n c e D i s n e g a t i v e f o r Germanium. The l o n g i t u d i n a l and t r a n s v e r s e e f f e c t i v e e x c i t o n masses, l i s t e d i n row 3 and 4 of Table 2 . 2 , have been c a l c u l a t e d u s i n g e q u a t i o n 2.31 and the q u a n t i t i e s g i v e n i n row 1 and 2 of Table 2 . 2 . These v a l u e s are then used t o c a l c u l a t e the averaged reduced e f f e c t i v e e x c i t o n mass, 00), l i s t e d i n row 5 , u s i n g e q u a t i o n 2 . 3 0 . These averaged reduced e f f e c t i v e e x c i t o n masses, f o r the i n f i n i t e s t r a i n case, can now be used i n equation 2 . 2 9 t o c a l c u l a t e the e x c i t o n ground s t a t e E(o°) f o r the i n f i n i t e s t r a i n case. T h i s i s t a b u l a t e d i n row 6 of the Table. In o r d e r t o c a l c u l a t e the q u a n t i t y Z Q, (equation 2 . 2 8 ) , t h e t r a n s v e r s e and l o n g i t u d i n a l e x c i t o n r a d i i have to be de-termined. These q u a n t i t i e s can be estimated by u s i n g the t r a n v e r s e and l o n g i t u d i n a l e x c i t o n e f f e c t i v e masses and equ a t i o n 2 . 2 5 . The q u a n t i t i e s so d e r i v e d are t a b u l a t e d i n 3 9 Table 2 . 2 Parameters used i n c a l c u l a t i n g the e x c i t o n b i n d i n g e n e r g i e s f o r the d i r e c t e l e c t r o n i c t r a n s i t i o n s i n GaSb which i s u n i -a x i a l l y s t r a i n e d i n the " ( 0 0 1 ^ and <^lli) d i r e c t i o n s . ROW PARAMETER v l BAND (J = %m=±/z ) v2 BAND (J= =%>m=±% ) T | | < 0 0 1 > T | | < 1 1 1 > T | | < 0 0 1 > T | | < 1 1 1 > 1 m„ A + B = - 1 7 . 0 ± 1 . 6 A + D/VT - 1 9 . 7 ± 3 . o A - B = - 5 . 0 ± 1 . 6 A - D/VJ = - 2 . 3 ± 3 . 0 2 m0 A - B/2 = - 8 o 0 ± 1 . 0 A - D/2Y3" = - 6 . 6 ± 1 . 5 A + B/2 = - 1 4 . 0 ± 1 . 0 A + D/2YJ = 1 5 . 4 ± 1 . 5 3 m^(eo) m„ 0 . 0 3 4 ± 0 . 0 0 2 0 . 0 3 6 ± 0 . 0 0 2 5 0 . 0 2 8 + 0 „ 0 0 1 0 . 0 2 7 + 0 . 0 0 1 5 4 mx/i(e°) n. 0 . 0 2 6 ± 0 . 0 0 1 5 0 . 0 2 4 ± 0 . 0 0 2 0 . 0 3 8 ± 0 . 0 0 3 0 . 0 4 2 ± 0 . 0 0 9 5 / m. 0 . 0 3 1 ± 0 . 0 0 2 0 o 0 3 1 + 0 . 0 0 2 0 . 0 3 1 ± 0 . 0 0 1 0 . 0 3 1 ± 0 . 0 0 1 6 E («») (meV) 1 . 7 1 ± 0 . 1 1 . 7 1 ± 0 . 1 1 . 7 1 ± 0 . 0 5 1 . 7 1 ± 0 . 0 5 7 ( A ) 2 4 5 ± 1 5 2 3 2 ± 1 6 2 9 8 ± 1 0 3 0 8 ± 1 6 8 a l l ( A ) 3 2 0 ± 2 0 3 4 7 ± 2 5 2 1 9 ± 2 0 1 9 8 ± 5 0 9 ( e V ) 2 0 . 6 7 x 1 0 " * ± 0 . 5 0 . 8 5 x 1 0 " 6 ± 0 . 5 0 . 7 1 x ior* ± 0 . 5 0 . 5 8 x 1 0 ' 6 ± 0 . 5 40 Table 2.2, row 7 and 8. The q u a n t i t y Z was then c a l c u l a t e d ' o from e q u a t i o n 2.28 and i s t a b u l a t e d i n row 9 of Table 2.2. For an e s t i m a t i o n of the e x c i t o n b i n d i n g energy, E ( T ) , f o r a f i n i t e s t r a i n , T, the experimental v a l u e s of the s t r a i n s p l i t t i n g i n the d i r e c t e x c i t o n t r a n s i t i o n can be s u b s t i t u t e d , i n t o e q u a t i o n 2.26, as the band s p l i t t i n g 2 A . T h i s has been done f o r the two bands and the v a r i o u s degree of s t r a i n , u s i n g t h e e n e r g i e s of the e x c i t o n s p l i t t i n g g i v e n i n Chapter 4 and the q u a n t i t i e s are t a b u l a t e d i n Table 2.3. Using the e x c i t o n s p l i t t i n g energy i n s t e a d of the a c t u a l v a l e n c e band s p l i t t i n g 2A i n e q u a t i o n 2.26, i n t r o d u c e s o n l y a very s m a l l e r r o r s i n c e , as can be seen from Table 2.3, the e x c i t o n e n e r g i e s f o r the two bands are o n l y s l i g h t l y d i f f e r e n t from each other f o r a g i v e n s t r a i n . Osipov, i n h i s c a l c u l a t i o n s f o r Germanium, found t h a t the dependence of the e x c i t o n b i n d i n g energy, on the s t r a i n , was most a p p r e c i a b l e f o r s m a l l band s p l i t t i n g i n the <C00r> d i r e c -t i o n . T h i s does not appear t o be the case f o r GaSb. For a g i v e n s t r a i n , the e x c i t o n b i n d i n g e n e r g i e s a l l appear to be equal i n view of the l a r g e e r r o r s a s s o c i a t e d w i t h the v a l u e s . T h e r e f o r e , even though the e x c i t o n b i n d i n g e n e r g i e s i n c r e a s e w i t h d e c r e a s i n g s t r a i n , the i n c r e a s e appears t o be approximately t h e same f o r both the v l and v2 bands. T h e r e f o r e , the assump-t i o n t h a t , the v a l e n c e band s p l i t t i n g 2A i s equal t o the e x c i t o n peak s p l i t t i n g , i s a good one. *0sipove uses a v a r i a t i o n a l method to determine the e x c i t o n r a d i i f o r Germanium. The above method, although not as a c c u r a t e , g i v e s v a l u e s of the e x c i t o n r a d i i f o r Germanium which are w i t h i n 20$ of Osipov's v a l u e s f o r Germanium. Table 2 . 3 The e x c i t o n peak energy s p l i t t i n g and the c a l c u l a t e d e x c i t o n b i n d i n g e n e r g i e s f o r the v a r i o u s s t r a i n s , T , a p p l i e d i n the ^001) and <(lll) d i r e c t i o n s . STRAIN T EXCITON PEAK ENERGY SPLITTING EXCITON BINDING ENERGY v l VALENCE BAND v 2 VALENCE BAND T ||<001> (meV) T||<111> (meV) T || <oor> (meV) T||<111> (meV ) T ||<001> (meV) T||<lll> (meV) - 4 . 1 x 10"* - 4 . 1 ± 0 . 1 5 - 3 . 8 ± 0 . 1 5 1 . 8 7 ± 0 . 1 5 1 . 9 3 ± 0 . 1 7 1 , 8 8 ± 0 . 1 6 1 . 8 6 ± 0 . 1 8 - 1 . 2 x 10"* - 1 . 7 ± 0 . 6 2 . 2 ± 0 . 4 2 . 04 ± 0 . 4 0 + 3 . 1 x 10'* + 2 . 6 + 0 . 3 5 2 . 0 4 ± 0 . 2 5 1 . 9 3 ± 0 . 2 5 + 5 . 5 x 10'* 4 - 7 + 0 . 3 5 4 . 6 ± 0 . 3 5 1 . 8 5 ± 0 . 2 5 1 . 8 9 ± 0 . 2 5 1 . 8 6 + 0 . 2 5 1 . 8 4 ± 0 . 2 5 + 1 0 . 0 x 10"" 8 . 7 + 0 . 7 1 . 8 1 + 0 . 4 1 . 7 8 ± 0 ; 4 1 . 7 1 + o . i 1 . 7 1 ± o . i 1 . 7 1 ± 0 . 0 5 1 . 7 1 ± 0 . 0 5 42 2.2.3 CX. /^63AND ^ABSORPTION PEAKS Johnson (I964) i - n h i s i n v e s t i g a t i o n of the i n t r i n s i c a b s o r p t i o n edge of u n s t r a i n e d GaSb, observed t h r e e sharp peaks, on the steep r i s e of the fundamental a b s o r p t i o n edge, which he l a b e l l e d y 9 and Y (0.8109 eV, O.8058 eV and O.7967 eV, r e s p e c t i v e l y ) . The oi-peak was a s s o c i a t e d w i t h the normal e x c i t o n a b s o r p t i o n , b e i n g the t r a n s i t i o n from the f o u r f o l d de-generate f^v alence band t o t h e e x c i t o n l e v e l at an energy E^ (equation 2.23) below the /"It c o n d u c t i o n band. Johnson i n t e r p r e t e d the /3 and $ peaks as a b s o r p t i o n peaks a s s o c i a t e d w i t h the c r e a t i o n of e x c i t o n - i m p u r i t y complexes. The p o s s i b i l i t y of the f o r m a t i o n of such complexes was f i r s t suggested by Lampert (1958) and e x p e r i -mental evidence f o r t h e i r e x i s t e n c e has been found i n a number o f semiconductors. In some cases i t has been p o s s i b l e t o c o r r e l a t e the observed peaks w i t h the c o n c e n t r a t i o n of known i m p u r i t i e s , however, Johnson was unable t o do t h i s f o r GaSb, s i n c e sharp peaks c o u l d o n l y be observed i n "undoped" samples. Sharma and Rodriguez (I967) t r e a t e d the t h e o r y of e x c i t o n s bound t o i o n i z e d i m p u r i t i e s i n semiconductors u s i n g t h e e f f e c t i v e mass approximation. They s t a t e t h a t the o p t i c a l measurements of Johnson e t a l . (1965) on GaSb c o u l d be i n t e r p r e t e d by assuming the e x i s t e n c e of e x c i t o n - a c c e p t o r 43 complexes,, They f i n d t h a t t he e x c i t o n - a c c e p t o r complex can e x i s t i f (m e/m) < 0.29« T a k i n g m e = 0o047mo and m^  = 0.23tn„, t h e n the r a t i o (m^/m^) = 0„20 f o r GaSb and an e x c i t o n -a c c e p t o r complex i s possible„ T h e i r c a l c u l a t i o n o f t h e b i n d i n g energy o f the complex i s 0„018 eV. Sharma e t a l . conclude t h a t t he ft peak, l y i n g 0.0082 eV below the edge o f the co n d u c t i o n band o f GaSb, cannot be regarded as b e i n g a s t a t e o f the e x c i t o n - a c c e p t o r complex s i n c e i t s b i n d i n g energy i s l e s s than t h a t o f the n e u t r a l a c c e p t o r . They say i t c o u l d be a s s o c i a t e d w i t h an i m p u r i t y s t a t e o t h e r t h a n t h e one r e s p o n s i b l e f o r the e x c i t o n - a c c e p t o r complex. THE EFFECT OF STRAIN ON ACCEPTOR LEVELS The ground s t a t e o f sha l l o w a c c e p t o r s i n GaSb has the f~Jv symmetry o f t h e v a l e n c e band edge and i s thu s 4-fold degenerate. The a p p l i c a t i o n o f a u n i a x i a l o r b i a x i a l s t r e s s l e a d s t o the s p l i t t i n g o f these a c c e p t o r s t a t e s i n t o two (Kramers) d o u b l e t s , i n a s i m i l a r manner t o the s t r a i n s p l i t t i n g o f the v a l e n c e band d i s c u s s e d i n s e c t i o n 2.1.3 . When t h e s t r e s s i s a p p l i e d a l o n g a <t>0l)> o r <xll)' axes, the s t r e s s s p l i t a c c e p t o r d o u b l e t s can be s p e c i f i e d by the quantum number mj = ±3/2 and mj = ±1/2 which i s analogous t o the l a b e l l i n g o f t h e s t r a i n s p l i t Vw v a l e n c e band l e v e l s . The d i p o l e t r a n s i t i o n s e l e c t i o n r u l e s , f o r the s t r a i n s p l i t a c c e p t o r l e v e l , are the same as f o r the s t r a i n s p l i t V%v v a l e n c e band which have been d i s c u s s e d i n s e c t i o n 2.1.3. I f t he % peak i s an e x c i t o n - a c c e p t o r complex, i t would be expected t h a t t h e f peak should s p l i t under the a p p l i c a t i o n of an a n i s o t r o p i c s t r e s s . The b i a x i a l s t r a i n measurements on t h e t peak f o r t h i s and o t h e r work w i l l be d i s c u s s e d i n s e c t i o n 4«4» B a i l e y ( 1 9 7 ° ) « u s i n g a g r o u p - t h e o r e t i c a l a n a l y s i s , has shown t h a t an a b s o r p t i o n peak, due to a bound e x c i t o n , (a f r e e e x c i t o n bound t o e i t h e r a n e u t r a l o r i o n i z e d donor o r a c c e p t o r ) i s expected t o s p l i t i n t o two o r more components and e x h i b i t p o l a r i z a t i o n e f f e c t s f o r a u n i a x i a l s t r e s s . The s p l i t t i n g and p o l a r i z a t i o n dependence can then be used t o i d e n t i f y t h e t y p e s o f complexes i n v o l v e d i n band-edge o p t i c a l t r a n s i t i o n s i n d i r e c t z i n c blende semiconductors. 45 2.3 REVIEW OF ABSORPTION DUE TO DIRECT TRANSITIONS The fundamental a b s o r p t i o n edge, of semiconductors and i n s u l a t o r s , corresponds t o the t h r e s h o l d energy f o r e l e c t r o n t r a n s i t i o n s between the h i g h e s t n e a r l y f i l l e d band (valence band) and the lowest n e a r l y empty band (conduction band). The photon a b s o r p t i o n i s very s m a l l ( i m p u r i t y a b s o r p t i o n , Johnson (I964)) f o r e n e r g i e s much l e s s than t h a t c o r r e s p o n d i n g t o the f o r b i d d e n energy gap and i n c r e a s e s by a f a c t o r of 1^04 or more at photon e n e r g i e s g r e a t e r than the f o r b i d d e n energy gap. The study of the fundamental a b s o r p t i o n p r o v i d e s i n -f o r m a t i o n about the e l e c t r o n s t a t e s near the band extrema. A d i r e c t way of o b s e r v i n g the fundamental edge i s by d e t e r m i n i n g the photon a b s o r p t i o n from o p t i c a l t r a n s m i s s i o n measurements. T h i s i s the method used i n t h i s i n v e s t i g a t i o n . The fundamental a b s o r p t i o n edge can a l s o be s t u d i e d by o b s e r v i n g the e l e c t r o m a g n e t i c r a d i a t i o n r e f l e c t e d from the sample s u r f a c e . However, f o r t h i s method, one must be v e r y c a r e f u l i n the p r e p a r a t i o n of t h e r e f l e c t i n g s u r f a c e s . The a b s o r p t i o n edge a s s o c i a t e d w i t h d i r e c t t r a n s i t i o n s and i t s a s s o c i a t e d e x c i t o n s has been reviewed by McLean (i960) and by E. J . Johnson ( W i l l a r d s o n & Beer V o l . I l l (1967), and t h e r e f o r e o n l y a b r i e f review w i l l be g i v e n here. The i n t e r a c t i o n between e l e c t r o m a g n e t i c r a d i a t i o n and e l e c t r o n s i s d e s c r i b e d by the H a m i l t o n i a n A • V (2.32) 46 where, A i s the v e c t o r p o t e n t i a l c h a r a c t e r i z i n g the r a d i a t i o n , e i s t h e e l e c t r o n i c charge and m0 i s the f r e e e l e c t r o n mass. I f we assume a simple band model, then the c o n d u c t i o n and v a l e n c e band extrema occur at k=0 and have s p h e r i c a l o energy s u r f a c e s g i v e n by E c ( k ) = E0 + h*k* ; E v ( k ) = - h*k* , . 2 m e 2m* \&»66) where m e and iah are i s o t r o p i c e l e c t r o n and h o l e e f f e c t i v e m a s s e s , r e s p e c t i v e l y . Then, f o r the r e g i o n c l o s e to t h e energy gap,E 0, t h e a b s o r p t i o n c o e f f i c i e n t , ^ , f o r allowed d i r e c t t r a n s i t i o n s , i s g i v e n by (McLean ( i 9 6 0 ) - ) nm. J L _ ( 2 z f l \ l T . p c v ( 0 ) | 2 ( h * > - E a f ( W E . ) . where hw i s t h e photon energy, n . i s the index o f r e f r a c t i o n , a i s a u n i t v e c t o r i n the d i r e c t i o n o f photon p o l a r i z a t i o n andJJL= (memA )/(m e + mj, ) i s the reduced mass o f t h e e l e c t r o n -h o l e p a i r . The q u a n t i t y p c y ( 0 ) i s the f i r s t term o f the ex-p a n s i o n o f t h e momentum m a t r i x about k=0 and i s non-zero f o r allowed t r a n s i t i o n s . I n t h e d e r i v a t i o n o f e q u a t i o n 2 . 3 4 i t was assumed t h a t t h e e l e c t r o n , r a i s e d t o the c o n d u c t i o n band, was f r e e t o move throughout t h e c r y s t a l u n a f f e c t e d by the remaining e l e c t r o n s i n t h e v a l e n c e band. However, E l l i o t t ( 1 9 5 7 ) has shown t h a t i f t h e coulomb i n t e r a c t i o n between the e l e c t r o n and t h e h o l e are taken i n t o account, t h e n the a b s o r p t i o n i s enhanced ov e r t h a t p r e d i c t e d by u s i n g t h e simple B l o c h wave f u n c t i o n s and i t s energy dependence i s changed. For allowed t r a n s i t i o n s t a k i n g e l e c t r o n - h o l e i n t e r a c t i o n i n t o account t h e 47 absorption v a r i e s as o<(haO >~ EJ 1(0) e £ sinh z where: z =7T / Ex (0) * (ha> - E e J ( 2 . 3 5 ) and E x ( 0 ) =• /A e4 i s the b i n d i n g energy o f the e x c i t o n o f zero wave v e c t o r i n i t s ground s t a t e . B e s i d e s the conti n u o u s a b s o r p t i o n , due to allowed t r a n s i -t i o n s 5 a b s o r p t i o n can a l s o be produced by t r a n s i t i o n s i n t o t h e bound e x c i t o n s t a t e s . T h i s produces a s e r i e s o f d i s c r e t e a b s o r p t i o n l i n e s at e n e r g i e s E^CO/n*" ( n = l , 2 , 3 — ) below the cont i n u o u s a b s o r p t i o n t h r e s h o l d . These d i s c r e t e a b s o r p t i o n l i n e s a re v e r y sharp and, i f the b i n d i n g energy o f the e x c i t o n can be c a l c u l a t e d , t h e s e a b s o r p t i o n l i n e s can be used t o determine the fundamental energy gap E 0 very p r e c i s e l y . The a b s o r p t i o n i n t o the e x c i t o n s t a t e , n, i s p r o p o r t i o n a l "k° / \ 1 however, s i n c e i t f a l l s o f f as l / r i 3 , g e n e r a l l y o n l y t h e n=l s t a t e i s observed. 48 2 . 4 TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS In order t o c a l c u l a t e the deformation p o t e n t i a l s of GaSb, at 2°K, the e l a s t i c c o n s t a n t s , Cjj , must be known f o r 2°K. The room temperature v a l u e of the e l a s t i c c o n s t a n t s have been d e t e r -mined by McSkimin e t a l . ( 1 9 5 6 ) and McSkimin e t a l . (1968 ) . The temperature dependence of the e l a s t i c c o n s t a n t s between, 3 0 0°K and 4 . 2°K, has r e c e n t l y been determined by L i n e t a l . ( 1 9 7 2 ) . P r i o r t o the p u b l i c a t i o n of L i n ' s r e s u l t s , an estimate o f t h e e l a s t i c c o n s t a n t s at 0°K was done u s i n g the t h e o r y developed by Keyes ( 1 9 6 2 ) . He has shown, i n terms of reduced e l a s t i c c o n s t a n t s , t h a t t h e second order e l a s t i c c o n s t a n t s of t h e I I I - V compounds f a l l i n t o a w e l l d e f i n e d p a t t e r n . Keyes argues, t h a t s i n c e the e l a s t i c p r o p e r t i e s of a m a t e r i a l are b a s i c a l l y determined by the i n t e r a c t i o n s of e l e c t r o n s , he d e f i n e s an e l a s t i c c o n s t a n t c g i v e n by Here q i s the e l e c t r o n i c charge i n statcoulombs and b i s the d i s t a n c e i n c e n t i m e t e r s between n e a r e s t n e i g h b o r s . For a z i n c blende c r y s t a l , the d i s t a n c e between n e a r e s t n e i g h b o r s i s r e l a t e d t o the l a t t i c e c o n s t a n t , a, by the r e l a t i o n h = 2a 4 I t i s then found t h a t , t o w i t h i n a few p e r c e n t , the d i m e n s i o n l e s s reduced e l a s t i c c o n s t a n t s d e f i n e d by have the same v a l u e s , f o r a g i v e n c. . and a g i v e n temperature 4 9 f o r a l l the I I I - V compounds so f a r measured. (see W i l l a r d s o n & Beer, V o l . 2 ( 1 9 6 6 ) page 1 1 2 ) . I t was t h e r e f o r e assumed t h a t the temperature dependence of the reduced e l a s t i c con-s t a n t s , c - t :, should be approximately the same f o r a l l I I I - V compounds. Table 2 . 4 l i s t s the second order e l a s t i c c o n s t a n t s , c ^ j , the reduced e l a s t i c c o n s t a n t s , c^-, and the q u a n t i t y , c Q , f o r v a r i o u s I I I - V compounds f o r which the temperature dependence i s a v a i l a b l e . The temperature dependence of the reduced e l a s t i c c o n s t a n t s f o r GaAs and InSb g i v e n i n Table 2 . 4 have been p l o t t e d i n F i g u r e 2 . 2 a,b,c. The temperature de-pendence of GaSb, i n d i c a t e d by a d o t t e d l i n e i n F i g u r e 2 . 2 , was estimated by assuming t h a t i t s dependence would be s i m i l a r t o t h a t of GaAs and InSb. The e s t i m a t e d v a l u e s of the e l a s t i c c o n s t a n t s at 0°K and L i n ' s v a l u e s f o r GaSb are t a b u l a t e d i n the l a s t two rows of T a b l e 2 . 4 . The e s t i m a t e d v a l u e s f o r c^^ and c ^ agree q u i t e w e l l w i t h L i n ' s v a l u e s . However, the v a l u e of c does not 12 agree very w e l l . In the c a l c u l a t i o n of the deformation p o t e n t i a l c o n s t a n t s j t h e e l a s t i c c o n s t a n t s determined by L i n e t a l . ( 1 9 7 2 ) w i l l be used. Table 2 . 4 The e l a s t i c c o n s t a n t s c^' and the reduced e l a s t i c c o n s t a n t s c|j f o r GaAs, InSb and GaSb f o r v a r i o u s temperatures c„ 1 C o COMPOUND TEMP. x l 0" Dyne xlO" Dyne xlO" Dyne xlO" Dyne cj °K cm cm cm cm GaAs a 300 1 1 . 8 1 5 . 3 2 5 . 9 4 6 . 4 1 I . 8 4 0 . 8 3 0 0 . 9 2 7 7 7 . 4 1 2 . 2 1 5 . 6 6 5 - 9 9 6 . 4 1 1. 90 0 . 8 8 3 0 . 9 3 4 0 1 2 . 2 6 5 . 7 1 6 . 0 0 6 . 4 1 1 . 9 1 0 . 8 9 1 0 . 9 3 6 I n S b b 273 6 . 4 8 3 . 2 7 3 . 0 9 0 . 3 7 2 1 . 4 7 2 0 . 8 7 9 0 . 8 3 1 0 6 . 7 5 3 . 4 7 3 .16 0 . 3 7 2 1 . 8 1 5 0 . 9 3 3 0 . 8 4 9 GaSb 3°°S 8 . 8 4 2 4 . 0 2 6 4 . 3 2 2 0 . 4 7 4 1 . 8 6 0 . 8 5 0 0 . 9 1 2 2 9 8 d 8 . 8 4 4 . 0 3 4 . 3 2 0 . 4 7 4 1 . 8 6 0 . 8 5 0 0 . 9 1 2 4 . 2 C 9 . 092 4 . 1 4 8 4 . 4 4 0 0 . 4 7 4 1 . 9 2 0 . 8 7 6 0 . 9 3 7 0 e 9-13 4 . 3 5 4 . 3 7 0'.474 1 . 9 3 0 . 9 1 7 0 . 9 2 2 (a) W i l l a r d s o n & Beer Volume I I (1966) page 112 (b) P o t t e r (1956) (c) L i n et a l . (1972) (d) McSkimin et a l . (1956) and McSkimin et a l . (1968) (e) v a l u e s estimated from F i g u r e 2.2 51 (a) + -c D 4— V) c o o o ••— V) _D L U a> o ZJ (L» CC 100 200 Temperature °K 300 Figure 2.4 (a) The temperature dependence of the reduced e l a s t i c constant cj for GaAs, InSb and the estimated dependence for GaSb. 52 + CVJ O 0.95 (b) c o to c o O u •D CD CJ =3 T J Q: 0.90 CO o 0.85 LU 0.80 _ ^ _ InSb ^12 — ' — •*— ~~~~~ * — — • ' — _ (Estimated) / ^ ^ - ^ *•«. Ga As i 1 . 1 1 0 100 200 Temperature °K 300 (c) o c o • 4 — V) o O o V> D LU T 3 a> u •a a> o r 0 0.95 ~~ Ga As ^ 0.90 Ga Sb (Estimated) / 0.85 InSb 0.80 - . 1 , 1 1 1 100 200 Temperature °K 300 F i g u r e 2.4 The temperature dependence of the reduced e l a s t i c c o n s t a n t s (b) c£ and (c) c ^ f o r GaAs, InSb and the e s t i m a t e d dependence f o r GaSb. 53 CHAPTER 3 EXPERIMENTAL DETAILS 3 . 1 INTRODUCTION In t h i s c h a p t e r the experimental t e c h n i q u e s w i l l be pr e s e n t e d and d i s c u s s e d . In order t o e x p e r i m e n t a l l y observe the e f f e c t of s t r a i n on the e x c i t o n spectrum o f GaSb, the f o l l o w i n g problems had t o be overcome. F i r s t , i n o r d e r t o observe the e x c i t o n s p e c t r a , samples 17 w i t h a c a r r i e r c o n c e n t r a t i o n of l e s s than 1 x 1 0 ' cm J had to be o b t a i n e d . The samples are d i s c u s s e d i n s e c t i o n 3 - 5 . 1 . The o r i e n t a t i o n and p r o d u c t i o n of v e r y t h i n (2 t o 5J/m) p a r a l l e l f a c e d samples i s d i s c u s s e d i n s e c t i o n 3 . 5 * 2 , 3«5«4 and 3 . 5 . 5 . A d i s c u s s i o n o f the g l a s s s u b s t r a t e s used and the c a l c u l a t i o n s of t h e i r thermal e x p a n s i v i t y i s presented i n s e c t i o n 3 « 5 « 3 . Secondly, s i n c e the e x c i t o n spectrum c o u l d o n l y be observed at v e r y low temperatures, the samples had to be c o o l e d . A Helium immersion dewar, d i s c u s s e d i n s e c t i o n 3«4«1, was designed and b u i l t t o f a c i l i t a t e the immersion of the sample d i r e c t l y i n a Helium bath. A u n i a x i a l s t r e s s dewar, d i s c u s s e d i n s e c t i o n 3«4«2, was a l s o used t o permit a d e t e r --\ m i n a t i o n of t h e o p t i c a l s e l e c t i o n r u l e s g o v e r n i n g the e l e c t r o n i c t r a n s i t i o n s i n v o l v e d i n the s t r a i n s p l i t e x c i t o n d o u b l e t . The i d e n t i f i c a t i o n of the t r a n s i t i o n s i n v o l v e d p e r m i t s a determin-a t i o n of the s i g n of the deformation p o t e n t i a l s . 54 T h i r d , a s u i t a b l e means of measuring the s p e c t r a l v a r i a t i o n of the photon a b s o r p t i o n had t o be employed. A d i s c u s s i o n of the monochromator used and i t s c a l i b r a t i o n procedures i s g i v e n i n s e c t i o n 3*2, 3.3 and Appendix A. F i n a l l y , a d i s c u s s i o n of the c a l c u l a t i o n of the a b s o r p t i o n c o e f f i c i e n t from the measured t r a n s m i s s i o n i n t e n s i t y i s g i v e n i n s e c t i o n 3.6. 55 3.2 OPTICAL SYSTEM The o p t i c a l system c o n s i s t s o£ a l i g h t source and chopper assembly, a f o r e - d i s p e r s i n g o p t i c a l system, a h i g h r e s o l u t i o n near i n f r a - r e d E b e r t monochromator, an image d i s t o r t i n g system, and a p h a s e - s e n s i t i v e d e t e c t i o n system. A schematic diagram of the o p t i c s i s shown i n F i g u r e 3.1. The f o r e - d i s p e r s i n g system and the monochromator were evacuated and then r e f i l l e d t o atmospheric p r e s s u r e , w i t h dry n i t r o g e n gas, i n order to remove water vapour a b s o r p t i o n . The c r y o s t a t - d e t e c t o r chamber c o u l d be evacuated t o a p r e s s u r e of l e s s than 10"? t o r r . 3.2.1 SOURCE AND CHOPPER ASSEMBLY The c r i t e r i a f o r c h o o s i n g a source are t h a t i t should produce a maximum l i g h t i n t e n s i t y i n the near i n f r a - r e d r e g i o n (1 t o 2 m i c r o n s " ) , and t h a t the s p e c t r a l d i s t r i b u t i o n of the i n t e n s i t y be continuous. The lamp found most s u i t a b l e i s a q u a r t z - i o d i d e S y l v a n i a "Sun-Gun" (type DWY). The lamp, which operates at a b r i g h t n e s s temperature of 3400°K, c o n s i s t s of a h e l i c a l t u n g sten c o i l e n c l o s e d i n a quartz tube c o n t a i n i n g an i o d i n e atmosphere. A minimum of a i r c o o l i n g i s r e q u i r e d f o r continuous o p e r a t i o n . No sharp peaks occur i n i t s emission spectrum. The p h y s i c a l dimensions of the lamp are s m a l l enough t o a l l o w i t t o be mounted v e r y c l o s e to the entrance s l i t s , S^, 1 micron (rim) = 10,000 Angstroms ( I ^ Detector - Cryostate Vacuum System Detector Image Distorting Optics Fore - Dispersing Optics •Chopper Light Source gure 3.1 Schematic diagram of the Ebert monochromator o p t i c s i n c l u d i n g the f o r e -d i s p e r s i n g o p t i c s , the image d i s t o r t i n g o p t i c s , and the d e t e c t o r - c r y o s t a t e vacuum s e c t i o n . 57 t h e r e b y m i n i m i z i n g t h e d i s t a n c e the l i g h t must t r a v e l i n a i r . F u r t h e r , s i n c e the lamp a c t s as an extended l i n e source, i t i s p a r t i c u l a r l y s u i t e d f o r use w i t h the l o n g s l i t monochromator. The source was operated on the r e g u l a t e d v o l t a g e l i n e which e l i m i n a t e d i n t e n s i t y f l u c t u a t i o n s . Between the source and the s l i t s , S^, a c i r c u l a r aluminum chopper blade i s mounted. The blade has 29 p r e c i s i o n c u t , e q u a l l y spaced s l o t s l o c a t e d near i t s rim. When d r i v e n at 1 8 0 0 r.p.m. (30 r e v . per s e c ) , t h e l i g h t beam i s chopped at 8 7 0 Hz. T h i s frequency was chosen because i t i s not a harmonic o f 60Hz, and i t i s a h i g h enough frequency f o r con-v e n t i o n a l i n f r a - r e d d e t e c t i o n t e c h n i q u e s t o be used. 3 . 2 . 2 FORE-OPTICS In o r d e r t o e l i m i n a t e o v e r l a p p i n g o r d e r s , which would be produced by t h e monochromator i t s e l f , i f i l l u m i n a t e d by a broad band o f wavelengths, a f o r e - d i s p e r s i n g c a l c i u m f l u o r i d e prism, P, was i n c o r p o r a t e d i n the entrance o p t i c s (Glass, 1 9 6 4 ) . The prism i s a l u m i n i z e d on the back f a c e , and the double passage o f r a d i a t i o n through i t 3 - produces a l i n e a r d i s p e r s o n o f a p p r o x i -mately 0.19 cm. per micron at the monochromator s l i t , S-^ . The p r i s m can be r o t a t e d manually from o u t s i d e the vacuum system t o i l l u m i n a t e the s l i t , S-^ , w i t h the d e s i r e d band o f wavelengths. 3 . 2 . 3 EBERT MONOCHROMATOR The main d i s p e r s i n g element o f the o p t i c a l system i s a h i g h r e s o l u t i o n E b e r t monochromator. In the Ebert arrangement, the entrance and e x i t s l i t s , (S-^ and S 2 r e s p e c t i v e l y o f f i g u r e 3 . 1 ) are s i t u a t e d on e i t h e r s i d e o f the g r a t i n g and e q u i d i s t a n t 5 8 from i t s c e n t r e , the a x i s o f the spectrometer,, T h i s system e x h i b i t s o n l y a s t i g m a t i c a b e r r a t i o n ; and even t h i s d e f e c t can be e l i m i n a t e d i f the s l i t s are made i n the form o f a r c s of a c i r c l e w i t h c e n t r e on the a x i s o f the spectrometer. (For a complete d i s c u s s i o n o f the Ebert system, see Ebert 1 8 8 9 and F a s t i e , 1 9 5 2 .a,b.) The g r a t i n g used i n the monochromator i s a Bausch and Lomb plane r e f l e c t i o n g r a t i n g r u l e d w i t h 3 0 0 l i n e s per mm. and b l a z e d at 1 . 4 / t m i n t h e second o r d e r . The g r a t i n g can be r o t a t e d about a v e r t i c a l a x i s (see Appendix A, f i g . A . l ) i l l u m i n a t i n g the e x i t s l i t w i t h quasi-monochromatic l i g h t o f d i f f e r i n g mean wavelength. I f cL i s the angle between the g r a t i n g normal and the spectrometer a x i s , then f o r the m^n o r d e r d i f f r a c t i o n f o r t h i s p a r t i c u l a r geometry the mean wave-l e n g t h , A , i n c i d e n t on t h e e x i t s l i t , i s g i v e n by mX = 6 . 6 5 0 s i n (°0 ( 3 . 1 ) where, A , i s measured i n microns. For a l l work r e p o r t e d here, the system was operated at a r e s o l v i n g power (A/AA) o f 1 6 , 0 0 0 c o r r e s p o n d i n g t o an energy r e s o l u t i o n , at 0 . 8 8 eV ( 1 . 4 y u m ) , o f 5 x l 0 ~ 5 e y 0 Higher r e s o l u t i o n i s p o s s i b l e but was found unnecessary f o r t h i s work. 3 . 2 . 4 IMAGE DISTORTING SYSTEM The s l i t s o f t h e Ebert monochromator are t h r e e i n c h e s l o n g . Thus, i n o r d e r t o fo c u s the image on the sample and on the d e t e c t o r an image d i s t o r t i o n system of m i r r o r s (M^ r-^-and M^ o f f i g u r e 3 . 1 ) was employed (Cobb, 1 9 6 1 ) . 59 3.2.5 POLARIZATION For the measurements r e p o r t e d here, the d i f f r a c t i o n g r a t i n g was s e t near i t s b l a z e angle. Thus the p o l a r i z a t i o n o f the emerging r a d i a t i o n , due t o the g r a t i n g , was s m a l l . I t was found e x p e r i m e n t a l l y t h a t t h e r e was a s m a l l degree of p o l a r i z a t i o n , of P = O.O9"'5", due t o a l l the o p t i c a l elements i n the spectrometer. For the u n i a x i a l s t r a i n , measurements used t o determine the s e l e c t i o n r u l e s f o r the v a r i o u s v a l e n c e band t o c o n d u c t i o n band t r a n s i t i o n s , a P o l a r o i d p l a s t i c l a m i n a ted Type HR l i n e a r polarizer""""" was i n t r o d u c e d i n t o the spectrometer between the sample and the d e t e c t o r . The p o l a r i z e r was mounted i n a s p e c i a l h o l d e r which p e r m i t t e d i t s r o t a t i o n by 90° from o u t s i d e the vacuum system. T h i s p o l a r i z e r was mounted i n the spectrometer f o r u n i a x i a l measure-ments and was removed f o r a l l o t h e r measurements. P o l l a k e t a l . (1971) observed i n t e r f e r e n c e f r i n g e s due t o the sheet p o l a r i z e r , however, i n t e r f e r e n c e f r i n g e s due t o the p o l a r i z e r were not observed i n t h i s work. The degree of p o l a r i z a t i o n , P, i s d e f i n e d as P = In - Ix where f o r the p r e s e n t work, In and Ij_ are the r a d i a t i o n i n t e n -s i t i e s measured p a r a l l e l and p e r p e n d i c u l a r t o the s l i t d i r e c t i o n r e s p e c t i v e l y . -x--x- The HR p o l a r i z e r used i n t h i s work has an e x p e r i m e n t a l l y determined p o l a r i z a t i o n r a t i o of 12.6 at 1.49 / t m . The p o l a r -i z a t i o n r a t i o (PR) i s d e f i n e d as PR = I | | / l j _ where In and Ij_ are the t r a n s m i t t e d i n t e n s i t i e s through two i d e n t i c a l p o l a r i z e r s f o r t h e i r t r a n s m i s s i o n axes p a r a l l e l and p e r p e n d i c u l a r t o each o t h e r r e s p e c t i v e l y . T h i s p o l a r i z a t i o n r a t i o i s t h e r e f o r e a measure of the e f f i c i e n c y of the p o l a r i z e r and i s d i f f e r e n t from the degree of p o l a r i z a t i o n p r e v i o u s l y d e f i n e d . 60 3.2.6 ELECTRONIC DETECTION SYSTEM ' R a d i a t i o n p a s s i n g through the sample i s f o c u s s e d on a PbS detector-"- "which has a s m a l l s e n s i t i v e a r ea. O p e r a t i n g at room temperature, the d e t e c t o r has a response time of 200 y / s e c , a dark r e s i s t a n c e o f 1 M-O- and a s i g n a l - t o - n o i s e r a t i o o f 600 as quoted by t h e manufacturer. A wire wound 1 M-fl- l o a d r e s i s t o r connected i n s e r i e s w i t h the detector, c o n v e r t s the photo c u r r e n t i n t o a s i g n a l v o l t a g e which i s a m p l i f i e d by a P r i n c e t o n A p p l i e d Research (PAR) low n o i s e p r e - a m p l i f i e r (type CR-4A) (see f i g u r e 3.2). The p r e - a m p l i f i e r output v o l t a g e i s f u r t h e r a m p l i f i e d by a PAR Lock-In A m p l i f i e r (type JB-5) and d e t e c t e d i n synchronism w i t h a r e f e r e n c e s i g n a l produced by a photodiode i l l u m i n a t e d by the chopped r a d i a t i o n from the so u r c e . S i n c e the output D.C. s i g n a l o f the Lock-In A m p l i f i e r i s superimposed on a -8 v o l t D.C. l e v e l , a b u c k i n g v o l t a g e i s i n t r o d u c e d t o balance out t h i s -8 v o l t s . The r e s u l t i n g s i g n a l i s d i s p l a y e d on a s t r i p - c h a r t r e c o r d e r . The e q u i v a l e n t bandwidth of the system i s 0.25 Hz. A schematic diagram of the system i s shown i n f i g u r e 3.2. The l i n e a r i t y o f the i n t e n s i t y response of the PbS d e t e c t o r was t e s t e d by changing the i n t e n s i t y o f the i n c i d e n t r a d i a t i o n by known amounts, u s i n g a s e r i e s o f n e u t r a l d e n s i t y f i l t e r s and measuring the s i z e o f the r e s u l t i n g s i g n a l . The response was found t o be l i n e a r w i t h i n the measurable accuracy. * I n f r a t r o n Type B-J - S A 1 0 0 + 4 5 Volts PbS Detector Load I Meg ft Photo - Diode Reference Channel PAR PAR C R - 4 A J B - 5 Bucking Pre - Amp Lock- In Amplifier Voltage Honeywell Brown Strip-Chart Recorder F i g u r e 3.2 Block diagram of the d e t e c t o r e l e c t r o n i c s . 62 3.3 CALIBRATION The wavelength standards used f o r the c a l i b r a t i o n o f the Eb e r t Monochromators are the wavelengths g i v e n by E. K. P l y l e r e t a l . ( 1952) f o r the 1.4ywm H 2 0 and 1.7 jxm. CH^ a b s o r p t i o n bands. These standards are r e l i a b l e and e a s i l y a c c e s s i b l e f o r c a l i b r a t i o n purposes. The wavelength r e g i o n o f i n t e r e s t f o r GaSb was 1.5 j^-m t o 1 .6 jj.m which does not o v e r l a p e i t h e r the 1 .4 a b s o r p t i o n band o f atmospheric water vapour (1.34y^m to I . 4 8 JZ-m.) or the 1.7yam a b s o r p t i o n band of methane (CH^) ( 1 .62 j/m t o 1 .74y/m). T h i s nonoverlap of r e g i o n s has the advantage t h a t t h e r e are no a b s o r p t i o n peaks superimpared on the GaSb s p e c t r a . However, t h e disadvantage i s t h a t l i n e a r d i s p e r s i o n t e c h n i q u e s f o r c a l i b r a t i o n cannot be a c c u r a t e l y used. T h e r e f o r e , more ac-c u r a t e c a l i b r a t i o n procedures were r e q u i r e d . The method used t o c a l i b r a t e the spectrometer i s d e s c r i b e d i n Appendix A. The g r a t i n g d r i v e was observed t o have a s y s t e m a t i c e r r o r , w i t h a p e r i o d equal t o one r e v o l u t i o n o f the g r a t i n g d r i v e screw, which caused the e r r o r i n the wavelength t o v a r y about 2A° ( 0 . 1 meV) over one complete r e v o l u t i o n of the d r i v e screw. However, i f the s y s t e m a t i c e r r o r was taken i n t o account, then the wavelength e r r o r was l e s s than 1 A ° and was l i m i t e d by the accuracy w i t h which the v a l u e s c o u l d be read o f f the c h a r t s . 63 3.4 CRYOGENICS In o r d e r t o observe w e l l d e f i n e d f r e e - e x c i t o n peaks i n GaSb, without a p p r e c i a b l e phonon broadening, the sample must be maintained at a v e r y low temperature ( l e s s than 15°K). Th i s n e c e s s i t a t e d m o d i f y i n g the e x i s t i n g d e s i g n t o accommodate a l i q u i d h e lium metal dewar i n the apparatus, (see f i g u r e 3.3) For t he b i a x i a l s t r a i n measurements, t h i s m o d i f i c a t i o n allowed t h e sample t o be immersed i n l i q u i d helium which was main-t a i n e d at a temperature of about 2°K. For the u n i a x i a l s t r a i n measurements, the sample was c o o l e d by thermal c o n d u c t i o n through a b r a s s s t r a i n j i g to a l i q u i d helium bath at 4.2°K. 3.4.1 HELIUM IMMERSION DEWAR The helium immersion dswar t a i l was c o n s t r u c t e d out o f pyrex w i t h 1" diameter f l a t pyrex windows on e i t h e r s i d e of t h e t a i l , (see f i g u r e 3.3a) The t a i l was j o i n e d to a 1" copper tube by means o f a g l a s s - t o - m e t a l s e a l . The immersion t a i l was j o i n e d t o the main helium dewar by means of a Wood Ts metal s e a l , (see Appendix B f o r Wood's metal procedure) The sample was h e l d l o o s e l y i n a copper h o l d e r t h a t was f a s t e n e d i n s i d e the immersion t a i l and was a p p r o p r i a t e l y masked, i f n e c e s s a r y , t o prevent s t r a y l i g h t from g e t t i n g p a s t the sample. S i n c e , w i t h the helium immersion t a i l , l i q u i d helium was i n the l i g h t path, the helium had to be maintained below i t s ]\-point t o e l i m i n a t e bubbles due to b o i l i n g . An a u x i l i a r y ' v pumping system was used to m a i n t a i n the helium bath temperatures 64 (a) A . Helium Dewar Wood's Metal Seal Glass-to | ^ Metal Seal Pyrex Sample F i g u r e 3.3 (a) The Helium immersion dewar. (b) The sample h o l d e r f o r the u n i a x i a l s t r e s s dewar. at about 2.0°K. The l i q u i d helium i n the l i g h t beam (path l e n g t h J»0.5") reduced the i n t e n s i t y by about 20% but d i d not i n t r o d u c e any s p e c t r a l peaks i n the r e g i o n investigated,, One disadvantage of t h i s p a r t i c u l a r dewar t a i l a rrange-ment i s t h a t w h i l e the dewar was mounted i n the system, th e i n c i d e n t r a d i a t i o n on the sample c o u l d not be monitored. However, s i n c e o n l y r e l a t i v e a b s o r p t i o n c o e f f i c i e n t s need be measured, t h i s was not c o n s i d e r e d a s e r i o u s disadvantage. 3.4.2 UNIAXIAL STRESS DEWAR One disadvantage of the b i a x i a l s t r a i n measurements i s t h a t the a b s o r p t i o n c o e f f i c i e n t f o r r a d i a t i o n i n c i d e n t normal to the sample s u r f a c e , i s not p o l a r i z a t i o n dependent and t h e r e f o r e the o p t i c a l s e l e c t i o n r u l e s c o u l d not be i n v e s t i g a t e d . However, by superimposing a u n i a x i a l s t r e s s on the sample a l o n g a p a r t i c u l a r c r y s t a l l o g r a p h i c d i r e c t i o n , t h e s t r a i n w i l l no l o n g e r be b i a x i a l l y homogeneous and the a b s o r p t i o n c o e f f i c i e n t i s then dependent on the d i r e c t i o n o f p o l a r i z a t i o n o f the i n c i d e n t r a d i a t i o n . The u n i a x i a l s t r e s s dewar (Figure 3.3b) was o r i g i n a l l y designed and d e s c r i b e d by Parsons (I968) and was not s i g n i f i c a n t l y a l t e r e d from h i s o r i g i n a l d e s i g n . U s i n g t h i s dewar, o n l y compressive s t r e s s e s c o u l d be a p p l i e d to t h e sample (see s e c t i o n 3.5.6 f o r d e t a i l s o f mounting i n s t r e s s dewar). 66 3.5 SAMPLES AND SAMPLE PREPARATION I t has been observed i n t h i s work and p r e v i o u s l y noted by Johnson (1964) t h a t i n GaSb the o b s e r v a t i o n of e x c i t o n a b s o r p t i o n peaks depends v e r y s t r o n g l y on the amount of s h a l l o w i m p u r i t i e s present, and are not observed f o r samples w i t h a room temperature c a r r i e r c o n c e n t r a t i o n g r e a t e r than 1-2 x 10x? cm""3. T h e r e f o r e , i t i s necessary t o o b t a i n samples w i t h i m p u r i t y c o n c e n t r a t i o n l e s s than the above l i m i t . Once s u i t a b l e samples are obtained, a v e r y important p a r t of the o p t i c a l i n v e s t i g a t i o n of b i a x i a l l y s t r a i n e d t h i n semiconducting samples, i s the p r e p a r a t i o n and mounting of the sample, on a s e l e c t e d g l a s s s u b s t r a t e . A s u i t a b l e procedure was developed which p e r m i t t e d the s u c c e s s f u l p r e p a r a -t i o n of t h i n samples. 3.5.1 SAMPLES The samples which e x h i b i t e d the d e s i r e d e x c i t o n spectrum were obt a i n e d from B e l l & Howell (Pasadena, C a l i f o r n i a ) . These samples were undoped s i n g l e c r y s t a l s w i t h a room temperature c a r r i e r c o n c e n t r a t i o n of 8.6 x lO-^cm -^, a m o b i l i t y of 918 cm^ —1 ( v o l t . s e c ) and a r e s i s t i v i t y of 0.094 ^cm. The g r i n d i n g and p o l i s h i n g of these samples i s d e s c r i b e d i n s e c t i o n 3.5*5 6 7 ,3. H. 2 CRYSTAL ORIENTATION Samples w i t h their- normals n o m i n a l l y i n the <^111^ d i r e c t i o n were o b t a i n e d p r e - s l i c e d from B e l l & Howell. The o r i e n t a t i o n s of these samples were checked u s i n g a back r e -f l e c t i o n Laue X-ray machine and were found to be w i t h i n 5 ° of the <(lll^ d i r e c t i o n . The c r y s t a l o r i e n t a t i o n was i d e n t i f i e d u s i n g the diagrams gi v e n by T a y l o r ( 1 9 6 l ) . Samples f o r the <£>0l) d i r e c t i o n were o r i e n t e d and c u t i n our l a b o r a t o r y , from a l a r g e p i e c e of the same i n g o t from which the <yLll/> samples had been c u t . The samples were o r i e n t e d w i t h t h e i r normals i n the <00L) d i r e c t i o n t o w i t h i n ± 1 ° . ,1.5..? GLASS SUBSTRATES AND THEIR THERMAL EXPANSION Si n c e the b i a x i a l s t r a i n i n the sample i s produced by t h e d i f f e r e n t i a l thermal c o n t r a c t i o n between the sample and t h e s u b s t r a t e , t o which i t i s permanently bonded, a v a r i e t y o f s u b s t r a t e s w i t h d i f f e r e n t thermal c o n t r a c t i o n s , over the temperature range of i n t e r e s t , have t o be used. Since the measurements t o be made are t r a n s m i s s i o n measurements, the s u b s t r a t e s chosen have t o be t r a n s p a r e n t and, e x h i b i t no s t r o n g a b s o r p t i o n peaks over the wavelength r e g i o n of i n t e r e s t . A l s o , they must have an i s o t r o p i c thermal expansion so t h a t t h e s t r a i n produced w i l l be u n i f o r m i n the plane of the sample. For b i a x i a l s t r a i n of GaSb, g l a s s s u b s t r a t e s prove to be a good c h o i c e . The s t a t i c b i a x i a l s t r a i n , T, imposed on the sample 68 by c o o l i n g , was assumed to be equal t o the d i f f e r e n c e i n r e l a t i v e thermal e x p a n s i v i t i e s (expansion per u n i t l e n g t h ) of the GaSb, (.AL/L) & a S A , and the g l a s s s u b s t r a t e , (AL/L)&iats . The v a l u e t h a t i s u s u a l l y quoted f o r m a t e r i a l s i s the average c o e f f i c i e n t of l i n e a r expansion, <^(t), d e f i n e d by the r e l a t i o n where L i s the o r i g i n a l l e n g t h and dL i s the change i n l e n g t h over the temperature i n t e r v a l oft. The thermal e x p a n s i v i t y (AL/L) can be determined by i n t e g r a t i n g o((t) over the temper-a t u r e range of i n t e r e s t . The b i a x i a l s t r a i n , T, on the sample The v a l u e s of T f o r v a r i o u s g l a s s s u b s t r a t e s was d e t e r -mined by n u m e r i c a l i n t e g r a t i o n , t a k i n g temperature i n t e r v a l s of 10°K or l e s s . For GaSb, the l i n e a r thermal expansion, g i v e n by Novikova e t a l . (1964) and Sparks et a l . (1967) was The thermal e x p a n s i v i t i e s as a f u n c t i o n of temperature f o r GaSb and the v a r i o u s g l a s s s u b s t r a t e s used are p l o t t e d i n F i g u r e 3.4- A d i s c u s s i o n of these curves f o l l o w . o<(t) = 1 dL L dt (3.2) i s then g i v e n by t h e ^ e x p r e s s i o n used t o c a l c u l a t e the thermal e x p a n s i v i t y (AL/L) & < i ii 3 69 0 50 100 150 200 250 300 Temperature °K F i g u r e 3-4 The thermal e x p a n s i v i t y between 294°K and 0°K f o r GaSb and the v a r i o u s g l a s s s u b s t r a t e s used. B - 2 6 0 : The v a l u e s f o r the thermal expansion, oi, of Zena ty p e B - 2 6 0 soda-lime g l a s s down t o a temperature o f 113°K were obtained from Zena D e u t s c h e s p i e g e l g l a s s A-G. I t was found t h a t t h e s e v a l u e s f o r B - 2 6 0 g l a s s were c l o s e t o the v a l u e s g i v e n by C o r n i n g G l a s s f o r t h e i r soda-lime g l a s s #0080 down t o 77°K. T h e r e f o r e , u s i n g these r e s u l t s , p l u s the r e s u l t s g i v e n by White ( I 9 6 3 ) f o r soda-lime g l a s s over the temperature r e g i o n 0°K t o 30°K, (between 0°K and 30°K t h e change i n the thermal e x p a n s i v i t y i s n e g l i g i b l e b e i n g ?=4 x 1 0 " ) the temperature v a r i a t i o n of B - 2 6 0 g l a s s , given i n F i g u r e 3 . 4 , was ob t a i n e d . # 7 7 4 0: Head and Laquer ( 1 9 5 2 ) measured the thermal expan-s i v i t y , from 3 0 0°K t o 0°K, f o r a b o r o s i l i c a t e g l a s s (pyrex) which they f e l t was C o r n i n g #7740 g l a s s . T h e i r data agreed v e r y w e l l w i t h the data, f o r #7740 down to 77°K, p r o v i d e d by C o r n i n g and t h e r e f o r e Head and L a q u e r f s data was used and e x t r a p o l a t i o n was not ne c e s s a r y . #7913: Thermal e x p a n s i v i t y data f o r Corning s i l i c a g l a s s #7913 ( 9 6 $ s i l i c a ) was not a v a i l a b l e . However, data f o r a v e r y s i m i l a r C o r n i n g g l a s s , #7900 (96% s i l i c a ) was a v a i l a b l e down to 120°K from Corning, and White ( 1 9 6 4 ) measured the l i n e a r thermal expansion of Vycor #7913 g l a s s over the temperature r e g i o n 0°K t o 30°K. A l s o , Head and Laquer ( 1 9 5 2 ) measured the thermal e x p a n s i v i t y of Z e i s s fused quartz ( s i l i c a ) from 360°K t o 0°K and t h e i r data p l u s the above mentioned data f o r #7900 g l a s s was used t o estimate the 71 thermal e x p a n s i v i t y o f #7913 g l a s s quoted i n Table 3 . 1 . The e r r o r of 5®% quoted i n the t a b l e i s the estimated e x t r a p o l a t i o n e r r o r . T h i s e r r o r however, o n l y i n t r o d u c e s an e r r o r of 5% i n t h e s t r a i n on the GaSb sample. # 7 0 5 9 : Thermal e x p a n s i v i t y data f o r #7059 g l a s s was o n l y a v a i l a b l e down t o 77°K, however, by comparing t h i s d a t a w i t h d a t a f o r v a r i o u s g l a s s e s o f s i m i l a r e x p a n s i v i t y , such as Jena 2 9 5 4 ^ g i v e n by Head and Laquer, the e x p a n s i v i t y was e x t r a p o l a t e d down to 0°K. #0211: For t h i s g l a s s , o n l y the room temperature expansion c o e f f i c i e n t o f 6 . 7 x 1 0 _ 0 was a v a i l a b l e (the v a l u e o f 7 . 2 x 1 0 " u s u a l l y quoted f o r #0211 i s an average over 0 to 3 0 0 ° c ) . However, by comparing t h i s g l a s s w i t h o t h e r s h a v i n g a s i m i l a r l i n e a r expansion c o e f f i c i e n t and a known e x p a n s i v i t y (e.g. Jena 1 6 ^ Keesom e t a l . ( 1 9 3 0 ) ) the approximate thermal e x p a n s i v i t y curve shown i n F i g u r e 3 . 4 was o b t a i n e d . T h i s v a l u e was o n l y approximate and was not h e a v i l y weighted i n the c a l c u l a t i o n s . Table 3 . 1 l i s t s the t y p e s of g l a s s e s used, t h e i r modulus o f e l a s t i c i t y , the t hermal e x p a n s i v i t y between 300°K and 0°K and the r e s u l t i n g s t r a i n on the GaSb sample. The e r r o r s quoted i n Table 3 . 1 i n c l u d e the estimated e r r o r due t o e x t r a p o l a t i o n p l u s an estimated e r r o r due to thermal h i s t o r y and the o r i g i n of the g l a s s ( 5 - 1 0 % ) . Table 3 . 1 The thermal e x p a n s i v i t y of the g l a s s s u b s t r a t e s used and the s t r a i n on the GaSb sample. GLASS ' TYPE MEAN LINEAR COEFF. OF EXPANSION ( io-7°c"') THERMAL EXPANSIVITY ( 3 0 0°K - 0°K) ( 10-") STRAIN ON GaSb ( io"") MODULUS OF ELASTICITY ( 1 0 6 p s i ) B - 2 6 0 a Soda-Lime 96 ( 2 0 - 3 0 0°C) 1 5 - 2 i 0 . 5 - 4 . 2 ± 0 . 5 008 ob Soda-Lime ( 0 - 3 0 0°C) 1 0 . 0 0 2 1 1 b 67 ( o°c ) 1 2 . 2 + 0 . 5 - 1 . 2 ± 0 . 5 1 0 . 8 GaSb — 1 1 . 0 7 0 5 9 b Barium-Aluminum B o r o s i l i c a t e 46 ( 0 - 3 0 0°C) 8 . 1 ± 0 . 5 + 2 . 9 ±'0.5 9 . 8 7 7 4 0 b B o r o s i l i c a t e 3 2 . 5 ( 0 - 3 0 0°C) 5 . 5 ± 0 . 5 + 5 - 5 ± 0 . 5 9 . 1 7 9 0 0 b 96% S i l i c a 8 ( 0 - 3 0 0°C) 1 0 . 0 7 9 1 3 b 96% S i l i c a 8 ( 0 - 3 0 0°C) 1 . 0 ± 0 . 5 + 1 0 . 0 ± 0 . 5 9 . 6 (a) Zena Glass, Deutsche S p i e g e l g l a s s AG (b) C o r n i n g Glass, Corning, N.Y. 73 3 . 5 . 4 SAMPLE MOUNTING In o r d e r to prepare t h i n , p a r a l l e l - f a c e d samples, a s p e c i a l sample h o l d e r and mounting te c h n i q u e had t o be d e v i s e d . The sample h o l d e r which produced the d e s i r e d r e s u l t s i s shown i n f i g u r e 3 ° 5 a . T h i s sample h o l d e r f a c i l i t a t e d h a n d l i n g , t h i c k n e s s measurements and l e v e l i n g adjustment d u r i n g the p r e p a r a t i o n stage o f the sample. A f t e r t h e i n i t i a l o r i e n t a t i o n and s e c t i o n i n g o f the sample ( d e s c r i b e d i n s e c t i o n 3 . 5 . 2 ) t h e sample was wax-mounted", t o the b r a s s h o l d e r "A" (see f i g u r e 3 . 5 a ) t o which a p i e c e o f g l a s s had p r e v i o u s l y been a f f i x e d . The sample h o l d e r "A" was t h e n secured by means o f a yoke "B" t o the b r a s s movable p i s t o n "C". The b r a s s s l e e v e "D" c o u l d be a d j u s t e d so t h a t the bottom s u r f a c e of the s l e e v e was even w i t h t h e bottom s u r f a c e o f the sample. By means o f a depth micrometer, ( S t a r r e t t , N0.44OM) and t h r e e l e v e l i n g screws, t h e g l a s s s u r f a c e c o u l d be made p a r a l l e l t o the s l e e v e edge and,provided t h a t t h e sample s u r f a c e next to the g l a s s was p a r a l l e l t o t h e g l a s s , the sample s u r f a c e s c o u l d be made p a r a l l e l t o each o t h e r . U s i n g t h i s procedure the sample The wax used to mount the samples t o the sample h o l d e r was Loc-wax-10, o b t a i n e d from Geoscience I n s t . Corp. T h i s wax i s a s p e c i a l l y compounded b l o c k i n g wax w i t h a low m e l t i n g p o i n t s p e c i a l l y compounded f o r t h i s purpose. 74 A - Sample Holder Full Scale B - Yoke C - Movable Piston D - Brass Sleeve Thickness Monitor Figure 3 . 5 (a) Detailed drawing of the adjustable sample holder, (b) Device used for monitoring the sample thickness during preperation. 75 s u r f a c e s c o u l d be made p a r a l l e l , w i t h a t h i c k n e s s v a r i a t i o n o f o n l y ± 1 ^tm0 A f t e r one s u r f a c e of the sample had been s u i t a b l y prepared ( s e c t i o n 3 . 5 . 5 ) ^ the sample was then epoxied, u s i n g Power Bond""" (#31Z) epoxy, t o the chosen g l a s s s u b s t r a t e . T h i s epoxy has a low v i s c o s i t y b e f o r e s e t t i n g , i s t r a n s p a r e n t i n the wavelength r e g i o n l^um to 2jum and does not c r a c k upon c y c l i n g from room temperature to l i q u i d Helium temperature. The epoxy l a y e r was estimated to be l e s s than 1 jum t h i c k . I t was found t h a t removing the sample from the sample h o l d e r b e f o r e b e i n g epoxied t o the g l a s s s u b s t r a t e caused t h e sample to buckle o r warp (even i f i t was 2 0 0 jum t h i c k ) t h e r e f o r e , the sample was epoxied t o the g l a s s s u b s t r a t e b e f o r e b e i n g removed from the sample h o l d e r . To prevent t h e two p i e c e s of g l a s s from b e i n g epoxied t o g e t h e r , a t h i n l a y e r of grease was p l a c e d on the g l a s s around t h e sample, t h e r e b y f a c i l i t a t i n g i t s l a t e r removal. A weight o f about 4 0 0 gm. was used t o p r e s s the g l a s s a g a i n s t the sample, t h e r e b y p r o d u c i n g a v e r y t h i n l a y e r of epoxy. A f t e r the epoxy had hardened, the sample was removed from the sample h o l d e r , t u r n e d over, and mounted on the sample h o l d e r A as p r e v i o u s l y d e s c r i b e d . The sample was t h e n l e v e l e d and the second s u r f a c e prepared. I n d u s t r i a l F ormulators o f Canada, Vancouver, B. C. 76 3 . 5 . 5 GRINDING AND POLISHING OF SAMPLES Of a l l t h e s t a g e s i n v o l v e d i n the o p t i c a l i n v e s t i g a t i o n o f a m a t e r i a l , probably the most important i s the i n i t i a l s t a ge i n v o l v i n g the p r e p a r a t i o n o f the sample and i t s s u r f a c e s . I f the samples are not p r o p e r l y prepared, the s p e c t r a which one wishes to study might be too s m a l l t o observe o r i t might be compl e t l y masked by o t h e r e f f e c t s i n t r o d u c e d by improper sample p r e p a r a t i o n . For the i n v e s t i g a t i o n of the t r a n s m i t i n g p r o p e r t i e s o f a d i r e c t gap semiconductor at i t s fundamental a b s o r p t i o n edge, t h e r e are t h r e e b a s i c requirements f o r a s u i t a b l e sample. F i r s t , s i n c e t h e a b s o r p t i o n c o e f f i c i e n t o f a d i r e c t m a t e r i a l i s v e r y h i g h ( « 5 0 0 0 cm"-1-), the samples have to be v e r y t h i n (1 t o 5yUm t h i c k ) . Second, the samples have to have plane p a r a l l e l s u r f a c e s so t h a t the t r a n s m i t t e d r a d i a t i o n w i l l not be dependent on the p o s i t i o n i n g o f the sample i n t h e l i g h t beam; and t h i r d l y s i n c e the samples are v e r y t h i n , t h e damaged s u r f a c e l a y e r , l e f t a f t e r p o l i s h i n g , must be as s m a l l as p o s s i b l e so as not to i n v o l v e an a p p r e c i a b l e percentage o f the sample t h i c k n e s s . The sample p r e p a r a t i o n , w i t h t h e s e t h r e e aims i n mind, can c o n v e n t i e n t l y be d i v i d e d i n t o f i v e stages; sample o r i e n t a t i o n and s e c t i o n i n g , ( s e c t i o n 3 . 5 . 2 ) , mounting ( s e c t i o n 3.5.4) g r i n d i n g , rough p o l i s h i n g , and f i n a l p o l i s h i n g . The g r i n d i n g and p o l i s h i n g s t e p s are o u t l i n e d i n T a b l e 3 . 2 i n c l u d i n g the amount of m a t e r i a l g e n e r a l l y removed, t h e approximate r a t e o f removal o f m a t e r i a l and an estimate 77 Table 3 . 2 The grinding compounds and lap used i n the sample preperation inc l u d i n g an estimate of the surface damage produced by each compound COMPOUND3 LAP AMOUNT REMOVED SURFACE DAMAGE (ESTIMATED) APPROX. RATE OF REMOVAL 3 /tm A l 2 ° 3 GLASS > 1 0 0 JJLTO. 5 /tm 1 0 juim/min 3 /tm A 1 2 0 3 NYLON 2 0 jum 3 jlirn 2 ^ un/min 0 . 3 /tm Linde-A NYLON 1 5 ju.ro. 0 . 5 jUm 3 / 4 /(m/min 0 . 0 5 /tm Linde-B POLITEX-D 5 ^am 0 . 5 a^m l / l O ^ tm/min a) A l l compounds were i n a s l u r r y made up of 1 part compound, 1 part Suspendex (Geoscience Inst. Corp. ), and 1 part water by volume. The Suspendex eliminates the s e t t l i n g and caking of the A1 20^ compounds. b) For 2 cm. diameter samples. 78 o f the damage l a y e r produced by the p a r t i c u l a r o p e r a t i o n . (A d i s c u s s i o n of s u r f a c e damage i s g i v e n i n s e c t i o n 3»5»5 E ) . A ; SAMPLE THICKNESS MONITORING In o r d e r to s u c c e s s f u l l y prepare t h i n samples the sample t h i c k n e s s has t o be monitored d u r i n g the v a r i o u s s t a g e s o f p r e p a r a t i o n . The apparatus which was used t o monitor t h e t h i c k n e s s i s shown i n f i g u r e 3 . 5 b . I t c o n s i s t s o f a S t a r r e t t "Last Word" d i a l micrometer mounted s e c u r e l y on a r i g i d b r a s s stand. T h i s micrometer has a f u l l s c a l e d e f l e c t i o n of ^00 ju.m d i v i d e d i n t o 2 jj.m d i v i s i o n s . I t was found t h a t i n o r d e r to a c c u r a t e l y monitor t h e sample t h i c k n e s s d u r i n g p r e p a r a t i o n , the sample f a c e s had t o be ground f l a t and p a r a l l e l p r i o r t o making the i n i t i a l t h i c k n e s s measurement. With the sample mounted on the b r a s s sample h o l d e r , measurements were then made be f o r e and a f t e r each p r e p a r a t i o n s t a g e and the amount o f m a t e r i a l removed was determined. T h i s procedure, i f c a r e f u l l y c a r r i e d out, enabled the sample t h i c k n e s s t o be monitored a c c u r a t e l y t o w i t h i n 2 jj.m d u r i n g the e n t i r e p r e p a r a t i o n s t a g e . B: GRINDING The i n i t i a l s e c t i o n i n g o f a sample wafer i s u s u a l l y c a r r i e d out on a diamond s e c t i o n i n g wheel o r a wire saw. The i n i t i a l wafer t h i c k n e s s was u s u a l l y about 40° yW«i w i t h i t s s u r f a c e s g e n e r a l l y b e i n g rough and not p a r a l l e l t o each o t h e r . The g r i n d i n g stage i s p r i m a r i l y t o f l a t t e n the s u r f a c e and remove any s u r f a c e damage due to the s e c t i o n i n g p r o c e s s . The g r i n d i n g was done by hand on a p i e c e o f p l a t e g l a s s u s i n g 3 jlkm Alumina powder ( A I 2 O 3 ) . The sample s u r f a c e was v e r y rough a f t e r t h i s stage o f p r e p a r a t i o n . C: ROUGH POLISHING The rough p o l i s h i n g i s done p r i m a r i l y to remove the s u r f a c e damage, l e f t by the rough g r i n d i n g stage, and t o l e a v e a smooth s u r f a c e s u i t a b l e f o r the f i n a l p o l i s h i n g s t age. The rough p o l i s h was done m e c h a n i c a l l y u s i n g a U n i p o l p o l i s h e r (Geoscience Instrument Corp.) which has a 10" - diameter p o l i s h i n g s u r f a c e . The p o l i s h i n g speed c o u l d be v a r i e d c o n t i n u o u s l y from 50 t o 675 rpm, although g e n e r a l l y , speeds l e s s than 150 rpm were used, to a v o i d o v e r h e a t i n g o f t h e sample s u r f a c e . The f i r s t stage o f the rough p o l i s h i n v o l v e d the use o f 3 JJLVO. AI2O3 on, a n y l o n c l o t h which was s t r e t c h e d over a c i r c u l a r p i e c e o f - j " p o l i s h e d p l a t e g l a s s , which had p r e v i o u s l y been affixed""" to the b r a s s p o l i s h i n g wheel. The p l a t e g l a s s p r o v i d e d a hard f l a t s u r f a c e which c o u l d e a s i l y be c l e a n e d . Hard n a p l e s s c l o t h s were used f o r a l l s t a g e s o f p o l i s h i n g i n o r d e r to minimize the rounding o f The g l a s s p l a t e was a t t a c h e d t o the b r a s s p o l i s h i n g wheel u s i n g P o l i t e x adhesive f i l m from Geoscience Instrument Corp. 8 0 the sample s u r f a c e s . ( P o l i s h i n g c l o t h s with a nap p r o v i d e a h i g h l u s t e r s u r f a c e but a l s o a l l o w the sample t o " s i n k " i n t o t h e c l o t h , thereby c a u s i n g the edges t o be removed f a s t e r than th e c e n t r e portion.) T h i s stage l e f t a smooth s u r f a c e w i t h a matte l i k e appearance and some s c r a t c h e s , which were g e n e r a l l y l e s s than 1 jj.ro. deep. The second stage of the rough p o l i s h was performed u s i n g Linde A ( 0 . 3 yum) Al^O^ p o l i s h i n g powder on a n y l o n ; c l o t h s t r e t c h e d over a c i r c u l a r p l a t e of g l a s s as p r e v i o u s l y de-s c r i b e d . T h i s stage produced a s u r f a c e with a r e l a t i v e l y h i g h l u s t e r and v e r y s m a l l h a i r l i n e s c r a t c h e s which were l e s s than \ jAm deep. D: FINAL POLISH The aim of the f i n a l p o l i s h i s t o produce a s p e c u l a r s u r f a c e f r e e from s c r a t c h e s , s t a i n s , i m p e r f e c t i o n s , and a l l t r a c e s of d i s t u r b e d m a t e r i a l . There are v a r i o u s methods of p o l i s h i n g semiconductors, among them being, e l e c t r o l y t i c and chemica l p o l i s h i n g . However, chemical o r e l e c t r o l y t i c p o l i s h -i n g w i l l g e n e r a l l y p r e f e r e n t i a l l y e t c h at the edge of the sample or at a d i s l o c a t i o n . T h i s w i l l produce a p i t t e d s u r f a c e which, f o r t h i n samples, c o u l d a c t u a l l y p e n e t r a t e through the sample. T h e r e f o r e , mechanical p o l i s h i n g was chosen. 81 The f i n a l p o l i s h u s i n g Linde-B (0.05 y/m) Alumina powder, was done m e c h a n i c a l l y on a r o t a t i n g wheel which was covered w i t h a P o l i t e x - D c l o t h (Geoscience Instrument C o r p . ) . T h i s i s a f l a t hard s u r f a c e c l o t h i d e a l f o r p o l i s h i n g f l a t s u r f a c e s . The P o l i t e x - D c l o t h , which has an adhesive backing, was a f f i x e d t o the c i r c u l a r p i e c e o f -|" p o l i s h e d p l a t e g l a s s which had p r e v i o u s l y been att a c h e d to the b r a s s p o l i s h i n g wheel. The f i n a l p o l i s h was done w i t h the wheel r o t a t i n g at 100 rpm. The p o l i s h i n g l a p was kept w e l l - l u b r i c a t e d w i t h d i s t i l l e d water to prevent any e x c e s s i v e h e a t i n g o f t h e sample. The s u r f a c e s a f t e r the f i n a l p o l i s h had a m i r r o r f i n i s h and, were found t o be f l a t t o w i t h i n one wavelength o f s o l i u m l i g h t over 9§% of i t s c e n t r a l s u r f a c e . U s i n g t h i s method o f sample p r e p a r a t i o n , sample t h i c k n e s s r a n g i n g from 2 t o 5 yWm were s u c c e s s f u l l y o b t a i n e d . The f a b r i c a t i o n o f t h i n ( l e s s t han 50 yWm) unsupported samples was not attempted. A l l p o l i s h i n g s t a g e s were c a r r i e d out w i t h the p o l i s h e r i n an a i r t i g h t p l e x i g l a s s box, which helped to e l i m i n a t e c o n t a m i n a t i o n o f the p o l i s h i n g l a p s . 82 E: SURFACE DAMAGE DUE TO SAMPLE PREPARATION Any a b r a s i v e s u r f a c e treatment i n v o l v e d i n the p r e p a r a t i o n o f a semiconductor sample, such as c u t t i n g , g r i n d i n g o r mechanical p o l i s h i n g , produces a damaged l a y e r on the s u r f a c e o f the sample. I f care i s not taken t o reduce t h i s damaged l a y e r t o a minimum, i t can have pro-nounced e f f e c t s on the e l e c t r i c a l and o p t i c a l p r o p e r t i e s o f the m a t e r i a l . The presence of a damaged l a y e r can cause s i g n i f i c a n t d i s t o r t i o n o f t h e shape o f the a b s o r p t i o n edge curve and the l o s s o f the e x c i t o n s t r u c t u r e ( L i s i t s a e t a l . 1968). For a g i v e n a b r a s i v e p r o c e s s , c a r e f u l l y c a r r i e d out, t h e depth o f damage i s dependent upon t h e c h a r a c t e r i s t i c s o f the a b r a s i v e p a r t i c l e such as i t s s i z e , shape, hardness and freedom o f motion and on the abraded m a t e r i a l s p r o p e r t i e such as c r y s t a l o r i e n t a t i o n o f the s u r f a c e and mechanical p r o p e r t i e s . Faust (I962) s t a t e s t h a t the damaged l a y e r c o n s i s t s o f a t h i n , h i g h l y fragmented l a y e r at the s u r f a c e and a deeper l a y e r o f p l a s t i c a l l y deformed m a t e r i a l which i s p r e s e n t , as a r e s u l t o f l o c a l h e a t i n g . Gatos e t a l . (I96I) found t h a t as a consequence o f the p o l a r n a t u r e (see s e c t i o n 2.1.1) of the I I I - V compounds, th e same treatment on a {11.1} s u r f a c e l e a d s t o a g r e a t e r depth of damage when t e r m i n a t i n g w i t h group V atoms (B s u r f a c e ) than when t e r m i n a t i n g w i t h group I I I -atoms (A s u r f a c e ) . U s i n g a chemi c a l e t c h d i s s o l u t i o n r a t e 83 method, Gatos determined the s u r f a c e damage produced by a b r a d i n g w i t h 20 yUm s i l i c o n c a r b i d e on g l a s s to be 10 jmm f o r the B s u r f a c e and 6 yum f o r t h e A s u r f a c e o f GaSb. S i n c e i t i s found t h a t , f o r t h e s e m a t e r i a l s , the s u r f a c e damage decreases l i n e a r l y w i t h t h e a b r a s i v e p a r t i c l e s i z e , i t can be assumed, t h a t the depth o f the s u r f a c e damage, f o r GaSb, i s of the o r d e r of the a b r a s i v e p a r t i c l e s i z e . T h i s r u l e o f thumb was used t o estimate the s u r f a c e damage r e p o r t e d i n Table 3.2. Due t o the f a c t t h a t v e r y l i t t l e i n f o r m a t i o n was a v a i l a b l e on the s u r f a c e damage o f p o l i s h e d GaSb, i t was f e l t n e cessary t o b r i e f l y i n v e s t i g a t e the e x t e n t of t h e s u r f a c e damage on the p o l i s h e d c r y s t a l s . Two methods were employed to e s t i m a t e t h e s u r f a c e damage: ( i ) o p t i c a l t r a n s m i s s i o n measurements, and, ( i i ) e l e c t r o n d i f f r a c t i o n i n v e s t i g a t i o n o f an etched and unetched sample. For the o p t i c a l t r a n s m i s s i o n measurement, a sample was prepared w i t h a m i r r o r l i k e s u r f a c e as p r e v i o u s l y d e s c r i b e d and then the s p e c t r a o f t h i s sample, at i t s fundamental a b s o r p t i o n edge, was o b t a i n e d . The sample was then t h o r o u g h l y c l e a n e d and c h e m i c a l l y p o l i s h e d w i t h a s o l u t i o n o f 1 cm^Bro and 10 cm ^ CH^OH (Methanol) f o r 2 0 seconds. T h i s removed a p p r o x i m a l l y 6 j/.m from each s u r f a c e . A f t e r t h i s p r o c e s s , the s u r f a c e was g e n e r a l l y o f a l e s s s p e c u l a r appearance h a v i n g s m a l l e t c h p i t s on t h e s u r f a c e w i t h one s u r f a c e b e i n g g e n e r a l l y o f a b e t t e r 84 q u a l i t y than the oth e r s u r f a c e , due to the p o l a r nature of No n o t i c a b l e d i f f e r e n c e between the s p e c t r a of the etched and unetched s u r f a c e s was observed, i n d i c a t i n g t h a t the s u r f a c e damage was not a f f e c t i n g the r e s u l t s . However, t h i s t e s t was not c o n c l u s i v e s i n c e the poorer s u r f a c e s of the etched samples i n t r o d u c e d a c o n s i d e r a b l e amount of s c a t t e r i n g t o the l i g h t beam and t h e r e f o r e a reduced t r a n s -m i t t e d i n t e n s i t y . The reduced l i g h t i n t e n s i t y r e q u i r e d a wider s l i t w idth f o r d e t e c t i o n which r e s u l t e d i n a r e d u c t i o n o f r e s o l u t i o n . For the e l e c t r o n d i f f r a c t i o n study, a sample was pre -pared w i t h a m i r r o r s u r f a c e i n the same manner as bef o r e and then cut i n h a l f . One h a l f was c h e m i c a l l y p o l i s h e d w i t h the Bromine-Methanol s o l u t i o n and the other h a l f was l e f t as o r i g i n a l l y prepared. The sample s u r f a c e s were then i n v e s t i g a t e d u s i n g h i g h energy e l e c t r o n s (100 K V o l t , /\«0.04 A) and a g r a z i n g i n c i d e n c e d i f f r a c t i o n method. G e n e r a l l y , f o r e l e c t r o n s of t h i s energy, th e s u r f a c e p e n e t r a t i o n i s onl y about 1000 A. The etched s u r f a c e s showed a d e f i n i t e spot p a t t e r n w i t h no evidence of any r i n g s . The unetched samples had both a d e f i n i t e spot p a t t e r n and a superimposed r i n g p a t t e r n , i n d i c a t i n g a c e r t a i n amount of amorphous or p o l y c h r y s t a l i n e m a t e r i a l on the s u r f a c e . However, i t should be p o i n t e d out t h a t a s u r f a c e contaminant o f the order of 100 A t h i c k n e s s can produce an observable r i n g p a t t e r n . Monolayers of a hydrocarbon compound on the the s u r f a c e s . 85 sample s u r f a c e can a l s o produce a w e l l d e f i n e d r i n g p a t t e r n . ( Y e a r i a n 1 9 5 9 page 2 6 0 ) T h e r e f o r e , i t i s concluded t h a t t h e r e i s minor s u r f a c e damage remaining, due t o the sample p r e p a r a t i o n but t h a t any gross damage i s probably l e s s than 5 0 0 A, and any p l a s t i c a l l y deformed l a y e r i s most l i k e l y l e s s than 5 0 0 0 A ( 0 . 5 Jjun) i n t h i c k n e s s . 3 . 5 . 6 MOUNTING OF PREPARED SAMPLE IN UNIAXIAL STRESS DEWAR A f t e r the samples were o r i e n t e d and c u t t o s i z e f o r the u n i a x i a l s t r e s s measurements they were mounted i n the u n i a x i a l s t r e s s dewar t a i l , as shown i n F i g u r e 3 . 6 . Applied Stress F i g u r e 3 * 6 Sample mounting d e t a i l f o r u n i a x i a l s t r e s s dewar. Indium metal was p l a c e d between the top of the sample and the dewar t a i l and at the f r o n t of the sample t o produce a good thermal c o n t a c t . S i n c e the v a r i o u s g l a s s e s used were of d i f f e r e n t t h i c k n e s s i t was necessary t o use b r a s s shims t o p o s i t i o n the g l a s s so t h a t the s t r e s s was a p p l i e d u n i f o r m l y a l o n g the bottom edge of the g l a s s . Using t h i s 86 mounting tec h n i q u e , the samples c o o l e d very q u i c k l y (as d e t e r -mined by the s h i f t i n g of the fundamental a b s o r p t i o n edge wi t h t e m p e r a t u r e s ) . Since the e x c i t o n peaks, f o r the samples mounted i n the u n i a x i a l s t r e s s dewar, were not n o t i c a b l y broadened as compared w i t h the same sample mounted i n the Helium immersion dewar, i t was estimated t h a t the sample tem-pe r a t u r e was l e s s than 10°K. 3.6 CALCULATION OF ABSORPTION COEFFICIENT FROM MEASURED  TRANSMISSION INTENSITY The a b s o r p t i o n c o e f f i c i e n t , o<(hv), i s the f r a c t i o n of energy, at frequency, )> , t h a t i s absorbed from a l i g h t beam p a s s i n g through a u n i t t h i c k n e s s of the m a t e r i a l and i s r e -l a t e d t o the change of i n t e n s i t y I per u n i t l e n g t h by the eq u a t i o n 91 = - o t ( h v ) l (3.4) dx The o b j e c t of t r a n s m i s s i o n measurements i s t o o b t a i n t h e s p e c t r a l v a r i a t i o n of the a b s o r p t i o n c o e f f i c i e n t over the range of photon e n e r g i e s of i n t e r e s t . E x p e r i m e n t a l l y we measure the a t t e n u a t i o n of the r a d i a t i o n i n t e n s i t y on t r a v e r s i n g a t h i c k n e s s , d, of the sample. The amount of a t t e n u a t i o n , or the t r a n s m i s s i o n c o e f f i c i e n t , T(hv'), i s the r a t i o of the t r a n s m i t t e d l i g h t i n t e n s i t y , 1^, t o the i n c i d e n t l i g h t i n t e n s i t y , I . The t r a n s m i s s i o n c o e f f i c i e n t , i n the absence of i n t e r -f e r e n c e e f f e c t s , f o r an ab s o r b i n g media (sample) c h a r a c t e r -i z e d by a complex index of r e f r a c t i o n , n.. - ik-, , and of 8 7 t h i c k n e s s , d 5 b o u n d e d on one s i d e by a s e m i - i n f i n i t e vacuum ( n Q = l ) and on the other s i d e by a s e m i - i n f i n i t e non-absorbing medium ( g l a s s s u b s t r a t e ) w i t h index of r e f r a c t i o n , n^, i s g i v e n by T(hv) JLJllii}. = T, T i e " " ' (or) •I l x - 2 < where the f a c t o r s T-p T 2, R^, and R 2 are the square of the magnitude of the F r e s n e l c o e f f i c i e n t s and are g i v e n i n Appendix C Equation C 9 f o r normal i n c i d e n c e . A review of the d e r i v a t i o n of Equation 3•5 i s g i v e n i n Appendix C. The a b s o r p t i o n c o e f f i c i e n t i s r e l a t e d t o the e x t i n c t i o n c o e f f i c i e n t , k^, by the r e l a t i o n s h i p = 4-n-k, ( 3 . 6 ) The problem o f c a l c u l a t i n g the a b s o r p t i o n c o e f f i c i e n t a c t u a l l y i n v o l v e s a t h r e e l a y e r problem, (sample, epoxy, g l a s s ) bounded on e i t h e r s i d e by a vacuum i n the case of the u n i a x i a l dewar, and by l i q u i d Helium ( n ^ l . O ) f o r the immersion dewar. The exact c a l c u l a t i o n of the formula f o r t h i s case i s a very l e n g t h y p r o c e s s . T h e r e f o r e , f o r the work r e p o r t e d here, v a r i o u s assumptions have been made. F i r s t , s i n c e the index of r e f r a c t i o n o f the epoxy ( n = 1 . 5 7 ) i s very near t o t h a t f o r g l a s s , i t i s assumed t h a t the epoxy and g l a s s can be t r e a t e d as one l a y e r . Second, s i n c e the r e f l e c t e d i n t e n s i t y at the glass-to-vacuum ( l i q u i d Helium) i n t e r f a c e i s o n l y 4%, i t i s assumed t h a t the g l a s s can be t r e a t e d as a s e m i - i n f i n i t e , non-absorbing l a y e r , where the e f f e c t of the r e f l e c t i o n at the glass-to-vacuum i n t e r f a c e 88 can be taken i n t o account i f necessary, by m u l t i p l y i n g the t r a n s m i s s i o n c o e f f i c i e n t by an a p p r o p r i a t e f a c t o r . With the p a r t i c u l a r experimental arrangement of the immersion and u n i a x i a l s t r a i n dewars used, i t was not p o s s i b l e t o s i m u l t a n e o u s l y measure the i n c i d e n t and t r a n s m i t t e d r a d i a -t i o n i n t e n s i t y under i d e n t i c a l c o n d i t i o n s . In order t o c a l c u l a t e the t r a n s m i s s i o n c o e f f i c i e n t f o r a sample the i n c i d e n t i n t e n s i t y p r o f i l e I (h>>) would f i r s t be re c o r d e d . The sample was then mounted i n the l i g h t beam and the t r a n s -m i t t e d l i g h t i n t e n s i t y I ^ ( h v ) was then r e c o r d e d . F i n a l l y w i t h the sample removed the i n c i d e n t i n t e n s i t y p r o f i l e was again r e c o r d e d . The t r a n s m i s s i o n c o e f f i c i e n t s at a l l e n e r g i e s were then n o r m a l i z e d so t h a t the a b s o r p t i o n c o e f f i c i e n t , on the low energy s i d e of the fundamental a b s o r p t i o n edge, had a p a r t i c u l a r v a l u e at a g i v e n photon energy. Using t h i s method o n l y r e l a t i v e , and not a b s o l u t e , a b s o r p t i o n c o e f f i c i e n t s c o u l d be c a l c u l a t e d . A second reason why ab s o l u t e a b s o r p t i o n c o e f f i c i e n t measurements were not attempted i s because when the sample, w i t h i t s h i g h index of r e f r a c t i o n , was p l a c e d i n the con-v e r g i n g l i g h t beam, i t a l t e r e d the o p t i c a l path l e n g t h t o th e d e t e c t o r . T h i s produced a change i n t h e e f f e c t i v e l i g h t i n t e n s i t y measured by the d e t e c t o r . ( I t was found e x p e r i m e n t a l l y t h a t a l t e r i n g the p o s i t i o n of the d e t e c t o r , a l o n g w i t h i t s f o c u s s i n g m i r r o r , by a h o r i z o n t a l displacement o f l/lOOO" (25 y^m) a l o n g i t s a x i s , produced a change i n i n t e n s i t y of 15% i n the measured l i g h t i n t e n s i t y . ) 89 In o r d e r to c a l c u l a t e the a b s o r p t i o n c o e f f i c i e n t from e q u a t i o n ( 3«5 ) the e x t i n c t i o n c o e f f i c i e n t , 1<%, has t o be known. An approximate e x t i n c t i o n c o e f f i c i e n t can be c a l c u l a t e d by f i r s t d e t e r m i n i n g an approximate a b s o r p t i o n c o e f f i c i e n t from the formula T ( h y ) = e«d ( 3 . 7 ) and then u s i n g e q u a t i o n ( 3 . 6 ) t o c a l c u l a t e k^. T h i s approx-imate e x t i n c t i o n c o e f f i c i e n t i s used then i n e q u a t i o n ( C 9 ) and ( 3 . 5 ) t o c a l c u l a t e the a b s o r p t i o n c o e f f i c i e n t . I f n e c e s s a r y , the e x t i n c t i o n c o e f f i c i e n t c o u l d then be r e -c a l c u l a t e d u s i n g e q u a t i o n ( 3 . 6 ) and the above pr o c e s s r e -peated. However, t h i s was not g e n e r a l l y necessary. The v a l u e s f o r the index of r e f r a c t i o n n-^  f o r GaSb were taken from W i l l a r d s o n & Beer Volume 3 ( 1 9 6 7 ) page 524-S i n c e the samples were of the o r d e r of 2 t o 5 /"n t h i c k and the o v e r a l l t h i c k n e s s v a r i a t i o n was of the order of ± 2 jam, i n t e r f e r e n c e f r i n g e s c o u l d not be observed. Thus, the sample t h i c k n e s s , d, used i n e q u a t i o n ( 3 . 5 ) was the v a l u e determined d u r i n g the p o l i s h i n g procedure and was g e n e r a l l y a c c u r a t e t o w i t h i n ± 2 yUm. T h i c k samples ( g r e a t e r than 50 ywm) c o u l d be measured u s i n g the d i a l micrometer d e s c r i b e d i n s e c t i o n 3 .5»5« The r e f l e c t i v i t y can be a f f e c t e d by the c o n d i t i o n of the s u r f a c e s of the sample. However, i f the s u r f a c e has a good p o l i s h the r e f l e c t i v i t y w i l l c l o s e l y f o l l o w the v a l u e g i v e n by the e q u a t i o n (C9). The s p e c t r a l v a r i a t i o n of the r e -f l e c t i v i t y of GaSb i s v e r y s m a l l r e l a t i v e t o the s p e c t r a l v a r i a t i o n of the a b s o r p t i o n c o e f f i c i e n t , c*, i n the wave 90 l e n g t h r e g i o n of i n t e r e s t . Even i f oi = 5 0 0 0 cm"-'-, then k-^  = 6 x 1 0 and the r e f l e c t i v i t y of the vacuum-to-GaSb i n t e r f a c e i s only a l t e r e d by 0 . 0 1 $ by i n c l u d i n g the e x t i n c t i o n c o e f f i c i e n t . A computer program was w r i t t e n which c a l c u l a t e d the a b s o r p t i o n c o e f f i c i e n t u s i n g equation (3-5) and (C9). The c a l c u l a t e d a b s o r p t i o n c o e f f i c i e n t s were then computer p l o t t e d as a f u n c t i o n of the photon energy, which had p r e v i o u s l y been c a l c u l a t e d from the c a l i b r a t e d wavelength (see Appendix A). 91 ) CHAPTER 4 PRESENTATION AND DISCUSSION  OF EXPERIMENTAL RESULTS 4.1 INTRODUCTION In t h i s c h a p t e r the experimental r e s u l t s w i l l be presented and d i s c u s s e d . In s e c t i o n 4«2 the b i a x i a l and u n i a x i a l s t r a i n measurements are d i s c u s s e d . The r e s u l t s o f the e x c i t o n peak s p l i t t i n g f o r the homogeneous b i a x i a l s t r a i n s are then used t o c a l c u l a t e the v a l e n c e band edge deformation p o t e n t i a l s i n s e c t i o n 4»3« The u n i a x i a l s t r a i n measurements are i n -t e r p r e t e d t o determine the s i g n s of the valence band deforma-t i o n p o t e n t i a l . The j3 and t a b s o r p t i o n peaks, f i r s t observed by Johnson (1964), are c o n s i d e r e d and the r e s u l t s of the s t r a i n measurements on the s e peaks are presented i n s e c t i o n 92 4.2 EXCITON ABSORPTION SPECTRA By c h o o s i n g a v a r i e t y of g l a s s s u b s t r a t e s w i t h thermal e x p a n s i v i t i e s g r e a t e r and l e s s than t h a t of GaSb, v a r i o u s amounts of compressive and t e n s i t e s t r a i n can be produced. For a l l of the b i a x i a l s t r a i n r e s u l t s r e p o r t e d here, the samples were r i g i d l y mounted on the g l a s s s u b s t r a t e at room temperature. The samples were then c o o l e d , t o below the A- p o i n t , by immersion d i r e c t l y w i t h i n the l i q u i d helium b a t h . The a b s o r p t i o n s p e c t r a of the b i a x i a l l y s t r a i n e d samples was found t o be independent of the d i r e c t i o n of p o l a r i z a t i o n of the i n c i d e n t l i g h t . T h i s i m p l i e s t h a t the b i a x i a l s t r a i n i s homogeneous i n the plane of the sample, as d i s c u s s e d i n s e c t i o n 2.1.3. A d i s c u s s i o n of the r e s u l t s o b t a i n e d f o r the b i a x i a l s t r a i n i n the {00l\ and f l l l ^ p l a n e s i s giv e n i n s e c t i o n 4.2.1. Using a s p e c i a l u n i a x i a l s t r e s s dewar (see s e c t i o n 3-4.2) a u n i a x i a l compressive s t r e s s was superimposed on the b i a x i a l l y s t r a i n e d sample. Using t h i s technique the homogeneity of the b i a x i a l s t r a i n was removed and, by u s i n g p o l a r i z e d l i g h t , t h e o p t i c a l s e l e c t i o n r u l e s c o u l d be determined. T h i s p er-m i t t e d a c o r r e l a t i o n of the e x c i t o n peaks w i t h the s t r a i n s p l i t v a l e n c e bands v l and v2 (see F i g u r e 2.3), A d i s c u s s i o n o f the u n i a x i a l s t r e s s r e s u l t s i s giv e n i n s e c t i o n 4.2.2. 93 4 . 2 . 1 BIAXIAL STRAIN MEASUREMENTS A v a r i e t y of g l a s s s u b s t r a t e s were choosen i n order t o produce, upon c o o l i n g t o 2°K, both compressive and t e n s i l e s t r a i n s i n the GaSb samples. G l a s s B - 2 6 0 and 0211 have a thermal e x p a n s i v i t y g r e a t e r than GaSb (see Table 3 . 1 ) , between room temperature and 2°K, and t h e r e f o r e produce a compressive s t r a i n i n the sample upon c o o l i n g . G l a s s e s 7 0 5 9 , 7 7 4 0 and 7913 have thermal e x p a n s i v i t i e s l e s s than GaSb and t h e r e f o r e produce a t e n s i l e s t r a i n i n the sample. F i g u r e 4 « 1 shows the a b s o r p t i o n c o e f f i c i e n t , at 2°K, (on a l i n e a r a r b i t r a r y s c a l e ) as a f u n c t i o n of photon energy f o r the v a r i o u s g l a s s s u b s t r a t e s used. The r e s u l t s shown are f o r the b i a x i a l s t r a i n a p p l i e d i n the [ i l l ] plane. The s p e c t r a f o r the b i a x i a l s t r a i n a p p l i e d i n the (00l] plane are q u a l i t a t i v e l y s i m i l a r t o those shown f o r the f i l l ] o r i e n t a t i o n . Each of the a b s o r p t i o n c o e f f i c i e n t s shown i n F i g u r e 4 - 1 have an a r b i t r a r y s c a l e so t h a t i n t e r c o m p a r i s o n o f t h e i r a b s o r p t i o n c o e f f i c i e n t s i s not p o s s i b l e . A d i s c u s s i o n of the c a l c u l a t i o n of the a b s o r p t i o n c o e f f i c i e n t i s g i v e n i n s e c t i o n 3 . 6 . The samples mounted on g l a s s B - 2 6 0 are under a com-p r e s s i v e s t r a i n when c o o l e d down t o 2°K (see Table 3 . 1 ) . The a b s o r p t i o n edge of these samples was found t o be s h i f t e d 94 F i g u r e 4.1 A b s o r p t i o n c o e f f i c i e n t at 2°K as a f u n c t i o n of photon energy f o r the v a r i o u s g l a s s s u b s t r a t e s used w i t h the sample normal i n the C L U ) d i r e c t i o n 95 t o h i g h e r photon e n e r g i e s r e l a t i v e t o the u n s t r a i n e d e x c i t o n peak p o s i t i o n o f 0 . 8 1 0 7 eV. There are two v e r y sharp absorp-t i o n peaks s i t u a t e d at the t o p o f the a b s o r p t i o n edge. The low energy peak, s i t u a t e d at 0 . 8 1 3 3 eV f o r a b i a x i a l s t r a i n i n the [ l l l j p lane, i s i d e n t i f i e d (see s e c t i o n 4 . 2 . 2 ) w i t h t h e t r a n s i t i o n between the ±l/ 2 ( v l ) v a l e n c e band and the e x c i t o n l e v e l and i s t h e r e f o r e l a b e l l e d the 06/* peak. The h i g h e r energy peak, s i t u a t e d at 0 . 8 1 7 4 eV, i s i d e n t i f i e d w i t h t h e t r a n s i t i o n between the mj= ± 3 / 2 ( v 2) v a l e n c e band and the e x c i t o n l e v e l and i s t h e r f o r e l a b e l l e d the o<yz peak. The spectrum f o r g l a s s 0211 i n d i c a t e s two peaks at a photon energy s l i g h t l y above the u n s t r a i n e d e x c i t o n peak p o s i t i o n . These two peaks were not v e r y w e l l r e s o l v e d and v e r y l i t t l e weight was put on t h e i r v a l u e s i n the deformation p o t e n t i a l c a l c u l a t i o n s o f s e c t i o n 4«3<> For the samples under a t e n s i l e s t r a i n , the a b s o r p t i o n edge was s h i f t e d t o lower photon e n e r g i e s , r e l a t i v e t o the u n s t r a i n e d e x c i t o n peak, when c o o l e d t o 2°K. The s p e c t r a o f samples on g l a s s 7 0 5 9 shows a sharp peak (<^'A) at the t o p o f the a b s o r p t i o n edge and a v e r y d e f i n i t e knee on the a b s o r p t i o n edge. U s i n g u n i a x i a l s t r e s s measurements, and the s e l e c t i o n r u l e s g i v e n i n s e c t i o n 2 . 1 . 3 , t h i s knee was found t o be due to the ^ 3 / 2 . e x c i t o n a b s o r p t i o n peak. For samples mounted on g l a s s 7 7 4 0 the s p e c t r a was q u a l i t a t i v e l y the same as f o r samples mounted on g l a s s 7 0 5 9 except t h a t a d e f i n i t e peak ( c*»/«. ) appears on the a b s o r p t i o n edge. For t h e samples mounted on g l a s s 7 7 4 0 a more p r e c i s e d e t e r m i n a t i o n o f the oCyx e x c i t o n peak p o s i t i o n was p o s s i b l e than f o r the o t h e r samples under a t e n s i l e s t r a i n . The samples mounted on g l a s s 7913 were under a t e n s i l e s t r a i n o f 1 0 ~ 3 . The s p e c t r a showed two v e r y broad e x c i t o n a b s o r p t i o n peaks, one on the a b s o r p t i o n edge and the o t h e r a t t h e t o p o f t h e a b s o r p t i o n edge. The e x c i t o n peaks c o u l d be i d e n t i f i e d but the u n c e r t a i n t y i n t h e i r energy p o s i t i o n was l a r g e r t h a n f o r the peaks observed f o r samples mounted on o t h e r s u b s t r a t e s . The s p e c t r a of the v a r i o u s samples was not found t o change upon repeated c y c l i n g between room temperature and l i q u i d h e lium temperature. T h i s i n d i c a t e s t h a t p l a s t i c f l o w was not o c c u r r i n g i n the s t r a i n e d samples. The photon e n e r g i e s o f t h e e x c i t o n peaks, f o r the v a r i o u s s t r a i n s a p p l i e d i n the [00l] and f i l l ] p l a n e s , are t a b u l a t e d i n Table 4 . 1 and p l o t t e d i n F i g u r e 4 . 2 and 4 . 3 r e s p e c t i v e l y . The l i n e a r i n t e r p o l a t i o n o f F i g u r e 4»2 i s done by assuming t h a t the zero s t r a i n p o i n t ( 0 . 8 1 0 7 eV) i s the same as f o r the [ i l l ] plane shown i n F i g u r e 4 . 3 . The o<-exciton peak c o u l d not be observed i n the t h i c k u n s t r a i n e d samples s i n c e the a b s o r p t i o n was too l a r g e and t h i n unmounted samples were not prepared. The u n s t r a i n e d e x c i t o n peak energy can however, be o b t a i n e d from F i g u r e 4 . 3 as the p o i n t at which th e l i n e drawn through the ab-s o r p t i o n peak p o s i t i o n s , c r o s s the zero s t r a i n p o s i t i o n . Table 4.1 / The o<% and o<i/z e x c i t o n peak energy p o s i t i o n f o r the v a r i o u s o r i e n t a t i o n s and g l a s s s u b s t r a t e s used. The e x c i t o n peak energy s e p e r a t i o n and the mean s h i f t , S E H , of the f o r b i d d e n energy gap i s a l s o t a b u l a t e d . SUBSTRATE SAMPLE ORIENTATION STRAIN ON SAMPLE PEAK ENERGY (eV) <=<i/z. PEAK ENERGY (eV) PEAK SEPERATION (meV) S E H b ' (meV ) B - 2 6 0 {00.1] - 4 . 2 x l 0 ~ 4 0 . 8 1 7 4 + 0 . 0 0 0 1 0.8133 + 0 . 0 0 0 1 - 4 . 1 ± 0 . 1 5 + 4 . 6 + 0 . 2 {111] ~ 4 . 2 x l 0 ~ 4 0 . 8 1 8 1 + 0 . 0 0 0 1 0.8143 + 0 . 0 0 0 1 - 3 . 8 ± 0 . 1 5 + 5 . 5 + 0 . 2 0211 {111} - 1 . 2 x l 0 " 4 0 . 8 1 2 7 + 0 . 0 0 0 5 0 . 8 1 1 8 + 0 . 0 0 0 5 - 0 . 9 + 0 , 7 +lo5±0.4 7059 {111} + 3 . 1 x l 0 " 4 0 . 8 0 5 1 + 0 . 0 0 0 3 0 . 8 0 7 7 + 0 . 0 0 0 2 + 2 , 6 ± 0 . 4 - 4 . 3 + 0 . 5 7 7 4 0 (001} + 5 . 5 x l 0 " 4 0.8022 + 0 . 0 0 0 1 0 . 8 0 6 8 . + 0 . 0 0 0 1 + 4 . 6 + 0 , 1 5 - 6 . 2 + 0 . 2 [111] + 5 . 5 x l 0 ~ 4 0 . 8 0 0 9 + 0 . 0 0 0 1 0 . 8 0 5 6 + 0 . 0 0 0 1 + 4 . 7 + 0 . 1 5 - 7 . 5 ± 0 . 2 7913 {111] +10 x l O " 4 0 . 7 9 2 9 + 0 . 0 0 0 3 0 . 8 0 1 3 ±0o0003 + 8 . 4 ± 0 . 4 - 1 3 . 6 + 0 . 5 (a) The s u b s c r i p t s 1/2 and 3/2 f o r the <K peak i n d i c a t e s t h a t the peak i s due to the t r a n s i t i o n from the mj=±l/2 and mj=±3/2 v a l e n c e band to the e x c i t o n l e v e l , r e s p e c t i v e l y . The i d e n t i f i c a t i o n of the v a l e n c e bands i s d i s c u s s e d i n s e c t i o n 4 * 2 . 2 (b) Taking the u n s t r a i n e d e x c i t o n peak energy t o be 0 . 8 1 0 7 eV .820 STRESS APPLIED IN (001) P L A N E F i g u r e 4.2 The e x c i t o n peak energy p o s i t i o n as a f u n c t i o n of the b i a x i a l s t r a i n a p p l i e d i n the {00l] plane. oo - 5 X I 0 " 4 0 + 5 X I C T 4 + I 0 x l 0 ~ 4 COMPRESSIVE STRAIN TENSILE STRAIN (—=•) ON SPECIMENS F i g u r e 4«3 The e x c i t o n peak energy p o s i t i o n as a f u n c t i o n of the b i a x i a l s t r a i n a p p l i e d i n the [ i l l } plane vO 1 0 0 From t h i s we determine t h a t t h e u n s t r a i n e d e x c i t o n peak has an energy of 0 . 8 1 0 7 ± 0 . 0 0 0 2 eV which i s v e r y c l o s e t o the v a l u e o f O.8IO9 eV r e p o r t e d by Johnson (I964) f o r th e cK peak i n u n s t r a i n e d GaSb. The o(,/L and o<yr e x c i t o n peak e n e r g i e s are found t o v a r y l i n e a r l y w i t h s t r a i n t o w i t h i n the e r r o r i n d i c a t e d . The e x c i t o n peak energy s e p a r a t i o n t a b u l a t e d i n T a b l e 4 » 1 has been p l o t t e d i n F i g u r e 4 » 4 f o r the b i a x i a l s t r a i n a p p l i e d i n the [ i l l ] plane. The s l o p e o f t h i s l i n e i s 8 . 7 ± 0 . 7 eV per u n i t o f s t r a i n and w i l l be used i n s e c t i o n 4 » 3 t o c a l c u l a t e t h e v a l e n c e band shear deformation p o t e n t i a l D u. A s i m i l a r graph o f the e x c i t o n s p l i t t i n g f o r the s t r a i n a p p l i e d i n the {OOlj plane had a s l o p e o f 9.1 ± 0 . 7 eV per u n i t o f s t r a i n . T h i s v a l u e w i l l be used t o c a l c u l a t e t h e v a l e n c e band defor m a t i o n p o t e n t i a l D u « The graph o f the mean s h i f t o f the f o r b i d d e n energy gap ( E - E 0 ) " , t a b u l a t e d i n Table 4 . 1 and p l o t t e d i n F i g u r e 4 » 5 , y i e l d s a s l o p e o f - 1 1 . 3 ± 1 . 0 eV per u n i t o f s t r a i n i n the [00l] plane and - 1 3 . 5 ± 0 . 7 eV per u n i t o f s t r a i n i n the [ i l l ] p l a n e . These s l o p e s w i l l be used i n s e c t i o n 4 . 3 to c a l c u -l a t e t h e h y d r o s t a t i c deformation p o t e n t i a l (D^ - ). The mean s h i f t o f the f o r b i d d e n energy gap i s (E - E Q ) where E = ( E Q ( 1 ) + E Q ( 2 ) / 2 i s the mean energy gap and E Q i s t he u n s t r a i n e d energy gap. E Q ( 1) and E Q ( 2 ) are the e n e r g i e s a s s o c i a t e d w i t h the t r a n s i t i o n s between the s t r a i n s p l i t v a l e n c e bands v l and v 2 r e s p e c t i v e l y and the con d u c t i o n band as shown i n F i g u r e 2 . 3 . 10.0 -5.0 gure 4«4 The e x c i t o n peak ene a p p l i e d i n the {ill} rgy s e p e r a t i o n as a f u n c t i o n of the b i a x i a l plane 102 F i g u r e 4.5 The mean s h i f t of the f o r b i d d e n energy gap,&EH, f o r the s t r a i n a p p l i e d i n the {00l] and [111] c r y s t a l l o g r a p h i c p l a n e s 103 4 . 2 . 2 UNIAXIAL STRAIN MEASUREMENTS As has been mentioned i n s e c t i o n 2 . 1 , 3 , when a sample i s cemented t o a g l a s s s u b s t r a t e , an i s o t r o p i c b i a x i a l s t r a i n i s produced i n the plane o f the sample upon c o o l i n g o When p o l a r i z e d l i g h t i s i n c i d e n t normal to the sample s u r -f a c e i t i s o n l y p o s s i b l e t o o b t a i n CT p o l a r i z a t i o n (see s e c t i o n 2 , 1 . 3 )<> By superimposing a u n i a x i a l s t r e s s on the b i a x i a l l y s t r a i n e d sample, the s t r a i n i s no lo n g e r i s o t r o p i c i n the plane o f the sample. The o p t i c a l s e l e c t i o n r u l e s , d i s c u s s e d i n s e c t i o n 2 . 1 . 3 , can then be used t o i d e n t i f y t he va l e n c e bands i n v o l v e d i n the t r a n s i t i o n s . Using the u n i a x i a l s t r e s s dewar, d i s c u s s e d i n s e c t i o n 3 « 4 . 2 , o n l y a compressive s t r e s s c o u l d be a p p l i e d t o the sample. I f the sample i s i n i t i a l l y under a t e n s i l e b i a x i a l s t r a i n , and a u n i a x i a l compressive s t r e s s i s a p p l i e d along a p r i n c i p l e c r y s t a l o g r a p h i c d i r e c t i o n , i t i s p o s s i b l e , i n p r i n c i p l e , t o comp l e t e l y remove one component o f the b i a x i a l s t r a i n . T h i s would l e a v e a p u r e l y t e n s i l e u n i a x i a l s t r a i n p e r p e n d i -c u l a r t o the a p p l i e d u n i a x i a l s t r e s s . I f l i g h t i s i n c i d e n t normal t o the sample and p o l a r i z e d p e r p e n d i c u l a r t o the a p p l i e d s t r e s s ( p a r a l l e l t o remaining s t r a i n ) then a c o n d i -t i o n o f Tf p o l a r i z a t i o n e x i s t s . A c o n d i t i o n o f (f p o l a r i z a -t i o n w i l l e x i s t i f the l i g h t i s p o l a r i z e d p a r a l l e l t o the a p p l i e d s t r e s s ( p e r p e n d i c u l a r t o remaining s t r a i n ) . F i g u r e 4 * 6 shows the change i n the s p e c t r a o f a sample, w i t h an i s o t r o p i c b i a x i a l t e n s i l e s t r a i n ( 7 7 4 0 s u b s t r a t e ) 104 i r 0.785 0.790 0.795 0.800 0.805 0.810 PHOTON ENERGY (eV) F i g u r e 4»6 The change i n the s p e c t r a of a sample, wi t h an i n i t i a l i s o t r o p i c b i a x i a l t e n s i l e s t r a i n i n the {00l} plane, due to a u n i a x i a l s t r e s s a p p l i e d a l o n g a <0Ol) d i r e c t i o n , (a) S p e c t r a f o r the l i g h t p o l a r i z e d p e r p e n d i c u l a r t o the t e n s i l e s t r a i n (©"polarization ) (b) S p e c t r a of the b i a x i a l l y s t r a i n e d sample u s i n g u n p o l a r i z e d l i g h t . (c) S p e c t r a f o r the l i g h t p o l a r i z e d p a r a l l e l t o the s t r a i n d i r e c t i o n ( 7 T p o l a r i z a t i o n ). 105 i n the f00l] plane, due t o a u n i a x i a l s t r e s s a p p l i e d a l o n g a <^00l) d i r e c t i o n . The example shown i s f o r a s t r e s s of 8 2 4 . 5 x 10 dynes cm" . C a l c u l a t i o n s i n d i c a t e t h a t the r e -maining s t r a i n , f o r t h i s case, i s approximately u n i a x i a l . In F i g u r e 4-6 the s o l i d l i n e (b) i n d i c a t e s the s p e c t r a of the b i a x i a l l y s t r a i n e d sample observed w i t h u n p o l a r i z e d l i g h t . The broken l i n e s (a,c) i n d i c a t e the s p e c t r a f o r CT and If p o l a r i z a t i o n as i n d i c a t e d . As the u n i a x i a l s t r e s s i s i n c r e a s e d from zero both of the e x c i t o n peaks are observed t o s h i f t t o h i g h e r e n e r g i e s (towards the u n s t r a i n e d e x c i t o n peak energy) the low energy peak s h i f t i n g more r a p i d l y w i t h s t r a i n than the h i g h energy peak. T h i s i s as expected s i n c e the sample i s b e i n g r e l i e v e d o f s t r a i n i n one d i r e c t i o n . A l s o , as the s t r e s s i s i n c r e a s e d the low energy peak i s observed t o i n c r e a s e i n i n t e n s i t y f o r <fpolarization and decrease i n i n t e n s i t y (almost d i s a p p e a r i n g ) f o r ftpolarization. The h i g h energy peak i s presen t f o r both p o l a r i z a t i o n s , but i s more i n t e n s e f o r " r e p o l a r i z a t i o n . As d i s c u s s e d i n s e c t i o n 2 . 1 . 3 the t r a n s i t i o n from the v l (J=3/2 mj=±l / 2 ) v a l e n c e band t o the co n d u c t i o n band i s allowed f o r both p o l a r i z a t i o n s , whereas, the t r a n s i t i o n from the v 2 (J=3/2 mj= ± 3 / 2) v a l e n c e band t o the con d u c t i o n band i s allowed f o r ( ^ p o l a r i z a t i o n o n l y . On the b a s i s of these s e l e c t i o n r u l e s we can t h e r e f o r e conclude t h a t f o r a t e n s i l e s t r a i n , the low energy e x c i t o n peak i s due t o the t r a n s i t i o n between the v 2 v a l e n c e band and the e x c i t o n l e v e l and the h i g h energy e x c i t o n peak i n v o l v e s the t r a n s i t i o n between the v l v a l e n c e band and the e x c i t o n l e v e l . 1 0 6 The e x c i t o n peaks can t h e r e f o r e be l a b e l l e d c<yL and c< ^ f o r the low energy and h i g h energy peaks, r e s p e c t i v e l y . I f the sample i s i n i t i a l l y under a compressive b i a x i a l s t r a i n , the a p p l i c a t i o n o f a compressive u n i a x i a l s t r e s s can o n l y remove the homogeneity o f the s t r a i n . F i g u r e 4<>7 shows how the s p e c t r a , of a sample wi t h an i s o t r o p i c com-p r e s s i v e b i a x i a l s t r a i n (B - 2 6 0 s u b s t r a t e ) i n t h e [ i l l ] plane, changes as a r e s u l t o f a p p l y i n g a compressive u n i -a x i a l s t r e s s a l o n g a (221) d i r e c t i o n , , The s o l i d l i n e (b) i s the s p e c t r a o f t h e b i a x i a l l y s t r a i n e d sample observed w i t h u n p o l a r i z e d l i g h t . The broken l i n e s (a,c) show the s p e c t r a o f the sample wi t h the inhomogeneous compressive s t r a i n f o r the two p o l a r i z a t i o n s . The l i n e s l a b e l l e d C and 7T i n d i c a t e t h a t the l i g h t i s p o l a r i z e d p e r p e n d i c u l a r and p a r a l l e l t o the a p p l i e d u n i a x i a l s t r e s s , r e s p e c t i v e l y (major u n i a x i a l s t r e s s d i r e c t i o n ) . As the u n i a x i a l s t r e s s i s i n c r e a s e d from z e r o , both of the e x c i t o n a b s o r p t i o n peaks are observed t o s h i f t t o h i g h e r photon e n e r g i e s . In F i g u r e 4 » 7 * f ° r "the inhomogeneously s t r a i n e d samples, bo t h a b s o r p t i o n peaks are p r e s e n t f o r cf p o l a r i z a t i o n w i t h t h e low energy peak b e i n g l e s s i n t e n s e . For ff p o l a r i z a t i o n , o n l y the low energy peak i s observed. T h e r e f o r e , u s i n g t h e s e l e c t i o n r u l e s from s e c t i o n 2 . 1 . 3 , the low energy peak i n v o l v e s the t r a n s i t i o n from th e v l v a l e n c e band to the e x c i t o n l e v e l and can be l a b e l l e d oi,^ , w h i l e the h i g h energy peak i n v o l v e s the t r a n s i t i o n from the v 2 valence band to the e x c i t o n l e v e l and i s l a b e l l e d °<J/I . 107 T r 0.805 0.810 0.815 0.820 0.825 PHOTON ENERGY (eV) F i g u r e 4.7 The change i n the s p e c t r a of a sample, wi t h an i n i t i a l i s o t r o p i c compressive b i a x i a l s t r a i n i n the f i l l ] p lane, due t o a u n i a x i a l s t r e s s a p p l i e d a l o n g a <221> d i r e c t i o n , (a) S p e c t r a f o r the l i g h t p o l a r i z e d p a r a l l e l t o the a p p l i e d s t r e s s d i r e c t i o n (V p o l a r i z a t i o n ) (b) S p e c t r a of the b i a x i a l l y s t r a i n e d sample u s i n g u n p o l a r i z e d l i g h t , (c) S p e c t r a f o r the l i g h t p o l a r i z e d p e r p e n d i c u l a r t o the a p p l i e d s t r e s s d i r e c t i o n ( ( ^ p o l a r i z a t i o n ). 108 S i m i l a r r e s u l t s were o b t a i n e d f o r a sample w i t h an i s o t r o p i c compressive b i a x i a l s t r a i n i n the fOOl] plane w i t h a u n i a x i a l compressive s t r e s s a p p l i e d a l o n g a <^00l) d i r e c t i o n . T h e r e f o r e , as a r e s u l t o f th e s e measurements and the b i a x i a l s t r a i n measurements, i t i s concluded t h a t f o r a t e n s i l e s t r a i n , the "centre o f g r a v i t y " o f the v l and v 2 v a l e n c e bands (see f o o t n o t e page 1 6 ) moves towards the c o n d u c t i o n band. Under a t e n s i l e s t r a i n the v 2 v a l e n c e band moves "up" (see f o o t n o t e page 3 0 ) w i t h r e s p e c t t o t h e "centre o f g r a v i t y " , a n d the v l v a l e n c e band moves "down" w i t h r e s p e c t t o the " c e n t r e of g r a v i t y " . The v l and v 2 v a l e n c e bands can be l a b e l l e d w i t h the magnetic quantum numbers mj — ±l/ 2 and mj = ± 3 / 2 , r e s p e c t i v e l y , w i t h t h e e x c i t o n peaks, i n v o l v i n g these bands, b e i n g l a b e l l e d o(,/L and c x ^ . Under a compressive s t r a i n , the "centre of g r a v i t y " o f the v l and v 2 v a l e n c e band moves away from the c o n d u c t i o n band w i t h the v l v a l e n c e band moving "up" w i t h r e s p e c t t o t h e " c e n t r e of g r a v i t y " , and t h e v 2 v a l e n c e band moving "down" w i t h r e s p e c t t o the " c e n t r e of g r a v i t y " o f the v a l e n c e bands. F i g u r e 4»8 shows the v a r i a t i o n o f the e n e r g i e s o f the o^)ju , and the ofy^ peaks as a f u n c t i o n o f the u n i a x i a l s t r e s s a p p l i e d t o the g l a s s s u b s t r a t e . The i n i t i a l b i a x i a l s t r a i n i s t e n s i l e ( 7 7 4 ° s u b s t r a t e ) i n the [ ° 0 l ] plane and t h e compressive u n i a x i a l s t r e s s i s a p p l i e d along a APPLIED MASS (kgm) 0 10 20 30 40 I 1 1 : 1 r APPLIED STRESS (Dynes/cm2 x I0"8) H o F i g u r e 4.8 The <*jk, °<s/x and ft a b s o r p t i o n peak energy p o s i t i o n as a f u n c t i o n of the ^ s t r e s s a p p l i e d t o the g l a s s s u b s t r a t e . The sample was i n i t i a l l y under a t e n s i l e b i a x i a l s t r a i n i n the {001} plane. 110 <(00l) d i r e c t i o n , , A t y p i c a l example of the s p e c t r a i s shown i n F i g u r e 4*6 • An attempt was made t o c a l c u l a t e t h e s t r a i n i n t h e sample u s i n g t h e Young Ts modulus and the P o i s s o n r a t i o o f the g l a s s , but the r e s u l t s were not con-s i s t e n t w i t h the b i a x i a l s t r a i n measurements. I t can be seen from F i g u r e 4 » 8 , t h a t the ^yi. , and cVj^peaks a l l s h i f t l i n e a r l y w i t h t h e a p p l i e d u n i a x i a l s t r e s s w i t h i n the e r r o r s i n d i c a t e d . I t has been observed by Langer e t a l . ( 1 9 7 0 ) and G i l l e o e t a l . ( 1 9 7 0 ) t h a t the energy o f the **y4 peak, f o r s e v e r a l z i n c blende type m a t e r i a l s , i s d i f f e r e n t f o r <T" and It p o l a r i z a t i o n . I n t h i s work no energy d i f f e r e n c e c o u l d be d e t e c t e d f o r t h e two p o l a r i z a t i o n s t o w i t h i n t h e e x p e r i -mental u n c e r t a i n t y (a d i f f e r e n c e of ± 0 . 2 meV c o u l d be d e t e c t e d f o r sharp peaks). I l l 4 . 3 CALCULATION OF DEFORMATION POTENTIALS The e f f e c t of a l a t t i c e d eformation, on the band s t r u c t u r e o f a semiconductor, can be c o n v e n i e n t l y t r e a t e d by the use of t h e deformation p o t e n t i a l t h e o r y reviewed i n s e c t i o n 2 . 1 . 4 . The equations g i v i n g the v a l e n c e band energy s p l i t t i n g (2A) at k= 0 , i n terms o f the u n i a x i a l deformation p o t e n t i a l s 0^ and D^ d e f i n e d by K l e i n e r and Roth ( 1 9 5 9 ) , have been summarized i n row 3 of Table 2 n l f o r t h e t h r e e p r i n c i p l e c r y s t a l l o g r a p h i c o r i e n t a t i o n s . The e q u a t i o n g i v i n g the s h i f t o f the fundamental energy gap (SE H = E - E0 ), i n terms o f the d i f f e r e n c e of the h y d r o s t a t i c deformation p o t e n t i a l parameters, ( D J - D j ) , has been summarized i n row 4 o f T a b l e 2 . 1 . I n s e c t i o n 2.2.2 i t has been shown t h a t , f o r a g i v e n s t r a i n , t h e e x c i t o n b i n d i n g e n e r g i e s f o r the v l and v2 v a l e n c e bands are approximately e q u a l , f o r the s t r a i n s used i n t h i s work. T h e r e f o r e , i t i s assumed t h a t , f o r a g i v e n s t r a i n , the v a l e n c e band s p l i t t i n g energy, (2£), i s equal to t h e e x c i t o n peak s p l i t t i n g energy. Any e r r o r i n t r o d u c e d by t h i s assumption w i l l be l e s s than t h e e r r o r i n v o l v e d i n d e t e r m i n i n g t h e e x c i t o n peak e n e r g i e s . In row 3 o f Table 2 . 1 , the e q u a t i o n s , r e l a t i n g t h e v a l e n c e band s p l i t t i n g t o the s t r a i n , show a q u a d r a t i c term ( %E00I f/2A6 and ( $E,„ j*~/2L\0 where SE„, and $E,„ , to a f i r s t approximation, can be equated t o t h e v a l e n c e band s p l i t t i n g and A0 i s the s p i n - o r b i t s p l i t t i n g [ e q u a l to 0 . 7 4 9 eV f o r GaSb (Reine e t a l . ( 1 9 7 0 ) ) ] . S i n c e t h e l a r g e s t e x c i t o n s p l i t t i n g , observed i n t h i s work, i s 0 . 0 0 8 4 eV, t h e s e 1 1 2 q u a d r a t i c terms w i l l be n e g l i g a b l e i n comparison t o the S E 0 o , and 5E„, terms and can t h e r e f o r e be n e g l e c t e d . T h i s assumption c o u l d a l s o be a r r i v e d at by o b s e r v i n g t h a t , over the r e g i o n of s t r a i n covered here, the e x c i t o n peak s p l i t t i n g v a r i e s l i n e a r l y w i t h s t r a i n , w i t h i n the experimental accuracy (Figure 4 * 4 ) * T h e r e f o r e , the v a l e n c e band s p l i t t i n g i s simply g i v e n by the q u a n t i t i e s 6E0OI and 6E„, f o r the b i a x i a l s t r a i n i n the [ 0 0 l } and the f i l l ] p i anes, r e s p e c t i v e l y . S u b s t i t u t i n g the s l o p e s of the e x c i t o n peak s p l i t t i n g energy f o r the b i a x i a l s t r a i n i n the {OOTj and f i l l ] p l a n e s , which were determined i n s e c t i o n 4 * 2 . 1 , i n t o the equations f o r &E00) and &Em y i e l d s a v a l u e of +3.5 ±0.3 eV f o r the deformation p o t e n t i a l , D u , a n d + 4 . 4 ±0.3 eV f o r the deformation p o t e n t i a l D^,. T h i s i s e q u i v a l e n t t o - 2 . 4 ± 0 . 2 eV and - 5 . 1 ±0.3 eV f o r the P i c u s and B i r deformation p o t e n t i a l s , b and d, r e s p e c t i v e l y (see f o o t n o t e page 21)o The s i g n s of the deformation p o t e n t i a l s have been determined from the u n i a x i a l s t r a i n measurements u s i n g p o l a r i z e d l i g h t (see s e c t i o n 2 . 1 . 4 ) . The d i f f e r e n c e of the h y d r o s t a t i c deformation p o t e n t i a l s of the c o n d u c t i o n and valence bands, ( D ^ - ), can be de-termined by s u b s t i t u t i n g the s l o p e of the mean s h i f t of the energy gap as a f u n c t i o n of a p p l i e d s t r a i n ( s e c t i o n 4 * 2 . 1 ) i n t o the e q u a t i o n f o r 6 E W g i v e n i n row 4 of Table 2 . 1 . Using t h e s l o p e s c a l c u l a t e d i n s e c t i o n 4 ° 2 . 1 , we o b t a i n a v a l u e of - 1 0 . 4 ±0.9 eV per u n i t of s t r a i n a p p l i e d i n the { 0 0 l ] plane and -8.9 ± 0 . 5 eV per u n i t of s t r a i n a p p l i e d i n the [ i l l ] plane f o r the deformation p o t e n t i a l parameter ( D j - ) . 113 The v a r i a t i o n of the mean width of the f o r b i d d e n energy-gap, 6E W , f o r the a p p l i c a t i o n of a h y d r o s t a t i c p r e s s u r e , P, i s g i v e n by $E„ = ( dE,/dP)-P = (Dd6 - Dj) (AV/V) U s i n g the d i l a t a t i o n , (A = AV/V), d e f i n e d by equ a t i o n D3 and g i v e n by e q u a t i o n D9 f o r a c u b i c c r y s t a l one o b t a i n s ( dE 0/dP) = 3(Dd - Dj)-(s„ + 2s,2) The v a l u e s f o r s„ and s a can be c a l c u l a t e d u s i n g equation D8 and the v a l u e s of c„ and c a g i v e n i n Table 2.4. Using the mean v a l u e of -9.6 ±1.0 eV f o r (D^ - Dj) we o b t a i n a v a l u e of (16.3 ±2.0)xl0"a (eV cm 2 dyne"' ) f o r (dE 0/3P). T h i s can be compared w i t h the two independent v a l u e s quoted, i n the review a r t i c l e by Paul ( l 9 6 l ) , of 16. 3xl0~'i and 12.2x10'" (eV cm 2 dyne ' ) and the va l u e o f 14.5x10 (eV cm 2 dyne"' ) g i v e n by K o s i c k i e t a l . (1968). The defor m a t i o n p o t e n t i a l s o b t a i n e d here, and those o f o t h e r i n v e s t i g a t o r s , are g i v e n i n Table 4.2. P o l l a k e t a l . (1971) determined the deformation p o t e n t i a l s by s t u d y i n g the e f f e c t of a compressive u n i a x i a l s t r e s s on the f r e e e x c i t o n i n GaSb, at 1.7°K, u s i n g wavelength-modulated r e -ef Z / f l e c t i v i t y and s t r e s s e s of up to 5x10 dyne cm" . (In the pr e s e n t work a t e n s i l e s t r a i n of 10 3 i n the [ i l l ] plane i s e q u i v a l e n t t o a s t r e s s of +1.3x10*' dyne cm~z and a compressive s t r a i n of -4.2x10* i n the { i l l ] plane i s e q u i v a l e n t t o a s t r e s s o f -0.5x10 dyne cm .) Benoit a l a Guillaume e t a l . (1970) s t u d i e d the e f f e c t o f a u n i a x i a l compressive s t r e s s on the photoluminescence of 114 Table 4.2 The deformation p o t e n t i a l parameters o b t a i n e d i n t h i s work and i n p r e v i o u s work. DEFORMATION POTENTIAL THIS WORK PREVIOUS WORK (Ddc -Ddv ) (eV) - 1 0 .4 ± 0. < 8 . 0 ± 0 . d - 8 .9 + 0 >5 b 7 7 .1 ± »6 + 0 . 0 . 4 b -4 C -d d 8 •5% 8 . 2 f Dtt (eV) 3< 5 ± 0 . 3 a D'r. (eV) 4 .4 ± 0 . 3 b b (eV) - 2 4 ± 0. 2 a -1 . 8 ± 0 . i d - 2 f - 3 . 3 ± 0 . 6 § 3 .Oh d (eV) - 5 .1 ± 0 . 3 - 4 . 2 ± 0 . 2^ - 4 . 6 f , -8 5 . 3 5 ± . l h 1. 7^ (a) f o r {OOli s t r e s s measurements (b) f o r {111} s t r e s s measurements (c) f o r [110] s t r e s s measurements (d) P o l l a k e t a l . ( 1 9 7 1 ) ( 1 . 7°K) (e) K o s i c k i e t a l . (1968 ) ( f ) B e noit a l a Guillaume e t a l . ( 1 9 7 0 ) ( 3 5°K) (g) G a v i n i & Cardona ( 1 9 7 0 ) ( 7 7°K) (h) Yu e t a l . ( 1 9 7 1 ) ( « 2 9 4°K) 115 GaSb ( P i e z o e m i s s i o n ) and determined the deformation p o t e n t i a l s . T h e i r i n v e s t i g a t i o n was done at 35°K, w i t h an energy r e s o l u t i o n o f 0 . 5 meV. No e r r o r s were quoted f o r t h e i r r e s u l t s and no temperature adjustment, t o 35°K, was performed f o r the e l a s t i c c o n s t a n t s used i n t h e i r c a l c u l a t i o n s . G a v i n i e t a l . ( 1 9 7 0 ) measured the deformation p o t e n t i a l s at 7 7 ° K by measuring the r e f l e c t i v i t y of n-type GaSb, u s i n g a s m a l l u n i a x i a l s t r e s s modulation ( P i e z o r e f l e c t a n c e ) . T h e i r s t r e s s was e s t i m a t e d t o be 1 0 6 dyne cm'l The e r r o r i n t h e i r v a l u e s was e s t i m a t e d t o be ±20% which i n c l u d e s an e r r o r of ±10% i n the v a l u e of the h y d r o s t a t i c deformation p o t e n t i a l . (The p i e z o r e f l e c t a n c e t e c h n i q u e determines the r a t i o of the shear deformation p o t e n t i a l s , b and d, t o the h y d r o s t a t i c d e f o r m a t i o n p o t e n t i a l , a, where, a, has t o he determined by an independent experiment.) Yu e t a l . ( 1 9 7 1 ) used the s t r e s s - i n d u c e d b i r e f r i n g e n c e due t o i n t e r - b a n d t r a n s i t i o n s ( i n t r i n s i c p i e z o b i r e f r i n g e n c e ) t o determine the deformation p o t e n t i a l s of GaSb at room temperature. Yu e t a l . ( 1 9 7 1 ) s t a t e s t h a t one of the two d e f o r m a t i o n p o t e n t i a l s o b t a i n e d by p i e z o b i r e f r i n g e n c e always agrees q u i t e w e l l w i t h v a l u e s o b t a i n e d by o t h e r methods. There i s a good agreement f o r the v a l u e o f , d, obtained f o r t h e p r e s e n t work and the v a l u e o b t a i n e d by Yu e t a l . R e c e n t l y , G a v i n i and Cardona ( 1 9 7 0 ) as w e l l as Lawaetz (Phys. Rev. B (to be p u b l i s h e d ) ) have developed t h e o r i e s f o r t h e shear d e f o r m a t i o n p o t e n t i a l s . (The r e s u l t s of G a v i n i and Cardona and of Lawaetz have been reviewed and summarized f o r 116 GaSb by P o l l a k e t a l . ( 1 9 7 1)«) G a v i n i and Cardona f i n d t h a t the change i n b and d, i n g o ing from the Group IV to I I I - V semiconductors, i s g i v e n by A b - - 0 . 2 3 eV ^ d = - 0 . 6 9 eV T a k i n g an average v a l u e of the deformation p o t e n t i a l s f o r t h e group IV compounds, P o l l a k e t a l . ( 1 9 7 1) p r e d i c t s t h a t f o r GaSb, b = - 2 . 7 + 0 . 3 5 eV and d = - 5 . 1 ± 1 . 4 eV. There i s a good agreement between t h i s v a l u e f o r the deformation p o t e n t i a l parameter, d, and t h a t o b t a i n e d from t h i s work. The t h e o r y of Lawaetz p r e d i c t s ( P o l l a k e t a l . ( 1 9 7 1 ) b = - 2 . 3 eV and d = - 5 . 2 eV f o r GaSb. These v a l u e s are i n v e r y good agreement wi t h t h e deformation p o t e n t i a l s o b t a i n e d i n t h e p r e s e n t work. From Table 4«2 i t can be seen t h a t t h e r e i s a l a r g e d i s c r e p a n c y between the d e f o r m a t i o n p o t e n t i a l s measured by the p i e z o r e f l e c t a n c e technique and the o t h e r t e c h n i q u e s . T h i s d i s c r e p a n c y was a l s o observed i n the a l k a l i h a l i d e s by G a v i n i and Timosk ( 1 9 7 0). They e x p l a i n the d i f f e r e n c e q u a l i t a t i v e l y as due t o the f a c t t h a t i n modulated p i e z o r e -f l e c t a n c e one measures the s m a l l - s t r e s s deformation p o t e n t i a l s o f the e x c i t o n s , i n s t e a d of those of the o n e - e l e c t r o n band edges. The deformation p o t e n t i a l parameter (Dd - T)d ) should be independent of the d i r e c t i o n of strain,, The v a l u e s r e p o r t e d here are d i f f e r e n t f o r the two o r i e n t a t i o n s s t u d i e d . The v a r i a t i o n of the v a l u e of (D^ - Dj) w i t h s t r a i n o r i e n t a t i o n has been noted p r e v i o u s l y f o r GaSb by P o l l a k ( 1 9 7 1) and f o r 1 1 7 a number of o t h e r m a t e r i a l s such as CdTe (Thomas ( I 9 6 l ) ) and Ge (Glass ( 1 9 6 4 ) , Osipov ( I 9 6 7 ) ) . Osipov ( 1 9 6 7 ) attempted t o e x p l a i n the d i s c r e p a n c y on the v a s i s of the v a r i a t i o n o f the e x c i t o n b i n d i n g energy w i t h s t r a i n . He found t h a t f o r Ge, the dependence of the e x c i t o n b i n d i n g energy on the s t r a i n was most a p p r e c i a b l e f o r smal l band s p l i t t i n g i n the <C00l) d i r e c t i o n . A s i m i l a r c a l c u l a t i o n c a r r i e d out, i n s e c t i o n 2 . 2 . 2 i n d i c a t e s t h a t t h i s i s not the case f o r GaSb. I t was found t h a t , f o r GaSb, the v a r i a t i o n o f the e x c i t o n b i n d i n g energy w i t h s t r a i n was r e l a t i v e l y independent of the o r i e n t a t i o n of the s t r a i n . For a l l o r i e n t a t i o n s the e x c i t o n b i n d i n g energy was found t o decrease as the s t r a i n i n c r e a s e d . However, when t h i s e f f e c t was taken i n t o account i n the c a l c u l a t i o n of (D^ - D/) the c o r r e c t i o n was n e g l i g i b l e i n comparison w i t h o t h e r energy e r r o r s . In t h i s work, two a b s o r p t i o n peaks, h a v i n g e n e r g i e s 0 . 7 9 6 2 ± 0 . 0 0 0 2 eV and 0 . 8 0 5 5 ± 0 . 0 0 0 5 eV, were observed i n t h i c k ( 1 0 0 to 2 0 0 jxra) u n s t r a i n e d samples of GaSb (see F i g u r e 4.9)o These two peaks occur on the steep r i s e o f the funda-mental a b s o r p t i o n edge. T h e i r e n e r g i e s c o i n c i d e v e r y c l o s e l y t o the e n e r g i e s o f the f and ^ peaks, observed i n GaSb by Johnson ( 1 9 6 4 ) , at O . 7 9 6 7 eV and O .8O58 eV, r e s p e c t i v e l y . These two peaks w i l l t h e r e f o r e be l a b e l l e d if (O . 7 9 6 2 eV) and jS (O .8O58 eV) i n accordance w i t h Johnson. A b r i e f d i s c u s s i o n o f these two peaks i s g i v e n i n s e c t i o n 2 .2.3. A d i s c u s s i o n o f the s t r a i n dependence o f the ^ and Y peaks f o l l o w s i n s e c t i o n s 4«4»1 a n d 4<>4o2, r e s p e c t i v e l y . 4 . 4 . 1 DEPENDENCE OF THE ft ABSORPTION PEAK ON STRAIN The j3 a b s o r p t i o n peak i s observed as a sharp peak i n t h i c k u n s t r a i n e d samples at a photon energy o f 0 . 8 0 5 5 ± 0 . 0 0 0 5 eV ( F i g u r e 4 . 9 ) . The peak i s a l s o observed i n t h i n b i a x i a l l y s t r a i n e d samples, but i s not as i n t e n s e as i n the t h i c k e r samples. The peak energy i s observed to s h i f t w i t h the induced s t r a i n . However, no s t r a i n - i n d u c e d s p l i t t i n g was observed. The energy o f the ^ peak, f o r the v a r i o u s b i a x i a l s t r a i n s , i s l i s t e d i n T a b l e 4 » 3 and i s p l o t t e d i n F i g u r e 4 » 1 0 . The p o s i t i o n o f the peak energy, f o r a g i v e n s t r a i n , i s observed at s l i g h t l y d i f f e r e n t e n e r g i e s f o r the b i a x i a l s t r a i n i n the fOOlj plane and the { i l l } p l a n e . Over the r e g i o n o f s t r a i n covered, the j3 peak appears t o s h i f t l i n e a r l y w i t h s t r a i n F i g u r e 4>9 The /S and t a b s o r p t i o n peaks i n u n s t r a i n e d samples of GaSb. Table 4 . 3 The energy of the /3 and X a b s o r p t i o n peaks f o r the v a r i o u s b i a x i a l s t r a i n s a p p l i e d i n the {001} and {111} c r y s t a l l o g r a p h i c planes SUBSTRATE SAMPLE ORIENTATION STRAIN ON SAMPLE ( 10-*) p PEAK ENERGY (eV) r PEAK ENERGY (eV) B - 2 6 0 {001} - 4 . 2 0 . 8 1 1 0 + 0 . 0 0 1 0 [111} - 4 . 2 0 . 8 1 0 5 + 0 . 0 0 1 0 0 . 8 0 1 5 + 0 . 0 0 1 0 ' 0211 {111} - 1 . 2 0 . 8 0 7 2 + 0 . 0 0 0 5 0 . 7 9 8 5 ± 0 . 0 0 1 5 NONE {111} 0 0 . 8 0 5 5 + 0 . 0 0 0 5 0 . 7 9 6 3 ± 0 . 0 0 0 2 7059 {111} + 3 . 1 0 . 8 0 1 5 + 0 . 0 0 1 0 0 . 7 9 2 7 ±0.0005 7740 {003} + 5 - 5 0 . 7 9 9 5 ± 0 . 0 0 0 5 0 . 7 9 1 0 ± 0 . 0 0 1 5 {111} + 5 . 5 0 . 7 9 8 5 ±0.0005 0 . 7 9 1 0 ± 0 . 0 0 2 7913 {111} + 1 0 . 0 0 . 7 8 5 0 ± 0 . 0 0 1 0 121 F i g u r e 4-10 The /S and Y a b s o r p t i o n peak e n e r g i e s as a f u n c t i o n of the a p p l i e d b i a x i a l s t r a i n . h a v i n g a sl o p e o f -10o9 ±1.0 eV per u n i t s t r a i n i n the JOO l j plane (dotted l i n e ) and -1208 +1.0 eV per u n i t s t r a i n i n the { i l l } plane ( s o l i d l i n e ) . The 1% peak was, i n some cases, d i f f i c u l t t o i d e n t i f y due t o the p r o x i m i t y o f the s t r a i n s p l i t ©<. peaks (the <=<yz peak f o a t e n s i l e s t r a i n and the c<\/x peak f o r a compressive s t r a i n ) . However, when a u n i a x i a l compressive s t r e s s was a p p l i e d , i n a d d i t i o n t o an a l r e a d y t e n s i l e b i a x i a l l y s t r a i n e d sample, and o b s e r v a t i o n was made w i t h If p o l a r i z e d l i g h t , the °< 3/z peak was suppressed and the ^ peak c o u l d then be e a s i l y i d e n t i f i e d (see F i g u r e A«5, curve c ) . I f the ^3 peak energy p o s i t i o n i s e x t r a p o l a t e d t o l a r g e r t e n s i l e s t r a i n s , i t s energy w i l l c o i n c i d e w i t h the energy o f the <=<y.,peak at a b i a x i a l s t r a i n o f approximately 10~3 . T h i s would make i d e n t i f i c a t i o n o f the ^3 peak v e r y d i f f i c u l t u n l e s s p u r e l y u n i a x i a l s t r a i n s were employed and o b s e r v a t i o n s were made w i t h 7T p o l a r i z e d l i g h t . When a compressive u n i a x i a l s t r e s s was a p p l i e d , t o an a l r e a d y b i a x i a l l y s t r a i n e d sample, the j3 peak was observed t o s h i f t t o h i g h e r photon energy as a f u n c t i o n o f a p p l i e d u n i a x i a l s t r e s s . F i g u r e 4.8 shows the j3 peak p o s i t i o n , o f a sample w i t h an i n i t i a l t e n s i l e b i a x i a l s t r a i n i n a }00l] pla n e , as a f u n c t i o n o f a p p l i e d u n i a x i a l s t r e s s a l o n g a <(001y> d i r e c t i o n . The peak i s observed t o s h i f t l i n e a r l y w i t h a p p l i e d u n i a x i a l s t r e s s w i t h i n t he experimental un-c e r t a i n t y . 123 4 . 4 * 2 DEPENDENCE OF THE t ABSORPTION PEAK ON STRAIN The If a b s o r p t i o n peak i s observed as a v e r y 'sharp peak, i n t h i c k u n s t r a i n e d samples, at a photon energy of 0 . 7 9 6 2 + 0 . 0 0 0 2 eV ( F i g u r e 4 . 9 ) . A t h i c k (175 ^m) sample was b i a x i a l l y s t r a i n e d , by cementing i t between two p i e c e s of 7 7 4 0 g l a s s . For t h i s sample the ^ peak was observed to s h i f t t o an energy of 0 . 7 9 1 0 eV (see F i g u r e 4 . 1 1 ) but no s p l i t t i n g was e v i d e n t . The peak was broadened s l i g h t l y ; p o s s i b l y i n d i c a t i n g a g r a d i e n t of s t r a i n through the sample. The y peak i s a l s o observed i n t h i n (2jixn. t o 10^/m) b i -a x i a l l y s t r a i n e d samples. However, the i n t e n s i t y of the peak i s c o n s i d e r a b l y s m a l l e r than f o r the t h i c k samples. The energy of the if peak, f o r the v a r i o u s b i a x i a l s t r a i n s , i s g i v e n i n Table 4 * 3 and p l o t t e d i n F i g u r e 4 . 1 1 . The t peak was o n l y observed i n one sample, b i a x i a l l y s t r a i n e d i n the [00l] p l a n e . The peak energy measured, f o r t h i s sample, i s the same as t h a t f o r the £LH} o r i e n t a t i o n t o w i t h i n the e x p e r i m e n t a l e r r o r . The fpeak was observed t o s h i f t l i n e a r l y w i t h s t r a i n , h a v i n g a s l o p e of - 1 1 . 7 ± 1 . 4 eV per u n i t s t r a i n . The i n t e n s i t y of the ^ or ^ peak under a u n i a x i a l s t r e s s , does not appear t o change as a f u n c t i o n of the p o l a r i z a t i o n d i r e c t i o n . However, p o l a r i z a t i o n e f f e c t s cannot be d e f i n i t e l y r u l e d out, s i n c e the u n i a x i a l s t r e s s measurements were c a r r i e d out f o r t h i n samples and the peak i n t e n s i t y was very s m a l l . I n F i g u r e 4 . 5 the ^ peak does appear t o be more i n t e n s e f o r the if p o l a r i z a t i o n . However, the apparent i n c r e a s e i n the 1 2 4 F i g u r e 4 . 1 1 The X a b s o r p t i o n peak f o r , ( a ) an u n s t r a i n e d GaSb sample and (b) a sample wi t h an i s o t r o p i c t e n s i l e b i a x i a l s t r a i n a p p l i e d i n the [ i l l } c r y s t a l l o g r a p h i c plane. 125 3^ peak c o u l d be due to the decrease i n i n t e n s i t y of the peak f o r 7T p o l a r i z a t i o n . Johnson ( 1 9 6 4 ) concluded t h a t these peaks were due t o bound e x c i t o n complexes. (A bound e x c i t o n complex c o n s i s t s o f a f r e e e x c i t o n bound t o e i t h e r a n e u t r a l or i o n i z e d donor o r a c c e p t o r . ) Krauze e t a l . ( 1 9 7 2 ) s t u d i e d the e f f e c t of u n i a x i a l compression on the photoluminescence of p-type GaSb. They concluded t h a t , s i l i c o n i m p u r i t i e s i n GaSb formed hydrogen-l i k e a c c e p t o r l e v e l s , w i t h an i o n i z a t i o n energy of 13 meV and t h a t the (f peak was due t o e x c i t o n s bound t o these s h a l l o w n e u t r a l a c c e p t o r s . They observed a s h i f t of the l i n e w i t h s t r e s s but d i d not r e p o r t any s p l i t t i n g of the l i n e . As has been d i s c u s s e d i n s e c t i o n 2 . 2 . 3 , B a i l e y ( 1 9 7 0 ) has shown t h a t an a b s o r p t i o n peak, a s s o c i a t e d w i t h any one o f the bound e x c i t o n complexes, would s p l i t i n t o two or more components. These components would e x h i b i t p o l a r i z a t i o n e f f e c t s under the a c t i o n of a u n i a x i a l s t r e s s . S i n c e s p l i t t i n g has not been observed f o r e i t h e r the j3 or ^peak, they can-not be a s s o c i a t e d w i t h a bound e x c i t o n complex. The main f e a t u r e s of the s t r a i n dependence of the p and y peaks, are t h e i r l a c k of observed s p l i t t i n g f o r e i t h e r o f the s t r e s s d i r e c t i o n s and the almost i d e n t i c a l energy s h i f t , w i t h s t r a i n , of both l i n e s . The f a c t t h a t the ^ and Y peaks do not s p l i t c o n t r a d i c t s the c o n c l u s i o n , r e g a r d i n g t h e s e peaks, reached by Johnson (I964) and by Krauze et a l . ( 1 9 7 2 ) . The l a c k of s t r a i n s p l i t t i n g and the almost i s o t r o p i c 126 s h i f t o f the V peak has a l s o been observed by P o l l a k e t a l . I t should be p o i n t e d out t h a t both the 3^ and % peak are due t o bul k e f f e c t s and not a s u r f a c e e f f e c t , s i n c e t h e i r i n t e n s i t y i n c r e a s e s as the sample t h i c k n e s s i n c r e a s e s . A t the p r e s e n t time no a l t e r n a t e e x p l a n a t i o n as to the A f t e r completion of t h i s t h e s i s C. Benoit a l a . Guillaume and P. L a v a l l a r d ( Phys. Rev. B, 4 9 0 0 , ( 1 9 7 2 ) ) p u b l i s h e d t h e i r P i e z o e m i s s i o n r e s u l t s f o r GaSb. I n t h e i r work they i n v e s t i g a t e d the e f f e c t of a u n i a x i a l s t r e s s on the /3 and X e x c i t o n l i n e s a t 2°K and 35"K. At 35*K they observed two p o l a r i z a t i o n dependent components of the/3 l i n e , however, a t 2°K they observed only one component of the /3 l i n e f o r any d i r e c t i o n of p o l a r i z a t i o n . For the l i n e a t 2° K they observed a s t r e s s induced s p l i t t i n g of the l i n e i n t o two components. T h i s d i f f e r e n c e i n behaviour of the two l i n e s , under an a p p l i e d s t r e s s , i s a t t r i b u t e d t o the j - j s p l i t t i n g of the two-hole s t a t e i n the * complex. They concluded t h a t the & and i complexes are combinations of an ( 1 9 7 1 ) . o r i g i n of the fi and $ peaks can be g i v e n . e x c i t o n w i t h an a c c e p t o r i n the n e u t r a l s t a t e . 127 CHAPTER 5 CONCLUSIONS In the pr e s e n t work, the o p t i c a l a b s o r p t i o n s p e c t r a of undoped GaSb h a v i n g a room temperature c a r r i e r c o n s e n t r a t i o n of 8 . 6 x 1 0 / f c cm-3, have been s t u d i e d w i t h the use o f : ( i ) t h i n samples (2 t o 5/m) mounted on g l a s s s u b s t r a t e s which were, when c o o l e d t o 2°K, i s o t r o p i c a l l y s t r a i n e d i n the plane of the samples. ( i i ) t h i n samples mounted on g l a s s s u b s t r a t e s w i t h an a n i s o t r o p i c s t r a i n i n the plane of the sample, produced by superimposing a u n i a x i a l s t r e s s on the i n i t i a l l y i s o t r o p i c b i a x i a l l y s t r a i n e d sample. ( i i i ) t h i c k (100 t o 200 yWm) u n s t r a i n e d samples at 2°K. Using t h e above sample mounting te c h n i q u e both t e n s i l e and compressive b i a x i a l s t r a i n s can be ob t a i n e d , whereas, w i t h other c o n v e n t i a l s t r a i n t e c h n i q u e s only compressive s t r a i n s can con-v e n i e n t l y be o b t a i n e d . The maximum t e n s i l e s t r a i n o b t a i n e d here i s 10'3 ( s t r e s s •» + 1 . 3 x 1 0 9 dyne cmz) and the maximum compressive s t r a i n o b t a i n e d i s - 4 . 2 x 10 ( ^ - 0 . 5 x 10 dyne cm"* ). The i s o t r o p i c b i a x i a l s t r a i n measurements on the t h i n samples were used t o determine the shear and h y d r o s t a t i c deformation p o t e n t i a l c o n s t a n t s . The shear component of the b i a x i a l s t r a i n causes the f o u r - f o l d degenerate ftv v a l e n c e band t o s p l i t i n t o two t w o - f o l d degenerate bands. The e x c i t o n a b s o r p t i o n peak, a s s o c i a t e d w i t h the t r a n s i t i o n from the va l e n c e band t o the e x c i t o n energy l e v e l , s p l i t s i n t o two 128 a b s o r p t i o n peaks as a r e s u l t of the s t r a i n s p l i t t i n g o f the v a l e n c e band» By showing t h a t , f o r the s t r a i n s used i n the p r e s e n t work, the s t r a i n dependence of the e x c i t o n b i n d i n g energy i s n e g l i g a b l e i n comparison to o t h e r energy e r r o r s , we can equate the e x c i t o n peak energy s p l i t t i n g t o the v a l e n c e band s p l i t t i n g . We can t h e r e f o r e use the e x c i t o n peak energy s p l i t t i n g t o c a l c u l a t e the v a l e n c e band shear and h y d r o s t a t i c d eformation p o t e n t i a l s . The v a l e n c e band shear deformation D w = + 3 . 5 ± 0 . 3 eV (b = - 2 . 4 ± 0 . 2 eV) and f o r a b i a x i a l s t r a i n i n the {ill} plane i s T>L = + 4 . 4 ± 0 . 3 eV (d = - 5 . 1 ± 0 . 3 eV). The s i g n s of the shear deformation p o t e n t i a l s have been determined from the u n i a x i a l s t r a i n measurements u s i n g p o l a r i z e d l i g h t and the o p t i c a l s e l e c t i o n r u l e s f o r the t r a n s i -t i o n s i n v o l v i n g the v a l e n c e bands. These v a l u e s of the de-f o r m a t i o n p o t e n t i a l s are i n good agreement w i t h the v a l u e s c a l c u l a t e d from Lawaetz Ts t h e o r y which p r e d i c t s ( P o l l a k e t a l . ( 1 9 7 1 ) ) b = - 2 . 3 eV and d = - 5 . 2 eV. There i s a l s o a good agreement f o r the v a l u e of d, p r e d i c t e d by the t h e o r y of G a v i n i and Cardona ( 1 9 7 0) which i s ( P o l l a k e t a l . ( 1 9 7 1 ) ) d = - 5 . 1 ± 1 . 4 eV and t o within e x p e r i m e n t a l e r r o r f o r t h e i r v a l u e of b = - 2 . 7 ± 0 . 3 5 eV. The h y d r o s t a t i c d e f o r m a t i o n p o t e n t i a l (D^ - ) has been found t o be - 1 0 . 4 + 0 . 9 eV per u n i t s t r a i n a p p l i e d i n the [OOl] p i ane and - 8 . 9 ± 0 . 5 eV per u n i t s t r a i n a p p l i e d i n the {ill} p i ane. I t was found, i n the p r e s e n t work, t h a t the p o t e n t i a l f o r a b i a x i a l s t r a i n i n the plane i s 129 v a r i a t i o n of the h y d r o s t a t i c deformation p o t e n t i a l , f o r d i f f e r e n t s t r a i n d i r e c t i o n s , cannot be e x p l a i n e d by the v a r i a -t i o n of the e x c i t o n b i n d i n g energy. S i n c e , f o r the s t r a i n s used here, the v a r i a t i o n of the e x c i t o n b i n d i n g energy wi t h s t r a i n i s l e s s than the e r r o r s a s s o c i a t e d with the determina-t i o n of the e x c i t o n energy peak p o s i t i o n . From the u n i a x i a l s t r a i n measurements u s i n g p o l a r i z e d l i g h t , i t was found t h a t f o r a t e n s i l e s t r a i n , the <^j4 e x c i t o n a b s o r p t i o n peak l i e s a t a h i g h e r photon energy than the e x c i t o n a b s o r p t i o n peak. T h i s means t h a t the v2 (J=3/2, ni j =+3/2) valence band i s c l o s e r , i n energy, t o the con-d u c t i o n band than the v l (J = 3/2, mj = +l/2) valence band (see F i g u r e 2.3). For a compressive s t r a i n the s i t u a t i o n i s r e -ve r s e d w i t h the e x c i t o n a b s o r p t i o n peak l y i n g a t a hi g h e r photon energy than the ^ e x c i t o n a b s o r p t i o n peak. The d e t e r m i n a t i o n of the d i r e c t i o n of motion of the valence bands, f o r an a p p l i e d s t r a i n , determines the s i g n s of the shear de-fo r m a t i o n p o t e n t i a l c o n s t a n t s and . The ^2 and Y a b s o r p t i o n peaks, f i r s t observed by Johnson (I964), were observed i n t h i c k u n s t r a i n e d samples at photon e n e r g i e s of 0.8055 ±0.0005 eV and 0.7962 ±0.0002 eV, r e s p e c t i v e These two a b s o r p t i o n peaks are a l s o observed i n the t h i n b i -a x i a l l y s t r a i n e d samples. However, t h e i r i n t e n s i t y i s con-s i d e r a b l y l e s s than f o r the t h i c k e r samples. The main f e a t u r e s of the s t r a i n dependence of the j3 and Y a b s o r p t i o n peaks are t h e i r l a c k of observed s p l i t t i n g f o r e i t h e r the f00l} or [ i l l ] s t r a i n d i r e c t i o n s and the almost i d e n t i c a l energy s h i f t w i t h 1 3 0 s t r a i n of both l i n e s . Over the r e g i o n of s t r a i n covered both the j$ and }f peaks are observed t o s h i f t l i n e a r l y w i t h a p p l i e d s t r a i n . These two peaks do not e x h i b i t p o l a r i z a t i o n e f f e c t s f o r e i t h e r a u n i a x i a l or a b i a x i a l s t r a i n . Johnson (1964) concluded t h a t the 3^ and Y a b s o r p t i o n peaks i n v o l v e d a bound e x c i t o n - i m p u r i t y complex. B a i l e y (1970) has shown t h a t an a b s o r p t i o n peak a s s o c i a t e d w i t h a bound e x c i t o n complex would s p l i t i n t o two or more components and would e x h i b i t p o l a r i z a t i o n e f f e c t s under the a c t i o n of a u n i a x i a l s t r e s s . The r e s u l t s r e p o r t e d here t h e r e f o r e con-t r a d i c t the c o n c l u s i o n reached by Johnson r e g a r d i n g these two peaks. At the p r e s e n t time no a l t e r n a t e e x p l a n a t i o n as to the o r i g i o n of the and Y a b s o r p t i o n peaks can be g i v e n . However, i t can be s t a t e d t h a t these two peaks are due to bulk and not s u r f a c e e f f e c t s s i n c e t h e i r i n t e n s i t y depends upon sample t h i c k n e s s . 131 APPENDIX A WAVELENGTH CALIBRATION OF THE EBERT MONOCHROMATOR I t was found necessary t o i n v e s t i g a t e the monochromator wavedrive mechanism i n d e t a i l ±x order t o determine the r e l a t i o n between the d r i v i n g mechanism and the c o r r e s p o n d i n g wave-l e n g t h . The wavedrive mechanism c o n s i s t s o f a synchronous motor, connected, through a s e r i e s o f v a r i a b l e speed gears, t o a p r e c i s i o n wavedrive screw. The wavedrive screw was then connected, by means o f a d r i v i n g nut, t o the g r a t i n g through a s e r i e s o f i n t e r c o n n e c t i n g r i g i d arms. A mechanical counter, connected t o the d r i v i n g screw, recorded each complete t u r n o f t he screw. A cam and m i c r o s w i t c h assembly, connected t o the d r i v i n g screw, was used t o produce an e l e c t r i c a l p u l s e output which c o u l d be recor d e d on the c h a r t r e c o r d e r t o i n d i c a t e one complete r o t a t i o n o f the d r i v e screw. A schematic diagram o f the g r a t i n g d r i v e mechanism i s shown i n F i g u r e A l . A c c u r a t e measurements were made of the wavedrive mechanism and the f o l l o w i n g c a l i b r a t i o n procedure was developed. The r e l a t i o n s h i p between the angle, ok, t h a t the normal t o the g r a t i n g makes w i t h the spectrometer a x i s (see F i g u r e A l ) and t h e wavelength, X, f o r t h i s p a r t i c u l a r spectrometer i s g i v e n by Cobb (106l) as, (A l ) F i g u r e A l Schematic diagram of the Ebert monochromator wave d r i v e mechanism. H 1 Co to 133 A l l measurements were made u s i n g the second o r d e r so t h a t m=2 and eq u a t i o n ( A l ) reduces t o TV = 3 . 3 2 5 s i n (c< ) ( jxm) ( A 2 ) I f a r e l a t i o n can be e s t a b l i s h e d between the number i n d i c a t e d on the wavedrive r e v o l u t i o n c o unter, attached t o th e d r i v i n g screw, and the angle, <X, then the wavelength can be c a l c u l a t e d u s i n g equation (A2). I t can be shown t h a t t h e angle, o<> i s r e l a t e d t o the o t h e r q u a n t i t i e s , i n d i c a t e d i n F i g u r e A l , by the f o l l o w i n g e q u a t i o n oi = 90 - X - s i n ^ / V + VD2 - A 2' I C I t was found e x p e r i m e n t a l l y t h a t the r e l a t i o n between the wavedrive cou n t e r number ( c a l l e d DRIVE i n the computer program) and t h e d i s t a n c e F i s F = 2 . 5 4 x ( DRIVE - 1 5 5 . 0 ) / 4 8.O Icm.) ( A 4 ) The d i s t a n c e A+B, C and D were measured u s i n g a micrometer and the f o l l o w i n g v a l u e s were o b t a i n e d C - 33.388 cm. D = 1 2 . 4 6 5 cm. ( A 5 ) G = A + B = 3 4 . 5 7 1 cm. The d i s t a n c e 5 A , i s r e l a t e d t o the o t h e r known q u a n t i t i e s by the eq u a t i o n C 2 - D 2 - F 2 - G 2 = 2F VD2 - A 2' - 2AG ( A 6 ) S o l v i n g f o r A g i v e s A = -WG ±Vw2G2 - (G 2+F 2)(W 2 - 4 F 2 D 2 ) ' 2 ( G 2 + F 2 ) (A7) ( A 3 ) where W = C 2 - D 2 - F 2 - G 2 1 3 4 The p o s i t i v e s i g n i n e q u a t i o n (A7) i s f o r the d i s t a n c e F > 0 ( i . e . DRIVE ^ 1 5 5 . 0 ) and the n e g a t i v e s i g n i s f o r F < 0 (DRIVE < 1 5 5 . 0 ) . E q u a t i o n ( A 7) was s o l v e d by the a u t h o r T s s u b r o u t i n e c a l l e d SOLVEA, which i s g i v e n at t h e end o f t h i s appendix. The o n l y remaining unknown q u a n t i t y i s the angle,K. T h i s angle can be determined by u s i n g a s e r i e s o f standard wavelengths t o determine the angle o< and then angle If can be found u s i n g equations A3 and A4. Once angle Y i s determined t h e n the r e l a t i o n between wavedrive number and wavelength i s e s t a b l i s h e d . Computer programs were w r i t t e n t o s o l v e the above equ a t i o n s and t o determine the angle X . The l e t t e r s A t o F, used t o i n d i c a t e the d i s t a n c e s shown on the diagram, are the same as those used i n the computor programs. In o r d e r to c a l i b r a t e the spectrometer the e x i t dewar system was f i r s t evacuated u s i n g the roughing pump. N a t u r a l gas (Methane, CH^) was then admitted t o the dewar system up t o a p r e s s u r e of about 4 5 cm. Hg. The 1 . 7 ^m Methane ab-s o r p t i o n band was r e c o r d e d , and the Methane was then immediately pumped out and the system was f l u s h e d w i t h a i r . The 1 . 4 / a m atmospheric water vapour a b s o r p t i o n band was t h e n r e c o r d e d . The standard wavelengths f o r these a b s o r p t i o n bands, g i v e n by P l y l e r e t a l . ( 1 9 5 2 ) , were 'read o f f the c h a r t s t o t h e n e a r e s t 0 . 0 2 of a wavedrive number u s i n g a t r a n s p a r e n t o v e r l a y . (The e r r o r i n r e a d i n g the wavelength 135 was l e s s t h an + 0.02 o f a wavedrive number which corresponds o t o approximately 1A at 1„4 jU1^. The e r r o r i n the f i t t e d wavelengths r a r e l y exceeded 1A). The stand a r d wavelengths are then run through the program c a l l e d WAVE4 which s o l v e s f o r the angle X ( c a l l e d ANGG) f o r each standard wavelength and wavedrive number. Normally, X , should be a constant f o r a l l wavelengths given, but i t was found t h a t V changed by about 0.1% over the r e g i o n o f i n t e r e s t . Angle X was t h e r e f o r e l e a s t squares f i t t e d by s u b r o u t i n e LSFT t o g i v e If as a f u n c t i o n o f the wavedrive number. I t was found t h a t angle X v a r i e d l i n e a l l y w i t h wavedrive number, and, u s i n g the computer program n o t a t i o n , was g i v e n by ANGG = SLOPE x DRIVE + GAMMA , (A8) Over t h e c a l i b r a t e d r e g i o n , t he wavelength can now be determined f o r any g i v e n wavedrive number by the f o l l o w i n g procedure (done by s u b r o u t i n e WAVLEN) (1) c a l c u l a t e F u s i n g e q u a t i o n (A4) (2) c a l c u l a t e A from s u b r o u t i n e SOLVEA (3) c a l c u l a t e a ngles f$ and Y from ji : BET = S I N " 1 ^ + D 2 - A 2 j X : GAM = SLOPE x DRIVE + GAMMA (4) c a l c u l a t e angle c* from: o< : AL = 1 . 5 7 0 7 9 6 3 - (GAM + BET) (5) c a l c u l a t e wavelength from e q u a t i o n (A2) 136 ft WAVE4 NEEDS SUBROUT IMES CA LCUL,LSOFIT,SOLVE A, 1 C IF ICODE = 1 THE PROGRAM WILL READ ALAMD AND DRIVE OFF C THE SAME CARD c c IF IF ICODE - 2 THE PROGRAM WILL READ THE CARDS WITH ALAMD FIRST AND THE CARDS WITH DRIVE SECOND SEVERAL RUNS ARE TO BE READ PUT ALAMD(LAST) = 0.0 r c c IF ONLY ONE RUN IS TO BE READ IN PUT ALAMD(LAST) = -1.0 D IMEMS 10N ALAMD( 100 ) .DRIVE( 100 ),A( 100 ) ,B( 100 ) ,E( 100 ) ,F( 100 ) , 2ANGB( 100 ) , ANGD ( 100 ) ,CARD(20)", " " " ~ " 3ANGA( 100 ) ,AMGG( 100 ) ,H(100 ) ,YF IT( 100 ) , YD I F F ( 100) YES=0.0 52 10 M0=1 ANO=NO 1=0 READ 10,CARD FORMAT ( 2 0 A 4 ) PRINT 11, CARD 11 1 2 FORMAT (1H1 ,20X,20A4,// ) RITE =YES READ ( 5 , 1 ) ICODE FORMAT ( 1 1 1 ) GO TO (40,2 ) , ICODE 1=1+1 3 4 ' 5 — READ ( 5 , 3 ) A LAMD(I ) FORMAT ( 1 F 1 0 . 4 ) IF (A LA MD( I ) ) 4,4,2 j = o ~"~ " " "~ ~ ' J = J+1 READ ( 5 , 7 ) D R I V E ( J ) 7 40 FORMAT ( 1 0 X , IF 10.4) IF ( D R I V E ( J ) .GT.0 .0 ) GO TO 5 IF ( J .ME . I ) GO TO 6 IF (ALAMD(I ) ) 50 , 51 ,6 1=1+1 READ ( 5 , 4 1 ) ALAMD(I ) ,D R I V E ( I ) 41 50 51 FORMAT ( 2 F 1 0 . 4 ) IF (ALAMD{I ) ) 50,51,40 R E P E A T = N 0 GO TO 4 9 RE PEAT= YES ' GO TO 4 9 49 13 1=1-1 C0MST=3.32 5 C=33.388 D=12 .465 G=34.571 CALL CALCUL (C,D,G,A LAMD,DRIVE,CONST,GAMMA,SLOPE,I,RITE) 2 4 WRITE ( 6 , 2 4 ) GAMMA,SLOPE FORMAT ( 6 X , 1 F 8 . 5 , 1 E 1 6 . 7 ) IF (REPEAT.EO.YES ) GO TO 52 6 GO TO 8 COMTIMUE WRITE ( 6 , 9 ) 9 8 FORMAT ( 10X,'EXECUTION HAS BEEN TERMINATED BECAUSE DATA UNEQUAL') CONT INUE STOP END ' " " 1 3 7 SUBROUTINE LS FT (OR IVE , ANGG,N,P,K,YF , YD IF F ) DI MENS ION D R I V E ( 1 0 0 ) ,ANGG(100),YF( 100 ) ,W( 1 0 0 ) , E 1 ( 5 0 ) ,E2( 5 0 ) , P ( 5 0 ) 1,YD IFF ( 100) COMMON M M = K N I = 1  EP = 0.000 1 DO 1 I=1,M P ( I ) = 0 . 0 •' _ _ _ 1 CONT INUE " " ' ~ """" " ~ " ~ EXTERNAL AUX CA LL- PPL OF (DRIVE,ANGG,YF,W,E1,E2,P,0.0,N,M,Nl,MP,FP,AUX )  WRITE ( 6 , 4 0 ) 40 FORMAT ( 64H ESTIMATES OF ROOT MEAN SQUARE STAT I S T I C A L FRROR IN PAR 1AMETERS ) _ _ _ _ _ _ WRITE ( 6 , 5 ) ( E l ( I ) ,1 = 1 , M T " " ~ ~ WRITE ( 6 , 4 ) 4 FORMAT (60H ESTIMATES OF ROOT MEAN SQUARE TOTAL ERROR IN THE PARAM 1ETERS ) WRITE (6,5 ) (E2 ( I ) ,I = 1,M ) 5 FORMAT (1X,RG15.5) WRITE ( 6 , 6 ) " " ' " ~ ~ " 6 FORMAT (43H VALUES OF X VALUES OF Y FITTED VALUES OF Y ) DO 7 I = 1,N  YD I FF( I ) =YF(I )-AMGG( I ) 7 WRITE ( 6 , 5 ) DRIVE( I ) ,ANGG( I ), YF( I ) W R IT E ( 6 , 8 ) 8 ' FORMAT (1H1) " " " - - - - - -10 CONTINUE RETURN  END FUNCTION AUX ( P , P , X , L ) DIMENSION P ( 5 0 ) , P ( 5 0 ) COMMON M ' " P( 1 ) = 1 .0 AUX= p( i ) ; . PO 10 J=2,M . " D ( J ) = D ( J - l )*X 10 AUX = AUX+P(J ) * D ( J ) RETURN " END 138 r S W R T J U T T ^ T T ^ L ^ F A ^ T ^ C , D , F , G , ITE L , R I I E l ' ' ' ' •DOUBLE PRECIS ION A , Z , C , D , F , G , D S O R T Y (A ) = 2 . 0 * F * D S O R T ( D * D - A * A ) - 2 . 0 # A * G A N O = - l . 0 IF ( R I T E . E Q . A N O ) GO TO 13 , WRITE ( 6 , 1 4 ) r P + FTJRMAT ( IETF) ; ' 13 CONTINUE X = ( C * C - D * D - G * G ) - F * F COUNT=0.0 ~" ~ " " " "' A = 0 . 0 Z=1.0  C C TEST TO SEE IF A IS P O S . OR N E G . C XY = DSORT( (D + F ) * (D+F)+G*G ) IF ( C . G T . X Y ) GO TO 3 _ C I A=A+Z CDUNT = C0UMT + 1 .0 IF (COUNT. G T . 1 0 0 . 0 ) GO TO 7 S = D * D - A * A " ~ " ' " IF ( S . L T . O . O ) GO TO 2 W = Y( A )  v=w-x : VA = ABS(V ) IF ( R I T E . E O . A N O ) GO TO 12 WRITE ( 6 , 1 1 ) C O I J N T , F , A , Z , S , W , V , V A ~ ~ " I I FORMAT ( 6 X , 1 F 6 . 1 , 2 D 1 4 . 7 , 2 X , 1 D 1 6 . 8 , 2 X , 4 D 1 6 . 6 ) IF ( V A . L T . D E L ) GO TO 5  12 CONTINUE IF (V ) 4 , 5 , 1 4 _ A = A - Z _ _ _ z = z / I O . O ' " ~ ~ ' • • A = A - Z GO TO 1  3 A=A-Z C0UMT = C0UMT + 1 .0 IF ( COUNT .GT . 100.0_)_ GO TO 7___ S = D * D - A * A ~ ~ " " " " " ~~ ~ " IF ( S . L T . 0 . 0 ) GO TO 2 W = Y(A )  v=w-x VA=ABS(V) WRITE ( 6 , 1 1 ) COUNT t F i A T Z t S t ^ i J ' ^ V A . _ _ ^ IF ( V A . L T . D E L ) GO TO 5 - - •• - - - • - - — •-. I F ( V ) 3 , 5 , 6 6 A=A+Z Z=T7ToTo A = A+Z GO TO 3 2 " ' WRITE ( 6 , 9 ) A , C , 0 , F , G " " 9 FORMAT ( 6 X , 4 8 H T R Y I N G TO TAKE SQUARE ROOT OF N E G . NUMBER FOR A= , 2 1 F 8 . 3 , 3 H C = , 1 F 8 . 3 , 3 H D = , 1 F 8 . 3 , 3 H F = , 1 F 8 . 3 , 3 H G = , 1 F 8 . 3 / )  GO TO 5 7 WRITE ( 6 , 8 ) A , C , D , F , G 1 3 9 fH FORMAT ( 6 X , , S 0 L V E A HAS E X C E E D E D 1 0 0 I T T E R A T I O M S FOR THE V A L M F "fl= 1 2 1 F 8 o 3 , 3 H C = , 1 F 8 „ 3 , 3 H D = , 1 F 8 . 3 , 3 H F = , 1 F 8 . 3 , 3 H G = , 1 F 8 . 3 / ) W R I T E ( 6 , 1 0 ) V , Z 10 FORMAT ( 1 0 X , 1 8 H T H E E R R O R IN A I S , IF 1 0 . 5 , 3 0 H W I T H AN I N C R E M E N T A L CH 2 A N G E OF , 1 F 1 0 . 8 / ) v 5 CONT INUE  f R E T U R N END SUB ROUT INE C A LCUL < C , D , G , AL AMD , DR I V E ,CONS T,GAMMA,SLO P E »'I ,RITE ) DIMENSION A(100 ) ,B( 100 ) ,E( 100),F( 100) VH( 100) , ANGA( 100) , ANGB( 100) , 2ANGD(100),ANGG( 100),ALAMD(100) ,DRIVE( 100 J,YFIT( 100) ,YDIFF( 100) 3 ,AFI T (100 ) ,ADIF (100 ) ,P (25 ) < AN0=1.0  ( YES=0.0 DO 60 J=1 , I F ( J ) = ( ( D R I V E ( J ) - 1 5 5 . 0 ) # 2 . 5 4 ) / 4 8 . 0 ! c ! c THIS SECTION TESTS TO SEE IF A REAL SOLUTION IS POSSIBLE I C  PV=SORT(F(J)*F(J)+G*G) IF ( C . G T . ( P V - D ) . A N D . C . L T . ( D + PV) ) GO TO 7 WRITE (6,1) C , D , G , J , F ( J ) , J , D R I V E ( J ) j l FORMAT (6X,52HAN IMPOSSIBLE SOLUTION HAS OCCURED FOR THE VALUES C= | 2 , 1 F 8 . 3 , 3 H D=,1F8.3 ,3H G = , 1 F 8 . 3 , 3 H F ( , 11 2, 2H ) = , 1F8 . 3 ,7H DRIVE( ,1 I2 j 3,2H) = , 1 F 8 . 3 / )  1 GO TO 72 ! C : 7 CONTINUE 1 CALL SOLVEA <A ( J),CT D , F(J),G»0.001,-1.0) B(J )=G-A(J ) : E( J ) = SQRT( D»»2-A( J )**2 ) [ H(J)=E(J ) + F(J ) ANGB(J)=ARSIN(H(J) /C) ANGD(J)=ARSIN(A(J) /D) I 60 CONTINUE : TJO 62 K = 1 , I AL=(ALAMD(K)*l .0 E-04 ) /CONST ... .  A N GA ( K ) = A R S IN ( A L ) ANGG(K) = 1 .57079 6-(ANGA(K)+ANGB{ K)) ANGG(K) = (ANGG(K5-0.75)*1 000.0 62 CONTINUE CALL LS FT < DRIVE , ANGG , I ,P ,2 , YF IT, YD IFF >-DO 63 JK=1,I ' ' . ANGG(JK)=(ANGG(JK)*0.001)+0.75 63 CONTINUE GA MMA=(P(1)*0.001)+0.75 SLOPE = P( 2 ) :^0.001 IF (RITE.EO.ANO) GO TO 72 WRITE (6 ,6)  6 FORMAT (10X , 10HWAVELENGTH, IX,9HWAVEDRIVE,4X,1HA,8X,1HB,8X,1 HE,8X, 21HF,6X ,5HALPHA,4X,4HBETA,5X,5HDELTA,5X,5HGAMMA) 00 70 L=1,I WRITE (6,5) L,ALAMD(L) ,DRIVE(L ) »A( L ) , B ( L ) » E ( L ) , F ( L ) , A N G A ( L ) , 2ANGB(L) ,ANGD(L ) ,ANGG(L) _5 FORMAT ( 6X ,1 13, 1 F U . 3 , I F 1 0 . 4 , 10F9 .4)  , 70 CONT INUE j WRITE (6 ,9) GAMMA,SLOPE 9 FORMAT (/6X,7HGAMMA =,1F12.7 ,6X,7HSL0PE =,1E16.8) ALDIF2=0.0 DO 71 M=1,1 GAM = SLOPE-DRIVE(M)+GA MMA  ! BET=ARSIN(H(M)/C) ! ALPH=1.5707963-(GAM+BET) j ALFIT=C0NST*SIN(ALPH)*1.0E+04 ! ALD IF=AL.AMD( M J-ALFI T I ALDIF2=ALDIF2+ALDIF**2 L WRITE (6 ,8) DRIVE(M) ,GAM,8ET,ALPH,ALAMD(M) ,ALFIT ,ALDIF 141 8 FORMAT (6X ,7F12 .4 ) XL aiNTJNUIF. WRITE (6,2) WRITE (6,3) GAMMA,SLOPE,ALDIF2 2 FORMAT ( / / , 1 0 X , ' R Y FITTING THE ANGLE GAMMA AND GETTING') 3 FORMAT (20X,'GAMMA = ' , 1 F 1 2 . 5 , 6 X , ' S L O P E = ' , 1 E 1 6 . 8 , / 1 5 X , ' T H E SUM 20F THE SQUARES OF THE DIFFERENCES Of THE WAVELENGTHS I S ' , 1 F 1 2 . 4 ) _^ CALL LSFT (DRIVE,ALAMD, I , P , 2 , A F I T , A D I F )  AD IF 2=6.0 DO 10 N=1,I ADIF2=ADIF2+ADIF(N)**2 10 CONTINUE WRITE (6,4) WRITE (6,3) GAM , SLO , AD I F2 :  4 FORMAT ( / / 1 0 X , ' B Y FITTING THE WAVELENGTH AND GETTING') 72 C O N T I N U E RETURN END 142 SUBROUTINE WAVI EN (DR IVE,SLOPE,GAMMA,WAVE•AL) DOUBLE PRECISION A , C , D , G , F C=33.388 0=12 .4 65 G = 34.571 F= (1 DRIVE-155 .0 )*2 .54 ) / 48 . 0 CALL SOLVEA ( A , C , D , F , G » 0 . 0 0 0 1 , - 1 . 0 ) E = DS 0RT(D*D-A*A ) EI = E-!-F BET=DARS IN( H/C ) GAM = SL 0PE »DR I V E + GAMMA AL = 1 .57079 63-(GAM + 3ET) WAVE=3.325*SIN(AL)* 1.OE + 04 RETURN END 1 4 3 APPENDIX B WOOD'S METAL SEAL ON HELIUM IMMERSION DEWAR The Helium immersion t a i l was j o i n e d t o the main Helium dewar by means of a Wood's Metal s o l d e r j o i n t . The Wood's Metal s e a l was used i n o r d e r t o reduce the amount of heat r e q u i r e d on the dewar t a i l , s i n c e g l a s s - t o - m e t a l s e a l s can be damaged by e x c e s s i v e o r uneven h e a t i n g . I Woods Metal Copper Glass-to Metal Seal Glass F i g u r e BI D e t a i l e d diagram o f the Wood's Metal s e a l . Before assembling, the Wood's Metal j o i n t was c a r e f u l l y t i n n e d w i t h Wood's Metal and a l i q u i d f l u x , c o n s i s t i n g o f 40 p a r t s ZnCl, 20 p a r t s NH^Cl and 40 p a r t s o f H20 by weight. Once the j o i n t was p r o p e r l y t i n n e d , o n l y o c c a s i o n a l c l e a n i n g was n e c e s s a r y . O c c a s i o n a l l y the Wood's Metal would d e t e r i o r a t e ( i t appeared t o become c r y s t a l i n e upon s o l i d i f i c a t i o n ) and would have t o be r e p l a c e d . A l s o , any f i l m on the Wood's Metal would have t o be skimmed o f f i n or d e r t o o b t a i n a good s e a l 0 At a l l times d u r i n g s o l d e r i n g , the g l a s s - t o - m e t a l s e a l was p r o t e c t e d by a damp c l o t h . 1 4 4 Poor s e a l s can sometimes be d e t e c t e d by pumping on the dewar at room temperature.However, they are u s u a l l y o n l y d e t e c t e d upon t r a n s f e r r i n g l i q u i d Helium. O c c a s i o n a l l y Helium I I l e a k s , through the Wood's Metal s e a l , were present which o n l y became apparent a f t e r the Helium bath was pumped below the }\- p o i n t . A slow t r a n s f e r of l i q u i d Helium i n t o the dewar i s a l s o a d v i s e d i n o r d e r t o reduce the p o s s i b i l i t y of t h e r m a l l y s h o c k i n g the g l a s s - t o - m e t a l seal» 1 4 5 APPENDIX C FORMULA FOR THE ABSORPTION COEFFICIENT FOR  AN ABSORBING MEDIUM ON A SUBSTRATE The problem, of c a l c u l a t i n g the a b s o r p t i o n c o e f f i c i e n t f o r an a b s o r b i n g m a t e r i a l f i x e d t o a non-absorbing s u b s t r a t e , can e s s e n t i a l l y be t r e a t e d as a t h r e e - l a y e r problem,. The a b s o r b i n g medium i s assumed t o be bounded on one s i d e by a vacuum" and on the o t h e r s i d e by an i n f i n i t e non-absorbing medium(glass s u b s t r a t e ) . The epoxy, which secures the sample t o the g l a s s , has approximately the same index o f r e f r a c t i o n as t h e g l a s s and t h e r e f o r e can be c o n s i d e r e d as an i n t e g r a l p a r t of t h e g l a s s . For a p a r a l l e l beam o f l i g h t , o f u n i t amplitude, i n c i d e n t normal t o the s u r f a c e o f a m a t e r i a l , bounded on e i t h e r s i d e by an i n f i n i t e non-absorbing medium, t h e r e f l e c t e d , R, and t r a n s m i t t e d amplitude, T, f o r the m a t e r i a l are g i v e n by (Heavens ( I 9 6 5 ) page 57) R = r x + r 2 e " 2 i 8 , 1 2 . c ( C l ) T = t j t 2 e- 1*' 1 + r l r 2 e " 2 i S ' The index of r e f r a c t i o n o f l i q u i d Helium i s ve r y n e a r l y e q u a l t o 1 . 0 at 1 . 4 /tm, and t h e r e f o r e , no d e s t i n c t i o n i s made as to whether t h e sample i s bounded by a vacuum, a i r o r l i q u i d Helium. 1 4 6 Here, r-^, r 2 , t-|_, and t£ are the F r e s n e l c o e f f i c i e n t s and °> i s the change i n phase of the beam on t r a v e r s i n g the m a t e r i a l Vacuum n 0 = 1.0 Incident Beam Reflected Beam A absorbing f media 1 nj = n,-ik, glass j n 2 r | r | t t , Vacuum j N 3 = Transmitted Beam F i g u r e C l Schematic diagram of a two l a y e r m a t e r i a l con-s i s t i n g of an a b s o r b i n g media and g l a s s s u b s t r a t e bound on e i t h e r s i d e by a vacuum. I f the m a t e r i a l i s an a b s o r b i n g medium, the r e a l index of r e f r a c t i o n , ( n ^ ) , of the f i l m , i s r e p l a c e d by a complex index of r e f r a c t i o n , n^ = n^ - i k ^ , r e p r e s e n t i n g the a b s o r b i n g medium where k^ i s the e x t i n c t i o n c o e f f i c i e n t and r e p r e s e n t s the a t t e n u a t i o n of the wave per vacuum wavelength. The F r e s n e l c o e f f i c i e n t s (Heavens ( 1 9 6 5 ) page 5 1 ) f o r an a b s o r b i n g medium, then become 1 4 7 1 = ( 1 - n x) + ik-L = ( 1 + n 2) - i l ^ ' 2 = ( n x - n 2) - i k x = (n-^ + n 2) - i k j * i -1 + n. ik. * 1 — 1 t 2 = 2(n - i k ) = t. = t si/4 .i<=<;t (C2) (nj^ + n 2) - i l ^ Here, the index of r e f r a c t i o n of the vacuum i s 1 . 0 , n^ and k^ are as defined above and n 2 i s the index of r e f r a c t i o n of the glass substrate. The phase angle, $ i , defined i n equation C l , f o r an absorbing medium, can then be written as S, - a - i b (C3) where a = 2 it n^ d-^  b = 2Tri<1 d± A the quantity d-^  i s the thickness of the absorbing medium, and A i s the free space wavelength of the incident r a d i a t i o n . S u b s t i t u t i n g equation C2 and C3 into equation C l , the trans-mission amplitude of the l i g h t beam becomes T = [ t j |t 2| e i ( 0 ' ' + ^ e - i ^ a - : L b ) 1 + IrJIr-l e i C / ^ . + ^ e - 2 ^ a - i b ) or (neglecting absolute value signs) .iA ^-b T = t j t 2 e" 1 + r i r 2 e where: A = <*,+ o(z -a B = f,+ ft -2a iB - 2 b ( C 4 ) 148 The intensity, E_, of the transmitted beam in the non-absorbing substrate i s given by E T = n 2 TT* where T" i s the complex conjugate of T. Thus E„ T = t ^2 t , 2 e~ 2 b n. 1 + r 2 r 2 e-4b + 2r r e~ 2 b cosB JL _w ( C 5 ) 1 ~2 This expression predicts interference maxima and minima in the transmitted spectra of the layer. However, when there are several interference fringes within the spectral bandwidth, i t i s necessary to integrate equation C5 over the bandwidth. When there are several interference fringes within the band width the effect i s equivalent to integrating over exactly one interference fringe. In order to perform the integration, equation C5 needs to be expressed in a more easily integrable form. Consider the sum S = f x mcos(m9) = A f f ( x e i e ) m 4- (xe~±e)m] Now using the relation £ (xe 1*)" 1 = (1 - xe 1*) e s-1 we get S = m _ I 1 - xe 1 8 1 - xe 1 - xcos & 1 + x z - 2xcos d Now consider -1 + 2 _£ x mcosm0 = m-o 1 + x 2 - 2x cos & from which we get the series 1 _ 1 1 + x' 2xcos 9 x 1 + 2 X xmcosm6> •TO : / (C6) 149 U s i n g e q u a t i o n C 6 , equation C5 can be expressed i n the f o l l o w i n g form E T 4 - 2 , 2 .~2b n 2 1 - r x 2 r 2 2 e - 4 b 1 + 2 __ ( - r 1 r 0 e ~2 b ) c o s m B (C7) I f we i n t e g r a t e over one complete c y c l e , the c o s i n e terms a l l v a n i s h and we get the f o l l o w i n g r e l a t i o n f o r the t r a n s m i t t e d energy i n the non-absorbing s u b s t r a t e . 2, 2 _ o ( H t l t 2 e 0 1 a l T 1 T 2 e - c * d i 2 1 - r L 2 r 2 2 e ~ 2 o < d l 1 - e ~ 2 < * d l where o( — 4 T T ' k - L and i s the a b s o r p t i o n c o e f f i c i e n t and k-^  i s t h e e x t i n c t i o n c o e f f i c i e n t s A l s o T l = 1*1 | 2 = -(1 + n i ) 2 + k x 2 = 4(n| 2+ k x 2 ) < n l + n 2 > 2 + k l 2 2 = ( 1 " n l ) 2 + k l 2 l ) 2 + k i : T — R l = R, (1 + n i ) ' + k n 2 | 2 = v " l ~ "_2_'_ T _1_ r  2  ( n l " n ? } + k ^ 2 ( n x + n 2 ) 2 + k x 2 ( C 8 ) (C9) 150 APPENDIX D  REVIEW OF STRESS AND STRAIN COMPONENTS  AND THEIR TRANSFORMATIONS The s u b j e c t of s t r e s s and s t r a i n i s e x t e n s i v e l y d i s c u s s e d by Nye (1957) and w i l l o n l y be reviewed b e i e f l y here. A body which i s acted on by e x t e r n a l f o r c e s i s s a i d t o be i n a s t a t e of stress,, I f the f o r c e a c t i n g on the element o f s u r f a c e AAj has components ^F, then we can d e f i n e the t e n s o r fT;f7 = f l i m 4Ft- 1 (Dl) The symmetric p a r t of t h e t e n s o r , [Tijj, i s c a l l e d t h e s t r e s s t e n s o r , [of-] , and t h e antisymmetric p a r t of [T^j i s the d e n s i t y o f r e s u l t a n t t o r q u e which may be n e g l e c t e d i n t h i s d i s c u s s i o n . The normal s t r e s s e s , <f[i , are p o s i t i v e f o r t e n s i o n s and n e g a t i v e f o r ' c o m p r e s s i o n . The s t r a i n t e n s o r [(HijJ i s d e f i n e d as the symmetrical p a r t of t h e t e n s o r [&tj] where, e^= bul/hxj , r e p r e s e n t s the v a r i a t i o n o f t h e displacement, u; , w i t h p o s i t i o n , xj , i n a body. T h e r e f o r e , the s t r a i n component £;j i s g i v e n by €tf = _(ecj + e J t) = £ / a u _ + <W) (D2) \dxj dxi) The d i a g o n a l components, , o f , €lj, are the t e n s i l e s t r a i n s and the o f f d i a g o n a l components are t h e shear s t r a i n s . The normal s t r a i n s , £il , are p o s i t i v e when the medium i s d i l a t e d . The change i n volume o f a u n i t cube i s c a l l e d the d i l a t a t i o n , A , and, i f the symmetrical t e n s o r , f ^ j j , i s r e f e r r e d t o i t s p r i n c i p a l axes, i s g i v e n by A =Av/v = ( l + e„)(i + fi_/y)(i + e»)-i <D3) Hooke's law s t a t e s t h a t f o r s u f f i c i e n t l y s m a l l s t r e s s e s (below the e l a s t i c l i m i t " ) t he amount o f s t r a i n i s p r o p o r t i o n a l t o the magnitude o f the a p p l i e d stress,, S i n c e , i n g e n e r a l , t h e homogeneous s t r e s s and s t r a i n are both second rank t e n s o r s , t h e n the g e n e r a l i z e d form o f Hooke's law i s 6ij = S s t > , (D4) o r u s i n g t h e E i n s t e i n summation con v e n t i o n 6'j= sijkt Ojj ( i , j , k , l = x,y,z) where are the e l a s t i c compliances and are elements o f a f o u r t h order t e n s o r (81 components). The s t r e s s may a l s o be expressed i n terms o f the s t r a i n by t h e r e l a t i o n <y = c;; w £ u (DS) where the terms ctyA^ are t h e 81 s t i f f n e s s c o n s t a n t s o f the c r y s t a l (of which o n l y 21 are d i s t i n c t ) . Both t h e s t r e s s and the s t r a i n components can be w r i t t e n i n a s i n g l e s u f f i x n o t a t i o n by making the f o l l o w i n g s u b s t i t u t i o n s : xx ( 1 1 ) — * 1, yy ( 2 2 ) - * - 2 , zz ( 3 3 ) — » • 3 , y z , zy ( 2 3 , 3 2 ) — * 4 , xz,zx ( 1 3 , 3 1 ) — - 5 , and xz,zx ( 1 2 , 2 1 ) — - 6 . In a s i m i l a r manner the s^kf and c[jk( can be reduced t o two s u b s c r i p t s by r e p l a c i n g i j and k l w i t h the s i n g l e s u b s c r i p t as above. * The e l a s t i c l i m i t i s the maximum u n i t s t r e s s which can be o b t a i n e d i n a m a t e r i a l without c a u s i n g permanent deformation. 152 (D6) Equations D4 and D5 then have the form < = c- 6j ( i , j = 1. 6) I t should be p o i n t e d out t h a t f o r the purposes o f t r a n s -f o r m a t i o n s sij and s t i l l have t o be t r e a t e d as f o u r t h o r d e r t e n s o r s and not as second order t e n s o r s . For t h e c u b i c Tj c r y s t a l , c i j , has o n l y 3 independent v a l u e s and has the f o l l o w i n g form , 0 0 0 ^  c n c„ c ; l 0 0 0 C n C a C « 0 0 0 0 0 0 c w 0 0 0 0 0 0 c ^ 0 0 0 0 0 0 c 0>7) w i t h a s i m i l a r m a t r i x f o r [s<j]. For the c u b i c system, c£y and s4y are r e l a t e d as f o l l o w s S)) + S , % (s„ - s,2)(s„ + 2 s l t ) 'It -H4 Ts7 = 1 - S q  s* )(s„ + 2s l t ) (D8) with a s i m i l a r s e t o f equ a t i o n s r e l a t i n g sty i n terms o f c^ y . The volume c o m p r e s s i b i l i t y i s the p r o p o r t i o n a l decrease i n volume o f a c r y s t a l , when s u b j e c t e d t o u n i t h y d r o s t a t i c p r e s s u r e . F o r a c u b i c c r y s t a l the volume c o m p r e s s i b i l i t y i s = 3(s„ + 2s / z ) ( D 9 ) - A P The l i n e a r c o m p r e s s i b i l i t y o f a c r y s t a l i s t h e r e l a t i v e decrease i n the l e n g t h o f a l i n e , when the c r y s t a l i s s u b j e c t e d t o a u n i t h y d r o s t a t i c p r e s s u r e . For a c u b i c c r y s t a l the 153 l i n e a r c o m p r e s s i b i l i t y i s i s o t r o p i c and i s g i v e n by j8 = s„ + 2s/z (D10) The volume change of a c u b i c c r y s t a l under u n i a x i a l t e n s i o n , T, i s independent o f the d i r e c t i o n o f the t e n s i o n and i s g i v e n by AV = (s„ + 2 s / z )T ( D l l ) The q u a n t i t i e s £ty, <7ij and s^^j , c ty A 7 are "second and f o u r t h o r d e r t e n s o r s r e s p e c t i v e l y . When a c o o r d i n a t e t r a n s f o r m a t i o n i n v o l v e s o n l y c a r t e s i a n c o o r d i n a t e s , under i n f i n i t e s i m a l d i s t o r t i o n , as i n the case o f l i n e a r e l a s t i c i t y t h e o r y , the t r a n s f o r m a t i o n of the t e n s o r s has the g e n e r a l form e«y = **V* 6k< S;/kJ = c*vM *Jn °b% Jm«n ( D 1 2 ) where the otft are t h e d i r e c t i o n c o s i n e s between the o l d and new c o o r d i n a t e systems axes. For the work r e p o r t e d here the sample i s f i x e d t o a g l a s s s u b s t r a t e . When the sample and g l a s s are c o o l e d the sample i s b i a x i a l l y s t r a i n e d (which i s assumed t o be homo-geneous i n the sample plane) by t h e d i f f e r e n t i a l thermal c o n t r a c t i o n between the two m a t e r i a l s . I f we assume t h a t t h e b i a x i a l s t r e s s i s a p p l i e d i n the z = 0 plane then t h e s t r e s s i n the z - d i r e c t i o n (#£_) must be zero, s i n c e t h e sample i s i n e q u i l i b r i u m and no e x t e r n a l s t r e s s i s a p p l i e d d i r e c t l y i n t h a t d i r e c t i o n . The s t r a i n £ i t i n the z - d i r e c t i o n has t o be such t h a t the s t r e s s i s z e r o . T h e r e f o r e , from 154 e q u a t i o n D6 and D7, and u s i n g the f a c t t h a t <f^ = 0 we o b t a i n t h e f o l l o w i n g requirement on t h e s t r a i n i n the z d i r e c t i o n 6 3 = (6.+ 0 ( D 1 3 ) I f the s t r a i n s 6,and 6_ are known then the s t r e s s can be determined from the f o l l o w i n g r e l a t i o n s °1 = |c« - <_J _; + |c/2. - cjlj £ z (D14) <K = I f we have a uniform b i a x i a l s t r e s s such t h a t £, = GL then When a s t r a i n i s a p p l i e d a l o n g a d i r e c t i o n o t h e r than a p r i n c i p l e c r y s t a l axes, i t i s necessary t o perform a t r a n s -f o r m a t i o n such t h a t the s t r a i n s are g i v e n i n terms of the c o o r d i n a t e system a s s o c i a t e d w i t h t h e p r i n c i p l e c r y s t a l axes. A treatment o f the t r a n s f o r m a t i o n has been g i v e n by G l a s s (I964) and t h e r e f o r e , o n l y a b r i e f summary o f the r e s u l t s w i l l be g i v e n h e r e . C o n s i d e r the plane o f t h e sample t o be t h e x,y plane w i t h the z - d i r e c t i o n normal t o the s t r e s s 0 The s t r a i n i n t h e x,y plane i s b i a x i a l and i s g i v e n by 6X>= £yy = T and the s t r a i n i n t h e z - d i r e c t i o n i s fejf= - A T where A i s determined by the c o n d i t i o n t h a t the normal s t r e s s must v a n i s h under e q u i l i b r i u m c o n d i t i o n s . F i r s t , c o n s i d e r the b i a x i a l s t r a i n a p p l i e d to a [OOlj plane and t a k e t h e x,y,z axes to be the < 1^00),<t)10^> and <J001> c r y s t a l l o g r a p h i c axes. For t h i s plane, the c o o r d i n a t e system and the c r y s t a l l o g r a p h i c p r i n c i p l e axes are c o i n c i d e n t and 155 t h e r e f o r e , no t r a n s f o r m a t i o n i s n e c e s s a r y . From equation D13 we f i n d t h a t e3= _ _ (2T) (D15) t h e r e f o r e /\ = 2c/t For a b i a x i a l s t r e s s a p p l i e d t o a [ l l O ] plane, the sample axes and the c r y s t a l l o g r a p h i c p r i n c i p a l axes are no l o n g e r c o i n c i d e n t , and a t r a n s f o r m a t i o n has t o be performed. 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