Direct determination of the phase of reflectivity in CdS and ZnO in the exciton region

Solid State C o m m u n i c a t i o n s . Vol

11, pp 1 6 5 1 - 1 6 5 3 , 1972

Pergamon Press.

P r i n t e d m Great Britain

D I R E C T DETERMINATION OF THE PHASE OF R E F L E C T I V I T Y IN CdS AND ZnO IN THE EXCITON REGION I Fllmskl C h e m i c a l Laboratorb B, T e c h n . Umv. of Denmark, 2800 L y n g b y , Denmark and T Skettrup P h y s i c s Laborator~y III, T e c h n . Univ. of Denmark, 2800 L y n g b y , Denmark

(Received28 September 1972 b> L Hedm)

T h e p h a s e of r e f l e c t l v l t y h a s been m e a s u r e d m CdS and ZnO in the e x c i t o n region at 4.2 K. The s p e c t r a were recorded at normal i n c i d e n c e by e l h p s o m e t n c m e t h o d s , the e l h p t m ~ t y b e i n g reduced by the c r y s t a l d m h r m s m . T h e p h a s e s p e c t r a for the 4--exclton in CdS and the Aand B- e x c i t o n s in ZnO d e v i a t e d c o n s i d e r a b l y from the c o r r e s p o n d i n g s p e c t r a computed by means of K r a m e r s - K r o m g a n a l y s i s . It was found that this kind of b e h a v m u r may be e x p e c t e d in the p r e s e n c e of e x c l t o n free s u r f a c e l a y e r s and m the c a s e of s a m p l e m h o m o g e n e l t l e s .

IN ORDER to d e t e r m i n e the o p t m a l c o n s t a n t s of a medium from r e f l e c t i o n m e a s u r e m e n t s both the a m p l i t u d e and the p h a s e of r e f l e c t l v l t y must be known.

the o p t i c a l axz" ( c - a x i s ) of the c r y s t a l . T h e component of h g h t p o l a r i z e d along the c - a x i s (E ' c) is r e f l e c t e d with dKferent a m p h t u d e and p h a s e m the s p e c t r a l region of d m h r o l s m than the component of light p o l a r i z e d p e r p e n d i c u l a r to the c-ax~s (E I c). H e n c e , the r e f l e c t e d h g h t is e l h p t l c a l l y p o l a r i z e d . The p h a s e d i f f e r e n c e b e t w e e n the components Ell c and K J_ c c a n then be measured by means of an o p t i c a l c o m p e n s a t o r . The ratio of t h e two r e f l e c t l v l t y a m p h t u d e s ~s determined from the s l o p e of the h n e a r l y p o l a r i z e d light emerging from the c o m p e n s a t o r .

However, m e a s u r e m e n t s of the p h a s e of r e f l e c t l v l t y are often a v o i d e d , s i n c e t h e s e , in g e n e r a l , m v o l v e s t e d i o u s l n t e r f e r o m e t r m methods. I n s t e a d , the p h a s e of r e f l e c t l v l t y is computed from the a m p h t u d e of r e f l e c t l v l t y by means of Kramers-Kromg relatmns. We report m this l e t t e r p h a s e m e a s u r e m e n t s m CdS and ZnO by a new method a v o l d m g m t e r ferometry. T h i s is to our knowledge the first time the p h a s e of r e f i e c t l v l t y for e x c l t o n s p e c t r a h a s b e e n m e a s u r e d d l r e c t i y . The p h a s e s p e c t r a obt a i n e d m t h e s e e x p e r i m e n t s d e v i a t e from the corr e s p o n d i n g p h a s e s p e c t r a computed by means of the K r a m e r s - K r o m g r e l a t i o n s .

T h e p a r a m e t e r s of the e l h p t m a l l y p o l a r i z e d h g h t were r e c o r d e d d i r e c t l y by means of an automatin e l h p s o m e t e r to be d e s c r i b e d e l s e w h e r e .

The r e f l e c t m n m e a s u r e m e n t s were performed with normal i n c i d e n t hght p o l a r i z e d 45 ° a w a y from

1651

S i n c e the a b s o r p t i o n for E!] c Is n e g h g l b l e m the s p e c t r a l region of the 4--exclton m CdS ( q and B- e x c l t o n s m ZnO), the p h a s e of r e f l e c t l v l t y for El * c m zero m t h i s region. Hence, t h e m e a s u r e d p h a s e d i f f e r e n c e dzrectly y i e l d s the p h a s e spectrum for the A- e x c l t o n m CdS and the A- and B- e x c l t o n s

1652

CdS AND ZnO IN THE E X C I T O N R E G I O N

in ZnO For higher energ5 excItons the recorded s p e c t r a are l e s s useful s i n c e both reflectl~,t~ components (E c and E ± c) ~ary w~th energ}

Vol

11, No 12

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FIG. 1. P h a s e s p e c t r a in the e x c l t o n region of ZnO at 4.2°K. The arrows m d m a t e the minima of r e f l e c t t w t y for the different e x c l t o n s . The full h n e is the measured spectrum which in the region of the 4- and B- e x m t o n s is the phase of reflect l v l t y , while it at higher e n e r g i e s yields the phase difference b e t w e e n the hght polarized with E i c and E ± c. The dashed h n e is the p h a s e spectrum o b t a i n e d from K r a m e r s - K r o m g a n a l y s m of the c o r r e s p o n d i n g reflection spectrum. The dots r e p r e s e n t the phase c a l c u l a t e d from the model i n v o l v i n g an e x c l t o n free s u r f a c e layer with a t h m k n e s s of 60A.

In the following we s h a l l only c o n s i d e r the s p e c t r a of the 4- e x c l t o n for CdS and the 4- and B- e x c I t o n s for ZnO, t a k e n at 4 2°K T h e s e s p e c tra exhibit an unexpected b e h a w o u r ( F i g s 1 and 2) T h e r e f l e c t i o n s p e c t r a are s t m d a r to those earlier reported, 1.2 and are not shown in the figures. For CdS the maxlmum of phase occurs near the r e f l e c t t v l t y minimum as expected, but the phase is c h a n g i n g s i g n , before returning to zero For ZnO the phase goes through aI1 v a l u e s bet w e e n 0 and 2~, and there is no maximum of p h a s e T h e K r a m e r s - K r o m g transforms are a l s o shown m t h e figures The largest d i s c r e p a n c i e s b e t w e e n the experimental and K - K transformed curves occur near the l o n g i t u d i n a l r e s o n a n c e energy where the phase is e x p e c t e d to be large. In this region the phase of CdS becomes n e g a t i v e , while the phase ot ZnO c o n t i n u e s to i n c r e a s e i n s t e a d of d e c r e a s i n g T h i s behavlour of the phase s p e c t r a of ZnO is observed

A

B

FIG 2 P h a s e s p e c t r a m the e x c l t o n region of CdS at 4 2°K The arrows i n d i c a t e the minima of reflecttwt~ for the different e x m t o n s T h e f u l l line is the measured spectrum which for the 4exclton is the phase reflect1~lt~ and for the Be x c l t o n the phase difference b e t w e e n E I c and E ± c The dashed h n e is the phase computed from a K r a m e r s - K r o m g a n a l s s l s The dots are c a l c u l a t e d from the model with an e \ c l t o n free surface layer of a t h i c k n e s s of 80A Howe~er, an exctton b r o a d e m n g of l- = 1 S meV is included to avoid the phase change oi 2 = as in F i g 1

only at very low temperatures At higher temperatures the phase of the B- exctton changes sign and the spectrum looks slmilar to the CdS- spectru,n It was a l s o observed that the reflected light cont a m e d a component of d e p o l a n z e d light s h o w i n g a pronounced peak near the minima of r e f l e c u v l t v The K r a m e r s - K r o m g relations are valid for a system with a causal linear response where the response function ts bounded at mflmte frequencte~ However, the K r a m e r s - K r o m g relations between phase and amplitude of refelctlon are not directly apphcable w h e n the crystal ts surrounded b~ a 3 medium with refractlve index different from umty, or m obhque mcldence, 3 m case of a laser structure 4 or an m h o m o g e n e o u s m e d m m s W e have been consldermg both the case of a [a~er structure and of an m h o m o g e n e o u s crystal Both models can explain the unusual features of the phase spectra W e have apphed the model of Hopfleld and T h o m a s " w h m h include an exc~ton free surface layer wlth a constant background dlelectrlc constant This layer was introduced m order to interpret a spike at the longltudmal energ? m the reflectlon spectrum. s.7 Such a layer In connection ~tth the small damping necessary for observing the spike '~ ~ii produce large phase shifts (up to 2,-) as observed

Vol 11, No 12

CdS AND ZnO IN THE E X C I T O N R E G I O N

1653

In c o n c l u s i o n , it may be s t a t e d that in the c a s e of e x c i t o n s p e c t r a the use of K r a m e r s - K r o n i g r e l a t i o n s may l e a d to incorrect r e s u l t s for the p h a s e Both the a m p l i t u d e and the p h a s e of ref l e c t i v i t y must be m e a s u r e d to obtain the correct ~alue of the complex r e f l e c t l v i t y Furthermore, it was found that the model involving an e x c l t o n free s u r f a c e la~er may e x p l a i n the o b s e r v e d p h a s e s p e c t r a T h i s method of measuring r e f l e c t i o n - , p h a s e - and d e p o l a r i z a t i o n s p e c t r a s i m u l t a n e o u s l y s e e m s to be a p r o m i s i n g method of d e t e r m i n i n g crystal properties directly

for ZnO. Using e x p r e s s i o n (30) of r e f e r e n c e 6 and the p a r a m e t e r s of r e f e r e n c e 8 we o b t a i n the p o i n t s In F i g . 1 for ZnO ~ h i c h a g r e e well with the measured cur~e In t h i s model the d r a s t i c p h a s e c h a n g e is a d i r e c t c o n s e q u e n c e of the e x c l t o n free s u r f a c e layer. In c a s e of CdS ( F i g . 2) the p h a s e c h a n g e is not s o pronounced and b r o a d e n i n g is therefore introduced into the model The p r e s e n c e of the d e p o l a r i z e d r e f l e c t e d light is not e x p l a i n e d by t h i s model and another model i n v o l v i n g i n h o m o g e n l t i e s introduced as a randoml~ s t r a t i f i e d medium with a d i s t r i b u t i o n of r e s o n a n c e f r e q u e n c i e s is a l s o c o n s i d e r e d P r e liminary calculattons involving Monte-Carlo methods for light r e f l e c t e d from this kind of medium i n d i c a t e s a large d e p o l a r i z a t i o n e f f e c t near the l o n g i t u d i n a l r e s o n a n c e frequency. T h e s e c a l c u l a t i o n s a l s o predict large p h a s e s h i f t s (up to 2~) as in the c a s e of the model with an e x c l t o n free s u r f a c e l a y e r . This is not s u r p r i s i n g both m o d e l s involve l a y e r s

-lcknowledo, ements - T h e authors w i s h to thank Professor R W Asmussenand Professor N I Meyer for their i n t e r e s t in the work

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THOMAS D G , J. Phys. Chem. Sohds 15, 86 (1960)

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THOMAS D G and H O P F I E L D J J ,

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LUPASHKO E A., MIKOSLAVSKII V K and SHKLYAREVSKII I N., Opt Spectrosc. 29, 419 (1970)

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F I L I N S K I I., Phys. Status Sohdt (b) 49, 577 (1972).

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S K E T T R U P T and B A L S L E V I., Phys

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Chem. Phys. 38, 612 (1963).

and THOMAS D G , Phys. Rev

132, 563 (1963).

Rev. Let! 27, 1644 (1971)

Rev B3, 1457 (1971)

F u r CdS und ZnO wurde der P h a s e n w i n k e l d e s komplexen R e f l e x l o n s k o e f f i z i e n t e n im E x c l t o n b e r e i c h bel 4,2 K g e m e s s e n . Bel l i n e a r p o l a r i s l e r t e m , s e n k r e c h t e l n f a l l e n d e m L i c h t wlrd wegen d e s D i k r m s m u s des K r i s t a l i e das r e f l e k t l e r t e L l c h t e l l i p t l s c h p o l a r l s l e r t , d i e Spektren wurden dutch e m e n a u t o m a t l s c h e n E l h p s o m e t e r aufgezelchnet. Die P h a s e n s p e k t r e n d e s 4- E x z l t o n s in CdS s o w l e die j e n l g e n der ,qund B- E x z l t o n e n in ZnO wlchen m e r k h c h yon den m i t t e l s der K r a m e r s - K r o m g s c h e n R e l a t l o n e n b e r e c h n e t e n Spektren ab M o g h c h e U r s a c h e d i e s e r A b w e l c h u n g 1st die N i c h t b e r u c k s i c h t i g u n g der K r i s t a l h n h o m o g e n l t a t und der E x l s t e n z e l n e r O b e r f l a c h e n s c h i c h t ohne exzitonen