Temperature Dependence Of The Phonons In Barium-Metatitanate And Some Effects Of The Exciton On The Raman-Scattering Of Cuprous-Oxide And Cadmium-Sulfide

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Content INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page{s)'\ If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being ‘ p h o tographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 A Xerox Education Company I 74-26,049 SCALABRIN, Artemio, 1942- TEM PERATURE DEPENDENCE OF THE PHONONS IN BaTi03 AND SOM E EFFECTS OF THE EXCITON O N THE RA M A N SCATTERING OF Cu20 AND CdS. University of Southern California, Ph.D., 1974 Physics, solid state University Microfilms, A X ER O X Com pany, Ann Arbor. Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. TEMPERATURE DEPENDENCE OF THE PHONONS IN BaTi03 AND SOME EFFECTS OF THE EXCITON ON THE RAMAN SCATTERING OF Cu20 AND CdS by Artemio S c a l a b r i n A D i s s e r t a t i o n P r e s e n te d to th e FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In P a r t i a l F u l f i l l m e n t o f the Requirem ents f o r th e Degree DOCTOR OF PHILOSOPHY (P h y sic s) June 1974 UNIVERSITY O F SO U TH ER N CALIFORNIA THE GRADUATE SCHOO L UNIVERSITY PARK LOS ANGELES. CALI FORNIA 9 0 0 0 7 This dissertation, written by A r te m io S c a la b r in under the direction of hi.S.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H IL O S O P H Y Dean DISSERTATION COMMITTEE t airman Dedicated to M aria E lena ii ACKNOWLEDGEMENTS I wish to express my g r a t i t u d e to P ro fe ss o r J. H. Parks for h is guidance in the experim ental a s p e c ts of the l a s e r physics in my work a t USC. The many d isc u ssio n s with P ro f e s s o r S. P. S. Porto and also Dr. J. H u rre ll on the v a rio u s a sp e c ts of the s o l i d s t a t e physics involved in my d i s s e r t a t i o n , are very much a p p re c ia te d . I want a ls o to acknowledge Dr. A. Chaves for h is help given in the experim ental p a rt and in the a n a ly s is of the d a ta of my work. I am also g r a t e f u l to the follow ing f r ie n d s fo r the many d isc u ssio n s and suggestions during my y e ars a t USC: Dr. F. Penna, Dr. P. Cervenka, Dr. D. B o z in is, Dr. J. V asconcellos and Dr, P. da R. Andrade. F in a lly , I acknowledge the Fundacao de Amparo a Pesquisa do Estado de Sao Paulo, B ra z il for the f i n a n c ia l support. TABLE OF CONTENTS Page ACKNOWLEDGEMENT i i i ABSTRACT vi LIST OF FIGURES v i i LIST OF TABLES xi CHAPTER I. INTRODUCTION .................................................................................... I 1.1 The L a t t i c e Dynamics o f BaTi03 ........................... 1 1.2 E x cito n s and the Raman S c a t t e r i n g o f CU2 O and C d S ............................................................. 5 I I . FLUCTUATIONS, DIELECTRIC FUNCTION AND RAMAN CROSS SECTION................................................................... 7 2.1 Response Function f o r M echanical and E l e c t r i c a l Forces ......................................................... 7 2.2 The Raman E f f e c t ....................................................................14 2.3 The L o re n tz ia n Response F u n c t i o n ..............................17 I I I . THE RAMAN MODES IN BaTi03 ........................................................... 24 3.1 E xperim ental Methods f o r D eterm ining L a t t i c e O p tic a l M o d e s .................................................... 24 3.2 E a rly Works on Barium T i t a n a t e .........................2 5 3.3 E xperim ental P r o c e d u r e ....................................................27 3.4 The Cubic P h a s e ....................................................................28 iv 3.5 The T etrago nal P h a se ................................................... 31 3 .5 .1 The E modes....................................................... 32 3 .5 .2 The m o d e s .................................................. 40 3.6 The D i e l e c t r i c C o n s t a n t ......................................... 57 3 .6 .1 The £ d i e l e c t r i c c o n s t a n t .................... 59 3 .6 .2 The e d i e l e c t r i c c o n s t a n t .................... 60 3.7 An A n a ly sis of the Spectrum Param eters in R e la tio n to the Theory of Ferro- e l e c t r i c i t y ....................................................... 66 3 .7 .1 A thermodynamic d e s c r i p t i o n of f e r r o e l e c t r i c phase t r a n s i t i o n . . 66 3 .7 .2 Dynamic models of the f e r r o e l e c t r i c phase t r a n s i t i o n ...................... 69 3 .7 .3 D isc u ssio n of the BaTiO, s p e c t r u m ............................................................ 70 IV. EXCITONS AND THE RAMAN SCATTERING OF Cu20 AND CdS............................................................................................ 79 4.1 The I s Yellow E xciton in Cu20 ......................... 79 4.2 The O rig in o f A n tireso n an ces in the D isp e rsio n of the Raman Cross Section of CdS...................................................................................... 88 APPENDIX The E O p tic-A c o u stic Phonon I n t e r f e r e n c e in B a T i O j ................................................................................................ 96 REFERENCES.......................................................................................................... 99 VITA............................................................................................................................... 103 v ABSTRACT Measurements of a l l phonon modes in Barium T ita - n ate are re p o rte d in the tem perature range 10-130 °C where the c r y s t a l i s in the te tr a g o n a l phase. A l e a s t squares f i t t i n g of the s p e c t r a l shapes is done and the r e s u l t i n g tem perature dependence of the f r e q u e n c ie s , 1 inew idths and coupling param eters is p re s e n te d . The d i e l e c t r i c c o n sta n t , e is c a l c u l a t e d and compared with the microwave measu rements of e . I t is shown t h a t th e re is no s o f t o p tic mode connected w ith the f e r r o e l e c t r i c - p a r a e l e c t r i c phase t r a n s i t i o n in Barium T ita n a te . Two e f f e c t s of the ex cito ns in the Raman cross s e c tio n are stu d ie d in Cuprous Oxide and Cadmium S u lf id e . In Cuprous Oxide the reso n an t enhancement of the 220 cm ^ Raman lin e due to the Is yellow ex cito n is i n v e s ti g a t e d . In Cadmium S u lfid e a model is proposed to e x p la in the c a n c e lla tio n e f f e c t in the Raman cross se c tio n by assuming an in te r f e r e n c e between th e sharp exciton s t a t e s and the broad e le c t r o n i c band s t a t e s . LIST OF FIGURES F igure Page 1. Imaginary p a r t of th e resp o n se f u n c tio n of a damped harmonic oscillation ................... 21 2. Raman Lineshape of a damped phonon mode. The phonon becomes overdamped f o r r/w Q> / T " . The d i f f e r e n c e between f i g . 1 and f i g . 2 is only in th e Bose f a c t o r ....................................................................22 3. Peak freq uen cy of th e im aginary p a r t of the respo nse f u n c t i o n of a damped harmonic o s c i l l a t o r as a f u n c tio n of the damping p a r a m e t e r ..................................................................................................................23 4. The u n i t c e l l o f BaTiOj...................................................................29 5. E symmetry transverse optical phonon spectrum in tetragonal BaTiO^ at room temperature. The arrow shows the p o s i t i o n o f the second peak no t r e s o l v e d in t h i s s p e c t r u m .....................................34 6 . E symmetry l o n g i t u d i n a l o p t i c a l phonon spectrum in tetragonal BaTiO^ at room temperature . . . . 36 7. Tem perature dependence o f the E modes of BaTi03 .............................................................................................................37 8 . Lineshape o f the overdamped E mode. Dots are ex p e rim e n ta l p o i n t s . Dashed l i n e s a re the l e a s t sq u a res f i t to th e damped o s c i l l a t o r m od el..................................................................................................................39 v i i I i i 9. Temperature dependence of the overdamped E phonon frequency and lin e w id th Tq in BaTi03 ................................................................................................................41 10. The ZZ A^TO Raman spectrum o f BaTiO^ a t room t e m p e r a t u r e .................................................................................................43 11. The ZZ A^TO spectrum of BaTiO-j in back s c a t t e r i n g c o n f i g u r a t i o n ...................................................................45 12. The ZZ A^ l o n g i t u d i n a l o p t i c a l spectrum of BaTiOj................................................................................................................46 13. Raman lin e sh a p e of the A-^TO phonons a t 3 c h a r a c t e r i s t i c te m p e ra tu re s. Dots a re e x p e r im ental p o i n t s . Dashed l i n e s are the l e a s t squares f i t of the d a ta p o in ts to th e coupled o s c i l l a t o r m odel.......................................................................................49 14. Temperature dependence of the A^TO mode p aram eters in BaTiOg.............................................................................51 15. Temperature dependence o f the A-^TO mode f r e quencies in BaTiO^; open c i r c l e s are the peak f re q u e n c ie s and d o ts are fre q u e n c ie s from f i t to coupled o s c i l l a t o r s m odel...............................................52 16. Temperature dependence of the A^LO phonon fre q u e n c ie s and e f f e c t i v e charges in BaTiO^ . . . 54 17. Temperature dependence of the A^TO p o l a r i t o n in B a T i O ^ ...................................................................................................... 55 viii i 18. Temperature dependence of the d i e l e c t r i c c o n s ta n t p e r p e n d i c u l a r to the f e r r o e l e c t r i c a x is in the t e t r a g o n a l phase of B a T i O ^ ........................... 61 19. Real p a r t o f the d i e l e c t r i c f u n c tio n e/, , in the Z d i r e c t i o n o f BaTiO^ u sin g coupled mode t h e o r y ...................................................................................................63 20. Temperature dependence of the d i e l e c t r i c c o n s ta n t p a r a l l e l to the f e r r o e l e c t r i c a x is in th e t e t r a g o n a l phase of BaTiO^. C ap acitance measurements from r e f e r e n c e ( 1 8 ) ...........................................64 21. Temperature dependence o f the phonon + e l e c t r o n d i e l e c t r i c c o n s t a n t in BaTiOg p a r a l l e l to th e f e r r o e l e c t r i c a x i s . The f u l l c i r c l e s a re th e LST v a lu e s from th e peaks of s p e c t r a . . 65 22. Spontaneous p o l a r i z a t i o n o f BaTiOj in t e t r a g o n a l f e r r o e l e c t r i c phase a f t e r Wemple e t a l . r e f e r e n c e ( 2 5 ) ................................................................68 23. Temperature dependence o f the frequ en cy of the f e r r o e l e c t r i c modes in t e t r a g o n a l BaTiO^ . . 74 24. CU2 O band s t r u c t u r e a t th e c e n te r of the B r i l l o u i n z o n e ............................................................................... 79 O 25. Spectrum o f CU2 O a t 5°K e x c ite d a t = 6046A. A i s th e quadru po le e x c ito n em ission. B is th e 220 cm”-* - Raman l i n e and C is th e phonon a s s i s t e d e x c i to n lu m in e sc e n c e ............................................82 ix o 26. Spectrum o f C ^O a t 80°K e x c ite d a t = 6085A. A i s the quadrupole e x c ito n em ission. B is the 220 cm-1 Raman l in e and C is the phonon a s s i s t e d e x c ito n luminescence .............................................. 82 27. Raman s c a t t e r i n g of the 220 cm-1 l i n e of CU2 O a t 5°K. The f i t t i n g param eters are: ue = 16386 cm"1 , A = 2.9, B = 1.02 103 ...................... 85 28. Raman s c a t t e r i n g o f the 220 cm-1 l i n e of Cu£0 a t 80°K. The f i t t i n g param eters a re : coe = 16292 c m '1 , A = 8 . 6 , B = 5.05 102 ...................... 86 29. CdS band s t r u c t u r e a t the c e n te r o f the B r i l l o u i n Zone............................................................................................91 x LIST OF TABLES Table Page 1. Frequency and l i n e w i d t h o f th e overdamped E mode..............................................................................................................42 2. Frequency o f th e phonons in CdS. Data tak en from r e f e r e n c e ( 5 2 ) .........................................................................90 xi CHAPTER I INTRODUCTION 1.1 The L a t ti c e Dynamics o f BaTiO^ The i n t e r e s t in th e ph ysics of barium t i t a n a t e (BaTiOj) is q u ite e x te n s i v e , mainly due to i t s rem arkable f e r r o e l e c t r i c p r o p e r t i e s . BaTiO^ shows th r e e phase t r a n s i t i o n s : a t 130°C i t s c r y s t a l s t r u c t u r e goes from the cubic high tem peratu re phase to the t e tr a g o n a l phase and t h i s s t r u c t u r a l phase t r a n s i t i o n is accompanied by a p a r a e l e c t r i c - f e r r o e l e c t r i c phase t r a n s i t i o n ; a t 6 °C and -80°C the s t r u c t u r e changes to orthorhombic and rhombo- h e d r a l , r e s p e c t i v e l y . In th e s e th r e e phases the f e r r o e l e c t r i c axis is s u c c e s s iv e l y along the (0 , 0 , 1 ) , (0 , 1 , 1 ) , (1 , 1 , 1 ) d i r e c t i o n s . 1 The importance of l a t t i c e dynamics in the u n d e r sta n d in g of s t r u c t u r a l phase t r a n s i t i o n s in f e r r o e l e c t r i c and a n t i f e r r o e l e c t r i c c r y s t a l s was recognized by Cochran 2 who developed a m icro sco p ic theo ry of phase t r a n s i t i o n s . C ochran's model assumes t h a t the f e r r o e l e c t r i c t r a n s i t i o n is the r e s u l t of an i n s t a b i l i t y a s s o c i a t e d with a c e r t a i n normal mode of v i b r a t i o n . In Cochran's th e o ry th e f e r r o e l e c t r i c p r o p e r t i e s and th e l a t t i c e dynamics are lin k e d through the 1 3 L yddane-Sachs-Teller (LST) r e l a t i o n given by o o i = i In t h i s e q uation the f e r r o e l e c t r i c p r o p e r t i e s are r e p r e sented by the d i e l e c t r i c c o n s ta n t, eQ, and the l a t t i c e dynamics by the freq u e n cie s of the tr a n s v e r s e and l o n g i tu d in a l o p t i c a l modes, and , r e s p e c t i v e l y ; e is the e l e c t r o n i c d i e l e c t r i c c o n stan t. Cochran sug gests t h a t the observed c r i t i c a l temperature dependence of eQ would m an ifest i t s e l f in the o p t ic a l phonon spectrum as a tr a n s v e rs e mode, the s o f t mode. As the frequency of t h i s mode s o fte n s or approaches zero, an i n s t a b i l i t y is generated in the l a t t i c e which i n i t i a t e s the phase t r a n s i t i o n . According to t h i s model one would expect the e x is te n c e of s o f t modes in BaTiO^ a s s o c ia te d w ith each of i t s th re e phase t r a n s i t io n s . The f i r s t c r y s t a l to which t h i s theory was ap p lied was BaTiO^.^ C ry s ta ls in which the phase t r a n s i t i o n is d i r e c t l y caused by a s o f t mode mechanism are r e f e r r e d to as d i s p l a c iv e f e r r o e l e c t r i c s . There i s an o th er mechanism which can lead to phase t r a n s i t i o n s . This mechanism a r i s e s when c e r t a i n ions in the c r y s t a l l a t t i c e have more than one p o s i t i o n of e q u ilib riu m and f l u c t u a t e from one p o s i t i o n to another LOi wTOi w ith a c e r t a i n r a t e . A p a r a e l e c t r i c phase in t h i s model c o rre sp o n d s to an equal p o p u la tio n o f a l l e q u iv a le n t p o s i t i o n s of th e io n. A f e r r o e l e c t r i c phase i s ach iev ed when one p o s i t i o n i s more p o p u la te d than o t h e r s which g e n e r a t e s , a non zero average d ip o le moment. The c r y s t a l s which can be d e s c r ib e d by t h i s mechanism a re r e f e r r e d to as o r d e r - d i s o r d e r type f e r r o e l e c t r i c s . The e a r l y d e v el- 5 6 opment o f t h i s model was p r i m a r i l y the work o f Mason. * Mason and M a tth ia s proposed an o r d e r - d i s o r d e r model f o r BaTiO^ in which the Ti ion has 6 e q u iv a le n t p o s i t i o n s in th e u n i t c e l l and p la y s th e c e n t r a l r o l e in 7 th e phase t r a n s i t i o n s . One of th e re a so n s f o r th e broad and e n t h u s i a s t i c r e c e p t i o n o f th e s o f t mode th e o r y among s o l i d s t a t e spec- t r o s c o p i s t s is t h a t i t p r e s c r i b e s a sim ple guide f o r e x p e rim e n ta l r e s e a r c h : one has only to look f o r the o p t i c a l s o f t mode r e s p o n s i b l e f o r a g iv en phase t r a n s i t i o n . This mode approaches zero a t th e phase t r a n s i t i o n tem p er a t u r e , T , a c c o rd in g to the r e l a t i o n a,2 « (T-Tc ) so t h a t i t has to have a sm all frequ en cy a t a te m p era tu re n e a r to th e t r a n s i t i o n . I t has become common p r a c t i c e to i d e n t i f y th e s o f t mode as th e low est freq u en cy mode, even though i t f r e q u e n t l y does n o t go to zero a t T . Of co u rse th e e x is te n c e o f an o p t i c a l mode w ith small frequency is in i t s e l f not a s u f f i c i e n t evidence f o r the c r y s t a l to be d i s p l a c iv e ; i t i s e s s e n t i a l to ex perim en tally dem onstrate t h a t the d i e l e c t r i c c o n sta n t obtained from the LST r e l a t i o n has the same value as the a c tu a l measured d i e l e c t r i c c o n stan t and t h a t i t d is p la y s the c o r r e c t tem perature dependence. In many c r y s t a l s which are thought to be d i s p l a c iv e th ese a d d itio n a l r e - g quirements are no t s a t i s f i e d . Several authors have i n d ic a te d a discrepancy between the cap acitan ce measured d i e l e c t r i c c o n sta n t in the f e r r o e l e c t r i c d i r e c t i o n of BaTiOj and the value determined from the phonon spectrum by th e use of the LST r e l a t i o n . There has r e c e n t l y been a s i g n i f i c a n t p ro g ress in r e l a t i n g c e r t a i n l a t t i c e dynamic param eters to an o rd e r- d is o r d e r mechanism inv olving a double well p o t e n t i a l . The phase transition in NaNO^ was consistently described by q t h i s approach. Although BaTiOj has g e n e ra lly been assumed to be a t y p ic a l d is p la c iv e c r y s t a l , the X-Ray d i f f r a c t i o n m easure ments of Comes e t a l . ^ show a d if f u s iv e n e s s c h a r a c t e r i s t i c of an o r d e r - d is o r d e r t r a n s i t i o n . This d ata has reopened the d isc u ssio n about the n a tu r e of f e r r o e l e c t r i c i t y in BaTiOg. The prim ary aim o f t h i s d i s s e r t a t i o n was to study the tem perature dependence of the l a t t i c e modes of BaTiO^ 5 throughout th e t e t r a g o n a l phase in c lu d in g T , and to i n v e s t i g a t e th e consequences o f th e s e f in d in g s in r e l a t i o n to the th e o ry of f e r r o e l e c t r i c i t y o f t h i s c r y s t a l . 1.2 E xcito n s and t h e Raman S c a tt e r i n g o £ Cu 2 0 and CdS The energy bands formed from e l e c t r o n i c s t a t e s in a c r y s t a l give r i s e to a con tin uo us a b s o r p t io n . In many c r y s t a l s i t i s a ls o observ ed an a d d i t i o n a l spectrum o f sharp l i n e s c lo s e to each band edge. These l i n e s have been a t t r i b u t e d to t r a n s i t i o n s w ith in an e l e c t r o n - h o l e p a i r , bound t o g e t h e r by coulombic f o r c e s and r e f e r r e d to 11 as an e x c ito n . In t h i s d i s s e r t a t i o n we have s t u d i e d examples in which e x c ito n s i n f lu e n c e th e Raman s c a t t e r i n g : one in Cuprous Oxide (C ^O ) and th e o t h e r in Cadmium S u lf id e (CdS). In CU2^ ob served the r e s o n a n t enhancement o f th e 220cm"-*- Raman l i n e due to the Is yellow e x c i to n . This i s i n t e r e s t i n g in th e f a c t t h a t t h i s Raman l i n e a r i s e s from a two phonon s c a t t e r i n g p ro c e ss and the in v o lv e d e x c ito n i s no t r a d i a t i v e in the d ip o le ap p ro x im atio n . In CdS we have s t u d ie d c a n c e l l a t i o n e f f e c t s in the Raman s c a t t e r i n g c ro s s s e c t i o n . This is a c h a r a c t e r i s t i c e f f e c t which has been observed in o th e r sem iconductor 12 c r y s t a l s as w e ll. The e x p la n a tio n g iven in th e l i t e r - 13 a tu r e assumes t h a t t h i s c a n c e l l a t i o n i s a r e s u l t o f the d e s t r u c t i v e i n te r f e r e n c e of re so n a n t and a n tir e s o n a n t s c a t t e r i n g am plitudes. W e show t h a t i t is p o s s ib le to e x p la in t h i s e f f e c t by c on sid ering the i n te r f e r e n c e between reso nan t c o n tr i b u ti o n s of the bands and sharp ex cito n s t a t e s . CHAPTER 2 FLUCTUATIONS, DIELECTRIC FUNCTION, RAMAN CROSS SECTION In t h i s c h a p t e r th e b a s i c concepts and fo rm u la s r e l a t e d w ith t h e i n t e r p r e t a t i o n o f e x p e rim e n ta l d a ta on Raman s c a t t e r i n g a r e d e r iv e d . 2.1 Response F u n c tio n f o r Mechanical and E l e c t r i c a l Forces. The fr e q u e n c y spectrum o f a medium can be r e l a t e d to th e f l u c t u a t i o n o f i t s p h y s i c a l q u a n t i t i e s . This s ta te m e n t i s f o r m a l iz e d in a u s e f u l way by th e N yquist theorem which r e l a t e s th e f l u c t u a t i o n o f a v a r i a b l e W to th e im aginary p a r t o f a re s p o n s e f u n c t i o n G by th e ex- 14 p r e s s i o n : 2 jWI = h / r . [n(w)+l] Im G((o) (2.1] 0) where W i s u s u a l l y a " d is p la c e m e n t" v a r i a b l e and to i s th e fre q u e n c y . I f th e g e n e r a l i z e d d isp lac em e n t i s cau se d by an e x t e r n a l f o r c e F th e d e f i n i t i o n o f G is G = W/F (2.2) i . e . , G i s th e d is p la c e m e n t p e r u n i t f o r c e . I f t h e system has more th a n one v a r i a b l e we can d e f in e a m a t r i x G in which th e m a t r ix elem ent Gij i s a s s o c i a t e d w ith th e p a i r 7 8 Wi and F j . A c r y s t a l is composed of an o rd ere d arrangement of ions and given th e number o f ions p er u n i t c e l l and i t s p o s i t i o n w ith r e s p e c t the o r i g i n o f th e c e l l one can determ ine th e number and symmetry o f th e v i b r a t i o n modes using group th e o ry . To determ ine the spectrum of v i b r a t i o n s we co nsid er the e x te r n a l fo r c e s on the ions and e l e c t r o n s . The thermal b a th can be r e p r e s e n te d by an e x te r n a l m echanical fo rc e F which c r e a t e s a mechanical p o l a r i z a t i o n . I f the p a r t i c l e s are charg ed, an e l e c t r i c a l f i e l d i s a ls o produced and the t o t a l p o l a r i z a t i o n is the sum of e l e c t r i c a l and mechanical components. The i n t e r n a l c o n s is te n c y i s o b tain ed by s o lv in g th e coupled fo rce and Maxwell’ s e q u a tio n s . A c o up lin g can e x i s t among the ions and a ls o among the e l e c t r o n s . The Raman e f f e c t a r i s e s from a coupling between the ion and th e e l e c t r o n motion in the c r y s t a l v ia the e le c tro n -p h o n o n i n t e r a c t i o n . ^ ^ This i n t e r a c t i o n is a n o n - l in e a r c ou pling and one can d e s c r ib e i t as a r i s i n g from a cubic term a 3 j (Wion - We£) in th e c r y s t a l p o t e n t i a l energy expansion. Let us c o n sid e r only the symmetry modes, i . e . , the modes having p o l a r i z a t i o n along th e p r i n c i p a l symmetry 9 a x i s . The e q u a tio n s o f m otion can th en be w r i t t e n a s : Ion M ^ C W ^ W ^ w ^ ♦ M M vM m^ C Y v/ v^ v/ v) * J 1% j(V 1 V 2 - Zul ' % - Fn v = 1 , 2 , . . . , 3N (2.3) E le c t r o n m . (W . + y . W. +oj2) - E ot.rw,, - W. ) 2 - Z.E*e.=F. J J J J J u IF F j j 3 3 P o l a r i z a t i o n P = (1/V) £ C Z .,W + z.W .] y,j M u J J In th e s e e q u a tio n s W a re d i s p la c e m e n ts , to a re f r e q u e n c i e s and y a re th e r e a l and im aginary c o u p lin g p a ra m e t e r s , r e s p e c t i v e l y ; y^ a re th e damping c o e f f i c i e n t s ; and nij th e a p p r o p r i a t e reduced m asses; Z and z. a re e f f e c t i v e c h a r g e s ; and a. are th e e le c t r o n - y 3 & 3y phonon i n t e r a c t i o n c o e f f i c i e n t s . The f i e l d E i s r e l a t e d to the p o l a r i z a t i o n P th ro u g h M axw ell’ s e q u a t io n s : v x S = - I f! V x S = i If v • 3 = o v • 5 = o and th e r e l a t i o n s 3 = 3+ 4ttP 3=3 10 These e q u atio n s lead to a second o rd e r d i f f e r e n t i a l e q u a tio n r e l a t i n g E and P: V x v x E + i y i f l . = - C2.4) 0 zt c 31 Even though the complete ion e q u a tio n is needed to i n t e r p r e t some of our r e s u l t s we d is c u s s th e p h y s ic a l con t e n t s o f eq. (23) by c o n sid e rin g an i s o l a t e d symmetry mode. In t h i s case only two e q uation s remain, one fo r the ion and one f o r the e l e c t r o n motion. Let us now i n v e s t i g a t e the l i n e a r respo nse (a = 0 ). F i r s t we make th e fo llow in g tr a n s fo r m a tio n s in o rd er to sim p l i f y th e e q u a tio n s: W v (IV V2 = Xv Zv % C2 ' 5) F (M ) 1/2= f v v v These tr a n s fo r m a tio n s leave th e p o l a r i z a t i o n i n v a r i a n t : P = Z W = Q X V V V X V V The r e s u l t i n g e q u a tio n s are: X„ + TX + w2X - QE = f ( t ) v o ( 2 . 6 ) x + yx + w*x = Q0E » f ( t ) in which X r e f e r s to th e ion and x r e f e r s to the e le c t r o n m o tio n . 11 Consider now the motion d riven by a plane wave fo rce for which we can w r ite : f ( t ) = f(w,q)exp(i?t*r - iwt) + c*c. E(t) = E (w,q)exp (icf*v - iwt) + c*c. X(t) = X(w,q)exp(iq*? - iwt) + c*c. x (t) = xCw,q)exp(i^*rr - iwt) + c*c. The r e s u ltin g a lg e b ra ic equations are: (-w2 - iTw + w2 )X(w,q) - QE(w,q) = f(w,q) ° (2.7) 2 2 (-w - iyw + w )x(w,q) - Q E(w,q) = f(w,q) e c W e now r e l a t e E(w,q) to X(w,q) using equation (2.4) n o tin g t h a t the d e f i n i t i o n of P in eq. (2.3) req u ire s th a t P (t) o s c i l l a t e s in the same way as X (t): P(t) = P(w ,q)exp(i$*? - iwt) The r e s u l t i n g r e l a t i o n between P(w,q) and E(w,q) is : -q(q*lf(w,q)) + (q2 - w2 / c 2 )E(w,q) = 4irw2P (w , q ) / c 2 (2.8) This shows th a t in general th ere is a tensor r e l a t i o n b e tween E and P, E = T P. In a c r y s t a l i t i s p o s s ib le to e x c ite e it h e r tr a n s - + 4 . . 4 v e rs e , (qJ.E) or lo n g itu d in a l (q ||E ) waves. For the case - b - b o f tr a n s v e r s e e x c ita tio n s we have q»E = 0 , E ||P and E (q,w) = 4TrQX(q,w)/(q2/w2 - 1) (2.9) where q now is measured in the same u n i ts as w (c = 1 ). 12 S u b s t i t u t i o n of eq. (2.9) in to the f i r s t eq. (2.7) g i v e s : 0) o 2 X(q,«) = f (q >to) ( 2 . 10 ) The response fu n c tio n to the m echanical force is then which is a p a r t i c u l a r case of th e more g e n eral te n s o r mechanical response fu n c tio n c o u p lin g , ana <P(.oo,qj = —j g — • lve see tn a r t o r cl'^00 • q /to - 1 This r e s u l t is v a l i d fo r the more general case and i t shows t h a t t h i s term in the response fu n c tio n i s e n t i r e l y r e s p o n s ib le fo r the q dependence o f the modes, i . e . , for the p o l a r i t o n and o b liq u e phonon d is p e r s i o n s . A non zero value o f $ im p lies a non zero value o f E and t h e r e f o r e the energy in th e medium i s p a r t mechanical and p a r t e l e c t r i c a l . As q^O the energy r e s i d e s in the e l e c t r i c a l f i e l d and as q -* -00 the energy becomes e n t i r e l y m echanical. c h a r a c t e r i s t i c s analogous to th e phonon response function. In p a r t i c u l a r i t shows a l s o a p o l a r i t o n d i s p e r s io n and this has been i n v e s t i g a t e d f o r the case t h a t the electron states ( 2 . 11 ) In the p re s e n t case T and are diagonal for no mode ( 2 . 12 ) The e l e c t r o n response f u n c tio n f ( q , « ) / x ( q , oj) shows 13 are e x c ito n s . Let us now c o n sid er the response function fo r an e x te r n a l e l e c t r i c f i e l d . The equations o f motion are; X + TX + w2 X = QE o (2.13) 2 x + yx + w x = 0 c e -eE The e f f e c t i v e charge Q accounts fo r the lo ca l f i e l d c o r r e c t i o n s . Considering small o s c i l l a t i o n s d riven a t frequency w we o b ta in the a lg e b r a ic eq uations: C - C D 2 - ifw + w2)X = QE (-w2- iyrn + w2) x = QeE (2.14) which lead to the response fu n ctio n s G(u) = | = Q/[ -w2 -ifo) + u 2] = QG(Wj«>) (a) (2.15) Ge (u) = | = Qe / [-m2- iyrn + m2] (b) where G(w,°°) is j u s t the response fu n ctio n given by eq. ( 2 . 1 1 ) in the l im i t q-*00. W e now d efine the e l e c t r i c s u s c e p t i b i l i t i e s x^ and x by e 1 P = XiE = QX = Q2G (w ,°°) E (a) Pel = xe E = Qex = Qe Ge (w)E (b) (2.16) The d i e l e c t r i c fu n ctio n of the medium is then defined as: e = 1 + 4irx = 1 + 47rxe + 4ttX^ (2.17) 14 From the d e f i n i t i o n s above one sees t h a t th e d i e l e c t r i c f u n c tio n due to the phonons can be d eterm ined i f we know G(w,°°) and th e e f f e c t i v e c h a rg e s . The f u n c t i o n G(w,<») can be o b ta in e d e x p e r im e n ta lly from the Raman spectrum of t r a n s v e r s e phonon modes. In o rd e r to determ in e th e e f f e c t i v e charges we have to c o n s id e r to the l o n g i t u d i n a l s o l u t io n s o f eq. ( 2 . 8 ) . In t h i s case q^E and eq. (2.8) g iv es ^ = - 4-rr?’ which means simply t h a t D = E + 4irP = 0. Since "V -y D = eE, t h i s im p lie s t h a t th e l o n g i t u d i n a l mode f r e q u e n c i e s a re th e s o l u t i o n s of: e(wL) = 0 (2.18) 2.2 The Raman E f f e c t Let us now d e riv e th e e q u a tio n s which d e s c r i b e the Raman e f f e c t . In t h i s case th e medium is d riv e n by f o r c e s a t two d i f f e r e n t f r e q u e n c ie s : m echanical f o r c e f ( t ) = f ( w ,q ) e x p ( iq * r - i w t ) -y l a s e r f i e l d E ( t) = E e x p (ik * r-iw t) L L L The l i n e a r resp o n se to the l a s e r f i e l d is g iv en by eq. (2.16-b) in which we r e p l a c e w by w . T his l i n e a r L resp o n se i s a ls o r e s p o n s i b l e f o r the R ayleigh s c a t t e r i n g . The l i n e a r re sp o n se to th e m echanical f o r c e s i s a lr e a d y known and i s g iv en by eq. (2.16b), The n o n l i n e a r f o rc e a p p ea rin g in th e e q u a tio n s f o r th e ion s and the e l e c t r o n s i s o f th e form: 15 n& e (x (t) - x ( t ) ) This gives r i s e to a p o l a r i z a t i o n at freq u e n cie s d i f f e r e n t than w and w Since X(t) X(w) X(coT)‘ = -iw t + J_/ _x(t) x(w)_ e x(wL) iwLt +c.c, we w i l l o b ta in a p o l a r i z a t i o n which in clu d es the follow ing frequency dependence: P (t) 'v 6 ( to D r e c t i f i c a t i o n P (t) ^ expCi2w^ t) harmonic g e n eratio n P (t) ^ exp (i frequency mixing (Raman) Now c o n sid er so lu tio n s o f the e l e c t r o n i c motion at frequency ws = w l " i . e . , where ws is the Raman component. The c o n tr i b u ti o n s to the p o l a r i z a t i o n a t th is frequency r e s u l t from the follow ing terms: X*(u)XCW L )e" iC(UL"C °:)t X(t) X(t) x ( t ) x ( t ) = x (w)x(w^)e X(t) x ( t ) = x ( t ) X(t) = x (w)X(m^)e i ( V “H X* ( w) x ( W L) e" 1 ^ w l t The e q u atio n of motion for xfw^) reduces to (-o)2 2 -iYea)s +to2) x (ws ) + 23 (x (w) - X(<jj) ( (x(coL)-X(uL)) = 0 (2.19) 16 U sing th e l i n e a r s o l u t i o n f o r x ( g o t ) we o b t a i n L xCus ) For th e u s u a l phonon, e l e c t r o n and l a s e r f r e q u e n c i e s , w^>>to0 , and w - , so t h a t th e ion term i n s i d e th e b r a c k e t s i s much s m a ll e r th a n th e e l e c t r o n i c term and can be n e g e l c t e d . The p o l a r i z a t i o n a t th e fre q u e n c y ws can th e n be w r i t t e n as W e can s i m p l i f y t h i s e x p r e s io n f u r t h e r by n o t in g t h a t in th e phonon fre q u e n c y r e g i o n th e c o n t r i b u t i o n o f X(co) i s much g r e a t e r th a n t h a t o f x(w) b e c a u se o f t h e phonon re s o n a n c e g o . A pplying th e N y q u ist theorem to th e f l u c t u a t i o n o f X(w) y i e l d s 2BQ; (x (u >)-X ( u >)) e l ( 2 . 21 ) The power sp ectru m o f t h i s Raman p o l a r i z a t i o n i s : < |P N l J 2 >W L_ 4 *B2Qe <|x*(o))-X ft(m)) |E l > ( 2 . 22 ) <|X(w) | 2> = | [n(w) + 1 ] ImG(oj) (2.23) where G(w) i s th e e x p r e s s io n g iv e n in e q . ( 2 .1 1 ) . 17 The d i f f e r e n t i a l Raman c ro s s s e c t i o n , a , can be * to ’ 14 w ritten as 3 9 t n u mt i r, i l r2 s s L < P > = V (2.24) n Lc lEL and, in c lu d in g the appro xim ation s above, t h i s e x p re s s io n becomes 4hd2n aijJw.Q'J ° u = T — 2 - ---------- 2 ° 2 f 2 .------------ 2 77 [ " W U nLc ' “ e - ' V s ^ s ) (“e ^ W V ( 2 ^ This form ula shows a c h a r a c t e r i s t i c re s o n a n t b e h a v io r a t l a s e r fre q u e n c ie s wT = w and w, = ui +w . When the J _ » 0 L 6 O l a s e r freq u en cy i s c lo s e to the e l e c t r o n i c reso n an ces one has to c o n s id e r th e n a tu re of th e s t a t e s in more d e t a i l . In g e n e ra l one has to sum over a l l p o s s ib l e e l e c t r o n i c s t a t e s in c lu d in g b o th the e x c ito n and th e continuum band s t a t e s . The phonon in fo rm a tio n i s c o m p lete ly c o n ta in e d in G(w) and when m easuring the Raman spectrum w ith a fix e d l a s e r freq u e n cy the e n t i r e e x p re s s io n m u ltip ly in g ImG(to) i s tak e n as a c o n s ta n t. 2.3 The L o re n tz ia n Response F un ction A p a r t i c u l a r case which i s im p o rta n t to study in more d e t a i l i s th e l i n e a r harmonic o s c i l l a t o r resp o n se f u n c tio n 18 g(w) = - r ------------ (2 .2 6 ) a) -to - iTo) o T h is f u n c t i o n d e f i n e s a mode fre q u e n c y ujo and a l i n e w i d t h T and i s th e phonon l i m i t o f eq. ( 2 .1 2 ) . In t h i s c a s e th e p e a k o f Im g(w) i s s h i f t e d to w a rd low er f r e q u e n cy . T h is s h i f t i s f u r t h e r i n c r e a s e d i n by th e p r e s e n c e o f th e Bose f a c t o r . These e f f e c t s a re i l l u s t r a t e d in f i g s . 1 and 2. F i g . 1 shows a p l o t o f th e im a g in a ry p a r t o f th e r e s p o n s e f u n c t i o n f o r v a r i o u s v a lu e s o f th e r a t i o r / w0 * In fig * 2 c o r r e s p o n d i n g p l o t s a re shown f o r th e phonon c r o s s s e c t i o n g iv e n by eq. ( 2 . 2 5 ) . W e n o t e t h a t a lth o u g h t h e peak o f Im(w) alw ay s o c c u rs a t f r e q u e n c i e s g r e a t e r th a n z e r o , t h e peak o f zero f r e q u e n c y f o r a f i n i t e v a lu e o f th e r a t i o r / t o . In t h i s c a s e th e phonon i s s a i d to become overdam ped. L e t us now d e te rm in e t h e peak v a lu e s o f th e Img(w) and cr ,. to W r itin g t h e re s p o n s e f u n c t i o n as g(w) = g 1 (to) + itog" (to) The im a g in a r y p a r t i s g iv e n by r Im g(w) = g" (to) = —— =—= --------------r (to^-to ) + (Tto) . ( 2 . 2 7 ) The maximum v a lu e o f g"(w ) i s found t o o c c u r a t t h e f r e q u ency to g iv e n by 19 r c»S- (D 2) 2 + era))2 n(w) - g Cu) ( 1 ) Taking iLH = 0 we get th e e q u a tio n 4 ? 4 3w - aw - a> 0 = 0 a ? 2 r 2 ■ 2“o " r • or 2 01 + ' J a 2 + 1 2 to 2 0 ( A ) “■ P 6 The c o r r e c t sig n of the square ro o t by n o tin g t h a t f o r r = 0 the peak is a t wq . r = 0 - » ■ a = 2 2 (i) o 9 2 2 _ 2 “ o ±J 16 it ■ a - o 3 O 3 P 6 T h e re fo re t h e sig n has to be + * The peak o f g” C w) is th e n 7 w2 n t 2 \ r 2 up - f y 1 — 2 + 3 + 1 — 7 • c2-28) P 3 M 2“ o / 2“ o L et us now examine the e f f e c t o f the Bose F a c to r , W e d e fin e B(w) = n(w) + 1. W e th u s h a v e : rj Ctu) = ------------------ B(to) = (2.29) ftw/kT , , -fiw/kT e - 1 1 -e In th e l i m i t i n g c a s e of v e ry sm a ll te m p e r a tu re s or h ig h p h o n o n .fre q u e n c y , su c h t h a t fia) >> kT , we have 20 B = 1 and the peak frequency o f c o in c id e s with the peak o f g ' 1 (w) and is given by eq. (2 .2 8 ). The o th e r lim itin g case o f h igh tem perature or sm all frequency such th a t hco>>kT a ls o has a sim ple so lu tio n . For t h i s case BCoOfb----------— r- — = l - o - | r ) ^ and ' °u = lAl2 X -----~ 2------ 2“ 2-------------1 f2-30’ " 1 1 (u2 V ) 2 + (T o j) so t h a t the c ro ss s e c tio n i s p r o p o r t io n a l to the temper- t a tu r e . The maximum of a i s found to occur a t u > given M P 6 by “i Z ■ “o ^ 2 - I C2- 31) W e n o te t h a t the approxim ate v a lu e f o r which th e phonon becomes overdamped, -> 0 , i s r = / 2uo (2.32) In f i g . 3 23 p l o t the norm alized peak frequency a)p/wo , o f g 1 1 (u>) given by eq. (2.28) and th e peak frequency o f a(u)) given by eq. (2 .3 1 ). Note t h a t in the low te m p e ra tu re l i m i t the peak o f th e phonon c ro ss s e c tio n o(w) c o in c id e s w ith the peak w o f g 1 1 (to) . Also fo r any a c t u a l phonon the v alu e w i l l ( i ) o be in between th e two curves o f f i g . 3, IftfAG Q{W) (ARBITRARY ) 21 0 .5 10 w/Wo F ig u re 1. Im ag in ary p a r t o f th e r e s p o n s e f u n c t i o n o f a damped harm onic o s c i l l a t o r . O ' { Y f t (ARBiTRAR*) 22 ------10 3 r . C 0 .5 to w/Wo L 5 2.0 F igure 2 Raman lin e s h a p e o£ a damped phonon mode. The phonon becomes overdamped f o r T/w /T7 The d i f f e r e n c e betw een f i g s . 1 and 2 i s only in th e Bose f a c t o r . Figure 3 low temperature high temperature 1.0 2.0 3.0 1.5 2 .5 r m Peak frequency o f the im aginary p a r t o f the response fu n c tio n o f a damped harmonic o s c i l l a t o r as a fu n c tio n o f the damping p aram eter. CHAPTER 3 THE RAM AN MODES IN BaTi03 3.1 E xperim en tal Methods f o r D eterm ining L a t ti c e O p tic a l Modes There a re th r e e w e ll e s t a b l i s h e d te c h n iq u e s f o r d e te rm in in g th e o p t i c a l phonons o f a c r y s t a l . They a re I n f r a r e d a b s o r p tio n or r e f l e c t i v i t y ( I . R . ) , Raman s c a t t e r i n g , and n e u tro n s c a t t e r i n g . Using th e I.R . and f i r s t o rd e r Raman e f f e c t we can d e te rm in e th e long w avelength v i b r a t i o n s o f th e c r y s t a l (q % 0 ) , i . e . , we can d eterm ine th e phonons c o rre sp o n d in g to the c e n t e r o f th e B r i l l o u i n Zone, th e so c a l l e d r - p o i n t ( 0 ,0 ,0 ) in q space. The p o s s i b i l i t y o f u sin g e i t h e r Raman or I.R . is determ ined by symmetry s e l e c t i o n r u l e s . The n e u tro n s c a t t e r i n g e x p e r im ents a re n o t r e s t r i c t e d to sm all q wave v e c t o r s , and can be used to determ ine th e e n t i r e d i s p e r s i o n o f th e phonon bran ch es in symmetry d i r e c t i o n s o f th e B r i l l o u i n Zone. In comparing th e s e te c h n iq u e s i t i s a p p ro p r ia te to say t h a t n e u tro n s c a t t e r i n g com plim ents Raman and I.R . N eutron s c a t t e r i n g may g iv e more in fo r m a tio n b u t i t cannot 24 25 com pete i n r e s o l u t i o n w ith Raman and I n f r a r e d te c h n iq u e s f o r d e te r m in in g q ^ 0 p h o n o n s. A lso th e e x tre m e c o l l i - m a tio n and d e g re e o f p o l a r i z a t i o n o f a l a s e r l i g h t s o u rc e makes Raman, when a p p l i c a b l e , by f a r th e b e s t m ethod f o r d e te r m i n i n g t h e symmetry o f th e mode. 3 .2 E a r l y Works on Barium T i t a n a t e The d e t e r m i n a t i o n o f t h e l a t t i c e o p t i c a l modes in BaTiO^ was a c c o m p lis h e d o n ly i n t h e l a t e 6 0 ' s when good s i n g l e domain c r y s t a l s were made a v a i l a b l e . P a rs o n s and 16 17 Rim ai and P in c zu k e t a l . r e p o r t e d th e f i r s t s a c c e p t a b l e Raman m easurem ent in s i n g l e domain c r y s t a l s . 18 19 DiDomenico e t a l . * have a ls o m easured t h e sp e c tru m o f BaTiO^. They have done m easurem en ts w ith v e ry s m a ll s p e c t r o m e t e r s l i t s , c l o s e to th e l a s e r l i n e and have c l e a r l y i d e n t i f i e d th e E overdam ped mode. The l i n e - sh a p e o f th e overdam ped mode was a l s o s t u d i e s a t s e v e r a l t e m p e r a t u r e s . 20 F le u r y and L a z a i have shown t h a t t h i s E mode i s s t r o n g l y c o u p le d to t h e E a c o u s t i c mode. They have m easu red th e B r illo u in -R a m a n s p e c tru m as a f u n c t i o n o f 21 te m p e r a t u r e . From t h e d a t a t h e y o b t a i n e d t h e te m p e r a t u r e d e p en d en ce o f t h e f r e q u e n c i e s , w i d t h s , and c o u p lin g p a r a m e t e r o f t h e c o u p le d m odes. T h e i r r e s u l t f o r t h e ETO mode a r e a c t u a l l y i n c o r r e c t b e c a u s e th e t e m p e r a tu r e d e p e n d e n c e o f t h e mode f r e q u e n c y i s d e r i v e d i m p l i c i t l y 26 from th e LST r e l a t i o n u s in g an in d e p e n d e n tly m easured d i e l e c t r i c c o n s t a n t as w i l l be shown in t h i s t h e s i s r e s e a r c h . I t i s a ls o w orth m entionin g t h a t t h e i r a n a l y s i s i n d i c a t e d th e lin e w i d t h o f th e ETO mode sh o u ld be i n d e p e n d e n t o f te m p e ra tu re th ro u g h o u t th e e n t i r e t e t r a g o n a l p h a s e . T h is r e s u l t i s n o t re a s o n a b le nor to be e x p ec te d f o r the b ro a d , overdamped phonon in q u e s t io n . 2 2 Burns m easured th e te m p e ra tu re dependence o f th e Raman s c a t t e r i n g o f the symmetry p o l a r i t o n from w hich he d e te rm in e d th e phonon c o n t r i b u t i o n to th e d i e l e c t r i c c o n s t a n t in th e Z - d i r e c t i o n of BaTiO^. In s e c t i o n s 3.5 and 3.6 h is r e s u l t s a re compared w ith th o se o f t h i s d i s s e r t a t i o n . The A^TO sp ectrum o f BaTiO^ shows a shape c h a r a c t e r i s t i c o f i n t e r f e r e n c e phenomena. T his i n t e r f e r e n c e was i n i t i a l l y s t u d i e d in Raman s c a t t e r i n g by Rousseau and 23 P o rto and th e d a t a was f i t t e d to a Auger model i n which a sh a rp l e v e l i n t e r a c t s w ith a c o n tin u o u s b ackg roun d. Chaves e t a l . ^ made a d e t a i l e d stu d y of th e A^ p o l a r i t o n lin e s h a p e . They have shown t h a t b e s i d e s th e known i n t e r f e r e n c e t h e r e i s a s tro n g c o u p lin g betw een th e m id dle and th e upper m odes. 27 3 .3 E x p e r im e n ta l P ro c e d u re A l l e x p e r im e n ts r e p o r t e d i n t h i s d i s s e r t a t i o n were p e rfo rm e d in s i n g l e domain c r y s t a l s grown by S an d ers A s s o c i a t e s I n c . , N ashua, New H am psh ire. The c r y s t a l s o f BaTiO^ w ere grown from an e x c e s s o f TiC^ m e lt in a r e s i s ta n c e f u r n a c e . C r y s t a l s grown by t h i s m ethod have o n ly 2 5 b e en a v a i l a b l e s i n c e 1965 and l i g h t a b s o r p t i o n m ea su re - 9 f \ m ents i n d i c a t e a h ig h d e g re e o f p u r i t y com pared t o th e 27 c r y s t a l s grown by th e Remeika f l u x m ethod a v a i l a b l e p r e v i o u s l y The c r y s t a l s were p o l i s h e d and t h e n p o le d f o r th e rem oval o f a l l 90° domains form ed d u r in g m a n u f a c tu r e . For t h e t e m p e r a t u r e dependence m e a s u re m e n ts , th e c r y s t a l was m ounted i n a h o l d e r i n s i d e an o v e n . The te m p e r a t u r e was m easu red w ith a c h ro m e l-a lu m e l th e rm o c o u p le a t t a c h e d to th e m e ta l b a s e o f t h e c r y s t a l h o l d e r . A se co n d th e rm o c o u p le was im m ersed in a d i s t i l l e d w a te r i c e m ix tu r e as a r e f e r e n c e and th e p o t e n t i a l d i f f e r e n c e betw een t h e two th e rm o c o u p le t e r m i n a l s was m easu red w ith a d i g i t a l v o l t m e t e r . The tem p e r a t u r e o f t h e c r y s t a l was i n c r e a s e d s lo w ly th e o r d e r o f 1°C eac h 10 m in u t e s . The Raman s p e c t r a o f BaTiOj was t a k e n w ith a s t a n d a r d Raman s e t u p . A l i g h t beam o f a b o u t 500 mw a t 514A° from t h e a rg o n io n l a s e r was fo c u s e d on th e c r y s t a l w ith a f = 30 cm l e n s . The sam ple h e a t i n g in d u c e d by t h e l a s e r 28 beam was found to have n e g l i g i b l e e f f e c t f o r th e s e mea su re m e n ts. The s c a t t e r e d l i g h t was c o l l e c t e d w ith an f /4 le n s and th e image o f th e c r y s t a l was fo cu sed on th e e n t r a n c e s l i t o f the s p e c tr o m e te r . The s p e c tr o m e te r was a Spex double d i s p e r s i o n s p e c tro m e te r equipped w ith two 1200 lines/m m g r a t i n g s . The r e s o l u t i o n w ith 10y e n tra n c e and e x i t s l i t s is about .5cm *. The l i g h t coming out o f th e e x i t s l i t o f th e s p e c tro m e te r was d e te c t e d by an ITT FW 130 p h o t o m u l t i p l i e r and th e anode c u r r e n t m easured by an e l e c t r o m e t e r . The e le c t r o m e t e r s i g n a l was r e c o r d e d in a p a p e r by a s t r i p - c h a r t r e c o r d e r . The s i g n a l to n o is e r a t i o in a l l e x p e rim e n ts was always e x tre m e ly good, S/N il 100. The geom etry o f s c a t t e r i n g was a p p r o p r i a t e to th e symmetry o f th e phonon we w ished to m easure and i s shown e x p l i c i t l y in th e s p e c t r a f i g u r e s . 3 .4 The Cubic Phase The c r y s t a l s t r u c t u r e o f BaTiO^ i s c u b ic above a p p ro x im a te ly 130°C ( 1 2 0 ° C f o r f lu x grown c r y s t a l s ) . The c e n t e r o f i n v e r s io n symmetry im p lie s t h a t no sp ontan eo u s p o l a r i z a t i o n e x i s t s and th e c r y s t a l i s p a r a e l e c t r i c . + 2 The u n i t c e l l i s a cube w ith Ba ions in th e face c e n t e r s . The s t r u c t u r e can a ls o be d e s c r i b e d as a system o f TiOg o c ta h e d r a jo in e d a t th e c o rn e r s w ith Ba io ns 7 Q p la c e d i n t e r s t i t i a l p o s i t i o n s betw een th e o c ta h e d ra , as shown in f i g . 4. 29 Figure 4. The u n i t c e l l of BaTiOg. 30 T here a r e 5 atoms p e r u n i t c e l l w hich r e s u l t in th e e x i s t e n c e o f 15 l a t t i c e v i b r a t i o n mode b r a n c h e s . S in ce we c a n n o t d i s t i n g u i s h X, Y and Z d i r e c t i o n s th e modes a re t r i p l y d e g e n e r a te and t h e r e rem ain o n ly 5 in d e p e n d e n t b r a n c h e s . The p o i n t group i s 0^ and th e i r r e d u c i b l e r e p r e s e n t a t i o n i s {4 F_ + F_ }. One o f th e F, i s an lu 2u lu a c o u s t i c b ra n c h and th e o t h e r t h r e e o p t i c a l b ran c h es a r e p o l a r and I.R . a c t i v e . The F i s non p o l a r and i s I .R . s i l e n t . None o f th e o p t i c a l modes in c u b ic phase i s Raman a llo w e d by symmetry c o n s i d e r a t i o n s . The f i r s t th o ro u g h i n v e s t i g a t i o n o f th e I n f r a r e d 2 9 sp e c tru m o f BaTiO^ was done by S p i t z e r e t a l . a t room t e m p e r a t u r e . They i d e n t i f i e d 3 I .R . a c t i v e modes and one o f them was found to be overdam ped. The s tu d y o f th e I .R . sp e ctru m in th e c u b ic phase 28 30 was done by L a s t and B a lla n ty n e . L a s t m easured th e F i u mode n e a r 500 cm”*. B a lla n ty n e m easured th e 500 cm”* mode, th e 182 cm * mode and th e overdam ped mode p r e v i o u s l y d i s c o v e r e d by S p i t z e r . The a s s ig n m e n t o f a fre q u e n c y to th e overdamped mode i s n o t a sim p le t a s k . B a lla n ty n e d i d a Krammers- K ronig a n a l y s i s o f th e I .R . sp e ctru m and computed th e im a g in a ry p a r t o f th e d i e l e c t r i c c o n s t a n t , e IT (to). He to o k th e peak o f e ’’ (to) as th e mode fre q u e n c y and t h i s value was w^ = 13 cm"* a t 200°C. 31 B arker in a t h e o r e t i c a l a n a l y s i s o f p re v io u s e x p e rim e n ta l work f i t t e d th e e x p e rim e n ta l d a ta to a harmo n ic o s c i l l a t o r model and c a l c u l a t e d a fo r c e c o n s ta n t fo r t h i s as th e p h y s i c a l l y m eaning fu l q u a n t i t y to c h a r a c t e r i z e th e mode. The o s c i l l a t o r p a ra m e te rs r e s u l t i n g from B a r k e r 's f i t t i n g o f th e I.R . r e f l e c t i v i t y I t is i n t e r e s t i n g to n o te t h a t even though Raman modes should not be p r e s e n t in th e c u b ic phase we measure two Raman bands in th e X(ZZ)Y c o n f i g u r a t i o n peaked a t th e phase t r a n s i t i o n te m p e ra tu re . 3.5 The T e tra g o n a l Phase BaTiOj is f e r r o e l e c t r i c betw een 130°C and 6 °C, The c r y s t a l s t r u c t u r e is t e t r a g o n a l w ith p o in t symmetry and th e i r r e d u c i b l e r e p r e s e n t a t i o n is {4A^ + 5E + 1B^}. The change i n the symmetry i s r e l a t e d in th e fo llo w in g t a b l e th e overdamped mode, = 40 cm“^ (200°C) . Barker c o n sid ers 42 182 BOO Yj(cm- 1 ) 168 47TZj / wj a t 200°C 1250 2. 2 .8 5.5 43 a p p ro x im a te ly 240 cm'-*- and 525 cm*^ a t 134°C, j u s t above 32 Symmetry P o in t Group I r r e d u c i b l e R e p r e s e n t a t i o n c u b ic 2u lu t e t r a g o n a l A ll th e o p t i c a l modes a r e now Raman a c t i v e and w ith th e e x c e p ti o n o f th e y a re a l s o I .R . a c t i v e . 3 .5 .1 The E modes The 4 o p t i c a l E modes in th e t e t r a g o n a l p h a se a r e d o u b ly d e g e n e r a t e , w ith t h e i r p o l a r i z a t i o n alon g th e X and Y a x is (a a x i s ) . The Raman p o l a r i z a b i l i t y t e n s o r s o f th e s e modes a r e ’ 0 0 d ’ "o 0 o" a ( E ( x ) ) = 0 0 0 a ( E ( y ) ) = 0 0 d _d 0 0 ^ _0 d 0 _ The E t r a n s v e r s e o p t i c a l Raman sp ectru m (ETO) a t room te m p e r a tu re i s shown in f i g . 5. I t i n d i c a t e s one overdam ped mode w hich o r i g i n a t e s from th e overdam ped mode i n th e c u b ic p h a s e , and t h r e e more modes w ith f r e q u e n c i e s 180 cm ^ 308 cm" 1 and 489 cm"1 . The 180 cm- 1 mode i s p o s i t i o n e d w i t h i n th e t a i l o f th e overdam ped mode and i s n o t r e s o l v e d i n t h i s s p e c tru m . I t s p o s i t i o n can 33 n e v e r t h e l e s s be d eterm in ed by the o b liq u e phonon d i s - T O p e r s i o n J ^ o r by p o l a r i t o n d i s p e r s i o n . This mode fo llo w s from th e c u b ic F mode a t 182 cm”l and i t s freq u en cy is i n s e n s i t i v e to te m p e ra tu re or th e phase t r a n s i t i o n . The E mode a t 489 cm ^ a r i s e s from the s p l i t t i n g o f th e rem aining c u b ic mode a t 500 cm"1 . The E mode a t 308 cm- 1 comes from th e s p l i t t i n g o f th e cubic i n a c t i v e F2U mode. The s c a t t e r i n g geometry n o t a t i o n which ap p ears in f i g . 5, X(ZY)Z has th e fo llo w in g meaning. The symbols o u ts id e th e b r a c k e ts i n d i c a t e th e d i r e c t i o n o f p ro p a g a tio n of th e i n c i d e n t photon a t l e f t , X, and th e s c a t t e r e d photon a t r i g h t , Z. The symbols in s id e th e b r a c k e ts giv e th e p o l a r i z a t i o n o f th e i n c id e n t photon a t l e f t , Z and th e p o l a r i z a t i o n o f th e s c a t t e r e d photon a t r i g h t , Y. The n o t a t i o n X(ZY)Z in f i g . 5 th u s r e p r e s e n ts 90° s c a t t e r i n g and in t h i s c o n f i g u r a t i o n the e x c ite d phonon, 3 , p ro p a g a te s in th e XZ p la n e d e fin e d by the l a s e r and s c a t t e r e d wave k as shown below In t h i s diagram momentum c o n s e r v a tio n has b een in d ic a t e d by showing q = k^ - kg . From th e above p o l a r i z a b i l i t y t e n s o r we can show t h a t th e e x c ite d E symmetry phonon is RAMAN INTENSITY (ARBITRARY) 8 0 70 x60 60 X(ZY)Z 50 40 308 30 180 489 0 100 200 FREQUENCY (cm1 ) F igure 5. E symmetry tr a n s v e r s e o p t i c a l phonon spectrum in t e tr a g o n a l B ’aTiO, a t 300 400 500 room te m p e ra tu re . The arrow shows p o s i t io n of second peak n o t re so lv e d in t h i s spectrum . 35 is p o l a r i z e d in th e Y d i r e c t i o n , i . e . , p e r p e n d ic u la r to the p la n e o f p ro p a g a tio n and r e f e r r e d to as a tr a n s v e r s e mode as i n d ic a te d in f i g . 5. The E l o n g i t u d i n a l Raman spectrum , ELO, is shown in f i g . 6 . I t c o n s i s t s o f 3 sharp l i n e s a t 180 cm”-*-, 308 cm and 466 cm"-*-. A band a t 722 cm"'*' is a ls o o b se rv ed , somewhat b ro ad e r and w ith s m a lle r Raman c ro s s s e c t i o n . The s c a t t e r i n g geometry is alm o st forw ard s c a t t e r i n g a t a sm all a n g le X(ZY)X + AY and an e x p e rim e n ta l arrangem ent c o l l e c t s l i g h t a t a p a r t i c u l a r a n g le 0 which d e fin e s th e d i r e c t i o n of p ro p a g a tio n o f th e phonon. T his forw ard s c a t t e r i n g and e x p e rim e n ta l d e t a i l s a re shown in th e diagram below The band a t ^ 270 and %525 cm"-*- which a ls o appear in t h i s spectrum r e s u l t from th e s c a t t e r i n g by th e phonons. The te m p e ra tu re dependence o f th e E TO and LO modes was m easured between 10°C and 120°C. The r e s u l t i n g mode fre q u e n c ie s in t h i s te m p e ra tu re range a re p l o t t e d in f i g . 7. Very l i t t l e change in th e f r e q u e n c ie s i s measured as th e te m p e ra tu re i s v a r i e d from n e a r T , to n e a r T . c l c 2 308 180 X3 722 466 FREQUENCY (cm-') Figure 6 . E Symmetry lo n g itu d in a l o p t i c a l phonon spectrum in te tr a g o n a l BaTiO a t room te m p e ra tu re . w 3 os FREQUENCY (cm* 37 ~ \ i i i i i i 1 -----1 ---- 1 ---- r i r 700- 600 500- 400 300 - 200- 100 - modes with E -symmetry OTRANSVERSE in tetragonal BaTiOj • LONGITUDINAL • f t * * 0 9* • -------------- -—•—— •-------------- ■ • • o « r 'c2 Figure 7. 20 40 60 80 TEMPERATURE CC) 1 0 0 1 2 0 'C | T em perature dependence o f th e E modes o f BaTiOj. 38 The frequency v a lu e s fo r th e overdamped E mode a r e o b a tin e d from th e l e a s t sq u a res f i t to the e x p e rim e n ta l d a ta u sin g a damped harmonic o s c i l l a t o r resp o n se f u n c tio n . The Raman S p e c tra was f i t t e d to form ula 2.25 u sin g G(w) given by 2.20. The agreem ent between th e c a l c u l a t e d v a lu e s and the e x p e rim e n ta l p o in ts i s very good. The e x p e rim e n ta l s p e c t r a and t h e o r e t i c a l curves c a l c u l a t e d a t fo u r d i f f e r e n t te m p e ra tu re s a re shown in f i g . 8 . The p l o t s were a r b i t r a r i l y n o rm aliz e d a t d i f f e r e n t tem p era t u r e s by ta k in g equal v a lu e s f o r the Raman s c a t t e r i n g a m p litu d e . A c tu a lly the Raman s t r e n g t h is d e c re a s in g 19 as th e te m p e ra tu re in c r e a s e s . Since th e o p t i c a l E mode and the a c o u s tic E mode 2 0 a re co u p le d , i t i s n e c e s s a ry to c o n s id e r under what c o n d itio n s i t i s v a l i d to a n a ly z e them in d e p e n d e n tly . T his i s c o n s id e re d in th e Appendix where i t i s shown t h a t for a l l fre q u e n c ie s fo r which th e c o u p lin g p a ra m e te r c s a t i s f i e s c ( u j ) << 1 , th e o p t i c a l spectrum can be app ro x im ated by gQ(w) given in e q . 2 .26. Using th e v a lu e s 2 2 o f Ua, Ta , A , c jq , r Q given in r e f e r e n c e (20) i t i s found t h a t f o r w>5 cm ^ t h i s c o n d itio n is s a t i s f i e d . The e x p e rim e n ta l d a ta has been f i t t e d to gQ(w) in the range 5 cm ^ to 100 cm ^ which i s more th an tw ice th e range p r e v io u s ly used i n r e f e r e n c e (19). RAMAN INTENSITY (ARBITRARY) 39 \ V \ \ \ h \ V \ \ \ \ A \ \ i > -x V \ \ \ \ \ ' t 1 ------- 1 ------- 1 -------r xl0\ \ OVERDAMPED \ E-PHONON in BnTi03 \ \ \ xlO\ ^ \ \ \ \ •\ * \ \ ' *\ xIO y • \ ^ \ \ ,n \ ' ^ > \ N \ X ,° ^ \ \ " \ \ \ \ \ \ X *2*^ W \ ^ \ \ \ \ X «\ \ 7 8 .0 °C ' X V v v __________ ____________ ____________________ V _______________ 4 0 .0 ° ^ Vn S . N N - " •* » I2.7°(T~- J L X X J L X 0 F ig u r e 8 . 2 0 00 X 120 4 0 6 0 8 0 FREQUENCY (cm -1) L in e sh a p e o f t h e overdam ped E mode. D ots a re e x p e r im e n ta l p o i n t s . Dashed l i n e s a re t h e l e a s t s q u a r e s f i t to th e damped o s c i l l a t o r m o d e l. 40 The f i t t i n g p a ra m e te rs w0 and r Q a re p l o t t e d as a f u n c t i o n o f te m p e ra tu re in f i g . 9. The s o l i d l i n e i s drawn only in o rd e r to i n d i c a t e th e tr e n d of v a r i a t i o n o f ioq . The s c a le somewhat e x a g e ra te s th e te m p e ra tu re dependence by ta k in g th e o r i g i n a t 30 cm- -* -. The f r e quency o)_ d e c re a s e s as th e te m p e ra tu re d e c re a s e s ap- o p ro a c h in g Tc 2 , from 53 cm- 1 a t 120°C to 35 cm' 1 a t 10°C. The lin e w i d t h a ls o d e c r e a s e s w ith te m p e ra tu re b u t s a t u r a t e s c lo s e to Tc 2 . In t a b l e 1 we compare our r e s u l t s w ith o th e r Raman and I n f r a r e d r e s u l t s . 3 .5 .2 The modes. The 3 o p t i c a l A^ modes in t e t r a g o n a l BaTiO-j a re p o l a r i z e d in th e Z d i r e c t i o n ( f e r r o e l e c t r i c a x is o f th e c r y s t a l a ls o r e f e r r e d to as th e c a x i s ) . The Raman p o l a r i z a b i l i t y t e n s o r shown below i s d ia g o n a l f o r th e mode a 0 0 a (Ax (Z )): 0 0 0 0 0 b The a Raman p o l a r i z a b i l i t y component in v o lv e s A1 phonons e x c l u s i v e l y . The ZZ A-^TO spectrum a t R.T. is shown in “ 1 - 1 f i g . 10 and i n d i c a t e s t h r e e peaks a t 178 cm , 272 cm and 519 cm '1 . Two re m a rk a b le c h a r a c t e r i s t i c o f t h i s s p e c trum a r e th e i n t e r f e r e n c e c l o s e to th e sh a rp 178 cm"1 41 5 5 Overdomped E-phonon in BaTiO, 5 0 180 170 ’g 4 5 160 o 150 UJ ui ft 4 0 u_ 140 130 120 35 110 10 0 2 0 4 0 60 8 0 100 120 TEMPERTURE CC) C2 F ig u r e 9. T em p eratu re d ep endence o f th e overdam ped E phonon f r e q u e n c y 0)o and l i n e w i d t h r o in BaTi03 . LINEWtDTH (cm"') 42 T°C “ T O ^nT 1) r T<fC’,' 1) R eferen ce 38 105 t h i s work 30 39 94 (18) Raman 36 (20) Raman 2 2 34 85 (29) I . R . 45 125 t h i s work 60 50 140 (18) Raman 41.8 92 (20) Raman 115 52 156 t h i s work 45 88 (2 0) Raman TABLE 1 Frequency and L in ew id th o f th e Overdamped E Mode 2 7 2 X < 2 Z) Y 178 FREQUENCY (tm-i) F igu re 10. The 1Z A^TO Raman spectrum of BaTiO^ a t room tem p era tu re . 44 phonon and th e pronounced assymmetry o f th e o t h e r two b ro a d e r p h o n o n s. The spectrum shows a n o th e r i n t e r e s t i n g and u n e x p ec te d c h a r a c t e r i s t i c : th e p o l a r i z a b i l i t y i s h ig h ly a n i s o t r o p i c . In f i g . 11 we compare th e a and XX a TO spectrum ta k e n in b a c k s c a t t e r i n g c o n f i g u r a t i o n . Z Z Note t h a t Y(ZZ)Y- p r e s e n t s t h e same g e n e r a l shape as Y(XX)Y. However, t h e r e i s an o b se rv e d s h i f t in th e peak p o s i t i o n s o f th e second and t h i r d phonons. The XX spectrum shows a background in th e low freq u e n cy p a r t and th e sh a rp i n t e r f e r e n c e o b se rv e d i n ZZ a p p a r e n t l y d i s a p p e a r s . Since we were p r i m a r i l y i n t e r e s t e d in th e te m p e r a t u r e dependence o f th e phonon spectrum a s s o c i a t e d w ith a phase t r a n s i t i o n , we d e c id e d to c o n c e n tr a te our a t t e n t i o n in th e ZZ spectrum w hich p ro v id e s a w e ll u n d e rs to o d s t r u c t u r e . The A^ l o n g i t u d i n a l ZZ sp e c tru m , A^ LO, i s shown in f i g . 12. I t has t h r e e peaks a t 189 cm \ 370 cm- ^ and 7 25 cm ^ . W e a ls o o b se rv e A^TO components in t h i s spectrum a r i s i n g from back s c a t t e r i n g fo llo w in g th e r e f l e c t i o n o f th e l a s e r beam a t th e e x i t s u r f a c e o f th e c r y s t a l . The i n t e n s i t y o f t h i s r e f l e c t i o n i s about 15% o f i n c i d e n t beam due to th e h ig h index o f r e f r a c t i o n (n % 2.2) o f BaTi03 . * n ChavesJU d id an e x te n s iv e stu d y o f th e ZZ A^ p o l a r i t o n shape and showed t h a t i t can be d e s c r i b e d v e ry a c c u r a t e l y by a model in which c o u p lin g among th e t h r e e 286 Y(XX)Y 276 180 524 178 F R E Q U E N C Y ( c m * ' ) F igure 11. The XX and ZZ TO spectrum of BaTi03 in baclc s c a t t e r i n g c o n f ig u r a tio n . 189 7 2 5 471 FREQUENCY (cm-') F ig ure 12. The ZZA^ l o n g it u d i n a l o p t i c a l spectrum of BaTiOj. av 47 phonons i s taken i n t o a c c o u n t. This c o u p lin g o f th e modes i s a c h a r a c t e r i s t i c found in many c r y s t a l s and is i n d i c a te d by modes which have a l a r g e s p re a d of fre q u e n c y . In BaTiO^, mode c o u p lin g was f i r s t p ro p o sed by B ark er and 31 H o p fie ld in the a n a l y s i s o f i n f r a r e d d a ta . The spectrum i s d e s c r ib e d by a 3x3 m a tr ix re sp o n se f u n c tio n G(u),q). In th e 90° s c a t t e r i n g c o n f i g u r a t i o n th e phonon wave v e c to r q i s s u f f i c i e n t l y l a r g e to be in th e f l a t re g io n o f th e p o l a r i t o n d i s p e r s i o n . In t h i s case the re s p o n se f u n c tio n can be assumed in d e p e n d e n t o f q. The in v e r s e G i s g iven by eq. 2.12 as G h w ) = - iuil^ +H' where "¥" = ( 6 . .) i s th e u n i t m a tr ix , 1 j ’ -«~2 2 ft = (ft . j ) i s th e fo rc e c o n s ta n t m a trix i s th e damping m a trix Even though i t i s n o t p o s s i b l e to e n t i r e l y d i a g o n al i z e V , one can d i a g o n a l i z e either"* ? o r 1 ^ . In many c a se s i t i s p o s s i b l e to f i n d a t r a n s f o r m a ti o n which le a d s from a p u r e - r e a l c o u p lin g to a p u re -im a g in a ry c o u p lin g o f th e modes. However, as was p o in te d out by C h a v e s , ^ t h i s i s n o t g e n e r a l l y tr u e f o r phonon modes 32 which a re I .R . a c t i v e . He has a ls o shown t h a t th e 48 p o l a r i t o n sh a p e s i n BaTiO^ a r e b e s t d e s c r i b e d by a p u r e - r e a l c o u p lin g o f t h e m odes. F or th e above m en tio n ed r e a s o n we ch o o se to d i a - g o n a l i z e T , i . e . , to c o n s i d e r th e in te rm o d e c o u p lin g as r e a l : ' r 0 0 r 2 wl W1 2 0 r = 0 r 2 0 ++ 2 fi = a)2 1 2 2 W2 2 23 0 0 r 3_ 0 U )2 23 “ L I t i s r e a s o n a b l e t o assum e = 0 b e c a u s e t h e r e i s no sp e c tru m o v e r l a p o f th e 1 s t and 3rc* phonon p e a k s . In f a c t when th e sp e c tru m i s f i t t e d assum ing a t r i a l i n i t i a l v a lu e o f d i f f e r e n t from z e r o , t h e o p tim iz e d f i n a l v a lu e i s a d j u s t e d t o z ero w i t h i n th e l i m i t o f t h i s p a r a m e te r e r r o r o f th e f i t t i n g p r o c e d u r e . The A^TO sp e c tru m was f i t t e d w ith 8 p a r a m e t e r s , a ^ , , r 2 , r 3 , and w23* A c t u a l l y , t o f i t th e Raman sp e ctru m t h r e e more p a r a m e t e r s a r e n eeded to s c a l e th e i n t e n s i t y , s i n c e th e i n t e n s i t y i s a m easure o f th e c r o s s s e c t i o n a £ [n(aO + l] I mB+cf % -V and B = (Bj» & 2 > ^ 3 ) a v e c t o r w hich d e te r m in e s th e Raman s c a t t e r i n g a m p l i t u d e s . 49 c l c l 128.5 °C m CL ROOM TEMPERATURE CL 10 0 O 200 3 0 0 4 0 0 5 0 0 6 0 0 700 FREQUENCY (cm"') F ig u re 13. Raman lin e s h a p e o f th e A^TO phonons a t 3 c h a r a c t e r i s t i c te m p e r a tu r e s . Dots a r e e x p e rim e n ta l p o i n t s . Dashed l i n e s a r e th e l e a s t sq u a re f i t o f th e d a ta p o i n ts to th e c o u p le d o s c i l l a t o r m o d el. 50 In f i g . 13 the c a l c u l a t e d spectrum and th e e x p e r im e n tal d a ta are shown f o r th r e e te m p e ra tu re s . N otice the b roaden ing and s h i f t o f th e m iddle and h ig h e r peaks as compared to the R.T. spectrum . Also shown is the ZZ spectrum a t 134°C, 2°C above phase t r a n s i t i o n . The f i r s t sh arp peak d isa p p ea red as expected b u t th e two broad peaks rem ain; t h e i r w idths in c re a s e d • the peak p o s i t io n s and the co u p lin g param eter do n o t change v e ry much. This v i o l a t i o n o f th e symmetry r u l e s appears to i n d ic a t e t h a t the symmetry seen by th e Raman e f f e c t i s n o t p e r f e c t l y c u b ic , or in o th e r words th e r e i s something remnant o f th e t e tr a g o n a l symmetry. W e a ls o n o tic e d t h a t in the p o l a r i z a t i o n ZY no Raman l i n e s a re p r e s e n t . The coupled mode p aram eters a re p l o t t e d as a fu n c t i o n o f tem p era tu re in f i g . 14. The frequ ency of the sharp mode i s rem arkably i n s e n s i t i v e to te m p e ra tu re , which s u g g e sts t h i s phonon is n o t a s s o c i a t e d w ith a s o f t mode. The freq u en cy d e c re a se s w ith te m p e ra tu re and w^ in c r e a s e s s l i g h t l y . This b e h av io r o f w^ i s an i n t e r e s t i n g consequence o f th e mode c o u p lin g and in th e a c tu a l s p e c tr a b oth peak fre q u e n c ie s appear to d e c re a s e . F ig . 15 i l l u s t r a t e s t h i s p o i n t. N otice t h a t th e uncoupled f r e quency w£ o b ta in e d from f i t t i n g i s much h ig h e r than th e peak w^. This is l a r g e l y due to th e p resen c e of th e id Bose f a c t o r as d isc u sse d in c h a p te r 2. 51 600| i i i i i i-------1 -------1 -------j-------j-------p Coupled transverse modes with Ai - symmetry in tetragonal BaTi03 cUf: frequencies 5 0 0 b C0UP|in gs T ( : linewidths -O ^ 4 0 0 - E u >- o w 300 O u Q L U . 200 tf A W i -O 0 O — o O -Q — O "..-0-"0" A .-A*-' . - - i r C d , 12. -i* r- -1 7 .-VO V -V- ■ J L J I I I I L 4 0 6 0 8 0 100 TEMPERATURE (°C) 1 2 0 'C| 140 F ig u re 14. T em perature dependence o f th e A^TO mode p a ra m e te rs in BaTiO^. 52 o o * o Oo o O o o oo 5 0 0 - ° O O O O O o, • • • 4 0 0 - o phonon peak from spectrum • phonon frequency from fitting I E u 3 0 0 - >- u Z U l 3 O Ui o c li. o O O o o 0 o o o o O O O O q o 200 - 6 6 6 p < 5 < 7 0 p o & o « o o°* Oo 100- 20 4 0 60 80 100 TEMPERATURE (°C) 120 Tc, 140 F igure 15. T em perature dependence o f th e A-^TO mode f r e q u e n c ie s in BaTiOg; Open c i r c l e s a re th e peak f r e q u e n c ie s and d o ts a re f re q u e n c ie s from l . s . f i t to cou pled o s c i l l a t o r s model. S3 A n o th e r c h a r a c t e r i s t i c w o rth o b s e r v in g i s th e c o u p lin g p a r a m e t e r s : w^^ i s more th a n tw ic e w and th e y do n o t change v e ry much w ith t e m p e r a t u r e . In f a c t t h e o n ly p a ra m e te r s r e a l l y s e n s i t i v e t o te m p e ra tu re a r e th e l in e w i d t h s and T . The f a c t t h a t T (T) is n o n l i n e a r m ight i n d i c a t e th e r e l a t i o n s h i p o f t h e s e phonons to th e p h a se t r a n s i t i o n . F ig . 16 shows th e te m p e r a tu r e dependence o f th e A^LO phonons w hich rem ain p r a c t i c a l l y c o n s t a n t th r o u g h o u t th e t e t r a g o n a l p h a s e . The e f f e c t i v e c h a rg e p a ra m e te rs shown in t h i s f i g u r e were o b ta in e d by s o l v i n g th e e q u a tio n e(wL) = 0 u s in g th e m easured v a lu e s f o r th e l o n g i t u d i n a l f r e q u e n c i e s . In f i g . 17 we show r e s u l t s o f m easurem ents o f t h e Aj, p o l a r i t o n mode as a f u n c t i o n o f te m p e r a tu r e f o r a f i x e d e x t e r n a l s c a t t e r i n g a n g le . At room te m p e ra tu re th e i n t e r n a l s c a t t e r i n g a n g le was 0 ^ 1 0 0 ’ w here th e i n t e r f e r e n c e d ip a ro u n d 200 cm 1 i s a maximum. N o tic e t h a t th e sp ectru m b ro a d e n s as th e te m p e r a tu r e i n c r e a s e s and the i n t e r f e r e n c e d ip d is a p p e a r s j u s t above t h e phase t r a n s i t i o n . At 134°C th e sp ectru m i s s i m i l a r t o th e one ta k e n a t r i g h t a n g le s c a t t e r i n g shown in t h e u p p e r p a r t o f f i g . 13. The room te m p e r a tu r e sp e ctru m o f t h e A ^, p o l a r i t o n can be a c c u r a t e l y d e s c r i b e d by th e c o u p le d r e s p o n s e m a t r ix 54 700 longitudinal modes with Aj_-symmetry in tetragonal BaTiO, 600 a), - LO frequencies Q - effective charges 500 C l ) Li- 200 00, 1 0 0 20 60 40 120 Tc, 80 1 0 0 TEMPERATURE (“C) F ig u re 16. Tem perature dependence of th e A^LO phonon f r e q u e n c ie s and e f f e c t i v e ch arg es in BaTiO^. 3 0 * C 6 0 0 5 0 0 4 0 0 3 0 0 200 100 F R E Q U E N C Y (c n ri) F igure 17. Temperature dependence of th e A-^TO p o l a r i t o n in BaTiO^. C n Cn 56 when th e q dependence i s e x p l i c i t l y i n t r o d u c e d . 2^ We d id n o t a tte m p t to f i t th e te m p e ra tu re dependence o f th e p o l a r i t o n b e cau se a l l in f o rm a tio n needed f o r our p u rp o se can be tak e n from th e a n a l y s i s o f th e phonon s p e c t r a . As a f i n a l p o i n t on th e phonon s p e c t r a o f t e t r a g o n a l BaTiO^ we m ention th e f a c t t h a t o nly E modes have been d e te rm in e d by th e I.R . r e f l e c t i o n m easurem ents 29 o f S p i t z e r e t a l . and o t h e r s . There a re v a r io u s re a s o n s why th e A^ modes were n o t s e e n . The main re a s o n is th e extrem e damping o f th e I n f r a r e d l i g h t which e x c i t e s th e E modes. Because th e p e n e t r a t i o n d e p th i s v e ry sm all what one m easures i s th e I n f r a r e d r e f l e c t i v i t y o f th e s u r f a c e where th e symmetry i s u s u a l l y n o t w e ll d e f in e d as in th e b u lk and we have a m ixing of th e E and A^ modes. S ince t h e r e a re two p o l a r i z a t i o n d i r e c t i o n s f o r th e E modes [X and Y] and o n ly one [Z] f o r th e A^ modes th e r e f l e c t e d spectrum h as e s s e n c i a l l y E symmetry c h a r a c t e r . Indeed th e I n f r a r e d r e f l e c t i v i t y o f powder and ceram ic BaTiO^ was found to be alm ost th e same as th e one m easured 34 f o r th e s i n g l e c r y s t a l sample o f S p i t z e r . I t would be v e ry i n s t r u c t i v e to r e p e a t th e I n f r a r e d r e f l e c t i v i t y o f BaTiO^ w ith th e l a r g e s i n g l e domain c r y s t a l s now a v a i l a b l e . By a p ro p e r chem ical e tc h in g i t sh ould be p o s s i b l e to remove th e f i r s t few l a y e r s and make th e s u r f a c e s i n g l e c r y s t a l . We a ls o want to p o i n t o u t t h a t th e 520 cm * 57 mode s p l i t t e d from th e cub ic mode was a c t u a l l y seen by Last in I n f r a r e d a b s o rp tio n o f a t h i n f ilm of BaTiO . However, he was n o t a b le to see th e o th e r two 3 rem aining modes because th e range o f h is in stru m e n t was l im i te d to the i n t e r v a l 300 cm"'*' to 600 cm"'*'. 3 . 6 The D i e l e c t r i c C onstant The p o l a r i z a t i o n in a medium, P(w), i s r e l a t e d to the e l e c t r i c f i e l d E (w) by P(w) = x(w) E(w) (3.1) where x(w) is th e e l e c t r i c s u s c e p t i b i l i t y . In a c r y s t a l we can u s u a lly s p l i t x (w) in to th re e major c o n tr i b u ti o n s : xO) = Xe le c O) + XrO) + XD(w) (3.2) where x n i s th e e l e c t r o n i c s u s c e p t i b i l i t y , xn is C J L 6C R re s o n a n t io n ic c o n t r i b u t i o n , Xp is a D eb ye-like non r e s o n a n t c o n tr i b u ti o n coming from any s t a t i s t i c a l d is o r d e r in th e c r y s t a l . The io n ic term a r i s e s from th e o p t i c a l phonons and, p o s s ib l y , from th e a c o u s tic phonons through th e p i e z o e l e c t r i c e f f e c t . These a c o u s t i c resonances a re sharp and depend on the sample dim ension, whereas the o p t i c a l phonon c o n t r i b u t i o n is in dependent of sample s i z e . 58 The d i e l e c t r i c f u n c tio n i s d e fin e d by e(w) = 1 + 4ttx(w) (3.3) W e a re i n t e r e s t e d in f r e q u e n c ie s in th e phonon range where th e e l e c t r o n i c c o n t r i b u t i o n v a r i e s slo w ly w ith fre q u e n c y and we can w r i t e : 1 + x i = £ % n e le c 00 Here n i s th e index o f r e f r a c t i o n o f th e c r y s t a l . The c o n t r i b u t i o n of th e o p t i c a l phonons can be c a l c u l a t e d i f we know th e re s p o n se f u n c t i o n G (w ): XR (w) = Q +1T(w)Q (3.4) - > ■ where Q = (Q^, Q2 , . . . , Qn ) and i s e f f e c t i v e charge a s s o c i a t e d w ith th e i ^ phonon mode. The d i e l e c t r i c c o n s t a n t in th e [Z] d i r e c t i o n o f BaTiO^ was computed w ith eq. 3.3 and eq. 3.4 d i r e c t l y by u sin g i n eq. 3.4 th e coupled mode re s p o n s e f u n c t i o n o b ta in e d from th e f i t t i n g o f th e phonon sp e ctru m . When th e re s p o n se f u n c t i o n can be a p p ro x im ated by a sum o f n o s c i l l a t o r s w ith no damping th e d i e l e c t r i c c o n s ta n t e (0 ) can be c a l c u l a t e d by u sin g th e LST(3) r e l a t i o n : 59 (3 .5 ) w here w *; TO ■ j ^ |_ and w a re th e i t r a n s v e r s e and l o n g i t u d i n a l o p t i c a l mode f r e q u e n c i e s r e s p e c t i v e l y . An i m p o r t a n t q u e s t i o n a r i s e s when we have h i g h ly damped modes as i n BaTiO^: Is t h e LST r e l a t i o n s t i l l v a l i d ? I f i t i s v a l i d , w hat f r e q u e n c i e s sh o u ld we u se in i t ? c a s e t h a t t h e r e a r e n damped, u n c o u p le d m odes. F i r s t , he d e f i n e s t h e mode f r e q u e n c y o f th e TO modes as t h e p o le s o f th e d i e l e c t r i c c o n s t a n t , and th e L0 f r e q u e n c i e s a s th e z e r o s o f t h e d i e l e c t r i c c o n s t a n t . Then he shows t h a t we can o b t a i n a fo rm u la s i m i l i a r to e q . ( 3 .5 ) b u t i n s t e a d o f th e TO mode f r e q u e n c i e s one s h o u ld u s e th e f o r c e c o n s t a n t s 0) . . l 3 . 6 . 1 The ex d i e l e c t r i c c o n s t a n t In t h e t e t r a g o n a l p h a s e , t h e d i e l e c t r i c c o n s t a n t in a d i r e c t i o n p e r p e n d i c u l a r to th e f e r r o e l e c t r i c a x is a r i s e s from th e E phonons and was c a l c u l a t e d from t h e d a t a u s i n g th e LST r e l a t i o n : 31 B a rk e r gave an answ er to t h i s q u e s t i o n in th e 60 The main c o n t r i b u t i o n comes from t h e f o r c e c o n s t a n t o f th e overdamped phonon . The r e s u l t i n g te m p e ra tu re dependence o f Ejl(O) i s p l o t t e d in f i g . 18. The e n c i r c l e d d o t i s th e d i e l e c t r i c c o n s ta n t v a lu e from th e p o l a r i t o n 35 m easurement o f Laughman e t a l . In th e same f i g u r e we have p l o t t e d the c a p a c ita n c e m easurem ents o f th e e x d i e l e c t r i c c o n s t a n t done by Wemple e t a l . ^ KHz and 250 MHz. The f i r s t th in g to n o t i c e i s th e fre q u e n c y dependence o f th e c a p a c ita n c e m easurem ents: The 100 KHz curve shows c r i t i c a l b e h a v io r a t b o th T_t and T w hereas the 250 MHz i n c r e a s e s mono- c i c 2 t o n i c a l l y as th e te m p e ra tu re goes from Tc ^ to T ^ • The phonon c o n t r i b u t i o n a p p e a rs to have a s lo p e s i m i l a r to c u rv e b b u t d i s p l a c e d by an alm ost c o n s t a n t f a c t o r th e o rd e r o f 7-8*10^. 3 .5 .2 The e ,, d i e l e c t r i c c o n s t a n t In th e Z d i r e c t i o n t h e d i e l e c t r i c c o n s t a n t i s d e te rm in e d by th e symmetry phonons. S in c e th e phonons a r e co up led we do n o t use t h e LST r e l a t i o n to compute I n s t e a d we u se d e f i n i t i o n 3 .3 d i r e c t l y w ith x ( w) g iven by 3.4 where G(w) i s th e co up led r e s p o n s e m a tr ix a lr e a d y d e te rm in e d from th e f i t t i n g o f th e TO sp e ctru m . The LO f r e q u e n c i e s o f f i g . 1 1 were used t o d e te rm in e th e e f f e c t i v e c h a rg e s which a re a l s o p l o t t e d in th e same f i g u r e . The 61 5 0 0 0 r 4 0 0 0 I 1 ----------- 1 ----------- 1 ----------- 1 ------ a - C apacitance m easu rem en ts at lOOKHz by Wemple et al b - idem a t 2 5 0 MHz • - r e s o n a n c e contribution (e le c tro n s + p h o n o n s) O - p o larito n m e a su re m e n ts by L a u g h m a n et al 3 0 0 0 \ a \ \ 2000 \ \ v \ X X © I / • • • • 1 0 0 0 - 0 T 2 0 4 0 6 0 8 0 100 120 T 140 c2 T E M P E R A T U R E (°C) c « F ig u re 18. Tem perature dependence o f th e d i e l e c t r i c con s t a n t p e rp e n d ic u la r to th e f e r r o e l e c t r i c a x is in the te tr a g o n a l phase o f BaTiO^. 62 r e a l p a r t o f th e d i e l e c t r i c c o n s t a n t a t room te m p e ra tu re i s a ls o p l o t t e d in f i g . 19. The l i m i t i n g v a lu e Lim e^(w) 0) o d e f i n e s th e d i e l e c t r i c c o n s t a n t which i s p l o t t e d as a f u n c t i o n of te m p e ra tu re in f i g . 20. T h is d a ta shows a v e ry sm all i n c r e a s e in e (0 ) w ith te m p e r a tu re , which i s alm ost im p e r c e p tib le on th e s c a l e o f th e f i g u r e . This 22 r e s u l t d e p a r ts somewhat from th e one o b ta in e d by Burns; even though th e o rd e r o f m agnitude i s th e same. W e b e l i e v e t h a t t h i s d is c re p a n c y i s due to th e Bose f a c t o r which was u naccoun ted f o r in th e p o l a r i t o n d i s p e r s i o n c u rv es o f B urns. The c a p a c ita n c e m easurem ents o f th e d i e l e c t r i c 25 c o n s t a n t by Wemple e t a l . a t 100 KHz and 250 MHz i s a ls o shown in f i g . 2 0 . In f i g . 21 we compare th e d i e l e c t r i c c o n s ta n t c a l c u l a t e d w ith coupled mode t h e o r y w ith th o s e o f Burns and th o s e c a l c u l a t e d by th e LST fo rm u la u s in g th e peak f r e q u e n c ie s o f the spectru m . I t i s seen from f i g s . 20 and 21 t h a t th e a c t u a l d i e l e c t r i c c o n s t a n t i s much l a r g e r th a n th e phonon c o n t r i b u t i o n . C o n s id e rin g th e v a lu e o f Burns or th e LST v a lu e u s in g peak f r e q u e n c i e s , t h e r e i s no way t h a t we can o b ta in a c r i t i c a l te m p e ra tu re dependence n e a r T i as e x p ec te d . T his c l e a r l y i n d i c a t e s t h a t a so urce 63 50 4 0 30 20 - 1 0 -20 100 200 300 400 500 600 700 800 F R E Q U E N C Y ( c m * ' ) -30 F ig u re 19. Real p a r t o f th e d i e l e c t r i c f u n c t i o n e ' , in th e Z d i r e c t i o n s o f BaTiO^ u s in g coup led mode th e o r y . 250 b - idem at 2 5 0 MHz 200 - 1 5 0 - 5 0 - 0 i 1 r a-C apacitance at 100kHz / / / c - Resonance contribution/ (phonons + electrons) j / / i / I / I / / n / T I I I / / / / / / 20 ± 64 • • • 4 0 6 0 8 0 100 120 T 140 'C2 TE M P E R A TU R E (°C) C | F igure 20. Tem perature dependence of th e d i e l e c t r i c c o n s ta n t p a r a l l e l to the f e r r o e l e c t r i c a x is in the t e t r a g o n a l phase o f BaTiO 3* C apacitance measurements from r e f . (18) DIELECTRIC CONSTANT ( Z - AXIS) 65 50i 40 30 20 10 — I -1— 1 1 1 - 1 1 1 1 " " 1 ' 1 ... 1 | > • • — • - 1 1 1 • • + • + • • i 1 1 _ 1 + + + + ° o ° o ° o O ° < » 0 [ 1 1 1 o 8 o o o o 1 ' 1 1 1 - 1 • from peaks of spectra 1 1 " 1 o from coupled modes theory fittings I 1 1 1 + from Burns ref (14) 1 I 1 1 1 1 1 1. 1 . 1 i 1 . 1 . 1 1 1 1 i 1 — fc _ i— T * 20 40 te m ^ ratu b IV o 1 0 0 1 2 0 T c ' 1 4 0 F ig u re 21. T em p erature dependence o f th e phonon + e l e c t r o n d i e l e c t r i c c o n s ta n t in BaTiO^ p a r a l l e l to th e f e r r o e l e c t r i c a x i s . The f u l l c i r c l e s a r e th e LST v a lu e s from th e peaks o f s p e c t r a . 66 o th e r than phonons, has to be found fo r t h i s d i e l e c t r i c beh av io r and the most pro bably o r i g i n i s a type X^ c o n t r i b u tio n . 3.7 An A n aly sis o f the Spectrum P aram eters in R e la tio n to th e Theory o f F e r r o e l e c t r i c i t y 3 .7 .1 A Thermodynamic D e s c rip tio n o f F e r r o e l e c t r i c Phase T r a n s itio n Let us f i r s t c o n sid e r a thermodynamic d e s c r i p t io n of the f e r r o e l e c t r i c phase t r a n s i t i o n . I t i s convenient to c h a r a c t e r i z e a f e r r o e l e c t r i c c r y s t a l by an e q u atio n o f s t a t e in term s o f the fre e energy In a d d itio n to th e en tro p y term , th e fre e energy f o r a f e r r o e l e c t r i c c r y s t a l in c lu d e s a sum o f g e n e ra liz e d fo rc e -d is p la c e m e n t p ro d u cts and the energy a s s o c ia te d w ith the spontaneous p o l a r i z a t i o n ? in th e p resen c e o f the f i r s t and second o rd e r . A f i r s t o rd e r t r a n s i t i o n occurs when th e r e i s a d i s c o n t i n u i t y in the en tro p y which i s given by the f i r s t d e r i v a ti v e o f the fr e e energ y , dF = -SdT - s x i dXi + £ dP i (3.6) f i e l d E. W e d i s t i n g u i s h two types o f phase t r a n s i t i o n s , Using th e fo llo w in g expansion o f F in powers of 67 th e p o l a r i z a t i o n F = F + 1 a P 2 + i*6 P4 + i « P 6 , (3 .7 ) 0 2 i* b ' th e change in e n tro p y a t th e t r a n s i t i o n te m p e ra tu re i s g iv en by AS = S -S = 4-P2 — + I P 4 H + i p 6 M (3 .8 ) o 7 sc 9T * » sc 8T e sc 3T where P i s th e spo ntan eo u s p o l a r i z a t i o n a t th e t r a n - sc s i t i o n te m p e r a tu re . From th e thermodynamic p o i n t o f view a d i s c r i p t i o n o f t h e b e h a v io r o f a f e r r o e l e c t r i c c r y s t a l i s d e te rm in e d by p a ra m e te rs l i k e a , 8 and 6 in e q u a tio n ( 3 .7 ) . Devonshire'*' u se d an ex p an sio n s i m i l a r to eq. (3 .7 ) in o r d e r to d e s c r i b e a l l t h r e e phase t r a n s i t i o n s in BaTiO^. He assumed th e s im p le s t p o s s i b l e te m p e ra tu re dependence o f th e c o e f f i c i e n t s a , 8 and 6 : a l i n e a r l y dependent on T, a = A(T - T ) , and 8 and 6 c o n s t a n t s . In t h i s case eq . (3 .8 ) re d u c e s to AS = L V 2 | | (3 .9 ) 2 S C o I Thus i f P f 0 as T T a f i r s t o rd e r p hase t r a n s i t i o n 3 0 C w i l l o ccu r w ith l a t e n t h e a t AQ = Tc AS. In f i g . 22 th e sp o n tan eo u s p o l a r i z a t i o n o f BaTiOj i s shown as a f u n c t i o n o f t e m p e r a t u r e . Note t h a t as T + Tc ^ 134°C th e p o l a r i z a t i o n i s f i n i t e which i s c h a r a c t e r i s t i c o f a f i r s t o r d e r 68 Z.25 E .20 a. .1 5 2 . 10 T E M P E R A T U R E ( • € ) F igu re 2 2 . Spontaneous p o l a r i z a t i o n of BaTiO^ in t e t r a gonal f e r r o e l e c t r i c phase a f t e r Wemple e t a l . r e fe r e n c e (25) . 69 f e r r o e l e c t r i c to p a r a e l e c t r i c p hase t r a n s i t i o n . In a second o r d e r f e r r o e l e c t r i c to p a r a e l e c t r i c ph ase t r a n s i t i o n , th e spon tan eo u s p o l a r i z a t i o n d e c r e a s e s m o n o to n ic a lly w ith i n c r e a s i n g te m p e ra tu re and becomes zero a t some te m p e ra tu re T = T . At T = Tc , th e r e i s a d i s c o n tin u o u s change in th e s p e c i f i c h e a t which i s a second o r d e r d e r i v a t i v e o f F c = T ( f § ) = " T • C3 - 1 0 ) \3T/XP \3T j xp 3 .6 .2 Dynamic m odels o f th e f e r r o e l e c t r i c phase t r a n s i t i o n . The m ic ro s c o p ic th e o r y o f th e p hase t r a n s i t i o n which has been p ro posed by Cochran i s b ased on th e e x is ta n c e o f a s o f t mode. Cochran used a s h e l l model f o r th e io n s in a d ia to m ic c r y s t a l to show t h a t th e fre q u e n c y o f th e q % 0 TO mode can be w r i t t e n as a d i f f e r e n c e o f two q u a n t i t i e s : .2 - „1 2 ) ( z ' e ) 2 y < = R - 4t t ^ (3.13) 1 0 9v where y is th e red u ced mass o f th e io n s , v the volume o f th e u n i t c e l l and Z 'e i s th e e f f e c t i v e io n ic c h a rg e . The f i r s t term on th e r i g h t r e s u l t s from th e r e s t o r i n g " s h o r t ra n g e " f o r c e on any one atom and th e second term 4 from th e L o ren tz l o c a l f i e l d . Since th e l a t t i c e v i b r a t i o n s a r e n o t c o m p le te ly harm onic, t h e q u a n t i t i e s a p p e a rin g in eq. (3,13) w i l l v a ry 70 w ith t e m p e r a t u r e and t h e r e i s a p o s s i b i l i t y o f c a n c e l a t i o n o f t h e two term s a t a c e r t a i n c r i t i c a l t e m p e r a t u r e T . T h is i s th e p h y s i c a l mechanism f o r th e s o f t mode l i m i t , 0 , in th e dynam ic m odel. C ochran p o s t u l a t e d th e f o l lo w in g C u rie law form f o r th e s o f t mode f re q u e n c y w2 =wqY(T - T^) . (3 .1 4 ) The f a c t t h a t a v i b r a t i o n a l mode f r e q u e n c y e x h i b i t s t h i s d e p en d en ce on t e m p e r a t u r e i s u s u a - l y t a k e n as s u p p o r t f o r 3 6 C o c h ra n ’ s t h e o r y . However, i t h a s b een shown by A ndrade g e t a l . t h a t f o r an o r d e r - d i s o r d e r ty p e t r a n s i t i o n , such as i n th e c a s e o f NaNC^, one can d e s c r i b e t h e mode f r e quency as < d2 = to2 [ 1 + y(T - T )] (3 .1 5 ) o c w here Tc i s t r a n s i t i o n t e m p e r a t u r e . I t i s a p p a r e n t t h a t a f i r s t o r d e r p h a se t r a n s i t i o n su c h as i n BaTiO^ can be d e s c r i b e d by e i t h e r e q . (3 .1 4 ) o r eq . ( 3 .1 5 ) s i n c e t h e s e f O form s a r e e q u i v a l e n t f o r Tc = Tc + • 3 . 6 . 3 D i s c u s s i o n o f t h e BaTiO^ sp e c tru m . 31 B a rk e r i d e n t i f i e d th e overdam ped F^u mode in th e c u b ic p h a se as t h e s o f t mode in BaTiO^ and r e i n t e r p r e t e d 30 th e r e s u l t s o f B a l l a n t y n e 's I .R . m ea su rem en ts to show t h a t th e f o r c e c o n s t a n t h as a t e m p e r a t u r e d ep en d en ce w hich 71 a g re e s w ith e q . ( 3 . 1 4 ) . We t h in k t h a t t h i s r e s u l t cannot be ta k e n as a p ro o f o f th e s o f t mode th e o r y in th e sense t h a t th e v a lu e o f Tc used to f i t w^(T) in e q .( 3 .1 4 ) i s the same as th e one which i s o b ta in e d from th e C urie Law f o r c(T) in th e p a r a e l e c t r i c p h a se . The re a s o n s f o r t h i s a re f i r s t l y becau se th e I.R . d a ta c o u ld n o t be a c c u r a t e l y f i t t e d , by B a rk e r, to an o s c i l l a t o r model and, se c o n d ly , be- cause th e to^,CT) curve c o n ta in s only t h r e e p o i n t s which is c e r t a i n l y n o t s u f f i c i e n t to s t a b l i s h a m ea n in g fu l v a lu e of T . c W e now ta k e th e two h ig h ly damped E and A1 modes and r e l a t e them w ith th e f e r r o e l e c t r i c i t y in BaTiO^. T h eir r e l a t i o n to f e r r o e l e c t r i c i t y i s o n ly i n th e se n se t h a t fo r th o se modes some atoms in th e c r y s t a l o s c i l l a t e in a p o te n t i a l which c o n ta in s a double w e l l. Then u s in g e q (3 .1 5 ) which a p p l i e s to an double w e ll o r d e r - d i s o r d e r mechanism we p l o t th e te m p e ra tu re dependence o f th e fre q u e n c y o f th e s e two modes in f i g . 23 u sin g Tc = 132°C f o r th e A1 mode and T = T = 6 °C f o r th e E mode. The agreem ent of expe- c cz rim e n t and th e o ry i s v e ry good. T his agreem ent does not g a r a n t e e s t h a t th e t r a n s i t i o n i s an o r d e r - d i s o r d e r one b ecau se as we have seen th e same d a ta can be r e i n t e r p r e t e d in term s o f th e s o f t mode t h e o r y . In deed th e re a so n t h a t e q .( 3 .1 5 ) or e q .( 3 .1 4 ) a g re e w ith e x p erim e n t depends on th e therm odynam ic f a c t th e a n h a rm o n ic ity o f th e p o t e n t i a l e n e r gy can be a p ro x im ated by th e l i n e a r term in th e s e r i e s ex p an sio n of U=U(T). A nother im p o rtan t p h y s ic a l q u a n t i t y i s the low f r e quency d i e l e c t r i c c o n s ta n t. E a rly m easurements o f th e tem p e r a t u r e dependence of £q(T) above determ ined t h a t i t fo llo w s the C urie Weiss Law. B a r k e r 's s o f t mode fre q u e n cy c o r r e l a t e s t h i s b e h av io r fo r e ( T ) , b u t in o rd er to be c o n s i s t e n t th e s o f t mode has to g iv e the c o r r e c t e(T) both above and below T . The d a ta o b ta in e d by c a p a c ita n c e mea surem ents shows t h a t ap p roachin g T ^ from below the quan t i t y 1 /e,, i s approach ing zero b u t we have shown t h a t t h i s b e h a v io r is n o t a r e s u l t o f s o f t p h o n o n s. In comparing measurements o f d i e l e c t r i c c o n s ta n t one has to d i s t i n g u i s h between d i e l e c t r i c c o n s ta n t a t con s t a n t s t r e s s and d i e l e c t r i c c o n s ta n t a t c o n s ta n t s t r a i n . The l a t t e r i s c a l l e d the clamped d i e l e c t r i c c o n s ta n t. Dev- 3 8 o n s h ire has shown t h a t th e f r e e and clamped p aram eter which ap p ear in h is expansion o f th e f r e e energy o f BaTiO^ can be v ery d i f f e r e n t . D e v o n s h ire 's c a l c u l a t i o n showed t h a t th e clamped d i e l e c t r i c c o n s t a n t should be about two o r d e r s o f m agnitude sm a lle r th an th e d i e l e c t r i c c o n s ta n t a t c o n s t a n t s t r e s s . The phonon d i e l e c t r i c c o n s t a n t t h a t one o b ta in s by m easuring the Raman s c a t t e r i n g or th e i n f r a r e d r e f l e c t i v i t y o f a c r y s t a l i s the clamped v a lu e and sh o u ld t h e r e f o r e be compared to th e c a p a c ita n c e d i e l e c t r i c c o n s ta n t which i s a ls o clamped. The d ata o f Wemple a t a l . in f i g . 18 and 73 f i g . 2 0 shows t h a t th e c a p a c i ta n c e v a lu e s o f and e„ a r e v e ry s e n s i t i v e to th e m ea su rin g f r e q u e n c y . The low er fre q u e n c y (100 KHz) v a lu e s o f C j. and e „ a r e much l a r g e r t h a n th e h igh fre q u e n c y (250 MHz) v a l u e s . T his d i s c r e pancy c an be a t t r i b u t e d to th e n o n -cla m p in g o f th e d i e l e c t r i c a c o u s t i c re s o n a n c e s and a l s o to t h e p o l a r i z a t i o n of th e i m p u r i t i e s in th e c r y s t a l . Wemple e t a l . n o t i c e d a v e r y sm a ll change in th e d i e l e c t r i c c o n s t a n t f o r f r e q u e n c ie s above 200 MHz and t h e r e f o r e i t i s r e a s o n a b le to assume th e 250 MHz d a ta to be clam ped. I f we ta k e t h e s e v a lu e s as r e p r e s e n t i n g th e t r u e i n t r i n s i c clam ped d i e l e c t r i c c o n s t a n t i t becomes c l e a r t h a t th e phonon c o n t r i b u t i o n i s j u s t a sm a ll p a r t o f e and i t d o e s n 't g iv e th e e x p e c te d c r i t i c a l dependence as Tc i s a p p ro a ch e d from below . One c o u ld a s s e r t t h a t t h e r e i s a phonon mode w hich a c c o u n ts f o r e (T) w hich i s n o t o b se rv e d in th e o Raman s p e c tru m . T h is i s n o t v e ry p r o b a b le s i n c e a l l th e p o s s i b l e o p t i c a l modes (q % 0) have b e en i d e n t i f i e d . Thus, we a r e f o r c e d to c o n c lu d e t h a t th e c r i t i c a l te m p e r a tu re d ep enden ce of eQ i s due to a mechanism o t h e r th a n th e o p t i c a l p h o n o n s. To e x p la in t h i s d i s c r e p a n c y one can s u g g e s t v a r i o u s e f f e c t s such as n o n s t o ic h io m e tr y d e f e c t s as p ro p o se d by B urns, f r e e c h a r g e s due t o i m p u r i t i e s , e t c . ; how ever, none o f t h e s e can e x p la in th e fo llo w in g anomalous f a c t s : 74 400 300 80 200 continuos line is the l.s.fitftng ot data to W = W o ( i+ y ( T - T c ) ) ^ 3 0 - 00 20 4 0 60 80 TEMPERATURE CC (00 120 Tci (40 F igure 23. Temperature dependence o£ the frequency of the f e r r o e l e c t r i c modes in t e t r a g o n a l BaTiO^ 75 odd even blue green yellow greer even bl ue r 7 -------- — Dipole even Dipole Forbidden Allow ed Figure 24. Cu90 band s t r u c tu r e a t the c e n te r of the I d B r i llo u in Zone 76 a) the s e l e c t i o n r u le v i o l a t i o n through which the two Raman a c tiv e modes in the t e t r a g o n a l phase at 300 cm ^ and 500 cm ^ a re a lso a c tiv e in th e cubic phase, b) the p e c u l i a r lin e w id th v a r i a t i o n o f the (300 cm mode and the E(35 cm *) mode, c) the a n is o tro p y of the D ebye-like d i e l e c t r i c c o n s t a n t . We propose an e x p la n a tio n o f the BaTiO^ spectrum c h a r a c t e r i s t i c s by assuming an i n t r i n s i c , a n i s o t r o p i c o r d e r - d i s o r d e r mechanism fo r the phase t r a n s i t i o n . In t h i s model th e d i e l e c t r i c c o n sta n t can be d e s c rib e d q u a l i t a t i v e l y by a D ebye-like term e e = --------------------------------------------------------------- (3.16) 1 + W T in which t = x e ^U/kT the r e l a x a t i o n time a s s o c ia te d o w ith the m icro sco p ic f l u c t u a t i o n s o f e in the d i r e c t i o n of i n t e r e s t . The r e l a x a t i o n depends on AU the p o t e n t i a l b a r r i e r a s s o c i a t e d w ith th ese f l u c t u a t i o n s . In the Z d i r e c t i o n , the p o t e n t i a l b a r r i e r is high and decreases as the tem p eratu re in c r e a s e s . t (T) i s then d e crea sin g and e(T) i s i n c r e a s in g as observed in the Raman s p e c tr a . In o rd e r to p r e c i s e l y account f o r the c r i t i c a l dependence a t T ^ i t w i l l be n e c e ssa ry to use a more complete formula f o r than eq. ( 3 .1 6 ) , perhaps a form 9 s i m il a r to t h a t used by Andrade e t a l . to e x p la in the 77 d i e l e c t r i c constant in NaNO^. In the X and Y d i r e c ti o n the b a r r i e r has to be lower so th a t t is approximately independent of tem perature except close to T • This would give a Debye-like c o n trib u tio n almost constant in the XY plane as observed. In a d d itio n to the d i e l e c t r i c c o nstant i t is p o s s i b l e to ex plain the following f a c t s in the framework of an o r d e r-d is o rd e r model. 1) The permanence of the modes at 300 cm”^ and 500 cm"^ in the cubic phase. A double well p o t e n t i a l d iso rd e r mechanism would imply the p o s s i b i l i t y of an instantaneous te tra g o n a l symmetry fo r the d iso rd e r mode even though the symmetry is cubic on the average. 2) This double well p o t e n t i a l d iso rd e r mechanism is c o n s is te n t with the X-ray s c a t t e r i n g measurements of 10 37 Comes, Lambert and Guinier. ’ They have shown th a t th e re i s d iff u s e X-ray s c a t t e r i n g in BaTiO^ and KNbO^. This d if f u s e s c a t t e r i n g shows an i n t e n s i t y d is c o n tin u ity a t the phase t r a n s i t i o n and t h e r e fo re was a sso c iated with f e r r o e l e c t r i c i t y in these c r y s t a l s . An ion c o n sta n tly changing i t s c r y s t a l p o s itio n would be c o n s is te n t with mixed symmetries as well as a d i f f u s e X-ray s c a t t e r i n g component. The d iffu s iv e n e s s is mainly a sso c ia te d with the Ti ions, a f a c t which confirms the o r i g in a l idea of Mason on f e r r o e l e c t r i c i t y in BaTiO^. 78 3) The dram atic in c r e a s e of the lin e w id th of th e 300 cnT^ phonon w ith in c r e a s in g tem p eratu re can a ls o be understood in t h i s model. The lin e w id th o f a phonon a s s o c ia te d w ith the d i s o r d e r f l u c t u a t i o n s can be w r i t t e n r = a + bT + — — ■ ■ - f (3.17) 1 + tO T in which th e d e p a rtu re from l i n e a r i t y a r i s e s from the l a s t term. Since t w i l l decrea se as th e tem perature i n c r e a s e s , t h i s l a s t term w ill le a d to i n c r e a s in g T observed for the mode. In the XY plane, assuming t to be weakly dependent on tem p eratu re w i l l g ive a l i n e a r T dependence f o r r, except c lo s e to T where i t s a t u r a t e s i n d i c a t i n g a more ra p id v a r i a t i o n of t near T ,,. x c Z . As a f i n a l comment we observe t h a t since KNbOj, which i s isomorphic w ith BaTiO^ in a l l fo u r phases, shows a d i f f u s e X-ray s c a t t e r i n g , i t would be i n s t r u c t i v e to measure i t s Raman s c a t t e r i n g in the cu b ic phase. One should d e t e c t the same A^ mode of th e t e t r a g o n a l phase a s s o c i a t e d w ith the d i s o r d e r mechanism. 79 CHAPTER EXCITONS AND THE RAMAN SCATTERING OF Cu20 AND CdS 4.1 The Is Yellow Exciton in Cu20 Cuprous Oxide c r y s t a l i z e s in the c u p r ite s t r u c t u r e with space group 0^. I t has th e follow ing zone c e n te r h v i b r a t i o n a l modes: 2F1 + iF + iE + 1A, + 1F0 lu 2g u lu 2u The two F^u are I.R. a c t iv e and t h e i r freq uen cies were shown to be a t 146cm~* and 609 cm '* ' f o r the TO' s and 149 cm"'*' and 661 cm"'*' for th e L0' s . The Raman mode was for some time thought to be 1 39 the 2 2 0 cm' 1 lin e but t h i s l i n e does not obey th e c o rr e c t p o l a r i z a t i o n s e l e c ti o n rule.^** The Raman mode was f i n a l l y i d e n t i f i e d to be a t 515 cm"*' c lo s e to the p r e d ic te d frequency c a l c u l a t e d by Carabatos.^® The oth er th ree modes a re not a c t iv e in Raman or I . R . , but they show up in a b so rp tio n and luminescence. The e x cito n bands in Cu20 are pro bab ly the b e st known example of h id ro gen ic l ik e s p e c tra in s o l i d s . They form fou r Ridberg S e ries known as b lu e , blue green, green and yellow exciton s e r i e s ^ which are shown in f i g . 24. 80 We are i n t e r e s t e d m ainly in th e Is yellow e x c ito n . The v alen ce and co nd uction bands a s s o c i a t e d w ith th e yellow e x c ito n have same p a r i t y which makes th e Is e x c i to n (n = 1 I - 0 ) a b s o r p t io n and em ission fo rb id d en in th e d ip o le ap p ro x im atio n . The main a b s o r p tio n (and em issio n ) is a s s i s t e d by th e Eu phonon whose frequ ency i s c lo s e to 1 1 0 cm The 220 cm ^ Raman l i n e r e s u l t s from a s c a t t e r i n g p ro c e s s in which two Eu 1 1 0 cm ^ phonons a re g e n e ra te d . The frequency o f the Eu phonon doubled g iv es c o r r e c t l y -1 44 2 2 0 cm sin c e i t s d i s p e r s i o n w(q) i s known to be f l a t . We have s t u d i e d the Raman enhancement o f the 220 cm ^ l i n e c lo s e to I s yellow e x c ito n . We have measured the r e l a t i v e Raman e f f i c i e n c i e s u sin g a S p e c tra - P h y sics CW dye l a s e r w ith Rhodamine 6 G /ethanol s o l u t i o n as th e l a s i n g medium, pumped by a CW a rg o n -io n l a s e r . The freq u en cy tu n in g i s ach iev e d by t i l t i n g a prism in the r e s o n a t o r c a v i t y and we get an o u tp u t beam o f .5 X l i n e w id th in the range 5600 R - 6200 R, which i s very a p p r o p r i a t e to cover th e Is yellow e x c ito n re g io n o f CU2 O. The measured s p e c t r a i s shown in f i g . 24 and 25. The sample was cooled u sin g an Air P rod ucts " c r y o t i p . " The i n t e n s i t y of th e l a s e r was checked b e fo re and a f t e r each measurement. A s e t of a t t e n u a t o r f i l t e r s was u sed to keep the power low (10-30mW) to p r e v e n t h e a t i n g . 81 Figure 25. Spectrum o f C ^O a t 5°K e x c ite d a t X^=6046 S. A i s the q u a d ru p o le . e x c ito n em ission. B is th e phonon a s s i s t e d e x c ito n lum inescence. 82 ft Figure 26. Spectrum of a t 80°K ex cite d a t A^=6085 ft A is the quadrupole exciton emission. B is the 220cm ^ Raman lin e and C is the phonon a s s i s t e d exciton luminescence. 83 This p rocedure had th e a d d i t i o n a l advantage o f reducing the broad band f l u o r e s c e n t em ission o f th e dye l a s e r . To a n a ly se th e d a t a we use th e quantum m echanical 44 approach of K lein. He worked out th e s c a t t e r i n g e f f i c ie n c y f o r t h i s s c a t t e r i n g p ro ce ss by c o n s id e r in g the fo llo w in g i n t e r a c t i o n scheme: > 0- - - - |i> ]c> | 0 > and |f> a re th e i n i t i a l and f i n a l s t a t e s ; |b> and |c> a re i n te r m e d ia te non r e s o n a n t s t a t e s ; |a> i s th e e x c ito n r e s o n a n t s t a t e . He showed t h a t i f one n e g l e c t s the c o n t r i b u t i o n of th e non re s o n a n t s t a t e s , i . e . , only re s o n a n t r e a l i n t e r m ediate s t a t e s a re allow ed then one g e ts f o r the s c a t t e r i n g e f f i c i e n c y . R = Kabs (a) x /x r (4.1) where Kabs (a) i s th e a b s o r p tio n c o e f f i c i e n t of th e r e s o n an t e x c ito n a, Tr = l / y r i s th e r a d i a t i v e l i f e t i m e of th e e x c ito n and x = l / y ( k ) , i s th e n e t l i f e t i m e o f th e a e x c ito n due to bo th r a d i a t i v e and non r a d i a t i v e p r o c e s s e s . 46 Campaan and Cummins measured the a b s o r p t io n of the Is yello w e x c i to n and found t h a t i t can be e x p re ssed by 84 Kabs(aj) = . 034 Cw-a)^s -110)z + 1.3 *10 " ^ (co-w^-150)2 (4.2) - 1 proving t h a t i t is mainly due to the 1 1 0 cm phonon and t h a t i t follow s c o r r e c t l y the square ro o t frequency law. 47 W e now assume as was done by Yu e t a l . in the a n a ly s is of t h e i r d a ta fo r the same s c a t t e r i n g of the Is e x c ito n , t h a t for l a s e r freq uencies c lo s e t o , or s l i g h t l y above th e Is ex cito n a b so rp tio n edge, the only non r a d i a t i v e c o n tr i b u ti o n to the exciton damping i s the ex in tra b a n d a c o u s t i c a l s c a t t e r i n g , so Y = y + yd (4.3) ex ac R Now i f we use the e x p re ssio n determined by 47 Toyzawa f o r the e f f i c i e n c y of the a c o u s t i c a l s c a t t e r i n g Y ^ T to = C(w-w -110) (4.4) ac ac ® we get fo r R: B(w - Up - 1 1 0 )^ R = — ---------- £----------- ----- (4.5) (w - a) - 110) + A where A ^ ant* ® a s c a ^e param eter. W e have used t h i s e x p re ssio n to f i t our d a ta . The r e s u l t s a re shown in f i g . 27 and f i g . 28. RAMAN INTENSITY (ARBITRARY) 3 2 0 1 ------------ r - Cuj>0 5 °K 220 \ \ 120 s O' -O - 20 J ___ 1 6 5 0 0 . 1__ 1 65 6 0 i I__ 1 6 6 2 0 i 16680 Figure 27, c m -1 Raman s c a t t e r i n g o f the 220cm l i n e F R E Q U E N C Y ng o f th e 221 The f i t t i n g p a ra m e te rs Cu20 a t 5°K. u =16386 cm"1 , A=2.9, B =1.02.10 3 of are 100 >- 7 0 4 0 10 (6400 16460 16520 16560 FREQUENCY c jn - Figure 28, Raman s c a t t e r i n g of the 220cm lin e of C^O a t 80 K. The f i t t i n g -1 2 parameters are: w = 16292 cm ,A=8 . 6 and B=5.05.10 . e oo o> 87 N o tice t h a t th e p o s i t i o n o f the e x c i to n energy is d e c re a s in g a l i t t l e w ith te m p e ra tu re . The e x c i to n energy used in th e f i t t i n g was tak en from th e measured quadrupole lu m in escen t em ission which was very sh a rp . The s c a l e param eter B does not r e p r e s e n t a c t u a l r e l a t i v e i n t e n s i t i e s o f s c a t t e r i n g s in c e they do not c o r respond to the same d e t e c t o r re s p o n se . The a c t u a l s c a t t e r i n g i n t e n s i t y was much h ig h e r a t 5° K th an a t 80° K. Even though th e 80° K Raman d i s p e r s i o n can be f i t t e d to th e e x p re s s io n (4.5) the v a lu e o f th e param eter B does n o t fo llo w the exp ected te m p e ra tu re dependence. 48 Since C i s p r o p o r t i o n a l to T one should have A ^ 1/T, so we e x p ect A to d e c re a s e , no t to in c r e a s e w ith tem p er a t u r e . This d is c re p a n c y means one o f two t h in g s : e i t h e r t h e a b s o r p t io n Kabs(w) changes a t 80° K as compared to 5° K where the r e p o r t e d measurement was done, o r th e r e are o t h e r iinfrOTtant c o n t r i b u t i o n s to th e r e l a x a t i o n o f the e x c ito n c l o s e to the band edge, a t th e h i g h e s t measured t e m p e r a t u r e . More e x p erim e n tal d a ta i s needed to r e s o l v e t h i s problem. One way o f checking the v a l i d i t y of eq. (4.5) i s to perform a d i r e c t measurement o f th e e x c i to n l i f e time as a f u n c t i o n of th e e x c i t i n g e n ergy . This would give d i r e c t l y t ( w ) and r R. 88 Such an experiment can be done, in p r i n c i p l e by looking a t the time dependence of th e d i r e c t emission of I s e x c ito n . By co olin g the sample in an immersion dewar to l i q u i d Helium tem perature the non r a d i a t i v e decay can probably be reduced enough to make i t p o s s ib l e to measure the l i f e t i m e w ith the now a v a ila b le pulsed dye l a s e r . Due to the quadrupole n a tu re of the t r a n s i t i o n t h i s e x cito n has a long r a d i a t i v e l i f e t i m e , which was c a l c u l a t e d to be 49 of the o rd e r of 10 sec. 4.2 The O rigin of the Antiresonances in th e D ispersion of Raman Cross Section of CdS. We want to tak e a look in the c a n c e l l a t i o n e f f e c t which is observed in the d is p e rs io n of the Raman cross s e c tio n o f Cds and propose a mechanism fo r i t d i f f e r e n t from t h a t used in the l i t e r a t u r e . Cadmium S u lfid e c r y s t a l has a w u r t z i t e s t r u c t u r e : the symmetry is C^v w ith 4 atoms per u n it c e l l . The 9 o p t i c a l branches have the fo llow in g i r r e d u c i b l e r e p r e s e n t a t i o n a t the zone c e n te r (q 0 ): 1A1 (z) + lE1 (x,y) + 2E2 Cx,y) + 1B1 E^ , E^ and B^ a re doubly deg enerate. The , E^ and 2E2 are Raman a c t iv e . The and E^ modes a re a lso I.R . a c t iv e . The modes are s i l e n t both in Raman and I.R . 89 The i o n i c d isp la c e m e n t is in th e Z d i r e c t i o n f o r th e and modes and in th e XY p la n e fo r th e and E2 modes. The f r e q u e n c i e s of th e symmetry modes a re given in t a b l e 2 . 5 0 » 5 1 >52 The e l e c t r o n i c s t a t e s which can be brought in to reso nan ce in a s c a t t e r i n g experim ent a re shown in f i g . 29. The s p l i t t i n g o f A and B v a le n c e bands i s due to s p i n - o r b i t i n t e r a c t i o n . The A v a le n c e band i s s t r o n g l y a c t i v e o n ly f o r l i g h t w ith S l C . The B and C bands i n t e r a c t s t r o n g l y w ith b o th modes of p o l a r i z a t i o n . A s so c ia te d w i t h each p a i r o f v a le n ce c o n d u ctio n band t h e r e i s an e x c i to n h id r o g e n ic type s e r i e s o f s t a t e s . P robably th e f i r s t evidence of a d e p a r tu r e of th e w^ law in th e Raman s c a t t e r i n g of CdS was the o b s e r v a t i o n 53 by L e i t e and P o rto o f an enhancement i n th e c r o s s s e c t i o n o f th e A^LO 305 cm ^ phonon when l a s e r l i n e s were c lo s e to the a b s o r p t io n b an d s. Besides t h i s enhancement in th e f i r s t o r d e r phonons o v erto n es o f th e 305 cm"l LO were observed up to n = 9 o r d e r . ^ Resonant b e h a v io r o f th e A^ and TO were a ls o r c ob serv ed which was n o t so pronounced as th e LO. R a ls to n e t a l . ^ extended th e measurements in CdS to e n e r g ie s w ell below th e band gap and observed a v e ry pronounced dip on Symmetry Frequency (cm‘ ^) E 2 43 E2 256 E1 (TO 243 Ei(LO) 307 A]. (TO) 234 Aj(LO) 305 TABLE 2 Frequency o f the Phonons in CdS. Data taken from r e f e r e n c e (52). 91 conduction band A valence bands B 7 Eg = 2-58 eV = 20 meV 7 Figure 29. CdS band s tr u c tu r e a t the c e n te r of the B rillo u in Zone. 92 The Raman e f f i c i e n c i e s of th e (A^,E^) TO modes p r i o r to the o n s e t o f r e s o n a n t enhancement. The same e f f e c t was 12 found in ZnS and o t h e r c r y s t a l s . Let us now c o n s id e r the e x p la n a t i o n given to t h i s a n tir e s o n a n c e dip in CdS."^ The s t a r t i n g p o i n t was the Loudon's quantum m echanical e x p re s s io n f o r the Raman s c a t t e r i n g am plitud e in terms o f momentum and d efo rm atio n p o t e n t i a l m a t r ix elem ents W = 2 aB P^ £ pk oB 3a ao < W t - W PR PR £ oB 3a ao (c»jD -wn) fw +w ) ^ 3 R a o J PLd£ q PR o3 3a ao + ( V V C ‘V “ Li pL pR ^ o3 3 a ao (a)3+a> L)C < V % ) (4.9) £ PR PR o3 3a ao ( - ‘ V"oH o V'V y pR oB 3a ao (ajg-ajrt) (uM +Up) a RJ where = wl " wo an<^ W L an(^ wo a re ^a s e r anc* phonon f r e q u e n c i e s , r e s p e c t i v e l y . In t h i s e x p re s s io n R a lsto n e t a l 56 and a ls o 57 Damen and S c o t t i s o l a t e d th e double re s o n a n t term o f the T? - Tj (a v a le n c e band) t r a n s i t i o n . The o t h e r terms were lumped t o g e t h e r to form the a n t i r e s o n a n t term. Assuming p a r a b o l i c bands and ta k i n g £ in dependent o f th e sum over a l l k s t a t e s can be computed: 93 l A a * - f a k \ a Ch * A f( « .g -o,L ) where i i f(w -a>L) = (wg+a)0 -(D L) z - (w -t^ ) 2 (4.7) 1 S as was determined by Loudon. The Raman e f f i c ie n c y was then w r i t te n as: R(co) = IA f(wg -<oL) + B | 2 (4.8) and B was assumed to have op po site s ig n of A in order to give, the c a n c e l la t io n . For the s t a t e s a = 6 in eq. (4.6) (w+u>^) (tjo+a)^) Since paa = 0 (because pa g = ' T^ e numerator of eq. (4.9) is the same as the c o rrespo nd ent reson an t term and t h e r e f o r e i t cannot give th e (-) sig n we are looking f o r . T herefore the c a n c e l l a t i o n in t h i s model has to come from a f 3 te rm s. The problem is then reduced to the c a l c u l a t i o n of the in te rb an d electron-phonon c o e f f i c i e n t » a f 3 . Also B can be assumed to be independent of frequency only i f a (or 8 ) come from h ig her energy s t a t e s then those co n sid ered in f i g . 29. 94 We propose as an a l t e r n a t i v e way of i n t e r p r e t i n g the d a ta to assume t h a t the c a n c e l l a t i o n in CdS is produced by an i n t e r f e r e n c e o f the r e s o n a n t c o n t r i b u t i o n s o f th e broad c o n d u ctio n band and th e s h a rp e x c i to n s t a t e s . The Raman i n t e n s i t y i s then given by Of course we have a ls o th e same problem of r e q u i r i n g t h a t the two c o n t r i b u t i o n s have d i f f e r e n t s i g n , b u t we can now giv e a p h y s i c a l j u s t i f i c a t i o n . The s i g n of th e e x c ito n - phonon c o u p lin g depends on the e f f e c t i v e masses of th e h o le and th e e l e c t r o n forming th e e x c i to n and in p r i n c i p l e i t can be p o s i t i v e on n e g a tiv e . For t h e case where b o th the hole and e l e c t r o n masses a re the same th e c o u pling is zero. In o rd er to apply e q .( 4 . 1 0 ) i t is n e c e s s a ry to c o n s id e r th e n a tu re o f the s t a t e s in v o lv e d . For W . we cond ta k e the slow v a ry in g c o n t r i b u t i o n g iv en by e q . ( For the e x c i to n s t a t e s we take R ■ K o n d + W exc I 2 (4.10) . (4.11) where i s th e photon frequency r e q u i r e d to c r e a t e th e 95 e x c ito n , y i s the width o f the ex cito n s t a t e , £ is the ’ 1 a aa exciton-phonon coupling. The experim ental data could in t h i s way be f i t t e d to e q .(4 .1 0 ) but not u n iv o c a lly : a n a l y t i c a l forms f o r wcon(j and ^ eXc o t ^ e r than those above men tio ned can give a good f i t to the data. The main d i f f i c u lty h e re is th e f a c t th a t th ere are only few experim ental p o in ts o f R(aO clo se to the c a n c e lla tio n frequency. Using the now a v a i la b l e high power C W dye l a s e r s i t w i l l be p o s s ib le to o b ta in a experim ental curve much more acc u ra te than the e x i s t i n g d a ta . This w ill perm it to check in d e t a i l th e a p p ro p ia te a n a l y t i c a l form W exc and t h e r e f o r e give an i n s i g h t about the exact n a tu re of the ex citon s involved. APPENDIX THE E OPTIC-ACOUSTIC PHONON INTERFERENCE IN BaTiO- The p o t e n t i a l energy d e n s i t y U o f a p o l a r i z e d and e l a s t i c a l l y deformed c r y s t a l can be w r i t t e n in th e form :**2 U = I c . . , - u. .u, 1 + . .P. P. + a . . , P .u ., (1) 2 l j k l i] kl 2 l j i j l j k i jk v J where th e c . ., , , U. . , P ., a . ., , a . . a re th e e l a s t i c con- l j k l ’ i j * i ’ i j k ' ij s t a n t s , s t r a i n , p o l a r i z a t i o n , p i e z o e l e c t r i c c o e f f i c i e n t s and p o l a r i z a b i l i t y t e n s o r r e s p e c t i v e l y . The e q u a tio n s of motion f o r o s c i l l a t o r y time 20 dependence are (-to2 + a)2 + ir u) X + 2L S. P = F (2) a a p x C-o)2 + in2 + iF w) P + X = F_ o o ' 4tt P where iqx = u , i s th e mass d e n s i t y , F and F a re the X i d r i v i n g f o r c e s and a = a is the c o e f f i c i e n t a p p r o p r i a te to th e E symmetry. We d e f in e v e c t o r s R = (X, P) and F = CFx ,F p ) 96 97 Then we can w r ite (2) using a compact from AX = F where A is the 2x2 m atrix formed w ith the c o e f f i c i e n t s of X,P in (2). The response m atrix G, th e in v e rse o f A G = A- 1 , i . e . , X = GF (3) Assume i n i t i a l l y th a t th e re is no c o u p lin g . Then ’O r 1-0 g % ) o G C o » ) = . 0 g°(uOj 'a where g ° (w ) = + i f id) ^ g° (u) = ((ojj - oj2 + i r aw) -1 a re the f a m i l i a r o p tic and a c o u s tic response fu n c tio n of a damped o s c i l l a t o r . Now l e t us go back to the o r i g i n a l problem. In o rd e r to solve eq. 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Core Title Temperature Dependence Of The Phonons In Barium-Metatitanate And Some Effects Of The Exciton On The Raman-Scattering Of Cuprous-Oxide And Cadmium-Sulfide

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Advisor Parks, Joel H. (committee chair), Hurrell, John P. (committee member), Porto, Sergio P.S. (committee member)

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