Ballabh, T K

Mukhopadhyay, S

IACS Institutional Repository

huiian J. Phys. 73A (5), 669-673 (1999)U P  A- an international journalDirect computation of structural parameter from diffraction pattern with multiple scatteringS M ukhopadhyay and T K BallabhCondensed Matter Physics Research Centre, Department of Physics,Jadavpur University, Calcutta-700 032. IndiaReceived 7 December I99H, accepted U  March 1999Abstract : In this work, we report an attempt for determination of surface structural parameter directly from simulated single beam LEED intensity pattern from a class ol possible stnicfure We have made a detailed analysis of electron diffraction pattern with multiple scattering for one dimensional model of surface It has been shown that with some realistic assumptions, structural parameter can be directly determined from intensity pattern only for single relaxed ovcrlayer structure.Keywords : LEED, surface relaxation, electron-solid diffraction PACSNos. ; 61 14.Dc.6I 14Lj,68.35.-p1. IntroductionLarge scattering cross sections of atoms for low Energy electrons make LEED (Low Energy Electron Diffraction) a very powerful tool for surface structure determination (1 j. But direct inversion of diffraction pattern to yield structure is difficult, as phase information is lost and strong m ultiple scattering introduces additional complexity. Therefore, surface structure are determ ined by method o f searching among the set of plausible structures by com paring experim ental and computed patterns [2]. In conventional LEED theory, a crystal is assum ed to be consist o f layers of ion cores parallel to the surface. Theoretical com putation o f LEED  pattern from a given structure, firstly involves iniraplanar multiple scattering, then layers arc stacked one by one taking into account interplanar m ultiple scattering [3], Here, we have performed our computation effectively for one dimensional model o f surface, a chain of point scattercr. Iniralaycr multiple scattering is taken care of by energy dependent reflection  and transm ission coefficient o f each scatterer. For one d im ensional situation , a single beam intensity pattern is obtained w hereas for three dim ensional model, m ultiple beams one found.670 S Mukhopadhyay and T K BallabhIn  ih is  p a p e r , w e e x a m in e  th e  p o ss ib il ity  o f  c o m p u ta tio n  o f  s tru c tu ra l p a ra m e te r  fro m  d if fra c tio n  p a tte rn  and  su cce ss fu lly  re tr ie v e  the  re la x a tio n  p a ra m e te r  fo r  so m e  c lea n  an d  a d so rb e d  sy s tem  w ith  sin g le  re lax ed  o v e rla y e r  from  s im u la te d  d if fra c tio n  p a tte rn . T o  th is  en d , lo ss o f  p h ase  in fo rm atio n  is co m p en sa te d  by in tro d u c tio n  o f  the c o n s tra in t tha t th e  o v e ra ll d if fra c tio n  p a tte rn  m u s t be co m p a tib le  w ith  co m p lex  re f le c tio n  co e ff ic ie n t o f  idea lly  te rm in a te d  b u lk  su b s tra te , w h ich  can  be c o m p u te d  from  th e  k n o w n  d a ta  o f  b u lk  la ttic e  c o n s ta n t. W e  sh o w  th a t w ith  th is  c o n s tra in t, re la x a tio n  p a ra m e te r  can  be  d ire c tly  o b ta in e d  from  th e  in ten s ity  patterft on ly  fo r the  case  o f  sin g le  re lax ed  o v e rla y e r  s tru c tu re  by in v e rtin g  th e  e q u a tio n s  o f  c o n v e n tio n a l L E E D  an a ly s is . F o r  tw o  la y e r  re la x e d  s tru c tu re , in te n s ity  p a tte rn  as w ell as p h a se  in fo rm a tio n  is re q u ire d  fo r d ire c t d e te rm in a tio n  o f  s tru c tu re . S o  reco n s tru c tio n  o f  p h ase s  o f  ov e ra ll p a tte rn  is n ecessary  fo r re triev a l o f  p a ram e te rs .In S e c tio n  2, w e p re se n t b as ic  e q u a tio n s  and  line o f  c o m p u ta tio n  an d  in S e c tio n  3, w e  p re s e n t n u m erica l re su lts  fo r so m e  sp ec ific  sy s tem s an d  d isc u ss  th em . In fin a l sec tio n , w e  c o n c lu d e  th e  p a p e r  w ith  a  su m m ary  o f  m ain  p o in ts  rev ea led  in th is  w ork .2. TheoryR e fle c tio n  c o e ff ic ie n t o f  a  m o d e l o f  su rface  w ith  o n e  re lax ed  la y e r [4] a t a  d is ta n c e  d ^ i  o v e r  th e  su b s tra te  h a v in g  co m p lex  re f lec tio n  c o e ff ic ien t isR, = r + 1 - ( 1)k is the w ave vector o f incident electron beam. Interaction o f incident electrons with relaxed scatterer can be sum m arised in terms o f its complex reflection and transm ission coefficients r  and t respectively and these depend on chem ical com position and two dim ensional geometry o f the layer. Rs can be com puted by layer doubling m ethod [3], converging the incident electron beam within a stack o f identical layers to the semi-infinite crystal limit, i.e., by iterating eq. ( 1 ) using the bulk lattice constant a for an d i? j for taking the com plex reflection coefficient and transm ission coefficient o f the bulk specim en r/, and t/, respectively. From eq. (1), it is clear that the relaxation param eter di2 for a single relaxed structure can be determined easily if both the phases and am plitudes o f  Rs and R i  are known, when chem ical com position and layer geom etry o f bulk and relaxed layer are given. But as phases of Ri are lost in experiment, we elim inate the phases o f R i from eq. ( 1 ) and obtaincos (2 fa/j2 + 0 + 6)where A is computed from the equations4sin6 s  1^2 sinj,when 0  =  arg (R^) + arg (r),2|/fj||r|/lAcosO -  \ - r ^  \ cosg,g -  a rg  ( / V z - ^ - l ) -(2)Direct computation of structural parameter etc 671|/?^ I is kn ow n  fro m  d iffra c tio n  in tensities and 0  and 0 c a n  be com puted Irom  a ,  the b u lk  la tt ic e  co n s ta n t and \ R f  \. T h e re fo re , the re la xa tio n  param eter c f .  can be de te rm in ed . F o r a f ix e d  va lue o f  righ t hand side o f  eq (2), tw o  cosine angles (say. a  and 2 7 t  — cx) can be ob ta ined . W ith  the physica l re s tric tio n  that is close to the bu lk  la ttice  constan t a^ fo u r poss ib le  values can be obta ined at each energy g r id  po in t. O f these lo u r poss ib le  d istances, tw o  arc con trac tive  (r /,.  < a )  i ^ d  other tw o  are expansive (J , .  > a )  T w o  va lues o l d \ j  are com puted Irom  the angle tx and the other tw o  iirc Irom  ( 2 ; r — (X). A t each e n c ig y , we ge l a set o l to u r c ii i le ie n t </|>, the ufiiciuc \a lu c  ol these sets g ives the actual va lue o f d \ i .  So fo r surfaces w ith  one relaxed ovx^laycr, c ither clean or adsorbed, re laxa tion  param eter can be de term ined from  in tensity  patn?rn on ly  so that it must reproduce subsirate re n e c lio n  c o e ff ic ie n t p ro p e rly  in fu ll range ol energyF o r surfaces w ith  tw o  re laxed ove rlave r, an in term ed ia te layer comes between to  la ye r and b u lk  p e rio d ic  substrate. Le t R^\  be the c o m p l e x  re nec tion  c o e ff ic ie n t o f  that in te rm ed ia te  laye r w ith  regu la r substrate u n d e ily in g . Th is  is not know n, hence d ^ i  can not be com p u te d  fro m  the in te n s ity  pattern alone. H ow ever, i f  the phases a long w ith  in te n s ity  is ava ilab le , then |/^y| | can be determ ined from  the equation\ u \ -  ~ ' (3)O nce the |/^5 i |  is kn o w n , d istance between second and th ird  o v e rla y c r can beob ta ined  by the same w ay as it was determ ined fo r single ove rlaycr structure. Then phases o f  R s \  can be com puted . T here fo re , in the next step, the re laxation param eter can also be ob ta ined.R e c e n tly , th e  p h ase  in fo rm a tio n  has been  reco rd ed  ex p e rim en ta lly  in h o lo g rap h ic  L E E D  m cthod ,s fo r so m e  sp ec ia l sy s tem  f5], bu t the real d ifficu lty  is tha t p h ases are  not a v a ila b le  in m o s t o f  the  ex p e rim en ts . O n the o th er hand , phases m ust be specific  so  that the re f le c t io n  c o e f f ic ie n t o f  th e  b u lk  can  be m a tch ed  w ith  tw o  real d is tan ce s  d ^ i  ^md ^ 2 .^  s u c c e s s iv e  a p p lic a tio n  o f  eq . (1 ). S o  re c o n s tru c tio n  o f  p h ase  in fo rm a tio n  o f  o v e ra ll re f le c tio n  c o e ff ic ie n t R i  is  w o rth w h ile  to  co m p u te  the  re lax a tio n  pa ram ete rs . In the nex t se c tio n , w e  p re s e n t th e  re su lts  an d  d iscu ss  th e ir im portance  and  im plica tion .3. Results and discussionO u r te c h n iq u e  su c c e s s fu lly  re tr ie v e d  the re laxa tion  p a ram ete r from  s im u la ted  experim en ta l d a ta  o f  c le a n  su rfa c e s  A1 (1 1 1 ), R h  (1 1 1 ), C u  (111), Pd (100) and oxygen  abso rbed  system  o f  A1 (1 1 1 ) , R h  (1 1 1 ). E n erg y  d e p e n d e n t re flec tion  and  tran sm ission  coeffic ien ts  o f  a p lane fo r  e a c h  ty p e  o f  sp e c ie s  a re  co m p u ted  from  stan d ard  d y n am ica l L E E D  code  14] fo r ~ (0 0 ) b ea m  o f in c id e n t e le c tro n . S tan d ard  energy  d ep en d en t a tom ic  phase sh ifts , bu lk  in tc rp lan a r d is ta n c e  a n d  tw o  d im e n s io n a l la t tic e  v ec to rs  o f  a p lan e , a re  tak en  as in p u ts  fo r th is  com putation. W e  h a v e  c o n s id e re d  in ten sity  p a tte rn  fo r d iffe ren t sing le 'O v erlay e r s tru c tu re  w ith  various values o f  d \ 2% bo th  co n trac tiv e  and  ex p an siv e  re lax a tio n  w ith  typ ica l varia tio n  o f  b u lk  in te r p la n a r  d is ta n c e  a n d  th e s e  s im u la te d  in te n s i t ie s  a re  ta k e n  as p se u d o672 S Miikhopadhyay and T K Ballahhe x p e r im e n ta l  d a ta  T h is  s im u la te d  in te n s ity  p a tte rn , b u lk  la ttic e  c o n s ta n t an d  e n e rg y  d e p e n d e n t r  and  t  a rc  taken  as inpu ts fo r th is d irec t co m p u la tio n . In T ab le  1, w e h ave  show n  so m e  ty p ic a l v a lu e s  o f  p a ra m e te rs  o f  su rfa c e s  fo r s im u la tin g  d if f ra c tio n  p a tte rn  an d  the re tr ie v ed  pa ram ete rs .Table 1. Typical values of parameters of surfaces for simulating intensity patterns and the retneved parameter valuesSurfacesAl ( l l l )C u ( l l l )Rh ( I l l )Pd (100)A K n i )  + (1X1)0 overlaycr R h ( i m  + (2X2)0 overlaycrBulk interplanar distance («) (A)2.338 2.0872.192 1 9452.3382.192Simulated value ofc/i2 (A)2.216 2 034 2.246 2 003 1 4621.238Retrieved value (A) of d\2 (A)2.2162 033 2.246 2 002 1.4621.237A s w e  h av e  p o in te d  o u t fo r  su rfa c e  w ith  tw o  re la x e d  o v e r la y e r , e x a c t s tru c tu ra l p a ra m e te r s  can  be d e te rm in e d  i f  th e  c o rre c t p h a se  in fo rm a tio n  o f  th e  to p  la y e r  c an  be so m e h o w  reco n s tru c ted . It seem s th a t u n d e rs ta n d in g  o f  p h ase  v a ria tio n  w ith  en e rg y  and  its re la tio n  w ith  is e s s e n tia l  fo r  m a k in g  a  p ro g re s s  in th is  d ire c tio n . W e  h av e  s tu d ie d  p re v io u s ly  [6 | th e  in te n s ity  e n e rg y  c u rv e  (I*~V c u rv e )  an d  p a ra m e tr ic  p lo ts  (re a l v,s im a g in a ry  p lo t)  o f  R i  a n d  R s  o f  so m e  o n e  d im e n s io n a l s tru c tu re s  w ith  se v e ra l re la x e d  o v e rla y e rs  ta k in g  e n e rg y  in d e p e n d e n t ty p ica l v a lu e s  o f  r  =  0.1 and  t =  0 .9  +  0 .4 / [3J for s im p lic ity . S tu d y  o f  th o se  in ten s ity  cu rv e  rev ea ls  th a t fo r s im p le  re lax ed  o v e rla y c r  s tru c tu re , p o s itio n  o f  B ra g g  p e a k  d o  n o t c h a n g e  a p p re c ia b ly  as th a t o f  b u lk  te rm in a te d  p e r io d ic  su b s tra te , b u t e x tra  fe a tu re s  ap p ea r in th e  reg io n  b e tw een  tw o  B rag g  p eak s. T h e re fo re , n ea r B ra g g  p e a k s  p a tte rn s  fo r m u lti la y e r  re la x e d  s tru c tu re  can  b e  c o n s id e re d  as an  e ffe c tiv e  s in g le  re la x e d  o v e rla y e r  s tru c tu re  w ith  a  m o d ified  R s. S o  an  ite ra tio n  m ay  b e  e m p lo y ed  fo r r e c o n s tr u c t in g  the  p h a s e s  o f  R^^ w ith  th e  h e lp  o f  eq . (2 ) an d  p ro b a b ility  o f  c o n v e rg e n c e  o f .such i te ra t io n  w ill be  m a x im u m  a t th e  e n e rg y  p o in ts  n e a r  B ra g g  p e a k s . F u r th e r  in v es tig a tio n  is req u ired  to  e s tab lish  th e  n a tu re  o f  th is  ite ra tio n .4. C onclusionW e hav e  sh o w n  h e re  th a t re lax a tio n  p a ra m e te r  fo r a s in g le  re la x e d  o v e rla y e r  s tru c tu re  can  be  d e te rm in e d  d ire c tly  fro m  d if fra c tio n  in ten s ity  p a tte rn  o n ly  e v en  in  p re se n c e  o f  m u ltip le  s c a tte r in g . A d d itio n a l c o n s tra in t th a t the  d if fra c tio n  in te n s ity  .should b e  c o m p a tib le  w ith  c o m p le x  re f le c tio n  c o e ff ic ie n t o f  the  b u lk  p e rio d ic  sub.strate, c o m p e n sa te s  th e  lo ss  o f  p h ase  in fo rm a tio n . T h o u g h  th is one  d im en sio n a l m o d el ap p ea rs  to  be  s im p le  in  c u rre n t fram ew o rk  o f  L F F D  th eo ry , bu t th is  w o rk  show.s a c le a r  ap p ro ach  o f  d irec t d e te rm in a tio n  o f  s tru c tu re  w ith o u t tria l and  sea rch  p ro c e d u re . T h is  w o rk  sh o u ld  be e x te n d e d  fo r g en e ra l m u lti la y e rD ire c t  com putation  o f  stru ctu ra l p a ra m eter etc 673re la x e d  s t ru c tu re  w ith  th re e  d im e n s io n a l m u ltib e a m  e ffe c t fo r d ire c t in v e rs io n  o f  rea l e x p e r im e n ta l L E E D  p a tte rn .A cknow ledgm entO n e  o f  th e  a u th o r s  (S . M .)  is g ra te fu l to  U m v e rs ily  G ra n ts  C o m m iss io n  (U .G .C .) . G o v e rn m e n t o f  In d ia  fo r p ro v id in g  a  re se a rc h  fell<)wship. A u th o rs  a ck n o w led g e  F^rof. A . N B asu , J a d a v p u r  U n iv e rs ity , C a lc u tta  fo r h is valuaM e su g g estio n s .References(J1 K Heinz. M Kotteke, U Loftier and R Doll Surface Sd. 357-358 I (1996)[2]f3J14]151[6]J B Pendry J Phys C13 937 (1980)J B Pendry Low Energy Electron Diffraction (London • Academic) (1974)M A Van Hove and S Y Tong Surface Crystallography b\ LEED  (Berlin . Springer-Verlag) (1979)K Reuter, J Bernhardt, H Wedlcr, J Schardt, U Starke and K Hcini: Phys Rev Lett 79 4818 (1997), U Starke, J Schandt, J Bernhardt. M Franke, K Reuter. H Wedler and K Hem? Phys Rev I  rii 80 7.38(1998)S Mukhopadhyay and T K Ballahh Proc D A E  Solid State Physics Symp (Cochin University, Ketala. India) 40C 381 (1997)

Direct Computation of Structural Parameter from Diffraction Pattern with Multiple Scattering

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Direct Computation of Structural Parameter from Diffraction Pattern with Multiple Scattering

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