The preparation and characterization of bismuth thin films on GaSb(110) using low energy electron
diffraction
by Scott Len Lantz
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Physics
Montana State University
© Copyright by Scott Len Lantz (1990)
Abstract:
The growth of thin bismuth films on the GaSb(110) surface has been studied using low energy electron
diffraction (LEED) and Auger electron spectroscopy (AES). The GaSb samples were single crystal bars
and the (110) surface was prepared by cleaving in situ. Bismuth was evaporated from a solid source,
and the evaporation was monitored with a quartz crystal oscillator. Diffraction spot profiles and
integrated intensity versus voltage (IV) curves were measured at temperatures ranging from 120 K to
350 C. Bismuth was found to grow epitaxially for monolayer coverages, preserving the p(lxl) LEED
pattern symmetry. Upon annealing a phase transition occurred such that the p(lxl) symmetry was
replaced by a p(lx2) symmetry, where the LEED pattern periodicity was doubled along the [001]
direction of the bulk GaSb crystal. Preliminary work has been done toward the determination of the
atomic structure of the Bi/GaSb(110) interface. Modifications were made to the existing computer
programs, including the incorporation of the simplex algorithm into the structure search
decision-making process. Using this method, the calculated atomic structure of the clean GaSb(110)
surface was shown to be in agreement with that obtained by previous methods. Searches were also
made for the atomic structure of the p(lxl) structure for a bismuth coverage of one monolayer. The
results indicate an ordered bismuth overlayer with an atomic structure differing markedly from the
truncated bulk or reconstructed clean GaSb(110) surface structure. T H E P R E P A R A T IO N A N D C H A R A C T E R IZ A T IO N OF
B IS M U T H T H IN F IL M S O N G aS b(IlO ) U S IN G
LO W E N E R G Y E L E C T R O N D IF F R A C T IO N
by
Scott Len L a n tz
A thesis su b m itte d in p a r tia l fu lfillm e n t
o f th e require m en ts fo r th e degree
of
M a ste r o f Science
in
Physics
M O N T A N A S T A TE U N IV E R S IT Y
Bozeman, M o n ta n a
M a rch 1990
Z /Z fZ
APPROVAL
o f a thesis su b m itte d by
Scott Len Lan tz
T h is thesis has been read by each m em ber o f the thesis com m ittee and has been
fo u n d to be s a tis fa c to ry re g a rd in g co n te n t, E n g lis h usage, fo rm a t, c ita tio n s ,
b ib lio g ra p h ic style, and consistency, and is ready fo r subm ission to the College o f
G rad uate Studies.
C h a irp e /so n , G rad uate C om m ittee
Date
Approved fo r the M a jo r D e p a rtm e n t
*2 3
f
o
m ent
Date
A pproved fo r the College o f G raduate Studies
Date
/
G raduate 'ibean
S T A T E M E N T OF P E R M IS S IO N TO USE
In p re s e n tin g th is thesis in p a r tia l fu lfillm e n t o f th e re q u ire m e n ts fo r a
m aster's degree a t M o n ta n a S tate U n iv e rs ity , I agree th a t th e L ib ra ry s h a ll m ake
i t a v a ila b le to borrow e rs u n d e r ru le s o f th e L ib ra ry .
th e s is
are
a llo w a b le
B r ie f q u o ta tio n s fro m th is
w ith o u t sp e cia l p e rm is s io n , p ro v id e d t h a t a c cu ra te
acknow ledgm ent o f source is made.
P e rm ission fo r extensive q u o ta tio n fro m o r re p ro d u ctio n o f th is thesis m ay
be g ra n te d b y m y m a jo r professor, o r in h is absence, b y th e D ean o f L ib ra rie s
w hen, in th e o p in io n o f e ith e r, th e proposed use o f the m a te ria l is fo r sch o la rly
purposes. A n y copying or use o f the m a te ria l in th is thesis fo r fin a n c ia l g ain sh a ll
n o t be allow ed w ith o u t m y w ritte n perm ission.
S ignature
Date.
IV
TABLE OF CONTENTS
L IS T O F F IG U R E S .................................................................................................................
v
A B S T R A C T .............................................................................................................................. v ii
1. IN T R O D U C T IO N ...........................................................................................................
2. CO NC EPTS I N L E E D C R Y S T A L L O G R A P H Y .........................................................
I
3
4. G R O W TH A N D K IN E T IC S OF B iZ G a S b (IlO ).....
8E3S3B
L E E D .......................................................................
A E S ..........................................................................
Sam ple P re p a ra tio n ............................................
QCO C a lib ra tio n ...................................................
N
3. E X P E R IM E N T A L T E C H N IQ U E S ..................: ......
£
The L E E D E x p e rim e n t.......................................
Surface C la s s ific a tio n s .....................................
G eom etrical T heory o f D iffr a c tio n ..................
D iffra c tio n In te n s itie s ........................................
83 8 3 I &
5. S TR U C TU R E A N A L Y S E S ...................................
&
D eposition and G ro w th o f B is m u th ................
G ro w th K in e tic s and A n n e a lin g ......................
D is c u s s io n ............................................................ .
The O rdered G ro w th o f B is m u th .......
The p ( l x l ) to p (lx 2 ) Phase T ra n s itio n
P ro g ra m O v e rv ie w ..........................................
D is c u s s io n ..........................................................
G a S b (IlO ) Clean Surface G eom etry
The I M L p ( l x l ) S tr u c tu re ....................................................................... 57
6. S U M M A R Y A N D C O N C L U S IO N S ........................................................................... 61
A P P E N D IC E S ....................................................................................................................... 62
A p p e n d ix A -S im p le x A lg o r ith m ............................................................................63
A p p e n d ix B -V ideo C am era T im in g C ir c u it........................................................70
R E F E R E N C E S ....................................................................................................................... 76
LIST OF FIGURES
F ig u re
Page
1 D isp la y detection system fo r L E E D ..............................................................................
3
2 T w o-d im ensional B ra va is la tt ic e s ...............................................................................
5
3 E xam ples o f surface c la s s ific a tio n s ............................................................................
7
4 Row re p re se n ta tio n o f surface la ttic e p o in ts .............................................................
9
5 T w o-d im e n sio n a l reciproca l la ttic e s ........................................................................... 13
6 E w a ld C o n s tru c tio n .......................................................................................................... 14
7 L E E D e q u ip m e n t............................................................................................................... 25
8 E le c tric a l connections fo r A E S ..................................................................................... 28
9 Sam ple p re p a ra tio n .......................................................................................................... 32
10 (0,1) d iffra c tio n beam dependence on b is m u th coverage........................................37
11 S ym m e try re te n tio n between the (2,1) and (-2,1) b ea m s......................................... 39
12 E nergy-dependent size e ffe c t....................................................................................... 40
13 (1,0) beam profile s a t d iffe re n t electron e n e rg ie s................................................... 41
14 R elative in te n s itie s o f the (0,1) and (1,0) b e a m s ....................................................... 43
15 F o rm a tio n o f h a lf-o rd e r d iffra c tio n s p o ts ................................................................ 45
16
B is m u th A u g e r sig n a l dependence on a n n e a lin g te m p e ra tu re ..................... 46
17 F lo w c h a rt d ia g ra m fo r s tru c tu re search p ro c e d u re s ............................................50
18 S ca tte rin g phaseshifts fo r (a) antim on y, (b) g a lliu m , and (c)b is m u th ..........52
19 S c a tte rin g cross-sections fo r (a) an tim o n y, (b) g a lliu m , and (c) b is m u th ........ 53
20 Side v ie w o f the I II - V ( I lO ) s u rfa c e ............................................................................ 56
21 C om parison o f calculated and e xp e rim e n ta l IV curves fo r
the clean G aS b(IlO ) surface......................................................................................... 58
22 S im p lex operations in tw o d im e n s io n s .................................................................... 65
Vl
LIST OF FIGI JRES-contimiftd
F ig u re
Page
23 F lo w c h a rt fo r th e S im p lex A lg o r it h m ...................................................................... 66
24 R esults o f the sim plex convergence te s ts .................................................................. 69
25 V ideo cam era o u tp u t s ig n a ls ....................................................................................... T l
26 B lo ck d iagram fo r th e video cam era tim in g c ir c u it ..............................................74
27 In te g ra te d o u tp u t and e xte rn a l trig g e r tim in g
75
ABSTRACT
The g ro w th o f th in b is m u th film s on the G aS b(IlO ) surface has been studie d
u s in g lo w ene rg y e le ctro n d iffra c tio n (L E E D ) and A u g e r e le ctro n spectroscopy
(AES ). The GaSb sam ples w ere sin gle c ry s ta l bars and th e (H O ) surface was
prepared by cleaving in s itu . B is m u th was evaporated fro m a solid source, and the
e va p o ra tio n was m o n ito re d w ith a q u a rtz c ry s ta l o s c illa to r. D iffr a c tio n spot
p ro file s and in te g ra te d in te n s ity versus vo ltag e (IV ) curves w ere m easured a t
te m p e ra tu re s ra n g in g fro m 120 K to 350 C. B is m u th was fo u n d to grow e p ita x ia lly
fo r m o nolayer coverages, p re se rvin g th e p ( l x l ) L E E D p a tte rn sym m etry. U pon
a n n e a lin g a phase tr a n s itio n occurred such th a t th e p ( l x l ) s y m m e try was
replaced by a p (lx 2 ) sym m etry, w here th e L E E D p a tte rn p e rio d ic ity was doubled
along the [001] d ire ctio n o f the b u lk GaSb crystal. P re lim in a ry w o rk has been done
to w a rd th e d e te rm in a tio n o f th e atom ic s tru c tu re o f th e BiZGaSb(HO) in terface.
M o d ific a tio n s w e re m ade to th e e x is tin g c o m p u te r p ro g ra m s, in c lu d in g th e
in c o rp o ra tio n o f th e sim plex a lg o rith m in to th e s tru c tu re search decision-m aking
process. U s in g th is m e th o d , th e c a lc u la te d a to m ic s tru c tu r e o f th e clean
G a S b (IlO ) surface was show n to be in agreem ent w ith th a t o b tained b y previous
m ethods. Searches w ere also made fo r th e atom ic s tru c tu re o f th e p ( l x l ) s tru c tu re
fo r a b is m u th coverage o f one m onolayer. The re su lts in d ica te an ordered b is m u th
o ve rla ye r w ith an atom ic s tru c tu re d iffe rin g m a rk e d ly fro m th e tru n c a te d b u lk or
reconstru cted clean G aSb(H O ) surface s tru ctu re .
I
C H A P TE R I
IN T R O D U C T IO N
The w o rk presented in th is thesis represents the re s u lts o f th e e xp e rim e n ta l
a n d th e o re tic a l in v e s tig a tio n s in to th e g ro w th o f t h in b is m u th film s on th e
G a S b (IlO ) surface. The m o tiv a tio n fo r such a stu d y is m any-fold. ■GaSb (g a lliu m
a n tim o n id e ) is an im p o r ta n t I I I - V
se m icon ductor.
I t has m a n y in d u s t r ia l
a p p lic a tio n s fo r th e c o n s tru c tio n o f e le c tro n ic devices such as e le ctro -o p tica l
sensors and lig h t e m ittin g diodes [ I ] .
The successful c o n s tru c tio n o f electron ic
devices, w h ic h are g e ttin g in c re a s in g ly s m a lle r, could b e n e fit g re a tly fro m the
know ledge o f atom ic and electronic s tru c tu re s o f sem iconductor-m etal interfaces.
T h is is im p o rta n t in s tu d y in g th e fo rm a tio n o f S c h o ttk y b a rrie rs and ohm ic
contacts [2], A t th e atom ic scale, however, lit t le is understood about the electronic
s tru c tu re a t these in te rfa c e s , due to com plex a to m ic re c o n s tru c tio n s o f th e
sem iconductor su b stra te and la c k o f s u ffic ie n t th e o re tic a l m odels to describe th e
in te rfa c e [2], A lth o u g h th e (H O ) surface o f th e I I I - V sem iconductors is less used
in in d u s tr ia l a p p lica tio n s (th a n , fo r exam ple th e (100) surface), i t is som ew hat
s im p le r in th a t th e surface is c h a rg e -n e u tra l, and th e s u b s tra te reco n stru ctio n s
are c o m p a ra tiv e ly w e ll understood [2,3].
C lean (H O ) surfaces hence provid e an
e x c e lle n t b asis fo r th e s tu d y o f a to m ic a n d e le c tro n ic s tru c tu re s a t m e ta l
in te rfa ce s.
B is m u th was selected as an absorbate because i t was fo u n d to fo rm an
ordered in te rfa c e on a s im ila r I I I - V com pound G a A s (IlO ) [4].
F o r th is reason i t
2
was believed th a t b is m u th w o u ld fo rin an ordered in te rfa c e on G a S b (IlO ), w h ic h
could th e n be s tu d ie d b y L E E D (lo w ene rg y e le ctro n d iffra c tio n ), w h ic h is a
p o w e rfu l and extensively-used tool fo r surface analysis.
The e xp e rim e n ta l techniques used in th e study o f th e BiZG aSb(IlO ) system
in clu d e L E E D and AES (A uge r electron spectroscopy). The g ro w th o f b is m u th was
in v e s tig a te d fo r m onolayer coverages, and th e effects o f a n n e a lin g w ere studied.
M easurem ents ta k e n w ith L E E D in c lu d e d d iffra c tio n spot in te n s ity p ro file s and
in te n s ity versus voltag e (IV ) curves.
A u g e r m easurem ents w ere p r im a r ily used
to m easure th e d e so rp tio n o f b is m u th th a t o ccurred w h e n th e sam ple w as
annealed.
T he th e o re tic a l a n a lysis o f th e d a ta in v o lv e d m o d e llin g o f th e a to m ic
s tru c tu re o f th e in te rfa c e .
T h is in c lu d e d d yn a m ica l L E E D ca lcu la tio n s fo r th e
gen eration o f I V da ta fo r the proposed m odel, w h ich could th e n be com pared to the
e x p e rim e n ta l I V data.
A best f i t s tru c tu r a l m odel was th e n searched fo r b y
m a k in g changes to th e m odel u n t il th e c a lc u la te d d a ta b e st m a tch e d th e
e x p e rim e n ta l data.
In the fo llo w in g chapters, th e re s u lts o f these studies o f th e
BiZG aSb(IlO ) system are presented fo llo w in g a description o f th e techniques used.
3
C H A P TE R 2
CO NC EPTS IN L E E D C R Y S T A LLO G R A P H Y
The L E E D E xp e rim e n t
The e sse n tia l elem ents o f a L E E D a p p a ra tu s co n sist o f an u ltr a - h ig h
vacuum cham ber, an electron gun to provide a m onoenergetic electron beam over
an energy range fro m 30 to 400 eV, a c ry s ta l holder w ith fa c ilitie s fo r m oving,
h e a tin g and cooling the sam ple, and a detection system fo r m e a su rin g the n u m b e r
o f e la s tic a lly scattered electrons in a prescribed d irectio n.
The type o f detection
system used in th is w o rk is the displa y system shown schem atica lly in F ig u re I.
Electron gun
Fluorescent
collector screen
Sample
Suppressor grids
Figure I: Display detection system for LEED
4
The e le ctro n g un produces a beam o f electrons w h ic h s c a tte r fro m th e c ry s ta l
surface.
The e la s tic a lly scattered electrons are filte re d by th e suppressor grid s,
th e n accelerated in to th e fluorescent screen b y a voltage o f 4-5 k V .
The in te n s ity
p a tte rn produced by th e electrons s tr ik in g th e screen can th e n be m easured b y
p h o to g ra p h ic o r video re co rd in g techniques.
F o r a w e ll-o rd e re d c ry s ta l surface,
th e L E E D in te n s ity p a tte rn con ta in s a w e a lth o f in fo rm a tio n co n ce rn ing th e
surface s y m m e trie s and surface a to m ic geom etries.
The e x tra c tio n o f these
surface p ro p e rtie s, th o u g h , ofte n re q u ire s extensive analyses.
The n o ta tio n and
m ethods used in such analyses are developed below.
Surface C la ssifica tio n s
W hen
c o n s id e rin g
an
o rd e re d
s u rfa c e ,
th e re
fu n d a m e n ta l concepts, th a t o f a la ttic e and a basis.
a re
tw o
im p o r ta n t
T he la ttic e is th e tw o
d im e n sio n a l a rra y o f reference points w h ic h possess the tra n s la tio n a l sym m etries
o f th e surface. The basis is th e arra n g e m e n t o f the surface atom s w ith respect to
th e la ttic e po in ts.
The geom etry o f a p a rtic u la r surface s tru c tu re is said to be
com pletely de te rm in e d once bo th the la ttic e and the basis are kn o w n .
There are
five possible d is tin c t tw o dim en sion al la ttic e s , called B ra v a is la ttic e s .
These are
show n in F ig u re 2. The p rim itiv e vectors, a i and a%, define th e surface u n it cell,
a nd are g e n e ra lly selected to fo rm th e s m a lle s t possible p a ra lle lo g ra m w h ic h
preserves th e surface sym m etries.
The c la ssifica tio n o f a surface s tru c tu re often relates th e tw o d im e n sio n a l
surface la ttic e to the thre e dim ensional substrate la ttice. One scheme, proposed b y
P a rk and M adden [5], relates the surface la ttic e p rim itiv e vectors ( a i,a 2 ) to
5
lattice
points
Square
ai = a 2
Rectangular
Primitive
at ^ a 2
Centered
Rectangular
a 14 a 2
~ -
Hexagonal
a 1= a 2
Oblique
ai ^ a 2
a ^ 90°
Figure 2: Two-dimensional Bravais lattices
6
those o f a p a ra lle l su b stra te plane ( b i, b 2 ) w ith a tra n s fo rm a tio n m a trix M , such
th a t
b i = M 11B 1 + M 12U2
bg = M 21R1 + M 22a 2
The s tru c tu re is called sim ple w hen a ll e n trie s o f M are in teg ers, and coincidental
i f M co n ta in s b o th in te g e rs and r a tio n a l num bers.
If M
c o n ta in s ir r a tio n a l
n u m b e rs, th e s tru c tu re is la b e lle d in c o h e re n t o r in co m m e n su ra te .
A lth o u g h
u n iv e rs a lly applicable, th is m ethod does n o t alw ays provide an in tu itiv e p ic tu re fo r
w h a t th e s tru c tu re looks lik e . O th e r m ethods have been proposed w h ich are m ore
descriptive, such as th e one proposed by W ood [6]. H ere, th e re la tio n between the
surface a nd s u b s tra te la ttic e s is expressed b y th e ra tio s o f th e le n g th s o f th e
p r im itiv e vectors, I b i / a i I and I bg/ag I , and th e ro ta tio n angle betw een th e tw o
la ttice s, cp. The surface u n it cell is la belled as e ith e r centered (c), o r p rim itiv e (p).
A centered u n it cell im p lie s th a t there is a la ttic e p o in t in the center o f the surface
u n it cell.
T h is m eth od is sim ple and applicable to a la rge n u m b e r o f stru ctu re s,
in c lu d in g th e BiZG aSb(IlO ) system studie d here. Some exam ples o f la b e llin g w ith
th is scheme are presented in F ig u re 3.
D ire ctio n s on th e surface are u s u a lly expressed in re la tio n to the u n it cell.
A s ta n d a rd approach is to use th e M ille r indices h' and k ', w here h ' and k ' are
b o th in te g e rs, such th a t (h ',k') refers to a d ire c tio n on the surface. T h is d ire c tio n
is defined to be p e rpe ndicu lar to the tra n s la tio n vector D given by
D = h 'a i + k'ag
( 2)
7
c( 2x2)
P(VSx^)SO0
Figure 3: Examples of surface classifications
8
As illu s tr a te d in F ig u re 4, th e surface la ttic e can be th o u g h t o f as p a ra lle l row s o f
p o in ts w ith sepa ra tion dh'k' given by
I
_
d^k'
h '2
k '2
a2 s in 2 y
a 2 s in 2 y
2 h 'k 'c o s y
a 1a 2 s in 2y
w here y is th e angle between a i and ag. T h is in te rp re ta tio n w ill become useful in
the n e x t section in discussing the geom etric th e o ry o f d iffra c tio n .
G eom etric T heory o f D iffra c tio n
The s ta rtin g p o in t fo r describing electron d iffra c tio n a t c ry s ta l surfaces is
the de B ro g lie re la tio n ,
A. = h /p
(4)
w h ic h re la te s th e lin e a r m om e ntu m o f th e electron p to its w a ve le n g th A. h is
P lanck's constant (4.13 x I O '15 eV sec).
W hen expressed in te rm s o f the k in e tic
energy o f the electron (E = p 2/2m ), the de B ro g lie re la tio n becomes
A = h /V 2 m E
(5)
W hen A is in A n gstrom s (A), and the energy E is in electron v o lts (eV), the re la tio n
is a p p ro xim a te ly,
A = V 150Ze
(6)
9
Surface unit cell
Translation vector
D
t
D irection
defined by
Rows of lattice points with
spacing dHk
ZtYKcosy
Q1B2Sin2Y
Figure 4: Row representation of surface lattice points
10
In te rfe re n c e effects occur w hen th e e le ctro n w a ve le n g th is o f th e same o rd e r o f
m a g n itu d e as th e atom ic spacing on th e c ry s ta l surface. W ith atom ic spacings on
the o rd e r o f a few A ngstrom s, the corresponding energy range is I to 500 eV. The
a ctu a l energy range used in L E E D e xpe rim e nts is som ew hat sm a lle r, 50-300 eV,
o w in g to in e la s tic s c a tte rin g and in s tr u m e n ta l effects, w h ic h are discussed in
la te r sections. In th e geom etric theory, a ll sca tte rin g is assum ed to be elastic, and in s tru m e n ta l effects are ignored.
F o r e la s tic w ave s c a tte rin g fro m a one d im e n s io n a l p e rio d ic a rra y ,
co n stru ctive in te rfe re n ce takes place w h en th e scattered waves fro m ne ig h b o rin g
la ttic e po in ts have p a th differences o f m u ltip le s o f the electron w avelength X. I f d
is th e a rra y spacing, a nd Oq is the angle o f incidence m easured fro m th e n o rm a l,
th e n c o n s tru c tiv e in te rfe re n c e o f the b a ck-scatte red waves occurs a t angles 0
when,
d(sin0 - sin0o) = n X
(7)
H ere n is an in te g e r denoting the order o f d iffra c tio n . T h is is kn o w n as the B ragg
equ ation in one dim ension. Since our L E E D experim ents are perform ed a t n o rm a l
incidence, i t w ill be assumed th a t Oq = 0, henceforth.
In e xte n d in g th e B ra g g equation to a tw o dim en sion al surface, re c a ll fro m
th e p re v io u s section th a t th e surface la ttic e p o in ts can be th o u g h t o f as an
ensemble o f p a ra lle l rows w ith d ire ctio n (h ',k') and sepa ra tion d h 'k'-
W hen each
ro w is considered a scatte rer, interfere nce m a xim a are expected fo r
sin0 = (l/d h 'k ') n ^
or w ith th e use o f equation (6),
(8)
11
sin 8 = (I / dh'k') • n • VTsoTW
(9)
F ro m th is equ ation, i t is e vident th a t an increase in electron energy w ill cause the
d iffra c tio n m a x im a to move closer to th e surface n o rm a l, a nd th a t th e la rg e r th e
u n it cell, th e closer th e f ir s t m a xim u m w ill be to th e surface norm al.
I n la b e llin g th e d iffra c tio n m axim a, the M ille r in dice s are com bined w ith
the o rder o f d iffra c tio n n, to y ie ld the Laue indices,
h = n h'
and
k = n k'
( 10)
W ith th is n o ta tio n , a d iffra c tio n m a xim u m is re fe rre d to as th e (h ,k) beam.
W ith m o st L E E D a p p a ra tu s, th e d iffra c tio n beam p a tte rn is d ire c tly
obse rva ble, w h ic h in
p rin c ip le
a llo w s fo r th e d e te rm in a tio n o f d h 'k '
and
consequently the geom etry o f the surface u n it cell. A m ore sophisticated approach
to a n a ly z in g a nd in te rp re tin g L E E D p a tte rn s makes use o f th e reciproca l la ttic e ,
described below.
F o r every tw o d im e n sio n a l B ra v a is la ttic e , th e re e xists a corre spond ing
reciproca l la ttic e whose p rim itiv e vectors, ( a ^ a ^ ) satisfy
a j • a j —2 jc5y
(ID
F ro m th is equation, the lengths and directio ns o f the reciproca l la ttic e vectors can
be determ ined.
12
*
2 TL
a i = ----- :---ai s m y
a -I a j
fo r
( 12)
i* j
(13)
H ere aga in y is th e angle between a% and a 2 . Reciprocal la ttic e s fo r the fiv e tw o
d im en sion al B ra va is la ttic e s are shown in F ig u re 5. The u t ilit y o f reciprocal space
analyses is th a t th e L E E D p a tte rn is a d ire c t re p re s e n ta tio n o f th e re cip ro ca l
la ttic e , as show n below.
The fo rm a tio n o f the L E E D p a tte rn can be explained u s in g the concept o f
re c ip ro c a l space w ith th e h e lp o f an E w a ld c o n s tru c tio n , show n fo r n o rm a l
in cide nce in F ig u re 6.
A set o f lin e s is d ra w n p e rp e n d ic u la r to th e surface
th ro u g h the reciproca l la ttic e points, w h ic h become the re cip ro ca l la ttic e rods. A
sphere o f ra d iu s I A, is th e n d raw n d ire c tly above one o f the reciprocal la ttice points
a t a distance I A, fro m th e surface. A n a rro w d ra w n fro m th e center o f the sphere
to the surface represents th e in c id e n t e lectron wave vector.
The in te rse ctio n s o f
th e re ciproca l la ttic e rods w ith the sphere define th e d ire ctio n s o f th e d iffra c tio n
beams.
The vectors k g and k re p re se n t th e wave vectors o f th e in c id e n t and
scattered beams, respective ly, and k - k g is th e m om e ntu m tra n s fe r.
I t can be
seen fro m th is co n stru ctio n th a t the d iffra c tio n beams com pletely characterize th e
sym m etries o f th e reciprocal la ttice , so th e L E E D p a tte rn is a d ire c t representation
o f the la ttice .
Once th e re cip ro ca l la ttic e o f a surface s tru c tu re is d e te rm in e d (from th e
L E E D p a tte rn ), the B ra va is la ttic e can be fo u n d by a tra n s fo rm a tio n from
13
Real Space Lattice
Reciprocal Space Lattice
a.
a;
a2
Figure 5: Two-dimensional reciprocal lattices
14
0 -3
0 -2
0 -1
00
01
02
03
Reciprocal Lattice points
F ig u re 6: E w a ld C o nstructio n
re ciproca l space to re a l space.
The surface reciprocal la ttic e , ( a i* ,a 2 * ) m ay be
correlated to th a t o f the substrate, ( b i* , b 2 *) in a m anner s im ila r to equation (I).
b x — M 11B 1 + M 12^2
(14)
bg = M 21B 1 + M 22H2
The M*j can be d e te rm in e d by an in s p e c tio n o f th e L E E D p a tte rn .
The
tra n s fo rm a tio n to fin d the m a trix M , fro m M * is [7]
M ij
r_ ± l _ >
< detM *y
(15)
15
T he advantages o f u s in g re c ip ro c a l la ttic e s in L E E D p a tte rn a n a lysis
becomes m ore e v id e n t w h e n s tu d y in g o v e rla y e r s tru c tu re s a n d p a tte rn s w ith
dom ains.
D e ta ile d d e s c rip tio n s o f these types o f analyses are discussed in
references [7,8].
So fa r o n ly th e p o sitio n s o f th e d iffra c tio n beams have been considered.
A n a ly s is o f th e beam p o sitio n s can be used to d e te rm in e th e geom etry o f th e
surface u n it cell, b u t provides no in fo rm a tio n fo r the atom ic positions w ith in th e
u n it ce ll, i.e. th e basis.
T h is in fo rm a tio n is contained in th e d iffra c tio n beam
in te n s itie s . The d iffra c tio n beam in te n s ity as a fu n c tio n o f electron energy is ve ry
sensitive to atom ic positions, and th u s provides a means o f deducing the com plete
atom ic geom etry. There does n o t exist, how ever, a means o f e x tra c tin g a geom etry
d ire c tly fro m the IV curves.
The m ost com m on approach is to em ploy a search
te c h n iq u e , w h e re b y ca lc u la te d IV curves fo r a proposed th e o re tic a l m odel are
com pared to th e e x p e rim e n ta lly o b tained curves, th e n changes are m ade to th e
m odel u n t il th e tw o sets o f IV curves are com patible. The techniques fo r s tru c tu re
searches are discussed fu r th e r in C h a p te r 5.
I n th e fo llo w in g section th e basic
e le m e n ts fo r a k in e m a tic a l th e o ry o f s c a tte rin g are discussed, th e n some
developm ent is m ade to w a rd describing a m ore com plete d yn a m ica l theory.
D iffra c tio n In te n s itie s
The d e te rm in a tio n o f the basis fo r a surface s tru c tu re re lie s on the analysis
o f th e d iffra c tio n in te n s itie s and th e ir dependence on the in c id e n t electron energy.
I p re se n t below a k in e m a tic a l deve lop m en t, in w h ic h th e effects o f m u ltip le
s c a tte rin g a n d in e la s tic s c a tte rin g are ig n o re d .
E xte n sio n s to m ore com plete
d y n a m ic a l th e o rie s w h ic h in c lu d e these e ffe cts are th e n discussed.
The
16
k in e m a tic a l th e o ry is accurate in p re d ic tin g th e general shape o f I V curves, and
provides a u se fu l s ta rtin g p o in t fo r m ore com plete d yna m ical theories.
T he in c id e n t e le c tro n beam is re p re s e n te d b y a p la n e w ave, w it h
w avele ngth X, and d ire ctio n given by u n it vector
sq,
by
V = V o-C lko r
(16)
w here k g = (27tA,) so is th e electron wave ve cto r w h ic h also represents th e c ry s ta l
m o m e n tu m o f th e electron.
W hen th e scatte red wave is also represented as a
plane wave, w ith wave vector k = (2 tiA,)s , th e scattered wave a m p litu d e a t position
r due to sca tte rin g fro m atom j is [8]
V = (x|/o 'e - ‘k r ) . f j ( k , k 0) .e ‘ |k- k - ,R i
(17)
The f ir s t te rm describes an outgoing wave, and f j is th e a tom ic s c a tte rin g fa c to r
w h ic h depends in general on th e in it ia l and fin a l wave vectors k g and k . The la s t
te rm describes th e p h a s e s h ift betw een th e wave scattered fro m th e jt h atom , a t
p o s itio n r j , a n d a wave s c a tte rin g fro m th e o rig in .
W h e n a ll th e atom s are
considered, th e a m p litu d e m u s t be sum m ed,
v « e -'k " . % f , ( k , k , ) - e'(k - k °) A'
(18)
In th e k in e m a tic a l a p p ro x im a tio n , th e sum over fj is replaced b y th e so-called
s tru c tu re fa c to r F [8], w h ic h depends on th e geom etry o f th e u n it cell, the types o f
17
atom s w ith in the u n it cell, and th e ir respective positions, i.e. th e basis. The vector
R j, can be w ritte n in term s o f the u n it cell as
Rj = n 1a 1 + n 2a2
(19)
w h ich, along w ith F, can be s u b stitu te d in to equation (18) to y ie ld
.
. Mg
y - F • e™£kr . ^ y niar(k-ko) • ^ e™2a2'(k™ko)
Hi
Hg
(20)
The u p p e r lim its o f these tw o sum m ations, M i and M 2 , depend on th e area o f the
c ry s ta l surface th a t th e electron beam scatte rs from .
som etim es ca lle d th e la ttic e fa c to r G.
The double su m m a tio n is
The sca tte rin g in te n s ity I, w h ic h is th e
p h y s ic a lly m easurable q u a n tity , is thus
H F ] 2 -IGl2
(21)
The sum m a tions in equation (20) can re a d ily be perform ed, g iv in g
'
2 _ Sin2^ M 1U1- ( k - k o ) )
sin 2 (-i a x • ( k - k 0))
s in 2( | M 2a 2 - ( k - k p ) )
s in 2
a 2 • ( k - k 0))
(22)
w here M i and M g are th e u p p e r lim its on th e sum m ations. T h is equation predicts
d irectio ns k fo r th e in te n s ity m a xim a such th a t
18
(23)
w here h i and hg are integers. A n e q u iva le n t expression, u sing k = (2jr/X)s, is
a 1 - ( s - s 0) = h 1X and
(24)
a 2 " ( S - S 0)
= I i2X
These are the Laue conditions fo r d iffra c tio n fro m tw o d im en sion al la ttice s, w h ich
are e q u iva le n t to th e B ragg equation, equation (8). The u n it vectors s w h ich sa tisfy
these re la tio n s th e n d e te rm in e th e s c a tte rin g angles fo r w h ic h th e d iffra c tio n
beams occur.
The in te n s ity o f a d iffra c tio n beam, a t a given energy, depends on the fir s t
te rm in equation (21), n a m e ly the square o f the stru ctu re factor, I F l 2. Recall th a t
(25)
w here th e fj are th e atom ic sca tte rin g factors. The fj can be expressed as a series
expansion in a n g u la r va ria b le s [7,8],
= X (2Z + 1JpZ(cosCpG)el8z s in 5,
w h ere (pG is th e angle betw een k and k g , and X is 7z/k.
(26)
The P ; are Legendre
p o ly n o m ia ls , and th e in d ice s I are th e same a n g u la r m o m e n tu m q u a n tu m
n u m b e rs t h a t are used in d e scrib in g th e a n g u la r p a r t o f an e le ctro n ic wave
19
fu n c tio n .
The pro p e rtie s o f s c a tte rin g are contained in th e phaseshifts 5/, w h ic h
are d eterm ined fro m th e s c a tte rin g p o te n tia l used in the m ethod described below.
The c ry s ta l sca tte rin g p o te n tia l is g e n e ra lly a p p ro xim a te d as a m u ffin tin
p o te n tia l, co n stru cte d fro m o ve rla p p in g in d iv id u a l a tom ic p o te n tia ls [9].
The
m u ffin tin s are s p h e ric a lly sym m e tric regions s u rro u n d in g th e ion-cores w ith in
th e c ry s ta l, w ith r a d ii defin ed b y th e distance fo r w h ic h th e p o te n tia ls fo r tw o
adjacent ion-cores is equal. The regions between the m u ffin t in spheres are given
a constant p o te n tia l, w h ic h contains in p a rt, th e o p tica l p o te n tia l o f th e crysta l.
The o p tica l p o te n tia l consists o f bo th a re a l and an im a g in a ry p a rt. The re a l p a rt,
called the in n e r p o te n tia l V 0, is re la te d to the in d e x o f re fra c tio n o f the crystal, in
th a t th e electron experiences a change in k in e tic energy in sid e th e crysta l re la tiv e
to its energy in th e vacuum .
The p rim a ry effect o f changes in V 0 is to s h ift the
peaks in th e IV da ta along the energy axis [7], The im a g in a ry p a r t o f th e o p tica l
p o te n tia l V j characterizes th e electron's energy loss th ro u g h in e la s tic collisions,
p r im a r ily w ith o th e r electrons. I t is d ire c tly re la te d to the in e la s tic m ean free p a th
Xee fo r an electron in the crystal, by [8]
(27)
'ee
The o p tica l p o te n tia l cannot, in general, be derived fro m th e o re tic a l principles, b u t
is de te rm in e d e m p iric a lly by choosing the set o f V 0 and V i w h ic h provide th e best
IV data.
A n o th e r im p o rta n t aspect o f th e sca tte rin g process is th e exchange effect,
due to the F e rm i s ta tis tic a l behavior o f the electrons. The effect o f exchange is th a t
electrons o f th e same spin states repel each o ther, and are a ttra c te d w hen th e ir
20
sp in states are d iffe re n t.
T h is effect can be m odelled by a p o te n tia l, called th e
exchange p o te n tia l. The exchange p o te n tia l and th e c ry s ta l p o te n tia l characterize
th e s c a tte rin g p o te n tia l, w h ic h is th e s ta rtin g p o in t fo r d e scrib in g electron -solid
s c a tte rin g w ith the H a rtre e -F o ck equation.
W hen w r itte n in the B o rn -O p p e n h e im e r a p p ro x im a tio n , th e H a rtre e -F o ck
equation can be w ritte n as [9]
(-H 2 / 2 m )V 2 + V s( r ) -
e
] • cp0( r ) = - V ex • (p0( r )
(28)
w here (p0 represents th e in c id e n t electron wave fu n c tio n , a n d E is the e lectron
energy. V s is the cry s ta l p o te n tia l, and V 6XfPo is the exchange c o n trib u tio n to th e
in te ra c tio n . The c ry s ta l p o te n tia l V s, fro m w h ich the phaseshifts can be com puted
and th e exchange p o te n tia l V ex are discussed below.
I n th e o v e rla p p in g charge d e n s ity m odel developed fo r th e s tu d y o f
com pound sem iconductors [9,10], th e p o te n tia l V a , fo r a p a r tic u la r io n core
la b e lle d by a, is expressed in te rm s o f th e e le ctro n ic charge d e n s ity p a (r), th e
atom ic n u m b e r Z a , and th e io n ic ity I a , as
(29)
The la s t te rm in th is expression contains th e e le ctrostatic M a d e lu n g sum M [11]
fo r th e c ry s ta l la ttic e , a nd d is th e n e a re s t n e ig h b o r d ista n ce .
T h is te rm
represents th e e le ctro sta tic in te ra c tio n o f one io n core w ith a ll th e others in th e
crysta l. The effect is th a t the anion p o te n tia ls are u n ifo rm ly s h ifte d negative b y a
s m a ll am oun t, and th e ca tio n p o te n tia ls are sh ifte d positive [10]. The in te g ra tio n
21
can be s im p lifie d b y assum ing th a t the charge d e n sity is s p h e ric a lly sym m e tric,
i.e.
P aM =
O aM
4 n r2
(80)
W ith th is assum ption, the expression fo r th e p o te n tia l nea r an ion-core is
V aM =
r
+f
j CaW t + e2j ^ d
r *
J t
t +
d
+ V exW r ) )
(31)
The la s t te rm is th e exchange p o rtio n o f th e p o te n tia l w h ich is g e n e ra lly m odelled
b y a fu n c tio n a l o f th e charge d e n s ity as w i ll be seen below .
re p re s e n ta tio n th a t is used to evaluate V a n u m e ric a lly .
T h is is th e
The m u ffin t in c ry s ta l
p o te n tia l can th e n be constructed b y superposing neig h b o rin g ion-core potentia ls.
The c a lc u la tio n o f th e phaseshifts fro m th e p o te n tia l is approached in a
sta n d a rd w a y [12]. The ra d ia l Schrodinger equation
M 2 I d2R K r) |
^2m J
d r2
Va(r) + V^(p(r)) +
1(1 + l)7z2
Rz (r) = (E + V 0)Rj (r)
(32)
2 m r2
is solved n u m e ric a lly o u t to th e b o u n d a rie s o f th e m u ffin t in spheres.
The
exchange p o te n tia l in th is equation is a fu n c tio n a l o f th e charge d e n sity p. Also
note th a t th e in n e r p o te n tia l, V 0 enters in to th is expression.
The lo g rith m ic
d e riv a tiv e o f wave fu n c tio n R ;(r) is m atched to th e so lution outside th e m u ffin tin ,
w h ic h consists o f sp he rical Bessel fun ctio n s j ; and n /, a t the m u ffin t in bou ndary
[ 12].
22
The phaseshifts also depend e x p lic itly on th e exchange p o te n tia l V ex. The
models fo r th e exchange p o te n tia l used in m ost L E E D ca lcu la tio n s are the S la te r
m odel [13] and th e H a ra m odel [14].
The c o n trib u tio n to th e pha se sh ifts w h en
u sing the S la te r exchange m odel is [10]
r MT
5slater( E ) ~ ( l / k ) J p 1/3(r)dr
(83)
and w hen u s in g th e H a ra m odel, i t is
r MT
5Hara( E ) ~ ( l / k 3) J p(r)dr
(34)
These tw o expressions are v a lid in the lim it th a t k rjviT » 1 . N ote th a t the S la te r
effect fa lls o ff w ith in c re a s in g energy as 1/k, o r 1/E ^ 2, w h ile th e H a ra m odel
decreases fa s te r, as 1/E 3/2.
Because th e H a ra m odel is m ore se n sitive to th e
charge den sity ( it depends on p in stea d o f p 1^ ) , the phaseshifts calculated w ith the
H a ra m odel are in general m ore sensitive to charge density v a ria tio n s . The H a ra
exchange m odel is used in th is w o rk because i t has been show n to provide th e best
agreem ent betw een th e o ry and e xp e rim e n t fo r com pound sem iconductors [10].
Once th e p h a seshifts, geom etry, and th e com position o f th e c ry s ta l have
been specified, th e d iffra c tio n in te n s itie s can be calculated.
One p o p u la r m ethod
fo r com pound sem iconductors is called th e T -m a trix m u ltip le s c a tte rin g m ethod
[3,8,9]. T h is m ethod u tiliz e s a la ye r-b y-la ye r approach in describing th e sca tte rin g
process. The calcula tions are perform ed in reciprocal space, w here th e s c a tte rin g
betw een layers and w ith in th e layers can be described w ith p ro b a b ility am plitu des
23
and m a tr ix equations, w h ich are m ore su ita b le fo r ite ra tiv e calcula tions [3]. The
details o f th is m ethod are described in [3,8,9].
24
C H A P TE R 3
E X P E R IM E N T A L T E C H N IQ U E S
TEED
The e xp e rim e n ta l apparatus used in th is w o rk is depicted in F ig u re 7. The
m a jo r com ponents consist o f th e reverse v ie w L E E D optics, m odel R V L 10-120,
fro m P rin c e to n Research, an u ltr a - h ig h vacuu m cham ber, a h ig h re s o lu tio n
P u ln ix m odel T M -8 4 0 N CCD cam era, and a M a cin to sh I I personal com puter. The
sam ple is h e ld b y th e v e rtic a l sam ple holder, w h ich can be m oved tra n s la tio n a lly
in a ll th re e dim ensions, and ro ta te d ab o u t th e v e rtic a l axis.
T he optics and
e le c tro n g u n are m o u n te d on a fo rm e d b e llo w s w ith a d ju s tin g screws fo r
a lig n m e n t n o rm a l to th e sam ple surface. H e a tin g to 500°C is possible by use o f a
b u tto n h e a te r located a t th e sam ple base. A liq u id n itro g e n cold fin g e r provides
sam ple cooling c a p a b ilitie s to 150 K.
The cam era is in te rfa c e d to th e com puter
th ro u g h a D a ta T ra n s la tio n Q uick C a p tu re ™ fram e grabber board. The softw are
package developed fo r th is system , called M a c L E E D , is w r itte n in the M P W C
p ro g ra m m in g language and u tiliz e s s ta n d a rd M a cin to sh to o l box fu n c tio n s and
m e n u -ty p e d isp la ys.
D a ta a c q u is itio n fo r L E E D w ith th is a rra n g e m e n t is
re la tiv e ly easy, and a b r ie f o u tlin e o f the process is presented below. M ore detailed
discussions o f th is system can be found in references [15] and [16].
The d iffra c tio n p a tte rn is recorded b y th e cam era and tra n s fe rre d to the
com puter as a sta n d a rd composite video signal. The video sig n a l is th e n d ig itiz e d
25
CCD camera
Adjusting
screws x
LEED
Formed
bellows
Button
heater
Sample
LEED Optics
Control Unit
LN2
coldfinger
Macintosh Il
Computer
Figure 7: LE E D equipment
Frame Grabber
Board
26
b y th e fra m e grabb er boa rd one fram e a t a tim e , and te m p o ra rily stored in the
board's R A M .
The stored im age can th e n be displayed on th e com puter m o n ito r,
o r placed in to th e com puter's m em ory fo r fu tu re analysis.
A sin g le fra m e , o r
video im age is tra n s fe rre d fro m th e cam era to the co m p u te r every 1/3O^h o f a
second.
T hu s, each fra m e corresponds to a lig h t sig n a l accu m u la te d in a v e ry
sh o rt tim e . I n o rder to get the best possible s e n s itiv ity w ith stored images, several
(u s u a lly 16) fra m e s are added to g e th e r a nd store d as one sin g le p ic tu re .
An
e x te rn a l tim in g c ir c u it has re c e n tly been co n stru cte d w h ic h a llow s th e video
sig n a l to be in te g ra te d in sid e the cam era ra th e r th a n th ro u g h com puter softw are
te ch n iq u e s.
A d iscussio n o f th is c ir c u it a nd its p e rfo rm a n c e are g ive n in
A p p e n d ix B.
F o r th e collectio n o f IV data, th e user m u s t specify th e in c id e n t electron
energy range (ty p ic a lly 40-300 eV), the energy step in te r v a l (u s u a lly 2 eV), th e
d iffra c tio n beams o f in te re s t, and the n u m b e r o f fram es over w h ic h each im age is
to be in te g ra te d .
th re e steps.
The specification o f th e d iffra c tio n beam s is accom plished in
F irs t, an im age is d ispla yed fo r th e L E E D p a tte rn a t an e lectron
energy n e a r th e b e g in n in g energy, th e n th e positions o f th e d iffra c tio n spots are
recorded. N e xt, an im age is displayed fo r an energy nea r th e top o f the range, and
the positions o f the beams are again recorded. The user m u s t th e n lin k the in it ia l
and fin a l p o s itio n s fo r each beam.
T he p o s itio n o f each d iffra c tio n beam is
im p o rta n t because IV da ta is m easured o n ly fo r a s m a ll area s u rro u n d in g each
spot (u s u a lly 20x20 pixels).
F ro m th e tw o g iven positions and energy values, th e com puter calculates
w here th e spots sh o u ld be fo r each energy va lu e in th e specified range.
The
electron gun is th e n set to th e s ta rtin g energy, and the com puter searches fo r an
in te n s ity m a x im u m w ith in a s m a ll re c ta n g le centered on th e expected spot
27
p o s itio n .
Once th e m a x im u m is fo u n d , th e re c ta n g le is ce n te re d on th is
m a x im u m , th e in te n s ity is in te g ra te d , and a backg ro und s u b tra c tio n is m ade by
s u b tra c tin g fro m th e in te n s ity the average va lu e a ro u n d th e p e rim e te r o f the
rectangle.
The com puter stores the in te n s ity data, th e n advances th e electron gun
energy and th e above steps are repeated.
F o r th e collection o f stored images, th e user specifies th e n u m b e r o f fram es
to accum ulate and th e electron gun energy. The com puter th e n reads th e fram es
successively fro m th e fram e grabber board, and stores th e da ta in a 16 b it fo rm a t,
w here th e in te n s ity values range fro m 0 to 255. The to ta l a ccum ulated sig n a l is
th e n d iv id e d by th e n u m b e r o f fram es. I n th is case, the e n tire im age (619 x 422
pixels) is saved.
AES
The system can be re a d ily converted to collect A u g e r d a ta b y changing a few
e le ctrica l connections, as show n in F ig u re 8.
T h is arra n g e m e n t is kn o w n as th e
re ta rd in g fie ld m ethod. Several oth e r m ethods fo r AES m easurem ents have been
developed, and are discussed in d e ta il elsewhere [7,17]. The fo llo w in g discussion
o f th e A E S techniques used in th is w o rk is applicable to re ta rd in g fie ld type AE S
systems in general.
A h ig h ly energetic (1-2 keV ) beam o f electrons is a im e d a t th e sam ple,
p ro d u cin g core-level e xcita tio n s in th e c ry s ta l atoms.
D e -e xcita tio n th ro u g h th e
A u g e r process causes em ission o f o u te r sh e ll electrons fro m th e excited atoms.
The k in e tic energies o f these e le ctro n s are c h a ra c te ris tic o f th e e le c tro n ic
c o n fig u ra tio n s o f th e e m ittin g atom s, and th u s provid e a m eans o f s tu d y in g the
electronic s tru c tu re o f th e atom s in the c ry s ta l [17]. The A E S technique is
28
Electron gun
Sample
V1 sin cot
L o c k -in
A m p lif ie r
F ig u re 8: E le c tric a l connections fo r A E S
applicable to surface analysis because electrons e m itte d fro m deep w ith in the b u lk
o f the c ry s ta l lose k in e tic energy to in e la stic collisions before escaping to the
vacuum . The effect o f these electrons on th e A u g e r spectrum is to increase the
background in te n s ity , le a vin g the m a in feature s o f the spectrum to re ly on atom s
on or near the cry s ta l surface. F or m ost m a te ria ls, the depth o f AE S s e n s itiv ity is
about 10 atom ic layers. W hen the A u ger spectrum is analyzed, several properties
o f th e c ry s ta l surfa ce can be deduced, such as e le m e n ta l co m p o sitio n and
chem ical bon din g [17].
R e fe rrin g again to F ig u re 8, the A u g e r electrons are filte re d by grids 2 and
3 w h ic h are h e ld a t a p o te n tia l V r , k n o w n as the r e ta rd in g p o te n tia l.
The
29
re ta rd in g p o te n tia l provides a lo w e r lim it on th e k in e tic energy o f th e electrons
a llo w e d to pass th ro u g h .
W hen V r is v a rie d fro m -V m a x to zero, th e c u rre n t
detected b y th e collector a t a p a rtic u la r value o f V r is
i( V r ) =
jN (E )d E
(35)
""eV m a x
w here eVm a x is th e in c id e n t electron beam energy, and N (E ) is th e d is trib u tio n o f
th e electrons (n u m b e r o f electrons detected).
T h is d is trib u tio n appears as s m a ll
steps on a la rge , s lo w ly v a ry in g b a ckg ro u n d [7,17].
A b e tte r re s o lu tio n o f th e
A u ger peaks can be achieved b y m easuring th e q u a n tity dN /dE [17]. T his q u a n tity ,
how ever, is n o t d ire c tly m easurable, b u t i t can be m easured in d ir e c tly in th e
fo llo w in g m a n n e r.
A s m a ll AC m o d u la tin g voltage is superim posed on th e re ta rd in g p o te n tia l,
V r = V r + V q sincot,
(36)
such th a t th e collector c u rre n t i, becomes a fu n c tio n o f V r , and th e T a y lo r series
expansion about i( V r ) is
i( V r + V 0 sincot) = i( V r ) + V0 sincot
^ d i(V r ) ^
+
V0 s in 2
d2i( V r )
+... (37)
The term s in th is expansion p ro p o rtio n a l to sincot and cos2cot are
f d i(V r ) l
A(CO) = V0
I dVr
J
r d 3i( v r ) )
8 I
dV3
J
(38)
30
B(2co)
V02 d^K V ,.)!
4 I dVf J
( d4i( V r ) )
48
I dV4 J
(39)
The second h a rm o n ic com ponent, B(2m) o f the collector c u rre n t is th e n m easured
by a phase-sensitive (lo ck-in ) a m p lifie r. W hen V q is sm a ll enough, the fir s t te rm
in B (2 go) is d o m in a n t, and th e m easurem ent o f B(2oo) is p ro p o rtio n a l to dN (E)/dE.
The re s o lu tio n w ith th is system is ~8 eV fo r a m o d u la tio n voltage o f 8-10 volts. F or
m ost types o f AE S surface analysis, a m uch b e tte r re so lu tio n is necessary [17], b u t
o u r re s o lu tio n is adequate fo r the w o rk presented here, in w h ic h A E S was used
p r im a r ily to m o n ito r th e re la tiv e changes in co n ce n tra tio n o f b is m u th on the
surfa ce.
D a ta collection in the AES mode consisted o f a few sim ple steps. The energy
range was specified fo r each scan, along w ith a fu ll scale s e n s itiv ity fa cto r fo r the
lo c k -in a m p lifie r.
The co m p u te r th e n c o n tro lle d th e r e ta rd in g p o te n tia l, and
re ce ive d th e s ig n a l B(2co), fro m th e lo c k -in a m p lifie r.
T h e n th e s ig n a l was
converted to an x-y a rra y and d isp la ye d on th e co m p u te r m o n ito r as a curve,
dN /dE versus E. Once p re lim in a ry AES scans were made, i t was o n ly necessary to
scan c e rta in regio ns o f th e energy sp e ctru m , co rre sp o n d in g to c h a ra c te ris tic
peaks fo r g a lliu m , a n tim o n y, and b ism u th . B y com paring th e changes in peak to
peak ra tio s fo r the th re e elem ents, th e re la tiv e changes in conce ntratio ns on th e
surface could be deduced.
31
Sam ple P re p a ra tio n
S ingle c ry s ta l GaSb sam ples w ere sin gle c ry s ta l bars w ith n -typ e doping,
and c a rrie r concentrations o f ~1.7 x
(cm -1). Sample sizes were n o m in a lly 1.5"
in le n g th w ith .25" square cross sections. The o rie n ta tio n o f th e c ry s ta l was such
th a t th e (H O ) plane was p e rp e n d ic u la r to th e lo ng axis o f th e cry s ta l.
A clean
surface was ob ta in e d b y cle avin g th e sam ple w ith a knife-edge along th e (H O )
plane.
To a lig n th e knife-edge p a ra lle l to th e (H O ) plane, and th u s m in im iz in g
steps and facets on the cleaved surface, sm a ll cuts w ere m ade in the sam ple w ith
a diam o nd c u ttin g w heel .045" deep, .020" w ide, and spaced .080" apa rt. T h is is
show n in F ig u re 9.
The sam ple was th e n cleaned u ltra s o n ic a lly in a b a th o f
acetone. In s id e th e cham ber, the sam ple was h e ld w ith th e lo n g axis v e rtic a l and
cleaved w ith th e h o riz o n ta l knife-edge.
The q u a lity o f th e cleaved surface was
im m e d ia te ly assessed by v is u a l o b se rva tion, a nd b y o b se rvin g th e d iffra c tio n
p a tte rn as th e sam ple was tra n s la te d h o riz o n ta lly w ith respect to th e e le ctro n
beam.
The surface was deemed acceptable, i f a b rig h t and cle ar p ( l x l ) p a tte rn
was observed.
To p o sitio n th e sam ple fo r an expe rim e nt, a regio n was searched fo r th a t
gave a b r ig h t and consistent L E E D p a tte rn over a .020" x .020" area. The sam ple
was th e n p o s itio n e d so th e e le c tro n beam s tru c k th e c e n te r o f th is area.
A lig n m e n t o f th e e le ctro n beam p e rp e n d ic u la r to th e surface was achieved b y
tu r n in g th e a d ju s tm e n t screws a tta ched to th e la rge b ello w s a t th e top o f th e
cham ber, see F ig u re 7. The a lig n m e n t procedure was m o n ito re d by th e com puter,
w h ic h d isp la ye d coordinates on the screen fo r b o th th e a c tu a l and th e desired
locations o f th e p rim a ry re fle c tio n (00) beam.
positions coincided.
A d ju stm e n ts were made u n t il th e
32
<110> Direction
.080
.015
1. 5 " - 2.0
Figure 9: Sample preparation
33
QCO C a lib ra tio n
The deposition o f b is m u th was achieved by evaporation fro m a solid piece o f
b is m u th h e ld in a tu n g s te n basket.
The cru cib le was heated to th e s u b lim a tio n
te m p e ra tu re o f b is m u th (~600 °C), b y p la c in g the basket in a low -voltage h ig h c u rre n t c irc u it.
The d e p o sitio n was m o n ito re d w ith a q u a rtz c ry s ta l m ic ro -
balance o s c illa to r h e ld n e a r th e sam ple d u rin g e v a p o ra tio n .
The change in
frequency o f the q u a rtz c ry s ta l was ca lib ra te d to b is m u th coverage on the sam ple
in th e fo llo w in g m anner.
A q u a rtz c ry s ta l has several modes o f v ib ra tio n , corre spond ing to several
re s o n a n t fre q u e n c ie s .
The
m o s t com m on m ode used in
m ic ro b a la n ce
ap p lica tio n s is th e shear mode along the A T -plane [18]. The resonant frequencies
o f the q u a rtz crystals used in these applications is ty p ic a lly 4-5 M H z [18], and the
freq uency change fo r a coverage o f a fe w m onolayers is u s u a lly 10-100 H z [19].
O ver such a s m a ll freq uency in te rv a l, th e re la tio n s h ip betw e en th e freq uency
change A f and th e mass o f the adsorbed m a te ria l Am is v e ry n e a rly lin e a r, a nd
can be stated as [19]
A f = C Am
(40)
w here C is a constant. T h is re la tio n rem a ins v a lid as long as A f is m uch less th a n
the resonant frequency (4-5 M H z).
F o r c a lib ra tin g th e m icro-balance fo r an e x p e rim e n t, A m is replaced by
pAAx, w here p is the mass d en sity o f th e adsorbed m a te ria l, At its thickness, and
A is th e area o f the q u a rtz crystal. E q u a tio n (40) th e n becomes
34
A f = C A p At
(41)
Since the area o f th e cry s ta l is also a constant, the equation can be w ritte n as
A f = (1/k) p A t
(42)
W ith th e e q u a tio n in th is fo rm , th e co n sta n t k is re fe rre d to as th e in s tru m e n t
s e n s itiv ity
[1 9 ].
I t is determ ined , in p a rt, b y th e geom etry o f th e m icro-balance
con stru ctio n . F o r the m icro-balance used in th is w o rk, the in s tru m e n ta l constant
was d e te rm in e d to be
3 .0 0 9 x 1 0 -8
in cgs u n its .
T h is was accom plished b y
co m parin g th e frequency change as com pared to th a t o f a co m m e rcia lly-o b ta in e d
o s c illa to r.
T he accuracy o f th is m e th o d has been v e r ifie d w ith R u th e rfo rd
ba cksca tte rin g experim ents, and the tw o m ethods agree to w ith in about 10% [15].
In o rd e r to c a lib ra te th e QCO fo r b is m u th adsorption on GaSb, th e s tic k in g
coefficie nt fo r b is m u th on the q u a rtz c ry s ta l was assumed to be equal to th a t fo r
GaSb. W ith th is assum ption, a change in th e frequency o f th e o scilla to r is d ire c tly
p ro p o rtio n a l to b is m u th coverage on the sam ple. The goal o f th e c a lib ra tio n was to
d e te rm in e th e a m o u n t o f freq uency change corresponding to a coverage o f one
m onolayer, w here a m onolayer is defined as one b is m u th atom p e r substrate atom
on the G a S b(IlO ) surface.
In reference to e q u a tio n (42), th e p ro d u c t
At
p is the
surface mass d e n sity (g/cm 2) o f the o ve rla ye r m a te ria l, w h ic h was a convenient
q u a n tity to calculate firs t.
The (H O ) surface o f a zincblende c ry s ta l has tw o atom s p e r surface u n it
cell, and a u n it cell area o f
35
A
(43)
w here a is th e c ry s ta l la ttic e constant. F o r GaSb, a=6.095 A , w h ic h gives a u n it
cell area o f 2 6 . 2 7 x 1 0 " cm^.
W ith tw o atom s per u n it cell, th e surface a tom ic
density pa is th e n
Pa
— = 7.61 x 1014atoms / cm 2
(44)
A
The surface mass d e n s ity was th e n fo u n d b y m u ltip ly in g p a b y the ra tio o f th e
mass n u m b e r o f b is m u th d ivid e d by Avogadro's constant,
Al; • p = p a -
'
209.0
N
^6.02 x 1024;
= 2.64 x 10“ 8 g / cm2
(45)
T hu s, th e fre q u e n cy change fo r one m o n o la ye r o f b is m u th on th e G a S b (IlO )
surface was de te rm in e d to be
A f = (2.64 x IO -8) --------------- = = 8.8Hz
3.01 xlO™8
(46)
36
C H A P TE R 4
G R O W TH A N D K IN E T IC S OF BiZGaSb(IlO)
D eposition and G ro w th o f B is m u th
The f ir s t e xp e rim e n t was an a tte m p t to reproduce th e re s u lts obtained b y
D uke, et.al. [20] fo r th e clean G aS b(IlO ) surface. IV curves w ere m easured a t lo w
te m p e ra tu re (150 K ) fo r v is u a l com parison w ith the previously-obtained data and to
ensu re th a t th e cle an surfa ce c h a ra c te ris tic s w ere c o n s is te n t w ith p re vio u s
experim ents. The IV curves m atched v e ry w e ll, in th a t a ll o f th e m a jo r features o f
th e curves w ere successfully reproduced.
The sym m etries o f th e clean surface
y ie ld e d , as expected, a c le a r p ( l x l ) L E E D p a tte rn w ith a m ir r o r re fle c tio n
sym m e try, (-h ,k) = (h,k).
N e x t, a s tu d y o f IV dependence on b is m u th coverage was made.
B o th
deposition and da ta collection were perform ed a t room te m p e ra tu re . IV da ta were
ta k e n fo r several coverages, ra n g in g fro m 0 to 2 m onolayers, to determ ine w h e th e r
lo ng range sym m e try existed in th e b is m u th overlayer.
beam are show n in F ig u re 10.
The re s u lts fo r the (0,1)
The o v e ra ll in te n s ity g ra d u a lly decreases w ith
coverage, except fo r a s lig h t increase n e a r a coverage o f about one m onolayer, and
th e shape o f th e curve changes co n tin u o u sly u p to abo ut one m onolayer.
The
changes in c u rre d b y th e re s t o f th e beams are s im ila r to th is , except fo r the (1,0)
and (-1,0) beams (th is effect is discussed below). Presented la te r in the discussion
section is th e conclusion th a t th is continuous change in IV d a ta to gether w ith the
37
(0,1) beam
Bi/G aSb(110)
Intensity (Arbitrary Units)
2.03 ML
1.40 ML
1.20 ML
' “ 1.00 ML
0.84 ML
0.66 ML
0.43 ML
' 0.20 ML
clean
Electron Energy (eV)
F ig u re 10: (0,1) d iffra c tio n beam dependence on b is m u th coverage
38
increase in o ve ra ll in te n s ity o f a ll b u t the (1,0) and (-1,0) beams is in d ic a tiv e o f long
range o rder in th e one m onolayer system.
The p ( l x l ) o rd e r and m ir r o r re fle c tio n sym m e try o f th e clean surface d id
n o t change w ith coverage. The re te n tio n o f sym m etry betw een th e (2,1) and (-2,1)
beams is show n in F ig u re 11. Since the in te n s itie s o f the tw o beams are equal in
m a g n itu d e , th is also v e rifie s th e a lig n m e n t o f th e electron beam p e rp e n d icu la r to
th e surface.
L o w te m p e ra tu re IV da ta was collected fo r th e clean surface a nd a t
I M L coverage fo r subsequent s tru c tu re analyses.
G ro w th K in e tic s and A n n e a lin g
The d iffra c tio n beam p ro file s w ere recorded a t each o f th e coverages fo r
d iffe re n t in c id e n t e le c tro n beam energies.
F o r su b m o n o la ye r coverages, an
energy-dependent broadening effect on th e d iffra c tio n beams was observed, b u t the
effect was n o t seen fo r coverages o f I M L o r more. The effect was th a t some o f th e
d iffra c tio n spots w e n t fro m being sharp a nd w e ll defined to diffuse and fuzzy,and
th e n became sh a rp aga in as th e in c id e n t e le ctro n energy was va rie d .
A v is u a l
re p re se n ta tio n o f th is effect a t a coverage o f 0.4 M L is given in F ig u re 12. F ig u re
12(a) is a d iffra c tio n p a tte rn im age ta k e n w ith an electron beam energy o f 52 eV.
F igures 12(b) and 12(c) correspond to 76 and 106 eV respectively. The effect can be
seen fo r th e (0,-1) and (1,0) beams. The lo cation o f the (0,-1) beam is a t th e lo w e r
r ig h t side o f the im ages, and th e (1,0) beam is located n ea r th e top o f the images.
The sizes o f these tw o d iffra c tio n spots b o th increase from 12(a) to 12(b), and th e n
decrease again in 12(c). The effect is v is u a lly m ore obvious fo r th e (0,-1) beam. To
fu r th e r su p p o rt th is size effect, th e spot in te n s ity p ro file s fo r th e (1,0) beam are
p lo tte d in F ig u re 13. The data fo r these plots were ta ke n d ire c tly fro m the d ig itize d
im ages in F ig u re 13, and th e p ro file m a x im a have n o rm a lize d .
N o te th a t th e
39
T
1
I
1
r
Intensity (Arbitrary Units)
(2,1) beam
(-2,1) beam
r
1.4 ML
1.1 ML
0.5 ML
clean
I
50
I
I
1 00
I
I
1 50
i
I
200
■
I
250
Electron Energy (eV)
F ig u re 11: S ym m etry re te ntion betw een the (2,1) and (-2,1) beams
40
(a) 5 2 e V
(b ) 7 6 e V
(c) 1 0 6 e V
F igure 12: Energy-dependent size effect
- 6.5 -I
52 eV
76 eV
106 eV
Distance (Screen Pixels)
Figure 13: (1,0) beam profiles at difierent electron energies
42
changes in th e f u ll w id th a t h a lf-m a x im u m (F W H M ) o f th e p ro file s agree w ith
F ig u re 12, in th a t the spot size increases, th e n decreases. A n in te rp re ta tio n o f th is
effect is given in the discussion section.
A n im p o rta n t fe a tu re o f the L E E D p a tte rn its e lf was th a t th e (-1,0) and (1,0)
beams va nished as th e b is m u th coverage approached one m onolayer. The re la tiv e
in te n s itie s fo r th e ( I , 0) and (0,1) beams are p lo tte d in F ig u re 14. The in te n s ity
values used in th is p lo t were obtained b y in te g ra tin g the I V curve over the energy
range. As in d ic a te d e a rlie r, the (0,1) beam experiences an increase in in te n s ity
n e a r a coverage o f about I M L , b u t the in te n s ity o f the (1,0) beam becomes v e ry
s m a ll a t a coverage o f ab o u t I M L , a n d re m a in s s m a ll as m ore b is m u th is
deposited.
I t w ill be show n th a t th is im p lie s th e surface s tru c tu re fo r b is m u th
coverages above I
M L is p la n a r, ca u s in g d e s tru c tiv e in te rfe re n c e in these
d irectio ns.
A n n e a lin g effects were th e n considered.
The sam ple was given an in it ia l
coverage o f b is m u th and th e n subjected to a series o f h e a tin g a nd cooling cycles.
S everal d iffe re n t a n n e a lin g e xpe rim e nts w ere perform ed, w ith in it ia l coverages
ra n g in g fro m 0.4 to 2.9 M L .
In each case, th e sam ple was ann ealed fo r about
th re e m in u te s , a t te m p e ra tu re in te r v a ls o f 30-50°C, u p to 350°C.
Auger
m easurem ents w ere m ade in it ia lly and a fte r every anneal cycle to e stablish th e
a m o u n t o f desorption o ccu rrin g a t each step.
The m ost notable effect o f ann ealing was a tra n s itio n fro m the p ( lX l) L E E D
p a tte rn to a p (lx 2 ) p a tte rn , in w h ich h a lf-o rd e r spots form ed along th e k-axis o f the
L E E D p a tte rn . The tra n s itio n began a t an a n n ealing te m p e ra tu re o f about 100°C.
F o r a n n e a lin g te m p e ra tu re s above 350°C, th e s y m m e try re tu rn e d to p ( l x l ) ,
suggesting com plete desorption o f the b is m u th . F o r coverages o f 1.5 M L o r more,
th e tra n s itio n became com plete a t an a n n e a lin g te m p e ra tu re n e a r 200°C.
For
43
1 OO
■ (0,1) beam
o (1,0) beam
80
c
Z)
o
CO
60
-e
<
CZ)
C
CD
40
■§
S5
Q
■
20
6
o
°
o
O
O
O
0
0.0
0.5
1.0
1.5
2.0
Bismuth Coverage (ML)
F ig u re 14: R elative in te n sitie s o f th e (0,1) and (1,0) beams a t various b is m u th
coverages
44
su bm on olaye r coverages, th e tra n s itio n was ne ve r q u ite com pleted, b u t fa in t
strea ks w ere seen to fo rm betw een th e d iffra c tio n spots along th e k -a x is o f the
L E E D p a tte rn . The tra n s itio n to th e p (lx 2 ) sym m e try is show n in F ig u re 15. The
th re e p ictu re s in th is F ig u re correspond to d ig itiz e d im ages ta k e n w ith an in it ia l
b is m u th coverage o f 2.9 M L , and th e electron beam energy fo r each o f th e im ages
was 77 eV. The a n n e a lin g te m p era tures are la belled fo r each o f th e images. The
tra n s itio n begins w ith strea ks fo rm in g betw een the d iffra c tio n spots along th e k axis, as in d ic a te d in F ig u re 15(a). F igures 15(b) and 15(c) dem onstrate th a t as the
a n n e a lin g te m p e ra tu re is increased, th e stre a ks coalesce in to w e ll defin ed 1/2
o rder d iffra c tio n spots.
S how n in F ig u re 16 is a p lo t o f th e A E S s ig n a ls ve rsu s a n n e a lin g
te m p e ra tu re fo r th e fo u r in it ia l coverages in d ic a te d e a rlie r. The A E S sig n a l used
was the peak to peak h e ig h t o f the 96-101 eV b is m u th A u ger feature. The portions
o f the p lo t corresponding to the p ( lx l ) and p (lx 2 ) phases are la b e lle d in the Figure.
W hen th e a n n e a lin g process was begun w ith a coverage o f m ore th a n 1.5 M L o f
b is m u th , th e tra n s itio n to th e p ( lx 2 ) s y m m e try was accom panied b y b is m u th
d e so rp tio n dow n 1.5 m o n o la ye r.
F o r i n it ia l coverages less th a n 1.5 M L , no
b is m u th desorption was observed. T his is also evident fro m F ig u re 16.
D iscussion
The O rdered G ro w th o f B is m u th
As in d ic a te d e a rlie r, the g ro w th o f b is m u th up to one m onolayer appears to
be an ordered process.
One reason fo r th is conclusion is th a t i f th e g ro w th was
p u r e ly d is o rd e re d , o r ra n d o m , i t w o u ld have c o n trib u te d o n ly in c o h e re n t
s c a tte rin g and hence no increase in th e d iffra c tio n in te n s itie s w o u ld have been
observed. A n o th e r reason is th a t the changes in the fe a tu re s o f IV curves m u s t
45
(a ) 1 2 5 ° C
(b ) 1 7 5 ° C
(c) 2 7 5 ° C
F ig u re 15: F o rm ation o f h a lf-o rd e r d iffra c tio n spots
♦ 0.4 ML
■ 1.5 ML
A 2.6 ML
# 0.9 ML
Bi/GaSb(110)
200
Annealing Temperature ( °C/3min. )
F ig u re 16: B is m u th A u ger sig nal dependence on a n n e a lin g tem perature
47
have come fro m changes in the long-range o rd e r o f th e surface atom ic geom etry.
These o bse rva tions w ith g ro w th o f b is m u th in d ic a te th a t ( I ) lo n g range o rd e r
exists in the b is m u th overlayer, or (2) th e b is m u th its e lf is n o t ordered b u t induces
changes in th e geom etry o f th e substrate a t th e in terface. The p o s s ib ility th a t the
changes in I V
d a ta are due solely to b is m u th -in d u c e d re c o n s tru c tio n o f th e
su b stra te , a n d n o t fro m lo n g range o rd e r in th e b is m u th o v e rla y e r it s e lf is
discussed in C h a p te r 5. Since there are no a d d itio n a l changes in th e feature s o f
the IV curves fo r h ig h e r coverages, th e m ethod o f g ro w th beyond one m onolayer is
no lo n g e r tw o -d im e n s io n a lly ordered.
T h is la c k o f o rd e r p re ve n ts fu rth e r L E E D
analyses in to the g ro w th process above I M L .
The energy-dependent size effect discussed above fu r th e r supports these
conclusions.
A s im ila r size effect has also been observed w ith th e SbZGaAs(IlO)
system [21], and m ore re ce n tly w ith the BiZG aA s(IlO ) system [4].
T his effect has
been dem onstrated to be in d ic a tiv e o f tw o dim ensional ordered is la n d g ro w th o f the
adsorbate [22]. The absence o f th e effect a t I M L and h ig h e r coverages in d ica te s
th a t th e g ro w th process beyond I M L is no lo nger v ia tw o -d im e n s io n a l ordered
islands. T h is is consistent w ith the changes observed in th e IV curves, m entioned
above.
A fe w com m ents can be m ade ab o u t th e a tom ic s tru c tu re o f th e I M L
in te rfa c e based on th e o b se rva tio n t h a t th e (1,0) a nd (-1,0) beam s became
in c re a s in g ly s m a ll fo r th is coverage.
I f th e surface w ere to have the tru n c a te d
b u lk s tru c tu re o f th e (H O ) plane, these beams, as w e ll as a ll (h,0) (h=odd) beams,
should n o t have appreciable in te n s itie s due to an interfere nce effect [3]. Since th is
effect was observed fo r the I M L coverage, i t is suggested th a t th e stru c tu re o f th e I
M L b is m u th overla yer system m ig h t resem ble th a t o f the tru n c a te d b u lk stru ctu re .
48
T h is c o n s id e ra tio n was used in c o n s tru c tin g a th e o re tic a l m odel fo r th e IM L
atom ic s tru c tu re , w h ic h is discussed in C h a p te r 5.
The p ( l x l ) to p flx 2 ) Phase T ra n s itio n
T h e phase t r a n s itio n observed u p o n a n n e a lin g , in w h ic h th e p ( l x l )
sym m e try was replaced b y a p (lx 2 ) sym m e try is now discussed. The k-axis o f the
L E E D p a tte rn , a lo n g w h ic h th e p e rio d ic ity changed, corresponds to th e [001]
d ire ctio n on the c ry s ta l surface. The fo rm a tio n o f h a lf o rder L E E D spots along th is
d ire c tio n in d ic a te s th e fo rm a tio n o f a surface s tru c tu re w ith lo ng-ra nge order,
th a t has a p e rio d ic ity o f tw ic e
d ire c tio n .
th e u n it cell o f th e s u b s tra te along the [001]
The g ra d u a l tra n s itio n o f th e L E E D p a tte rn s y m m e try in d ic a te s a
continuous change in th e re -o rd e rin g o f th e surface atoms. S im ila r fo rm ation s o f
s u p e rstru ctu re s have been observed w ith b is m u th on o th e r I II - V ( I lO ) substrates
[23].
49
C H A P TE R 5
S T R U C TU R E A N A LY S E S
Propram O verview
Searches w e re c a rrie d b u t fo r tw o d iffe r e n t s tru c tu re s , th a t o f th e
re c o n s tru c te d cle an surface, and th e I M L p ( l x l ) b is m u th o v e rla ye r.
T he
param eters used in these tw o searches consisted o f the bond lengths, bond ro ta tio n
angles, and th e re a l and im a g in a ry p a rts o f th e o p tica l p o te n tia l. The ju s tific a tio n
fo r u s in g these param eters w ill become e vid e n t in th e discussion section o f th is
C hapter, w here i t w ill be show n th a t th e use o f these param eters lead successfully
to the d e te rm in a tio n o f th e reconstru cted clean surface s tru c tu re , w h ic h was in
agreem ent w ith the re su lts obtained by D uke, et.al. [20].
A general o u tlin e o f th e procedures used in th e B iZ G aS b(IlO ) s tru c tu re
searches is presented in F ig u re 17. The f ir s t step is the c a lc u la tio n o f the atom ic
p o te n tia ls. T h is was accom plished b y a s e lf consistent ca lcu la tio n , w h ic h s ta rte d
w ith th e s p e c ific a tio n o f an in it ia l charge d e n sity.
A n a to m ic p o te n tia l was
co n stru cte d fro m th e charge d e n sity, th e n th e r e la tiv is tic H a rtre e -F o c k -S la te r
e q u a tio n [9] was solved fo r th e e le ctro n w ave fu n c tio n s , u s in g a local S la te r
exchange [13].
The w ave fu n c tio n s w ere th e n used to ca lcu la te a new charge
d e n sity, and th e process was co n tin ued u n t il convergence c r ite ria were m et, i.e.
the p o te n tia l and charge d e n sity were consistent w ith each other.
50
Compute
atomic
potentials
Compute
crystal
potentials
Compute
scattering
phaseshifts
Define model
parameters
Compute
structure file
Calculate
LEED intensities
Find
r-f actor
Has >
termination
criteria
been
met? j
Change model
parameters according
to simplex algorithm
Figure 17: Flowchart diagram for structure search procedures
51
The c ry s ta l p o te n tia l was th e n com puted by o ve rla p p in g atom ic p o te n tia ls
in a m a n n e r described in C h apter 2. The atom ic p o te n tia ls and a cry s ta l data file ,
w h ic h specifies th e geom etry o f the ,crystal and its atom ic com position were used
as in p u t fo r th is step. F o r th e I M L p ( l x l ) overla yer system , tw o separate c ry s ta l
p o te n tia ls were calculated, one fo r the overlayer, and one fo r th e substrate.
The s c a tte rin g phaseshifts were th e n obtained fro m th e c ry s ta l p o te n tia l,
u s in g th e H a ra m odel fo r th e exchange p o te n tia l. The phaseshifts were calculated
fo r sca tte rin g fro m each atom ic species in th e crysta l. F o r th e clean surface o n ly
tw o sets o f phaseshifts were needed, b u t i f sca tte rin g fro m th e b is m u th o ve rla ye r
was to be in clu d e d , th e n th re e sets o f phaseshifts were needed.
The phaseshifts
w e re c a lc u la te d fo r Z-values up to Z=6, and are show n in F ig u re 18.
T he
p h a se sh ifts fo r h ig h e r a n g u la r m o m e n tu m states w ere v e ry s m a ll, and hence
re la tiv e ly in s ig n ific a n t in co m p u tin g th e sc a tte rin g in te n s itie s . T h is is som ew hat
e vid e n t fro m F ig u re 18, w here the phaseshifts fo r Z=5 and Z=6 re m a in close to zero
over m ost o f th e energy range.
S c a tte rin g cross-section versus in c id e n t energy
plots were constructed fro m the phaseshifts in F ig u re 18, and are shown in F ig u re
19.
The dashed lin e s in these plots rep re se n t th e to ta l s c a tte rin g cross section,
ob ta in e d fro m th e in d iv id u a l cross sections show n w ith so lid lines.
These p lo ts
provide a m ore in tu itiv e p ic tu re fo r the s c a tte rin g dependence on in c id e n t energy.
M ost o f th e sc a tte rin g is seen to occur fo r th e energy in te r v a l 100-130 eV, fo r a ll
th re e species.
The n e x t step is to specify th e in it ia l m odel p a ram eters, such as surface
bo n d le n g th s , bon d ro ta tio n angles, etc.
T h e same p h a s e s h ifts w ere used
th ro u g h o u t th e search, even as th e ge o m e tric p a ra m e te rs are v a rie d , w h ic h
in tro d u c e d some e rro r because th e p h a se sh ifts w ere c a lc u la te d fo r a specific
52
O
10O
200
300
Electron Energy (eV)
Figure 18: Scattering phaseshifts for (a) antimony, (b) gallium, and (c) bismuth
53
Antimony
Gallium
0.0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 300 0.0 30.0 60.0
90.0 120.0 150.0 WC
Energy (eV)
Energy (eV)
Bismuth
0.0 30.0 60.0
90.0 120.0 150.0 180.0 210.0 240.0 270.0 300.0
Energy (eV)
Figure 19: Scattering cross-sections for (a) anitmony, (b) gallium, and (c) bismuth
54
geom etry. The e rro r should have been sm all, however, a nd th is m ethod has been
successfully u tiliz e d in w o rk s im ila r to th a t presented here [3,9].
The energy
range and th e d iffra c tio n beam in dice s w ere also specified a t th is stage.
T h is
in fo rm a tio n was th e n read in to a stru ctu re file , w h ich is a da ta file th a t specifies
the m odel geom etry, energy range, beam indices, in n e r p o te n tia l, etc. o f the m odel
in a s ta n d a rd fo rm a t, w h ic h can th e n be re a d b y th e m a in p ro g ra m d y n le e d ,
described next.
The s tru c tu re file , together w ith the pha seshift file s com prised the in p u t fo r
th e m a in p ro g ra m , ca lle d d y n le e d .
D yn le e d was used to com pute th e L E E D
in te n s itie s fo r th e d ire ctio n s defined by th e Laue conditions, e q u a tio n (24) a t the
energy values specified in th e s tru c tu re file .
The program was w ritte n to u tiliz e
the T -m a trix m ethod m entioned in C h apter 2.
The re s u ltin g IV data was com pared to th e e xp e rim e n ta l d a ta by the x -ra y
r-fa c to r m ethod [9], o rig in a lly developed fo r use w ith x -ra y d iffra c tio n studies. The
r-fa c to r m ethod compares tw o sets o f curves, x e and y e, such th a t
Z (%e- C i Y e f
(47)
ii
w h e re r i is th e r-fa c to r fo r th e i^h p a ir o f IV curves, a n d th e s u m m a tio n is
perform ed over the discrete energy value in d e x, e. ci is a n o rm a liz a tio n constant
defined by
E x 6Ye
Ci
(48)
55
These sum m a tions are perform ed fo r each o f th e p a irs o f IV curves, and th e n an
o v e ra ll r-fa c to r is d eterm ined fro m
N
R
2
3
X ri Ae
3 J_ i=l
2N
(49)
i=l
In th is e q u a tio n , N is th e n u m b e r o f p a irs o f IV curves, and A e i is th e energy
range fo r th e i ^ 1 beam. T h is equation puts an emphasis on th e size o f the data sets
b e in g com pared and is, essen tially, ta k in g th e average o f th e in d iv id u a l r-fa c to rs
rp
The re s u ltin g r-fa c to r R describes th e degree to w h ic h th e tw o sets o f curves
m atch .
A lo w e r r-fa c to r m eans th e proposed m odel provides a b e tte r f i t to th e
e x p e rim e n ta l data.
The m ethod o f v a ry in g th e param eters in searching fo r a best f i t s tru c tu re
was based on th e sim p le x a lg o rith m (see A p p e n d ix A). W ith th is m ethod, th e rfa c to r is used as th e c o n tro llin g fa c to r in th e search, and th e sim p le x m e th o d
converges on a m odel, i.e. th e set o f o p tim a l p aram eters,' w ith th e m in im u m r facto r.
The search was s ta rte d several tim e s fo r each proposed s tru c tu re w ith
d iffe re n t in it ia l sets o f param eters to ensure th a t a s tru c tu re was found th a t gave
th e absolute m in im u m r-fa cto r. C rite ria fo r deciding how lo w an r-fa c to r m u s t be
has been discussed b y D u ke [20].
In general, the L E E D s tru c tu re analyses are
used in c o lla b o ra tio n w ith o th e r surface se n sitive te ch n iq u e s, such as ang leresolved photoem ission fo r c o n firm a tio n o f th e re su lts [3,7,8,9].
56
D iscussion
G a S b fl 10) Clean Surface Geom etry
U s in g the m ethod described above, fo u r d iffe re n t searches were in itia te d fo r
the re co n stru cte d clean surface s tru c tu re , each s ta rtin g w ith a d iffe re n t set o f
in it ia l param eters. The d iffra c tio n in te n s itie s were com puted fo r sca tte rin g fro m
the fir s t fo u r layers o f the m odel, and the s c a tte rin g from the fir s t three layers was
com puted exactly. S ix phaseshifts were used fo r each atom ic species. The re su lts
o f these searches were in good agreem ent w ith the re su lts p re vio u sly obtained by
D uke, et.al. [20]. The bond le n g th param eters converged to values w ith in 1% o f the
u n re co n stru cte d le ngths, and the top la y e r ro ta tio n angle converged to 30° ± 1°,
such th a t the top la y e r anion (a n tim o n y) was ro ta te d away fro m th e surface and
the top la y e r cation (g a lliu m ) to w a rd the surface. Shown in F ig u re 20, is a side
vie w o f the (H O ) surface w ith the bond lengths labelled as A, B, C, and the top la ye r
ro ta tio n angle is la belled as coi. T his bond le ngth-conserving ro ta tio n is id e n tic a l
to th a t observed by D uke, et.al. [20]. W hen the values fo r the re a l and im a g in a ry
S anion
O cation
F ig u re 20: Side view o f the I II - V ( I lO ) surface, show ing the
bond lengths A, B, C, and the top la yer ro ta tio n angle coi.
57
p a rts o f th e in n e r p o te n tia l were also used as param eters, th e y converged to values
o f 10.1 eV fo r th e re a l p a rt and 4.5 eV fo r th e im a g in a ry p a rt, w h ic h also agrees
w ith previous re su lts [20].
In each o f th e fo u r searches, the r-fa c to r values converged n ea r 0.27, w h ich
is som ew hat h ig h e r th a n w h a t was p re vio u sly obtained [20]. I t took an average o f
25 cycles fo r b o th th e r-fa c to rs and th e p a ra m e te rs them selves to converge to
w ith in 1%.
Since a sin gle cycle o f th e sim p le x loop re q u ire d abo ut one h o u r o f
co m p u te r C P U tim e , th e clean surface s tru c tu re search w as n o t p u rsu e d any
fu rth e r .
I t w ill be show n in A p p e n d ix A th a t th e s im p le x m e th o d converges
q u ic k ly to th e neighborhood o f the fin a l values, b u t th e n u m b e r o f re q u ire d cycles
increases ra p id ly fo r a tig h te r convergence.
T h e set o f I V curves fo r th e converged s tru c tu re are com pared to th e
e x p e rim e n ta l curves in F ig u re 21.
The e x p e rim e n ta l d a ta are dep icted w ith
dashed lin e s and s o lid lin e s are used fo r th e proposed m odel data.
There are
s lig h t discrepancies betw een the data sets, b u t these re s u lts c le a rly in d ic a te th e
v a lid ity o f u s in g th e sim plex m ethod fo r L E E D s tru c tu re analyses.
W ith th is in
m in d , th e same approach was used in a n a lyzin g the surface s tru c tu re associated
w ith th e I M L p ( lx l ) stru ctu re .
The I M L p ( l x l ) S tru ctu re
A nalyses fo r tw o d iffe re n t I M L p ( l x l ) m odel s tru c tu re s were a tte m pted,
w h ic h b o th fa ile d to converge w ith in good agreem ent w ith th e e xp e rim e n ta l data.
I t th u s appears th a t these tw o m odels are b o th in c o rre c t fo r th is p a r tic u la r
s tru c tu re , b u t th e re s u lts o f these calcula tions are s t ill in fo rm a tiv e and im p o rta n t
in th a t these tw o m odels can be ru le d out.
The f ir s t m odel w h ic h I c a ll the
disordered m odel, assumes no long range o rder in the b is m u th overlayer. In ste a d
170.0
233.0
300.0
170.0
233.0
F ig u re 21: Com parison o f calculated and expe rim e nta l IV curves fo r the clean
G a S b (IlO ) surface. Dashed lines correspond to experim ental data and the solid
lin e s represent th e o re tica l data.
300.0
59
i t assumes th a t th e changes in th e IV da ta in d ic a te d in F ig u re 10 are due to
b ism u th -in d u ce d re co n stru ctio n o f th e GaSb substrate, w ith no c o n trib u tio n to the
I V ch a ra cte ristics fro m th e b is m u th except fo r an o ve ra ll decrease in in te n s ity due
to in c o h e re n t s c a tte rin g .
Hence, no b is m u th p h a se sh ifts w e re used in th e
analysis. A s fo r th e clean surface analysis, fo u r d iffe re n t sim plex-based searches
w ere m ade.
I n each case th e sim p le x search converged w ith an r-fa c to r va lue
betw een .43 a n d .46, and upon a d iffe re n t set o f p a ra m e te rs each tim e .
The
converged geom etric stru ctu re s were w id e ly d iffe re n t, as m uch as ±20° fo r th e top
la y e r bond ro ta tio n angle and 12% v a ria tio n fo r th e bond lengths. The convergence
o f the r-fa c to r a t such h ig h values, together w ith the fa ct th a t an e n tire ly d iffe re n t
geom etric s tru c tu re was reached in each case suggests th a t the disordered m odel
is in a p p ro p ria te fo r th is stru ctu re .
The second m odel used in th e I M L p ( l x l ) ana lysis was one in w h ic h th e
b is m u th o ve rla ye r atom s occupy the va ca n t anion and ca tio n sites a t the surface,
and th e s tru c tu re a t th e surface resem bles th a t o f th e tru n c a te d b u lk s tru c tu re .
The e x p e rim e n ta l re su lts in d ic a te d th a t th is was a favorable m odel, as m entioned
in C h a p te r 4. T h is m odel is ofte n called th e G oddard m odel [25], and the I M L
S bZG aA s(IlO ) system has been show n to have th is s tru c tu re [21].
Since th e
geom etry o f th is a rra n g e m e n t is som ewhat o f an extension o f th e tru n c a te d b u lk
s tru c tu re , th e same geo m e tric p a ra m e te rs w ere used as in th e clean surface
a n a ly s is , see F ig u re 20.
The b o n d in g c o n fig u ra tio n o f th is a rra n g e m e n t is
discussed b y S ke a th e t.a l [25].
A g ain , fo u r d iffe re n t searches w ere made.
The
re su lts were s im ila r to those obtained fo r the disordered m odel, in th a t th e r-fa c to r
values converged a t h ig h values (.41-.50), and the converged geom etric param eters
corresponded q u ite d iffe re n t stru ctu re s in each case. A n average o f 24 cycles were
com pleted in th e sim p le x loop fo r th e search u s in g th is m odel.
These re s u lts
60
in d ic a te , as w ith th e d iso rd e re d m odel, th a t th is m odel is in a p p ro p ria te fo r
describ ing th e I M L p ( l x l ) stru c tu re , w h ic h consequently m u s t be q u ite d iffe re n t
fro m th e tru n c a te d b u lk stru ctu re .
61
C H A P TE R 6
S U M M A R Y A N D C O N C LU S IO N S
F ro m th e re s u lts o f the e xp e rim e n ta l observations presented in C h a p te r 4,
th e pow er and u t ilit y o f surface analysis w ith L E E D has become evident. Several
conclusions ab o u t th e q u a lita tiv e c h a ra c te ris tic s o f th e g ro w th and k in e tic s o f
b is m u th on G a S b (IlO ) were reached.
C u rre n tly , developm ents are being made
to w a rd m ore q u a n tita tiv e approaches to th e analysis o f the e xp e rim e n ta l data. F o r
exam ple, a co llab ora tive e ffo rt is c u rre n tly in progress w ith m em bers o f the M S U
c h e m is try d e p a rtm e n t fo r a m ore rig o ro u s a n d s y s te m a tic a n a ly s is o f th e
d iffra c tio n spot p ro file s, w h ic h w ill lead to a b e tte r u n d e rs ta n d in g o f th e k in e tic s
in vo lve d in the g ro w th and a n n ealing processes.
The re s u lts o f th e clean surface ca lc u la tio n s in d ic a te d th a t th e sim p le x
a lg o rith m is q u ite adequate fo r m in im iz in g th e r-fa c to r and fin d in g th e correct
m odel param eters. T h is fu rth e r supports th e conclusion in A p p e n d ix A , th a t the
s im p le x a lg o rith m can be su cce ssfully used in th e d e te rm in a tio n o f a to m ic
stru c tu re s w ith L E E D calculations.
The e ffo rt to determ ine the stru ctu re s o f th e b is m u th o ve rla ye r systems is
being contin ued and developed fu rth e r a t th is tim e. C u rre n tly , w o rk is being done
in
d e v e lo p in g o th e r m odels fo r th e I
M L p ( l x l ) s tru c tu re , based on th e
considerations o f the e xp e rim e n ta l observations fo r th is and s im ila r systems, and
on th e re s u lts o f s im ila r s tru c tu re searches [26].
62
A P P E N D IC E S
A P P E N D IX A
S IM P L E X A L G O R IT H M
The sim plex a lg o rith m , fir s t proposed b y N e ld e r and M eade in 1965 [27], is a
m in im iz a tio n ro u tin e fo r m u ltiv a r ia te fu n c tio n s based on geom etric p rin c ip le s .
A d va n ta g e s o f u s in g th is a lg o rith m are th a t i t does n o t re q u ire n u m e ric a l
calculus o r m a tr ix m a n ip u la tio n s , and th a t i t m akes no use o f one d im e n sio n a l
m in im iz a tio n techniques.
One disadvantage is th a t i t is n o t v e ry e ffic ie n t in the
n u m b e r o f fu n c tio n e valuatio ns th a t i t requires.
A d e scrip tio n o f th e a lg o rith m ,
and how i t was in co rp o ra te d in to th e s tru c tu re search process is presented below.
A sim p le x is defined as the geom etrical fig u re consisting, in N dim ensions,
o f N + l vertices.
C onnecting lin e s and polygonal faces are considered p a rt o f th e
sim plex. I n tw o dim ensions, a sim plex is a tria n g le , and in th re e dim ensions i t is
a te tra h e d ro n . A necessary re s tric tio n on the sim plex is th a t i t be non-degenerate,
i.e. i t m u s t co ntain a fin ite volum e o f d im ension N . W ith o u t th is re quire m en t, the
sim p le x w o u ld be confined to a space o f dim en sion less th a n N , and th e fu n c tio n
p a ram eters w o u ld be s im ila rly constrained.
F o r th e m in im iz a tio n o f fu n c tio n o f N param eters, an in it ia l sim p le x is
constructed b y a r b itr a r ily choosing N + l sets o f fu n c tio n param eters. These sets o f
p a ram eters th e n define th e in it ia l vertices. The response o f th e fu n c tio n is found
fo r each o f th e vertices, and a search is begun fo r a p a ra m e te r set w h ich gives the
lo w e st fu n c tio n va lu e .
T h is is achieved b y "m oving " th e sim p le x th ro u g h th e
space d e fin e d b y th e fu n c tio n p a ra m e te rs, th ro u g h a series o f o p e ra tio n s and
64
d e cisio n s w h ic h exchange old vertices fo r new ones th a t give b e tte r, i.e. low er,
fu n c tio n response values.
The f ir s t steps in "m oving" th e sim p le x are to d e te rm in e th e best, w o rst,
and second w o rs t vertices, and th e n to fin d th e centroid o f th e hype rp lane defined
b y a ll th e ve rtices except th e w o rst.
T h e n one o f 4 operations is perform ed, as
show n in F ig u re 22 fo r th e tw o d im e n sio n a l case.
The boldface tria n g le s in th e
F ig u re rep re se n t th e in it ia l sim plexes, and the new vertices fo u n d b y each o f the
operations are la b e lle d accordingly. Associated w ith each op e ra tio n is a coefficient
a, w h ic h represents th e le n g th ra tio defined in th e Figure. These coefficients are
decided upon before th e m in im iz a tio n ro u tin e is started. The choice o f values fo r
these coefficients is a rb itra ry , and u s u a lly based on in tu itio n and experience. The
va lu e s used in th e p ro g ra m described in C h a p te r 5 are:
a r = l, a e=2, and
(Xq—CCg -i/2 .
The sim p le x d e cisio n-m aking a lg o rith m is depicted in a flo w c h a rt fo rm a t
in F ig u re 23.
H ere, a p roced ura l o u tlin e is given fo r deciding w here to lo o k fo r
new ve rtice s, and w hen to exchange o ld ve rtices fo r new ones.
n w o rs t represents th e second w o rs t verte x.
I n th is F ig u re ,
N ote th a t th e v e rte x fo u n d b y th e
re fle c tio n o p e ra tio n is n o t k e p t unless th e fu n c tio n response is b e tte r th a n th e
second w o rs t value. T h is is because th e n e w ly acquired v e rte x w o u ld th e n be the
w o rs t ve rte x, and th e n e x t ope ra tion w o u ld be a re fle c tio n b a ck to th e previo us
w o rs t ve rte x, re q u irin g a re p e titio u s fu n c tio n evaluatio n.
E xam ples o f com puter
program s u tiliz in g th e sim plex a lg o rith m can be found in references [28] and [29].
The sim p le x a lg o rith m was in c o rp o ra te d in to th e B iZG aS b(IlO ) s tru c tu re
searches b y u s in g m odel param eters such as bond lengths, bond ro ta tio n angles,
Figure 22: Sim plex operations in two dimensions
66
Read input data
Create N vertices and rank
according to response value
Find reflected vertex
r response^
at expanded
. vertex <
V n w o rs V y
r responses
at reflected
k vertex < = j
Find expanded
v e rte x
Xbest’X
Find contracted
ve rte x
^ re s p o n s e \
at contracted
k vertex < a
X .w o r s t? X
response >
at expanded
^vertex <=y
X .b e s t ’ X ^
Accept reflected
vertex and reject
w orst vertex
Accept contracted
vertex and reject
worst vertex
Accept expanded
vertex and reject
worst vertex
Contract all but
best vertex toward
best vertex
X
Has
X
term ination'
criteria been
Vreached^y
Figure 23: F low chart for the Simplex A lg o rith m
67
and th e in n e r p o te n tia l as th e fu n c tio n param eters, and le ttin g th e r-fa c to r fo r a
given m odel ca lcu la tio n represent the fu n c tio n response. The a p p lic a b ility o f th is
approach was tested w ith a t r ia l s tru c tu re fo r several d iffe re n t s ta rtin g sim plexes,
fo r b o th corre ct and in c o rre c t s tru c tu ra l models. The e x p e rim e n ta l d a ta used in
these te s ts w ere th e clean G a S b (IlO ) I V d a ta , fo r w h ic h th e co rre ct a to m ic
s tru c tu re has been determ ined [20].
The convergence o f th e r-fa c to r, fo r these tests is show n in F ig u re 24(a).
The re s u lts fo r s ix d iffe re n t searches are show n here, w ith th e s o lid lin e s
corresponding to th e correct m odel, and th e dashed lin e s re p re s e n tin g th e use o f
an in c o rre c t m odel. W hen the correct m odel was used in th e search, the r-fa cto rs
converged v e ry near to the same value a ll th re e tim es, b u t th e use o f an in co rre ct
m odel caused th e r-fa c to rs to converge to q u ite d iffe re n t values fo r each search.
F o r the searches made w ith the correct m odel, about 20-30 cycles w ere re q u ire d fo r
th e r-fa c to r to converge to w ith in a c o m p a ra tive ly s m a ll d e v ia tio n fro m th e fin a l
value. A d d itio n a l convergence o f th e r-fa c to r values re q u ire d m a n y m ore cycles,
w h ic h is e vid e n t fro m th e F igure.
A com parison o f th e r-fa c to r convergence to th e convergence o f th e m odel
pa ra m e te rs fo r one search is show n in F ig u re 24(b). The dashed lin e represents
the percent sta n d a rd d e via tio n o f the r-fa c to r value, and th e solid lines correspond
to th e m odel param eters. F o r th is p a rtic u la r search, the param eters converged at
a p p ro xim a te ly the same ra te as the r-factor. T h is is representative o f a ll o f the te st
searches, even w hen the in c o rre c t m odel was used.
In th is F ig u re , a reasonable
convergence was reached a fte r about 30 cycles o f the sim p le x loop, and a d d itio n a l
convergence was co m p a ra tive ly slow. The o th e r te st searches converged s im ila rly
a fte r 20-30 cycles.
68
The re s u lts o f these te s t searches in d ic a te th a t th e sim plex-based search
converges q u ic k ly u p to 20-30 cycles, a n d m u ch slow er th e re a fte r.
T h e y also
in d ic a te th a t w hen th e correct m odel was used in th e search, th e sim plex search
converged to w a rd th e same s tru c tu re each tim e , and th a t th e use o f an in co rre ct
m odel led to d iffe re n t stru ctu re s each tim e . Based on these re su lts, the sim plexbased search m ethod appears to have m e rit fo r use in L E E D s tru c tu re analyses, in
th a t a d is c rim in a tio n can be m ade b etw e en co rre ct a n d in c o rre c t s tru c tu r a l
m odels.
69
F ig u re 24: Results o f s im p le x convergence tests, (a) Average r-fa c to r values, (b)
Convergence o f m odel p a ra m e te rs com pared to r-factor convergence.
70
A P P E N D IX B
V ID E O C A M E R A T IM IN G C IR C U IT
I p re se n t in th is section a discussion o f th e tim in g c ir c u it designed to
fa c ilita te u s in g the video cam era in the ra n d o m -in te g ra tio n mode, b u t f ir s t some
fu n d a m e n ta l features o f th e video sig nal its e lf are presented.
The o u tp u t sig n a l fro m th e video cam era follow s th e s ta n d a rd in te rla c e d
video fo rm a t w here one fram e, or com plete im age, is composed o f an odd a nd an
even fie ld . The fields correspond to th e h o riz o n ta l lines o f th e video signal. The
odd fie ld contains th e video sig n a l fo r th e odd-num bered lin e s and th e even fie ld
contains th e even-num bered lines.
The tw o fie ld s to g e th e r fo rm th e in te rla c e d
fram e. F ig u re 25(a) shows the cam era o u tp u t w hen used in th e n o rm a l op e ra tin g
mode.
I t is composed o f a lte rn a tin g odd a n d even fie ld s, each separated b y a
v e rtic a l sync.
The v e rtic a l syncs occur w ith a frequency o f 60 H z so a com plete
fra m e is tra n s fe rre d every 1 /3 0 ^ o f a second. As seen in th e F ig u re , the video
o u tp u t u tiliz e s a I V peak-to-peak tim in g signal. W ith in th e cam era a single CCD
a rra y is used to m easure the lig h t sig nal fo r bo th fields. In th e n o rm a l o p e ra tin g
mode, th e C C D a rra y reads th e lig h t s ig n a l fo r 1 /6 0 ^ second fo r each fie ld ,
tra n s fe rs th e fie ld signal, th e n reads th e sig n a l fo r the n e x t fie ld , and so on. T h is
p a r tic u la r cam era (P u ln ix M odel T M -8 4 0 N ) is equipped fo r ra n d o m -in te g ra tio n
o u tp u t, in w h ic h th e CCD a rra y can be in s tru c te d to read th e lig h t sig n a l fo r any
desired le n g th o f tim e (longer th a n l/6 0 th sec). T h is fe a tu re is use fu l in lo w -lig h t
Tl
Alternating
odd and even
output fields
(a)
A
::nnnnnnn
\
Normal
output
signal
Vertical sync
-------- >
Time
Normal output
Integrated field output
+1 V
0V
(b)
+5 V
Integration
control
signal
0V
Figure 25: Video camera output signals
72
a p p lic a tio n s , w h e re an accu m u la te d , i.e.
in te g ra te d s ig n a l is necessary fo r
adequate se n s itiv ity .
In o rd e r to use th e in te g ra tio n c o n tro l feature , an e x te rn a l T T L o r CM OS
b in a ry s ig n a l m u s t be su p p lie d to a p in connection on th e b ack o f th e cam era.
W hen th is sig n a l is h e ld low (0 V olts), th e o u tp u t from th e CCD a rra y is in h ib ite d .
W hen th e sig n a l is sw itched to h ig h (+5 V o lts), o u tp u t begins a t the n e x t v e rtic a l
sync. T his is show n in F ig u re 25(b). N ote th a t the cam era o u tp u t always contains
th e basic tim in g sig nal, and o n ly th e superposed lig h t sig n a l is affected by u s in g
th e in te g ra tio n control. A lso note th a t w h e n u s in g the in te g ra tio n control, o n ly
one fie ld can be in te g ra te d a t a tim e, and the in te g ra te d fields are separated by the
tim e o f in te g ra tio n .
T he im p o rta n ce o f these fe ature s w i ll become obvious in
discussing how th e tim in g c irc u it was constructed.
The Q u ic k C a ptu re ™ fra m e g ra b b e r boa rd w h ic h is used to d ig itiz e th e
video sig n a l is designed to read one com plete fram e, or tw o consecutive fields, a t a
tim e. B u t w hen the cam era is used in th e in te g ra tin g mode, th e in te g ra te d fields
do n o t appear consecutively.
R a th e r th a n tr y in g to separate and recom bine th e
in te g ra te d o u tp u t s ig n a l, an a lte rn a te approach was used w h ic h is described
below.
The d ig itiz in g boa rd has th e p ro v is io n fo r an e x te rn a l trig g e r, to in itia te
d ig itiz a tio n . Use o f the e x te rn a l trig g e r require s a T T L sig n a l also.
D ig itiz a tio n
th e n begins a fte r th e v e rtic a l sync w h ic h fo llo w s a lo w to h ig h tra n s itio n o f th e
e x te rn a l trig g e r connection.
So w h en th e e x te rn a l tr ig g e r is p ro p e rly used,
d ig itiz a tio n can be in itia te d fo r each o f th e fram es co n ta in in g th e in te g ra te d fields.
Each o f these fram es contains o n ly one in te g ra te d fie ld , so w hen th e y are d ig itiz e d
and read in to m em ory, th e y m u st be added together to m ake a com plete in te g ra te d
im age. Presented n e x t is the T T L c irc u it developed fo r these applications.
73
A schem atic d e scrip tio n o f th e c irc u it is presented in F ig u re 26. The base
tim in g sig n a l fro m th e cam era is used to provid e th e in it ia l signal. The I V sig nal
is adequate fo r use in T T L c irc u its .
T h is sig n a l is passed th ro u g h a series o f
tim in g c irc u its , w h ic h change th e tim in g o f th e sig n a l to a llo w fo r in te g ra tio n
tim e s corre spond ing to an in te g ra l n u m b e r o f fram es, i.e. m u ltip le s o f 1/3O^h o f a
second. The design o f the c irc u it allow s fo r in te g ra tio n tim es o f 33, 66, 99, 233, and
363 microseconds. The cam era o u tp u t re s u ltin g fro m the use o f th is tim in g c irc u it
is show n in F ig u re 27. A lso show n in F ig u re 27 is the sig n a l to be used fo r the
e x te rn a l trig g e r, w h ic h is the same as th e in te g ra tio n c o n tro l sig nal. The tim in g
o f th is s ig n a l is such th a t o n ly the fram es c o n ta in in g th e in te g ra te d fie ld s are
d ig itize d .
T h is tim in g c irc u it represents th e f ir s t step to w a rd u s in g th e video cam era
in th e in te g ra tin g mode.
C om plete in c o rp o ra tio n o f th is c ir c u it w i ll re q u ire
fu rth e r re fin e m e n ts in th e da ta a cq u isitio n softw are so th e in te g ra te d fram es can
be com bined and stored correctly.
74
Video
camera
Integration control
<r
Video output
Video input < Frame
grabber
board
External trigger e—
Comp sync out
Sync
separator
33 msec
66 msec
Individual circuits
for specified
integration
tim es
99 msec
233 msec
363 msec
Figure 26: Block diagram for the video camera tim in g c irc u it
75
Integrated video field signal
even
odd
even
odd
even
odd
vA
e ven
Odd/even field
designation
External trigger
//
/Z j
Figure 27: Integrated o utput and external trig g e r tim in g
76
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