Computation of surface states on bcc transition metals
by Mizuho Kawajiri
A thesis submitted in partial fulfillment of the requirement for the degree of DOCTOR OF
PHILOSOPHY in Physics
Montana State University
© Copyright by Mizuho Kawajiri (1978)
Abstract:
The resolvent method Is combined with a tight-binding model in order to obtain layer-by-layer
densities of states at low index surfaces of semi-infinite bcc transition metals. Shockley-type localized
surface states or resonances are found in all hybridization gaps in the three symmetry directions
studied. Prominent structures in the angle-resolved polarization dependent photoemission spectra are
discussed in the light of the present calculation. in assigning peaks of an observed EDC to calculated
initial state features, a knowledge of the final state symmetry and selection rules is extremely
important. ©
1978
M I Z U H O KAWAJ I R l
ALL RIGHTS RESERVED .
. COMPUTATION'
ON BCC TRANSITION
of surface st a t e s
by
MIZUHO KAWAJI RI
A t h e s i s s u b m i t te d i n p a r t i a l f u l f i l l m e n t
o f t h e r e q u ir e m e n t f o r t h e d e g re e
o£
'
.
DOCTOR OF PHILOSOPHY
In
.
P h y s ic s
A p p ro v ed :
. MONTANA STATE UNIVERSITY
Bozem an, M ontana
M arch , 1978
metals
ill
ACKNOWLEDGEMENT
I am v e r y g r a t e f u l t o my t h e s i s a d v i s o r . D r. Jo h n H erm anson, f o r
s u g g e s t i n g t h i s p ro b le m a n d f o r many h e l p f u l d i s c u s s i o n s th r o u g h o u t
t h i s w o rk .
W ith o u t h i s p a t i e n c e and e f f i c i e n t and t im e l y a d v ic e t h i s
w ork w o u ld n o t h a v e b e e n c o m p le te d .
A ls o I w o u ld l i k e t o e x p r e s s my
th a n k s t o D r. L a p e y r e 's g ro u p f o r p r o v i d i n g us w i t h u n p u b lis h e d d a t a
and o t h e r v a r i a b l e i n f o r m a t i o n .
In p a r t i c u l a r ,
I e x te n d my s p e c i a l
th a n k s t o D r. Jam es A n d e rso n w ho, a t t h e e a r l i e s t p o s s i b l e c h a n c e ,
p e rf o r m e d an e x p e r im e n t s u g g e s t e d b y p r e l i m i n a r y r e s u l t s o f o u r w o rk .
I w o u ld a l s o l i k e t o th a n k D r. H erm an so n , D r. A n d e rso n and o t h e r
members o f my t h e s i s c o m m itte e f o r c o n s c i e n t i o u s p r o o f r e a d i n g o f
th is
th e s is .
D a v id S tro m , when h e w as i n h i s s e n i o r y e a r a t Bozeman
H igh S c h o o l, h e lp e d w i t h i m p o r ta n t p a r t s o f o u r c o m p u te r p ro g ram m in g .
My h u s b a n d . B i l l Schwalm h a s b e e n a f a i t h f u l c o -w o rk e r and h a s b r o u g h t
up num erous i n t e r e s t i n g i d e a s .
The f i n a n c i a l s u p p o r t o f t h e N a t i o n a l
S c ie n c e F o u n d a tio n an d M ontana S t a t e U n i v e r s i t y a r e d e e p ly ack n o w led g e d
TABLE OF CONTENTS
LIST OF TA B L E S................................................................................................................................ v i
LIST OF F I G U R E S ...................................................................................................................
v ii
ABSTRACT.......................................................................................................................................
Ix
C h a p te r
I.
P age
INTRODUCTION
A.
. ................................................. ' .............................................
P r e v io u s S t u d i e s o f W an d Mo Low In d e x F a c e s
. . . .
(0 0 1 ) f a c e ..................................................... ......................................
(H O ) and (1 1 1 ) f a c e ..............................................
II.
III.
4
4
10
PHOTOEMISSION SPECTROSCOPY AND THE INTERPRETATION OF
ELECTRON ENERGY DISTRIBUTION CURVES . . . . . . . . . . .
13
A.
..........................................................
13
M e asu re m e n t o f e n e r g y .......................
M easu rem en t o f s u r f a c e w ave v e c t o r
..................................
S e l e c t i o n o f p o l a r i z a t i o n ....................................................
13
15
15
P h o to e m is s io n S p e c tr o s c o p y
B.
E n e rg y D i s t r i b u t i o n C urve . . . .
C.
S e l e c t i o n R u le s
............................................
fo rT r a n s itio n P ro c e ss
CALCULATION OFDENSITY OFSTATES
A.
. . . . . . .
. ......................................................
B u lk Band S t r u c t u r e and S l a t e r - K o s t e r P a r a m e te r s
B.
C.
. .
L o c a li z e d D e n s i ty o f S t a t e s i n te rm s o f R e s o l v e n t
M e t h o d ...............................................
R e l a t i o n b e tw e e n th e B u lk a n d S u r f a c e R e s o lv e n ts
17
22
B l o c h 's t h e o r e m .................................................
Group o f t h e wave v e c t o r ..........................................................
S l a t e r - K o s t e r p a r a m e t e r s ..........................................................
IV .
I
27
27
27
30
31
32
. .
35
D. ' C a l c u l a t i o n o f B u lk G reen F u n c t i o n .............................
38
ONE-DIMENSIONAL DENSITIES OF STATES AT LOW INDEX .
SURFACES
...................................... ’ ............................................. ........................
41
y
V.
A.
W(OOl) F a c e ................................................
. ............................. . . 4 1
B.
M o(IlO ) a n d W ( I l l ) F a c e ................................................................... 54
DISCUSSION .................................................................. ..
............................. 65
A.
G e n e r a l R e m a r k s .......................................................... .... ........................65
B.
S u g g e s te d C a l c u l a t i o n s ..................................I ............................ 70
APPENDIX
COMPUTATIONAL PROCEDURE
. : ...........................................
A.
H a m ilto n ia n a n d SK P a r a m e te r s
B.
The C o e f f i c i e n t s o f D e te r m in a n t P o l y n o m i a l s ................... 78
C.
R o o ts o f D e te r m in a n t P o ly n o m ia ls
D.
O u t l in e o f P r o g r a m s ................................................ .... ....................... 83
REFERENCES
71
.................................................72
............................................79
88
vi
LIST OF TABLES
Table
Page
I.. DipoIe-allowed initial statessymmetries for normal
e m is s io n fro m lo w - in d e x f a c e s o f c u b ic m e ta ls ........................
2.
25
O b s e rv e d EDC/c a l c u l a t e d SDOS p e a k l o c a t i o n s fro m lo w in d e x f a c e s o f W an d M o ............................................................................. 66
A p p e n d ix t a b l e I : The s y m m e trie s and c o r r e s p o n d in g b a s e s
i n t h e c o n v e n ti o n a l c o o r d i n a t e s y s te m a n d .w i th t h e s u r f a c e
n o r m a l i n th e z d i r e c t i o n .........................................................
74
A p p e n d ix t a b l e 2 : P a s c a l 's t r i a n g l e
........................... . . . .
82
A p p e n d ix t a b l e 3; I n v e r t e d P a s c a l 's t r i a n g l e ....................................82
I
I________ I___________________________
U______________ I
_________ 1H 1
-
I l _________________________
v ii
LIST OF FIGURES
F ig u r e
P ag e
1.
R e l a t i v e gap a n d a b s o l u t e g a p ..............................................................
3
2.
EDC e m i t t e d n o rm a l to th e (00 1 ) f a c e o f t u n g s t e n a t
1 0 .2 eV p h o to n e n e r g y ................................................................... ....
6
3.
. .
S u r f a c e - s ta te peak i n t e n s i t y as a fu n c tio n o f p o la r
a n g le ( o r k „ ) f o r th e two p r i n c i p a l a z im u th a l d i r e c t i o n s
7
4.
N orm al e m is s io n AREDC fro m W(OOl) .......................................................... 11
5.
A n g l e - r e s o lv e d p h o to e m is s io n s p e c t r o s c o p y shown
s c h e m a t i c a l l y ......................................
14
6.
S a m p le /e m is s io n co n e o r i e n t a t i o n w i t h sa m p le n o rm a l
4 2 .3 ° fro m CMA a x i s ............................................................................................16
7.
H y p o t h e t i c a l e n e rg y b a n d an d d i r e c t and i n d i r e c t
t r a n s i t i o n s shown s c h e m a t i c a l l y ..........................................................
19
F i r s t B r i l l o u i n zone o f t h e b o d y - c e n t e r - c u b i c l a t t i c e
w i t h s p e c i a l sym m etry p o i n t s l a b e l e d .......................................'.
28
T w o -d im e n s io n a l B r i l l o u i n zone f o r a (001) f a c e o f b c c
c ry s ta l . . . . . .
.................................................................... . . . .
29
S c h e m a tic s u r f a c e and b u l k r e g i o n f o r s e c o n d n e a r e s t
n e ig h b o r i n t e r a c t i o n
................................................
37
8.
9.
10.
11.
APW a n d RAPW e n e rg y b a n d s a lo n g A sym m etry d i r e c t i o n
f o r tu n g s te n
. . ................................................................................................ 42
12.
U n h y b r id iz e d e n e rg y b a n d a lo n g A sym m etry c a l c u l a t e d by
P i c k e t t - A l l e n - C h a k r a b o r t y 1s SK p a r a m e te r s ..................................
43
13.
O n e - d im e n s io n a l DOS f o r A^ s y m m e t r y ............................
44
14.
O n e - d im e n s io n a l DOS f o r A^ sym m etry f o r t h e f i r s t e i g h t
l a y e r s and b u l k ........................................................................................... .
46
I I Il
viii
15.
T w o -d im e n s io n a l b a n d s t r u c t u r e f o r th e p r o m in e n t s u r f a c e
s t a t e s / r e s o n a n c e s on t h e c l e a n W(OOl) s u r f a c e ...................
;
47
16.
APW b u l k b a n d
50
17.
O n e - d im e n s io n a l DOS f o r
18.
W e ig h te d sums o f t h e l a y e r DOS f o r A^ and A^ sym m etry
53.
19.
APW and RAPW e n e rg y b a n d s a lo n g t sym m etry d i r e c t i o n
f o r t u n g s t e n ................................................................................................
55
20.
O n e - d im e n s io n s I DOS f o r
56
21.
APW and RAPW e n e r g y b a n d s a lo n g A sym m etry l i n e f o r
t u n g s t e n ................................................................................................ ....
22.
e n e rg y p a r a l l e l t o A(OOl) sym m etry l i n e
sym m etry . . . . . . . . . .
b a n d s o f M o(IlO )
52
. . . . . .
.
58
O n e - d im e n s io n a l DOS f o r th e A b a n d s and th e A b a n d s
o f W ( I l l ) ........................................... ..........................................
60
23.
W e ig h te d sums o f t h e l a y e r DOS f o r
61
24.
EDC e m i t t e d n o rm a l t o W ( I l l) c l e a n and H c o v e re d s u r f a c e
fo r s -p o la riz e d lig h t
.................................................•..........................
63
EDC e m i t t e d n o rm a l to W ( I l l ) c l e a n and H c o v e r e d s u r f a c e
fo r p - p o la riz e d lig h t
.............................................................................
64
The s l a b c a l c u l a t i o n and G reen f u n c t i o n c a l c u l a t i o n o f
DOS i l l u s t r a t e d s c h e m a t i c a l l y
..........................................
69
A p p e n d ix f i g u r e I : A flo w
86
25.
26.
A^ and A^ sym m etry
c h a r t o f MAIN p ro g ra m
. . . .
A p p e n d ix f i g u r e 2: A flo w c h a r t o f b u l k G reen f u n c t i o n
p r o g r a m ........................................................................................................... .
87
ix
ABSTRACT
The r e s o l v e n t m ethod I s com bined w i t h a t i g h t - b i n d i n g m odel in o r d e r t o o b t a i n l a y e r - b y - l a y e r d e n s i t i e s o f s t a t e s a t low in d e x
s u r f a c e s -of s e m i - i n f i n i t e b c c t r a n s i t i o n m e t a l s .
S h o c k le y - ty p e
l o c a l i z e d s u r f a c e s t a t e s o r r e s o n a n c e s a r e fo u n d i n a l l h y b r i d i z a t i o n
g aps i n t h e t h r e e sym m etry d i r e c t i o n s s t u d i e d . P r o m in e n t s t r u c t u r e s
i n t h e a n g l e - r e s o l v e d p o l a r i z a t i o n d e p e n d e n t p h o to e m is s io n s p e c t r a a r e
d is c u s s e d in th e l i g h t o f th e p r e s e n t c a lc u la t io n .
i n a s sig n in g peaks
o f an o b s e r v e d EDC to c a l c u l a t e d i n i t i a l s t a t e f e a t u r e s , a k n o w led g e o f
th e f i n a l s t a t e sym m etry a n d s e l e c t i o n r u l e s i s e x tr e m e ly i m p o r t a n t .
11
I.
INTRODUCTION
The t r a n s l a t i o n sym m etry o f c r y s t a l s f a c i l i t a t e s
s o li d s t a t e p h y s ic s .
Thus s o l i d s t a t e p h y s i c s i s
th e th e o ry o f
l a r g e l y d e v o te d to
t h e s tu d y o f e l e c t r o n s i n t h e b u l k c r y s t a l o f r i n g to p o lo g y i g n o r i n g
su rfa c e s.
H o w ev er, i n p r a c t i c e t h e r e a r e many phenom ena w h ic h a r e
e n t i r e l y d e te r m in e d by t h e i n t e r a c t i o n o f s u r f a c e atom s w i t h atom s
\.
im p in g in g on t h e c r y s t a l s su c h a s h e te r o g e n e o u s c a t a l y s i s i n w h ic h th e
r o l e o f d - e l e c t r o n s • o f t r a n s i t i o n m e t a ls i s i m p o r t a n t , s e m ic o n d u c to r
d e v ic e s o r c o r r o s i o n .
The f i r s t s t e p
to w a rd t h e m ic r o s c o p i c u n d e r ­
s t a n d i n g o f s u c h s u r f a c e co m p lex es i s t h e s tu d y o f s t a t i c
e le c tro n
d i s t r i b u t i o n a t th e s u r f a c e b o th i n e n e rg y and i n s p a c e .
When t h e s u r f a c e i s i n t r o d u c e d t h e t r a n s l a t i o n sym m etry o f th e
c r y s t a l i s b r o k e n an d s u r f a c e b o n d s a r e d i s r u p t e d ; t h e s e a r e known a s
d a n g li n g bonds"*".
The s u r f a c e r e l a x e s e i t h e r in w a rd o r o u tw a r d .
Due to
t h e s e c h a n g e s th e, e l e c t r o n s n e a r th e s u r f a c e a r e r e d i s t r i b u t e d s p a t i a l l y .
R a t h e r a r t i f i c i a l s t a t e s p r o v id e d by o n ly b r e a k i n g t h e b o n d s a r e c a l l e d
S h o c k le y s t a t e s
2
.
On t h e o t h e r h a n d i f s u r f a c e r e l a x a t i o n and c h a rg e
r e d i s t r i b u t i o n a r e c o n s id e r e d th e c o r r e s p o n d in g e l e c t r o n s t a t e s a r e
c a l l e d Tamm S t a t e s
3
.
T h e re a r e r a t h e r w e l l - e s t a b l i s h e d common f e a t u r e s a s s o c i a t e d w i t h
s u r f a c e s t a t e s w h ic h a r e c o n v e n ie n t t o know b e f o r e a t t e m p t i n g to
p e rf o r m m ore d e t a i l e d a n a l y s i s
4
.
F i r s t o f a l l , a d a n g li n g b o n d p i c t u r e
o f t h e LCAO ty p e w o rk s f o r t r a n s i t i o n m e t a l s .
I f b u lk e le c tr o n s t a t e s
1I
2
s t i c k s h a rp ly
e x p e c te d .
out
of
th e s u r f a c e th e n a p ro m in e n t s u r f a c e e f f e c t i s
S e c o n d ly s u r f a c e s t a t e s show up i n a b s o l u t e g ap s i n th e b u l k
e n e rg y b a n d s j u s t as i n t h e im p u r it y p ro b le m i n s e m ic o n d u c to r s .
gap i s
I f th e
to o n a rro w a w e l l l o c a l i z e d s u r f a c e s t a t e w o n 't b e form ed b u t i t
becom es a r e s o n a n c e .
A r e l a t i v e gap i s an e n e rg y r e g i o n w h e re th e
i n i t i a l d e n s i t y o f s t a t e s o f a g iv e n sym m etry s u d d e n ly becom es much
lo w e r b u t d o es n o t go s t r i c t l y
to z e ro .
e x a m p le , by an 1s '- s h a p e d b a n d ( F ig .
Such a gap may b e c a u s e d , f o r
I-a ).
What i s n o t m ean t by a
r e l a t i v e gap i s a gap b e tw e e n b a n d s o f one sym m etry em bedded i n a
c o n tin u u m due t o a b a n d o f d i f f e r e n t sym m etry.
c a l l e d a b s o l u t e , s i n c e we a r e -co n c e rn ed
Such a gap w ould be
o n ly w i t h one sym m etry ty p e a t
o n c e (F ig .1 - b ) .
I n t h i s t h e s i s s u r f a c e d e n s i t y o f s t a t e s (SDOS) o f S h o c k le y ty p e f o r
n o rm a l p h o to e m is s io n fro m low in d e x f a c e s ( 0 0 1 ) , (H O ) and (1 1 1 ) o f
c l e a n Mo and W a r e c a l c u l a t e d .
o f c r y s t a l momentum k,,=0 i s
d im e n s io n a l B r i l l o u i n z o n e .
The p o i n t w h e re th e p a r a l l e l ,component
th e m ost i n t e r e s t i n g p o i n t i n t h e e n t i r e two
The h i g h sym m etry a t su c h a p o i n t rem oves
u n n e c e s s a r y c o m p l i c a t i o n , and g ro u p t h e o r y becom es a p o w e r f u l t o o l f o r
t h e a s s ig n m e n t o f s u r f a c e b a n d s t r u c t u r e d ^
I n t h e f o l l o w i n g s e c t i o n a h i s t o r y o f b o th e x p e r im e n ts and t h e o r y
re la te d
to t h i s s u b j e c t i s o u t l i n e d .
A n g le R e s o lv e d P h o to e m is s io n
S p e c tr o s c o p y (ARPS) w i t h t h e s y n c h r o t r o n r a d i a t i o n s o u r c e i s
th e b e s t
t o o l t o p r o b e e n e rg y b a n d s t r u c t u r e o f e l e c t r o n s i n t h e c r y s t a l
F ig u r e I .
R e l a t i v e gap ( a ) and a b s o l u t e gap ( b ) .
4
In C h a p te r I I we lo o k a t t h e e x p e rim e n t and t r y t o s e e th e ty p e s o f
i n f o r m a t i o n o b t a i n e d and how t o r e l a t e e x p e r i m e n t a l d a t a w i t h t h e o r y .
The m eth o d o f c a l c u l a t i o n i s o u t l i n e d i n C h a p te r I I I .
The r e s u l t s o f
th e c a l c u l a t i o n and i n t e r p r e t a t i o n o f th e r e s u l t s i n c o m p a ris o n w i t h
e x p e r im e n ts a n d t h e p o s s i b l e e x p e r im e n ts w h ic h w o u ld t e s t t h e i n t e r ­
p r e t a t i o n a r e t o p i c s o f C h a p te r IV .
th e same t e c h n i q u e i . e .
c a l c u l a t i o n and
I)
T h re e im m e d ia te e x t e n s i o n s u s in g
a p p l i c a t i o n t o f e e m e ta ls
2) k^O
3) th e c h e m is o r p tio n p r o b le m , a r e d i s c u s s e d i n C h a p te r
V.
The d e t a i l s o f c o m p u t a ti o n a l p ro b le m s a r e g i v e n . i n th e A p p e n d ix .
A.
P r e v io u s S t u d ie s o f W and Mo
(0 0 1 ) f a c e .
2 3
th e 1930s * .
Low In d e x F a c e s *2
The e x i s t e n c e o f s u r f a c e s t a t e s was p r e d i c t e d a s e a r l y a s
More r e c e n t l y , F o r s tm an, P e n d ry and H e in e
8 -1 1
p re s e n te d
a c a l c u l a t i o n o f s u r f a c e s t a t e s i n th e s p d h y b r i d i z a t i o n g a p .
peak lo c a te d j u s t
A sh a rp
b e lo w th e F erm i l e v e l w as f i r s t o b s e r v e d i n f i e l d
e m is s io n e n e rg y d i s t r i b u t i o n s p e c t r a (FEED) o f W(OOl) b y Swanson and
G ro u se r" ^ ’
Plum m er and G a d z u k ^ fo u n d t h a t t h i s s h a r p p e a k a t . 4eV
a n d a n o t h e r a t 1 .5 e V b e lo w th e F erm i l e v e l (Ep) w e re e x tr e m e ly s e n s t i v e
to s u r f a c e c o n ta m in a tio n and a s s i g n e d t h e o r i g i n o f t h e s e p e a k s to
s u r f a c e s t a t e s r e s u l t i n g fro m s p i n - o r b i t s p l i t b a n d s .
F e d e r and S tu rm
p e rf o r m e d a G reen f u n c t i o n ty p e c a l c u l a t i o n b a s e d on t h e f a c t t h a t t h e
s p i n - o r b i t c o u p lin g m ix e s s t a t e s o f sym m etry
A^,
A’ ^ a n d
A^
I
5
a n d p r o d u c e s one r e l a t i v e
bands o f
& sy m m etry .
gap a n d one a b s o l u t e gap b e tw e e n t h r e e d
T h e i r n u m e r i c a l r e s u l t s r e v e a l a v e ry
p ro n o u n c e d s u r f a c e r e s o n a n c e i n th e r e l a t i v e gap 0 .4 e V b e lo w E^1 and a
s u r f a c e s t a t e i n th e a b s o l u t e gap 1 .5 e V b e lo w E .
The s u r f a c e s e n s i t i v e F erm i p e a k a t 0 .4 e V b e lo w E
a ls o has been
o b s e rv e d i n p h o to e m is s io n s p e c t r a b u t n o t t h e s e c o n d p e a k 1 .5eV b e lo w
E
The m e a su re m e n ts o f th e momentum o f e m i t t e d e l e c t r o n s a s
w e l l a s t h e i r e n e rg y p r o v id e d a c l e a r p i c t u r e 0 f th e e l e c t r o n e n e rg y
band s tr u c tu r e .
F e u e r b a c h e r and h i s co-w orkers"*"^’ ^
c o l l e c t e d n o rm a l
e m is s io n d a t a fro m W t o com pare w i t h t h e b a n d c a l c u l a t i o n s .
p o l a r i z a t i o n dependence
k,,
The s t r o n g
■19 7
’ ( F i g . 2) a n d t h e i n t e n s i t y a s a f u n c t i o n o f
20 7
9 ( F i g . 3) o f t h i s F erm i p e a k w e re a l s o s t u d i e d .
The AEPS
t e c h n iq u e h a s b e e n im p ro v e d c o n s i d e r a b l y ,w ith th e u s e o f
s y n c h r o t r o n r a d i a t i o n a s th e p h o to n s o u r c e , w h ere due t o t h e c o n tin u u m
n a t u r e and w e l l - d e f i n e d p o l a r i z a t i o n p r o p e r t i e s o f s y n c h r o t r o n
ra d ia tio n ,
I ) p h o to n e n e r g y ,
2) o p t i c a l p o l a r i z a t i o n a n d
3) t h e momentum o f e m i t t e d e l e c t r o n s a r e u n d e r th e c o n t r o l o f th e
„ 7 ,2 1 - 2 3
e x p e rim e n te r
.
C o n tr a r y to t h e e a r l i e r in te r p r e ta tio n " * " ^ "*"^
w h ich i s
p r e v a i l i n g v ie w , i n p a r t i c u l a r among e x p e r i m e n t a l i s t s ,
p h y s i c a l o r i g i n o f th e F e rm i p e a k , K aso w sk i
o f a l i n e a r c o m b in a tio n o f m u ff in t i n
24
o r b ita ls
h a s shown
s till
th e
r e g a r d i n g th e
on t h e b a s i s
(LCMTO) t e c h n i q u e
N(E) (R E L U N IT S /A B S . PHOTON)
6
E BELOW Ecr (eV)
F ig u r e 2 .
EDC e m i t t e d n o rm a l t o th e (001)
t u n g s t e n a t 1 0 . 2eV p h o to n e n e r g y .
p o la riz e d lig h t;
fac e o f
U pper c u r v e , p -
lo w e r c u r v e , s - p o l a r i z e d l i g h t
( r e p r o d u c e d fro m R e f. 1 9 ) .
(1 Ol AZI MUTH
[11) AZI MUTH
RELATI VE
I NT E NS I T Y
W(OOI)
-.5
F i g u r e 3.
0
K11CA"1)
.5
-.5
0
K11CA'1)
.5
S u r f a c e - s t a t e p e a k i n t e n s i t y as a f u n c t i o n o f p o l a r a n g le ( o r k „ ) f o r th e two
p r i n c i p a l a z im u th a l d i r e c t i o n s ,
(O l) and (1 1 ) o f th e W(OOl) s u r f a c e ; h v = 1 0 .2 eV .
The e r r o r
b a r s r e p r e s e n t t h e s t a n d a r d d e v i a t i o n o f 10 s e p a r a t e m e a su re m e n ts ( re p r o d u c e d from R e f. 2 0 ) .
8
a p p l i e d to. s l a b s o f W, t h a t th e e n e rg y a s s o c i a t e d w i t h t h e f o r m a tio n o f
an s —d^2 s u r f a c e s t a t e i s much l a r g e r th a n s p i n - o r b i t s p l i t t i n g .
S m ith a n d M a t t h e i s s
25
p e rf o r m e d an LCAO ty p e c a l c u l a t i o n t o o b t a i n t h e
k ,,- d e p e n d e n t e l e c t r o n e n e rg y b a n d s o f a c l e a n W(OOl) s u r f a c e f o r a
s ix te e n l a y e r s la b u s in g S la te r - K o s te r p a ra m e te rs f i t t e d
r e l a t i v i s t i c APW b u l k b a n d s .
A t k,,=0 t h e r e i s o n ly one p r o m in e n t
s u r f a c e s t a t e a b o u t 2 . IeV b e lo w t h e F e rm i l e v e l due t o
o rb ita ls .
to pon-
ty p e
E n e rg y b a n d s o f A^ a n d A^ ty p e sym m etry c r o s s a lo n g th e A
l i n e b u t a r e s e p a r a t e d b y a gap away fro m t h e A l i n e
( F i g s . 11 and 1 5 ) .
Thus f o r 'k „ f 0 a n o t h e r s u r f a c e s t a t e a p p e a r s , l o c a t e d b e tw e e n IeV and
t h e F e rm i l e v e l .
T h is s t a t e due t o t h e A^ and A^ m ix in g w as i d e n t i ­
f i e d a s t h e F e rm i p e a k .
L . S . Weng a n d c o - w o r k e r s '^ ^
FEED and
ABPS e x p e r im e n ts an d DOS c a l c u l a t i o n seem t o a g r e e w i t h S m ith a n d .
M a t t h e i s s ' r e s u l t and i n t e r p r e t a t i o n .
e l e c t r o n e n e rg y d i s t r i b u t i o n
w e re s e e n f o r k „ = 0 .
I n t h e i r a n g le r e s o l v e d p h o to -
c u rv e (ABPEDC)
28
, two s u r f a c e r e s o n a n c e s
The h i g h - l y i n g r e s o n a n c e i s l o c a t e d 0 . 3 ( 0 . 4)eV
b e lo w t h e F e rm i l e v e l f o r Mo(W).
A l o w - l y i n g r e s o n a n c e a p p e a r s 3 .3 ( 4 . 2 )
eV b e lo w t h e F erm i l e v e l f o r Mo(W).
i n t o two p e a k s f o r 6 > 2 °.
The f i r s t r e s o n a n c e p e a k s p l i t s up
I n s t e a d o f a s l a b c a l c u l a t i o n , w h ic h am ounts
t o j u s t a m a t r i x d i a g o n a l i z a t i o n p r o b le m , Weng p e r f o r m e d , f o r f i x e d k M
a G reen f u n c t i o n c a l c u l a t i o n s t a r t i n g fro m a b a s i s s i m i l a r t o t h a t
u s e d b y S m ith a n d M a t t h e i s s .
H is s u r f a c e l o c a l d e n s i t y o f s t a t e s
9
r e v e a ls
k„=0.
two p ro n o u n c e d p e a k s , one a t 0 .6 e V and one a t 3 .3 e V b e lo w E
F
at
B a se d on t h i s r e s u l t o f th e c a l c u l a t i o n and on th e r e s u l t o f
e x p e r im e n ts Weng e t a l .
up o f
c o n c lu d e d t h a t , t h e F erm i p e a k .w a s m a in ly made
and s o r b i t a l s and was r e l a t e d to th e c r o s s - o v e r o f the.
Ag and A^ b a n d s a t k ,,= 0 . The s e c o n d lo w - ly i n g e x p e r i m e n t a l p e a k was
a s s i g n e d t o t h e one due to th e A^ g a p .
The m a jo r d i f f i c u l t y w i t h th e ab o v e i n t e r p r e t a t i o n i s
e x p e r i m e n t a l l y o b s e rv e d F erm i p e a k h a s maximum i n t e n s i t y
(n o rm a l e m is s io n )
fo rb id d e n .
20 7
1 (F ig .
t h a t th e
a t k,,=0
3) w h e re p h o to e m is s io n fro m su c h a s t a t e i s
M o reo v er t h i s p e a k i s known t o d i s a p p e a r f o r s - p o l a r l z e d
l i g h t ( F i g . 2 ,4 ) w h ic h d oes n o t c o n t a i n a s u r f a c e n o rm a l com ponent o f
th e A v e c t o r .
I f th e F erm i p e a k i s
a s s o c i a t e d w i t h a Ag and A^
c r o s s - o v e r th e i n i t i a l s t a t e s h o u ld h a v e sym m etry Ag+A^ o r p o s s i b l y
Ag+A^+s w h e re s i s L o w d in ' s s s t a t e ^ ’ " ^ .
s tric tly
Among them Ag sym m etry i s
f o r b i d d e n f o r any p o l a r i z a t i o n o f th e l i g h t
5
.
The c o n t r i ­
b u t i o n fro m th e p a r t o f th e i n i t i a l s t a t e w i t h A^ sym m etry w o u ld o c c u r
b o t h f o r s - an d p - p o l a r i z e d l i g h t b u t s h o u ld b e s t r o n g e r i n s - p o l a r i z e d
l i g h t w h ic h d o e s n o t a g r e e w i t h o b s e r v a t i o n .
th e s - l i k e
com ponent s a t i s f i e s
The c o n t r i b u t i o n from
th e s e l e c t i o n r u l e and o b s e r v a t i o n ,
h o w e v e r t h e i n t e n s i t y o f su c h a p e a k i s e x p e c te d t o b e v e r y w e a k .
seco n d s u rf a c e s t a t e in th e
The
(001) sym m etry d i r e c t i o n a t - 4 .3 e V was f i r s t
o b s e r v e d by F e u e r b a c h e r an d F i t t o n
19 7
’ ( F ig .2 ,4 ) .
T h is p e a k i s n o t as
10
s t r o n g a s t h e F e rm i p e a k .
Jam es A n d e r s o n 's r e c e n t d a t a show s th e
i n s e n s i t i v i t y o f t h e p e a k s i z e t o th e k„ v a lu e
31
.
th is seco n d p eak i s s tro n g f o r p - p o la r iz e d l i g h t .
L ik e t h e F e rm i p e a k ,
S m ith and M a tth e is s
a s w e l l a s Weng a s s i g n th e p h y s i c a l o r i g i n o f t h i s p e a k to t h e
or
Ag gap s t a t e .
(H O ) an d (1 1 1 ) f a c e .
I n c o n t r a s t w i t h th e (0 0 1 ) f a c e w h ic h e x h i b i t s
a d r a m a tic s u r f a c e s e n s i t i v e p e a k b e lo w
e x p e rim e n te rs '
c u rio u s ity
s ta n d in g ), v e ry l i t t l e
(1 1 1 ) f a c e .
and w h ic h h a s a r o u s e d many
( b u t h a s n o t y e t l e d to a c o n c lu s i v e u n d e r ­
a t t e n t i o n h a s b e e n p a i d to e i t h e r th e (H O ) o r
The c o m p le x ity o f th e b u l k b a n d s t r u c t u r e i s a b o u t th e
same a s t h a t o f t h e (0 0 1 ) f a c e , b u t s o f a r n o s t r o n g s u r f a c e p e a k h a s
b e e n o b s e r v e d on e i t h e r o f t h e s e f a c e s .
s tu d y o f th e W (HO) and
C in ti e t a l.
peak n e a r E
32
P e r h a p s th e o n ly p u b l is h e d
(1 1 1 ) f a c e s i s F e u e r b a c h e r ' s 1974 AEPS w ork
18
p e rfo rm e d a s i m i l a r e x p e r im e n t on M o(H O ) a n d fo u n d a
l i k e F e u e r b a c h e r 's p e a k A i n F i g . 13 o f E e f . 1 8 .
a u t h o r s s p e c u l a t e t h a t due t o th e i n s e n s i t i v i t y
B oth
o f t h i s p e a k to s u r f a c e
c o n ta m in a tio n and p h o to n e n e r g y , i t may h a v e a d i f f e r e n t p h y s i c a l
o r i g i n fro m th e F e rm i p e a k i n W(OOl) o r M o(O O l).
C i n t i ' s g ro u p a l s o
o b t a i n e d a w eak s u r f a c e - s e n s i t i v e p e a k 4 .5 e V b e lo w E^ i n th e s - d
h y b r id iz a tio n gap.
B e c a u se o f th e s i m i l a r i t y
i n b u l k e n e rg y b a n d s
b e tw e e n Mo a n d W, in d e e d a s i m i l a r p e a k s h o u ld b e o b s e r v e d i n W (IlO ).
F e u e r b a c h e r 's p h o to n e n e rg y d o es n o t seem l a r g e eno u g h t o e x c i t e t h e
W (O O I)
hv = 1 8 eV
NORMAL
EMISSION
-
6
0= E
Ei (eV)
F ig u r e 4 .
N orm al e m is s io n AREDC from W(OOl) f o r p - p o l a r i z a t i o n
s -p o la riz a tio n
(d ash ed lin e )
s u rfa c e s ta t e s a re sh ad ed .
ta k e n by Jam es A n d e rso n .
( f u l l lin e )
E m is s io n s due t o th e
and
12
d e e p e r s t a t e s . . T u rn in g t o th e W ( I l l )
peak s iz e i s
fa irly
f a c e , s i n c e p e a k l o c a t i o n and
c o n s t a n t w i t h r e s p e c t t o p h o to n e n e r g y , F e u e rb a c h e r
a s s i g n e d t h e s t r u c t u r e A an d B i n F i g . 18 i n R e f. 18 t o s u r f a c e
e m is s io n p r o c e s s e s .
LI H
II.
PHOTOEMISSION SPECTROSCOPY AND THE INTERPRETATION
OF ELECTRON ENERGY DISTRIBUTION CURVES
A.
P h o to e m is s io n S p e c tr o s c o p y
F i g . 5 show s a s c h e m a tic d ia g ra m o f p h o to e m is s io n s p e c t r o s c o p y .
The e x p e r i m e n t e r s h i n e s t h e UV l i g h t w i t h known f r e q u e n c y on th e
c r y s t a l and m e a s u re s t h e r e s p o n s e i n t h e fo rm o f t h e num ber o f e j e c t e d
e le c tr o n s as a fu n c tio n o f f i n a l s t a t e k i n e t ic e n e rg y .
The a n g le s o f
i n c i d e n c e ( 6 ^ , <j>^) and e m is s io n ( 0 ^ , cfi^) c a n b e m e a s u re d w i t h s u i t a b l e
a lig n m e n t o f t h e a p p a r a t u s .
The r a d i a t i o n from t h e s y n c h r o t r o n i s
p o la r iz e d i n th e p la n e o f th e s to r a g e r in g
21 33
’
w h e re a s t h e 1 l i g h t fro m
a c o n v e n ti o n a l i o n d i s c h a r g e lamp i s u n p o l a r i z e d .
The c o n tin u o u s
n a t u r e o f t h e s p e c tr u m p r o v id e s a n e n e r g y - t u n a b l e p h o to n s o u r c e .
The .
l i g h t c u r v e , o r s p e c t r a l d e p e n d e n c e o f th e i n t e n s i t y o f t h e l i g h t beam
a f t e r m o n o c h r o m a tiz a tio n , i s g iv e n e ls e w h e r e
23
.
The s t r o n g e s t i n ­
t e n s i t y a v a i l a b l e fro m 240MeV e l e c t r o n s t o r a g e r i n g a t t h e U n i v e r s i t y
o f W i s c o n s in 's S y n c h r o tr o n R a d i a t i o n C e n te r w i t h t h e m o n o ch ro m ete r u s e d
by.M S U 's p h o to e m is s io n g ro u p i s a b o u t 18eV.
e n e rg y i s
IOeV < hv < 30eV
M easu rem en t o f e n e r g y .
The u s e f u l l i m i t o f p h o to n
23
Ig n o rin g i n e l a s t i c s c a t t e r i n g p ro c e s s e s th e
k i n e t i c e n e rg y o f t h e e m i t t e d e l e c t r o n E^ i n te rm s o f i n i t i a l e n e rg y
E.,
I
t h r e s h o l d E , a n d i n c i d e n t p h o to n e n e rg y hv i s
tn
Ek = hv + E1 - E t h .
C RYSTAL
F ig u r e
5.
A n g l e - r e s o lv e d p h o to e m is s io n s p e c t r o s c o p y shown s c h e m a t i c a l l y .
15
The e l e c t r o n e n e rg y d i s t r i b u t i o n c u rv e (EDC) i s o b t a i n e d a t f i x e d h v ,
and t h e n um ber o f e j e c t e d e l e c t r o n s i s m e a s u re d a s a f u n c t i o n o f E^.
T h is ty p e o f m e a su re m e n t i s done w i t h b o th a m o n o c h ro m a tic l i g h t s o u r c e
a n d th e s y n c h r o t r o n r a d i a t i o n s o u r c e .
M easu rem en t o f s u r f a c e w ave v e c t o r .
B e c a u se o f t h e sym m etry o f an
i d e a l s u r f a c e p o t e n t i a l , k„ i s a good quantum n um ber.
Thus i f k„ i s
m e a su re d t o g e t h e r w i t h E ^ , s u c h a s p e c tr u m s h o u ld map o u t th e two
d im e n s io n a l s u r f a c e e n e rg y b a n d s t r u c t u r e .
w h ic h c o n t a i n s i n f o r m a t i o n on k,, i s
s p e c t r o s c o p y (A EPS).
P h o to e m is s io n s p e c t r o s c o p y
c a l l e d a n g le r e s o l v e d p h o to e m is s io n
The f o l l o w i n g s im p le r e l a t i o n h o l d s b e tw e e n k„
a n d t h e a n g le o f e m e rg e n c e ( F ig . 5 ) .
S in c e k„ i s c o n s e r v e d (we w r i t e
k ,,' i n s i d e an d k „ o u t s i d e o f th e c r y s t a l )
k,, = k ,,' = /ZmE^/fi. s i n 8^
i g n o r i n g s u r f a c e um klapp s c a t t e r i n g .
I f th e l a t t e r i s p r e s e n t, a
s u r f a c e r e c i p r o c a l l a t t i c e , v e c t o r may b e a d d e d t o k „ ’ .
S e le c tio n o f p o la r iz a tio n .
p h o to e m is s io n g ro u p i s
The e x p e r i m e n t a l s e t up u s e d b y th e MSU .
d i s c u s s e d i n d e t a i l i n Jam es K n a p p 's t h e s i s
23
F o r th e " s " - p o l a r i z a t i o n m e a s u re m e n t, t h e l i g h t i s i n c i d e n t a lo n g t h e
sa m p le n o rm a l ( F ig . 6 ) .
The E v e c t o r i s i n t h e c r y s t a l s u r f a c e p l a n e .
D i s r e g a r d i n g t h e m e c h a n ic a l d e t a i l s ,
one s e l e c t s t h e e m is s io n a n g le by
m oving a s m a l l c i r c u l a r a p e r t u r e a ro u n d t h e l i p o f t h e c o n e .
b e s e e n fro m F i g . 6 , any p o l a r a n g le o f e m is s io n B b e tw e e n 0
As can
O
and 8 4 .6
O
16
Zs
n
(CMA
emission
cone
F ig u r e 6 .
SampI e / e m is s i o n co n e o r i e n t a t i o n w i t h sam p le n o rm a l
4 2 .3 ° from CMA a x i s ( r e p r o d u c e d from R e f. 2 3 ) .
17
can b e s e l e c t e d , w h i l e th e a z im u th a l a n g le can b e v a r i e d by r o t a t i n g th e
sa m p le a b o u t i t s n o r m a l.
i n d e p e n d e n tl y o f
su rfa c e is
<j> .
I n t h i s c o n f i g u r a t i o n , one c a n n o t v a ry A
S in c e A i s f i x e d i n s p a c e , i t s
d i r e c t i o n i n th e
d e te r m in e d by t h e r o t a t i o n o f t h e s a m p le .
In th e p -p ol a r i z a t io n
c o n fig u ra tio n ,
th e sa m p le i s
ro ta te d about
th e a x i s o f th e cone s o t h a t th e sa m p le n o rm a l r o t a t e s a b o u t th e l i p o f
the! cone by 9 0 ° .
I n t h i s c o n f i g u r a t i o n th e A - v e c to r i s r a i s e d o u t o f
th e s u r f a c e by 4 2 .3 ° a n d , i n te rm s o f t h e p l a n e o f i n c i d e n c e ,
is
o f 57% p - p o l a r i z a t i o n and 43% s - p o l a r i z a t i o n , , a n d t h e r e e x i s t s
same r e l a t i o n s h i p b e tw e e n
B.
th e l i g h t
6,
th e
an d A a s i n th e s - p o l a r i z a t i o n c a s e .
E n e rg y D i s t r i b u t i o n C urve
F ig .
7 - a shows a h y p o t h e t i c a l e n e rg y b a n d as a f u n c t i o n o f k a lo n g
some sym m etry l i n e i n k s p a c e .
The d e n s i t y o f s t a t e s D(E) i s th e
num ber o f a llo w e d s t a t e s p e r u n i t e n e r g y d e f i n e d as
D(E) =
I 6 (E - E )
k
.
We s e e fro m t h i s d e f i n i t i o n
fla t.
a band,
s ta te ,
t h a t D(E) h a s a s h a rp p e a k w hen th e b a n d i s
When p h o to n s o f s u f f i c i e n t e n e r g y i n t e r a c t w i t h t h e e l e c t r o n s i n .
th e e l e c t r o n may a b s o rb a p h o to n and jump up t o some e x c i t e d
th e n l e a v e t h e c r y s t a l s u r f a c e t o th e vacuum .
Thus th e i n f o r ­
m a tio n t h a t e s c a p i n g e l e c t r o n s c a r r y i s n o t th e i n i t i a l d e n s i t y o f
s t a t e s b u t r a t h e r th e j o i n t d e n s i t y o f s t a t e s (JDOS) d e f i n e d a s
18
D ™ (E) = f
6 (E - E % )
.
J
k
I f th e f i n a l s t a t e d e n s i t y i s sm o o th , h o w e v e r, th e s i n g u l a r i t y
s t r u c t u r e o f t h e JDOS and i n i t i a l s t a t e DOS a r e s i m i l a r ( F i g .
The p h o to n momentum i s
7 -b ).
l e s s th a n 1% o f a t y p i c a l e l e c t r o n momentum.
Thus i f s c a t t e r i n g among e l e c t r o n s o r b e tw e e n e l e c t r o n s a n d phonons
i s n e g l i g i b l e , t h e e l e c t r o n f i n a l s t a t e m u st h a v e th e same c r y s t a l
momentum a s th e one i n t h e i n i t i a l s t a t e ,
to w ith in th e r e c ip r o c a l
l a t t i c e v e c t o r G ( a b u l k p h o t o e x c i t a t i o n p r o c e s s i s assu m ed h e r e ) .
kf = L
+ G
.
Then t h e t r a n s i t i o n i s v e r t i c a l i n t h e e n e rg y b a n d d ia g ra m ( F ig .
th is
7 -c);
ty p e o f m o d el i s c a l l e d t h e d i r e c t m o d el. . On t h e o t h e r h a n d ,
when s c a t t e r i n g p r o c e s s e s i n v o l v i n g a p h o to n ------ w h ic h a r e i n f a c t
s e c o n d o r d e r p r o c e s s e s -----a r e n o t n e g l i g i b l e ,
th e t r a n s i t i o n w ould
n o t be v e r tic a l ( in d ir e c t p ro c e s s ).
I n f a c t th e m e a s u re d EDC o r e v e n AREDC a r e n o t a d i r e c t m apping o f
t h e d e n s i t y o f s t a t e s w h ic h may b e co m p ared t o t h e c a l c u l a t i o n .
We
n e e d t o know m ore a b o u t t h e b e h a v i o r o f e l e c t r o n s i n th e s o l i d when th e
c r y s t a l i s exposed to th e r a d ia tio n f i e l d .
We s h o u ld h a v e a p r e c i s e
c a l c u l a t i o n o f t h e p h o t o e l e c t r o n c u r r e n t w h ic h i n c l u d e s v a r i o u s
s c a tte rin g p ro cesses.
A c l a s s i c a l m odel w h ic h i s s t i l l u s e d t o
i n t e r p r e t t h e e x p e r im e n ts i s
c a l l e d t h e t h r e e - s t e p m o d e l, wap
o r i g i n a l l y p u t f o rw a rd b y S p i c e r an d W o o te n ^ ^ ^ .
A c c o rd in g t o t h i s
PAIR BAND
F ig u r e 7.
a ) A h y p o t h e t i c a l e n e rg y b a n d a s a
f u n c t i o n o f k and c o r r e s p o n d in g DOS.
th e b a n d i s f l a t a s h a r p DOS r e s u l t s ,
When
b ) The
p a ir band has a s im ila r s in g u la r ity s tr u c tu r e
if
th e f i n a l s t a t e e n e rg y b a n d i s s m o o th .
c) D i r e c t and i n d i r e c t t r a n s i t i o n s .
20
m o d e l, when, p h o to n s o f s u f f i c i e n t e n e r g y a r e i n c i d e n t on t h e s o l i d ,
e l e c t r o n s a b s o rb t h e p h o to n s a n d , i f
w ork f u n c t i o n o f th e s o l i d ,
away t o t h e s o l i d s u r f a c e ,
t h e i r e n e rg y i s
l a r g e r th a n th e
t h e e l e c t r o n s i n th e e x c i t e d s t a t e s w a n d e r
th e n le a v e t h e s o l i d t o t h e vacuum .
The
m o st i m p o r ta n t a s s u m p tio n i n t h i s m odel i s t h a t t h e s e p r o c e s s e s a r e
in d e p e n d e n t.
U s u a l ly th e p h o t o a b s o r p t i o n p r o c e s s i s
JDOS and th e p r o p a g a t i o n p r o c e s s i s
w a lk c a l c u l a t i o n " ^ ’ ^ .
c o r r e l a t e d w i t h th e
d e s c r i b e d by a c l a s s i c a l random
P e r h a p s C a r o l ! e t al.^"*" w e re t h e f i r s t to
p e r f o r m s y s t e m a t i c s t u d i e s on th e a p p r o p r i a t e n e s s o f t h i s m o d el.
They
came t o th e c o n c lu s i o n t h a t i n g e n e r a l t h i s m odel d oes n o t p r o v id e a ,
c o r r e c t p i c t u r e e v e n when th e s c a t t e r i n g e v e n ts can b e n e g l e c t e d .
A
c a l c u l a t i o n e q u i v a l e n t t o th e l a t t e r c a s e was p e rfo rm e d by S c h a ic h
and A s h c ro ft
4 2 -4 4
u s in g t h e q u a d r a t i c r e s p o n s e th e o r y i n w h ic h th e
tim e d e p e n d e n t p e r t u r b a t i o n e x p a n s io n t o t h e s e c o n d o r d e r i n th e
e l e c t r i c v e c to r p o t e n t i a l A i s in c lu d e d .
o f C a ro l! e t a l .
The m ost i m p o r ta n t c o n c lu s io n
i s t h a t one c a n n o t s im p ly s e p a r a t e th e o p t i c a l
t r a n s i t i o n s t e p fro m th e p r o p a g a t i o n o f t h e e x c i t e d e l e c t r o n and i t s
e s c a p e i n t o th e vacuum , e v e n i n th e a b s e n c e o f i n e l a s t i c s c a t t e r i n g
e v e n ts .
T h is s i t u a t i o n
re m in d s us o f t h e two s l i t s
q u an tu m m e c h a n ic a l p a r t i c l e s ^ " * ;
in te rfe re n c e .
e x p e r im e n ts f o r
t h e r e one c a n n o t i g n o r e th e e f f e c t o f
I n th e UV e n e rg y ra n g e h o w e v e r, th e ab o v e a u th o r s
e x p e c t t h a t th e c o r r e s p o n d in g EDC w i l l r e f l e c t r a t h e r c l o s e l y th e
I
21
s p e c t r a l f e a t u r e s o f t h e i n i t i a l s t a t e a l o n e , i n p a r t i c u l a r t h e ones
c lo s e to th e s u r f a c e .
A lth o u g h t h e r e a r e e f f e c t s
due t o t h e i n e l a s t i c s c a t t e r i n g s o f
e l e c t r o n s o r e l e c tr o n - p h o n o n e f f e c t s w h ic h b r o a d e n s t r u c t u r e s i n th e
AKEDC, s o f a r we h a v e s e e n t h a t t h e o b s e r v e d s p e c tr u m maps t h e s i n g u l a r ­
i t y s t r u c t u r e o f th e i n i t i a l d e n s ity o f s t a t e s .
Some o f th e e m i tt e d
e l e c t r o n s may h a v e t h e i r o r i g i n deep i n t h e b u l k and some o f them a r e
fro m t h e s u r f a c e r e g i o n .
t h a t an o b s e r v e d p e a k i s
t e s t ‘d ’
The m o st p o w e r f u l c r i t e r i o n
due t o th e s u r f a c e i s
f o r d e te r m in in g .
th e c o n ta m in a tio n
I f t h e p e a k p o s i t i o n i s a t f i x e d i n i t i a l e n e rg y f o r
v a r i o u s in c o m in g p h o to n e n e r g i e s t h i s a l s o c o r r o b o r a t e s t h e e v id e n c e
f o r th e s t a t e 's s u r f a c e n a tu r e .
f o r s u r f a c e p r o b le m s , t h e kj»
n o t w e ll-d e fin e d .
S in c e kj.
i s n o t a good quantum num ber
d e p e n d e n c e ( k M f ix e d ) o f t h e e n e rg y i s
A c c o r d in g ly a s u r f a c e e n e rg y l e v e l i s s m e a re d in a
s t r a i g h t l i n e o v e r th e k j.
d ire c tio n .
I f a tra n s itio n
is in itia te d
fro m s u c h a s t a t e t h e l o c a t i o n o f th e i n i t i a l s t a t e e n rg y w i l l n o t
d e p e n d on t h e in c o m in g p h o to n e n e r g y .
peak i n te n s it y
rea so n is
Som etim es t h e s e n s i t i v i t y o f t h e
t o th e p o l a r i z a t i o n i s a l s o u s e d a s a c r i t e r i o n
19
.
The
t h a t t h e s u r f a c e p o t e n t i a l i s a p p r o x im a te ly , o n ly a f u n c t i o n
o f z (z n o rm a l t o t h e s u r f a c e ) .
.
.
^
^ 9V(z)
i n te rm s o f A • z —~ ^------ :
a z
The A-p o p e r a t o r c a n b e e x p r e s s e d
,
22
A .S(f|
" fi
av
3Z
Now A -z=0 i f A d o es n o t c o n ta i n th e z co m ponent; th u s s u r f a c e e m is s io n
v a n is h e s i n s - p o l a r i z a t i o n .
T h is r e a s o n i n g , w h ic h seem s t o b e some­
w h a t q u e s t i o n a b l e i n g e n e r a l , was d e v e lo p e d t o s tu d y s u r f a c e p h o to ­
e m is s io n fro m s im p le m e t a l s , w h e re th e s u r f a c e p o t e n t i a l g r a d i e n t i s
s t r o n g e s t i n th e d i r e c t i o n n o rm a l t o th e s u r f a c e .
C.
S e l e c t i o n R u le s f o r T r a n s i t i o n P r o c e s s *2
W ith in t h e o n e - e l e c t r o n m odel th e e l e c t r o n - r a d i a t i o n f i e l d i n t e r -
a c t i o n H a m ilto n ia n i n th e l o w - i n t e n s i t y a p p r o x im a tio n (A
me
2
~ 0) becom es
(A* p + p*A)
A lth o u g h one d o es n o t n e c e s s a r y h a v e V•A=O, n e v e r t h e l e s s th e
i n c l u s i o n o f t h e p •A te r m d oes n o t b r i n g a n y th in g new i n th e r e s u l t a n t
s e l e c t i o n r u l e s a n d we i g n o r e i t i n t h e f o llo w in g d i s c u s s i o n .
G olden R u le f o r th e t r a n s i t i o n
F e r m i's
r a t e Wjrj f o r e x c i t a t i o n from an i n i t i a l
fi
d e n s i t y o f s t a t e s N _ a t i n i t i a l e n e rg y E . - E - hv t o f i n a l d e n s i t y
o f s t a t e s Njr a t f i n a l e n e rg y E^ = E i s
.f i = , < f | A . p | i )
W„
w h e re | i ^
I
Ni (E-NV)Nf (E)
an d | f ^ a r e t h e i n i t i a l and f i n a l s t a t e s o f t h e e l e c t r o n .
S in c e t h e v e c t o r p o t e n t i a l A does n o t c o n t a i n th e e l e c t r o n c o o r d i n a te
i n th e d i p o l e a p p r o x im a tio n , th e m a t r i x e le m e n t can b e e x p r e s s e d as
A* ( f I p I i ) .
T a k in g t h e a p p r o x im a tio n A^~0 and u s in g t h e e q u a ti o n o f
23
m o tio n f o r th e o p e r a t o r x l e a d s to
A- < f |p | i )
w h e re
~ A* ( f I - ^ s -
[
h,x
] |i)
A* ( f I
^ Ax < f | x | i >
+ Ay < f | y | i )
+
x |i)
(f|z|i)
E f - Ei
OJ_e
fi
I t h a s b e e n shown
t h a t f o r th e e m is s io n n o rm a l t o th e s u r f a c e
(k „= 0 ) t h e e l e c t r o n f i n a l s t a t e s h o u ld b e i n v a r i a n t u n d e r c r y s t a l
o p e r a t i o n s w h ic h
The
fin a l
le a v e
s ta te
fo r
th e
n o rm al
u n d e r th e f a c t o r g ro u p s C ^ ,
d ir e c tio n s , re s p e c tiv e ly
49-52
53
e m is s io n
C^v and
m u st
unchanged.
be
in v a ria n t
f o r (0 0 1 ),
(H O ) and (111)
and
A^.
I n c o n t r a s t to e a r i e r
t h i s d o es n o t mean t h a t t h e f i n a l s t a t e
s i n g l e p l a n e w av e.
d iffic u lty
n o rm a l
( s u r f a c e n o rm a l i s a lo n g t h e z a x i s ) , and
s h o u ld t r a n s f o r m a c c o r d in g t o
w ork
su rfa c e
.
c o n s is ts o f a
'In f a c t i f t h i s w e re s o , im m e d ia te ly one s e e s a
The n o n - s i n g l e p l a n e wave n a t u r e o f th e f i n a l s t a t e h a s
a l s o e x p e r i m e n t a l l y b e e n o b s e r v e d i n n o b le m e ta ls
53
.
What g ro u p t h e o r y
r e a l l y p r e d i c t s i s r a t h e r a l i n e a r c o m b in a tio n o f p l a n e w aves w h ic h
r e t a i n s t h e sam e sym m etry o f t h e s i n g l e p l a n e wave s t a t e -------t h a t i s
an o u tg o in g wave s a t i s f y i n g in c o m in g b o u n d a ry c o n d i t i o n s ------- th e t im e r e v e r s e d LEED s t a t e
The r e p r e s e n t a t i o n f o r t h e p r o d u c t o f d i p o l e
moment o p e r a t o r on th e i n i t i a l s t a t e can b e decom posed i n t o i r r e d u c i b l e
re p re s e n ta tio n s .
By t h e m a t r i x e le m e n t th e o re m , t h e sym m etry o f th e
24
i n i t i a l s t a t e m u st th u s b e th e same a s t h a t o f th e d i p o le , o p e r a t o r .
F o r e x a m p le , f o r
A sy m m etry , th e com ponent z t r a n s f o r m s l i k e A
w h e re a s com p o n en ts x an d y tr a n s f o r m l i k e
A ^.
Thus i f
th e e l e c t r i c
f i e l d v e c t o r o f t h e i n c i d e n t l i g h t d o e s n o t c o n ta i n an Ez com ponent th e
t r a n s i t i o n fro m a A^ ty p e i n i t i a l s t a t e
is s tr ic tly
f o r b i d d e n and s o o n .
The s e l e c t i o n r u l e s f o r p u r e E ^ , E^ o r E^ p o l a r i z e d l i g h t f o r th e ab o v e
t h r e e low in d e x sym m etry f a c e s a r e r e p r o d u c e d i n T a b le I fro m r e f e r e n c e
5.
When
one d i s c u s s e s e x p e r i m e n t s , u s u a l l y th e e l e c t r i c v e c t o r E i s
decom posed i n t o t h e com ponent p a r a l l e l (p ) t o th e p l a n e o f i n c i d e n c e ,
w h ic h i s d e f i n e d by th e z a x i s an d the- d i r e c t i o n o f in c o m in g l i g h t ,
an d th e com ponent p e r p e n d i c u l a r ( s ) t o i t
th e E v e c to r l i e s
a lm o s t i n th e xy p l a n e .
th e E v e c t o r can h a v e a lm o s t p u r e E ^ .
is
( F ig . 5 ) .
F or s m a ll
On th e o t h e r h a n d i f
0
I
0 .- 9 0 °
I
N o te t h a t w h a te v e r t h e a n g le 0^
(O <0.< 9 0 °) th e s - p o l a r i z e d l i g h t n e v e r c o n ta i n s any E co m p o n en t.
I
z
T h is i s n o t t h e l a s t w ord i n d i c a t i n g t h e im p o r ta n c e o f
p o la riz a tio n
s tu d ie s .
H e rm a n s o n 's r e c e n t l y p ro p o s e d 'p o l a r i z a t i o n
s ig n a t u r e *
d e m o n s tr a te s th e f u r t h e r pow er o f g roup t h e o r y by w h ic h an
e x p e r i m e n t e r now i s a b le t o u n i q u e ly d e te r m in e th e sym m etry ty p e o f
t h e i n i t i a l s t a t e f o r any s e m i - i n f i n i t e s y s te m w ith o r w i t h o u t
a d s o rb a te s .
The d e t e r m i n a t i o n r e q u i r e s m easu rem en t o f n o r m a l e m is s io n
Ckll=O) an d a l s o m e a s u re m e n ts f o r k ,,f0 d i r e c t i o n s on t h e m i r r o r p l a n e s .
T a b le I .
D i p o l e - a ll o w e d i n t i a l s t a t e s y m m e trie s f o r n o rm a l e m is s io n fro m lo w - in d e x
f a c e s o f c u b ic m e t a l s .
F o r e a c h f a c e th e d i r e c t i o n s o f C a r t e s i a n c o o r d i n a t e a x e s
a r e r e f e r r e d t o th e c o n v e n t i o n a l c u b ic a x e s , a n d i r r e d u c i b l e r e p r e s e n t a t i o n s a t k„=0
a r e l i s t e d i n s i n g l e g ro u p n o t a t i o n ^ .
C ry s ta l
I rre d u c ib le
C o o r d in a te Axes
Face
x
y
Z
(00 1 )
100
010
001
(H O )
001
110
HO
(11 1 )
110
112
111
R e p re s e n ta tio n s
A llo w ed I n i t i a l S y m m e trie s
E ||x
E ||y
E Hz
Ag Ag' Ag
A5
A5
A1
S1 Z2
Z3 Z4
Z3
E4
Z1
A1 A2
A3
A3
A3
A1
h
V
O b s e rv e d EDCs c a r r y c o p io u s, i n f o r m a t i o n b e y o n d o y r m o d el,
is
Thus i t
r a t h e r i m p r e s s i v e t o s e e good a g re e m e n t b e tw e e n t h e e x p e r im e n ts and
th e c a l c u l a t i o n w i t h th e h e l p o f t h e g ro u p t h e o r y .
The m ethod o f
c a l c u l a t i n g th e l a y e r - b y - l a y e r DOS w h ic h may b e c o m p a r e d .to m e a su re d
EDCs i s shown i n t h e n e x t c h a p t e r .
III.
CALCULATION OF DENSITY OF STATES
When th e S h o c k le y ty p e s u r f a c e i s p r e s e n t t h e t r a n s l a t i o n sym m etry
i n th e d i r e c t i o n
p a r a l l e l t o th e s u r f a c e i s p r e s e r v e d ; i . e . , th e
s u r f a c e p o t e n t i a l does n o t m ix e i g e n s t a t e s o f t h e b u l k w h ic h h a v e
d i f f e r e n t com ponents k „ o f wave v e c t o r p a r a l l e l t o th e s u r f a c e . Thus
g iv e n k,, one can c a l c u l a t e l a y e r - b y - l a y e r d e n s i t y o f s t a t e s .
S in c e o n ly
th e p a r a l l e l com ponent o f w ave v e c t o r i s a good quantum num ber f o r th e
s u r f a c e p ro b le m
a tw o - d im e n s io n a l B r i l l o u i n zone s c h e m e " ^ 9^ ( F ig . 9)
i s i n t r o d u c e d w h ich p r o j e c t s t h e t h r e e - d i m e n s i o n a l B r i l l o u i n zone
( F i g . 8). o n to two d im e n s io n s p e r p e n d i c u l a r t o th e d i r e c t i o n o f i n t e r e s t .
N o t i c e t h a t i n t h i s schem e A, E, arid A l i n e s a r e p r o j e c t e d to th e T '
p o i n t f o r th e (O O l), (H O ) and (1 1 1 ) f a c e s r e s p e c t i v e l y .
We com bine t h e r e s o l v e n t m eth o d w i t h a t i g h t - b i n d i n g i n t e r p o l a t i o n
schem e f o r t h e b u l k b a n d s .
The l a y e r DOS f o llo w d i r e c t l y
fro m th e
im a g in a r y p a r t o f t h e l a y e r d e p e n d e n t G reen f u n c t i o n w h i c h . i s com puted
fro m th e b u l k r e s o l v e n t w i t h o u t i n t r o d u c i n g any a d j u s t a b l e p a r a m e t e r s .
We b e g in t h i s c h a p t e r by r e v ie w in g t h e o n e - e l e c t r o n p ro b le m i n t h e
b u l k ; th e n we c o n s i d e r th e s e m i - i n f i n i t e s y s te m .
A.
B u lk Band. S t r u c t u r e an d S I a t e r - K o s t e r P a r a m e te r s
B l o c h 's th e o re m .
th e c r y s t a l i s
The S c h r o d in g e r e q u a t i o n f o r ah e l e c t r o n m oving i n
■
"■
28
F ig u re 8.
F i r s t B r i l l o u i n zone o f b o d y - c e n t e r - c u b i c
l a t t i c e w i t h s p e c i a l sym m etry p o i n t s l a b e l e d .
29
K2
—
*
K1
F ig u r e
9.
bcc c ry s ta l.
T w o -d im e n s io n a l B r i l l o u i n
zone f o r a (OOl) f a c e o f a
30
[-!“ - +
V(X)J^ (x)
=
(I)
Eip(x)
w h e re t h e c r y s t a l l i n e p o t e n t i a l V (x) h a s t r a n s l a t i o n sym m etry
V (x +
w h e re
a)
= V (x );
a =
+ ma^ + n a ^
Z, m and n a r e i n t e g e r s , a ^ , a^ and a^ a r e B r a v a is l a t t i c e v e c t o r s
T h is m eans t h e g ro u p o f S c h r q d in g e r
e q u a tio n c o n ta in s a t r a n s l a t i o n
g ro u p a s w e l l a s a r o t a t i o n g ro u p and th e wave f u n c t i o n h a s t o s a t i s f y
th e B lo c h c o n d i t i o n
iK x + a) = e ^
a ip(x)
(2)
w h e re k i s a new q u an tu m num ber c a l l e d c r y s t a l momentum w h ic h i s
a s s o c i a t e d w i t h th e t r a n s l a t i o n sym m etry o f t h e c r y s t a l w ith p e r i o d i c
b o u n d a ry c o n d i t i o n .
tr a n s la tio n g ro u p .
N o te t h a t e " ^ a i s a r e p r e s e n t a t i o n o f th e
E q . (2 ) i s s a t i s f i e d i f
ip(x) = u (x ) e ^ k x ;
u ( x + a) = u (x )
Group o f t h e w ave v e c t o r .
S u b s titu tin g
iJj ( x)
i s i n th e form
.
(3)
(3 ) i n t o
( I ) and c a n c e lin g o u t
th e e x p o n e n t i a l f a c t o r we g e t an e q u a t i o n f o r u (x )
- r —-V ^ u (x ) +
/m
The 'e f f e c t i v e *
T h is t e l l s
k on
+
m
k -p - k^
u (x ) = E u (x )
.
(4)
p o t e n t i a l c o n t a i n s te rm s w h ic h d epend on k o r k * p .
us t h a t u (x ) d e p e n d s on k .
ijj(x) o r. u (x )
^ (x )
f V(x)
I
ik » a
From now on we p u t th e s u b s c r i p t
to re m in d o u r s e l v e s o f t h i s k d e p e n d e n c e :
U^(X)
And o f c o u r s e E = E ( k ) .
I f a p o i n t g ro u p o p e r a t i o n
th e B lo c h f u n c t i o n one o b t a i n s '
i s a p p l i e d on
31
_ IR k eX
PR^ k ^ ' = U ,Rk^X^ e
’
The e f f e c t o f o p e r a t i n g
on
<5)
ijj- i s t o p ro d u c e an e i g e n f u n c t i o n a l s o
i n B lo c h fo rm b u t w i t h t h e k v e c t o r r o t a t e d t o Bk.
Thus i n g e n e r a l
i s n o t t h e g ro u p o f t h e r e d u c e d S c h r o d in g e r e q u a ti o n (4 ) i . e . (p r , h] fO .
F o r g iv e n k , h o w e v e r, t h e r e i s a su b g ro u p o f
w h ic h l e a v e s th e wave
v e c t o r i n v a r i a n t and h e n c e t h e k - p te r m i n E q . ( 4 ) .
T h is g ro u p i s c a l l e d
t h e g ro u p o f t h e wave v e c t o r .
Know ing t h e g ro u p o f g iv e n k i n t h e B r i l l o u i n zo n e we c a n f i n d t h e
num ber o f r e p r e s e n t a t i o n s and t h e d i m e n s i o n a l i t y o f e a c h r e p r e s e n t a t i o n .
Then we may f i n d t h e c h a r a c t e r t a b l e . The u £ ( x ) p a r t o f th e e ig e n f u n c t i i n s h o u ld t r a n s f o r m a c c o r d in g t o t h i s c h a r a c t e r t a b l e .
The
d im e n s io n a lity o f a r e p r e s e n ta tio n t e l l s us i f th e r e i s a d e g e n era cy .
E ach r e p r e s e n t a t i o n c o r r e s p o n d s t o an e i g e n v a lu e E = E ( k ) .
Thus we can
t e l l q u a l i t a t i v e l y how t h e e n e rg y b a n d s s p l i t i n E (k ) s p a c e by s tu d y i n g
th e c h a r a c t e r t a b l e s .
F i n a l l y we may a s s i g n t h e sym m etry o f th e e ig e n ­
s t a t e a s s o c i a t e d w i t h a c a l c u l a t e d e n e rg y b a n d .
S la te r - K o s te r p a ra m e te rs .
The s t r u c t u r e o f t h e c r y s t a l l i n e p o t e n t i a l
V (x ) i n E q . (4 ) i s i n g e n e r a l unknow n.
U s u a l ly th e fo rm o f V (x) i s
a ssu m e d , b a s e d on some p h y s i c a l c o n s i d e r a t i o n .
In th e tig h t- b in d in g
m e th o d , i n s t e a d , o f a c t u a l l y s o l v i n g E q . ( 4 ) we s t a r t w i t h t h e f o ll o w i n g
w ave f u n c t i o n w h ic h s a t i s f i e s t h e B lo c h c o n d i t i o n ( 2 ) .
|c t ,k ) = — —
-
I e x p ( ik - R ) |R , a )
( 6)
w h e re N ^,
N3 a r e t h e num ber o f l a t t i c e p o i n t s i n e a c h d im e n s io n ; ct
r e p r e s e n t s t h e a to m ic sym m etry s , x , xy e t c .
The sum i s
o v e r t h e atom s
i n e q u i v a l e n t p o i n t s i n a l l th e u n i t c e l l s o f t h e c r y s t a l .
a Low din f u n c t i o n l o c a t e d a t
| Rj^,
is
atom l L , w h ic h h a s t h e sam e sym m etry
p r o p e r t y a s a to m ic o r b i t a l s b u t i s o r t h o g o n a l i z e d ; i . e . ,
(R 1 , a | R . , $ ) =
Si j Sag ^
The m a t r i x e le m e n t o f t h e o n e - e l e c t r o n H a m ilto n ia n b e tw e e n d i f f e r e n t k
v a n i s h e s ; t h e d i a g o n a l e le m e n t i s
< k , a |H . |k , 3 ) - «
=
----12
I
e x p ( i k .( R 1 -R j ) ) ( R j , UjHlR1 , g )
3„ M
I e x p ( i k .R ) ( 0 , a | H | R , g ) ,
N o tic e t h a t th e i n t e g r a t i o n h as th r e e c e n te r s .
R1-R
.
(8 )
U s u a lly t h e m a t r ix
e le m e n ts o v e r t h i r d n e ig h b o r s a r e n e g l i g i b l y s m a l l.
id e a " ^ i s to f i t
= R
S l a t e r and R o s t e r '
t h e a b o v e m a t r i x e le m e n ts t o a m ore a c c u r a t e e l e c t r o n
e n e rg y b a n d c a l c u l a t i o n su c h a s APW, th e n u s e them a s 'd i s p o s a b l e '
p a ra m e te rs .
In th e p r e s e n t c a lc u la tio n I have used S la te r - R o s te r
p a r a m e t e r s o b t a i n e d by P i c k e t t , A l l e n a n d C h a c k r a b o r t y ^ ’ ^
by f i t t i n g
t o th e n o n - r e l a t i v i s t i c APW b a n d s o f P e t r o f f and V i s w a n a t h a n ^ .
B.
L o c a li z e d D e n s ity o f S t a t e s i n te rm s o f R e s o lv e n t M ethod
We c o u ld lo o k a t t h e b u lk c r y s t a l a s b e in g c o n s t r u c t e d from a s e t
o f e q u i d i s t a n t p l a h e s p a r a l l e l t o t h e c r y s t a l f a c e a lo n g w h ich , th e
s u r f a c e i s t o b e fo rm e d .
—
til
Then t h e B r a v a is l a t t i c e v e c t o r R on th e n
33
l a y e r i s s p l i t i n t o P, p a r a l l e l t o t h e s u r f a c e p l a n e an d th e com ponent
na^:
R = n a ^ + P , w h e re
p =
n , A, and m a r e i n t e g e r s .
+ ma^
In t h i s v iew i t
(9)
is
c o n v e n ie n t t o w r i t e an
o r th n o r m a l s e t o f l o c a l i z e d o r b i t a l s a s
|a , R )
= |n a ; p )
,
(1 0 )
w h e re a l a b e l s t h e a to m ic s y m m e try .
When t h e s u r f a c e i s p r e s e n t o n ly t r a n s l a t i o n sym m etry p a r a l l e l t o t h e
s u r f a c e h o l d s and a two d im e n s io n a l B lo c h th e o re m r e s u l t s w here k M i s
a good q u an tu m n u m b er.
As i n t h e t h r e e d im e n s io n a l c a s e , we may w r i t e
two d im e n s io n a l B lo c h sums o r l a y e r o r b i t a l s ,
|n a ,k „ ') = ■~ 1
g e x p ( i k „ 'P )
/ N 1N2 p
|n a ; p \
(11)
The sum i s o v e r t h e atom s i n e q u i v a l e n t p o i n t s i n a l l t h e u n i t c e l l s on
th e l a y e r .
T h is l a y e r o r b i t a l tr a n s f o r m s i n a s im p le way u n d e r th e
s u r f a c e t r a n s l a t i o n g ro u p and u n d e r th e s m a l l g roup o f sym m etry
r o t a t i o n s a b o u t t h e k ^ a x i s w h ic h le a v e k,, u n c h a n g e d .
The m o st s im p le
c a s e i s k„=0 ( s u r f a c e u n i t c e l l s a r e i n p h a s e ) w h ere t h e s m a l l group
becom es i d e n t i c a l w i t h t h a t o f t h e sym m etry l i n e s A, E o r .A f o r c u b ic
(0 0 1 ),
(H O ) o r (1 1 1 ) r e s p e c t i v e l y .
Then t h e i n t r o d u c t i o n o f a s u r f a c e
d o es n o t m ix d i f f e r e n t i r r e d u c i b l e r e p r e s e n t a t i o n s f o r t h e s e g r o u p s .
F o r f i x e d k M, a s s o c i a t e d w i t h e a c h l a y e r , t h e r e i s o n ly one o r b i t a l .
Then t h e p ro b le m becom es v e ry s i m i l a r t o th e l i n e a r c h a in p ro b le m w i t h
34
an i m p u r i t y atom .
I n te rm s o f t h e l a y e r o r b i t a l s , t h e V
quantum
s t a t e w h ic h i s a n
e i g e n s t a t e o f th e s u r f a c e H a m ilto n ia n H=HjfV can b e
e x p r e s s e d as
IV jk 11 y
=
I
I n a ,k „ ^ n a ,k „ | v ,k „ )
.
(1 2 )
na
The r e s o l v e n t o p e r a t o r G can b e e x p r e s s e d as
V
G(E ,k „ ) = I
V
V * k,, |
(13)
E + ip - E v K . ||
The t o t a l d e n s i t y o f s t a t e s
f o r f i x e d k„ i s
d e f i n e d as
N (E ,k „ ) = £ 6(E - E r )
V
K"
S in c e
P ( 1 /x )
-
I
ttS ( x )
x + in
i n te rm s o f th e G reen f u n c t i o n we may w r i t e
N (E ,k „ ) = - TT 1 Im £
V
- TT
Im T r G(E+ Jk 11) .
(1 4 )
E + in - Ev tit
On t h e o t h e r h a n d , th e d e n s i t y o f s t a t e s p r o j e c t e d o n to e a c h a to m ic
o r b i t a l and e a c h l a y e r , i . e . , l o c a l d e n s i t y o f s t a t e s
na
(LDOS) i s
I 2 ,
( E jk 11) = % | ( n a j k , , IV jk 11^ | <5(E - Ev ^ )
-
-I
TT
I I ^ n d jk 111V jk 11^ I 2 Im
E + i n - Ev k „
= - TT 1 Im ^nttjk111 ^
IVjic11^ :------------------------- - ( v jk ,,
E + lT1 "
5V k 11 •
35
(15)
not, net
I n t h e n e x t s e c t i o n a s im p le r e l a t i o n b e tw e e n t h i s s u r f a c e r e s o l v e n t
G and b u l k r e s o l v e n t g w i l l b e show n, w h ere
I
H i s t h e b u l k H a m ilto n ia n
o
(1 6 )
o
C.
R e l a t i o n b e tw e e n th e B u lk arid S u r f a c e R e s o lv e n ts
Due t o th e b reak d o w n o f th e p e r i o d i c b o u n d a ry c o n d i t i o n by t h e
s u r f a c e th e e le c tr o n s t a t e s
a r e d i s t o r t e d n e a r th e s u r f a c e and th e
e n e rg y l e v e l s a r e s h i f t e d .
The B lo c h f u n c t i o n s do n o t fo rm e ig e n ­
s ta te s
o f th e new H a m ilto n ia n .
,
The o f f - d i a g o n a l e le m e n ts w h ich
com m unicate w i t h th e s u r f a c e a r e c a l l e d th e S h o c k le y p o t e n t i a l ^ .
may lo o k a t t h e p ro b le m i n a l i t t l e
d iffe re n t lig h t^ ,
We
a s f o ll o w s .
A a n d B d e n o te t h e s u r f a c e r e g i o n a n d c r y s t a l r e g i o n i n F ig . 10.
L r e p r e s e n t s th e l a y e r n u m b er.
We d e s c r i b e t h e s u r f a c e p o t e n t i a l i n
te rm s o f t h e m a t r i x
As th e c r y s t a l i s
( A I i|j)= 0 w h e re
c u t i n h a l f and t h e l o c a l i z e d o r b i t a l s a t i s f i e s
^A | i s
any l a y e r o r b i t a l s i n r e g i o n A an d | ^
s o l u t i o n t o th e S c h r o d in g e r e q u a t i o n s .
b u l k r e s o l v e n t i s w r i t t e n as
i s th e
I n t h i s r e p r e s e n t a t i o n th e
36
8AA
8AB
8BA '
8BB
:- ^ A A
ft = I - gVT
“ s BAvAA
1/
As V.
( 18 )
^BA = ^AB =
Then th e L ip p m a n -S c h w in g e r e q u a ti o n G =ft
g g iv e s us f o r th e s u r f a c e
G reen f u n c t i o n
GBA
or
and
^
—]_
BA8AA
—I
—I
BB8BA
—I
BA ~ ~
—I
GBB =
or
-I
BB8BA 1
—1
BA8AB
GBB
BB8BB
BB^8BB ~ 8]
= I , s o we h a v e th e
u" 1BB :
ft
GBB
8BB
8BAg
AA8AB.
Knowing t h e b u l k G reen f u n c t i o n we can im m e d ia te ly o b t a i n
th e S h o c k le y
s u r f a c e G reen f u n c t i o n w i t h o u t kno w in g t h e S h o c k le y p o t e n t i a l .
i s n o t s u r p r i s i n g s i n c e t h e S h o c k le y p o t e n t i a l
T h is
s im p ly b r e a k s down
t h e p e r i o d i c b o u n d a ry c o n d i t i o n a h d d o es n o t c a r r y any f u r t h e r
F ig u re
10.
S c h e m a tic s u r f a c e
n e a r e s t n e ig h b o r i n t e r a c t i o n .
(A) and b u l k (B) r e g i o n s f o r se c o n d
L r e p r e s e n t s l a y e r n um ber.
38
in fo rm a tio n .
D
C a l c u l a t i o n o f B u lk G reen F u n c tio n
F i r s t we lo o k a t t h e b u l k G re e n ia n p r o j e c t e d o n to t h e l a y e r
o rb ita ls .
s c + .w
- I
M
. E
X
m
'
e Xk
( 20 )
u
w h e re
The m a t r i x e le m e n t i n t h e l a y e r o r b i t a l r e p r e s e n t a t i o n i s
^ m a ,k „ I g(E+ , k „ ) |n g , k „ ^
^
^ m a ,it,, I a ' k "XT a 1. k | A k ^
a8Akj_
A k | g ’k ^
g 'k [ n g , k „ ^
E+ - =XE
Now
a kN = ——
/
/¥ 3
(m a ,k |a '
k \
e ^ k - k c^l m a ,k „ \
7
i_ _ J
/ N3
/ N
Thus
m'
, from (6) and (1 1 )
e ^ ' k j L d ^m1a ' ,kv „ |m a ,k „ ^
; th e n
39
g.'met n@
%+%)
= %
I
Kl ^ k
k j_ A
. I
ei(m -n )k ju
d
E+ - e Xk
E+ - H (k)
I
■ l
A k| g k }
I
N3
ag
i(m -n )k x d
e
The sum i s r e p l a c e d by an i n t e g r a l
-I
i ( m - n ) k JL d
[ e + - H (k)j
gmc« „6 <E+- E" ) m - ~ Z
a3
dkj_
w h ere th e d e n s i t y o f a llo w e d w a v e - v e c t o r s k J. i s N3^dZZTT
S e t t i n g m-n=£» kju d=x.
TT
^mOt n3
(E+ ,k „ )
(*+ - H0® ]
-TT
o r, d e fin in g e
^ma n3
i£ x
= z , d z = iz d x we h a v e t h e f i n a l form
z
(E+ ^ ll)
2'TTi
2-1 I +
"I
l-l
I E*1" -
0
^-I
(k )j
’
' 1 a 3
Ac.B(E + ’ z)
( 22 )
det(E+ - H (z))
2TTi
w h e re
ag
_
( e , z ) i s t h e c o f a c t o r o f (E
+
- Hq (Z ))
.
The s p e c i f i c a t i o n o f Icll=O i s done when t h e b u l k H a m ilto n ia n
k a lo n g t h e a p p r o p r i a t e sym m etry l i n e ,
The d e te r m i n a n t o f [ e + - Hq (Z )j
o f o r d e r tw e lv e o r s o .
/
(k ) f o r
( e . g . , A, Z o r A) i s e v a l u a t e d .
u s u a l l y i n v o lv e s s y m m e tric p o ly n o m ia ls
The s y m m e tric p r o p e r t y comes fro m t h e c u b ic
s tru c tu re of c ry s ta ls :
p u r e im a g in a r y .
th e m a t r ix e le m e n t o f Hq (Ic) i s a lw a y s r e a l o r
-f
Know ing th e r o o t s o f d e t( E -H q ( z ) ) a n d u s in g th e
r e s i d u e th e o re m th e n u m e r i c a l e v a l u a t i o n o f t h i s i n t e g r a t i o n i s
p e rf o r m e d s t r a i g h t f o r w a r d l y .
The N ew ton-R aphson m ethod w as fo u n d t o
b e a d e q u a te t o f i n d t h e r o o t s o f th e above p o ly n o m ia l.
fo r d e ta ils .
See A p p e n d ic e s
IV .
A.
ONE-DIMENSIONAL DENSITIES OF STATES AT LOW INDEX SURFACES
W(OOl) F ace
F i g . 1 1 - a shows th e n o n - r e l a t i v i s t i c APW e n e rg y b a n d s ' o f W(OOl)
u s e d t o o b t a i n th e S l a t e r - R o s t e r p a r a m e t e r s .
R e l a t i v i s t i c APW (RAPW)
b a n d s u s e d b y C h r i s t e n s e n a r e shown i n F i g . 11-b t o i l l u s t r a t e
th e
e s s e n t i a l d i f f e r e n c e b e tw e e n th e two c a l c u l a t i o n s f o r t h e p u r p o s e o f
fin d in g s u rfa c e s t a t e s o r re s o n a n c e s.
bands th e re a re th re e
A lo n g th e A
l i n e o f th e APW
A b a n d s made o f s , p and d . 2 2 h y b r i d s .
I
z
3z - r
U sin g t h e s e S l a t e r - R o s t e r p a r a m e te r s u n h y b r id i z e d b a n d s a r e r e p r o d u c e d
i n F ig .
12.
We s e e t h a t t h e p o r b i t a l s h a v e a ^c o n s i d e r a b l e e f f e c t on
th e r e s u l t a n t b a n d s .
o f th e s u rf a c e
B o th p
z
and d„ 2 2 o r b i t a l s s t i c k s h a r p l y o u t
3z - r
( s u r f a c e n o rm a l i s i n t h e z d i r e c t i o n ) .
T h is i n d i c a t e s
t h e s u r f a c e w i l l h a v e a p ro n o u n c e d e f f e c t on th e s e s t a t e s .
a b s o l u t e gap w id th p r o d u c e d by h y b r i d i z a t i o n i n th e
The l a r g e
b a n d s p e r m i ts
a l o c a l i z e d s t a t e t o s p l i t o f f fro m th e b u l k b a n d e d g e .
O n e - d im e n s io n a l DOS f o r
A^ u s in g th e r e s o l v e n t m eth o d d e s c r i b e d
i n C h a p te r I I I a r e d i s p l a y e d i n F i g .
t o th e f l a t p a r t o f t h e b a n d s .
13
65
.
The BDOS h a s f o u r p e a k s due
A s m a l l p o s i t i v e im a g in a r y p a r t . 27eV
(FWHM) h a s b e e n a d d e d t o t h e e n e rg y t o s im u l a t e c o l l i s i o n a l b r o a d e n in g .
I n t h e a b s e n c e o f t h i s b r o a d e n in g th e BDOS p e a k s a p p e a r a s i n v e r s e
sq u a re ro o t s i n g u l a r i t i e s .
The f i r s t l a y e r d e n s i t y o f s t a t e s
d i f f e r s d r a m a t i c a l l y fro m t h e BDOS.
(SDOS)
A l l b a n d ed g e s i n g u l a r i t i e s i n
ENERGY
(RYDBERGS)
42
F ig u r e 1 1 .
APW ( a ) and RAPW (b ) e n e rg y b a n d s a lo n g
t h e A sym m etry d i r e c t i o n f o r t u n g s t e n .
43
6
4
2
CD E f = O
O
2
£
4
6
8
F ig u r e 1 2 .
U n h v b id iz e d e n e rg y b a n d s a lo n g th e A sym m etry l i n e
c a l c u l a t e d by u s in g P i c k e t t - A l l e n - C h a k r a b o r t y s ' SK p a r a m e t e r s .
The f u l l l i n e s a r e t h e h y b r i d i z e d b a n d s .
TUNGSTEN (OOI)
ONE-DIMENSIONAL
DENSITIES OF STATES
------ B(E)
E( eV)
F ig u re 13.
O n e - d im e n s io n a l d e n s i t i e s o f s t a t e s f o r
sym m etry: b u lk (d a s h e d c u r v e ) , f i r s t
l a y e r ( f u l l c u r v e ) ; a l l c u rv e s a r e n o r m a liz e d t o t h r e e e l e c t r o n s p e r atom .
i n d i c a t e th e e n e rg y l o c a t i o n o f th e c r i t i c a l p o i n t s
The arro w s
A^ ( m id p o in t o f th e l i n e )
BDOS a r e w ashed o u t an d one v e r y s h a r p p e a k 2 .3 e V b e lo w
i n th e a b s o lu te spd h y b r id i z a t io n g ap .
t h e s e c o n d l a y e r DOS.
show s up
T h is p e a k i s much w e a k e r i n
By t h e e i g h t h l a y e r , t h e DOS i s v e r y s i m i l a r
t o t h e BDOS ( F ig . 1 4 ) .
One o f t h e m o t i v a t i o n s f o r t h i s w ork was t o p r o v id e an e x p l a n a t i o n
f o r t h e e x i s t e n c e o f t h e s u r f a c e p e a k s bey o n d t h a t o f t h e n a rr o w in g o f
t h e b u lk e n e rg y b a n d s n e a r t h e s u r f a c e * ^ , i n a c c o r d a n c e w i t h th e
re d u c e d a to m ic c o o r d i n a t i o n , w h ic h a p p e a r s t o , s h i f t t h e b u l k c r i t i c a l
p o i n t s i n g u l a r i t i e s c l o s e t o s e v e r a l o b s e rv e d s u r f a c e f e a t u r e s .
In
c o n t r a s t t o t h i s t h e r e s o l v e n t m ethod show s a d r a m a tic m o d i f i c a t i o n
o f b a n d ed g e f e a t u r e s w h i l e t h e b a n d e d g e s a r e n o t s h i f t e d i n e n e rg y .
Our r e s u l t seem s t o a g r e e w i t h K aso w sk i
M a tth e is s
25
'.
c a lc u la tio n
25
and w i t h S m ith and
The f u l l - l i n e c u rv e i n F i g . 15 i s a tw o - d im e n s io n a l e n e rg y
b a n d gap s t r u c t u r e
c a lc u la tio n *
24
a lo n g A p ro d u c e d by SK p a r a m e te r s u s e d i n t h e ab o v e
The s u r f a c e s t a t e s / r e s o n a n c e s o b t a i n e d b y t h e s l a b
show up i n t h e b a n d g a p s .
I n some AKPS e x p e r i m e n t a l r e s u l t , b e s i d e s t h e F e rm i p e a k , a n o th e r .
s u r f a c e p e a k a b o u t 4 .3 e V b e lo w E a lo n g A
I
4 ).
h a s b e e n o b s e rv e d
B o th p e a k s a r e s t r o n g i n p - p o l a r i z e d l i g h t .
19 7
'
( F ig .
To e l u c i d a t e th e
p h y s i c a l o r i g i n o f t h e two s u r f a c e ,- .s e n s i t i v e p e a k s , s p i n - o r b i t c o u p lin g
Ay ty p e gap s t a t e s o r Ag an d A^ c r o s s - o v e r s t a t e s
25—28
w e re p ro p o s e d ■
f o r t h e F e rm i p e a k a n d a A^ o r A^ gap s t a t e was a s s i g n e d f o r t h i s s e c o n d
T U N GS T E N f OOn : A 1
ONE-DIMENSIONAL
DENSITIES OF STATES
BULK
EI GHTH
LA YER
FOURTH
LA Y ER
S E C ON D
LAYER
FIRS T
LA Y E R
E NE RGY(eV)
F ig u r e 14.
th e b u lk .
O n e - d im e n s io n a l DOS f o r
sym m etry f o r 1 s t , 2 n d , 4 t h , 8 th l a y e r s and
The s t r u c t u r e i d e n t i f i e d a s a s u r f a c e f e a t u r e i s s h a d e d .
47
ELECTRON E NERGYleVI
W(OOl)
F ig u r e 1 5 .
Two d im e n s io n a l b a n d s t r u c t u r e f o r th e
p r o m in e n t s u r f a c e s t a t e s / r e s o n a n c e s on th e c l e a n W(OOl)
s u r f a c e o b t a i n e d by a s l a b
c a lc u la tio n
(R e f.2 5 ) .
48
peak.
I n b o t h c a s e s t h e c o r r e s p o n d in g gap s t a t e s w e re o b t a i n e d
a p p r o x im a te ly 0 .4 e V b e lo w E^1 w h ic h w ould e x p l a i n t h e l o c a t i o n o f th e
F e rm i p e a k .
H ow ever, s u c h i n t e r p r e t a t i o n s f i n d a s e r i o u s d i f f i c u l t y i n
e x p l a i n i n g t h e s h a r p c o n t r a s t i n t h e o b s e rv e d p o l a r i z a t i o n d e p e n d e n c e :
The s p i n - o r b i t c o u p lin g m ix e s A^, A^' and A^ sym m etry and fo rm s A^
s ta te s .
I m p o r t a n t l y enough t h i s A^ s t a t e d o e s n o t c o n t a i n A^ sym m etry.
H ow ever, due t o t h e s e l e c t i o n r u l e d i s c u s s e d i n C h a p te r I I , i f
th e peak
i s n o t o b s e rv e d i n t h e EDC f o r s - p o l a r i z e d l i g h t t h e r e l e v a n t i n i t i a l
s t a t e m u st b e A^ t y p e .
B e s id e s t h e a b s o l u t e A^ ty p e gap t h e r e i s a
A-j ty p e r e l a t i v e gap i n t h e RAPW b a n d s ( F i g . 1 1 -b ) b u t t h e SDOS
p ro d u c e d b y s u c h r e l a t i v e gap s t a t e s a r e e x p e c te d t o b e s m a l l .
g a p s p r o d u c e d by r e l a t i v i s t i c
e f f e c t s a r e n a rr o w .
T h ese
I f th e h y b r id iz a tio n
gap i s n a r r o w , th e n t h e b i n d i n g e n e rg y o f t h e gap m u st a l s o b e s m a l l.
The b i n d i n g e n e rg y o f t h e gap s t a t e d e te r m in e s t h e d e c a y l e n g t h , w h ic h
i s e s s e n t i a l l y t h e im a g in a r y p a r t o f t h e c r y s t a l momentum.
Thus a
s t a t e i n a n a rro w gap m u st e x te n d d eep i n t o th e c r y s t a l and c a n n o t h a v e
a l a r g e a m p litu d e on t h e s u r f a c e l a y e r s .
p ro d u c e a l a r g e p e a k i n t h e SDOS.
Such a s t a t e c o u ld n o t
A s i m i l a r a rg u m e n t h o ld s a g a i n s t t h e
A^ and A^ c r o s s - o v e r s t a t e i n t e r p r e t a t i o n ^ ^
c ase a ls o f a i l s
M o re o v e r t h e l a t t e r
t o e x p l a i n t h e o b s e rv e d a n g le d e p e n d e n c e o f th e
i n t e n s i t y , a s m e n tio n e d ab o v e
20 7
' .
LCAO ty p e c a l c u l a t i o n ^ ’ ^
o r K a s o w s k i's c a l c u l a t i o n ^ c o n s i s t e n t l y
49
f i n d one s u r f a c e s t a t e a t
p a lo n g t h e (0 0 1 ) f a c e .
T h is f a c t seem s t o
i n d i c a t e a common f e a t u r e o f su c h m odels w h ich e x c lu d e s t h e phenom ena
a s s o c ia te d w ith th e seco n d p e a k .
N o g u e ra e t a l . ' s
w h ic h i n c l u d e s t h e f i n a l s t a t e s , e x p l i c i t l y
r e l a t e d EDC s t r u c t u r e s .
c a lc u la tio n ^ ,
r e v e a l s two s u r f a c e -
They i n t e r p r e t th e s e c o n d p e a k a s o r i g i n a t i n g
i n a v e r t i c a l t r a n s i t i o n b e tw e e n an o c c u p ie d B lo c h wave v e ry n e a r
t h e s p d -2 b a n d ed g e and an em pty e v a n s c e n t B lo c h w a v e .
z
do n o t p r o p a g a t e i n t o th e c r y s t a l ;
Such w aves
th u s i t i s a l s o a s u r f a c e e f f e c t .
A t t h i s moment I am n o t a b le t o g iv e any c o n c lu s i v e s t a t e m e n t on th e
p h y s i c a l o r i g i n o f t h i s p e a k , h o w e v e r, N o g u e ra e t a l . ’ s i n t e r p r e t a t i o n
seem s t o a g r e e w i t h Jam es A n d e r s o n 's r e c e n t ARPEDC d a t a i n w h ic h th e
second peak i s
ra th e r in s e n s itiv e
s u r f a c e s t a t e i s a llo w e d o n ly a t
t o t h e change i n k „ : th e
gap
p w h e r e a s t h e t r a n s i t i o n from th e
lo w e s t b a n d e d g e t o an e v a n e s c e n t w ave s t a t e i s a llo w e d a l l o v e r
a n d t h e ch an g e i n t h e l o c a t i o n o f s u c h e d g e i s
One d i f f i c u l t y
A
c o n tin u o u s ( F i g . 1 6 ) .
re m a in s h o w e v e r, i n a s s i g n i n g th e p h y s i c a l o r i g i n
o f t h e F e rm i p e a k t o s p d (A^) ty p e h y b r i d i z a t i o n gap s t a t e ,
t h e r e i s a l a r g e d i s c r e p a n c y i n th e l o c a t i o n i n e n e r g y .
in th a t
However
t h e r e a r e some c a l c u l a t i o n s and e x p e r i m e n t a l e v id e n c e w h ic h i n d i c a t e
a l a r g e e f f e c t o f s u r f a c e r e l a x a t i o n o r c h a rg e r e d i s t r i b u t i o n
t h is p a r t i c u l a r p la n e :
re la x a tio n
up
in to
e ffe c ts
th e e n e rg y
in
K a s o w s k i's c a l c u l a t i o n ^ ^ shows t h a t s u r f a c e
c o u ld
re g io n
c a u se
w h e re
th e
th e
gap s t a t e
o b s e rv e d
to s h i f t
peak
o c c u rs.
Ln
O
r
F ig u r e 1 6 .
H
G
G
APW e n e rg y b a n d s p a r a l l e l to t h e (0 0 1 ) sym m etry l i n e ,
a) k „ a = 0 , b ) ktla = 1 / 4 it, c) k „ a = 1 /2 tt .
- s y m m e trie s e x i s t .
N o te t h a t
o p e n s an a b s o l u t e g a p .
and
When k„
4 0 o n ly + and
c r o s s - o v e r a t k,,
4 0
51
A m o d el c a l c u l a t i o n done b y N o g u e ra e t a l . a l s o shows t h a t 13% in w a rd
r e l a x a t i o n f o r a w id e r a n g e o f t h e s l o p e p a r a m e t e r o f t h e s u r f a c e
p o t e n t i a l b r in g s th e s u rf a c e peak to E ^.
o f s u r f a c e r e l a x a t i o n Lee e t a l . ^
F o r t h e e x p e r i m e n t a l e v id e n c e
r e p o r te d t h a t t h e i r e x p e rim e n ta l
d a t a t o g e t h e r w i t h a LEED c a l c u l a t i o n show s th e c le a n s t a t e
o f W(OOl)
su rfa c e is
c o m p re s se d b y a b o u t 11%, s u c h t h a t t h e to p l a y e r s p a c in g i s
o
r e d u c e d b y 0 .1 8 ± 0 .0 3 A .
The d o u b l e t A r. b a n d s a r e made b y th e h y b r i d i z a t i o n o f p and d
5
.
.
x
xz
( o r Py a n d d
).
The m ix in g i s r e a s o n a b ly s m a l l .
i s o b t a i n e d i n t h i s sym m etry ( F i g . 1 7 ) .
re fle c ts
fla t
b a n d ed g e n e a r F ^ ^ ’ .
No l o c a l i z e d s t a t e
N o te th e p ro n o u n c e d p e a k w h ich
B o th A ^ and A^ s t a t e s a r e
a llo w e d i n i t i a l s t a t e s d e p e n d in g on t h e i n c i d e n t l i g h t p o l a r i z a t i o n ^ .
T r a n s i t i o n s fro m o t h e r s i n g l e b a n d s a r e n o t a llo w e d .
F o r c l o s e r c o m p a ris o n o f th e p r e s e n t c a l c u l a t i o n w i t h e x p e rim e n ts
an a t t e m p t h a s b e e n made t o i n c l u d e t h e e f f e c t s o f e l e c t r o n - e l e c t r o n
s c a t t e r i n g on t h e DOS c u r v e s .
I n a c c o r d a n c e w i t h t h e t h r e e s t e p m odel
th e DOS o v e r th e f i r s t e i g h t l a y e r s a r e w e ig h te d a c c o r d in g t o th e
f o l l o w i n g e q u a ti o n
8
’w e ig h te d sum ’ =
£ e ^
A=I
8
x DOS ( f t ) /
Ji
e ^
A=I
The a t t e n u a t i o n c o e f f i c i e n t s w e re o b t a i n e d by e x t r a p o l a t i n g t h e d a t a o f
S te m
a n d S i n h a r o y * ^ * ^ : W(Mo) = 0 . 4 a n d W(W) = 0 . 3 p e r l a y e r .
The
w e ig h te d sums f o r n o rm a l e m is s io n f o r A^ a n d A^ a r e i n F i g . 18 w h ic h
TUNGSTEN (00!): A.
ONE-DI MENSl ONAL
DENSITIES OF STATES
BULK
FOURTH
LA Y E R
SECOND
LAYER
/
Fl R S T
LA Y E R
E NE RGYIeV)
F i g u r e 17.
O n e - d im e n s io n a l DOS f o r
a r e fo u n d b e lo w E
F'
sym m etry.
No s u r f a c e s t a t e s
53
W EIGHTED
TUNGSTEN
SUM
(OOl)
__ I________________ I _________ "
-G
-4
ENERGY
F ig u r e 18.
I------------------T - = I
-2
( e V)
W e ig h te d sums o f th e l a y e r DOS f o r
A5 C dashed c u rv e ) sy m m etry .
O
f u l l c u rv e ) and
The f i r s t e i g h t l a y e r s a r e in c l u d e d
a c c o r d in g t o th e e x p o n e n t i a l e s c a p e p r o b a b i l t y due t o e l e c t r o n e le c tro n s c a tte r in g .
la y e r.
The a t t e n u a t i o n c o e f f i c i e n t y (W )= 0 .3 p e r
The s h a d e d p e a k c o rr e s p o n d s t o t h e
su rfa c e s ta t e .
h y b r i d i z a t i o n gap
54
c o n t a i n o n ly c o n t r i b u t i o n s fro m s t a t e s o f
A1 o r
Ac sy m m etry .
In
o r d e r t o com pare t h e s e c u rv e s w i t h t h e r e s u l t s o f p h o to e m is s io n
e x p e r im e n ts w h ic h do n o t d i f f e r e n t i a t e b e tw e e n s y m m e tr ie s , t h e s e two
c u rv e s m u st b e a d d e d .
B.
M o (IlO ) a n d W ( I l l ) F ace
T h e re a r e f o u r t i g h t - b i n d i n g e n e rg y b a n d s o f
d e r i v e d fro m a to m ic o r b i t a l s o f t h e ty p e s , p
z
^ sym m etry
, d 2 and d 2 2 ( th e
z
x -y
z - a x i s i s n o rm a l t o t h e s u r f a c e p l a n e ) .■ T h ese b a n d s i n M o(or W) show
w id e h y b r i d i z a t i o n gaps b e lo w E^1 ( F ig .
1 9 ).
Our r e s u l t s f o r
o n e - d im e n s io n a l l a y e r DOS on M o(IlO ) a r e shown i n F i g . 2 0 .
th e
The f i r s t
l a y e r DOS h a s a p e a k A due t o t r u e s u r f a c e s t a t e n e a r th e b o tto m o f
th e sp d g ap .
The e n e rg y l o c a t i o n o f t h i s p e a k , 4 . 22eV b e lo w E ^,
to g e th e r w ith i t s
l a r g e a m p litu d e on th e s u r f a c e l a y e r , s u p p o r t s t h e
i n t e r p r e t a t i o n o f t h e 4 . 5eV p e a k i n t h e n o rm a l e m is s io n d a t a
b e in g due t o t h i s h y b r i d i z a t i o n gap s t a t e .
32
, as
A seco n d c a lc u la te d peak
l a b e l e d B, due t o q u a s i - d e g e n e r a t e b a n d e d g e s I . IeV b e lo w E , c o u ld
c o n tr ib u te to th e t r i p l e t s tr u c t u r e o f C in ti e t a l .
32
.
i m p o r ta n t e x c e p t io n t o t h e r u l e t h a t b p l k s i n g u l a r i t i e s
th e s u r f a c e .
a r e rem oved a t
I t s p e r s i s t e n c e seem s t o b e due to t h e n o n -m o n o n ic s h a p e
o f th e seco n d band n e a r th e
but lie s
P e a k B i s an
ab o v e E
F*
F ^ ^ 'e d g e .
Peak C h as a s im ila r o r ig in b u t
ENERGY (RYDBERGS)
55
----- I--- 4 - . i -
I
N x
F ig u r e 19.
i_
I
MAm
■
I
r
APW ( a ) and RAPW (b ) e n e rg y b a n d s a lo n g
Z sym m etry d i r e c t i o n f o r t u n g s t e n .
One-Dimensional
Densities of State
MOLYBDENUM (HO)= I 1
BULK
EIGHTH
FOURTH
LAYER
SECOND
LAYER
/
FIRST
LAYER
ENERGY (eV)
F ig u r e 2 0 .
O n e - d im e n s io n a l DOS f o r th e
b a n d s o f M o (I lO ) .
i d e n t i f i e d a s s u r f a c e f e a t u r e s a r e l a b e l e d A, B, C.
n o r m a liz e d t o 4 e l e c t r o n s p e r atom .
S tru c tu re s
The c u r v e s a r e
57
F o r tH e
A I t h e , t h e twx> b a n d s y m m e trie s o f i n t e r e s t a r e
A ^( F ig . 2 1 ) .
fro m
The sym m etry l i n e a p p r o p r i a t e t o n o rm a l e m is s io n e x te n d s
T to P a lo n g
A. a n d fro m P t o H a lo n g F i n th e B r i l l o u i n z o n e .
The a to m ic o r b i t a l s s . p^ and d ^ z 2 _ r 2 fo rm a b a s i s f o r
p , d
and d
fo r
x
xz
xy
d ire c tio n .
and 1
A^ (F ^) and
A ( F ) , i f t h e z a x i s i s a lo n g t h e (11 1 )
u
j
F i g . 22 a) shows c l e a r l y t h e re m o v a l o f b u l k s i n g u l a r i t i e s
n e a r t h e s u r f a c e a s w e l l a s th e e m e rg en c e o f a s u r f a c e r e s o n a n c e in
th e sp d gap.
T h is p e a k , l a b e l e d A, i s d e g e n e r a te w i t h b a n d s t a t e s o f
F^ sym m etry i n t o w h ic h i t
can d e c a y , so i t i s n o t a t r u e s u r f a c e s t a t e .
P e a k B i n F i g . 22 b ) , on th e o t h e r h a n d , i s due t o a s u r f a c e s t a t e i n
th e a b s o l u t e gap fo rm ed by t h e
A^ b a n d s .
We a s s i g n p e a k A and B, a t
I . 36eV a n d . 75eV b e lo w E^1, r e s p e c t i v e l y ^ t o th e n o rm a l e m is s io n
s t r u c t u r e s a t 1 .2 and . 7eV on W ( I l l ) ;
a w id e r a n g e o f p h o to n e n e r g y .
th e l a t t e r a re s t a t i o n a r y ' o v er
The DOS p e a k s C an d D i n F i g . 22b) a l s o
h a v e s u b s t a n t i a l a m p litu d e on th e s u r f a c e p l a n e .
w i t h q u a s i - d e g e n e r a t e b a n d e d g e s a n d do
c u r r e n t s i n c e th e y l i e
They a r e a s s o c i a t e d
n o t c o n t r i b u t e t o t h e p h o to ­
ab o v e E ^.
The w e ig h te d sums o f l a y e r DOS f o r n o rm a l e m is s io n a r e g iv e n i n
F i g .23.
The m a jo r s p e c t r a l f e a t u r e s o f th e SDOS i n F i g . 20 and 22 a r e
r e t a i n e d i n t h e s e sum s.
The s t r o n g p e a k i n F ig . 23 a) I . IeV
'below E^1 i s m a in ly due t o t h e b u l k b a n d s i n g u l a r i t y .
The b r o a d p e a k a t
- 2 . 7eV i n F i g . 23 c) may b e a s s o c i a t e d w i t h th e p r o m in e n t e m is s io n
(eV)
ENERGY (RYDBERGS)
F ig u re 2 1 .
APW ( a ) and RAPW (b ) e n e rg y b a n d s a lo n g
A sym m etry f o r t u n g s t e n .
59
F ig u re 2 2 .
O n e - d im e n s io n a l DOS f o r
Ag b a n d s o f W ( I l l ) .
a) t h e
b a n d s and
b ) th e
B and e d g e s a lo n g th e (1 1 1 ) sym m etry l i n e
(A + F) a r e i n d i c a t e d .
The c u rv e s a r e n o r m a liz e d t o 3 (A^) and
6 (Ag) e l e c t r o n s p e r ato m .
60
(a)
tu n g sten
(in) a ,
One-Dimensional
Densities of States
BULK
EIGHTH LAYER
FOURTH LAYER
FIRST LAYER
-2
O=Ef
ENERGY (eV)
(b) TUNGSTEN (lll); A
A3. F;
4 \
BULK
EIGHTH
LAYER
FOURTH
\L A Y E R /
SECOND
LAYER
FIRST
LAYER
-2
O=Ef
ENERGY (eV)
I
WEIGHTED SUMS
(a) MOLYBDENUM
(IIO) Z, A
(b)TUNGSTEN (III): A
(c) TUNGSTEN (III): A
-6
F ig u r e 2 3 .
sy m m etry .
-4
-2
ENERGY (eV)
O=Ef
W e ig h te d sums o f th e l a y e r DOS f o r
The f i r s t e i g h t l a y e r s a r e i n c l u d e d .
Ag and
The
a t t e n u a t i o n c o e f f i c i e n t s y(Mo) = 0 .4 and y(W) = 0 . 3 p e r
la y e r.
62
s t r u c t u r e n e a r t h i s e n e rg y on W ( I l l ) .
The
(Ag) s t a t e s h o u ld b e s t r o n g e r i n p - ( s - ) p o l a r i z a t i o n ,
f o r n o n - n o rm a l i n c i d e n c e o f th e e x i s t i n g l i g h t .
The Ag p e a k i s
th e
o n ly c a s e among t h e t h r e e lo w - in d e x f a c e s w h ic h i s e x p e c t e d po b e
s tro n g e r in s -p o la riz e d l ig h t .
I n o r d e t t o make a d e f i n i t i v e t e s t
o f t h e i s s u e , Jam es A n d e rso n r e c e n t l y p e rf o r m e d an e x p e r im e n t u s in g
p o la riz e d lig h t
31
.
He fo u n d a s u r f a c e s e n s i t i v e p e a k - 0 .4 e V i n s -
p o l a r i z e d l i g h t w i t h p h o to n e n e rg y hv =
13.5eV ( F i g . 24) an d - 0 . 8eV
f o r p - p o l a r i z e d l i g h t w i t h p h o to n e n e r g y hv = 16eV ( F i g . 2 5 ) .
D e f i n i t e l y t h e f i r s t p e a k s h o u ld b e due t o a Ag s t a t e on t h e b a s i s o f
■5
th e s e l e c t i o n r u le s
p r e d ic tio n w e ll.
.
Thus h i s e x p e r im e n ts seem t o s u p p o r t th e
63
EMI SS I ON
N l E , , hv)
C O U N T S / I NCI DENT PHOTON
NORMAL
F ig u r e 2 4 .
EDC e m i t t e d n o rm a l t o th e W ( I l l ) c le a n ( f u l l l i n e )
H c o v e re d ( d a s h e d l i n e ) s u r f a c e f o r s - p o l a r i z e d l i g h t .
w e re ta k e n b y Jam es A n d e rs o n .
and
The d a t a
N (Ei, hv) C O U N T S / I N C I D E N T PHOTON
64
W( I I I ) hv =1 6eV
NORMAL EMI SSI ON
E, (eV)
F ig u r e 2 5 .
EDC e m i t t e d n o rm a l t o th e W ( I l l) c l e a n ( f u l l l i n e )
H c o v e re d ( d a s h e d l i n e ) s u r f a c e f o r p - p o l a r i z e d l i g h t .
w e re ta k e n by Jam es A n d e rs o n .
and
The d a ta
V.
A.
DISCUSSION
G e n e r a l Rem arks
So f a r o n e - d im e n s io n a l d e n s i t i e s o f s t a t e s fro m c le a n S h o c k le y
s u r f a c e s o f b c c t r a n s i t i o n m e t a ls on th e t h r e e lo w - in d e x f a c e s h a v e
been o b ta in e d .
The c a l c u l a t i o n h a s b e e n done f o r t h e s e m i - i n f i n i t e
s y s te m u s in g t h e r e s o l v e n t m ethod t o g e t h e r w i t h S l a t e r - R o s t e r t i g h t b in d in g p a ra m e te rs f i t t e d
t o n o n - r e l a t i v i s t i c APW e n e rg y b a n d s . .
The m o n o to n ic b u l k b a n d s i n g u l a r i t i e s a r e c o m p le te ly rem oved i n th e
f i r s t la y e r d e n s ity o f s ta t e s
(SDOS).
Only one S h o c k le y s u r f a c e
s t a t e / r e s o n a n c e w as fo u n d i n e a c h h y b r i d i z e d g a p , n a m e ly , A ^ (O O l),
2 ^ (1 1 0 ), A ^ ( I ll)
and A ^ ( I l l ) .
A c c o rd in g t o th e f i n a l s t a t e sym m etry
and t h e s e l e c t i o n r u l e s d i s c u s s e d i n C h a p te r I I
, th e t r a n s i t i o n
Ag ty p e i n i t i a l s t a t e s h o u ld b e s t r o n g e r i n s - p o l a r i z e d l i g h t .
from
T h is i s
th e o n ly ex am p le we h a v e fo u n d , w h ic h p r e d i c t s a s u r f a c e s t r u c t u r e
in s - p o la r iz e d l i g h t .
T a b le 2 su m m a riz e s o u r w ork ( i n d i c a t e d by HR), and im p o r ta n t
e x p e r im e n ts and c a l c u l a t i o n s done b y o t h e r s .
s t a t e e n e r g i e s i n E ( IlO ) and A ( I l l )
± 0 .6 eV .
Our c a l c u l a t e d s u r f a c e
a g r e e w i t h e x p e r im e n ts w i t h i n
We a rg u e d t h a t t h e o b s e rv e d F e rm i p e a k i n A(OOl) s h o u ld be
due t o a A^ gap s t a t e a lth o u g h th e A^ p e a k l o c a t i o n i s
s h ifte d
( ~ - 1 .9 e V ) .
c o n s id e r a b l y
The s e c o n d o b s e r v e d p e a k - 4 . SeV i n A(OOl)
come fro m o u t s i d e o f o u r m o d e l.
s h o u ld
Our n u m e r i c a l v a lu e o f t h e A^ gap
Mti V/ I
.66
T a b le 2 .
O b se rv e d EDC/c a l c u l a t e d SDOS p e a k l o c a t i o n s fro m lo w -in d e x
f a c e s o f W an d Mo; p ( s ) i n s i d e t h e b r a c k e t i n d i c a t e s s t r o n g p - ( s - )
p o l a r i z a t io n dependence o f th e p eak .
E ach c a l c u l a t i o n p r e d i c t s t h e
n a t u r e o f t h e p e a k a s d e s c r i b e d i n t h e f o u r t h colum n.
S u rfa c e
EDC p e a k s
C a lc u la tio n
Rem arks
W(OOl)
- 0 . AeV(p)
-O.AeV15
Ay gap s t a t e s , two p e a k s ,
p a r a m e te r d e p e n d e n t. G reen
fu n c tio n c a lc u la tio n .
- I e V 25
k M^ 0 , Ag and A^ c r o s s - o v e r
gap s t a t e , s l a b c a l c u l a t i o n .
.
- 2 . IeV 25
A^ gap s t a t e .
- 2 . 3eV (HK)
A^ gap s t a t e , p - p o l a r i z a t i o n .
M o(IlO )
- 4 .5 e V 32
-4.2eV (H K )
W ( I l l)
- 0 .7 e V 18
- 0 . 7 SeV(HK)
. Ag gap s t a t e , s - p o l a r i z a t i o n .
- 1 .2 e V 18
- 1 . 36eV(HK)
gap s t a t e , p - p o l a r i z a t i o n .
31
- 0 . AeVj i
,
-0 .8 e V
31
gap s t a t e p - p o l a r i z a t i o n .
s -p o la riz a tio n .
p -p o la riz a tio n .
67
s t a t e e n e r g y a g r e e w i t h o t h e r c a l c u l a t i o n s f o r th e u n r e l a x e d s u r f a c e .
The i n t e r p r e t a t i o n o f t h e p e a k n e a r E
g iv e n by u s seem t o b e th e m ost
n a t u r a l one and r e a d i l y e x p l a i n s th e o b s e rv e d p o l a r i z a t i o n d e p e n d en c e
a s w e l l a s t h e p e a k i n t e n s i t y a s a f u n c t i o n o f k M.
I t i s r a t h e r s u r p r i s i n g t o s e e s u c h good a g re e m e n t o f th e
c a l c u l a t i o n w i t h e x p e r im e n ts i n s p i t e o f th e o b v io u s s i m p l i f i c a t i o n
made i n t h e ab o v e m o d e l.
o n e - e l e c t r o n m o d e l;
M ost i m p o r t a n t l y , p r e s e n t c a l c u l a t i o n assum es
t h i s e l e m i n a t e s a l l m any-body e f f e c t s s u c h a s
s u r f a c e r e c o n s t r u c t i o n o r c h a rg e r e d i s t r i b u t i o n .
t h i s a s s u m p tio n seem s t o a p p e a r in t h e
lo c a tio n o f th e
o b s e rv e d o n e .
An in a d e q a c y o f
A sym m etry w h e re th e e n e rg y
b a n d gap s t a t e i s c o n s i d e r a b l y lo w e r th a n th e
As' m e n tio n e d i n C h a p te r IV , s u r f a c e r e c o n s t r u c t i o n
seem s t o b e n o n - n e g l i g i b l e i n th e (0 0 1 ) f a c e and i f t h i s i s i n c l u d e d ,
t h e r e s u l t a n t p e a k l o c a t i o n seem s t o b e p u s h e d up c l o s e t o t h e one
o b se rv e d .
The o n ly p o s s i b l e way t o i n c l u d e a Tamm p o t e n t i a l w i t h i n
t h e t e c h n i q u e u s e d h e r e i s t o i n t r o d u c e some ad h o c p a r a m e t e r i n t o
t h e c a l c u l a t i o n s i n c e we do n o t know t h e p r e c i s e fo rm o f th e Tamm
p o t e n t i a l o r o f th e b a s is f u n c tio n s .
F i r s t p rin c ip le
c a lc u la tio n s
i n c l u d i n g m any-body e f f e c t s an d s e l f - c o n s i s t e n c y w i l l becom e
m ore c r u c i a l f o r th e c a s e o f c h e m is o rb e d s u r f a c e s .
F o r c le a n s u r f a c e s ,
on th e o t h e r h a n d , m any-body e f f e c t s seem to i n t r i d u c e some q u a n t i t a t i v e
d i f f e r e n c e i n th e p o s i t i o n o f th e s u r f a c e s t a t e s b u t do n o t.s e e m to
68
c h a n g e t h e q u a l i t a t i v e f e a t u r e s s u c h a s t h e num ber o f s u r f a c e s t a t e s .
The s p i n - o r b i t c o u p lin g i s a l s o i g n o r e d i n t h e p r e s e n t
c a lc u la tio n .
The g a p s i n t r o d u c e d by t h e s p i n - o r b i t c o u p li n g a r e an
o r d e r o f m a g n itu d e s m a l l e r th a n t h e h y b r i d i z a t i o n g a p s .
A lth o u g h t h e
t r u e e n e rg y b a n d s s h o u ld r e s e m b le m ore c l o s e l y t h e o n e s w i t h s p i n o r b i t c o u p lin g i n c l u d e d , i t seem s t h a t t h e e f f e c t o f t h e i n t r o d u c t i o n
o f s p i n - o r b i t c o u p lin g i s s l i g h t .
The s u r f a c e s t a t e s due t o su c h
s m a ll g a p s w o u ld seem t o b e b u r i e d b e lo w t h e l i m i t s o f e x p e r i m e n t a l
r e s o lu tio n .
F o r ex am p le a lo n g A t h e r e a r e two g ap s j u s t b e lo w th e
F e rm i l e v e l i n t h e RAPW b a n d s ( F ig . 1 1 ) ; s t r i c t l y
s p e a k in g one m ig h t
e x p e c t two p e a k s ; b u t a c t u a l l y o n ly o n e p e a k h a s b e e n o b s e r v e d .
S i m i l a r s i t u a t i o n s e x i s t i n A w h e re n o n - r e l a t i v i s t i c
c ro ss-o v e rs
A^ o r F ^ , F^
( F i g . 20) p ro d u c e e x t r a g a p s i n t h e RAPW b a n d s .
t h e s e g a p s , h o w e v e r, a r e to o n a rro w t o s p l i t th e p e a k s
A ll
31
One o f t h e a d v a n ta g e s o f t h e s e m i - i n f i n i t e r e s o l v e n t c a l c u l a t i o n
p re s e n te d h e re i s
shown i n F i g . 1 4 .
t h e c o n tin u o u s n a t u r e o f th e r e s u l t a n t s p e c tr u m as
The s l a b c a l c u l a t i o n on t h e o t h e r h a n d i s done
by d ia g o n a liz in g a la r g e m a trix .
The s i z e o f th e m a t r i x i s d e te r m in e d
by t h e num ber o f l a y e r s ta k e n and t h e num ber o f s t a t e s a s s o c i a t e d w i t h
each la y e r .
The s p e c tr u m o b t a i n e d fro m su c h c a l c u l a t i o n i s E v s k„
i n s t e a d o f DOS v s E.
As F i g . 26 i l l u s t r a t e s ,
i s o l a t e d fro m o t h e r e n e rg y c o n ti n u e .
th e s u r f a c e s t a t e s a re
69
K11= O
DENSI TY OF
STATES
F ig u r e 2 6 .
a t k ,,= 0.
a) The d i s c o n t i n u o u s n a t u r e o f t h e s l a b c a l c u l a t i o n s
The num ber o f e n e rg y l e v e l s i s
tim e s th e num ber o f s t a t e s p e r l a y e r .
shows up in th e b a n d g a p .
band s t r u c t u r e ,
th e num ber o f l a y e r s
The s u r f a c e e n e rg y Eg
b ) A c o r r e s p o n d in g t h r e e - d im e n s i o n a l
c) F o r th e s e m i - i n f i n i t e g e o m e try t h e G reen
f u n c t i o n c a l c u l a t i o n g iv e s a c o n tin u o u s DOS f o r g iv e n k „ .
11
70
B.
S u g g e s te d C a l c u l a t i o n s
A d i r e c t n o n - t r i v i a l e x t e n s i o n o f t h i s ty p e o f c a l c u l a t i o n w o u ld
b e an a p p l i c a t i o n
to th e fe e t r a n s i t i o n
.......... among w h ic h
K. S . Sohn e t a l . ^
m e ta ls su c h a s N i, Cu, Ag, Au,
h a v e a l r e a d y c a l c u l a t e d th e
S l a t e r - R o s t e r p a r a m e t e r s (tw o c e n t e r e d i n t e g r a l s o n ly ) f o r Cu.
th e a p p l i c a t i o n i s im m e d ia te .
Thus
S in c e th e p o l a r i z a t i o n t e s t s a r e m o st
e f f e c t i v e a t h ig h sym m etry p o i n t s , k„=0 c a l c u l a t i o n s a r e t h e m o st
in te re s tin g .
H ow ever, a s one l e a v e s R11=O m o st o f th e b a n d s w h ich
c r o s s e d a t k,,=0 now open i n t o g ap s and h e n c e more s u r f a c e s t a t e s o f
S m ith -M a tth e is s -W e n g
shown i n E q . (2 2 )
ty p e a r e e x p e c t e d .
i n C h a p te r I I I .
The k„ d e p e n d e n c e o f SDOS i s
T h is ty p e o f c a l c u l a t i o n s h o u ld
a l s o b e a c h ie v e d .
The t h i r d ty p e o f a p p l i c a t i o n o f t h i s t e c h n i q u e ,
c h e m is o r p tio n
w o u ld b e l e s s s t r a i g h t f o r w a r d b u t c o u ld p o s s i b l y b e d o n e .
We
c a l c u l a t e d t h e s u r f a c e G reen f u n c t i o n u s in g th e b u l k G reen f u n c t i o n i n
C h a p te r I I I .
B ased on th e same p h i lo s o p h y , t h e n , one s h o u ld b e a b le
t o c o n s t r u c t t h e G reen f u n c t i o n f o r th e c h e m is o rb e d s u r f a c e .
H ow ever,
t o p e r f o r m t h i s c a l c u l a t i o n , we f i r s t n e e d a c o m p le te k n o w led g e o f th e
G reen f u n c t i o n f o r c l e a n s u r f a c e i n c l u d i n g th e c o n t r i b u t i o n from th e
Tamm p a r t o f t h e c l e a n s u r f a c e p o t e n t i a l .
S e c o n d ly we h a v e to know
th e i n t e r a c t i o n b e tw e e n t h e c le a n s u r f a c e a n d th e i s o l a t e d
a d s o rb a te
7 2 ,7 3
APPENDIX
APPENDIX.
COMPUTATIONAL PROCEDURE .
We c a l c u l a t e th e b u l k G reen f u n c t i o n f i r s t ,
f u n c t i o n as o u t l i n e d i n C h a p te r I I I .
The p ro g ra m c o n s i s t s o f one m ain
p ro g ra m a n d m ore th a n t e n s u b r o u t i n e s .
r e q u i r e s a p p r o x im a te ly 24 g r a n u l e s
th e n th e s u r f a c e G reen
In t y p ic a l c a se s th e c a lc u la tio n
(12K) i n th e form o f l o a d m o d u le s .
CPU tim e f o r one h u n d r e d e n e rg y p o i n t s i s 1 .9 4 m in . on t h e SIGMA7.
The p ro g ra m s t o f i n d t h e c o e f f i c i e n t s o f d e te r m in a n t p o ly n o m ia ls a r e
w r i t t e n i n BASIC and t h e r e s t i n FORTRANIV.
FORTRAN- IV i s u s e d s i n c e
th e H a m ilto n ia n and G reen f u n c t i o n s a r e co m p lex .
I n th e f o ll o w i n g
s e c t i o n s th e a c t u a l c o m p u t a ti o n a l p a r t o f th e w ork i s
A.
d e s c rib e d .
H a m ilto n ia n and SK P a r a m e te r s
The e n e rg y r e g i o n o f i n t e r e s t i s b e lo w th e vacuum l e v e l .
e le c tro n s ta te s
When th e
c o n s i s t o f h y b r i d i z e d s p d a to m ic o r b i t a l s , once a
sym m etry d i r e c t i o n i s
c h o s e n f o r s tu d y i n th e B r i l l o u i n z o n e , s a y A,
one can w r i t e a 9x9 H a m ilto n ia n m a t r i x .
I f l i n e a r c o m b in a tio n s o f
a to m ic s t a t e s a r e s e l e c t e d w h ic h t r a n s f o r m a c c o r d in g t o i r r e d u c i b l e
r e p r e s e n t a t i o n o f A sy m m e try , t h e m a t r ix can b e e x p r e s s e d i n b lo c k
d i a g o n a l fo rm .
Then t h e 9x9 p ro b le m i s r e d u c e d t o one 3x3 (A ^ ), two
( d e g e n e r a t e ) 2x2 (A^) a n d two I x l (A ^, A ^ ) p r o b le m s .
The m a t r i x
e le m e n ts f o r s im p le c u b ic c r y s t a l s i n g e n e r a l d i r e c t i o n s i n k s p a c e
a r e g iv e n i n T a b le I I o f S l a t e r a n d R o s t e r ’ s p a p e r (SR)
30
.
We can
73
a p p ly t h i s t a b l e t o th e b c c s t r u c t u r e i f we lo o k a t th e n e a r e s t
n e ig h b o r s
lik e
(NN) o f a g iv e n atom a s b e in g l o c a t e d a t t h e e i g h t p o i n t s
( 1 1 1 ) , t h e s e c o n d NN t o b e a t t h e s i x p o i n t s l i k e
t h i r d NN to b e a t tw e lv e p o i n t s l i k e
l a t t i c e p a ra m e te r = 2 a .
Eq.
(2 1 ) i s k ^ a f o r
SS
A and
(0 0 0 ) + SE.
-SS
+ 4E
0,
w h ere C
( 2 0 0 ) , w h e re we c h o o se th e
N o te t h a t th e q u a n t i t y kj_ d a p p e a r in g in
A s y m m e trie s b u t 2 k ^ a f o r Z sy m m e try . .
Then f o r e x a m p le , a lo n g A, ( s / s )
(s /s ) = E
(2 0 0 ) a n d th e
i n th e n o t a t i o n o f SK becom es
( I l l ) c o s ( 2 0 + 2E
SS
(20 0 ) (2 + c o s (2 0 )
(220)(1 + Zcos(ZG)),
G = k a.
The b a s i s s e t f o r A^ c o n s i s t s o f s , z a n d 3z
2
- r
2
,
So th e s u b - m a t r i x becom es
/■
(s /s )
H(A1 )
(s /z )
(s /3 z ^ -r2)
(z /z )
( z / 3 z 2- r 2 )
7
2
=
7
7
(3 z - r /3 z - r )
A
IR
R
-IP .
B
iQ
R
-iQ
C
,
2
N o te t h a t ( s / z )
k^.
2
an d ( z / 3 z - r ) i n H(A1) a r e p u r e im a g in a r y f o r r e a l
The co m plex m a t r i x can b e tr a n s f o r m e d t o a r e a l one by a u n i t a r y
m a t r i x U,
1 0
U =
0
O
i
0
0
0
1
A p p e n d ix t a b l e I :
The s y m m e trie s and c o r r e s p o n d in g b a s e s i n t h e c o n v e n ti o n a l
c o o r d i n a t e s y s te m and w i t h t h e s u r f a c e n o rm a l i n t h e z d i r e c t i o n .
H a m ilto n ia n B a s is F u n c tio n s
S u r f a c e n o rm a l i n z d i r e c t i o n
C o n v e n tio n a l c o o r d i n a t e s y s te m
H
>
Sym metry
A5
s,
,2 .2
z , 3z - r
s,
Q 2-r 2
z , 3z
X,
XZ
X,
XZ
2 2
x -y
A0
2
2 2
x -y
xy
y
xy
2
2
s , x + y , 3z - r , xy .
.h
(x z -y z )// 2
M
LO
z , xz+ yz
2 2
x -y , x -y
I
A3
•
s , xd-y+z, xz+ yz+ xy
2 2
x -y , x -y , y z -x z
s,
■
„ 2 2
2 2
z , 3z - r , x - y
xy
X,
XZ
.y, yz
s,
z , 3z - r
x , x z , xy
”75
I n t h e a c t u a l c a l c u l a t i o n I c h o s e th e s u r f a c e n o r m a l and b a s i s
f u n c t i o n s a s su m m a riz ed in th e s e c o n d colum n o f A p p e n d ix t a b l e I .
U s in g Van V le c k -'s b a s i s f u n c t i o n g e n e r a t i n g m achine
55
we can show
t h a t t h e s e b a s e s t r a n s f o r m e x a c t l y l i k e t h e o n e s i n t h e t h i r d colum n
i f t h e s u r f a c e n o rm a l i s
s y m m e tr ie s .
c h o se n i n th e z d i r e c t i o n f o r a l l t h r e e
The s u b - m a t r i c e s o f t h e H a m ilto n ia n i n th e t h r e e sym m etry
d i r e c t i o n s o b t a i n e d u s in g th e b a s e s i n A p p e n d ix t a b l e I a r e sum m arized
i n th e f o ll o w i n g p a g e s .
U n d e r t h e u n i t a r y t r a n s f o r m a t i o n t h e m a t r i x H(A^) becom es a . r e a l
s y m m e tric m a t r i x
LR
-Q
CJ
We b e g in th e c o m p u ta tio n by c h e c k in g t o s e e i f th e SK p a r a m e t e r s
fitte d
t o APW b a n d s a c t u a l l y r e p r o d u c e t h e o r i g i n a l APW b a n d s when t h e
above H a m ilto n ia n i s d i a g o n a l i z e d .
F o r th e m a trix d ia g o n a liz a tio n
s u b p ro g ra m s SEIGEN o r DEIGEN i n th e MATHSTAT l i b r a r y a r e u s e d .
(s/s )
H( A1)
(s /z )
( s / 3 z 2- r 2 )
(z /z )'
( z / 3 z 2 - r 2)
H( A5 )
.
( x /x z )
( x z /x z )
( 3 z 2- r 2 / 3 z 2- r 2 )
N
H( A2 )
( x /x )
( x 2- y 2 / x 2- y 2 ) ,
(s/s )
H( A2 ' )
( x y /x y )
;
2 (s /x )
( s / 3 z 2- r 2)
(s /x y )
(x /x )+ (x /y )
2 ( x / 3 z 2- r 2 )
2 ( x /x y )
( 3 z 2- r 2 / 3 z 2- r 2)
( 3 z 2- r 2 /x y )
H( Zx )
(x y /x y )
H( Zg) = ( x z / x z ) —( x z / y z )
(z /z )
,
2 (z /x z )
(x /x )-(x /y )
,
(x z /x z )+ (x z /y z )
H( E ) =■
4
2 (z /x 2-y 2)
(x2-y 2/x2-y 2)
(s/s )
3 (s/x )
3 (s /x y )
( x /x ) + 2 ( x / y )
2 (x /x y )+ (x /y z )
( x y /x y ) + 2 ( x y /x z )
(x /x )-(x /y )
2 (x /x 2-y 2 )
(x /y z )-(x /x y )
(x2-y?x2-y 2)
-2 (x z /x 2-y 2)
(x y /x y )-(x y /x z )
78
B.
The C o e f f i c i e n t s o f D e te rm in a n t P o ly n o m ia ls
r— - r ;
~
We c a l c u l a t e th e b u l k G reen f u n c t i o n i n E q. (2 2 ) i n C h a p te r I I I
u s in g t h e r e s i d u e th e o re m .
5ma n@
f-1
( E ,k „ )
r/p (E , z)
2 TT i
m-n= £ .
detCE"1"- Hq Cz ) )
L et
P (z ) = z
£-1
Aag
q(z)
( E »z )»
d e t(E +- Hq Cz) )
th e n
^m cx
I
^
,
w h e re q ( z J
= 0,
| z^ | < I
q'(z.)
F i r s t we h a v e t o w r i t e th e d e te r m i n a n t o f t h e 3x3 m a t r i x w h ic h by
i t s e l f i s s e v e r a l p a g e s lo n g .
One way t o a v o id t h i s r a t h e r le n g th y
a l g e b r a and t h e e r r o r s w h ic h w o u ld f o llo w from i t ,
t h e i d e a o f p o ly n o m ia l f i t t i n g .
w h e re z = e x p ( i £
is
I f we w r i t e cos £ =
) th e d e te r m i n a n t d ( z ,E )
1 / 2 (z + 1 / z)
can b e e x p r e s s e d as
d ( z ,E ) = P q + p ^ ( z + l / z ) + p2 ( z 2 + 1 / z 2 ) +
w h ere
t o a p p ly
^ p ^ j a r e f u n c t i o n s o f E and t h e SK p a r a m e t e r s .
+ pn (z 11 + I / z11)
By l o o k in g a t
t h e m a t r i x we can e a s i l y f i n d t h e te r m o f h i g h e s t o r d e r n i n z.
M u l t i p l y i n g by z11 on d ( z , E ) , we th e n o b t a i n a p o ly n o m ia l o f d e g re e 2 n ,
Q (z ,E ) = P
+ P z + P z2 + ....................+ P
z2 n .
I n s t e a d o f f i n d i n g th e | P ^ j a l g e b r a i c a l l y we can t r e a t them a s unknow ns.
79
A s s ig n in g ( 2 n + l) a r b i t r a r y z num bers f o r g iv e n E;
P g + '' . . . . _ Zn
+ Z1 P2 „
Z1
d(.z1,E; =
P q+ z P +z
QCz2 , E) - z2
d ( z 2 ,E ) -
V z2pIfz2 V"'
Q("Z2 n + l ,E )
Z2n+1
+ z2
d ( z 2 n + l ,E ) ■ P o+Z2 n + lP l + z 2 n + lP 2+ '
• • • ' + z
N u m e ric a l v a l u e s o f d(z_^,E) a r e im m e d ia te ly
s t a t e m e n t i n BASIC.
Zn,
P 2„
^nP
2n+l
2n
’
com puted u s in g MAT INV
So b y f e e d i n g i n th e 3x3 H a m ilto n ia n m a t r i x we
can o b t a i n th e c o e f f i c i e n t s
.
S i n c e |p ^ |a r e f u n c t i o n s
o f E and
SK p a r a m e t e r s , we h a v e t o r e p e a t t h i s p r o c e d u r e f o r a r b i t r a r y E v a lu e s
A s i m i l a r p r o c e d u r e can be a p p l i e d f o r co m p u tin g th e m in o rs o f th e
m a trix E -H (k ).
C.
R o o ts o f D e te rm in a n t P o ly n o m ia ls
A t y p i c a l d e g re e o f th e d e te r m in a n t p o ly n o m ia l Q (z ,E ) i s tw e n ty
o r so.
The c o e f f i c i e n t s a r e com plex.
The z e r o s o f t h i s d e te r m in a n t
a r e s i n g u l a r p o i n t s o f th e i n t e g r a n d i n E q .( 2 2 ) .
su c h a r e l a t i v e l y
n ess.
F in d in g r o o t s o f
l a r g e com plex p o ly n o m ia l r e q u i r e s a l i t t l e
c le v e r­
E x tre m e c a r e m u st b e ta k e n t o a v o id e v e ry n u m e r i c a l i n s t a b i l i t y
A c l e a r an d q u i t e d e t a i l e d e x p l a n a t i o n b o th o f n u m e r i c a l p ro b le m s
i n v o lv e d an d o f th e p ro g ra m s i s g iv e n i n a m onograph r e c e n t l y p r e p a r e d
X
80
b y W. Scbwalm
.
Thus I am n o t r e p e a t i n g i t h e r e .
In th e c u rre n t
v e r s i o n o f th e r o o t f i n d i n g p ro g ra m , t h e N ew to n -R ap h son m ethod i s
a p p lie d .
As m e n tio n e d i n th e m onogrph t h e N ew ton-R aphson m ethod does
/
n o t w ork w e l l f o r th e r e c i p r o c a l p o ly n o m ia ls w h ic h a lw a y s o c c u r i n th e
c a s e o f t h e d e te r m i n a n t p o ly n o m ia ls o f i n t e r e s t .
r e d u c t i o n can b e done a s i l l u s t r a t e d b e lo w .
H ow ever, a sym m etry
F o r e x a m p le , c o n s id e r
a r e c i p r o c a l p o ly n o m ia l o f d e g re e s i x
x^ + AjX^ +
+ AgX^ + A^
+ A^x = O
N o te t h a t i f x = x ^ i s a r o o t o f t h e above a l g e b r a i c e q u a ti o n th e n s o
i s 1 /x ^ .
Thus we can w r i t e t h e ab o v e e q u a ti o n as
(x^ + a x + I ) (x^ + g x + I ) (x^ + y x + I ) = O
E q u a tin g c o e f f i c i e n t s o f x11 we f i n d th e r e l a t i o n b e tw e e n th e two s e t s
o f c o e ffic ie n ts
A1 = a + g + y = CT1
A2 = ag + gy + y a + 3 = CT2 + 3
A^ = agy + 2 ( a + g + y ) = a 3 + 2 a 1
I n s t e a d o f f i n d i n g r o o t s o f th e o r i g i n a l e q u a ti o n we f i n d ct, gs 7;
(y - ct) (y - 6 ) (y - y ) = o
i.e .,
y
3
- (a + g + y )y
2
+ (a0 + By+ y a ) y - agy = O
or
y
3
2
- cr-jy + a2y- - a 3
O
81
F i n a l l y k n o w in g a ,
g , y we s o l v e t h r e e q u a d r a t i c e q u a t i o n s t o f i n d
th e o r i g i n a l s y m m e tric e q u a ti o n :
x
2
+ ax + I = 0
x^ + 3x + I = 0.
and
x
2
+ Yx + I = 0 .
I t tu rn s o u t to be th e c a se t h a t th e g e n e ra l f e a tu r e s r e v e a le d f o r ,th is
s p e c i a l exam ple, a r e t r u e f o r h i g h e r d e g re e r e c i p r o c a l e q u a t i o n .
So
a way t o a p p ro a c h t h e s o l u t i o n w o u ld b e t o
1) f i n d t h e r e l a t i o n b e tw e e n |
j
and
|
j
;
2) s o l v e t h e r e d u c e d e q u a t i o n ;
th e n
3) s o l v e a s e t o f q u a d r a t i c e q u a t i o n s .
r e l a t i o n b e tw e e n |
j an d | CT^ j
fo rm a P a s c a l 's t r i a n g l e .
f o r d e g re e tw e lv e ( s i x c o e f f i c i e n t s )
2.
I n t e r e s t i n g l y eno u g h th e
is illu s tr a te d
An ex am p le
i n A p p e n d ix t a b l e
A c t u a l l y , w h a t we r e a l l y n e e d i s t h e r e l a t i o n (h =
{ A^
j
The i n v e r t e d fo rm o f P a s c a l 's t r i a n g l e i s a l s o a u t o m a t i c a l l y o b t a i n e d
(A p p e n d ix t a b l e 3 ) .
The re d u c e d p o ly n o m ia l i s n o n - r e c i p r o c a l .
one m u st b e c a r e f u l
a b o u t t h e o r d e r of, c o e f f i c i e n t s .
Thus
A ls o th e
c u r r e n t v e r s i o n o f t h e r o o t - f i n d i n g p ro g ra m i s w r i t t e n s o t h a t th e
c o e f f i c i e n t o f th e h i g h e s t o r d e r n e e d n o t b e o n e .
82
A p p e n d ix t a b l e 2 :
P a s c a l 's t r i a n g l e .
Ccmst
Og
A1
0
I
A2
6
0
A.
4
Ac
o
°2
■
a3
°4
a5
a6
'
o<.
I
0
5
0
1
15
0
4
0
1
0
10
0
3
0
I
20
0
6
0
2
0
A.
4
Ac
5
A p p e n d ix t a b l e 3:
aI
a,
I
I n v e r t e d P a s c a l 's t r i a n g e .
C o n st
A1
I
A
2
0
I
—6
0
I
0
-5
0
I
9
0
-4
0
I
0
5
0
-3
0
I
-2
0
2
0
-2
0
'
A
3
A,
6
83
D. O u t l in e o f P ro g ra m
>
M ain P ro g ra m : I n p u t fro m t h e t e r m i n a l t o t h i s p ro g ra m a r e l i s t e d
b e lo w .
1) E le m e n t nam e, sym m etry o f i n t e r e s t and any o t h e r i n f o r m a t i o n
p e r t i n e n t t o th e c a l c u l a t i o n .
I n p u t FORMAT o f t h i s
10A4; t h u s any c h a r a c t e r s up t o 40 l e t t e r s
c a te g o r y i s
a re read in .
2) D a ta i n p u t
E l , E 2: The lo w e s t and h i g h e s t e n e rg y l i m i t o f i n t e r e s t
STEP
: The e n e rg y i n t e r v a l
ETA
: The s m a l l im a g in a r y p a r t o f th e e n e rg y
DMPG
: The dam ping p a r a m e te r u s e d i n th e i n c o h e r e n t sum c a l c u l a t i o n
3) O p tio n s t a t e m e n t s
Any num ber
C
4 0 ( y e s ) o r 0 (n o ) f o r th e f o ll o w i n g o u t p u t s .
: C o e f f i c i e n t s f o r th e d e te r m i n a n t p o ly n o m ia l a r r a n g e d i n o r d e r
of
[ c ( l ) E 3 + C(2)E2 + C(3)E + C (4)J
z 2n + ............
T h ese c o e f f i c i e n t s a r e o b t a i n e d i n a s e p a r a t e p ro g ra m by d o u b le
p o ly n o m ia l f i t t i n g
and s t o r e d i n l o g i c a l d e v ic e u n i t num ber 39.
MAIN p ro g ra m r e a d s t h e c o n t e n t s o f d e v ic e 39.
The
U nder t h e n o n - z e r o
o p t i o n t h e c o m p u te r w r i t e s th e c o e f f i c i e n t s r e a d i n b y MAIN p ro g ram
on th e l i n e p r i n t e r .
F:
! C o e f f i c i e n t s f o r th e m in o r p o ly n o m ia ls o f th e m a t r i x E - H ( k ) ,
84
c a l c u l a t e d s e p a r a t e l y and s t o r e d i n th e l o g i c a l d e v ic e u n i t
num ber 4 9 .
The f i r s t two i n d i c e s o f F ( I , J , K )
c o rre sp o n d s to
t h e I j t"*1 e le m e n t o f t h e m in o r .
G
: B u lk G reen f u n c t i o n G ( I 5J 9L) i s c a l c u l a t e d i n th e
su b p ro g ra m GREEN
RES
: The r o o t s o f t h e d e te r m i n a n t p o ly n o m ia l a r e fo u n d i n t h e G reen
fu n c tio n s u b ro u tin e .
RES i s t h e r e s i d u a l s w hen t h e r o o t s a r e
s u b s t i t u t e d i n t h e o r i g i n a l p o ly n o m ia l.
E (k)
: A t l e a s t one p a i r o f r o o t s z = e x p (ik i.d ) , g iv e n E 9 s h o u ld
r e p r o d u c e th e o r i g i n a l b u l k b a n d E ( k i ) .
W ith n o n - z e r o
o p t io n a l l r o o t s o f t h e p o l y n o m ia l, th e c o r r e s p o n d in g k i d
v a l u e s and a b s o l u t e v a lu e s o f t h e r o o t s a r e p r i n t e d .
A l l o u t p u t o f t h e s e o p t i o n s i s p r i n t e d on t h e l i n e p r i n t e r .
I n p u t / o u t p u t d e v ic e u n i t num bers u s e d i n t h i s p ro g ra m a r e a s
f o ll o w e s .
39
: F i l e , BASIC o u t p u t o f
c o e f f i c i e n t s o f d e te r m i n a n t p o ly n o m ia l
49
■: F i l e , BASIC o u t p u t o f
c o e f f i c i e n t s o f m in o r p o ly n o m ia ls
100
: F i l e , SDOS f o r l a y e r s L = I th ro u g h L=4
101
: F i l e , BDOS
103
: F i l e , w e ig h te d sum
104
: F i l e , SDOS f o r L=5 th ro u g h L=8
108
: T e le ty p e in p u t
85
The c a l c u l a t i o n o f t h e s u r f a c e G reen f u n c t i o n i s done b y a s e r i e s
o f m a trix m a n ip u la tio n s .
I n FORTRANIV u n f o r t u n a t e l y no s i n g l e m a t r i x
o p e r a t i o n s t a t e m e n t s s u c h a s a r e fo u n d i n BASIC a r e a llo w e d .
each m a trix o p e ra tio n i s
M a tr ix P a c k a g e p ro g ra m .
Thus
done b y a s u b r o u t i n e a s s e m b le d i n th e Com plex
86
(
START )
INPUT E l,E 2 ,S T E P ,E T A ,DMPG
READ
SELECT OPTIONS
CALL GREEN
' WRITE BDOS
ON 101
y
CALCULATE BDOS
CALL SCREEN
r WRITE SDOS ON
100(L=1 t o 4)
104(L=5 to 8 ) /
CALCULATE SDOS
WRITE ISUM ON
ON 103
/
CALCULATE WEIGHTED SUM
E = E+STEP
E=E2
A p p en d ix f i g u r e I .
A flo w c h a r t o f MAIN p ro g ra m .
87
CLEAN ARRAY G ( I ,J ,K )
CONSTRUCT COEFFICIENTS
NORMALIZE THE COEFFICIENTS
P (z )
CALL ROOT-FINDING WITH
SYMMETRY REDUCTION
ROOT CHECK I
SUBSTITUTE ROOTS TO THE
ORIGINAL POLYNOMIAL BY
NESTING MULTIPLICATION
ROOT CHECK I I
SUBSTITUTE ROOTS TO
z=e
d RELATION GIVEN
E AND SET IF E=E(k d) IS
REPRODUCED
SORT ROOTS INSIDE THE UNIT
CIRCLE FROM THE REST
CALCULATE DERIVATIVES OF
DENOMINATORS Q' ( z , ) , |z .
I EVALUATE~g~~i
A p p en d ix f i g u r e 2 .
A flo w c h a r t f o r GREEN f u n c t i o n p ro g ra m
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to
D378
K179
cop. 2
DATE
A#
K a v a j i r i , M izuho
C o m p u tatio n o f s u r f a c e
s t a t e s on b c c t r a n s i t i o n
m e ta ls
I S S U E D TO