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Structure factor in photoemission from valence band
Journal of Electron Spectroscopyand Related Phenomena76 (1995) 487- 492
STRUCTURE H.Daimon,
FACTOR
IN PHOTOEMISSION
S.Imada, H.Nishimoto
FROM
VALENCE
BAND
aald S . S u g a
Department of Material Physics, Faculty of Engineering Science, Osaka University, Toyona k a , O s a k a 560, J a p a n Two-dimensional angular distribution of photoelectrons from valence band was analyzed using a tight-binding initial state. In simple cases, the angular distribut i o n c a n b e u n d e r s t o o d as a p r o d u c t o f " o n e - d i m e n s i o n a l d e n s i t y o f s t a t e s ( O D D O S ) " , " photoemission structure factor", and " angular distribution from atomic orbital". This n e w l y i n t r o d u c e d " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " is a n i n t e n s i t y d i s t r i b u t i o n in a r e ciprocal space similar to the X-ray or electron diffraction structure factor replacing scatt e r i n g f a c t o r b y t h e c o e f f i c i e n t for e a c h a t o m i c o r b i t a l in t h e L C A O w a v e f u n c t i o n . T h e r e m a r k a b l e s y m m e t r y - b r o k e n p a t t e r n s o b s e r v e d f r o m n b a n d of s i n g l e - c r y s t a l l i n e g r a p h i t e w a s c l e a r l y e x p l a i n e d . T h e t w o - f o l d s y m m e t r y w a s o b v i o u s as t h e r e s u l t o f t h e " a n g u lar d i s t r i b u t i o n f r o m t h e p~ a t o m i c o r b i t a l " f o r t h e ~- a n d l i n e a r l y - p o l a r i z e d s y n c h r o t r o n r a d i a t i o n . T h e d i f f e r e n c e o f t h e i n t e n s i t y in d i f f e r e n t B Z s w a s c l e a r l y e x p l a i n e d b y t h e p h o t o e m i s s i o n s t r u c t u r e f a c t o r . H e n c e , t h e t w o - d i m e n s i o n a l m e a s u r e m e n t c a n g i v e tts information not only (1)about the ODDOS, but also (2)about the symmetry of the initial s t a t e , a n d e v e n m o r e ( 3 ) a b o u t t h e c o e f f i c i e n t s in t h e initial s t a t e w a v e f u n c t i o n . 1. I N T R O D U C T I O N Recently observed two-dimensional angular distribution of photoelectrons from the valence band of graphite[l] showed remarkable symmetry-broken patterns. Figures la-f show the patterns measured at the binding energies EB of 0.1(a), and f r o m 0.7 t o 4.7 e V b y 1 e V s t e p ( b - f ) using a d i s p l a y - t y p e h e m i s p h e r i c a l m i r r o r a n a l y z e r [ 2 ] . T h e i n c i d e n t l i g h t w a s a line a r l y p o l a r i z e d s y n c h r o t r o n r a d i a t i o n (hu = 54 e V ) , o f w h i c h e l e c t r i c v e c t o r lies a l o n g t h e x - d i r e c t i o n in F i g . 2. T h e s e p a t t e r n s show non six-fold but two-fold symmetry, and also non-equivalence among different Brillouin zones (BZ). The result by Sant o n i et al.[3] u s i n g a p - p o l a r i z e d l i g h t e x c i t a tion showed almost six-fold patterns. They d i d n o t o b s e r v e s t r o n g i n t e n s i t y in t h e s e c ond BZ because their acceptance cone was not enough. In this paper observed asymmetries are analyzed using a tight-binding initial state. 0368-2048/95 $09.50 © 1995 ElsevierScience B.V. All rights reserved SSD! 0368- 2048(95) 02478-6
2. A N A L Y S I S We calculate the angular dependence of photoemission intensity from tight-binding initial s t a t e s w i t h i n t h e d i p o l e a p p r o x i m a tion. Using a Golden Rule formula[4], the i n t e n s i t y d i s t r i b u t i o n I(0, ¢) is e x p r e s s e d as
I(0, ¢) r. 1) 1(kll)lMI2
(l)
w h e r e D l(kH) is O D D O S a t t h e d i r e c t i o n (0, ¢) w h e r e t h e p a r a l l e l c o m p o n e n t o f t h e w a v e v e c t o r o f p h o t o e l e c t r o n k is kll, D j (kll) represents the shape of cross section of the b a n d s t r u c t u r e a t a r b i t r a r y El~ in t h e c a s e o f a n i d e a l t w o - d i m e n s i o n a l s y s t e m . 0 a n d d) are polar and azimuthal angles of the photoelectrons referred to the surface normal. H e r e , we e x p r e s s t h e m a t r i x e l e m e n t M using the momentum operator p and the vect o r p o t e n t i a l o f t h e i n c i d e n t l i g h t A as M cv< f l P " A[i >,
(2)
488
Obs.
Calc.
Obs.
EB -~
EB =
O.leV
2.7eV
0.7eV
3.7eV
1.7eV
4.7eV
Calc.
/ O t h e r band ((r-band) F I G . 1. T h e c r o s s s e c t i o n o f t h e b a n d s t r u c t u r e o f K i s h g r a p h i t e m e a s u r e d b y t h e t w o - d i m e n s i o n a l d i s p l a y type spherical mirror analyzer. White lines show the boundary of the Brillouin zones, a - j show the c r o s s - s e c t i o n a l p a t t e r n s a t t h e e n e r g y s h o w n left. C a l c u l a t e d p a t t e r n s f o r e a c h p a t t e r n a r e s h o w n r i g h t side.
a w a v e v e c t o r q, a n d r e p r e s e n t e d t i g h t - b i n d i n g a p p r o x i m a t i o n as
li > =
~
1
in
~ Z ~,q (R~.~,).,~¢~(r_Rj_~), (3) j
iv
w h e r e t h e s p i n is n e g l e c t e d b e c a u s e it is c o n s e r v e d in t h e d i p o l e e x c i t a t i o n , ri is a p o s i t i o n v e c t o r o f t h e i - t h a t o m in t h e j - t h u n i t cell a t R j . ¢i~(r) is t h e v - t h o r b i t a l o f t h e i - t h a t o m , w h i c h is e x p r e s s e d u s i n g a s p h e r i c a l h a r m o n i c Ylm(0, ¢) as
¢(r) = n.~ (,')~m(0, ¢). F I G . 2, S t r u c t u r e o f g r a p h i t e .
w h e r e ]f > is a final s t a t e a n d li > is a n initial s t a t e . T h e initial s t a t e li > is w r i t t e n h e r e b y an eigenfunction of the Bloch state with
(4)
P~t(r) is its r a d i a l f u n c t i o n , l a n d m s t a n d for t h e o r b i t a l a n g u l a r m o m e n t u m . The final s t a t e If) is e x p a n d e d a r o u n d a n a t o m
as/5] • l'
If) = 47r ~--~(z) e llfn I
-i6
~
*
, Yt,m,(0k,¢k)Yt,,,-,,(0, ¢)Rl,(r).(5)
489
U s i n g t h e a b o v e f o r m u l a for ]i >, M is w r i t t e n as M c~ ~
1
Z
Z
3
eiq(R3 h")aw
iu
< f(r)[p. A[Oi,(r- Rj H e r e , r in t h e b r a k e t r ' q R a 4 r~. T h e n M c( ~
1
Z
~
j
ri) > .(6)
is r e p l a c e d b y r =
e i q ( R ' ~-"')('+"
M o( Z bk Q,G ~ (~ iv
iu
< / ( r ' +R: +~dlP" Al¢~(r') > .(7) From Bloch's theorem band structure,
for
t i m e q u i v a l e n t l a y e r s p a r a l l e l to tile s u r face, a n d k~_ h a s a n i m a g i n a r y p a r t t o c o n sider absorption. The idea of "weightedi n d i r e c t - t r a n s i t i o n " , w h i c h is t h e c o m b i n a tion of ODDOS and this factor was introd u c e d b y G r a n d k e et al.[6] T h e s e c o n d t e r m 1/(1 - c' i(k~ q~).c) is v e r y close t o t h e d e l t a s y m b o l bk, q~,Cl:" H e n c e , M call b e w r i t t e n as
the
complex
< f(r' ~ Rj t ri)] = e ik.(R, ~,,) < f ( r ' ) ] .
aiv{e-iG'"":liv
(1'2) 1(0, O) is
Hence, the intensity distribution e x p r e s s e d as
1(0, ¢) ~../)'(k,) I~ 5k -q,G G
(8)
Then
~-~ a.i, e iG".-l,vl2.
(13)
iv,
M oz ~
~--'ai'e-i(k-q)~ e" i(k q).Rj 3
iv
< f(r')]p. Al@.(r' ) > .
(9)
We define the "angular distribution i n d i v i d u a l a t o m i c o r b i t a l " A as
from
(10) where r and e are a dipole operator and a p o l a r i z a t i o n u n i t v e c t o r o f t h e light, r e s p e c t i v e l y . H e r e we i n t r o d u c e d t h e d i p o l e representation of the transition operator r . e in a p p r o x i m a t e m a n n e r , b e c a u s e t h e a t o m i c o r b i t a l is n o t a n e x a c t e i g e n s t a t e o f the Hamiltonian. T h e r a n g e o f t h e s u m o v e r j is u n l i m i t e d for p a r a l l e l c o m p o n e n t Jl[ a n d s e m i - i n f i n i t y for p e r p e n d i c u l a r c o m p o n e n t j±. oo
oc
311=-o~j±:0
1
= Z6kll Gtl
l(0,4))~Dt(kll)lZ~k
q,GZa~c
iG"]~[..l,,]2
G
Az. =< f(r)[r.el¢i.(r) >o<< f(r)lp.Al¢~.(r ) >,
1
W h e n t h e initial s t a t e is c o m p o s e d o f o n l y o n e v - t h a t o m i c o r b i t a l , we c a n p u l l t h i s f a c t o r o u t o f t h e a b s o l u t e v a l u e as
q , , G , 1 - e-i(k;- - q ± ) c
H e r e , Gll is 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 , a n d c is t h e l a t t i c e v e c t o r b e t w e e n
= D~(klI)II;'I2IA~I 2.
(bU
We introduced here s t r u c t u r e f a c t o r " F as
v-- Z G
a "photoemission
,G,,.
q,G
(l;-,)
i
T h e t e r m F is a n i n t e n s i t y d i s t r i b u t i o n in a r e c i p r o c a l s p a c e , w h i c h is i n d e p e n d e n t of t h e k i n e t i c e n e r g y o f p h o t o e l e c t r o n s or incident angle of photons. This factor originates from the interference of the waves f r o m d i f f e r e n t g r o u p s o f a t o m s in a u n i t cell a n d is s i m i l a r t o t h e X - r a y o r e l e c t r o n diffraction structure factors replacing scatt e r i n g f a c t o r s b y t h e c o e f f i c i e n t s for e a c h a t o m i c o r b i t a l in t h e L C A O w a v e f u n c t i o n . N o t e t h a t t h e s i m i l a r i t y is n o t a n e x a c t o n e . T h e t e r m F is i m p o r t a n t w h e n we c o m p a r e t h e i n t e n s i t i e s b e t w e e n t h e 1st a n d 2nd BZs.
490
When there are more than one kind of atomic orbitals, this term can be expressed as
F=
E 5 k - q , G E a~e-~G~'A~°' G i
(16)
w h e r e A.0 is a m e a n v a l u e o f A., a n d A~ in (14) s h o u l d b e c h a n g e d t o A . / A . o . T h i s formula seems useful because the variation o f A. is r a t h e r s m o o t h . H e r e we s h o w e d t h a t t h e a n g u l a r d i s t r i b u t i o n c a n b e u n d e r s t o o d in s i m p l e c a s e s as a p r o d u c t o f O D D O S , " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " F, a n d " a n g u l a r d i s t r i b u t i o n f r o m a t o m i c o r b i t a l " A. F r o m n e x t s e c t i o n we will see h o w t h e s e t e r m s F a n d A a c t in r e a l s y s t e m s a p p l y i n g t h e m t o t h e d a t a for single c r y s t a l g r a p h i t e . 3. A N G U L A R D I S T R I B U T I O N FROM INDIVIDUAL ATOMIC ORBITAL The operator r •e can be represented by t h e s p h e r i c a l h a r m o n i c s as r.e=
~
~/~--~e~,Yl~(O,¢)'r,
(17)
where C:t:l = (qze. + ieu)/vf2, Co = e~. Then the photoemission intensity distribution from atomic orbital having angular m o m e n t u m l a n d m is r e p r e s e n t e d u s i n g e q . ( 4 ) , (5), (10), a n d (17)as
IA,~I2 = I ~
I n t h e c a s e o f n b a n d o f g r a p h i t e , ¢(r) is a p, o r b i t a l , a n d t h e s p h e r i c a l h a r m o n i c s is w r i t t e n as Ylo(O, ¢) = ~ c o s 0 . I n t h e exp e r i m e n t [ l ] t h e e l e c t r i c v e c t o r is a l o n g t h e x-axis, a n d o n l y e~ is n o t z e r o . T h e r e f o r e , we h a v e Ip~(k) c
R(I ---, I,:l'){-)~,l(Ok,¢k)C'(l'l, 10)
P = l -t-1
+]~, l ( 0 k , ¢ k ) c l ( l ' -- 1, 10)}12.
(19)
I n a c c o r d w i t h t h e d i p o l e s e l e c t i o n rule, l' c a n b e e i t h e r 2 o r 0. I n t h e p r e s e n t case, h o w e v e r , o n l y t h e I' = 2 s t a t e is r e a l i z e d bec a u s e t h e final s t a t e h a s e i t h e r m' = +1 or -1. Since the two Gaunt coefficients have the same value, eq.(19) becomes
lvz(k ) c( ] - Y2~(0k, Ck ) +Y2_~(Ok:~Pk)] 2 [sin 0k c o s 0k cos Ckl z .
(20)
(21)
T h e w a v e f u n c t i o n s o f l o w e r b a n d s (03 a n d 02 b a n d s ) at t h e F p o i n t are a l m o s t p u r e py a n d p:~ a t o m i c o r b i t a l , r e s p e c t i v e l y . Therefore, their angular distributions from a t o m i c o r b i t a l a r e d e r i v e d f r o m eq. (18) as
i ; : (k)
[sin2 ok sin 2¢ k 12 ,
I~':~(k) oc
IR2{V ~-~nsiu0kCOS2¢ k
3
(22)
2
R(l -, kl')
P =l-4-1
-~-lT~(3cos20k-
1)}-~ e '-i(~ 60) R0 [2 (23)
1 E
eg YI,l m+,(0 k, Ck)C 1 ( l!m + p,, lrn)[ 2.
(18)
T h i s is a g e n e r a l f o r m u l a o f t h e t r a n s i t i o n probability to the direction k from atomic o r b i t a l h a v i n g a n a n g u l a r p a r t ]~,,(0:¢). H e r e R(l ~ M ~) is t h e i n t e g r a l o f t h e r a dial p a r t , w h i c h is a c o n s t a n t c o m p l e x n u m b e r d e p e n d i n g o n I a n d l ~ a n d t a b u l a t e d as Rl+le ~5'±~ b y G o l d b e r g e t al.[7], cl(l'rn ',lm) is t h e C l e b s c h - G o r d a n o r t h e G a u n t coefficients[8].
T h e v a l u e s u s e d h e r e a r e R2--0.645, R 0 = 0 . 1 8 5 , 62 = 2.996,and 60=6.16017].
F I G . 3. a, b, a n d c s h o w s t h e p h o t o e l e c t r o n a n g u l a r d i s t r i b u t i o n p a t t e r n s e m i t t e d f r o m Pz, P~, a n d px a t o m i c o r b i t a l s .
491
These angular distributions of the phot o e m i s s i o n f r o m t h e p~(n), pv(a3), a n d P.~(a2) o r b i t a l s a r e c a l c u l a t e d w i t h e q . ( 2 1 ) , (22), a n d (23) a t EK = 50 e V a n d s h o w n in F i g . 3a, b, a n d c, r e s p e c t i v e l y . F r o m eq. (21) we find t h a t p h o t o e l e c t r o n s a r e hOe e m i t t e d w i t h i n y-z p l a n e ( ¢ = ± 2 ) , w h i c h is t h e v e r t i c a l line o f t h e Fig. l a - f t h r o u g h t h e c e n t e r . T h e b r i g h t r e g i o n in Figs. 3a is in g o o d a g r e e m e n t w i t h t h o s e a r e a in Fig. 1a-e, w h e r e we c o u l d o b s e r v e p a t t e r n s . F o r I~'~(k), t h e a z i m u t h a l a n g l e d e p e n d e n c e s h o w s n e g l i g i b l e i n t e n s i t y a t Ck = 0, ± ~ a n d n. l ~ ( k ) is s t r o n g o n t h e c e n t r a l v e r t i c a l line in c o n t r a s t t o I~a(k). T h i s a r g u m e n t strongly suggests that the Fermi surface a n d t h e 7r b a n d a r e c o m p o s e d o f p= o r b i t a l b u t n o t o f t h e p:,. o r Pv o r b i t a l . H e n c e , we c a n d i s c u s s a b o u t t h e initial s t a t e s y m m e t r y f r o m o b s e r v e d e x t i n c t i o n rule. F r o m a b o v e d i s c u s s i o n , it is c l e a r t h a t t h e p e a k s n e a r t h e c e n t e r ( c e n t r a l axis) o f t h e 1st B Z in Figs. I f is n o t t h e o n e f r o m n b a n d b u t t h a t f r o m t h e 02 b a n d s .
f a c t o r I;' i n c l u d i n g g a n d G, i.e.
1 g t;2(0,¢) O( 2(1 ~ %p.l:jF)l]~
t
~ ~G.d, 12. ~
(27)
SomeofG
a r e G0 = (0,0.0), 4 "G x ,- - ( ~ , 0 , 0 ) , 0) T h e v e c t o r dl is (d., 0.0). " ' T h e v a l u e o f k is p r i n c i p a l l y d e t e r m i n e d acc o r d i n g t o t h e k i n e t i c e n e r g y a n d t h e det e c t e d a n g l e 0. T h e initial s t a t e w a v e vect o r is c o n n e c t e d to k b y t h e d e l t a f u n c t i o n as q = k - G . T h e initial w a v e f u n c t i o n o f e q . (24) d o e s n o t c h a n g e e x c e p t for t h e p h a s e e v e n if q c h a n g e s to q t G. B e c a u s e it is n o t n e c e s s a r y t h a t q is c o n f i n e d to 1st B Z , t h e r e a r e m a n y w a y s t o c h o o s e t h e p a i r of q a n d G. F o r e x a m p l e , w h e n we c a l c u l a t e in a 2 n d B Z w i t h t h e r e c i p r o c a l v e c t o r o f G1 , w h i c h m e a n s k is in t h e 2 n d B Z , we c a n set G = 0 a n d q is in t h e 2 n d B Z , or set G = G1 a n d q in t h e 1st B Z , etc. B e c a u s e o f t h e p e r i o d i c i t y , e v e r y c h o i c e will r e s u l t in t h e s a m e r e s u l t . G2
=
(2n 2,r 3d: ~
4. PHOTOEMMISSION STRUCTURE FACTOR T h e e i g e n f n n c t i o n o f n b a n d q~(q,r) is o b t a i n e d as
1 P
ql(q, r )
eJq R ' '
,27', ~2(~ , .>,,,, Iol)
( g p A ( r -- R,,) t c*qd~p-t'(r - R,,. - d i l l , (2-1) I.q~ " w h e r e .q is d e f i n e d as g = c ~qd~ t c iqd~ ~ e iqd:~.
(25)
dl, d2 a n d da a r e p o s i t i o n v e c t o r s o f n e a r e s t
n e i g h b o u r c a r b o n a t o m s s h o w n in The total angular distribution band neglecting ODDOS, U(0,¢), t e n u s i n g eq. (14) , (21) a n d (24)
Fig. 2. from n is w r i t as
[I-~ + e - ' G d ' 12 1 ~ (0, ¢) c~ 2(1 +
spp~ Igl)
[sin0kcOS0kCOS¢k ]2 • (26)
The inequivalency between different BZs comes from the photoemission structure
F I G . 4. P h o t o e m i s s i o n
structure
factor
l:'~(kr, k u ) ,
F2(0k, ~bk) is p l o t t e d in Fig. 4. T h e p a r a l l e l o g r a m in t h e f i g u r e s h o w s t h e u n i t cell in t h e r e c i p r o c a l s p a c e . W h e n we p l o t in (k.~,ky) p l a n e , t h e a n g u l a r d e p e n d e n c e o f F 2 (k:~, kv) is e n e r g y i n d e p e n d e n t b e c a u s e t h e y a r e f u n c t i o n s o f o n l y (k~,kv). T h i s f u n c t i o n is a p e r i o d i c f u n c t i o n w i t h a p e r i o d of v~ × v ~ R 3 0 ° o f t h e n e t w o r k o f BZs,
492 w h i c h m e a n s it r e p e a t s e v e r y s e c o n d n e a r est neighbor BZs. This difference of the periodicity between the bulk BZs and this f u n c t i o n is t h e o r i g i n o f t h e n o n e q u i v a lence of the i n t e n s i t y d i s t r i b u t i o n b e t w e e n 1st a n d 2 n d B Z s . T h i s d i f f e r e n c e is o r i g i nated by the interference between the photoelectrons from the group of A atoms and that of B atoms. W h e n t h e u n i t cell o f r e a l s p a c e is c o m p o s e d o f m o r e t h a n t w o atoms, this difference always occurs. In o t h e r w o r d s , t h e d i s t r i b u t i o n in F i g . 4 is a reciprocal space intensity distribution of photoelectrons.
bution, where the density of states are multiplied by a weight of the intensity distribut i o n o f F i g . 5 c c o n s i d e r i n g its e n e r g y d e p e n d e n c e . T h e c a l c u l a t e d p a t t e r n well r e p r o d u c e s t h e s t r o n g i n t e n s i t y w i t h i n t h e 1st B Z in F i g s . l b - c . H e n c e t h e " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " can r e p r o d u c e well the experimental non-equivalent intensity d i s t r i b u t i o n in t h e 1st a n d 2 n d B Z s . ACKNOWLEDGEMENT The authors are grateful to Prof. W. Schattke a n d P r o f . R . S a i t o for t h e i r v a l u able discussion. This study was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. REFERENCES
FIG. 5 . a : Photoemission structure factor F2~ b: angular distribution from atomic orbital A2, and c: the total intensity I at E K e 5 0 eV. Figure 5 shows how the total intensity I~(0k, Ck) ( F i g . 5c) is c a l c u l a t e d b y a p r o d u c t o f F 2 ( F i g . 5a) a n d A 2 ( F i g . 5 b w h i c h is t h e s a m e as F i g . 3 a ). C o m p a r i n g F i g s . 5a, b, a n d c, it c a n b e r e c o g n i z e d t h a t t h e gross feature of the inteusity distribution (the disappearance along the central vertical line) is d e t e r m i n e d m a i n l y b y t h e a n g u lar d i s t r i b u t i o n f r o m a t o m i c o r b i t a l o f F i g . 5b, a n d m o d i f i e d b y t h e s t r u c t u r e f a c t o r in photoemission of Fig.5a. The most promin e n t f e a t u r e in F i g . 5c is t h e d a r k a r e a o v e r t h e K p o i n t s in t h e 2rid B Z . T h e r e a s o n why the intensity suddenly changes from 1st B Z t o 2 n d B Z is t h e s u d d e n c h a n g e of the phase difference between two waves from A atom groups and B atom groups. The phase changes suddenly at K point bec a u s e g c r o s s e s z e r o a n d ~] c h a n g e s f r o m -t-1 t o -1. P h y s i c a l l y s p e a k i n g , t h e w a v e s from A atom groups and B atom groups i n t e r f e r e p o s i t i v e l y in t h e 1st B Z b u t t h e y interfere negatively beyond K point. T h e r i g h t s i d e f i g u r e s o f Fig. l a - f s h o w the simulated photoelectron angular distri-
[1] H.Nishimoto, T.Nakatani, T.Matsushita, H.Daimon, S.Suga, H.Namba, and T.Ohta to be published. [2] H.Daimon, Rev.Sci.Instrum. 59, 545 (1988), .ib.id. 61~ 205 (1990), H. Nishimoto et al.: Rev. Sci. Instrum. 64 2857 (1993). [3] A.Santoni, L.J.Terminello, F.J.I-timpsei, and T.Takahashi~ Appl.Phys. A 52~299(1991). [4] P.J.Feibelman, and D.E.Eastman, Phys. Rev. B 10 (1974) 4932. [5] J.B.Pendry, Low energy electron d.iffruction~ Academic Press, London (1974). [6] T. Grandke, L. Ley~ and M. Cardona, Phys. Rev. B 18 (1978) 3847. [7] S.M.Goidberg, C.S.Fadley~ and S.Kono~ J.Electron Spectroscopy and Related Phenomena, 21~ 285(1981).
[a]
....
¢)Yt m (0, ¢) sin OdOd~
STRUCTURE H.Daimon,
FACTOR
IN PHOTOEMISSION
S.Imada, H.Nishimoto
FROM
VALENCE
BAND
aald S . S u g a
Department of Material Physics, Faculty of Engineering Science, Osaka University, Toyona k a , O s a k a 560, J a p a n Two-dimensional angular distribution of photoelectrons from valence band was analyzed using a tight-binding initial state. In simple cases, the angular distribut i o n c a n b e u n d e r s t o o d as a p r o d u c t o f " o n e - d i m e n s i o n a l d e n s i t y o f s t a t e s ( O D D O S ) " , " photoemission structure factor", and " angular distribution from atomic orbital". This n e w l y i n t r o d u c e d " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " is a n i n t e n s i t y d i s t r i b u t i o n in a r e ciprocal space similar to the X-ray or electron diffraction structure factor replacing scatt e r i n g f a c t o r b y t h e c o e f f i c i e n t for e a c h a t o m i c o r b i t a l in t h e L C A O w a v e f u n c t i o n . T h e r e m a r k a b l e s y m m e t r y - b r o k e n p a t t e r n s o b s e r v e d f r o m n b a n d of s i n g l e - c r y s t a l l i n e g r a p h i t e w a s c l e a r l y e x p l a i n e d . T h e t w o - f o l d s y m m e t r y w a s o b v i o u s as t h e r e s u l t o f t h e " a n g u lar d i s t r i b u t i o n f r o m t h e p~ a t o m i c o r b i t a l " f o r t h e ~- a n d l i n e a r l y - p o l a r i z e d s y n c h r o t r o n r a d i a t i o n . T h e d i f f e r e n c e o f t h e i n t e n s i t y in d i f f e r e n t B Z s w a s c l e a r l y e x p l a i n e d b y t h e p h o t o e m i s s i o n s t r u c t u r e f a c t o r . H e n c e , t h e t w o - d i m e n s i o n a l m e a s u r e m e n t c a n g i v e tts information not only (1)about the ODDOS, but also (2)about the symmetry of the initial s t a t e , a n d e v e n m o r e ( 3 ) a b o u t t h e c o e f f i c i e n t s in t h e initial s t a t e w a v e f u n c t i o n . 1. I N T R O D U C T I O N Recently observed two-dimensional angular distribution of photoelectrons from the valence band of graphite[l] showed remarkable symmetry-broken patterns. Figures la-f show the patterns measured at the binding energies EB of 0.1(a), and f r o m 0.7 t o 4.7 e V b y 1 e V s t e p ( b - f ) using a d i s p l a y - t y p e h e m i s p h e r i c a l m i r r o r a n a l y z e r [ 2 ] . T h e i n c i d e n t l i g h t w a s a line a r l y p o l a r i z e d s y n c h r o t r o n r a d i a t i o n (hu = 54 e V ) , o f w h i c h e l e c t r i c v e c t o r lies a l o n g t h e x - d i r e c t i o n in F i g . 2. T h e s e p a t t e r n s show non six-fold but two-fold symmetry, and also non-equivalence among different Brillouin zones (BZ). The result by Sant o n i et al.[3] u s i n g a p - p o l a r i z e d l i g h t e x c i t a tion showed almost six-fold patterns. They d i d n o t o b s e r v e s t r o n g i n t e n s i t y in t h e s e c ond BZ because their acceptance cone was not enough. In this paper observed asymmetries are analyzed using a tight-binding initial state. 0368-2048/95 $09.50 © 1995 ElsevierScience B.V. All rights reserved SSD! 0368- 2048(95) 02478-6
2. A N A L Y S I S We calculate the angular dependence of photoemission intensity from tight-binding initial s t a t e s w i t h i n t h e d i p o l e a p p r o x i m a tion. Using a Golden Rule formula[4], the i n t e n s i t y d i s t r i b u t i o n I(0, ¢) is e x p r e s s e d as
I(0, ¢) r. 1) 1(kll)lMI2
(l)
w h e r e D l(kH) is O D D O S a t t h e d i r e c t i o n (0, ¢) w h e r e t h e p a r a l l e l c o m p o n e n t o f t h e w a v e v e c t o r o f p h o t o e l e c t r o n k is kll, D j (kll) represents the shape of cross section of the b a n d s t r u c t u r e a t a r b i t r a r y El~ in t h e c a s e o f a n i d e a l t w o - d i m e n s i o n a l s y s t e m . 0 a n d d) are polar and azimuthal angles of the photoelectrons referred to the surface normal. H e r e , we e x p r e s s t h e m a t r i x e l e m e n t M using the momentum operator p and the vect o r p o t e n t i a l o f t h e i n c i d e n t l i g h t A as M cv< f l P " A[i >,
(2)
488
Obs.
Calc.
Obs.
EB -~
EB =
O.leV
2.7eV
0.7eV
3.7eV
1.7eV
4.7eV
Calc.
/ O t h e r band ((r-band) F I G . 1. T h e c r o s s s e c t i o n o f t h e b a n d s t r u c t u r e o f K i s h g r a p h i t e m e a s u r e d b y t h e t w o - d i m e n s i o n a l d i s p l a y type spherical mirror analyzer. White lines show the boundary of the Brillouin zones, a - j show the c r o s s - s e c t i o n a l p a t t e r n s a t t h e e n e r g y s h o w n left. C a l c u l a t e d p a t t e r n s f o r e a c h p a t t e r n a r e s h o w n r i g h t side.
a w a v e v e c t o r q, a n d r e p r e s e n t e d t i g h t - b i n d i n g a p p r o x i m a t i o n as
li > =
~
1
in
~ Z ~,q (R~.~,).,~¢~(r_Rj_~), (3) j
iv
w h e r e t h e s p i n is n e g l e c t e d b e c a u s e it is c o n s e r v e d in t h e d i p o l e e x c i t a t i o n , ri is a p o s i t i o n v e c t o r o f t h e i - t h a t o m in t h e j - t h u n i t cell a t R j . ¢i~(r) is t h e v - t h o r b i t a l o f t h e i - t h a t o m , w h i c h is e x p r e s s e d u s i n g a s p h e r i c a l h a r m o n i c Ylm(0, ¢) as
¢(r) = n.~ (,')~m(0, ¢). F I G . 2, S t r u c t u r e o f g r a p h i t e .
w h e r e ]f > is a final s t a t e a n d li > is a n initial s t a t e . T h e initial s t a t e li > is w r i t t e n h e r e b y an eigenfunction of the Bloch state with
(4)
P~t(r) is its r a d i a l f u n c t i o n , l a n d m s t a n d for t h e o r b i t a l a n g u l a r m o m e n t u m . The final s t a t e If) is e x p a n d e d a r o u n d a n a t o m
as/5] • l'
If) = 47r ~--~(z) e llfn I
-i6
~
*
, Yt,m,(0k,¢k)Yt,,,-,,(0, ¢)Rl,(r).(5)
489
U s i n g t h e a b o v e f o r m u l a for ]i >, M is w r i t t e n as M c~ ~
1
Z
Z
3
eiq(R3 h")aw
iu
< f(r)[p. A[Oi,(r- Rj H e r e , r in t h e b r a k e t r ' q R a 4 r~. T h e n M c( ~
1
Z
~
j
ri) > .(6)
is r e p l a c e d b y r =
e i q ( R ' ~-"')('+"
M o( Z bk Q,G ~ (~ iv
iu
< / ( r ' +R: +~dlP" Al¢~(r') > .(7) From Bloch's theorem band structure,
for
t i m e q u i v a l e n t l a y e r s p a r a l l e l to tile s u r face, a n d k~_ h a s a n i m a g i n a r y p a r t t o c o n sider absorption. The idea of "weightedi n d i r e c t - t r a n s i t i o n " , w h i c h is t h e c o m b i n a tion of ODDOS and this factor was introd u c e d b y G r a n d k e et al.[6] T h e s e c o n d t e r m 1/(1 - c' i(k~ q~).c) is v e r y close t o t h e d e l t a s y m b o l bk, q~,Cl:" H e n c e , M call b e w r i t t e n as
the
complex
< f(r' ~ Rj t ri)] = e ik.(R, ~,,) < f ( r ' ) ] .
aiv{e-iG'"":liv
(1'2) 1(0, O) is
Hence, the intensity distribution e x p r e s s e d as
1(0, ¢) ~../)'(k,) I~ 5k -q,G G
(8)
Then
~-~ a.i, e iG".-l,vl2.
(13)
iv,
M oz ~
~--'ai'e-i(k-q)~ e" i(k q).Rj 3
iv
< f(r')]p. Al@.(r' ) > .
(9)
We define the "angular distribution i n d i v i d u a l a t o m i c o r b i t a l " A as
from
(10) where r and e are a dipole operator and a p o l a r i z a t i o n u n i t v e c t o r o f t h e light, r e s p e c t i v e l y . H e r e we i n t r o d u c e d t h e d i p o l e representation of the transition operator r . e in a p p r o x i m a t e m a n n e r , b e c a u s e t h e a t o m i c o r b i t a l is n o t a n e x a c t e i g e n s t a t e o f the Hamiltonian. T h e r a n g e o f t h e s u m o v e r j is u n l i m i t e d for p a r a l l e l c o m p o n e n t Jl[ a n d s e m i - i n f i n i t y for p e r p e n d i c u l a r c o m p o n e n t j±. oo
oc
311=-o~j±:0
1
= Z6kll Gtl
l(0,4))~Dt(kll)lZ~k
q,GZa~c
iG"]~[..l,,]2
G
Az. =< f(r)[r.el¢i.(r) >o<< f(r)lp.Al¢~.(r ) >,
1
W h e n t h e initial s t a t e is c o m p o s e d o f o n l y o n e v - t h a t o m i c o r b i t a l , we c a n p u l l t h i s f a c t o r o u t o f t h e a b s o l u t e v a l u e as
q , , G , 1 - e-i(k;- - q ± ) c
H e r e , Gll is 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 , a n d c is t h e l a t t i c e v e c t o r b e t w e e n
= D~(klI)II;'I2IA~I 2.
(bU
We introduced here s t r u c t u r e f a c t o r " F as
v-- Z G
a "photoemission
,G,,.
q,G
(l;-,)
i
T h e t e r m F is a n i n t e n s i t y d i s t r i b u t i o n in a r e c i p r o c a l s p a c e , w h i c h is i n d e p e n d e n t of t h e k i n e t i c e n e r g y o f p h o t o e l e c t r o n s or incident angle of photons. This factor originates from the interference of the waves f r o m d i f f e r e n t g r o u p s o f a t o m s in a u n i t cell a n d is s i m i l a r t o t h e X - r a y o r e l e c t r o n diffraction structure factors replacing scatt e r i n g f a c t o r s b y t h e c o e f f i c i e n t s for e a c h a t o m i c o r b i t a l in t h e L C A O w a v e f u n c t i o n . N o t e t h a t t h e s i m i l a r i t y is n o t a n e x a c t o n e . T h e t e r m F is i m p o r t a n t w h e n we c o m p a r e t h e i n t e n s i t i e s b e t w e e n t h e 1st a n d 2nd BZs.
490
When there are more than one kind of atomic orbitals, this term can be expressed as
F=
E 5 k - q , G E a~e-~G~'A~°' G i
(16)
w h e r e A.0 is a m e a n v a l u e o f A., a n d A~ in (14) s h o u l d b e c h a n g e d t o A . / A . o . T h i s formula seems useful because the variation o f A. is r a t h e r s m o o t h . H e r e we s h o w e d t h a t t h e a n g u l a r d i s t r i b u t i o n c a n b e u n d e r s t o o d in s i m p l e c a s e s as a p r o d u c t o f O D D O S , " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " F, a n d " a n g u l a r d i s t r i b u t i o n f r o m a t o m i c o r b i t a l " A. F r o m n e x t s e c t i o n we will see h o w t h e s e t e r m s F a n d A a c t in r e a l s y s t e m s a p p l y i n g t h e m t o t h e d a t a for single c r y s t a l g r a p h i t e . 3. A N G U L A R D I S T R I B U T I O N FROM INDIVIDUAL ATOMIC ORBITAL The operator r •e can be represented by t h e s p h e r i c a l h a r m o n i c s as r.e=
~
~/~--~e~,Yl~(O,¢)'r,
(17)
where C:t:l = (qze. + ieu)/vf2, Co = e~. Then the photoemission intensity distribution from atomic orbital having angular m o m e n t u m l a n d m is r e p r e s e n t e d u s i n g e q . ( 4 ) , (5), (10), a n d (17)as
IA,~I2 = I ~
I n t h e c a s e o f n b a n d o f g r a p h i t e , ¢(r) is a p, o r b i t a l , a n d t h e s p h e r i c a l h a r m o n i c s is w r i t t e n as Ylo(O, ¢) = ~ c o s 0 . I n t h e exp e r i m e n t [ l ] t h e e l e c t r i c v e c t o r is a l o n g t h e x-axis, a n d o n l y e~ is n o t z e r o . T h e r e f o r e , we h a v e Ip~(k) c
R(I ---, I,:l'){-)~,l(Ok,¢k)C'(l'l, 10)
P = l -t-1
+]~, l ( 0 k , ¢ k ) c l ( l ' -- 1, 10)}12.
(19)
I n a c c o r d w i t h t h e d i p o l e s e l e c t i o n rule, l' c a n b e e i t h e r 2 o r 0. I n t h e p r e s e n t case, h o w e v e r , o n l y t h e I' = 2 s t a t e is r e a l i z e d bec a u s e t h e final s t a t e h a s e i t h e r m' = +1 or -1. Since the two Gaunt coefficients have the same value, eq.(19) becomes
lvz(k ) c( ] - Y2~(0k, Ck ) +Y2_~(Ok:~Pk)] 2 [sin 0k c o s 0k cos Ckl z .
(20)
(21)
T h e w a v e f u n c t i o n s o f l o w e r b a n d s (03 a n d 02 b a n d s ) at t h e F p o i n t are a l m o s t p u r e py a n d p:~ a t o m i c o r b i t a l , r e s p e c t i v e l y . Therefore, their angular distributions from a t o m i c o r b i t a l a r e d e r i v e d f r o m eq. (18) as
i ; : (k)
[sin2 ok sin 2¢ k 12 ,
I~':~(k) oc
IR2{V ~-~nsiu0kCOS2¢ k
3
(22)
2
R(l -, kl')
P =l-4-1
-~-lT~(3cos20k-
1)}-~ e '-i(~ 60) R0 [2 (23)
1 E
eg YI,l m+,(0 k, Ck)C 1 ( l!m + p,, lrn)[ 2.
(18)
T h i s is a g e n e r a l f o r m u l a o f t h e t r a n s i t i o n probability to the direction k from atomic o r b i t a l h a v i n g a n a n g u l a r p a r t ]~,,(0:¢). H e r e R(l ~ M ~) is t h e i n t e g r a l o f t h e r a dial p a r t , w h i c h is a c o n s t a n t c o m p l e x n u m b e r d e p e n d i n g o n I a n d l ~ a n d t a b u l a t e d as Rl+le ~5'±~ b y G o l d b e r g e t al.[7], cl(l'rn ',lm) is t h e C l e b s c h - G o r d a n o r t h e G a u n t coefficients[8].
T h e v a l u e s u s e d h e r e a r e R2--0.645, R 0 = 0 . 1 8 5 , 62 = 2.996,and 60=6.16017].
F I G . 3. a, b, a n d c s h o w s t h e p h o t o e l e c t r o n a n g u l a r d i s t r i b u t i o n p a t t e r n s e m i t t e d f r o m Pz, P~, a n d px a t o m i c o r b i t a l s .
491
These angular distributions of the phot o e m i s s i o n f r o m t h e p~(n), pv(a3), a n d P.~(a2) o r b i t a l s a r e c a l c u l a t e d w i t h e q . ( 2 1 ) , (22), a n d (23) a t EK = 50 e V a n d s h o w n in F i g . 3a, b, a n d c, r e s p e c t i v e l y . F r o m eq. (21) we find t h a t p h o t o e l e c t r o n s a r e hOe e m i t t e d w i t h i n y-z p l a n e ( ¢ = ± 2 ) , w h i c h is t h e v e r t i c a l line o f t h e Fig. l a - f t h r o u g h t h e c e n t e r . T h e b r i g h t r e g i o n in Figs. 3a is in g o o d a g r e e m e n t w i t h t h o s e a r e a in Fig. 1a-e, w h e r e we c o u l d o b s e r v e p a t t e r n s . F o r I~'~(k), t h e a z i m u t h a l a n g l e d e p e n d e n c e s h o w s n e g l i g i b l e i n t e n s i t y a t Ck = 0, ± ~ a n d n. l ~ ( k ) is s t r o n g o n t h e c e n t r a l v e r t i c a l line in c o n t r a s t t o I~a(k). T h i s a r g u m e n t strongly suggests that the Fermi surface a n d t h e 7r b a n d a r e c o m p o s e d o f p= o r b i t a l b u t n o t o f t h e p:,. o r Pv o r b i t a l . H e n c e , we c a n d i s c u s s a b o u t t h e initial s t a t e s y m m e t r y f r o m o b s e r v e d e x t i n c t i o n rule. F r o m a b o v e d i s c u s s i o n , it is c l e a r t h a t t h e p e a k s n e a r t h e c e n t e r ( c e n t r a l axis) o f t h e 1st B Z in Figs. I f is n o t t h e o n e f r o m n b a n d b u t t h a t f r o m t h e 02 b a n d s .
f a c t o r I;' i n c l u d i n g g a n d G, i.e.
1 g t;2(0,¢) O( 2(1 ~ %p.l:jF)l]~
t
~ ~G.d, 12. ~
(27)
SomeofG
a r e G0 = (0,0.0), 4 "G x ,- - ( ~ , 0 , 0 ) , 0) T h e v e c t o r dl is (d., 0.0). " ' T h e v a l u e o f k is p r i n c i p a l l y d e t e r m i n e d acc o r d i n g t o t h e k i n e t i c e n e r g y a n d t h e det e c t e d a n g l e 0. T h e initial s t a t e w a v e vect o r is c o n n e c t e d to k b y t h e d e l t a f u n c t i o n as q = k - G . T h e initial w a v e f u n c t i o n o f e q . (24) d o e s n o t c h a n g e e x c e p t for t h e p h a s e e v e n if q c h a n g e s to q t G. B e c a u s e it is n o t n e c e s s a r y t h a t q is c o n f i n e d to 1st B Z , t h e r e a r e m a n y w a y s t o c h o o s e t h e p a i r of q a n d G. F o r e x a m p l e , w h e n we c a l c u l a t e in a 2 n d B Z w i t h t h e r e c i p r o c a l v e c t o r o f G1 , w h i c h m e a n s k is in t h e 2 n d B Z , we c a n set G = 0 a n d q is in t h e 2 n d B Z , or set G = G1 a n d q in t h e 1st B Z , etc. B e c a u s e o f t h e p e r i o d i c i t y , e v e r y c h o i c e will r e s u l t in t h e s a m e r e s u l t . G2
=
(2n 2,r 3d: ~
4. PHOTOEMMISSION STRUCTURE FACTOR T h e e i g e n f n n c t i o n o f n b a n d q~(q,r) is o b t a i n e d as
1 P
ql(q, r )
eJq R ' '
,27', ~2(~ , .>,,,, Iol)
( g p A ( r -- R,,) t c*qd~p-t'(r - R,,. - d i l l , (2-1) I.q~ " w h e r e .q is d e f i n e d as g = c ~qd~ t c iqd~ ~ e iqd:~.
(25)
dl, d2 a n d da a r e p o s i t i o n v e c t o r s o f n e a r e s t
n e i g h b o u r c a r b o n a t o m s s h o w n in The total angular distribution band neglecting ODDOS, U(0,¢), t e n u s i n g eq. (14) , (21) a n d (24)
Fig. 2. from n is w r i t as
[I-~ + e - ' G d ' 12 1 ~ (0, ¢) c~ 2(1 +
spp~ Igl)
[sin0kcOS0kCOS¢k ]2 • (26)
The inequivalency between different BZs comes from the photoemission structure
F I G . 4. P h o t o e m i s s i o n
structure
factor
l:'~(kr, k u ) ,
F2(0k, ~bk) is p l o t t e d in Fig. 4. T h e p a r a l l e l o g r a m in t h e f i g u r e s h o w s t h e u n i t cell in t h e r e c i p r o c a l s p a c e . W h e n we p l o t in (k.~,ky) p l a n e , t h e a n g u l a r d e p e n d e n c e o f F 2 (k:~, kv) is e n e r g y i n d e p e n d e n t b e c a u s e t h e y a r e f u n c t i o n s o f o n l y (k~,kv). T h i s f u n c t i o n is a p e r i o d i c f u n c t i o n w i t h a p e r i o d of v~ × v ~ R 3 0 ° o f t h e n e t w o r k o f BZs,
492 w h i c h m e a n s it r e p e a t s e v e r y s e c o n d n e a r est neighbor BZs. This difference of the periodicity between the bulk BZs and this f u n c t i o n is t h e o r i g i n o f t h e n o n e q u i v a lence of the i n t e n s i t y d i s t r i b u t i o n b e t w e e n 1st a n d 2 n d B Z s . T h i s d i f f e r e n c e is o r i g i nated by the interference between the photoelectrons from the group of A atoms and that of B atoms. W h e n t h e u n i t cell o f r e a l s p a c e is c o m p o s e d o f m o r e t h a n t w o atoms, this difference always occurs. In o t h e r w o r d s , t h e d i s t r i b u t i o n in F i g . 4 is a reciprocal space intensity distribution of photoelectrons.
bution, where the density of states are multiplied by a weight of the intensity distribut i o n o f F i g . 5 c c o n s i d e r i n g its e n e r g y d e p e n d e n c e . T h e c a l c u l a t e d p a t t e r n well r e p r o d u c e s t h e s t r o n g i n t e n s i t y w i t h i n t h e 1st B Z in F i g s . l b - c . H e n c e t h e " p h o t o e m i s s i o n s t r u c t u r e f a c t o r " can r e p r o d u c e well the experimental non-equivalent intensity d i s t r i b u t i o n in t h e 1st a n d 2 n d B Z s . ACKNOWLEDGEMENT The authors are grateful to Prof. W. Schattke a n d P r o f . R . S a i t o for t h e i r v a l u able discussion. This study was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. REFERENCES
FIG. 5 . a : Photoemission structure factor F2~ b: angular distribution from atomic orbital A2, and c: the total intensity I at E K e 5 0 eV. Figure 5 shows how the total intensity I~(0k, Ck) ( F i g . 5c) is c a l c u l a t e d b y a p r o d u c t o f F 2 ( F i g . 5a) a n d A 2 ( F i g . 5 b w h i c h is t h e s a m e as F i g . 3 a ). C o m p a r i n g F i g s . 5a, b, a n d c, it c a n b e r e c o g n i z e d t h a t t h e gross feature of the inteusity distribution (the disappearance along the central vertical line) is d e t e r m i n e d m a i n l y b y t h e a n g u lar d i s t r i b u t i o n f r o m a t o m i c o r b i t a l o f F i g . 5b, a n d m o d i f i e d b y t h e s t r u c t u r e f a c t o r in photoemission of Fig.5a. The most promin e n t f e a t u r e in F i g . 5c is t h e d a r k a r e a o v e r t h e K p o i n t s in t h e 2rid B Z . T h e r e a s o n why the intensity suddenly changes from 1st B Z t o 2 n d B Z is t h e s u d d e n c h a n g e of the phase difference between two waves from A atom groups and B atom groups. The phase changes suddenly at K point bec a u s e g c r o s s e s z e r o a n d ~] c h a n g e s f r o m -t-1 t o -1. P h y s i c a l l y s p e a k i n g , t h e w a v e s from A atom groups and B atom groups i n t e r f e r e p o s i t i v e l y in t h e 1st B Z b u t t h e y interfere negatively beyond K point. T h e r i g h t s i d e f i g u r e s o f Fig. l a - f s h o w the simulated photoelectron angular distri-
[1] H.Nishimoto, T.Nakatani, T.Matsushita, H.Daimon, S.Suga, H.Namba, and T.Ohta to be published. [2] H.Daimon, Rev.Sci.Instrum. 59, 545 (1988), .ib.id. 61~ 205 (1990), H. Nishimoto et al.: Rev. Sci. Instrum. 64 2857 (1993). [3] A.Santoni, L.J.Terminello, F.J.I-timpsei, and T.Takahashi~ Appl.Phys. A 52~299(1991). [4] P.J.Feibelman, and D.E.Eastman, Phys. Rev. B 10 (1974) 4932. [5] J.B.Pendry, Low energy electron d.iffruction~ Academic Press, London (1974). [6] T. Grandke, L. Ley~ and M. Cardona, Phys. Rev. B 18 (1978) 3847. [7] S.M.Goidberg, C.S.Fadley~ and S.Kono~ J.Electron Spectroscopy and Related Phenomena, 21~ 285(1981).
[a]
....
¢)Yt m (0, ¢) sin OdOd~