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Zero-slope points and electronic energy bandmapping
Solid State Coummnications, Vol.49,No.3, pp.217-221, Printed in Great Britain.
ZERO-SLOPE
POINTS AND E L E C T R O N I C
M. WShlecke, Fachbereich
Physik,
A. B a a l m a n n
Universit~t
(Received
ENERGY
BANDMAPPING
and M. N e u m a n n
OsnabrUck,
29 S e p t e m b e r
0038-I098/84 $3.00 + .00 Pergamon Press Ltd.
1984.
D-4500
1983 by P.H.
OsnabrGck,
W-Germany
Dederichs)
A n g l e r e s o l v e d p h o t o e m i s s i o n spectra of Au(111) h a v e b e e n m e a s u r e d as a f u n c t i o n of polar angle• The w a v e v e c t o r comp o n e n t s of the e l e c t r o n s have b e e n d e t e r m i n e d by s y m m e t r y a r g u m e n t s only. The m e t h o d a p p l i e d is b a s e d on z e r o - s l o p e points and e x t r e m a in the b i n d u n g e n e r g y c a u s e d by symmetry. R e s u l t s o b t a i n e d from a p r e d o m i n a n t l y n o n r e c o n s t r u c t e d Au surface are p r e s e n t e d for the Z direction.
I. I n t r o d u c t i o n E x p e r i m e n t s in angle r e s o l v e d phot o e m i s s i o n s p e c t r o s c o p y (ARUPS) have b e e n p e r f o r m e d in the near past with c o n s i d e r a b l e i m p r o v e m e n t s for e n e r g y and a n g u l a r resolution. Thus c o m p a r i s o n of e x p e r i m e n t a l data with c a l c u l a t e d electronic energy b a n d s should be a much more c r u c i a l test. U n f o r t u n a t e l y this c o m p a r i s o n is impeded by the fact that the d e t e r m i n a t i o n of the w a v e v e c t o r of the B l o c h states of the e l e c t r o n in the c r y s t a l is not s t r a i g h t forward. A l t h o u g h the e x c i t i n g UV p h o t o n adds n e g l i g i b l e m o m e n t u m to the e l e c t r o n m o m e n t u m not all the c o m p o n e n t s of ~ are c o n s e r v e d for d i r e c t t r a n s i t i o n s , bec ause of m i s s i n g p e r i o d i c i t y normal to the c r y s t a l surface. Thus the m o m e n t u m p e r p e n d i c u l a r to the surface k~ is not conserved. In the f o l l o w i n g we c o n c e n trate on a b r i e f r e v i e w of the m e t h o d s of d e t e r m i n i n g k ~ w h i c h are k n o w n so far and a p p l i e d to d - b a n d m e t a l s in p a r t i cular. The i n t r o d u c t i o n of the zeroslope m e t h o d (ZSM) is c o m p l e t e d by exp e r i m e n t a l results on Au(111) for an e l e c t r o n i c e n e r g y b a n d m a p p i n g along Z.
bandmapping parabolic free-electronlike b a n d s in a c o n s t a n t inner p o t e n tial V_ served as an a p p r o x i m a t i o n for E f ( k f ) ~ l ] . T h i s simple a p p r o x i m a t i o n fails at the b o u n d a r y of the B r i l l o u i n zone, b e c a u s e gaps open up if a real p o t e n t i a l is considered. On the other hand the f o r m a t i o n of gaps has b e e n used to d e t e r m i n e band e n e r g i e s at critical points e x p e r i m e n t a l l y . This has been done by v a r y i n g the p h o t o n e n e r g y ~ (symmetry m e t h o d [ 2 ' 3 ] ) . B a n d m a p p i n g is e x t e n d e d to other points of the B r i l l o u i n z o n e by an i n t e r p o l a t e d s e m i e m p i rical final band b a s e d on the results for c r i t i c a l points using the p r o c e d u r e d e s c r i b e d above. A c c o r d i n g to results on c o p p e r some i n d i c a t i o n s of gad closing for final states are found [~-6]. Thus the e x a c t i d u d e of the s y m m e t r y method at c r i t i c a l points is no more bey o n d all doubt. A more d e t a i l e d desc r i p t i o n and v a l u a t i o n of the d i f f e r e n t m e t h o d s can be found e l s e w h e r e [7] . B e s i d e s some g e n e r a l ideas a b o u t the shape of the e n e r g y surface the app e a r e n c e angle m e t h o d [8] needs no further a s s u m p t i o n s . R o u g h l y speaking, the m e t h o d is c l o s e l y r e l a t e d to the symmetry method, b e c a u s e in b o t h cases the a p p e a r e n c e of a t r a n s i t i o n is s e a r c h e d for. This is done by v a r y i n g the p h o t o n energy ~ (symmetry method) or the polar angle e ( a p p e a r e n c e a n g l e method) w h i l e k e e p i n g the r e m a i n i n g p a r a m e t e r constant• The a c c u r a c y of these m e t h o d s is b a s e d on the d e t e r m i n a t i o n of app e a r e n c e angles• Peaks d i s a p p e a r or appear not a b r u p t l y but w i t h i n a c e r t a i n a n g u l a r range• T h e r e f o r e it is d i f f i c u l t to d e f i n e w i t h i n this range an a c c u r a t e a p p e a r e n c e angle• The same h o l d s for the d e t e r m i n a t i o n of t r a n s i t i o n s from the Fermi surface[ 9] A ÷ m e t h o d w h i c h is e x a c t for arbitrary k v e c t o r s and thus not r e s t r i c t e d
II. R e v i e w and the z e r o - s l o p e m e t h o d The m e t h o d s u s e d so far may be c l a s s i f i e d as a p p r o x i m a t e and exact methods. The a p p r o x i m a t e m e t h o d s rely on the a s s u m p t i o n that the final state band d i s p e r s i o n E f ( k f ~ ) i s k n o w n to some degree. An i n v e r s i o n of the r e l a t i o n E f ( k f ~ ) i n c o n n e c t i o n with the m e a s u r e d e n e r g y E of the e l e c t r o n in the v a c c u m allows the d e t e r m i n a t i o n of k i ~ according to k~l = k ~ = k~(Ef)
= k~(E)
(1)
w h e r e the s u b s c r i p t i d e n o t e s the initial state. In the very b e g i n n i n g of 217
218
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
to c r i t i c a l points (symmetry m e t h o d in c o n n e c ~ p n with extrema in the binding e n e r g y T M ) has been p r o p o s e d very early. In 1964 Kane[ I0] has s u g g e s t e d to determine k~ by t r i a n g u l a t i o n of the same final state via e m i s s i o n from two different crystal faces or from one surface but w i t h d i f f e r e n t emission angles. If an energy c o i n c i d e n c e E f ( k l ) = E f ( ~ 2) is o b s e r v e d for ~ = c o n s t , the w a v e v e c t o r s kl, ~2 are given by k2=R-~l w h e r e R is an element of the symmetry group G. Tne normal c o m p o n e n t of the w a v e v e c t o r of the e l e c t r o n follows from R and the measured parallel c o m p o n e n t s of the wavevectors. A l t h o u g h the t r i a n g u l a t i o n method is exact i~ has not been applied very often[4, 11] b e c a u s e of experimental p r o b l e m s already p o i n t e d out in [7] . A new and exact way to o v e r c o m e the u n c e r t a i n t y in k m is to use the zeroslope b e h a v i o r of energy bands caused by symmetry. We are i n t e r e s t e d in ~ a v e v e c tors k w h e r e the d e r i v a t i v e 8E(k)/~(~-~) v a n i s h e s f o r + d i r e c t i o n s s p e c i f i e d by the unit vector n. The general idea of this m e t h o d is e x p l a i n e d as follows. Usually electronic energy bands E(~) are plotted along straight lines as ~ func~ ~i n o~ < w i t h d i r e c t i o n s g i v e n by n and k=ko+
Fig.
Vol. 49, No. 3
1: B r i l l o u i n - z o n e of the fcc structure
ture. From these general ideas the following e x p e r i m e n t a l p r o c e d u r e is deduced. One has to chose a certain crystal face in order to record spectra at d i f f e r e n t polar angles 8. The angle 8 has to be varied in a plane p e r p e n d i c u l a r to the p r o m i n e n t lines or m i r r o r planes of the fcc structure. T h e r e f o r e symmetry arguments p o i n t e d out in [7,12] c o n c e r n i n g earlier b a n d m a p p i n g of gold do not hold in the strict sense, that an e x t r e m u m in the b i n d i n g energy should be caused n o n a c c i d e n t a l l y by symmetry. A l t h o u g h the (FWL) plane is p e r p e n d i c u l a r to the (FLK) plane, the first one exhibits no m i r r o r symmetry and thus does not ensure extrema caused by symmetry. III. E x p e r i m e n t and Results As an example we have m e a s u r e d spectra (ARUPS) of Au(111) with the polar angle 8 varied in the (I~0) plane and m e a s u r e d w i t h respect to the normal d i r e c t i o n [111]. The m i r r o r plane (FKWX) and the plane (FLK) form an angle of 90 ° with each other. The m i r r o r plane behavior of (FKWX) ensures extrema in the b i n d i n g energy when e is varied. Although we do not a p p r o a c h Z on a line normal to E, the zero-slope b e h a v i o r is f u l l f i l l e d in the limit of small ~. Thus b a n d m a p p i n g along Z is p o s s i b l e w h i l e using a (111) plane, w h i c h avoids rec o n s t r u c t i o n almost completely. A n g l e r e s o l v e d p h o t o e l e c t r o n spectra of Au(111) have been m e a s u r e d for d i f f e r e n t polar angles 0 using synchrotron r a d i a t i o n 15.4 < ~w < 22.5 eV from the BESSY b e a m l i n e 42.91 [14] and a commercial e l e c t r o n s p e c t r o m e t e r (beamline ADES 400, VG). Fig. 2 shows a series of spectra for ~ = 20.5 eV, Seven prO-
Vol.
Table
2]9
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
49, No. 3
I: zero-slope
points ~o and d i r e c t i o n
reference structure
~
F,X,L,W £,H,P,N
fcc bcc
A,Z,S,Z A,Z,D,G
fcc
bcc fcc
direction
O
fcc bcc
bcc
of ~. The n o t a t i o n
of
arbitrary
perpendicular
to lines
A
A
[1TO],
[[01],
[0~1]
F
[110],
[011] , [~01]
(FLK), (FKWX) (FXUL), (XUW) (FNP), (FNH)
bcc
is that of
[12]
(FPH),
perpendicular
n o u n c e d f e a t u r e s can b e seen. In Fig. the b i n d i n g energy of four of t h e s e peaks (A,B,C,G) is p l o t t e d as a f u n c -
3
h w = 2 0 . 5 eV
,,, .
to planes
(PNH)
~:
tion of 0. As e x p e c t e d due to the symm e t r y a r g u m e n t s p r e s e n t e d above e x t r e m a in the b i n d i n g energy show up. O t h e r f e a t u r e s of Fig. 2 w h i c h are not plotted in Fig. 3 are p r o b a b l y due to u m k l a p p processes. For ~ = 20.5 eV the b i n d i n g e n e r g y E b and the c o r r e s p o n d i n g p o l a r angles 8 for the extrema are g i v e n in Table II.
'..,.../'~,"
.,J
..,
,,/'k.,,,~...
,'.Ji
~8 o
i 53°
58°
~.~/
53< 48<
"-x
_
~.. E F =0
/1".,. .p...,,,,:
/k,/--,,
...,-,jr -
-:- i '.....
jBi"',J,..
11".,.
C'., :,,,',j
-2
,-,
,, " " : : . " # " ~ #,'-~'- '.,,."'.,,,,~
..i--'--: / #"i :/"~'~."'~' "
"
"
"#'~%"
i"- i /
#:
43'
,~--~" /
..,',, "": ~""~-.,J /-: \-~
c_ m
38< 33< . j
,_..l"
i "O~ O''"O""O"--O--...~O.,_O..~ G
28 < J I
EF
I
I
I
-2
-4
-6
I -8
-8
Binding energy (eV)
Fig. Fig.
2: A n g l e r e s o l v e d p h o t o e m i s s i o n of Au(111) for d i f f e r e n t p o l a r angles 8
I 30
I
I
40 50 PoLar a n g l e
I
I
60
70
3: A n g u l a r d e p e n d e n c e of the binding e n e r g y for h ~ = 2 0 . 5 eV. A r r o w s indicate e m i s s i o n from the Z d i r e c t i o n
220
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
Table II: 4 ~ ° E b (eV) 1.8
41 ° 2.5
e k(~ -1 )
2.39
2.37
I n c l u d e d are the w a v e v e c t o r s w i t h an unc e r t a i n t y of 5-10% of zone b o u n d a r y momentum, d e p e n d i n g on the curvature. As already p o i n t e d out electron e m i s s i o n at the extrema c o r r e s p o n d s to e l e c t r o n s travelling along Z in the crystal, k" is c a l c u l a t e d using ktl=~-l(2m(~m-~-Eb)) 1/2sin8 with the w o r k f u n c t i o n ~=5.4 eV deduced from the spectra. The w a v e v e c t o r k ~ along E is c a l c u l a t e d using k ~ = k , ° ( s i n 35.270) -I . The lines A and E form an angle 35.27 ° with each other. The emission is m a i n l y due to w a v e v e c t o r s from a region close to the X-point (k=2.18 ~-I). A comparison of e x p e r i m e n t a l data (Fig. 4) shows within the e x p e r i m e n t a l u n c e r t a i n t i e s of about 5 % good agreement with c a l c u l a t e d RAPW-values[15]. To avoid a m i s i n t e r p r e tation of the results p l o t t e d in the lower part of Fig. 4 we want to point out, that the lowest black dots correspond to the c a l c u l a t e d energy band that crosses the X point at 7 eV. These new results c o n f i r m our p r e l i m i n a r y results on Au(111) [16] and clearly d e m o n s t r a t e the u s e f u l l n e s s of the method. Very recently the zero-slope method has been applied to Ag with good success[17]. In c o n c l u s i o n we want to point out that the e x t e n s i v e use of symmetry arguments may serve as a powerful tool for e x p e r i m e n t a l bandmapping. We have shown that b a n d s t r u c t u r e m a p p i n g along Z is possible by using the almost n g n r e c o n structed Au(111) face. The zero-slope method does not depend on the ~nowledge of the inner p o t e n t i a l Vo, but V o can easily be c a l c u l a t e d assuming a certain m* if it is needed for further investigations based on more approximate methods as r e v i e w e d in section II. F i n a n c i a l support by the Bundesm i n i s t e r i u m fur F o r s c h u n g und T e c h n o l o gie (BMFT) is g r e a t f u l l y a c k n o w l e d g e d (grant NUP 230).
C
G
3.0
7.5
50 °
58 ° 2.06
2.35
Vol. 49, No. 3
EF=O AU
•
0 o
e-
.~_ e.m
-8
Fig.
K
X E' E' Wave vector
K
4: E l e c t r o n i c energy b a n d dispersion of Au along Z at the X point. E x p e r i m e n t a l points have been d e t e r m i n e d with the zeroslope method using different photon energies: h~=15.4 eV, 16.8 eV, 18.5 eV, 20.5 eV and 22.5 eV.
REFERENCES [1]
[2]
[3]
J. St~hr, P.S. Wehner, R. S. Williams, G. Apai, and D.A. Shirley, Phys. Rev. B17, 587 (1978) F.J. Himpsel and D.E. Eastman, Phys. Rev. B18, 5236 (1978) E. Dietz and D.E. Eastman, Phys. Rev. Lett. 4!1, 1674 (1978) F.J. Himpsel and W.E. Eberhardt, Solid State Commun. 3__!1,748 (1979)
[4] [5] [6]
[7]
P.O. N i l s s o n and N. Dahlb~ck, Solid State Commun. 2_99, 303 (1979) J.K. Grepstad, B.J. S l a g s v o l d and I. Bartos, J. Phys. F12, 1979 (1982) A. Baalmann, M. Neumann, H. Neddermeyer, W. Radlik and W. Braun, Ann. Israel Phys. Soc. 6, 351 (1983) F.J. Himpsel, Adv. Phys. 3_22, 1 (1983)
Vol. 49, No. 3 [8]
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
N.E. Christensen, Solid State Commun. 38, 309 (1981) [9] D.E. Eastman, J.A. K n a p p and F.J. Himpsel, Phys. Rev. Lett. 4_!, 825 (1978) [10] E.O. Kane, Phys. Rev. Lett. 12, 97 (1964) [11] P. Heimann, H. M i o s g a and H. Neddermeyer, Solid State Commun. 29, 463 (1979) [12] A.P. Cracknell, J. Phys. C6, 826 (1973)
[13] H. Neddermeyer, Solid State Commun. 40, 809 (1981) [14] W. B r a u n and G. J~kisch, Ann. Israel Phys. Soc. 6, 30 (1983) [15] N.E. C h r i s t e n s e n and B.O. S e r a p h i ~ Phys. Rev. B4, 3321 (1971) [16] M. W6hlecke, A. B a a l m a n n and M. Neumann, Ann. Israel Phys. Soc. 6, 318 (1983) [17] P.O. Nilsson, priv. c o m m u n i c a t i o n
221
ZERO-SLOPE
POINTS AND E L E C T R O N I C
M. WShlecke, Fachbereich
Physik,
A. B a a l m a n n
Universit~t
(Received
ENERGY
BANDMAPPING
and M. N e u m a n n
OsnabrUck,
29 S e p t e m b e r
0038-I098/84 $3.00 + .00 Pergamon Press Ltd.
1984.
D-4500
1983 by P.H.
OsnabrGck,
W-Germany
Dederichs)
A n g l e r e s o l v e d p h o t o e m i s s i o n spectra of Au(111) h a v e b e e n m e a s u r e d as a f u n c t i o n of polar angle• The w a v e v e c t o r comp o n e n t s of the e l e c t r o n s have b e e n d e t e r m i n e d by s y m m e t r y a r g u m e n t s only. The m e t h o d a p p l i e d is b a s e d on z e r o - s l o p e points and e x t r e m a in the b i n d u n g e n e r g y c a u s e d by symmetry. R e s u l t s o b t a i n e d from a p r e d o m i n a n t l y n o n r e c o n s t r u c t e d Au surface are p r e s e n t e d for the Z direction.
I. I n t r o d u c t i o n E x p e r i m e n t s in angle r e s o l v e d phot o e m i s s i o n s p e c t r o s c o p y (ARUPS) have b e e n p e r f o r m e d in the near past with c o n s i d e r a b l e i m p r o v e m e n t s for e n e r g y and a n g u l a r resolution. Thus c o m p a r i s o n of e x p e r i m e n t a l data with c a l c u l a t e d electronic energy b a n d s should be a much more c r u c i a l test. U n f o r t u n a t e l y this c o m p a r i s o n is impeded by the fact that the d e t e r m i n a t i o n of the w a v e v e c t o r of the B l o c h states of the e l e c t r o n in the c r y s t a l is not s t r a i g h t forward. A l t h o u g h the e x c i t i n g UV p h o t o n adds n e g l i g i b l e m o m e n t u m to the e l e c t r o n m o m e n t u m not all the c o m p o n e n t s of ~ are c o n s e r v e d for d i r e c t t r a n s i t i o n s , bec ause of m i s s i n g p e r i o d i c i t y normal to the c r y s t a l surface. Thus the m o m e n t u m p e r p e n d i c u l a r to the surface k~ is not conserved. In the f o l l o w i n g we c o n c e n trate on a b r i e f r e v i e w of the m e t h o d s of d e t e r m i n i n g k ~ w h i c h are k n o w n so far and a p p l i e d to d - b a n d m e t a l s in p a r t i cular. The i n t r o d u c t i o n of the zeroslope m e t h o d (ZSM) is c o m p l e t e d by exp e r i m e n t a l results on Au(111) for an e l e c t r o n i c e n e r g y b a n d m a p p i n g along Z.
bandmapping parabolic free-electronlike b a n d s in a c o n s t a n t inner p o t e n tial V_ served as an a p p r o x i m a t i o n for E f ( k f ) ~ l ] . T h i s simple a p p r o x i m a t i o n fails at the b o u n d a r y of the B r i l l o u i n zone, b e c a u s e gaps open up if a real p o t e n t i a l is considered. On the other hand the f o r m a t i o n of gaps has b e e n used to d e t e r m i n e band e n e r g i e s at critical points e x p e r i m e n t a l l y . This has been done by v a r y i n g the p h o t o n e n e r g y ~ (symmetry m e t h o d [ 2 ' 3 ] ) . B a n d m a p p i n g is e x t e n d e d to other points of the B r i l l o u i n z o n e by an i n t e r p o l a t e d s e m i e m p i rical final band b a s e d on the results for c r i t i c a l points using the p r o c e d u r e d e s c r i b e d above. A c c o r d i n g to results on c o p p e r some i n d i c a t i o n s of gad closing for final states are found [~-6]. Thus the e x a c t i d u d e of the s y m m e t r y method at c r i t i c a l points is no more bey o n d all doubt. A more d e t a i l e d desc r i p t i o n and v a l u a t i o n of the d i f f e r e n t m e t h o d s can be found e l s e w h e r e [7] . B e s i d e s some g e n e r a l ideas a b o u t the shape of the e n e r g y surface the app e a r e n c e angle m e t h o d [8] needs no further a s s u m p t i o n s . R o u g h l y speaking, the m e t h o d is c l o s e l y r e l a t e d to the symmetry method, b e c a u s e in b o t h cases the a p p e a r e n c e of a t r a n s i t i o n is s e a r c h e d for. This is done by v a r y i n g the p h o t o n energy ~ (symmetry method) or the polar angle e ( a p p e a r e n c e a n g l e method) w h i l e k e e p i n g the r e m a i n i n g p a r a m e t e r constant• The a c c u r a c y of these m e t h o d s is b a s e d on the d e t e r m i n a t i o n of app e a r e n c e angles• Peaks d i s a p p e a r or appear not a b r u p t l y but w i t h i n a c e r t a i n a n g u l a r range• T h e r e f o r e it is d i f f i c u l t to d e f i n e w i t h i n this range an a c c u r a t e a p p e a r e n c e angle• The same h o l d s for the d e t e r m i n a t i o n of t r a n s i t i o n s from the Fermi surface[ 9] A ÷ m e t h o d w h i c h is e x a c t for arbitrary k v e c t o r s and thus not r e s t r i c t e d
II. R e v i e w and the z e r o - s l o p e m e t h o d The m e t h o d s u s e d so far may be c l a s s i f i e d as a p p r o x i m a t e and exact methods. The a p p r o x i m a t e m e t h o d s rely on the a s s u m p t i o n that the final state band d i s p e r s i o n E f ( k f ~ ) i s k n o w n to some degree. An i n v e r s i o n of the r e l a t i o n E f ( k f ~ ) i n c o n n e c t i o n with the m e a s u r e d e n e r g y E of the e l e c t r o n in the v a c c u m allows the d e t e r m i n a t i o n of k i ~ according to k~l = k ~ = k~(Ef)
= k~(E)
(1)
w h e r e the s u b s c r i p t i d e n o t e s the initial state. In the very b e g i n n i n g of 217
218
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
to c r i t i c a l points (symmetry m e t h o d in c o n n e c ~ p n with extrema in the binding e n e r g y T M ) has been p r o p o s e d very early. In 1964 Kane[ I0] has s u g g e s t e d to determine k~ by t r i a n g u l a t i o n of the same final state via e m i s s i o n from two different crystal faces or from one surface but w i t h d i f f e r e n t emission angles. If an energy c o i n c i d e n c e E f ( k l ) = E f ( ~ 2) is o b s e r v e d for ~ = c o n s t , the w a v e v e c t o r s kl, ~2 are given by k2=R-~l w h e r e R is an element of the symmetry group G. Tne normal c o m p o n e n t of the w a v e v e c t o r of the e l e c t r o n follows from R and the measured parallel c o m p o n e n t s of the wavevectors. A l t h o u g h the t r i a n g u l a t i o n method is exact i~ has not been applied very often[4, 11] b e c a u s e of experimental p r o b l e m s already p o i n t e d out in [7] . A new and exact way to o v e r c o m e the u n c e r t a i n t y in k m is to use the zeroslope b e h a v i o r of energy bands caused by symmetry. We are i n t e r e s t e d in ~ a v e v e c tors k w h e r e the d e r i v a t i v e 8E(k)/~(~-~) v a n i s h e s f o r + d i r e c t i o n s s p e c i f i e d by the unit vector n. The general idea of this m e t h o d is e x p l a i n e d as follows. Usually electronic energy bands E(~) are plotted along straight lines as ~ func~ ~i n o~ < w i t h d i r e c t i o n s g i v e n by n and k=ko+
Fig.
Vol. 49, No. 3
1: B r i l l o u i n - z o n e of the fcc structure
ture. From these general ideas the following e x p e r i m e n t a l p r o c e d u r e is deduced. One has to chose a certain crystal face in order to record spectra at d i f f e r e n t polar angles 8. The angle 8 has to be varied in a plane p e r p e n d i c u l a r to the p r o m i n e n t lines or m i r r o r planes of the fcc structure. T h e r e f o r e symmetry arguments p o i n t e d out in [7,12] c o n c e r n i n g earlier b a n d m a p p i n g of gold do not hold in the strict sense, that an e x t r e m u m in the b i n d i n g energy should be caused n o n a c c i d e n t a l l y by symmetry. A l t h o u g h the (FWL) plane is p e r p e n d i c u l a r to the (FLK) plane, the first one exhibits no m i r r o r symmetry and thus does not ensure extrema caused by symmetry. III. E x p e r i m e n t and Results As an example we have m e a s u r e d spectra (ARUPS) of Au(111) with the polar angle 8 varied in the (I~0) plane and m e a s u r e d w i t h respect to the normal d i r e c t i o n [111]. The m i r r o r plane (FKWX) and the plane (FLK) form an angle of 90 ° with each other. The m i r r o r plane behavior of (FKWX) ensures extrema in the b i n d i n g energy when e is varied. Although we do not a p p r o a c h Z on a line normal to E, the zero-slope b e h a v i o r is f u l l f i l l e d in the limit of small ~. Thus b a n d m a p p i n g along Z is p o s s i b l e w h i l e using a (111) plane, w h i c h avoids rec o n s t r u c t i o n almost completely. A n g l e r e s o l v e d p h o t o e l e c t r o n spectra of Au(111) have been m e a s u r e d for d i f f e r e n t polar angles 0 using synchrotron r a d i a t i o n 15.4 < ~w < 22.5 eV from the BESSY b e a m l i n e 42.91 [14] and a commercial e l e c t r o n s p e c t r o m e t e r (beamline ADES 400, VG). Fig. 2 shows a series of spectra for ~ = 20.5 eV, Seven prO-
Vol.
Table
2]9
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
49, No. 3
I: zero-slope
points ~o and d i r e c t i o n
reference structure
~
F,X,L,W £,H,P,N
fcc bcc
A,Z,S,Z A,Z,D,G
fcc
bcc fcc
direction
O
fcc bcc
bcc
of ~. The n o t a t i o n
of
arbitrary
perpendicular
to lines
A
A
[1TO],
[[01],
[0~1]
F
[110],
[011] , [~01]
(FLK), (FKWX) (FXUL), (XUW) (FNP), (FNH)
bcc
is that of
[12]
(FPH),
perpendicular
n o u n c e d f e a t u r e s can b e seen. In Fig. the b i n d i n g energy of four of t h e s e peaks (A,B,C,G) is p l o t t e d as a f u n c -
3
h w = 2 0 . 5 eV
,,, .
to planes
(PNH)
~:
tion of 0. As e x p e c t e d due to the symm e t r y a r g u m e n t s p r e s e n t e d above e x t r e m a in the b i n d i n g energy show up. O t h e r f e a t u r e s of Fig. 2 w h i c h are not plotted in Fig. 3 are p r o b a b l y due to u m k l a p p processes. For ~ = 20.5 eV the b i n d i n g e n e r g y E b and the c o r r e s p o n d i n g p o l a r angles 8 for the extrema are g i v e n in Table II.
'..,.../'~,"
.,J
..,
,,/'k.,,,~...
,'.Ji
~8 o
i 53°
58°
~.~/
53< 48<
"-x
_
~.. E F =0
/1".,. .p...,,,,:
/k,/--,,
...,-,jr -
-:- i '.....
jBi"',J,..
11".,.
C'., :,,,',j
-2
,-,
,, " " : : . " # " ~ #,'-~'- '.,,."'.,,,,~
..i--'--: / #"i :/"~'~."'~' "
"
"
"#'~%"
i"- i /
#:
43'
,~--~" /
..,',, "": ~""~-.,J /-: \-~
c_ m
38< 33< . j
,_..l"
i "O~ O''"O""O"--O--...~O.,_O..~ G
28 < J I
EF
I
I
I
-2
-4
-6
I -8
-8
Binding energy (eV)
Fig. Fig.
2: A n g l e r e s o l v e d p h o t o e m i s s i o n of Au(111) for d i f f e r e n t p o l a r angles 8
I 30
I
I
40 50 PoLar a n g l e
I
I
60
70
3: A n g u l a r d e p e n d e n c e of the binding e n e r g y for h ~ = 2 0 . 5 eV. A r r o w s indicate e m i s s i o n from the Z d i r e c t i o n
220
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
Table II: 4 ~ ° E b (eV) 1.8
41 ° 2.5
e k(~ -1 )
2.39
2.37
I n c l u d e d are the w a v e v e c t o r s w i t h an unc e r t a i n t y of 5-10% of zone b o u n d a r y momentum, d e p e n d i n g on the curvature. As already p o i n t e d out electron e m i s s i o n at the extrema c o r r e s p o n d s to e l e c t r o n s travelling along Z in the crystal, k" is c a l c u l a t e d using ktl=~-l(2m(~m-~-Eb)) 1/2sin8 with the w o r k f u n c t i o n ~=5.4 eV deduced from the spectra. The w a v e v e c t o r k ~ along E is c a l c u l a t e d using k ~ = k , ° ( s i n 35.270) -I . The lines A and E form an angle 35.27 ° with each other. The emission is m a i n l y due to w a v e v e c t o r s from a region close to the X-point (k=2.18 ~-I). A comparison of e x p e r i m e n t a l data (Fig. 4) shows within the e x p e r i m e n t a l u n c e r t a i n t i e s of about 5 % good agreement with c a l c u l a t e d RAPW-values[15]. To avoid a m i s i n t e r p r e tation of the results p l o t t e d in the lower part of Fig. 4 we want to point out, that the lowest black dots correspond to the c a l c u l a t e d energy band that crosses the X point at 7 eV. These new results c o n f i r m our p r e l i m i n a r y results on Au(111) [16] and clearly d e m o n s t r a t e the u s e f u l l n e s s of the method. Very recently the zero-slope method has been applied to Ag with good success[17]. In c o n c l u s i o n we want to point out that the e x t e n s i v e use of symmetry arguments may serve as a powerful tool for e x p e r i m e n t a l bandmapping. We have shown that b a n d s t r u c t u r e m a p p i n g along Z is possible by using the almost n g n r e c o n structed Au(111) face. The zero-slope method does not depend on the ~nowledge of the inner p o t e n t i a l Vo, but V o can easily be c a l c u l a t e d assuming a certain m* if it is needed for further investigations based on more approximate methods as r e v i e w e d in section II. F i n a n c i a l support by the Bundesm i n i s t e r i u m fur F o r s c h u n g und T e c h n o l o gie (BMFT) is g r e a t f u l l y a c k n o w l e d g e d (grant NUP 230).
C
G
3.0
7.5
50 °
58 ° 2.06
2.35
Vol. 49, No. 3
EF=O AU
•
0 o
e-
.~_ e.m
-8
Fig.
K
X E' E' Wave vector
K
4: E l e c t r o n i c energy b a n d dispersion of Au along Z at the X point. E x p e r i m e n t a l points have been d e t e r m i n e d with the zeroslope method using different photon energies: h~=15.4 eV, 16.8 eV, 18.5 eV, 20.5 eV and 22.5 eV.
REFERENCES [1]
[2]
[3]
J. St~hr, P.S. Wehner, R. S. Williams, G. Apai, and D.A. Shirley, Phys. Rev. B17, 587 (1978) F.J. Himpsel and D.E. Eastman, Phys. Rev. B18, 5236 (1978) E. Dietz and D.E. Eastman, Phys. Rev. Lett. 4!1, 1674 (1978) F.J. Himpsel and W.E. Eberhardt, Solid State Commun. 3__!1,748 (1979)
[4] [5] [6]
[7]
P.O. N i l s s o n and N. Dahlb~ck, Solid State Commun. 2_99, 303 (1979) J.K. Grepstad, B.J. S l a g s v o l d and I. Bartos, J. Phys. F12, 1979 (1982) A. Baalmann, M. Neumann, H. Neddermeyer, W. Radlik and W. Braun, Ann. Israel Phys. Soc. 6, 351 (1983) F.J. Himpsel, Adv. Phys. 3_22, 1 (1983)
Vol. 49, No. 3 [8]
ZERO-SLOPE POINTS AND ELECTRONIC ENERGY BANDMAPPING
N.E. Christensen, Solid State Commun. 38, 309 (1981) [9] D.E. Eastman, J.A. K n a p p and F.J. Himpsel, Phys. Rev. Lett. 4_!, 825 (1978) [10] E.O. Kane, Phys. Rev. Lett. 12, 97 (1964) [11] P. Heimann, H. M i o s g a and H. Neddermeyer, Solid State Commun. 29, 463 (1979) [12] A.P. Cracknell, J. Phys. C6, 826 (1973)
[13] H. Neddermeyer, Solid State Commun. 40, 809 (1981) [14] W. B r a u n and G. J~kisch, Ann. Israel Phys. Soc. 6, 30 (1983) [15] N.E. C h r i s t e n s e n and B.O. S e r a p h i ~ Phys. Rev. B4, 3321 (1971) [16] M. W6hlecke, A. B a a l m a n n and M. Neumann, Ann. Israel Phys. Soc. 6, 318 (1983) [17] P.O. Nilsson, priv. c o m m u n i c a t i o n
221