Inverse photoemission from Cu(001)

~

Solid State Communications, Printed in Great Britain.

Vol.47,No.5,

INVERSE

pp.329-332, 1983.

PHOTOEMISSION

G. T h ~ r n e r Fachbereich

Physik,

Universit~t (Received

and G.

Osnabr~ck,

FROM CU(001) Borstel

D-4500

28 F e b r u a r y 1 9 8 3

OO38-1098/83 $3.00 + .OO Pergamon Press Ltd.

OsnabrHck, Fed.

Republic

of Germany

by M. Cardona)

B r e m s s t r a h l u n g u l t r a v i o l e t spectra for Cu(001) are calculated for a photon e n e r g y ~ = 9 . 7 eV w i t h i n the framework of an inverse o n e - s t e p model of p h o t o e m i s s i o n for several angles of incidence of the e l e c t r o n beam. The c o m p a r i s o n with r e c e n t l y r e p o r t e d e x p e r i m e n t a l data shows that both the c a l c u l a t e d e n e r g y d i s p e r s i o n of the peak in the spectra and the v a r i a t i o n of the peak i n t e n s i t y are in reasonable a g r e e m e n t with the experiment.

B r e m s s t r a h l u n g i s o c h r o m a t i c spect r o s c o p y in the v a c u u m u l t r a v i o l e t t s p e c t r a l region (UBIS) so far has been used to probe the d e n s i t y of u n o c c u p i e d e l e c t r o n i c states in solids I-3. Applying i m p r o v e d e x p e r i m e n t a l t e c h n i q u e s 5 W o o d r u f f and Smith 4 and D e n n l n g e r et al. r e c e n t l y were able to i n t e r p r e t UBIS data on Cu, Ni and Pt single c r y s t a l s in terms of d i r e c t optical interband t r a n s i t i o n s b e t w e e n u n o c c u p i e d states w i t h i n the g r o u n d state bulk band structure. A similar c o n c l u s i o n has been d r a w n in the o b s e r v a t i o n of an u n o c c u pied surface state on ~d 6 and in spinp o l a r i z e d UBIS from Nil. Since bremss t r a h l u n g is the t i m e - r e v e r s e d process of p h o t o e m i s s i o n 8,9, the p h o t o e m i t t e d e l e c t r o n current and the photon flux in UBIS are d i r e c t l y p r o p o r t i o n a l to each other9, 10. This fact allows for calculating UBIS spectra w i t h i n w e l l - k n o w n t h e o r i e s of p h o t Q ~ m i s s i o n like the t h r e e - s t e p m o d e l "-i or o n e - s t e p m o d e l s b a s e d on a t i m e - r e v e r s e d LEED formalism12, 13 Using an inverted t h r e e - s t e p model for d e s c r i b i n g the inverse p h o t o e m i s sion process, W o o d r u f f et al. 14 recently p e r f o r m e d c a l c u l a t i o n s for Cu and Ni with i n c l u s i o n of t r a n s i t i o n m a t r i x elements w h i c h were c a l c u l a t e d by a comb i n e d i n t e r p o l a t i o n scheme. In such a m odel l i f e t i m e effects may be a c c o u n t e d for o n l y in a crude way by c o n v o l u t i n g the final s p e c t r u m in form of a histog r a m m in some p l a u s i b l e way w i t h a L orent z i a n . As one of the c o n s e q u e n c e s , band gaps e x i s t i n g in the g r o u n d state band s t r u c t u r e will u s u a l l y show up c l e a r l y in the c a l c u l a t e d spectra, in c o n t r a s t to e x p e r i m e n t a l results and more r e a l i s t i c o n e - s t e p m o d e l s w h e r e such gaps may be w a s h e d out by the finite l i f e t i m e of the e l e c t r o n states 15. The fact that the inverted t h r e e - s t e p model does not a l l o w for the i n c i d e n t e l e c t r o n s to o c c u p y e v a n e s c e n t band gap states 16 r e p r e s e n t s a serious limita-

tion of the model. UBIS c a l c u l a t i o n s w i t h i n the framework of a o n e - s t e p model so far have been r e p o r t e d for (111) s u r f a c e s of Ni, Pd and Pt by L a r s s o n and N i l s s o n 17. The c o r r e s p o n d e n c e of their Pt(111) results to the s u b s e q u e n t e x p e r i m e n t a l spectra of D e n n i n g e r et al~ is not clear at the moment. This might be due to the c o m p l e t e n e g l e c t i o n of r e l a t i v i s t i c effects in the calculation, w h i c h for a material of high Z like Pt (Z=78) should be important. For the u n o c c u p i e d surface state Qn Pd(111) the a g r e e m e n t between theory 17,18 and a recent UBIS e x p e r i m e n t v seems to be reasonable, a l t h o u g h r e l a t i v i s t i c corrections have been n e g l e c t e d in these c a l c u l a t i o n s , too. In this c o n t r i b u t i o n we present a series of theoretical UBIS spectra for Cu(001), c a l c u l a t e d w i t h i n the inverse o n e - s t e p model of p h o t o e m i s s i o n I0, and c o m p @ r e them with recent e x p e r i m e n t a l data 4. Copper is well suited for comput a t i o n a l p u r p o s e s since reliable band s t r u c t u r e p o t e n t i a l s 19'20 exist in lit e r a t u r e and r e l a t i v i s t i c c o r r e c t i o n s are small. In c o m p a r i n g t h e o r e t i c a l results with the p u b l i s h e d e x p e r i m e n t a l UBIS data, one e n c o u n t e r s the p r o b l e m that the p u b l i s h e d e x p e r i m e n t a l spectra so far are angle integrated with respect to the o u t g o i n g p h o t o n flux. The s i m u l a t i o n of this e x p e r i m e n t a l situation w i t h i n the c o m p u t a t i o n a l scheme w o u l d be very time consuming. Fortunately in the case of Cu(001) the electric d i p o l e selection rules for d i r e c t i n t e r b a n d t r a n s i t i o n s b e t w e e n bulk states a l l o w for reducing the p r o b l e m to an e s s e n t i a l l y angle r e s o l v e d one: For a t r a n s i t i o n energy ~ = 9 . 7 eV and e l e c t r o n p r o p a g a t i o n along the [001] d i r e c t i o n in k - s p a c e there is only one t r a n s i t i o n of type AI~A1 e n e r g e t i c a l l y p o s s i b l e ~. In the n o n r e l a t i v i s t i c limit this t r a n s i t i o n is a l l o w e d for light p o l a r i z e d a l o n g the [001] direction, but f o r b i d d e n for all p o l a r i z a t i o n s p e r p e n 329

330

INVERSE PHOTOEMISSION

d i c u l a r to that direction. For e l e c t r o n s incident n o r m a l l y onto the (001) surface these s e l e c t i o n rules are in fact somewhat relaxed, since owing to the b r o k e n t r a n s l a t i o n a l s y m m e t r y a l o n g [001] and lifetime e f f e c t s the wave v e c t o r c o m p o nent k z p e r p e n d i c u l a r to the surface is no more a good q u a n t u m number. One thus expects for this c o n f i g u r a t i o n that the c o m p o n e n t s Ax,y of the v e c t o r p o t e n t i a l a s s o c i a t e d with the e m i t t e d e l e c t r o m a g netic r a d i a t i o n field will be nonzero, but much smaller than the c o m p o n e n t Az, and as a f u n c t i o n of the e l e c t r o n e n e r gy E will show e s s e n t i a l l y no structure. These e x p e c t a t i o n s are c o r r o b o r a t e d by our UBIS c a l c u l a t i o n s for e l e c t r o n s normally incident o n t o Cu(001) , as may be seen from Fig. I. R e l a t i v i s t i c effects, which are not i n c l u d e d in our c a l c u l a tions, may c a u s e a further r e l a x a t i o n of s i ~ l e - g r o u p dipole selection rules ='. But this e f f e c t will be neg-

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ENERGY ABOVE EF (eV) Fig. I. UBIS c o n t r i b u t i o n s (~w=9.7 eV, normal e l e c t r o n incidence) for light polarization p e r p e n d i c u l a r (Az) and parallel (Ax,y) to the Cu(001) surface. ligible in the p r e s e n t case, since the bands of interest are well s e p a r a t e d in e n e r g y from each o t h e r and their h y b r i d i z a t i o n due to s p i n - o r b i t i n t e r a c t i o n 21 will t h e r e f o r e be very small. F r o m the results of Fig. 1 it follows, that a primary UBIS spectrum, w h i c h is a n g l e i n t e g r a t e d with r e s p e c t to the d i r e c tion of p h o t o n flux, will be d o m i n a t e d by that c o n t r i b u t i o n w h i c h is a s s o c i a t e d with A z. The c o n t r i b u t i o n s from A x will p r o d u c e a smooth b a c k g r o u n d o~ low intensity in the p r i m a r y spectrum, which p r e s u m a b l y will be c o m p l e t e l y o v e r r i d d e n by the inelastic b a c k g r o u n d o r i g i n a t i n g from second an d h i g h e r o r d e r b r e m s s t r a h lung p r o c e s s e s j. Since the latter processes are not t a k e n into a c c o u n t in our c a l c u l a t i o n s , w h i c h focuse u p o n the e l a s t i c d i r e c t c o n t r i b u t i o n s , the in-

FROM CU(OOI)

Vol. 47, No. 5

fluence of A~ ,, s h o u l d be n e g l i g i b l e for a c o m p a r ~ 6 n with the e x p e r i m e n t a l UBIS data. Note, h o w e v e r , that A x , v ~ 0 as well as the fact that the d i e l e c t r i c c o n s t a n t ¢I for c o p p e r is less than u n i t y for h ~ = 9 . 7 e V 22 allows the emitted e l e c t r o m a g n e t i c r a d i a t i o n to leave the sample and b e c o m e d e t e c t a b l e . These a r g u m e n t s s h o w that for a m e a n i n g f u l c o m p a r i s o n w i t h the e x p e r i m e n t a l Cu(001) data it is not n e c e s s a ry to c a l c u l a t e an angle i n t e g r a t e d spectrum, but it is s u f f i c i e n t to take into a c c o u n t the c o n t r i b u t i o n from A_ only, at least in the case of n o r m a l ~ y i n c i d e n t electrons. For o f f - n o r m a l l y i n c i d e n t e l e c t r o n s up to an a n g l e of i n c i d e n c e e=29 o, the s i t u a t i o n turns out to be b a s i c a l l y the same, since owing to the d i f f r a c t i o n of the electron b e a m at the surface the i n t e r i o r angle with respect to the surface n o r m a l is smaller than 8. Due to the fact that the e x p e r i m e n t a l data r e f e r to a fixed p h o t o n energy ~ = 9 . 7 eV, s c r e e n i n q of the p h o t o n field inside the m e t a l 23 can safely be n e g l e c t e d in the p r e s e n t case. In Fig. 2 we show the c a l c u l a t e d v a r i a t i o n of the 9.7 eV i s o c h r o m a t spectra from Cu(001) with angle of e l e c t r o n incidence 8 and c o m p a r e it with the exp e r i m e n t a l data of W o o d r u f f and S m i t h 4. The t h e o r e t i c a l spectra in Fig. 2b are those a s s o c i a t e d with the c o m p o n e n t A z of the e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l p e r p e n d i c u l a r to the surface. For the c a l c u l a t i o n s we used the Cu p o t e n t i a l as g i v e n by Burdick 19 since it turned out to be superior to that of M o r u z z i et al. 20 with respect to the e n e r g e t i c p o s i t i o n s of the o c c u p i e d bands obtained in angle r e s o l v e d U V - p h o t o e m i s sion 24. The a b s o r p t i v e part V i of the p o t e n t i a l was -0.4 eV for the lower uno c c u p i e d band states and -1.0 eV for the upper ones. The full w i d t h at half m a x i m u m F W H M = 2 * V i for those band states lying about 10-12 eV above the Fermi level E F then c o r r e s p o n d s a p p r o x i m a t e l y to that d e r i v e d from angle r e s o l v e d p h o t o e m i s s i o n data 25. For s i m p l i c i t y we have i n c o r p o r a t e d the e f f e c t of instrumental b r o a d e n i n g into the value of V i for the lower u n o c c u p i e d bands 12 This p r o c e d u r e will o v e r e s t i m a t e the b a c k g r o u n d signal for EE F. The surface b a r r i e r ~=4.59 eV 26 was at 0.8 i n t e r l a y e r d i s t a n c e s outside the o u t e r m o s t atomic layer. The c o m p a rison between the e x p e r i m e n t a l and t h e o r e t i c a l spectra in Fig. 2 r e v e a l s a good overall agreement. The c a l c u l a ted peak positions show a shift &E towards lower e n e r g i e s w h i c h i n c r e a s e s up to 0.5-0.6 eV for i n c r e a s i n g a n g l e 8. Part of this d i s c r e p a n c y m a y be exp l a i n e d by the n e g l e c t i o n of second and h i g h e r order b r e m s s t r a h l u n g p r o c e s s e s 3 in the calculation. If one takes into a c c o u n t c o n t r i b u t i o n s from these processes a p p r o x i m a t e l y by a d d i n g to the pri-

VoI. 47, No. 5

INVERSE PHOTOEMISSION FROM CU(OOI)

331

Cu(O01)[lOO]

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e = 29 °

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ENERGY ABOVE EF(eV) ENERGY ABOVE EF(eV) Fig. 2. V a r i a t i o n of the ~ = 9 . 7 eV i s o c h r o m a t spectra from Cu(001) as a f u n c t i o n of angle of e l e c t r o n incidence e, a) : e x p e r i m e n t a c c o r d i n g to Ref. 4, b) : p r e s e n t theory.

mary spectra in Fig. 2b the c o r r e s p o n d i n g 8 i n t e g r a l of the p r i m a r y c o n t r i b u tion all peak p o s i t i o n s will shift to s l i g h t l y h i g h e r energies. This is dem o n s t r a t e d for e=O ° in Fig. 3. In o r d e r to o b t a i n a good a d j u s t m e n t to the exp e r i m e n t a l s p e c t r u m (c) it turned out to be n e c e s s a r y to add the integral of the p r i m a r y s p e c t r u m (a) with a factor of five to the s p e c t r u m (a). Note, however, that this e f f e c t m a y not a c c o u n t for e n e r g e t i c shifts of the order of 0.5 eV. The ratio of the c a l c u l a t e d peak h e i g h t and the h e i g h t of the backg r o u n d signal varies from nearly 7 for 8=0 ° to n e a r l y I for e=29 ° , w h i c h is c l e a r l y not in a g r e e m e n t with the e ~ p e rimental data of W o o d r u f f and Smith ~. Since more recent e x p e r i m e n t a l UBIS data on Cu(001) due to Dose et al. 27 differ s i g n i f i c a n t l y from those given in Ref. 4, we p o s t p o n e a more d e t a i l e d d i s c u s s i o n of these points until the e x p e r i m e n t a l s i t u a t i o n has become more clear. W i t h i n the o n e - s t e p model of inverse p h o t o e m i s s i o n we arrive at a sim ilar i n t e r p r e t a t i o n in terms of transitions b e t w e e n u n o c c u p i e d states in the band s t r u c t u r e as in the inverted t h r e e - s t e p model 14 . In the n o t a t i o n of Ref. 14 one t r a n s i t i o n &1~&1 from band No. 7 to band No. 6 c o n t r i b u t e s to the s p e c t r u m for 8=O °. For o f f - n o r m a l l y incident e l e c t r o n s the initial state wave function must show even p a r i t y with respect to the FKWX m i r r o r plane. This c o n d i t i o n allows for one a d d i t i o n a l transition, if e i n c r e a s e s towards 29 ° .

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(eV)

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Fig. 3. UBIS c o n t r i b u t i o n s ( ~ = 9 . 7 eV, normal e l e c t r o n incidence) from Cu(001), a) : c a l c u l a t e d p r i m a r y spectrum, b) :calc u l a t e d s p e c t r u m with h i g h e r order proc e s s e s included, c): e x p e r i m e n t according to Ref. 4.

332

INVERSE PHOTOEMISSION FROM CU(O01)

Owing to final state lifetime effects and i n s t r u m e n t a l b r o a d e n i n g these two t r a n s i t i o n s are not r e s o l v e d in the spectra of Fig. 2. In summary we have found that an inverted o n e - s t e p m o d e l of p h o t o e m i s sion d e s c r i b e s UBIS s p e c t r a of Cu(001) r e a s o n a b l y well both w i t h regard to the

Vo[. 47, No. 5

e n e r g e t i c p o s i t i o n and the i n t e n s i t y of the o b s e r v e d peaks. Since UBIS probes the u n o c c u p i e d band states, it m a y be used as a c r i t i c a l test for band structure c a l c u l a t i o n s just above the Fermi level, a r e g i o n not a c c e s s i b l e in ordinary photoemlssion.

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