Vectorial volume effect in interfacial photoemission from Cu(111) and Au(111) at low photon energies

Surface Science 62 (1977) 106-118 © North-HoUand Pubhshmg Company

VECTORIAL VOLUME EFFECT IN I N T E R F A C I A L PHOTOEMISSION FROM Cu(111 ) AND A u ( l 11) AT LOW PHOTON ENERGIES H LAUCHT, J.K. SASS, H.J. LEWERENZ and K L KLIEWER * Fntz-Haber-lnstttut der Max-Planck-Gesellschaft, Faradayweg 4 - 6, D-1000 Berhn 33, Germany Received 7 July 1976, manuscript recewed m fmal form 14 October 1976

Pronounced angle-of-incidence and polarization dependences of the photoemlsslon yields from Cu(111) and Au(ll 1) were observed in the photon energy range of 3 4-5 0 eV with the mterfaclal photoemlsslon-mto-electrolyte techmque Strong evidence was obtamed that these vectorial features are due to the amsotropic bulk' excitation of photoelectrons and restnctwe escape condition of parallel k-vector conservation Calculations, based upon a simple model of amsotroplc bulk photoemlsslon, show good agreement with the experimental results

1. Introduction There have been numerous experimental observatxons [ 1 - 1 0 ] that p-polarized hght is considerably more effectwe than s-polarized hght m producing photoelectrons from sohds. However, the origin o f this effect is stdl a subject of considerable controversy [11]. This seems, at first glance, somewhat surprising since photoexcited electrons must have a sufficiently large velocity component normal to the surface to escape the sohd and, thus, an electric field component normal to the surface, which occurs only for p-polarized hght, would appear to be particularly effectwe m producxng photoelectrons The actual situation ~s, however, not as simple as that represented by the precedxng, Lmphcltly classical comment, the reasons for this are twofold. First, for bulk photoemlsslon, the matrix elements which govern the photoexcltatlon processes lead to excitations which cannot be viewed m a simple classical fashion Second, p-polarized hght offers the possibility of "surface" photoemlsslon which involves mechamsms umque to p-polarxzatlon [12-14]. Bulk photoemlsslon is customardy thought of as resulting from momentumconserwng, &rect transitions which can, in general, occur for any field orientation For a cubic crystal, the optical absorptance resulting from such transitions is * A vlsltmg scientist, permanent address Ames Laboratory - ERDA and Department of Physics, Iowa State Umverslty, Ames, Iowa 50010, USA 106

H Lauch t et al / Vectorial volume effect m mterfactal photoemtsslon

107

dependent on the angle of incidence of the light but not on the crystal orientation If one construes this fact as implying that the resulting photoexclted electrons are distributed umformly in momentum space over all crystallographlcally equivalent states or. for slmphclty, distributed lsotropically, a simple theory for bulk photoemission results in which the yield is determined essentially by the optical properties Indeed, the excess in the photoyleld observed with p-polarized hght above that expected from the optical characteristics has often been attributed to a surface effect [2,3,7]. It has been established that the assumption that photoexcited electrons are distributed uniformly over equivalent crystallographic states is incorrect Gobeh et al [15] and Becker et al [16] have shown that electrons excited by directtransitions are not distributed evenly among the different, energetically degenerate arms of the star of k. The strong anisotroplc occupation of equivalent final states is a consequence of the polarization dependent operator of the transition matrix element given by (@f IA P[ qJ~), with ~b~ and ~f the lnltial'and final states and p the momentum operator As the vector potential A is different for p- and s-polarized light, the resulting distribution of excited electrons In momentum space is also different The vectorial features of bulk photoemisslon arising from the inclusion of polarizatlon effects in the matrLx elements have recently been discussed by Schaich [ 17] for single-crystal surfaces, and corresponding anlsotroples have been described also for polycrystalhne material [9]. In fact, the simple picture alluded to above, that is, that p-polarized light, with an electric field component normal to the surface, excites more electrons with large wavevector component normal to the surface is not totally without merit. Identifying contributions from volume and surface excitations in the yield with p-polarized light is therefore the main task in the interpretation of experimentally observed vector effects In a recent study on silver [10], large vectorial effects were observed in a photon energy range where no direct transitions occur, i e below 3 85 eV [16] The interpretation is then much simpler and direct exndence for a new nonlocal surface photoemisslon theory [13,14] was obtained In the present paper we report the observation of vectorial effects in photoemlsslon from the (111) face of copper and gold. As in our study of silver [10], we employ the mterfaclal technique of photoemission into an electrolyte, by which we obtain a threshold for electron emission significantly lower than that in vacuo. In the range of low excitation energies available with this technique, the optical properties of gold and copper are not as strilong as those of Sliver and are dominated by groups of well-identified direct transitions which are very similar for the two materials Thus these materials In this low-energy region appear to be excellent systems for investigating the vectorial character of bulk photoemission In section 2 we discuss the capability of the electrolyte technique in photoemlssion studies. We describe the experimental arrangement and present the photon energy, angle-of-incidence, and polarization dependence of the yields for copper

108

H Laucht et al / Vectorial volume effect m mterfactal photoemlsston

and gold in section 3. In interpreting our results in sechon 4 we will use a bulk, direct-transition model to examine to what extent the results are consistent with a reasonably complete treatment of such a model This procedure is suggested by recent conclusions of Becker et al [16] as well as Ilver and Nllsson [18] Finally, m section 5, we discuss the lmphcatyons o f this analysis and point out ItS hmltatlons

2. The photoemission-into-electrolyte technique Interfaclal photoemlsslon at an electrolyte contact [19] has recently been employed m studies relating to bulk and surface properties of metals [9,10,20,21] Due to the reduced threshold for electron eImsslon, an additional range of photon energies, roughly from 5 to 3 eV for Cu and Au, can be utlhzed There is also the possibility to adjust the threshold for different metals to the same value One aspect of photoemlssl'on into electrolyte warrants further comment Instead of being emitted into vacuum, electrons with energy and momentum appropriate for escape are confronted with a condensed medium Although experimental evidence that electrons do escape into the electrolyte IS unequivocal, we do not know a priori the escape condmons Electron escape into vacuum is governed by conservation of the crystal momentum parallel to the surface if the periodicity of the bulk lattice parallel to the surface is maintained m the immediate surface region [11] As has recently been shown [22] an electron excited in the bulk can receive discrete amounts of parallel m o m e n t u m from a perturbed surface The effect was observed m angle-resolved photoemlsslon studies and termed chemlsorptlon-mduced surface umklapp. In our opinion, two different types of surface umklapp processes may be thought of, depending on whether the photoelectron receives the a d d m o n a l parallel momentum "Inside" or "outside" the sohd, that is, within the region where the mean potentml normal to the surface is such that the region should be thought o f as part of the sohd or outside of thls region This separation is, of course, qualitative but nonetheless useful and is dlustrated m fig 1 Momentum exchange inside the solid will alter the escape characteristics and may change the total yield whereas the addition of parallel momentum to an electron that has already escaped will merely alter its direction o f motion and not the total emission In general, on a surface with an adsorbate, one may have a situation where both of the processes can occur Nevertheless, surface umklapp "Inside" the potential barrier should be slgmflcant only when there is a strong interaction with the adsorbate The interaction of water molecules with the noble metals at an electrolyte contact is weak [23] Consequently the parallel k-vector conservation should also represent the escape condition into an electrolyte Experimental confirmaUon of this expectation can be deduced from the large ratios o f the p-polarized yield to the s-polarized yield at metal-electrolyte raterfaces [9,10] As the quahty of a surface deteriorates, this vector ratio decreases

H Laucht et al / Vectorial volume effect m mterfactal photoemtsston

109

J I

',

(a)

I

I

t I

i I

inside

outstde

Fig 1 Mean potential curves normal to the surface illustrating the separation into regions "inside" and "outside" the sohd (a) Clean surface, (b) surface including adsorbate with which it strongly interacts, (c) surface including adsorbate with which it weakly interacts

[24] This is not surprising since the vectorial character of the excitation matrix elements indicates that s-polarized light produces many more electrons moving at large angles to the surface. The relaxation of the strict escape condition at a distorted surface will then mean a significantly larger increase in the s-yield as compared with that for p and the ratios will decrease In our studies of silver [10] using photoemIsslon Into electrolytes, these ratios were ~10. As an alternative to the electrolyte technique, the surface of a sample is frequently cesiated to obtain a lower threshold The interaction with the adsorbate is then very strong and the escape condition of parallel k-vector conservation IS likely to be relaxed to a considerable extent This would mean that large vector ratios associated with the escape restriction tend to decrease on cesiated surfaces In the work of Nilsson et al [25] on thin ceslated silver films, the vector ratios are five to ten times smaller than those obtained in electrolyte [26]

3. Experimental methods and results Our experimental arrangement for measuring photocurrents at a metal-electrolyte contact, modified somewhat from that of our previous work [9,20], is shown in fig. 2 The reference electrode and the counter electrode have been replaced by a single normal hydrogen electrode Reference properties of this electrode are excellent for the small currents typically observed In our photoemlsslon studies The electrode potential of the sample, i e , its threshold, is simply controlled by an appropriate voltage between the two electrodes Compared to the usual three-electrode

110

H Laucht et al / Vectortal volume effect m mterfactal photoemtsston

~

NHE RM

Flg 2 Experimental set-up for photocurrent measurements at metal-electrolyte Interfaces (XL xenon lamp, M monochromator, CH hght chopper, GPP glan polarizatmn prism, NHE normal hydrogen electrode, ME measuring electrode, SM stepping motor, R M measuring

resistance) arrangement with electronic control of the potential via a potentlostat, the signalto-noise ratio for the photocurrents has been improved by at least an order of magnitude This enabled us to obtain better spectral resolution than In earlier studies The spectral half-width of the triangular transmission function of the monocharomator wag 5 nm, corresponding to 0 05 and 0 1 eV at photon energies of 3 and 5 eV, respectively Single crystal f'flms of copper and gold (approximately 1500 A thick) with (111) faces were obtained by directly evaporating the metal onto air-cleaved mica substrates heated to -..400°C Reflection high-energy electron diffraction (RHEED) analysis of the films confirmed the (111) orientation. Transferring the metal films from the evaporation chamber to the electrochemical cell is the most critical step for obtaining reproducible results. An Investigation of this problem [24] led to the procedure of producing a nitrogen glow discharge in the vacuum chamber shortly before rapid transfer to the cell This procedure protects the films during their exposure to air, presumably as a result of a chemlsorbed nitrogen layer on the surface Similar protective features have been found in a study of the resistance of thin silver films [27] In the electrochemical cell this chemlsorbed nitrogen will be desorbed With samples treated in this manner, the highest yields and vector ratios as well as the best reproducibility (10-15%) were obtained Additional evidence that the exposure to air has no seriously adverse effects on the photoemlssion properties was obtained by in-situ electrochemical dxssolutlon of up to 50 layers of the sample [26] Yield changes of less than 5% resulted from such treatment The electrode potential of both the Au(111) and Cu(111) electrodes was kept at - 0 1 V with respect to the normal hydrogen electrode This corresponds to a threshold for electron emission of about 3.0 eV for both metals The yields were measured for four angles of incidence (0 °, 22 5°, 45 0 ° and 67 5°) with p- and spolarized hght, the intensity of which had been determined with a calibrated bolometer [20] The photon energy dependence of the yields Yp,Ys and the vector ratios Yp/Ys are shown in figs 3 and 4 Yields are quite different for copper and gold and in-

H Laucht et al

/ Vectorml volume effect m mterfactal photoemtsslon

111

)0-3

15

I I

I

10.~-

10

o

~ 10s

Cu(111) 4" "7

P~~

• f

o

o

I

rj .Q -o

o

o

10-6

3.0

35

4.0

4.5

50 35

40

45 photon energy

50 htoleV

Fig 3 Photon energy dependence of the yields Yt~ and Ys for p- and s- polarized light and t h e e ratios Yp/Ys for C u ( l l l ) m I n H2SO 4, angles of incidence 0 ° ( - - ) , 22 5 ° (. ), 45 ° ( - - - ) , 67 5 ° ( ), approximate threshold 3 eV

\

10-3

~r

"6 i0-~

~0

//

?./Y

i

',

, Cz/

_o

\ >

A/

>~ )0-s -m

.

/

E

""

__

Au[111)

•

p

g 10-6 30

3,5

~0

4S

S,013S

40

~S S0 photon energy h•leV

Fig 4 Photon energy dependence of the yields Yp and Ys for p- and s-polarized hght and their ratios Yp/Ys for A u ( l l l ) an In H2SO4, angles of incidence 0 ° ( - - ) , 22 5 ° ( ), 45 ° ( - - - ) , 67 5 ° ( ..... ), approximate threshold 3 eV

H Laucht et al / Vectorial volume effect m mterfactal photoemtsston

112

crease strongly with increasing angle-of-incidence for p-polanzed light A decrease of the yield with increasing angle is observed for s-polarized light. Specific vectorial features are much more apparent in the ratms Yp/Ys, which differ markedly for the two metals Above 4 1 - 4 . 2 eV, in the case of copper (fig 3), the yields suddenly increase more rapidly * and the vector ratio shows &stmct structure in this energy range.

4 V o l u m e effect analysis

Bulk photoemisslon from Au and Cu, In the low energy range of interest here, originates predominantly from conduction band to conduction band transitions in the vicinity of the L point [16] For copper the relevant energy contours of initial and final states as well a.s the resulting contours of constant lnterband energy for k-conserving transitions in the (110) plane that contains the F - L direction are shown in fig 5 The energy elgenvalues were calculated from a combined interpolation scheme [28,29]. Because of the near-perfect rotational symmetry of the energy values, fig 4 is appropriate for any plane containing the F - L direction. Due to the fcc crystal structure, transitions of the land shown m fig. 5 can occur in eight equivalent regmns m k-space The extent to which electrons are excited in these different regions depends on the orientation of the electnc field e of the light wave Applying the usual escape condition, we expect "elastic" (or unscattered) photoelectrons [16] only from that single region where the final states have momentum nearly perpendicular to the surface and are outwardly directed Thus even total yield measurements m a photoemlssmn experiment can probe a hnuted region in k-space, a possibility usually thought of in connection with angular resolved photoemlssmn Consequently, the number of emitted electrons is expected to depend significantly on the onentatmn of e If we consider plane-wave expansions of the initial and final states for a particular direct optical transltmn within this contributing region,

~, = ~

O,(k + Gv) exp(l(k + Gv)r ) ,

(1)

vf(k + Gv) exp(l(k + Gv)r),

(2)

l)

~f = ~ v

with the number of G's chosen to reflect adequately the momentum space environment, the vectorial features pointed out by Schaich arise naturally when we analyze the transition probabihty ~t/if in the volume-effect limit [ 17], the result IS

W,f ,x le" ~

(k + G~,) of (k + Gv)b](k + Gv)[ 2 .

(3)

v

An Important consequence of eq (3) can be deduced from symmetry considera* This increase is somewhat obscured by the loganthmm scale

H Laucht et al / Vectorial volume effect m mterfactal photoemzsston

113

Cu K~

(1111 L

~U

TV

K~

(111) L

'I'V

~U

K~

(111} L

~U

~V

Flg 5 Contours o f constant initial (El), constant final (Ef) and constant mterband energy (Jlto) for mterconductlon-band transitions m Cu in the vicinity of the L-point for photon energms between 4 2 eV and 5 0 eV

tions The states along F - L belong to the C3v symmetry class Thus for k parallel to G i i i = (2/a) (111), 1 e perpendicular to the surface, the sum in eq (3) points in the same direction [4,16,17], no matter how many plane waves are included Because of the scalar product In eq (3), excitation of such transitions will occur only with a component el (30) o f e perpendicular to the surface. Such a component is present only with p-polarized light. The resulting vector ratio for any such transition along F - L should thus be very large With increasing component kll of k perpendicular to G 1 il this strict selection rule is rapidly relaxed due to the increasing devmtlon from symmetry of the coefficients v, and vf m eq (3) (see also fig. 1 in ref [ 17]). A calculation by Animalu [31 ] indicates that the increase of the transition probabdlty goes roughly as k~ On the basis of these arguments, we now proceed in a slmphfied fashion to calculate the photon-energy dependence of the vector ratios We write the transition probabIhty m the form [17] W,f = .lie 2 + Jlle~ ,

(4)

and assume that

]1 = c , , J,I = C 2 k ]

(5)

(6)

,

where C] and C2 are taken to be constants Here, J± and Jll are quantmes that are averaged over a constant f'mal-energy circle on the optical transition surface [17]. For reasons discussed below we evaluate as a "normalized" measure of the vector ratio the expression fS(Ef

- E 1 - Ii¢o) 6 ( k f - k , ) O ( 2 m ( E f - Vo)/h 2 - k~l) d3k

re±Cls2

Yell Ca

(7)

fkl21 ~ ( E f - E , - ~¢o) (~(kf - k,) O ( 2 m ( E f - V o ) / ~ 2 - k 2 ) d 3 k I2

114

H Laucht et al / Fectonal volume effect in mterfacud photoemzsslon

with the help of the combined interpolation scheme Because of the rotational symmetry the analysis can be performed in any plane containing G l l i, for the twodimensional k-space integration we use a modified Gilat-Raubenheimer [32] method adapted to two dimensions The two 5-functions in eq (7) insure energy and m o m e n t u m conservation and the step function 0 ( 2 m ( E f - V 0 ) / ] I 2 - k~l) represents the escape condition of parallel k-vector conservation. Calculating yield ratios of the form given in eq. (7) has the advantage that a number of effects such as light penetration, d-band optical absorption, escape depth, and refraction of hght, which comphcate absolute yield interpretations, wdl approximately cancel m the ratios. Results of the calculation are shown in figs 6 and 7 for Cu and Au The angleof-incidence dependence is here eliminated as it has been assumed in eq (7) that e2-11 -e±.2 If one takes into account that, with p-polarized light, both field components are present and that the p-yield can then be written as Yp = ote2p±+ fie211 ,

(8)

where %z and %11 are the electric field components perpendicular and parallel to the surface, respectwely, the experimental vector ratios can also be represented in a

I

4

Cu(111)

O

N O

E

~f

..4

A

J 0

35

~0

~5

50

photon energy h~/eV

Fig 6 C o m p a n s o n of experimental ( - - - ) and calculated ( ) n o r m a h z e d (e 2 = e~/) vector ratios Ye±/Ye// for C u ( l 11), e x p e n m e n t a l values are gwen for 22 5 ° (o), 45 ° (+) and 67 5 ° (A) angles of incidence

H Laucht et al I Vectorial volume effect in tnterfactal photoemtsston ~U-)I

J

A

115

1 Au (111)

0

0

\

"6

\

'\ o E

30

35

~0

~5

photon energy

50 1~w/eV

Fig 7 Comparison of experimental ( . . . . ) and calculated ( ) normahzed (e2 = e 2). vector ratios Yet~Yell for Au(111), experimental values are gwen for 22 5° (o), 45 ° (+) and 67 5° (1) angles of mcldence

normalized, i.e. angle-of-incidence-independent form [6]

Ye, Yell

(Yp 3

-

Ysepllles)lep± Ysle2s

(9)

The term (Yp - Ys e p2 J e s2) l n eq (9) extracts from the experimental data the yield due to the electric field perpendicular to the surface The electric fields epll, ep± and es are calculated classically and the results are essentially independent of wtuch set of optical constants available in the hterature [33,34] IS used. The experimental, normalized vector ratios thus obtained are also shown in figs 6 and 7 Experimental values for Ye±lYeii are nearly the same for the three angles o f incidence. For comparison, the calculated ratios have been scaled to the magnitude o f the experimental results. Consistent with the sImdarity o f Au and Cu in this energy region, the same scaling factor (C1/C2 in eq. (7)) can be used for both Cu and Au in figs. 6 and 7. In view o f the uncertainty in the gap at the L point (L~-+ L~,) for gold [35,36], we have used an intermediate value o f 3.9 eV in the calculation. Compared with fig 7, the agreement between the experimental and calculated results can further be improved by "fine tuning" of the parameters in the Interpolation scheme. This, however, does not appear to us to be justified at the moment.

116

H Laucht et al /Vectortal volume effect m mter]actal photoemlsston

5 Discussion The results of the calculation, based on eq (7), may be understood qualitatively by considering the location In k-space o f the transitions involved at various photon energies The onset of lnterband transitions occurs at the zone-boundary (fig 5) and is associated with a small kll corresponding to the neck radius of the Fermi surface Upon slightly Increasing the photon energy the transmons start to spread out in k-space, particularly towards A The average kll thus decreases and, according to eqs (4) and (6), the excitation probability for s-polarized light gets smaller. This in turn increases the vector ratio as excitation with p-polarized hght does not depend on k / / i n our approximation (eq (5)) With further increase o f / i w , that part o f the optical energy surface associated with large kll starts to dominate and the ratios again decrease In gold only the tugher energy decrease is experimentally observed as Ys cannot be measured with sufficient accuracy below 3 4 eV Nevertheless, indications of the maximum expected theoretically are seen In copper the full variation of the vector ratios is observed and excellent agreement between the experimental and calculated results is obtained One prediction of the model, which is experimentally verified, is the lack o f angles-of-incidence dependence of the vector ratios in normalized form (eq (1), figs 6 and 7) Such a result indicates that it is not necessary, usually to measure a whole range of angles of incidence The present success of the volume-effect model seems somewhat surprising m view of the results of Nllsson and Eastman [4] or Becker et al [16] These authors suggested that a sizeable fraction of the photoelectrons emerge as a result of scatterlng processes at the surface Any structure due to elastic photoemlsslon is then expected to sit on top of whatever background o f scattered electrons occurs Diffuse scattering of electrons at a surface can be viewed as a relaxation of the escape condition of parallel k-vector conservation With s-polarized light many electrons, originating from equivalent regions in k-space (section 4) where e is nearly parallel to a G l l r v e c t o r and excitation is strong, might escape at a diffusely scatterlng surface On the other hand such scattered electrons would tend to wash out structure and decrease the magnitude of the vector ratios Such effects have been observed in our experiments, but only with samples of poor surface quality One aspect of the copper results brings the entire model into question The experimental vector ratio extends beyond the threshold energy for lnterband transitions (4 15 eV) and remains high down to the lowest energies that were investigated This indicates very clearly that the direct transition volume theory cannot be the complete story Experimental indications o f this problem have been discussed previously [4,16,37]. Basically, the problem is this for energies below the Interband threshold on a (111) face, there occur no bulk final states which can couple to the states in the vacuum if the condition of conservation of crystal momentum parallel to the surface is retained Thus the question which needs answering is, what are the final states from which the electrons emerge~ The breakdown of kll con-

H Laucht et al /Vectortal volume effect m mterfactal photoemtsszon

117

servatlon reqmred to acineve contmmty of the yields towards the threshold is so complete that it appears to us wholly inconsistent with the quahty of the surfaces as judged by their appearance m the large vector yield ratios A similar problem appears in sliver In this case we were able to describe the highly structured energy dependence of the yield ratio with a surface photoemlsslon theory [10,13]. Smce the mgredlences for such a theory for copper, m particular, a longitudinal chelectrlc function, are not avadable, a corresponding attempt cannot be made Even for such a surface effect, the question of winch final states are involved remains. Thus we do not feel that the yield can be attributed to "scattered electrons" In this low energy region, populatmn of states near the photoem~ssmn threshold can occur through a surface effect for p-polanzatmn or, for p- and s-polarization, through excitations from the Drude absorptmn [38], both of which mvolve nondirect trans~tlons We think the most reasonable explanation of low energy emission is direct couphng vaa these non-direct processes into vacuum states winch have evanescent character inside the solid At present, the extent to which these addmonal excitations contribute to the yield m the direct-transition regime above 4 1 eV, and thus render our volume effect analysis mvahd, is not known However, judging by the discontinuity of the experimental results m the vicinity of 4.1 eV and by the close relatlonsinp of the calculatmns to the direct-transition model, an alternatwe mterpretatmn of the vector ratios based upon non-d~rect processes such as surface effect, seems quite unlikely within the photon energy range above 4.1 eV We conclude by remarking that recent energy- and angle-resolved photoemisslon studies with variable polarization by Hansson et al. [39] seem to confirm the evidence presented here In addition, calculatmns are needed where, m contrast to the work of Janak et al [40], the polarization dependence of the transmon probability ~s taken into account. The nearly free-electron apprommatlon for the matrix elements [16,17] cannot be used for quantitative purposes when absolute vector ratios are to be computed [4,40] As can be deduced from the results ofJanak et al [40], our constant matrix-element approximation for the normal field component (eq (5)) may also be too rough m the vicinity of the L-pomt

Acknowledgements The authors would hke to thank Prof. Dr. H. Genscher for continuing support and many stimulating discussions. The expert technical assistance of E. Plltz was of great benefit to the experimental investigations One of the authors (H L.) gratefully acknowledges the financial support of tins work by the Deutsche Forschungsgememschaft Another of the authors (K L K ) would hke to thank Prof Dr. H Genscher for the opportumty to work m Ins Instltut and the members of the Instltut for their hospltahty

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H Laueht et a l / Vectortal volume effect m mterfactal photoemtsston

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] 12] 13] 14] 15] 16] 17] 18] 19] [201 [21] [22] [23] [24] [25] [26] [27] [28] [29]

[301

[31] [32] [331 [34] [351 [36] [37] [38] [39] [40]

RM Broudy, Phys Rev B3(1971) 3641 S A Flodstrom and J G Endnz, Phys Rev Letters 31 (1973) 893 P O Gartland, S Berge and B J Slagfold, Phys Rev Letters 30 (1973) 916 P O Nllsson and D E Eastman, Phys Scnpta 8 (1973) 113 B Feuerbacher and B Fltton, Sohd State Commun 15 (1974) 295 P Munz, Solid State Commun 17 (1975) 627 S A Flodstrom, G V Hansson, S B Hagstrom and J G EndrlZ, Surtace Scl 53 (1975) 156 J Hofmelster and R Schwarzer, Phys Letters 53A (1975) 283 J K Sass, Surface Scl 51 (1975) 199 J K Sass, H Laucht and K L Khewer, Phys Rev Letters 35 (1975) 1461 See for example B Feuerbacher and R F Willis, J Phys C9 (1976) 169, and references therein J G Endrlz, Phys Rev B7 (1973) 3464 K L Khewer, Phys Rev Letters 33 (1974)900 P J Felblman, Phys Rev Letters 34 (1975) 1092 G W Gobeh, F G Allen and E O Kane, Phys Rev Letters 12 (1964) 94 H Becker, E Dletz, U Gerhardt and H Angermueller, Phys Rev B12 (1975) 2084 W L Schalch, Phys Status Sohdl 66 (1974) 527 L Ilver and P O Nllsson, Solid State Commun 18 (1976) 677 A M Brodsku and Yu V Pleskov, In Progress in Surface Science, Ed S G Davlson (Pergamon, 1972), and references therein J K Sass, R K Sen, E Meyer and H Gerlscher, Surface Scl 44 (1974) 515 V A Bendersku, S D Babenko, Ya M Zolotovlttsku, A G Krlvenko and T S Rudenko, J Electroanal Chem 56 (1974) 325 J Andersson and G J Lapeyre, Phys Rev Letters 36 (1976) 376 S Trassattl, J Electroanal Chem 33 (1971) 351 H Laucht, Thesis, Technical Umv Berlin (1976) P O Nllsson, l Lmdau and S B M ttagstrom, Phys Rev B1 (1970) 498 S Stuckl and J K Sass, to be published K H Becker and G H Comsa, Metalloberflaeche 29 (1975) 241 N V Smith and L F Matthels, Phys Rev B9 (1974) 1341 For copper we used the parameters of ref [16], for gold the data of N V Smith, Phys Rev B9 (1974) 1365, and a modification described m section 4 We are very grateful to H Becker for sending us his computer program for the combined interpolation scheme The indices of the e's, as of the k's, are chosen (In contrast to the work of Schalch (17)) with respect to the surface This assures the notation of k// conservation during electron escape through the surface A D E Anlmalu, Phys Rev 163 (1967) 557 G Gllat and L J Raubenhelmer, Phys Rev 144 (1966) 390 G B Iram, T Huen and F Wooten, J Opt Soc Am 61 (1971) 128 P B Johnson and R W Christy, Phys Rev B6 (1972) 4370 N E Chrlstensen and B O Seraphm, Phys Rev B4 (1971) 3231 N V Smlth, Phys Rev B5 (1972)1192 P J belblman and D E Eastman, Phys Rev B10 (1974) 4932 K L Khewer and K N Bennemann, to be pubhshed G V Hansson, S A Flodstrom and S B M Hagstrom, in Proc Photoemlsslon from SurIaces Conf, Noordwljk, 1976 (to be published) J F Janak, A R Wllhams and V L Moruzzt, Phys Rev B l l (1975) 1522