---

~'--"::J

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11 6J. Phys. C: Solid State Phys. 19 (1986) 5437-5451. Printed in Great Britain

~ Plasmon effects on image states at metal surfaces

P M Echeniquet:;: and J B Pendry§t The Cavendish Laboratory. University of Cambridge. Cambridge CB3 OHE. UK§ Imperial ColJege of Science and Technology. London SW7 2BZ. UK

Received 4 December 1985

Abstract. We discuss the formation of Rydberg states at surfaces. presenting models for the

many-body effective potential outside a surface. In the light of these effects the bindingenergy and effective mass of electTons in these surface states is studied. as welJ as the many.

body contributions to lifetimes. We speculate that the disagreement with experiment ofsome of our conclusions may be due 10 resolution problems.

l. Introduction

It has long been recognised that electrons can be trapped at surfaces. As early as the1930's Tamm (1932) and Shockley (1939) showed that electrons ~ be localised nearthe 'end' of a semi-infinite Kronig-Penney potential. The electrons in such a state donot have enough energy to escape into the vacuum and cannot penetrate into the bulkwhen the solid shows a negative electron affinity. The electron can be trapped by its ownimage potential (Le. the potential created by the polarisation charge it induces at thesurface). In fact the Coulombic tail of the image potential of l/z (z being the distancefrom the surface) allows an infinite Rydberg-type series to existo

These states are of importance in a very wide range of phenomena in physics.Hydrogenic states of this kind were studied on theoretical grounds by Sommer (1964),Cole and Cohen (1969), Cole (1970), and Shikin (1970) who noted that liquid heliumshowed total reflectivity to electrons near the vacuum zero of energy and thereforeshould trap electrons in Rydberg states. Brown and Grimes (1972) and Grimes andBrown (1974) have studied the properties of electrons bound in image-potential-inducedsurface states on the surface of liquid helium. Garcia and Solana (1973) studied the roleof the shape of the surface potential in a one-dimensional Kronig-Penney model.Another example of this sort of state is provided by surface polarons. electrons boundto the surface regio n of a dielectric by coupling to the phonon fields (Sak 1972. Evansand Milis 1973, Clark 1975, Milis 1977). Positrons can also get localised at surfaces(Hodges and Stott 1973a, b, Mills 1978, 1979, Pendry 1980, Nieminen alld Manninen1974. Nieminen and Hodges 1977,1978. Barberan and Echenique 1979). In metals suchstates can arise if a gap in the direction normal to the surface contains the vacuum leve\.

A clear picture of surface states may be obtained in terms of scattering theory, assuggested by Pendry and Gurman (1975). A surface state is created by multiple reflec-

:j: Also at: Euskal Herriko Unibertsitatea. Kimika Fakultatea, Donostia, Euskadi. Spain

0022-3719/86/275437 + 15 $02.50 @ 1986 The Institute of Physics 5437

5438 P M Echenique and J B Pendry

tions between the atomic potentials of the crystal and the surface barrier, i.e. reflectionat the crystal gap and the surface potential. If rcei<f>cand rbei<f>bare the reflectioncoefficients of the crystal and the barrier respectively, the conditions for a surface stateare

--..---.....

._.\':~Y'

rcrb = 1 <1>b + <1> e = 2lfn.

In figure 1 we draw the band structure for Cu(OOI) and Ag(OOI), both of which show agap near the vacuum zero of energy, implying a reflection coefficient of unit amp!itude.

The observation of barrier-induced surface states is probably,most readily done bymeans of low-energy electron diffraction (LEED)and inverse photo-emission experi-ments. In LEEDat energies above the vacuum zero the surface states always have somecoupling to beams that can escape from the surface and cause characteristic peaks andminimain the LEEDreflectivity (Anderson 1970, McRae and Caldwe111976a, b, McRae1971. 1979, Rundgren and Malmstron 1977). Alternative descriptions, however, havebeen forwarded in terms of interference effects (Dietz el al 1980, Bosse el al 1982).

'--'

~_.-I

I

>-IiI

!

0.7

,0.8i lb)

t;.f~::,!'~~;.7

t;r

'"e::J

~I

I

_.: ~~ I- .2 r

'3 ~

>.

~~

I~1.

:0I E I ~, >- I _OS ~ ! >-

~~=0.48~ °

~

I ~vo ~I

~"~_ .1

II ~. I10.4 - -2' , I

I r I :

4l' '

i -3r r 0.3

'f,..31' I J'J- EF=0.2405r x r x

Figure 1. The band strueture of(a) Cu(OOI) and (b) Ag(OOI). The sea le on the left-hand si derepresents the energy relative to the vaeuum zero.

~

.,."

¡'"t

An analysis of inelastic scattering effects on the Rydberg spectrum has been madeby Echenique (1976). A trapped electron beam can lose flux via scattering with the bulkcrystal, by diffraction processes scattering it into another beam, and by interaction withsurface excitations. Echenique and Pendry (1978) ca1culated the binding energy andlifetime broadening of these states and proved that they could, in principIe, be resolvedfor all members of the Rydberg series.

Johnson and Smith (1983) pointed out that these states could be observed by angle-resolved inverse photo-emission. Recently Altmann el al (1984), Straub and Himpse1(1984), Rheil el al (1984), Hulbert el al (1985a), Woodruff el al (1985) and Goldman elal (1985) have reported inverse photo-emission experiments measuring the bindingenergy and width of these states.

~

(=-::,

~~

"

- - ---

/mage states at metal surfaces 5439

CaIculations of binding energies of image-potential-induced surface states have beenmade using simplified models for the image potential. Echenique and Pendry weremainly interested in the possibility of resolving the Rydberg series and not in the actualbinding. They evaluated the binding energy using a z-dependent local image potentialwhich was approximated by its asymptote (-4Z)-1 for distances greater than a certaindistance Zofrom the surface and by a constant potential (-4Z0)-1 for distances smalIerthan Zo. Theoretical studies of barrier-induced states have used the classical imagepotential for the surface barrier (Straub and Himpsel 1986, Garrett and Smith 1986,Smith 1985a, b, Hulbert et al 1985b, Goldmann et al 1985).

In this paper we shaIl study the effect on the binding energy of electrons at surfacesincluding plasmo n dispersion and single particle effects in the medium response, bothwithin the point charge approximation for the image potential and also using a formalismincluding the bound-state nature of the electron creating its image potential. We shalIconcern ourselves with the dependence of the binding energy on the electronic density ofthe medium and we shalI employ the Hedin and Lundqvist (1969) self-energy formalismoNieminen and Hodges (1978) have applied this method, together with an undispersedfunction to represent the medium response, to study positron surface states at metalsurfaces. Identical results can be obtained by a different approach to the self-energyproblem recently proposed by Manson and Ritchie (1981). In § 3 we shall be concernedwith the variation of energy with momentum paralIel to the surface while in § 4 simplearguments derived from the formalism of Echenique and Pendry (1978) have been usedto predict that the widths of the image states should be narrower than the ones reportedup to now by Straub and Himpsel (1984). (However, recent results by Giesen etal 1985,have confirmed this prediction.)

2. Semic1assical approximation

In this model we look for the solutions of the one-electron Schrodinger equation

(-O.5V2 + Veff(Z»<I>O(z) = Eo<l>o(z) (1)

where Vef¡(z) is ca1culated as the image potential induced by a point charge electro n andis given for a charge approaching the solid from vacuum, in the specular reftection modelby

-Vzf 2 f exp[i(Q' vp - w)z/vz - 2Qlzl]

Veff(Z)= 4Jl"2 d Q dw (Qvz)2 + (Q. vp _ w)2 R(Q, w)(2)

where

R(Q, w) = (1 - é(Q, w»/(l + é(Q, w». (3)

Here Vz and vp are the components of the electron velocity normal and parallel tothe surface respectively and the position at time t is (vp!, - Vzl).This expression isvalidwhen the charge is outside the solido

The quantity é(Q, w) can be related to the bulk dielectric function by (Ritchie andMarusak 1966)

- Qf

oc dkzE(Q, w) = Jl" -oc(k; + Q2)E(k, w)

(4)

5440 P M Echenique and J B Pendry

where k: = (k; + Q2). In this model an approximation for the bulk dielectric function

implies an approximation for É(Q, w) and therefore for the image potential. We couldemploy any of the well known dielectric functions for the bulk and thus evaluate theimage potential. Alternatively we could use our knowledge of the surface response topostUlate a model for R(Q, w). This was suggested by Echenique et al (1981) and issometimes referred to as the surface plasmo n pole approximation. In this model

R(Q, w) = w~/[w(w + i.) - w;Q] (5)

where Wsis the surface plasmo n frequency, Ws= Wp/21/2,wpis the bulkplasma frequency,Tis an infinitesimal damping constant and wsQis given by

wsQ = (w; + aQ + {3Q2 + Q4/4)1/2. (6)

Here a = (3/5)1/2VFWs;VFis the Fermi velocity.This response function reproduces the results of Ritchie and Marusak (1966) and

Inkson (1971, 1973) for the surface plasmon dispersion relation for small Q and takesinto account the single particle character of the response through the presence of the Q4termo The Q2-term is added in equation (6) to force the surface plasmon dispersionrelation to joint the single particle continuum at the same point as the bulk line does. asprescribed by Wikborg and Inglesfield (1977). A good approximation to {3satisfying thecriterion

Wp(Qc) = Qc(Qc + 2kF)/2 (7)

where wp(k). the volume plasmon frequency at wave-vector k, is obtained as a functionof the electron gas density parameter by a fit of the form

{3=a+b/r~.

Here a = 0.0026, b = 2.6798 and e = 1.85.This model for the surface response has been used to study the interaction between

two semi-infinite planes and the results obtained agree to surprisingly good accuracywith the ones derived by other workers using the full random phase approximation(Ashley et al 1981).

For an electron having momentum kOpparallel to the surface the model of equation(5) gives for the image potential

_ -w; Ix exp( -2Qz)8(wsQ - QkOp)Veff(z) - 2 ( 2 _ Q2k2 )1/2 dQ.o WsQ WsQ Op

(8)

(9)

8 is the step function.For large values of Zonly small Q contributes to the integral and one recovers as an

asymptotic limit the classical image potential

Veff(Z) = -1/(4z). (10)

If the negative affinity character of the material is represented by an infinite barrierand the surface potential by the classical image potential, the bound-state wavefunctionwill be the hydrogenic one and the binding energy is the appropriately scaled hydrogenIs energy.

Eb = _!(!)2 Hartrees = -0.85 eVo (11)

We have evaluated, for kOp= O, the semiclassical image potential using the surfaceresponse model of equation (5) and in figure 2 compare the results obtained with the

---

-'~~

~~

....,"

~~.

-,

- ----- - - - - -- --

lmage states at metal surfaces 5441

ones evaluated from the asymptotic expression. For small values of z the image potentialdiffers significantly from its cIassicallimit (-1/4z). In the region where the probabilitydensity of the first state is appreciable. this is a noticeable effect. In figure 2 we draw for

/---

~...

/ ..../ '

/ "/ '

I "I "

/ 'I ......

II

//

--.;~

(-<.~;

z (au)

Figure 2. The point particle image potential Vez) in eV and first-state probability density!<1>,,(:>I"for an undispersed (-) and a dispersed (---) (equation 5) model for thesurface response as a function of the distance to the surface.

.c.'~""'"--=y

comparison the ground-state wavefunction. both in the dispersed and in the undispersedmodel for the surface response. For the second state and higher ones the cIassicalapproximation should be a good one. Our approximation for the surface response leadsto a finite potential at the origin; its value can be approximated by

Veff(O) = -O.38/r~.622. (12)

This gives -6.57 eV for the case of an electron interacting with a solid at the electrondensity of aluminium. We can estimate the importance of the deviation of the cIassicalpotential from its asymptote, still within the infinite barrier model to represent thenegative affinity metal, by determining variationally the energy using a trial wavefunctionof the form

<l>o(Z)= (4f3)1/2Z e-rz. (13)

-1

-2-> ,-3 . / r,=3.0-...s= -4

5442 P M Echenique and J B Pendry

w,0.8

.~...c.~-;"3'

0.7

~""

"-"-

............

....

W,Q (~=Ol

........--- ---0.5

0.1,

2 3 1, 5 6r.

Figure 3. The binding energy of the first image state in eV, within the point charge semi-classical approximation for an undispersed (label w,) and a dispersed model against thesurface response (label w,Q).

C:¡:~~~~~

Then the r-dependent binding energy is given by

r2 w~Ix

[r

]3 d Q

Eb(f)="2-TO r+Q w~Q'

Figure 3 shows the results of a caIculation of the binding energy as a function of rs'If the Q-dependence in wsQis neglected one tinds an rs-independent binding energycorresponding to the classicallimit of equation (9); the binding energy being, of course,the appropriately scaled hydrogenic limit. The Q-dependence of wsQ, representingphysicalIy the surface plasmon dispersion and single particle effects reduces the bindingenergy, introducing a slight dependence on rs'

Note that equation (6) appears to yield values of wsQthat are somewhat too largecompared with experiment (Feibelman el a/1972) thus leading to an image plane positioninside the crystal and therefore underestimating the image potential and binding energy.It would be a simple matter to tit observed curves using a form similar to equation (6).However the use of the self-energy formalism that we present in the next section preventsthe unphysical divergence of the point-particle image potential alIowing us to redetinean image potential, tinite everywhere even if the undispersed plasmon model is used forthe surface response.

(14)

.../-'

,~~).~

3. Self-energy formalism

The semiclassical approximation uses the potential induced by a point charge as the onefelt by the electron in a bound state. This can be taken into account by using a self-energyformalism, detining the effective potential felt by the electron as (Hedin and Lundqvist

~

",

lmage states at metal surfaces 5443

1969. Nieminen and Hodges 1978).

Veff(r)<I>o(r)= f ~(r, r', Eo)<I>o(r') d 3r'(15)

where the self-energy of an electron in an eigen-state of eigenvalue Eo and wavefunction<l>o(r)is given by

~(r, r', Eo) = 2~ J W(r, r', w)G(r, r', Eo + w) d w (16)

where W(r, r', w) is the screened interaction given, for z, z' outside the solid by

-1J

R(Q,w). ,W(r, r', w) = 2Jr Q exp[-lQ. (rp - rp) - Q(z + z'») d2Q.

R(Q, w) is defined in equation (5) and the one-particleGreen function is

, 1J

, . , ~ <1>7(z')<I>I(z)G(r,r.Q)=

(2 )2 d-kpexp[lk.(rp-rp»),",- Q E .Jr I - 1+ IT

(17)

(18)

where by <l>lwedenote the set of eigenstates of the effective potential associated withthe normal motion and

¡;;":.-; El = k~/2 + El: (19)

where Elz is the energy associated with the normal motion and r = (rp, z). The potentialenergy of an electron in the ground state <l>o(r)is then

D= 2~J dw J d2r J d3r'<I>ó(r)W(r.r', w)G(r.r', Eo + w)<I>o(r').(20)

Carrying out the integration in rpand kpwe obtain

D= :2,L Jdw

J d2Q R(Q.w). I(Ole-QzlfW(2Jr)- I Q w - DE+ IT

(21)

where

DE = DEz + Q2/2 - kOp .Q (22)

and

~

DEz = Elz - Eoz

is the energy difference between the fth state and the ground state associated with motionnormal to the surface. kOpis the electron's parallel momentum in the ground state <l>o(r).Note that equation (21) can be used (Manson and Ritchie 1981) to define a local Veff(Z)

D = f <l>ó(Z)Veff(Z)<I>O(z)dz (23)

where

Veff(Z)=~L J dw J dQ R(Q,w) <1>7(z)e-Qz(2Jr) I Q w - DE + iT <l>ó(z)

x J <1>7 (z') e -Qz' <l>o(Z')dz'(24)

"

-- - --

5444 P M Echenique and J B Pendry

which is of course, in general. different from the point partic1e resulto In fact if we alsotake a basis set oí plane waves to describe the electron motion in the normal direction,equation (24) reproduces the result of Manson and Ritchie for Vcff(Z)(with wsQ= ws)

Ws Ix

Ix Q cos(kz) e-Qz8(d)

VefC(Z)= - -;- o dk o dQ (Q2 + k2)d1:2 (25)

- 7-

where

d = [ws + (Q2 + k2)/2] - Q2k5p.

Since we are concerned with the binding energy of the image state. and not with thedetails of the potential we shall for the moment focus our attention on the energy shift,D. We will return later to the question of the effective potential. In our model ofequation (5) and for low enough kp (as is always the case in the inverse photo-emissionexperiments) D is real and is given by

D = - w~ ~ J1(0Ie-Qzlt>128(A)

2 f W A 1/2 d QsQ

(26)

where

A = (wsQ + Q2/2 + bEJ2 - Q2k~ (27)~i!~~'

and 8 is the step function.A lower limit to D and thus to the binding energy is obtained by taking the f = °term

of equation (26) and neglecting the sum over the complete set of states. One obtains

w~J

I(Ole-QZ!0)128(B)DL =- _2 B1/2 dQ (28)wsQ

where

B = (wsQ + Q2/2)2 - Q2k~.

An upper limit can be obtained by neglecting bEz in the denominator of equation(26) and summing approximately over intermediate states by invoking c1osure. One willexpect this upper limit to be quite a reasonable approximation to the actual valuesince for high f-values the hydrogenic wavefunction overlap with the ground-statewavefunction will be small, and in any case for the discrete states bEz ~ WsQ.We obtain

- _ w~J

(0Ie-2QzIO)8(B)Du - 2 B1/2 dQ. (29)wsQ

As in equations (23) and (24) we could define from equations (27) and (28)

L _ _ w~J

e-QZ(0Ie-QzIO)8(B)VeCC(z)- 2 B1/2 dQwsQ

...

~(30)

andW2

Je-2Qz

V~f(Z) = - i wsQBl/2 8(B) dQ. (31)

This last expression only differs in the two-dimensional recoil term, Q2/2, from theone that could be obtained by taking the average value of Veff(z)(equation (9» over theground state wavefunction, <l>o(z).

--

l~<';;:;

{:~

...

lmage states at metal surfaces 5445

~~ 03cS':::. 0.2es~ 0.1<:>

i-j

i-1

+I

0.6

:¡¡;05

......

... ,...

...

" ,...-----

'i

0.4

2 3 5 64r,

Figure 4. The upper (u) and lower (L) boundaries to the binding energy calculated vari-ationally using the upper DI; (broken curve) and lower DL (full curve) limits to the meanpotential energycalculated with the dispersed model against thesurface response. The uppercurve indicates the per cent difference in binding energy depending on whether the upperor lower limit is used for the mean potential energy.

---------------------w,

0.8 l~III

_ 0.6>~

.:.:I

"1

05

0.4

2 3 5 64r,

Figure5.The self-energycalculationofthe bindingenergyobtainedbyusingtheundispersed(-, w,) and the dispersed(-, w,Q)model against the surfaceresponsedefined inthe texto The resul! obtained in the undispersed point charge model for the image potential(---, w,) is also shown for comparison.

O,.

5446 P M Echenique and J B Pend,y

Note that the two expressions. (30) and (31). obtained from the first term in thesummation oí equation (26). have different asymptotes for large z: -1/(2z) and-l/( 4z). the second having the correct asymptotic behaviour.

We have evaluated the binding energy variationally as a function of the electron gasdensity parameter. 's, for both the dispersed and undispersed model for the surfaceresponse using equations (28) and (29). The results of the calculations for kop = O areshown in figures 4 and 5. Figure 4 shows the upper and lower limits to the bindingenergies as a function of,s for the response model of equation (5).

--..-.-. .,..;

4. Effective mass

"

Recent jnverse photo-emission experiments (Reihl el al 1984) seem to indicate that theeffective mass of Rydberg surface states in motion parallel to the surface may be as highas m* = 1.3 or even m* = 1.6 (Borstel el al 1985). Garcia el al (1985) have suggestedthat the effective masses and the observed image-state binding energy can be explainedby a strong surface corrugation effect. Recent theoretical work using the multiplescattering theory of photo-emission (Pendry el a/1986) shows that the structure of theRydberg surface states massively weights m* to the free-space value. They conc1ude thatno physically reasonable corrugation of the Ag(001) surface can produce the requiredenhancement. We shall focus our attention on many-body and plasmon effects on theeffective mass.

From the detinition of a local effective potential we can go on to define a local, z-dependent. effective mass through the shift in potential energy bV(z)

\:ii~~

bV(z)kf¡p/2 = Vez. kop)- V(z, O)

so that the effective mass correction is(32)

bm(z) = m(z) - 1 = -bV. (33)

A rough estimation of the effective mass correction can then be obtained by taking thevalue of bm(z) at the value of maximum probability density, bm(zmax)' This procedurefor the semic1assical image potential with an undispersed surface plasmon field leads to

bm(z) = 1/(8w~z3) (34)

and for Zmax= 1/r this gives r3/ (8wD, yielding a 2.5% correction for r = 0.25 and 's =2.67, and a 3.5<7ccorrection for,s = 3. The unphysical divergence at the origin of theequation can be corrected by using as our effective potentials the ones defined via theself-energy formalism which. for the effective potentials defined, and using c10sureto sum approximately over intermediate states, leads to the following effective masscorrection:

~i1......

bmEP(z) = Ws f Q2 e-2Qz

2 l..., /"\2 ,.,,1 dQ (35)

or using Manson and Ritchie's definition

Ws Ix

Ix cos(kz) e-QzQ3bm:-.tR(Z) = -;- dk 1/'00.. hl . ,~, . . h 1_" dQ.'o o

(36)

----

Image states al metal surfaces 5447

Both ÓmEPand c5mMRare finite at the origin

ÓmEP(O)=;r/[8(2ws) 1/2] (37)

and

ÓmMR(O)= 1/[6(2ws)1/2j. (38)

They have the same asymptotic behaviour for large Z, given by equation (34). In figure6 we show for rs = 3 the ratio of the z-dependent effective mass given by equation (35)to the asymptotic limit of equation (34). In figure 7 we show the ratio between c5mEP(z)and ÓmMR(z).

0.5

5z (au)

10

e

8-e""o-1;.o

o

Figure 6. The ratio of ÓmEP(z) given by equation (35) 10 the point partic1e limit, equation

(34), as a function of the distance of the partic1e from the surface (w.Q = w" r. = 3).

1.0

l':;:~

o-e~

~Q5.o

o 5z (au)

10

Figure 7. The ratio of ómEP(z) and ómMR(z)as a function of the distance from the surface(w.Q = w" r. = 3).

.,

5448 P M Echenique and ] B Pendry

However the physical quantity is the average value of bm(z), i.e.

bm = (Olbm(z)!O). (39)

We have evaluated this quantity from equations (35) and (36) using a variationalwavefunction to minimise the binding energy for the undispersed response model,equations (29) and (31), and the results are given in table 1. For comparison we also give

.~~;

Table lo Effective mass correction as a function of the electro n gas density parameter usingformulae (35). (36), and (39). The first cotumn gives Óm with a f determined variationallyfor each r,o whilst the second gives óm for f = 0.25.

,

bmEPand c5mMRcalculated with a hydrogenic wavefunction (i.e. keeping r = 0.25 andnot determining it variationally). These results show that the increase in effective massdue to plasmon-related many-body effects is at the most 2%. Equation (35) gives anupper limit to the effective mass correction. A more exact calculation could be performedusingequation (26). Taking wsQ= Ws

t?¡i:;:;;~..

bm = w, 2: J 1(0Ie-QzltWQ2 dQ2 r (w, + Q2/2 + bEz)2

but the sum over intermediate states would be c1early an enormous effort not justifiedby the already very small value of the upper limit on bm.

So many-body effects associated with plasmons cannot produce any significant devi-ation of the free electro n mass. Inc1usion of small-energy electron-hole pairs in themedium response increase this contribution up to at most 5% (Echenique and Pendry1985), but cannot explain the experimentally observed effective masses. Pendry el al(1986) have suggested it might be worthwhile to re-examine the experimental situationagaino The difficulty of estimating effective masses using low-resolution detectors isgreat, and a high-resolution study which reduces the error bars may well resolve thisdisturbing discrepancy between theory and experimento

(40)

'"

"':

'- 5. Lifetimes~

So far we have concentrated on the binding energy and effective mass of the state, andwe have assumed that the modulus of the crystal reflectivity reis unity. In fact flux canescape from the stateo There are three effects ofwhich we must take account: real surfaceexcitations, the absorption in the crystal due to decay of the state into excitations of thecrystal, and, in a general case, the fact that flux can also be lost by diffraction into otherbeams which escape from the surface, thus reducing reto being less than unity even inthe absence of absorption due to inelastic effectso These processes were discussed by

r, ÓmEP ómEpo ómIR (f = 0.2) ÓmMR(f =0.25)

1 1.85 x 10-3 2.19xlO-3 1.80 X 10-3 3.34 X 10-32 7.15 X 10-3 9.81 X 10-3 4.42 X 10-3 8.00 X 10-33 1.53 X 10-2 2.23 X 10-2 7.26 X 10-3 1.30 X 10-24 2.52 X 10-2 3.90 X 10-2 1.00 X 10-2 1.80 X 10-25 3.90 X 10-2 5.90 X 10-2 1.30 X 10-2 2.30 X 10-26 5.10 X 10-2 8.26 X 10-2 1.60 X 10-2 2.80 X 10-2

('\

~~,..~...,,

"

- - - ---

lmage states at metal surfaces 5449

l

Echenique (1976) and by Echenique and Pendry in their 1978paper. They gave 0.32 eVfor the width of the first state which agrees well. as remarked by Straub and Himpsel.with their experimental finding of 0.3 eV for the width of the state at the bottom of thebando This agreement, however, is a fortuitous one since, at the energies of the inversephotOemission experiments no diffraction into beams escaping into the vacuum canoccur and no real surface excitations are possible. At the energies of the experiment ofStraub and Himpsel and for the state of the bottom of the bando the only contributionto the width comes from decay to the crystal states and is small beca use of the smalloverlap between the wavefunctions. Following Echenique and Pendry we can write forthe width of the state

1/2T = - Vo¡(d<l>c/dE)/[(d<l>c/dE) + (d<l>b/dE)] (41)

where <1>eand <l>bare the phases of the crystal and barrier reflectivities respectively. Evenassuming that the electron feels the total imaginary part of the seIf-energy in the bulkthe deviation of the phase of the barrier reflectivity is bigenough to lead to a small energybroadening. In other words, the Rydberg state wavefunctions are effectively decoupledfrom the crystal wavefunctions and thus the small broadening. Typical values at theseenergies are

V()j = 1eVd<l>c JT---d E gap width

(.t2)

and if we use the approximation to the barrier reflectivity given by McRae (1979) thereis an upper limit of 50 meV for the width of the state. or even as low as 30 meV if oneuses the value for d<l>b/dEcalculated by Echenique and Pendry in their 1978 paper. Amore detailed calculation by Echenique etal (1985) has shown that Auger processes. viaelectron-hole pair excitation. give a contribution to the linewidths of 20 meV. Recentexperimental data by Giesen et al (1985) show. in agreement with the above. that thewidth of the first two Rydberg states on the (111) silver surface is at most 80 meV. Thefact that the width of the first two states is the same indicates that the value may be stilldetermined by the experimental resolution.

6. Conclusions

A study of many-body effects on the binding energies. effective masses and linewidthsof image states has been made. Explicit results for these quantities are given in figure 5and table l.

A self-energy formalism together with a plasmon model for the surface response hasbeen used. Our results show that the binding energies are very cIose to the appropriatelyscaled hydrogenic ones. Many-body effects result in a slight density dependence. thebinding energy increases with increasing density. Some materials. such as Cu and Ni donot have a well defined plasmon and as Giesen et al (1985) have suggested moresophisticated caIculations including effects associated with d-band transitions arerequired for a reliable assessment of many-body effects. Bausells and Echenique (1986)have studied the influence of many-body effects still within the self-energy formalismbut using experimentally measured undispersed E(w) response functions.

Our caIculations for the effective mass, summarised in table 1, show that many-bodyeffects. associated with plasmons. produce an increase of the effective mass of the order

"

---

5450 P M Echenique and J B Pendry

of 2% and therefore cannot explain the experimentally observed effective masses thatshould be very cIose to unity. Recent experimental work (Hulbert el al 1985a. b) showsthat in copper the image potential state displays a free-electron-like (m* = 1)dispersionthus confirming our predictions.

An upper limit for the width of the first image state. as seen in the inverse photo-emission experiments has been estimated to be 30 meV. Note that because of this finitewidth a tunnelling current can be established throughout these states in spite of theirlocalised character (Binning el al 1985. Becker el al 1985. Louis el al 1986).

Finally it should be stated that our calculation neglects the p'enetration of thewavefunction in the crystal. Although the penetration of the wavefunction in the crystalis gene rally small (Weinert el al 1985. Hulbert el al 1986) a quantitative estimationdepends on the position of the state at the gap and on the particular surface face. Thiseffect resu1ts in an extra contribution to the effective mass and life time of the state(Bauseils el al 1986) but does not change our conclusions. Increasing experimentalreso1ution wou1d provide information about the details of elastic and inelastic processesoccurring at surfaces. In fact a systematic study. using various surfaces. of surface statesboth image and crystal induced. in the terminology of Echenique and Pendry. couldbring much experimental information about the real and imaginary part of the surfacepotential.

. .~:j..:~

Acknowledgments¿';~~;f¡w..,""

This research was sponsored in part by a grant from the USA-Spain Joint Committeefor Scientific and Technologica1 Corporation and by the Spanish CAICT. We acknowl-edge N V Smith for communication of his work before publication. We thank F Flores,R H Ritchie. F J Himpsel. D Straub. S Huffner and N Garcia for stimulating discussions.One of the authors (PME) gratefulIy acknowledges help and support from IberdueroSA and the March Foundation.

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lmage states at metal surfaces

1

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