.,

Surfa<:.: S<:ience 247 (1991) 125-132Nonh-Holland

Image potential states at surfaces *

125

P.M. EcheniqueDepartamento de Física de '''fateriales. Facultad de Química. Universidad del País Vasco. Apdo /072. 20080 San Sebastián. Spain

and

M.E. UrangaDepartamento de Matemática Aplicada. Escuela Técnica Superior de Ingenieros Industriales. Universidad del País Vasco. Bilbao. Spain

Received 21 may 1990; acc.:pted for publication 30 May 1990

~Tbe main physics of image potential induced surface states is presented in a simple manner. Recent experimental data on .Iifetime

is compared with theoretical predictions. The role of image states in STM. as a two-dimensional electron gas ando as a source of a 1//noise at disorded surfaces is discussed.

1. Introduction

-

That electrons can be localized at surfaces hasbeen recognized for a long time. The electrons insuch state cannot penetrate into the bulk when thematerial (solid or liquid) shows a negative electronaffinity and do not have enough energy to escapeinto the vaccum because'its own image potential.i.e.. the potential created by the polarization chargeit induces at the surface. In fact the Coulombictail of the image potential of l/z aIlows an in-finite Rydberg-type series to exist.

Hydrogenic states of this kind were studied inliquid helium [1]. in the case of a surface polaron[2]. and in metals were such states can arise if agap in the direction normal to the surface containsthe vacuum level.

Johnson and Smith [3] pointed out tha't thesestates could be observed by angle-resolved inversephotoemission. Inverse photoernission (or brems-strahlung spectroscopy) [4,5] is well suited forprobing bound stat~s directly by measuring the

* 1nvited lecture presented by P.M. Echenique.

energy and momentum of an incident electron andthe energy of the emitted photon. In the last yearstwo photon experiments have provided good reso-lution data bn binding energies and effectivemasses [6]. and even lifetime data [7.9] has beenextracted. Energy loss experiments have produced[10] some data on the binding energy of imagestates on simple surfaces.

2. Phase shift model

A simple way of understanding the physicsinvolved in image states is provided by the phaseshift model [11,\2]. This model has been widelyemployed to give a transparent picture of surfacestates wbich can be used either for precise numeri-cal calculations. or for simple analytic approxima-tions. In this picture the surface state is consideredas a wave trapped between the bulk crystal and

. the surface barrier. This assumes a region of con-stant potential between the crystal and the surfacebarrier but since tbis need only be infinitesimallywide. this places no restriction on the argumentswe are about to make.

0039-6028/91/$03.501) 1991 - Els.:vier Science Publishers B.V. (North-Holland)

:1¡

¡

126 P.M. Echenique. M.E. Uranga / Image pOlenlialslales al surfaces

Summing the repeated scattering, taking intoaccount all paths via the scattering formalism ofPendry and Gurman [13] the condition for asurface state becomes

det{l- ReRo} =0, (1)

where I is the identity matrix and Re and R o arethe crystal and barrier reflection matrices.

A simple. approximation containing the essen-tial physics consists in reducing the problem to aone-dimensional one; then the sum of the re-peated scattering gives for the total amplitude ofthe wavefunction at the surface

1

1 - rore ~i(4)B+4>c)'(2)

where rco; If>coare the modulus and the phase ofthe crystal and the surface barrier reflectivity re-spectively. The conditions for a surface state arethem

rero=l, If>o + If>c= 27Tn. (3)

Because of flux conservation re and ro must beequal to unity. This means that the energy (E)and momentum parallel to the surface (kp) of thewaves must be such that there are no propagatingstates within the crystal and no possibility of fluxescaping from the crystal into vacuum, i.e., a gapin the projected band structure and energy belowthe vacuum leve!.

This still leaves condition (3) for the phases tobe satisfied and this will only happen for certainvalues of energy and momentum. In particular thecrystal phase If>ewill vary at most by 7Tfor theenergy values we are concerned with here. Clearlyif this would also hold for the barrier phase reflec-tivity, condition (3) on the phases will have one orno solutions (this is actually the case for a stepmodel of the surface barrier). Ir If>o,however,varies rapidly with energy, as is the case for theimage potential, there is a good chance that manysolutions can be found.

The variation of the phase with energy is di-rectly related to the probability of the electron

spending time on the crystal or the barrier side[11]:

dlf>o1-o:: 1'1'12dz,dE ..barner s.de

d If>c ¡- o::1'1'

12 dz

dE 'cryslal side

Thcsta tic ti

magnituliquid hclose tothus theten ths (

away ( -an electsurfaceand Eb= -0.8.is reasol

The (

the cryspenmenso-calleó

g?'-"':'~p.tht gap,v~ fnga'fF.'íf tgap, asmetals,eV/n2;proximatsituation

metals q\-0.85 e'for such

more tigtfaces. Th

Manycated mIthe barri,results C0discussed

o ¡thtreII..~ th

ener:Asurfa~1

applied tigaps assonot paraIl

We sh(review 01

importan,model re;

systematil

(4)

'1' being the wavefunction describing the elec-tronic state. It is worthwhile to stress the pointthat variation of either If>cor If>ois an importantelement in producing a surface state: the greaterthe variation the greater ihe chance of sweepingthrough the 27Tcondition. Thus Echenique andPendry referrred to a surface state induced prim-arily by a rapid variation of If>c as a crystalinduced sta te, and that induced by a rapid varia-tion of If>oas a barrier induced state. We stressthe point that the two sorts of states are describedby condition (3).

3. Binding energy

A useful expression for If>Bis the WKB solu-tion [14] for the classical image potential - 21/z,

(

221

)lf>o(E)= V-2E ~1 7T,

where E is the energy measured from the vacuumlevel, as E approaches the vacuum leve!, the phasevaries very rapidly with energy.

In a real crystal, when image states exist, onehas usually a band gap several electron volts widewhile the Rydberg series lies within an eV belowthe vacuum level (21 = ~ in this case) thereforebecause of the rapid variation of the barrier phasereflectivity we can take 1>e as a constant andobtain for the binding energy

(5)

_ 2Eb = _ 2\

2(n+a)'; 11=1.2.....(6)

where

1 Es- 12\ = "4 Es + 1 '

a = i (1- ~e).

(7)

(8)

j¡.,

I

'-- - - -

- ---

P.M. Echenique, M.E. Vranga / Image potential states at surfaces

side The coupling constant Z\ and therefore thestatic dielectric constant Es fixes the order ofmagnitude of the binding energy. In the case ofliquid helium the static dielectric constant is veryclose to unity (1.0572 for 4He and 1.0428 for 3He),thus the binding energy is of the order of a fewtenths of meV (- 3.73/n2 meV); the state is faraway (- 100 Á) from the surface. For the case ofan electron attracted by its image potential to thesurface of an ionic crystal, such as LiF Es= 8.65and Eb::::-0.5/n2 eVo For metals Es= - 00; Eb= -0.85/n2 eVoIn the last two cases the electronis reasonably close to the surface - 2.3Á.

The details of the binding are fixed by (])c,i.e.,the crystal scattering. In metals and in most ex-perimental cases the image states appear in theso-called Shockley inverted gap case, where thegap is p-like at the bottom and s-like at the top ofthe gap, then a is greater than zero since (])cvariés from O at the bottom to 17at the top of thegap. Ir the vacuum level occurs at the top of thegap, as it occurs in the [111] faces of some fccmetal s, then (])c::::17, a = O and Eb = -0.85eV/n2; while if the vacuum level occurs ap-proximately at the rniddle of the gap, such asituation arises in the [100] faces of the same fccmetals quoted above, (])c= 117,a = 0.25 and Eb =-0.85 eV/(n + 0.25)2. Therefore one expects thatfor such fcc "metals the image states would bemore tigthly bound in the [111] than in the [100]faces. This has been found experimentally [6,7,15).

Many more calculations using more sophisti-cated models both for the crystal scattering andthe barrier one have been presented [16,30). Theirresults corrobora te the basics of the physical modeldiscussed above.

Srnith [31) has presented an elegant study of thetrends that should be expected for the bindingenergies and effective masses for a variety ofsurface states on different surfaces. He has alsoapplied the phase shift formalism to the case ofgaps associated with a reciprocal lattice vector gnot parallel to the surface normal.

We should hke to emphasise as stressed in thereview of Echenique and Pendry [32) that theimportance of the one-dimensional phase shiftmodel relies on the correct description of thesystematics and trends of the binding energies of

(4)

elec-Jointrtan teater:ping

and>nm-)'stalana-.tressribe~

e

solu-?:l/Z,

(5)

:uumJhase

.onewide>elow-efore:>hase

_ ar'-..-

~~(6)

(7)

(8)

m

both crystal induced and image potential inducedsurface states. For accurate quantitative calcula-tions it is well established [31] that surface-stateenergies (and the work-function) are correctly re-produced only in calculations which have beenperformed self-consistently. The scattering for-malism of Pendry and co-workers [13), or equiv-alent ones, can still be used for accurate calcula-tions via eq. (1). We insist, however, that theimportance of the one-dimensional phase-shiftmodel is to provide a simple explanation of sys-tematic trends. Even within this simple modelsome further assumptions have to be made tojustify the use of a two-band nearly-free-electronmodel, outside the gap region (Au(111» [33), orfor cases in which the states at the gap are pre-dorninantly d-like. Perhaps it is not surprising thatthe biggests discrepancies reported between thesimple phase shift model (with a nearly free elec-tron (])c) and the experimental values have been,in the n = 1 state of Au [27,31) where the state isnearly 2 eV above the top of the gap; and inNi(lll) [34-36] where the states at the bottom ofthe gap are predominantly d-like. As pointed outby Steinmann [15] even if a better calculation of(])c could resol ve the disagreement about the n = 1state, it is questionable whether it could accountfor the binding energy of the n = 2 state, which ishigher than the hydrogenic value. A careful analy-sis of the positioos of the image plane rnight havesome bearing on these cases since one would ex-pect that the d-baod character will tend to dis-place the ceotroid of the induced charge densitytowards the material compared with the results ofthe jellium model. A similar conclusion has beenreached by Pitarke et al. [37) io their analysis ofthe influence of the shape and image potential 00the oscillations of current versus voltage in STM.

4. Effective mass

lo the preceding section we have coocernedmainly with surface slates at the bottom of thesurface band, and \Ve have nol takeo into account

aoy possible variation (other Ihan the implicit tk;free-electron-like one) with the direction parallelto the surface. Within the one-cleclron picture of a

A.

I

128 P.M. Echenique. M.E. Uran/{a / Ima/{e potential.<tates at surfaces

solid the dispersion of Eh(kp) can only be due toa scattering with the ion cores of the surface, i.e.,

variation of the crystal reflectivity with k p or avariation of the dynamic image potential with theelectron parallel velocity.

4.1. Crysta/ effects

Echenique and Pendry suggested that for smallkp the effective mass should be very close tounity. The qualitative reason is that even for themost tightly bound Rydberg states the wavefunc-tion lies mainly in the vacuum, and the electronmoves in a regíon with no potential variationparallel to the surface. In variance with this ideaGarcia. et al. [38] suggested that image-potentialsurface states are strongly affected by surface cor-rugations. This implies a relation between Eh andeffective mass m* which does not agree with theaccurate two-photon photoemission measurementsof Giesen et al. [7,34] showing that the bindingenergy is independent of m* within error bars of- 0.03 eVoMoreover the [100] surfaces orientationexhibits a more pronounced surface corrugationbut as mentioned above the binding energy ofimage states in such surface is less than in the[111].

A multiple scattering calculation for the energydispersion of the first image state ~inding energyin the [100] silver surface performed by Pendry etal. [39] gíves an effective electron mass equal tounity. When effective masses are defined outsidethe region of small kp' significant deviations fromunity appear: Giesen et al. have explained [7]satisfactorily the experimentally observed effectivemasses using a one-dimensional phase-shift modelwith explicit inclusion of the shrinking width ofthe gap and the behaviour approaching bandedges. A comparison between the results of thesimple phase shift model and the experimentalones has recently presented by Steinmann [15].

4.2. Dynamica/ image potentia/ effects

From the definition of local effective potentialone could for operational purposes define a local

effective mass through the shift in potential energy8V( z).

8V(z)k;V(z. kp) = V(z, O) + '). (9)

these tIallowsmomen

leadingsuch el'for a rHof Ih.: (value ofor Ihe

possibltcentre e

crystalrapidlycertainbroader

governeabsorpt

A rough estimate of the effective mass correc-tion (m * = 1 + 8m) can then be obtained by usingthe undispersed surface plasmon to represent thesurface response and taking the value of 8V(z) atthe position of maximum probability density8V( zmax)' This gives, for a hydrogenic wavefunc-tion to represent the first image state at 2.5%correction for rs = 2.67 (Cu) and 3.5% correctionfor r, = 3 [40,41].

The use of a more adequate surface responsefunction including particle-hole excitation doesnot change the above conclusions [22,42]. Analo-gous result is obtained when explicit inclusion ofthe wavefunction penetration into the crystal istaken into account [22].

8E a:

~J

5. LifetimesTherefopace WIn. AII nare in ¡:memberfind agwavefurcrystal ,fected \

crystal rl/S. A t:

So far we have concentrated on the bindingenergy and effective mass of the sta te. and wehave assumed that the modulus of the crystalreflectivity re is unity. In fact flux can escapefrom the state. The electron in such a state is not

infinitely long-lived but can in principIe, make aIransition to other states, i.e.. to an emptycurrent-carrying state of the crystal. For finitemomentum parallel to the surface additional in-elastic channels are present !eading to an extrabroadening. In principie one could think that inmetals electron-electron interaction w0uld result

in decay processes that broaden surface states toresonances. This interaction may be so se\"ere dS todestroy all but the lowest member of Ihe Rydhergsenes.

As the energy normal to the surface. ap-proaches the vacuum leve!. Ihe stales get closer inenergy6.EJ. a: l/n} but at the same time the centreof gravity of the resonant sta te charge distrihutionmoves further away from the surface, having lessprobability of coupling to surface excilations.hence the energy broadening of the levels becomessmaller. Echenique and Pendry [11] proved th:1I

8E a:-

.-.....

r~.by b:fieseparalethe ~ta:,the enerIhe initi

the em~When tilO Ihe ':wiJth .!

......

I

--- --

:rgy

P.M. Echenique, M.E. (jranga (/muge PQlenlial Slales al surfa('es 129

(9)

these two effects compensa te in such a way thatallows resonant levels to be resolved. For themoment we shall not discuss in detail the processesleading to an energy broadening, and representsuch effects by an absorptive potential 1:;, whichfor a bulk free electron of the same energy can beof the order of 0.5-1 eV (depending on the actualvalue of the energy). ClearIy if these values heldfor the image states, no Rydberg series will bepossible. What saves the day is the location of thecentre of gravity of the surface state outside thecrystal because' these inelastic' processes decayrapidly in strength outside the surface. Beyond acertain range. zC' they can be neglected, and thebroadening, SE.J. of an image state wiII begoverned by its overIap with this region of finiteabsorption:

rec-

,;Ingthe

.) at1sityunc-2.5%;tion

onsedoes1alo-n ofal iSI

jZC 2 a4lB 1

SE .J.'a:. I'1'I dz a:. aE a:.3"-00 n(10)

~~Therefore the broadening scales as 1/n3 and keepspace with the leve! separation in the limit of largen. All resonances converging on a given thresholdare in principIe resolvable. If we can see the firstmembers of a series the rest should follow. Wefind again the same idea. the Rydberg state'swavefunctions are effectively decoupled from thecrystal wavefunctions and therefore would be af-fected very little by scattering processes at thecrystal region. This, of course, is exact only for highn's. A typical value of SE.J. can be obtained:

Idingj we

)'stal;capes notIke a

mptyfinite11 in-extraat inresult¡es to

as l,

~.

.g~

-1: a4lclaEI

8E.J. a:. a4lB/aE

-1:¡32(gap width)n3

50 meVn3

(11)

More detailed calculations have been presentedby Echenique. Flores and co-workers [43,44]. Theyseparate two contributions. First they calcmate forthe state at the bottom of the surface state bandthe energy broadening related to transitions fromthe initial image state to a final state defined bythe empty. current-carrying states of the metal.When the Rydberg states have momentum parallelto the surface, an extra contribution to the energywidth arises from a process in which the initial

. ap-ser ID;en tre)ution

g lessltions,::omesj that

state decays into a state localized in the surfacehaving the same z-dependent wavefunction butwith a different k-parallel component (intrabandtransition). The results of two calculations of theenergy broadening r(1); r(2) for the two processesdescribed above are given in ref. [43.44] and in thereview of Echenique and Pendry [32].

Typical values of model calculations of Pl)and r(2), with no wavefunction penetration are ofthe order of 5 meV for the case of a silver surface[43]. When penetration of the wavefunction istaken into account the halfwidth increases, whilethe one associated to the momentum parallel re-mains of the same order. For a 30% penetration ofthe wavefunction de Andres et al. [44] quote 33meV for !r(1) in Ag, which is about a factor 10greater than the value obtained with no wavefunc-tion penetration. We should like to emphasize thatin spit(: of their complexity the calculations of thelifetime of image states use very simplified modelsfor the surface response that might not be veryadequate tó describe appropriately the responsefunction of materials such as Cu or Ag. Thecalculations [43,44] use simplified models for thesurface response, based either on the specularmodel of Ritchie and Marusak [45] for the surfaceresponse, together with the small w expansion fothe bulk RPA dielectric response function of Pers-son and Andersson [47] for the case of an electrontotally outside the crystal [43]. At the energiesinvolved preliminary analysis [48] indicates thatthe use of the experimentally derived undispersedresponse functlon may lead to energy widths.greater by a factor of 2 ar more than the onesobtained using the respon se functions quotedabove. Recent experimental data by Schoenlein etal. [8] and Nielsen et al. [49] on Ag(lOO) .havemeasured a lifetime of about 25::!: 10 fs [8] or20::!:5 fs [49]. These correspond to a line width ofabout 35::!:la meV which is more than twice thecalculated' value with no wave-penetration [43].Nielsen et al. [49] and Steinmann [15] have sug-gested that the discrepancy is presumably due toan additional decay channel, e.g., surface imper-fections. This is .an interesting suggestion thatshould be explored. but also one should take intoaccount the abbve comments on the wavefunctionpenetration and on the innuence of the different

~,

I

I

130 P.M. Echenique. M.E. Uranga / /mage po/ential sta/es a/ surfaces

models of the response function. A calculationalong these lines is in progress [48].

Very recently Schoenlein et al. [50] have re-ported the first time-resolved studies of higherorder image potential states on Ag(lOO).

They measure the lifetime of the n = 2 imagestate on Ag(100) to be - 180 fs, substantiallylonger (- a factor of 7) than that of the n = 1state. This result is in agreement with the originalprediction of Echenique and Pendry [11] (see eqs.(14) and (15». The n3 scaling prediction is exactonly for high n 's. More recent calculations by deAndres et al. [51] using a many-body self-energyformalism predict n3 lifetime scaling for statesn ~ 5. For lower order states, the model predicts alifetime scaling less than n3, with

('Tn_2/'Tn= 1) =:::4.

The hydrogenic model of de Andres et al. asstressed by Schoenlein et al. [50] overestimates'Tn_1 since it neglects wavefunction penetrationinto the bulk. Thus one expects the ratio of life-times 'Tn_2/'T"=1to be between four and eight,which is in good agreement with the experimentalmeasurements of Schoenlein et al. [50]. In the caseof silver (and copper) (111) surfaces the n = 2image state lies in the continuum and thereforethe surface state presents an elastic linewidth,originating from the lack of total reflectivity, Le.,re < 1. This elastic half width for the n = 2 state ofCu and Ag has been estimated to be around 25meV [44].

6.STM

Louis et al. have presented an analysis of theconduction process through image surface statesin scanning tunneling microscopy [52,53]. For lowelectric fields the Auger inverse lifetime discussedabove, becomes larger than the tunneling lifetimeando therefore, the tunneling process controls thecurrent while for large fields a saturation occurs,since the Auger process is not fast enough as toallow new electrons to come from the tip into thecrystal, thus further increases of the field do notlead to an increase in the tunneling current. Thetransition occurs at fields around 0.40 V/ Á in the

simplified model of Louis et al. [52], but becausethe exponential dependence of the tunneling life-time this should be a good value for any realsituation.

cient (efunctiolanother

phase iexpericIbefort: .

high dedelay irtor is c(surfacedisordelof enco;

whose pthe elecldisordertum to

producew- reswOI1cers

h~m~~re

7. Two-dimensional electron gas

Image states represent a very simple and funda-mental class of surface states which is useful instudying the two-dimensional (20) electron gas[54]. To obtain a 20 Ferrni electro n gas a highelectron density is needed. In liquid helium thebinding energy is weak (- meV)and an instabilityof the 20 electron gas develops when the densityis increased [55]. In metals the binding energy is ofthe order of 1 eV but a problem arises due to theprocesses that contribute to the energy width ofthe image state. The interband transition (decayinto allowed bulk states) contribution to the en-ergy width is high and comparable to the in-traband (decay into image sÜltes having smallerparallel momentum within the band) contribution.This means that electrons may not be able tothermalize to the bottom of the 20 band becausethey decay into bulk states and the high densityneeded to obtain a Ferrni electron gas would notbe achieved. In insulators, such as diamond orLiF, relatively strong binding appears (a largeenough dielectric constant) and the escape outsidethe band into buIk states is prevented by theelectronic structure. Amau and Echenique [56]have evaluated variationaIfy that the intrabandenergy halfwidth is 150 meV for parallel momen-tum of 0.15 (Á -1). They conclude that if onecould populate the first image-state electron bandup to parallel momentum of k p = 0.13 Á -1 then adensity of about 1013 cm - 2 could be a.ttained,enough to think in terms of a20 Ferrni electrongas [57].

(d4>B('

dE

The conlifetimestions th.

For quitcut-off fmeasureinsulatOl

Aclrnowlt

The :

Flores, J. ~Sl( an.

Rese3:ilibert~'J.Guipuzk,CAICYl

8. Disordered surfaces

Pendry [58] has suggested toat image states atthe surfaces of insulators may playa central rolein the generation of 1/1 noise at the surfaces ofthese materials. The phase of the reflection coeffi-

Referenc

[1]W.T.~M.("..V.B.S

..

~!,¡.¡~j

~

cause~ life-! real

ltes atal rolelces ofcoeffi-

.

P.M. Echenique, M.E. Uranga I lmage po/en/ia} sta/es a/ surfaces

cient (eq. (4» is related to the amount of wave-function Iying inside the material. It has alsoanother significance [58]: the derivative of thephase is also directly proportional to the delayexperienced by a pulse of electrons, energy E,before the reflection process is complete, thus, ahigh density of states in the material will trap anddelay incident electrons. The surface of an insula-tor is covered by a potentially conducting sheet ofsurface states. As an electron runs around the

. disordered surface, it will have a finite probabilityof encountering a crystal reflection coefficient rewhose phase, <Perises sharply in energy, and thenthe electron will be trapped inside the solid in thedisorder induced localized state to eventually re-tum to the surface state bando This process willproduce fluctuations in the electrostatic field andwill result in noise. Pendry, Kirkman and co-workers [59] have shown that the trapping stateshave a characteristic distribution of lifetimes asmeasured by

jd<PB(E)d<PB(E+W) )_.!. ( )\ dE dE w. 12

The consequence of this distribution of trappinglifetimes is that the trapped charge shows fluctua-tions that have a "1/1 noise" power spectrum.For quite modest thickness of insulator the inversecut-off frequency (wmin::=exp(-52L/2); 52 is ameasure of disorder and L is the thickness of theinsulator) will be of the' order of days or longer.

Acknowledgements

. The authors gratefully thank Professors F.Flores, J.B. Pendry, and R.H. Ritchie for discus-sions and Iberduero S.A. for help and support.Research sponsored jointly by Euskal Herriko Un-ibertsitatea, Hezkuntza Saila, Eusko Jaurlaritza,Guipuzkoako Foru Aldundia and the ~SpanishCAICYT.

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