JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 2 8 JANUARY 2004

Measurement and dynamics of the spatial distribution of an electronlocalized at a metal–dielectric interface

Ilya Bezel,a) Kelly J. Gaffney,b) Sean Garrett-Roe, Simon H. Liu,c) Andre D. Miller,d)

Paul Szymanski, and Charles B. Harrise)

Department of Chemistry, University of California and Chemical Sciences Division,E. O. Lawrence Berkeley National Laboratory, Berkeley, California 94720

~Received 30 July 2003; accepted 5 September 2003!

The ability of time- and angle-resolved two-photon photoemission to estimate the size distributionof electron localization in the plane of a metal–adsorbate interface is discussed. It is shown that thewidth of angular distribution of the photoelectric current is inversely proportional to the electronlocalization size within the most common approximations in the description of image potentialstates. The localization of then51 image potential state for two monolayers of butyronitrile onAg~111! is used as an example. For the delocalizedn51 state, the shape of the signal amplitude asa function of momentum parallel to the surface changes rapidly with time, indicating efficientintraband relaxation on a 100 fs time scale. For the localized state, little change was observed. Thelatter is related to the constant size distribution of electron localization, which is estimated to be aGaussian with a 1564 Å full width at half maximum in the plane of the interface. A simple modelwas used to study the effect of a weak localization potential on the overall width of the angulardistribution of the photoemitted electrons, which exhibited little sensitivity to the details of thepotential. This substantiates the validity of the localization size estimate. ©2004 AmericanInstitute of Physics.@DOI: 10.1063/1.1622386#

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I. INTRODUCTION

In the past decade, significant experimental and theoical efforts have been applied to electron dynamics at infaces. The nature of the environment changes drasticacross the interface and so does the electron’s behaviorexample, electrons, which are described by delocaliBloch waves inside a crystalline metal, localize in a solvcage in a bulk solvent.1 Various spectroscopic techniqueemploy electrons as a structureless probe of their loenvironment.2–6 Here, time- and angle-resolved two-photophotoemission~TPPE! is used to estimate the dynamicsthe electron localization size in the plane of a metal–solvadsorbate interface.

The electronic states under study are the image potestates~IPS!. These states have been well characterized tretically and experimentally for a variety of substrates aadsorbates.7–24 Briefly, IPS are caused by an electron’s atraction to the polarization it induces at a mesurface.10,11,13,25This gives rise to a 1/zimage potential~z isthe distance to the surface!and a hydrogenlike progressioof states residing in the vacuum outside of the adsorblayer. Here, we restrict the discussion to the first (n51) state

a!Also at Miller Institute for Basic Research in Science, 2536 Channing W#5190, Berkeley, California 94720.

b!Present address: Stanford Synchrotron Radiation Laboratory, 2575Hill Road, Menlo Park, California 94025.

c!Present address: The Aerospace Corporation, El Segundo, Calif90245.

d!Present address: Intel Corporation, Hillsboro, Oregon 97124.e!Author to whom correspondence should be addressed. Electronicharris@socrates.berkeley.edu

8450021-9606/2004/120(2)/845/12/$22.00

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of the IPS progression. The structure of this state in therection normal to the plane of the surface is wunderstood.11,13,25Being confined within a few angstroms ithe direction normal to the surface by the electrostatic inaction, the IPS electron is a sensitive probe of changes inadsorbate layer.

The IPS electrons are free to move parallel to the sface, in the (x,y) plane. In other words, the electrons canviewed as Bloch waves propagating parallel to the surfaand excitation of such free ‘‘delocalized’’ electrons has bedocumented for many systems.10,14,18,19,23,24If interactionswith the surface adsorbate are strong enough, electronsknown to localize in the plane of the interface.

Angle-resolved TPPE has a unique ability to distingubetween the localized and the delocalized Ielectrons.7,10,21,24,26–29In these experiments, a short laspulse excites electrons from the states below the Fermi leof the metal to the unoccupied IPS. The electrons are phemitted by the second~probe! laser pulse, and their kineticenergy is measured in the direction of emission. The protion of the electron momentum on the surfacepi is con-served during the photoemission step.9,30,31From the classi-cal perspective, the electrons that were initially at restphotoemitted normal to the surface, and those that wmoving parallel to the surface are photoemitted at an angutoward the surface normal. The photoejected electron kinenergyEf and momentump are related to those at the inteface ~Fig. 1!,

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846 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Bezel et al.

Ei5pi

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2m*2Ebind, ~2!

pi5p sin~u!, ~3!

whereme is the free electron mass,Ei is the initial energy ofthe IPS electron moving parallel to the surface,m* is theeffective mass for such motion,\v is the probe photon energy, andEbind is the energy ofn51 IPS electron binding tothe surface. Thus, the signature of a delocalized electronparabolic kinetic energy increase withpi and the effectivemasses are typically close tome . The dependence of kinetienergy on the momentum of the state is referred to asstate’s dispersion. The states localized in the (x,y) plane donot move parallel to the surface~or have a very large effective mass!and thus they have nondispersive, angle indepdent kinetic energies. Both localized and delocalized stawere observed in TPPE experiments, sometimes simuneously~Fig. 2!.7,10,26,27,29

Nondispersive features in two-photon photoemissspectra can have different origins for different systems.example, nondispersive states were observed for low coages of thiolates on Ag~111!. There, the electrons werthought to be localized on the sulfur atoms of tadsorbate.27 For benzene on Ag~111!, unannealed layers localize electrons, which was attributed to localizationdefects.32 Harris et al. have reported the localization of IPelectrons above two monolayers ofn-heptane on

FIG. 1. Schematic of photoemission from the localized~a! and delocalized~b! states. Top panels: artistic representation of the localized and delocaIPS states~shaded shapes! and electron momenta upon photoemission~ar-rows!. Delocalized states have well-defined values ofpi which are con-served. The localized state does not have a defined value ofpi due to finitesize in the plane of the interface. The widths of the probability distributin the momentum and coordinate space are denoted byDpi andDx, respec-tively. Vectorsp and p' are the final state momentum and its componenormal to the surface, respectively. Electric fieldE is directed normal to thesurface. Bottom panels: dispersion curves. Thick solid curves on the boshow IPS populated by the pump pulse\vpump from the occupied statesunderneath the Fermi levelEF of the substrate~not shown!. They are photoemitted by the probe pulse photons\v ~vertical arrows! resulting in non-dispersive~very large massM! and dispersive features in the spectra for tlocalized and delocalized states, respectively. Dashed lines show thetion of the vacuum levelEvac(50).

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Ag~111!.10,24 Unlike the thiolate or benzene systems, locization at the alkane interface is dynamic in nature. Fotwo-monolayer coverage of butyronitrile on Ag~111!, elec-tron localization is accompanied by solvation which was oserved for both the dispersive and nondispersive state7,8

Wolf and co-workers report the solvation and dynamic locization of electrons for multilayer coverages of D2O onCu~111!, which was assigned to electrons trapped in the bof the adsorbate layer.26,29 We limit the discussion to thestates that localize while maintaining their IPS character

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FIG. 2. Experimental kinetic energy spectra~dots! for n51 IPS at twomonolayers of butyronitrile on Ag~111! and the result of a typical global fit~thick solid line!. ~a!–~c! Dispersions taken at different delay times. Spectaken at different observation angles are vertically offset within each spanel. ~A! Delocalized state: the rightmost peak, disperses with anglehigher energy.~B! Localized peak: the nondispersive feature at;1.1 eV.~C!A small amplitude nondispersive peak at;0.8 eV that appears at highobservation angles is not assigned. For a better description of the propof this peak refer to the Appendix. Each spectrum was normalized tomaximum amplitude and a linear background was subtracted. Experimerror bars~not shown!were used in the fitting.

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847J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 An electron localized at a metal–dielectric interface

was suggested for the cases ofn-heptane and butyronitrileLocalization is thought to be due to small polaron formatiwhere the electron localizes at the electron induced latdistortion.10,24 We suggest that TPPE can be used to makquick, simple, and robust estimate of the IPS electron loization size distribution and its dynamics in many cases.

A signature of the IPS electron localization size in t(x,y) plane is encoded in the dependence of the photoization current on the observation angle. Namely, thoseelectrons that are tightly localized in the plane of the intface must give rise to a signal which decays more slowith an increasing angle of observation than those thatlocalized to a large surface area. Changes of the localizasize must result in different angular distributions of the phtoemitted electrons.

The paper is organized as follows: First, we describebasis for relating the electron localization size to the widththe angular distribution of the photoemitted electrons. Trobustness of this estimate is tested by using a simstraightforward model. The next section uses an examplIPS electron localization at a two-monolayer butyronitrlayer on Ag~111! to test this approach. This paper is focuson an amplitude analysis only; a complete description ofnitrile dynamics will be presented elsewhere.33 An analyticalform of the apparatus function of a time-of-flight~TOF! de-tector while observing the delocalized and the localized etrons has been derived.34 Methods used in obtaining the signal amplitudes for the localized and the delocalized stafrom the experimental data are discussed in the Append

II. METHOD

For the delocalized states, the physical mechanismsulting in photoemission along directions away from the sface normal is given by Eqs.~1!–~3!. Upon localization, themomentum parallel to the surface is not a good quannumber. An electron localized to a small size in the planethe interface has a large spread inpi due to the HeisenbergUncertainty Principle. The spread ofpi in the localized stateis transferred to the projections of the photoemitted electmomenta giving rise to a broad angular distribution. If tspread in momentum space is known, the shape ofground state wave function in position space can be obtaby using a Fourier transform. The localization length isversely proportional to the wave packet width in momentspace.

The above can be formally justified as follows. If thelectron was initially in a stateCi with the energyEi , thephotoemission rate into a unit energy interval~and thus theintensity of the measured photocurrent! is given by Fermi’s‘‘golden rule,’’

P~Ef !52p

\r f~E!u^Cf udHuCi&u2d~Ef2Ei2\v!, ~4!

wherer f is the energy density of the final states for phoemission into a unit solid angle, and the perturbation is givby

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with e and c, the electron mass and the speed of light,spectively,p, the electron momentum operator, andA, thevector potential of the field. In first-order perturbation theothe final state wave functionCf is given by35,36

Cf~R!5~11G0* T* !expS ipf "R

\ D , ~6!

whereG0 is the Green’s function of free electron motion,Tdescribes the electron scattering by the surface, andR is thevector (x,y,z).

Proper description of the final state wave function ismajor complication in the traditional photoemission expements such as photoemission from the core electronic stof surface adsorbates or from the bulk of the metal. Fexample, in the three-step model of photoemission an etron photoexcited in the bulk experiences scattering bysurface, which leads to a rich structure in the photoemissprofile.31,35,37,38Electrons photoemitted from the adsorbatecores leave behind molecular ions that gives rise to a renance structure in the continuum of the final states, whaffects photoelectron distributions.35,37 Image potential statesseem to be a simpler case, where the most basic phydescription can be sufficient. Indeed, the excess IPS electare already positioned outside the surface and the strapping energyEST ~the energy between the bottom of thdelocalized band and the localized state! is small.

To proceed, a few approximations are made:

~i! The in-plane components of the vector potential cbe neglected. This is a good approximation for a sface highly reflective at the probe pulse wavelengtThe only component of the vector product in Eq.~5!operates along thez direction, i.e., normal to the surface.

~ii! The scattering operatorT also acts only along thezdirection. This is the operator responsible for electrdiffraction in low energy electron diffraction~LEED!experiments. The visible wavelengths of the photoioizing pulses that are used in the TPPE experimeyield the electronic energy on the order of 1 eV. Tde Broglie wavelength for such motion is;10 Å,which is comparable to the lattice constant of smmolecular adsorbates. Thus, the diffraction effeshould scatter the electrons to large angles fromsurface normal, away from the range of interest. Tcan be verified by LEED, which shows no diffractiospots at the photoelectron energies and angles ofterest.Scattering on the potential well where the electronlocalized can also affect the final state wave functioThe influence of the localization potential on the mesured angular distributions is discussed in the nsection. We show in the model calculations below ththis influence can be neglected to a good approximtion when making estimates of the localization sifor the n51 IPS electrons.

~iii! The initial and the final wave functions are separaon the in-plane and out-of-plane components,

C~R![C~x,y,z!5x~z!c~x,y!. ~7!

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848 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Bezel et al.

The first two approximations ensure that separability islost during the photoemission process. For most of phoemission experiments, the separability assumption is antreme oversimplification of the problem as noted abohowever, this assumption is frequently usedIPS.7,10,11,13–16,23,24

Within these approximations, the initial delocalized athe final state wave functions become Bloch waves in tdimensions~2D!,

c i~x,y!5ai~x,y!expS ipi ,xx

\ DexpS ipi ,yy

\ D , ~8a!

c f~x,y!5af~x,y!expS ipf ,xx

\ DexpS ipf ,yy

\ D , ~8b!

where the subscriptsx andy signify projections on the corresponding direction anda(x,y) is a function with the trans-lational symmetry of the lattice describing variation of twave function amplitude within each unit cell. Lateral corugation of the potential energy surface due to eithersubstrate or the adsorbate is generally minimalIPS,10,39–41 so the shape of the initial state wave functiwithin each unit cell is constant,ai(x,y)51, and the finalstates are free particles, soaf(x,y)51.

Due to assumption~i! above, the perturbation does nact on thex andy components of the separable wave funtion, and thus the photoemission rate in Eq.~4! can be writ-ten as

P~Ef !52p

\kzr f~Ef !u^c f~x,y!uc i~x,y!&u2

3d~Ef2Ei2\v!, ~9!

where

kz5u^x f~z!udHux i~z!&u2. ~10!

Note thatkz does not depend on the shape of the initial wafunction in the plane of the interface. Thekz term couplesthe initial and final wave functions in thez direction, and isthe reason that the normal wave vector component isconserved during photoemission.

It would be erroneous to use the 3D density of the fistates in Eq.~9! for a separable system. According to thassumptions, the transition takes place inz space only and a1D density of final states must be used instead,

r f~Ef !5r1D}1

pf ,z, ~11!

wherepf ,z is the final-state momentum in thez direction@p'

in Fig. 1~a!#.The photoionization rate takes the form

P~Ef !}kz

pf ,zU E E e2 i @~pf ,x2pi ,x!x/\#

3e2 i @~pf ,y2pi ,y!y/\#dxdyU2

3d~Ef2Ei2\v!. ~12!

Two limiting cases correspond to localized and delocized states, respectively. For delocalized states, the scatt

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events which destroy the initial state phase are neglectedthe c i in Eq. ~8a! are the true eigenstates of the system.this case, the value of the overlap integral in Eq.~12! is zerounless the initial and the final sates have the same momtum parallel to the surface,pi . This is the justification of thestatement that the projection of the momentum on the suris conserved in the photoemission. The probability of trantion, kz, does not depend onpi and thus the signal intensitis just proportional to the initial state populationndel,

Pdel~pi!}kz

pf ,zndel~pi!dpi ,pi

, ~13!

wheredpi ,piis the Kronecker symbol~one if pi5pi and zero

otherwise!.Alternatively, the scattering processes can be so str

that the phase is destroyed very fast. This correspondslocalized state. This localized state is composed of all ofmomentum components and can be expanded as

c loc~x,y!5(pi

apicpi

, ~14!

wherecpiare the delocalized wave functions given by E

~8a!. The photoemission rate thus becomes

Ploc~E!}kz

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api

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3e2 i @~pf ,y2pi ,y!y/\#dxdyU2

d~Ef2Ei2\v!.

~15!

Similarly, exponentials integrate to zero unlesspi5pf , andthe matrix element becomes a delta function which elimnates the sum from the expression leaving only the coecient for the momentum term of the localized state thatthe same parallel momentum as measured in the final st

Ploc~pi!}kz

pf ,zapi

2 . ~16!

The photoemission intensity as a function ofpi thus reflectsthe momentum distribution in the initial state, i.e., the initstate wave function in the momentum space.

For a delocalized state with an effective mass neaequal to the free electron mass, the denominator of Eq.~13!,pf ,z , is independent ofu. In this case, with nearly identicadispersions for the initial and final states, the final perpdicular momentum depends only on the energy of the inistate relative to the photon energy and does not depenthe position within the final-state band. For the localizstate with a flat dispersion being photoemitted into a fistate with a parabolic dispersion, the projection of the finstate momentum in the perpendicular direction varies wthe parallel momentum of the final state. There is thusadditional weak dependence inpf ,z on u,

Ploc~pi!}kz

p cosuapi

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Ap22pi2

api

2 . ~17!

The experimental measurements are additionally weighby 1/cos4 u to account for the decreasing magnitude of t

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849J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 An electron localized at a metal–dielectric interface

field perpendicular to the surface at anglesuÞ0 for bothpump and probe pulses and by the apparatus function odetector.34 Once Ploc is measured, the electron localizatiosize can be estimated by using a Fourier transform. Assing that the wave function has a Gaussian envelope, thelationships between the full widths at half maximu~FWHM! of the wave function’s squared amplitudes in tcoordinate and the momentum spaces is

DxFWHM54 ln 2

DpFWHM /\. ~18!

This is a rule of thumb. Although rough and lacking detait is extremely robust and simple to use. Robustness ofestimate given by Eq.~18! can be verified on a simple buillustrative model which carries primarily pedagogical valu

III. ILLUSTRATIVE MODEL

In making estimates of electron localization size telectron scattering by the localization potential wasglected and strictly speaking Eq.~18! is an estimate frombelow. How strongly does the scattering on the potenwell, where the electron is localized, affect this result? Texact answer can only be obtained by performing a multmensional calculation of the final state wave functionsformidable, system-specific task. Textbook examples of ptoemission, however, suggest that for sufficiently largenetic energies the effect of the potential on the final stwave functions can be neglected and the rate of photoesion becomes simply proportional to the Fourier componeof the initial state as in Eq.~17!.35,36 Can this approximationbe applied to the localized IPS? Obviously, the image pottial cannot be neglected. The image potential acts only althez direction, however, and preserves the separability ofproblem. The localization potential destroys separability, bin many cases, it is small in comparison to the kinetic engies of photoemission. Indeed, the electron stabilizationergy upon localizationEST was measured to be,100 meVfor many systems, whereas the typical kinetic energy ofphotoemitted electrons is;1 eV.

It is wrong to assume that the eigenstates do not chawhen a localization potential is introduced. For a given potive energy, the electronic spectrum is infinitely degenerand, however small, any localization potential will mix thdegenerate states and form new eigenstates$c f%. A weaklocalization potential, however, does not lift the degeneraand, thus, any combination of$c f% can be used. Calculationof photoemission probability may be performed in arbitrabasis, however; the unperturbed final states are chosenconvenience.

This qualitative result can be verified by using a simp2D particle-in-a-box model. These calculations were carrout for illustrative purposes. Infinite potential walls weplaced atz50 Å, z5400 Å, x52300 Å, andx5300 Å anda basis of 3200 plane-wave states with the lowest quannumbers was chosen. It resulted in the density of statesthe order of 2500 eV21 at the region of excitation~;0.5 eVabove zero!. The image potential was modeled as

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wheree is the electron charge,e0 is the vacuum permissibil-ity, and offsetz053 Å was chosen to imitate the thicknessthe adsorbate. The localization potential was taken to be

Vloc~x,z!52V0

cosh2~axx!

1

~z1z0!3~20!

with ax50.1 Å21 to imitate localization to;10 Å. The pa-rameterV0 was used to control the energy of localizatioEST. The resulting potential is shown in Fig. 3. In the asence of the localization potential, the problem is completseparable and the solution to the system in thez direction isthe set of IPS. In thex direction the solution is a set of planwaves. As expected, introduction of a modest localizatpotential strongly mixes the eigenstates, which lose thseparability and plane-wave character.

As stated above, this calculation neglects any lateral crugation of the potential due to the adsorbates, as this cogation is minimal.10,39–41It is the dynamic self-trapping potential that dominates the physics of localization ainvalidates the assumption of separability. This calculatitherefore, focuses on the effect of a localizing potential suas Eq.~20!.

Note that all particle-in-a-box models are fundamentadifferent from true, unbound space. Upon photoexcitationmonochromatic light, a continuum of states is excited inunbound case, each state with a different direction of progation. There is no time evolution upon such excitation bcause all states have the same energy, but it is still propeask about the angular distribution of the photocurrent. Inbox, however, monochromatic light excites one eigenstatea few eigenstates if there is a degeneracy, but never atinuum. Thus, monochromatic excitation cannot be usedobtain a meaningful angular distribution; formally, one cwrite an angular distribution for a single eigenstate but sua distribution would change erratically from one eigenstto another.

Instead, a wave packet should be prepared consistina number of eigenstates, and, thus, the angular distribubecomes a function of time. Monochromatic excitation implies a long time limit in the interaction with the light fieldThe observable is calculated by its projection onto each fistate averaged over all time, e.g., Eq.~4!. Each calculated

FIG. 3. Sample potential for the particle-in-a-box model. Only the cenregion of the box is shown.

850 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Bezel et al.

FIG. 4. ~Color! Wave packet propagation. Wave packet squared amplitude is shown by using colors from blue~zero amplitude! to dark red~maximumamplitude!. From left to right: propagation timet50, 24, 42, and 63 fs. The initial conditions areDxFWHM553, 26, and 8 Å for rows~a!–~c!, respectively.Dynamics of the corresponding angular distributions are given in Fig. 6~a!. The potential is the same for all panels,EST572 meV.

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matrix element is sensitive to the form of each final stawhich poses problems for a particle-in-a-box model, as mtioned above. An alternative is the impulsive limit, in whican instantaneous interaction with the light field transfersinitial state onto a superposition of final states, formingwave packet. This impulsive limit is nearer the pump–procharacter of the TPPE experiment than the single final spicture used by Bovensiepenet al.29 and avoids the fundamental limitations posed by a particle-in-a-box model of ecitation to a continuum.

Consequently, in these calculations, a wave packetexcited, and the angular distributions were calculated apropagated. The wave packets were prepared by startinga function,

y0~x,z!5exp~2ax2!exp~2bz!, ~21!

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whereb50.2 Å21 and the parametera was used to controthe size of initial wave function localization. The transitiomatrix elements were then determined,

u^yf uzuy0&u2, ~22!

whereyf are the calculated eigenstates. The excitation sptrum was centered at 0.5 eV kinetic energy and was 100 mwide. The position and the width of the excitation spectruare not the same as used in the experiment. Namely,generated wave packet has lower kinetic energy in ordeavoid the effects of the lack of high energy eigenstates inbasis and has a larger bandwidth in order to make it tightethe coordinate space and thus enhance visualizationpropagation. Typical dynamics of the wave packet propation are presented in Fig. 4. After a propagation time,t, the

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FIG. 5. ~Color! Wave packets after63 fs propagation time. The initiacondition is the same withDxFWHM

517 Å, the localization potential re-sults in ~a! EST572 meV, ~b! EST

51.2 eV. Angular distributions ateach propagation time were obtaineby integrating the wave packet squaramplitude within narrow angular bins~white rays!.

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851J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 An electron localized at a metal–dielectric interface

wave packet amplitude squared was integrated inside naangular bins, which gave the probability of photoemissinto different solid angles@Fig. 5~a!#. The maximum propagation time corresponded to the electron departing;350 Åaway from the surface, i.e., almost to the box boundarythe first set of tests, the localization potential was fixed~itresulted inEST572 meV) and the initial conditiona wasvaried to provideDxFWHM58 – 53 Å @Figs. 4 and 6~a!#. Theangular distributions were found to follow Eq.~18! quitewell. In the second set of tests, the initial state wDxFWHM516.7 Å was chosen and the depth of the localiztion potentialV0 was varied in a broad range, resulting in tlocalization energyEST of up to 1.2 eV. Even at the largesvalues ofV0 , the overall width of the angular distributiostayed close to the ‘‘theoretical’’ value ofDpFWHM /\50.16 Å21, although a pronounced structure was pres@Figs. 5 and 6~b!#.

Given the nature of the exercise, little attention was pto making sure that the model potentials and the initial cditions correspond to a particular physical system. Insteasample potential was chosen which only gives approximacorrect values of binding energies, asymptotic behaviorthe potentials, extent of the localization potential, initial eletron distributions, etc. Neither the presence of the imagetential nor the depth of the localization potential significanaffects the width of the angular distributions which justifi

FIG. 6. Dynamics of angular distributions obtained from the particle-inbox model. Solid lines show the evolution of the angular distributions wpropagation timet524– 71 fs, the thickest line corresponding to the lattime. Dots, Eq.~18! predictions.~a! The potential is fixed to provideEST

572 meV and the localization size of the initial conditions are variSample wave packets are shown in Fig. 4.~b! The width of the initial stateis fixed toDxFWHM516.7 Å, andV0 is varied leading to different localization energies.

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the crudeness of the model. The estimate of a localizasize from an experimentally determined angular distributis therefore a robust one and can be applied even whenprecise shape and magnitude of the localization potentialnot known.

IV. EXPERIMENTAL EXAMPLE

Localization of IPS electrons resulting in the populatiof a nondispersive state has been observed for many mollar adsorbates. One-monolayer coverages of metha1-propanol, 1-butanol, and 1-pentanol, as well as twmonolayer coverages ofn-pentane,n-hexane, cyclohexanen-heptane,n-octane, acetonitrile, and butyronitrile, westudied by our group.7,8,10,24,33 Following photoexcitation,electron dynamics in these systems are quite complicasimilar trends, however, can be distinguished. For shpump–probe delays, a dispersiven51 state is observed. It isattributed to population of the delocalized, extended, frelectron-like band. The signal amplitude of the dispersstate decreases gradually due to various relaxation proceand a nondispersive state grows in at later delay times~200–500 fs!. At even larger delays~*1 ps! only the nondispersivestate is observable. A common feature of all of these systis a small value of the energy difference between the botof the delocalized band and the localized state,EST ~Fig. 2!.The typical energy separation between the lowest delocaland the localized peaks is very small, on the order50 meV, which is less then the peak widths, and thuspeaks completely overlap at low observation angles~smallpi).

The energies of the localized, nondispersive statescrease with time. This is attributed to solvation of the locized electrons, a process that involves rearrangement oadsorbate molecules to accommodate the excess electroalternative mechanism that can contribute to lowering oflocalized state energy with time is hopping from sites whigher binding energy to those with lower energy. The decalized, dispersive electron states were also found to solprovided that the adsorbate molecules are polar.7,8,33No sig-nificant change in energies of the delocalized states wascorded for nonpolar adsorbates. Solvation of the delocalielectrons was explained by collective reorientation of thesorbate molecules underneath, which reduces the local wfunction.7,8

In addition to the energy relaxation through solvatiand localization, population relaxation takes place. One pof it, electron recombination back to the metal substrate,already been mentioned. Given proper conditions it can blong as a few picoseconds. Intraband relaxation occurs osubpicosecond time scale. It results in cooling of the decalized electrons, i.e., population relaxation from the initiapopulated high-pi states to the bottom of the delocalizeband. The evidence of efficient intraband relaxation is a hrate of population rise and decay for high-pi states and slowrise and decay of low-pi states as shown in Fig. 7~b!.7,10,24

Detailed description of these processes can be foelsewhere.10,16,18,23They, however, complicate the measurments of the localization size and need to be taken intocount. A global fitting routine was applied to an entire da

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852 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Bezel et al.

set, which allowed extracting angular distributions of decalized and the localized peak amplitudes. The Appenpresents this approach to data analysis using a data setwo monolayers of butyronitrile on Ag~111! as an exampleThe results obtained for this system are qualitatively vsimilar to those reported previously.7,10,24 Figure 7 showsthat although amplitude dynamics for the delocalized sdepend on the observation angle, the localized state hasnamics which are almost the same regardless of the angobservation. This automatically means that the shape ofgular distribution for the localized state is time-independ~Fig. 8!.

Since the amplitude of then51 delocalized peak fordifferent values ofpi is proportional to the population of thdelocalized band, the width of the angular distributionflects the effective electronic temperature,42

kT5Dpi2/2m* , ~23!

wherek is the Boltzmann constant. Figure 8~b! shows thatthe electronic temperature is initially very hot,;2000 K, butrapidly approaches the temperature of the substrate~130 K!.

For the case of small polarons, the mass is much heathanme and the band is narrow, giving rise to a nondispsive state in the experiment. Formally, one may considestate which is in actuality delocalized but whose disperscannot be measured within the available energy and momtum resolution, i.e., a state with an effective mass grethan ;10me . In this case, the angular distribution wou

FIG. 7. Time evolution of signal amplitudes obtained in a typical global~a! Localized state~integrated amplitude!. ~b! Delocalized state. Differentobservation angles are shown by using different line styles. All dynamicsnormalized to unit maximum amplitude.

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also reflect the polaronic temperature as per Eq.~23!. Theexperimental observations are at odds with this model,cause the observedDpi;0.0860.02 Å21 @Fig. 8~a!#gives apolaronic temperature of,30614 K, which is much smallerthan the substrate temperature. This observation, theredoes not favor the assignment of the nondispersive featura delocalized Bloch wave with a high effective mass.

A better agreement with the experiment can be achieupon description of the electron as physically localized inplane of the interface, as suggested above. In this caseangular distribution of the signal amplitude tells about ttightness of localization. A Gaussian angular distributiwith FWHM of 0.1960.05 Å21 corresponds to the localization to 1564 Å FWHM in the (x,y) plane and does nochange significantly throughout the entire range of detimes~Fig. 8!. This localization size is comparable to the sof a single butyronitrile molecule. It matches well with thoretical predictions for charge localization in twodimensional crystals where if a charge localizes, it localizto a single lattice site.43,44

When the width of the localized peak is broad, apparnegative dispersion has been observed.26,29Its origin is easilyunderstandable within our model: large linewidths originafrom a strongly inhomogeneous environment. Electronsdifferent environments have different localization energand thus they have different localization sizes. Those etrons which are more strongly bound are more tightly locized. They give rise to a broader angular distribution atlower kinetic energy side of the line. The result is an appent negative dispersion which can be prominent for lines;0.5 eV width and are typically negligible for narrower lineof ;0.1 eV.

.

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FIG. 8. Dynamics of the widths of the angular distributions in the momtum space for the~a! localized and~b! delocalized peaks. Different symbolcorrespond to different global fits. They are slightly offset horizontallyavoid congestion. Error bars are derived from the fit of the distributionsGaussian function ofpi .

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853J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 An electron localized at a metal–dielectric interface

The estimate given by Eq.~18! by no means is claimedto be exact. In fact, caution must be taken when applyinto other systems. For example, this estimate is a gross osimplification of photoemission from the bulk states or phtoemission from the core electron levels of the surfaadsorbates.35 The validity of the assumptions~i!–~iii! mustbe verified before using such an estimate. For the IPS etrons that are already outside of the substrate and oweakly localized, this estimate was found to be sufficientan educated guess of the extent of the electron localizaMore importantly, by using this estimate, the dynamics ofelectron size can be traced. The size of electron localizais an important parameter in description of electron strapping for many systems. Electron self-trapping is extsively discussed in physics of crystalline bodies whereelectron can localize by distorting the lattice formingpolaron.43–50Localization phenomena can also be causedirregularities of the potential~Anderson localization!, nonlin-ear interactions with the environment~intrinsic localizedmodes!, etc.17,32,43,45,51,52Experimental and theoretical wordevoted to this subject is overwhelming; especially, in retion to low-dimensional systems, e.g., electron localizationconjugated polymers and nanowires.5,17,43–48,50,51,53,54Pro-cesses of electron localization are of interest for studiesbulk liquids, where extensive attention was given to electphotoinjection and solvation. In these systems, the excelectrons localize quickly either forming anions or remainiin the intermolecular cavities.2–6,55,56There, electron local-ization is a self-consistent process in which the shape ofelectronic wave function is dictated by the solvent cagechanges as the solvent rearranges. Despite this interesdirect question of the electron localization size has mobeen addressed in theoretical works and indeed such msurements are complicated and obscured by other dynamrelaxation processes involving, for example, solvation.believe that angle-resolved TPPE experiments can provitesting ground for these theories.

Most of the theoretical studies of electron localizatiwere focused on obtaining stationary solutions, i.e., obsetion of the equilibrium state of the electron; observation acalculations of dynamics of polaron formation is signicantly harder. Recently, more attention was given to dynaics of these processes.5–7,10,45–47,57Starting from electron in-jection into an undistorted lattice, one can visualize differpathways for electron localization. In systems of reducedmensionality, one can construct a barrierless adiabatic pStarting from an undistorted lattice, the system procethrough a large polaron transient state, in which bothelectron and the lattice distortion span over a number oftice sites, and ends in a small polaron stationary state thacollapsed to a single lattice site. Evidence for the adiabpath was found in semiclassical calculations for Holsteimodel by Kalosakaset al.57 For moderate values of thelectron–phonon coupling in 2D there is a barrier to locization and the electrons relax to the bottom of the delocized band instead of localizing to the ground small polastate.58 Alternatively, a small polaron can be formed directin a nonadiabatic process where the delocalized electronstantaneously hops to a localized electronic state. An

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ample of explaining all electron relaxation processes bynonadiabatic polaron formation mechanism is the workGeet al.24 Fully quantum-mechanical calculations of polaroformation were done in Trugman’s group for a small ondimensional lattice.46,59 The polaron formation time wasfound to be on the order of a lattice vibration. The localizstate rise time measured in our experiments is long, onorder of a few hundred femtoseconds@Fig. 7~a!#. There islittle change of the electron localization size on this timscale, which favors the hypothesis of the direct localizatto a small polaron, i.e., the nonadiabatic path. A compariof the experimental data with ongoing model calculatioshould lead to a better assessment of the electron localizaprocess.

V. CONCLUSIONS

In conclusion, the ability of TPPE to estimate the spatextent of localized electrons via the momentum-dependsignal intensity has been demonstrated. Wave packet calations employing simple model potentials show that theformation contained within the momentum-space photoetron distribution is preserved for a wide range of localizatiwell depths. The amplitudes of the dispersive and the ndispersive peaks from the spectra of then51 IPS for twomonolayer butyronitrile Ag~111! provide an example of theutility of angle-resolved TPPE. The amplitude of the dispsive peak was related to the relative population of the decalized, free electron band. Following excitation, the eletronic temperature was found to change rapidly with timwhich is evidence of fast intraband relaxation. The amplituof the nondispersive peak was related to the localizationin the dimensions parallel to the interface, which was fouto be almost independent of time.

ACKNOWLEDGMENTS

We would like to thank M. Wolf, U. Bovensiepen, WMiller, S. Trugman, G. Kalosakas, L. Ku, V. Mandelshtamand C. Wittig for inspiring discussions. Supported by tdirector, Office of Energy Research, Office of Basic EnerSciences, Chemical Sciences Division for the U.S. Depment of Energy, under Contract No. DE-AC03-76SF000We acknowledge NSF support for specialized equipmused in these experiments. I.B. is grateful to the Miller Instute for Basic Research in Science for their sponsorshipfinancial support.

APPENDIX: DATA FITTING AND ANALYSIS

Here, we present experimental data analysis for tmonolayers of butyronitrile on Ag~111! taken at 130 K whichserves as a model system that has many common trendsother previously reported systems. The detailed descripof the experimental setup and data acquisition systembeen given elsewhere.60 Briefly, in order to judge the local-ization size dynamics, signal amplitudes must be extracIn an angle-resolved TPPE experiment the photoelectronergy spectra are collected at different observation anglespump–probe delay times and spectral evolution is studieda function of these two parameters~Fig. 9!. Spectra collected

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854 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Bezel et al.

at the same delay time but at different angles will be referto as ‘‘dispersion’’ measurements since they are used totract the dispersive characteristics of the delocalized etronic bands. Figure 2 shows typical dispersions takendifferent pump–probe delay times. A complete data set csists of approximately 200 spectra taken at different deland observation angles. It has been an extreme challengkeep all other experimental parameters unchanged duringentire time of data collection,;10 h per data set. Long-timdrift of experimental setup has been accounted for by nmalizing dynamics taken over a period of;1 h to match awell calibrated dispersion at a reference delay time.

The dispersive ‘‘delocalized’’ state, peak~A! in Fig. 2, isthe dominant feature of the spectra at early delay timLater, both the dispersive and the nondispersive ‘‘localizestate~B! can be observed. After a picosecond delay, onlnondispersive state survives although the presence of thelocalized state cannot be ruled out for low observation angwhere the peaks overlap. Energies of both the localizedthe delocalized states are shifting to lower values withcreasing delay; this has been attributed to two-dimensiosolvation. Spectra taken at low observation angles contasmall dispersiven52 peak of the IPS progression at highenergy. It has been cut out to facilitate fitting.

A nondispersive peak~C! with a small amplitude appearat lower energies and high observation angles. Its originnot assigned. Based on an extensive survey of differentsorbates on Ag~111! the following properties are revealed:

~i! The peak appears only when the localized IPS sappears. It is present for all systems that localizeelectron and is absent for all systems that do not.

~ii! For different systems it is located 0.3–0.4 eV belothe localized state; the energy separation betweenlocalized state and the satellite peak remain conswith time.

~iii! Its amplitude grows with observation angle; in striing contrast with other states, whose amplitudes dwith the angle; the ratio of the its amplitude to th

FIG. 9. Time delays, observation angles, and approximate values ofpi ; ateach point of the grid experimental spectra were recorded. Large negdelays~not shown!were used in the background subtraction. Typical specand fits for four dispersions taken at different delay times~a!–~d! are pre-sented in Fig. 2. Localized and delocalized peak amplitudes obtainedifferent global fits are presented in Fig. 10. Shading shows the area wpeaks in individual spectra are distinguishable.

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amplitude of the localized state stays constantgiven angle of observation~but may change from system to system!.

~iv! Its width for different systems is 0.2–0.5 eV and itlarger than the width of the localized state.

~v! Complete deuteration of the adsorbate~heptane!doesnot significantly change neither positions, widths, ndynamics of this peak compared to the protonatedsorbate.

It is tempting to assign peak~C! to an alternative photoionization route from the same localized state, e.g., photoiontion with emission of a phonon, which carries energy amomentum recoil. However, indifference to deuteration alarge peak separation energies rule out the phonon-assprocess. This peak was fit in the global fitting procedudescribed below along with the delocalized and the localipeaks; discussion of its origin and properties is left for mothorough investigation.

The dispersive~delocalized!and the nondispersive~lo-calized! states can be resolved at intermediate delay timand sufficiently high observation angles as shown byshaded area in Fig. 9. Outside of the shaded area the pcannot be separated. In order to judge the amplitudes oflocalized and the delocalized peaks outside of the shaarea, one cannot rely on fitting individual spectra wherefit parameters describing the positions and the widths ofoverlapping lines become excessively correlated. This lited the ability to interpret the data at low angles whenethe localization was observed and only results for hangles were previously discussed for such systems.10,24,32Toglimpse beyond the shaded area, a method of interpolatiothe fitting parameters has to be used. In the previous repdispersions taken at the same delay time were fitted simuneously with the positions and widths of the peaks at langles dictated by those at higher angles.7 This approachallowed to fit dispersions at the intermediate delay timwhere both the localized and the delocalized states gaveto comparable signal amplitudes. Here, a global fit ofwhole data set is used: the parameters describing the enpositions and the widths of the peaks were assumed tosmooth functions of the observation angle and time.

Approximately 1–2 thousand nonlinear fitting parameters would be necessary to fit individual spectra for a typdata set~;200 spectra, three peaks each!. In the global fit-ting routines, 20–40 nonlinear parameters were used. Linfitting parameters, e.g., peak amplitudes, are much mstrongly influenced by laser power fluctuations and attemto parameterize them globally resulted in poor fit. Conquently, their values were varied independently for easpectrum. Here, we discuss the peak amplitude variationfunction of the observation angle and the delay time.

Figure 7 shows dynamics of the peak amplitudesdifferent angles; the results obtained in a typical globalSimilar to other systems9,10,23 the delocalized state showsstrong dependence of the rise and decay time on the obvation angle~i.e., onpi). The localized state exhibits angleindependent dynamics. Note that the rise and the decay tiof the localized signal do not coincide with those for t

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855J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 An electron localized at a metal–dielectric interface

delocalized state. Unlike in the former studies, the glofitting allows one to trace these dynamics even for the lowobservation angles. Angle-independent dynamics of the pamplitudes means that the shape of the signal amplitudefunction of angle stays constant throughout all delay timTypical dependencies of the localized and the delocalisignal amplitudes as a function ofpi are shown in Fig. 10.

The main challenge in global fitting of the data set psented in Fig. 2 is in the assumption of functional dependcies of the fitting parameters on angle and delay time.example, the Lorentzian linewidth for the delocalized statelinear with pi according to Wonget al.;9 quadratic~linearwith Ei) according to Bertholdet al.,61 or has a more complex form, depending on the relaxation processinvolved.16,18,23The line shape itself cannot be easily paraeterized. For example, the simplest assumption is a symric line shape; it is a good approximation for some systebut is clearly not the case for others.9,10,17,23,26,27,29Similarproblems arise with the time dependencies of the parame

FIG. 10. Amplitudes of the localized~right column!and delocalized~leftcolumn! states as a function ofpi at four different delay times;~a!–~c!,matching Figs. 2 and 9. Different symbols show results of different glofits. They are slightly offset horizontally to avoid congestion. Amplitudesthe localized and of the delocalized states are strongly anticorrelated avalues ofpi . Typical error bars are shown~95% confidence interval!. Theerrors are obtained in a Monte Carlo simulation of the data sets.

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To resolve this problem, a few different functional dependecies were tested.

Confidence limits of the fit parameters were calculaby using Monte Carlo simulation of data for each globalOften, these errors are significantly less then the variatiothe fit parameters when different dependencies were assufor the global parameter variations. The spread of theresults obtained with different models does not have a gstatistical meaning, rather it is a measure of one’s imagition. There is no guarantee that a different inquisitive depdence would not produce a very different result. Ultimateif the peaks overlap no amount of fitting can tell them apwith 100% confidence without bringing in additional knowedge regarding the peak shapes. An example of such sysatic change of line shape is presented by Bovensiepenet al.The line shape of the localized signal peak was foundchange with the observation angle due to a large~;0.5 eV!inhomogeneous broadening.29 This resulted in systematicallyincreasing skewness of the peaks and to apparent negdispersion. For butyronitrile on Ag~111! the lines are sharpand this effect is expected to be small. Without any regardthe underlying physical reasons, symmetric line shapes wlinear or quadratic dependencies of their parameters onpi

were used to account for angular variation of the parameand an exponential decay or a convolution of a Gaussianexponential decay was chosen for the time variation~all con-voluted with the experimentally measured pump–procross-correlation!. This parameterization of line shapeslowed to formalize the knowledge that the line shape paraeters are smooth functions ofpi and delay time.

The initial values of the fitting parameters~e.g., the poly-nomial coefficients or the time constants! were chosen baseon the fit to individual spectra in the shaded area~Fig. 2!. Iflittle variation was observed as a result of the global fit ax2 merit function was near unity, then constant values ofparameters were used throughout all delays~angles!and thefit was repeated. As expected, the largest errors appear inpeak amplitudes at low observation angles. Amplitudesthe localized and the delocalized peak are strongly anticolated and increase of one can be compensated by thecrease of the other. Despite that and regardless of the ftional dependencies of the parameters, the amplitvariations of the localized and delocalized states stay qutatively the same.

The delocalized and localized signal amplitudes shoin Fig. 10 were fit to Gaussian functions ofpi ,

exp~2pi2/2Dpi

2! ~A1!

at each delay time. The dynamics ofDpi for each state areshown in Fig. 8.

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