Mechanisms producing inelastic structures in low-energy electron transmission spectra

PHYSICAL REVIEW B VOLUME 38, NUMBER 16 1 DECEMBER 1988

Mechanisms producing inelastic structures in low-energy electron transmission spectra

R. M. Marsolais' and L. SancheGroupe du Conseil de Recherches Medicales en Sciences des Radiations Faculty ofMedicine, University ofSherbrooke,

Sherbrooke, Quebec,Canada JlHSN4(Received 26 February 1988)

Electron-energy-loss mechanisms producing structures in the low-energy electron transmission(LEET) spectra of condensed molecular films are examined within the framework of two-streammultiple-scattering theory. A phenomenological model is developed and applied to specific process-es occurring in the course of the propagation of low-energy electrons through thin films. Electroni-cally excited states having strong excitation cross sections near threshold or oscillations in their ex-citation functions are found to produce sharp oscillatory features in LEET spectra. Analysis of thedc and doubly differentiated spectra of N2 and CO films reveals that threshold effects and reso-nances (i.e., transient anions) present in the gas-phase cross sections of these molecules reemerge in

the condensed phase provided that they involve valence orbitals.

I. INTRODUCTION

During the last two decades, low-energy electrontransmission (LEET) spectroscopy has been applied tothe study of the interaction of very-low-energy (0—20 eV)electrons with metallic and molecular solid films. ' Ina typical experiment, a thin film (-5—100 A) vacuum de-posited on a metal substrate is exposed to low-intensitymonochromatic electron beam in ultrahigh vacuum. ALEET spectrum is obtained by measuring the totalcurrent transmitted at the substrate, I, (E), as a functionof the incident energy.

The transmitted current is sensitive to both the elasticand inelastic interactions occurring in the film and at itsinterfaces. Analysis of LEET spectra is therefore notnecessarily straightforward since more than a single pro-cess can be responsible for the spectral features. Despitethis drawback, several theoretical' ' ' ' ' ' and experi-mental' ' 'i efforts have been made to identify theorigin of the structures found in LEET spectra and con-sequently to develop simple procedures to analyze thedata. Today, the underlying mechanisms in LEET spec-troscopy are beginning to be understood. Those responsi-ble for the elastic features are essentially related to theelectronic conduction-band structure above the vacuum

level ' ' ' ' ' ' and hence sensitive to structural or-der ' ' and film thickness. ' ' ' For very thin orderedfilms. quantum interferenccs of the electron wave be-tween the vacuum-film and film-substrate boundaries (i.e.,quantum size effects) ' ' appear in the elastic portion ofthe transmitted current. The inelastic features in LEETspectroscopy usually appear as broad maxima' ' result-ing from a convolution of inelastically scattered currentdistributions created by electrons having lost most oftheir energy in producing excitations and band-to-bandtransitions. With this knowledge, LEET spectroscopyhas been applied to a diversity of problems.

By monitoring the thickness and energy dependencesof the "elastic'* features, it was possible to characterizefilm growth including the determination of their thick-

ness, ' ' orientation, ' layer-by-layer construc-tion, ' ' and phase changes. ' The formation of de-fect ' and quantum well structures in thin films hasalso been detected by LEET spectroscopy. Studies in-volving the inelastic features served to identity spin-forbidden transitions having strong threshold cross sec-tions within molecular solids. ' ' ' ' These measure-ments also led to estimates of inelastic cross sections nearthreshold' and of the energy of the lowest conductionlevel Vo (i.e., the position of the band edge near the vac-cum level) in several molecular ' ' and organicsolids. " Furthermore, in experiments where the pri-mary particles are low-energy electrons incident fromvacuum on a solid surface, LEET spectroscopy can beused on a routine basis to monitor film charging' andchemical degradation and to calibrate the electron-energy scale with respect to the vacuum level.

Determination of electronic transition energies andtheir amplitudes and of Vo values by LEET spectroscopyrely on the mechanism iohich is assumed to be responsiblefor the formation of the inelastic structures and theirproper assignment to specific electronic states. Excita-tion of a bound electron in a film to an unoccupied stateby an incoming electron scatters this latter to a lower-energy state. Near the electronic excitation threshold,such inelastically scattered electrons "fall" to the lowest"conduction" level of the film and their transmissionprobability becomes unity when the band edge (i.e., Vo)lies below the vacuum level. As the electron energy is in-creased, excitation of the same transition produces elec-trons of correspondingly increasing energy up to a pointwhere they have nonzero probability to escape in vacu-um. From then on, I,(E) usually diminishes progressive-ly unless another electronic transition with comparablemagnitude becomes energetically possible. This mecha-nism, referred to here as the "Vo mechanism, " is expect-ed to cause an increase in transmission followed by a tail-ing decrease as the electron energy is swept across the en-ergy of an electronic transition. So far, the Vo mecha-nism has been considered qualitatively by many au-

38 11 118 1988 The American Physical Society

38 MECHANISMS PRODUCING INELASTIC STRUCTURES IN. . . 11 119

thors' '" ' ' ' ' ' and introduced in calculations viaSnell's law ' ' ' but no systematic and quantitative as-sessment of its validity and influence on the magnitudeand shape of inelastic features has been made. As firstpointed out by Harrigan and Lee, the sensitivity ofLEET to electronic energy levels of solids is not straight-forward, involving a convolution of a joint density ofstates (DOS). For example, in recent experiments sharpstructure has been observed near excitonic thresholds ofN2 films. Since these latter have a positive bulk electronaffinity (Vo &0), the results indicate that the Vo mecha-nism is not needed in this case to explain the presence ofsharp inelastic structures in LEET spectra. Obviously,our present description of the mechanisms responsible forthe inelastic features in LEET needs to be furtherdeveloped if we are to rely on such spectra to determineVQ values and investigate electronic transitions nearthreshold produced by electron impact on solid films.

The purpose of the present paper is to give a quantita-tive assessment of the mechanisms responsible for inelas-tic features in LEET spectroscopy. We base our discus-sion on a semiphenomenological model which is appliedto the inelastic region of the LEET spectra of Nz and COfilms. In Sec. II, we consider general basic concepts re-garding the theory of LEET spectroscopy and lay thegrounds for our discussions. We describe the quasi-single-step thermalization processes producing inelasticstructures and distinguish between three types of contri-butions: processes related to the Vo mechanism (i.e., final

DOS), the amplitude of the transition, and the presenceof boundaries. Full description of the model is given inSec. III, with emphasis on cross sections and reflectioncoeScients. A model calculation is presented to illustratethe properties and the main predictions of the model. InSec. IV the model is applied to the spectra of condensedN2 and CO. We first justify the applicability of the modeland some of the approximations made in the calculations.Then, on the basis of specific calculations, we argue thatthe individual transition probabilities for the electronicexcitations observed ought to have large values nearthreshold or exhibit strong oscillations as a function ofelectron energy. We summarize our results and their im-plications for LEET spectroscopy in Sec. V.

II. BASIC CONCEPTS

A. Probability and energetics of inelastic transitions

Consider the simplified potential-energy-level diagramof Fig. 1 for an electron of energy E entering a solid filmfrom vacuum along the z direction. The horizontal heavyline and double line represent the potential energy withrespect to the vacuum level (VL) and the total electronenergy, respectively. The vertical dashed line at z=Odenotes the film-vacuum interface. V~ is the potential en-ergy due to electronic polarization of the dielectric by thenegative charge of the electron. The dotted horizontalline represents the energy of the lowest conduction levelVp of the solid. After having produced an electronictransition of energy E the electron is found at a lowerleuel in the Plm "conduction band" with energy

4PC

0e~CQ0I

VL

I

II

Vacuum ' Film

E,

y —"—E'

FIG. 1. Simplified potential-energy-level diagram for an elec-tron of energy E entering a solid film from vacuum along the zdirection. The horizontal heavy line and the double line

represent the potential energy with respect to the vacuum level

(VL) and the total electron energy, respectively. The verticaldashed line denotes the film-vacuum interface. V~ is the poten-tial energy due to electronic polarization of the dielectric by thenegative charge of the electron. After exciting a level of energy

E, the electron has a final energy E'. The dashed horizontalline represents the energy of the lowest conduction level Vo ofthe solid.

B. Effects of multiple elastic scattering on inelastic transitions

Since we are interested in energy losses which are ofthe order of the incident electron energy, only multipleelastic or quasielastic scattering needs to be considered(i.e., double electronic transitions are not energeticallypossible). Multiple elastic or quasielastic (i.e., includingsmall energy losses to phonons) scattering can occur be-fore and after a given energy-loss event by the electron.A priori, these effects can be best described by micro-scopic multiple-scattering theories such as the LEED for-malism. In these theories, the electron-film interaction isassumed to be appropriately given by a one-electronenergy-dependent local potential. The latter is furtherdecomposed into an "optical" or "polarization" uniformbackground potential punctuated by limited-range

E'=E —E . An inelastic event can occur only if a freeenergy level is available to accommodate the lower-energy electron. Hence, at the electronic excitationthreshold E' = Vp =E—E . . The probability amplitude Pof an electronic transition j for an electron of initial ener-

gy E and wave vector k and final energy E' and wave vec-tor k' is represented by

P) ~(1/k)( A', E', k'iT

iA, E,k),

where A and A' are quantum numbers specifying thestate of the medium before and after the inelastic event.T indicates a transition operator which cannot be ex-pressed in simple form except in the Born approxima-tion where expression (1) reduces to the Fermi goldenrule. This latter, however, is not expected to be validfor electronic transitions near threshold, since in this casethe final wave vector k' is necessarily quite different fromthe initial wave vector k.

11 120 R. M. MARSOLAIS AND L. SANCHE 38

scattering potentials at lattice sites, but the validity offormal multiple-scattering theory does not depend on thisfurther simplification. The optical potential V of theLEED theory is usually complex in order to take into ac-count the loss processes which deplete the diffractedbeams. Here, we must consider V to be real to take ex-plicitly into account the inelastic channels. The scatter-ing at lattice sites, whether elastic or inelastic, is assumedto be faithfully given by local T matrices which formallyresult from multiple scattering within the scattering po-tential. Multiple scattering at the scale of the lattice im-mediately proceeds from that step on.

When the propagation is purely elastic, then the result-ing "ensemble averaged" cross section can be approxi-mated by the local cross section, do, ldQ~

~ T„,~

times a structure factor S(q), which introduces, withinthe Born approximation, the effects of the band struc-ture. As one or more inelastic channels are included inthe scattering process elastic scattering prior to and aftereach inelastic event still contributes to modulate the crosssections (i.e., the scattering strengths,

~ T~„~ ) but theextent of the modulation is a function of the number ofelastic collisions. As the local inelastic transitionstrength increases, or as several inelastic channels be-come available, the effective number of elastic collisionsdecreases, yielding diffuse band-structure effects.

C. Transmission through a film held between two interfaces

The presence of the film-vacuum and film-substrate in-terfaces changes the boundary conditions of the problemdiscussed in the previous section. The effect of the film-metal interface adds a current reflected in the film, asdoes the film-vacuum interface; but more relevant to theproblem of inelastic transitions near threshold is theeffect due to the finite film thickness. In an infinite soliddestructive interferences lead to a gap at a precise elec-tron momentum. For a thin film held by a metal sub-strate, electrons can still tunnel though this gap to themetal, since the bulk band structure is not fullydeveloped. Thus, near surfaces, and quite apart from thesurface relaxation of excitonic levels, inelastic events canproduce electrons having final energies below Vp. In fact,for E'&0, transitions below Vp are purely due to thepresence of the metal. The probability of such events isrelatively small and should decrease rapidly with energysmaller then Vp. For thick enough films, transitions al-lowed by the metal substrate are still forbidden awayfrom the interface when E' is below the band gap and theVp level is well defined. When E' is above vacuum, how-ever, this restriction is suddenly removed and leakage tovacuum becomes possible.

D. Transmission in real films

In real films, the situation is further complicated by theexistence of small energy-loss (i.e., phonons) channels andby the presence of localized states in the gap due to de-fects and impurities throughout the lattice. The formertends to blur the band structure, even below vacuum,whereas the latter represent non-negligible exit channels

for energy losses by electrons, especially in the event ofscattering processes which strongly localize the incidentelectron. Once trapped, the inelastically scattered elec-tron can diffuse, via hopping processes or tunneling, andeventually escapes the film. Through these trappingsites, it can be transmitted in the gap.

For most LEET spectra which are recorded with poly-crystalline or amorphous films, multiple scattering withinthe film and reflections at both interfaces give rise tostreams of elastic (E'=E) and inelastic (E' &E) currentsflowing in all directions. The essence of the present semi-classical model is to consider all of these currents to beincoherent. This is particularly justified when largeenergy-loss events are to be described since theyefficiently create incoherent electrons. The resulting sta-tistical current distribution can symbolically be writtep as

J(E',r)=J(E', r;X der IdQ„R, R'), (2)

where r is any point within the film, X~,d o~

/d 0represents the relevant differential cross sections, and Rand R' are effective reflection coefficients at the film-vacuum and film-metal interfaces, respectively. All ofthese quantities are functions of energy.

The action of the metal is to sample out, at all energies,a fraction of these currents according to the reflectioncoefficient at the metal boundary. The transmittedcurrent is given by

I,(E)=f ds f dE'(J(E', L) n[1 —R'(E')])s, (3)

where J.n is the component of the current density at en-ergy E' normal to, and evaluated at, the metal surface(denoted by L), and ( ) s represents a statistical angu-lar average. The upper limit of the integral over energy isE because we neglect superelastic events, while the lowerlimit is —~ for the sale of generality, since it usually liesnear Vp ~ The surface integral form. ally extends to the en-tire metal surface S although it is practically confined toa small area of the order of the beam size.

The basic action of thermalization is to transfer frac-tions of the elastic current to lower energies, in a cascadeprocess, until the electrons have reached thermal ener-gies. In a thin film, the thermalization process is usuallyincomplete because an incident electron can escape thefilm after only a few cascades or no cascade at all. Thetransmitted current given by Eqs. (2) and (3) thereforepredominantly results from a limited number of energy-loss processes. As the electron energy is swept whenrecording a LEET spectrum, electron propagation ener-gies lower than the first electronic transition are quasi-elastic until a large energy loss occurs. After the elec-tronic transition, the energy-loss electron is again in aquasielastic energy region since no other electronic exci-tations are possible. As the energy increases further,double- and multiple-exciton losses become energeticallypossible as well as interband transitions. LEET spectrain these higher-energy regions exhibit very faint andbroad features which would be extremely difficult to in-terpret due to the large number of inelastic channels. Itis for this reason that LEET spectra are usually recordedat very low energies where quasielastic scattering and sin-


gle losses dominate. In the present work, our attention isconfined to single energy losses to excitons.

can be used to define an effective cross section 0 throughthe relationship

III. THEORETICAL ANALYSIS Q =no, (4)

A. Mathematical formulation

We have based our theoretical analysis on the transferequations of Michaud and Sanche ' and Chandrasekharwith the following assumptions. First, the film-vacuumand film-metal interfaces are considered to be perfectparallel planes. The z axis (Fig. 1) is normal to the planesand the film extends from z =0 (vacuum) to z =L (metal).Second, the two-stream approximation ' is introduced;i.e., electron currents flow only in the forward and thebackward z directions. The angular dependence of thescattering is taken into account by a phenomenologicalparameter, e, which we set equal to unity in order to sim-plify the algebra. The main consequence of this latter ap-proximation is to renormalize the thickness of the film orthe cross sections by a factor of the order of unity. Third,the medium is considered to be homogeneous and thescattering, whether elastic or inelastic, is described byposition-independent scattering (or transition) probabili-ties per unit length: Q (E,E' E). The—latter are definedas the probability per unit length that an electron of ener-

gy E loses an amount of energy E —E'. This quantityI

where n is the number density of molecules. In order toshow explicitly the effect of the anisotropy of thediffusion, we have kept the distinction between forward(Qf) and isotropic (Q, ) scattering in the transfer equa-tions (i.e., Qf+Q„ is the total forward scattering crosssection and Q„ the total backscattering cross section). '

The mean free path is then simply given by

I '= I [2Q, (E,E' E)+—Qf(E,E' E)]dE—' .

Finally, we have restricted the analysis to the case of adiscrete excitation spectrum for the medium. Thescattering probabilities then take the form

Q (E E' E)—Q (E)fi(E' E)

+ g Q„', (E)5(E'—E+E,),j=l

where E is the electron energy loss in channel j. Wehave a similar expression for Qf (E,E' E). —

The transfer equations of Michaud and Sanche ' thenbecome

Q„(E)J—+(z,E)+Q„(E)J (z,E)-Z

[—2QJ(E) —Qf', (E)]J+(z E)+[Q&(E+E )+Qj~ (E+E )"]J+(ZE+EJ)J

+QJ;(E+E )J (a,E+E )I,

=Q„,(E)J (z, E)+Q„,(E)J+(z,E)Z

+ g {[—2Q„;(E)—Qf;(E)]J (z, E)+[Q„;(E+E)+Qf;(E+E )]J (z,E+E )

J

+Q„',(E+E, )J+(z,E +E, ) I,

where J+(z,E) and J (z,E) represent the current densityin the forward (+z) and the backward ( —z) directions,respectively. These equations describe an infinite set ofcoupled currents where electrons cascade down throughthe various discrete energy levels. We are interested inthe simple case where electrons can lose energy only oncewithin a finite number N of inelastic channels. Equations(5) then reduce to

dJo+ = —(Q„,+Q;)Jp+ +Q,,Jp

I

where j= 1,2, . . . , X, and where we have definedJp =J*(z,E) for the elastic currents, JJ+—=JJ*(z,E EJ )— —for the N inelastic channels, Q J, =Q„,(E E)—, —QJ; =Q„;(E~E EJ), and Qfj;—=Qf;(Eq~E EJ). The-total inelastic transition probability is given by

N

Q, = g (2Q„, +Qf;)= g Q~ .j=1 j=l

Equations (6) are solved analytically with the followingboundary conditions:

—dJO = —(Q,', +Q;)Jp +Q,'Jp+

dJj+Q,',J, +Q!,J;+(Q,', +Qf—', )J;+Q,', Jp .

—dJ:Q„',J,-+Q—,',J,++(Q,', +Qf, )J, +Q„';J,+,

(6)

Jp+ (0)=J;„T;„+RpJp (0),Jp (L)=R Pp+ (L),J+(0)=R J, (0),J (l)=R'J+(L),

where J;„ is the current density of incident electrons, L isthe thickness of the film, R —=R (E E )and— .


I( Io+——(L)(1—Ro)+ g I+(L)(1 R')—

Po

(1—R')X (1—R')+ g (A, 'CJ —2C~+CJA, )

4p) Q'

where

C(~) =(1+RJ )Qji [(1—Ro)Q, —(1+Ro )Q]

+(1—R, )Q '[(1+Ro )Q; —(1—R o)Q],C~—:Q;Q J(1+R' )(1—R )+Q, Q/(1 —R' )(1+R )

+2Q~, (1—R )(1—R o )Q, Q(,

and

C]=(1+RJ}Q([(1—Ro)Q, +(1+Ro)Q]+(1—RJ. )Q J[(1+Ro )Q;+ (1—R o )Q ],

Rj =R'(E E—J) (j =0, 1, . . . , N) are the reflectioncoefficients from within the film at energy E —E (withEo=0) for the film-vaccuin interface and the film-metalinterface, respectively, and T;„—:1 —R;„ is the transmis-sion coefficient for the incident electrons. R;„differsfrom R because the angular distribution is not the samefor the incident electron beam as far for the backscat-tered electrons. The exact solution for the transmittedcurrent is, using Eq (3.},

IoT,„(1—Ro)

if E)0.We have considered only two angular distributions: uni-directional (for the incident electron beam) and isotropic(for electrons within the film). These are clearly onlyconvenient approximations. In the classical limit, i.e.,when R(E,8,$)=0 for 8&8,„and R(E,8,$)=1 for8 & 8,„,we obtain simply

(R &=0

for the incident electron and

(9a)

This simplified expression is used to estimate the scatter-ing probabilities in the quasielastic regime just below aninelastic threshold.

Even though the three-dimensional problem has beenconveniently reduced to one dimension, we still have totake explicit account of the strong dependence of thereflection coefficients on the angle of incidence withrespect to the planes of the boundaries. Indeed, even ifan electron may have sufficient energy to overcome theinner potential from within the film, its velocity normalto the surface (k, /m ') may be too small to allow the es-cape. Classically, an electron will escape the film only if8&8,„, where 8,„=

IV

I/(E+

IV

I)=C . In order

to take into account the angular distributions of the elec-trons' velocities, we defined the mean reflectioncoefficient as ( R ) to be unity for E & 0 and

(R ) = f dP f d8sin(8) f(8$)R (E8$)

Qi =2Q'. +Q;

Q =-(Q, Q, }'" E+I v, I

' 1/2

(9b)

for

A,—=exp(QL),

Q '—:(2Q,'.Q//Q; }+Qj;,1

Po= {~ '[——(1—Ro}Q' —Q;+Q]2QQ,',

X [(1—Ro)Q,,—Ro(Q;+ Q)]

+A [(1—R o)Q„', +Q,. +Q]x[(1—R, )Q„', —R,(Q, —Q)]j,

Q; ~0, po =(1—RoR o)+(1 Ro)(1—R )Qo„,L—,

p~ =(1 RJR,')+(1—RJ )—(1 R')QJ, L, j =0, 1—, . . . , N .

Physically, Q is an effective (convoluted) inelastic transi-tion probability per unit length which vanishes in thepurely elastic case and approaches Q,. in the highly inelas-tic case; po and p- are renormalization factors whichcharacterize the elastic propagation at energy E —E- ac-cording to the geometry of the film (through L in oursimple case} and the reflection properties of its boun-daries.

For purely elastic scattering (N =0 and Q, ~0), Eq. (7)reduces to

1

(1+0)'10 1

3 (1+}8)'

4 8 1 16+ (1+0)' 5 (1+P)' »—= 1 —

I 1 —exp[(E/I

VI

)'~ ]I (lob)

for electrons from within the film, where we have definedP=tan( —,'cos 'C).

for electrons from within the filin. The other extreme isthe quantum-mechanical reflection coefficients for a two-dimensional step function. Assuming plane waves andusing

4k, k,'

T=(k, +k,')'

where A k' =2m' (E—I

V I), fi2k =2m*E, k—:k,+ k )), and k —=k, +k )„we obtain

4i/E (E—I

VI

)'~(10a)E+« —

I v, I

}is&i~

for the incident electron and


As required, both the classical and the extremequantum-mechanical expressions for electron refraction[Eqs. (9) and (10)] go continuously to one near E =0.The optical potential V has its full meaning in the ap-proximation of free propagation between scatteringevents at lattice sites. As long as this approximation isvalid, and provided the scattering is predominantly elas-tic, the coherent multiple scattering gives rise to effective(energy-dependent) reflection coefficients consistent withthe band structure, i.e., Bragg reflection for E' & Vo andreflection due to mismatches between Bloch and vacuumwave vectors for E') Vo. In the limit of strongly inelas-tic propagation, the reflection properties do not reflectthe band structure, but are still governed by themismatches between bulk and vacuum wave vectors.When the electron gets trapped and then propagatesthrough hopping or tunneling processes, the optical po-tential approach breaks down. Caron et al. have shownthat in this case the electron motion is diffusion con-trolled.

The present phenomenological model can take into ac-count these various effects through the energy depen-dence of the reflection coefficients and scattering proba-bilities, which lose somewhat their original physicalmeaning to become adjustable parameters. When Vo isnegative and the classical version of refraction is used, itis adequate to have V =Vo=const, since, within theclassical picture, electrons with E'= Vo are considered tohave zero kinetic energy. When Vo is positive, however,a more complicated situation prevails owing to the coex-istence of various propagation modes. V~ should then beexplicitly energy dependent. We will nevertheless takeVz to be constant again, since any refinement on V (E)must be accompanied by appropriate changes in Q„,(E).

Finally, we take into account the finite resolution ofthe electron beam, small phonon losses, and potentialfluctuations due to imperfections in the film by perform-ing a Gaussian average on the ideal transmitted current:

-cI It

cIE2

{ar b.units}

lU LU W 4J UJI I I I

I'

0.0 0.2 O.I

E-E-Q (eV)j

(~)I I I I I

E) E2 E3 E4 Es

2-

vanishes when E &E, + Vo. Curves (a) —(c} in Fig. 2 arethe results for Vo ——0.4, —0.6, and —1.6 eV and E] ——6eV. The vibrational energy levels are considered to beequidistant and separated by 0.25 eV. Curve (d) is thenegative value of the second energy derivative of curve(c). The slow decrease in transmitted current up to thefirst 1oss, at E, +Vo, is due to refraction. A steplikestructure appears above the first threshold (i.e., above en-

ergy E, + Vo) in curve (c). It results from a transfer of afraction of the incident current to lower energies causedby the inelastic processes as expected from the Vo mecha-nism. One finds, however, that the steps have differentamplitudes despite the mutual equality of the inelasticcross sections Q . This saturation reflects the nonlineari

ty between the total inelastic cross section and transmit-ted current. Inelastically scattered electrons can escape

(1,(E))= J dE I,(E )-"-""

~~swhere S is related to the full width at half maximum(FWHM) of the energy distribution: dE =2S&ln2.

B. Application to vibronic excitation near threshold

(nA)

0-

5.4

I

6.4

5 6 7

ELECTRON ENERGY (eV)As an example, we consider an electron of initial ener-

gy E having a finite probability to excite near thresholdthe lowest five vibrational levels of an electronically excit-ed state of a molecule within a film. The energy of eachvibronic level is E (j= 1 —5) and the inelastic scatteringprobability into these channels, Q, has the energy depen-dence shown in the inset (e) of Fig. 2.

For simplicity, the elastic scattering probability[Q„,(E)=0.1 A ] and the reflection coefficient at thefilm-metal interface [R'(E}=0.5] are chosen to be in-dependent of energy. The reflection coeScient at thefilm-vacuum interface obeys classical electron refraction(Eq. 9). The calculation is performed in the idealized sit-uation where inelastic electrons have no access to levelsin the gap, i.e., the transition probability in channel j

FIG. 2. Model calculations reproducing low-energy electrontransmission (LEET) spectra of hypothetical films. The incidentcurrent is Io ——3 nA, the film-metal reflection coefficient isR'=0. 5, and the thickness of the film is 22.5 A. The reflectioncoefficient at the film-vacuum interface (z =0) obeys classicalelectron refraction. Five inelastic channels have been includedat energies E, (j = 1 —5) equal to 6.00, 6.25, 6.50, 6.75, and 7.00eV, respectively. Their excitation function near threshold has

othe shape shown in {e), with Q, X0.2 A =Qf', ——2Q,', . Theelastic scattering probability is constant, Q,,(E}=0.1 A . Thecalculation has been done for three dift'erent films with Vo ——(a)+ 0.4 eV, (b) —0.6 eV, and (c) —1.6 eV. The second energy

derivative of (c) is given in {d) and was obtained after smoothingwith a Gaussian distribution having a full width at half max-imum AE of 80 meV.


the film to vacuum whenever E )E, giving rise to theseries of curves downward steps beginning at 6 eV. Thisstructure due to a "reflectivity" mechanism is character-ized by peaks of fairly even amplitude positioned at F. .It can easily overlap the previous one [Fig. 2(b)] for morepositive Vo. It is absent in Fig. 2(a} because in this calcu-lation Vo is positive and transitions to levels in the gapare forbidden. Finally, Fig. 2(d) shows the differences inthe doubly differentiated (DD) LEET signatures of thetwo mechanisms just discussed. The structure due totransitions to the bottom of the conduction band (the Vo

mechanism) is not necessarily representative of the transi-tion strengths but the minima in the DD LEET spectraprecisely define the energy E, + V0. Such minima shouldtherefore allow a precise determination of Vo values (i.e.,

Vo=E E/} si—nce E~ can be known from spectroscopicdata and the electron energy in vacuum can be deter-mined within +0. 1 eV from the LEET spectrum. Com-pared to negative Vo values [Figs. 2(b) and 2(c)] thetransmission is only marginally enhanced by energylosses when Vo is positive.

It turns out that in most of the published LEET andDD LEET spectra of condensed mole-cules'" ' ' ' ' ' the vibronic structure is notresolved. The energy dependence of the transmittedcurrent exhibits broad maxima with no superimposedvibronic features. We show as examples in Fig. 3 the

LEET spectra for hexadiene, ' fluorobenzene, cyclohep-tatriene. ' Above 2 eV the spectral features are created bythe mechanisms just described but individual vibrationallevels belonging to the excited electronic states are notobserved in these curves and their energy derivatives. 'Still, the energy of the first vibronic state (i.e., E, + Vo)should lie close to the onset of the lowest-energy inelasticstructure. Bader et al. ' have used this criterion to ob-tain V0 values in molecular N2 CO, and 02 films whereasother authors have taken maxima in the first energyderivative of the transmitted current (dI, /dE) torepresent E, + V0. If no vibronic structures were presentin the dc curves of Fig. 2, we would expect the maxima indI, /dE to lie somewhere between the threshold energy[i.e., 6.4, 5.4, and 4.4 eV for curves (a), (b), and (c), respec-tively] and the following maximum in I, . Thus the max-imum in dI, /dE is not located at E, + Vo. For example,in the fluorobenzene spectra in Fig. 3, the first inelasticonset lies around 3.1 eV whereas the maximum slope inthe rising portion of the following peak (i.e., the max-imum in dI, /dE) lies at 3.8 eV, with respect to the energyof the vacuum level. Thus, in this case, taking the max-imum in dI, /dE to represent the onset of the first elec-tronic state would result in a value of 0.7 eV too high inthe determination of Vo.

IV. APPLICATION TO N2 AND CO FILMS

A. General considerations

0

I

In a recent high-resolution LEET experiment, weresolved the vibronic structure of several exciton states ofcondensed Nz and CO by measuring the second energyderivative of the transmitted current. The dc and DDLEET spectra recorded with ten-monolayer films areshown in Figs. 4 and 5, respectively. The rise near 7 eV

ZOCCC

VCl

!

Qal

Z

QI-

zz~~Ktx: KUJ Du H—Qde

g) ~iK

gM0~zQQ t-z

0 2 4 6 8 10 12 14 16 18ELECTRON ENERCY (eV)

FIG. 3. LEET spectra of hexadiene, fluorobenzene, and cy-cloheptatriene films taken from Refs. 1 and 35. The film thick-

0ness and temperature were approximately 100 A and 100 K, re-spectively. Above 2 eV, the structure in these spectra is due toelectronic transitions having a strong probability to be pro-duced by electron impact near threshold.

0 1 2 3 4 5 6 7 8 9 10 11 12 13ELECTRON ENERGY teVj

FIG. 4. LEET spectrum (bottom curve) of N2 on Pt togetherwith its doubly differentiated (DD) LEET spectrum (top curve).The numbers indicate the amplitude factor of the various sec-tions of this latter spectrum. The modulation amplitude was 60mV. The film thickness and temperature were ten monolayersand 18—20 K, respectively (Ref. 5).


l- I

ZUJKKO0

KI-O

z

!

ClDOQOiz

0

x50

't 2 3 4 5 6 7 8ELECTRON ENERGY (eVj

FIG. 5. LEET spectrum (bottom curve) of CO on Pt togetherwith its DD LEET spectrum (top curve). The modulation am-plitude was 75 mV. The film thickness and temperature wereten monolayers and 18-20 K, respectively (Ref. 5).

in N2 and 5 eV in CO in the dc curves is due to energy-loss electrons which increase the transmitted current.The vibronic structure is clearly apparent in the DDcurves shown in the upper portion of Figs. 4 and 5.Features below the onset of electronic excitation (i.e., 6eV for CO and 6.5 eV for N2) have been discussed previ-ously. In the 0.5-2-eV range they are due to transientanion formation leading to strong vibrational excitationof the ground state of the molecules. ' Above 2 eV, upto the electronic excitation threshold, the structure hasbeen ascribed' to quantum size effects indicating that inthis region, the coherence of the electron wave ispreserved, at least partially. When the excitonic channelsbecome accessible, quasielastic scattering competesdirectly with electronic transitions, loses its preponder-ance, and the coherence is more efBciently eliminated.Thus the foregoing arguments favor the applicability ofour model to the case of CO and N2, since the electronpropagation is quasielastic below the inelastic thresholdand highly inelastic above.

In view of the fact that the electron-energy loss (EEL)spectra of physisorbed N2 and CO are similar to thegas-phase EEL spectra up to about 30 eV, but for the to-tal absence of Rydberg states, it seems reasonable to usecross sections consistent with gas-phase values. It is evenmore so, since the structure observed in the DD LEETspectra originate from the same electronic transitionsknown to possess sharp thresholds in the gas phase. Theanisotropy of the diffusion is another important parame-ter which cannot be uniquely determined from a simple

investigation. We have therefore arbitrarily set2Q&~, ——Q~, in order to compensate for the fact that weused @=1. For consistency, all electronic transitionswere given the same anisotropy.

Since we are investigating mechanisms for which themetal substrate plays no special role, its reflectioncoeScient was taken to be constant [R'(E)=0.5]. Thisapproximation is not critical and can easily be remedied.The choice of the 61m-vacuum reflection coeScient, onthe other hand, is more critical, especially in the vicinityof the vacuum level (E'—=0). We have systematically usedthe extreme quantum-mechanical version for refraction;i.e., R;„and R were respectively given by Eqs. (10a) and(10b). Whether we take the classical instead ofquantum-mechanical version, influences mostly the mag-nitude of the high-frequency fluctuations in the transmit-ted current. The quantum-mechanical version appears tobe closer to reality even if it disfavors the appearance offine structure, thus adding credibility to our analysis.

B. Inelastic threshold of CO

The inelastic regime in CO (Fig. 5) starts at about 5.5eV and the inelastic threshold is dominated byX 'X+(v =0)~a II(v =0, 1, . . . ) transitions for energiesup to about 7 eV. The data also indicate that V0 is neg-ative (Va = —0. 5 eV) and that the shape of the DD LEETstructure is consistent with the V0 mechanism. UsingEq. (8), we find from the dc LEET spectrum thatQ„,=-0.01 A ' in the 0—2-eV energy region. Theanalysis is not so simple above 5.5 eV owing to the pres-ence of the inelastic channels. To estimate Q„, in that re-gion, we have used the fact that the propagation is quasi-elastic up to VD+ E& ——5.5 eV and extrapolated to higherenergies the Q„, obtained in V0+E, with the aid of Eq.(8), namely Q„,=0. 1 A '. In the calculations, we havetherefore posed Q„,(E')=0.1 A ' for E between 5 and 7eV. There is no need to know Q„, between 2 and 5 eV.Such a crude approximation for Q„, exploits the fact thatthe second energy derivative is not sensitive to slow vari-ations of Q„. Moreover, at energies below the vacuumlevel, Q„, has no effect whatsoever on the transmissionsince all inelastic electrons transmit to the metal at suchenergies.

The inelastic transition probabilities were representedby the following simple analytical expression:

Q~=QQa (1—expI —[(E E, —V0)/Es]~) )—for E & E + VQ and Q/=0 for E ~E, + V0 where p, E&,Q0, and a. are adjustable parameters: p governs the cur-vature at threshold, Ez controls the steepness of the riseabove threshold, and Qaa gives the strength of the tran-sition in each channel (aj & 1). This convenient form hasthe advantage of being consistent with the cross sectionsobserved near threshold in the gas phase whichbehave as Q~ =—QDQ (E E —VDY. A starti—ng estimatefor Q0 is provided by the thickness dependence of theLEET spectra and by the known values in the gas phase:Q0=0.005 A ' [i.e., crJ, =3X10 ' cm using Eq. (4)]. Inorder to further reduce the number of adjustable parame-


ters, we have kept a constant ratio between the a, whichwe adjusted to match the gas-phase values near thresh-old 38,39

A direct inspection of the CO dc LEET spectrumshows that the inelastic threshold is smooth; so, thenear-threshold excitation functions for the excitonic lev-els cannot fluctuate rapidly on the scale of a vibronicquantum. The weakness of the DD LEET structure isalso inore consistent with a relatively slow rise of Q~i

above threshold. The calculations indicate that the twotypes of LEET spectra (i.e., dc and DD) are partiallydecoupled. On one hand, the inelastic threshold currentgenerated by the model varies slowly with Q, and itsmagnitude (i.e., the first maximum in transmitted currentabove the onset of the inelastic threshold}, which depend-ed upon the asymptotic values for Q, , was reduced by thesaturation effect discussed earlier (Sec. III) and by theleakage to vacuum of an appreciable fraction of theinelastically scattered electrons whenever E & E in anygiven channel. On the other hand, the second energy-derivative structure was governed by the shape of QJi in

the vicinity of each threshold. The most satisfactory re-sults were obtained with excitonic thresholds spread overseveral hundreds of meV, as in Figs. 6(a) and 6(b). Thetransmitted current, and its second energy derivative,corresponding to curve (a) is depicted in Fig. 7. Curve (b)has been found to yield very similar results. Even thoughthe DD LEET spectrum turns out to be sensitive mainlyto the opening of new inelastic channels, it is worth em-phasizing that the onset of a given channel has to be pro-nounced in order to yield any observable structure.

The structure due to the reflectivity mechanism did notgive rise to strong features in the calculated spectra, inaccordance with the data. These weak features may pos-sibly explain the weak structure beyond 6.5 eV in Fig. 5.Indeed, the observed spacing (=0.21 eV) (Ref. 5) is inmuch better agreement with the vibronic spacing of the

01"

O t 2RESIDUAL ENERGY E' {eV)

FIG. 6. Cross sections used to calculate LEET spectra forCO in the neighborhood of the a 'H inelastic threshold. (a) En-ergy dependence of the inelastic transition probabilitiesQ/=5Qfj;=5Q, ', /2. (h) Another Q~ which yields results similarto those of curve (a). Both (a) and (b) are given in terms of theresidual energy E'=E —E, .

+Q ~ e

LIIa

I

0

~ ~

a'l1 V=O & 2 3~5

~ w e y ~ s T % ~

S 6ELECTRON ENERG&

FIG. 7. LEET spectra for CO in the vicinity of the a H in-elastic threshold obtained with the parameters of Fig. 6 and oth-ers given in the text. The DD LEET spectrum was obtainedafter a Gaussian smoothing of the transmission spectrum withhE =80 meV. The film thickness was taken to be 15 A.

a Il state (=0.21 eV) than that of the next excited state,a' X+, with Ace=—0. 14 eV.

C. Inelastic threshold of N2

According to the example considered in Sec. III B, thedominant features in the DD LEET spectra of N2 be-tween 6 and 10 eV in Fig. 4 appear to be caused by thereflectivity mechanism. It is, however, not possible tospecify whether the structure originates in transitions inthe gap (i.e., incomplete destructive interference due totunneling} or in the trapping of the inelastic electron fol-lowed by diffusive propagation. The former process per-tains to the near-surface region while the latter occursthroughout the bulk of the film. The bottom of the"conduction" band in solid N2 is suSciently far abovevacuum (Vo =+0.8 eV) (Ref. 16) for transitions to traplevels near E'=0 to occur with a much reduced probabil-ity with respect to transitions in the neighborhood of Vp.In other words, the modulation of the amplitude of thetransition caused by the band structure results in a veryweak transition probability to levels near E'=0 unlessthe transition amplitude

~

T~

takes large values nearthreshold. The fact that electronic transitions responsi-ble for the DD LEET structure are known to possessstrong cross sections near threshold in the gas phase sug-gests that the mechanisms involved persist in the con-densed phase.

The finite transition probability to levels below Vp andits rapid decrease is taken into account in our calcula-tions by a Q J of the form

Q/(E) =(Qoa )/I ]+exp[( Vo+E E)/Eq] I, —

where Qo, a, , and Es are again adjustable parameters

MECHANISMS PRODUCING INELASTIC STRUCTURES IN. . . 11 127

having the same meaning as in the calculation for CO.This expression is clearly highly approximative, but it hasthe merit of appealing to a minimum number of parame-ters. In order to take some account of the fact that thepropagation in the gap is not the same as in the "conduc-tion" band, we have used an elastic scattering probabilitygiven by

Q„,(E')= A (1—1/I 1+exp[( Vo E'—)/Ez] j )+8at the final energies, E'=E E, .—We have verified thatthe exact values for constants 3 and B have little efFecton the results and have taken B =0.01 A ' and A =0.5A '. As already mentioned, Q„,(E') no longer representsa scattering probability when E'& Vo, but rather adiffusion parameter. We have used V~(E) =const= —0.8eV. Thus all propagation modes are described by asingle parameter.

The elastic cross section [Q„,(E)=0.2 A ] at the ini-tial energies (E&6 eV) has been determined using thesame procedure as for CO. The calculations included 29transitions: X 'Xz+(v=0)~ A X+(u =0, . . . , 15) andX'X+(v =0)~8 II (u =0, . . . , 12). These are thestates identified in the experimental DD LEET spectrawhich are known to posses strong threshold cross sec-tions in the gas phase. ' The a, were again kept con-stant and close to gas-phase values near threshold. ' Theshapes of the relevant excitation functions are shown inFig. 8.

The result of a calculation with Qo ——0.006 A ' andEz ——0.2 is shown in Fig. 9. The major discrepancy be-tween this result and the experimental spectra is the pres-ence of a trough near 8.5 eV caused by the limited num-ber of inelastic transitions included. Indeed, the vibronicprogression of the 8' h„state overlaps the 8 H se-

4 Fs s s

0 0.5 18 1.5RESIDUAL ENERGY E'(eV)

FIG. 8. Cross sections for calculating the LEET spectra ofN2 in the neighborhood of the 8 Hg inelastic threshold. (a) En-ergy dependence of the inelastic transition probability with

Q, XO 01 A =2Q. j; /a, = Q,';/as, r expressed as a function of theresidual energy, E'=E —E, . (b) Energy dependence of the elas-tic scattering probability Q,, between 0 and 1.5 eV.

( ll

~Op. 3 l I I I t Ill/tB T7s v= 0 1 2 3 4 5 6 7 89

0"6 7 8 9

ELECTRON ENERGY (eV)

FIG. 9. LEET spectra obtained from calculations for N& inthe vicinity of the 8 H~ inelastic threshold using curves (a) and(b) in Fig. 8. The DD LEET spectrum was obtained after aGaussian smoothing of the spectrum with EE=SO meV. Thethickness of the film was taken to be 25 A.

quence, and contributes to the total inelastic cross sec-tion. The 8' h„state is therefore a necessary ingredientto fit the data. The trough is also partially caused by theconstant V used in the calculation, giving rise to a (R )above Vo smaller than expected considering thesignificant mismatch between the long-wavelength Blochfunctions near E'= Vo and the vacuum free-electronwave function.

D. Effect of electron resonances on LKET spectra

The number of inelastic vibronic channels is a rapidlyincreasing function of energy above the first exciton,especially above 9.5 eV (more than 100 above 11 eV) inN2. In view of the saturation effect discussed previously,we would expect no detectable fine structure beyond 10eV in Fig. 4. Surprisingly, evenly spaced structure is ob-served between 11 and 14 eV at all thicknesses investigat-ed. The beginnin~ of this structure was correlated withX 'X+(v =0)~C II„(v =0, 1,2, 3) transitions, whichare known to possess large cross sections near thresholdin the gas phase. ' The re6ectivity mechanism maystill be the basis for the appearance of this structure. Itrequires strong transition amplitudes to be operative inthat energy range, in accordance with the gas-phase data.

The structure appearing beyond 11.5 eV, however, can-not be correlated with any of the excited states probed byhigh-resolution EEL measurements in Nz 61ms. Thus itdoes not seem to originate from the opening of new vib-ronic channels and is more likely to be caused by rapidfluctuations of the cross section in one or a few of the in-elastic channels available. Such rapid variations of thecross section at energies difFerent from those of electronictransitions in N2 can be caused by the formation of atransient anion N2 (i.e., an electron resonance). In thiscase, the oscillatory structure is due to vibrational motionof the anion. Since Rydberg states are absent in con-


densed N2, the transient anion has to be formed in thefield of a valence excited state. The resogance is thencomposed of the scattered electron trapped by the posi-tive electron affinity of a valence excited state (i.e., acore-excited resonance). Shape or single-particle reso-nances involving no electronic excitation at the target sitecannot be considered since their lifetimes in the 11—14eV energy range are too short to produce vibrationalstructure. Gas-phase data indicate the presence of acare-excited resonance in the 13-eV region which decaysinto a weak unidentified valence state near the C 0„state. The spacing between the peaks in the vibrationalstructure of this resonance agrees with the spacing ofthe oscillatory structure observed in the DD LEET spec-tra between 11.5 and 14 eV.

To investigate the effect of a long-lived resonance de-caying in an inelastic channel on the line shape of DDLEET spectra, we have performed the following idealizedcalculation. We included 51 loss channels to account forthe saturation effect. All channels were separated by 200meV, beginning at 6 eV and ending at 16 eV. The 50nonresonant inelastic channels (j&25} had constant-magnitude excitation functions above a well-definedthreshold [dashed line in Fig. 10(a)], whereas the resonantchannel (j =25, E =11 eV) only possessed resonancestructure starting 2 eV above threshold [solid line in Fig.10(a)]. The resulting DD LEET spectrum is shown inFig. 10(b). As expected, each nonresonant channel givesrise to a narrow peak (t'} whenever E'=E, while the res-onance channel does not (t). However, the absence of awell-defined threshold at E'= Vo in the resonant channel

leads to the addition of a broad maximum [s, Fig. 10(b}]which modulates the sharp threshold peaks t'. This weakartifact (s) disappears when the resonant channel is givenan excitation function similar to that of the nonresonantchannels. Finally, Fig. 10(b) indicates that the resonancestructure in Q; easily reemerges in the DD LEET spec-trurn, in spite of the saturation effect.

The existence of a core-excited resonance associatedwith a valence state in the condensed phase is not totallysurprising. In recent electron stimulated desorption ex-periments, many resonances of this type have been ob-served and found to decay via dissociative attachement(DA) in condensed 0& ' CO ' NO, , C12, N&O,and C„Ht„+z (n =1, 2, 4, 5, and 6). ' Moreover, thenumber of resonances decaying by DA was found to belarger in the condensed phase due to the violation of theselection rule X ~

~

~X+ which forbids ( —)~~(+)transitions in the electron-single-target frame of refer-ence. This is the first time, however, that a core-excitedresonance is found to decay into an excitonic state. Thenegligible change in the spacing of the vibrational struc-ture of the transient anion in going from gas to con-densed phase implies that its lifetime is only marginallyaffected by the neighboring molecules. Nevertheless, weexpect some energy relaxation owing to the dielectricresponse of the medium. If the first well-defined peak inthe experimental DD LEET spectra, at 12.2 eV in Fig. 4,is identified with the first peaks of the resonance struc-ture, one finds a relaxation shift of 0.8 eV. This value isin agreement with the relaxation energy (0.7 eV) of theIIg shape resonance near 2 eV in N2 films.

0,5)

-d2 ~I

dE2 I, o

(V ')-0,4

1

I I I I I I I

11 12 13 14

ELECTRON ENERGY (eV)

(0)/

/

//

0 I I I I I I

0 1 2 3RESIDUAL ENERGY E' (eV)

0,01-J

FIG. 10. Calculation of the inelastic portion of LEET spectrafor an idealized multiple-inelastic-channel system where one in-

elastic channel has a resonance above threshold. The parame-ters are similar to those of the previous calculations (Figs. 6—9)except that the classical version of electron refraction has beenintroduced in the formulation. (a) Energy dependence of thetransition probability of the resonant channel (solid line) and ofthe normal channels (dashed line). (b) Second energy derivativeof the transmission spectrum, obtained after a Gaussiansmoothing with hF =80 meV.

V. CONCLUSIONS

We have developed a two-stream multiple electronscattering model to describe LEET features associatedwith electronic transitions. Such transitions were foundto create broad maxima in transmission over which step-like structure is superimposed. This latter, which is usu-ally not resolved in transmission experiments, is presentwhen the excited state involves strong vibronic transi-tions near threshold. The spacing of the vibronic levelsallow an unambiguous determination of the electronicstate involved. The visibility of these sharp vibronic tran-sitions is strongly enhanced by recording the second ener-gy derivative of the transmitted current. The amplitudeof the structure is both the dc and DD curves, however,is not representative of the transition amplitudes due to asaturation mechanism. When the Vo level is well definedfor a given film, the onset of the lowest-energy vibronicfeature faithfully represents the onset of the first electron-ic transition. Thus it can serve to obtain accurate Vovalues provided that the energy of the vacuum level,which establishes the energy scale, can be accuratelydetermined in a high-resolution experiment. ' When novibronic structure appears in the spectrum, the onset ofthe first broad inelastic maxima can still serve as a refer-ence to determine the energy of the lowest-energy elec-tronic transition but the accuracy is lower. As a generalrule, the inelastic features in a LEET spectrum must be


properly correlated with the corresponding electronictransitions to obtain meaningful Vo values. This is usual-

ly done by comparison"' ' with gas-phase thresholdexcitation spectra. Such a correlation is not necessarilyobvious since only transitions with a strong amplitudenear threshold produce observable features. The visibili-

ty of sharp features is enhanced in DD LEET spectra,but in this case care must be exercised in distinguishingbetween phenomena other than sharp threshold capableof causing sharp current variations. For example, theformation of electron-exciton complexes ' or core-excited resonances has been found to produce sharpfeatures in transmission. Indeed, with the modeldeveloped in this article, we have shown that such transi-tory negatively charged states decaying into excitonicstates produce sharp oscillatory structure in LEET when

they possess vibrational structure. When their lifetime istoo short for the vibrational structure to develop, theystill can produce sharp features in LEET spectra as previ-ously observed in hydrocarbon films. '

More specifically, the present work has shown that theexcitation function for the X ' X+~a H transitions incondensed CO and the X 'Xg+ ~ A X+ andX 'X+ ~8 II in condensed Nz possess large values nearthreshold (Qj&0. 1 A ' corresponding to o'; &5X10cm ) which are consistent with those found in the gasphase. Vibrational structure in the excitation functiondue to the formation of a transient anion reemerges in the12-14-eV portion of the LEET spectrum of N2. This res-onance is shifted to lower energy by 0.8 eV with respectto its gas-phase value but its lifetime does not appear tobe appreciably modified. This anion state could easily beidentified as of the core-excited valence type since onlyvalence states survive as possible parent neutral states inthe condensed phase.

More thorough investigations of inelastic low-energy

electron interactions in condensed matter as extensions ofgas-phase behaviors could be undertaken with the presentmodel. An obvious first step would be to perform a simi-lar analysis which would include various refinements onR (E), R'(E), cross sections, electron self-energies, andanisotropy. These improvements are nevertheless thwart-ed by intrinsic limitations such as the double-stream ap-proximation or the approximation of depth-independentinelastic scattering cross section, which are appropriatefor transitions to localized levels but not for transitions toevanescent waves. More stringent is the problem of thecross sections which represent a large number of un-known physical quantities, especially at very low energies(i.e., in the quasielastic energy region) where coherentphenomena and collective effects are expected to greatlyperturb the gas-phase mechanisms. More quantum-mechanical models should ultimately be applied. Thesemicroscopic models should include, at least an energy-dependent self-energy (or equivalently a relaxation shiftof the scattering strengths) and have built in the effect ofcoherent and incoherent multiple scattering on the ob-servable cross sections. Account of the collective aspectof the excitations, such as the dispersion of the excitons,and of the anisotropy of the scattering, which certainlydiffers from the gas phase, should be introduced in theformalism. Systematic measurements of the excitationfunction of various vibrational and excitonic levels ofcondensed molecules could provide a valuable guide tothe development of more sophisticated theoretical mod-els.

ACKNOWLEDGMENTS

We wish to thank Thomas Goulet and Mare Michaudfor helpful comments and valuable discussions. Thisresearch is sponsored by the Medical Research Council ofCanada.

'Present address: Asea Brown Boveri Corporate Research,Baden-Dattwil, CH-5405, Switzerland.

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L. G. Caron, G. Perluzzo, G. Bader, and L. Sanche, Phys. Rev.B 33, 3027 (1986), and papers cited therein.

G. Perluzzo, G. Bader, L. G. Caron, and L. Sanche, Phys. Rev.Lett. 55, 545 (1985).

4E. Keszei, J.-P. Jay-Gerin, G. Perluzzo, and L. Sanche, J.Chem. Phys. 85, 7936 (1987).

5R. M. Marsolais, M. Michaud, and L. Sanche, Phys. Rev. A 35,607 (1987), and citations therein.

T. Goulet, V. Pou, and J.-P. Jay-Gerin, J. Electron Spectrosc.Relat. Phenom. 41, 157 (1986), and citations therein.

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B. T. Jonker, N. C. Bartelt, and R. L. Park, Surf. Sci. 127, 183(1983).

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' K. Hiraoka and M. Nara, J. Phys. Chem. 86, 442 (1982).' K. Hiraoka and M. Nara, Bull. Chem. Soc. Jpn. 57, 2243

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For application to high-resolution electron-energy-loss spec-troscopy, see L. Sanche and M. Michaud, Phys. Rev. B 30,6078 (1984). For application to electron stimulated desorp-tion see L. Sanche, Phys. Rev. Lett. 53, 1638 (1984).T. Goulet and J.-P. Jay-Gerin, Radiat. Phys. Chem. 27, 229(1986).M. E. Harrigan and H. J. Lee, J. Chem. Phys. 60, 4909 (1974).U. Fano and J. A. Stevens, Phys. Rev. B 34, 438 {1986).See, for example, A. Messiah, Mecanique quantique (Dunod,Paris, 1964), Vol. II.

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34P. A. Wolff, Phys. Rev. 95, 56 (1954).L. Sanche, J. Chem. Phys. 71, 4860 (1979).L. Sanche, G. Perluzzo, and M. Michaud, J. Chem. Phys. 83,3837 (1985).H. H. Brongersma, A. J. H. Boerboom, and J. Kistemaker,Physica 44, 449 (1969).N. Swanson, R. J. Celotta, C. E. Kuyatt, and J. W. Cooper, J.Chem. Phys. 62, 4880 (1975).J. Reinhardt, G. Joyez, J. Mazeau, and R. I. Hall, J. Phys. B5, 1884 (1972).

~M. Michaud (private communication).

R. I. Hall, J. Mazeau, J. Reinhardt, and C. Schermann, J.Phys. B 3, 991 (1970); P. Hammond, G. C. King, J. Jureta,and F. H. Read, ibid. 20, 4255 (1987).

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Mechanisms producing inelastic structures in low-energy electron transmission spectra