Kuhr Fitting 1999

8/3/2019 Kuhr Fitting 1999

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Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 257273

www.elsevier.nl/locate/elspec

Monte Carlo simulation of electron emission from solids

*J.-Ch. Kuhr, H.-J. Fitting

Physics Department, Rostock University, Universitatsplatz 3, D-18051 Rostock, Germany

Received 3 May 1999; received in revised form 16 July 1999; accepted 17 August 1999

Abstract

Electron emission and spectroscopy has been simulated by a new Monte Carlo program which has been adapted especially

to low energy electron scattering. The underlying physic model in based on elastic Mott cross sections and inelastic losses

with full dispersion DE5 hv(q). Charge carrier multiplication by secondary electron creation and subsequent cascading

processes have been included. Surface effects like surface plasmons and the quantum mechanical surface transmittivity have

also been taken into account. Results are obtained for the materials Be, C, Al, Si, Ag, Au, and SiO . They include energy2

spectra of secondary and characteristic electrons. We find that attenuation lengths and related escape depths approach the

inelastic mean free path l only in higher electron energy regions; below 100 eV they drop down to roughly 20% of l . Thein in

results should find application in spectroscopic microscopy by means of low energy electrons in the sub-keV range. 1999

Elsevier Science B.V. All rights reserved.

Keywords: Electron emission; Low energy scattering; Attenuation length; Monte Carlo simulation

1. Introduction second and third step the transport and the surface

escape processes of free electrons.

An essential quantity in electron spectroscopy isElectron emission and spectroscopy is distin-

the inelastic mean free path l (E) depending on theinguished by and named mainly after its first step ofemitter material and the electron kinetic energy E.

excitation, the kind of free electron generation.Experimentally, the overlayer method has often been

However, afterwards, the motion and the transport ofused to determine this mean distance between two

electrons still within the emitter is subject of com-inelastic interactions, [14]. In this method, the

mon interaction of the excited electrons with specific

interesting material A is deposited with varyingscattering modes of the material. The emission overthickness d onto the bulk substrate B. Electrons with

the surface barrier into the vacuum follows as thecharacteristic energy E are excited in the substrate B,

third and final process. We want to focus oure.g. Auger (AES) or photoelectrons (XPS, UPS), and

attention mainly to the common processes of theare counted by an energy-selective analyzer under a

fixed take-off angle a. Then the signal intensity is

given as a function of the overlayer thickness d in*Corresponding author. Tel.: 149-381-498-1646; fax: 149-381-

form of an exponential law498-1667.

E-mail address: [email protected] ( H.- d]]]N(d) 5N(0)exp 2 (1)J. Fitting) H Jl (E)cosaat

0368-2048/ 99/ $ see front matter 1999 Elsevier Science B.V. All rights reserved.

P I I : S 0 3 6 8 - 2 04 8 ( 9 9 ) 0 0 0 8 2 - 1


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258 J.-C. Kuhr, H.-J. Fitting / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 257273

where, in general, l is the attenuation length getting more information about the secondaryatincluding either the inelastic as well as the elastic electron excitation in experimental comparison

scattering events and containing the projections of with coincidence spectroscopy.

the respective straggling paths in a certain direction,

given here by a. Thus the program could be a powerful tool forIn Eq. (1) the surface barrier influence is neglected modelling spectroscopic microscopy with charac-

and the attenuation is referred to the stream of teristic electrons.monoenergetic (characteristic) electrons with energy

E. In the limit of vanishing elastic coupling we may

insert the inelastic mean free path l into Eq. (1).inTaking into account the escape over the surface 2. Theoretical background

barrier we get the mean escape depth l of emittedeselectrons. However, the mean escape depth l is The theoretical background is described in morees cmostly understood as the mean escape depth of all detail in Refs. [18,19]. For the elastic scattering we

emitted electrons integrated over the hemisphere 2p, use Mott quantum cross sections from Czyzewski et

i.e. over all emission anglesa

, see e.g. [5]. al. [20], which the authors have calculated from theThus, the interaction of elastic and inelastic scat- partial wave phase shifts of Ref. [21]. The results

tering events becomes very complex and should be confirm the well-known fact that the wave me-

solved numerically, e.g. by Monte Carlo calculations. chanical influence becomes stronger with increasing

Werner and coworkers [6,7] have performed this task atomic number Z and decreasing electron energy E.

analytically as well as by Monte Carlo simulations Moreover, the backscattering is increased with re-

and present a comprehensive data set for 45 elements spect to the conventional 1st Born approximation.

[8]. A continuation and survey of that work is given The inelastic scattering is schematically demon-

in [9]. strated in Fig. 1 with the energy loss and related

Including the full dispersion relation in the low momentum change of the primary electron and, on

energy electron scattering only very few papers have the other hand, the energy and momentum transfer to

been published: a very early Monte Carlo simulation the excited secondary electron.

for secondary electron emission of Ganachaud andcoworkers [10,11], then a fundamental series of MC

papers of Ding and Shimizu [1216], later on MC

calculations of the Salvat group [17].

In the present paper we show the usefulness of the

MC method to obtain spectroscopic information in

conjunction with spatial resolution, predominantly

demonstrated in the previous first part [18]. There,

the program has been developed especially for low

energy electron scattering. Apart from relativistic

Mott cross sections for the elastic scattering the main

feature of the inelastic scattering is the momentumdependent dynamic form factors S(q, v). Thus, the

electron energy range extends from several keV to

energies of about 10 eV, i.e. just above the vacuum

level and covers the spectra of elastic characteristic

electrons as well as of inelastically emitted electrons.

Next application steps of the present low energy MC

program are scheduled for:Fig. 1. Allowed inelastic interactions in a 1-dimensional energy

loss-momentum space presentation for the primary (left) and checking the dispersion relation of different ma- excited secondary (right) electron, respectively. hv 2maximum

max

terials energy loss and transfer.


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J.-C. Kuhr, H.-J. Fitting / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 257273 259

An analysis of the density of states of occupied

bands shows groups of differently strong bound

electrons [22,23]. Within the framework of the

Lorentz model we distinguish groups of electrons nj

with nearly the same binding energy hv whichojcorrespond each to a certain oscillator.

Then they contribute to the complex electronic

susceptibility

2vpj

]]]]]]x q,v 5s d j 2 2v q 2 v 2 ivG qs d s doj j

29e n fj j

]] ]]]]]]5 ? (2)S D 2 2e m v q 2 v 2 ivG q0 s d s doj jHere n is the electron concentration involved andj9f their oscillator strength of the interband transition.j

9The parameters f , G and v may be found by j j ojapproximation to the (experimentally) given optical

loss function for q 5 0. The phase-conserving com-position of the several oscillators leads to the energy

loss function

2 1 2 1]] ]]]]Im 5 Im (3)H Je q,vs d 1 1Ox q,vs d5 6j

j

Fig. 2. Optical loss function (q 5 0) of amorphous carbon accord-The loss function (3) has its maxima at fre-ing to the Lorentz model fitted with three oscillators to thequencies where the complex dielectric permittivityexperimental function of Ref. [24] (above), and its dispersion

vanishes, i.e. e v 5 1 1 x v 5 0. This justifies thes d s d [Eqs. (4) and (5)].interpretation of the interaction as a collective longi-

tudinal density oscillation of bound electrons, i.e.i.e. creation of Compton-like electrons with "v(q) 5bound plasmons. In Fig. 2 as an example, the energy

2 2" q / 2m. Thus a5 1/2 is to be chosen.loss function of carbon [24] is presented approached

Moreover, a further method is given by theby three oscillators and demonstrating the appear-modified Drude Lindhard modelance of a p-plasmon at 5 eV and a (p1 s)-plasmon at

22 eV. The momentum dependence is obtained from 1 1]] ]]]Im 2 5O Im 2 (6)the optical loss function (q 5 0) and its quadratic H J H Je q,v 1 1 x q,vs d s djjdispersion

or the Lindhard function e (q,v) and the Mermin2 L" 2 formalism]"v q 5 "v 0 1 a q (4)s d s d0 0 m

e q,v,t 5 1s dMwith the spreading1 1 i/vt e q,v1 i/t 2 1s d s df gL2

]]]]]]]]]]]]]1G q 5 G 0 f1 1 bq g (5)s d s d1 1 i/vt e q,v1 i/t 2 1 / e q,0 2 1s d s d s df g f gL L

as has already been described in Ref. [18]. (7)In single electron collisions this quadratic disper-

sion guarantees the approach to the so-called Bethe Here the peak width G5 "/t is given by the meanridge for large q and relatively great energy losses, decay time t.


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Finally, the optical data model (ODM) of Ashley The q-dependent matrix element M(q,v) contains[25,26] the specific scattering information [34]. Experimental

correlation to the Bethe ridge is given in [35].`

We may summarize that the dynamic form factor1 v9]] ]

S DIm2 5

Edv9

H JS(q,v) expressed in terms of the energy loss functione q,v vs d

0 Imh 2 1 /e(q,v)j for conduction and valence band1 " electrons as well as in terms of the generalized2]] ]S D3 Im 2 d v2 v9 2 q (8)H Je 0,v9 2m oscillator strength GOS for core electron excitations d

allows us to include all interaction channels as acan be applied to all groups of materials and to weak single two-dimensional function in one and the same(valence band) and tight (core) bound electrons as q 2 DE-plane. Thus a direct access to the DE( q)has been shown in Table 1 of [18]. Two advantages pairs becomes possible when processing the Montebecome obvious: Carlo procedure.

Furthermore we may add that surface plasmons

are taken into account as well as secondary electrons1. The ODM model enforces a quadratic dispersion

being treated like primary ones. Details have been2. The f-sum rule is automatically fulfilled for each given in [18].q. The surface loss probability is obtained by the

modified volume loss function ~Im 2 1 / e v 1h f s d1 . In Ref. [18] (Ch. 2.3) we have shown that theg jBut thirdly a disadvantage appears: Eq. (8) estab-mean coupling depth z increases slightly withlishes a form-invariant dispersion missing the in- 0

electron energy E, e.g. in Ag from nearly 2 A atcreasing spread with q as given in Eq. (5). Thus theE5 100 eV to about 5 A at E5 1 keV. This isBethe ridge does not possess the increasing width

comparable with the inelastic mean free path l ofwith q. But this disadvantage has been improved inslow electrons. Therefore, surface plasmons willupon by Ding and Shimizu [13].strongly affect the surface-near electron emission,In our MC calculations, for the metals Be, Al, Ag,especially the secondary electron generation andand Au we have used as far as possible the Mermim

transport.theory Eq. (7) or the optical data of [27,28] with aSecondary electrons (SEs) are created by core anddispersion of quasi-free electrons a5 0.5 and a peak

2 valence band ionization, free electron excitation,spreading with b|1 A according Eqs. (4) and (5),volume and surface plasmon decay. Their initialrespectively. The optical data of carbon were dis-

2 energy E immediately after exciation is given bypersed with a5 0; 0.5; 0.5 and b|6 A for the three SEthe exact energy balance of the process, i.e. E 5plasmon peaks with increasing energy, respectively SE

2 hv2 E (E , binding energy); their initial momen-[29]. For silicon the data a5 0.4 and b|6 A where b b2 tum q after excitation is distributed equally over SEchosen [30], for SiO a|0, b|6 A [31,32]. Al-2

the full solid angle 4p.though the magnitude of the dispersed loss functionThe position of SE creation is given by thewas fixed for each q by sum rules; [18], some more

interaction position, i.e. also the plasmon decay withinvestigation is needed as already mentioned in

SE generation is located at the preceding position ofSection 1.plasmon creation. The latter assumption seems to beOn the other hand, core ionization is performedsomewhat questionable since a plasmon is onlymainly by generalized oscillator strengths (GOS),weakly localized. On the other hand, it is known that[33], given in spectral densities:plasmons tend to decay at inhomogeneities, e.g.

E surface roughness.d f q,v 1s d n 2]]] ]] ]5 O ? uM q,vu E 2 "v (9)s s dn After their creation the SEs are usually treated as2d "v Rnqa ys d0

scattered primary electrons. For a SE to overcome

the surface barrier, the basic conditions of SectionHere the summation includes all final states E . Rn y3.3 must be fulfilled.means the Rydberg energy and a the Bohr radius.0


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3. Monte Carlo procedure

We have developed a numerical algorithm which

allows direct access to the inverse transformation of

a given probability function. The method has beentermed method of direct access to the inverse

transformation. Each time when a random number is

generated, the algorithm finds the corresponding

scattering value (e.g. scattering angle, energy loss,

free flight step length) directly by looking up a

multidimensional table, which has been calculated

and stored in material-specific matrices prior to the

run. This means that we do not have to perform

numerical integrations or elaborate computations of

cross sections during run time. As a consequence, theFig. 3. Partial and total inelastic electron mean free paths as a

required CPU time for picking up the individual function of the kinetic energy E for Al; the interaction with thescattering data of an electron trajectory is indepen-conduction band electrons is based on the Mermin dielectric

dent of the mathematical complexity of the underly-function, Eq. (7), the K and L impact ionization is calculated with

ing model. For more detailed information we refer to GOS, Eq. (9).Refs. [18,19].

low energy region the valence band contribution is3.1. Mean free path

the dominating one.

In Fig. 4 the inelastic mean free path l is showninThe elastic as well as the inelastic mean free pathsfor the metal silver in comparison with results ofl and l , respectively, are calculated from theel inother authors [13,31,36,37]. Fig. 5 presents l forinrespective cross sections described Section 2. Thethe compound insulator SiO in comparison with2

calculation of l (E) requires the integration of thein results of [31,32,38] and in context with the energydifferential inverse mean free path:loss function consisting of valence band and core

21"vmax level contributions as well as of optical phonon1 dl]] ]]5E d "vs dl E d "vs d s d"vmin

"v q1 E,vs dmax 1 1]] ]5E d "v E dq ?s dpa qE"v q2 E,vs d Bmin

2 1]]? Im (10)H Je q,vs d

The area of integration is limited by the conserva-

tion rules of energy E and momentum q as well asby non-allowed transitions are marked in Fig. 1 by

the hatched area. The maximum energy loss for

metals is hv 5E2E . For semiconductors andmax Finsulators the minimum loss is limited by thehv 5E .min g

In Fig. 3 the total and the partial inelastic meanFig. 4. Inelastic mean free path vs. electron energy in Ag; the data

free paths of the K-shell, L-shell and the valence of this work are compared with those of Ding and Shimizu [13]band interactions are shown in dependency on the (quadratic dispersion: broken line; fourth order dispersion: full

line), with Ashley [36], Tanuma et al. [31] and with Kanter [37].electron energy in the metal Al. Obviously, in the


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Fig. 5. Loss function and corresponding inelastic mean free path l of electrons with kinetic energy E in SiO ; the interaction with LOin kin 2

phonons is also included leading to a second characteristic minimum of l below 1 eV. Comparison with data of Tanuma [31], Ashley et al.in[32] and Reich et al. [38].

losses. Secondary electrons with low kinetic energies scattering events may be determined as the ratio of

below the band gap in SiO : E(9 eV, will undergo the respective cross section to the total one, e.g.:2 gscattering with the longitudinal optical LO phonons

s Es delhv(150 meV and hv(60 meV, see e.g. [39]. It is ]]]]p E 5s d1 2 el s E 1 s Es d s del inremarkable that the inelastic mean free path obtained211 1 1by means of Eq. (10) and presented in Fig. 5 fully

]] ]] ]]5 ? 1 (11)S Dcorresponds to those calculated by the Frohlich l E l E l Es d s d s del el intheory of electron polar-optical phonon interaction in Such a splitting of elastic and inelastic prob-[39]. Hence the energy losses are only of the order

abilities is demonstrated in Fig. 6 for the elements AlDE|0.1 eV the dissipative transport, e.g. of sec-and Au.ondary electrons, is much less affected than the

non-dissipative elastic transport of characteristic

electrons. Emission experiments can be followed 3.2. Trajectoriesdown to energies of the electron affinity x 50.9Si O 2

A forceless straight line movement of actualeV leading to high SE escape depths of l |160 A,SElength s between two adjacent interaction sites r ,[39]. n n

r is assumed with the polar and azimuth flightThe probability either of the elastic or the inelastic n11


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of the electron reference frame by means of the

matrix operation [18,19]

T Ts 5 O ?z9 5 W O ?z9 (12)s dn n n n21

Twhere O 5 W O is an orientation matrix and O

n n n21 n

is its transposed matrix.

The interaction matrix W represents the scatteringn

process as a twofold angular rotation of the electron

reference coordinate system. Prior to the first inter-

action, the flight direction is defined by the initial

matrix W , which is given by the incident angles0

Q 5a and C 50 of the electron probe beam, and0 0 0the unity matrix O 51 of the sample frame. Details

21

of the angular transformation have been shown

already in [18].Fig. 6. Probabilities of elastic and inelastic scattering as function In case of angular resolved electron detection, seeof electron energy in Al and Au targets.

Fig. 7b (e.g. in overlayer experiments) only a very

small number of electrons will reach the analyzer

angles a and b with respect to the laboratory and a high number of electron trajectories is neces-n nsystem, Fig. 7a. On the other hand, the actual polar sary in order to get a sufficient statistical certainty.

Q and azimuth C scattering angles are related to Therefore, in such cases the trajectory reversaln nthe actual electron flight system z95(0,0,1). Then method [40,6,8] will be used. The reverse electron

the flight direction vector s relative to the laboratory path is initiated from the detector to the interior ofnframe is calculated from the unique flight vector z9 the sample and traced back to the origin of the real

Fig. 7. (a) Electron trajectories with solid sample coordinates (x, y, z, a, b) and moving electron frame (x9, y9, z9, Q, C). The corresponding

transformation is given by Eq. (12); (b) emission of electrons under a take-off angle a where the scattering-off as well as the elasticesc

scattering-in are demonstrated.


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path, in general, to the position of the first inelastic refracted or even reflected at such barriers. This

collision. Of course, this method is applied only for process depends on the kinetic energy E of the

characteristic, i.e. monoenergetic electrons in a electron, its incident angle a with respect to the

quasi-elastic energy regime. On the other hand, the surface normal, and on the potential barrier height,

dissipative emission of secondary electrons SE is i.e. the electron affinity x, Fig. 8. Then the quantumregistered in the conventional way, i.e. in forward mechanical transmission probability is given by

path direction. The efficiency of the reverse trajec-1 / 2x

]]tory method is much higher than that of the real path 4S1 2 DEcos a

]]]]]]]]T5 (13)procedure. The validity is quite good, [9]. Some- 1 / 2 2x]]F1 1 S1 2 D Gtimes, certain deviations for large take-off angles Ecos a

a .608 are possible.es c 2for E cos a$x, otherwise T5 0 and the electron isFig. 7b makes obvious the problem of multipletotally reflected at the barrier.elastic scattering and its influence to the electron

Furthermore, the transmitting electron is refractedmean escape depth l and the mean attenuationes caccording tolength l . In the case that the scattering-off eventsat

approach the scattering-in ones, thenl

is nearly 1 / 2at E

]]

sin a9 5 sin a (14)equal to l . S Din E2 xwhere a9 is the final emission angle into the vacuum.3.3. Surface escape processBoth the transmission probability T and the refrac-

tion angle a9 are demonstrated in Fig. 8.Due to the discontinuity of the crystal potential in

the vicinity of surfaces, a slow electron may be

4. Results

To cover a variety of different materials, first

applications of the MC program have been focused

on metals Be, Al, Ag, Au, then on semi-4 13 47 79conductors C, Si as well as on the insulator SiO .6 14 2Each material has been considered with its collective

as well as individual excitation mechanisms of free

electrons, valence band and core electrons as de-

scribed in Section 2. The optical dielectric loss

functions used for the considered materials are

experimental data taken from [27,28].

Since experimental optical loss functions with

q(0 serve as input quantities, even materials with a

rather complicated energy loss structure, such as

transition elements and noble metals, can be treatedmore realistically [18].

In order to estimate the validity and quality of the

present Monte Carlo model we propose and recom-

mend a general test of theory by means of an h2R

presentation in a related functional plane [18]. Thus

backscattering coefficients h(E ) and the maximum0electron ranges R(E ) have been simultaneously0Fig. 8. Emission of an electron with energy E and momentum kchecked against experimental values. A reasonableacross the surface barrier with an electron affinity x (upper left);agreement is obtained. Furthermore, we were suc-its refraction [Eq. (14)] (upper right) and its transmission prob-

ability T(E) under an escape angle a [Eq. (13)], (below). cessful in modelling the anomalous backscattering


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for heavy elements, e.g. Ag, Au, when their back- initial electron energy E . The maximum range R is0scattering fraction h at energies E , 500 eV drops defined as that target depth reached in general by 1%0below the values of light elements, e.g. h(Au, 100 of the incident PE, [41]. That is in the limit of the

eV)(0.1. Other first applications have been directed measurement accuracy. Whereas h and R are in

to electron penetration, energy transfer profiles, and relatively good agreement with the experimental datadepth distributions for X-ray generation in multiple [42,41], the secondary electron yield shows some

layer targets [18]. quantitative deviation in the low energy region E ,0500 eV. From momentum-resolved coincidence elec-

4.1. Electron penetration and backscattering tron spectroscopy [43,44] it is known that secondary

electrons are produced primarily by the decay of

The backscattering ratio h, the secondary electron valence band excitations resulting from large

yield d and the maximum electron range R in Si and momentum transfer inelastic scattering. We have

Au targets are presented in Fig. 9 as a function of the tried to enforce such mechanism in our program but,

Fig. 9. Maximum range R (reached still by 1% of incident PE), backscattering coefficient h and secondary electron yield d of electrons with

initial energy E in Si and Au targets; present MC results in comparison with experimental data h, d of Bronshtein and Fraiman [42] and0

R(E ) of Fitting [41].0


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nevertheless, a sufficient approach could not be range, and accordingly, the SE escape depth l . Anes creached. Similar deviations of Monte Carlo SE arbitrary variation of the related cross sections did

creation were obtained by Ding and Shimizu [15]. change drastically the SE rates but did not substan-

Therefore, we suppose that firstly the mechanism of tially improve the simulation. Here, some more

volume and surface plasmon decay, which are im- theoretical investigation seems to be necessary.portant SE excitation channels, may play a decisive On the other hand, the anomalous decrease of the

role. backscattering coefficient for low energies and the

A second sensitive influence on the SE emission is heavy elements Ag and Au is well described by our

the inelastic mean free path l in the low energy MC calculations. Thus we could use the appropriatein

Fig. 10. Polar angular distributions of backscattered (BE) and elastically backscattered electrons (EBE) from Ag and Au targets and for two

initial energies E . (SEs, in general, follow a pure cosine distribution indicated by a cycle-like shape.)0


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data of PE backscattering h and maximum range R by an energy loss DE5 hv of the exciting primary

as a validation of the theoretical models used in the electron PE is shown.

MC calculation [18]. For a given energy loss, the cumulative probabilityIn Fig. 10 the angular distributions of backscat- for a SE to be emitted from an energy state E may be

tered (BE) and elastically backscattered electrons expressed by the joined density of initial and final(EBE) are presented for Ag and Au target and initial states [45]

energies E 5 100 eV and 3 keV. Similar MC simula-0 DsEdDsE 1 "vdtions in comparison with experimental angular dis- ]]]]]]]psE, "vd 5 (15)EFtribution measurements have been carried out already E DsEdDsE 1 "vddE

earlier by Ding et al. [14]; a good agreement was Eminobtained.

For noble metals like Au and Ag, the density ofThe deviations from the usual cosine distributionsstates DsEd, differs considerably from those of a free(e.g. characteristical for diffusive SE emission) be-

electron metal. Thus, they were taken from bandcome obvious especially for the EBEs and low E .0structure calculations of [46].Therefore the anisotropic angular distributions reflect

We may recognize in Fig. 11 that with losses of

mainly the elastic Mott cross sections. 10hv$ 4 eV the great number of 4d electrons in Agbecomes accessible for the SE excitation. Lower4.2. Secondary electron emissionlosses can generate SE only from the 5s state.

After generation, straggling to the surface andIn context with Fig. 9 we have already discussedemission into the vacuum the SEs appear with anthe discrepancies between calculated and experimen-energy distribution presented in Fig. 12 for Al, Si,tally found SE rates. One problem is the origin ofAg, and Au emitter targets. The calculated dis-secondary electrons, i.e. their initial bound states Etributions are compared with experimental ones ofin the band structure of the solid emitter before[47,48]. For the simple metal Al the so-calledexciation, their spatial position of generation andplasmon shoulder becomes visible caused by volumetheir kinetic initial energy immediately after excita-plasmon decay and subsequent excitation of elec-tion. In Fig. 11 the origin density of SE states created

trons from the Fermi level E . The SE energyFdistribution of Au has been improved when having

taken into account the surface plasmon decay.

4.3. Energy distributions

Two typical reflection energy loss spectroscopy

(REELS) experiments are chosen to demonstrate the

usefulness of the present low energy MC program. In

Fig. 13a overall energy distribution of 100 eV

electrons backscattered from silver are plotted in

comparison with experimental distributions of Chaseet al. [49]. For this reason the cylindrical mirror

analyzer (CMA) take-off geometry with a 5es c42,3668 was assumed. A surprising good agreement

is obtained. Then in Fig. 13b the energy distribution

of backscattered electrons distinguished in backscat-

tered primary electrons BE and true SEs loosen from10 the target material. We see that both partial dis-Fig. 11. Cumulative probability of SE creation from initial 4d

1 tributions are mixed and the common criterion with aand 5s states of the Ag conduction band depending on the relatedPE loss hv, see also Eq. (15). discrimination E , 50 eV and E $ 50 eV be-SE BE


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Fig. 12. Energy distributions of secondary electrons; present MC results in comparison with experimental distributions of Pillon et al. [48]

for Al, of Goto et al. [47] for Ag and for Au given in [45].

comes invalid for low initial energies E 5 100 eV. the characteristic low core level N loss, especially,0 2.3Obviously the SE function is overestimated due to in the single BE scattering distribution near E5

prevailing scattering of BE into low energy SE E 2 DE(40 eV, i.e. DE [Ag(N )]$60 eV. A0 2,3

regions. second analysis of REELS is made in Fig. 14af.In Fig. 13c, electrons that are inelastically back- The total loss spectrum of 250 eV electrons from Al

scattered from Ag are decomposed into electrons is recorded near the elastic peak and compared with

having suffered pure surface losses and those of pure experimental data of [50,51]. Single surface and bulk

volume losses. Expectedly, the pure surface losses modes are marked by s and b, twofold losses by 2s

have an information depth close to the elastically and 2b, the bulk-surface loss combination by b 1 s,

BEs. A similar picture is obtained for the number n respectively. All these peaks are clear distinguished

of interactions, Fig. 13d, where the single inelastic in the MC simulation but smeared in experimentally

scattering (n 51) also contributes more to the nearly recorded spectra.

elastic parts of the energy spectra. We also recognize On the right hand side, Fig. 14b, we see the depth


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Fig. 13. Energy distributions of backscattered electrons (E 5 100 eV) from Ag. The total distribution (a) is compared with experimental0data of Chase et al. [49], decomposed in BEs and true SEs (b), as well as in pure volume and surface interaction losses (c); the contribution

of n-fold interactions is shown in (d). Data are obtained for normal PE incidence and CMA take-off geometry a 5 (42,366)8.esc

distribution of elastically and inelastically backscat- with the initial energy E of PE. This escape depth0

tered electrons. In the following sub-figures the behaviour is predestined to be used in a combinationenergy distribution of pure volume losses (Fig. 14c) of spectroscopy and microscopy.

and pure surface losses (Fig. 14e) are presented in

context with the respective escape depth distributions 4.4. Escape depth

for the single (n 51), double (n 52) and triple (n 53)

energy loss. Surface losses of BEs are extended In the introduction we have already considered thebeneath the surface to depths of about 10 A. On the problem of different mean free paths like the elastic

other hand, the volume loss electrons escape still one l , the inelastic one l as well as theirel infrom a doubled depth range. Their escape depth is influence on the mean attenuation length l and theatincreasing with the number n of losses suffered and mean escape depth l , see Eq. (1). Whereas in reales c


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Fig. 14. Energy distributions (left) and depth distribution (right) of electrons (E 5 250 eV) backscattered from Ag; (a) total energy0distribution near the elastic peak with bulk b, surface s losses and their combinations 2b, 2s and (b 1 s), respectively. Our MC data are

compared with experimental distributions taken from Pellerin et al. [50] and Massignon et al. [51]; (b) backscattering depths of elastic and

inelastic interaction as well as of all electrons (total); a similar separation has been done for single n 51 and multiple n 52; 3 volume

interactions (c and d) as well as for n surface losses (e and f), respectively. All data are obtained for normal PE incidence and CMA take-off

geometry a 5 (42,366)8.esc


15/17


Fig. 15. Theoretical elastic (el.), inelastic (inel.), and total (tot.) mean free paths l in comparison with our MC-calculated mean attenuation

lengths (atten.) of electrons with kinetic energy E in Be, Al, Au and SiO samples, respectively. The total mean free path defined bykin 2

1 /l 5 1 /l 1 1 /l remains almost meaningless.tot el in


16/17


experiments mostly l or l are measured, the about 10 eV just above the vacuum level. Secondaryat esctheoretical understanding is associated with the pure electron creation and transport has been treated in the

elastic and inelastic mean free paths l and l , same way as the scattering of primary electrons.el inrespectively. The correlation of l and l has been Surface plasmons losses are included as well asel in

evidenced in [52] and [8]. Most of the analytical refraction processes at the surface barrier. The MCdescription succeeds only for the limiting case of the calculations are obtained for the materials: metals

straight line approximation when weak elastic cou- Be, Al, Ag, Au, semiconductors C and Si4 13 47 79 6 14pling is assumed or for strong elastic coupling with a and the compound insulator SiO and to the respec-2diffusion like model. The analytical model [53] holds tive energy spectra, the angular distributions, and

for any ratio of elastic to inelastic scattering. Never- escape depths of BEs, EBEs as well as of SEs.

theless, MC simulations should be an appropriate Plasmon shoulders in the SE distribution, core

tool for the extraction of the theoretical relevant losses in REELS as well as single and multiple

quantities l and l from the combined complex losses near the elastic peak could be modelled inel indata l and l found in experiments. We have good agreement with experimental data. Moreover,at esccarried out this task and present the respective results we get the escape depths and dispersion relations of

for Be, Al, Au and SiO in Fig. 15ad. Indeed, as these characteristic electrons. For electron energies2expected, the attenuation length l is always smaller much greater than the electron affinity x of theatthan the inelastic mean free path l . However, in the surface, (E4x), the escape depth l approachesin escenergy range E. 100 eV the deviation is smaller the attenuation length l which is, in agreement toatthan reported by Jablonski [47], who found devia- Werner et al. [40], only slightly smaller than the

tions of more than 30%. Thus we may state that the inelastic mean free path l . However, for smallinnumber of electrons scattered-off elastically is nearly electron energies E, 100 eV the elastic scattering

equal to the number of electrons scattered-in from dominates and the attenuation length passes a mini-

other directions to the given take-off direction a , mum and drops below 30% of the inelastic mean freees csee Fig. 7b. Our results correspond to those of path. The results suggest applications in a combina-

Werner et al. [40,6] who also confirms that the mean tion of electron spectroscopy and microscopy.

attenuation length l approaches the inelastic meanatfree path l . However, for low energies below 100ineV the scattering is more and more dominated by the

Acknowledgementselastic scattering and the attenuation length latpasses a minimum and approaches roughly only 20% We are indepted to Dr. Andreas von Czarnowskiof the inelastic mean free path l as demonstrated inin for performance of additional computer runs andFig. 15. illustrations.

5. SummaryReferences

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