P HYSICAL RE VIEW B VOLUME 9, NUMBER 7 1 APRIL 1974

Photoelectron fieter~f~stion of the attenuation of lowwnergy electrons in Al,Os

F. L. Battye, J. G. Jenkin, J. Liesegang, and R.C.G. LeckeyPhysics Department, La Trobe University, Bundoora, Victoria, Australia 3083

(Received 5 October 1973)

The attenuation length for electrons in Al,O, has been determned to vary from 6.7 A at 157 eV to16.7 A at 1404 eV by means of an x-ray-photoelectron spectroscopic technique. The attenuation ofphotoelectron and Auger-electron lines from a substrate was monitored as the thickness of an overlayerof Al,O, was incremxi, the results being corrected for the asymmetry of photoelectron angulardistributions from various. subshells and for effects due to the geometry of the spectrometer. The resultshave been compared with a number of theoretical models of electron scattering. The major energy-lossprocess has been demonstrated to be due to inelastic scattering involving valence-band electrons.

I. INTRODUCTION

An accurate knowledge of the attenuation lengthfor electrons in solids with energies in the range0. 15-1.5 keV is of the utmost importance in thequantitative interpretation of a wide range of elec-tron spectroscopic measurements. This is theenergy range of interest in x-ray-photoelectronspectroscopy, for example, and many Auger tran-sitions also result in the emission of electronswith kinetic energies of this order, The scarcityof reliable measurements of attenuation lengths forsuch low-energy electrons is directly attributableto the shortness of the attenuation length, which istypically 15 A or less in this energy region.

We report here, by means of a technique whichdirectly monitors the attenuation of photoelectrons,the determination of the energy dependence of theattenuation length of electrons in A1~03 coveringthe energy range 150-1400 eV. The results so ob-tained are compared with the predictions of twotheoretical models: one in which all energy statesin Altos (including the valence band) are consideredto be tightly bound, and the other extreme situationin which the valence-band electrons are consideredto be free.

II. EXPERIMENTAL METHOD

A. General

Some measurements of attenuation lengths insolids have recently been made by an Auger-elec-tron spectroscopic technique ' which monitors theattenuation of substrate Auger signals as the thick-ness of a thin covering film of some dissimilarmaterial is increased (or, alternatively, the in-crease in an Auger signal cha, racteristic of the thinfilm is monitored). Such measurements sufferfrom the serious disadvantage that the primaryelectron beam which is used to cause inner-shellionization, which then leads to Auger recombina-tion, must itself be scattered and attenuated by thepresence of the thin film. This effect can clearlynot be eradicated without prior knowledge of the

attenuation length for electrons of energy equal tothat of the primary beam.

The use of an x-ray-photon beam as the primaryprobe obviates this difficulty because of the verymuch larger attenuation length for x-ray photonsthan for electrons of similar energies, s and be-cause the mass-absorption coefficients for soft-x-ray photons are generally well known. By suitablechoice of photon energy and substrate material,photoelectrons and photo-Auger electrons of well-defined energies may be generated over a wide en-ergy range; and by monitoring the attenuation ofsuch electrons in their passage through a thinfilm of some dissimilar material which is essen-tially transparent to the incident photon beam, aninelastic mean free path (X) or attenuation lengthfor electrons of each energy may be obtained fromthe relation

I=Ioe ' ",

where I and Io represent the intensity of a particu-lar electron line with and without a covering thinfilm of thickness t.

The x-ray-photoelectron spectrometer used inthis work is shown schematically in Fig. 1 and isdescribed in some detail in Refs. 7 and 8. In or-der to obtain high photoelectron luminosity the x-ray source and sample rod are of cylindrical sym-metry, with electrons emitted from the sample in-to a cone of half-angle 45' ~ 2' being accepted intothe spherical-sector electrostatic energy analyzer.The use of this particular geometry is advanta-geous in terms of over-all spectrometer sensitivityand signal-to-noise ratio, but leads to a number ofcomplications for the purposes of the present ex-periment. The use of a cylindrical sample rodposes certain difficulties in the controlled evapora-tion of very thin and uniform films (see Sec. IIB)and, in addition, results in the sample being ir-radiated by x rays at a variety of angles of inci-dence and in the emission of electrons over a rangeof angles of emergence which are still within theangular acceptance of the energy analyzer. These

2887

BATT YE, JENKIN, LIE SEGANG, AND LE CKE Y

latter considerations are, however, capable ofstraightforward analytical treatment a,s given inSec. IIC.

After energy analysis, electrons are detected bya Cu-Be dynode electron multiplier (EMI type9643/2B} and the data acquired in a Nuclear Data1024-channel multiscalar whose channel-advancemechanism is synchronized with the scan voltageof the spectrometer.

8. Preparation and control of thin-film parameters

In order to deposit a uniformly thick film foreither substrate or sample overlayer purposes, thecylindrical sample rod could be raised from themeasurement position (as shown in Fig. l) and

rotated about a horizontal symmetry axis, approxi-mately 20 cm above a pair of evaporation boats. Inclose proximity to the sample when in this position,a quartz-crystal-detector head was positioned inorder to monitor and control automatically the rateof deposition by means of an Ultek thin-film depo-sition system which was coupled to the powersource driving the evaporation boat in use. By ro-tating the sample rod at a constant rate and operat-

J I

I i1) I

t

SH

ing the evaporation system at a constant evapora-tion rate, a uniform film of thickness t could bedeposited on the sample rod of radius a. Thistechnique is essentially that used previously bySteinhardt et al. ' A Varian A-scope (multiple-beam interferometer} with an accuracy of +25 A

for films of thickness & 200 A was used for thecalibration of the quartz-crystal oscillator bymeans of a series of controlled evaporations. Weestimate that the average thickness of each of thefilms used in the present work has been determinedto an accuracy of +2%. Because the pressure inthe sample preparation area of the spectrometerrose to - I x10 torr during evaporation, theevaporation of 99.999'p&-pure Al at a rate of 0. 25 A

per second resulted in the deposition of completelyoxidized films as observed by Klasson et al.

Films of Alz+ were produced with thicknessesin the range 5-30 A (i. e. , of the order of an elec-tron attenuation length at the energies of interest)and consequently it was important to ensure thatno islanding or other irregular deposition patternswere present. Sample films of AlzQ3 were exam-ined with this in mind in a scanning electron mi-croscope (SEM) equipped with an energy-dispersivex-ray detector. Although the surface topographyof the sample was clearly seen to be complex thedistribution of Al over the sample surface was ob-served to be essentially isotropic as seen by thedetection of Al Ea radiation caused by the incidentelectron beam in the SEM. From this evidence,and from the evidence of results to be discussedlater, we are satisfied that the thin films used herewere uniform and of constant thickness.

X-RAY

SOURCE

FIG. 1. Sample preparation area and part of the spec-trometer. Positions of sample 5) are shown both formeasurement and for evaporation. The cylindrical shield(SH) encloses the sample during evaporation from one oftwo evaporation boats (E) and is equipped with a shutter.(A is anode; F is filament. )

C. Geometrical corrections

As mentioned earlier, because of the geometryof the spectrometer shown in Fig. 1, x rays areincident on the cylindrical sample rod at a varietyof angles P (Fig. 2). Although the attenuation ofthe x rays used in this work is very small, a com-plete mathematical treatment must include thiseffect, which will clearly be different for differentangles of incidence (t}. We consequently define anattenuation length A.„ for x rays entering the sub-strate, and ignoring surface and interface refrac-tion, we relate the x-ray flux per second in an ele-ment of the substrate to the incident x-ray flux persecond 5C„(0) emitted from a small element ofanode surface via

54„= 64„(0)exp[-v 2(a —r)/X„cos(q —«)j, (2)

where the a,zimuthal angle of the substrate volumeelement is g and that of the anode surface elementis approximately g. Since a is experimentallymuch smaller than the x-ray-anode radius and thespectrometer-entrance-slit radius, the polar anglesof k„and 5, are taken always to be v/4 (see Fig. 2).

PHOTOELECTRON DETERMINATION OF THE ATTENUATION. . . 2889

al angle for the emitted photoelectron (= ]). FromRefs. 13-15 we have that

tj

II

IIIlI

I

II

I

I

t NI

A II I/ IgI Q

iI :., ~4

Jw

LHPHI = gPM= ePH = f1

)(a-rg2cos(x- 4)

MN-—cos(e-4) exp[- v 2(a - r)/&, cos(K —&)], (6)

P=2. 0 for s shells

=1.0-1.5 for p shells

=1.5 for d~@ shells

= 1.2 for f~ + shells,

whereas for an isotropic distribution, as may beassumed for Auger-electron emission, P=0.

We assume that the photoelectrons are attenu-ated in the substrate according to the factor

FIG. 2. Geometrical arrangement of x-ray source,sample, and spectrometer. The small volume elementof substrate (~br 6z 6&) at the point P is irradiated by xrays of wave vector Q and emits electrons of wave vectork.

Thus the number of photoeclectrons per second(5I) which are produced within an element of sub-strate (of volume 5A 6r located a-r from the sur-face) by photons incident at an angle P, and whichare emitted into a solid angle 50 at 8 to the sub-strate normal (Fig. 2), is

5I=64 cosp —N.dr606Ado'

dA

where N is the atomic density of the substrate anddo/dQ is the differential photoelectron cross sec-tion of the relevant subshell for photons of energyhv. If an overlayer of dissimilar material coversthe substrate this will cause an additional x-rayattenuation given by exp[- Wt/X„'cos(g —a)] if theoverlayer has a photor attenuation length describedby ~„',

The photoelectrons so generated within a smallelement of substrate will be emitted at a varietyof angles to the direction of the incident-photonwave vector depending on the atomic shell fromwhich they are ejected. Since this angular distri-bution determines the path length in the samplewhich electrons must travel before reaching thesurface, it is essential that this factor be incor-porated in the theoretical interpretation. The dif-ferential cross section may then be written+

with the corresponding factor

exp[- &2t/X, cos(s —t') ] (6)

in the overlayer, A., and ~, being the electron at-tenuation lengths in substrate and overlayer, re-spectively, and e the angle of emergence of thephotoelectron from the sample element with re-spect to the element normal.

We may thus write that the intensity of photo-electrons, due to photon flux 5$„(0)(cc6g), whichemerge at angle 8 into solid angle M (cc6$) fromthe substrate element which is a- r below the sur-face of the overlayer is

6I6Y -=—

6A

cc cos&exp[- v 2 (a —r)/& „cos(g —K)]

x exp[- &2t/X» cos('g —K )]exp[- v 2(a —r)/

X, cos(s- $}]

x exp[-&2t/&, cos(s —()]—5g 5(5r . (7)

The total number of photoelectrons emerging persecond from the sample and entering the spectrom-eter is thus obtained by integrating over r and theangles g and $. It should be noted (i) that the in-tegrand is essentially constant over the width of theentrance aperture of the spectrometer and (ii) thatEq. (7) suffers no loss of generality by assumingthe element in Fig. 2 has ~=0.

In evaluating the integral it may be seen f romFig. 2 that the angles 8, P, and g are related tothe angles z, g, and ( through the following rela-tionships:

d—=4—[(1——,'p)+ —,'psin g(g, f)], (4)

where g is the angle between the incoming x-raydirection k„and the direction of the photoemittedelectron S„Pis the asymmetry parameter, and o

is the relevant total subshell photoelectron crosssection. The angle g is a function both of the x-ray azimuthal direction angle (= g) and the azimuth-

r=(coss, sirvc, 0},

5„= (-cosg, —sing, - I)/W2,

5, = (cosh, sfn(, —I)/v 2,

whence

cosg= k, 5„—[1—cos(g —$)]/2,

cosP= —k„%=cos(g- z)/v 2,

(8)

(9)

(10)

(11}

(12}

2890 BA TTYE, JENKIN, LIE SE GANG, AND LE CKE Y

cos8=5, f =cos(t' —K)/v 2

The anisotropic factor in Eq. (4) may then be re-arranged so that

—=—(1+ [1+6cos($ —r})der a'

dQ 4m

which assumes that all photoelectrons leave thesample at 8= v/4, i.e. , )=0. As may be seen in

Fig. 3, the effect of the photoemission angularasymmetry parameter P is clea, rly small but thecorrection which allows for the geometry of thespectrometer is significant.

—Scos'($ —r))]P/16j (14) III. EXPERIMENTAL RESULTS

T cc f'&zf;&~&~f~' epx[v 2(r- a)(1/&„cosy

+ 1/).,cost')]

x exp[- v 2f(1/X„+ 1/X'„)/cos0 + 1/X, cost'1

x(1+ [1+6cos($ —0) —3 cos (5 —'8)] P/16)

&& cosg dry d$. (15)

In the present experiment we make the followingassumption concerning the magnitudes of the meanfree paths: X„-X„'» X„X,'. Integration over &

and g then yields

T ~ f~' exp[- &2t/&,'

cosh][2 —(1+cos2(

—Svcos()P/16] cos$dt'.

This integral has been numerically evaluated for—s/2 ~ t' ~ v/2, but because of the exponential

term, the effective range of $ is actually much

less. The result may be compared with the rudi-

mentary form of the attenuation law, Eq. (1),

and the observed photoelectron intensity as obtainedby integration of Eq. (7) then becomes

The applicability of Eq. (16) is illustrated in Fig.4 which shows the attenuation of the 4f, &~ levelfrom a gold substrate illuminated by Al Kn pho-tons, as a function of the overlayer (AjeO~) thick-ness t. Representative spectra of the Au»z 5&&

doublet are included to illustrate the signal-to-noise ratio achieved.

Gold, copper, and germanium substrates wereemployed in the present work, together with AlKe (1467 eV) and C Ea (277 eV) x rays, as indi-cated in Table I, which also lists the numerical val-ues of the attenuation lengths. The attenuationlengths were obtained by adjusting the value of &,in Eq. (16) so as to match the experimental datain the manner illustrated by Fig. 4 for the partic-ular case of the Au 4f7~z peak.

Results for the attenuation length of 150-1500-eVelectrons in AQO~ are displayed in Fig. 5, whichalso includes data by Klasson et c/. for 1389 and2856 eV, "Gesell and Arakawa for 20 eV, '6 andKanter' at 5, 1500, and 2000 eV. Kanter's resultsand those of Gesell and Arakawa refer to largelyunoxidized samples of Al. The present results aresignificant in that they extend previous measure-ments into the region between 100 and 1 keV, whichis of particular interest for the interpretation ofmany current electron- spectroscopic measure-ments.

1.0

0$

I-t7) 0.6zLU

z

UJ 0.4O

XIX

z 0.2

FIG. 3. Attenuation ofelectron intensity by cover-ing films of relative thick-ness t/Q as given by Eq.(16) for the cases P=O andP=2, and as given by therudimentary form~ e~&/&e

OD 1.0 2.0

PHOTOELECTRON DETERMINATION OF THE ATTENUATION. . . 2891

~'t~7.1 A ts141A t=21.2AI

of incident energy E0 interacting with a systempossessing a number of inner-core energy levelslabeled by the subscript i, as

o&s -Qs&q&[1 P&e ~Foist]ln(EO/E()/EOE(, (1V)

1.0

5 0.6

x

Lito OA,

N

g 02Oz

I I I

0.0 1.0 2.0 &0',I I I I I I I l j I I I I I I I

0.0 1,0 2.0 go t (R)

FIG. 4. Data from attenuation of the Au 4f&g2 photo-electron peak by A1203 films of thickness t fitted to thecurve of attenuation vs relative thickness t/Q from Eq.(16) for the case P=1.2.

IV. THEORETICAL CONSIDERATIONS

A theoretical treatment of electron scattering insolids at these energies is fraught with difficulties.We have, however, attempted to delineate the yrob-lern by calculating the attenuation lengths to be ex-pected on the basis of some extreme theoreticalassumptions.

A. Tight-binding model

The most important scattering mechanisms forfast photoelectrons as they traverse an A1203 sam-ple towards the surface are likely to be inner-shellionizations and individual scattering events involv-

ing a valence-band electron. For the particularcase of A1203 we have chosen to ignore the possi-bility of collective interactions with valence-bandelectrons (plasmon generation) and have consideredyhonon collisions to be insignificant as an energy-loss mechanism.

Inner-shell ionization may be treated with somedegree of accuracy by means of a scheme intro-duced by Lotz'8 wherein the Bethe inelastic scatter-ing cross-section formula' is modified to includean empirical correction factor which is significantfor low-energy incident electrons. Lotz's method,as detailed by Pessa and Newell, also allows fora relativistic correction term to be incorporatedwhich is, however, of no significance in the pres-ent work.

Following Pessa and Newell we may thus writethe inelastic collision cross section, for electrons

s = 1/fly ere(AlsOq)] (18)

where ors(All03) = 2o»(A1) + 3ors(O) and where N„is the molecular density of A1~0,.

The result of this calculation is displayed ascurve 1 in Fig. 5. The agreement may be seen to begood for energies less than about 250 ev, which isapproximately 20 times the average binding energy

TABLE I. Values of the electron attenuationAl&03 for energies in the range 150-1400 eV.

X-ray Binding Electronenergy substrate energy kinetic

X Ray (eV) level (eV) energy (ev)

Al Xe 148B.7 Au Qgl g 83 1404

Au 4tg, ig 334 1153

Cu ELL Auger 814

CEe 277

Cu 2p312

Ge 2p21g

Cu 3$112

$31

1217 270

length in

Mean freepath (L)

1B.7*0.8

14.8+ 0.8

12.8+ 1.010.1+ 0.8

8, 5k 1.58.7+ 0.5

where we assume E0~ E, , q& is the number of elec-trons in the ith subshell and the constants a„b„and c& have been determined by Lotz and are givenin the article by Pessa and Newell. In the interestsof clarity, the small relativistic correction termsapplicable here have been omitted from Eq. (17).

The major difficulty in calculating a total inelas-tic collision cross section concerns the method tobe used in treating collisions involving the valence-band electrons. Such interactions constitute themost probable energy-loss mechanisms, as may beinferred from the fact that the contribution to thecross section from all inner-shell ionizations cal-culated on the basis of the formula discussed aboveis an amount ranging from 4% at Eo = 100 eV to 14%at E0= 1500 ev.

One extreme theoretical assumption consists oftaking the view that the Lotz tight-binding formal-ism may also be applied to the valence-band elec-trons. It is clear that such a treatment is mostlikely to be relevant for the lowest-energy incidentelectrons for which the small binding energy of thevalence-band electrons will be of most significance.It is unlikely that this approximation will remainvalid for even moderately energetic incident elec-trons. In order to test this assumption we haveapplied Eg. (17) (including relativistic correctionterms) to all the energy states of A110~ and havederived an estimate of the energy dependence of thetotal inelastic mean free path &Ts in this tight-bind-ing approximation from eT via

BATT YE, JE NKIN, LIE SE GANG, AND LE CKE Y

30—

(2 20-Z

10—

I I I I I ilI10

I

I

1

I

l

1

1

I I 1 I I I if100 1000

ELECTRON KINETIC ENERGY (eV)10000

FIG. 5. Attenuationlengths for electrons inA1203 from the present work(0) together with results ofKlasson et al. (&). The re-sults of Gesell and Arakawa(+) and of Kanter (0) forelectron attenuation in Alare also shown. Solid linesrepresent attenuation lengthscalculated from 1, the Lotzequation, and 2 and 3, theMott formalism with mini-mum allowed energy lossesof I and 6.3 eV, respective-ly. Curves labeled 4 and 5have been derived using ascreened-Coulomb-interac-tion mechanism with mini-mum energy loss set at 1and 6.3 eV, respectively.

of valence-band electrons, but the application ofthis model clearly underestimates the inelastic col-lision cross section for energies higher than 250eV.

B. Mott scattering model

At the other extreme we may investigate the re-sults of the assumption that the valence-band elec-trons in A1203 are entirely free. One well-knownmodel applicable to this situation is that of Mottscattering, which treats the elastic scattering ofidentical spin-~ particles within the framework ofa phase-shift analysis. ' Thus

d Mo~t /l'sin4(-,'8) cos4( —,'8}

2 cos(f in[tan (28)]]~sin (—,'8)cos'(&8) f

where t; =e /16meo. A calculation of the mean freepath for electrons in A120o using Eq. (19}for allvalence electrons and Eq. (17) for all other energylevels results in curve 2 shown in Fig. 5, whichclearly overestimates the magnitude of the total in-elastic collision cross section at all energies ofinterest. This overestimate is due to the use of anunscreened Coulomb interaction potential implicitin the Mott formalism, which results in the predic-tion of large numbers of small-angle (small-ener-gy-loss) collisions. It seems likely that the basicMott model may be improved by the introduction ofa minimum energy loss (6. 3 eV, being the band gapin AlzO, ) and by the use of a screened Coulomb in-teraction potential. As a first step, the predictionof the Mott formula whenever a minimum energyloss of 6. 3 eV is imposed is shown in Fig. 5 as

In the first Born approximation the scatteringamplitude A.„for a particle of mass p, to be excitedfrom state P, to state P~ due to an interaction de-scribable via a potential V is '

Aao=(-u »/'k)'4 loVlda& (20)

which reduces to

Ao, =(- p, /2vk )V(ko- k, ) (21)

in the plane-wave approximation. For elastic scat-tering Ik, I

= Ik, I= Ikol and Ik, —k, I

= 2kosin(-,'8),where 8 is the scattering angle. Using a screenedinteraction potential of the form

V(r) = (e'/4weor) e o, (22)

where ro is a screening radius, it follows that

Ao, (8) = —4e /{16wego[4sin (o8)+ 1/k or op (23).Allowing for the spin states of the interacting sys-tem we find that

do s c =A(8)'+ A(w —8}'—A(8)A(m —8)

g2 4 2 2

E~~ 4 sino(~o8) + (koro) 4 coso(—'8) + (koro) o

16[4 sino( —,'8}+ (koro) o][4 coso(—'8) + (koro) o]

(24)where g =eo/16m'o as before and 0 ko=2pEo.

For the present purposes, the screening radius

curve 3. The agreement with experiment is seento be considerably improved over that obtained us-ing the basic Mott formalism.

C. Screened-Coulomb-potential model

PHOTOELECTRON DETERMINATION OF THE ATTENUATION. . . 2893

ro has been determined using a formula which isknown to be reliable for the description of plasmongeneration in simple metals~3

ko ——(b,E/E J,)kr, (25)

Precision measurements of the variation in thetotal inelastic scattering mean free path for elec-trons in A1~0, have been made covering the impor-tant energy range 150-1500 eV by a photoelectronspectroscopic technique. It has been demonstrated

where ko=-2w/ro, kz is the wave vector of an elec-tron at the Fermi energy E~, and bE is the plas-mon energy. Although Eq. (25) is of uncertainvalidity for the case of A1~0„a characteristic en-ergy loss of approximately 20 eV has been observedin this material and identified as due to plasmoncreation. The value of ro obtained in this manneris =3 A.

The results of calculating a mean-free-path func-tion via Eq. (24) is indicated in Fig. 5 as curve 4,where the minimum energy loss has been taken as1 eV to allow for instrumental resolution. If theminimum energy loss is taken equal to the band gapin A120, (6. 2 eV), this prediction becomes curve5. In all cases the inner-core levels have beentreated as before via the Lotz method.

V. CONCLUSIONS

that the major process contributing to the stoppingpower of A1~0, for such electrons is connected withcollisions involving valence-band electrons, and avariety of theoretical models have been investigatedin an attempt to describe these collision processes.A model in which the valence electrons are treatedas if they were tightly bound has been shown to givereasonable agreement with experiment for electronenergies ~ 250 eV but to be a poor approximationat higher energies. Imposing a minimum-energy-loss condition on free-free collision models, there-by accounting for the band gap in A120„has beendemonstrated to give good agreement at higher en-ergies. %e consider that the agreement betweenthe Mott scattering model and experiment is some-what fortuitous, being the result of the introductionof the minimum-energy-loss condition, which isequivalent to a truncation of the unscreened Cou-lomb potential used in this model. The introductionof such a minimum energy loss would not be possi-ble in the treatment of a metal for which the Mottformula would clearly overestimate the collisioncross section. Despite some uncertainty in thevalue to be used for the screening parameter ro inthe screened Coulomb model we consider this mod-el to represent a reasonable treatment of electronscattering for energies in the region 1-5 keV inA1203.

E. Bauer, Vacuum ~22 539 (1972).M. P. Seah, Surf. Sci. ~32 703 (1972).K. Jacobi and J. Holzl, Surf. Sci. 26, 54 (1971).J. W. T. Ridgeway and D. Haneman, Surf. Sci. ~24 451(1971).

D. W. Palmberg and T. W. Rhodin, J. Appl. Phys. 39,2425 (1968).

B. L. Henke, R. L. Elgin, R. E. Lent, and R. B.Ledingham, Norelco Reporter 14, 112 (1967).

P. C. Kemeny, A. D. McLachlan, F. L. Battye, R. T.Poole, R. C. G. Leckey, J. Liesegang, and J. G. Jen-kin, Rev. Sci. Instrum. 44, 1197 (1973).

A. D. McLachlan, R. C. G. Leckey, J. G. Jenkin, andJ. Liesegang, Rev. Sci. Instrum. 44, 873 (1973).

Ultek model 60-915, deposit thickness monitor, Ultekmodel 60-925, deposit rate control, and Ultek model60-940, power controller.R. G. Steinhardt, J. Hudis, and M. L. Perlman, Phys.Rev. B 5, 1016 (1972).M. Klasson, J. Hedman, A. Berndtsson, R. Nilsson,C. Nordling, and D. Melnik, Phys. Scr. 5, 93 (1972).J. E. O. L. scanning electron microscope, model JSM-

U3; Nuclear Diodes EDAX, model 505.J. W. Cooper and S. T. Manson, Phys. Rev. 177, 157(1969).S. T. Manson, J. Electron. Spectrosc. Related Phe-nomena 1, 413 (1972-3).M. O. Krause, Phys. Rev. 177, 151 (1969).T. F. Gesell and E. T. Arakawa, Phys. Rev. Lett. 26,377 (1971).H. Kanter, Phys. Rev. B 1, 523 (1970); 1, 2357 (1970).W. Lotz, Z. Phys. 206, 205 (1967).H. Bethe, Ann. Phys. (Leipz. ) 5, 325 (1930).V. M. Pessa and W. R. Newell, Phys. Scr. 3, 165(1971).A. S. Davydov, Quantum Mechanics (Pergamon, NewYork, 1965), p. 407.See Ref. 21, p. 376.H. Raether, in SP&,nI,er Tracts in Modern Physics,edited by G. Hohler (Springer, Berlin, 1965), Vol. 38,pp. 84-157.M. Suleman and E. B. Pattinson, J. Phys. F 1, L21(1971).