Application of a Semi .. Empirical Sputtering Model to Secondary Electron Emission - Schwarz

8/6/2019 Application of a Semi .. Empirical Sputtering Model to Secondary Electron Emission - Schwarz

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secondary escape depth A which are employed in both models. The present model is derived in Sec. III, an d it is demonstrated that the parameters B, Ed' an d A are essentiallyequivalent in the two models. Section IV compares bothmodels to experimental data as a function of primary energy,incident angie, and material parameters. Trends in the escape probability B and the secondary creation energy Ed areanalyzed here. In Sec. V, the limitations and approximationsof the present model are discussed. A brief comparison tomore refined models of SE emission is also presented. Thepaper concludes with a detailed summary.

II. STANDARD MODELThe standard SE emission model developed by Salow3

and Bruining,4 and refined by Lye and Dekker8 and others,is briefly described here in its simplest form.5 The yield b oflow-energy electrons is given by

8 = C' n(x,Ep )/(x)dx,_0

(1 )where n (x,Ep ) is the number of secondary electrons at depthx produced by a primary electron of energy Ep, and/(x) isthe probability that a secondary electron generated at depthx travels to and escapes from the sample surface. The function n (x,Ep ) is assumed proportional to the stopping power(energy loss rate)

1 dEn(xE)=-- - (2 ), p Ed dx 'where Ed is the average energy required to create a secondary electron. The function/ex) is given by

lex) =Bexp( -X /A) , (3 )where the surface escape probability B is less than 1 and A- isthe SE escape depth. If it is further assumed that the energyloss rate in the near-surface region is constant, H such that

dE Ep- dx =R ' (4)

where R is the primary range, then solution ofEq. ( 1 ) yieldsE Ao= B _P [1 - exp ( - R / A) ] .Ed R (5)

It is of interest to insert some standard approximationsfor the range R and escape depth A into the yield equation inorder to examine th e yield dependence on material parameters. 9 A typical approximation for the rangelO.]I R (in em )which provides a good description of various energy dissipation experiments is

(6)where N is the atomic density (cm - 3) and Z is the atomicnumber. (The prefactor in this equation is approximatelyconstant with Ep expressed in units of keY. Energies aretherefore expressed in keY in all of the equations.) The energy exponent n, as determined from energy dissipation experiments, is 4/3 in the energy range of nterest here (nearEmax ) , and -5/3 at higher primary energies.5,! 2 A usefulformula for the escape depth developed by Ono and Kan-

2383 J. Appl. Phys., Vol. 68, No.5, 1 September 1990

aya'l which provides a good empirical description of measured escape depths is

(7)where, in a crude but effective approximation, the averageionization energy is assumed proportional to the first ionization potential J of the isolated atom. With n = 4/3, Eq. (5 )may be rewritten as

onorrn = 1.11 E n ~ l f : : ' 3 [1 - exp( - 2.34 E ; ~ ~ m ) ], (8 )where the yield and energy are normalized to their maximum values given by

Emax = 10.6 I J l4Z \/ 4 (9)and

8max =0.39 B(E",ajEd ). (10)Equations (9 ) and (10) are similar to the Emax relationsderived by Ono and Kanaya'l which they demonstrate provide a good description of experimental maxima in metals.Substituting typical values of I and Z into these equations,one finds that Emax is approximately 0.5 keY, as required,and lim", is on the order of 1, provided that Ed-lOO eV(assuming B,-O.S). This large value of Ed is comparable tothe energy loss for the first collision of the primary. 9 On theother hand, it is frequently assumed that Ed is the energybarrier for secondary emission (a few eV), represented by q;(the work function) or q; + EF (the Fermi energy) in metals, and by Eg + X (energy gap plus electron affinity) ininsulators. l3 This discrepancy will be examined in the following sections.

Th e normalized energy dependence given by Eq. (8 ) forn = 4/ 3 is plotted in Fig. 1 (open circles) along with thecorresponding equation for n = 5/3 (squares), essentiallyreproducing the curves in the analogous figure fromDionne. 7 It is clear that the n = 4/ 3 curve provides a goodfit at low energies while the 5/3 curve works well at highenergies, consistent wi th the observed power dependence ofthe range. Vaughn 14 has discussed the need for a more accurate equation for the purpose of vacuum electronic devicesimulation. He proposed the following empirical formula:

(11 )where k = 0.25 for E norm > 1 and 0.62 for E norm < 1. Theempirical equation is also plotted in Fig. I (x's).

For off-normal incidence of the primary electrons, theyield is observed to increase slightly at low energies (wherethe range R is on the order of the escape depth A) and moresubstantially at higher energies. This effect is treated in thestandard model 15 by replacing R by R cos ewhere e= 0 atnormal incidence. With this substitution, the following relations are obtained:

- l/ n ( 12a)

8 ( e ) ~ 1 (E E )0 ( 0 ) - p ma x , ( 12b)

8(0) ( O)" (E ' E )- - ; : : : : ; cos p max 8(0) , (12c)

S. A. Schwarz 2383

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Vaughn14 recommends an empirical fit based on a first-order Taylor expansion of the cos function (which accounts tosome extent for the large fraction of primary energy lostthrough the surface at high e) given by

Emax ({1) = 1 + k e2,Emux (0) s- (13a)

("nax (8 ) = 1 + k ()2/2, ( l3b)8max (0 ) swhere k, is a smoothness factor varying from 0 for a roughsurface to ~ O . 5 for a smooth surface. The magnitude of k,and the factor of 2 in Eq. (13b) will be discussed in thecontext of the present model.

iiI. PRESENT MODELMonte Carlo simulations ofSE emission16 or sputtering

frequently generate an image of the primary collision cascade. On viewing such an image, it becomes evident thatparticle ejection can occur late in the collision cascade. Th ecomputed yield in these simulations is basically the numberof displaced particles generated within the escape depth ofthe surface with momentum directed outward. A similar vi-sualization motivates the semi-empirical model of Schwarzand Helmsl,2 in which properties of he collision cascade as awhole, rather than of the primary-surface interaction, areemphasized. The derivation proceeds as foHows:

The number of secondaries created in the collision cascade is approximated as

(14)where Ed is the average energy loss in the SE creation eventand in transit between creation events, This relation is simila r to other standard approximations for li d 17 and is equal tothe integral of n(x,Ed ) in Eg. (2 ) over the range R, so thatthe parameter Ed is essentially equivalent in the present andstandard models.

Assume, momentarily, that the primary electron travelsa straight line path of length R into the material, creating acylindrical affected volume of radius b. Th e actual primarypath and collision cascade shape are not critical, since the SEyield is determined primarily in the near surface region. Th ecylinder indicates that the probability of secondary creationis a function only of the distance from the path of the primary. The volume of the cylinder is defined as

(15)where ). 3 is the average volume within the collision cascadecontaining one secondary electron. Th e parameter A, is assumed to be the SE escape depth, which is approximately themean vertical distance between collisions traveled by a secondary anywhere within the actual collision cascade. Thisassumption is discussed further in Sec. V. The number ofsecondaries within a surface disk of thickness )., i.e., withinone escape depth of the surface, is rrb 2,1, 1,1,3 which, multiplied by the escape probability B and utilizing Eqs, (15) and(4 ) gives

2 E8=B rrb =B-,1,2 Ed R2384 J. Appl. Phys., Vol. 68. No, 5, 1 September 1990

(16)

This equation is identical to Eq. (5 ) of the standard model inthe limit R ).). and therefore indicates that the parameter A, isalso essentially equivalent in the two models. Th e stoppingpower proportionality in Eq. (16) is common to many mod-els and theories. Roughly half of the secondaries within oneescape depth of the surface are directed outward, suggestingthat the escape probability B is ~ O . 5 in the absence of anyextraneous surface barrier.

For a range R which is small compared to the cylinderradius, a cylindrical affected volume is no t reasonable. In thesemi-empirical sputtering model, J a spherical affected volume centered at the surface is assumed at very low primaryenergies, so that the probability of SE creation is a functiononly of the distance from the point of impact. The volume ofthis sphere, in analogy to Eq. (15), is given by

(17)At higher primary energies, the affected volume must become elongated, approaching a cylindrical geometry. Thesimplest analytical approach is to perturb the sphere into anellipsoid. 1 The relevant formula for the area of intersectionwith the surface (1TC2 ) of an ellipsoid of major radius a andminor radius b, centered at a depth d, is [from the ellipsoidrelation Cd 10)2 + (c/b)2 = 1J

(18)The volume of the ellipsoid jrrab 2 is again fixed by Eq. (17)so that

(19)The ellipsoid is centered at depth d = R 12 with major radiusa = r + R /2 , so that the ellipsoid is symmetric with respectto the assumed straight line path of the primary electron.This geometrical construction is illustrated in Fig. 2, wherethe dimensions are those obtained at energy Em,x for then = 4/3 case. Defining r = R 12r and using Eq, (19), wehave

d=rr,a = ( l + r)r, (20a)(20b)

1 r3 ,2=-=-- . (20c)a 1+yFrom Eq. (18), the following simple formula for the SEyield is readily obtained:8 B rrr2 _1_ [1 _ _r_Yl

..12 l+r l + r ) J (21a)

( E )2/3 1 + 2= 1.211 _P r .Ed (1 + r)3 (2Ib)Substituting Eq. (6) for the range (with n = 4/3) and Eq.(7 ) for the escape depth gives

" 2 8 E2 I3 1 + 1.44 E norm (22)unorm: := O norm J '(1 + 0.72 Enorrn ) -where

Emux = 9.0(lZ I!'IE Y3)and

S, A. Schwarz

(23)

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rvacuum

solidR/2

--------------v----------- ~

R range). escape depth

"

I'- -

R/2

r

FIG. 2. The geometric construction employed in the present model. R is therange of he primary electron, /l is the mean secondary escape depth, and risproportional to ),E ~ i ) [Eg. (22) J. The illustrated dimensions occur at energy Em", (n =0 4/3) where R /2 r = 0.72 and R / A ,= 2.3.

Dmax =0.58 BCEnmJEd)ZI:'. (24)Equations (23) and (24) for the maxima resemble those ofthe standard model, Eqs. (9) and (10), however, Emax nowdepends on Ed while Dmax is proportional to E ~ ; ~ ratherthan Emax ' independently of the value of n assumed. Theellipsoidal perturbation producing Eq. (21) does not approach the cylindrical limit at high energies as expressed inEq. (16) but is reasonable within the energy range of interesthere (El\orm < -10).For off-normal incidence, a rather complex expressionwas obtained in the semi-empirical sputtering model by tilting the ellipsoid along the direction of incidence. 1 Similarresults may be obtained however, as in the standard medel,by replacing the range R by R cos (} [ r by ]I cos (J in Eq.(21 ) J. In other words, the lateral motion of the primaryelectron is assumed irrelevant; only the rate of energy losswith depth is important. In this case

E ma , (8 )......:.::..::.:..__ = (cos 8) - [ [ / (n - V:I)IEmax (0 ) C2Sa)

2385 J< Appl. Phys., Vol. 68, No.5, i September 1990

(25b)(25c)

bee) ( { J ) ~ ~ 2 - - ; : : : ; ; cos (l0(0) (25d)These equations have significant implications as discussedshortly.

IV. COMPARISONSA. Energy dependence

The energy dependence of the SE yield in Eq. (22) isindicated in Fig. 1 by the solid circles. Fo r E > 10 theellipsoidal perturbation is inaccurate and o n ; n ; ' ~ s t in;teademploy Eg. (16). Equation (22) is also inaccurate whenEnorm < -0,5 where the approximations for Rand l1d [Egs.e6) and ( 14) Jare no longer valid. Appropriate correctionsare considered in Sec. V, Despite these limitations in the lowand high-energy regimes, the simple equation derived fromthe present model clearly provides a superior fi t to experiment over the entire illustrated energy range. The angle dependence results described below suggest that this good fit isnot accidental, but rather reflects the importance of the cascade geometry.

B. Angle dependenceA wealth ofdata for the effect ofthe angle of ncidence of

the primary electron on the magnitude of the SE yield provides the most important supporting evidence for the presentmodel. Of particular interest are three observations whichare not explained by the standard modeL First is the relatively weak dependence of Oma, , as compared to E max ' on incident angle, as noted empirically in Eq. (13). Second is therapid rise of the yield at high primary energies, where0(8)18(0) is approximated by (cos e) -' I with 1] > 1. Thirdis the apparent dependence of the cosine exponent "17 onatomic number Z.Figure 3 shows data from Salehi and Flinn 5 for theincident angle dependence of the yield ratio omax (B ) /omax (0) and the energy ratio Em"x Ce)/Emax (0). For constant fl , the standard model predicts that the yield and energy ratios should have the same dependence on incident anglewhile the present model predicts that the yield ratio shouldvary as the 2/ 3 power of the energy ratio [Eq. (25b) J. Thepresent model would therefore indicate that the slope of theyield curve in the log-log plot of Fig. 3 should be 2/ 3 theslope of the energy curve, rather than equal to it. The data infact indicates a slope reduct ion of approximately a factor of2, accounting for this factor in the empirical equation, Eq.( 13b). As suggested by Eq. (13b), this effect is generallyevident! 8 20 and receives a simple explanat ion in the presentmodel. The present model dictates that, for n = 4/3 in Eq.(25a), the energy ratio should vary as (cos 8) \ which, in aTaylor expansion as in Eq. (13a), would yield a k, value of0.5. This is consistent with the smooth surface value of ksuggested by Vaughn. 14 s

S. A. Schwarz 2385



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l.J- ~ ~ ~ ~ : ~ ~ : - - - - - - - - - - - ~ o J

1.a

~ I I ~ IJ J 0.6i'----'I: CAlC . 2 ~ I

a ~ O . 2

- ~ - ' - - - I - - ~ - - - r - - - r - - - . _ J . 0o 0.2 0.4 O.S 0.8 1.0- In cos 9

FIG. 3. Experimental data of Salehi and Flinn" for the relative increase ofthe energy maximum (open circles) and yield maximum (open triangles)as a function of cos f) (log-log) for the amorphous glass(V 0,l07 (P,O,)()3 . The present model predicts that the slope of the lowercurve should be 2/3 that of the upper curve; the standard model predictsequal slopes for the two curves; a strict stopping power proportionalitywould predict zero slope for the upper curve.

A typical plot of the yield variation with incident angle,with primary energy as a parameter, is given in Fig. 4(a)(from the data of Yang and Hoffman20 for the polymerKapton). The predictions of the present model are shown inFig. 4(b). Also shown are dotted curves indicating a simple(cos 8) - I dependence. Th e standard model, Eq. (12c). indicates that the plotted data should lie below the dottedcurve, while the present model, Eq. (25d), shows that the(cos 0) - t dependence may easily be exceeded. This largeangular dependence is also indicated by the observation [3 Jthat

0(8) = (cos (1) - 1).5(0) (26)

where 1] varies from 1.3 for light elements to 0.8 for heavyelements. The explanation for the atomic number Z dependence of the exponent Tf lies in the fact that, at a given energyEp. Enorm = EplEmax is much larger in the low Z materialowing to the Z dependence ofEmax [Eq. (23)]. InFig.4(b),Tf values of 0.8 and 1.3 correspond roughly to Eaorrn values of2 and 4, respectively.

In summary, the relatively slow variation of Om ax withincident angle () as compared to Emax , the (cos 8)- 'I dependence of the yield with 71 exceeding 1, and the dependence of1/ on atomic number and energy, are accounted for in thepresent model.

c. Material dependenceTabulations of experimental data for 8max an d Emax inmetals and insulators are examined below to demonstrate

that Dmax has a weaker dependence than Emax on materialparameters. Values of the secondary creation energy Ed andthe surface escape probability B are then deduced from thedata an d trends in these parameters are assessed.2386 J. App\. Phys" Vo\. 68, No, 5, i September 1990

'"] ""'" ;;.;;, ...-;.;;;.,.;;---------- ~ - ( 2.0 1 , .... &(6) ~ 6(0)/ooS6 : I1.1 - . - - - - - 3o I ~ ~ . : . . . : - - - - -

(a)-1.0 ------,-----,---'1- , - - -o 0.2 0.4 D.S O.B 1.0

1 - cos aa)

ate) = 8 (O)/cos e

o 0,2 0.4 0.6 O.S 1.01 - cos eb)

FIG. 4. (a) Experimental data from Yang and Hoffmall20 for the relativeyield increase vs 1 _00 cos O. in the polymer Kapton. The parameter is theapproximate value of the primary energy EI' normalized t o Em",' (b) Prediction of the present model (n ,= 4/3) from Eqs. (22) and (25). The dotted curves (llcos B) would not be exceeded in the standard model.

If it is assumed that the range R is proportional to E;with n held constant, then the standard model predicts that8""" is propor tional to Em", while the present model predictsthat 8max is proportional to E ;:;,. Figure 5(a ) shows a plotof om", vs. Emax from the data compiled by SeHer. l (Where arange of values were listed, the mean value is plotted.) Figure 5(b ) is a similar plot for insulating compounds based onthe data compilation of Grais and Bastawros,10 where thenumbers in the figure correspond to the numbering systememployed in that publication. Curves for 8 max


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1.5 - (a) ,78794931 47 75

26 50 2 '_1 741iI 5 72 1 9CE 1.0 40 ;t l.., '4 30 e' 8013 ,34 6

38 J!56 57E ~ x , '37lA' 55,1 9 metals

0.5 0 0.5 i.OEmax (keV)

20 I - 10(b) - 9,,15-1 ,1' G ~ 7

Eiiflax , ,, ,, --,- X> 2eV-- ~ : : i : ~ : : ~ J r insulators,-0 ,0 0.5 1.0 1.5 2.0 2.5

Emax (keV)FIG. 5. Experimental mean values of the yield maximum and energy maximum: (a) Data compilation for metals, from Seiler,' with atomic numbersindicated. (b) Data compilation for insulators, from Grais and Dastawros,'O numbered in order of increasing energy gaps (1::g ) as follows: I-PbS,2-Sb2 Te" 3-GeS, 4-PbO, 5-TeO" 6-Sb1 0" 7-NaI, 8-KI, 9-CsI, lO-CsBr,II-CsCl, 12-NaBr, 13-KBf, 14-RbCI, 15-LiBr, 16-KCI, 17-NaCl, 18-LiCl,19-NaF, 20-KF, and 21-LiF. The dashed curves ( a: En"" from the standardmodel) and solid curves (0 : E ; : ; ~ from the present model) approximatelybisect the data points (the insulators with electron affinity t> 2 eV excluded). Assuming a 50% surface escape probability ( B ' ~ 0.5). the secondary creation energiesE" indicated by these fits are -90eV [in (a) Jand-20eV (in (b)] for the standard model, and -60eV [in (al] and-5eV[in (b) J OI the present model. The high Ed value in metals suggests significant energy loss to conduction band electrons.

value in metals suggests that the development of the collisioncascade is limited by energy loss to the conduction bandelectrons. (This point is further discussed in Sec. V.) According to Eg. (23), the twelvefold reduction in Ed leads tomore than a factor of two increase in Emax which is quiteconsistent with the experimental data in t he figure [note thefactor of 2.5 energy scale change between Figs. 5(a) and5 (b ) J. The standard model requires a four-fold reduction inEd' ifB is assumed constant [Eq. ( 10) ], but does not predictany dependence of Emax on Ed [Eq. (9)]. The higher Emaxvalues in insulators would therefore be attributed solely tolarger escape depths. 21 While large escape depths are observed experimentally, the present model suggests thatgreatly increased secondary creation has an equally important effect on the magnitude of Emax .

The vertical displacement of any data point from thesolid curves in Fig. 5 determines the value of B / E;/3 in Eq.(24). This value is independent of the assumed relations for2387 J. Appl. Phys . Vol. 68, No.5, 1 September 1990

the escape depth and range. The values of the escape probabilityB deduced from Fig. 5 (b) using Eq. (24) are shown inFig. 6, where Ed is held constant at 5 eV. These values areplotted as a function ofthe electron affinityX as tabulated byGrais and Bastawros. o Th e plot clearly shows a sa turationof the escape probability at 0.5 for low X values and a 1I;ydependence for X> 1 eV. An inverse dependence on the surface barrier is predicted by the transport theory of sputtering. 22 The saturation effect is expected when X is less thanthe average emission energy of a few eV. Grais and Bastawros 10 note that no dear trend for B is observed in the standard model when Ed is assumed equal to Eg + X.

Th e insulators with high electron affinity are observedto have relatively low Ernax values in Fig. 5 (b). Since thesurface barrier is high, th e emitted secondaries arise from ahigher energy portion of the internal secondary energy distribution. The inelastic mean free path is a strong function ofenergy and decreases rapidly in the few eV region as theenergy is raised. A small mean free path or escape depth, inturn, leads to a smaller EmaK in eithe r model. Th e low valueof Ed for insulators in the present mode! is correlated withthe large observed escaped depths.

The analogous exercise of determining the surface factor B from the experimental data for metals has, to the author's knowledge, not been reported. The values of B deduced by application of Eq. (24) to the data of Fig. 5(a),with Ed = 60 eV, are shown in Fig. 7 versus atomic numberZ. An average value for B 0[0.5 is maintained over the entireZ range. The apparent scatter ofB values disguises some realtrends. The group IA elements, for example, form a lowerbound to the plot with an average B value of ~ O . 3 . This is atfirst surprising since these elements have very low workfunctions if; and would, by analogy to the X effect in insulators, be expected to have high yields. This effect was recognized, however, long ago and an elegant explanation waspresented by Baroody.6 Briefly, Baroody obtained the num-

0.75insulators

13

ill 0.50$"


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11,Or IJ' ~ - , AI Ga I> . 0 --.- 1------. - - - - - - - ~ _ /iIA I

0.5 Cl, (t 0 0 0 0 0 0' C - - - - - - - - C - - o - ~ 1i l I " I) c c ~ ~ " A ....O o ~ .. :. ...... . . o., . o 0 I..... :a---K--- - - - - - R b ~ -- A9-- - .. 0 II-B c. AyIII I-A Cs J'

oL t t t --,-_.o 20 40 60 00 100zFIG. 7. Surface escape probabilities (with E J fixed at 60 eV) fort he data ofFig. S(a), from Eq. (2 4), plott ed as a function of atomic number Z. Trendsfor the group I-A, II-B, (with valencies ofl) and III-A (with valencieso(3)elements are indicated.

ber of conduction electrons exceeding the surface bindingenergy by shifting the Fermi sphere in momentum space andcalculating the volume of the sphere lying above the threshold momentum plane (a geometric construction not unlikethe present model). An appropriate surface binding energyis EF + ; (Refs. 23 and 24) where EF is the Fermi energy.Inserting this energy in to Baroody's analysis, the followingexpression is obtained for th e number of secondaries generated per unit distance traveled by the primary:

N(Ep + cp) = 9.3 X 106(keV ll 2 em - 1)(E}.!2IEp cp),(27)With typical values inserted, this expression, multiplied byan escape depth A. of 10 A, predicts a yield t5 of 1. Thisequation also indicates that lower energy secondaries in thesurface region may be even more effective in exciting conduction band electrons. In the present model, Ep in Baroody's formula would be replaced by a mean cascade energy.Noting that E i{2 is directly proportional to the conductionelectron densityNv, where v .s th e valency, Eq. (27) predictsthat the contribution of he conduction band electrons to th eyield is proportional to vi p. This relation is appealing because an inverse dependence on work function d> is maintained, which can, for example, explain trends ~ i t h cesiation. With regard to Fig. 7, the illustrated trends are nowseen to be reasonable. The group I-A and U-B elements havea valency of 1and consequently low B values. The group IlIA elements, with a valency of 3, form an upper bound in thefigure with B values of ~ O . 6 . A more detailed analysis ofthese B values is unwarranted, in consideration of the largeexperimental error and the crude assumptions employed.The assumption that Ed is constant is further discussed inSec. V.

Finally, it remains to show that the approximations forthe range R and escape depth A. lead to reasonable predictions for omax and Emax . The predictions of Eg. (23) for Emuxand Eg. (24) for omax are shown in Figs. 8(a) and 8(b)respectively, and compared to the experimental data fromFig. 5(a). Constant values of Ed = 60 eV and B = 0.45 areemployed, so that the observed yield variation is due only toa geometrical effect, while variations ill the surface escape2388 J. Appl. Phys., Vol. 68. No.5, i September 1990

l.!E60

1.0

2.0

1.5

1.0'b

0.5

5 experimenta mooel

. experimentQ model

0o:fr.'" .Q 0'Q

r1fJa.Q'1"

04----- , - - - - ,0 20 40

100z

' \d '...ca-I' 0Q 1

(b)

If'd'b ."" .e' Q..

w..

B = 0.45 ',-----160 SO 100Z

FIG. 8. (a ) Eq. (23) of he present model for En"" (squares) is compared toexperimental data from fig. 5(a) (hullets) as a function of atomic numberZ, in the manner of Dno and Kanaya.9 (b) Eq. (24) of the present modelfor (j"h" (squares) compared to experiment. Ed is fixed at 60 eV and B isfixed at 0.45.

probability or secondary production are ignored. Nevertheless, the rise of the maxima with atomic number Z is wellaccounted for. The correlation with ionization potential I isweaker but still evident. (Ono and Kanaya obtain a superior fit to Om ax data with a similar formula, derived from adiffusion model in which the relatively weak Z dependenceof omax results from electron reflection effects.) Values of Bcould be derived from Fig. 8(b) to force an exact fit, however, these values would depend on the assumed materialdependences for R and A. while the values derived from theexperimental 0lllax / Emax ratios in Fig. 7 do not. Values of Edcould similarly be chosen to force a fit to the Emax data inFig. 8(a), however, these values would also be affected bythe assumed material dependencies, the large experimentaluncertainties, and the weak dependence of Emax on Ed'

V. DISCUSSIONIt is appropriate to consider whether the present modelcan be applied to ion-induced SE emission. In most cases of

interest, the ratioRIA. is too large (r 1 ) and Eq. (21 ) of thepresent model is therefore inapplicable. Figure 9 shows a fitof the standard model [Eq. (16)] to experimental data24--28

S. A. Schwarz 2388



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3 I --- 6 h B ~ : IJ '" _A uI "\ . "". . . . " i


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maximum is observed to occur near the energy corresponding to the maximum in the stopping power. For electroninduced SE emission, the total stopping power generallypeaks at energies wen below the observed Emax values, although the energy loss to inner shell excitations does peak inthe vicinity of Em x 30 The energy dependence of the stopping power, to the extent that it influences the parameter R,plays an important role in the present model. At high primary energies, as discussed, the stopping power proportionality is dominant. At lower energies, however, a strict proportionality to stopping power would not lead to an energymaximum Emux which varies with incident angle, nor wouldit predict an energy dependence for the relative increase inyield 8(e)/8(0).

Reflection or backscattering of primary ions has notbeen specifically accounted for in the present model. In thediffusion model,9 backscattering results in the relativelyweak Z dependence of Dmax as observed in Fig. 8. In thetransport modeV4 backscatter ing reduces the cosine exponent '1 in Eq. (26). I t is clear that backscattering should beexplicitly accounted for in a detailed theory. The effect ofbackscattering is, to some extent, implicit in the presentmodel. In the near spherical low energy limit, backscatteringis relatively unimportant since a large fraction of he primaryenergy is deposited near the surface. At higher energies, thereflection probability affects the probability of energy loss inthe near-surface region and can be approximated by asmooth etfece l on the Z dependence of the range approximation. Since the range approximation Eq. (6 ) already contains a strong Z dependence, which leads to a reasonable fit,the omission of the reflection coefficient is not considered tobe a major source of error. Indeed, the Z dependence of thereflection coefficient and the primary range have a commonorigin. In addition, backscattering may playa role in energydissipation measurements of the primary range, such asthose which produced the approximation for the range givenin Eq. (6). Ono and Kanaya9 have demonstrated the effectof variations in the backscat tering coefficient or the rangeexponent n on the shape of the universal curve. In the presentmodel, the experimental results could be mimicked by usingslightly increased n values in high Z materials.

There are several important theoretical approaches tothe problem of SE emission such as the diffusion models, 9the transport models,24 and Monte Carlo simulations, 16which account, ei ther implicitly or explicitly, for the nonlinear depth distribution of secondaries. These theories addressseveral issues, such as SE energy distributions, lateral distributions, and transmission yields, which are outside the scopeof the present model. The Boltzmann transport theory ofSEemission developed by Schou24 closely parallels the sputtering theory of Sigmund. 22 The spatial distribution for energyloss to ionization, D(E,x) , typically peaks at R 13 in thetransport theory, as compared to R /2 for the maximum ellipsoid width in the present model. The review by Schou24provides useful criticisms of assumptions in the semi-empirical models such as the use of mean values for Ed and },. Thepresent model should not be viewed as a substitute for existing theories, but rather as a useful means by which to visualize important experimental trends. A particular strength of


the present approach is the ability of the resulting simpleequation to simultaneously describe energy, mass, and angleeffects with good accuracy.

VI. SUMMARYA semi-empirical model was described which estimates

the SE yield as the fraction of displaced particles generatedin an ellipsoidal disturbed volume within the escape depth ofthe surface. The most commonly employed ("standard")semi-empirical model, in its simplest form, focuses on theinteraction of the primary electron with the near-surface region. Both models lead to simple formulas which predict thedependence of the yield on primary energy, incident angle,and material parameters. A detailed comparison of hese formulas to experimental data in the literature led to the following results: 0) The present model provides a superior fit tothe universal energy dependence curve over a wide energyrange; (ii) The present model predicts that the maximumenergy Emax is more severely affected by a change in incidentangle than the corresponding maximum yield Dmax , in accord with experiment; (iii) The present model predicts thatthe yield ratio O(tJ)/D(O) can significantly exceed(cos tJ) -- 1, in accord with experiment; (iv) The presentmodel provides an explanation for the (cos f) - 'I dependence of the yield ratio, where 7] is observed to vary from 1.3for low Z elements to 0.8 for high Z elements. I t also provides a reasonable estimate of 7]; (v) The present model provides a superior description for the material dependence ofthe experimental ratio omaJEmax; (vi) Experimentalomax 1Emux ratios determine the approximate value of thesecondary creation energy Ed' Ed is ~ 6 0 eV in metals and- 5 eV in insulators, suggesting that a large fraction of theprimary energy Ep is dissipated by the conduction-bandelectrons; (vii) The order of magnitude reduction of Ed ininsulators results in more than a factor of 2 increase in Emaxin the present model, as observed experimentally. This suggests that an increased production of bulk secondaries, aswell as an increase in the escape depth, is responsible for highinsulator yields; (viii) The ratio B 1 E ~ 1 3 , where B is the surface escape probability, may be deduced from experimentaldata without knowledge of he escape depth 11. orthe range R;(ix) The deduced surface escape probabilities B in insulatorsare inversely dependent on electron affinity..t, and saturatefor low values of X; (x) The deduced B values in metalssuggest a trend with metal valency, in accord with the modelof Baroody;6 (xi) Material trends are reasonably reproduced in either model with the range R and escape depth 11.approximations employed; (xii) Ion-induced SE emissiondata are generally outside the region of applicability of thepresent model, however, the values of the escape probabilityB and secondary creation energyEd deduced from one set ofdata are in good agreement with the values obtained for electron-induced SE emission; (xiii) A method to extend themodel to higher ( E ~ > lOEmax) and lower (Ep < O.5Emax)energies is described.

The above results arise in the present model from thenonlinear depth distribution of created secondaries. Themodel provides a useful visualization of several experimental

S. A. Schwarz 2390



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trends which are not well described in the standard model.The simple equation for the SE yield simultaneously describes variations with primary energy, incident angle, andmaterial parameters with sufficient accuracy to be of use inprocess simulations.ACKNOWLEDGMENTS

The author is grateful to C. C. Chang, D. M. Hwang, R.B. Marcus, and N. G. Stoffel for useful discussions.

1S. A. Schwarz and e. R. Helms. J. App!. Phys. 50, 5492 (1979).l s. A. Schwarz, J. Vac. Sci. Tech. A 6,2069 (1988).3 H. Salow, Z. Tech. Phys. 21 8 (1940).4 H. Bruinillg, Physics and Applicarions of Secondary Electron Emission

(McGraw-Hill, New York, 1954).'H . Seiler, J. App!. Pbys. 54, R 1 (1983).tE . M. Baroody, Phys. Rev. 71!, 780 (1950).7 G. F. Dionne, J. App!. Phys. 44, 536 t (1973).8 R. G. Lye and A. J. Dekker, Phys. Rev. 107, 977 (1957).oS. Ono and K. Kanaya, J. Phys. D. 12,619 (1979).IC K. L Grais and A. M. Bastawros, J. App!. Phys. 53, 5239 (1982)."J . R. Young, J. App!. Phys. 27,1 (1956).


12K. Kanaya and S. Ono, J. Phys. D. 11, 1495 (1978)."G . F. Dionne, J. Appl. Phys. 46,3347 (1975).14J. R. M. Vaughn. IEEE Trans. EI. Dev. 36,1963 (1989)."M. Salehi and E. A. Flinn, J. App!. Phys. 52, 994 (1981).16 Z. J. Ding and R. Shimizu, Surf. Sci. 197, 539 (l9B8).17 P. Sigmund and S. Tougaard, in Inelastic Particle-Surface Collifions,edited by E. Tauglauer and W. Heiland (Springer, Berlin, 1981), pp. 2-37.IS I. M. Bronshtein and V. A. Dolinin, Sov. Phys. Sol. St. 9, 2133 (1968).!"C. Bouchard and 1. D. Carette, Surf. Sci. 100, 241 (1980).1 0K . Y. Yang and R. W. Hoffman, Surf. Int. Ana!' 10,121 (1987).21 K. Kanaya, S. Ono, and F. Ishigaki, J. Pbys. D 11, 2425 (1978).22 P. Sigmund, in Sputtering by Particle Bombardment I (Springer, Berlin

1981) pr. 9-67.LJ M. S. Chung and T. E. Everhart, J. Appl. Phys. 45, 707 (1974).24 J. Schou, Scann. Micros. 2, 607 (1988).25R , A. Baragiola, E. V. Alanso, and A. Oliva Florio, Phys. Rev. B 19,121

(1979).2 A. Koyama, T. Shikata, and H. Sakairi, lpn. J. App\. Phys. 21, 65 (198!).27 D. Hasselkamp, Die ioneninduzierte Kinetische Elektronen-emission von

Metallen bei Miuier!!n und Grossen Projektil-energien (University ofGiesscn-Hll.bititationsschrift, Giessen, 1985).'8 H. H. Andersen and J. F. Ziegler, Hydrogen Stopping Powersand Rangesin all Elements (Pergamon, New York, 1977) pp. 1-317.'"e. J. Tung, 1. C. Ashley, and R. H. Ritchie, Surf. Sci. 81, 427 (1979).

J(J J. C. Ashley, C. J. Tung, and R H. Ritchie, IEEE Trans. Nucl. Sci. 222533 (1975).

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Application of a Semi .. Empirical Sputtering Model to Secondary Electron Emission - Schwarz