DEPTH OF ORIGIN AND ANGULAR SPECTRUM OF ... - Science Direct

Nuclear Instruments and Methods in Physics Research B36 (1989) 124-136 North-Holland, Amsterdam

124

DEPTH OF ORIGIN AND ANGULAR M. VICANEK Fysisk I~titut,

SPECTRUM

**§, J.J. JIMENFiZ RODRIGUEZ

Odense ~n~uers~t~t, LX-5230

OF SPUTI’ERED

ATOMS

* *& and P. SIGMUND

Odense M, Denmark

Received 13 September 1988 and in revised from 31 October 1988

A theoretical analysis is presented of the depth of origin of atoms sputtered from a random target. The physical model aims at high energy sputtering under linear cascade conditions and assumes a dilute source of recoil atoms. The initial angular dist~bution of the recoils is assumed isotropic, and their energy distribution is E- 2 like without an upper or tower cutoff. The scattering medium is either infinite or bounded by a plane surface. Atoms scatter according to the m = 0 power cross section. Electronic stopping is ignored. The sputtered flux, differential in depth of origin, ejection energy and ejection angle has been evaluated by Monte Carlo simulation and by five distinct methods of soluti,on of the linear Boltzmann equation reaching from continuous slowing down neglecting angular scattering to the P, approximation and a Gram-Charlier expansion going over spatial moments. The continuous slowing down approximation used in previous work leads to results that are identical to those found from a scheme that only ignoresangularscatteringbut allows for energy loss straggling. Moreover, these predictions match more closely with the Monte Carlo results than any of the approximate analytical schemes that take account of angular scattering. The results confirm the common assertionthat the depth of origin of sputteredatoms is determinedmainlyby the stopping of low energy recoil atoms. The effect of angular scattering turns out to be astonishingly small. The distributions in depth of origin, energy, and angle do not depend significantly on whether the scattering medium is a halfspace or an infinite medium with a reference plane. The angular spectrum comes out oniy very slightly over cosine from the model as it stands, in agreement with previous experience, but comments are made on essential features that are not incorporated in the physical model but might influence the angular spectrum. An improved set of constants matching the standard power cross section to Born-Mayer scattering brings the depth of origin of sputtered atoms into close numerical agreement with results from more reatistic simulation models.

1. Introduction It is generally accepted that atoms or molecules ejected from a solid surface in a particle-induced sputter event originate in a very shallow surface layer of the target [l]. At typical emission energies in the lower eV range, the depth of origin is at most a few monolayers, although theoretical estimates differ in detail [2-81. Early work by one of us [2] indicated an average depth of origin of - 5 A for atoms sputtered from a typical metal target, rather independent of bombardment conditions. Triggered by the need to estimate near-surface composition changes in alloy sputtering, this estimate was later expanded into a depth distribu-

* Work supported by the Danish Research Academy and tbe Deutsche Forschungsgemeinschaft.

8 Present address: Institut fur Theoretische

Physik, Technische UniversitLt, D-33 Braunschweig, FRG. ** Work supported by the Danish Natural Science Research Council. tati Permanent address: Universidad Complutense de Madrid, Faculty of Physical Sciences, Department of Electricity and Electronics, 28040 Madrid, Spain.

0168-583X/89/$03.50 Q Elsevier Science Publishers B.V. (~o~h-Ho~~d Physics publishing Division)

tion which turned out to be close to exponential [3]. That result was reasonably consistent with those found by Monte Carlo simulation [4,5], while lattice simulation codes indicated atoms to be ejected mainly from the top two surface layers [6,7,10]. There are several reasons for considering the topic once more. Most of all, the issue has turned up to be more important than anticipated earlier 1111. There is ampie evidence to demonstrate strong composition gradients in alloy sputtering, such as the fact that different surface-sensitive techniques give different signals [12-141. Such composition gradients in turn affect the angular distribution of sputtered particles 114-161. Earlier analytical estimates 13,151 were based on continuous slowing down of recoil atoms moving toward the’ surface. Evidently, the neglect of scattering and energy straggling is plausible for qualitative orientation but not necessarily for a detailed analysis of experimental data as attempted recently [17,18]. The very fact that the above simple estimate disregarding angular scattering yielded [3] the standard expression for the sputter yield [2] could be considered as a surprise, but may also indicate an effective compensation of scattering events into and out of a given angular

M. Vicanek et al. / Depth of origin and angular spectrum of sputtered atoms interval. In that connection, the role of angular scattering has been misunderstood [19]. Although little attention has been paid to the point, one Monte Carlo simulation indicated the existence of a pronounced maximum in the distribution of sputter depth a few A inside the target [4], in obvious contrast with the simple analytical estimates [3,15] based on a similar physical model. Since that maximum appears to be statistically si~fic~t, it would, if real, seem to be an essential feature for the analysis of experimental data. The present study was started some time ago with the purpose of clearing up the points mentioned above. A closer study of the literature revealed that the differences in apparent sputter depth predicted from different simulation codes actually exceeded the difference between predictions from analytic transport theory and those from simulation. It was found desirable, therefore, to carry out a round robin simulation of a sputter ejection event at the same time, in order to pin down systematic differences between different simulation codes and their possible causes [9]. The two studies address different aspects of the same problem and are submitted as companion papers. Ref. [9] is based only on simulation techniques and aims at a comparison between the different techniques. The event considered in [9] is relevant to sputtering, but the depth distributions evaluated there are not depth distributions of sputtered atoms, because no averaging was performed over the energy spectrum of recoil atoms as well as their distribution in source depth. In the present work, we consider actual, theoretical depth-of-origin distributions of sputtered atoms, found on the basis of well-defined, although admittedly simp~~ng assumptions, by applying several schemes of solving the transport equation. Moreover, Monte Carlo simulations have been performed that match as closely as possible the underlying physical model. The Monte Carlo code used here [20] was also part of [9]. In this way, there is a direct link between the two papers. Since the results quoted here concern a physical quantity that differs from the one considered in 191, it was found appropriate to report them separately.

2. The model The depth of origin of sputtered atoms is determined by low-energy recoil atoms. The pertinent physical parameters are the cross sections for energy loss and scattering of such atoms, their initial distributions in depth, energy and angle, the surface barrier and, of course, the detailed structure of the surface. Notwithstanding the important role played by the latter, it appears appropriate for the present purpose to keep to the standard model of a halfspace, filled homo-

125

geneously with atoms, and limited by a perfect, planar surface. We shall also keep within the standard model for a planar surface potential, although the way the calculations are done, that assumption enters only at a very late stage. For the initial distribution of recoil atoms we aim at a situation characteristic of high-energy ion bombardment where many recoil atoms (primary and secondary ones) are generated over a wide range in depth and energy, and where the angular distribution is not far from isotropic. Then, within the depth range pertinent to sputtering, we may consider the following ideal situation: The entire halfspace is a dilute source of recoil I) atoms. II) The angular distribution of the recoils is isotropic. III) The energy distribution of the source is a Em2 for O~Ecoo.

IV) The source may, but need not, be constant in depth. Assumption II) cannot be strictly valid within the shallow surface layer that is of interest here, but taken together with IV), it provides a useful standard model that can be improved systematically on the basis of more specific knowledge from energy deposition theory or computer simulation. Assumption III) appears less problematic. The Ev2 recoil spectrum seems more robust than any of its theoretical derivations, and extending the upper limit to infinity is known to be immaterial to the sputter yield [2]. Dropping a possible lower limit implies in practice that the spectrum is valid down to the surface binding energy. This assumes the underl~g collision cascade to be linear 11). With regard to energy loss and scattering, calculations have been performed on the basis of the standard power cross section (21,221,

da(E,

T) = C,E-mT-‘-”

dT,

O 1 which determine the spectrum of sputtered particles while the total cross section, Jda, claimed to be relevant in 1191 (despite being divergent!) does not enter. For this reason alone, the conclusions of [l9] must be invalid. More severely, if energy loss is ignored, all particles of the source must eventually emerge; therefore, the sputter depth becomes identical with the source depth, which happens to be infinite in the present model (and very large in ref. [19]).

-(l-i)

l+zmg(5(1-

t)“-)]

= S(5).

(12)

In the limit of m= 0, this equation simplifies drastically. Not only does it reduce to a differential equation since the g-function goes in front of the integral; more importantly, the remaining integral over t reduces to the stopping cross section, i.e., straggling and all higher moments over the cross section drop out. Therefore, the limit of (11) for M = 0, g(.$, cos 8) = @(t/cos

0) ]l/cos

61 e-t’cose,

(13)

actually requires only angular scattering to be excluded while straggling does not even enter. 3.3. Legendre po~nomial

expansion

Allowing now for angular scattering we expand the flux density in terms of Legendre polynomials, noting that eq. (IO) remains valid, so g(E, cos @) = : (21+ l)&(OPI(== f-0

(3).

(14)

-P,((l-

-(l-t)

(17)

3.5. General case for m = 0 Going back to the general case including both scattering and energy loss, we note that also eq. (15) reduces to a set of differential equations in the limit of m = 0. Since this, at the same time, is by no means an unrealistic limit [2], all actual numerical estimates will be given for this case. We write eq. (1.5) in the form j$&-I

+ (1+

l)g,+,l+ P+1)4&=4oS(5)> 08)

where

This yields the following system of equations for the coefficients g,( .$), a

t)“‘)].

(19) or

1+2mP,(COS+)g,(EQ

- t>2m)],

(15)

where cos Q,= (1 - t)l12. 3.4. Energy loss ignored Before going on, we may try to apply an assumption complementary to the one made in 3.2. Let us, for a

-W)l9

(20)

II/being the logarithmic derivative of the gamma function. Numerical values of the A, are given in table 1. The remaining difficulty is the structure of eq. (18) which does not readily allow a determination of the g, recursively. Approximate solutions can be found by truncation (see below).

M. Vicanek et al. / Depth of origin and angular spectrum of sputtered storm

128

1221, each function g,(E) was constructed separately from its moments g;. Since g; is nonv~s~g only for n = f, I + 2, I + 4 ,.. . , the width of the Gaussian base functions was fixed by the convention that in the expansion in modified Hermite polynomials He,,,

Table 1 The coefficients A,, eq. (20) I

-4

0

1.ooo 1.280 1.750 2.2808

1 2 3

gl( L, with L = 1 or 3. Results have been listed in table 3. It will show up below that the Pt approximation yields even less credible results than the moments method. Its predictions have been included below mainly for comparison with the Ps approximation.

In order to have a reference for comparison with the approximate analytical solutions we also solved the transport problem by numerical simulation. We applied an existing Monte Carlo code 1201 and the cross section (1) for m = 0. The influence of the surface was investigated by simulation of both the halfspace and the infinite medium. The code [20] has been designed from the outset to simulate as closely as possible the physical situation specified by the linear Boltzmann equation. In addition to the features described in ref. [20], we introduced two modifications in order to make the code more efficient for this specific problem. Firstly, since only the position x - x0 of the surface relative to the source enters, we may keep the starting position of the particles fixed and vary the position of the surface instead of the reverse. This is what also was done in the above analytical scheme. Moreover, by placing detectors at various depths, we may extract more than one piece of stochastic information from any individual trajectory. For an infinite medium, this

2.5,

8 = 0” -

,,

,

,

1 ,

,

(

1 ,

I

,

)

,

,

,

I

I

e= SO”

r

bj

1

h4Onte Cart0 (half) 2

-‘-.

t

a

Straight

Ime

1.5

_

‘:

6;

9 Stmght

Y2 E

Mcnte Carlo (ho(f) Monte Carlo (infInite)

ltne

1.0

0.5

0.0 0.0 NC;X

0.5

1.0

1.5

2.0

NC;x

Fig. 2. Distribution in depth of origin of ejected particles. Surface barrier not included. Solid line: Monte Carlo, halfspace; dotted line: Monte Carlo, infinite medium; short-dashed line: PI approximation; long-dashed line: Ps approximation; dot-dashed line: straight line approximation. a) Normal ejection. b) Ejection at 60 ‘.

130

M. Vicanek et al. / Depth of origin and angular spectrum of sputtered atoms

scheme is straightforward. For a halfspace, only firstpassage times may be (and have been) counted. The second modification introduced is more specific to the present task and serves to directly simulate the scaled quantity g(& cos e), i.e., to eliminate the energy variable. Note that this particular feature is a direct consequence of the source spectrum (5) which has no upper or lower limiting energy Ea. If we introduce a cutoff energy EC, below which particles are not generated, the simulation yields the correct emitted spectrum only for E > EC. Because of the scaling property, this information is sufficient to determine the spectrum at any energy.

!i

1’

15 G a, g

1.0 --.-.-.-._._. \ \ \

0.0

1.0

0.5

cos 0

5. Results Fig. 2 shows calculated profiles for the depth of origin of the emitted particle flux at a fixed emission energy. No surface barrier has been imposed. Because of the scaling properties of the underlying physical model, the same profile holds for all emission energies. Fig. 2a concerns normal ejection. We first observe that the two Monte Carlo curves for the half space and the infinite medium coincide very closely. This is not surprising, since double crossings of the surface must be rare events after normal ejection. Next, the straight line (or continuous slowing down) approximation (12) comes surprisingly close to the Monte Carlo results up to about 6 = 1.5. Note that in that approximation, the average emission depth is (4) = 1 [3,15]. The Pi approximation is seen to yield poor results at all depths while the Ps approximation breaks down at small but is accurate at large depths. The fact that the latter curve shows a maximum at nonzero depth is unquestionably an artefact of the Ps approximation. Fig. 2b shows the corresponding information for oblique ejection at 8 = 60 “. The profile is steeper, roughly by a factor of cos t9= 0.5, cf. eq. (12), since particles emitted at oblique directions have a shallower depth of origin [3,15]. The Ps approximation turns out to be a more appropriate scheme to describe these processes than for normal ejection, presumably since emission in oblique directions allows for more scattering events. Again, the straight line picture appears accurate up to slightly above the average ejection depth but overestimates the emission probability at larger depths. The Pi approximation appears better than for normal ejection but is still unsatisfactory at all depths. Note that there is no indication of a maximum at a nonzero depth. Most strikingly, the Monte Carlo results for the halfspace still coincide with those for the infinite medium within the accuracy of the drawing. The latter observation confiis the qualitative argument put forward in [l] which is based on the steep recoil spectrum, a EL’. Indeed, an atom passing the

-

,

1.5-

,

,

,

,

)

,

I

,

_ NC;x = 0.3 -

MonteCarlo Monte

I

Carlo

I

I

0.0

,:’

(half 1 (infinite)

I

I

I

I

b ,‘. ,,*’

I 1.0

0.5 case

0’15 p-z-F-q -

0.10 -

”

Mcnte

Carlo

(half)

"".'Monte Carlo(lnflnlle)

al 8 G

Fig. 3. Angular distribution of ejected particles recoiling from a fixed depth of origin. Surface barrier not included. Notation as in fig.2. a) Depth of origin zero. b) Depth of origin 0.3/NCo. c) Depth of origin 2.O/NC,.

131

A4. Vicanek et al. / Depth of origin and angular spectrum of sputtered atoms

plane x with some energy E, may be scattered back into the halfspace which it had left and pass the plane once more in the same direction, if the medium is infinite. Such a sequence of events requires several large-angle scatterings that are associated with substantial energy loss. Consequently, on the second crossing, the energy has degraded to some value E2 that must typically be 4 E,. Since EC2 GZ ET2, the contribution of these trajectories to the total emitted spectrum at E, is insignificant, and first passages dominate. This argument must break down at grazing ejection. Fig. 2b demonstrates that it is still valid at 60 O. Fig. 3 shows similar information, now plotted as a function of ejection angle with the depth of origin fixed. The function cos @g([, cos 8) has been plotted since according to the text to eq. (6), that expression represents the angular dependence of the sputtered flux. Fig. 3a shows emission from depth zero. The straight line approximation is a convenient reference standard since it implies isotropic emission. The Monte Carlo results show a slight enhancement at oblique ejection due to atoms recoiling away from the surface but deflected toward the surface subsequently. Even though the difference between the halfspace and the infinite medium is now noticeable, it is still strikingly small. The Pi and the Ps results deviate drastically from the numerical solution. Evidently, surface emission requires an accurate solution in the vicinity of the sources, and this requirement clashes with the physical picture underlying the P, approximations. Fig. 3b shows similar information for the depth 5 = 0.3. The straight line picture now predicts an exp( -.$/cos 0) law according to eq. (12), i.e., a depletion at oblique exit because of the longer way to the surface. It describes the overall behavior reasonably well, whereas the Pi result turns out to be rather poor. The Ps approximation follows quite closely the numerical curves at almost all angles. Particles coming from a sizeable depth E = 2.0 (fig. 3c) are strongly collimated in the normal direction [15]. The P3 approximation characterizes this situation very accurately whereas it is evident that the straight line picture cannot handle emission from twice the average depth of origin or more. Somewhat surprisingly, even the Pi approximation is too crude to describe the angular distribution. 6. Application to sputtering According to eqs. (6) and (lo), the particle flux through the surface plane, differential in depth of origin, energy, and angle is given by J(x,

E, cos l?) dx dE d2Q

=g($, cos 8) dx -ddE Q &TE=

where no allowance has yet been made for a surface barrier. In high energy sputtering, the source density is given approximately by [l] Q = FF,(O), (31) where F,,(x) is the average energy deposited per unit depth at depth x per bombarding ion. With this, the differential flux (30) becomes the emitted flux per bombarding ion. In writing (31), we performed a switch in coordinates from a source at depth 0 and a surface at x to a source at -x and surface at depth 0. The main physical assumption underlying the present picture is a slow variation of the source density over the depth range pertinent to sputtering. This assumption is never fulfilled rigorously, but it is less questionable at high than at low ion energies. The constant F in (31) has the value 6/n2 for m = 0 WI.

Integration over all depths yields the differential sputter yield, with the surface barrier still being disregarded, J(E,

COS e)

dEd=Q=$$-z

d29,

eg(COS

e),

(32) where g(c0s

Jomdgg(5.

e)=

(33)

cos e).

For straight line motion, eq. (12) yields [3] g(cOs

(34)

e)=i,

while in the Pi approximation, we find g(cOS 8) z 0.5 + 0.764 COS8.

(35)

The Ps approximation yields g(cos 0) = 0.5 + 1.242 COSfI - 0.910 cOS38.

(36) Fig. 4 shows angular dependences cos Bg(cos 8), cf. eq. (32), still disregarding surface binding. Again, we find a negligible difference between the numerical results for the halfspace and the infinite medium. Both angular distributions are very slightly over cosine. This feature is overestimated by the Pi solution but is better described by the Ps curve. A solution found by the moments method based on moments 1 I 3 and n I 7 has been included in this graph. It appears to overestimate the over cosine character of the distribution. In addition, it lies a bit low. The straight line approximation agrees most closely with the numerical result, although it predicts a strict cosine distribution. Integration of (32) over energy and angle with a planar surface potential, E cos2e>

(37)

u,

leads to the sputter yield Y= $&/id

COS 8

%coS

0

cos e c0s3eg(c0s

e) =

0

(38)

132

M. Vicanek et al. / Depth

of origin

and angular spectrum of sputtered atoms

0.3

,,1,,,,,,,,,,,,1,,,

all angles -

Monte .’ Monte

-..-

Corio (half) Carla ( InfInIte)

MonteCarlo Monte Carlo ...__.p

(half) (infinite)

Moments

NC; x Fig. 4. Angular distribution of emitted particles from all depths. Surface barrier not included. Notation as in fig. 2. Double-dot-dashed line: Moments expansion, I < 3 and Y I 2 in eq. (24). F$. (38) demonstrates the familiar fact that the planar surface barrier favors sputtering of particles approaching the surface close to normal. Values of the normalized sputter yield y as defined by eq. (38) are given in table 4. We observe that all values agree to better than 20%. If we exclude the two poorest values (P1 and moments), the scatter is about 1%. We find it astonishing that there is no more than a 1% difference between a model taking into account the angular scattering of low energy recoils and another one disregarding it. As was found previously 19, the value y = l/8 obtained from the straight line picture coincides with the standard result found from the asymptotic solution of the Boltzmann equation, where an assumption such as straight line motion of any particle does not enter explicitly. It is, however, a matter of fact that the scattering angle between low energy recoil atoms does not enter any of the results of that work. This input is Table 4 Sputter yields and mean sputter depth Scheme Continuous slowing down Straight line PI Ps Moments Monte CarIo, infinite med. Monte Carlo, halfspace

Sputter yield

Sputter depth

.Y>eq. (38)

(xv%,

0.325 0.125 0.139 0.122 0.111

0.800 0.800 0.510 0.658 0.693

eq* (3%

Fig. 5. Distribution in depth of origin of particles ejected at ail angles. Planar surface baker included. Notation as in fig. 2. A result from the moments expansion has been included (longdashed line), similar to fig. 4.

thrown away at the point where the asymptotic assumption of high ion energy compared to the energy of sputtered atoms is introduced. This, of course, explains the finding of [3] that the continuous slowing down picture is able to reproduce the sputter yield found in 121. Fig. 5 shows the depth dist~bution of sputtered atoms integrated over all emission angles, i.e., y( 5) with /zdEy(= ~~d~jdEjd’~J(~, j-dxjdE/d’QJ(x,

x

S&m-

1

NG

0.1235

0.639

0.1235

0.639

E,cosB)

14 dtld

E, ws 8)

cos

e c0s38g(E,

’

cam

8)

.

(39)

d< d ws 3 eos38g(&, ws 8) I J

Table 4 shows calculated values for the mean sputter depth in units of (NC,)-‘, (0 = NC,,(x). It is seen

M. Vicanek et al. / Depth

of origin and angular

that the two Monte Carlo simulations for the halfspace and the infinite medium yield identical values within three significant digits, i.e., well witbin the accuracy of the simulation. The straight line approximation overestimates the mean depth of origin because of the neglect of angular scattering, yet the error is no more than 20%. On the other hand, the P1 approximation overestimates the angular scattering and gives a smaller mean depth. The value obtained from the P3 scheme is in good agreement with the Monte Carlo result. It appears that the present calculations confirm the validity of the theoretical scheme applied in [2], in

spectrum

of sputteredatoms

133

particular the adoption of an infinite medium for the emission stage and the continuous slowing down approximation for ejected low energy atoms [3]. Why then is the depth of origin of sputtered atoms estimated in [2,3] about a factor of two larger than that found in Monte Carlo simulations based on a very similar interatomic potential [9]? In order to relate the dimensionless quantity (t) to a length, one has to assign an appropriate value to the constant C,. The value recommended in ref. [2] was

c, = &12,

a = 0.219 A

(4Oa)

Born - Mayer

I

id’

f (t”Z) -2 IO

,J = AdrJo

/!’

1

S,(M,‘M*) 4M,

a*

A

ld3

ld4

/ / I If

Id5

r

IO

6

1oc

lo-’

Id3

lo-’

b

,

, ,

-

”

1

t ‘12 0.5

I

I

1

m 0.4 0.3 0.2 -

0.0

,(js

I 10-4

I 1()-s

C to-2

10-l

E,IA Fig. 6. Classical scattering theory for Born-Mayer interaction potential, V = A exp( - r/a).

0

01

0.2

0.3

0.4

0.5

m

a) Differential cross section in modified Lindhard variables where t’j2 = c sine/Z, C= M2E/[(Ml+M2)A], and fI the scattering angle in the cm. system. f(t’j2) = (2f3’2,/~~2) do/dt. Solid lines: Numerical results for 6 =lO-‘-lo-‘. The window contains that part of the diagram which was utiIized in ref. [Z] to establish the scaled cross section with the constants (@a). That cross section is also shown (dashed line). b) Stopping cross section versus relative energy. Solid line: Numerical result. Dashed line: Scaled cross section from [2]. c) Exponent m in power law (l), versus relative energy E, = M,E/( MI + M2). found from the slope of fig. 6b. d) Numerical constant A, versus power exponent nz. A,,, is defined by s(c) = h,/[2(1m)] f’--2m [22], where s(c) = (kf~ + M2)/(4M,).&/(na2A) is the reduced stopping cross section.

134

M. Vicanek et al. / Depth

of origin and

and X, = 24,

(4Ob)

which gives C,, G 1.81 A2. The value for ha stems from a fit of the power law (1) to a calculated cross section for Born-Mayer scattering [27]. The fit was performed in an energy region which was somewhat above the typical ejection energies. The reason was a lack of easy access to more appropriate data. Fig. 6a shows an extended version of fig. 4 in ref. [2]. It is seen that the m = 0 power law is a reasonable average in the range of energy and scattering angles analyzed in [2], but it underestimates the cross section toward lower values of the energy-angle variable t (defined in the caption). Fig. 6b accentuates this feature with regard to the stopping cross section. Note that for copper, the energies of ejected atoms are typically in the range of E = 10-4-10-3, where the scaled stopping cross section underestimates the numerical result by about a factor of two. Note also that the scaled cross section corresponding to m = 0 is significantly steeper than the numerical result. Fig. 6c shows an effective m-value, determined by matching a power law to the stopping cross section shown in fig. 6b. It is seen that at energies pertinent to ejected atoms, m lies around 0.1 or slightly higher for Born-Mayer interaction. Finally, fig. 6d shows the numerical constant X, which is connected with the constant C,,, in (1) [21,22]. We note that the actual curve is not universal but somewhat dependent on the potential (here Born-Mayer potential).

7. Discussion Measured angular distributions of sputtered atoms are typically cosine or over cosine shaped, except at ion energies near threshold. Some systematics is known. For example, impurity layers due to poor vacuum tend to cause narrower angular distributions [28,29]. There has also been found a pronounced tendency towards narrower distributions with increasing ion energy [30]. While the former feature is to be expected, the origin of the latter has been discussed [25,30] but is still not entirely understood. The present work, up till now, predicts the sputtered flux to be very close to cosine distributed according to fig. 4. While this is by no means a new result of the theory of random collision cascades, the present calculation is less restrictive than previous ones with regard to simplifying assumptions that enter. Recall that we have allowed for a halfspace, for angular scattering at all energies, and for fully stochastic energy loss.

angular spectrum of sputtered atoms

Without going into details, we briefly mention several features that have not been included, and which tend to narrow angular distributions. We recall that the source of recoils has been assumed homogeneous. For sputtering at very low ion energies, one may readily envisage a source profile that decreases rapidly with depth, while for high energies, the reverse is to be expected since the surface is a sink for energetic recoil atoms. This means that energy cannot be dissipated to recoil atoms with full efficiency in the vicinity of the surface. As a consequence, the deposited energy profile Fo( x) occurring in eq. (31) must show a sharp decrease very near the surface [24,31]. It is partly a matter of definition whether FD approaches zero or a nonzero value at x = 0. At any rate, eq. (31) can hardly be maintained in a theoretical attempt to accurately predict a fully differential sputtered flux. For qualitative orientation, we note that a source profile behaving like Q(x) a xa immediately leads to an angular distribution o cos1+‘V3, according to eqs. (12) and (33). An exponential profile, Q(x) = Q,(l exp( -x/b)), with some characteristic depth b, likewise leads to an over cosine distribution with (YG 2 bNC,, for bNC, -=z 1. Evidently, a noticeable deviation from the Knudsen cosine law requires a noticeable variation of the source within the average sputter depth (NC,,-‘, if the deviation is to be explained exclusively by spatial variations in the recoil source. A quantitative discussion would have to incorporate the dependence on ion energy of the gradient of the deposited energy profile. The treatments in [24,25] show some basic features but are not meant to yield accurate numbers. Intimately connected to the inhomogeneity of recoil sources, but an even more direct cause of a non-cosine sputtered flux, is cascade anisotropy [1,2]. It is known that the deposited momentum is directed toward the surface until a certain depth within the target [32] resulting in a narrowing of the angular distribution of sputtered particles. The pertinent correction, however, is decreasing with increasing bombarding energy, thus it does not explain the finding in ref. [30]. Another approach which takes into account the presence of a surface [23,25] yields a narrowing of the angular flux which is increasing with increasing bombarding energy, attaining a certain asymptotic magnitude for high energy. This seems to agree with the finding of ref. [30] for not too high energies. However, some caution is appropriate since the results of [23,25] have been obtained with an energy independent cross section. In particular, the fact that the sputtered flux becomes less over cosine for the highest energies [30] might be due to the combined effect of the cascade intersection with the surface and the energy dependence of the primary cross section. There are at least two more items to be mentioned in connection with the angular distribution of the sputtered flux. An atom passing the surface at an oblique direc-

M. Vicanek et al. / Depth oforigin and angular spectrum of sputtered atoms tion experiences a net deflection toward the (outward) surface normal since scattering centers are located preferrably on the target side of the trajectory [30,11]. This effect contributes to a further narrowing of the angular distribution. On the other hand, every atom leaving the surface must overcome binding forces which, in the case of planar binding, result in a deflection away from the surface normal. The magnitude of both effects is increasing with decreasing ejection energy, thus the resulting deflection might not be very drastic. In any case, neither of these effects can be responsible for the dependence of the angular distribution on the bombarding energy since both effects are determined by target properties alone. Finally, we remark that a proper adjustment of the constant C, (or, more precisely, A,) not only influences the depth of origin of sputtered atoms but also, through eq. (38), the total yield. With the conventional value C, = 1.81 AZ, yields from refractory metals (bee lattices) are generally overestimated. The same holds for light-ion yields. On the other hand, heavy ion yields for fee metals are commonly well-described or underestimated. With a larger value for C, as proposed here, the calculated yield becomes smaller. The full consequences will have to be discussed separately.

8. summary a) We have calculated the sputtered flux, differential in depth of origin, emission energy and emission angle. The physical model assumes a homogeneous source of recoils over a halfspace, distributed isotropically in direction and like EC2 in energy. Full allowance is made for recoil atoms to lose energy and get deflected by random collision events on their way to the surface. Scattering is governed by the standard power cross section with m = 0. b) Calculations have been performed by means of five different appro~mations for the solution of the linear Boltzmann equation. Moreover, a Monte Carlo simulation code has been utilized that was designed to reflect the same physical model as closely as possible. c) The Monte Carlo calculations show that in the present context, the agreement between predictions found under the assumption of an infinite scattering medium and those for a halfspace is very good to excellent, dependent on the considered quantity. d) Comparison between Monte Carlo results and those found by approximating particle trajectories by straight lines shows that the effect of angular scattering of the recoil atoms on the distribution in depth and emission angle is insi~ficant. e) The effect of angular scattering has also been evaluated analytically by three analytical schemes, none of which yielded results that were comparable in accu-

135

racy with those found by approximating trajectories by straight lines. The P, approximation was found unsatisfactory, while the moments method and the P3 approximation showed good agreement with the simulations in some cases and poor agreement in others. f) These approaches lead in several instances to distributions in depth of origin that showed a maximum at a nonzero depth. It appears that these maxima are artefacts introduced by the calculational method. g) The depth of origin of sputtered particles was found to be determined primarily by the stopping of the recoil atoms on their way to the surface, in agreement with what was asserted previously without proof. h) The angular distribution of the total emitted particle flux was found to be close to cosine shaped in the present model. This result is consistent with previous findings on the basis of the theory of linear collision cascades, yet here it has been found under less restrictive assumptions. i) Near-surface gradients in the distribution in depth of the recoil source may cause drastic deviations from the cosine law. j> Surface processes may evidently affect the angular spectrum but have largely been ignored. The results shown in fig. 5 and table 4 incorporate a conventional planar surface barrier, while all other results reflect the purely collisional properties of the particle flux. k) An improved estimate of the constants determining the power approximation to the Born Mayer scattering cross section leads to a smaller value of the mean depth of origin of sputtered particles than what was reported previously.

References

Ill P. Sigmund, Top. Appl. Phys. 47 (1981) 9. PI P. Sigmund, Phys. Rev. 184 (1969) 383. 131 G. Falcone and P. Sigmund, Appl. Phys. 25 (1981) 307. 141 J.P. Biersack and W. Eckstein, Appl. Phys. A34 (1984) 73. 151T. Ishitani and R. Shimizu, Appl. Phys. 6 (1975) 241. 161 M.T. Robinson, J. Appl. Phys. 54 (1983) 2650. 171 M. Rosen, G.P. Mueller and W. Fraser, Nucl. Instr. and Meth. 209/210

(1983) 63.

181 For further references, cf. ref. 191. 191 P. Sigmund, M.T. Robinson, MI.

Baskes, M. Hautala, F.Z. Cui, W. Eckstein, Y. Yamamura, S. Ho&a, T. Ishitani, V.I. Shulga, D.E. Harrison, I.R. Chakarov, D.S. Karpuzov, E. Kawatoh, R. Shimizu, S. Valkealahti, R.M. Nieminen, G. Betz, W. Husinsky, M.H. Shapiro, M. Vicanek and H.M. Urbassek, Nucl. Instr. and Meth. B36 (1989) 110. 1101 D.E. Harrison, Radiat. Eff. 70 (1983) 1. 1111 P. Sigmund, Nucl. Instr. and Meth. B27 (1987) 1. 1121 G. Betz and G.K. Wehner, Top. Appl. Phys. 52 (1983) 11.

136

M. Vicanek et al. / Depih of origin and angular spectrum of sputtered atoms

[13] M.F. Dumke, T.A. Tombrello, R.A. Weller, R.M. Housley and E.H. Cirlin, Surf. Sci. 124 (1983) 407. 1141 T. Okutani, M. Shikata and R. Shim& Surf. Sci. 99 (1980) L410. [15] P. Sigmund, A. Oliva and G. Falcone, Nucl. Instr. and Meth. 194 (1982) 541. [16] M.R. Weller, Nucl. Instr. and Meth. 212 (1983) 419. [17] R. Kelly and A. Oliva, Nucl. Instr. and Meth. B13 (1986) 283. [18] R. Kelly, Nucl. Instr. and Meth. B14 (1986) 421. [19] G. Falcone, Phys. Rev. B33 (1986) 5054. [20] M. Vicanek and H.M. Urbassek, Nucl. Instr. and Meth. B30 (1988) 507. 1211 J. Lindhard, V. Nielsen and M. Scharff, K. Dan. Vidensk. Mat. Fys. Medd. 36 (1968) no. 10. [22] K.B. Winterbon, P. Sigmund and J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37 (1970) no. 14. [23] K.T. Waldeer and H.M. Urbassek, Nucl. Instr. and Meth. B18 (1987) 518.

[24] H.M. Urbassek and M. Vicanek, Phys. Rev. B37 (1988) 7256. [25] K.T. Waldeer and H.M. Urbassek, Appl. Phys. A45 (1988) 207. [26] P. Sigmund, Appl. Phys. Lett. 14 (1969) 114. [27] M.T. Robinson, Tables of Classical Scattering Integrals for the Bohr, Born-Mayer, and Thomas-Fermi Potentials, Oak Ridge National Laboratory, ORNL-3993 (1963). [28] F.R. Vozzo and G.W. Reynolds, Nucl. Instr. and Meth. 209/210 (1983) 555. [29] M. Szymonski, W. Huang and J. Onsgaard, Nucl. Instr. and Meth. B14 (1986) 263. [30] H.H. Andersen, B. Stenum, T. Sorensen and H.J. Whitlow, Nucl. Instr. and Meth. B6 (1985) 459; 209/210 (1983) 487. [31] K.B. Winterbon, Ion Implantation Range and Damage Deposition Distributions, vol. 2 (Plenum, New York, 1975). [32] U. Littmark and P. Sigmund, J. Phys. D8 (1975) 241.