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Angular distribution and escape probability of sputtered atoms
Vacuum/volume 42/numbers 8/9/pages 537 to 542/1991 Printed in Great Britain
0042-207X/91 $3.00+.00 © 1991 Pergamon Press plc
Angular distribution and escape probability of sputtered atoms Departamento de Electric/dad y Electronic& Facultad de CC. F/sicas, Universidad Complutense, 28040-Madrid, Spain
A M C P ~ r e z - M a r t i n and J J J i m ~ n e z - R o d r i g u e z ,
received for publication 26 September 1990
To quantify the importance of the role played by the inherent discontinuity due to the surface, depth of origin, angular and energy distributions of sputtered atoms are calculated, for various mass ratios, on the basis of a Monte Carlo simulation code. Calculations are performed, on a random target, assuming an initially isotropic and Eo2-like energy distribution. Surface effect is treated by a mean free path length which depends on depth and direction of the emerging atom, its effect being a decrement in the collision rate and a net average deflection towards the normal to the surface, this being more pronounced as the moving atom becomes closer to the surface. This surface effect plus the role of a planar surface barrier increase the escape probability to the peak below the surface position, particularly for light atoms. Mass dependence of the ejected atoms, with respect to the mass of the matrix, is also analyzed in terms of the angular and energy distributions of the ejected atoms, from a given depth. The transfer energy cross-section plays a role in the opposite direction to that of the angular scattering but differences between a light and a heavy atom can be found. From shallow regions, heavy atoms are more likely to be ejected, for glancing emerging directions but, the opposite happens from greater depths. Energy distribution shows clearly that heavy atoms get to the surface, from greater depths, after having spent more energy.
I. Introduction Apart from the sputtering yield, which is the magnitude that quantifies the sputtering process, there are at least three important features which may help us to understand the process itself. These are the escape probability and the angular and energy distributions, all of which are extremely important in analytical techniques in which sputtering processes are involved. In the analysis of multi-component materials the knowledge of the escape probability and the angular distribution might be an indirect way to determine the depth from which the atoms have emerged. This topic has been recently re-investigated and the effect of the angular scattering turns out to be surprisingly small. A rather good agreement between the Monte Carlo simulation and the linear Boltzmann equation has also been shown ~. There is, however, a point which has been left out. This is the asymmetry introduced by the presence of the surface which inevitably causes either the recoil flux in the cascade to be anisotropic at the surface, or the distributions are strongly influenced by outward scattering from neighbouring atoms 2. This latter effect is known as the "missing layer" or "missing plane' effect 3 and it is evidently incorporated in computer simulation codes 4'~, which operate with an initial target configuration and might also be included in the Monte Carlo codes ~' ~ The aim of this paper is to describe a study of the escape probability and angular and energy distributions by means of a Monte Carlo (MC) simulation code, in which the missing layer effect is taken into account by means of an analytical expression 2, for the mean path length, as a function of depth, x, and the direction, 0, of the recoil, together with the concomitant restriction on the azimuthal angle which can also be expressed analytically.
Escape probability, angular and energy distributions are calculated by the Monte Carlo simulation. Results, accounting for (or not) the missing layer effect, are calculated and compared. Finally, the mass dependence for the cases of a light or a heavy ejected atom from a solid, is also investigated. 2. Description of the calculations Let us consider a random and monoatomic solid. Let M2 be the atomic mass of the atoms of the solid and No its density. In order to calculate depth of origin, angular and energy distributions of sputtered atoms, an atom of mass M~ will be placed at the depth x, the x-axis being perpendicular to the plane surface of the solid and positive in the inward direction. The atom M ~will be allowed to move with an initial energy distribution proportional to Eo 2 and isotropic in direction and a uniform recoil flux will be also assumed. Only binary collisions will be considered and inelastic energy losses will be neglected. The trajectories of the recoils are taken as straight lines between the points where the collisions take place, this point being the one in which the distance between the partners in the collision equals the impact parameter. The collisions are characterized in terms of the well-known powerlaw differential cross-section 9h°, da(E,T)
= CE
'nT
l mdT "
0
1,
(la)
where I i, for m ~< 1/4,
A = 52(zig2)3/4eV;
a = 0.219A.
(lb)
For this particular application in which a mass dependence is 537
A M C P#rez-Martin a n d J J Jimenez-Rodrfguez . Sputtered atoms
also investigated, an m-value different from zero should be used in order to preserve the mass dependence in the differential crosssection. It is clear that we are interested in the very low energy region so that a power-law a p p r o x i m a t i o n to the Born M a y e r potential is the most suitable' '. The proper m and 2 wilues which should be taken to fit the numerical Born M a y e r stopping crosssection have recently been e v a l u a t e d ' in the energy region ~ : - It) " 10 '. where ~ ; - M f / [ ( M , + M , _ ) A ] . F r o m those we have chosen m - 0.1 I, ;. = I0. The M o n t e Carlo m e t h o d always implies working with r a n d o m variables. % ranging in the [0,1] interwtl, which properly introduced, are able to reproduce the corresponding distribution. On this line, the length of flight between collisions is selected Irom the normalized distribution,
where the integral accounts for the n u m b e r c)l" a t o m s within a cylinder of base area a, and height /. A t o m s equally distributcd anywhere inside this cylinder imply a c o n s t a n t mean frec path. ,;-u. given by 2. = I/(Na,), which is a c o m m o n vahle in s t a n d a r d M C simulations. This assumption is not, however, valid close to the surface where the cylinder is split by the surface planc ( v - 0), into two different regions, OnE of which is helox\, tile stlrl;.ice. o f atomic density N, and the other, above the stlrlace plane, completely empty of atonls. A c c o u n t i n g for .iust those ;.tlonls inside the cylinder, an effective mean free path can be obtained<':
1 F~(/) = . e x p (
,;.(.\-,0)
I
p ...... sin d 2
I
/' ...... s i n # ( 2 sin ~:, 7r.v + 3 + ~ cos ~
";-,,
//2).
A
x.\
3
.\ > p ....... sin f) ) sin :~. ,
(2)
l
The transferred energy, 7", in a collision is determined according to.
v ~< p ...... s m (J (ga)
with. '"7'
('E
Fe(7') =
i ,,, (3)
Us
where a, = ~p,{,,, is the total scattering cross-section and p ...... is the m a x i m u m impact parameter. A widely used value J: l\)r p ...... is, p ...... d,'\.rt, d = N. x ~ being the mean distance between a t o m s in the solid. F l o m e q u a t i o n (3), the energy transfer is r a n d o m l y determined and then the scattering angle is calculated from well-known relationships. This is equivalent to r a n d o m l y selecting the impact parainctcr so that the scattering angle is determined in ten]as o f the impact p a r a m e t e r and the scattering potential. The initial energy distribution of the recoil a t o m s is "
G(E.E,,) =
E
m
m) E,I,~"'(E,,+ I/) I
ip(1)-i#(I
E>>E, >> F
"'"
(4)
where l" is the binding energy in the bulk of the solid and E is the energy of the primary which generates the cascade. This function after normalizing reads : mI" ,q(E.) -
(5) EdJ
1]"
where there is no upper cut-off for E,> lind E a is the displacement Energy. A p h i m u surface barrier, lit x - 0. is also mchlded which means thal it recoil a t o m of energy 1':. and direction (/(0 measured with respect to the x-axis), will o\.ercome the surface il': E . c o s e 0 ~> U.
(6)
{,' being the surface binding energy, Note that a barrier ph/ced at .v - 0 implies that no collisions are allowed for .v < 0. The probability, tk)r an atom, to travel a length l. in a solid e l density N. without undergoing tiny collision is :
Q(l) 538
exp
FC q -
,)o
.v COS2 =
Na, d!
]
= cxp[-NaJ],
(7)
{Sb)
p ...... sin 0"
turning out to bc d e p e n d e n t on x-dcpth and d-dircction. This dependence is relevant in the region close to thc surface (x ~< i>...... sin d). Deep inside the solid, ;.(.\.0),5,. tends to unity but, in thc region of interest, I'cu from the surface ).(.v J)) is greater than )... but not significantly dLle to a nor>zero probability i o r an alom to reach the surlhce, placed anywhere except at infinity. Once / and p have been r a n d o m l y selected Irom equations (2), (3) and (8), the point .v. where the collision takes place, h, determined its a function of [ and the position of the previous collision. The azimuthal tingle, d). is then randomly selected in such a way that the colliding target atom is located inside lhc target (.v < 0). so that c/, is r a n d o m l y selected \erifying t h a t : .v +/,,
:q~ cos ~/~ /~ sin d)
>~ 0.
~c))
where ~., fi and ;' are the direction cosines of the trajcctor b previous to the collision. Note that, its collisions arc only allowed lot x > 0, there is no extra restriction for the impact p a r a m e t e r p. equation (9) being the unique restriction that ensures that the colliding atom really belongs to the solid. Note that. al'ler the correction mcntioned above, atoms nloving close to the surface will decrease their collision rate. and scattcring deflection towards the normal of the surlktce will bc enhanced. REsults obtained froln equations (S) and (9) ~cl-c successfully c o m p a r e d with those obtained from a sinmlation code in which till a t o m s arc initially located in ~ell-known coordinatcs <'.
3. Results and discussion Results are s h o ~ n in this section for three difl;arcnl mass con> bmations
the s t a n d a r d equal-mass case, H
,1/~ J,l,
1, a light
atom in a heavy matrix. H ~ 3 and tinally, a heavy, atom in a light nmtrix, P - 0.3. When results IBr different mass ratios arc con]pared with each other, some ten]arks need to bc made concerning the cut-off" parameters usually introduced m c o m p u t e r simuhltion codes ~:. A standard one consists of introducing a
A M C Porez-Mart/n and J J JimOnez-Rodr/guez Sputtered atoms
maximum impact parameter, which corresponds with neglecting collisions whose scattering angles are smaller than, ~om~o, the value corresponding to Pma*. This means introducing a cut-off at different ~0mmangles, depending on p. Another possibility is to keep ~Ommconstant for all the cases but this leads to the use of a different Pm~,* and, what is equivalent, a longer mean path length for heavier atoms. F r o m both alternatives, we choose to take, p .... = d / x / r c , the same for all the three mass ratios, because we prefer, for this particular application, to keep the mean path length constant. The calculations reported here have been performed for an atomic density No = 0.0845 atoms/A 3 and E(~ = V = U = 3.56 eV, which correspond to a copper target, whereas light and heavy impurities neon and gold, respectively, have been taken. First of all, in order to check our own calculations and for comparison with results previously reported ], escape probability and angular distributions have been computed, in the case # = 1, for m = 0 and 2 = 24. Figure l(a) shows good agreement between the results reported here and those of ref (1), if the surface correction is not taken into account. When the surface missing layer effect is considered the escape probability shows a peak very near to the surface and the whole curve turns out to be higher because the mean path length is longer close to the surface and atoms are more likely to be deflected towards the normal to the surface so, collisions make it easier to overcome the surface barrier. Figure l(b) shows the comparison, between both calculations, for the angular distribution of sputtered atoms. Three different depths have been plotted and the agreement for all of them is more than acceptable. Figure l(b) also shows, in the window, the changes observed on the angular distributions when the surface correction is included. As expected, glancing emerging atoms are now more probably ejected from shadow regions. For instance, at x = 0, the atoms emerging, when a correction is applied (MCcorrected), are those previously obtained (uncorrected) plus those specifically backscattered due to the asymmetry produced by the surface. Note that when results are normalized, each of them is, respectively, normalized to its corresponding escape probability and this value is higher for the MC-corrected case. Figure 2(a) shows the depth of origin of sputtered atoms, for various atomic mass ratios, when no barrier surface is considered.
It is clear that atoms undergoing no collisions have a probability to escape from the surface (x = 0) equal to 0.5. Any value, at the surface, larger than 0.5 has to be due to backscattered collisions. The values at the surface, in Figure 2(a), are explained by their being larger when corresponding to the lighter system (/~ ~- 3) and smaller when corresponding to the heavier system. When curves are normalized to unity [window in Figure 2(a)] no differences are observed between a light or a heavy atom because the two effects are competing with each other: the scattering angle on the one hand and the cross-section on the other ; a light atom colliding with a heavy matrix atom scatters at wider angles than a heavy atom in a light matrix. Consequently the free path travelled (along the x-axis), between collisions, by the lighter atom, is shorter and the chances to reach the surface are expected to be smaller, but the differential cross-section, equation (1), is larger for the heavy atom so this effect is compensated. Calculations have also been performed for m = 0 (not shown) in order to hide the mass dependence of the differential cross-section and under these conditions heavy atoms are more likely to leave the solid from greater depths, which corroborates what has been previously stated. The escape probability for the equal mass case (# = 1) is lower than any other case at greater depths because now the mean transfer energy per collision is larger so it becomes more difficult to escape. Figure 2(b) shows the same cases as Figure 2(a) but when a planar surface barrier is considered. At great depths, although all curves behave similarly to the free-surface case, clearer differences can be observed between the light and the heavy atoms. N o w the heavy atom is more likely to escape from greater depths and it might be due to the fact that the scattering angle is always narrow. Novelties also appear in the shallow region where the case (# ~- 3) shows a clear peak close to the surface, while the other two cases have a constant probability to escape roughly in the small region where the peak appears for the light atom. This fact can be understood in the light of the role played by the surface barrier. Atoms emerging at grazing angles, with respect to the surface plane, will not be allowed to overcome the surface barrier unless, owing to collisions, they change direction in such a way as, to be closer to the surface normal. Consider, in fact, atoms close
c-i d
a)
**
4
~, X3 CJ
-BEST +++++ MC ***** MC-corrected
~**
O
CL-
i-(1)
b) free surfoce - -
L
MC
....
Me-corrected
•2_-~ ~--, __. 03
,+**
C3_
t
+**
ca 2 - (2)
03 O_ © U?
E
< I
h
I
×/d
I
I 5
0
-1.0
i
t
i
I
-0.8
i
i
i
I
-0.6
i
i
i
I
-0.4 cos 0
i
i
r--~
-0.2
L_
i
0.0
Figure 1. Calculations made with rn = 0, 2 = 24. (For details see the text.) (a) Escape probability vs relative depth. (b) Angular distributions of ejected atoms from: x = 0, label (1) ; x = 1.97/~, label (2) : x = 13/~, label (3). 539
,4 M C P e r e z - M a r t / n a n d J J Jim#nez-Rodr/guez.- Sputtered atoms
a) free surface
i,
~
XD CJ
\\
~
LO ,,
d
--- /~,~.3
b) planar barrier
2~ ~Z C
2
/3_
(o k\\
©
N N
(/9 LJ
\\
!
CL 0 c~ co
L
/
LJ
L
O:.. \"'"
i 5
dO
5
×/d
×/d
Figure 2. Escape probability \s depth. (a) Frcc-surl~.tcc. (b) Planar surface-barrier. { ' - 3.56 cV. ('urxcs normalized Io tmity arc plotted in the x~indox~,
enough to the surface, whose direction of m o t i o n is so glancing that they have little chance to overcome the surface. F o r these atoms, it will be easier to overcome the surface if their distance to the surface is long enough, but not too long, to have an appreciable probability to undergo one collision that changes its direction in a p r o p e r way. Such a collision is more likely to happen to a light t h a n to a heavy atom. This explains why the peak is more p r o n o u n c e d for lighter atoms. Heavy atoms, whose m o t i o n is almost along straight lines, because scattering angles are very small, do not show such a peak. A n g u l a r distributions of a t o m s sputtered from a given depth in b o t h cases, free-surface and p l a n a r barrier, are sketched in Figures 3 5. In addition to the a n g u l a r distribution from x - (L two depths have been chosem one in the shallow region a n d the other more deeply. For comparison, the depth values are those taken in ref ( 1 ). All a n g u l a r distributions are normalized to unity. Figures 3(a) and (b) show angular distribution of the a t o m s ejectcd from the surface (x = 0), for various mass ratios when, respectively, the barrier surface is not or is considered. Figure 3(a) clearly shows that heavy a t o m s arc more p r o b a b l y ejected
at glancing emerging directions than any others. The small peak is shifted towards the n o r m a l direction the lighter the atom is. this is entirely due to backscattering. At the surfacc most a t o m s emerge without undergoing collisions but if a collision happens the lighter atom will be deflected more than any other, so hs emerging direction will be closer than normal to the surface. When a cut is imposed by the p l a n a r surface-barrier this elleel is almost negligible and all the three curves merge. Figures 4(a) and (b) s h o ~ the angular distribution o l the c.iected atoms, for wuious mass ratios, from a near-surface depth. Figure 4(a), tbr the free-surface case, shows that when a t o m s emerge nearly perpendicular to the surface, all curves are almost fiat, keeping the initial isotropic distribution, because the chances to leave the solid without undergoing collisions are higher. This effect is more p r o n o u n c e d lbr heavy a t o m s because the scattering angle is smaller, if collisions occur. Figure 4(b) points out how Ihe p h m a r barrier x~,orks: closc to the normal to the surface ahnost no changes can be appreciated because the barricr is almost t r a n s p a r e n t and at glancing emerging directions the surface refraction slightly increases the xalues rendering the ~ h o l c
a) free surface ,
b) p!anar barrier
x~O
x=O E 0
E O
--/*=1 . . . . /s~,3 ..... fx-~.3
i
~D "U_
! i
*d
La=I
k
~s
k
cS L (D
2 C~
2t_ '
Oh
ED E
<£
<1-
0
-1.0
-0.8
-0.6
-0.4 cos
-0.2
0.0
o[~ ............. - 1.0
-0.8
0
FiRure 3. Angular distributions o f t ¢clcd atoms h-am .\ - 0. (a} Frcc-surl'acc. (b) Planar surl'accd~arricr, ~ ' 540
0.6
0.4 ,SOS ~9
3.56 cV.
~ -0.2
:7.,7
A M C Pdrez-Martfn and J J Jim#nez-RodHguez: Sputtered atoms
/ a) free s u r f a c e
I|
x = 1.96)~
c
4 C 0
b) p l a n a r b a r r i e r x = 1.96,~
4I
0 L~
- ,u,=l .... ~ ....... ,u~.3
"C. 4~ . -CO -
- ,u,=l .... ~-~3 ....... ,u~.3
d~
"c
~
03 ,(-' ~-
C3
2
~_ 2 Cr~ c-
0
0
-1.0
-0.8
-0.6
-0.4 cos 8
-0.2
0.0
-1.0
-0.8
-0.6 cos
-0.4 8
-0.2
0.0
Figure 4. Angular distributions o f ejected atoms from x - 1.97 ,~. (a) Free-surface. (b) Planar surface-barrier, U = 3.56 eV.
spectrum more under-cosine. The differences between angular distributions for heavier or lighter atoms are partially hidden by the barrier which converges both distributions closer to each other. Figures 5(a) and (b) also display angular distributions of ejected atoms, but from a greater depth. F r o m the comparison between both figures it can be observed, as a general rule, that a planar barrier tends to diminish the over-cosine effect. Two features are now remarkable: firstly that heavy atoms seem to be more sensitive to the barrier for normal ejection and secondly that, unlike the shallow region, at glancing angles light atoms are more probably ejected. This is just the opposite to that found in Figures 3 and 4. This is because at 13 ,~ either the heavy or the light atoms undergo a few collisions before reaching the surface. The heavy atom spends more energy per collision and the scattering angle is narrow, so the chances of reaching the surface with a glancing emerging direction are very small. However, this is not the case for the light atom, which spends less energy per collision and the chances of reaching the surface, with a glancing
emerging direction, are higher because the scattering angle is wider. The fact that heavy atoms collide with smaller angles may illustrate why, at normal direction, they are more likely to reach the surface but as they spend more energy per collision they are more sensitive to the barrier. This argument is corroborated by Figures 6(a) and (b) where the energy distributions of ejected atoms are displayed when they emerge, respectively, from x = 1.97 ,~ and x = 13 A. Whilst no remarkable differences are observed in Figure 6(a), Figure 6(b) shows clearly that heavy atoms spend more energy to reach the surface when they emerge from greater depths. It has also been observed that, if an energy cut-off is introduced, there are no changes above 50 eV with respect to the case where there is no cut-off. However, below 20 eV, the peak of the distribution shifts about 1 eV, as has been recently reported '4. We do not present here any of these results because in order to compare them with other previously reported calculations ~, the sputtering yield needs to be evaluated and this is not the aim of the present work.
b) p l a n a r b a r r i e r x = l 3A
a ) free s u r f a c e x=13A 4
4
C 0
cO "7
....
c~ k_
~,=1
- -
#~,3
:
,u,=l
O3
~5 k~D
r-h 2 (3
-5
C~ C
O~
<
C
.<
,..
0 -1.0
,
,
,
~
-0.8
~
~
I
-0.6
~
K
J
I
-0.4 COS 0
~
"
~
-0.2
0.0
o -1.0
-0.8
-0.6
-0.4 cos
-t_
t
-0.2
0.0
e
Figure 5. Angular distributions of ejected atoms from x = 13 A. (a) Free-surface. (b) Planar surface-barrier, U = 3.56 eV. 541
A M C Porez-Martin and J J Jimenez-Rodr/guezSputtered atoms
0.15
0.15 b) p!arar barrier x=lSk
a) planar barrier x= 1.96A C O
C 0
XD
-
.Q
.....
-
xD
~.-~3
--- ~3
"U_
#-~.3
co
4:3 >,
~5
6) c /£]
@ C/_ LJ
i. . . . i
is! <
>-,
I 0.00
. . . .
~
A
0
2O
0
E(eV)
Figure 6. Energy distribution of ejected atoms from : (a) .x = 1.96 A. (b) x = 13 A, ( ¢
Most o f the calculations shown here (Figures 2-6) have been carried out by accounting lk)r the surlacc corrcclion, except for Figure 1, in which results without such a correclion have also been included. This is because no important changes have been lbund, on the angular spectra o f cjccted atoms, as a rcsult o f taking into account the surface correction. Clear changes have been found, however, ik)r the escape probability, showing an cnhancenaenl, all along thc curve, and even more lcnmrkablc very close to the surface. It cannot be inferred, from our results, that the missing layer effect produces a more over-cosine angular distribution, as had been predicted previously,~. This is entirely due to the fact that the surface-barrier has been located at .v = O. so that there are no collisions at all. l\)r .v < 0. II is obvious that thc location o1" the surface-barrier at .v d pro-educes two effects: the escape probability decreases and the anguhtr distribution becomes more over-cosine. The addition o f an e x h a layer, l'rom . v = 0 to . v = d, even though there is no atom present, makes it possible to lose energy so that the escape probabili( 5 has to decrease and the fact that this extra layer is empty of atoms makes such collisions tend to detlecl the atoms towards the n o l n l a ] t(/ the SUl'l'tlce ( o v e r - c o s i n e ) . 4. Conclusions
Most o f the features predicted from sputtering yield simulation codes can be obtained by placing an atom al a given depth and allowing it to collide, starting from an isolropic and an E, ~-like energy distribution. It is clear that a surface correction is rather necessary to account l\)r the lack o f symmetry introduced by the existence o f the surface. This fact has been evaluated analytically locating the surface at .v = 0. The features observed are that the escape probability increases and peaks somewhere very close to the surface while the angular spectra show slight changes. If a surface-barrier had been applied, at .v = d, the escape probability would not have increased so much, meanwhile the angular spectra would be clearly over-cosine.
542
52
~-
0.00
=
E( V)
2C
3.56 eV.
Different mass ratios show different angular distributions, the distribution being more under-cosine the heavier the atom in the near-surface region, and this efl'ect being inverted at deeper depths. This effect is masked by the reverse role played by the planar surface-barrier. h n p o r t a n t differences have also been found, as a function of mass o f the ejected atom. in the energy spectra. The very wellknown peak found belween 2 and 4 eV is remarkably higher for heavy atoms when they emerge from large depths.
Acknowledgments
Very helpful c o m m e n t s concerning the manuscript made by Dr H Urbassek and M Vicanek are gratefully appreciated. We would also like to thank P r o f A G r a s - M a r t i for his valuable commcnls.
References I M Vicanek, J J .limdnez-Rodriguez and P Signmnd, 3,ml h>zrum Alcdl, B36, 124 (1989). "H 1t Andersen. 13 Stenum, T Sorcnsen and H J Whilla\~. ?¢m'l Inwrum Alelh. B6, 459 (1985). ;11 I1 Anderson. : \ m l h>trum 3h'HI. B33, 466(19,~N) a I) E Harrison, Ra~ga/ Ell. 711, [ (1983). "M lieu and M T Robinson. Appl Phy.~. 18, 381 ([979). "A M (' Pdre×-Martin and J J Jimdncz-Rodrigucz. l),wum. 39. 71) ([989). :M Vicanek and tt M Urbassck. Nucl Invtrum Met~t, B30, 5(17 (I9b;8). ~.1 P Biersack and W Eckstein, Appl Ph3's, A34, 73 (1984). ".l Lindhard, V Nielsen and M SharJE Mater Fv.~ Mudd Dan l i d .%,Ld,. 36(10), (1968). "~K B Winterbon, P Sigmund and J B Sanders, Mamr lq's Mudd /)(1/; f'it/SeZ~'k. 37(14). (1970). ~ P Sigmund, Phv.s Rer, 184, 383 (1969). ~"If H Andersen, Nucl hl,slrutll Met/I, BIB, 32l (19871. ~P Sigmund, R¢'1'Roum Phvs, 17, 969 (I972). J4 R A Brizzolara, C B Cooper and T K Olson, ,\,'ucl ln.wrum Mclh, B35, 36 (19881. ~'P Sigmund, Nm'/ln,vlrum Me/h. B27, 1 (1987).
0042-207X/91 $3.00+.00 © 1991 Pergamon Press plc
Angular distribution and escape probability of sputtered atoms Departamento de Electric/dad y Electronic& Facultad de CC. F/sicas, Universidad Complutense, 28040-Madrid, Spain
A M C P ~ r e z - M a r t i n and J J J i m ~ n e z - R o d r i g u e z ,
received for publication 26 September 1990
To quantify the importance of the role played by the inherent discontinuity due to the surface, depth of origin, angular and energy distributions of sputtered atoms are calculated, for various mass ratios, on the basis of a Monte Carlo simulation code. Calculations are performed, on a random target, assuming an initially isotropic and Eo2-like energy distribution. Surface effect is treated by a mean free path length which depends on depth and direction of the emerging atom, its effect being a decrement in the collision rate and a net average deflection towards the normal to the surface, this being more pronounced as the moving atom becomes closer to the surface. This surface effect plus the role of a planar surface barrier increase the escape probability to the peak below the surface position, particularly for light atoms. Mass dependence of the ejected atoms, with respect to the mass of the matrix, is also analyzed in terms of the angular and energy distributions of the ejected atoms, from a given depth. The transfer energy cross-section plays a role in the opposite direction to that of the angular scattering but differences between a light and a heavy atom can be found. From shallow regions, heavy atoms are more likely to be ejected, for glancing emerging directions but, the opposite happens from greater depths. Energy distribution shows clearly that heavy atoms get to the surface, from greater depths, after having spent more energy.
I. Introduction Apart from the sputtering yield, which is the magnitude that quantifies the sputtering process, there are at least three important features which may help us to understand the process itself. These are the escape probability and the angular and energy distributions, all of which are extremely important in analytical techniques in which sputtering processes are involved. In the analysis of multi-component materials the knowledge of the escape probability and the angular distribution might be an indirect way to determine the depth from which the atoms have emerged. This topic has been recently re-investigated and the effect of the angular scattering turns out to be surprisingly small. A rather good agreement between the Monte Carlo simulation and the linear Boltzmann equation has also been shown ~. There is, however, a point which has been left out. This is the asymmetry introduced by the presence of the surface which inevitably causes either the recoil flux in the cascade to be anisotropic at the surface, or the distributions are strongly influenced by outward scattering from neighbouring atoms 2. This latter effect is known as the "missing layer" or "missing plane' effect 3 and it is evidently incorporated in computer simulation codes 4'~, which operate with an initial target configuration and might also be included in the Monte Carlo codes ~' ~ The aim of this paper is to describe a study of the escape probability and angular and energy distributions by means of a Monte Carlo (MC) simulation code, in which the missing layer effect is taken into account by means of an analytical expression 2, for the mean path length, as a function of depth, x, and the direction, 0, of the recoil, together with the concomitant restriction on the azimuthal angle which can also be expressed analytically.
Escape probability, angular and energy distributions are calculated by the Monte Carlo simulation. Results, accounting for (or not) the missing layer effect, are calculated and compared. Finally, the mass dependence for the cases of a light or a heavy ejected atom from a solid, is also investigated. 2. Description of the calculations Let us consider a random and monoatomic solid. Let M2 be the atomic mass of the atoms of the solid and No its density. In order to calculate depth of origin, angular and energy distributions of sputtered atoms, an atom of mass M~ will be placed at the depth x, the x-axis being perpendicular to the plane surface of the solid and positive in the inward direction. The atom M ~will be allowed to move with an initial energy distribution proportional to Eo 2 and isotropic in direction and a uniform recoil flux will be also assumed. Only binary collisions will be considered and inelastic energy losses will be neglected. The trajectories of the recoils are taken as straight lines between the points where the collisions take place, this point being the one in which the distance between the partners in the collision equals the impact parameter. The collisions are characterized in terms of the well-known powerlaw differential cross-section 9h°, da(E,T)
= CE
'nT
l mdT "
0
1,
(la)
where I i, for m ~< 1/4,
A = 52(zig2)3/4eV;
a = 0.219A.
(lb)
For this particular application in which a mass dependence is 537
A M C P#rez-Martin a n d J J Jimenez-Rodrfguez . Sputtered atoms
also investigated, an m-value different from zero should be used in order to preserve the mass dependence in the differential crosssection. It is clear that we are interested in the very low energy region so that a power-law a p p r o x i m a t i o n to the Born M a y e r potential is the most suitable' '. The proper m and 2 wilues which should be taken to fit the numerical Born M a y e r stopping crosssection have recently been e v a l u a t e d ' in the energy region ~ : - It) " 10 '. where ~ ; - M f / [ ( M , + M , _ ) A ] . F r o m those we have chosen m - 0.1 I, ;. = I0. The M o n t e Carlo m e t h o d always implies working with r a n d o m variables. % ranging in the [0,1] interwtl, which properly introduced, are able to reproduce the corresponding distribution. On this line, the length of flight between collisions is selected Irom the normalized distribution,
where the integral accounts for the n u m b e r c)l" a t o m s within a cylinder of base area a, and height /. A t o m s equally distributcd anywhere inside this cylinder imply a c o n s t a n t mean frec path. ,;-u. given by 2. = I/(Na,), which is a c o m m o n vahle in s t a n d a r d M C simulations. This assumption is not, however, valid close to the surface where the cylinder is split by the surface planc ( v - 0), into two different regions, OnE of which is helox\, tile stlrl;.ice. o f atomic density N, and the other, above the stlrlace plane, completely empty of atonls. A c c o u n t i n g for .iust those ;.tlonls inside the cylinder, an effective mean free path can be obtained<':
1 F~(/) = . e x p (
,;.(.\-,0)
I
p ...... sin d 2
I
/' ...... s i n # ( 2 sin ~:, 7r.v + 3 + ~ cos ~
";-,,
//2).
A
x.\
3
.\ > p ....... sin f) ) sin :~. ,
(2)
l
The transferred energy, 7", in a collision is determined according to.
v ~< p ...... s m (J (ga)
with. '"7'
('E
Fe(7') =
i ,,, (3)
Us
where a, = ~p,{,,, is the total scattering cross-section and p ...... is the m a x i m u m impact parameter. A widely used value J: l\)r p ...... is, p ...... d,'\.rt, d = N. x ~ being the mean distance between a t o m s in the solid. F l o m e q u a t i o n (3), the energy transfer is r a n d o m l y determined and then the scattering angle is calculated from well-known relationships. This is equivalent to r a n d o m l y selecting the impact parainctcr so that the scattering angle is determined in ten]as o f the impact p a r a m e t e r and the scattering potential. The initial energy distribution of the recoil a t o m s is "
G(E.E,,) =
E
m
m) E,I,~"'(E,,+ I/) I
ip(1)-i#(I
E>>E, >> F
"'"
(4)
where l" is the binding energy in the bulk of the solid and E is the energy of the primary which generates the cascade. This function after normalizing reads : mI" ,q(E.) -
(5) EdJ
1]"
where there is no upper cut-off for E,> lind E a is the displacement Energy. A p h i m u surface barrier, lit x - 0. is also mchlded which means thal it recoil a t o m of energy 1':. and direction (/(0 measured with respect to the x-axis), will o\.ercome the surface il': E . c o s e 0 ~> U.
(6)
{,' being the surface binding energy, Note that a barrier ph/ced at .v - 0 implies that no collisions are allowed for .v < 0. The probability, tk)r an atom, to travel a length l. in a solid e l density N. without undergoing tiny collision is :
Q(l) 538
exp
FC q -
,)o
.v COS2 =
Na, d!
]
= cxp[-NaJ],
(7)
{Sb)
p ...... sin 0"
turning out to bc d e p e n d e n t on x-dcpth and d-dircction. This dependence is relevant in the region close to thc surface (x ~< i>...... sin d). Deep inside the solid, ;.(.\.0),5,. tends to unity but, in thc region of interest, I'cu from the surface ).(.v J)) is greater than )... but not significantly dLle to a nor>zero probability i o r an alom to reach the surlhce, placed anywhere except at infinity. Once / and p have been r a n d o m l y selected Irom equations (2), (3) and (8), the point .v. where the collision takes place, h, determined its a function of [ and the position of the previous collision. The azimuthal tingle, d). is then randomly selected in such a way that the colliding target atom is located inside lhc target (.v < 0). so that c/, is r a n d o m l y selected \erifying t h a t : .v +/,,
:q~ cos ~/~ /~ sin d)
>~ 0.
~c))
where ~., fi and ;' are the direction cosines of the trajcctor b previous to the collision. Note that, its collisions arc only allowed lot x > 0, there is no extra restriction for the impact p a r a m e t e r p. equation (9) being the unique restriction that ensures that the colliding atom really belongs to the solid. Note that. al'ler the correction mcntioned above, atoms nloving close to the surface will decrease their collision rate. and scattcring deflection towards the normal of the surlktce will bc enhanced. REsults obtained froln equations (S) and (9) ~cl-c successfully c o m p a r e d with those obtained from a sinmlation code in which till a t o m s arc initially located in ~ell-known coordinatcs <'.
3. Results and discussion Results are s h o ~ n in this section for three difl;arcnl mass con> bmations
the s t a n d a r d equal-mass case, H
,1/~ J,l,
1, a light
atom in a heavy matrix. H ~ 3 and tinally, a heavy, atom in a light nmtrix, P - 0.3. When results IBr different mass ratios arc con]pared with each other, some ten]arks need to bc made concerning the cut-off" parameters usually introduced m c o m p u t e r simuhltion codes ~:. A standard one consists of introducing a
A M C Porez-Mart/n and J J JimOnez-Rodr/guez Sputtered atoms
maximum impact parameter, which corresponds with neglecting collisions whose scattering angles are smaller than, ~om~o, the value corresponding to Pma*. This means introducing a cut-off at different ~0mmangles, depending on p. Another possibility is to keep ~Ommconstant for all the cases but this leads to the use of a different Pm~,* and, what is equivalent, a longer mean path length for heavier atoms. F r o m both alternatives, we choose to take, p .... = d / x / r c , the same for all the three mass ratios, because we prefer, for this particular application, to keep the mean path length constant. The calculations reported here have been performed for an atomic density No = 0.0845 atoms/A 3 and E(~ = V = U = 3.56 eV, which correspond to a copper target, whereas light and heavy impurities neon and gold, respectively, have been taken. First of all, in order to check our own calculations and for comparison with results previously reported ], escape probability and angular distributions have been computed, in the case # = 1, for m = 0 and 2 = 24. Figure l(a) shows good agreement between the results reported here and those of ref (1), if the surface correction is not taken into account. When the surface missing layer effect is considered the escape probability shows a peak very near to the surface and the whole curve turns out to be higher because the mean path length is longer close to the surface and atoms are more likely to be deflected towards the normal to the surface so, collisions make it easier to overcome the surface barrier. Figure l(b) shows the comparison, between both calculations, for the angular distribution of sputtered atoms. Three different depths have been plotted and the agreement for all of them is more than acceptable. Figure l(b) also shows, in the window, the changes observed on the angular distributions when the surface correction is included. As expected, glancing emerging atoms are now more probably ejected from shadow regions. For instance, at x = 0, the atoms emerging, when a correction is applied (MCcorrected), are those previously obtained (uncorrected) plus those specifically backscattered due to the asymmetry produced by the surface. Note that when results are normalized, each of them is, respectively, normalized to its corresponding escape probability and this value is higher for the MC-corrected case. Figure 2(a) shows the depth of origin of sputtered atoms, for various atomic mass ratios, when no barrier surface is considered.
It is clear that atoms undergoing no collisions have a probability to escape from the surface (x = 0) equal to 0.5. Any value, at the surface, larger than 0.5 has to be due to backscattered collisions. The values at the surface, in Figure 2(a), are explained by their being larger when corresponding to the lighter system (/~ ~- 3) and smaller when corresponding to the heavier system. When curves are normalized to unity [window in Figure 2(a)] no differences are observed between a light or a heavy atom because the two effects are competing with each other: the scattering angle on the one hand and the cross-section on the other ; a light atom colliding with a heavy matrix atom scatters at wider angles than a heavy atom in a light matrix. Consequently the free path travelled (along the x-axis), between collisions, by the lighter atom, is shorter and the chances to reach the surface are expected to be smaller, but the differential cross-section, equation (1), is larger for the heavy atom so this effect is compensated. Calculations have also been performed for m = 0 (not shown) in order to hide the mass dependence of the differential cross-section and under these conditions heavy atoms are more likely to leave the solid from greater depths, which corroborates what has been previously stated. The escape probability for the equal mass case (# = 1) is lower than any other case at greater depths because now the mean transfer energy per collision is larger so it becomes more difficult to escape. Figure 2(b) shows the same cases as Figure 2(a) but when a planar surface barrier is considered. At great depths, although all curves behave similarly to the free-surface case, clearer differences can be observed between the light and the heavy atoms. N o w the heavy atom is more likely to escape from greater depths and it might be due to the fact that the scattering angle is always narrow. Novelties also appear in the shallow region where the case (# ~- 3) shows a clear peak close to the surface, while the other two cases have a constant probability to escape roughly in the small region where the peak appears for the light atom. This fact can be understood in the light of the role played by the surface barrier. Atoms emerging at grazing angles, with respect to the surface plane, will not be allowed to overcome the surface barrier unless, owing to collisions, they change direction in such a way as, to be closer to the surface normal. Consider, in fact, atoms close
c-i d
a)
**
4
~, X3 CJ
-BEST +++++ MC ***** MC-corrected
~**
O
CL-
i-(1)
b) free surfoce - -
L
MC
....
Me-corrected
•2_-~ ~--, __. 03
,+**
C3_
t
+**
ca 2 - (2)
03 O_ © U?
E
< I
h
I
×/d
I
I 5
0
-1.0
i
t
i
I
-0.8
i
i
i
I
-0.6
i
i
i
I
-0.4 cos 0
i
i
r--~
-0.2
L_
i
0.0
Figure 1. Calculations made with rn = 0, 2 = 24. (For details see the text.) (a) Escape probability vs relative depth. (b) Angular distributions of ejected atoms from: x = 0, label (1) ; x = 1.97/~, label (2) : x = 13/~, label (3). 539
,4 M C P e r e z - M a r t / n a n d J J Jim#nez-Rodr/guez.- Sputtered atoms
a) free surface
i,
~
XD CJ
\\
~
LO ,,
d
--- /~,~.3
b) planar barrier
2~ ~Z C
2
/3_
(o k\\
©
N N
(/9 LJ
\\
!
CL 0 c~ co
L
/
LJ
L
O:.. \"'"
i 5
dO
5
×/d
×/d
Figure 2. Escape probability \s depth. (a) Frcc-surl~.tcc. (b) Planar surface-barrier. { ' - 3.56 cV. ('urxcs normalized Io tmity arc plotted in the x~indox~,
enough to the surface, whose direction of m o t i o n is so glancing that they have little chance to overcome the surface. F o r these atoms, it will be easier to overcome the surface if their distance to the surface is long enough, but not too long, to have an appreciable probability to undergo one collision that changes its direction in a p r o p e r way. Such a collision is more likely to happen to a light t h a n to a heavy atom. This explains why the peak is more p r o n o u n c e d for lighter atoms. Heavy atoms, whose m o t i o n is almost along straight lines, because scattering angles are very small, do not show such a peak. A n g u l a r distributions of a t o m s sputtered from a given depth in b o t h cases, free-surface and p l a n a r barrier, are sketched in Figures 3 5. In addition to the a n g u l a r distribution from x - (L two depths have been chosem one in the shallow region a n d the other more deeply. For comparison, the depth values are those taken in ref ( 1 ). All a n g u l a r distributions are normalized to unity. Figures 3(a) and (b) show angular distribution of the a t o m s ejectcd from the surface (x = 0), for various mass ratios when, respectively, the barrier surface is not or is considered. Figure 3(a) clearly shows that heavy a t o m s arc more p r o b a b l y ejected
at glancing emerging directions than any others. The small peak is shifted towards the n o r m a l direction the lighter the atom is. this is entirely due to backscattering. At the surfacc most a t o m s emerge without undergoing collisions but if a collision happens the lighter atom will be deflected more than any other, so hs emerging direction will be closer than normal to the surface. When a cut is imposed by the p l a n a r surface-barrier this elleel is almost negligible and all the three curves merge. Figures 4(a) and (b) s h o ~ the angular distribution o l the c.iected atoms, for wuious mass ratios, from a near-surface depth. Figure 4(a), tbr the free-surface case, shows that when a t o m s emerge nearly perpendicular to the surface, all curves are almost fiat, keeping the initial isotropic distribution, because the chances to leave the solid without undergoing collisions are higher. This effect is more p r o n o u n c e d lbr heavy a t o m s because the scattering angle is smaller, if collisions occur. Figure 4(b) points out how Ihe p h m a r barrier x~,orks: closc to the normal to the surface ahnost no changes can be appreciated because the barricr is almost t r a n s p a r e n t and at glancing emerging directions the surface refraction slightly increases the xalues rendering the ~ h o l c
a) free surface ,
b) p!anar barrier
x~O
x=O E 0
E O
--/*=1 . . . . /s~,3 ..... fx-~.3
i
~D "U_
! i
*d
La=I
k
~s
k
cS L (D
2 C~
2t_ '
Oh
ED E
<£
<1-
0
-1.0
-0.8
-0.6
-0.4 cos
-0.2
0.0
o[~ ............. - 1.0
-0.8
0
FiRure 3. Angular distributions o f t ¢clcd atoms h-am .\ - 0. (a} Frcc-surl'acc. (b) Planar surl'accd~arricr, ~ ' 540
0.6
0.4 ,SOS ~9
3.56 cV.
~ -0.2
:7.,7
A M C Pdrez-Martfn and J J Jim#nez-RodHguez: Sputtered atoms
/ a) free s u r f a c e
I|
x = 1.96)~
c
4 C 0
b) p l a n a r b a r r i e r x = 1.96,~
4I
0 L~
- ,u,=l .... ~ ....... ,u~.3
"C. 4~ . -CO -
- ,u,=l .... ~-~3 ....... ,u~.3
d~
"c
~
03 ,(-' ~-
C3
2
~_ 2 Cr~ c-
0
0
-1.0
-0.8
-0.6
-0.4 cos 8
-0.2
0.0
-1.0
-0.8
-0.6 cos
-0.4 8
-0.2
0.0
Figure 4. Angular distributions o f ejected atoms from x - 1.97 ,~. (a) Free-surface. (b) Planar surface-barrier, U = 3.56 eV.
spectrum more under-cosine. The differences between angular distributions for heavier or lighter atoms are partially hidden by the barrier which converges both distributions closer to each other. Figures 5(a) and (b) also display angular distributions of ejected atoms, but from a greater depth. F r o m the comparison between both figures it can be observed, as a general rule, that a planar barrier tends to diminish the over-cosine effect. Two features are now remarkable: firstly that heavy atoms seem to be more sensitive to the barrier for normal ejection and secondly that, unlike the shallow region, at glancing angles light atoms are more probably ejected. This is just the opposite to that found in Figures 3 and 4. This is because at 13 ,~ either the heavy or the light atoms undergo a few collisions before reaching the surface. The heavy atom spends more energy per collision and the scattering angle is narrow, so the chances of reaching the surface with a glancing emerging direction are very small. However, this is not the case for the light atom, which spends less energy per collision and the chances of reaching the surface, with a glancing
emerging direction, are higher because the scattering angle is wider. The fact that heavy atoms collide with smaller angles may illustrate why, at normal direction, they are more likely to reach the surface but as they spend more energy per collision they are more sensitive to the barrier. This argument is corroborated by Figures 6(a) and (b) where the energy distributions of ejected atoms are displayed when they emerge, respectively, from x = 1.97 ,~ and x = 13 A. Whilst no remarkable differences are observed in Figure 6(a), Figure 6(b) shows clearly that heavy atoms spend more energy to reach the surface when they emerge from greater depths. It has also been observed that, if an energy cut-off is introduced, there are no changes above 50 eV with respect to the case where there is no cut-off. However, below 20 eV, the peak of the distribution shifts about 1 eV, as has been recently reported '4. We do not present here any of these results because in order to compare them with other previously reported calculations ~, the sputtering yield needs to be evaluated and this is not the aim of the present work.
b) p l a n a r b a r r i e r x = l 3A
a ) free s u r f a c e x=13A 4
4
C 0
cO "7
....
c~ k_
~,=1
- -
#~,3
:
,u,=l
O3
~5 k~D
r-h 2 (3
-5
C~ C
O~
<
C
.<
,..
0 -1.0
,
,
,
~
-0.8
~
~
I
-0.6
~
K
J
I
-0.4 COS 0
~
"
~
-0.2
0.0
o -1.0
-0.8
-0.6
-0.4 cos
-t_
t
-0.2
0.0
e
Figure 5. Angular distributions of ejected atoms from x = 13 A. (a) Free-surface. (b) Planar surface-barrier, U = 3.56 eV. 541
A M C Porez-Martin and J J Jimenez-Rodr/guezSputtered atoms
0.15
0.15 b) p!arar barrier x=lSk
a) planar barrier x= 1.96A C O
C 0
XD
-
.Q
.....
-
xD
~.-~3
--- ~3
"U_
#-~.3
co
4:3 >,
~5
6) c /£]
@ C/_ LJ
i. . . . i
is! <
>-,
I 0.00
. . . .
~
A
0
2O
0
E(eV)
Figure 6. Energy distribution of ejected atoms from : (a) .x = 1.96 A. (b) x = 13 A, ( ¢
Most o f the calculations shown here (Figures 2-6) have been carried out by accounting lk)r the surlacc corrcclion, except for Figure 1, in which results without such a correclion have also been included. This is because no important changes have been lbund, on the angular spectra o f cjccted atoms, as a rcsult o f taking into account the surface correction. Clear changes have been found, however, ik)r the escape probability, showing an cnhancenaenl, all along thc curve, and even more lcnmrkablc very close to the surface. It cannot be inferred, from our results, that the missing layer effect produces a more over-cosine angular distribution, as had been predicted previously,~. This is entirely due to the fact that the surface-barrier has been located at .v = O. so that there are no collisions at all. l\)r .v < 0. II is obvious that thc location o1" the surface-barrier at .v d pro-educes two effects: the escape probability decreases and the anguhtr distribution becomes more over-cosine. The addition o f an e x h a layer, l'rom . v = 0 to . v = d, even though there is no atom present, makes it possible to lose energy so that the escape probabili( 5 has to decrease and the fact that this extra layer is empty of atoms makes such collisions tend to detlecl the atoms towards the n o l n l a ] t(/ the SUl'l'tlce ( o v e r - c o s i n e ) . 4. Conclusions
Most o f the features predicted from sputtering yield simulation codes can be obtained by placing an atom al a given depth and allowing it to collide, starting from an isolropic and an E, ~-like energy distribution. It is clear that a surface correction is rather necessary to account l\)r the lack o f symmetry introduced by the existence o f the surface. This fact has been evaluated analytically locating the surface at .v = 0. The features observed are that the escape probability increases and peaks somewhere very close to the surface while the angular spectra show slight changes. If a surface-barrier had been applied, at .v = d, the escape probability would not have increased so much, meanwhile the angular spectra would be clearly over-cosine.
542
52
~-
0.00
=
E( V)
2C
3.56 eV.
Different mass ratios show different angular distributions, the distribution being more under-cosine the heavier the atom in the near-surface region, and this efl'ect being inverted at deeper depths. This effect is masked by the reverse role played by the planar surface-barrier. h n p o r t a n t differences have also been found, as a function of mass o f the ejected atom. in the energy spectra. The very wellknown peak found belween 2 and 4 eV is remarkably higher for heavy atoms when they emerge from large depths.
Acknowledgments
Very helpful c o m m e n t s concerning the manuscript made by Dr H Urbassek and M Vicanek are gratefully appreciated. We would also like to thank P r o f A G r a s - M a r t i for his valuable commcnls.
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