Theory of redeposition of sputtered flux on to surface asperities

Nuclear Instruments and Methods 182/183 (1981) 275-281 © North-Ilolland Publishing Company

THEORY OF REDEPOSITION OF SPUTTERED FLUX ON TO SURFACE ASPERITIES J. BELSON and I.H. WILSON Department o f Electronic and Electrical Engineering, University of Surrey, Guildford, Surrey, England GU2 5XH

This paper models the topographical evolution of features on amorphous surfaces under ion bombardment. Specifically, evolution due to accretion of material sputtered from areas adjacent to a feature has been investigated in terms of the flux density redeposited on to an arbitrary prof'de y =J(~) from a linear emitter. Analytical solutions have been found for the early ("first burst") evolution of linear and sinusoidal surface features in cases where the enfitter radiates isotropically or anisotropicaUy (cosine law) from each point of its length. The predictions of models based on these two types of emitter are compared. Both types produce enhanced deposition near the foot of a linear slope but the effect is much greater for isotropic emission. Above the foot of a linear slope there is a point beyond which the redeposition due to an anisotropic emitter is greater than that due to an isotropic emitter of identical luminance. For a 90° slope (step or groove of rectangular section) the point is about 0.4 times the emitter length (i.e. 0.4 Xgroove width) above the base. Sinusoidal asperities which are present in a high surface density are expected to receive significant redeposited flux only near their bases. By contrast, widely separated asperities would receive flux over almost all of their profiles. In this latter situation the magnitude of the redeposited flux density is found to be relatively insensitive to position on a profile.

1. Introduction This paper is concerned with the topographical evolution of features on amorphous surfaces under ion bombardment. Descriptions of the development of such features have previously been based on analytical arguments [1 ], computer simulation [2,3] and geometrical construction [4]. All of the above approaches describe the evolution as an erosion process based on just one e f f e c t - t h e variation in sputtering yield S with angle of ion incidence 0 (to the surface normal). It is realised however [5,6] that material wh_ich is sputtered from regions adjoining an asperity standing proud of a surface can be redeposited on to that asperity. This effect has so far been ignored in the theories mentioned above. It is clear that a detailed theory of surface evolution in general must treat the two simultaneous processes of sputtering and redeposition. With the growth of interest in the use of ion beam etching to produce micron sized features in microelectronic device structures, redeposition is becoming a problem of increasing technological importance. In this etching process redeposition can be the major cause of departures from the rectilinear geometries required for optimum device performance. A numerical model of groove evolution has recently been developed by Makh et al. [7] using an approach based on kinematic wave theory.

A first attempt at describing the early evolution of any surface asperity was made by the present authors in an earlier paper [8]. The redeposited material was assumed to be ejected from the surrounding planar surface with an isotropic spatial distribution. In the present work, analytical expressions are obtained and compared for cases where the ejected flux is isotropic and anisotropic (cosine).

2. Generalized flux integral Consider a line OL which lies along the positive x axis such that 0 < x < l, as shown in fig. la. Let every point of OL emit particles at a constant rate n per unit length per unit time into the half plane above the x axis. The trajectories of aU emitted particles are assumed to be straight lines. We define and determine a flux density due to OL in some region such as the element PQ of arc on the profile y =f(~) in fig. l a where ~ < 0 and f(0) = 0. The flux density in such a region is defined as: Lt IL,,Q

no. of particles traversing arc PO/s length of arc PQ

(1)

We assume no a priori physical reason for the profile f and its disposition to be related to the configuration of the emitter. It is thus in order to describe emitter II. MA'IERIAL BUILD-UPAND REMOVAL PROCESSES

276

J.

Belson, 12t. Wilson/Redeposition of sputtered flux and if this is inserted into eq. (5) we have

dF-- I

"%~\

F ( x l , x2) =

Y= - a sin T

=

%Y

(7)

If particles emitted only from points lying between certain limits (e.g. xl < x < x 2 ) reach PQ the integrated flux density is

o ! MRN

_,,_

x) _ l [ + (x : l (1 +f~)~/2 [(x - ~)2 +f2j dx.

t

~2 j x1 o

__ u(4~, x) I J'+_(x- ~)f~ / dq~ dx (8) (1 + f ~ ) ' / 2 t ( x - ~)2 + ~ j •

>

Fig. 1. Def'mition o f terms used in the derivations. (a) A substrate of arbitrary profile, y =f(~). (b) A linear substrate, y = - ~ tan a. (c) A sinusoidal s u b s t r a t e , y = - a sin ~.

and profile by different variables;x is reserved for distances along the emitter, whilst ~ is the independent variable the function of which describes the profile. Consider the emission from an element MN of length dx in fig. la. Let MN emit v dx particles per second from its mid-point R where, in general v = u(~),x). ¢ is the angle between the emitter axis and the line joining R to the element PQ (length ds) of the asperity profile. From the definition (1) the flux density at PQ due to MN is

3. lsotropic emission The simplest relevant form of v is one appropriate to isotropic emission with a constant number of particles n emitted per unit length, v is then independent of x and ¢ and the normalization condition is written as 7f

.J" v dq~ = n , i.e. v = n/zr.

(9)

o

The integrated form of eq. (8) appropriate to the isotropic case is discussed next, with linear and sinusoidal profile functions f = (-tan c0 ~ a n d f = - a sin respectively. 3.1. Isotropic emission on to a linear slope

If the emitter radiates n particles per unit length the appropriate normalization condition is

where

Fig. 1b shows a linear slope whose foot is at O, the origin of coordinates, and which is inclined at an angle c~ to the negative ~ axis. At a point on the slope where the height above the horizontal axis is Hl, (H is a dimensionless height), the ratio of incident flux density to emitted flux per unit length of emitter (which we call the normalized flux density) is found from eq. (8), with xl = 0 and x2 = l, to be

f~ = df/d~,

F

(3)

f v($, x) dck dx = n dx .

Since ds = -(1 +f~) d~,

(4)

we may rewrite eq. (2) as

d F = - (1 + ~ ) " ~

ax.

(s)

It is straightforward [8] to show that

a~ _ f + (x - ~)f~ a~ ( x - ~)~ +/a ,

sina l n ~ ( l + H c o t a ) 2 + H 2 q --iU L ~cosec~ J

(6)

The variation of Fin with distance up the slope H/sin o~ is shown as a series of dashed curves in fig. 2 for values of slope angle c~ between 30 ° and 180°. As

Z Belson, LH. Wilson /Redeposition ofsputteredflux

277

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430*-\X

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~

--~

: ~ zs. ?. ~• ~- .'~-~:7)-........... ~ "-5 "7.~---'-: i"

i

" ~ .<

.....

"\

L...":;-~

'?\~

COSINE

[[: "

0.6 0.8 1.0 RATE FLUX DENSITY

EMITTER

ISOTROPIC

EMITTER

F/n)

Fig. 2. Linear substrate; variation o f flux density w i t h d i s t a n c e up the slope and slope angle. C o n t i n u o u s curves: cosine emitter. Dashed curves: isotropic emitter.

might be expected the deposition rate is very high close to the foot of the slope and falls away rapidly with increasing distance up the slope. The curves for a > 90 ° serve to show that as the receiving surface closes down on to the emitter Fin ~ I for tt/sin a < 1 and Fin -+ 0 for H/sin a > 1.

3.2. Isotropie emission on to a sinusoidal profile Consider next the deposition on to a sinusoidal asperity (fig. lc) which is adjacent to the linear emitter. The asperity (half of which is shown) has a fixed base dimension o f rr units and a parametrically variable amplitude a. In this way a range of asperity shapes between tall spikes (a >> 7r) and shallow bumps (a,~Tr) may be configured for different emitter lengths l. In this case the normalized flux density is again given by eq. (8) with v=n/zr but this time with f = - a sin ~. We find [8] F=

n

A l cos p

27r(l +A2l 2

I--. (l + p)2 -- + H 2 1 2 - ]

cos2p)l/~'lnL(d-+p)2+H212j

I 7r(l + A 2 1 2 c o s 2 p ) I/2

×[tan-,[l+P]_ L

\-H~-J

[d+p~q tan-' 1~7-JJ

'

(11,

where HI (=h) is, as before, the height of the point of interest above the base and AI (=a) is the amplitude o f the sinusoid. Because the profile is non-linear, part of the emitter does not contribute to the flux density at a given point. The flux density integral has an upper limit x2 = l and a lower limit xl = d (see fig. lc) given by d = tan p - p ,

(12)

where p = sin -1 (H/A) (=-~). The dashed curves of figs. 5 and 6 show the normalized flux density Fin at the sinusoidal profile plotted against the height of the point of interest (expressed as a fraction of the amplitude a) for the isotropic type of emitter. Fig. 5 represents the case A = 10 (amplitude a = 1 0 × e m i t t e r length /) and fig. 6 is for A = 10 -3. The upper and lower dashed curves in each figure correspond to tall (a = 10) and shallow (a = 1) asperities respectively referred to the same emitter. Fig. 5 shows that when the emitter is small compared to the amplitude (A = 10) the flux II. M A T E R I A L B U I L D - U P A N D R E M O V A L P R O C E S S E S

278

3".Belson, LH. Wilson / Redeposition of sputtered flux

density decreases rapidly with increasing height but that substantial redeposition can be expected at the foot of the feature. By contrast the dashed curves of fig. 6 indicate that in the large emitter case (A = 10 -3) the flux density falls off relatively slowly with increasing height. Of the two asperity shapes represented in fig. 6, the shallow (a = 1) shows more height sensitivity than the tail (a = 10).

point on the substrate line, F(xl ,x2) is F(0, l) where

4. Anisotropic emission

f = ( - tan a)~ = h ,

X {(__-tana)~+(x-~)(--tana)} -. - - ( x ~ - ~ +-?dr.

(16)

If the point at which F ( 0 , / ) is being evaluated is at a height h above OL, then (17)

and as may be verified from fig. lb

4.1. The cosine emitter Let an atom within an amorphous planar surface be moving with velocity v in a direction making an angle 0 with the normal to the surface. We assume that the greater the value of v cos 0 the greater is the probability of escape from the surface. Such a model may be expected to be useful when treating surface atoms which are ejected as a result of multiple collision processes subsequent to the impact of an incident particle. In this type of event there is no " m e m o r y " of the incident particle's direction in the distribution of ejected a t o m s - t h e so-called cosine distribution-which merely reflects the increased probability of ejection with increased component o f momentum normal to the surface. Let emission from a point be independent of position on the emitter line OL (fig. 1) but dependent upon the direction 9. We take the emission function v = v(9) as v = no cos(½~" - 9)

t 1 F(O,/) = - fo ~n 1 sin 9 (-i+ t'an2a) '/2

(13)

The emission rate is thus zero along the axis OL and a maximum (no) normal to OL. Retaining the number of particles radiated per unit length as n the normalization condition is

h sin 9 - [h 2 ÷ ( x - ~)211/2 h = [h 2 +(h cot a +x)2] 1'2 '

(18)

which when inserted into eq. (16) yields the flux density integral dx

F = F(0, l) - nh sin a

2

~[h2+(hcota+x)2] o

3/2 " (19)

Upon integration, the ratio of incident flux density to emitted flux per length of source is

Fn- 21{ 1 -

[~q~l h c °(shecc°at+a + l cl°-_)~.-~.. s a ) J~ ~1. L.~-;-:_--.-.--.

(20)

Transforming h to a dimensionless height gives n

I,

2[

+ (l + H cot a)2} ,/z

l}

'

H (=hfl)

(H~=O) (21)

The variation of k'[n with distance up the slope 71

no sin 9 d9 = n , orn0 = g n . ~=o, The emitter function is now

f

(14)

P=~r/ 1 sin 9

(15)

h

\

4.2. Cosine emission on to the linear slope _

Consider the emission from the source line OL on to the linear profile (fig. lb) which passes through the origin 0 and which has slope ( - t a n cz) so that the profile function is f(~)= ( - t a n a)~. Since the entire length of OL contributes to the flux density at any

.--k

E

0

/

.....

f

"~

"" "

k

I

Fig. 3. Geometry relating to section 4.2. treating the deposition rate at point P in terms of its angle of elevation 3' from the end L of the cosine emitter.

d. Belson, LH. Wilson /Redeposition of sputtered flux 90*

80 °

70"

60 °

50 °

40"

279

30"

20*

g~

I°*8~ O* 0.1

0.3

0.2

DEPOSITION RATE (Fill)

IOO°

90 °

80 °

70 °

60 °

50 °

40 °

i i

1tooi

'

o!"",,,

-3o

!, " , ~.,.f~

150 °.

~

17oo ~ r ~ k . - ~ . . ~ 180 ° '

, 0.2

o

-" __~" . . . . . . . . 0

< , 0.2

",, 014

DEPOSITION

, ", 0.6

,~ 0.8

~

I, 'I.0

RATE(F/n)

Fig. 4. Linear substrate with cosine emitter; variation of deposition rate with angle of elevation ('r) from end L of emitter and slope angle (a). (a) a = 30;'-90 °. (b) a = 90°-170 ° .

(H/sin a ) for this emitter is shown in fig. 2 for various slope angles a, as a series o f c o n t i n u o u s curves. It is instructive to consider the geometry relating to eq. (20). This is depicted in fig. 3 which is based on fig. l b . Let the deposition rate be required at some point P whose angle of elevation from the end L o f the emitter is 7. PE and LF are perpendiculars o n t o LO and PO produced respectively. Since PE = h, PF=(hcoseca+lcosre) and EL = ( l + h cot a) we may write eq. (20). as * n

2 1

PL]'

(22)

* The quantity F on the left-hand side of eqs. (22) and (23) represents flux density and is not to be confused with the geometrical point F in fig. 3.

i.e. F 1 n - ~[1 - cos(re - 7)] •

(23)

Thus for cosine emission o n to a linear slope we have particularly straightforward access to the deposition rate ( b / n ) at any point in terms o f its angle ofelcvation fi'om a given point (L) and the inclination (a) o f the slope o n which it lies. For a n y given a the curve o f (Fin) vs. 7 takes the form o f a partial cardioid. Figs. 4a and 4b show (b/n) vs. 7 plotted for a = 30 ° 90 ° and 90 ° - 1 7 0 ° respectively. The deposition rate at a point whose elevation is re is represented by the length o f the radius vector from the origin o f polar coordinates to the appropriate curve for the a-value in question. II. MATERIAL BUILD-UP AND REMOVAL PROCESSES

J. Belson, LH. Wilson/ Redeposition of sputtered flux

280

4.3. Coshre emission on to a sbmsoidal profile

Small emitter, A

= a l l = 10

~: f.O-

In this section we consider the application of the emitter function v(q~) = ½n sin q~ to a sinusoidal substrate of amplitude a

f=

- a sin ~,

'~ 0.8; '

-- COSINK ---- ISOTROPIC "\~=10 o.,o

(24)

shown in fig. 1c. We wish to evaluate the integral (8) for a point on the substrate at a height h above OL. The lower limit of the integral is no longer zero. Thus F(xl,x~) = F(d, l) where d is the smallest value of x from which emitted particles can strike the point of interest. Using eq. (22) with eq. (8) we have

F(d, l)

I

nsin~ rl - J 2(1 + a 2 cos2~) 1/2

O.001 00t 0.| DEPOSITION RATE (NORMAUZED FLUX DENSITY Fin)

Fig. 5. Sinusoidal proffflc; variation of flux density with

height above the emitter for amplitude a = 10 and a = 1 (upper and lower pairs of curves respectively) with A = 10. Continuous curves: cosine emitter. Dashed curves: isotropic emitter.

d

h + (x - O(--a cos ~) ] X

(x

~)2 +h2

: dx.

(25)

Substituting for sm q~ from eq. (18) and integrating we have

F_ n

1

V d+p+A2l 2sinpcosp

2(1 +A212 cos2p) w2 L{A-~2-sin2p + ( d + p ~ } ~/2

l + p +A : l 2 sin p cos p q {A212 ~:mTp + (l-+ p)2 i i-/2J ,

(26)

where ~ has been replaced by - p (p > 0) and the amplitude a = Al (A dimensionless) as before.

5. Predictions of isotropic and cosine emitter models Fig. 2 shows that isotropic and cosine emission near the foot o f a linear slope,gives rise to a deposition rate (F/n)is, which is much greater than for cosine emission (Fin)cos. In the cosine model there is a lack of flux along directions close to the emitter axis and this suppresses the build-up evident in the isotropic model. Both types of emitter produce a reduction in (Fin) with increasing distance up a linear slope, ltowever, the rate of reduction is less for cosine than for isotropic emission. This is explained by the angular dependence of the emission in the cosine model increasing the effective luminance of the emitter with increasing distance up the slope. Increasing remoteness from the emitter is thus counteracted, to some extent, by increasing luminance to give a deposition rate which is less height dependent. In

fig. 2 the curves for slope angles a < 70 ° show that (Fin)cos is less than (F/n)is if the distance up the slope is smaller than about one emitter length. However on each curve there is a point above which (F/n)cos is greater than ( F / n ) i s. The point occurs progressively lower down the slope as a increases. For the especially important case where a = 90 ° (groove milling) the critical point is at a distance up the wall approxhnately equal to 0.4 times the emitter length (i.e. 0.4 times the groove width). Consider next the case of the sinusoidal asperity with a topography in which the emitter length l is small compared to the amplitude a, (A = 10) represented in fig. 5. The fall-off in flux density (Flu) with height is seen to be very rapid due to the shadowing effect of the curved profile. Both the isotropic and cosine models show that where the emitter is small only the foot of the profile will be significantly affected by redeposition. The lower pair of curves (a = 1) of fig. 5 depict the deposition from a small emitter on to a shallow bump. Deposition rates are ahnost identical for isotropic and cosine emitters at medium and large fractional heights, but near the base the isotropic model predicts a greater deposition rate. For the upper pair o f curves (a = l 0) the asperity is sharper and the cosine emitter gives rise to a consistently higher rate than the isotropic emitter. Fig. 6 shows curves for a sinusoidal asperity exposed to a large emitter (A = 10-3). Here Fin is relatively insensitive to height except near the very top of the asperity. The lack of height dependence is more marked for a sharp asperity (a = 10) than for a

J. Belson, LH. Wilson / Redeposition of sputtered flux ~ ..c

--.,,

Large e m i t t e r , A =

1.O-

0.8-

= 10 -3

\

\ a=lO

(:1=~0 0.6

-~

0.44

\\~=1

=1

....

|

,so-rRo ,c

I

~

-

ot

l OA DEPOSITION

RATE

(NORMALIZED

6. Conclusions

,

\

~

all

281

1.0 FLUX DERRITY

F/t3)

Fig. 6. Sinusoidal profile: variation of flux density with height above the emitter for amplitudes a = 10 and a = 1 with A = 10 -3. Continuous curves: cosine emitter. Dashed curves: isotropic emitter.

shallow bump (a = 1). In addition the cosine emitter results in an absence of enhanced deposition at the foot o f the profile and this serves to reinforce the insensitivity to height. Surface development may thus be expected to occur in such a way as to increase the diameter of a sharp spike without change of shape, apart from a flattening-off at the top. The profile would in fact tend to a form similar to the shape of the Fin vs. height curve itself. As shown in fig. 6, absolute values of Fin for the cosine model are less than for the isotropic model. This is reasonable in view of the fact that the emitter OL is a factor 103 greater than the amplitude a. Most of the emitted flux striking the asperity arrives on trajectories close to the axis OL. In the cosine model it is at these small angles to OL that the emission is weakest and so the resulting flux density is low.

Both types of emitter are predicted to give rise to significant deposition near the foot of linear slope but the effect is expected to be much greater for the isotropic type. Above the foot of a linear slope there is a point beyond which a cosine emitter would give rise to a greater flux density than an isotropic emitter. For a 90 ° groove wall this point is at a height equal to ca. 0.4 times the groove width. Sinusoidal asperities exposed to short emitters (i.e. high surface density o f asperities) are expected to be influenced by redeposition only near their bases. If the emitter is much larger than the asperity height (low surface density) the redeposited flux density should be relatively insensitive to position on the protile except near the very top.

References [1] A.D.G. Stewart and M.W. Thompson, J. Mat. Sci. 4 (1969) 56. [2] C. Catana, J.S. CoUigon and G. Carter, J. Mat. Sci. 7 (1972) 467. [3] T. Ishitani, M. Kato and R. Shimizu, J. Mat. Sci. 9 (1974) 505. [4] C.J. Barber, F.C. Frank, M. Moss, J.W. Steeds and I.S.T. Tsong, J. Mat. Sci. 8 (1973) 10. [5] A.R. Bayly, J. Mat. Sci. 7 (1972) 404. [6] W. Hauffe, 1978 Int. Conf. on Ion beam modification of materials, Paper F8. [7] S.S. Makh, R. Smith and J.M. WaUs, 1980 Int. Conf. on Low energy ion beams, vol. 2, Paper 48. [8] J. Belson and I.H. Wilson, Rad. Eft., 51 (1980) 27.

II. MATERIAL BUILD-UP AND REMOVAL PROCESSES