Laser-pulse sputtering of atoms and molecules Part II. Recondensation effects

Nuclear Ins~rnen~ and Methods in Physics Research B 91 (1994) 682-691 North-Holland

Laser-pulse sputtering of atoms and molecules Part II. Recondensation effects **

Beam tnteractions with Materials 8 Atoms

Roger Kelly * and Antonio Miotello I&mt. di Ftiica, Universitli di Trerzto and Consorzio I~ter~iver~tario Nation& Fisica della Materia, I-38050 For;o, Tremio, Italy

When solids are sputtered with sufficiently intense laser pulses, particle release takes place, and a terminating expansion then develops. The resulting gas-dynamic processes depend on whether backscattered particles are subject to recondensation or reflection. We here wish to examine the problem further with emphasis on recondensation effects. (a) We show that, under conditions of recondensation, the surface region permits simple analytical appro~mations. (b) These simple forms enable the recondensed quaff, Qrec. to be evaluated as the integral over time of the product of flow velocity and density. iteratively Q” can be deduced by numerical solution of the flow equations and in either case it is found to increase signi~c~t~ with the complexity of the particles involved. Thus it ranges from a fraction N 0.10 for atoms to _ 0.43 for large molecules. (c) Recondensation effects are particularly important when the polymer polyimide is laser sputtered, as the recondensation occurs in massive amounts beyond the bombarded spot and the patterns of recondensation show remarkable symmetry changes. (d) Closely related effects occur when targets are laser sputtered to produce films, even if formal recondensation is not involved.

1. ~t~u~tion

One-dimensional expansions which do root ter~j~ate have been we& understood since 19.59 [2] but have taken on a new significance in view of current work on laser sputtering of solids [3]. We have noted [4] that the solutions have two basic forms, depending on whether the expansions originate from the removal of the wall of a semi-infinite reservoir or from an effusion-like process. From an experimental point of view, waZZ renaoval will occur when, for example, particles inci- dent on a solid cause inst~taneous bond breaking at and beneath the surface 15-71, while e~~~-l~ke behavior will occur when, for example, a laser beam heats the surface sufficiently to cause vaporization [3].

One-dimensional expansions which terminate are probably more important from an experimental point of view, because most work with lasers is done under conditions of pulsed release. Only a few examples exist where the behavior is that appropriate to wall removal, of ~damental ~~rtance here being the sideways expansion of the plume of laser-sputtered polyimide [8] (Fig. 1) or Cti [9] (Fig. 3 of ref. 191). Most aspects of laser sputtering are better described in terms of ef@- &z-like behavior, for example the initially one-dimen-

* Corresponding author. ** For Part I, which deals with rigorous solutions, see ref. [ll.

sional expansion of the plume of laser-sputtered ~ly(me~y~ethac~late) OPT) [3,10] (Fig. 2). The low surface density seen in Fig. 2 has been taken as the fundamental characteristic of effusion-like behavior.

A further characteristic of terminating expansions is that there are two limiting boundary conditions. One must consider that particles scattered towards the back of the reservoir (or towards the effusing surface) can either recondense or reflect. This leads to a total of four basic types of terminating expansion: wall removal with recondensation or reflection and e~sion-ljke behavior with re~ndensation or reflection.

We wish here to extend the available information for expansions which terminate beyond what is available in previous work [l,ll-141 in order to examine recondensation effects. (a) We show that, under conditions of recondensation, the surface region permits simple analytical approximations. (b) These simple forms enable the recondensed quantity, Qrec, to be evaluated analytically. Alternatively Q” can be deduced by numerical solution of the flow equations. (c> Re~ndensation effects are particularly impo~ant when polyimide is laser sputtered, since the recondensation occurs in massive amounts beyond the boinbarded spot and the patterns of recondensation show remarkable symmetry changes [8,14] (Fig. 3). (d) Closely related effects occur when targets are laser sputtered to produce films [9] (Fig. 3 of ref. [9]), even if formal recondensation is not involved.

0168-583X/94/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved SSL?Z 0168-583X(94)00084-9

2. Solution of the problem of a terminating, effision- like expansion with recondensation

2.1. Underlying rela tiom

The variables in the present gas-dynamic problem will be written unitlessly:

A=a/uKI U=u/uK, P=p/p,,

x=x/uKr,, T = t/7,.

Here a is the sound speed, u is the flow velocity, p is the gas density, x is distance normal to the effusing surface, and t is time. uK (= as> and pK characterize the outer boundary of the Knudsen layer (KL,), formed as the effusing particles come to equilibrium with each other [15-171. The KL will be taken as having zero thickness and therefore becomes a boundary condition. Finally, TV is the length of the release process, often much greater than the laser pulse length.

The real expansion geometry is, of course, three-dimensional as in Fig. 4a. Since such a geometry tends to be prohibitively complex, however, we will take the expansion to be one-dimensional, as for the ejecta of PMMA seen in Fig. 2. This is shown schematically in Fig. 4b.

The flow equations relevant to the problem being considered thus become [l]:

aA ar+LJ;+ (Y;l)AJ!$O

(continuity eqaution); (la)

av av 2A aA

z’“-’ - - = 0 (Euler equation). (lb)

ax y-iax

Likewise, the relation between density and sound speed changes from the well-known form p a a’/(Y-‘) to

p =Az/(Y--1), (24

where we note an equality instead of proportionality and where y = C,/Cv is the heat capacity ratio. Val- ues of Z/(-y - I> are included in Table 1.

For y -+ 1 Srbold and Urbassek [12] have argued that there is a physical problem in that the gas particles leave the surface with a uniform T. But since there is a large number of degrees of freedom when y + 1, the energy tends to infinity. This requires the relation P = A2/(y-1) to be replaced with

A :p(Y-W= 1 9 PI

Fig. 1. Polyimide targets with a thickness of 125 urn were exposed to singie laser pulses (308 nm, - 20 ns, pulse diameter 800 pm, 2.3 J/cm’, normal incidence) in air [gl. The emitted heavy partictes, the contact front of the emitted light particles, and the shock wave caused by the light particles were photographed by firing parallel to the target surface a second (“probe”) laser with a delay of 410 ns (596 nm, - 1 ns). The lower of the two contact fronts is marked “CF” and the shock wave “SW”. The arrows represent

the recondensation process.

1X. SURFACE AND SPUTTERING

684 R. Kelly, A. Mtitello /Nucl. Instr. and Meth. in Phys. Res. B 91 (1994) 682-691

the expansion becomes isothermal, and A ceases to be a variable. Eqs. (la) and (lb) must therefore be rewritten in terms of P and U instead of A and U [1,121.

At the moment that an effusion-like expansion terminates, there is an abrupt change of boundary condition at the effusing surface. When particles backscattered to the surface undergo recondensation, A falls abruptly from 1 to A,,, and U from 1 to a negative

value -U,. As already noted El], the expansion, as a result of these changes, evolves from that for a non- terminating situation (region III of Fig. 5) to one with two parts (regions II and III of Fig. 5). Region II is shown as beginning at the surface at X = 0 and ending with an abrupt change of slope at the second l&ze of con&~ (LOC) at X,,. Alternatively region II could be defined as beginning at the virtual expansion front or minimum X at X. Region III begins unambiguously at Xu and, for an ambient vacuum,nends at the real expunsion front or maximum X at X.

Fig. 2. PMMA targets with a thickness of 700 km were exposed to five laser pulses (248 nm, - 20 ns, pulse diameter 700 pm, 2.1 J/cm2, normal incidence) in air El]. The emitted heavy particles (“ejecta”), and the shock wave caused by the emitted light particles, were photographed by firing parallel to the target surface a second (“probe”) laser with a delay of 4.9 I.LS (596 nm, N 1 ns). The contact front of the heavy particles is marked “CF” and the shock wave “SW”. The arrows represent the recondensation

process.

Fig lase the

R. Kelly, A. Miotello /Nucl. Instr. and Meth. in Phys. Res. B 91 (1994) 482-691 685

..3. Recondensed material on the surface of a polyimide target with a thickness of 8 pm which has been exposed to rough :I pulses (308 nm, - 20 ns, - 0.25 J/cm’, normal incidence) in air. The light region near the center is a hole sputtered thx entire target. The dark region beyond the hole is recondensed material. (a) Square mask with hole having width of 270

(b) Triangular mask (530 km).

ily 50

1 pm.

IX. SURFACE AND SPUTTERING

686 R. Kelly, A. Miotello /Nucl. Instr. and Meth. in Phys. Rex B 91 (1994) 682-691

REAL PROBLEM ONE-DIMENSIONAL TWO-DIMENSIONAL SIMULATION WITH SIMULATION WITH RECONDENSATION SIDEWAYS MOTION

SURkACE .._

SURFACE SURFACE

(aI (b) (C)

Fig. 4. The geometries relevant to the problems considered. (a) The real problem relevant to laser sputtering, whether with polyimide [8] or Cn [9] or any other system: combined outwards and sideways expansion. (b) The idealized problem used to analyze the expansions (section 2) and the recondensed quantity (section 3) for laser sputtering of materials for which the plume is reasonably one-dimensional. (c) The idealized problem used to analyze the recondensed geometry for laser sputtering of materials for which the plume shows

significant sideways expansion (section 4).

Region I seen in Fig. 5 exists only under conditions of reflection. As treated elsewhere [l], this is because the disturbance at the surface which is triggered by the termination of the expansion is reflected.

We now discuss the expansion as it proceeds from J? to J? in terms of rigorous expressions (when these are of simple form) and otherwise in terms of approximations.

The virtual expansion front at _%? (all T). The only unknown quantity at the virtual expansion front at 2 is l?, which is given exactly for all y by [ll

ii= -(3-y)/(y-1) (all r). (3)

c does not, however, enter into the discussion of recondensation effects to be undertaken here.

The surface at X = 0 (T = 1). Unknown quantities at the surface at X= 0 (Y = 1) include both A, and

00 I ’ 8 ’ ’ ’ * t -8 -4 0 4 8 12 16

MSTANCE RATIO, X = x/kq

0

Fig. 5. Sound speed vs distance for y = 5/3, ‘r = 4, and an ambient vacuum. Shown are the expansion fronts, lines of contact (LOCI, and regions. The branching at the left of the curve corresponds to conditions of reflection (upper branch, dashed) and recondensation (lower branch, solid). For recondensation region II could also be regarded as extending from

the virtual expansion front (2) to the second LOC (X,,).

U,. A particularly simple way to derive A,, and U, is as follows. We note that, at the moment when the expansion terminates abruptly at Y = 1 and recondensation sets in, the change at the surface can be regarded as occurring in two steps. First the effusing surface is sealed and we have the same situation as for reflection

111: A = Arf = (3 - y)/2;

U=UFf=O (ally;Y=l). (4a)

The sealed surface is now removed and the factor 2/(y + 1) is applied to Arf giving the exact result,

A,=(3-y)/(y+l) (all y;Y=l). (4h)

The factor 2/(y + 1) relates the sound speed at X= 0 to that of the reservoir under conditions of wall re-

Table 1 Various numerical constants relevant to terminating expansions under conditions of recondensation

Species Heat The exponent The value of A, The value of K capacity in the relation, when there is when there is ratio, y p = AZ/(?’ - 1) recondensation, recondensation,

(Eq. (2a)) A, = (3 - y)/(y + 1) U= - A,TK @. (4bN (Eq. (5bN

Atoms 5/3 3 l/2 0.59 < K < 0.61 Rotating diatomics l/5 5 213 0.50 < K < 0.51 Diatomics which both rotate and vibrate 9/7a 7 3/4 0.46 < K < 0.47

Intermediate molecules 11/9a 9 4/5 - 0.40 < K < u 0.41

Large molecules which both rotate and vibrate 1 CO 1 0.86 < - U, < 0.89 b

a Useful also for hot molecules (Table 2 of ref. [171). b Eqs. (7~) and (8).

R. Kelly, A. Miotello /Nucl. Instr. and Meth. in Phys. Res. B 91 (1994) 682-691 687

moval [l]. The corresponding value of Us now follows by analogy with what happens when a wall is removed

U,= -A,= -(3-y)/(y+l) (yzl;r=l),

(4c)

a result which is shown below to fail for Y = 1. (This failure, to within - lo%, was not realized previously

[ll.) Approximate results for region II (all T). We are not

aware of a rigorous description valid for region II, even for X= 0, usable for all r. Simple approximate results are readily obtained, however, by noting that, for y # 1, the following expressions are useful at X= 0:

A =As’$-(‘-‘)/a; P =A20/(Y-1)T-1 (y # 1); (5a)

U= -A,!T’-K (yf 1). (5b)

The exponent in Eq. (5a) is taken to be the same as that under conditions of reflection [l] whereas that in Eq. (5b) is wholly empirical. (But the factors *A,, are exact.) Values of A, and K are included in Table 1. To treat all of region II it is sufficient to make a linear interpolation between X = 0 and X,, . We indicate only the result for U, which is of rather high accuracy because of the general tendency of U to vary linearly with distance [1,12]:

U= -A,TK+ (U1l+AoZ’-K)(X/XII). (6)

For U,, and XII, see Eqs. (9). For Y = 1 we were able to establish several general

guidelines. The following initial condition, proposed by Sibold and Urbassek [12] (see their Eqs. (B12) and (B13)), appears to be exact:

U,=l+ln Pa. (7a)

By conserving particles in the limit 2’ = 1 + AT (appendix A) we obtain an approximate relation

u, = - 1/(3P,) + 4/3, (7b)

and there is therefore sufficient information to give

A, = 1; Pa = 0.150; U, = -0.89. (7c)

The form for A,,, which is exact, is that dictated by Eq. (2b). PO as derived here for conditions of recondensation (0.150) is seen to be less than that for reflection (l/3 [l]), as must be true.

Numerical solutions of the flow equations appropriate to Y = 1 [1,12] were next obtained for a series of values of r, namely 1.05, 1.1, 1.5, 2, 4, 8 and 16, and it was possible to show that the values of A, P, and U at the surface were well represented by the following:

A =A, = 1; P = 0.156r-1.32;

u=u,= -0.86 (Y = 1; all r>. (8)

Eqs. (8) are reasonably consistent with the initial conditions of Eq. (7~). The unusual time-independent form of the relation for U is consistent with the time-inde-

pendent form for A. However, why the relation is not precisely U= -A is not understood. (Interestingly, were the relation U= -A correct, then Eq. (7a) by itself would yield P,, = 0.135.)

In work which is not reported, the approximate forms for A and U seen in Eqs. (5) and (8) are compared with analytical or numerical results. General agreement to within - 6% is found.

The second LOC at X,, (all 2’). Region II terminates with an abrupt change of slope at the second LOC at X,, governed by dX,,/dr = U,, + A,,. In view of the existence of simple, explicit solutions for region III for Y # 1 [l], it is easy to evaluate dX,,/dZ’ with the well-known final results [l]

Y+l x,, = - y _ 1 (T - T--y)/(y+l)) (y # 1);

A,, = ~-2(~--l)/(~+l); p,, = rp4/(Y+l) (all y)*

y+l 2 j-J,, = - - -

Y-l*- 2(Y--l)/(Y+r)

Y-l (Y + 1). (9c)

For y = 1 the velocity of the second LOC is again dXu/dZ’ = U,, +A,,, but the final results are in part different [ 11:

X,,=2r lnr (y=l); (lOa)

A,,=l; P,,=T-’ (y=l); (1Ob)

U,=1+21nr (y=l). (1Oc)

Region III and the real expansion front at J! (all T). Region III is the remanent of the initial release process and would be the only region present were the release non-terminating. For an effusion-like expansion which proceeds into vacuum [l] or air ([ll], espe- cially Figs. 3 and 4) it is described by relations of surprisingly simple form. For example, for y # 1 and at ambient vacuum A decreases linearly to zero as in Fig. 5, the limiting distance being termed the expansion front. The position of the latter is given, for all y, by

X=(y+l)T/(y-1) (ally). (11)

3. Recondensed quantity

The recondensed quantity is easily obtained since the flow velocity @I) and density (P) are known at the surface, even if in approximate form. For Y # 1 P is given to within 12% by Eq. (5a) and U to within 6% by Eq. (5b). The recondensed quantity at time T follows as

Q”” = - /,%P dr

+‘+r)/(Y--1&--l(l _ T-K) (y + 1). (12a)


688 R. Ke&, A. Miotello /Nucl. Ins&. and h&h. in Phys. Rex B 91 (1994) 682-691

Similarly, for y = 1 one obtains

Q =‘= 0.427(1- Y-o.32) (y = 1). (12b) Eqs. (12) should be regarded as analytical descriptions of recondensation.

Q ret can also be evaluated ~~~e~caZ~. For this purpose we solve the flow equations for A and U (if y f 1) or for P and U (if y = l), and deduce Q”” as

Q ==l-J;p,dX. (13)

A comparison is made in Table 2 between Eqs. (12) and (13). Five different values of y were used in order to simulate the behavior of particles ranging from atoms, y = 5f3, to large molecules, y = 1. The results show (a) that Q’“” increases sign~~cantly with the complexity of the particles involved, and (b) that there is reasonable agreement between Eqs. (12) and (13).

4. Recondensed geometry

We finally treat the recondensed geometry, a topic closely related to the recondensed quantity but not amenable to approximation. Singleton et al. [i&19] have studied the laser-pulse sputte~ng of five poly- mers, including PMMA and polyimide. In each case two groups of particles were identified, a first group with small molecular weight and a second, heavy group (“ejecta”). With PMhJA the ejecta dominated, while with pol~mide the light particles dominated. In related work it was shown that with PMMA the ejecta under- went an initially one-dimensional expansion away from the target surface (Fig. 2), whereas with polyimide [14,18,20,21] the one-dimensional 0~~~~~~ expansion was coupled to a sid~ays expansion (Fig. 1). This behavior could be rationalized in terms of the force exerted on the heavy particles by the light ones [Xl.

Recondensation can be expected to give totally different results for laser sputtering of PMMA and polyimide. With PMMA, it will show precisely the same geometry as the laser-beam. This in turn means that the recondensed material will not persist from pulse to pulse but, rather, only that due to the final pulse will be present. According to Table 2, this will amount to - 43% of the material ejected per pulse provided the ejecta are describable by y + 1. A completely different situation will occur with polyimide. In fact, it is then found that the recondensed material accumulates beyond the bombarded spot and the symmetry of this material differs from that of the spot in such a way as to imply that the flow pattern involves a rotation (fig. 3).

Closely related to the pattern of recondensed material found on polyimide is the symmetry of a film deposited on a substrate for laser sputtering of essentially any material. For example, with an elliptical bombarded spot the symmetry shows a 90“ rotation [9,22,23] resembling that found with polyimide [14]. The reason for the similarity is simple. It is true that a deposited film has nothing to do with recondensation. The outwards expansion of this material towards the substrate will, however, always be coupled to a sideways expansion.

An obvious question to ask is whether the rotations, be they of recondensed material or of deposited films, occur through the laws of flow without extraneous effects such as charging 1241, temperature gradients, etc. We therefore examine the flow equations for two beam shapes, that of a square bombarded spot and that of a rectangular spot. The real geometries are tree-dimensional, involving a simultaneous outwards and sideways expansion into an ambient gas coupled with recondensation onto the target surface (Fig. 4a). These geometries are prohibitively complex, however,

Table 2

Q let as would occur for different vahres of Y if the gas expanded outwards into vacuum as in Fig. 4b instead of three-~mensiona~ly as in Fig. 4a. “analyt.” refers to the use of Eqs. (12) and “numer.” to the use of Eq. (13)

CT = t/r, Q- as a fraction of emitted particles

y=5/3 y=7/5

(uumer.) (anaIyt.1 (numer.)

1 0 0 0 2 0.024 0.036 0.040 4 0.052 0.059 0.080 8 0.072 0.075 0.110

14 0.083 0.084 0.13 20 0.089 0.088 0.14 30 0.095 0.092 0.15 m 0.104 0.174 a

’ Analytical values from Eqs. (12).

y=9/7

(numer.)

0 0.048 0.094 0.13 0.15 0.17 0.18 -

(analyt.1

0 0.059 0.103 0.134 0.153 0.163 0.172 0.215

y = II/9

(numer.)

0 0.048 0.102 0.14 0.17 0.18 0.21

a _ 0.27

y=l

(numer.)

0 0.070 0.14 0.19 0.23 0.26 0.29 -

(analyt.)

0 0.083 0.15 0.20 0.24 0.26 0.28 0.43

I x DIRECTION (arb. units)

I I I I I I I I I I I, , , 1 x DIRECTION (arb. units)

Fig. 6. Density contours calculated numerically with the flow equations for an initially square reservoir of gas which begins to expand in two dimensions when the four confining walls are removed at t = 0. The contour intervals are indicated by

A(p/p,,). (a) r = 0.048. (b) Z’= 0.32.

so we will consider expansion in only two ~mensions (Fig. 4~1 and without ambient gas.

To resolve the flow relevant to the recondensed geometry, Eqs. (la) and (lb) must be rewritten in two dimensions 1141. Furthermore, the expansion is not effusion-like such as that considered in sections 2 and 3 but rather of the type due to wall removal. The unitless density is given by p/p@, where pa is the initial density of the gas in the reservoir. The unitless time will continue to be written !I’ but now represents a ratio somewhat different Corn t/~, [14].

We first consider numerical calculations for the sideways expansion of a square reservoir of gas. The original configuration was an 8 x 8 array of points with density p/pa = 1. The spatial steps were AX= AY = 0.05 and the time step was Al’ = 0.008. At t = 0 the

Table 3 The extent to which a reservoir expanding sideways into vacuum as in Fig. 4c is exhausted for different values of Y as determined by numerical solution of the flow equations in two dimensions [14]. Note that !I” in section 4 is unitless time with a form appropriate to wall removal [14]

T Exhaustion of a Exhaustion of a square reservoir rectangular reservoir

0 1 1 0.024 0.980 0.974 0.048 0.933 0.916 0.072 0.868 0.833 0.16 0.601 0.498 0.32 0.302 0.214 0.64 0.073 0.050

four confining walls were removed and expansion as in Fig. 6 occurred. For 2” = 0.048 (Fig. 6a) we recognize an expansion with largely unrotated fete next to the sides of the reservoir though with an incipient loss

L’ ’ ““““““‘1 L T= 0.072 1

AQlp,) = 0.05 -

z

a, II I I I I I I t I I I I I I I I

x DIRECTION (orb. units)

T =0.32

A’p/p, ) = 0 01 -

x DIRECTION farb. units)

Fig. 7. Density contours calculated numerically with the flow equations for an initially rectangular reservoir of gas which

expands in two dimensions. (a) r = 0.072. (b) 2’ = 0.32.


690 R Kelly, A. alter / Nucl. Imtr. and Meth. in Phys. Rex 3 91 (1994) 682491

of density at the comers. For T = 0.072 the profiles (not reported) are largely circular, while for 2’ = 0.32 (Fig. 6b) there is a well-defined 4.5” symmetry rotation closely similar to that of Fig. 3a. Table 3 summarizes the extent to which the reservoir is exhausted for each value of Y. The rotations are apparent when the reservoir is roughly half empty.

Fig. 7 shows c~cula~ons for the sideways expansion of a rectangular reservoir of gas. The original configuration was a 4 X 16 array of points with P = 1, and the same spatial and time steps were used as for the square reservoir. Rectangular symmetry is found to be preserved only for the shortest times (Fig. 7a), for T = 0.16 the profiles (not reported) are largely circular, while for longer times there is a well-defined 90” symmetry rotation (Fig. 7b). Such a symmetry change is well known e~e~mentally [9,14,22,23]. Table 3 again summarizes the extent to which the reservoir is exhausted for each value of 7’.

The apparent rotation of the flow pattern seen in Fig. 3 is, therefore, a purely gas-dynamic effect, i.e. it obeys the same laws of flow that have been used throughout sections 2-4. Particles released from the corners can be regarded as moving into a two dimensional region, so that they rapidly lose their density as in Fig. 6a. That is, the flow is strongly divergent. Those emitted from the sides move more nearly in one di- mension, so that the flow is less divergent.

5. Discussion

We have considered term~ating expansions as are relevant when solids are sputtered with intense laser p&es. We go beyond what was developed in previous work [l,ll-141 in that we emphasize recondensation effects. Some of our results are rigorous but many are approximations good to within N 12%.

As overall conclusions we wish to indicate the following:

(a) When an expansion with y # 1 terminates and there is ~ecu~~~u~~~~, A falls ab~ptly from 1 to A,, and U from 1 to -A,, with A, given by (3 - y)/(y + 1). For y = 1 the final values are A = 1 and U = - 0.86. As time increases, the solutions evolve into two well- defined regions, here designated II and III. Depending on the definition chosen re_gion II begins either at the virtual expansion front at X or at the surface at X = 0, while region III terminates at the real expansion front at X (Fig. 5).

(b) The somewhat complicated analytical solutions for region II under conditions of recondensation [1,12] can be approximated at X= 0 by the remarkably simple forms seen in Eqs. (5) and (8). These forms are exact for X= 0 (r = 1) and otherwise are valid to within N 12%.

(c) The simple forms enable Q”” to be evaluated analytically as in Eqs. (12) but a numerical argument based on Eq. (13) can also be made. It is found that

Q ret increases significantly with the complexity of the particles involved, ranging from a fraction - 0.10 for atoms to N 0.43 for large molecules (Table 2).

(d) When the polymer polyimide is laser sputtered, r~ndensation occurs in massive amounts booed the ~mb~rded spot and the patterns of recondensation show remarkable symmetry changes as if the flow involved a simple rotation (Fig. 3). The behavior of polyimide is shown to be a purely gas-dynamic effect (Figs. 6 and 7) and is closely similar to symmetry changes which occur when essentially any material is laser sputtered to produce films [9,22,23].

Appendix A: An approximate relation between PO and U, for y = I

We here establish a relation between PO and U,, for y = 1 valid under conditions of recondemation by conserving the number of particles for a time near Y = 1, namely 2’ = 1 + A?‘. We approximate P for region II as being quadratic to the “right” of the effusing surface, while instead of a virtual region to the “left” we allow for a recondensing flux. The latter aspect of the argument is rigorous.

We now consider the contributions to the quantity Q of expanding material. The recondensed part, et,““, is

QEi’” = - U,P,AZ’.

Since P is given approximately by

P=P,t-(P,,-P,)(X2/X,2,),

the contribution to Q in region II to the “right” of the surface is just

Qfr = ixlIP dX = (2Ar,/‘3)( 1 + 2P,).

Finally, the region from X,, to X makes a contribution the form of which is rigorously

QIII = /X2 dX= /exp( -X/r) dX= Z’-’ = 1 - AX.

The relation P = exp( -X/X) was established previously [l], P,, and Xn are given by Eqs. (lo), while for y = 1 we have according to Eq. (11) X-+ ~0. By requir- ing that the total number of particles be Q = 1, it follows without difficulty that the following is true:

u, = -1/3P, + 4/3.

This is the origin of Eq. (7b).

R. Kelly, A. Miotello /Nucl. Instr. and Meth. in Phys. Rex B 91 (1994) 682-691 691

References

[l] R. Kelly and A. Miotello, Appl. Phys. B 57 (1993) 145. [2] K.P. Stanyukovich, Unsteady Motion of Continuous Me-

dia (Pergamon, London, 1960) pp. 121, 147, 151. [3] R. Kelly, A. Miotello, B. Braren, A. Gupta and K.G.

Casey, Nucl. Instr. and Meth. B 65 (1992) 187. 141 R. Kelly, J. Chem. Phys. 92 (1990) 5047. [5] S.I. Bozhevolnyi and I.V. Potemkin, J. Phys. D, to be

published. [6] H.M. Urbassek and J. Michl, Nucl. Instr. and Meth. B 22

(1987) 480. [7] R. Kelly, Nucl. Instr. and Meth. B 46 (1990) 441. [S] R. Kelly, A. Miotello, B. Braren and C.E. Otis, Appl.

Phys. Lett. 60 (1992) 2980. 191 J.C.S. Kools, T.S. Baller, S.T. de Zwart and J. Dieleman,

J. Appl. Phys. 71 (1992) 4547. [lo] B. Braren, K.G. Casey and R. Kelly, Nucl. Instr. and

Meth. B 58 (1991) 463. 1111 R. Kelly and B. Braren, Appl. Phys. B 53 (1991) 160. 1121 D. Sibold and H.M. Urbassek, Phys. Fluids A 4 (1992)

165.

[13] R. Kelly, Phys. Rev. A 46 (1992) 860. [14] A. Miotello, R. Kelly, B. Braren and C. Otis, Appl. Phys.

Lett. 61 (1992) 2784. [1.5] T. Ytrehus, in: Rarefied Gas Dynamics, vol. II, ed. J.L.

Potter (AIAA, New York, 19771 p. 1197. 1161 C. Cercignani, in: Rarefied Gas Dynamics, vol. I, ed. S.S.

Fisher (AIAA, New York, 1981) p. 305. [17] R. Kelly and R.W. Dreyfus, Surf. Sci. 198 (1988) 263. [18] D.L. Singleton, G. Paraskevopoulos and R.S. Irwin, J.

Appl. Phys. 66 (1989) 3324. [19] D.L. Singleton, National Research Council of Canada,

Ottawa, personal communication, 1992. 1201 R.S. Taylor, K.E. Leopold, D.L. Singleton, G.

Paraskevopoulos and R.S. Irwin, J. Appl. Phys. 64 (1988) 2815.

[21] S. Kiiper and J. Brannon, Appl. Phys. Lett. 60 (1992) 1633.

1221 C.N. Afonso, R. Serna, F. Catalina and D. Bermejo, Appl. Surf. Sci. 46 (1990) 249.

[23] R.K. Singh and J. Narayan, Phys. Rev. B 41 (1990) 8843. 1241 R.J. von Gutfeld and R. Srinivasan, Appl. Phys. Lett. 51

(1987) 15.


Documents

Laser-pulse sputtering of atoms and molecules Part II. Recondensation effects