Comments on explosive mechanisms of laser sputtering

applied

surface science ELSEYIER

Applied Surface Science 96-9X (1996) 205-215

Comments on explosive mechanisms of laser sputtering Roger Kelly

*, Antonio

Miotello

Dipartimento di Fisica, .!Jniwxit& di Trento and Istituto Nazionale per la Fisica della Materia, I-38050 POUO(TN), Italy Received 22 May 1995

Abstract Laser sputtering differs from ion sputtering mainly in the important role played by thermal effects. These include not only vaporization from the extreme outer surface and boiling from an extended near-surface region, but also two additional effects which are important at high fluences and short pulse lengths. The first is phase explosion (also termed “explosive boiling”) in the sense analyzed by Martynyuk and by Fucke and Seydel. For high fluences and short pulses the target is unable to boil because the time scale does not permit the necessary heterogeneolds nuclei to form. It therefore approaches more or less closely to T,, (the thermodynamic critical temperature), homogeneous nuclei form at a high rate, and the near-surface region relaxes explosively into a mixture of vapor and equilibrium liquid droplets. An alternative explosive mechanism, the subsurface heating model, has, however, also been postulated for high fluences and short pulses. The idea began in 1972 with work by Dabby and Paek, and continued in a large number of articles to the present day. In this model vaporization from the surface causes the target to lose the ideal exponential temperature profile (Ta exp(-q), p being the absorption coefficient) and to develop a modified profile such that the target is much hotter (up to 3000 K) just beneath the surface. We point out that the surface conditions chosen by Dabby and Paek are wrong. We then present a numerical solution to the problem and find that the subsurface region is indeed hotter but only by a few degrees. Explosive release of material therefore cannot occur by this mechanism but only by phase explosion.

1. Introduction We give in Table 1 a brief resume of the mechanisms of ion-beam and laser-pulse sputtering. There are both similarities and differences. For example, it is not surprising that with laser pulses collisional effects are minimal, being due mainly to secondary effects caused by plume-surface interaction. (Remember that the plume can contain energetic parti-

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author. Fax: + 39 461 88 1 696.

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cles.) Also it is no surprise that, again with laser pulses, thermal effects can be very important. Less well appreciated is that there are basically four types of thermal sputtering: (a) Normal uaporizution. In the first type of thermal sputtering, which can operate at essentially any fluence and pulse length, the target undergoes what we will term normal vaporization from the extreme outer surface. Nucleation does not enter. The flux (atoms cm-’ SC’) is governed by the Hertz-Knudsen equation [l-3], which if multiplied by m/p (equivalent to h3, m being the particle mass, p the target density, and A the mean atomic

R. Kelly, A. Miotello/Applied Surface Science 96-98 (1996) 205-215

206

spacing of the target) gives the velocity of surface recession:

ax t

= ~p,(2nmk,T)-1’2A’

cm s-‘,

(la)

x=0

(ax/at)lX==rJ =rrp,exp[F($-i)] X(2mnk,T)-“*A3.

(lb)

Here a! is the condensation (or vaporization) coefficient [4], p0 is the equilibrium (saturated) vapor pressure, pb is the “boiling pressure” (i.e. the ambient gas pressure, often similar to lo5 Pa), T,, is the boiling temperature appropriate to pb, and it is assumed that there is no vapor present in the ambient ( Pamb = 0) and no recondensation [3]. (In the contrary case one writes PO -pamb instead of p. in Eq. (la).) Eq. (lb) e xpresses p,, in terms of the Clausius-Clapeyron equation and is shown as being approximate because it assumes a precisely exponential relation between p0 and l/T. (A better approximation is given in Ref. [3].) Since the vapor pressure is non-zero at all temperatures exceeding 0 K, it follows that for normal vaporization the surface temper-

ature (T,) is not fixed. Claims to the contrary, that is that a “vaporization temperature” (TV) exists, are therefore wrong except in a limited sense. For a given time scale and a given sensitivity of measurement, vaporization will not be sensed until a particular temperature is reached. However. additional heat input will always cause T, to exceed this temperature and a true TV will not exist. We note that the appearance of Tb in Eq. (lb) does not mean that there has been confusion between vaporization and boiling. It is simply a mathematical device to express p,(T) as a function of p&T,). (b) Normal boiling. In the second type of thermal sputtering, which requires that the pulse length be sufficiently long for heterogeneous bubble nucleation to occur, the target undergoes what we will term normal boiling from a zone extending from the surface to a depth related to the absorption length (l/p, /J being the absorption coefficient). In this case the surface temperature is fixed at Tb and the temperature gradient at and beneath the surface is aT/ax = 0. We note, however, that even if the necessary heterogeneous nucleation sites exist, their density (- lo6 kg-’ [3]) may be too low for boiling to be important.

Table 1 Comparison of ion-beam and laser-pulse sputtering Ion-beam sputtering

Importance

Laser-pulse spattering

Importance

Collisional due to primary collisions (“prompt collisional”)

High

Collisional due to plume

Low

Collisional due to cascade (“slow collisional”~

High

(No analog)

_

Thermal due to thermal-spike (“prompt thermal”)

Probably low

Thermal due to normal vaporization or normal boiling

High

Thermal due to phase explosion

High

(No analog) (No analog)

_

Thermal due to subsurface heating

Assumed high

Thermal due to decomposition of surface (“slow thermal”)

Sometimes high a

Thermal due to decomposition of surface

(Not yet clarifIeded)

Random motion of residual defects

Sometimes higha

Random motion of residual defects

(Not yet clarified)

Chemically guided motion of residual defects

(Not yet clarified1

Chemically guided motion of residual defects

(Not yet clarified)

Electronic

Sometimes high’

Electronic

Sometimes high b

a For example with halides [40]. b A group of processes including, for example, the “hole-pair”

mechanism, defect formation, and surface plasmon excitation [41]

R. Kelly, A. Miotello/Applied

Surface Science 96-98

At this point we note that there appears to be a problem in the description by Morse [5] of what was termed “evaporation”. It was argued that if a liquid is confined by a solid piston which exerts pressure p,, then “evaporation” will begin as the temperature is raised when p0 reaches p, and the temperature will then remain fixed at what was termed 7”. We prefer to think that boiling was intended and that the relevant temperature was Tb. (c) Phase explosion. The third type of thermal sputtering requires that the laser fluence be sufficiently high and the pulse length sufficiently short that the target reaches _ 0.9OT,, (T,, being the thermodynamic critical temperature) at and beneath the surface. Homogeneous bubble nucleation therefore occurs, and the target makes a rapid transition from superheated liquid to a mixture of vapor and equilibrium liquid droplets. As with boiling one can expect that at and beneath the surface the condition aT/ax = 0 will apply. Much of the historical work was done by Martynyuk [1,2,6] and by Fucke and Seydel [3,7], and the terms phase explosion and explosive boiling were both introduced. Since the rate of homogeneous nucleation rises catastrophically near T,,, such nucleation does not constitute an obstacle (Fig. 1). The treatment of Ready [8] for short pulses (“Qswitched laser”) is basically of the same type. The

$

CRITICAL NUCLEATION (WATER)

40

090 REWED

0.92

ONSET OP ANOMALIES

0.0 1 0.4

I,

0.0

0.8

REDUCED TEMPERATURE,

1.0

I 1.2

T/T,,

Fig. 2. Phase diagram of a metal in the neighborhood of T1,. Equilibrium uuporization refers to experimental equilibrium vapor pressures (p,,) obtained for Cs when both liquid and vapor are present. Vaporization in vacuum refers to a rapid heating in vacuum such that a Knudsen layer forms [25]. These pressures were here taken as 0.55~. T,, is the limiting temperature for equilibrium heating and Tvacuum is the limiting temperature for rapid heating in vacuum. Anomalous behavior is found for a number of properties for T/Q > _ 0.80, including the density (very low), Cp (very high), and the resistance (very lngh). Also, major fluctuations in density and enthalpy are found to occur [I ,121. Normai heating refers to heating the system at the ambient gas pressure pmb until p,, equals pamb and the system (provided it is liquid and provided also that the heating is slow enough) undergoes normal boiling at Th. Superheating refers to a heating which is carried out sufficiently rapidly that the system passes beyond Tb and is therefore me&table. The spinodal is the limit to which the metastable liquid can be. heated. It was here calculated from the condition (~?p/3V), = 0 using the Berthelot equatlon of state. Due to Martynyuk [6].

i

CRITICAL NUCLEI/cm’s

0.88

207

(1996) 205-215

0.94

0.96

TEMPERATURE, T/Tte

Fig. 1. Temperature dependence of the rate of homogeneous nucleation for water as in Eq. (4) and of the lifetime of the corresponding metastable liquid phase at pamb = 1 atm. Due to Martynyuk [IZ].

main difference is that some authors [1,7] invoked a short time scale to justify superheating beyond Tb, others [3] invoked the small density of heterogeneous nucleation sites, whereas Ready invoked pressure-induced raising of T,. We prefer arguments based on time scale or site density. The treatment of Dyer et al. [9,10] is also similar to that of Refs. [ 1?7]. His argument is that, because of the short time scale, the system passes beyond r, and reaches a temperature (“lower than T,,“) where the tensile strength of the liquid falls to zero and in addition where density (or pressure) fluctuations occur. We see every reason to identify the temperature in question with that for the onset of anomalies as in Fig. 2. Density anomalies should occur near (rather than “at” !> T, [ 1,121. Unfortunately the authors accepted a role for subsur-

208

R. Kelly, A. Miotello/Applied Surface Science 96-98 (1996) 205-215

face heating, which, as will be clear in what follows, was a mistake. (d) Subsurface heating model. Whether or not a fourth process, the subsurface heating model, exists is less clear. Starting with work by Dabby and Paek [ 1 l] and Gagliano and Paek [ 131 there have been repeated claims for an explosive mechanism based on heating the target surface either to T, (the melting temperature) [14], or to Tb [15], or to an ill-defined T, [11,13,16]. Rapid vaporization then occurs, the surface is subject to cooling, and the subsurface region therefore retains a higher temperature with maximum f. As a result the pressure is much greater beneath the surface and a type of explosion, leading to similar results as with phase explosion, takes place. For example both vapor and liquid droplets are claimed to be expelled. We will analyze this process in Section 2.2, but point out here that we do not agree with the analysis given in Refs. [11,13-161. In essence, if the surface boundary condition is taken, for example, as: q = T,,

(2)

then it follows that either normal or explosive boiling is taking place. It was not appreciated, however, that under these circumstances the condition U/ax Z=0 must apply at and beneath the surface and, therefore, that there cannot be a subsurface maximum temperature. In other work the validity of this model was accepted but without additional analysis [8,9,17]. We would finally note that thermal effects have also been advocated for ion-surface interactions. These include thermal sputtering (also termed thermal-spike sputtering), as well as the closely similar thermal-spike mixing. With sputtering the claims ceased about 15 years ago owing to a number of studies which failed to find a predicted temperature dependence of the yield 118,191. With mixing, on the other hand, the claims persist to this day [20,21]. Not all authors (e.g. Ref. [22]) accept the correctness of thermal-spike mixing, however, and it is interesting that one of the arguments against such mixing, namely that the temperature dependence is too marked [23], is the precise opposite to that used to support sputtering. In other words, the ubsence of temperature dependence served to disprove thermalspike sputtering, whereas the presence of tempera-

ture dependence served to disprove thermal-spike mixing! We discuss these problems elsewhere [24].

2. The explosive mechanisms 2.1. The phase explosion mechanism of Martynyuk and of Fucke and Seydel Already by 1974 Martynyuk had laid down the principles of phase explosion, also termed explosive boiling, brought about normally by discharging a condenser into a wire [l]. He also recognized that bombardment of a surface with laser pulses would lead to similar behavior [2]. Important extensions to the argument were made later by Fucke and Seydel [3,71. The argument is best understood in terms of a p-T diagram as in Fig. 2 [6]. The curve marked “equilibrium vaporization” is that corresponding to liquid Cs in equilibrium with saturated Cs vapor at pressure p,,. That marked “vaporization in vacuum” differs mainly in that a so called Knudsen layer (KL) forms at the liquid surface [25]. This is the region in which the vaporized particles, initially having only positive velocities normal to the surface (u,), develop negative velocities, but, in order that momentum be conserved, the particles also develop a positive center-of-mass (or flow) velocity (u). The distribution function then changes from the usual Maxwell-Boltzmann form [25] to the following:

=n

m

/

K! 2rrk,T,

j3j2 I

E//2- ’ r( j/2)(

k&#”

+E,

11 Here E, is the total internal energy of the gas, j is the number of internal degrees of freedom, and r is the gamma function. Also, nK, uK, and Tk are the number density, the flow velocity, and the temperature at the “boundary” of the KL. (In fact, a KL is

R. Kelly, A. Miotello/Applied

Surface Science 96-98 (19961 205-215

really infinitely thick, but is nearly fully developed after only 2-3 mean free paths [26,27]. This is why we speak of a boundary.) The curve marked “onset of anomalies” indicates the approximate temperature at which such effects as a rapid fall of density and a rapid rise of electrical resistance begin. It lies at - O.SOT,, [6]. “Normal heating” refers to heating the system at the ambient gas pressure parrlb until p0 equals P,,,,~ and the system (provided it is liquid and provided also that the heating is slow enough) undergoes normal boiling at T,,. “Superheating” refers to a heating which is carried out sufficiently rapidly (roughly < 1000 ns according to Ref. [l] but perhaps a shorter time is preferable) that the system passes beyond Tb and is therefore metastable (superheated). The “spinodal” is the limit to which the metastable liquid can be heated. The heart of the argument now follows if one recognizes that normal boiling involves heterogeneous nucleation at a temperature only minimally higher than T,, [I]. But if superheating occurs and the temperature reaches - 0.9OT,, phase explosion (explosive boiling) occurs by homogeneous nucleation. As a result the hot region near the surface breaks down in a very short time into vapor plus equilibrium liquid droplets. The rate of homogeneous nucleation (I,) is calculable, being given approximately by [l]:

AGC

I, = 1.5 X lO’*exp

- k~ i\

B

nuclei cmm3 s -’

i

(4) Here L\G, is the free energy change associated with the formation of a spherical critical nucleus. Mar tynyuk [6] argues that Z, is numerically significant (i.e. Z, 2 1) only near T,, and gives as an example the values for Cs: Z, = 1 nucleus cm-’ s-’ at T = 0.874& and Z, = 10z6 nuclei cm-j s-’ at T= 0.9057;,. Values for water, due again to Martynyuk [12], are shown in Fig. 1. The time constant for explosive boiling is discussed by Martynyuk [ 1,121, and we here give a modified version of the argument. We would suggest that the time constant can be identified with filling the near-surface heated zone with critical nuclei. Suppose that the target is heated to near q, to a

209

depthof7X10-7cm(asforthevalue~=1.5X106 cm -I of Al [28]), so that the heated volume per unit area is V’ = 7 X 10M7 cm’. If the critical nucleus has a radius of - 10e6 cm then - 10” nuclei could bc contained in V *. The number of nuclei which would form during t = T,_ is I,V *7,acer. For I,, = 10z6 nuclei crne3 s-l as above, V * as above, and r,aser = 3-30 ns this works out to 2 X lo”-2 X 10” nuclei. We conclude that the nucleation process has a time constant comparable to the chosen values of rlaserr and that the necessary nuclei would therefore form. As already stated above, the final result is that the near-surface region of the target relaxes explosively into a mixture of vapor and equilibrium liquid droplets. The vapor has been imaged explicitly by Mele et al. [29] while the droplets have been imaged explicitly by Gagliano and Paek [ 131 and by Geohegan [30]. 2.2. The subsurj&e

heating mechanism

of Dabby

and Paek

The subsurface heating model was apparently first postulated in 1972 in work by Dabby and Paek [l 11, was first applied by Gagliano and Paek [13]. and was reinvestigated numerically by various authors [1416]. It has a broad acceptance in contemporary work (e.g. Refs. [9,10,17]). The basis is that a laser pulse heats the near surface region but, as a result, atoms are vaporized from the surface and carry away heat. The target therefore loses the ideal exponential temperature profile (T a exp(- Z.LX>)and develops a modified profile such that it is hotter by as much as 3000 K [ 161 just beneath the surface. As a result the pressure is much greater beneath the surface and a type of explosion, leading to similar results as phase explosion, occurs. At this point problems arise, not with the underlying idea (which is fully valid) but with the quantification. It was suggested that the temperature profile would appear as in Fig. 3 [ 131 by using a calculation which had the following basis: (a> The velocity of surface recession was in all cases [ 11,13- 161 identified with the ability of the system to transport heat to the surface: (5a)

210

R. Kelly, A. Miotello/Applied

, -

!

8

1

1

TEMPERATURE (QAQLIANO

3.0 -

I

Surface Science 96-98 (1996) 205-215

0

the vaporizing particles [31]. On the other hand, the inclusion on the left side of Eq. (5a) of C,AT and AH, [32], where CP is the heat capacity and AH, is the heat of melting (Table 2), is probably wrong. The sign of the right side of Eq. (5a) is appropriate when x is positive inside the target. In the absence of vaporization Eq. (5a) reduces to:

PROFILE -

AND PAEK) _

i

(U/ax)],=,

0.0

0

NORMALIZED

4

a DISTANCE

12

I 16

I

4 20

FROM SURFACE,

s

Fig. 3. Temperature

profiles appropriate to the analysis of Dabby and Paek [I 11. It will be noted that the condition T = TV(where r, is an ill-defined “vaporization temperature”) has been imposed at the surface as for normal or explosive boiling. Furthermore the velocity of surface recession was identified with Eq. (5a) even though the condition U/8.x = 0 at and benefth the surface holds for a boiling mechanism. The values of T-T,, which are a measure of the subsurface temperature maxima, will be noted to be enormous and it was concluded that explosive expulsion of material (both vapor and liquid droplets) would occur. The parameter B is given by B = (K+AH,)/(I,,,,,C,), i.e. effectively we The parameters s and l3 are given respectively have B a l/I,,,,,. by (I,,,,,C,x)/(KAH,) and T/Z’,. K, p, AH,, and C, are defined in Table 2. Due to Gagliano and Paek [13].

Here p is the density of the target, AH, is the heat of vaporization, and K is the thermal conductivity, each with units as in Table 2. AH, can, if desired, be corrected for such effects as the kinetic energy of

=o.

Eq. (5b) was used, for example, by Wood and Giles [34] although it is unclear why they neglected vaporization for Si which was raised to temperatures between 3000 and 5000°C. It is our opinion, however, that, in the presence of normal evaporation, the use of Eq. (5a) to define (a~/at)],,~ is wrong. As recognized by various authors [l-3,12,33] this velocity is given instead by Eq. (la) or (lb), two commonly used forms of the Hertz-Knudsen equation. (At least this is true if the system is in vacuum.) Likewise, in the presence of either normal or explosive boiling, Eq. (5a) fails to account for the physical requirement that in a boiling mechanism, which is a “ volume process’ ’, dT/ax is given by dT/an=O at and beneath the surface! Still a further definition of (ax/at>l.= 0 has been proposed under conditions of long laser pulses, i.e. under conditions when a steady-state temperature profile is reached [8,33]. This is discussed in Appendix A. It will be seen that also this definition is made without involving Eq. (5a). (b) In addition to the problem contained in using Eq. (5a) to define (6’x/at>l X=0, the analysis in all

Table 2 Parameter values employed for numerical resolution of Eq. (7) assuming the target to be Al Melting temperature Boiling temperature at 1 atm Thermodynamic critical temperature

933.5 K 2740 K 5720 K [6]

Heat of melting at T,, Heat of vaporization at r,

396 J g-’ 10500 J g-1

Density Specific Thermal Thermal

2.7 g cm- ’ 0.94 J g-’ K- ’ 2.25 J s-’ cm-’ K-t IJs-‘cm-‘K-l

(liquid or solid) heat (liquid or solid) conductivity (solid) conductivity (liquid)

Reflection coefficient Absorption coefficient

(5b)

0.79 5.0 x lo5 cm-’

R. Kelly, A. Miotello/Applied

Surface

cases [ 11,13-161 used a wrong surface boundary condition, namely either T, = T, [14] or T, = Tb [15] or r, = TV [11,13,16]. (The meaning of TV, the I I

3803

I

I

LASER

3743

-

3723

-

3703

-

3683

M 0

-

_

TIME

PULSE

I

,

1 I

a 20

DISTANCE

30

40

FROM SURFACE

“vaporization

(1996)

21

205-215

temperature”,

I

is unclear, but possibly

Tb was intended.) The problem is, of course, that the surface temperature, T,, will be fixed only for very slow heating, when normal boiling occurs at T, = Tbr

and for very fast heating, when phase explosion boiling) occurs at T, = 0.9OT,, . Let us therefore assume that the various authors meant either normal or explosive boiling. But then the condition dT/Jx = 0 must apply at and beneath the surface whereas it was unrestricted in Refs. [11,13- 161. On the other hand if we assume that they meant normal aaporization then T, is in no sense fixed! We now show that, in fact, for normal vaporization the quantity (dT/d.~)l,=~ is unrestricted. The argument will not be rigorous, being based on conserving energy in the outermost atom layer of the target as if this region constituted a reservoir of finite size. If the heat input due to a single laser pulse which has power I, is described integrally by: (explosive

PARAMETERS

= 22 ns

IO

Science 96-98

50

(nm )

I ,aser=(l

-R)Z,exp(-PX) Wcm-‘, (6a) where R is the reflectivity, then the heat input to the reservoir is just: &JO < I < A =(I

-R)Z,,[l

-exp(-PA)]

=(l-R)I,ph. TIME :

3.5

ns

\

6493

1

0

(6b)

At the same time the heat loss is given by the sum of two terms, corresponding, respectively, to conduction into (or out of) the bulk and to loss to vaporized particles [32]: -&?

I 4

DISTANCE

I

’

8

’

’

12

’

’ 16

FROM SURFACE

’

20 (nm

)

+ p&Y,2 (6c) 3.X x=A at X=(j’ By equating Eqs. (6b) and (6~) one can solve for (dT/&t)l,=,: 8T z

&=,j

z--

(1 -R)I,@ K

+ pAHH, ax -K dt i=o’ (64

0

4

DISTANCE

8

12

16

FROM SURFACE

20

(nm)

Fig. 4. Numerical solution of the equation of heat diffusion (Eq. (7)). The recession velocity (ax/dt)~,=o wasevaluated with Eq. (lb) and not (as in Refs. [11,13]) with Eq. (5a). The source term was evaluated (in agreement with Refs. [11,13]) with Eq. (6a). A surface boundary condition of the type T = TV(as in Refs. [I I ,I 31) was avoided. The pulse energy was taken as 4.0 J/cm’, 7,aser was t&en variously as 30 (Fig. 4a), 6.0 (Fig. 4b), or 3.0 ns (Fig. 4c), and times were chosen which in each case maximized f - c. Subscript “1” means “laser”. The spatial step in the calculation was 2.0 nm and the time step was 1 X IO-” s. Other parameter values are as in Table 2.

R. Kelly, A. Miotello/Applied

212

Surface Science 96-98

With parameters as in Table 2, together with I, = 4.0/(6.0 X lo-‘) = 7 X 10’ J cmp2 s-l as in Fig. 4b, the first term on the right of Eq. (6d) has the value - 1.8 X lo6 K cm-‘. To evaluate the second term on the right we take the mean atomic spacing (A) to be 2.5 X 10-s cm and the condensation coefficient to be (Y= 1, so that Eq. (la) becomes:

Herepat,is the equilibrium vapor pressure in units of atmospheres. The final results are given in Table 3. We find that for low values of (ax/at>l,= a the first term on the right dominates and the temperature profile is everywhere decreasing. But for high values of (a~/at>].=~ the gradient (U/~X)].=~ becomes positive and a subsurface maximum temperature (?) develops. The question now enters as to whether the subsurface temperature maximum inherent to Eq. (6d) is a major effect (as in Fig. 3) or a minor effect. To this end we next solve the problem numerically.

3. Numerical solution to evaluate the subsurface heating under conditions of normal vaporization In view of the problems associated with the analysis of Dabby and Paek [I l] and its subsequent wide acceptance [9,10,13-171, we have solved the relevant equation of heat diffusion numerically under conditions of normal vaporization, this being the only case in which subsurface superheating is possible. (Remember: both normal and explosive boiling lead to aT/dx = 0 at and beneath the surface.) The one-dimensional equation of heat diffusion for a

(1996) 205-215

target heated by a laser pulse can be written [32] in a form equivalent to that of Ref. [ 1 l]:

where C, is the heat capacity with units as in Table 2. Here the reference frame moves with the surface, so that the surface is always at x = 0. The recession velocity (a~/&)],=~ was evaluated with Eq. (1 b) and not (as in Ref. [ill) with Eq. (5a). The source term was evaluated (in agreement with Ref. [ll]) with Eq. (6a). incorrect surface boundary conditions of the type T, = T,, [14], T, = T,, [15], or q = TV [ 11,13,16] were avoided. The pulse energy was taken as 4.0 J cmm2, T,,_ was taken variously as 30, 6.0, or 3.0 ns, and times were chosen which in each case maximized ? - T,, ? being the subsurface maximum. The spatial step in the calculation was 2.0 nm and the time step was 1 X lo-l3 s. Other parameter values were as in Table 2. We would point out that the logic used in integrating Eq. (7) included the following. (a) Starting with a condition in which the temperature was uniformly equal to that of the ambient, Eq. (7) was integrated to yield T = T(t,x). (b) Then Eq. (lb) was evaluated to give (ax/&)]._ o. (c) Finally Eq. (5a) was evaluated to give (dT/dx)l,=o. Thus, unlike Ref. [ll], the primary definition of (iYx/i?t)i.=, was made with Eq. (lb). The results are shown in Figs. 4a-4c. For 7,ascr= 30 ns the maximum temperature is at x = 0 and is distinctly below T,,, the temperature profile being everywhere decreasing. For 6 ns, whence 5 times as great a power density, the maximum temperature is beneath the surface and is similar to q,, namely v 5720 K for Al [6]. It is important to note, however, that the difference ? - 7” has the negligible

Table 3 Values of (U/8x)(,=

0 calculated with Eqs. (6d) and (6e) using, where necessary, parameter values as in Table 2

Temperature (K)

Equilibrium vapor pressure a (atm)

2790 3400 4400

_ 10 N 100

1

’ From the relation -RTlnp = A,G’(gas) b With the value f,, = 7 X lo8 J cm ’ s

(ax/atl,=o (cm s-‘1

Second term on the right of Eq. (6d) (K cm-‘) b

(aT/ax)l,=o (K nm-‘1

1.5 14 120

4.3 x 104 3.9 x 105 3.4 x 106

-0.18 -0.14 CO.16

- AfGO(liquid) [42].

’ the first term on the right of Eq. (6d) works out to - 1.8 X IO6 K cm- ‘.

R. Kelly, A. Miotello/Applied

SurJke Science 96-98 (1996) 205-215

value - 2 K, and that it would be even less if the condition (U/ax)] Xc0 = 0 had been imposed. For 3 ns, the subsurface temperature maximum is distinctly greater than T,,. This, of course, is not physically possible. The conditions T = q, and (U/ax)l,= a= 0 should have been imposed so that the value - 20 K for T - T, would be greatly reduced. It will be noted that the “distance scale of the information” is of the order of 20-40 nm. This scale is acceptable only if it exceeds sufficiently the electron mean free path. For Al at very high temperatures, the latter quantity is similar to 4.0 nm (Appendix B) and there is therefore no problem. The basic question posed in Section 2.2 has thus been answered. The subsurface temperature maximum inherent to Eq. (6d) is a very minor effect provided it is calculated correctly, This remains true even at temperatures above T,,, temperatures which are not permitted at the sur$ace of a condensed phase. The mechanism based on explosion due to subsurface heating (thence a high pressure) is therefore wrong. It follows that phase explosion (explosive boiling) remains the only physically sound mechanism to explain laser sputtering at high flucnces and short pulse lengths.

4. Importance

of phase explosion

Phase explosion has been extensively studied by Martynyuk [ 1,2,6,12] and by Fucke and Seydel [3,7] as occurring when condensers with a sufficient charge are discharged into wires. In such experiments they determined the critical temperatures of many elements (Fig. 5). In other work bombardment with high fluence laser pulses was used to provide the necessary heat [2] and in this case T,, constitutes the upper limit to which the temperature of the target surface can be raised. We note that a similar upper limit does rwt apply to the particles in the plume, so that there is no contradiction in claims for plume energies such as 1.5-3 eV per particle [29]. T, may possibly also be a relevant upper limit to a thermal spike induced by ion impact provided the hot zone is continuous to the surface. But if the hot zone is internal it is not clear what is the relevance of qc. It is interesting, however, that phase explosion appears to play a role even in much gentler situations

0 0

200

213

I

I

I

1

400

600

600

1000

HEAT OF SUBLIMATION,

AH,,,

(kJ/mol )

Fig. 5. Critical temperatures CT,) of various metals versus the heat of sublimation (AH,,,) at absolute zero. The symbols have the following significance: (0) measurements by slow heating; (X, A, V) measurements based on discharging a condenser and thereby making wires explode; (@) calculated from the results of laser heating experiments. Due to Martynyuk [6].

involving water, the value of T,, here being only 647 K [35]. For example sand particles on Al telescope mirrors were removed by causing phase explosion in an interfacial water film [36]. Tissue was removed from human corneas by causing phase explosion in the water which constitutes the main component of the tissue [37]. A tentative example lies in work where the laser sputtering yield of porous (i.e. sintered) Al,O, was increased by a factor of about 10 when the Al,O, was immersed in water [38]. Maybe even the work on cleaning the bricks of an important Palacio in Vslladolid involved phase explosion in interfacial water films [39]!

5. Conclusions Laser-pulse sputtering differs from ion sputtering in that thermal processes can be very important. They include the following. (a) Normal vaporization based on the Hertz-

214

R. Kelly, A. Miotello/Applied

Surface Science 96-98 (1996) 205-215

Knudsen equation (Eq. (1 a) or (lb)) and which would lead to temperature profiles as in Figs. 4a-4c. (b) Normal boiling at T = Tb which is caused by heterogeneous nucleation and which would lead to a temperature profile with the form aT/dx = 0 at and beneath the target surface. Normal boiling will, however, often be bypassed for two reasons: because the time scale is too short for heterogeneous nucleation [ 1,7] and because the density of nucleation sites is too small [3]. (c) Phase explosion, also termed explosive boiling, of the type analyzed by Martynyuk [1,2,6,12] and by Fucke and Seydel [3,7]. This occurs at T0.9OT,,, is caused by homogeneous nucleation, and gives a temperature profile with the form dT/ax = 0 at and beneath the target surface. As a result there is a more or less violent expulsion of both vapor and equilibrium liquid droplets. The rate of homogeneous nucleation rises catastrophically near T, and therefore does not constitute an obstacle (Fig. 1). (d) Explosive release due to a subsurface temperature maximum (T^). This was analyzed incorrectly in early work [l 11, although it has been repeatedly invoked in later work [9.10,13-171. A more nearly correct analysis is given here in Section 3 and we find that the subsurface temperature maximum is a very minor effect provided it is calculated correctly. This remains true even at temperatures above L&, temperatures which are not permitted at the sulfate of a condensed phase.

T( x = 0) = constant = TV) and integrating with respect to x from 0 to infinity [33]. This is possible whenever the pulse is very long. We are not prepared to judge between the two definitions of TV [8,33].

Appendix B. The electron mean free path We here indicate briefly how the electron mean free path (I,) of Al at very high temperatures was estimated. Ashcroft and Mermin (p. 52 of Ref. [43]) give a relation for I, of the form:

03.1) where r, is the radius of the “free electron sphere”, a0 is the Bohr radius (the ratio r-,/a, is found on p. 5 of Ref. [43]), and pp is the resistivity (in pf2. cm; p. 8 of Ref. [43]). We assume that the thermal conductivity (K) scales as 1, (p. 22 of Ref. [43]) and note that the ratio of K between 5000 and 300 K is

WI: K(5000)

0.818

K(300)

= - 2.37 .

Using Eq. (B.2) to correct (B.l) I, = 4.0 nm.

P-w the result is

References Appendix

A. ( ax/

at)1 X= O for long laser pulses

A different definition of (ax/at)i.= 0 than that seen in Eqs. (la), (lb), and (incorrectly) (Sa) has been proposed under conditions of long laser pulses, i.e. under conditions when a steady-state temperature profile is reached [8,33]. This is:

(A.11 where TV was taken as Tb in one case [8] but as an unknown to be determined by the Hertz-Knudsen equation in the other case [33]. Eq. (A.l) is obtained from the equation of heat diffusion, Eq. (7), by imposing steady state (aT/at = 0, ax/Lb = constant,

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