Thermal-spike theory of sputtering: The influence of elastic waves in a one-dimensional cylindrical spike

PHYSICAL REVIEW B, VOLUME 65, 165425

Fluid dynamics calculation of sputtering from a cylindrical thermal spike

M. M. Jakas,1,* E. M. Bringa,2,† and R. E. Johnson2

1Departamento de Fı´sica Fundamental y Experimental, Universidad de La Laguna, 38201 La Laguna, Tenerife, Spain2Engineering Physics, University of Virginia, Charlottesville, Virginia 22903

~Received 12 September 2001; published 10 April 2002!

The sputtering yieldY from a cylindrical thermal spike is calculated using a two-dimensional fluid-dynamicsmodel which includes the transport of energy, momentum, and mass. The results show that the high pressurebuilt up within the spike causes the hot core to perform a rapid expansion both laterally and upwards. Thisexpansion appears to play a significant role in the sputtering process. It is responsible for the ejection of massfrom the surface and causes fast cooling of the cascade. The competition between these effects accounts for thenearly linear dependence ofY with the deposited energy per unit depth that was observed in recent molecular-dynamics simulations. Based on this we describe the conditions for attaining a linear yield at high excitationdensities and give a simple model for this yield.

DOI: 10.1103/PhysRevB.65.165425 PACS number~s!: 79.20.Rf, 47.40.Nm, 83.85.Pt

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I. INTRODUCTION

The ejection of atoms from the surface of a solid duriion irradiation is well documented both experimentally atheoretically.1 This phenomenon, known as sputtering, is dto the energy transferred to the atoms in the target byincident ion. This produces a cascade which can cause satoms to overcome the surface’s attractive barrier and esto vacuum.

In previous theoretical work the mean number of ejecatoms per incoming ion, or sputtering yieldY, is related tothe energy deposited the ion per unit thickness at the surof the targetFD , asY}FD

n . The value of the powern de-pends on the type of collision cascade produced by thenamely, linear and nonlinear cascades. For linear cascawhen the density of moving atomsNmov within the cascade issmall compared to normal densityN0 , one hasn51,2,3

whereas in the nonlinear case,Nmov;N0 , theoretical workpredicts thatn must be greater than 1.4,5 These results are sfirmly established that the consensus among workers infield is thatn.1 and nonlinear cascades are to some exsynonymous.6 Similar results have been found for sputteriin response to electronic energy deposited in a solid,7 buthere we refer to work on collision cascade sputtering.

Recent molecular-dynamics~MD! studies,8,9 however,cast doubt on this relationship. According to these pappurposely prepared nonlinear cascades can give rise totering yields which depend linearly onFD ~see Figs. 2–4!.Further evidence is found in Ref. 10. After modifying thstandard thermal spike theory~STST! to include the trans-port of mass, the sputtering yields calculated here appeto be much closer to a linear function ofFD than to theFD

2

predicted by the STST.Although the results in Ref. 10 show the importance

having a target which can change its specific volume afluid, it is not a full fluid-dynamics calculation. Since thtarget was assumed to be infinite, the sputtering yields habe calculated in the same manner as in the STST. That isexpression for the evaporation rate was used that wasrowed from the kinetic theory, and the sputtering yields w

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obtained by integrating it along a plane representing an oerwise nonexistent surface. Further, the transport was oradial, but the MD calculations showed the importanceenergy transport along the track.

In order to circumvent these difficulties our previous cculations are extended to a target which, in addition to becompressible, has a solid-vacuum interface. To this end,target density, velocity, and internal energy are all assumto vary with time in a manner which is described by tfluid-dynamics equations. Consequently, sputtering emenaturally, as that part of the target that succeeds in escafrom the condensed to the gaseous phase.

The aim of this paper is to show the most relevant aspeof the fluid-dynamics model, from the underlying mathemics to the results and implications of the proposed modAlthough this model can be applied to a variety of ioinduced thermal spike geometries, we have purposely limourselves to the idealized case described in previoussimulations.8,9 Therefore, the results in this paper only dscribe cylindrical thermal spikes, as does the STST. Theagreement between the present results and those experimin which Y exhibits a quadratic dependence onFD suggeststhat the connection between ‘‘real’’ spikes produced byincident ion and simple cylindrical spikes might not bstraightforward.

This paper is organized as follows. In Sec. II we introduthe fluid-dynamics equations as well as the various expsions used along the present calculations. Results and dissions are presented in Sec. III. Finally, the conclusionssuggestions for further studies are presented in Sec. IV.

II. THEORY

We assume that the target is a continuous medium wcylindrical symmetry, and it is completely characterizedits atomic number densityN, velocity v, and internal energye ~per atom! defined as

e5U13

2kBT, ~1!

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M. M. JAKAS, E. M. BRINGA, AND R. E. JOHNSON PHYSICAL REVIEW B65 165425

wherekB is the Boltzmann’s coefficient,T the temperatureandU is the potential energy per atom. By using the equatabove the heat capacity at constant volumeCV is assumed tobe that of a dilute gas, i.e., 3kB/2. This approximation, how-ever, is acceptable for the purpose in this paper, sinceshown in Ref. 11, the quadratic dependence ofY with FDdoes not appear to be connected toCV . MoreoverU is ob-tained from the expression10

U5~N0Mc02/m!~N/N0!n1m21F 1

n1m212

~N/N0!2

n1m11G ,~2!

whereM is the mass of the target atom,c0 is the speed ofsound atT50 K, andN0 is the normal atomic number density. m andn are two numerical constants which, as we eplained in Ref. 10, are not independent. Thus we setm52,thenn5A11Mc0

2/U0 , U0 being the potential energy at nomal density, i.e.,U052U(N0).

Using the same notation as in Ref. 12, we write the fludynamics equations as follows:

]N

]t52

]~vkN!

]xk, ~3!

]v i

]t52vk

]v i

]xk2

1

NM S ]p

]xi1

]s ik8

]xkD , ~4!

]e

]t52vk

]e

]xk1

1

N S Qcon1Qvis2p]vk

]xkD , ~5!

where the subscripts stand for ther and z coordinates,p isthe pressure, ands ik8 is the viscosity tensor12 defined as

s ik8 5hS ]v i

]xk1

]vk

]xiD , ~6!

whereh is the dynamic viscosity coefficient andQcon andQvis account for the heat produced by thermal conductand viscosity per unit volume and time, namely,

Qcon5¹~kT¹T!, ~7!

wherekT is the thermal conductivity and

Qvis5s ik8]v i

]xk. ~8!

The heat conduction coefficient is replaced by thatRef. 5:

kT525

32

kB

pa2AkBT

pM, ~9!

wherepa251.151 Å2. This form was also used in order tcompare results to previous work and because there seembe no reason for using a more ‘‘realistic’’ one since, as shoin Ref. 11,kT and the quadratic dependence of the sputteryield appear not to be connected.

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Making use of the fact that, for dilute gases,h andkT arerelated through the equationh5kTM /(3kB), we introducethe dimensionless viscosity coefficient

h* 53kBh/~MkT!. ~10!

Similarly, the pressurep is assumed to be a function oboth temperature and density. Here, we follow the appromation in Ref. 13 and splitp into two terms

p5pT1pC , ~11!

where thethermal pressure pT can be obtained from theexpression

pT5lNkBT, ~12!

l being a numerical constant. The so-called crystal presspC can be obtained from the potential energy Eq.~2! usingthe equation13

pC5N2]U

]N. ~13!

For computational purposes, Eqs.~3!–~5! are applied to afinite system, which is defined by inequalities 0<r<Rmaxand 0<z<zbot ~see Fig. 1!. Furthermore, the top wall, i.e.z50, is assumed to be made of a perfectly absorbent mrial, whereas the boundary at the bottom is perfectly cloas far as to the exchange of mass, momentum, and enerconcerned. The lateral wall can be made either closed, sas the bottom surface, or partially open. That is, closedmass transport but open to energy and momentum exchaResults in this paper were obtained using the latter opt

FIG. 1. Sketch illustrating the frame of reference and grid ulized in the present calculations. Att50 the ‘‘fluid’’ occupies theregion defined by inequalitieszsurf,z,zbot and 0,R,Rmax, andthe hot spike is confined to a cylinder of radiusRcyl .

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FLUID DYNAMICS CALCULATION OF SPUTTERING . . . PHYSICAL REVIEW B65 165425

Otherwise one would need an exceedingly large targeminimize the effects of energy and momentum reflectiowhen the deposited energy is large. A more detailed desction of this program will be published elsewhere.14

At t50 the target is at rest and within a range ofz definedby inequality (z>zsurf). For numerical reasons, however, wassume that the region that would normally be a vacuumfilled with a low-density fluid, i.e.,Nmin510233N0 . Ex-change of energy, momentum, and matter is forbidden influid, as well as in any other piece of a fluid with denslower thanNmin . The possible net flux of matter is continuously checked along the fluid, and the restrictions aboverelaxed as soon as the density of an element of the flincreases aboveNmin .

To energize the spike, all the fluid elements within a cinder of radiusRcyl are given an average energyEexc abovetheir initial energye052U01(3/2)kBT0 , whereT0 is thebackground temperature, often assumed to be 10 K. Thconsistent with the initial conditions used in a number ofMD simulations,8,9 again allowing direct comparison witthe results here. The initial conditions for Eqs.~3!–~5! thusbecome

v r ,z~0,r ,z!50,

N~0,r ,z!5H N0 if z>zsurf,

Nmin otherwise,

e~0,r ,z!

5H Eexc1e0 if r<Rcyl and z>zsurf,

U~Nmin!1~3/2!kBT0 if 0<z,zsurf,

e0 otherwise.

~14!

With the assumptions above, the deposited energycomes

FD5pRcyl2 N0Eexc.

As is customary, in solving the fluid dynamics equatiothe functionsN, v, and e are defined over a discrete setNR3NZ points, whose mesh size is determined byDr andDz ~see Fig. 1 and Table I!. A compromise has to be madabout target size since a large target implies a fairly lasystem of coupled equations with fairly long running timeWhereas too small a target gives rise to boundary effects

TABLE I. Value of the parameters used in the present calcutions.

Property Symbol Value

Atomic mass M 40.0 a.m.u.Atomic number density N0 0.026 at/Å3

Speed of sound c0 17 Å/psBinding energy U0 0.08 eVLennard-Jones distance s 3.405 Å

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would make calculations meaningless. Similarly, in chooszsurf one has to take into account that during ejection notthe matter that crosses the surface will be ejected. Therethe distance between the initial surface and the top wshould be large enough to not ‘‘collect’’ matter that, othewise, would not be ejected. Finally, the piece of matter reresenting the target must be thick enough. The condenphase is assumed to be 10s thick, which means thatzsurf'10s and zbot'20s. NR540 andNZ520 were found tobe adequate for all the cases studied in this paper.

When integrating the fluid-dynamics equations~3!, ~4!,~5! from t50 to t f , the total flux of matter passing througthe top boundary is also calculated. In this way the sputteryield is obtained as a function of timeY(t). This is used toverify if t f was long enough so that no matter remains withthe system that may significantly contribute to the sputteryield. We use theY(t)’s for t,t f to extrapolateY(t) toinfinity, i.e., Y`5 limt f→`Y(t f). Only runs for whichY`

2Y(t f)'0.1Y` are accepted. Normally,t f ranging from 10up to 50 ps are required.

Since calculations in this paper are meant to be compawith those in MD simulations, which often use LennarJones~LJ! potentials, the various parameters characterizour system correspond to those of argon~see Table I!. M540 a.m.u. and U050.08 eV have become standaparameters8,9 although the LJ calculations fully scale witU0 and M. Therefore, the results apply to a broader setmaterials as shown also using a Morse potential.15 Consistentwith this, for most cases we usedDr 5Dz5s, wheres isthe LJ distance. However, as several approximations wintroduced, we cannot ensure that the fluid in our calcutions accurately describes the potentials used in thesimulations. Similarly, we do not want to leave this sectiwithout mentioning that although the fluid representing ttarget is assumed to be compressible, Eq.~6! looks the sameas that of an incompressible fluid because we assumedthe Stokes’ condition holds, namely, that the so-called bviscosity coefficient is zero.

III. RESULTS AND DISCUSSIONS

We calculated the sputtering yield for different valuesl, h* , and the speed of soundc0 , and the results are depicted in Figs. 2–4. We observe that in all the cases the yincreases with increasing excitation energyEexc. Similarly,Eexc'U0 is an effective threshold for ejection for the initiaradius used, since the yields rapidly decrease forEexc com-parable to or less thanU0 . Whereas the MD requires varyinpotential types to obtain different material properties, hwe do this by directly varying the material properties. In thmanner the relationship between different materials candescribed.

We observe thatl has a great influence on the sputteriyield. The larger thel the greater the yield.l54 appears toreproduce MD simulations quite well, whereasl52 and 1underestimate the yields at small excitation energies. Thresults are, to some extent, easy to understand: with all oparameters remaining the same, asl becomes larger the ther

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M. M. JAKAS, E. M. BRINGA, AND R. E. JOHNSON PHYSICAL REVIEW B65 165425

mal pressure build up within the spike increases and mejection is expected.

FIG. 2. Sputtering yield as a function of the excitation enerand different values of parameterl. MD simulations are fromRef. 9.

FIG. 3. Sputtering yield as a function of the excitation enerand different values of viscosity coefficienth* .

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~1!, does not depend onl. Increasingl only increases thethermal pressure and speeds up the conversion of themotion into directed kinetic energy. Therefore, thermal coductivity has less time to take energy away from the spand the ejection of matter increases.

The effect of viscosity on the sputtering yield is illustratein Fig. 3. For the cases studied here, viscosity has a negainfluence on the ejection process, as yields are seen tosmaller with an increase of the viscosity coefficient. At smexcitation energies the viscosity appears to play a major rFurthermore, calculations usingh* 50.1 produced a goodagreement with MD simulations while those withh* 50.2and 0.4 resulted in significantly smaller yields. The fact ththe best agreement with MD simulations corresponds toculations withh* 50.1 is not unexpected sinceh* values ofapproximately that order have been calculated for a LennJones fluid.16

Finally, modifying the speed of sound does not producsignificant change in the sputtering yield. Figure 4 shoresults for the speed of sound both above and below itsmal value. The change in the sputtering yield is very smcompared with that produced by changing either the viscity coefficient or the thermal pressure coefficientl. We ob-serve that, for high excitation energies, an increase inspeed of sound leads to a slightly greater yield, but this tris reversed asEexc/U0 becomes smaller than 3.

As we mentioned in the Introduction, the most interesti

FIG. 4. Sputtering yield as a function of the excitation enerand different values of the speed of soundc0 .

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FLUID DYNAMICS CALCULATION OF SPUTTERING . . . PHYSICAL REVIEW B65 165425

FIG. 5. These plots illustrate the density anmass-flux vectors at different times for a spikwith dE/dX54 eV/Å, l54 @see Eq.~12!#, h*50.1, c0517 Å/ps, and Rcyl52s. The scaleused for translating from relative density (N/N0)into the gray scale is shown up in the figure. Nothat the scale is nonlinear, as more gray levelsused at both small densities and aroundN/N0

51. Furthermore, due to interpolation in the ploting software, details of the order of the grid sizor smaller, might not be accurately copied. Thhorizontal line denotes the initial position of thsurface.

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result in this paper is our ability to explore the material prameters that lead to the near linearity exhibited in the yiin our MD calculations even though the sputtering is a nolinear process. By exploring the parameter space we canter explain that phenomenon and assess its relevance.calculated yields in Figs. 2–4 clearly show that a lineargion is attained forEexc.U0 using a set of materials parameters. Therefore, nonlinear sputtering does not necessimply nonlinear yields. From these figures, it also appethat linearity is approached at higher energy densities tthose studied here for other materials parameters. Belowdescribe this phenomenon.

To understand the change in the dependence of the ywith increasing excitation density for fixedRcyl , we ana-lyzed the time evolution of the spike paying particular attetion to those aspects of the energy and momentum transthat are related to the ejection of matter. To this end, in F5 and 6 we have plotted the density, the mass-flux vector,temperature in the fluid at different times after the onsetthe spike. These cases correspond to a deposited energyeV/Å, l54, h* 50.1, c0517 Å/ps, andRcyl52s; and, inthe three figures, the initial surface is located at 10s, i.e.,zsurf510s.

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One readily observes that the temperature within the spdrops below 500 K in approximately 1 ps, and that the fluimmediately surrounding the spike hardly reaches temptures higher than, say, 100 K. This is in agreement withMD results and our earlier fluid-dynamics calculations10

These studies already showed that, due to the quick, abatic expansion of the fluid, the temperature of the spdecreases much more rapidly than it would due to therconduction. In addition, for times greater than 1 ps the thmal energy is converted into an elastic wave~seen in Fig. 5!that travels in the radial direction at approximately the speof sound. The reader must be aware of the nonlinear sused in Fig. 5 where the gray scale was purposely choseas to change rapidly around bothN0 and at low density. Dueto this, even the rather small relaxation of the surface denappears as a stripe, which extends to the right of the spand gets thicker with increasing time. These figures showthat the whole process would be better described as anplosion’’ rather than a smooth, thermally diffusive releaseenergy as proposed in the STST.5

Note that, in contrast to material further away from tsurface, the fluid that is near the surface and within the spappears to follow a spherical, rather than a cylindrical exp

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M. M. JAKAS, E. M. BRINGA, AND R. E. JOHNSON PHYSICAL REVIEW B65 165425

FIG. 6. Temperature and mass-flux vectorsdifferent times within the fluid for the spike described in Fig. 5. The gray scale used to plot teperature appears up in the figure.

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sion. That is, if one interpolates the mass-flux vector afigures out the streamlines of the fluid, then, one can reasee that near the open boundary of the spike, they seeradiate out from a point located on the spike axis somewhbelow the surface. In order to understand this, one harealize that the momentum acquired by any particle witthe fluid results from the fast, though small, displacementthe lateral and top boundaries which takes place at an eastage of the aforementioned explosion.

The forces produced by such displacements propagathe speed of sound which, within the hot spike, is faster tc0 .17 Therefore, by the time all the fluid within the spike hbeen set into motion, i.e.,t5Rcyl /c after the onset of thespike, a particle at~r,z! with 0<z<Rcyl and 0<r<Rcyl willhave acquired a velocity that is proportional to the time it hbeen exposed to such forces, namely,v r}r andvz}2(Rcyl2z). Therefore, asvz /v r'2(Rcyl2z)/r this particle willappear as moving away from a point located exactly onaxis atRcyl below the surface. By the same token, any pticle at a depth greater thanRcyl within the spike, will remainunaware of the presence of the surface and its velocitybe directed along the radial direction~see Fig. 7!. With in-

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creasing time our description above will become less aless accurate. However, as the forces acting within the stake the largest values during the earliest stage of the ‘plosion,’’ the velocities achieved by the fluid during that timessentially determine the subsequent dynamics of the sp

Another aspect of the velocity field which deserves atttion is that around the rim, on the cold side of the spikContrary to what happens deep in the target, where theside is compressed and subsequently displaced along thdial direction, the rim is partially wiped out. This not onladds more matter to sputtering, but also clears the wayfurther ejection as it widens the radius from within whicparticles are ejected.

From this simple picture one can readily calculate tsputtering radius. To this end, we define the excess energthe total energy per particle relative to the bottom of tpotential well, i.e.,e5e1 1

2 Mv21U0 . If one assumes thathe elastic wave in the upper part of the spike propagaisentropically along the streamlines, one may write

eA /dA25eB /dB

2, ~15!

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FLUID DYNAMICS CALCULATION OF SPUTTERING . . . PHYSICAL REVIEW B65 165425

FIG. 7. Close-ups of plots in Fig. 5 illustrating the dynamics of the fluid within the ‘‘core’’ ofthe spike and near the surface in more detail.

e

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ouruidlidre. Inim-omshe

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where dA and dB are the distance from the center of thspherical expansion to pointsA andB, respectively~see Fig.8!; similarly, eA and eB are the corresponding excess engies. Therefore, ase>U0 is a necessary condition for ejection, eA5Eexc and Rcyl /dA5RB /dB , one can obtain thesputtering radius (Rs) as18

Rs'Rcyl~Eexc/U0!1/2. ~16!

FIG. 8. Schematics used to obtain the sputtering radius.

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Accordingly, the sputtering yield can be calculated asamount of mass contained within a cone of heightRcyl andbase radiusRs , i.e.,

Y'p

3NRcyl

3 Eexc

U0. ~17!

In order to verify this simple expression, we calculate tsputtering yield for different spike radii. The results, thappear in Fig. 9, show that our fluid-dynamics calculatiocompare fairly well with the MD yields, and that Eq.~17!accounts reasonably well for the yields at high-excitatenergies. Discrepancies between MD and fluid dynamiclow excitation energies and for small spike radii do appeadetailed analysis of such deviations was not carried out.previously mentioned, the various quantities enteringmodel do not accurately account for the Lennard-Jones flin the MD simulations. In addition, having assumed the sotarget is a fluid, effects arising from the crystalline structuand the atomic nature of the target cannot be describedthe MD simulations focused collision sequences play anportant role at carrying energy and momentum away frthe spike, particularly for small spike radii. MD simulationalso show that the yield in this region is sensitive to tinitial energy distribution,9 which here is a Maxwellian. Fi-nally, it is worth noticing that Eq.~17! predicts a linear de-pendence of the yield with the excitation or deposited ene

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M. M. JAKAS, E. M. BRINGA, AND R. E. JOHNSON PHYSICAL REVIEW B65 165425

A result that was derived in Ref. 19 using a simple, intuitimodel rather than well supported, rigorous calculation.

Although we have chosen not to address the problemcrater formation, late in our calculations craters do appand they are all surrounded by a rim severals high ~thereader is referred to Ref. 20 for additional information abocrater formation!. The pit left by the spike is normally greatethan the initial radius of the hot core. It is formed as tresult of the net displacement produced by the elastic walong the radial direction. Near the edge of the pit, the ramomentum is less than it is in the material below. As a resa kind of cantilever is formed which is pushed upwardsthe fluid below. See the case oft52 ps in Fig. 7. However,for Eexc smaller thanU no pit is formed.

IV. CONCLUSIONS

Sputtering at relatively high excitation densities is an obut unsolved theoretical problem in ion solids interactioAnalytic diffusive thermal spike models are commonly usto interpret data at high excitation densities, although thmodels were never tested against more detailed calculatIn addition, there is a history of applying ideas from fludynamics to explain sputtering at high excitation densThese models are called by a number of names~gas flow,21

shock,19 pressure pulse,22 etc.! and attempt to account for thfact that sputtering at high excitation density does not ocon an atom by atom basis. These models also require a m

FIG. 9. Sputtering yield for different spike radii. MD calculations appear as symbols whereas hydrodynamics results are pas thick lines. Thin, straight lines show the sputtering yield obtaiusing Eq.~17!.

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detailed theoretical justification.Establishing a theoretical basis for sputtering models

high excitation density has been addressed recently bysimulations using model materials and simplified descrtions of the initial conditions. It was shown that standaspike models break down at precisely those high excitadensities which they were intended to treat. In additionsputtering regime was found. On increasing the energy dsity in the spike for fixed spike radius, the yield changfrom a nonlinear dependence on the excitation density tlinear dependence even though the ejection process cleremained nonlinear. This is contrary to the conventional wdom and suggests saturation occurs in the sputtering. Toamine this result we first showed that such a regime is neattained for any set of material properties using the diffusthermal spike model.11 Since the standard spike model involves solving only the energy equation, we then numecally integrated the full set of fluid equations for a ondimensional~1D! model of a cylindrical spike.10 Differenceswith the MD result remained which we attributed to the laof a surface. Here we use a 2D fluid-dynamics model witsurface to confirm that when the full set of equationstreated the MD result at high excitation density can betained. Therefore, as pointed out earlier, the principal dciency of the standard spike model is the assumption thattransport is diffusive.

We have calculated the sputtering yield from a cylindricthermal spike by directly integrating the full 2D fluiddynamics equations. The transport of mass and momentuseen to play a significant role in the ejection process. Sithe conversion of the thermal energy into kinetic/potenenergy within the spike occurs very early, the ejection pcess at high-energy densities is much more closely relatean ‘‘explosion’’ rather than to the thermal diffusion anevaporation models5 typically used to describe sputteringhigh-energy densities. Comparisons with MD simulationsing appropriate material parameters, show that our fludynamics description can account for the main featuresthe cylindrical thermal spike. These calculations also confithe MD result that transport along the cylindrical axis isimportant as radial transport and, therefore, a 2D moderequired. We show the reported nearly linear yield comabout because of the competition between pressurized etion and the transport of energy away from the spike bypressure pulse.

Using the evolution of the streamlines seen in theseculations we obtain a simple expression for the yield at hexcitation density for a reasonable set of material parameBringa and co-workers15,23had shown that in this regime thyield could be written in the form Y'C@Rcyl / l #

m$@dE/dx#eff(l/U0)%p, where @dE/dx#eff is the

energy deposited that ends up fueling the spike~herepRcyl

2 NEexc! and m and p are close to 1. They gaveC'0.18 for an LJ solid, which also appeared to apply tosults for other pair potentials.15 Here we use a picture of thejection attained from the 2D fluid dynamics model to estlish the theoretical basis for the value ofC. That is, theinternal pressure in the spike determines a critical rad@Rs'Rcyl(Eexc/U0)1/2# and a depth;Rcyl , leading to the

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FLUID DYNAMICS CALCULATION OF SPUTTERING . . . PHYSICAL REVIEW B65 165425

ejection of a conical volume of materialY' 13 RcylpRs

2N.This givesC5 1

3 , which is larger than the MD result. Thdifference is due in part to the fact that the material propties are not exactly those of the LJ solid and transport alcrystal axes removes energy from the spike as discushowever, all the principal features of the transport and etion are the same. This model resembles that of Yamamand co-workers19 but disagrees with the ‘‘so-called’’ pressurpulse model used for molecular materials.22

Several points need further investigation. The disagrment between the results in this paper and those experimwhereY}FD

2 suggests that the connection between a simspike, as the one studied in this paper, and those produceincident ions is not straightforward. The formation of crateat normal incidence is a topical problem that can bedressed by the model developed here. Further, the connebetween the sputtering yield and the time used to heatspike needs to be studied. In this paper, as in most of thesimulations, we assumed it to be negligibly small. This mbe correct for spike formation by a collision cascade, buknown to fail for electronic sputtering of rare-gas solids.7

Finally, it must be noted that the fluid-dynamics descrtion of the spike is a useful complement to MD. In the flumodel a broad range of material properties and types careadily studied, whereas complicated potentials are needeMD calculations of different materials. In fact it is seen

mv

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y

.

s

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r-gd,

c-ra

e-ntsleby

s-

ionheDys

-

bein

Figs. 2 and 3 that the saturation leading to the linear regis not simply dependent on the cohesive energy (Eexc'U0)and the initialRcyl , as found in the MD simulations usinpair potentials, but also depends on the material parametlandh* . In addition, local equilibrium chemistry, which caplay an important role in many of the materials of interesus, can be readily included in the fluid models. HowevMD has the advantage that non-normal incidence cantreated easily, the state of the ejecta~clusters vs atoms! canbe studied, and nonequilibrium chemistry can be introduTherefore, a program in which complementary calculatiusing fluid-dynamics and MD simulations is underway. Hwe have shown that a new linear sputtering regime is seeboth models and we have developed a simple analytic mfor the yield at normal incidence.

ACKNOWLEDGMENTS

Part of this work was carried out during a visit by onethe authors~M.M.J.! to the School of Engineering and Aplied Science, University of Virginia. Financial aids from tAstronomy and Chemistry Divisions of the National ScienFoundation~U.S.A.! and the Consejerı´a de Eduacio´n, Cul-tura y Deportes del Gobierno Auto´nomo de Canarias~Spain!are acknowledged.

m.

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es.

*Corresponding author. E-mail address: [email protected]†Current address: Lawrence Livermore National Laboratory, Cheistry and Material Sciences Directorate, P.O. Box 808 L-353, Liermore, CA 94550.1See, for example, C. T. Reimann, K. Dan. Vidensk. Selsk. M

Fys. Medd.43, 351 ~1993! and references therein.2M. W. Thompson, Philos. Mag.18, 377 ~1968!.3P. Sigmund, Phys. Rev.184, 383 ~1969!.4R. E. Johnson and R. Evatt, Radiat. Eff.52, 187 ~1980!.5P. Sigmund and C. Claussen, J. Appl. Phys.52, 990 ~1981!.6H. H. Andersen, Phys. Rev. Lett.80, 5433~1999!.7R. E. Johnson and J. Schou, K. Dan. Vidensk. Selsk. Mat. F

Medd.43, ~1993!.8H. M. Urbassek, H. Kafemann, and R. E. Johnson, Phys. Rev

49, 786 ~1994!.9E. M. Bringa and R. E. Johnson, Nucl. Instrum. Methods Phy

Res. B152, 167 ~1999!.10M. M. Jakas and E. M. Bringa, Phys. Rev. B62, 824 ~2000!.11M. M. Jakas, Radiat. Eff. Defects Solids152, 157 ~2000!.12L. D. Landau,Fluids Mechanics2nd ed.~Butterworth, Heine-

mann, 1995!.13Yu. B. Zel’dovich and Yu. P. Raizer,Physics of Shock Waves

~Academic, New York, 1969!.

--

t.

s.

B

.

14M. M. Jakas~unpublished!.15E. M. Bringa, M. M. Jakas, and R. E. Johnson, Nucl. Instru

Methods Phys. Res. B164–165, 762 ~2000!.16J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, inMolecular

Theory of Gases and Liquids~Wiley, New York, 1964!.17If the temperature of the spike is high, then, the speed of so

~c! is greater thanc0 sincec5(p/MN)1/2, therefore according toEq. ~11!, c5(lkBT1c0

2)1/2.18It must be noticed that, except for a numerical factor, Eq.~16!

coincides with theeffective sputtering radiusderived by H. Ur-bassek and P. Sigmund@Appl. Phys. A: Solids Surf.33, 19~1984!# using a Gaussian thermal spike. As they were obtainusing different models such an agreement appears to be amarkable coincidence for which we have no feasible explation.

19Y. Kitazoe, N. Hiraoka, and Y. Yamamura, Surf. Sci.111, 381~1981!.

20Z. Insepov, R. Manory, J. Matsuo, and I. Yamada, Phys. Rev61, 8744~2000!.

21H. M. Urbassek and J. Michl, Nucl. Instrum. Methods Phys. RB 22, 480 ~1987!.

22D. Fenyoand R. E. Johnson, Phys. Rev. B46, 5090~1992!.23E. M. Bringa, R. E. Johnson, and M. M. Jakas, Phys. Rev. B60,

15 107~1999!.

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