Bogaerts 2003

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Spectrochimica Acta Part B 58 (2003) 18671893

0584-8547/03/$ - see front matter 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.sab.2003.08.004

Laser ablation for analytical sampling: what can we learn frommodeling?

Annemie Bogaerts *, Zhaoyang Chen , Renaat Gijbels , Akos Vertesa, a a b

University of Antwerp, Department of Chemistry, Universiteitsplein 1, Wilrijk-Antwerp, B-2610, Belgiuma

George Washington University, Deparment of Chemistry, Washington, DC 20052, USAb

Received 4 July 2003; accepted 22 August 2003

Abstract

The paper is built up in two parts. First, a rather comprehensive introduction is given, with a brief overview of thedifferent application fields of laser ablation, focusing mainly on the analytical applications, and an overview of thedifferent modeling approaches available for laser ablation. Further, a discussion is presented here about the laserevaporated plume expansion in vacuum or in a background gas, as well as about the different mechanisms for particleformation in the laser ablation process, which is most relevant for laser ablation as solid sampling technique forinductively coupled plasma (ICP) spectrometry. In the second part, a model is presented that describes the interaction

of an ns-pulsed laser with a Cu target, as well as the resulting plume expansion and plasma formation. The resultspresented here, include the temperature distribution in the target, the melting and evaporation of the target, the vapordensity, velocity and temperature distribution in the evaporated plume, the ionization degree and the density profilesof Cu atoms, Cu and Cu ions and electrons in the plume (plasma), as well as the resulting plasma shielding of0 q 2q

the incoming laser beam. Results are presented as a function of time during and after the laser pulse, and as afunction of position in the target or in the plume. The influence of the target reflection coefficient on the abovecalculation results is investigated. Finally, the effect of the laser pulse fluence on the target heating, melting andvaporization, and on the plume characteristics and plasma formation is studied. Our modeling results are in reasonableagreement with calculated and measured data from literature. 2003 Elsevier B.V. All rights reserved.

Keywords: Laser ablation; Modeling; Evaporation; Plume expansion; Plasma; Inductively coupled plasma

1. Introduction

1.1. Laser ablation application fields

Laser ablation (LA) is used for a growingnumber of applications, such as pulsed laser dep-

*Corresponding author. Tel.: q3238202377; fax: q3238202376.

E-mail address: [email protected] (A. Bogaerts).

osition w13x, nanoparticle manufacturing w4x,

micromachining w5,6x, surgery w7x, as well as

chemical analysis w810x. Several analytical tech-

niques make use of the mechanism of lasersolid

interaction, in different regimes of laser irradiance,

including, among others, matrix assisted laser

desorption ionization (MALDI) w11x, laser micro-

probe mass spectrometry (LMMS) w12x, laser-

induced breakdown spectrometry (LIBS) w13x.

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1868 A. Bogaerts et al. / Spectrochimica Acta Part B 58 (2003) 18671893

Last but not least, in inductively coupled plasmamass spectrometry and optical emission spectrom-etry (LA-ICP-MS and LA-ICP-OES), laser abla-

tion is used as solid sample introduction methodw1420x. In spite of the many applications, theexact mechanisms of laser ablation (e.g. thermalvs. mechanical effects, mechanisms of particleformation,) are not yet fully understood. More-over, different kinds of mechanisms can play arole, depending on the type of material, the laserirradiance, the laser pulse length, etc. In particular,LA-ICP-MS, which is todays most suitable tech-nique for direct solid analysis, is still sufferingfrom a number of drawbacks, which are related to

the ablation process, the aerosol transport and thevaporization, atomization and ionization of thelaser-induced product within the ICP. A number ofexperimentally gained insights into the fundamen-tal processes demonstrate the need of modelshelping to localize such drawbacks and to defineregimes where these problems can be minimizedor overcome.

Therefore, we try to develop a numerical model,describing the various processes related to thelaser-solid interaction, as well as the behavior ofthe laser-ablated material. As a first step, we would

like to describe the most important parameters toimprove the technique in general. We will focusin first instance on the application of nanosecondlaser ablation of metals, as sample introductiontechnique for the ICP, to define the ablation pro-cess and to gain further insights into which partof the sample is removed, converted into atomsand ions, and to describe its expansion dimensionsabove the sample surface. Further studies andmodels will incorporate the ablation cell, gasflows, aerosol transport and ICP conditions. At

this stage, however, the model will also be usefulfor other applications, operating at similarconditions.

1.2. Overview of different modeling approaches

for LA

In the literature, a variety of models for laserablation are available, operating in differentregimes of wavelength (UV, Vis, IR), laser irradi-ance (10 10 Wycm ), pulse length (fs, ps, ns),4 10 2

target material (metals, organic solids, ), gasenvironment (vacuum, ambient gas), and appliedto the various applications mentioned above. A

large number of models for lasersolid interactionare based on thermal processes: heating of thesolid, followed by melting and evaporation. Theydescribe the lasersolid interaction on a macro-scopic scale, i.e. by the thermal heat conductionequation w2138x. This assumption is justified forns-pulsed laser interaction, especially for metals.Indeed, in metals, light is absorbed by interactionwith electrons. A quantum of optical energy isabsorbed by an electron, which is raised to ahigher energy state in the conduction band. The

excited electrons collide with lattice phonons,thereby transferring the absorbed energy to thelattice. Since the energy relaxation time is of theorder of 10 s for metals w1,39x, the opticaly13

energy can be regarded as being instantaneouslyturned into heat for ns-pulsed laser ablation. There-fore, the concepts of temperature and heat conduc-tion are applicable w39x. For ps-pulsed andfs-pulsed lasers, the electron energy cannot beregarded as instantaneously turned into heat, anda two-temperature model is needed, whichdescribes the coupling of the electron temperature

and the lattice temperature w4047x. Further, thereexist also a few kinds of microscopic models,based on the Boltzmann transport equation fortransport of electrons and for electronlattice inter-actions w48x, or based on molecular dynamicssimulations, for metals w49,50x or for organic solidsw5155x. However, these models are limited to ashort time-scale, because of the long computationtime. A mesoscopic model w5659x provides alink between the microscopic and macroscopicphenomena. The time-scale is less limited than in

molecular dynamics simulations, and the highlynon-equilibrium behavior can be adequatelydescribed. Finally, a number of analytical modelsfor laser ablation have been reported w6066x.

Several models do not only focus on the lasersolid interaction, but they describe also the expan-sion of the evaporated material plume, or theyconcentrate solely on the expansion process,assuming certain input from the evaporated mate-rial. The plume expansion can be investigatedeither by hydrodynamic models w2538,6781x,

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1869A. Bogaerts et al. / Spectrochimica Acta Part B 58 (2003) 18671893

by Monte Carlo simulations w8292x, o r b y acombination of both w93,94x. Most of the hydro-dynamic models for plume expansion also consider

the formation of a plasma of the evaporated mate-rial w2938,6774,76,77x. A few models focusmainly on the plasma, either from the evaporatedmaterial w66,95,96x, or the early-stage plasma dueto electron emission from the target w9799x,because this plasma can absorb a fraction of theincoming laser intensity before it reaches the target(so-called plasma-shielding).

1.3. Laser-evaporated plume expansion in vacuum

or in a background gas

Nearly all models describing lasersolid inter-action followed by plume expansion apply toexpansion in vacuum or in a low pressure back-ground gas (up till a few 100 Pa), which is easierto model (e.g. only one type of material to follow,and no interactions with the surrounding gas), andwhich is close to reality for many applications(e.g. pulsed laser deposition). Some papers haveinvestigated the effect of background gas on theplume expansion with hydrodynamic models w2732,6981x, but the pressure is mostly limited till

approximately 100 Pa or lower w3032,6975x,with a few exceptions till 1 atm w2729,7681x.The general effect of the background gas is report-ed to be the spatial confinement and slowing downof the expanding plume. Moreover, the materialcan even move backward, and several reflectedshocks within the plume were observed w31x. Inmost of the papers investigating the effect of thebackground gas, no additional conservation equa-tions (for the background gas) or extra terms inthe equations for the material plume (describing

interactions with the background gas) are present-ed. This would, however, be necessary for arealistic description of the behavior of the expand-ing plume in a background gas, especially atatmospheric pressure. Wood et al. have includedsome scattering terms in the hydrodynamics equa-tions, to account for the transfer of momentumfrom the plume to the background gas w7274x,but the model is only applied to low pressure(below 50 Pa). Gnedovets et al. have reported ahydrodynamic model with two distinct species

(material plume and background gas) and inter-actions between them w7880x. This model isapplied to expansion in a background gas at 1

atm, but for a long laser pulse (ms-range) at verylow laser irradiance (10 10 Wycm ), so that no4 5 2

plasma is formed. A similar model was, however,also applied recently to higher laser irradiance, butagain with the assumption of no plasma formationw81x. Finally, the effect of the background gas isalso investigated in several Monte Carlo simula-tions w8892x or combined hydrodynamicMonteCarlo models w93,94x, but they are again onlyapplied to pressures below a few 100 Pa.

However, when laser ablation is used as sample

introduction technique for the ICP, the ablationcell is filled with a gas (typically helium or argon)at 1 atm, and this gas is used to transfer the ablatedmaterial to the ICP (e.g. Ref. w16x). Also, in LIBS,the laser ablation typically takes place at 1 atmw76,96,100x, although a wide pressure range(10 1 atm) is also possible w100,101x. I t isy5

reported from experiments w102x that the expansiondynamics of the evaporated plume is highlydependent on the background gas pressure. Freeexpansion, plume splitting and sharpening, hydro-dynamic instability and stagnation of the plume

were observed at different pressure levels between10 and 100 Torr w102x. Other experiments havey6

revealed that the gas medium significantly affectsthe laser ablation efficiency w103,104x. Indeed, theICP emission intensity increases when using abackground gas with a higher ionization potential(e.g. He), due to less plasma shielding, and hence,more efficient laser sampling w103,104x. Therefore,the effect of the background gas should in principlebe incorporated for realistic modeling of the laserablation process.

1.4. Particle formation in the LA process

It is generally known that the laser ablatedmaterial consists not only of evaporated atoms,but particles are also formed, either directly by thelasersolid interaction, or later, through conden-sation in the expanding plume (see below). Theseparticles are the subject of many methodologicalinvestigations in LA-ICP-MS and LA-ICP-OES,because they are considered to be the main source

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of elemental fractionation (i.e. non-stoichiometricablation: the analytical signal does not representexactly the sample composition) w20,105110x.

Elemental (or isotopic) fractionation may occurduring the ablation process (e.g. preferential vola-tilization of lighter elements (or isotopes), perhapscoupled with preferential condensation), during theaerosol transport (e.g. preferential transport ofsmaller particles), as well as in the ICP, i.e. dueto incomplete vaporization (atomization, ioniza-tion, excitation) of the aerosol. Whereas over thepast 17 years, the lasersample interaction wasconsidered to be the major source of fractionation,nowadays, aerosol transport, and especially the

ICP are believed to have an important effect aswell w20,110x. Recent studies have proven that theaerosol structure, i.e. the particle size distribution,has significant effect on elemental fractionationw20,106110x. The laser produced aerosol can besubdivided into three fractions: (a) the largestparticles are too heavy to be transported; (b) theparticles with intermediate size can be transportedto the ICP, but cannot completely be vaporized;and (c) the fraction consisting of vapor and smallparticles can be completely transported and vapor-ized in the ICP. The degree of elemental fraction-

ation will depend on the ratio of these threefractions. If the majority of particles are smallenough, elemental fractionation will not be signif-icant. However, if the fraction of larger particlesincreases, an increasing number of particles willnot be transported or fully vaporized, and elemen-tal fractionation becomes increasingly important.Therefore, controlling the particle size distributionis of uttermost importance w108,109x.

In spite of the large variety of models availablefor laser ablation (see above), not many do con-

sider the formation of particles. Nevertheless, par-ticle formation is not only important for LA assample introduction technique for the ICP, but ithas also major consequences in the other applica-tion fields of laser ablation. Indeed, the particlespresent in the vapor plume scatter the incidentlaser light, which leads to a shielding of the targetsurface w77,111x. Further, the quality of micro-structures manufactured by laser ablation can beconsiderably reduced due to the formation ofdebris in the surrounding area, as a result of the

presence of nanoparticles w77x. However, nanopar-ticle synthesis is also a promising application oflaser ablation w78,112116x.

A number of models describe particle formationand growth in the expanding vapor plume, basedon condensation and nucleation theories w38,77x

or effusion theory w111x. Using simple formulas,information is obtained about particle size distri-bution and rate of cluster growth. A one-dimen-sional particle formation model, based oncondensation and nucleation theory, is developedby Gnedovets et al. w7880x and Blair et al. w81x,and it is coupled to the expansion dynamics of theevaporating plume and the background gas, assum-

ing no plasma formation. This model is applied tolaser ablation in 1 atm background gas. Indeed,Wood et al. suggest that the background gas playsa dominant role in particle formation w74x. Han etal. drew similar conclusions, because the clusterformation efficiency is determined by the rate ofcollisions between clusters, vapor atoms and back-ground gas, which is controlled by the density ofvapor atoms and buffer gas atoms w92x. I t i ssuggested that shock wave formation and reflectionphenomena due to the buffer gas play an importantrole in the nanoparticle formation w92x. Zhigilei

also states that efficient particle formation isexpected when the background gas pressure issufficiently high, so that many collisions of thevapor species can occur, although condensationcan take place under vacuum conditions as wellw117x. In general, condensation droplets are typi-cally formed in long-pulse (ms) and low-intensity(10 10 Wycm ) regimes w78x; however, they4 5 2

can also be generated at higher laser intensity andshorter pulse (e.g. 10 10 Wycm and few ns8 10 2

pulse) under slow expansion of the vapor plume

into the background gas w118x.The average particle size predicted with the

above models, ranges from approximately 1 nmw81x to tens w38,77x and even a few hundrednanometers w38,78,79x, depending on the appliedtheory and the conditions. Normally, condensationis used to explain the observation of small parti-cles, composed of tens to thousands of atomsymolecules. Formation of larger (sub-micrometerand micrometer sized) particles would require anunrealistically high number of collisions to occur

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in the expanding plume. Therefore, the largerparticles are probably the result of direct ejectionfrom the target w117,118x.

Different mechanisms can play a role for directcluster ejection from the target, depending on thetarget material. The ejection of large liquid dropletsandyor solid particulates can be caused by photo-mechanical effects, as a result of laser-inducedstress w117x. When the laser-induced stressesexceed the dynamic tensile strength of the targetmaterial, i.e. in the regime of stress confinement,when the laser pulse duration is shorter than thetime needed for mechanical equilibration, cavita-tion and disruption of a liquid surface region, or

mechanical fractureyspallation of a solid target cantake place w5355,117,119,120x. The importanceof this mechanism depends on the mechanicalproperties of the target material, as well as themicrostructure and surface roughness. This mech-anism can probably play a significant role fororganic materials (such as polymers) w5355,117,119x as well as for geological materials(e.g. silicates) w121x.

For metals, however, mechanical fragmentationis expected to be less important, and particles areassumed to be formed by liquid (large droplet)

ejection, which can be the result of several pro-cesses, in general termed hydrodynamic sputteringw117x. Large droplets can be ejected as a result ofa transient melting and motion of a liquid causedby steep gradients and the vapor plume recoilpressure w1,118,122,123x. The expulsion of meltby the impact of recoil pressure of the expandingvapor plume in a background gas is also calledliquid splashing w120x. This recoil pressure willbe higher when the background gas is at higherpressure w123x. Due to this local pressure, surface

hydrodynamic instabilities will be formed, whichresult in the ablation of liquid in the form ofdroplets w123x. The hydrodynamic mechanism oflarge droplet ejection in metals is strongly sup-ported by the analysis of surface morphologyw122,123x. Moreover, the splashing is expected tooccur in the radial direction, which is confirmedby measured crater profiles w110x.

At high laser irradiance, i.e. above 2.2=1010

Wycm in the case of silicon w124126x and above2

5=10 Wycm for aluminum w127x, or for short10 2

laser pulses w55,116x, ejection of large particulatesis found to be very important, and is attributed tophase explosion, or explosive boiling of the target

material beneath the surface layer w124127x.Indeed, bubbles are formed in this superheatedliquid layer. Once these bubbles obtain a criticalsize, they can expand rapidly, initiating explosiveboiling and mass ejection in the form of large(micrometer-sized) particulates w125,126x. Recent-ly, evidence was found that explosive boilingoccurs after completion of the laser pulsew126,127x. The importance of this mechanism athigh laser irradiance is confirmed by the strongincrease in crater volume and depth at a given

threshold irradiance w124126x.

1.5. Present (basic) model for LA

The above discussion shows that the entireprocess of laser ablation and the subsequent behav-ior of the ablated material (e.g. expansion into abackground gas, plasma ignition, shock waves,particle formation,) can be very complicated,and that it cannot readily be described with onesingle model. Therefore, it is our intention to

describe the entire process step by step.As a first step, we will describe in the present

paper a model for the laser interaction of a coppertarget, yielding heating, melting and vaporizationof the target material, followed by plume expan-sion, as well as plasma formation and shielding ofthe laser intensity. We will limit ourselves here toexpansion in vacuum, so that only the evaporatedmaterial needs to be considered. Although thismight not directly reflect the reality of laser abla-tion for the ICP, it is of interest to other applica-

tions (see above), and it can provide already someinsight in the mechanisms. Moreover, preliminarycalculations in our group have revealed that mostof the modeling results are not fundamentallydifferent for expansion in vacuum and in 1 atmbackground gas w128x. Also, particle formationwill be neglected in this first step. This modelwill, however, serve as a basis for the later descrip-tion of laser ablation in 1 atm helium or argon,and for the modeling of the formation and behaviorof particles. However, even this basic model is

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Table 1Input data used in our model, for Cu as target material, as well as the references where the data are taken from. Some parameters,related to the target, are given both for solid and liquid (molten) phase, when the values are different

Parameters Values for Cu Ref.

Thermal conductivity, k (W m K )y1 y1 380 (solid), 170 (liquid) w34xSpecific heat, C (J kg K )y1 y1p 420 (solid), 494 (liquid) w34xMass density, r (kg m )y3 8960 (solid), 8000 (liquid) w1,34xAbsorption coefficient, a (m )y1 7.44=107 w34xReflectivity used in the model, R 0.34 w95xMelting point, T (K)m 1358 w1xBoiling point, T (K)b 2836 w1xHeat of fusion, DH (Jymol)sl 1.3=10

4 w1xHeat of vaporization, DH (Jymol)lv 3.048=10

5 w1xFirst ionization potential, IP (eV)1 7.73 w133xSecond ionization potential, IP (eV)2 20.29 w133xElectronic work function, f (eV) 4.5 w133x

already quite complicated and considers alreadymany processes.

Our modeling approach is based on a modelthat was actually developed more than a decadeago for another application, i.e. laser ablationmicroprobe mass analysis (LAMMA) w34x. It willbe applied here to laser ablation conditions typicalfor LA-ICP-MS and LA-ICP-OES, and for LIBS,

i.e. a laser pulse duration of several nanoseconds,and laser irradiance in the order of 10 Wycm .9 2

Cu will be taken as the target material. In Section2, the different aspects of the model will beoutlined. Section 3 will show some typical results,for a copper target. The effect of target surfacereflectivity and of laser intensity will be discussedin Sections 4 and 5, respectively. A comparisonwith other calculated or experimental results ismade in Section 6. Finally, a conclusion andoutlines for future work will be given in Section

7.

2. Description of the model

The model presented in this paper describesseveral mechanisms, related to the laser solidinteraction and the behavior of the evaporatedmaterial. The different parts of the model will bedescribed in detail below, and the couplingbetween the various parts will be outlined.

2.1. Target heating, melting and vaporization

As a result of the laser impact onto the solidtarget, the temperature at the target surface willrise, and eventually, the target can melt, and evenvaporize. The temperature distribution in the targetis calculated with the heat conduction equation.The absorption length in the target (l s1ya w1x,abs

with a defined as the absorption coefficient, as7.44=10 m ; see Table 1) is in the order of 107 y1

nm. Because this is much smaller than the laserbeam diameter (typically in the order of 100 mm,e.g. w16x), the heat conduction can be solved inone dimension w22,24,28,39x:

w zT x,t T x,tB E . . k aC Fx |s q I x,t (1) .D Gt x C r x C ry ~p p

where T represents the temperature inside the

target, x is the position from the surface, t is thetime, k, C , r and a denote the thermal conductiv-pity, heat capacity, mass density and absorptioncoefficient of the target material, respectively. Thecalculations are performed here for a Cu target,and the corresponding data used in the model, aresummarized in Table 1. In the current model, non-linearities in material parameters, e.g. absorptioncoefficient, are neglected, and the thermal proper-ties of the target material are assumed to betemperature-independent. This is not really true,

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but for most metals the variations are relativelysmall over fairly wide temperature ranges w39x.

The first term of the right-hand side represents

the heat conduction, whereas the second termdenotes the energy source by laser energy absorp-tion. I(x,t) stands for the laser irradiance as afunction of time and position in the target, andcan be written as:

w xI(x,t)sI (t)exp(yax) 1yR (2)0

I (t) is the incident laser irradiance at the surface,0and R is the surface reflectivity. The latter isgenerally close to 1 for metals, but it can drop to

values as low as 0.1 during laser ablation, whenthe irradiance is high enough (approx. 10 Wy8

cm ) w1,39,122,129,130x. This is attributed to2

roughening due to the temperature increase of thesurface, to removal of surface films, and to meltingw39,122,129x, although the details of the mecha-nism might not yet be fully understood w130x. Weused a constant value of 0.34, as indicated in Table1, because of lack of more accurate data, and inorder not to further complicate the solution of theheat conduction equation. This is a reasonableintermediate value, which has also been reported

in Ref. w95x, albeit for a somewhat differentwavelength, and it was not very clear where thisvalue was adopted from. We have, however, inves-tigated the effect of this parameter on the modelingresults, as will be shown later in the paper, inSection 4.

When the temperature at a certain depth in thetarget exceeds the melting point of Cu, the targetstarts melting, and the local temperature remainsconstant during the time that phase transition takesplace. Furthermore heating of the molten target is

calculated in the same way, but using data formolten Cu (see Table 1). In the molten phase,convection is induced by the thermal gradient.Although this effect leads to additional heat trans-port, it is not part of the model. Melting normallystarts at the surface, but the melt front (i.e. theinterface between solid and molten phase) canextend inside the target when the temperature risesfurther. The heat conduction equation is solved asa function of time during and after the laser pulse,by an explicit finite difference method.

When the temperature at the surface becomesvery high, vaporization becomes significant. Thevapor pressure (p ) is calculated from the surfacevap

temperature, by integrating the Clausius-Clapeyronequation w131x:

w zDH T yT .lv s bx |p T sp exp (3) .vap s 0RT Ty ~s b

where T and T are the surface temperature ands bthe normal boiling point at pressure p s1 atm,0DH is the heat of vaporization, and R is the gaslvconstant. The data for a Cu target are also pre-sented in Table 1.

From the vapor pressure, the vapor density atthe surface (r ) is calculated from the ideal gasvap,slaw:

pvapr s (4)vap,s

kTs

where k is the Boltzmann constant.Further, assuming that the vapor atoms leaving

the surface follow a one-dimensional Maxwellianvelocity distribution, the flow velocity of the vapor

atoms above the surface(

v)

can be approxi-vap,smated by the average of the normal velocitycomponent at temperature T w132x:s

2kTsv s (5)vap,s y pmwhere m is the mass of a Cu atom.

The vapor density (r ), vapor velocityvap,s(v ) and vapor temperature at the surface (T )vap,s sare used as input (i.e. boundary conditions) in thesecond part of the model.

2.2. Expansion of the evaporated material plume

The expansion of the evaporated material intovacuum is described by the Euler equations ofhydrodynamics, expressing the conservation ofmass density, momentum and energy w34x:

rv .rsy (6)

t x

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rv . 2w xsy pqrv (7)

t x

2 2w z w zB E B E v p vC F C Fr Eq sy rv Eq qx | x |D G D Gt 2 x r 2y ~ y ~

qa I y (8)IB laser rad

Here r is the mass density, rv is the momentum,rE is the internal energy density, rv y2 is the2

kinetic energy density, and p is the local pressure.Assuming that the evaporated material follows theideal gas law, the pressure and internal energydensity can be expressed as Ref. w34x:

rkTps 1qx (9) .e

m

w zr 3rEs 1qx kTqIP x q IP qIP xx | . .e 1 i1 1 2 i2

m 2y ~

(10)

Here, x , x and x stand for the fraction ofe i1 i2electrons, singly charged (Cu ) and doublyq

charged (Cu ) ions in the vapor, and IP and2q 1IP represent the first and second ionization poten-2tial of Cu (values taken from w133x, see Table 1).These extra terms, and the factor (1qx ), areeneeded to account for the partial (or full) ioniza-tion of the vapor (see below). In this work, weassume that the Cu vapor can be ionized into bothCu and Cu ions w34x. However, it is clear thatq 2q

this formula can be extended to higher chargemultiplicity, when needed. In this formula, it isassumed that the electrons can be treated as idealgas, following MaxwellBoltzmann distribution.

Further, I is the laser irradiance and alaser IBstands for the absorption coefficient due to inverseBremsstrahlung, which is a function of position inthe plume (see below). The product of I andlasera gives the local laser energy absorption, whichIBrepresents a gain for the internal energy of thevapor. Finally, is the amount of energy emittedradby the vapor per unit volume and time in theBremsstrahlung process, which means a loss forthe internal energy of the vapor. Assuming thatthe electrons are characterized by a Maxwellian

velocity distribution, can be expressed asradw134x:

1y2 6B E2pkT 32pe 2 2C F s n Z n qZ n (11) .rad e 1 i1 2 i23D G3m 3hm ce e

where m and e are the electron mass and charge,erespectively, c is the velocity of light, n , n , ne i1 i2represent the densities of electrons, Cu andq

Cu ions, and Z and Z stand for the charges of2q 1 2the singly and doubly charged ions (Z s1 and1

Z s2). Again, this formula can be extended if2more multiply charged ionic species are present inthe evaporated plume.

The second and third terms in the right-handside of Eq. (8) represent non-linear terms, becausethey depend on the electron and ion densities,which in turn depend on the vapor density. More-over, the vapor density can show large variationsas a function of position, from near solid densitytill vacuum. This large discontinuity can give riseto shock waves. In order to accurately reflect thisshock formation, an appropriate solution methodneeds to be used to solve the three conservationequations. We used a finite difference method,based on the first order Godunov scheme, whichdescribes the shock wave as a series of analyticallysolvable mini-shocks (see Refs. w67,135139x formore information). It is worth to mention that wehave checked this Godunov solution method ofthe equations with another solution method, i.e.the second order MacCormack method (see, e.g.w139x), and both solution methods yielded thesame results. The Godunov method was, however,preferred for the present application, because ofthe shorter computation time (i.e. ca. 5 times faster,mainly because a larger timestep could be used).

2.3. Plasma formation

Because the temperature of the evaporated mate-rial is very high, the vapor is ionized (see above)and a plasma is formed. Since many collisionstake place between the various particles, the plas-ma can be considered in local thermal equilibrium(LTE). This means that in a sufficiently smallregion of the plume, thermal equilibrium is estab-lished between the electrons, ions and neutrals,

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and they are characterized with a common tem-perature. Moreover, the ionization degree (i.e.fraction of electrons and ions) can be described

by the SahaEggert equation w9x. Because previ-ous calculations have shown that not only Cuq

ions, but also Cu ions are formed at the typical2q

conditions under study here w34x, two differentSahaEggert equations are incorporated in themodel, i.e. for the ratio of Cu ions to Cu atoms,q 0

and for the ratio of Cu to Cu ions:2q q

3y2B E B Ex x 1 2pm kT IPe i1 e 1C F C Fs exp y (12)2

D G D Gx n h kT0 vap

3y2B E B Ex x 1 2pm kT IPe i2 e 2C F C Fs exp y (13)2

D G D Gx n h kTi1 vap

Here, n represents the total vapor number den-vapsity (n srym), x , x , x and x stand for thevap e i1 i2 0fraction of electrons, singly charged (Cu ) andq

doubly charged (Cu ) ions and neutral Cu atoms,2q

respectively, which are defined as x sn yn , etc.e e vapFurther, h is the Planck constant, and the othersymbols have been explained above. Note that thedifference between the electronic partition func-

tions of the two ion stages was neglected in theabove formulas.

The two SahaEggert equations are combinedwith two other equations, i.e. for the conservationof matter, and the conservation of charge,respectively:

x qx qx s1 (14)0 i1 i2

x q2x sx (15)i1 i2 e

Finally, the vapor temperature (T) in the SahaEggert equations is adopted from the internalenergy density (see Eq. (10) above).

These five equations (Eqs. (10), (12)(15)) aresolved together to calculate the five unknownvalues (x , x , xi , x and T), from which thee i2 2 0number densities of electrons, Cu , Cu andq 2q

Cu can be obtained. The NewtonRaphson meth-0

od w140x was applied to solve this strongly non-linear system of equations. As discussed above,this system of equations can be extended with

additional SahaEggert equations, when moremultiply charged ions need to be considered. Forevery new unknown, an extra equation is added,

so that the system remains solvable, and the newunknown value is also included in the other equa-tions (conservation of matter and charge, andequation for internal energy density).

This system of equations is solved at everyposition in the plume, except in the first positionnear the target surface, where the main source ofions and electrons is given by thermionic emissionfrom the heated surface w39x. This is described bythe LangmuirSaha equation w39,141143x:

B Ex fyIPi1 1C Ffexp (16)D Gx kT0 s

where f is the electronic work function (w133x;see Table 1), and the other symbols are declaredabove. Assuming that only singly charged (Cu )q

ions are formed, and combining the LangmuirSaha equation with the equations for conservationof matter (x qx s1) and conservation of charge0 i1(x sx ), yields three equations for threei1 e

unknowns. From the fractions (x), the numberdensities of electrons, Cu and Cu can beq 0

obtained, based on the total vapor density (seeabove).

2.4. Laser beam absorption in the plasma

Because of the formation of a plasma in frontof the target, the laser beam will be partiallyabsorbed before it reaches the target, i.e. so-calledplasma shielding w95x. Laser radiation is

absorbed primarily by inverse Bremsstrahlung,which involves the absorption of a photon by afree electron. The electron is raised to a higherstate in the continuum. This process must occurwithin the field of a heavy particle (ion or neutral),so that momentum is conserved w39x. Photo-ioni-zation of excited states contributes only for ener-getic photons impinging on easily ionized gasmixtures w9x, and is neglected here. The twocontributions to inverse Bremsstrahlung are elec-tron-neutral and electron-ion inverse Bremsstrah-

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Fig. 1. Laser intensity-time profile assumed in the model. It isa Gaussian-shaped pulse with 10 ns full-width at half-maxi-mum (FWHM), and maximum irradiance of 10 Wycm . The9 2

dashed line represents the calculated laser intensity arriving atthe target, after passing through the plume (plasma).

lung. Their absorption coefficients are given byw9x:

w zB EhcC Fa s 1yexp y Qn n (17)x |IB,eyn e 0D GlkTy ~

6 3w zB Ehc 4e l neC Fa s 1yexp yx |IB,eyi 4D GlkT 3hc my ~ e

1y2B E2p2 2C F= Z n qZ n (18) .1 i1 2 i2

D G3m kTe

Here, l is the laser wavelength, Q is the crosssection for photon absorption by an electron, dur-ing a collision with the neutrals (taken here as10 cm ) w9x, and the other symbols have beeny36 5

explained above. The common factor in bothformulas (1yexp(yhcylkT)) accounts for stim-ulated emission w9x.

2.5. Coupling of the different parts of the model

The different parts of the model are stronglycoupled. On one hand, the lasertarget interactionleads to evaporation, and the resulting vapor den-

sity, velocity and temperature are used as input forthe plume expansion and plasma formation. Onthe other hand, the absorption of the laser beamin the plasma represents an important couplingback, both to the plume expansion (gain and lossterms in the equation for conservation of energy),and to the lasersolid interaction, because the laserenergy can be considerably attenuated (plasmashielding) before it reaches the target, leading toless efficient target heating, melting andvaporization.

Hence, the different parts of the model need tobe solved simultaneously as a function of time, inorder to obtain an overall picture of the mechanismof laser ablation of a Cu target, with plumeexpansion and plasma formation.

3. Results of the model

The calculations are performed for a laser withwavelength of 266 nm, and a Gaussian-shapedlaser pulse, illustrated in Fig. 1 (solid line). The

pulse has a width (full-width at half-maximum,FWHM) of 10 ns, and the peak laser irradiance(or intensity) is 10 Wycm . Integrated over the9 2

entire pulse, this yields a fluence of 10.6 Jycm .2

In our calculations, we follow only one laser pulse.The dashed line in Fig. 1 shows the laser intensity

arriving at the target, after passing through theplume (plasma). It is clear that the laser intensityat the target drops significantly after approximately10 ns, as a result of plasma shielding of theoriginal laser pulse. After this significant drop,which results in a weaker plasma, and hence lessplasma shielding, the laser intensity at the targetrises again, but as time evolves further, the laserintensity at the target continues to decrease. Thisillustrates that a significant fraction of the originallaser intensity is absorbed in the plasma, before it

can reach the target. In other words, the laserintensity, which is really used for the ablation canbe much lower than the initial laser intensity.

3.1. Target heating, melting and vaporization

Initially, the Cu target is at room temperature.However, as a result of the incoming laser beam,the target will be heated. The calculated tempera-ture distribution in the target is depicted as afunction of time during and after the laser pulse

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Fig. 2. Calculated temperature distribution in the target (a),and calculated solidyliquid regions in the target as a result of

melting (b), as a function of time, for the conditions shown inFig. 1. The original target surface is at 0 mm.

Fig. 3. Calculated temperature at the target surface (a), meltdepth in the target (b), target surface recession rate due toevaporation (c) and depth of evaporation (d), as a function oftime, for the conditions shown in Fig. 1.

in Fig. 2a. The temperature is at maximum at thesurface of the target, as expected. The maximumtemperature of approximately 6750 K is reachedbetween 10 and 20 ns, i.e. when the laser irradi-ance is at maximum. When time evolves, theregion with elevated temperature (i.e. above roomtemperature) expands, due to thermal conduction

(see Eq. (1)). For the conditions under study, theheating is more or less limited to the first 10 mmof the target. As far as melting is concerned, onlya few micrometer is influenced, as appearing inFig. 2b, which shows the solid and liquid (molten)phases in the target, as well as the transition region,which is partly molten (with temperature equal tomelting point). Although the surface temperaturedrops significantly after the laser pulse, the first 2mm of the target still remains in molten phase forquite a long time. Only after approximately 60 ns,

resolidification of the melt starts to occur, and themelt depth starts to drop slowly.

The surface temperature and the melt depth are

also plotted as a function of time in Fig. 3a, brespectively). Fig. 3a shows a little dip in the

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surface temperature at approximately 10 ns, whichcorresponds to the dip in the laser intensity at thetarget (see Fig. 1). After 1020 ns, the surface

temperature begins to drop, but it is still approxi-mately 2000 K at 100 ns, for the conditions understudy. The target melting starts to occur at approx-imately 8 ns (see Fig. 3b), when the temperaturebegins to rise significantly. The melt depth increas-es to approximately 2 mm at 4060 ns, i.e. clearlyafter pulse termination. After a longer time, itstarts decreasing slowly, but it is still approximate-ly 1.5 mm after 100 ns.

Fig. 3 also gives information on the targetevaporation. The surface recession rate due to

evaporation (Fig. 3c) is calculated from the fluxof evaporated atoms, in the following way w34x:

dx j Mevap evapR s s (19)recession

dt r

M and r are the molecular mass of the atoms (kg)and the mass density (kgym ), respectively. The3

flux of evaporated atoms (m s ) is obtainedy2 y1

from the vapor pressure at the surface temperature(p(T ); see Eq. (3) above):s

Ap T .sj T s (20) .evap s

y2pMRTs

Here, A represents a sticking coefficient, which isusually taken equal to one for metals. The otherparameters have been defined before.

As shown in Fig. 3c, the calculated surfacerecession rate due to evaporation reaches a maxi-mum of almost 10 mys, and it shows a verysimilar profile as the laser intensity at the target

(Fig. 1), including the dip at approximately 10 ns.This means that evaporation also ceases at pulsetermination (ca. 30 ns). Indeed, from ClausiusClapeyron equation it follows that evaporationdramatically slows down when the surface tem-perature drops below the boiling temperature.

From the surface recession rate due to evapora-tion, the depth of evaporation can be obtained, andthe latter is plotted as a function of time in Fig.3d. The evaporation depth rises as a function oftime, starting from approximately 10 ns, and it

reaches a constant value of ca. 70 nm, afterapproximately 2530 ns, when the surface reces-sion rate has dropped to zero. Comparing Fig. 3b,d

tells us that the evaporation process affects onlythe surface layer, till a depth which is a factor of30 smaller than the depth at which melting takesplace.

In our model, material removal is solely attrib-uted to vaporization. Hence, the thickness of theevaporated layer, or evaporation depth (ca. 70nm), represents a lower limit of the crater depth,i.e. in case that vaporization is the only materialremoval mechanism. However, in laser ablationexperiments, splashing of melt due to the plume

recoil is observed (see Section 1.4 above). If weassume that the melt is completely ejected beforeresolidification occurs, the melt depth (ca. 2 mm)can be considered as an upper limit for the craterdepth. In the book of Ready (Table 3.8, p. 111)w39x, a measured crater depth of 2.2 mm wasreported, for laser ablation by a 44 ns laser pulsewith irradiance of 10 Wycm , i.e. similar condi-9 2

tions as under study here. This would indeedsuggest that splashing of molten material is respon-sible for a great deal of the material removal.

3.2. Plume expansion and plasma formation

Fig. 4 shows the calculated vapor density in theevaporated plume (a), the plume velocity (b) andvapor temperature (c), at different times, for theconditions mentioned above. It is worth mention-ing that the initial conditions (i.e. before the laserpulse starts) assumed in our calculations, are avapor density of 10 m (which corresponds to18 y3

a residual pressure of 10 atm), zero vapory7

velocity, a vapor temperature of 300 K, and a

residual ionization degree in the plume of 10 .y10

These assumed initial conditions were not foundto be critical for the calculation results.

It is clear from Fig. 4, for the different timesshown in this figure, that the vapor density ishighest at the target (i.e. almost 5=10 m ; not27 y3

so clearly seen in the figure). It drops graduallyas a function of distance in the plume, and veryrapidly at the plume front. As time evolves, theplume becomes longer. For the laser ablationconditions under study, the plume length is calcu-

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Fig. 4. Calculated density (a), velocity (b) and temperature (c)distributions in the vapor plume, at different times, for the con-ditions shown in Fig. 1.

lated in the order of 0.1 mm after 20 ns, andapproximately 2 mm after 100 ns. Note that thesevalues are calculated for plume expansion in vac-

uum. For expansion in a background gas at 1 atm,the vapor plume will be considerably shorter (seeSection 1.3 above). The plume velocity (Fig. 4b)increases towards the end of the plume, and reach-es maximum values of 20 00030 000 mys. Closeto the target, the vapor velocity is calculated to benegative (ca. y4500 mys at 20 nm, and approxi-mately y6000 till y6300 mys at 40 till 100 ns.This probably represents the plume recoil back tothe target, which can give rise to target liquidsplashing (see Section 1.4 above). The vapor

temperature distribution in the plume is presentedin Fig. 4c. It is calculated in the order of 10 00050 000 K, and decreases slightly as a function oftime. This temperature drop is most pronouncedafter 20 ns. Indeed, at 20 ns, the calculatedtemperature is much higher than at later times,because the laser pulse yields heating of theevaporated plume. At 40 ns, the laser pulse isalready terminated, which explains the lower tem-perature. The further drop as time evolves repre-sents cooling due to plume expansion. It shouldbe noted that at the plume front a peak in the

temperature was calculated, but this has no physi-cal meaning, because the calculated vapor densityat this position was very low, indicating that theplume was terminated.

Fig. 5 shows the plume characteristics in detailat 15 ns, i.e. at the maximum of the laser pulse.The vapor density in the plume (Fig. 5a) decreasesaway from the target. The plume length at thistime is calculated at approximately 0.1 mm. Thecorresponding vapor temperature (Fig. 5b) isapproximately 50 000 K at small distance (near

the target), and it drops to approximately 15 000K at the plume front. From this vapor temperature,the ionization degree, or the fraction of Cu atoms,0

Cu and Cu ions and electrons, is calculatedq 2q

with the SahaEggert equation (see above; Eqs.(12) and (13)). Fig. 5c illustrates that the Cu0

atoms are only present near the target. The rest ofthe plume is more or less fully ionized at the laserconditions under study. In the first 0.03 mm fromthe target, where the vapor temperature is veryhigh, the majority of Cu atoms is even doubly

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Fig. 5. Calculated density distribution (a), temperature distri-bution (b), fraction of Cu atoms, Cu and Cu ions and0 q 2q

electrons (c), density distributions of these species (d) in thevapor plume, and laser intensity as a function of position inthe plume (decreasing towards the target, as a result of plasmashielding) (e), at 15 ns, for the conditions shown in Fig. 1.

ionized (Cu ). At further distance, when the2q

vapor temperature drops, the Cu ions becomeq

dominant. The fraction of electrons is equal to the

sum of Cu ions and two times the Cu ionsq

2q

(see above, Eq. (15)). This explains why it isalmost equal to two in the beginning of the plume,when the Cu ions are dominant (see dash-dotted2q

line and right axis in Fig. 5c). From the fractionof the species, multiplied with the vapor density,the species densities in the plume are obtained,and the results are plotted as a function of distancein the plume in Fig. 5d. The Cu atom density is0

very high near the target (order of 5=10 m ;27 y3

note that the y-axis in Fig. 5d is truncated at

4=

10 m , in order not to hide all other density

26 y3

values), but it drops significantly in the first fewmicrometers from the target. The densities of thecharged species (electrons and ions) reach a max-imum at approximately 1 mm from the target, i.e.where the vapor density is still very high (see Fig.5a) and the plasma is almost fully ionized. TheCu and Cu ion densities are very comparableq 2q

here. After this maximum, the densities drop veryquickly, which is most pronounced for the Cuq

ions, and which is due to the steep decrease invapor density (see Fig. 5a). From these densities,

the absorption of the incoming laser beam in theplasma is calculated, i.e. plasma shielding, due toinverse Bremsstrahlung (IB). It appears that elec-tron-ion IB (Eq. (18) above) is negligible in theentire plasma. Indeed, the electron and ion densi-ties are too low (in spite of the fully ionizedplasma, but the plume density is too low), tocompensate for the very small constant factor inEq. (18). Electron-neutral IB (Eq. (17) above),however, is important in the first 10 mm near thetarget, where both electron and Cu atom densities0

are still sufficiently high. This absorption processyields a drop in the laser intensity in the first 10mm adjacent to the target, as is clear from Fig. 5e.The calculated laser intensity arriving at the targetis approximately 1.7=10 Wycm , which corre-8 2

sponds to the value shown in Fig. 1.Fig. 6 shows details of the plume at 100 ns. It

is clear that the plume has become much longernow (note the different scale of the x-axis!), aswas also illustrated in Fig. 4a. However, the shapeof the plume density is very similar (see Fig. 6a),

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Fig. 6. Calculated density distribution (a), temperature distri-bution (b), fraction of Cu atoms, Cu and Cu ions and0 q 2q

electrons (c), and density distributions of these species (d) inthe vapor plume, at 100 ns, for the conditions shown in Fig.1.

with a maximum near the target (of almost5=10 m ), a slight decrease as a function of27 y3

distance, and a steep drop at the plume front. Thevapor temperature (at maximum 25 000 K; seeFig. 6b) is somewhat lower than at 15 ns, as was

explained above (i.e. no more heating by the laser,and cooling due to plume expansion). Consequent-ly, the plasma is somewhat less ionized. More

specifically, the Cu ions are of less importance2q

than the Cu ions in the entire plume. The Cuq 0

atoms are still only present near the target, asappears from Fig. 6c, d. Indeed, the Cu atom0

density near the target is almost 3=10 m (the27 y3

y-axis of Fig. 6d is now truncated at 5=1024

m , so that the other density values remainy3

visible), but it drops very rapidly as a function ofdistance from the target. The Cu ion density,q

however, decreases very slowly towards the plumefront, and the Cu ion and electron density reach2q

their maximum between 0.5 and 1 mm from thetarget. Plasma shielding of the laser beam by IBdoes not play a role anymore at 100 ns, since thelaser pulse is already terminated at 30 ns (seeabove).

4. Effect of target surface reflectivity

The target surface reflectivity to be used in themodel (see above, Eq. (2)) is not known veryaccurately. It is in general close to 1 for metals,but it can drop to values as low as 0.1 during laser

ablation (see Section 2.1). We have used anintermediate value of 0.34 (see above). However,the choice of this parameter will affect the calcu-lation results, because it determines the amount oflaser energy that can be absorbed in the target,and that gives rise to target heating, melting andvaporization, and hence to the plume and plasmaformation. Therefore, we have investigated theeffect of this parameter on the calculation results,as is presented in Fig. 7. When the target surfacereflectivity, R, increases, less energy can go into

the target, leading to a lower temperature in thetarget (Fig. 7a), less melting (Fig. 7b) and evap-oration (Fig. 7c,d), and consequently a lowervapor density (Fig. 7e), a shorter plume (Fig. 7f),a lower plume velocity and temperature (Fig. 7g),and a less ionized plasma (Fig. 7h). The effect islimited for low values of R, but it becomessignificant for values approaching 1. Indeed, whenR is equal to 1, no energy can enter the target,and there will be no heating, melting, vaporization,plume and plasma formation. When R is equal to

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Fig. 7. Calculated target surface temperature (a), melt depth (b), surface recession rate due to evaporation (c), evaporation depth(d), average vapor density in the plume (e), plume length (f), vapor velocity and average temperature in the plume (g), and typicalfraction of Cu atoms, Cu and Cu ions and electrons in the plume (h), assuming different values of the target reflectivity, R.0 q 2q

0.9, 10% of the laser energy can enter the target,and this can result already in a significant heating(to a temperature of almost 4000 K, see Fig. 7a)and melting (to a depth of more than 1 mm, seeFig. 7b). However, the evaporation is still of minorimportance (Fig. 7c,d), so that only a short vaporplume is formed (Fig. 7f), at low temperature andtraveling with low velocity (Fig. 7g). Also, theplume is composed only of neutral Cu atoms, i.e.no plasma is formed yet (Fig. 7h). When R isequal to 0.8, 20% of the laser energy can be used

for target heating, yielding a higher temperature,more melting, as well as evaporation. Consequent-ly, a longer plume will be formed, and the plumevelocity and temperature will be higher, leading toalmost fully ionized plasma. The vapor densitywas calculated to be a bit lower than for R equalto 0.9 (see Fig. 7e), which is attributed to thefaster expansion of the plume. When R is furtherdecreased, the target temperature, melting andvaporization still increase, and the vapor plumebecomes longer, with higher density, velocity and

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temperature, and the electron density in the plasmaincreases. However, the effect becomes less pro-nounced when R approaches 0. Indeed, the differ-

ence between 100 and 90% laser energy enteringthe target (i.e. R equal to 0 or 0.1) is much lessthan the difference between 10% and 0% (i.e. Requal to 0.9 or 1). Since R is expected to berather small during laser ablation (see Section2.1), our choice ofR equal to 0.34 w95x is probablya good choice, and the calculation results can beconsidered reliable.

5. Effect of laser intensity

We have also performed some calculations atdifferent laser intensities, in the range 10 107 10

Wycm . The same Gaussian-shaped laser pulses2

were assumed in all cases; only the intensitiesvary, as is presented in Fig. 8. The solid lines inthis figure represent the original laser intensities,whereas the dashed lines denote the laser intensi-ties arriving at the target, after plasma absorption.It appears from this figure that plasma shieldingonly takes place for laser intensities above2=10 Wycm . Further, the plasma shielding8 2

becomes more pronounced at higher laser intensi-

ties. Consequently, the laser intensity arriving atthe target was calculated to be rather similar, inspite of varying original laser intensities, as canbe deduced from Fig. 8.

Fig. 9 illustrates some calculated quantities fordifferent incoming laser intensities (or irradi-ances). As is clear from Fig. 9a, the laser fluenceincreases linearly with laser irradiance, which islogical, because the pulse shapes are kept thesame. Further, at rising laser irradiance, the targetheating (Fig. 9b), melting (Fig. 9c) and vapori-

zation (Fig. 9d,e) increase, the plume becomessomewhat more dense (Fig. 9f), longer (Fig. 9g)and hotter (Fig. 9i), it travels with higher velocity(Fig. 9h), and it becomes more ionized (Fig. 9j).At a laser irradiance of 10 Wycm , the heating is7 2

very moderate, yielding a surface temperature ofless than 500 K. Consequently, neither melting norvaporization take place, and no plume is formed.At 10 Wycm laser intensity, the heating becomes8 2

more important, and melting occurs. However, thevaporization is still very limited, leading to a very

short and rather cool vapor plume (temperatureapprox. 800 K), consisting only of neutral Cu0

atoms (hence, no ionization takes place). For a

laser intensity of 2=10 Wycm , the target heating8 2and melting are still more significant, and vapori-zation starts playing a more important role. How-ever, the vapor plume is still rather short and cool(approx. 900 K), and plasma is not yet created.Upon further increasing the laser irradiance, targetheating, melting and vaporization grow in impor-tance, and the plume becomes longer and muchhotter, leading to plasma formation. At 5=10 Wy8

cm , the plume is already almost fully ionized into2

Cu ions (approx. 85%). For higher laser irradi-q

ance, the fraction of Cu ions rises, and exceeds

2q

the Cu fraction at 5=10 Wycm , because ofq 9 2

still higher plume temperature. Consequently, alsothe electron density in the plume increases withlaser irradiance, so that more plasma shielding canoccur, as was indeed illustrated in Fig. 8.

6. Comparison with other calculated or experi-

mental results

Comparison with other data from the literature,obtained either by models or experiments, is not

straightforward, because the laser ablation condi-tions in the literature (e.g. laser wavelength, pulseduration, irradiance, target material, expansion invacuum or in a background gas) are not alwaysclearly described, or they are not exactly the same,and the results will be different for differentconditions. However, we have attempted to makeat least some qualitative comparisons, to check ourmodeling results.

6.1. Comparison with other calculation results

In spite of the multitude of papers in theliterature describing models for lasersolid inter-action, plume expansion and plasma formation(see Section 1.2 above), not many of these papersshow detailed calculation results for conditionscomparable to the ones under study here. However,a few papers gave results that can be comparedwith our data. Amoruso w36x calculated the plumetemperature, neutral atom and electron density fora 6 ns laser pulse at 350 nm on an Al target. For

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Fig. 8. Laser intensity-time profiles for different values of laser irradiance, assuming the same pulse shape of 10 ns (FWHM). Thedashed lines represent the calculated laser intensities arriving at the target, after passing through the plasma.

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Fig. 9. Laser fluence corresponding to different values of laser irradiance (a), as well as calculated target surface temperature (b),melt depth (c), surface recession rate due to evaporation (d), evaporation depth (e), average vapor density in the plume (f), plumelength (g), plume velocity (h), average temperature in the plume (i), and typical fraction of Cu atoms, Cu and Cu ions and0 q 2q

electrons in the plume (j), as a function of laser irradiance.

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an irradiance of 10 Wycm , the electron density9 2

was in the order of 10 m , which was found to26 y3

be higher than the neutral density. The plume

temperature was in the order of 20 000 K. Bothelectron density and plume temperature increaseddrastically with rising laser intensity up to a valueof 1.5=10 Wycm . For higher laser intensities,9 2

the rise was not so pronounced. These results arein satisfactory agreement with our data. Kar andMazumder w38x calculated results for a 30 ns laserpulse of 248 nm on a Nb target, at a laserirradiance of 5=10 Wycm . The melt depth was8 2

reported to be 12 mm. The dimensions of theplume were in the order of 0.11 mm, increasing

with time. The plume temperature was approxi-mately 10 000 K, resulting in an ion fraction ofseveral%. This is lower than our results, but themelt depth and plume dimension are in goodagreement. Finally, Zhang et al. w37x performedcalculations for a 10 ns laser pulse at 248 nm ona graphite sample, for a laser irradiance of3=10 Wycm . Results are reported after 2, 5 and8 2

10 ns. At 10 ns, the plume length was approxi-mately 0.10.2 mm. The plume velocity was inthe order of 10 mys, and the plume temperature4

was approximately 10 00020 000 K. At the end

of the plume, a pronounced maximum of approx-imately 100 000 K was shown, similar to ourobservations. However, as discussed above (Sec-tion 3.2), this temperature peak corresponds to aregion where the plume density has dropped tovery low values, so it has no real physical meaning.In general, these calculation results were in goodagreement with our data.

6.2. Comparison with experimental data

Beside these modeling studies, several papersreport measurements in the laser-ablated plume,mainly for the plume temperature and electrondensity. Sometimes these experiments were carriedout in a background gas. However, preliminarycalculations from our group show that the plumetemperature and hence the electron density are nottoo much different for expansion in vacuum or in1 atm background gas w128x. The major differenceis expected in the rate of change of physicalproperties, which will be much faster in vacuum,

because of the faster expansion. Russo andcoworkers w144x studied a 3 ns laser pulse at 266nm on a Si target, for laser irradiance in the range

of 2=10 8=10 Wycm . At 2=10 Wycm ,9 10 2 9 2the plume temperature was approximately 15.000K, and the electron density was ca. 5=1024

cm , at a distance of 1.8 mm and a delay timey3

of 30 ns. Doyle et al. w145x also reported electrondensity values in the order of 110=10 cm ,24 y3

decreasing with distance from the target. The laserablation conditions in this case were a pulse of 30ns (FWHM), at 248 nm, on a Mg target, for anirradiance of 1.253=10 Wycm . Hence, the8 2

electron density was in the same order of magni-

tude, but for lower irradiance, which is probablyattributed to a difference in laser conditions (laserpulse, wavelength), or target. However, in generalthe agreement is reasonable. Moreover, the plumelength appeared to be in the order of 0.5 mm, at30 ns, and it increases with time, which is incorrespondence with our results. An electron den-sity in the order of 10 m was also measured24 y3

in Ref. w146x for a laser irradiance of 10 Wy10

cm , a pulse of 7 ns, at 355 nm, and an Al target.2

Monge et al. w147x measured a plume temperatureof approximately 10 000 K at maximum, and an

electron density approximately 10 10 m .22 23 y3

This is lower than our calculated results, but thedelay time of measurement was several microse-conds, which is much longer than in our study,and it is illustrated, both in our calculation results,as well as in Ref. w147x that the plume temperatureand hence the electron density drop as timeevolves, as a result of plume expansion. Hafez etal. w148x also measured electron densities in theorder of 10 m , for a laser pulse of 7 ns, at22 y3

335 nm, an irradiance of 10 Wycm , and a Cu10 2

target, both for expansion in vacuum and in 5 Torrargon. These reported electron density values arealso lower than our calculation results, but theywere probably also measured at a later time (notreported in the paper), because the plume lengthwas in the order of 14 mm. Finally, in Ref. w149x

it was reported that for pulsed ns UV lasers onmetal targets, a dense and highly ionized plasmais created at a laser irradiance in the order of4=10 Wycm . The measured electron density at8 2

approximately 50100 ns was in the order of

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10 m for a laser irradiance of 5=10 Wycm ,24 y3 8 2

and it dropped over several orders of magnitudeas a function of time (i.e. two orders of magnitude

at 1 ms).Beside electron density measurements, the ions

have also been investigated in laser ablation exper-iments. Dreyfus w150x has reported laser-inducedfluorescence measurements of Cu , Cu and Cu ,0 q 2as a function of fluence, to elucidate the mecha-nisms playing a role in laser ablation. Amoruso etal. w151,152x measured energy distributions of ionsin ns laser produced plasmas of Al and Cu targetsat visible and UV wavelengths, for a laser fluencein the range of 360 Jycm , to study the impor-2

tance of various plasma processes (i.e. inverseBremsstrahlung absorption, photo-ionization, elec-tron impact ionization and excitation). Franghia-dakis w153x also measured ion kinetic energydistributions in the laser-ablated plume, for UVns-pulsed lasers on various materials, includingCu. It was shown that not only Cu ions, but alsoq

Cu ions were present. The Cu intensity was2q q

typically a factor of 510 times higher than theCu signal, but the laser fluence was somewhat2q

lower than in our case (i.e. 18 Jycm ). The facts2

that multiply charged ions were detected, not onlyin Ref. w153x, but also in several other papersreferred to in Ref. w153x, are at least a qualitativevalidation of our calculations. From the measuredenergy distributions, an effective plasma tempera-ture in the range of 20.000100.000 K wasdeduced, which also correlates well with our data.

Finally, the phenomenon of plasma shielding isalso studied in a number of papers. Russo andcoworkers w105,154x found that the mass removalrate depends strongly on the laser irradiance, but

a roll-off in this dependence was observed between1=10 and 5=10 Wycm (depending on the8 8 2

material), which is attributed to the onset of plasmashielding. Indeed, it is generally stated (e.g. w22x)that for ns-pulsed lasers, with laser irradiancebelow 2.5=10 Wycm , only vaporization takes8 2

place, and a plasma is only created for a laserirradiance above 2.5=10 Wycm . The plasma8 2

becomes fully ionized at approximately 5=108

Wycm , and it becomes opaque for the incoming2

laser energy at approximately 10 Wycm . This9 2

correlates with our calculations for the onset ofplasma formation and plasma shielding.

7. Conclusion

We have presented a model for ns-pulsed laserablation of a Cu target that describes:

1. Lasersolid interaction, yielding heating, melt-ing, and vaporization of the target material;

2. expansion of the evaporated material plume invacuum;

3. plasma formation, generating electrons, Cuq

and Cu ions, beside the Cu atoms;2q

4. plasma shielding of the incoming laser light.

The results include the temperature distributioninside the target, the melt depth, the amount ofevaporation, the vapor density, velocity and tem-perature distribution in the plume, as well as theresulting ionization degree, and the density profilesof electrons, Cu atoms, Cu and Cu ions in0 q 2q

the plasma. All these results are presented as afunction of time during and after the laser pulse.

For a typical laser irradiance of 10 Wycm (i.e.9 2

fluence of approx. 10 Jycm for the pulse under2

study), it is found that the temperature in the target

rises till a few 1000 K, and is at its maximumnear the surface. The melt depth is calculated tobe in the order of 2 mm. The evaporation depth,however, is estimated to be only approximately 70nm, for one laser pulse. The vapor plume expandsas time evolves: the plume length is in the orderof 0.1 mm after 20 ns, and it increases to ca. 2mm after 100 ns. The vapor density in the plumeis very high near the target (approx. 5=10 m ),27 y3

but it decreases rapidly to typical values of 10 24

10 m in most of the plume. Moreover, the25 y3

vapor density also decreases slightly upon plumeexpansion. The vapor velocity in the plumeincreases towards the end of the plume, where itis in the order of 30 000 mys. The vapor temper-ature in the plume is calculated between 10 000 Kand 40 000 K. At this high vapor temperature, theplume is more or less fully ionized, and in someregions of the plume, i.e. where the vapor temper-ature reaches its maximum, the Cu ions can2q

even become dominant. Since the vapor plume isconverted into plasma, part of the incoming laser

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beam will be absorbed before reaching the target.For the laser ablation conditions under study, itwas found that electron-neutral inverse Brems-

strahlung is mainly responsible for the plasmashielding.

Because the target surface reflectivity is subjectto uncertainties (i.e. it is typically close to 1 forpolished metals, but it can drop to values as lowas 0.1 during laser ablation), we have explicitlyinvestigated the effect of this parameter on thecalculation results. When R was increased from 0to 1, the target heating, melting and vaporizationwere found to decrease, as well as the plumelength, the vapor density, velocity and temperature

in the plume, and the ionization degree andcharged species densities in the plasma. This islogical, because when R increases, the laser energythat can enter the target and give rise to targetheating, melting and vaporization drops. The effectofR is found to be much more pronounced at Rclose to 1, and is only small for R between 0 and0.8. Since this parameter is probably rather smallduring laser ablation, our choice of 0.34 is proba-bly a good choice.

Finally, we have also performed calculations fordifferent laser intensities (or irradiance), in the

range 10 10 Wycm . A higher laser irradiance7 10 2

gives rise to more target heating, melting andvaporization, resulting also in a longer plume, ahigher vapor density, velocity and temperature inthe plume, and more charged species (i.e. higherionization degree of the plasma), as is expected.For a laser irradiance of 10 Wycm , the target7 2

heating is calculated to be very moderate, andneither melting nor vaporization takes place. Whenthe laser irradiance increases to 10 Wycm or8 2

2=10 Wycm , target heating and melting become8 2

more pronounced, but the evaporation is still lim-ited, so that the vapor plume is still rather shortand cool, and no plasma is formed. At higher laserirradiance (5=10 Wycm ), target evaporation8 2

becomes much more significant, the plumebecomes longer and hotter, and a plasma is created.This results also in plasma shielding of the incom-ing laser light. For laser irradiance between5=10 and 10 Wycm , the target heating, melt-8 10 2

ing and vaporization still increase, the plumebecomes longer, and the vapor density, velocity

and temperature, as well as the plasma ionizationdegree still rise. However, the difference is not sopronounced anymore, because a higher original

laser irradiance yields more plasma shielding, sothat the laser irradiance that actually reaches thetarget does not rise to a large extent.

The present model does not yet describe theexpansion of the evaporated material into a back-ground gas, and it does not consider the formationof particles. Hence, it is not directly suitable forthe description of laser ablation as sample intro-duction method for the ICP, where the laser abla-tion cell is typically filled with gas at 1 atm, andwhere the particles are the main constituents of

the flow transported to the ICP. In the next step,the present model will be used as a starting pointfor a more realistic description of laser ablationfor the ICP, including the expansion in 1 atmbackground gas, and the formation of particles.

However, the goal of the present paper was todemonstrate that this basic model could alreadygive insight in the various aspects playing a rolein the laser ablation mechanism. From the satisfac-tory correlation between our calculation resultsand several other papers in the literature, reportingeither modeling or experimental studies, even for

expansion in a background gas, it can be concludedthat our model presents already a realistic pictureof the mechanisms playing a role in laser ablation.

Acknowledgments

A. Bogaerts is indebted to the Flemish Fund forScientific Research (FWO) for financial support.Z. Chen is financed by the IUAP-V project. A.Vertes acknowledges the support of the US Depart-ment of Energy under grant DE-FG02-

01ER15129.The authors also thank D. Gunther, S.Mao, J.-M. Mermet and B. Garrison for the inter-esting discussions and A. Okhrimovskyy for thetranslation of the Russian papers.

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