Gas dynamics of laser ablation: Inï¬‚uence of ambient atmosphere

Gas dynamics of laser ablation: Influence of ambient atmosphereAndrey V. Gusarov and Alexey G. GnedovetsBaikov Institute of Metallurgy, Russian Academy of Sciences, Leninsky Prospect 49, 117911 Moscow,Russian Federation

Igor Smurova)

Ecole Nationale d’Ingenieurs de Saint-Etienne, 58 rue J. Parot, F-42023 Saint-Etienne Cedex 2, France

�Received 15 November 1999; accepted for publication 9 April 2000�

A two-stage two-dimensional �2D� gas-dynamic model of laser ablation in an ambient gasatmosphere is proposed. The initial one-dimensional stage of the process is related to the ablationplume formation under the action of a laser pulse �duration of the order of 10 ns; fluence aboutseveral J/cm2; laser spot diameter about 1 mm� and describes heating, melting, and evaporation ofthe target, the target–vapor interaction in the Knudsen layer, and the vapor dynamics. The final 2Dstage is responsible for the formation of the energy and angular distributions of the ablated material.Considerable compression of the ambient gas around the expanding plume of the laser-evaporatedmaterial and a shock front propagating through the undisturbed ambient gas are found. The pressureof the compressed ambient gas behind the shock may be much higher than the ambient one.However, at the investigated ambient pressures below 100 Pa, it remains still much lower than thevapor pressure during laser evaporation. Therefore, the initial stage of laser ablation is essentiallyindependent of the ambient atmosphere. Once the laser pulse is over, the vapor pressure eventuallydrops down to a value comparable to the compressed ambient gas pressure. From this time on, thegas considerably suppresses vapor expansion. There is a noticeable difference between the vapordistribution in vacuum and the one in the ambient atmosphere: the vapor fills the entire plumevolume in vacuum while in the presence of ambient atmosphere it is accumulated near the plumeboundary and tends to form a thin shell. The angular and energy distributions of the ablated materialare especially sensitive to the nature and pressure of the ambient gas. Both the kinetic energy of theablated atoms and the width of their angular distribution decrease with the ambient pressure.© 2000 American Institute of Physics. �S0021-8979�00�02914-5�

I. INTRODUCTION

The pulsed laser deposition �PLD� technique, based onlaser ablation phenomena, offers universality for the deposi-tion of dielectrics, superconductors, semiconductors, andother materials. Its transient nature affords high controllabil-ity of the deposition thickness. The high kinetic energy of theablation plume promotes surface mobility in the growingfilm. However, there are certain drawbacks associated withcurrently available PLD techniques. Foremost of these arethe inhomogeneities of deposition rates resulting from thecosn � profile of the ablation plume and possible nonstoichio-metric material transfer of multielemental targets such as inthe case of ceramic high-temperature superconductors.

The quality of the deposited film critically depends onthe range and profile of the kinetic energy of the ablatedplume. While it is to the advantage of PLD to have highkinetic energy, the plumes that are too energetic cause filmdamage. In this connection the interaction of the laser-ablated plumes with background gases is of ever increasingattention. The low-pressure buffer gas reduces the ablationplume energy without reducing the ablation yield. The en-ergy is spent to involve the gas into motion and generating ashock wave, which is detected by emission spectroscopy,1,2

fast schlieren photography, and shadowgraphy.3 This reducesselective sputtering/desorption from the growing film and isuseful for manufacturing stoichiometric films.4,5 The tech-nique of PLD in a buffer gas also allows one to vary the filmstoichiometry.6

To understand the fundamental physics of laser ablation,experimental methods �for example, see Ref. 7� are devel-oped along with mathematical models. One of the early the-oretical investigations of the laser ablation in a backgroundgas was undertaken by Knight.8 He solved the one-dimensional �1D� transient self-similar gas-dynamic problemdescribing instantaneously starting vaporization with steadymass flux. The Knudsen layer at the target surface, vapor–gas contact surface, and shock in the gas were considered. Asimilar problem was recently numerically solved for a pulsedvapor source.9 Rather than examine the Knudsen layer indetail, the authors predefined the vapor temperature and ve-locity at the target surface as functions of time. If these func-tions are properly chosen, the approach adequately describesPLD conditions. The comparison with the conventional free-expansion model confirmed vapor slowing down by the gas.Among the consequences of finite duration of the vapor ejec-tion a possible backward motion at the later stages was re-vealed. Some analytical approaches to 1D gas dynamics ini-tiated by an unsteady vapor source were reported byBellantone and Hahn.10 The most complete solution for thea�Electronic mail: [email protected]

JOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 7 1 OCTOBER 2000

43520021-8979/2000/88(7)/4352/13/$17.00 © 2000 American Institute of Physics

1D problem was probably given by Breslavsky andMazhukin.11 They presented a self-consistent numericalmodel which took into account processes in both the con-densed and gaseous phases.

The 1D models listed above describe only the initialstage of laser ablation that is very important but not suffi-cient for PLD simulation because the lateral expansion of thevapor at the later stages plays a key role in the formation ofenergy and especially angular distribution. One of the sim-plest ways to overcome this restriction is to consider the 1Dspherical rather than 1D planar plume expansion.12 In thespherical model12 it is impossible to specify the initial vaporflow away from the surface and the authors had to use theinitial conditions as a half sphere filled with a stagnant vapor.The initial vapor density and temperature were estimatedfrom the experiment. The spherical expansion model is prob-ably better than the planar expansion one at the later stagesof laser ablation but gives only a qualitative view of theablation phenomenon as well. The analogy between the ab-lation plume and an underexpanded jet13 explains, for ex-ample, the ablated material focusing in an ambient gas.

A numerical method applied for modeling of laserplasma evolution in two dimensions14 and its interaction withlaser beam is based on the radiative gas dynamic equations.It was shown that the ionization wave is the dominantmechanism of plasma expansion during its interaction withthe laser beam.15 The dynamics of the energy fluxes, laserradiation passed through the plasma plume, and properplasma radiation reaching the target surface was found.16

However, this model was applied to study the influence ofambient pressure14 without detailed analysis of the evapora-tion kinetics.

Recently, the Monte-Carlo simulation technique hasbeen applied to laser ablation.17–19 The method takes intoaccount the actual three-dimensional geometry and is suit-able for the rarefied gas flows with considerable deviationsfrom the local equilibrium. These features are particularlyuseful for the PLD modeling. It was proposed19 to divide thelaser ablation process into two stages: �i� evaporation andformation of the laser plume and �ii� transport of noninterac-tive ablated particles through the background gas. At thesecond stage the number density of the ablated particles wassupposed to be low enough to neglect the collisions amongthem in comparison with the ones between the ablated par-ticles and the ambient gas molecules. The collective motionof the background gas induced by the collisions with theablated species was disregarded.

Thus, the theory of the laser ablation in a backgroundatmosphere is currently developed for the early 1D gas-dynamic stage and the later stage of rarefied gas described ingeneral by the linearized Boltzmann equation. The interme-diate stages with pronounced lateral motion and considerablevapor density are poorly studied. To fill partially this gap atwo-dimensional �2D� gas-dynamic model of laser ablationin an ambient gas atmosphere is proposed in this article.

The model, along with the general gas-dynamic ap-proach, is based upon the assumption of local equilibrium,which is useful until the mean free path is much less than thelaser plume dimension. The effects of diffusion, viscosity,

and heat conduction are disregarded. The mean free path,which is inversely proportional to the gas density, increasesas the plume dimension in cube and eventually becomes ofthe order of the plume dimension. The conventional gas-dynamic analysis becomes invalid at this moment. Thus, thegas-dynamic model describes the initial ‘‘dense’’ stage ofplume expansion. Duration of the gas-dynamic stage is esti-mated to be about several microseconds. Collisions betweenthe ablated atoms essentially cease at the end of this stage.Therefore, formation of the angular-energy distribution isnearly completed. In the presence of background gas, how-ever, further slow evolution of this distribution is expecteddue to the collisions between ablated atoms and the gas mol-ecules. In the case of the rarefied background gas the evolu-tion may be considered using the linearized Boltzmann ap-proach to rarefied gas dynamics. For example, the Monte-Carlo model18,19 is useful.

II. MATHEMATICAL MODEL

Taking into account laser pulse duration equal to about10�8 s and ablated material velocity equal to about103 – 104 m/s, the axial expansion of the ablation plume forthis period is estimated as 10�2 – 10�1 mm. This value ismuch less than the laser spot size �typically about 1 mm�.Thus, during laser pulse action the ablation plume dynamicsis essentially one dimensional. This fact allows one to con-sider temporal evolution of the ablation plume as consistingof two stages: At the initial stage the basic processes are thelaser radiation absorption, the target heating, melting, evapo-ration, and the 1D axial expansion of the vapor. At the finalgas-dynamic stage the basic process is the 2D axial and ra-dial expansion of the vapor and interaction thereof with theambient gas.

The laser radiation is considered to be uniformly distrib-uted over a circular spot of radius R0 at the target surface.The incident laser energy flux Ki �per unit area� and theabsorbed energy flux Ka�(1�r)Ki (r is the reflectivity� arethe functions of time t . Absorption of the laser radiationresults in the heating, melting, and evaporation of the target,thus only thermal processes are considered. To analyze thevapor cloud �the ablation plume� expansion in the ambientatmosphere, a two-step gas-dynamic model is applied. Noplasma effects are taken into account, such as the ionizationcontribution to the enthalpy9 and the laser energy absorptionin the gas phase.14,15 This is justified only at a low ionizationdegree. The experimental measurement of the plasma forma-tion threshold20 and the optical breakdown simulation16 indi-cate that the plasma effects can be disregarded under nano-second laser pulses at low fluence of less than or aboutseveral J/cm2.

The duration of the initial 1D stage is about several laserpulse durations �t . The initial stage is responsible for theablation plume formation. The target–vapor interaction inthe boundary layer is of primary interest here. The gas-dynamic fields at the end of this stage give initial conditionsfor the final 2D stage. At the final stage the target is consid-ered to be cold enough to have the simplified boundary con-dition of complete vapor absorption at the target surface ap-

4353J. Appl. Phys., Vol. 88, No. 7, 1 October 2000 Gusarov, Gnedovets, and Smurov

plied. Further expansion of the plume both in the axial andradial directions and its interaction with the ambient gas de-termine evolution of the vapor atoms angular and energydistributions. To take into account the presence of the ambi-ent gas, two-component gas dynamic equations are applied.

A. Initial stage

Target heating by laser radiation is described by the heattransfer equation in a moving reference frame attached to theevaporated surface,

� �

�t�Ue

�

�Zc� �NcHc��

�

�Zc� �c

�Tc

�Zc� , �1�

where Ue is the velocity of the evaporation front, Hc theenthalpy per atom, Nc the number density, Tc the tempera-ture, �c the thermal conductivity, and Zc the coordinatealong the inward normal to the target surface. The thermalconductivity dependence on temperature is approximated bya piecewise constant function: �c��s at Tc�Tm ; �c�� l atTc�Tm . The thermal equation of state takes into account thesolid–liquid phase transition at the melting point Tm with thelatent heat of fusion Qm and assumes the specific heats ofsolid Cs and liquid Cl phases to be constant but of differentvalues,

Tc�Hc�

�� Hc /Cs , HcCsTm ,

Tm , CsTm�Hc�CsTm�Qm ,

Tm��Hc�CsTm�Qm�/Cl , HcCsTm�Qm.

Thus, Eq. �1� actually describes the Stefan problem: heatconduction in both the solid and liquid phases and motion ofthe phase boundary, which is implicitly defined by Tc

�Tm .The gas phase consists of monatomic vapor of number

density N� and atomic mass m� and the inert ambient gascharacterized by the density Ng , molecular mass mg , andpolytropic index �. The equations of mass, momentum, andenergy conservation determine the gas phase dynamics:

�N�

�t�

��N�UZ�

�Z�0, �2�

�Ng

�t�

��NgUZ�

�Z�0, �3�

�� UZ�

�t�

�� UZ2�P �

�Z�0, �4�

��E� UZ2 /2�

�t�

��UZ�E� UZ2 /2�P ��

�Z�0, �5�

where UZ is the axial velocity, �m�N��mgNg the massdensity, P�(N��Ng)kT the pressure, E��3N�/2�Ng /(��1)�kT the volumetric internal energy, and Z the coordinatealong the outward normal to the surface.

The heat transfer equation, Eq. �1�, is accompanied byone initial and two boundary conditions,

Tc�Ta , t�0, �6�

Tc→Ta , Zc→� , �7�

UZ�N�Q��E� UZ2 /2�P ��c

�Tc

�Zc�Ka� t �, Z�0,

�8�

where Ta is the ambient temperature and Q� the latent heatof evaporation. The last equation is the energy balance at thetarget surface, which couples the heat conduction and thegas-dynamic equations. The evaporation front velocity Ue isdefined by the mass balance,

NcUe�N�UZ , Z�0. �9�

The initial conditions for gas-dynamic Eqs. �2�–�5� aredetermined by the ambient pressure Pa and temperature Ta ,

�T�Ta ,P�Pa ,N��0,UZ�0�, t�0. �10�

To obtain a number of the required boundary conditions, thecharacteristic equations following from Eqs. �2�–�5� shouldbe considered. The number of boundary conditions is equalto the number of independent variables transferred along thecharacteristics incoming into the volume from the boundarysurface under consideration.21 There are three sets of charac-teristics: C� , dZ/dt�UZ�C; C� , dZ/dt�UZ – C; C0 ,dZ/dt�UZ . Here C is the sound velocity. In the two-component case under consideration there are four indepen-dent characteristic variables. They may be introduced as twoRiemann invariants �one transferred along C� characteristicsand the other one transferred along C� characteristics�, en-tropy, and relative vapor concentration �both transferredalong C0 characteristics�. At infinity only C� characteristicsare incoming and therefore one condition is sufficient,

P→Pa , Z→� . �11�

At the evaporated target surface C0 and C� characteristicsare incoming ones and so three boundary conditions are to bespecified. One of them is the inert gas flux,

NgUZ�0, Z�0, �12�

and the two others are to be determined by the vaporizationkinetics.

To derive these conditions it is necessary to consider athin Knudsen layer adjacent to the surface. Jump conditionsrelate the vapor parameters at the outer edge of the Knudsenlayer, i.e., Nk�N�(0, t), Tk�T(0, t), Pk�P(0, t), and Uk

�UZ(0, t) with the surface temperature Ts�Tc(0, t) and thesaturated vapor pressure, which is given by the Clausius–Clapeyron equation Ps�P0�exp�Q� /(kTb)–Q� /(kTs)�,where Tb is the boiling temperature at normal pressure P0

�1 atm. In the case of intensive vaporization which is con-sidered here, Anisimov22 proposed the half-range Max-wellians approximation of the velocity distribution function,

f �v�� Ns

�2�kTs /m��3/2 exp� �m�v2

2kTs� , vZ0.

�Nk

�2�kTk /m��3/2 exp� �m��vÀUk�2

2kTk� , vZ�0.

The parameters Ts , Ns�Ps /Ts , and Pk are assumed to beknown. The other three parameters, �, Tk , and Nk , may be

4354 J. Appl. Phys., Vol. 88, No. 7, 1 October 2000 Gusarov, Gnedovets, and Smurov

determined from the conservation laws of mass, momentum,and energy. Thus, the relations between the surface and thebulk vapor parameters are derived without solving the Bolt-zmann equation and determining the detailed structure of theKnudsen layer. The results are reduced to8

� Tk

Ts� 1/2

�� 1��s2

64 � 1/2

��1/2

8s , �13�

Pk

Ps�� Tk

Ts� 1/2� � s2�

1

2 � erfc�s �exp�s2��s

�1/2��

1

2�1��1/2serfc�s �exp�s2�� , �14�

where s�Uk /(2kTk /m�)1/2 is the speed ratio. At theChapman–Jouguet point where the vaporization speed ap-proaches the sound speed: Uk�Ck��5kTk /(3m�)�1/2, thegas-dynamic disturbances cannot reach the target surface andtherefore influence vaporization kinetics. Further increasingthe vapor velocity and reducing the pressure below this pointappear to be impossible.

On termination of the laser pulse the target temperaturedrops down and back condensation may start. In this case theboundary value of the gas velocity is negative, only C� char-acteristics are incoming, and one boundary condition is re-quired. The condensation flux is defined by the Hertz–Knudsen equation,

NkUk�Ps�Pk

�2�mkTs�1/2 , �15�

which is applied when the saturated vapor pressure Ps be-comes lower than the gas pressure Pk .

B. Final stage

When the ablated material is ejected to distances com-parable with the laser spot size, the pronounced lateral ex-pansion starts. At the final stage the gas phase is describedby the conservation laws in 2D cylindrical frame (R ,Z),

�N�

�t�

��N�UZ�

�Z�

1

R

��RN�UR�

�R�0, �16�

�Ng

�t�

��NgUZ�

�Z�

1

R

��RNgUR�

�R�0, �17�

�� UZ�

�t�

�� UZ2�P �

�Z�

1

R

��R UZUR�

�R�0, �18�

�� UR�

�t�

�� UZUR�

�Z�

1

R

��R� UR2 �P ��

�R�0, �19�

��E� U2/2�

�t�

��UZ�E� U2/2�P ��

�Z

�1

R

��RUR�E� U2/2�P ��

�R�0, �20�

where UR is the radial velocity, U2�UZ2�UR

2 . The initialconditions for the system, Eqs. �16�–�20�, are the gas-

dynamic fields resulting from the initial stage at R�R0 andthe undisturbed gas parameters, Eqs. �10�, at R�R0 .

The boundary conditions for Eqs. �16�–�20� are definedby the pressure at infinity,

P→Pa , �R ,Z�→� , �21�

by the rigid wall �mirror� conditions at the regions of thetarget surface contiguous to the inert gas (N��0 and Ng

�0 at Z�0),

�P

�Z�UZ�0, Z�0, �22�

and by the Chapman–Jouguet condition at the regions con-tiguous to the vapor (N��0 and Ng�0 at Z�0),

UZ��C�� 5kT

3m�� 1/2

, Z�0. �23�

Equations �22� and �23�, respectively, specify the reflectionof the inert gas and complete absorption of the vapor whichmoves toward the surface with the maximum allowable ve-locity of local sound speed. The normal velocity component,Eq. �23�, is the only one parameter that is to be specified atthe completely absorbing surface, because in this case noneof the gas-dynamic disturbances can propagate upstream.

C. Energy and angular distributions

The energy and angular distributions of the ablated ma-terial are of practical interest and moreover may be found viaexperiment. In the framework of the proposed model thedistributions are obtained in the following way: the velocitydistribution function is derived from the gas-dynamic fieldsby integrating the local Maxwell velocity distribution func-tion over all the possible radius vectors r,

f �v�� N��r�

�2�kT�r�/m��3/2 exp� �m��v�U„r…�2

2kT�r� � dr.

�24�

In the case of axial symmetry it is reasonable to specify thevelocity vector v by the polar � and lateral � angles and thekinetic energy ��m�v2/2. This transformation defines theintegral energy-angular distribution of the vapor atoms:F(� ,�)d�d�� f (v)dv is the number of vapor atoms withenergies from � to ��d� within solid angle d��d cos �d�. The distribution is actually independent of �. Itmay be expressed as

F�� ,��0

�

dZ�0

� N��1/2

�1/2�kT �3/2

�exp� �2��m�U2�2�2m��1/2UZ cos �

2kT ��I� �2m��1/2UR

kTsin � �R dR , �25�

where the function I(x) is defined as

I�x ��1

2� �0

2�

exp�x cos ��d� .


Integration of Eq. �25� over energy results in one-dimensional angular distribution,

F��0

�

F�� ,��d� . �26�

D. Numerical procedure

The proposed mathematical model consists of threeproblems for partial differential equations: �a� 1D nonlinearheat transfer Eq. �1� with the initial condition, Eq. �6�, andboundary conditions, Eqs. �7� and �8�; �b� 1D gas-dynamicEqs. �2�–�5� with the initial conditions, Eqs. �10�, andboundary conditions, Eqs. �11�–�15�; �c� 2D gas-dynamicEqs. �16�–�20� with the boundary conditions, Eqs. �21�–�23�. The first and the second problems are linked by theenergy, Eq. �8�, and mass, Eq. �9�, balances. The latter twoproblems are linked by the fact that the gas-dynamic fieldsresulting from the 1D gas-dynamic problem are used as theinitial conditions for the 2D one.

2D gas dynamics is rather cumbersome for numericalsolution and therefore the choice of an effective numericalmethod for this problem is of primary interest here. Usually,the main difficulties arise from numerical treatment of vari-ous discontinuities �shocks, contact surfaces�. The first at-tempts to apply conventional finite-difference methods re-vealed strong oscillations of the gas parameters near thediscontinuities due to numerical errors but a number of ef-fective methods are now developed.23,24 The basic principlesare derivation of the conservative finite-difference equationsand the artificial dissipation behind the shocks. In addition,the present problem is characterized by an important drop ofdensity �about several orders of magnitude� across the con-tact surface between the vapor and the ambient gas. Theapproximation errors of the finite-difference methods maycause formation of regions with negative density that leads tonumerical instability. The main effort was to develop mo-notonous finite-difference equations,25 which preserve mo-notonous profiles across the discontinuities and thus preventformation of nodes with a negative density.

In the present work finite-difference equations for gasdynamics are derived in the framework of the large-particlemethod.26,27 The characteristic feature of this method is theEuler–Lagrange approach. A linear artificial viscosity isused to smear shocks. The advection terms are approximatedby upwind differences to obtain monotonous differenceschemes. The heat transfer equation is solved by means ofthe explicit difference scheme.23 The finite-difference equa-tions are written for the uniform grids, which can be rear-ranged if the disturbed zone reaches the border. The typicalgrid size used in the present calculations is 500 nodes for the1D heat conduction problem, 2000 nodes for the 1D gas-dynamic problem, and 800�800 nodes for the 2D gas-dynamic problem.

III. RESULTS AND DISCUSSION

The above model is applied to analyze ablation of graph-ite and Au targets by nanosecond laser pulses in vacuum andbackground gas atmosphere. The shapes of the laser pulses

used in simulation are shown in Fig. 1. Graphite ablation isanalyzed for the bell-shaped pulse �curve 1� with 25 ns fullwidth at half maximum �FWHM� that corresponds to theexperiment1 with a Nd/YAG 1.06 �m laser. The experimen-tal profiles28 of a KrF excimer laser with 248 nm wavelength�curve 2� and the bell-shaped pulse of a 532 nm YAG laser29

with 11 ns FWHM �curve 3� are used for the Au target. Thetarget material properties are listed in Table I.

The transient heat phenomena in the target are presentedin Fig. 2. Surface temperature Ts �curve 1� rises steeply andhas a maximum near the middle of the laser pulse. The maxi-mum value of surface temperature exceeds the boiling �sub-limation� point at normal pressure. The condensed phase ismaintained in this state because of the reactive pressure ofevaporation. A plateau of Au surface temperature �Fig. 2�b��at �180– 230 ns indicates the melt solidification. Dynamicsof evaporation is characterized by the evaporated thicknessDe��0

t Uedt �curve 2�. Evaporation starts with the notice-able delay relative to laser pulse and it is completed with thefall of the surface temperature. Back condensation, which istaken into account by the model, is insignificant. The meltthickness Zsl �curve 3 in Fig. 2�b�� reaches its maximumZsl�2 �m at t�70 ns which is more than twice as much thepulse duration, and the melt lifetime is about 7 times asmuch as the pulse duration.

The initial gas-dynamic stage of the laser ablation is pre-sented in Fig. 3. The evaporated material forms the ablationplume in the gas phase �see the N� curves� with initial tem-

FIG. 1. Temporal profiles of the laser pulses used in the simulation: curve 1,bell-shaped pulse with 25 ns FWHM and 4 J/cm2 incident fluence; curve 2,30 ns full width KrF eximer laser pulse with 248 nm wavelength and3 J/cm2 incident fluence; curve 3, bell-shaped pulse with 11 ns FWHM and4.9 J/cm2 incident fluence.

TABLE I. Summary of the target material properties used in simulation.

Quantity Symbol Au Graphite

Reflectivity r 0.33;a 0.67b 0Number density Nc 5.91�1028 m�3 1.13�1029 m�3

Temperature: melting Tm 1336 K ¯boiling Tb 3150 K 5100 Kc

Latent heat: fusion Qm 0.131 eV ¯boiling Qb 3.45 eV 9.13 eVc

Specific heat: solid Cs 28.8 J/�mol K� 24.9 J/�mol K�liquid Cl 31.3 J/�mol K� ¯

Thermal conductivity: solid �s 284 W/�m K� 1.6 W/�m K�d

0.81 W/�m K�e

liquid � l 155 W/�m K� ¯aAt 248 nm.bAt 532 nm.cSublimation parameters.dMean value accepted at T�2000 K.eMean value accepted at T�2000 K.


perature around the boiling point of the target material �sev-eral thousand K�. The vapor at Z�0 is significantly colderthan the target surface �compare with Fig. 2� because of theconsiderable temperature drop within the Knudsen layer. Thevapor expands from the surface with the velocity about sev-eral thermal velocities at the boiling point �see the UZ

curves�. On termination of the laser pulse the surface tem-perature drops and the saturated vapor pressure Ps drops aswell. The latter becomes lower than the vapor pressure in the

gas phase at t�50 ns and consequently the back condensa-tion starts that gives rise to the negative velocity near thesurface. At the initial stage the pressure inside the ablationplume is much higher than the typical background pressureused in PLD �below 100 Pa�. Therefore, the backgroundgases in the typical pressure range cannot seriously influenceboth the gas dynamics and the evaporation of the target at theinitial stage.

A. Ablation into vacuum

An example of the ablation plume evolution in vacuumat the two-dimensional stage is shown in Fig. 4. The initialconditions for the 2D gas-dynamic equations correspond tothe one-dimensional fields at t�50 ns shown in Fig. 3. Thehigh initial pressure of the vapor �about 50 atm� acceleratesits axial expansion and gives rise to its radial expansion �seeFig. 4�a��: a rarefaction front propagates through the vapor inthe radial direction from the laser spot boundary R�R0 to-wards the center R�0. The vapor flow remains one dimen-sional with UR�0 ahead of the front. Behind it the radialvelocity increases while the density and the pressure drop.

For the first 50 ns the 1D equations are used and theradial expansion is therefore suppressed, which introduces anerror into the calculations. To estimate it, a fraction of thevapor involved into radial expansion could be evaluated. Thevelocity of the rarefaction front is equal to the sound velocityC , which is about 1860 m/s in the carbon vapor at 3000 K.Thus, the front displacement for 50 ns is of the order of�R�100 �m. The thickness of a zone with pronounced ra-dial motion is about �R while the dimension of the regionoccupied by the vapor is about the size of the laser spot.Therefore the relative error introduced during the initial 1Dstage is estimated as �R/R0 .

The rarefaction front attains the axis R�0 at t�500 nsthat is shown in Fig. 4�b� and the region of 1D flow disap-pears at this moment. The expansion leads to vapor cooling�see Figs. 4�c� and 4�d��. At t�1 �s the vapor temperaturebecomes lower than the ambient one. Back condensation hasno considerable effect since the condensation speed is equalto the sound velocity, which is low because of the low tem-perature. The pressure P falls faster than the dynamic pres-sure U2. After t�1 – 2 �s the pressure forces become neg-ligible and the vapor motion is nearly inertial. The densityand velocity fields may be approximated by a self-similarsolution of Eqs. �16�, �18�, and �19� with P�0:

UZ�Z/t , UR�R/t ,�27�

N��R ,Z ,t �� t0 /t �3N�� t0 /t �R ,� t0 /t �Z ,t0�,

where N�(R ,Z ,t0) is the vapor density field at any instant t�t0 beyond 1–2 �s. The gas-dynamic fields at 1 and 2 �sare of different scales but have the similar shapes in accor-dance with Eqs. �27�. Further vapor expansion follows thesimilarity laws, Eqs. �27�.

The energy and angular distributions of the vapor atomsare shown in Fig. 5. They change with the vapor expansionfollowing the evolution of gas-dynamic fields. The fractionof the fast carbon atoms with energies about 3–6 eV in-creases until t�500 ns. This corresponds to the vapor accel-

FIG. 2. Evolution of heat processes in the target: �a� graphite �25 ns FWHM4 J/cm2 fluence laser pulse�; �b� Au (30 ns 3 J/cm2 fluence laser pulse�.Curve 1, surface temperature, Ts ; curve 2, evaporated thickness, De ; curve3, melt thickness, Zsl .

FIG. 3. Gas-dynamic fields at the initial 1D stage of laser ablation �graphitetarget; 25 ns FWHM 4 J/cm2 laser pulse�: vapor number density, N� ; gas-dynamic velocity, UZ ; pressure, P; temperature T . Dashed line curve, t�20 ns; solid line curve, t�50 ns. The velocity and temperature curves for20 ns are truncated 100 �m from the surface where the vapor density isnegligibly low.


eration in the axial direction because of its high pressure.Starting from this moment the energy distribution remainsalmost unchanged.

The width of the angular distribution decreases until t�500 ns but then increases; after t�1 �s it remains practi-cally the same. This nonmonotonous dependence may bequalitatively explained by the evolution of the gas-dynamicparameters. The atom velocity is the sum of the gas-dynamicvelocity and the thermal one. Therefore, the width of theangular distribution must increase with both the temperatureT and radial velocity component UR . At t�500 ns not allthe vapor is yet involved in the radial expansion and the

behavior of the angular distribution is determined generallyby T: its width decreases since the temperature falls. Att�500 ns the temperature is so low that the angular distribu-tion depends mainly on UR and becomes broader since UR

increases.Both the energy and angular distributions are frozen af-

ter t�1 �s when the gas flow is nearly self-similar. This isbecause the temperature is low and the thermal velocity isnegligible as compared to the gas-dynamic velocity. There-fore, the integral velocity distribution Eq. �24�, may be ex-pressed as

FIG. 4. Gas-dynamic fields at the final 2D stage of laser ablation in vacuum �graphite target; 25 ns FWHM 4 J/cm2 laser pulse; 0.5 mm laser spot radius�:vapor number density, N� ; axial velocity component, UZ ; radial velocity component, UR ; pressure, P; temperature, T . t� �a� 200 ns; �b� 500 ns; �c� 1 �s;�d� 2 �s.


f �v�� N��r��v�U„r…�dr. �28�

The distribution, Eq. �28�, with the self-similar density andvelocity fields, Eqs. �27�, does not depend on time t . Theenergy and angular distributions are derived from the veloc-ity distribution and, therefore, they also become steady-stateones.

The width of the steady-state angular distribution de-creases with the laser spot radius as shown in Fig. 6. Thus,the ablation plume is more focused if the laser spot radiusincreases. The angular distribution is determined by the com-petition between the axial and the radial expansions. Theaxial expansion starts at the early stage of evaporation and itis practically independent of the spot radius. The radial ex-

pansion starts later, at the 2D stage, and therefore it is moresensitive to the geometrical parameters such as the spot ra-dius. The rarefaction front separates the zones of purely axialflow in the center of the plume and 2D axial and radial flowat its periphery �see Fig. 4�a��. The larger the laser spot is thelonger the time required for the rarefaction to reach the cen-ter of the plume and to involve the entire vapor in the radialexpansion.

B. Ablation in a buffer gas

The ablated material with high pressure and velocitypushes the surrounding buffer gas and generates a shock. Theshock front propagates through the gas and separates the un-disturbed buffer gas from the gas compressed by the expand-

FIG. 4. �Continued�.


ing vapor. Thus, three flow regions are considered: the un-disturbed gas, the compressed gas, and the vapor. Thepressure of the compressed gas may be much higher than theundisturbed gas pressure and therefore may considerably af-fect the vapor expansion.

The peculiarities of the resulting gas-dynamic flow areillustrated in Fig. 7. The early stages of the vapor expansionin the buffer gas at t�500 ns do not significantly differ fromthe ones in vacuum because the vapor pressure remainsmuch higher than the compressed buffer gas pressure �com-pare the vapor parameters in Figs. 7�a� and 4�b��. At t�500 ns the vapor pressure becomes comparable with theexternal one and continues to drop. Therefore, starting fromthis time the vapor flow is influenced by the buffer gas �com-pare Fig. 7�b� with Fig. 4�c� and Fig. 7�c� with Fig. 4�d��.The ambient atmosphere limits the vapor expansion: the ab-lated material is stemmed by the gas and forms a thin layerwith elevated density near the entire gas–vapor contact sur-face.

The influence of the ambient pressure and the nature ofthe ambient gas on the vapor expansion is shown in Fig. 8.The vapor distribution in the buffer atmosphere significantlydiffers from the one in vacuum: the vapor fills the entirevolume of the ablation plume in vacuum �i.e., smooth den-

sity gradients� while it forms a thin shell at the plume bound-ary �sharp density gradients� in the presence of the ambientgas. At t�5 �s the volume occupied by the vapor and theexpansion velocity as well are reducing with both the gaspressure and its molecular mass.

Figure 9 shows the energy-angular distributions of thecarbon atoms ablated in the He atmosphere of various pres-sures and in vacuum at t�5 �s. In vacuum the distributionsformed at this instant are practically independent of time. Inthe buffer atmosphere, pressure forces do not fall to zero andthe distributions continue to vary slowly. The interaction ofthe plume with the ambient gas reduces the maximum kineticenergy of the ablated atoms and the width of their angulardistribution reduces as well. The vapor–gas interaction isrelatively more marked in the lateral direction than in theradial one that leads to preferential suppression of the radialexpansion. Note that the narrowing of the angular distribu-tion in the buffer atmosphere is a qualitative 2D effect. Thecorresponding 1D model predicts the increase of its widthsince the axial velocity component of the ablated material isreduced by the gas and at the same time the lateral compo-nent remains unchanged.

The energy distribution of the ablated material invacuum may be fitted to the Boltzmann distribution F(� ,�)�(�)1/2 exp��/kTeff(�)� with the effective temperatureTeff(�) used as an approximation parameter. This fact is wellknown and confirmed by the experiments, for example, inRefs. 28 and 29. The present simulation shows that thebuffer gas seriously influences the shape of the energy dis-tribution: the distribution along the normal to the surface �at��0) changes from the Boltzmann one to the nearly trian-gular distribution at higher ambient pressure �see Fig. 9�. Indirections different from the normal, a more complicatedshape of the energy distribution, a bimodal one, can befound. The relatively sharp high-energy edge of the distribu-tion is formed because the most energetic vapor atoms pen-etrate into the forward region of the ablation plume wherethey are slowed down by the buffer gas. So, the high-energytail of the distribution is cut at the energy corresponding tothe contact surface velocity.

Analysis of the results for various ambient gases �seeFigs. 10 and 11� shows that an impact of the gas molecularmass on the energy distribution is qualitatively the same asthe pressure impact: the maximum energy decreases with themolecular mass. The behavior of the angular distribution de-pends on the target material: as shown in Fig. 10�b�, theangular distribution of the ablated carbon atoms becomesbroader with the molecular mass of the buffer gas. On thecontrary, in the case of the Au target the width of the vaporatoms angular distribution decreases with the molecular massof the buffer gas �see Fig. 11�b��. These opposite tendenciesare probably caused by the large difference between theatomic mass of C �12 a.m.u.� and Au �197 a.m.u.�. One maynote that the analogy with the underexpanded jet13 predictsunconditional narrowing of the angular distribution with thebuffer gas molecular mass.

FIG. 5. Evolution of the ablated material energy and angular distributions invacuum �graphite target; 25 ns FWHM 4 J/cm2 laser pulse; 0.5 mm laserspot radius�: �a� energy distribution along the normal to the target surface,F(� ,0); �b� angular distribution, F�(�).

FIG. 6. Influence of the laser spot radius R0 on the angular distribution ofablated material F�(�) in vacuum �graphite target; 25 ns FWHM 4 J/cm2

laser pulse; t�5 �s).


C. Comparison with the experiments

The above simulation results are obtained under the as-sumption that the ablation plume consists of single atomsonly. However it may in fact contain compound species suchas molecules,30 clusters,31 and nanoparticles.32,33 In the caseof gold ablation considered here, the only neutral speciesdetected by mass spectroscopy was monatomic gold.28 Theablation plume formed by the interaction of a 4 J/cm2 laserpulse with a graphite target which is studied in this work inreality contains a certain amount of carbon molecules as in-dicated by the optical emission.1 There is no quantitativeexperimental information on the plume composition in thiscase. The ratio of C:C2:C3�56:35:100 �Ref. 30� was ob-tained at much lower laser fluence, below 1 J/cm2, and thegraphite surface temperature below or slightly above thegraphite melting point. Such a high fraction of the molecules

was explained by the mechanism of the cluster emissionfrom the edges of graphite sheets. In our simulation of graph-ite ablation, the maximum surface temperature is estimatedto be about 8000 K �see Fig. 2�a��. We do not expect thiscluster emission mechanism to take place because the graph-ite structure hardly remains at such high temperature. Weexpect, therefore, that most of the ablated species are singlecarbon atoms.

The possible clustering in the gas phase would beginwith dimers. It is well known that dimer formation requires athree-body collision,34 which is rather a rare event. The prob-ability that an atom is subjected to a three-body collisionduring time �t may be estimated as p3��5N2v�t where �is the atom diameter, N the atom number density, and v theaverage thermal velocity. According to our simulation re-sults, the characteristic values for the graphite ablation are

FIG. 7. Laser ablation of graphite in 100 Pa He buffer atmosphere �25 ns FWHM 4 J/cm2 laser pulse; 0.5 mm laser spot radius�. Gas-dynamic parameters:vapor �carbon� number density, N� ; ambient gas �helium� number density, Ng ; axial velocity component, UZ ; radial velocity component, UR ; pressure, P;temperature, T . t� �a� 500 ns; �b� 1 �s; �c� 2 �s. The small dents that appear at the sharp ridges are the result of low resolution of the graphical representation.


�t�1 �s, N�1�1018 cm�3, v�2000 m/s, and ��2 Å.This gives p3�6.4�10�4�1. Similar estimations for thegold ablation give p3�1 as well. Thus, the cluster formationin the Au and carbon ablation plumes at a �s time scale isnegligible.

The simulation results are compared with the experimenton optical emission of ejected material under laser ablationof graphite.1 In Fig. 12 the calculated maximum distance�along the Z axis� attained by the contact surface is com-pared with the experimental maximum distance attained bythe radiating carbon species. Both the model and the experi-ment reveal that the ambient gas slows down expansion ofthe ablation plume. The comparison with the experimentalresults clearly indicates the presence of the fast carbon spe-cies that are not taken into account by the gas-dynamicmodel. They penetrate into the buffer atmosphere deeperthan predicted by the model. However, the difference be-tween the experiment and the model decreases with the am-bient pressure. It may therefore be deduced that the fast spe-cies density is not significant in influencing the gas-dynamicflow as a whole.

The flux-weighted energy distribution along the normalto the surface, P(�)�(�)1/2F(� ,0), calculated in the frame-work of the gas-dynamic model is compared in Fig. 13 withthe ones derived from the time-of-flight �TOF�measurements.28 Both the gas-dynamic model and the ex-periment exhibit a rise of the ablated atoms’ mean energywith laser fluence. The calculated and experimental distribu-tions have similar shape but correspond to significantly dif-

FIG. 8. Laser ablation of graphite in vacuum and the noble gas atmosphere�25 ns FWHM 4 J/cm2 laser pulse; 0.5 mm laser spot radius�. Influence ofthe gas type and the ambient pressure, Pa �indicated�, on the vapor densitydistribution, N� , formed at the end of the gas-dynamic stage, t�5 �s. Thesmall dents that one can see for 10 Pa helium are the result of low resolutionof the graphical representation.

FIG. 9. Buffer gas pressure influence on the �a� energy-angular and �b�angular distributions of the ablated carbon atoms at the end of the gas-dynamic stage �graphite target; He ambient atmosphere; 25 ns FWHM4 J/cm2 laser pulse; 0.5 mm laser spot radius; t�5 �s). The angular distri-bution corresponding to 10 Pa pressure practically coincides with the 100 Padistribution and therefore it is not shown.

FIG. 10. Buffer gas molecular mass influence on the �a� energy-angular and�b� angular distributions of the ablated carbon atoms at the end of the gas-dynamic stage, t�5 �s �graphite target; 10 Pa ambient pressure; 25 nsFWHM 4 J/cm2 laser pulse; 0.5 mm laser spot radius�. The angular distri-bution in Ar practically coincides with the one in He and therefore it is notshown.


ferent values of laser flux. This is probably because themodel does not take into account the real complicated geom-etry of the target surface formed during the action of thou-sands of laser pulses used in the experiment.28 It is knownthat the amount and mean energy of the ablated materialconsiderably increase with an accumulated number of laserpulses.28 The accumulation effect may change the geometryand physical properties of the surface that leads to its over-heating in comparison to the virgin surface and the ablationthreshold decrease. This could be taken into account, prob-ably, by introducing the effective properties of a damagedsurface.

In Fig. 14 the flux-weighted velocity distribution alongthe normal to the surface, F�(v)�v2F�(m�v2)/2,0� , is com-pared with the experiment.29 There is good agreement in themean velocity of the ablated Au atoms. The calculated dis-

tribution is, however, narrower than the experimental one.The difference may be caused by a nonuniform energy fluxdistribution over the laser spot that is not taken into accountby the model. The high-energy tail of the experimental dis-tribution is formed probably because of the additional plumeheating by the incident laser beam. This effect is also disre-garded in the model.

It is known that the ablation plume is focused in thepresence of the ambient gas. The narrowing of the ablatedspecies angular distribution with the ambient pressure wasobserved, for example, for laser ablation of the ceramictargets.35 The simulation results shown in Fig. 9 qualitativelyagree with this experimental fact.

The emission rates of atoms and molecular species fromthe irradiated surface and their energy-angular distributionsdetermine the processes in the gas phase. The investigationsof the near-threshold laser ablation of gold28 and graphite30

under 248 nm laser radiation revealed pronounced nonther-mal effects. The mechanism of nonthermal evaporation isstill not clear, therefore the results can hardly be extrapolatedto another laser wavelength and to surface temperatures wellabove the melting point. The velocity distribution of goldatoms ablated by 532 nm laser pulses29 may, however, beregarded as consistent with the assumption of purely thermalevaporation: our gas-dynamic simulation �see Fig. 14� basedon the thermal evaporation model agrees with the above ex-perimental results. In this case, the energy spectrum modifi-cations �for example, the shift of its maximum to higherenergies� may be ascribed to collisions in the gas phase.

FIG. 12. Dynamics of the contact surface between the ablated material andthe buffer gas �25 ns, 1.06 �m laser pulse ablation of graphite in He atmo-sphere; 0.5 mm laser spot radius; 4 J/cm2 fluence�. Comparison of the gas-dynamic simulation �solid lines� with the experiment of Ref. 1 �symbols�.

FIG. 13. Flux-weighted energy distribution of ablated Au atoms along thenormal to the target surface, P(�) �Au target; 30 ns, 248 nm laser pulse with0.5 mm spot radius; vacuum; t�5 �s): the gas-dynamic simulation �lines�and the experiment of Ref. 28 �symbols�.

FIG. 14. Flux-weighted velocity distribution of ablated Au atoms along thenormal to the target surface, F�(v) �Au target; 11 ns FWHM 532 nm laserpulse with 25 �m spot radius; 4.9 J/cm2 fluence; vacuum; t�500 ns): thegas-dynamic simulation �solid line� and the experiment of Ref. 29 �sym-bols�.

FIG. 11. Buffer gas molecular mass influence on the �a� energy-angularand �b� angular distributions of the ablated Au atoms at the end of thegas-dynamic stage, t�5 �s �Au target; 10 Pa ambient pressure; 30 ns,3 J/cm2 laser pulse; 0.5 mm laser spot radius�.


IV. CONCLUSION

The vapor formed by laser ablation compresses the sur-rounding buffer gas where the shock wave is formed. Theablated material expansion is slowed down because of itsinteraction with the ambient gas. The vapor/gas interaction ismore pronounced in the radial direction that results in thepreferential suppression of the radial expansion and focusingof the ablation plume. Another effect of ambient atmosphereis an essential vapor redistribution inside the plume: the ab-lated material accumulates near the contact surface with thebuffer gas and tends to form a thin shell. The buffer gaseswith typical PLD pressures below �100 Pa cannot seriouslyinfluence the kinetics of target evaporation. The cluster for-mation in the Au and carbon ablation plumes at a �s timescale is negligible.

The model predicts the decrease of the kinetic energy ofablated atoms with both the ambient pressure and the mo-lecular mass of the buffer gas. The width of their angulardistribution reduces with the ambient pressure but may in-crease with the molecular mass of the buffer gas. The shapeof the energy distribution is close to the Boltzmann one invacuum but it changes considerably at higher ambient pres-sure or higher molecular mass of the buffer gas. The angulardistribution becomes narrower with an increase of the laserspot radius.

The comparison of the ablation plume expansion, pre-dicted by the model, with the experiment on opticalemission1 clearly indicates that there are fast ablated speciesthat are not taken into account by the model. However, theirdensity seems to be low and does not influence the gas-dynamic flow considerably. The energy distribution of theablated material given by the gas-dynamic model has thesame shape as the distributions derived from the TOFmeasurements28 but quantitative agreement is attained at dif-ferent values of laser fluence. This discrepancy may becaused by the accumulation effect �i.e., the dependence ofthe ablation rate on the number of laser pulses� that changesthe geometry and properties of the irradiated surface. Boththe hydrodynamic model and experiments exhibit a rise ofthe ablated atoms’ mean energy with laser fluence. The meanvelocity of the ablated Au atoms predicted by the model isconfirmed by the experiment.29

One of the possible reasons to explain the differencebetween the theoretical and experimental results on the abla-tion of graphite1 and gold28 could be the contribution of non-thermal mechanisms of material removal that are completelyoutside the framework of the present model. On the otherhand, gold ablation by 532 nm laser pulses29 is well de-scribed by the proposed thermal model.

ACKNOWLEDGMENTS

The authors would like to thank Dr. V. Yu. Fominskiiand Professor V. I. Mazhukin for fruitful discussions.

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Gas dynamics of laser ablation: Inï¬‚uence of ambient atmosphere

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