Adam S Foster

Clemens Barth

Claude R Henry

Chemical Identification of Ions in Doped NaCl by Scanning Force Microscopy

Keywords: numbers: 68, 37, Ps, 68, 35, Dv

A quantitative comparison between experiment and theory is presented, which shows that all ions of the Suzuki structure on (001) surfaces of Mg 2þ or Cd 2þ doped NaCl crystals can be identified despite the tipsurface distance, differences in impurity chemistry, and surface termination. The identification can be used to calibrate the potential of the tip's last atom, and it is proposed to use these surfaces for better characterization of deposited nano-objects.

Alkali halides have remained of intense scientific interest, due to their role in, for example, studies of molecular electronics [1][2][3][4][5] and cluster science [6][7][8], where they serve as insulating substrates. For the characterization of these surfaces, the dynamic scanning force microscope (SFM) [START_REF]Noncontact Atomic Force Microscopy[END_REF] is increasingly becoming the tool of choice due to its universal applicability. A key ambition in studies is to obtain atomic resolution on both the substrate and adsorbed nano-object, providing a complete characterization of the system. Since the substrate geometry is generally known, interpretation of image contrast of the substrate should greatly aid in understanding the nanoobject configuration. For ionic surfaces, atomic resolution imaging is mainly determined by the sign of the electrostatic potential from the tip apex [START_REF] Hofer | [END_REF], and for specific systems the contrast pattern observed directly identifies the character of the tip and ions on the surface. However, for the symmetric (001) surfaces of cubic crystals, images with tips of opposite potential are almost identical and cannot be distinguished [11,12]. In most cases, force spectroscopy at different atomic sites on the surface is needed for the identification of the surface ions [13,14], which demands a time-consuming comparison between experiment and theory-direct interpretation from conventional experimental images would certainly be strongly preferred.

It is well known that doping NaCl crystals with divalent impurity cations results in the precipitation of the so-called Suzuki phase [15,16]. Recent dynamic SFM studies of NaCl:Mg 2þ confirmed that Suzuki precipitates are embedded in the NaCl matrix on the (001) surface [17]. Two different types of surfaces regions can be found, which are well separated [17,18] [Fig. 1(a)]. The atomic unit cell of the Suzuki phase is twice as large as the one of NaCl(001) and is composed of three different ions including Na þ vacancies [Fig. 1(b)]. Because of the special geometry of the Suzuki structure, each type of ion can be resolved on the surface by dynamic SFM [17]. In this Letter, we show by comparison between experiment and theory that chemical identification can be provided independently of the tip-surface distance and chemistry of the tip. By further comparison to the Suzuki phase of the NaCl:Cd 2þ (001) surface, we show that this identification is consistent for impurities with dissimilar chemistry and different surface terminations.

Czochralski-grown NaCl crystals were doped with 1 mol % MgCl 2 or 2 mol % CdCl 2 [18]. The surfaces were prepared in UHV by cleavage at room temperature and annealing at 220 C. Experiments were performed at 3 Â 10 À10 mbar, at room temperature with a dynamic SFM with a conductive silicon cantilever (p-Si, 0:015 cm, 315 kHz resonance frequency, 37 N=m spring constant, 15 nm peak-to-peak amplitude). All images were acquired in the constant height mode and represent the detuning Áf of the cantilever resonance frequency. Dark contrast belongs to more negative detuning values (larger tip-surface attraction) [17].

Calculation of the tip-surface interaction and simulated images were performed via the general approach outlined in previous works [START_REF] Hofer | [END_REF], combining two-dimensional periodic atomistic simulations [19] and macroscopic van der Waals to provide the total force field across the surface. In simulated images, experimental values for cantilever parameters were used, and the tip radii (NaCl:Mg 2þ $20 nm and NaCl:Cd 2þ $100 nm) were determined via In order to fit the remaining potentials and to check the accuracy of the atomistic simulations in general, calculations of the structures have also been performed using the spin-polarized density functional theory, projected augmented wave potentials (including semicore states for Na, Mg, and Cd), and the Perdew-Burke-Ernzerhof functional [21]. A kinetic energy cutoff of 300 eV (350 eV for NaCl:Cd 2þ ) and a k-point sampling of 3 Â 3 Â 3 (3 Â 3 Â 1 for slab calculations) were found to converge the total energy of our systems to within meV. Surface relaxations were well converged for a slab four layers deep with a vacuum of 1.2 nm. Our calculated structures agreed well with previous experimental and theoretical studies [22,23]. The agreement between ab initio and atomistic results for the Suzuki bulk and surface equilibrium structures was excellent in all cases. However, since the contrast in SFM imaging has been previously shown to strongly de-pend on tip-induced displacements [START_REF] Hofer | [END_REF], we also calculated the energy vs displacement curve for atoms of each sublattice of every surface in both approaches. The potential parameters were then tuned to reproduce the ab initio displacement curves, while maintaining agreement in basic structures.

In Fig. 2 Simulations were best able to reproduce this contrast pattern and its dependence on tip-surface distance by using a Cl-terminated NaCl tip [24]. The overall quantitative agreement confirms the surface structure shown in Fig. 1(a). The main component in the distance dependence of the images is the increasing attraction between the tip and the Mg 2þ sites in the surface. At long range, greater than 0.4 nm, the large displacement of Na þ ion towards the tip [Fig. 3 In comparison to pure NaCl(001), the development of contrast as a function of tip-surface separation reflects a much richer variety of atomic processes. The presence of the vacancy sublattice plays an especially important role in increasing the possibility and magnitude of atomic displacements. Note that the limit of approach to the surface was the onset of instabilities due to atomic jumps, and contrast inversion was never observed.

Figure 4 shows the contrast development of experimental images and scan lines of the NaCl:Cd 2þ Suzuki surface as the detuning is increased and the tip approaches the surface. At largest distance, the contrast is formed by large dark spots, which form a sublattice equal to the one formed by either the Cd 2þ ions or the Na þ vacancies and do not change with decreasing distance. At closest distance, four additional bright spots appear around each dark spot forming a sublattice of either Cl À or Na þ ions. Additional dark spots appear more pronounced in between the large dark spots.

Initially, we considered that the NaCl:Cd 2þ Suzuki surface termination includes both Cd 2þ ions and Na þ vacancies, as in the case of NaCl:Mg 2þ . However, simulations were unable to reproduce the observed contrast with any tip, including tips based on the Suzuki structure itself [Fig. 3(g) and also see Ref. [24]]. Note also that the contrast differs considerably from that obtained on NaCl:Mg 2þ (001). In order to understand this, we analyzed in detail the possible surface terminations of both NaCl:Mg 2þ and NaCl:Cd 2þ by varying the vacancy and impurity location in surface and subsurface layers while maintaining the size of the Suzuki unit cell. For NaCl:Mg 2þ , we find that the lowest surface energy is for the Suzuki termination (0:18 J=m 2 ; NaCl termination is 0:22 J=m 2 ), as shown in Fig. 1(a). However, for NaCl:Cd 2þ the surface energy for the Suzuki termination (0:29 J=m 2 ) is higher than the minimum energy configuration with the uppermost layer consisting of a pure NaCl monolayer and the vacancies/impurities in the subsurface layer (0:21 J=m 2 ). This surface termination is shown in Fig. 1(b). Using the latter surface in SFM simulations, we were able to best reproduce the experimental images and scan lines with a Na-terminated NaCl tip, demonstrating that, in this case, the NaCl:Cd 2þ Suzuki region most likely has a NaCl termination.

The development of contrast as the tip approaches the surface (see Fig. 4) can again be understood by considering the role of tip-induced atomic displacements. In the absence of the tip, Cl À ions over vacancies displace strongly outwards from the surface [0.04 nm; see Fig. 1(c)], while those over Cd 2þ remain close to the ideal surface plane. This results in the strong difference in contrast over neighboring Cl À ions seen at long range and is strongly enhanced due to large tip-induced displacements [see Fig. 3(d)]. The change in the profiles shown in Fig. 4(b) is due to two components: the reduction in repulsion over Na þ ions due to displacements of neighboring Cl À ions towards the tip at close approach [see Fig. 3(e)] and the increased attraction to the rigidly held Cl À ion over a Cd 2þ ion. The increase in the contribution of the Cl À over Na þ ion as the tip approaches the surface, as shown in Fig. 4(c), is also due to the displacement of the Cl À ion towards the tip [see Fig. 3(f)]. The quantitative agreement between experiment and simulations highlights the ability of SFM to resolve the influence of impurities and defects, even subsurface, through its sensitivity to atomic displacements.

Our results show that all ions in the Suzuki structure can be identified by dynamic SFM imaging, thereby characterizing the tip's potential. This identification is possible due to the doping of NaCl with divalent impurity ions and thanks to the geometry of the atomic Suzuki structure, which contrasts the situation on pure NaCl [11]. It is shown that this can be done independently of the tip-surface distance, of the chemical nature of the divalent impurity ions, and despite differences in surface termination. Simulations show that the development of contrast patterns as a function of tip-surface separation on both surfaces is strongly linked to tip-induced atomic displacements, which is significantly different to the contrast formation on other cubic ionic surfaces.

The ready access to interpretation of atomically resolved images makes Suzuki surfaces an attractive choice for investigations of deposited molecules and nanoparticles. The chemical identification makes it much easier to characterize deposited nano-objects and reaction products at the atomic scale. An additional benefit is the ability to nanotemplate the surface with Suzuki regions of different electronic and reactive character. Several other Suzuki systems in NaCl exist, including the magnetic impurities Mn 2þ and Fe 2þ [25], which our calculations predict would form an antiferromagnetic ordering particularly suited to exchange force microscopy [26]. Other types of host system for the Suzuki phases such as oxides (MgO:Mn 4þ [15] and NiO:Mn 4þ [27]) might also be interesting in nanocatalysis. We believe that doping of insulators, in combination with high resolution dynamic SFM, provides a general tool for characterizing surfaces at the atomic scale. The wide use of doping with impurities of different valences in many insulators such as ZrO 2 and CeO 2 [START_REF]Catalysis by Ceria and Related Materials[END_REF][START_REF] Ou | [END_REF] suggests that the method could also be used in high resolution studies of their surfaces [30].

FIG. 1 ( 1 Ó

 11 FIG. 1 (color online). (a) Topography image of Suzuki precipitates on the NaCl:Cd 2þ (001) surface. Atomic structure of the (b) Suzuki terminated NaCl:Mg 2þ surface and (c) NaClterminated NaCl:Cd 2þ surface. The shaded and unshaded boxes indicate surface and subsurface vacancy sites, respectively. The dashed line shows the plane of the side view.

  (a), three experimental images acquired inside a Suzuki precipitate are shown, along with corresponding profiles along the h100i and h110i surface direction [Figs. 2(b) and 2(c)]. The contrast pattern changes from an array of dark spots to a second, in-between, additional array of dark spots. At closest distance, the second type of dark spots is most pronounced. The size of the array formed by the latter dark spots equals the size of the sublattice formed by either the Mg 2þ ions or Na þ vacancies. Around each of the latter dark spots, four bright spots can be seen which match the sublattice formed by the Cl À ions.

  FIG. 2 (color online). Comparison of experimental and simulated data for NaCl:Mg 2þ . The simulations used a Cl-terminated NaCl tip. (a) Images (color: experiment, gray scale: theory), (b) scan lines (red is simulated) in the h100i direction, and (c) scan lines in the h110i direction. The experimental data correspond to an average detuning of À16, À22, and À24 Hz (top-bottom, with a more negative detuning corresponding to imaging closer to the surface). The simulated data correspond to tip-surface heights of 0.45, 0.325, and 0.275 nm, resulting in average detunings of À16:4, À21:8, and À23:0 Hz, respectively.

FIG. 3 (

 3 FIG. 3 (color online). Simulation snapshots showing key atomic displacements for a Cl-terminated NaCl tip imagingNaCl:Mg 2þ at a tip height of (a) 0.4 nm over an Na þ site, (b) 0.4 nm over an Mg site, (c) 0.26 nm over a Cl À site, and for a Na-terminated NaCl tip imaging NaCl:Cd 2þ at a tip height of (d) 0.46 nm over a Cl À site above a vacancy, (e) 0.31 nm over an Na þ site, and (f) 0.4 nm over a Cl À site above a Na þ . (g) Comparison of experimental images of NaCl:Mg 2þ and NaCl:Cd 2þ with simulated images of NaCl:Cd 2þ using seven different tips X:Y (gray scale), where Y is the tip material and X is the terminating atom.

We thank A. L. Shluger for useful discussions. A. S. F acknowledges support from the Academy of Finland, the CSC, and ESF FANAS.


M E Povarnitsyn

T E Itina

M Sentis

K V Khishchenko

P R Levashov

Material decomposition mechanisms in femtosecond laser interactions with metals

Keywords: numbers: 61.80.Az, 79.20.Ds, 64.70.Dv, 64.70.Fx

A numerical hydrodynamic study of femtosecond laser ablation is presented. A detailed analysis of material decomposition is performed using a thermodynamically complete equation of state with separate stable and metastable phase states and phase boundaries. The lifetime of the metastable liquid state is estimated based on the classical theory of homogeneous nucleation. In addition, mechanical fragmentation of the target material is controlled based on available criteria. As a result, several ablation mechanisms are observed. A major fraction of the ablated material, however, is found to originate from the metastable liquid region, which is decomposed either thermally in the vicinity of the critical point into a liquid-gas mixture, or mechanically at high strain rate and negative pressure into liquid droplets and chunks. The calculation results explain available experimental findings.

I. INTRODUCTION

Femtosecond laser interactions have attracted a particular interest of researchers during the last decade. It is not surprising since these new ultra-short laser pulses provide a unique opportunity for the fast development of numerous very promising and exciting applications, such as laser machining, micro and nano-structuring, laser-induced plasma spectroscopy, nanoparticle synthesis in vacuum, in gas or in a liquid solution, medical imaging, laser surgery, etc. [START_REF]Short Pulse Laser Interactions with Matter: An Introduction[END_REF][START_REF]Femtosecond Laser Spectroscopy[END_REF][START_REF]Femtosecond Technology for Technical and Medical Applications[END_REF] Many experimental studies were performed demonstrating the unique possibilities of femtosecond laser systems. [START_REF] Nolte | [END_REF]5,6] At the same time, several theoretical investigations were focused on the better understanding of physical processes involved in ultra-short laser interactions. [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] The classical approach typically used for these interactions is based on a two-temperature model. [13] In addition, several hydrodynamic simulations were carried out to describe the target material motion. [10,11,14,15,16] The formation of a thermal wave, pressure gain and shock wave propagation have been observed in these calculations. Molecular dynamics (MD) simulations were furthermore performed that provided insights into the mechanisms, such as phase explosion, fragmentation, evaporation, and mechanical spallation. [17,18,19,20,21] In spite of a significant effort aimed at the understanding of the ablation mechanisms, several issues stay unclear. For example, theoretically predicted ablation depth was frequently underestimated with respect to the typical experimental values. This result is an indication that not all the ablation mechanisms were taken into account. In addition, numerous terms used to describe these mechanisms are confusing or misused. For instance, when explosive boiling term is used, which reflects the overheated (metastable) liquid decomposition into a liquid-gas mixture, it is often associated with material heating. The metastable liquid state, however, can be also achieved during fast material expansion and cooling. The proper term for this process is unclear. In this case, critical-point phase separation was proposed in Ref. 15, but only to describe material decomposition in the vicinity of the critical point.

In addition, up to now, it was difficult to handle this process correctly in the framework of hydrodynamic models, because most of the available equations of state (EOS) did not describe the metastable regions. Moreover no kinetics of gas bubble formation or/and condensation was previously introduced to determine the lifetime of these states. Besides, there is still a lack of information about the photomechanical failure of the material at extremely high strain rates (up to 10 9 s -1 ), which can be achieved. [7] Nevertheless, even these high strain rates do not always satisfy the fragmentation criteria [22,23] because of their short duration (tens of picosecond). All these issues need more analysis.

II. MODEL

To bring more light on the mechanisms of material decomposition in femtosecond laser ablation of metals, we performed a detailed hydrodynamic modeling. Remind, that compared to atomistic techniques, hydrodynamic models allow calculations for much larger systems and take much shorter computer time. Our numerical model is based on the solution of a system of Eulerian hydrodynamic equations by a high-order multi-material Godunov's method.

[24] The equations were extended to the case of one-fluid two-temperature hydrodynamics with laser energy absorption source, electron heat conductivity and energy exchange between electrons and heavy particles (atoms, ions, nuclei). [25] A Gaussian temporal profile is used to simulate the laser energy deposition. The heat conductivity of electrons is calculated according to model. [START_REF] Anisimov | Proc. SPIE Int. Soc. Opt. Eng[END_REF] Energy exchange between electrons and heavy particles, the reflectivity coefficient R and the optical penetration depth λ opt are derived by means of the wide-range frequency of electron-phonon/ion collisions. [14] For completeness of our model, we use a semiempirical multi-phase EOS for aluminum with separate description of subsystems of heavy particles and electrons. The specific Helmholtz free energy has a form F (ρ, T i , T e ) = F i (ρ, T i ) + F e (ρ, T e ), composed of two parts, which describe the contributions of heavy particles and electrons, respectively. Here ρ is the material density, T i and T e are temperatures of heavy particles and electrons. The

first item F i (ρ, T i ) = F c (ρ) + F a (ρ, T i ),
in turn, consists of the electron-ion interaction term F c (calculated at T i = T e = 0 K) and the contribution of thermal motion of heavy particles F a . Analytical form of F i has different expressions for the solid F (s) i , as well as for both liquid and gas phases F (l)

i . [START_REF] Khishchenko | Physics of Extreme States of Matter-2005[END_REF] Using these thermodynamic functions, the solid, liquid, and gas phases equlibrium boundaries are determined from the equality conditions for the temperature T i , pressure P i = ρ 2 (∂F i /∂ρ) T i , and Gibbs potential G i = F i + P i /ρ of each phases pairs. [START_REF] Landau | Statistical Physics[END_REF] The tables of thermodynamic parameters are calculated taking into consideration the information about phase transitions and metastable regions. [START_REF] Khishchenko | [END_REF]30] The free energy of electrons in metal F e has a finite-temperature ideal Fermi-gas form. [START_REF] Landau | Statistical Physics[END_REF] The model of our EOS contains about 40 adjustable constants so that the EOS meets the following requirements: (i) to describe experimental results on compression and expansion for a wide range of densities and temperatures including data on critical and triple points; (ii) to contain separate information about electron and ion/lattice subsystems; (iii) to represent changes of thermodynamic parameters during phase transitions. To account for kinetic processes, an estimation of the realistic lifetime of metastable liquid state is introduced.

Using this lifetime-based information we switch between two different modifications of the EOS: (i) with metastable states; and (ii) without metastable states.

In our model, when the liquid branch of the binodal curve was crossed and the matter turned into metastable state, we include a particular treatment for each of the following two competitive effects: (i) for the thermal decomposition, a criterion of the metastable liquid lifetime is used based on the theory of homogeneous nucleation; [START_REF] Frenkel | Kinetic Theory of Liquids[END_REF][START_REF] Skripov | Metastable Liquids[END_REF][START_REF] Tkachenko | [END_REF] (ii) for the mechanical fragmentation, a failure criterion of Grady is applied. [22] In the first case, we estimate the metastable liquid lifetime as τ nucl = (CnV ) -1 exp(W/k B T i ), where C = 10 10 s -1 is the kinetic coefficient, [START_REF] Tkachenko | [END_REF] n is the concentration, V is the volume under consideration (the volume of numerical cell in our case), W = 16πσ 3 /3∆P 2 is the work required on a gas bubble formation in a liquid phase, k B is the Boltzmann constant, ∆P is the difference between the saturated vapor pressure, known from EOS, and the pressure of substance. The temperature dependence of the surface tension σ is described in the form σ = σ 0 (1 -T i /T c ) 1.25 , see Ref. 33, 34. Here T c = 6595 K is the temperature in critical point (CP) for Al, [START_REF] Khishchenko | Physics of Extreme States of Matter-2005[END_REF] σ 0 = 860 g/s 2 is the surface tension of Al at melting temperature. [START_REF] Semenchenko | Surface Phenomena in Metals and Alloys[END_REF] In this case, as soon as the lifetime τ nucl in the volume V is expired, the metastable one-phase state separates into two-phase stable liquid-gas mixture. This phase separation process is accompanied by abrupt changes of thermodynamic parameters such as pressure, temperature, sound speed, compressibility, heat capacity, etc. The EOS with metastable phase states is therefore no more relevant in this volume, so that we continue to calculate the thermodynamic properties by using the EOS without metastable states.

To account for the second effect, a fragmentation criterion [22] is used for the liquid phase with the spall strength P spal = (6ρ 2 c 3 σ ε) 1/3 and the time to fracture τ spal = (6σ/ρ ε2 c 3 ) 1/3 , where ε is the strain rate and c is the sound speed. When the pressure of matter drops below the negative value -P spal and duration of this event is longer than the time τ spal , a criterion of fragmentation is satisfied and voids and new free surfaces can appear in the substance. To describe this phenomenon numerically we let a "broken" substance to shrink back until the pressure comes to zero value. The difference between old and new volumes we compensate by introducing void fraction into a numerical cell. Both thermal and mechanical criteria described above are used simultaneously and each of them can prevail in a given computational cell depending on the substance location on the phase diagram.

III. RESULTS

For the analysis of the target material evolution after the laser irradiation, it is convenient to use a phase diagram in the temperature-density plane (T, ρ) given by the EOS (Fig. 1).

The binodal curve corresponds to the liquid-vapor coexistence line (saturated vapor curve).

The spinodal curve (dashed) gives a limit of thermodynamic stability of the liquid and gas phases. The regions between the binodal and the spinodal curves show the metastable states of matter: the superheated liquid (l) and the supercooled vapor (g) states.

Initially, the aluminum target is in a solid state with ρ = 2.71 g/cm 3 , T i = T e = 300 K, and is subjected to a laser pulse with fluence F = 5 J/cm 2 , pulse width τ L = 100 fs and wavelength λ L = 800 nm. The reflectivity coefficient and the optical penetration depth are calculeted [14] and their values are R = 0.92 and λ opt = 13 nm, respectively. The conductionband electrons absorb laser energy in a skin-layer (a very thin region of the order of the optical penetration depth λ opt ). Then, these hot electrons transmit absorbed energy into the bulk owing to heat conduction and warm up the lattice due to electron-phonon collisions.

The time of electron-lattice relaxation (equilibration of temperatures of electron and heavy particle subsystems) is of the order of several picoseconds for aluminum and thus the lattice pressure has time to grow up before the heat energy of electrons dissipates into the bulk of the target. Formation of a narrow zone with a sharp pressure profile results in generation of intensive shock wave, which moves into the bulk, and rarefaction wave that causes a spread of the matter out of the target. The process of the shock wave propagation is clearly visible as the consecutive right-hand deviations of the 30, 50, 80, 130 nm-trajectories in Fig. 1.

Process of compression of different target layers changes into expansion one (decreasing of

the density along trajectories, Fig. 1).

The maximal temperature of heavy particles in the vicinity of the target surface (depth ≤ 15 nm) is high enough (up to T i ∼ 30 kK), so that the phase trajectories from this layer go above the CP and the target material is directly transformed into the gas phase (atomization, 5 and 15 nm trajectories in Fig. 1). Then, these trajectories cross the gas branch of the binodal and penetrate into the supercooled vapor region, where condensation starts and liquid-gas mixture forms. Note, however, that this layer represents a very small fraction of the ablated material, so that the condensation degree is insignificant.

Next layer in the target (depth from 20 to 30 nm) is first transformed into a metastable liquid state. This layer is heated to high temperatures (up to T i ∼ 25 kK), so that the corresponding phase trajectories cross the liquid branch of binodal in the vicinity of the CP and enter the metastable liquid region. Under these conditions, the target material is thermodynamically unstable. The lifetime of this state is estimated as described above.

Thermodynamic instabilities are known to occur near the CP (particularly, at 0.9T c < T i < T c ) leading to a rapid decomposition (several picoseconds) of the matter into a liquidgas mixture. This process is similar to the phase explosion, [17,18] though occurs during simultaneous material expansion and cooling. The other term used for this mechanism is critical-point phase separation proposed in Ref. 15.

This thermal mechanism, however, concerns only quite a thin slice (∼ 10 nm thickness), which trajectories enter the metastable liquid region near the critical point. In fact, the melted layer is much thicker (∼ 200 nm at F = 5 J/cm 2 ). The rest of the trajectories (5-7

in Fig. 1), which originate from the deeper lying melted layers of the target, also enter the metastable liquid region, but at temperatures much smaller than the critical temperature T c .

In this case, the matter can stay in the metastable state much longer. As a result, mechanical effects start to dominate over the thermal ones and material fragmentation (cavitation in liquid) occurs under the action of a tensile pressure wave. In other words, the time to mechanical fracture is shorter than the nucleation time (τ nucl < τ spal ). As soon as the time to fracture is exceeded, the material is decomposed into droplets and chunks (separated by vacuum). This mechanical decomposition, described also in Ref. 17, 20, 21, concerns the major part of the ablated material in our simulations.

The dynamics of the target surface layers decomposition is presented in Fig. 2. Initially, the solid-vacuum interface is located at x = 0 nm and target occupies x ≥ 0 nm half space. During the collisions of heated electrons with the lattice its temperature and pressure grows, and thus shock wave forms and propagates into the target (pressure peaks in Fig. 2).

Pressure release on the free surface of the target results in material expansion in the opposite direction through a rarefaction wave. At a delay t ∼ 10 ps after the beginning of the laser pulse, the density profile is still continuous [Fig. 2 The fractions of the target material ablated due to the described mechanisms depend both on the material and on laser parameters. At very small fluences (F < 0.25 J/cm 2 for aluminum) only melting occurs. When laser fluence is slightly above this value, material fragmentation takes place, in agreement with the previous MD simulations. [12,20] If we further increase laser fluence, both mechanical and thermal mechanisms play a role. At larger F , all three mechanisms (i) atomization; (ii) thermal decomposition (critical-point phase separation); and (iii) mechanical decomposition (dynamic fragmentation) occur in different layers of the target. The third mechanism, however, is found to be dominant in our simulations, accounting about 80% of ablated mass at all considered laser fluences (from 0.1 to 5 J/cm 2 ).

Finally, we propose several predictions from our simulations and compare our results

with the available experimental findings [5,7] and simulations. [11,15] We use a single-shot experimental data for the numerical model verification, though oxidation of aluminum target surface and usage of different methods of measurement result in essential discrepancy in single-shot laboratory findings, see Fig. 3. It turns out, that the consideration of only criticalpoint phase separation effect in simulations [15] gives rise to a significant underestimation of the ablation depth (several times) with respect to the experimental values (Fig. 3). This underestimation can be explained by the fact that only 10-20% of the target material is ablated due to these critical effects, whereas most of the ablated material is ejected due to the mechanical fragmentation of the liquid phase. These results confirm that the decomposed melted zone contributes strongly into the estimation of the ablated depth.

IV. CONCLUSION

In conclusion, we have presented the results of the hydrodynamic calculations of femtosecond laser ablation with a thermodynamically complete equation of state and with interplay between both kinetic lifetime of metastable liquid state and mechanical time to fracture.

Ablation mechanisms have been investigated as a function of laser fluence. A correlation between the melted and ablated depths has been revealed. Mechanical decomposition of a metastable liquid layer has been shown to be the dominant ablation mechanism. In the present work we have considered only aluminum ablation in one dimension. Similar analysis can be performed for different materials. In addition, an extension of the model to a two-dimensional case with multi-layer materials is underway.

FIG. 1 :

 1 FIG.1: Phase diagram of aluminum and time evolution of density and temperature of heavy particles for different layers of the target. Dashed curve: the spinodal; g: stable gas; l: stable liquid; s: stable solid; l+s: stable melting; g+l: liquid-gas mixture; g+s: solid-gas mixture; (g): metastable gas; (l): metastable liquid; (l+s): metastable melting; (s): metastable solid. The phase trajectories 1-7 correspond to depths of 5, 15, 20, 30, 50, 80, 130 nm from the initial target surface, respectively. Arrows along the trajectories show the flow of time. Here, laser pulse parameters are τ L = 100 fs, λ L = 800 nm, and F = 5 J/cm 2 .

FIG. 2 :Fig. 1 .

 21 FIG. 2: Contour plots of density (solid) and pressure of heavy particles (dashed) for different time delays after irradiation: (a)-10 ps, (b)-20 ps, (c)-30 ps and (d)-80 ps. Here, 1-is thermal decomposition zone, 2-is mechanical fragmentation zone. Laser parameters are the same as in Fig. 1.

2 )FIG. 3 :

 23 FIG. 3: Present simulation: melted (△, solid line) and ablated ( ) depths as a function of laser fluence for aluminum. Experimental data are taken from Ref. 5-( ) and Ref. 7-( ). The errors of the experimental data were reported to be of the order of 5-10%. Results of simulation Ref. 11-(⋆) and Ref. 15-( ) are also shown.

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project No. 06-02-17464) and the Council on Grants from the President of the Russian Federation (project No. NSh-3683.2006.2). M.E. Povarnitsyn gratefully acknowledges the financial support from