1 Introduction

One of the most intensively developing laser application fields at the present is short pulse laser nanostructuring of materials. The possibility of laser energy deposition in a tightly localized area on the material surface on femto and picoseconds time scales to pattern its surface is widely used in bio- and IT-technologies [1, 2]. Among the approaches proposed for the achievement of highly localized laser energy deposition, one can mark out the use of tightly focused laser beams with the peak laser fluence close to the threshold for material modification [3, 4], utilization of the optical near field effects, e.g., a local field enhancement in the vicinity of a probe tip [5–8], or a deposition of nano/microparticles [9, 10]. Following the first approach, the direct writing of ordered arrays of hollow bumps, jet, and nanobarrier-like structures on metal films of different thickness have been demonstrated in a number of works [11–14]; see Fig. 1a and b.

Fig. 1
figure 1

Side-view SEM images of a nanobump (a) and nanojet (b) structures fabricated on a surface of 60 nm gold film by single 30 fs laser pulses with Gaussian intensity spatial distribution. Laser beam with diameter of 4 mm was focused onto the sample surface by a 100× oil immersed microscope objective with 1.4 NA (Zeiss, Plan APOCHROMAT 100x). Laser pulse energy was E p=18 nJ (a) and 23 nJ (b). Side-view SEM image of a nanobarrier (c) and a broken nanobarrier (d) structures fabricated by a single 30 fs laser pulse with a flat-top rectangular intensity distribution. Laser fluence on the sample surface was 0.22 J/cm2 (c) and 0.25 J/cm2 (d). All images are taken at an angle of 45∘ [12]

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While experimentally, the applications of materials nanostructuring have achieved a significant success, especially in bio-technologies [15], the mechanisms of surface restructuring after short laser pulse excitation are yet meeting variety controversial explanations. In the particular case of thin metal films, possible mechanisms discussed in literature include Marangoni convection [11], peeling and expansion of the film due to the pressure of evaporating material [13, 14], as well as thermo-elastic and plastic deformation of the film, suggested based on recent continuum simulations [16]. To elucidate the mechanism responsible for nanostructuring, significant efforts were made in [12], where full or partial combination of the mechanisms mentioned above was proposed. Experimentally, nevertheless, the analysis of nanofeatures formation mechanism is commonly limited to posteriori observations of structures formed after the laser pulse that, however, does not reveal the complete kinetics of the process. Recently a new mechanism of the nanostructure formation on a thin metal film surface based on laser-induced formation of a liquid bubble and its collapse due to surface tension forces has been proposed [21]. This mechanism was supported by the first time-resolved experimental study of these processes at microscale [21, 22]. However, in order to justify these findings an adequate modelling of the real experimental situation is required.

Alternatively, the modeling of nanostructuring processes for the purpose of theoretical studies is not a straightforward task as well. The essential difficulty here is the necessity to describe several transient concurrent processes in a single computational cell. Moreover, those processes are taking place on wide time and spatial scales. Since working with pulses shorter than the electron–phonon relaxation time (∼10 ps), one has to account for the laser-generated electron–phonon nonequilibrium, fast electron heat conduction, and laser light absorption due to free electrons. Ordinarily, the Two Temperature Model (TTM) is used to describe free carrier’s temperature dynamics [17]. However, in short laser pulse experiments, the size of nuclei of the liquid phase during fast nonequilibrium melting at high degrees of overheating and in the presence of strong temperature and pressure gradients can be as small as several interatomic distances, and the phase transformation process itself can be as short as 1 ps and faster [18, 19]. The applicability of the continuum methods is therefore hampered under such extreme conditions [20, 23].

As it was demonstrated in [24], an attractive alternative to the continuum computational methods is provided by atomistic-continuum MD-TTM method (Molecular Dynamics coupled to Two Temperature Model), which not only accounts for the kinetics of fast nonequilibrium phase transformations at an atomic scale with MD, but also in the continuum includes the description of a fast electron heat conduction mechanism as well as the laser-induced electron–phonon nonequilibrium and laser light absorption with the help of TTM. However, the range of laser-generated elasto-plastic deformations in the solid part of material can extend to several microns [25], which eventually limits the application of Molecular Dynamics (MD) based approaches for the real experimental scales. Recently, based on the MD-TTM model, the large scale simulation on the short pulse surface nanostructuring process was performed on a simplified, scaled down system subject to isolate the essential mechanisms of nanostructurization [26]. It was shown that the relaxation of compressive stresses generated under conditions of the inertial stress confinement [27] is the main driving force responsible for the acceleration of a transiently melted region of the film resulting in the nanojet grow; see Fig. 1a. Fast electron heat conduction, on the other hand, in the presence of 2D or 3D (radial or spherical) temperature gradients provides the conditions for the fast cooling of the melted region and rapid solidification of the surface feature generated in the process of hydrodynamic motion of the liquid material. Nevertheless, a poor description of thermophysical properties of the material [24], a simplified physics of free carrier’s dynamics [28], and eventually the system size scaled down to one order of magnitude as compared to the experimental sizes [26] made a direct applicability of the modeling results for the experimental data treatment impossible.

In this paper, we report on results obtained by super large scale modeling of short laser pulse nanostructuring of a 60 nm Au film with the MD-based model that implements a realistic interatomic potential and description of free carrier’s dynamics. We apply a particular geometry for the MD-TTM simulation, which mimics the experimental sample and allows for the first time accessing the experimental scale and enables the direct comparison of the model and experimental results. The complex dynamics of interrelated processes involved into nanostructuring is revealed on a real time scale as well.

2 Experiment

In our experiments, we use a commercial 1 kHz femtosecond laser system (Femtolasers Produktions GmbH Femtopower Compact Pro) delivering 0.9 mJ, 30 fs laser pulses at average wavelength of 800 nm. Structures shown in Fig. 1 have been fabricated by single 30 fs laser pulses with different intensity distributions. Gold films with a 60 nm thickness coated onto silica glass substrates were fabricated by Layertec GmbH using magnetron sputtering. In order to generate nanobump and nanojet structures shown in Fig. 1a and b, a laser beam with the diameter of 4 mm was tightly focused onto the surface of a 60 nm gold film using 100× oil immersed microscope objective with a numerical aperture (NA) of 1.4 (Zeiss, Plan APOCHROMAT 100x). Laser pulse energy was E p=18 nJ (a) and 23 nJ (b). Structures shown in Fig. 1c and d were fabricated using an image transfer scheme with a 50× demagnification, which has been realized applying 50× Nikon long-distance microscope objective with NA of 0.45 and working distance of 13.8 mm and an additional lens with a focal distance f=200 mm. Focus distance of this lens is the same as of the tube lens used in the microscope that allows the corresponding image transfer. To generate the line-shaped flat-top microintensity distribution on the sample surface with sizes 1 μm×50 μm (Fig. 1c and d), image of a rectangular-shape mask with sizes of 50 μm×2500 μm has been transferred with a 50× demagnification. This mask has been fabricated by femtosecond laser ablation of a 100 nm chromium film deposited on a silica glass substrate [12]. Calculated laser fluence on a gold film surface was 0.22 J/cm2 (Fig. 1c) and 0.25 J/cm2 (Fig. 1d). The nanostructures formed in the case of lower laser fluences (Fig. 1c) look like nanobarriers with material redistributed from the edges toward the center of the irradiated region. At slightly higher laser fluences, the irradiated part of the film is detached and removed from the sample surface (Fig. 1d). Laser-fabricated nanostructures have been analyzed by Scanning Electron Microscopy (SEM). A more detailed description of the experiment and the setup along with the experimental analysis can be found elsewhere [12, 25].

3 Model

The idea of modeling on the experimental scale is based on symmetry utilization in the experiments described above; see Fig. 1c and d. A flat top pulse elongated in one direction was used for generation of a nanobarrier kind of structure on the surface of 60 nm Au film situated on a silica substrate. Since we do not expect any temperature and pressure gradients in one of the lateral directions, for an identical simulation/experiment comparison, we constructed the computational cell schematically shown in Fig. 2. The Au sample consisting of 150,000,000 atoms with dimensions of 10×5000×60 nm was taken in X, Y, and Z axes correspondingly, representing therefore a 60 nm thickness Au film on a silica substrate. The MD-TTM domain is covering 5 μm in Y direction ending with Nonreflective Boundaries (NRB) as they were described in [29]. Consisting of 3 perfect fcc planes on each side, NRB boundaries are devoted to behave dynamically to mimic bulk material behavior beyond MD domain and to absorb pressure waves resulting from relaxation of the laser-induced stresses inside the material. The number of planes corresponds to the distance they cover in crystal structure within the cut off radius for the chosen interatomic potential (described below). The total force applied to each of NRB planes and directed outward as the central XZ plane was measured as a function of their velocity in preliminary simulations of plane pressure wave propagation in a bigger sample containing these testing NRB planes inside. For the modeled Au material, the additional force F NRB per atom of NRB is then approximated with a linear fit to the measured points and given by

$$ F _{\mathrm{NRB}} = F _{0} -\mu V $$
(1)

where F 0=0.065 eV/Å is the compensating force acting on each atom constituting the NRB plane in the absence of any perturbation, the material parameter μ=0.368 eV Å/ps is the impedance coefficient, and V is the velocity of a corresponding plane. The NRB boundaries are composed in such a way absorb up to 99 % of the pressure wave energy and let us perform laborious MD calculations only in the area where the most intensive processes take place and atomic resolution is justified. The NRB boundaries, however, are transparent for the heat flux and ordinary TTM model was solved beyond them subject to the electron and phonon temperatures dynamics on the scale up to 30 μm from the center of the irradiated area in Y directions. In the direction X, we apply Periodic Boundaries (PB) conditions for a safe simulation of the material slice of 10 nm width and Free Boundaries (FB) are imposed at the top, where material is irradiated with a laser pulse. At the bottom, the Au sample interacts with an immovable silica substrate, represented by an immovable plane in XY.

Fig. 2
figure 2

Nanobarrier generated on the surface of a 60 nm Au film with a 30 fs flat top pulse in the experiment (left) and schematic representation of the computational cell used in the super large scale modeling of this process on the experimental dimensions (right). PB and NRB are periodic and nonreflective boundary conditions, respectively. The NRB are transparent for HEAT fluxes

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The MD-TTM domain was divided into a 3D mesh with 1 nm3 unit cell and the thermophysical properties of the atomic subsystem (such as temperature and pressure) were averaged over the closest 26 neighboring cells (that accounts for ∼2500 atoms) to suppress their value fluctuations due to the MD nose [24]. The choice of mesh size is justified by its minimization for efficient neighbor list search and forces calculations in the MD part on one hand and making the cell big enough for the collection of a large group of atoms to introduce their statistical description (under assumption of quasiequilibrium) for their macroscopic parameters determination.

In order to have the possibility for a direct simulation/experiment comparison, the MD-TTM model must implement the most realistic interatomic potential together with full complexity of the dependences of the free carriers system parameters. For this purpose, we choose the Embedded Atom Method (EAM) implementing the interatomic potential by Zhakhovskii [30], optimized for Au:

$$ E _{\mathrm{tot}} = \sum_{i} F ( n _{i} ) + \sum_{i<j} V ( r _{ij} )\quad\mbox{and}\quad n _{i} = \sum_{k\neq i} n ( r _{ik} ) $$
(2)

where V(r ij ) is a pair potential in eV units, F(n i ) is an embedding energy of i-atom, and n(r ik ) is a density function. The analytical EAM potential for Au is

(3)
(4)

where r c =0.6875 nm (for T=P=0) is chosen as a cut off radius and the rest parameters for calculations of E tot in electron-volts are listed in Table 1.

Table 1 Parameters for optimized EAM potential for Au by Zhakhovskii [30]
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For T=P=0, the potential gives −367.609 kJ/mol for cohesive energy, 179.4 GPa for the bulk modulus, and 0.4065 nm for the lattice constant. Also, this potential represents the thermophysical properties of the modeled material such as equilibrium melting temperature, heat capacity, volume of melting, and linear thermal expansion coefficient with more than 99.5 % of accuracy in comparison with their experimental values [31]. Thus, such an important property in our model as melting temperature T m =1343 K was measured in the series of liquid-crystal coexistence simulations [32] and showed a good agreement with its experimental value for Au material T exp=1337 K.

At the bottom of the sample, the atoms interact via the Lennard–Jones potential V LJ (in eV units) with an imaginary immovable silica substrate:

$$ V _{\mathrm{LJ}} ( r _{is} ) =4\varepsilon\biggl( \biggl( \frac{\sigma}{r _{is}} \biggr) ^{12} - \biggl( \frac{\sigma}{r _{is}} \biggr) ^{6} \biggr) $$
(5)

where ε=0.04415 eV, σ=0.257321 nm, and r is indicates a distance between atom i and position of silica substrate s (in Z direction). The parameter ε was chosen in accordance with an assumption that the adhesive energy between Au sample and the silica substrate is on the order of 10 % from that in Au bulk material given by Lennard–Jones potential (ε=0.04415 for Au bulk). This estimation was adopted from the statistics in the experimental measurements presented by Laser Zentrum Hannover (Germany). The initial distance between silica substrate and sample was set to the value at which the sample is resting in equilibrium and weekly bounded to the substrate.

The properties of electronic system were also described according to the most recent theoretical works [33]. Using the Density Functional Theory (DFT), Lin et al. has determined in his calculations the electron–phonon coupling and the electron heat capacity as functions of the electron temperature and the Density of States (DOS) (Fig. 3a and b), presented in tabulated form in [34] and used in our modeling. Also, the complex behavior of the electron heat conductivity K el was approximated as it was suggested by Anisimov [35], (Fig. 3c) as a function of the electron and lattice temperature T e and T l respectively:

(6)

with parameters ϒ=353 W m−1 K−1 and δ=0.16 for Au material. This dependence exposes linear behavior due to electron–phonon collisions with T e at low excitation, a significant decay due to electron–electron collisions at the excitations comparable with Fermi energy, E F, and a steep increase like plasma conductivity at higher excitations.

Fig. 3
figure 3

The electronic temperature dependences (T l =300 K) of the electron heat capacity (a), electron–phonon coupling parameter (b) [34], electron heat conductivity function (c) [35], and deviation of the reflectivity function dR/R from its equilibrium value (d) [41]. Dashed lines in (a) and (b) are previously suggested values in simplified approximation [36–39] for relatively low excitations (shaded electron temperature ranges in (b)). Dashed vertical lines in (c) and (d) are the Fermi temperature (T F=64,000 K) and maximum electron temperature reached in our simulation (T e_max=10,340 K) correspondingly

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The laser source S(y,z,t) of duration τ=30 fs originated at the center sample surface was taken with a real experimental geometry with the width of w=1 μm and inclusion of the ballistic transport due to nonthermalized electrons [39] in the directions of laser pulse propagation and along the wider axes of the sample, Y. The characteristic range of ballistic transport, λ ball∼75 nm, was estimated based on the electron–electron and electron–phonon collision rates, 1/τ rel=1/τ e–e+1/τ e–ph [40]:

(7)

where F is the incident fluence, R(T e ) is the reflectivity function with T e dependence, λ is a characteristic penetration depth of the electromagnetic wave (14.4 nm for Au at 800 nm wavelength), and parameter ω=4ln2. Next to the last multiplier in (7) accounts for the flat top pulse profile with the intensity redistribution along Y axis due to ballistic transport, whereas the last multiplier accounts for the finite film thickness d=60 nm as compared to the effective depth of the laser energy deposition, λ+λ ball [39].

Thus, using the laser source described by (7), we simulate a free sample surface exposed to the irradiation of a flat top pulse at the incident fluence of 275 mJ/cm2. The fluence was taken slightly higher than it was in the experiments, 125–250 mJ/cm2, [12] in order to speed up the process of the nanobarrier formation for its efficient observation during our simulation on a shorter time scale. For a qualitative comparison of our simulation results with the experimental data, it is important to carefully relate the absorbed fluence, which is an input energy into the computational cell, with the incident fluence, which is the laser output in the experiment. This task implies the knowledge of a reliable model for the reflectivity function. In the case of Au material, for the chosen wavelength of 800 nm and polished surface, it has a well-known value at the equilibrium conditions, R eq=0.945, which is frequently used as is in a number of theoretical and experimental works [11, 31]. In pico and femtosecond laser pulse experiments, however, due to the high temperature of the photo-excited carriers, the reflectivity becomes a function of the laser intensity as well. Thus, according to Hopkins [41], when the electronic temperature reaches the level of ∼10,000 K, the deviation of reflectivity function from its equilibrium value can reach already 15 %; see Fig. 3d. As a result, the actual value of the reflectivity averaged during the irradiation process in our modeling, R act=0.8998, leads to the resulting absorbed fluence twice as large as compared to that assuming equilibrium conditions. The thermoreflectance model by Hopkins, therefore, was carefully implemented in our atomistic-continuum model and yielded the of absorbed fluence value of 30.2 mJ/cm2 in our simulation.

The complexity of the model presented here requires careful analysis of a possible error source and a control scheme to ensure accuracy over the whole simulation. Ordinarily, as a means of precision control in MD-based numerical simulations one can use the energy conservation law. In [24], the calculation error in the MD-TTM model, which is essentially utilized in our present research, was estimated on the order of 0.1 % from the laser deposited energy. This was confirmed here in our preliminary test runs for a free standing sample.

In Fig. 2, we schematically show the system of 150,000,000 atoms, which was run across 16,384 processors on the Juelich Super Computer Facility on JUGENE machine (Germany) for 30 days resulting in a simulated experimental time of 1800 ps on the nanostructuring process [12]. In order to perform such a super large scale simulation, the MPI version of the atomistic-continuum model MD-TTM [26] was used. The processors space geometry 8×2048×1 in X, Y, and Z axes correspondingly was chosen for a better load balance during the whole simulation. This choice is governed by our expectation that the material laser-assisted expansion process will proceed upward and the number of atoms to be processed by each processor therefore will roughly stay the same. Thus, since each processor is devoted to treat equal volumes in space, we avoid the underuse of some processor and overloading others. The MPI algorithm in turn was optimized subject to remove any sequential parts of the code. Only the information related to a particular space volume was addressed with a corresponding processor. Also, instead of activation of collecting and broadcasting subroutines through a single processor, the reading and writing of information were also performed by each processor out and to its own file unit with data addressed to its particular volume in space. This approach allowed us to reach the super linear scalability of the model algorithm and the use of 16,384 processors resulted in speeding-up of ∼20,000 times in calculations.

4 Results and discussions

The results of our simulations are presented in Figs. 4–7. In Fig. 4, several snapshots from the modeling are shown up to 1800 ps for visualization of the atomic configuration of the sample. The atoms are colored according to the Central Symmetry Parameter (CSP) that was used here to distinguish between atoms with crystal structure (ordered surrounding) from those submerged in the liquid (disordered ambient) [42, 43]. The value of 0.11 for CSP was empirically chosen as a cut off between liquid if lower and solid if higher. Opposite to the Local Order Parameter (LOP) used in previous works and targeting the same purpose [24, 26, 27, 46], the CSP is crystal orientation independent and more efficient in observation of homogeneous solidification. The LOP however, has a better sensitivity to a rapid solid-liquid structure transformation realized during short pulse laser nanostructuring processes. During the quantitative analysis, therefore, both parameters were utilized for the data treatment.

Fig. 4
figure 4

Atomic configurations from the simulation of 30 fs nanostructuring of a 60 nm Au film on a silica substrate. A slice of 0.5 nm thickness of MD domain covering ∼2 μm width is shown in each snapshot in YZ plane. The atoms are colored according to the central symmetry parameter for distinguishing between crystalline surrounding (CSP<0.11) and liquid ambient (CSP>0.11)

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From Fig. 4, we can observe how the nanobarrier-like structure is gradually forming atop of a wider micron bump. The real time scale of the nanostructure formation (>2 ns) is much longer as compared to the nanobump formation in our previous work (∼250 ps) [26]. The experimental values of the pulse width and its energy lead to a formation of significant amount of molten material that accelerates in the upward direction. Additionally, the character of heat conduction, due to establishment of temperature gradient in 1D, is limited in this particular experiment and assumes much weaker rate of cooling and slower resolidification processes as a result. Thus, at the time of 1800 ps, we can see how the liquid part of the material, yet possessing a strong momentum, is about to detach from the solid part of the film. A closer view on the competition between the kinetics of fast homogeneous crystallization and hydrodynamic motion of the melt can be observed in Fig. 5, where a magnified part of the sample squared in the last snapshot of Fig. 4 is shown.

Fig. 5
figure 5

Zoomed area, squared in the last snapshot in Fig. 4, is shown for a thin (one atomic layer) central slice of material in YZ plane. The atoms are colored according to their CSP values and orientations of new-formed crystals are indicated by dashed arrows. The necking process leaves only 17 nm of molten film shortly before the spallation. Due to resolidification under conditions of elasto-plastic deformations, a part of the crystal is not visualized within the slice geometry, which is reflected as an empty space in the right bottom part of the figure

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The movement of the molten material in upward direction, obtained as a result of laser-induced stresses relaxation [26, 27], does not seem to be suppressed by increasing toughness of the overcooled liquid and new-formed solid structures growing inside the liquid phase. As a result, the process of necking acts as the onset of spallation, leaving ripped off sample surface behind as it was also detected in the experiment; see Fig. 1d. Thinning/necking of the gold film at the edges of the nanobarrier structure is also experimentally observed at lower laser fluences below the ablation/spallation threshold. Apparently, this process leads to redistribution of the material from the edges toward the center of the irradiated region as it can be seen in Fig. 1c and discussed [12]. In the case of Gaussian laser beams, similar processes lead to formation of spherical nanodroplets, which are ejected from the molten film surface. Using such the laser-induced transfer approach, therefore, these droplets can be caught on another substrate (receiver) and applied for fabrication of different structures consisting of metal nanoparticles [1, 44, 45].

The presence of homogeneous nucleation of a solid phase implies a high degree of overcooling of the molten material. The crystal structure regrowth is not fast enough to allow for propagation of the solid–liquid interface, which characterizes heterogeneous resolidification. Therefore, massive homogeneous liquid–solid transformation mechanism becomes a dominant process. Figure 5 demonstrates that both processes of resolidification (homogeneous and heterogeneous) are involved. Thus, the resulting nanobarrier-like feature, if formed, will have a polycrystalline character. This is confirmed by the experimental observations [11, 12] and our theoretical predictions found in [46].

In this particular simulation, the energy input was taken slightly higher than the one necessary for a completely recrystallized final structure. This is in agreement with the experimental value of ablation threshold absorbed fluence of 20 mJ/cm2 obtained in nanostructuring experiments [25], and can be also confirmed when considering the contour plots of the system where pressure, lift-off velocity, and density fields are showed in color for MD domain at the time t=75 ps; see Fig. 6. Due to relaxation of extensively high laser-induced stresses (on the order of 10 GPa), the formation of voids takes place, reflected in Fig. 6c as an abrupt drop in local density. The intensive lattice heating by hot electrons and proper sample geometry provides the conditions for inertial stress confinement in this experiment. According to [27, 47], the conditions have a great impact on the melting, spallation, and ablation mechanisms. Namely, upon building up laser-induced compressive stresses inside the film, their subsequent relaxation is followed by the propagation of unloading pressure waves. The relaxation of tensile stresses at a sufficient temperature can result in onset of the spallation in the material that is reflected in the pressure contour (Fig. 6a), as the formation of spherical areas with zero pressure and tiny waves propagating outward. The volumes where liquid material undergoes to the rapture process are also reflected in the lift-off velocity contour plot as local jumps in its values; see Fig. 6b. All the formed voids, however, collapse in this particular simulation. The applied laser fluence, therefore, happens to be the threshold value at which the short-time formation of voids leads to relaxation of tensile stresses without the internal crack development, whereas a higher fluence would lead to disintegration of the film.

Fig. 6
figure 6

Contour plots of pressure (a), lift-off velocity (b), and density (c) fields are shown for the central part of the MD-TTM domain at t=75 ps. The formation of voids can be observed everywhere as abrupt color changes

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Following the same procedure, as it was demonstrated in [26], by means of considering the time evolution of pressure, lift-off velocity, and density fields along with data on the dynamics of electron and lattice temperatures, we can thus reconstruct all the physical mechanisms involved into nanostructures formation on the experimental scale. Figure 7a and b show electron and lattice temperatures, pressure, and lift-off velocity, averaged within distance of 1 nm from the central XZ plane, as functions of time. In general, the dynamics of the plotted quantities confirms two essential mechanisms responsible for the nanostructures formation as they were isolated in a simplified model described in [26]. Namely, the relaxation of the laser-induced stresses, due to strong laser heating under conditions of inertia stress confinement, leads to acceleration of the molten matter in the upward direction and the established hydrodynamic motion is responsible for the nanostructure growth. The resolidification of this structure, on the other hand, is associated with a strong cooling rate provided by thermal energy flux outward the central part due to a fast electron heat conduction process. Despite the fact that the mechanism of nanostructuring in general holds the same as it was found in [26], the real complex dynamics of thermophysical quantities is completely addressed in our present simulation. On one hand, the utilization of experimental symmetries in nanobarrier-like structure formation helped us to model its process on the experimental scale. But on the other hand, the use of this symmetry has limited the thermal energy flux due establishment of temperature gradient in 1D only. This results in slower electron temperature decay, slower lattice cooling (Fig. 7a), and a much slower retardation and crystallization of the growing liquid nanobarrier (Fig. 7b).

Fig. 7
figure 7

Data averaged within 1 nm distance from the central XZ plane of the sample are shown in time evolution for electron and lattice temperatures (a), and pressure and lift-off velocity (b). Averaged in the same manner, lattice (c) and electron (d) temperatures represented as contour plots reveal their interrelated fluctuations with a gradual decay in time

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Based on the simulations reported in [26] and in this work, we can conclude that the lift-off velocity of hydrodynamic motion of liquid, 200–300 m/s, due to internal stresses relaxation, is a characteristic value in the nanostructuring process for its successful finalization. This value holds for a range of fluences beginning from those resulting in the smallest possible features formed (Fig. 1a), and up to the threshold values leading to spallation without final structure formed; see Fig. 1d. The fact of possessing such a high velocity after almost 2 ns in our simulation indicates that the process of formation of the nanobarrier-like structure can therefore take up to tens of ns. This is in a very good agreement with recent time-resolved pump–probe measurements in the experiments on the nanostructuring [12, 22, 25]. Another firm justification of our modeling results can be viewed when considering the bump formation on which the nanostructure grows as in the experiments. According to [25], whichever laser fluence was applied, the width/diameter of this bump does not change after 10 ns. Apparently, this can result from different material phases constituting the jet/barrier-like and bump structures. If the former is being developed under hydrodynamic motion of liquid, the latter is likely being in a solid phase under elastoplastic deformations, as it also suggested in [16]. Indeed, in our simulation, one can clearly identify these two different phases in Fig. 4. Considering the sequence of contour plots for lift-off velocity similar to that shown in Fig. 6b, we have found that the solid part of the structure due to plastic deformations is about to stop in motion by the end of simulation (1.8 ns) and would mainly remain in the same shape and size eventually forming the bump on a micron scale; whereas, the central part of the film, being in liquid phase, due to its high velocity would continue growing toward the formation of jet- or barrier-like structure or could completely spall off as it likely to be in our present simulation. On the other hand the complex hydrodynamic processes discussed in [21, 22], which involve collapse of the liquid bump and generation of two opposite targeted jets happen only in the central liquid part of the nanostructure at longer timescales, which are not considered in the present calculations.

Finally and the most cognitive, in Fig. 7c and d, one can observe the electron and lattice temperature periodic fluctuations, the unnoticed previously effect in the simplified simulation on nanostructuring [26]. Due to a real experimental geometry in our present modeling, however, these fluctuations became visible in a complex dynamics and were also studied in animated sequence of contour plots for the whole MD domain as they are constructed for Fig. 6. Obviously, the laser-induced stresses relaxation process leads to the film’s mechanical vibrations [24, 27]. As an isentropic contraction/expansion process, the lattice temperature, therefore, must obey same fluctuations in time as pressure [27]. And since phonon heat capacity is much greater upon the electron–phonon equilibration, the electron temperature in turn merely follows the lattice one due to the electron–phonon energy exchange process. In modeling of laser melting of nanocrystalline Au, similar oscillations were observed [48]; the frequency of which can be measured in thin film thickness determination in pump–probe experiments [49]. The amplitude of these temperature fluctuations in our nanostructuring experiments, however, reaches more than 200 K at the average value of 1500 K, and thus can affect some other materials properties. For example, the reflectivity function and the laser light absorption, according to Hopkins model [41], can be a strong function of the electron temperature. Although in our modeling the temperature fluctuations would result in the change of absorptivity value by ∼20 % only, there could be much greater changes (more than 100 %) at stronger degrees of excitation. If so, the areas with higher electron temperature would absorb the laser light stronger and applying a multi pulse regime synchronized with the electron temperature fluctuations, one could possibly obtain some fine surface subpatterning in the process of nanostructure formation. Future detection of these temperature oscillations in the experiment could also useful in a partial interpretation of the ripples formation process in multipulse surface patterning of targets of different thickness [50].

5 Conclusion

For the first time, the process of short laser pulse nanostructuring of thin metal films was modeled in molecular dynamic simulations on the scale directly accessible in the experiments. In order to perform such the super large scale simulation, the utilized MD-based model accounted for a realistic interatomic potential together with the advanced description of free carrier dynamics. The application of the MD-TTM model on the experimental scale revealed two essential mechanisms responsible for the short laser pulse surface nanotexturing. The relaxation of the laser-induced stresses inside of the molten target serves as a main driving force for the nanostructure growth. The electron heat conduction, on the other hand, is providing the conditions for its fast resolidification. In addition, the super large scale simulation in multiprocessing regime allowed for the direct comparison of our modeling results with the experimental data and found a very good agreement. We succeed in obtaining the quantitative consistence with experiment of our modeling results on the absorbed threshold fluence of 20 mJ/cm2, until we have nanostructures finalized on the surface as well as on their formation time for the smallest features, 10 ns. The model also allowed to describe the formation of the micron bump on which the nanostructures grow and its constant size (several μm), after few ns as it was detected in the experiments. Finally, the model predicted an interesting effect of the electron temperature oscillation due to electron–phonon coupling and mechanical vibration of the film under nanostructuring process. Although, the laser-induced mechanical vibrations were previously detected in both theory and experiment, the range of the electronic temperature oscillation at least indicates a change in optical properties of the material. Thus, a complex dynamics of the thermophysical properties of the material observed on the experimental scale in our simulation opens up new opportunities in both fundamental physics investigation on the short laser pulse surface nanomodification and its possible technological applications.