1 Introduction

Over the past years, there has been increasing interest in the interaction of ultra-short laser pulse with matter, because it provides precise microprocessing and fabrication of nanostructures on the surfaces of dielectrics, semiconductors, and metals [1, 2]. Such type of the laser irradiation significantly reduces the laser fluence required for surface modification. In addition, the interaction of mater with short laser pulse is of fundamental interest due to the fact that the high nonequilibrium states of matter are produced under strong optical excitation conditions [3–8]. The advantages of laser surface modification were recently demonstrated by using short pulse transient collisional soft X-ray laser (SXRL) [7, 9–18] and free electron (FEL) beams [19–22] irradiation of matter.

One of the main advantages of using the soft X-ray pulse for metals is high absorption coefficient. Typically, for optical pulses, the absorptance of metals is in the range between 10 and 30 %, and it is close to 100 % for normal incidence of soft X-ray pulses. The use of the soft X-ray pulses allows for direct comparison between the theoretical and the experimental data of the threshold energy of surface modification. In this case, the nontrivial question about calculation of the absorption coefficient can be omitted. It should be also noted that the laser ablation thresholds of absorbed laser energy for some metals irradiated by IR or visible and SXRLs are quite close to each other (see for example [16, 18]).

In spite of progress in the description of laser ablation for the metals [13–16], the mechanism of surface modification remains unresolved, because the existing experimental data are contradictory. For example, the threshold of modification for absorbed fluence F abs and the crater depth d for Au targets were measured as ~9 mJ/cm2 and 10 nm, correspondently [23]. On the other hand, in work [24] different values (F abs ~ 150 mJ/cm2 and d ~50 nm) were obtained with the same short optical laser pulses. The discrepancy exists also between experimental data of ablation thresholds for Al targets, which were measured in [25] as F abs ~ 18 and ~70 mJ/cm2 in [26]. The other peculiarity of visible laser ablation observed in experiments for some metals is existence of second threshold fluence when the slope of the dependence d(F abs) changes [28, 29]. However, this peculiarity occurs only with femtosecond laser pulses. The possible explanation of this phenomenon was given in [18].

In the case of soft X-ray pulse, the threshold fluences of surface modification F mod lay between the modification thresholds presented above. For example, F mod of gold was measured as 21 ± 5 mJ/cm2 [18] and F mod of Al equals 14 ± 5 mJ/cm2 [27]. In work [27], one of the main differences between metals at surface modification is the profile of the surface structure. At identical pulses, the changes in Au surface look like nanoscale ripples, while in the case of Al the formation of deep crater on the surface takes place. This fact confirms the existence of different mechanisms of surface modification.

It is important to note that the difficulties are partly associated with varied methodology of fluence measurement. For instance, in work [24] the fluence is calculated as a maximum fluence in the center of the laser spot. In work [29], the fluence was measured as an averaged value in the laser spot. In [18, 27], the fluence was measured as an averaged value in the modification zone. Therefore, significant part of this work deals with the description of a threshold characteristic at surface modification using distribution of fluence.

In this work, we study the nanometer scale modification of solid gold and aluminum surfaces irradiated by single X-ray laser pulse with wavelength λ = 13.9 nm. In experimental part of the work, the surface profiles and the averaged fluences in modification zone for different energies of the laser pulse are measured. To describe the ablation and melting of gold near the threshold fluence, we use two-temperature atomistic model with electron–temperature-dependent (ETD) interionic potential. The analysis of the results shows that melting and ablation play different roles in the surface modification. The melting determines the size of the modification zone. The ablation determines the crater depth in central part of the modification zone. The detailed comparison of the simulation results with experimental data is given in final part of the article. It is shown that description of laser pulse-like distribution of fluence is more productive than using the average fluence.

2 Experimental setup and results

The experiment with SXRL irradiation was carried out with use of the SXRL facility at JAEA [30–32]. The spatially coherent SXRL pulse was generated from the silver (Ag) plasma mediums using an oscillator-amplifier configuration with Ag double targets. The Ag tapes were used as the plasma targets, and the tape targets were irradiated by the linearly focusing laser pulses, generated from a chirped pulse amplification glass laser system with zigzag slab Nd:glass amplifiers. The SXRL generated from Ag double targets had following characteristics: wavelength equal to 13.9 nm, bandwidth was narrower than 10−4, duration time equal to 7 ps, and beam divergence was 0.35 mrad (H) × 0.3 mrad (V). The SXRL system worked in 0.1 Hz regime, and the output energy achieved was of the order of 1 μJ. The SXRL pulse was focused (see Fig. 1) on the sample surface by using a spherical Mo/Si multilayer-coated mirror with a radius of 1,000 mm in curvature.

Fig. 1
figure 1

Schematic diagram of experimental setup for irradiation of a sample surface by SXRL pulses. Sample stages with LiF crystal and Al or Au targets move in a plane. X-stage was used for delivery fresh targets, and Y-stage was used for changing focusing position of SXRL

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The output energy of the SXRL pulse was varied in each shot, but the average pulse energy value was estimated to be about 200–300 nJ. The spherical Mo/Si multilayer coating was optimized for soft X-rays with wavelength equal to 13.9 nm at an incidence angle of 2°. The mirror was placed at a distance of 2,637 mm from the SXRL output. Either one or two Zr filters having a thickness of 0.2 μm were placed in front of the spherical mirror to reduce the scattered optical radiation from the laser-produced Ag plasma. The transmittance of the single Zr filter and the reflectivity of the Mo/Si mirror for soft X-rays of 13.9 nm were about 48 % and about 50 %, respectively, which allowed to illuminate the sample surfaces with SXRL energy of 48–72 nJ. The Al plate with a thickness of 2.5 mm or the Au plate with a thickness of 0.05 mm was used as the target samples. Both Al and/or Au samples were mounted on the surface of LiF crystal, and such sandwich of targets was fixed on the holder having two movable directions (see Fig. 1). The targets on the holder were moved after each laser shot along the SXRL beam propagation direction and also perpendicular to it, in order to record the beam patterns at different focusing distances and to use the fresh LiF crystal or Al/Au surfaces. All target surfaces were well synchronized to assure equal focusing conditions. Al or Au targets were illuminated in different focusing positions of SXRL beam. For each focusing position, the same illumination was done on LiF crystal surface. Such procedure was carried out to apply LiF crystal detector property for determination of SXRL intensity distribution in each focusing spot [33]. All irradiation process described above was carried out in a vacuum chamber.

After irradiation of the LiF crystal by the SXRL, the photoluminescence patterns from the color centers (CCs) in LiF were observed by using a confocal fluorescence laser microscope (OLYMPUS model FV300). The 488 nm Ar laser was used in the microscope to excite CCs, which then emit luminescence in the spectral range of 500–800 nm with the peaks at 530 nm for F 2 and 670 nm for F +3 . Typical luminescent images of the XRL beam recorded with the LiF crystal irradiated by XRL beam with energy equal to ~48 and 24 nJ are presented in Figs. 2 and 3. Such images allow to clearly resolve fluence distribution in the SXRL beam focusing spots. The shape of the structures produced on the surfaces of the LiF crystal, Al and Au was observed and measured by use of a visible microscope (BX60, OLYMPUS Corporation) with the differential interference contrast (DIC) mode (see images in Figs. 2, 3).

Fig. 2
figure 2

The luminescent image of LiF surface and the visible (in DIC mode) images of LiF crystal, Al and Au surfaces nanomodification by irradiation of SXRL beam with pulse energy of ~48 nJ. The fluences are estimated like averaged values in modification zone

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Fig. 3
figure 3

The luminescent image of LiF surface and the visible (in DIC mode) images of LiF crystal, Al and Au surfaces nanomodification by irradiation of SXRL beam with energy of ~24 nJ. The fluences are estimated like averaged values in modification zone

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The detailed structures of modified surface which could not be seen with a visible microscope were observed by a SEM (JSM-6380LVN, JEOL Ltd.) (see Fig. 4). The depths of the produced nanoscale patterns were measured by an atomic force microscope (AFM, Explorer, TopoMetrix Corporation), see Fig. 5.

Fig. 4
figure 4

SEM images of Al and Au surfaces after their irradiation by a single laser shot of the SXRL beam with pulse energy of ~48 nJ. Magnified red boxes present parts of SEM images with higher spatial resolution. Strong difference in morphology of surface nanostructuring for Al and Au targets is seen

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Fig. 5
figure 5

AFM images of aluminum (a, c) and gold (b, d) surfaces after their irradiation by a single laser shot of the SXRL beam with different energies (48 and 72 nJ). Strong difference in ablation depth of surface nanostructuring for Al and Au targets is seen

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To find SXRL nanostructuring threshold fluences of Al and Au the SXRL, pulse energy was changed by placing of additional Zr filter (with a thickness of 0.2 μm) in front of the spherical mirror. This allowed to halve the SXRL energy. The schematic distribution of laser fluences is presented in Figs. 2 and 3 along with the images of the obtained modification zone. The results show that modification zone size depends on the type of material and on the pulse energy. It was found that in the case where the samples were irradiated by quasi Gaussian SXRL pulse with the energy about 48 nJ (Fig. 2), LiF crystal had the largest damaged spot (the size is about ~280 μm2), Al had the middle size of damage spot (~170 μm2), and the Au target had smallest one (~50 μm2). We estimate fluences F mod like averaged values in the modification zone. In this case, LiF has the smallest fluence and Au has the largest one. Taking into account the distribution of SXRL fluence on the surface, the averaged F mod could be estimated as follows: for LiF crystal F mod ~10 ± 5 mJ/cm2, for Al F mod ~13 ± 5 mJ/cm2, and for Au F mod ~21 ± 5 mJ/cm2. These values may be interpreted like the threshold fluences of the surface modification, because they are very close to the values of nanostructuring fluences observed in case of twice SXRL beam energy reduction (see Fig. 3). Moreover, the irradiation does not cause changes on Au surface when the pulse energy is equal to 24 nJ.

Figures 4 and 5 show SEM and AFM images of the irradiated Al and Au surfaces after a single laser shot with the pulse energies equal to 48 and 72 nJ. The typical size of the obtained surface irregularities is about 50–200 nm for both targets, but the morphological structures are rather different for them. The nanoscale ripples (conical-like structures) are formed on Al surface—at the periphery of the modification zone, where the absorbed laser energy is smaller. We have estimated that the conical structures have a diameter of around 50–100 nm, and they are formed in the shallow region having an average depth of ~10–20 nm (Fig. 5a). For pulse with the energy equal to 48 nJ, the modification zone of Au target completely consists of similar nanoscale ripples with more chaotic structure. However, when the same pulse energy is applied for Al, it produces the deep ablation craters with micron scale diameter at the center of modified zone. So we could conclude that the surface modification of Al at averaged laser fluence F mod ~14 mJ/cm2 (pulse energy equals to 48 nJ) is composed of two zones: the nanoscale ripples at the periphery and the deep craters at the center. With increasing of the F mod up to 30 mJ/cm2 (pulse energy equals to 72 nJ), the crater with diameter of a micron and the depth of about 100 nm was produced at the center of irradiated Al surface. It should also be noted that the imperfect focusing properties of the used spherical mirror have some influence on the inhomogeneity of the modification zone.

As already noted, the structures formed on Au surfaces at pulse energy equal to 48 nJ were different from the structures appearing on Al surface. Such difference could be seen both for surface morphology (Fig. 4) and for the ablation depths (Fig. 5). The ripple nanostructures formed on Au surfaces are similar to the structures observed at the periphery of the modification zone on Al surface. There are some differences in the topography of the structures for Al and Au (for example, nanoscale ripples for Au are more chaotic), but the averaged depths for these structures are very close. These changes in metal surface are probably caused by the splash of the molten gold under intensity gradient (i.e., pressure/temperature gradients) across the laser beam. However, this modification of Au surface is not accompanied by the crater formation, like it was found in the case of Al sample. As one can see from the traces of AFM image shown in Fig. 5b, the averaged fluence F mod ~21 ± 5 mJ/cm2 forms the modification on the Au target with the depth equal to only ~15 ± 10 nm all over the structure region. This modification is essentially small in comparison with the craters in center of the irradiated spot of Al. The depth of surface modification of Au sample increases with the increase in pulse energy, but not as significantly as for Al. Under average fluence F mod ~60 mJ/cm2, the depth of Au surface modification reaches the value of 15–80 nm (see Fig. 5d).

3 Model

The absorption of laser pulse initially leads to the excitation of electron subsystem (ES). In this case, initial state of the system is two-temperature (2T) state, and the electron temperature (T e) may be several orders higher than the ion temperature (T i). The time of electron–ion relaxation is comparable to the time of relaxation processes taking place in ion subsystem (IS): heat transfer, phase transitions, etc. One way for modeling is a continuum approximation with a two-temperature equation of state [34, 35]. Such approach is a powerful tool for investigation of laser ablation. However, this method does not take into account the phenomena existing at the atomistic level (such as metastable phase decay or nucleation) that are essential for the description of surface modification at low fluencies. The other way to consider the problem is molecular dynamics (MD). MD models of the 2T-system are proposed in works [36, 37]. In this model, the ions are simulated in the framework of classical molecular dynamics, while the electrons are treated as continuous. The energy transfer from the ES to the IS is implemented using Langevin thermostat. In this work, the advanced molecular dynamics 2T-model [18] is used for the simulation of melting and ablation processes in aluminum and gold. One of the main differences from the previous models is that our model takes into account the influence of electron pressure (P e) on the ion dynamics.

Au is a metal with one s-electron and ten d-electrons in the valence shell per atom. A combined description of the localized d- and delocalized s-electrons is needed in order to account for the actual electron pressure of this material. The total energy of ES also contains localized and delocalized parts [18, 38]. The delocalized energy of ES is determined only by the electron density and temperature, such as the energy of an ideal gas of FELs. The pressure P deloce of the delocalized electron energy may be taken into account in the ion dynamics by the blast force ~∇P deloce  ~∇T 2e [39, 40]. On the contrary, the localized energy of ES is determined by T e and positions of the ions. This fact may be interpreted like the dependence of interionic forces on the electron temperature. Therefore, for gold, the change in interionic forces plays the key role in dynamics of ions at the initial 2T-stage. One of the ways to describe such process is to use ETD interionic potential in a simulation. Such potential for gold was previously developed in work [18]. The ETD potential allows us to take into account the electron pressure of the localized electron energy in addition to the blast force of the delocalized electron energy.

For creation of ETD potential for gold, three EAM potentials (potentials in the embedded atom method form) at different electron temperatures were created. Construction of the EAM potentials was performed with force-matching method [41, 42]. This technique was used as implemented in the Potfit code [42]. This method provides a way to construct physically justified interparticle potentials without referring to experimental data. The idea is to adjust the interparticle potential to optimally reproduce per-atom forces computed at the ab initio level (e.g., with density functional theory) for a fine-tuned set of small reference structures. The reference data were calculated using the VASP code [43]. In our case, the calculations of the reference data were performed at three different T e (0.1, 3 and 6 eV) that are set as a parameter of the Fermi–Dirac distribution for partial occupancies of electron bands. All the three developed EAM potentials reproduce forces calculated by the VASP code to within 15–20 %, which is quite a good accuracy for the given method. The EAM potential has three independent functions: pair interaction φ(r), effective electron density ρ(r), and embedding energy term Φ. The Φ-function was identical for created potentials. Figure 6 presents the functions of the EAM potentials. The ETD potential is created using quadratic polynomial interpolation: φ(r) = φ 0 + φ 1 T e + φ 2 T 2e and ρ(r) = ρ 0 + ρ 1 T e + ρ 2 T 2e . The ETD potential coincides with the EAM potentials at three reference temperatures. With the use of the ETD potential, the potential energy U of an ion is given by:

$$\begin{aligned} U& = \frac{1}{2}\sum\limits_{j}^{N} {\left( {\varphi_{0} (r_{j} ) + \varphi_{1} (r_{j} ) \cdot \left[ {\frac{{T_{\text{e}} + T_{{{\text{e}},j}} }}{2}} \right] + \varphi_{2} (r_{j} ) \cdot \left[ {\frac{{T_{\text{e}} + T_{{{\text{e}},j}} }}{2}} \right]^{2} } \right)} + \hfill \\ &\quad + \varPhi \left( {\sum\limits_{j}^{N} {\left\{ {\rho_{0} (r_{j} ) + \rho_{1} (r_{j} ) \cdot \left[ {\frac{{T_{\text{e}} + T_{{{\text{e}},j}} }}{2}} \right] + \rho_{2} (r_{j} ) \cdot \left[ {\frac{{T_{\text{e}} + T_{{{\text{e}},j}} }}{2}} \right]^{2} } \right\}} } \right), \hfill \\ \end{aligned}$$
(1)

where the first sum is a pair potential interaction; r j is a distance between given ion and jth ion; T e,j is electron temperature where jth ion is located. The summation is performed over all Nth neighboring ions in a sphere with the radius r cut = 0.56 nm. The multibody nature of the potential is a result of the embedding energy term. Thus, the seven functions (Φ, φ 0, φ 1, φ 2, ρ 0, ρ 1, ρ 2) determine the ETD potential. When calculating the ion forces, the differentiation of only the ion positions is relevant since gradients of the potential functions are more considerable than gradient of T e.

Fig. 6
figure 6

The initially created EAM potential functions: φ(r) and ρ(r) for gold at various electron temperatures T e = 0.1 eV (solid curves), 3 eV (dash-dot curves), and 6 eV (dashed curves). The “embedding” function Φ(ρ) is independent of T e

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At low T e, the created ETD potential reproduces well the thermodynamic and mechanical properties of gold that are known from experiments (equation of state, melting curve, elastic constants, etc.). At increasing of T e, the ETD potential may describe the features of 2T-system. In particular, one of the phenomena that can be reproduced by the ETD potential is an increase in the ion melting temperature T melt at heating of the ES [44].

Another phenomenon that is reproduced by the ETD potential is increase in the pressure P loce of localized electron energy with increase in T e. In fact, as T e increases from 0 to 6 eV, the pressure in the system described by the ETD potential increases from 0 to 105 GPa. Together with the value of P deloce , the total pressure in gold at T e = 6 eV reaches 200 GPa like in works [44, 45]. Figure 7 shows plots of the total pressure P (ab initio calculations using the VASP code) and P loce (MD calculation with the ETD potential) versus T e. The difference between the two pressures determines the dependence P deloce (T e).

Fig. 7
figure 7

The dependence of pressure on T e: 1 total pressure P (ab initio calculations with using the VASP code), 2 the P loce reconstructed with the virial theorem (molecular dynamics calculation with the ETD potential). The difference between the two pressures determines the dependence P deloce (T e)

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The ion equations of motion are as follows:

$$m \cdot \frac{{{\text{d}}{\varvec{\upupsilon}}}}{{{\text{d}}t}} = {\mathbf{F}}({\mathbf{R}}_{1} ,{\mathbf{R}}_{2} \ldots \, {\mathbf{R}}_{j} ,\ldots {\mathbf{R}}_{N} ,T_{\text{e}} ) + {\mathbf{F}}_{\text{Lang}} (T_{e} ,T_{i} ) - \frac{{\nabla P_{\text{e}}^{\text{deloc}} }}{{n_{\text{i}} }},$$
(2)

where the first force is interionic force of the ETD potential (R j are ion coordinates), F Lang is the stochastic Langevin force simulating the electron–ion thermal energy transfer [37], the last term is the blast force of delocalized electron energy [40], and n i is the ion density. The value of the absorbed energy (e.g., fluence absorbed at interaction with laser pulse) is determined by the change in the total energy of the ion subsystem during the two-temperature stage. The use of the ETD potential and the blast force (mechanical influence of ES on IS) leads to nonconservation of total energy of the system. The nonconservation of the energy is determined by mechanical influence of the electron subsystem on the ion subsystem. However, this nonconservation is insignificant in comparison with energy given to the system during simulation. The main contribution in the energy exchange between the subsystems is given by Langevin force (thermal influence of ES on IS which is included in the equation for ES). Figure 8 shows the change in energies during the atomistic simulation of laser irradiation of gold.

Fig. 8
figure 8

The dependence of energy per unit of free surface area on time during the atomistic simulation of the laser irradiation: ES, the energy of electron subsystem; IS, the change in total energy of ion subsystem; IS + ES, the change in total energy of system. The absorbed fluence and pulse width equal 72 mJ/cm2 and 7 ps respectively

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One of the main difficulties in creation of the model is description of the gradient ∇P deloce near the surface. Since it is assigned that T e = 0 for vacuum, this expression formally gives infinity. We use the approximation P deloce  = B·T 2e and the calculation formula:

$$\frac{{\nabla P_{e}^{\text{deloc}} }}{{n_{\text{i}} }} = \frac{B}{{n_{\text{i}} }}\left[ {\frac{{T_{\text{e}}^{2} (x)\lambda }}{{(x + \lambda )^{2} }} + \frac{x}{(x + \lambda )}\frac{{T_{\text{e}}^{2} (x + {\text{d}}x) - T_{\text{e}}^{2} (x)}}{{{\text{d}}x}}} \right],$$
(3)

where x is the coordinate counted from the surface position into the depth of metal, and λ is the effective electron mean free path. The first term in (3) is the surface blast force. If T e = const (spatially homogeneous case), the summation/integration on all ions in the system with the use of formula (3) gives correct pressure acting on surface (P = B·T 2e  = P deloce ). In addition, the expression (3) gives right formula for a bulk ~∇T 2e at x ≫ λ.

Al is a metal with s-, p-electrons in the valence shell. In this case, gas of FELs is a good approach for description of the ES. Thus, the interionic forces do not depend on value of T e. Therefore, the ETD potential is not necessary for the simulation of surface modification of aluminum. Only the blast force is used for the description of a mechanical influence of ES on IS.

In the model used, the evolution of the ES is described by the following equation:

$$C_{\text{e}} (T_{\text{e}} ) \cdot \frac{{\partial T_{\text{e}} }}{\partial t} = \nabla \left( {K_{\text{e}} (T_{\text{e}} ) \cdot \nabla T_{\text{e}} } \right) - G \cdot (T_{\text{e}} - T_{\text{i}} ) - \frac{{I(t)e^{ - x/l} }}{l},$$
(4)

where C e is the electron heat capacity, K e is the electron thermal conductivity, G is the coupling constant for the electron–ion interaction, and I(t) is the absorbed laser intensity of the rectangular pulse of width equals 7 ps, and l is the attenuation length. The value of T e depends only on the x-coordinate. The Eq. (4) has additional condition: T e = 0 for vacuum. The electron energy does not overflow through the surface. For simulation, we use the following parameters in Eqs. (3) and (4): B = 4.2 GPa/eV2 for Au and 4.1 GPa/eV2 for Al (for the best description of the difference between total electron pressure and P loce at T e = 3 eV); λ = 2 nm for Au; and λ = 4 nm for Al; C e is taken from work [46]; K e is from works [36] and [47] for Au and Al, respectively; G is from work [46]; l = 18 nm for Au and l = 36 nm for Al.

The size of the simulation box is 2,100 × 28.6 × 8.2 nm in the x, y, and z directions. The periodic boundary conditions in y and z directions are used. The ions form a crystal in one half of the simulation box (at 1,050 < x < 2,100 nm). This simulation is similar to the modeling of ablation and melting of gold presented in [18], with some revision of the description of P deloce .

4 Results of the simulation

The atomistic simulations of laser ablation and melting of the metals have been performed. The dependence of the modification depth (ablation or melting) on absorbed fluence is calculated for the width of the laser pulses τ = 7 ps. The data obtained for gold agree well with previous results from work [18]. Figure 9 shows main results of the simulation. For gold, the mechanism of ablation is associated with shock wave propagation as well as with electron pressure relaxation. In case of aluminum, the mechanism of ablation is fully determined by propagation of shock wave because the value of electron pressure for Al is significantly lower than for Au at the identical absorbed fluences. In addition, the time of electron–ion relaxation for Al is smaller than for Au. However, the electron pressure may play significant role in ablation of aluminum at higher fluencies.

Fig. 9
figure 9

Results of the atomistic simulation. The dependence of ablation/melting depth d on absorbed fluence F for simulated metals at τ = 7 ps: 1 ablation depth for Au, 2 ablation depth for Al, 3 melting depth for Au, 4 melting depth for Al

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Our results for aluminum ablation agree with the hydrodynamic simulation [35] and two different experiments with optical subpicosecond pulses [26, 48]. However, there are two experiments [49, 50] with optical subpicosecond pulses showing considerably higher ablation threshold ~200 mJ/cm2 for absorbed fluence (at absorption coefficient equals to 15 %). We believe that the reason of this discrepancy between the experimental data for subpicosecond pulses and our simulation results is impossibility of the direct comparison of the experiments with pseudo-one-dimensional calculations. Such approach does not take into account the peculiarity of the surface distribution of laser fluence. The total description of the experimental data may be given by the full-scale atomistic simulation of such process with μm length scales in all three dimensions. At the present time, all similar simulations are performed with one-dimension (hydrodynamic model) or quasi three-dimension (atomistic simulation with periodic boundaries) approach, and the latter was adopted in this work. The surface mechanical/thermodynamic relaxation [18, 50] and nucleation process [51] are not taken into account during the simulations. Thereby, the significant features of surface modification are not considered in models. The first steps in this direction were taken recently in the work [52, 53]. However, one-dimensional calculation still remains the powerful tool for investigation and may describe some general features of surface modification.

Along with ablation process, we investigate the melting of metals at the irradiation by laser pulse. Results of the simulation are shown in Fig. 9. One can notice that the values of melting depth calculated for Al, as a function of absorbed fluence, are just above the dependence obtained from the one-dimensional hydrodynamic simulation [35]. However, the calculated threshold fluence of melting (about 6.5 mJ/cm2) is close to the value, which was obtained in work [35]. The threshold fluences of melting for the metals are different by several times. The reason of the lower threshold fluence of melting for aluminum is in a small value of K e and a high value of G, compared to gold. Aluminum accumulates the absorbed energy of laser pulse near surface while in gold the energy spreads from surface into bulk. In this case, the maximum of ion temperature for aluminum is higher than the maximum of temperature for gold at identical fluences. We believe that threshold fluence of melting determines the area of surface modification. Therefore, the size of modification area for aluminum is bigger than for gold. However, profile of the modification is determined by the threshold fluence of ablation and the type of ablation. The experiment carried out in this work shows that the surface modification may occur without ablation. In this case, the modification depth is much smaller than at ablation.

5 Comparison of simulation result with experiment

In this work, the four model situations are considered according to experimental measurements. The laser pulse interacts with gold or aluminum surface. The total energy E of pulse equals to 24 or 48 nJ. The profile of absorbed fluence is accepted as Gaussian distribution on the surface. The performed analysis provides the qualitative description of the modification process, and it allows to find the general features of the modification at laser irradiation.

We can apply the following equation for E:

$$E = \int\limits_{0}^{ + \infty } {F \cdot 2\pi r} {\text{d}}r = \int\limits_{0}^{ + \infty } {F_{0} \exp [ - r^{2} /R^{2} ] \cdot 2\pi r} {\text{d}}r = F_{0} \pi R^{2} ,$$
(5)

where F 0 and R are parameters of Gaussian distribution of absorbed fluence F. For best description of experimental data, we suppose that the value of R is equal to 4.43 μm. Figure 10 illustrates the analysis performed for the four model situations. The threshold fluence of melting determines the size of modification area on a surface.

Fig. 10
figure 10

The simulated profiles of absorbed fluence F on the sample surface at absorbed energy E = 48 nJ and E = 24 nJ. Numbered lines in the figure indicate the threshold values: 1 threshold fluence for melting of Al, 2 threshold fluence for melting of Au, 3 threshold fluence for ablation of Al, 4 threshold fluence for ablation of Au. Average sizes of the modification area (estimated from the experiment) are shown by the bold horizontal lines

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For pulse with E = 48 nJ, the radii of modification are equal to 3.9 and 7.2 μm for Au and Al, respectively. It is necessary to note that the maximum fluence F 0 in center of the laser spot reaches the threshold value of ablation for Al. The simulation performed for aluminum gives about 50 nm for the crater depth in the center of the laser spot. These results agree well with experimental data. However, the full-scale modeling (3D simulation with μm length scales in all three dimensions) is necessary for description of the full surface modification. For gold, the modification occurs without ablation. The modification is probably due to the splash of the molten gold under intensity gradient (i.e., pressure/temperature gradients) across the laser beam.

For pulse with E = 24 nJ, the radius of the surface modification for Al equals to 6.1 μm. The modification occurs without ablation. For gold, the modification of surface is practically absent since modification radius equals 0.2 μm. These results agree well with the experimental data obtained in this work. Good agreement with the experiments is also observed in case of the pulse energy E = 72 nJ.

We can apply the analysis for the search of the threshold fluences of modification and photoluminescence of LiF crystal. For the accepted approximation (Gaussian profile of fluence and R equals 4.43 μm for both energies of pulses), we obtain threshold values equal to 3.5 mJ/cm2 for fluencies of modification (i.e., melting) and to 0.15 mJ/cm2 for photoluminescence of LiF.

For comparison with the experimental part of the work, we use the following formula with the integration of F(r):

$$F_{\bmod } = \frac{1}{{\pi r_{\text{m}}^{2} }}\int\limits_{0}^{{r_{\text{m}} }} {F(r) \cdot 2\pi r{\text{d}}r}$$

where r m is a radius of the modification. For pulse with E = 48 nJ, r m equals to 3.9 μm for Au and to 7.2 μm for Al. It is worth noting that the calculated average fluences F mod are higher than the values found at initial estimation in experimental part of the work. The calculated F mod is about 50 and 30 mJ/cm2 for Au and Al, respectively. For pulse with E = 24 nJ, the calculated F mod of Al equals to 20 mJ/cm2. From the experimental part, the following estimations were obtained: 21 ± 5 and 14 ± 5 mJ/cm2. Some discrepancy between the experiment and the simulation is the consequence of the uncertainty in the surface profile of fluence. In what follows, the form of profile is supposed to be specified with more accuracy.

6 Conclusion

The modification of the metal surface under SXRL pulse is a complex process. The general properties of the surface modification obtained from the experiment and from the theoretical modeling are in close agreement. In this work, the values (maximum depth and modification size) measured directly agrees well with the results of the theoretical modeling. It should also be noted that the comparison has been performed for pulses with identical energy, which is measured with good accuracy. However, the calculated average fluences F mod are higher than the values found from the initial analysis of the experimental data. The possible reason of this discrepancy is lack of knowledge of the surface distribution of the SXRL fluence. We distinguish the following features of metal surface modification by laser pulse:

  1. 1.

    The analysis of surface modification by laser pulse must be based on description of the profiles of absorbed fluence and the energy of pulse. Such description is more productive than use of the average fluence.

  2. 2.

    The threshold fluence of melting determines the zone of surface nanostructure modification. The boundary between the undisturbed zone and the modification zone may be calculated as a place where fluence equals threshold value for melting.

  3. 3.

    The modification of surface may occur without ablation. In this case, the depth of modification is small and close to attenuation lengths of irradiated metal for specific SXRL wavelength.

  4. 4.

    The ablation is nonlocal process as opposed to melting. The full-scale modeling with the micron size of the simulation box is necessary for calculation of full profile of the surface modification. However, the maximum depth of crater may be estimated as ablation depth at quasi three-dimensional simulation (atomistic modeling with periodic boundaries).

  5. 5.

    The average fluence of surface modification F mod for aluminum is lower than for gold, because of the larger size of the modification zone. The reason for this is that aluminum accumulates the absorbed energy of laser pulse near surface while, in gold, the energy spreads from surface into bulk.

The modified structure formed on Al and Au surfaces induced by the SXRL pulse exposure is quite particular. The nanoscale ripples are formed on Al surface in the relatively low fluence region at periphery of the modification zone. It means that for pulse with E = 48 nJ, it might be possible to increase the area of conical structures by focusing SXRL beam in larger spots, which allowed reducing the averaged fluence on Al surface and avoid the ablation in center. These ripple-like nanostructures created on the metals are interesting for use as a small catalysts for chemical reactions, when the surface is coated with catalysis materials such as molybdenum, palladium, platinum, and so on. A periodical structure could be produced on metal surfaces if soft X-ray interferometer with double pulses interference will be used. This periodical structure would be expected to work as a diffraction optic for X-ray region.

In the future, when control of the modification process by the SXRL pulse irradiation on metal surfaces will be well established, the SXRL beam would be an attractive candidate as a tool of micromachining, which enables fabrication of the three-dimensional nanostructures.