Classical Molecular Dynamics Simulations of Surface Modifications Triggered by a Femtosecond Laser Pulse

Abstract

This work is devoted to classical molecular dynamics simulations of surface modifications (craters) drilled by single femtosecond laser pulses in silicon and diamond, materials relevant for numerous industrial applications. We propose a methodology paving the way towards a significant decrease in the simulation computational costs, which could also enable a precise estimation of the craters’ size and shape.

1. Introduction

Modern ultrashort laser pulses can deposit a large amount of energy into a very small volume of a solid material. This enables a precise local control of radiation-induced material damage, with numerous applications for femtosecond laser machining [1,2,3] and nanostructuring [4,5,6]. Moreover, advanced techniques like Femtosecond Laser Direct Writing (FLDW) enable efficient and precise material processing in industrial applications [7], with large flexibility. For such applications, guiding computer simulations, performed before experimental measurements, may allow us to avoid some of the expensive experimental ‘blind’ tests, in this way broadening the applicability and decreasing the overall cost of the laser processing. It would, therefore, be beneficial to have a predictive and computationally efficient simulation tool: (i) enabling to follow the material evolution up to an equilibrium state after the laser irradiation, and (ii) allowing for a flexible experimental geometry at a reasonable computational cost.

The existing computational models developed to describe the evolution of laser-irradiated solid materials are numerous. The approaches widely used due to their simplicity and efficiency are continuum models, such as Two-Temperature Model [8,9] and various hydrodynamic approaches [10,11,12,13]. They are relatively simple in their implementation, have low computational cost, and are capable of being implemented at the experimentally relevant spatial and temporal scales. However, their applicability is restricted in the regime of femtosecond timescales and μm-nm spatial scales due to the assumption of near-equilibrium conditions during the ongoing ultrafast phase transition processes (melting, spallation, ablation). Such assumption becomes even less justified under intense laser irradiation, when strong photoabsorption within the investigated material drives it into extreme transient states [14]. Non-equilibrium phase transition processes become then predominant [15]. The phase transition timescales can then be shorter than 1 ps, and the new-phase nuclei can reach a size corresponding to several interatomic distances only [16].

In contrast, atomistic modeling does not rely on the assumption of equilibrium as it always tracks particle trajectories. Recently, the ab-initio molecular dynamics (MD) approaches became more widespread thanks to the increasing computational power of modern computers and the ongoing advances in their algorithmic components. By relying on the quantum-mechanical description of a many-electron system, they provide high precision and versatility [17,18,19,20,21]. High computational cost, however, still limits their applicability to smaller systems (typically, with up to 2000 atoms). Even if utilizing supercomputer facilities [22], the size of simulated systems is much smaller than the typical size of many systems studied experimentally. The same problem of high computational costs also affects the simulations using a simplified Tight-Binding Hamiltonian [23,24], although to a lesser degree.

The classical molecular dynamics (MD) then remains the simplest and computationally most efficient atomistic approach. It is based on the solution of Newtonian equations of motion for each particle in the system, using empirical or ab initio interatomic potentials [25]. In MD, an interatomic potential fully defines the material properties. Therefore, a reliable interatomic potential is crucial to achieving a quantitative description of laser-induced ultrafast processes and eventual material damage. It was already demonstrated in several cases that one could describe successfully the experimental findings in the laser-processing experiments with properly defined interatomic potentials [26,27]. However, those works usually relied on a specific geometry of the setups and took advantage of their intrinsic symmetry, in order to decrease the computational costs. If one cannot rely on such symmetries, a full 3D simulation is necessary to reproduce the experimental results. If the temporal and spatial scales considered are large, such simulation lies beyond the capacities of the existing computational tools, even if using the classical MD.

In this paper, we propose a simulation scheme based on the classical MD approach that should overcome the computational bottleneck. We will demonstrate its performance on the example of two widely used and studied semiconducting materials: silicon and diamond. The simulations are based on the free and open-source classical MD code, LAMMPS [28,29], which enables efficient and reliable atomistic simulations of various materials and of their response to laser-induced heating. In the next section, we describe in detail the proposed simulation scheme and its assumptions.

2. Simulation Scheme

In order to improve the computational efficiency of classical MD simulations, we propose to scale down the simulated volume by decreasing all its three dimensions proportionally (by the same factor), while keeping the geometry intact. This approach is based on the principle of similitude [30] which states that two systems exhibit similar behavior if they have a geometric, kinematic, and dynamic similarity. I.e., these conditions are fulfilled if the two systems are made geometrically similar, and if the ratios of all the pertinent forces are the same in the two systems. In our case, this, in particular, implies that the “scaled-down” simulation volume should still represent a sufficiently large volume in the material, and not be a conglomerate of a few atomic layers which have very different properties than bulk material.

We apply this approach to predict crater shapes in silicon and diamond created by a single femtosecond laser pulse arriving at normal incidence to the material and focused onto a micrometer spot. For simplicity, we assume that the absorbed laser energy is distributed homogeneously in the irradiated volume, which is a cylinder extending from the material surface down to some depth in the material. We will call this volume the “primary interaction volume”. The cylinder diameter corresponds to the laser spot size. Its depth can be chosen arbitrarily. Note that according to Ref. [7], the FLDW allows us to ablate a specified amount of material with extensive dimensional flexibility.

In what follows, we will demonstrate that performing long-timescale simulations for the laser-heated scaled-down primary interaction volume can provide accurate predictions for the primary interaction volume—of a much larger size. The results are then obtained at a much lower computational cost than that estimated for the full-scale calculation. For this purpose, one should: (i) decrease the size of the laser spot and of the primary interaction depth by the same factor, (ii) perform a simulation to predict the size of the created crater, i.e., its width and depth, and (iii) scale them up with the same factor, in order to obtain predictions for the original laser spot and the original primary interaction depth. We expect that this scheme could provide a way to significantly reduce computational efforts in future studies of laser-induced damage in solid materials.

For the simulations, we utilize a classical MD code, LAMMPS, using well-established interatomic potentials [31,32,33]. Figure 1 shows a schematic representation of the simulation setup. The laser irradiation comes along the Z axis, perpendicular to the target surface. The primary interaction volume has a cylindrical shape. The LAMMPS simulation supercell (shown with blue lines) is a square box of sufficiently large size to account for the subsequent cooling of the system due to the atomic heat diffusion. As we use periodic boundary conditions (PBC) in the (X,Y) plane, the lateral size of the simulation supercell should also be large enough to guarantee that the supercell and its images do not influence each other significantly.

Below we list all assumptions currently applied to simulate the evolution of the laser-heated primary interaction volume:

• A. Primary interaction volume is a cylinder with a homogeneous distribution of absorbed energy inside.
• B. Classical interatomic potentials (MD) are applied.
  As discussed above, the classical approach is optimal due to the high computational cost of other atomistic approaches and the limited applicability of the continuum methods. Here, we also do not perform explicit modeling of non-thermal effects which could induce a change in the interatomic potential due to a strong electron excitation (cf. [34]). This is justified, as, on long timescales, we expect that the nonthermal effects will lead to a similar atomic structuring as after thermal melting.
  The classical interatomic potential for silicon, which we mainly utilize in the paper [31], has been first introduced in 1985 and is still widely used to describe this material (see, e.g., [35]). It reproduces accurately not only the stable, diamond-like crystal structure of silicon but also its thermophysical properties such as equilibrium melting temperature, melting curve slope, and heat capacity [35]. Therefore, we decided to base our study on the Stillinger-Weber potential. However, in Section 3.3 we will also demonstrate that using an alternative potential [32] still supports the correctness of the similitude approach.
• C. Thin-film supercell
  The thin-film supercell has two free surfaces (parallel to the XY plane) and four periodic boundary surfaces (Figure 1). Such boundary conditions provide an efficient method to decrease the number of atoms in the simulation. Convergence studies performed by varying the supercell size guarantee to exclude the final-size effect.
• D. Pressure wave cannot leave the supercell and is reflected back from the boundaries.
  In principle, one could implement dynamic non-reflective boundary conditions [36,37] in order to remove the reflected pressure waves. However, in all the cases analyzed in this work, the supercell was large enough to absorb most of the pressure waves (or the waves were weak enough to neglect them). Those observations were confirmed by the dedicated convergence studies.
• E. Absorbed laser energy remains within the supercell.
  We have verified that the supercell is large enough to absorb most of the heat without changing the overall atomic temperature or final crater shape in a noticeable way. If necessary for future applications, one could still add a heat sink at the boundaries, like it was done in [27], in order to fully exclude this effect.
• F. Putting the total momentum of the crystal to zero at each time step.
  This is necessary to avoid the drift of the computational layer supercell, caused by the momentum conservation and the escape of the ablated atoms. The ablated atoms are not affected by the change in the total momentum of the crystal.
• G. Approximate treatment of electron-ion coupling
  We approximate it as atomic heating with a constant rate, taking ∼0.5 ps in total. The description can be improved by using a combined MD-TTM model, similar to that published in [35]. Here, a three-dimensional treatment would be necessary. However, as we are interested in late-time predictions, the exact modeling of the heating mechanism is not necessary.
• H. Neglecting electron transport
  Long-timescale diffusive transport of electrons may lead to an additional electron cooling of atoms. The improvement proposed in the preceding item would also allow to include electron transport.
• I. Actual simulation timescales (<10 ns) are shorter than those necessary to accomplish a full resolidification of the material (μs timescales).
  Dedicated test simulations performed by us indicate that, for all the cases studied here, the resolidification should not significantly change craters’ shapes. It is then not necessary to run the simulations up to μs. This simplification allows us to reduce the overall computational costs. However, an ultimate validation of this assumption would require experimental verification.

3. Results

3.1. Craters in Silicon Obtained for the Primary Interaction Depth of 590 nm

Let us first consider a silicon film irradiated by a laser beam with the spot size of 1 μm diameter in the (X,Y) plane and with the primary interaction depth of 590 nm, extending along the Z axis. As explained above, we assume that the energy absorbed from the laser is homogeneously distributed within a cylinder with a diameter of 1 μm and a depth of 590 nm. It is computationally not feasible to simulate the evolution of such a large material volume with an atomistic model. Therefore, we use the hypothesis of similitude: we scale the cylinder size down by 80 times in all three dimensions. The simulated interaction volume then becomes a cylinder with a diameter of 12.5 nm and a depth of 7.4 nm, containing about 46 thousand atoms. The whole MD simulation supercell has the dimensions of 54 nm by 54 nm by 27 nm along the X, Y, and Z axes, respectively, and consists of $100\times 100\times 50$ crystal unit cells, containing 4 million atoms in total. Such volume can be simulated on a modern supercomputer within a reasonable time. The cylinder is located centrally within the whole MD simulation supercell. For ultrashort laser pulses, the laser energy is first absorbed by the electronic subsystem in the crystal, on timescales of a few tens fs. For simplicity, we assume in our model that it happens instantaneously, i.e., at $t=0$ ps when the lasing starts. Afterward, the electrons heat up the atoms via the electron-ion energy exchange. The corresponding atomic heating occurs in our simulation between the times, $t=0$ ps and $t=0.5$ ps. The latter corresponds to the timescale of the electron-ion thermalization in silicon, i.e., the temperature equilibration between the electronic and atomic subsystems.

Figure 2 shows snapshots of the section through the irradiated silicon crystal, recorded at different time instants. The absorbed dose within the primary interaction volume was 4 eV/atom. All the snapshots in this paper were generated with the help of GNU Parallel code [38]. The displayed part covers the primary interaction volume and its close surroundings in the (XZ) plane within a 3-Å thick layer extending along the Y axis. The orange rectangle corresponds to the primary interaction volume. Different coloring of the atoms indicates their different kinetic energy (given in eV). The times shown are: $t=0$ ps (equilibrium material), $t=50$ ps and $t=450$ ps.

As seen in Figure 2, the dose of 4 eV/atom leads to a creation of a well-shaped crater in silicon with a diameter of 13 nm (at the top) and a depth of about 7 nm (in the center). The two last snapshots show that the size of the crater does not change for at least 400 ps, indicating that the crater geometry has already become stable. Longer-timescale simulations are then not necessary.

Now, let us study the convergence of the above results with respect to the size of the primary interaction volume (i.e., the size of the cylinder), in this way testing the similitude hypothesis. Figure 3 shows the atomic snapshots for the same average absorbed dose as in Figure 2 (4 eV/atom), but with a twice larger cylinder: 25 nm in diameter, 15 nm height, consisting of about 363 thousand atoms. The cubic MD supercell is chosen to have the dimensions of 54 nm by 54 nm by 54 nm (consisting of $100\times 100\times 100$ crystal unit cells, with 8 million atoms in total), in order to provide a larger environment for cooling down of the primary interaction volume (for more details, see Appendix A).

Figure 3 shows a smoothly-shaped crater with a diameter of about 27 nm (at the top) and a depth of 14 nm (in the center). These values are approximately twice larger than those obtained for the crater shown in Figure 2. This is exactly what we would expect if the hypothesis of similitude worked: for a primary interaction volume twice larger in all dimensions, we should get a crater twice larger in all dimensions, and of a similar shape.

Note that, in this case, it takes a longer time to achieve a stable crater than in the case of the smaller cylinder (Figure 2). However, the final crater shape is very similar to the final crater shape obtained for the small cylinder.

The actual result indicates that the hypothesis of similitude works for this simulation setup. To emphasize, scaling back the predicted size of the silicon craters by 80 times (Figure 2) or 40 times (Figure 3) indicates that the original laser-induced crater should have had the diameter of 1.04–1.08 μm and the depth of about 560 nm, i.e., comparable with the size of the original interaction volume. However, for full verification of the hypothesis, one would need to perform a dedicated experiment.

3.2. Craters in Silicon Obtained for the Primary Interaction Depth of 295 nm

Below we will test the similitude hypothesis for a different primary interaction depth. We perform a similar series of simulations, corresponding to the case when the laser energy has been homogeneously absorbed in a cylinder with a diameter of 1 μm and a depth of 295 nm. The corresponding simulated primary interaction volume, scaled down by 40 times, had a diameter of 25 nm, and a depth of 7.4 nm. It contains ∼182 thousand atoms. The cylinder is scaled down by 80 times and is then a cylinder with a diameter of 12.5 nm and a depth of 3.7 nm, containing ∼23 thousand atoms.

Figure 4 demonstrates that the similitude hypothesis also works for this primary interaction depth. It shows the snapshots of the section through the irradiated silicon crystal for two different laser-irradiated volumes, corresponding to the primary cylinder scaled down by 80 or by 40 times. In both cases, the craters created are well-shaped, and their sizes scale by two in all dimensions, as expected: 13 and 27 nm in width and 3 and 6.5 nm in depth, respectively.

Similarly, by scaling up the size of the silicon craters (Figure 4) by 40 or 80 times, back to the expected experimental setup size (1 μm beam spot and 295 nm depth), we predict that the original laser-induced crater should have the width of 1.04–1.08 μm and the depth of 240–260 nm. Note that the crater geometry does not change significantly after 50 ps, similarly, as for the results described above, so that we may conclude that a stable crater geometry was reached. As in the previous section, these results indicate that our suggested simulation scheme, based on the hypothesis of similitude, is applicable to the considered setup.

3.3. Testing Similitude for Silicon Using the Tersoff Interaction Potential

In this section, we repeat the simulations from Section 3.1 (for the primary interaction depth of 590 nm), using Tersoff interaction potential [32]. Figure 5 shows the results for the same computational setup as in Figure 2 and Figure 3 (compare left and right snapshots respectively). The resulting prediction for the experimental crater is noticeably different from that obtained with the Stillinger-Weber potential. This can be expected as the two potentials have very different thermophysical properties. However, the similitude principle still holds in these simulations: the size of the primary interaction volume scaled down by 80 or 40 times yields the dimensions of the craters created in silicon scaled down in the same way. The simulation set, therefore, predicts a crater of about 1.12–1.16 μm width and 640 nm depth in the center.

3.4. Craters in Diamond Obtained for the Primary Interaction Depth of 181 nm

Finally, we performed the same tests for a diamond using the available LCBOP interaction potential [39]. In this case, we again assumed that the laser spot size had a 1 μm diameter, but we chose the primary interaction depth to be smaller (181 nm) than in the previous cases, in order to test how the similitude hypothesis applies for such a short depth. Figure 6 shows the snapshots obtained for the 120 times smaller primary interaction volume (left) and 80 times smaller one (right). Again, we verified that the depicted geometries did not change over 100 ps timescales. Despite single random carbon chains appearing in the small crater, which are probably not detectable, the overall shape and size of the crater obey the similitude principle quite well: scaling of both the simulated craters back to the original size yields the same result of 960 nm crater width and of about 140–170 nm depth. We conclude that the hypothesis of similitude seems reliable also in this case. However, an experimental verification would be necessary for the final validation.

4. Conclusions and Outlook

In order to improve the computational efficiency of classical MD simulations on laser-drilled craters in silicon and diamond, we proposed to scale down the size of the simulated volume, applying the principle of similitude. The results of our simulations indicate that this methodology can indeed significantly decrease computational costs for the considered computational setups, at the same time providing scalable predictions for the created crater’s dimensions. We checked that after downscaling the primary interaction volume by 60 times for silicon and 100 times for diamond (not shown), we also obtained proportionally scaled craters, in accordance with the similitude hypothesis. Still, dedicated computational studies are required to test the applicability of the hypothesis for more complex systems than the homogeneous materials considered here. In addition, experimental validation is required to ultimately confirm the applicability of the similitude principle for the description of laser-created craters. We hope that this work will inspire dedicated experimental efforts.

In summary, we believe that this study paves the way to more efficient computer simulations of craters created in silicon and diamond by an ultrashort laser pulse. If successful, our computational approach could be utilized in the future to guide material processing with ultrafast laser pulses, minimizing or, in certain cases, even eliminating the need for expensive experimental testing.