Large Scale Molecular Dynamic Simulation (MD-TTM)

A MD method is crucial for understanding the creation of complex surface mor-phologies and to study the influence of the energy deposition and transfer. In this collaborative work an atomistic-continuum MD-TTM approach, run by D.S. Ivanov, is used allowing a macroscopic modeling of material transport resulting mainly from laser energy induced phase changes. A direct comparison of experiment and simula-tion is obtained by utilizing the special geometry of one dimensional periodic nano structures induced by a sinusoidal grating-shaped intensity distribution, allowing a minimized simulation volume, depicted in Figure 2.11(a).

The MD-TTM treats the atomic motion in a MD simulation which is combined

sinusoidal fluence profile

continum approximation (TTM)

MD-TTM simula�on (PBC) (PBC)

MPI

r_cut-off

S(r,t)

E _e-ph

P, ρ T ,T_{e a}

X Y

Z

F_inc.

gold sample

X Y

(a) (b) (c)

Figure 2.11 Sketch of the sinusoidal intensity distribution, depositing energy in the slice of the simulated volume in (a), in (b) the boundary conditions applied around the atomistic-continuum MD-TTM simulation are shown. In (c) a single simulation core is depicted in 2-D, connected by a massage passing interface (MPI) with the neighboring processor. Within one core sub-cell parameters like the temperature and the pressure are averaged over a cut-off range ofrcut-off, including 26 neighboring cells and connected to the TTM.

with the description of the laser excited free electrons by a TTM. The principle of a MD simulation is solving Newton’s equation for each atom described by a potential.

Therefore MD simulations inherently include material properties like phase transi-tions, mass transport, surface tension, latent heat (for solid-liquid transition energy to separate bonds and for a free movement) and therefore have great advantage over other methods like finite element, or hydrodynamic simulations, which lack some of the above mentioned properties, especially at non-equilibrium conditions [72, 82].

The drawback of a MD simulation is the immense calculation expense, therefore some effort is undertaken to minimize the MD simulated volume and needed calcu-lation time:

A slice geometry is specifically chosen in this atomistic-continuum MD-TTM ap-proach, utilizing the sub-µm (270−500) nm periodicity along one direction and only a few nm in one orthogonal direction (thin slice), as depicted in Figure 2.11(a). Perpendicular to the modulation direction the grating has no difference in the intensity profile, thus the slice thickness is set to a minimal value of a few tenths of nm in order to allow the formation of bubbles in the overheated melt, an important driving force for the material lift off, besides the relaxation of the laser induced stresses.

None reflective boundary conditions (NRC) at the bottom of the simulated vol-ume to the continuum mimic a bulk material in depth, absorbing the pressure wave energy, avoiding reflecting it and thus reducing the simulation of MD volume [72, 83].

Linear scalibilty of the model is achieved by parallelizing the calculation by dividing the simulation volume in cells, depicted in Figure 2.11(b), which are then calculated by a single processor. The information about material transport is realized by a message passing interface (MPI) as described in [82].

An Embedded atom model (EAM) is used to describe the potential used for solv-ing the Newton’s equation for each atom. An EAM potential is implemented in the MD part, as described by V. V. Zhakhovsii et al. [84] for gold. The EAM potential describes material parameters like the melting temperature or the lattice constant with high precision.

The energy coupling from the electronic system to the lattice, described by MD, is now realized by the energy transfer given by ∆Ee−ph, determined in sub cells with a cut-off range r_cut-off of roughly one nm, depicted in Figure 2.11(c). The energy

∆Ee−ph enters the Newton equation in form of a velocity scaling, increasing the temperature. This atomistic-continuum approach is than described by the following coupled differential equations:

Ce(Te)∂Te

∂t = ∇(K_e(Te,Ta)∇T_e)−G(Te) [Te−Ta] +S(r,z,t,Te) , (2.66) mi

d²~ri

dt² = F~_i^EAM+ξmi

d~r^T_i

dt , i= 1,2,...,N_atoms , (2.67) withξmid~r^T_i

dt describing the thermal velocity of the atoms andF~_i^EAMthe force result-ing from the EAM potential [85]. The source term of the induced intensity gratresult-ing is, incorporating the function f_z(z) from Eq. (2.63) andf_t(t) from Eq. (2.64), given by,

S(r,z,t,T_e,ω) =E_inc(1−R(T_e,ω))f_p(y)f_z(z,T_e,ω)f_t(t) , (2.68) with the sinusoidal variation in y-direction, given by

fp(y) = cos² yπ d_p

!

, (2.69)

with the periodicity d_p. The coordinate in y-direction is defined from −d_p/2 to dp/2. In the x-direction no change in the intensity is present. To the sides periodic boundary conditions (PB) in a distance of the periodicity dp, are applied. In the simulation a free movement of the atoms is allowed in the top direction, described by free boundary conditions (FB). At the bottom the above introduced NRB are applied. The largest simulation volume in x, y and z-direction is (40×500×200) nm, consisting of roughly 240 million atoms. For this simulation volume, when 480 cores are running parallel, one hour calculation time computes ∼1 ps in the model. All molecular dynamic simulations presented in in this work are conducted by D.S.

Ivanov [36, 72, 85].

Laser pulse generation principles and the laser systems used for the experiments in this work are shortly presented and the pulse and sample characterization meth-ods are described. A mode locked Ti:sapphire laser system serving as front-end is described in Section 3.1. Its pulses are launched into a hollow core fiber setup delivering sub 4 fs pulses, after compression, as described in Section 3.1.5. These are utilized for pump-probe experiments with a broad spectral bandwidth. For the structuring the frequency tripled pulses of the same Ti:Sapphire system are ampli-fied in a special ultrashort pulse excimer amplifier to high energies, which utilizes the short wavelength and high pulse energy delivered by KrF excimer discharges as, described in Section 3.1.4. The measurement of these versatile pulses requires dif-ferent setups for pulse characterization which are shortly introduced in Section 3.2.

The method utilizing these ultrashort UV pulses for a nano-structuring of surfaces, by a two beam interference method is described in Section 3.3. In the last part of this chapter the methods required for detailed sample characterization of periodic nano-structures are presented in Section 3.4.

3.1 Generation of Ultrashort Laser Pulses

Within ultrashort laser pulses extreme intensities can be obtained since the ampli-tude maximum of different spectral properties are temporally in phase, construc-tively interfering. Mathematically, when using an infinite number of wavelengths, a delta function in time is obtained, by applying a Fourier transform to a constant and infinitely broad spectral intensity. Both conditions can not be found in reality.

This characteristic leads to the frequency time relation which is in its nature similar to the Heisenberg‘s uncertainty principle [86], this time bandwidth product (TBP) can be described by

∆~ωp·∆tp ≥ ~

2, (3.1)

∆ν_p·∆t_p ≥ T BP_p, (3.2)

∆λp

c

λ² ·∆tp ≥ T BPp, (3.3)

with the index p describing a simplified square pulse in temporal and spectral do-main. The pulse width ∆tp =τp therefore is limited by the bandwidth and a great effort is undertaken in laser science to create broader laser spectra to obtain shorter laser pulses. The methods used in this work to obtain sufficient pulse energy and

spectral bandwidth to structure surfaces and measure events on the fs time scale are described in the following [45].