Role of the temperature dynamics in formation of nanopatterns upon single femtosecond laser pulses on gold

In this paper we investigate the role of two-temperature heating dynamics for formation of periodic structures on metal surfaces exposed to single ultrashort laser pulses.The results of two-temperature model (TTM) two-dimensional simulations are presented on the irradiation of gold by a single 800-nm femtosecond laser pulse the intensity of which is modulated in order to reproduce an initial electron temperature perturbation, which can arise from incoming and scattered surface wave interference. The growing (unstable) modes of the lattice temperature distribution along the surface may be significant in the laser induced periodic surface structures formation. After the end of the laser pulse and before the complete coupling between lattice and electrons occurs, the evolution of the amplitude of the subsequent modulation in the lattice temperature reveals different tendencies depending on the spatial period of the initial modulation. This instabilitylike behavior is shown to arise due to the perturbation of the electronic temperature which relaxes slower for bigger spatial periods and thus imparts more significant modulations to the lattice temperature. Small spatial periods of the order of 100 nm and smaller experience stabilization and fast decay from the more efficient lateral heat diffusion which facilitates the relaxation of the electronic temperature amplitude due to in-depth diffusion. An analytical instability analysis of a simplified version of the TTM set of equations supports the lattice temperature modulation behavior obtained in the simulations and reveals that in-depth diffusion length is a determining parameter in the dispersion relation of unstable modes. Finally, it is discussed how the change in optical properties can intensify the modulation-related effects.

Figure 1

Electron microscope image of a gold surface exposed to a single femtosecond laser pulse, with normal incidence, peak incident fluence $F_i\approx 4\mathrm{J}{\mathrm{cm}}^{-2},{\lambda }_{\mathrm{las}}=800\mathrm{nm}$, and ${\tau }_p\approx 100\mathrm{fs}$.

Figure 2

(a) Sketch of the numerical modeling geometry. (b) Evolution of the electron (dashed line) and lattice (solid line) temperatures on the surface of the gold slab. The laser pulse is shown by the dotted line (arbitrary units). The melting temperature is outlined by the horizontal dot-dashed line. The time moments of the laser-pulse termination $t_0=0.4\mathrm{ps}$ and electron-lattice thermalization $t_c\approx 30\mathrm{ps}$ are marked by vertical dashed lines. Absorbed peak fluence is $F_0=110\mathrm{mJ}{\mathrm{cm}}^{-2}$.

Figure 3

2D distribution of the gold lattice temperature with isolines at $t=1.7\mathrm{ps}$. The laser pulse comes from the left. Darker areas correspond to higher lattice temperature. Only one perturbation period along the surface and the first 100 nm from the surface toward the bulk are shown. Laser parameters are the same as for Fig. 2. The intensity modulation period is $\mathrm{\Lambda }=0.8\mu \mathrm{m}$.

Figure 4

Amplitude of the modulations of the electron (a) and lattice (b) temperatures on the surface of gold for different modulation modes (numbers correspond to those in Table 2). Simulations were performed with parameters summarized in Table 1.

Figure 5

Dispersion relation obtained from the averaged growth rate of single mode simulations performed with the same parameters as in Fig. 2. Numbering corresponds to the modes listed in Table 2. The cutoff for the growth of the modulation amplitude over the period $[t_0;t_c]$ is $k=k_{\mathrm{c}}\sim 20\mu {\mathrm{m}}^{-1}$.

Figure 6

Calculated in-depth profile of the electron (dashed line) and the lattice (solid line) temperatures in a gold sample at $t=20.2\mathrm{ps}$. The laser parameters used are the same as in Fig. 2. The thin dotted line is the ratio between the electron and lattice temperatures (right axis).

Figure 7

Two eigenvalues of the system (1) calculated for Au with ${\xi }^{-1}=0.15\mu \mathrm{m}$. Solid line, the real part of the eigenvalues; dashed line, the imaginary part of the eigenvalues. Blue and red lines correspond to different eigenvalues.

Figure 8

Evolution of the surface pattern, which emerges on gold surfaces upon single laser-pulse irradiation (Ti:sapphire, ${\lambda }_{\mathrm{las}}=800\mathrm{nm}$, pulse duration ${\tau }_p\approx 100\mathrm{fs}$, normal incidence) at different incident fluence values. (a) $F_i\approx 2\mathrm{J}{\mathrm{cm}}^{-2}$, (b) $F_i\approx 3\mathrm{J}{\mathrm{cm}}^{-2}$, (c) $F_i\approx 4\mathrm{J}{\mathrm{cm}}^{-2}$, (d) $F_i\approx 5\mathrm{J}{\mathrm{cm}}^{-2}$. The images correspond to the center of the laser spot; the laser beam profile was close to Gaussian.

Figure 9

Temporal evolution of the normalized amplitude of the modulations of the lattice temperature on the surface for different modulation periods (numbers correspond to Table 2). (a) Thermophysical parameters from Table 1. (b) Ten times smaller thermal diffusivity. The laser intensity (black dotted line) is shown in arbitrary units.

Figure 10

Temporal evolution of the normalized amplitude of the modulation in the lattice temperature for different modes at a higher peak fluence $F_0=400\mathrm{mJ}{\mathrm{cm}}^{-2}$. The normalization has been performed with respect to the amplitude value just after the end of the laser pulse, $t_0$. Electron-lattice thermalization takes $\approx 55$ ps at this fluence.

Figure 11

Comparison of the modulation dynamics of the lattice temperature for $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\exp }=0.8\mu \mathrm{m}$ calculated with the constant (dot-dashed lines) and temperature-dependent (solid lines) optical properties. The initial laser intensity amplitude is $\eta =0.05$. (a) Time evolution of the modulation amplitude, $a_l$, on the surface of gold at $F_i=110\mathrm{mJ}{\mathrm{cm}}^{-2}$ (thin lines) and $F_i=400\mathrm{mJ}{\mathrm{cm}}^{-2}$ (thick lines). (b) Lattice temperature profiles along the surface at different time moments for $400\mathrm{mJ}{\mathrm{cm}}^{-2}$.