# Nonequilibrium electronic properties and stability consequences in metallic crystalline binary alloys under ultrafast laser excitation

Djafar Iabbaden, Arshak Tsaturyan, Jean-Marc Raulot, Jean-Philippe Colombier

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Djafar Iabbaden, Arshak Tsaturyan, Jean-Marc Raulot, Jean-Philippe Colombier. Nonequilibrium electronic properties and stability consequences in metallic crystalline binary alloys under ultrafast laser excitation. Journal of Alloys and Compounds, 2024, 1010, pp.177175. 10.1016/j.jallcom.2024.177175. ujm-04839267

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Submitted on 15 Dec 2024

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# Nonequilibrium electronic properties and stability consequences in metallic crystalline binary alloys under ultrafast laser excitation

Djafar Iabbaden<sup>a, *,</sup>, Arshak Tsaturyan<sup>b</sup>, Jean-Marc Raulot<sup>a</sup>, Jean-Philippe Colombier<sup>b, *</sup>

## ARTICLE INFO

**Keywords:**

- Crystalline metallic alloys
- Electronic properties
- Density Functional Theory
- Molecular dynamics
- Laser-matter interaction
- Ultrafast phenomena

## ABSTRACT

This study undertakes an exhaustive exploration of properties under electron-phonon nonequilibrium for a series of ten industrially pertinent crystalline alloys, namely AlCu, AlNi, AlTi, AuCu, CuNi, CuTi, NbZr, NiTi, NiZr, and ZrCu. This comprehensive investigation encompasses electronic heat capacity, electron-phonon coupling factor, excited electron density, electronic pressure, dielectric function, refractive index, extinction coefficient, reflectivity, and optical penetration depth. Employing *ab initio* simulations, we meticulously analyze the nonequilibrium optoelectronic attributes inherent to these alloys change under electronic excitation across the temperature range going from 0 to 50,000 K. Additionally, to contextualize the findings, we employ classical Molecular Dynamics coupled with a Two Temperature Model simulation, notably to deliver a deep comprehension of the intricate mechanisms of the CuTi alloy ablation induced by femtosecond laser irradiation.

## 1. Introduction

Ultrafast laser technology has become a hot topic in the field of material science by providing a powerful tool for investigating the intricate interactions between intense subpicosecond focalized laser pulses and crystalline materials with unprecedented precision [1-3]. These advances are found to be relevant in applications such as heteroepitaxy [4], welding crack [5,6], micromachining, nanostructuration, and surface functionalization [7-13]. Experimentally, a tremendous work has been garnered toward alloys to determine their optoelectronic properties in both crystalline and amorphous phases, especially at low temperatures [14-18]. Additionally, from a purely fundamental point of view, the femtosecond laser interaction community focus mainly on studying single metallic alloys [19-22] or gaped materials [23,24] for which the electronic properties are known and describe by free-electron gas approximation. However, many alloy structures are poorly explored mainly due to the lack of information about their optoelectronic properties. For multicomposition alloys to obtain these properties more sophisticated approximation need to use such as Density-Functional or Density-Functional-Perturbation theories.

The dynamics initiated by ultrafast lasers enable precise control during energy deposition. This leads to observation of phenomena such as melting [25-28], ablation [29-32], complex local structural

modifications primarily associated with the unique pattern of rapid heating-cooling cycles [33-36] such as wells or voids [12,37], vacancies [38,39], twinning [40], and devitrification [41-43]. These modifications can significantly impact the surface morphology by generating protrusions, textures, and periodic surface structures [44-49]. Moreover, all these alterations can be exploited to enhance wettability, reduce friction, or improve light trapping through the control of the laser parameters, namely absorbed fluence, pulse duration, number of pulses, and repetition rate. In this context, the nonlinear optical absorption mechanism dynamics produced by the ultrafast laser interaction with specific crystalline alloys are unique [50]. Therefore, to address the effect of femtosecond lasers on alloys, accurate and reliable atomistic simulations need to be achieved through a hybrid model combining Molecular Dynamics (MD) and Two Temperature Model (TTM), as proposed in the works of Anisimov et al. [51] and Ivanov et al. [52].

The knowledge of optoelectronic properties under electron-phonon nonequilibrium is primordial. By examining the concurrent experimental, theoretical, or even the plenteous available proposed approximations of the studies alloys nobody provided accurate parameters. Therefore, this work aims to fill this gap and provide complete set of parameters by using first principles methods to enable the modelling of femtosecond laser-matter interaction with these alloys. As mentioned

<footnote><sup>a</sup> LEM3, CNRS - Université de Lorraine - Arts et Métiers ParisTech, 7 rue Félix Savart, Metz 57070, France</footnote>

<footnote><sup>b</sup> UJM Saint-Etienne, CNRS, IOGS, Laboratoire Hubert Curien, UMR CNRS 5516, F-42023 St-Etienne, France</footnote>

<footnote>* Corresponding authors.
E-mail addresses: djafar.iabbaden@univ-lorraine.fr (D. Iabbaden), jean.philippe.colombier@univ-st-etienne.fr (J.-P. Colombier).</footnote>
previously, the electronic properties of single metallic target are well studied theoretically [24,53–60]. However, no such work exist yet in literature for multicomponent crystalline alloys apart from few experimental works that deal with the computation of electronic specific heat, thermal conductivity, and density of states at room temperature for alloys such as ZrCu, CuTi, and NiAl [61–64].

The paper is organized as follows. First, we provide a complete description of both *ab initio* and TTM-MD simulations. Then, we examine the electronic temperature dependence of electronic heat capacity, excited electrons, and electronic pressure. The electron-phonon coupling factor for 10 binary crystalline alloys and pure metals is then analyzed at different levels of laser excitation. In the final stage, optical properties such as the dielectric function, complex refractive index (n, k), reflectivity, and laser penetration depth are computed to understand how the alloys respond to incident radiation at different wavelengths. The logical continuation of the research is the application of the calculated parameters in practice, as reflected in the section devoted to the study of the ablation process of the CuTi alloy, accompanied by explosive boiling and fragmentation. The last section sums up the discussion in the conclusion.

## 2. Computational details

### 2.1. First-principles simulation

The computational investigation focuses on ten distinct binary crystalline alloys with equiatomic compositions, each consisting of elemental pairs as outlined in Table 1. These systems are studied using the framework of Density Functional Theory (DFT), extended to nonequilibrium conditions at nonzero electronic temperatures ($T_e$), to explore the behavior of these solid solutions under varying thermal and electronic environments. This approach provides a robust way to simulate the effects of ultrafast laser excitation, where electrons are excited to high temperatures while ions remain cold. The calculations are conducted using the ABINIT package [65,66], a well-established tool for plane-wave-based DFT simulations. The choice of Ultrasoft Pseudopotentials (PSP) [67,68] allows for an efficient and accurate treatment of the core electrons that significantly reducing computational cost while maintaining precision for transition metals. A Monkhorst – Pack method [69] is employed to sample the Brillouin zone with an $8 \times 8 \times 8$ k-point grid, ensuring accurate electronic structure calculations. A plane-wave cutoff energy of 30 Ha is applied which is sufficient for resolving the wave functions of these systems. The electronic states are broadened using a smearing parameter of 0.001 Ha to account for partial occupancies at finite temperatures. Convergence is rigorously ensured when the force on each atom falls below $10^{-9}$ eV/Å, providing a highly accurate description of the relaxed structures. Full geometric structure optimization, including cell volume and atomic positions is carried out

<table><caption>**Table 1**

Space group of the initial alloys structures constructed by the Atomsk package [72]. The structures contain either 2a or 4b atoms in the unit cell.</caption><thead><tr><th>Binary alloy</th><th>Space group</th></tr></thead><tbody><tr><td>Al<sub>50</sub>Cu<sub>50</sub></td><td>P 1</td></tr><tr><td>Al<sub>50</sub>Ni<sub>50</sub></td><td>P m – 3 m</td></tr><tr><td>Au<sub>50</sub>Cu<sub>50</sub></td><td>P 1</td></tr><tr><td>Cu<sub>50</sub>Ti<sub>50</sub></td><td>P 4/n m m:1</td></tr><tr><td>Ni<sub>50</sub>Ti<sub>50</sub></td><td>P m – 3 m</td></tr><tr><td>Zr<sub>50</sub>Cu<sub>50</sub></td><td>P 1</td></tr><tr><td>Al<sub>50</sub>Ti<sub>50</sub></td><td>P 4/m m m</td></tr><tr><td>Nb<sub>50</sub>Zr<sub>50</sub></td><td>P 1</td></tr><tr><td>Cu<sub>50</sub>Ni<sub>50</sub></td><td>P 1</td></tr><tr><td>Ni<sub>50</sub>Zr<sub>50</sub></td><td>C m c m</td></tr></tbody></table>

at 0 K using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization algorithm [70]. For the structural models, two subgroups of cubic B<sub>2</sub>-like crystalline supercells are selected from the Crystallography Open Database [71] as summarized in Table 1.

The ABINIT software package, designed to analyze electron-phonon coupling, provides a diverse set of capabilities for computing this property. It facilitates ground state calculations, evaluates electron-phonon matrix elements, and derives Eliashberg function using Density-Functional Perturbation Theory (DFPT) [73–76–79]. Moreover, the extraction of transport quantities properties such as electronic thermal conductivity, electronic resistivity, or McMillan critical temperature is possible [80,81]. The electron-phonon matrix elements are calculated for all binary alloys [82] using an unshifted k-point grid with dimensions of $32 \times 32 \times 32$ points with a subset of q-point mesh defined as $8 \times 8 \times 8$ at different $T_e$ excitation. In order to calculate the electron-phonon coupling strength ($G_{e-ph}$) within the electronic excitation range of $T_e=0–50000$ K, we used Simpson algorithm to evaluate integrals appearing in Equation (5).

### 2.2. Hybrid simulation: combination of two temperature model with molecular dynamics

The investigation focuses on local cavitation within CuTi triggered by ultrafast irradiation on a picosecond timescale. This exploration is crucial for understanding material’s dynamic response during the voids formation and liquid stretch. Thereby, a hybrid approach is employed by combining TTM model introduced by Anisomov et al. [51] with MD implemented in LAMMPS package [83,84]. The initial CuTi BCC-like structure supercell which has a box dimensions of $L_x \times L_y \times L_z=478.00 \times 46.57 \times 46.57$ nm³ having an orientation of [100], [010], and [001] is constructed using ATOMSK [72,85]. Segments of 191.20 nm (0.4$L_x$) and 47.80 nm (0.1$L_x$) are removed from the left and right sides along $L_x$ of this supercell respectively. This results in a final structure containing 33,750,000 atoms for laser-matter simulation. Free boundary conditions are set along the laser propagation direction (x), while periodic boundaries are imposed following y and z directions. The simulation domain is divided into two distinct regions: TTM-MD, where TTM and MD equations are solved simultaneously on a mesh of $100 \times 3 \times 3$, and the bulk region with a thickness of 4.5 µm, where only TTM equations are solved on a grid of $450 \times 1 \times 1$ to ensure heat dissipation. These TTM-MD and TTM regions are separated by an absorbing boundary condition region as described in the work of Schäfer et al. [86]. The electronic properties required to feed TTM-MD hybrid model, namely $C_e(T_e)$, $G_e(T_e)$, $R(T_e)$, and $l_p(T_e)$ are collected from the previous sections except $\kappa_e(T_e)=\kappa_0 T_e^\zeta$ that is computed using a linear approximation [87] where the value of $\kappa_0$ is used as reported in the work of Rogachev et al. [88]. The laser energy source term $S(x, t)$ follows a Gaussian-shape function as:

$$S(x, t) = \left( \frac{2F}{l_p l_p} \sqrt{\frac{\ln(2)}{\pi}} \right) e^{-4\ln(2)\left(\frac{t-t_0}{l_p}\right)^2} e^{-x/l_p} \quad (1)$$

where $t_p, t_0$ represent the pulse duration, and maximum time of the laser pulse at the surface of the target respectively. The absorbed fluence $F$ is defined as $F_{abs}=F_0[1 – R(T_e)]$, where $F_0$ represents the incident fluence and $R(T_e)$ is the reflectivity for a laser wavelength of $\lambda=800$ nm.

The initial supercell is first minimized using the conjugate gradient algorithm with a force norm stopping criterion of $10^{-6}$ eV/Å. Then, it was thermalized to 300 K within the isothermal-isobaric ensemble (NPT) during 50 ps. The conditions for laser-matter interaction, namely fluence and pulse duration are set to $F_{abs}=208$ mJ/cm² and $t_p=100$ fs respectively. After the energy deposition, the thermodynamics evolution of the system is monitored for 200 ps. An MD integration time step of 1 fs is maintained throughout all TTM-MD simulations, an Embedded Atom Method (EAM) interatomic potential as reported in the work of

<footnote>a 2 atoms.</footnote>

<footnote>b 4 atoms.</footnote>
Rogachev et al. [88] is employed, and the local atomic structures analysis is performed with OVITO [89].

# 3. Electronic properties

## 3.1. Electrons dynamics under excitation

The excited electrons are computed according to the equation mentioned in the work of Winter et al. [90]. In Equation (2), electrons above the Fermi level are considered excited and expressed as:

$$Z_{excite}(T_e) = \int_{\epsilon_F}^{\infty} g(\epsilon, T_e) f(\epsilon, T_e) d\epsilon \quad (2)$$

where $Z_{excite}(T_e)$ is the number of excited electrons and $\epsilon_F$ is the Fermi level.

It is important to highlight that in metallic alloys, s and p electrons are less tightly bound to the nucleus compared to d electrons. Thereby, they can easily delocalize, meaning they do not belong to any specific atom and are free to move throughout the entire metal. The free electrons ($Z_{free}(T_e)$) in contrary to $Z_{excite}(T_e)$ are computed according to equation mentioned in the work of Bévillon et al. [91] as:

$$Z_{free}(T_e) = \int_0^\infty g_{deloc}(\epsilon, T_e) f(\epsilon, T_e) d\epsilon \quad (3)$$

where $Z_{free}(T_e) = \sum_{i=s,p,f,g} Z_{free}(T_e)(i)$ and $g_{deloc} = g_s + g_p + g_f + g_g$ is the delocalized part of the DOS where only electrons belonging to s, p, f and g bands are considered.

The detailed analysis of these curves emphasizes different interesting trends where the individual elements of each alloy governs the excited electron evolution with $T_e$. Notably, each alloy exhibits a specific response to $Z_{excite}(T_e)$ as depicted in Fig. 1.

Fig. 1(a) illustrates $Z_{excite}$ dependence on $T_e$ for each crystalline alloy. As a first insight, $Z_{excite}$ value and trend for both AlCu and AlNi are almost identical. The curves display a gradual, linear and moderate increase in $Z_{excite}$ as $T_e$ rises, till reaching a maximum value of 2.2 at 50,000 K. AuCu follows linear trend as well with a lower value of $Z_{excite}$ which is 2.8. This behavior suggests that these alloys have a robust Fermi surface that prevents electrons from transitioning to higher energy states. For CuTi, NiTi, and ZrCu where the curves of $Z_{excite}$ exhibit identical linear behavior with increasing $T_e$. This behavior suggests a substantial impact on the excited electron population to the presence of Ti and Zr elements in these alloys. The $Z_{excite}$ curve of CuNi exhibits a linear behavior with $T_e$ where the value of 5.0 is reached at maximum excitation. Three crystalline alloys namely NiZr, AlTi, and NbZr have linear dependence at low excitation level ($T_e < 15,000$ K). However, at high excitation level (above 15000 K) the trend becomes non-linear with different saturation values of 7.1, 7.4 and 9.4 respectively. This behavior occurs when electron excitation remains modest and the number of accessible states near Fermi energy increases linearly with $T_e$. As $T_e$ surpasses 15000 K, $f(\epsilon, \mu, T_e)$ broadens, allowing more states to become thermally populated, but the number of accessible states begins to saturate. This results in a plateau in the growth of accessible states where most available states are already occupied.

Fig. 1(b) illustrates that as soon as $T_e$ increases to reach a high value of $5 \times 10^4$ K, the crystalline alloys exhibit distinct behaviors of $Z_{free}$. For AlNi, AlCu, CuNi, AuCu, the $Z_{free}$ increases to a final value of 1.6, 1.6, 1.75, and 1.3 respectively. Conversely, CuTi, NiTi, ZrCu, AlTi, NbZr, and NiZr alloys show relatively stable behavior with a smaller increase in free electrons where the number of free electrons remains nearly constant throughout the temperature range with a values of 1.0, 0.9, 0.9, 0.35, 0.3, and 0.2 respectively.

By excluding the free electrons from the total number of excited electrons, we can estimate the contribution from the more tightly bound d electrons. Therefore, alloys such as CuTi, NiTi, ZrCu, AlTi, NbZr, and NiZr do not exhibit a strong temperature dependence in their free

<figure><img src="image_3.png" /><figcaption>Fig. 1. Dependence of $Z_{excite}$ (a) and $Z_{free}$ (b) on $T_e$ ranging from 0 to 50,000 K for various crystalline alloys from Table 1.</figcaption></figure>

electrons. In AlCu, AlNi, and AuCu, the majority of excited electrons come from s and p electrons which explains the relatively small slope of $Z_{excite}(T_e)$. In contrast, more than 50 % of the excited electrons in CuNi, AlTi, NiZr, and NbZr originate from d electrons which means that $Z_{excite}(T_e)$ trends in these alloys are primarily driven by d electron contributions.

## 3.2. Electronic heat capacity

As mentioned earlier, the computation of electronic heat capacity has been extended to encompass 10 distinct crystalline alloys across a $T_e$ from 0 to 50,000 K. The electronic heat capacity, denoted as $C_e(T_e)$ is computed using Equation (4):

$$C_e(T_e) = \int_{-\infty}^{\infty} \frac{\partial f(\epsilon, \mu, T_e)}{\partial T_e} g(\epsilon, T_e) d\epsilon \quad (4)$$

where $g(\epsilon)$ is the DOS at energy level $\epsilon$, $\mu$ is the chemical potential at $T_e$ (See Figs. S1 and S2 shown in the supplementary material), and $f(\epsilon, \mu, T_e)$ is the Fermi-Dirac distribution function defined as $f(\epsilon, \mu, T_e) = \frac{1}{e^{(\epsilon-\mu/T_e)} + 1}$.

The results of this analysis are illustrated in Fig. 2. Upon initial inspection, it becomes evident that the values exhibited by these alloys surpass those of the individual metals comprising them. The $C_e(T_e)$ behavior of AuCu and CuNi demonstrate a linear progression across all temperature ranges. For all remain alloys $C_e(T_e)$ can be roughly divided
<figure><img src="image_4.png" /><figcaption>Fig. 2. Dependence of $C_e$ on $T_e$ ranging from 0 to 50,000 K for various crystalline alloys from Table 1.</figcaption></figure>

to two main stages, a linear part within the range of $T_e=0$ to ~ 10,000 K and a saturation regime for $T_e > 10,000$ K. Beyond this point, an asymptotic tendency following a polynomial equation emerges signifying a saturation phenomenon at higher $T_e$ (See Table S1 in supplementary material).

### 3.3. Electronic pressure

At lower temperatures, most electrons occupy states near the Fermi level, with lower energy states filled and higher ones vacant. As $T_e$ rises, more electrons acquire sufficient thermal energy to transition to higher energy states, altering the population distribution across the energy bands. Furthermore, this excitation leads to a substantial change in the lattice structure of a material due to the increase of the electronic pressure $P_e$. It can also alter the electronic properties, bond strengths, band gaps, and electronic transport properties [92,93]. Specifically, the results unequivocally establish that with the increase of electronic temperature, the system's pressure increases playing a significant role [94]. To investigate this characteristic, the following procedure is adopted for each alloy at every $T_e$. Beginning with the optimized unit cell volume and lattice parameter, the volume and total energy of the systems are artificially computed by changing the lattice parameter by magnitudes of ± 2, ± 1 %.

Fig. 3 reveals a consistent linear increase in $P_e$ across all alloys. The high values of pressures is attributed to CuNi with a peak value of 163 GPa, while the lowest value is observed for AlTi alloy at 50,000 K with a magnitude of 75 GPa.

### 3.4. Electron-Phonon coupling factor

The electron-phonon coupling factor ($G_{e-ph}$) assumes a pivotal role in delineating relaxation times of nonequilibrium states especially during the excitation. In this regime, energy transfer between the “hot” electron system and the cooler phonon system is governed by electron-phonon interactions, driving the relaxation of the nonequilibrium state towards thermal equilibrium. Since, the energy transfer from the electronic to the ionic subsystem during ultrafast laser excitation is dictated by the value of the electron-phonon coupling strength denoted as $G_{e-ph}$. In literature several approximations exist but the most widely used is based on free electron gas model which was introduced by Kaganov et al. [95]. This approximation is valid at lattice and electron temperatures significantly higher than the Debye temperature as mentioned in the

<figure><img src="image_5.png" /><figcaption>Fig. 3. Dependence of $P_e$ on $T_e$ ranging from 0 to 50,000 K for various crystalline alloys from Table 1.</figcaption></figure>

work of Lin et al. [53]. As reported in the work of Zhang et al. [96] the $G_{e-ph}$ depends on both electronic and lattice temperatures. However, at high temperature conditions this value is computed as:

$$G_{e-ph}(T_e) = \frac{\pi \hbar k_B \lambda(\omega^2)(T_e)}{g(\epsilon_F, T_e)} \int_{-\infty}^{\infty} -g^2(\epsilon, T_e) \frac{\partial f(\epsilon, T_e)}{\partial \epsilon} d\epsilon \quad (5)$$

where $\lambda(\omega^2)(T_e)=2 \int_0^\infty \omega a^2 F(\omega) d\omega$ is the second moment of the Eliashberg function with $\lambda$ representing the electron-phonon coupling constant, $\omega^2$ is the second moment of the phonon spectrum, $k_B$ is the Boltzmann constant, $\hbar$ is the reduced Planck constant, and $\epsilon_F$ is the Fermi level. The incorporation of finite electronic temperature effects is accomplished by utilizing the Fermi – Dirac distribution function of electrons f.

To bolster the foundation these investigations, we first turn our attention to the $G_{e-ph}$ of the individual metals (Al, Cu, Ni, Ti, Au, Nb, and Zr). The idea was to identify the element in each alloy that has the most significant contribution to the values of $G_{e-ph}$ in the binary alloys. This is useful in practice, as used programs package allows the activation of specific contributions of each element during the computation of $\lambda$. This reduces computational time while maintaining accuracy for materials with large supercells, such as high-entropy alloys or amorphous phases. Furthermore, in contrast to the work of Lin et al. [53] where $\lambda$ is not temperature dependent which leads to a significant underestimation of $G_{e-ph}$. The originality of the current research lies in systematically considering $\lambda$ as $T_e$-dependent in all computations. It is important to highlight that for both Ti and Zr the two possible crystallographic phases are considered, the Body-Centered Cubic (BCC) and Hexagonal Close Packed Crystal (HCP) structures are considered.

Fig. 4(a) shows that the $G_{e-ph}$ of AlCu alloy monotonously increases with $T_e$. As a first insight, we can notice that the contribution of Al remains contestant and neglected. However, the main trend of the curve follows the contribution from Cu. The obtained values of $G_{e-ph}$ for pure Al and Cu are consistent with the work of Zhang et al. [96] (See Fig. S3). Furthermore, in Fig. 4(e), it is observed that for NiZr alloy the evolution of the $G_{e-ph}$ is divided into two main ranges. It increases to ~ 10¹⁸ Wm⁻³K⁻¹ as it reaches 10,000 K. Second, it oscillates around the attained value of $G_{e-ph}$. In this case, the d band of both Ni and Zr have an influence on the $G_{e-ph}$. Thus, it is not trivial to correlate the contributions of each pure element to the evolution of the $G_{e-ph}$. In Fig. 4(h), the evolution of $G_{e-ph}$ in CuTi alloy is shown. For $T_e=0-5000$ K, the $G_{e-ph}$ decreases to a value of ~ 10¹⁸ Wm⁻³K⁻¹, then for $T_e > 5000$ K a monotonous increase is observed and a maximum value of ~ 3.0 × 10¹⁸
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Evolution of G<sub>e-ph</sub> at different T<sub>e</sub> ranging from 0 to 50,000 K for various crystalline alloys and their component elements: AlCu (a), AlNi (b), AlTi (c), AuCu (d), CuNi (e), CuTi (f), NbZr (g), NiTi (h), ZrCu (i), NiZr (j).</figcaption></figure>
$Wm^{-3}K^{-1}$ is reached at $T_e=50,000$ K. Fig. 4(i), the evolution of the NiTi alloy is illustrated. There is an increase in $G_{e-ph}$ within the temperature range of 0–5000 K. Beyond this range, specifically at higher temperatures ($T_e=5000–50,000$ K), the curve levels off and stabilizes, maintaining a constant value of ~ 2.0 × $10^{18}$ Wm$^{-3}K^{-1}$. In Fig. 4(j), we presented the evolution of the ZrCu alloy. At low temperature below 10,000 K there is a discernible reduction in $G_{e-ph}$, settling at an approximate value of ~ 0.2 × $10^{18}$ Wm$^{-3}K^{-1}$, closely following the Zr contribution trend. However, as the temperature exceeds 10,000 K, a marked strong increase in $G_{e-ph}$ becomes evident, achieving a targeted magnitude of ~ 2.5 × $10^{18}$ Wm$^{-3}K^{-1}$. Finally, Fig. 4(g), presents an increasing trend of the $G_{e-ph}$ in AuCu to a target value of ~ 1.75 × $10^{18}$ Wm$^{-3}K^{-1}$ at $T_e=50,000$ K. It is noteworthy that at $T_e=25000$ K, an abrupt rise of the $G_{e-ph}$ to ~ 1.7 × $10^{18}$ Wm$^{-3}K^{-1}$ is recorded, likely due to the high contribution of the $G_{e-ph}$ of Cu at this specific $T_e$, our values at 0 K match with previous work of Giri et al. [57].

Fig. 4(b) confirms that the estimated $G_{e-ph}$ for AlTi at low temperature below 10,000 K), a mixed and equivalent contribution from both Ti and Al. However, at higher temperature (above 10,000 K) the impact of Ti (in both BCC and HCP phases) on $G_{e-ph}$ values of the alloy becomes dominant, as reflected in the matching of $G_{e-ph}(T_e)$ curves of Ti and alloy. It is important to note that in AlTi, the $G_{e-ph}$ is almost saturated and remains flat at a constant value of ~ $10^{18}$ Wm$^{-3}K^{-1}$ especially above 10,000 K possibly due to a computed DOS that is not $T_e$ dependent (See Fig. S1(b)). As for the variation of the $G_{e-ph}$ in AlCu alloy. Moreover, Fig. 4(f), provides an estimation of the $G_{e-ph}$ for AlNi with a trend dictated by the contribution of Ni. Depending on $T_e$ three distinct regions exist: below 20,000, from 20,000 to 30,000 and above 30,000 K where $G_{e-ph}$ increases to ~ 4.0, and after decreases to ~ 2.0 × $10^{18}$ Wm$^{-3}K^{-1}$, and finally stays constant.

Fig. 4(c), the $G_{e-ph}$ curve of CuNi alloy exhibits a slow and monotonous increase from a values of ~ 0.5 × $10^{18}$ at $T_e=300$ K to ~ 4.0 × $10^{18}$ Wm$^{-3}K^{-1}$ at $T_e=50,000$ K. For this alloy it is hard to establish a clear relation between the $G_{e-ph}$ values of the individual Cu, and Ni elements. This difficulty arises due to several drops in the $G_{e-ph}$ of Ni at 5000, 30,000 and 40,000 K. This drops are directly correlated to the abrupt change of $g(\epsilon_F)$ at high values of $T_e$. In contrast to previous trends, Fig. 4(d) shows that for NbZr alloy, there is a slight increase of $G_{e-ph}$ from ~ 0.8 to ~ 1.2 × $10^{18}$ Wm$^{-3}K^{-1}$ at $T_e=300$, and 5000 K. In this temperature range, we observed that the trend corresponds exactly to the one of Zr-HCP. However, the trend of $G_{e-ph}$ is reversed with a decrease in the temperature above 5000 K where the $G_{e-ph}$ gradually drops from ~ 1.2 to ~ 0.25 × $10^{18}$ Wm$^{-3}K^{-1}$. Moreover, these values fall exactly between those of Nb and Zr, and the trend follows the behavior of $G_{e-ph}$ presented for Nb and Zr, allowing us to conclude an equal contribution from both elements.

## 4. Optical properties

In the context of studying the impact of ultrafast laser on crystalline alloys, the optical characteristics hold pivotal significance. In this scenario, they dictate the alloys response to the incident light at different wavelengths. A comprehensive first-principles methodology is used to compute the optical properties based on Kubo-Greenwood formalism [97]. Consequently, these optical attributes namely the optical penetration depth ($l_p$), and reflectivity ($R$) are computed at various $T_e$. The dielectric function denoted as $\epsilon(\omega)$ serves as a key property that characterizes the response of a material to the external electromagnetic field. It quantifies the material's reflection, absorption, and refraction where the real part represents energy storage, while the imaginary part signifies energy dissipation. Herein, we focus on the square of the optical function defined as $\tilde{n} = n + ik$. Where the real part $n$ represents the refractive index that describes attenuation of light and the imaginary component $k$ is extinction coefficient that measures how strongly the material absorbs light at a specific wavelength. The changes in the behavior of $n$ and $k$ during the electronic excitation of each crystalline

alloy material are explored in Fig. 5.

We investigated the optical properties by computing $n$ and $k$ considering three specific $T_e$ that correspond to different laser excitation regimes: low (0 K), moderate (15,000 K), and strong (30,000 K) for laser wavelengths 400, 800, and 1030 nm respectively. The values of $n$ and $k$ change depending on the perturbation, as shown in Fig. 5. For $\lambda=400$ nm, the refractive index exhibits a radically different responses as illustrated in Fig. 5(a).

For NiTi, NiZr, and AlCu the increase of $T_e$ leads to the drop in $n$ from 3.2, 2.4, 2.0 at 0 K to 2.1, 1.6, 1.4 at 30,000 K respectively. In addition, for ZrCu, AuCu, and CuTi as the $T_e$ rises, the $n$ undergoes oscillations, remaining consistently at values of 2.8, 2.2, 2.3, and 2.6 respectively. This indicates a lack of sensitivity to excitation effects. For AlTi, the $n$ increases from ~ 1.1 at $T_e=0$ K to ~ 2.7 at $T_e=30,000$ K. However, for AlNi as the $T_e$ rises, the $n$ remains consistently constant at a value of 2.8. Furthermore, for both NbZr and CuNi the $n$ increases from ~ 0.5 and ~ 1.3 at 0 K to ~ 3.1 and ~ 2.1 at 30,000 K respectively. This high value of $n$ indicates strong reflection effect that leads to slower light propagation. Moreover, to understand the full correlation between $n$ and the laser wavelength is fundamental since it has a broad implications for designing materials with tailored optical properties. Therefore, an investigation of $n$ and $k$ parameters at $\lambda=800$ nm is presented in Fig. 5(b).

For ZrCu, AlCu, and CuTi display a notable increase in $n$ as the temperature increases from 0 to 30,000 K. It is important to highlight that the shape of $n$ curve is almost unchanged in the case of the ZrCu and CuTi for $\lambda=400$ and 800 nm. However, for AuCu we observed oscillations of $n$ as $T_e$ increases at consistent value of 1.5. Moreover, compared to the previous case where $\lambda=400$ nm, a shift to lower value of $n$ is observed. Therefore, for binary alloy the increase of laser wavelength tends to decrease $n$. It is important to note that when transitioning from $\lambda=400–800$ nm, the curve tendency is completely reversed and $n$ shifts to higher values in the case of AlCu as observed at $T_e=30,000$ K where $n$ rises from ~ 1.4 to ~ 3.6. Lastly, for NiTi and NiZr at a wavelength of 800 nm, the elevated $T_e$ results in a decrease and a slight shift towards higher values in the $n$. The refractive index remains constant at $\lambda=400$ nm, and significant decreases at $\lambda=800$ nm. For AlNi, at a wavelength of 800 nm, the elevated $T_e$ results in a decrease and a shift towards higher values of $n$. While, for AlTi an increase in $n$ as $T_e$ increases from 0 to 30,000 K is noticed. For AlTi, there is an increase of $n$ at both wavelengths (400 and 800 nm). However, a shift toward a high values namely from ~ 1.1 to ~ 2.8 at 800 nm wavelength. For CuNi an increase in $n$ as the temperature increases from 0 to 30,000 K is observed. This increase of $n$ is preserved at 800 nm, nevertheless the $n$ shifts from ~ 1.2 to ~ 3.0 at 800 nm wavelength. Furthermore, for NbZr an increase of $n$ is observed within the $T_e$ range of 0–30,000 K but the trend and the shape of $n$ remains slightly changed for both $\lambda=400$ and 800 nm.

The evolution of $n$ for studied binary alloys was computed at $\lambda=1030$ nm as shown in Fig. 5(c). The majority of these alloys at this wavelength namely: CuTi, ZrCu, and NiZr exhibit saturation at values of ~ 3.6, 4.0, and 4.3 respectively. However, for AuCu the trend of $n$ is filled from constant evolution at $\lambda=400$ nm to an increase at $\lambda=1030$ nm. In contrast, the evolution in the $n$ seems not being affected by the increase of the laser wavelength in the cases of AlCu, NiTi, and NiZr. For AlNi, the value of $n$ is saturated at a value of ~ 4 and for AlTi, the trend of $n$ is preserved with a shift to slightly higher values. For CuNi the $n$ exhibits a saturation at value of 4.0. In contrast, the evolution of $n$ is preserved by the increase of the laser wavelength for NbZr. The evolution of $k$ is studied across a wide range of $T_e$ to understand the optical properties of the binary alloys summarized in Table 1. The behavior at $\lambda=400$ nm is depicted in Fig. 5(d).

The AlCu, AuCu, CuTi, and ZrCu demonstrate a stable values of ~ 2.4, 1.5, 1.6, and 1.8 respectively throughout the studied temperature range. Conversely, the NiTi and NiZr show a systematic decrease in $k$ as the $T_e$ rises. The AlNi alloy a decrease is noticed at $T_e=0–25000$ K but at $T_e=50,000$ K an irregular fluctuation is marked by an increasing trend of
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Dependence of *n* and *k* at *T*<sub>e</sub>=0–30,000 K for three laser wavelength λ=400, 800, and 1030 nm. These properties are computed for various crystalline alloys from Table 1.</figcaption></figure>

k. However, the AlTi demonstrates a stable value that oscillates roughly at 2.2. Finally, CuNi, a slight increase of *k* is observed with a maximum value of 2.9 at *T*<sub>e</sub>=50,000 K. On the other hand, the NbZr exhibits a stronger absorption of light with a high gradual increase values of the *k* under electronic excitation. Furthermore, Fig. 5(e), (f) show the evolution of *k* for 800 and 1030 nm respectively. For both laser wavelengths,

the *k* displays saturation with a shift toward the higher values for almost all the binary alloys. Exceptions to this rule is observed for AlCu and AlTi alloys where the *k* is flipped to a non-monotonic trend.
4.1. Reflectivity

The influence of T<sub>e</sub> on reflectivity (R) at normal incidence is inves-
tigated by the equation defined by R = [(1-n)<sup>2</sup>+k<sup>2</sup>]/[(1+n)<sup>2</sup>+k<sup>2</sup>]. The evolution of R at
various T<sub>e</sub> and different laser wavelength are presented in Fig. 6.

Fig. 6(a) shows that for ZrCu, R increases with T<sub>e</sub> at λ=400 nm.
However, at 30,000 K, a decrease of R to approximately 35 % was

observed indicating an effect of T<sub>e</sub> on this alloy’s optical response. The
alloy’s R exhibited distinct patterns for NiTi, NiZr, and AuCu, showing a
consistent decrease in reflectivity across all temperature ranges. For
AlCu, R initially decreases but then (above 15000 K) increases. Finally,
CuTi and CuNi showed saturation in R to values of approximately 35 and
45, respectively. For both AlNi and AlTi a decrease is observed then this
trend is reversed where an increase is marked at T<sub>e</sub> > 15000 K. However,

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Dependence of R, and $l_p$ on $T_e$ ranging from 0 to 30,000 K at three laser wavelength, namely $\lambda$=400, 800, and 1030 nm for various binary crystalline alloys from Table 1.</figcaption></figure>
for NbZr R exhibited an increase with at all values of $T_e$.

Fig. 6(b) suggests the evolution of R at $\lambda$=800 nm for all alloys mentioned in Table 1. Furthermore, we obtained a saturation of R with $T_e$ for AlCu, CuTi, NbZr, ZrCu, and NiZr at values of ~ 80, 45, 45, 50, and 60 % respectively. On one hand, for NiTi a decrease of R from ~ 40–20 % is noticed. On the other hand, a slight increase of R for AuCu from ~ 62–80 % with $T_e$ is observed. For AlNi a decrease of R to a value of ~ 20 %. However, for AlTi the electronic excitation $T_e$ has a minimal influence on R since its value oscillates at a constant value of ~ 50. Finally, both CuNi and NbZr a saturation values of R are reached with almost no influence of $T_e$ where a values of ~ 50, 45 are observed respectively. Furthermore, Fig. 6(c) provides valuable insights into the evolution of R at $\lambda$=1030 nm. The analysis highlights that the dynamic response of R to variations in $T_e$ and changes in laser wavelength does not have a significant impact compared to the scenario with $\lambda$=800 nm.

## 4.2. Penetration depth

The $l_p$ defines the distance over which the intensity of incident light decreases by a factor of $1/e$ (~ 37 %). The dependence of $l_p$ on $T_e$ and its correlation to $k$ is determined as $l_p = \frac{\lambda}{4\pi k}$. Herein, we shed light on the relationship between $l_p$ and $T_e$, previously unknown for the cited binary alloys summarized in Table 1. The results are shown in Fig. 6.

Fig. 6(d), shows the evolution of $l_p$ which is computed at $\lambda$=400 nm. The trend of $l_p$ demonstrates a constant saturated values for CuTi, ZrCu, and AlCu that ranges between ~ 10 and 20 nm, suggesting a limited influence of $T_e$. However, the NiTi, AuCu, and NiZr alloys demonstrate a monotonic variation in $l_p$ with $T_e$. Notably, for NiTi where a maximum value of $l_p$ ~ 40 nm is reached at maximum excitation (30,000 K). For AlNi alloy, we observed a monotonous increase of $l_p$ when $T_e$=0–15000 K, followed by a sudden decrease beyond this range. In addition, for AlTi $l_p$ shows a constant value that oscillates within ~ 10–20 nm with $T_e$. Moreover, For CuNi, the $l_p$ trend is almost constant to a value of 12 nm. The NbZr alloy uniquely exhibits a strong decrease with varying of $T_e$ with values of ~ 40 nm and 15 nm at 0 K and 15000 K respectively. However, beyond $T_e$ > 15000 K, this trend is reversed with a non-monotonous increases.

Fig. 6(e), displays the behavior $l_p$ at 800 nm. For almost all studied alloys cited in Table 1 a constant value of $l_p$ is obtained within the range of ~ 10–40 nm with $T_e$. Two exceptions can be noted, the $l_p$ of AlNi and NiTi increases with $T_e$ to higher values of ~ 120, and 60 nm respectively. Similarly, Fig. 6(f) demonstrates saturation of $l_p$ for major part of the

alloys summarized in Table 1 at $\lambda$=1030 nm. Same exception are noted. The NiTi exhibits a consistent increase trend that signifies a limited influence of $T_e$ or laser wavelength on $l_p$. However, the AlNi displays distinct trend in $l_p$ at different $T_e$, with an increase to approximately 110 nm at $T_e$=0–15000 K, followed by a decrease to approximately 80 nm within the temperature range of 15000–30,000 K.

In summary, for a laser wavelength of $\lambda$=800 nm, AlCu alloys have the lowest penetration depth, making them an interesting choice. In contrast, NiAl is ideal for applications requiring a high penetration depth. For applications where low reflectivity is essential, NiTi is the best option, while AlCu or AuCu are suitable for high reflectivity applications.

# 5. Ablation of CuTi: explosive boiling and fragmentation

Fig. 7 illustrates the cavitation and liquid stretch through a solid-liquid phase transition in CuTi alloy following laser energy deposition significantly above the ablation threshold, this process leads to explosive boiling, fragmentation, ejection, and the deposition of multiple liquid nanodroplets, similar to the findings of Lorazo et al. [98–100]. The spatial rearrangement of the atomic local structure is tracked within the time range of t=0–200 ps. At t=10 ps, the energy deposited is sufficiently high to completely melt the zone of interest. Nevertheless, at later time when t=110 ps, the significant part of the material is removed this is caused by elevation of the initial surface. This is accompanied by significant expansion of the system to a position of x ~ 30 nm. Additionally, t=150 ps beyond melting region, debris are ejected and nanoparticles are formed with different sizes that marks the transition from a regime of photomechanical spallation to the regime of phase explosion. Furthermore, at t=200 ps the examination of this new formed melted surface reveals a substantial part of the surface remained as a stretched liquid with a high of ~ 15 nm.

Fig. 8 shows the evolution of void formation and surface roughness in a CuTi alloy during the laser ablation process. The snapshots are taken at intervals from 0 to 200 ps. The initial stage takes place within the time interval of 0–30 ps, where at 0 ps the initial state shows a smooth surface with no visible voids. Then, at 10 ps small voids start appearing. However, at 30 ps the voids become bigger and more pronounced as the laser energy is transferred to the lattice. The second stage is identified at t=50–90 ps, that is characterized by void coalescence and growth. At t=50 ps, both the density and volume of voids increase due to rapid heating that results to boiling in the zone of interest. Moreover, voids

<figure><img src="image_9.png" /><figcaption>Fig. 7. CuTi local structure evolution under ultrafast laser irradiation as a function of time during the ablation explosive decomposition process for a laser conditions of fluence and pulse duration of $F_{abs}$=208 mJ/cm² and $t_p$=100 fs respectively. The region of interest is located in 0≤x≤40 nm for the times of t=0–200 ps. The atoms are colored according to the common neighbor analysis algorithm.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** CuTi surface manifold evolution during the ablation process made to detect the voids and surface roughness at the times of $t=0–200$ ps for a laser conditions of fluence and pulse duration of $F_{abs}=208$ mJ/cm² and $t_p=100$ fs respectively.</figcaption></figure>

coalesce creates a porosities in the alloy structure. At $t=70$ ps, the voids continue to merge to form a larger cavities. Finally, at $t=90$ ps the melted part appears with several interconnected voids and an increase of the surface roughness. At $t=110$ ps, the voids seem to reach a critical size in form of bubbling but a process of liquid stretch and confinement take place at $t=130–150$ ps especially in regions between larger voids mainly due to high tensile stresses, as reported in previous works [101]. At $t=170$ ps, the zone of interest shows significant fragmentation where both voids and thin liquid regions stretch to the point of breaking. In addition, the surface is highly irregular with multiple fragments. This process finishes at $t=200$ ps, where the fragmentation results on formation of isolated clusters and nanoparticles.

The behavior of CuTi serves as a case study to understand the thermodynamic origins of this ablation mainly the formed roughness and liquid stretch. Despite its importance this phenomenon where observed in previous works [25,102], but this aspect was not well addressed in alloys [39,103,104]. Consequently, the thermodynamic conditions driving these processes during 200 ps laser ablation are illustrated in Fig. 9.

Fig. 9(a) illustrates that at $t=0–25$ ps, the temperature rise instantaneously at the surface (0 nm) reaches a temperature peak of ~ 3000 K which is significantly above the theoretically computed melting point ($T_m=1456$ K) obtained by the EAM potential. Then, the heat diffuses deep into the alloy that causes the propagation of to region located at ~ 100 nm. Moreover, the zone of interest within the top 50 nm heats up to above 2000 K which means that a solid-liquid phase transition is initiated. At $t=25–75$ ps, the melting front progresses deeper and causes

more extensive melting and subsequent void formation. Furthermore, at time range from 75 to 150 ps the sustained high temperature at the surface maintain the boiling process that is frequently proposed based on observing liquid droplets in the plume [105,106]. Consequently, the void formed earlier expand and merge within the liquid part for a temperature above 2000 K. The formed supercritical fluid breakup fragmentation and phase explosion. Finally, in the time range of 150–200 ps a cooling process takes place where the surface temperature begins to decrease and reached a value of ~ 2000 K due to the heat dissipation.

Fig. 9(b), reports that concomitant with the temperature evolution, the simulation reveals a dynamic pressure response in the CuTi alloy due to the temperature gradient. Before 50 ps, an expansion and a strong compression wave that passes through the sample where the system experiences a high positive pressure up to ~ 12 GPa at the surface due to rapid heating and thermal expansion. Furthermore, at $t \sim 50–100$ ps a rarefaction wave that is characterized with a tensile stresses of maximum negative front head value around ~ -4 GPa are more prominent that favors both the liquid stretching and void formation. Finally, during the time interval of 100–200 ps the system evolves to relaxation where the pressure drops quickly leading to the promotion of void coalescence, stabilization of voids and fragments.

## 6. Conclusion

The present study provides the optoelectronic attributes of ten industrially significant crystalline alloys by a diverse methodologies

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Spatiotemporal evolution of thermodynamic quantities of CuTi sample during the 200 ps of simulation: a) lattice temperature, and b) pressure.</figcaption></figure>
namely, DFT and MD where it has yielded profound insights into their
dynamic behavior across a broad range of electronic excitation.
Furthermore, it discerns the preponderant role of the Cu element and
d orbital in dictating the electronic properties of these alloys. As a
practical demonstration, CuTi ablation under femtosecond laser pulse is
successfully investigated by employing TTM-MD approach with the
parameters delivered by this study. The findings suggest that under this
laser operating conditions above the threshold temperature, the CuTi
sample is overheated and it becomes thermodynamically unstable.
Therefore, it undergoes an explosive decomposition which consists of a
mixture of vapor, melted layers, and liquid nanodroplets. It has been
established that void formation results from the boiling process, void
coalescence is driven by the rarefaction wave, and liquid stretching
which leads to surface roughening is directly linked to both cooling and
pressure relaxation.

CRediT authorship contribution statement

**Djafar Iabbaden:** Writing – review & editing, Writing – original draft, Investigation, Formal analysis, Data curation, Conceptualization. **Arshak Tsaturyan:** Writing – original draft, Methodology, Investigatio- n, Conceptualization. **Jean-Marc Raulot:** Writing – review & editing, Supervision, Project administration. **Jean-Philippe Colombier:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization.

Declaration of Competing Interest

The authors declare no conflict of interest.

Data Availability

Data will be made available on request.

Acknowledgments

We gratefully acknowledge the ANR projects MEGALIT (ANR-18-
CE08) and INTRALAS (ANR-19-30-0036) for supporting this work. Nu-
merical computations were performed on the HPC resources from
GENCI-TGCC, project gen7041.

**Appendix A. Supporting information**

Supplementary data associated with this article can be found in the
online version at doi:10.1016/j.jallcom.2024.177175.

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