# Optical and Thermal Behavior of Germanium Thin Films under Femtosecond Laser Irradiation

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## Abstract

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^{−2}and 0.35 J cm

^{−2}, respectively. An ultrafast change in both optical and thermal properties was detected upon laser irradiation. Results also indicate that thermal melting occurs after germanium takes on a metallic character during irradiation, and that the impact ionization process may have a critical role in the laser-induced thermal effect. Therefore, we suggest that the origin of the thermal modification of germanium surface under femtosecond laser irradiation is mostly due the impact ionization process and that its effect becomes more important when increasing the laser fluence.

## 1. Introduction

- Stage 1 ($t<\tau $) photoionization, impact ionization;

- Stage 2 ($t<t$
_{eq}) electron–electron scattering, electron-phonon scattering and carrier recombination;

- Stage 3 ($t\ge t$
_{eq}) thermal equilibrium;

- Stage 4 ($t>>t$
_{eq}) thermal diffusion and re-solidification,

_{eq}is the time when thermal equilibrium is reached, corresponding to electron–phonon relaxation time.

## 2. Theoretical Model

- z is the direction perpendicular to the surface;
- $D$ is the ambipolar diffusion coefficient. It represents the mobility coefficient of charge carriers when there is a diffusion driven by an electric field, such as the diffusion of electron–hole plasma induced by the laser-related electric field;
- I is the laser intensity;
- $\alpha $ is the one-photon absorption coefficient;
- $\beta $ is the two-photon absorption coefficient, which can be ignored when the photon energy $h\upsilon $ is higher than Ge bandgap (E
_{g}≈ 0.66 eV at room temperature); - $\theta $ is the impact ionization factor, related to valence electrons excited by collisions with free electrons, occurring when the free electron energy exceeds the material bandgap (avalanche ionization);
- $\gamma $ is the Auger recombination coefficient. This phenomenon is very important in fs laser interaction with semiconductors. It refers to excited free electrons recombining again with holes, and transferring their energy to other electrons in the same band by electron–electron collisions.

- $({C}_{e,}{C}_{l})$,$({T}_{e,}{T}_{l})$, and $({k}_{e,}{k}_{l})$ are the heat capacity, the absolute temperature, and the thermal conductivity of electrons and lattice, respectively;
- G is the electron–lattice coupling coefficient;
- $S$ is the heat delivered by the laser source, defined as:

## 3. Results and Discussion

^{−2}. The laser heat source (black line) is also plotted. As can be seen, electrons are heated up to $3\times {10}^{4}$ K in less than 0.5 ps, when the lattice is still cold. This happens because the time interval during which the energy is delivered to the material (in the fs range) is shorter than the electron–phonon relaxation time (in the ps range). Therefore, similarly to other solid-state materials, the interaction of the fs laser with Ge is non-thermal, resulting in a minimum heat-affected zone (HAZ) after processing; the thermal wave (set of phonons) indeed has not enough time to propagate deeper inside the material in the timescale of irradiation, in contrast to the case of longer pulses, where a large melted zone can be produced [28]. This unique advantage of the femtosecond laser makes it the most effective tool for high-precision micromachining [29]. We notice that the electronic and lattice subsystems need more than 2 ps to reach equilibrium, because electron–phonon scattering time ${\tau}_{D}$ is higher than electron–electron scattering time ${\tau}_{ee}$ (see Appendix B); this means that electrons first tend towards equilibrium under the Fermi–Dirac distribution, and then transfer their energy to the lattice until relaxation [1,14]. Therefore, the laser-induced modification can be considered as a purely thermal phenomenon. Note here that the equilibrium temperature (T

_{eq}= 3464 K) is higher than the critical temperature (T

_{c}= 3104 K) of thermal ablation for Ge. This implies that the fluence value of 0.37 J cm

^{−2}exceeds the ablation threshold, and that Ge reaches a superheated state in about 2 ps; as a consequence, Ge clusters can be ejected under the phase explosion in liquid, vapor, and aggregates on the surface.

^{−2}. We note that free electron density increases dramatically during irradiation, reaching a maximum value at the same time as the pulse peak, and then it decreases together with the laser power. During the irradiation, the valence electrons get excited to the conduction band by one photon absorption, because the photon energy used (1.2 eV) is larger than the Ge bandgap (0.66 eV at T = 300 K). At the end of irradiation, electrons start to return to the valence band via Auger recombination. According to Equation (1), the Auger recombination process is more effective when the density of electrons excited by photoionization and impact ionization is high, because it is proportional to ${n}^{3}$. As we mentioned above, during this non-radiative recombination process, an electron and a hole recombine, and the excess energy excites an electron to a higher energy-state in the conduction band. Therefore, free electrons’ density decreases while the kinetic energy of the newly generated electron–hole pairs increases, and the total energy in the electronic subsystem remains constant. This phenomenon induced by Auger recombination is called the “energy accumulation effect” and has been proposed by Zhang et al. [12].

^{−2}, the free electron density greatly exceeds the critical density ${n}_{cr}$, which is the density corresponding to the metallic state where the real part of the dielectric function is zero, i.e., ${\epsilon}_{1}\left({n}_{cr}\right)=0$. Note that the critical density (${n}_{cr}\approx 0.57\times {10}^{21}$ cm

^{−3}) is reached in an ultrafast timescale (<1 ps). Therefore, we can deduce directly that any phenomenon induced after reaching ${n}_{cr},$ such as the formation of laser-induced periodic surface structures (LIPSSs) [15,30], is a non-thermal process.

^{−2}for the melting threshold and $F{\left(abl\right)}_{\mathrm{th}}=0.35$ J cm

^{−2}for the ablation threshold. These values fairly correspond to those experimentally measured and reported in the literature. For instance, Manickam et al. [31] measured the ablation depth using atomic force microscopy (AFM), deducing an ablation threshold of 0.32 J cm

^{−2}. Cavalleri et al. [32], by using ultrafast X-ray measurements of laser-heated depths, reported 0.22 J cm

^{−2}and 0.4 J cm

^{−2}as the melting and the ablation thresholds, respectively. It is worth highlighting here that our results are in agreement with the previous experimental results, despite the ablation and melting thresholds both being highly dependent on the laser parameters, the sample thickness, and the surrounding environment. We can observe from Figure 3 that our thin sample (200 nm) can be completely melted or completely ablated if the laser fluence exceeds 0.25 J cm

^{−2}or 0.55 J cm

^{−2}, respectively. By using Raman spectroscopy, it was found that at high fluence, melting could even reach the substrate (Si), leading to the formation of alloys with Ge [33], thus demonstrating that fs laser treatments can be profitably used for producing Si–Ge alloys.

^{−2}to 0.34 J cm

^{−2}. As can be readily seen, reflectivity always decreases at the beginning of irradiation. This implies that there is an increasing absorption of laser photons producing electron–hole pairs, as confirmed by several experimental works based on the pump–probe technique [22,23,24]. Then, in the same manner as the free electrons’ density, reflectivity increases, reaching a maximum value at the same time as the pulse peak, implying that there is a free electron plasma being built. Finally, reflectivity decreases after 300 fs (at the end of the pulse) due to Auger recombination, as mentioned above. However, this temporal evolution is not followed in the case of the lowest laser fluence investigated (0.01 J cm

^{−2}), where reflectivity always decreases during irradiation and then increases very slightly after the end of irradiation, indicating that there is a different electronic behavior of the conduction band with respect to higher fluences. Moreover, it is worth noting that peak reflectivity increases with increasing laser fluence, ranging from about 0.45 at 0.028 J cm

^{−2}to 0.75 at 0.34 J cm

^{−2}, most likely indicating that the material gains metallic properties upon irradiation. To evaluate this quantitatively, we calculated the evolution of Ge dielectric function $\epsilon $, as shown in Figure 5.

^{−2}to 0.34 J cm

^{−2}. It is worth recalling that the real part ${\epsilon}_{1}$ is relative to the characteristic of the solid state, whereas the imaginary part ${\epsilon}_{2}$ refers to photon absorption. With the sample thickness being only 200 nm and the optical penetration depth at 1030 nm wavelength higher than 400 nm, our Ge sample can be initially considered as a pseudo-transparent material. Note that the dielectric function of non-irradiated Ge at 1030 nm wavelength is $19.37+i1.42$ [34].

^{−2}, indicating that the Ge sample starts behaving like a metal. Also of note is the fact that this change of state is more and more pronounced with increasing laser fluence (the grey band in Figure 5 is larger). Conversely, ${\epsilon}_{1}$ is always positive when the fluence is 0.01 J cm

^{−2}, meaning that the free electron plasma has not yet reached the critical density ${n}_{cr}$, which explains why reflectivity always decreases at very low fluences as shown in Figure 4.

_{eq}, the maximum density of excited electrons n

_{e}, and the maximum kinetic energy of the free electrons E

_{e}as a function of the laser fluence varying from 0.01 J cm

^{−2}to 0.8 J cm

^{−2}. As can be clearly seen, there are two different regimes of equilibrium temperature evolution, which necessarily indicate two different types of responses. More precisely, at low fluences (<0.12 J cm

^{−2}), T

_{eq}increases very slowly, whereas at higher fluences, it starts to increase very rapidly. This result was confirmed experimentally by several teams working on different semiconductors. For instance, Salihoglu et al. [33] studied the morphology of Ge under fs laser pulses, 800 nm in wavelength with a repetition rate of 1 kHz, and identified three regimes with increasing laser fluence: (1) no modification, (2) formation of nanoparticles on the surface, and (3) ablation with micro-droplets around the crater, clearly indicating a rapid increase in the equilibrium temperature. Cai et al. [45] distinguish between two different ablation processes of a diamond surface when the fs laser switches from low to high fluence: (1) at low fluence, no ablation cracks are produced around craters and the diamond surface quality is good; (2) at high fluence, the material is completely removed under thermal effect and the surface quality gets significantly worse. The exact same results were observed by Wu et al. [46] on monocrystalline diamond. Therefore, we can generally conclude that minimization of thermal effects, leading to a more controllable and spatially resolved microstructuring of the treated material, is always achieved in the low-fluence regime. At high fluences, strong ablation occurs on large areas, possibly resulting in a higher rate of damaged material with surface cracks.

^{−2}(point “a” in the figure) the free electron density exceeds the critical density ${n}_{cr}$, whereas the bulk material is hardly heated, indicating that the optical behavior of Ge is metal-like, and thus the surface morphology can be modified with minimum thermal effect [8,16]. When the laser fluence reaches 0.12 J cm

^{−2}(point “b”), we observe that the kinetic energy of the free electrons exceeds the bandgap energy; electrons have now enough energy to excite valence electrons by impact ionization (avalanche ionization process), transferring more thermal energy to the lattice. When the fluence reaches the melting threshold 0.14 J cm

^{−2}(point “c”), lattice temperature strongly increases, as expected, and thermal modification can be observed on the Ge surface in the form of liquid or solid nano-microparticles, leading to surface damage and decreasing precision of microstructuring.

^{−2}), it is the lack of electrons excited by impact ionization that minimizes thermal effect. Free electrons, initially excited by photoionization, have insufficient kinetic energy to ionize electrons of the valence band by electron–electron scattering, and the energy transferred to the lattice by electron–phonon scattering is minimal, which is why the lattice temperature increases very slowly in this regime.

- Photoionization;
- Metallic state induced by photoionization (F > 0.028 J cm
^{−2}); - Impact ionization (F > 0.12 J cm
^{−2}); - Thermal modification (F > 0.14 J cm
^{−2}).

## 4. Conclusions

^{−2}), Ge takes on a metallic characteristic. If laser fluence is higher than a second threshold (0.12 J cm

^{−2}), the impact ionization process starts, increasing the thermal effect, which is triggered on a large scale when laser fluence reaches the melting threshold (0.14 J cm

^{−2}). Therefore, the impact ionization process can be considered as the key starting point of the thermal modification of Ge surface under a single fs laser pulse irradiation. This represents a significant contribution to understanding the fundamental physics of ultrafast laser–matter interaction. Of course, further applied and theoretical research is needed to confirm our results, and to answer many other questions in the emerging field of fs laser–matter interaction at the nanoscale.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Ge properties (T

_{l}is the lattice temperature, m

_{e}is the electron mass, n

_{r}is the real part of the refractive index, e is the electronic charge, c is the velocity of light, ${\epsilon}_{0}$ is the electric constant, ω is the laser angular frequency, n is the free electron density, k

_{B}is the Boltzmann constant, T

_{m}is the melting temperature).

Ambipolar diffusion coefficient $D$(m^{2} s^{−1}) | $65\times {10}^{-4}{\left({T}_{l}/300\right)}^{-3/2}$ | [11,26] |

One-photon absorption coefficient $\alpha $(m^{−1}) | $1.4\times {10}^{6}\left(1+{T}_{l}/2000\right)$ | [20] |

Bandgap E_{g} (eV) | $0.743-\frac{0.456\times {10}^{-3}{T}_{l}{}^{2}}{210+{T}_{l}}$ | [47] |

Auger recombination coefficient $\gamma $ (m^{6} s^{−1}) | $2\times {10}^{-43}$ | [11,26] |

Electron-electron scattering time ${\tau}_{ee}$(fs) | $~1$ | see Appendix B |

Optical effective electron mass ${m}_{opt}^{*}$ | $0.22{m}_{e}$ | [26] |

Cross section coefficient $\sigma $ (cm ^{2}) | $\frac{{n}_{r}{e}^{2}{\tau}_{ee}}{{m}_{opt}^{*}c{\epsilon}_{0}\left(1+{\omega}^{2}{\tau}_{ee}^{2}\right)}$ | [48] |

Impact ionization coefficient $\theta $(cm ^{2} J^{−1}) | $\frac{\sigma}{{n}_{r}^{2}{E}_{g}}$ | [48] |

Electronic heat capacity ${C}_{e}$ (J m ^{−3} K^{−1}) | $3n{k}_{B}$ | [11] |

Electron-phonon scattering time ${\tau}_{D}$(s) | $4\times {10}^{-13}\left(1+{\left(n/{10}^{27}\right)}^{2}\right)$ | [11,20] |

Electron-phonon coupling coefficient$G$(J m^{−3} K^{−1} s^{−1}) | ${C}_{e}/{\tau}_{D}$ | [20] |

Density $\rho $ (kg m^{−3}) | $\{\begin{array}{c}-0.1085{T}_{l}+5409{T}_{l}{T}_{m}\\ -0.4529{T}_{l}+6124{T}_{l}\ge {T}_{m}\end{array}$ | [20] |

Lattice heat capacity ${C}_{l}$ (J kg ^{−1} K^{−1}) | $\{\begin{array}{c}\left(1.256\times {10}^{6}+900{T}_{l}-0.644{T}_{l}^{2}+1.61\times {10}^{-4}{T}_{l}^{3}\right)/\rho {T}_{l}{T}_{m}\\ 3.194\times {10}^{-8}{T}_{l}^{3}-1.287\times {10}^{-4}{T}_{l}^{2}+0.1774{T}_{l}+259.4{T}_{m}\le {T}_{l}\le 1500K\\ 343.7225{T}_{l}\ge 1500K\end{array}$ | [20] |

Electron thermal conductivity ${k}_{e}$(W m ^{−1} K^{−1}) | $28{e}^{-2936/{T}_{e}}$ | [20] |

Lattice thermal conductivity ${k}_{l}$(W m ^{−1} K^{−1}) | $675{T}_{l}{}^{-1.23}$ | [11] |

Free-carrier absorption cross-section $\Theta $ (m^{2}) | $6.6\times {10}^{-24}$ | [11] |

Melting temperature ${T}_{m}$ (K) | $1211$ | [20,22] |

Critical temperature ${T}_{c}$ (K) | $3104$ | [20,22] |

Initial free electron density n_{c0} (cm^{−3}) | $2.33\times {10}^{13}$ | [11] |

## Appendix B

^{−3}is the number of Ge ions per unit volume ($\rho \approx 5.5$ g cm

^{−3}is the density, ${N}_{A}$ is the Avogadro number, and ${M}_{Ge}$ is the molar mass). The term ${v}_{e}^{*}$ is the mean velocity of electrons, and can be considered equal to Fermi velocity: ${v}_{e}^{*}={\left(2{E}_{F}/{m}_{e}\right)}^{1/2}$ where ${E}_{F}$ is the Fermi energy and ${m}_{e}$ is the electron mass. Therefore, ${\tau}_{ee}=0.94\times {10}^{-15}$ s $\approx 1$ fs. The electron–electron scattering and the electron–phonon scattering processes in several materials have indeed a typical timescale of $~1-100$ fs and $~0.1-1$ ps, respectively, as shown by Mazur et al. [14].

^{2}J

^{−1}. This factor strongly depends on the energy bandgap of materials; for example, for silicon it is $\theta \approx 21.2$ cm

^{2}J

^{−1}[27].

## Appendix C

^{−3}.

_{r}$=\frac{1}{\sqrt{2}}{\left[{\epsilon}_{1}+{\left({\epsilon}_{1}{}^{2}+{\epsilon}_{2}{}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is the real part of the refractive index, and $\kappa =\frac{{\epsilon}_{2}}{2{n}_{r}}$ is the extinction coefficient [15].

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**Figure 1.**Electron and lattice temperature evolution as a function of time for a Ge thin film under a single 300 fs laser pulse irradiation, at 1030 nm wavelength and a fluence of 0.37 J cm

^{−2}. The laser pulse heat profile (black line) is plotted in arbitrary units.

**Figure 2.**Free electron density evolution (blue dashed line) as a function of time for a Ge thin film under a single 300 fs laser pulse irradiation, at 1030 nm wavelength and a fluence of 0.37 J cm

^{−2}. The laser pulse heat profile (black line) is plotted in arbitrary units.

**Figure 4.**Temporal evolution of the reflectivity of the Ge thin film surface under a single 300 fs laser pulse irradiation, at 1030 nm wavelength and at different laser fluences ranging from 0.01 J cm

^{−2}to 0.34 J cm

^{−2}.

**Figure 5.**Temporal evolution of the dielectric function of the Ge thin film under a single 300 fs-laser pulse irradiation, at 1030 nm wavelength and at different laser fluences ranging from 0.01 J cm

^{−2}to 0.34 J cm

^{−2}. Grey bands indicate a negative value of the real part ${\epsilon}_{1}$ of the dielectric function.

**Figure 6.**Electron–phonon equilibrium temperature T

_{eq}, maximum density of free electrons n

_{e}, and maximum kinetic energy of the electrons E

_{e}, as a function of the laser fluence ranging from 0.01 J cm

^{−2}to 0.8 J cm

^{−2}.

**Figure 7.**Visual summary of the four stages of Ge modification under fs laser pulses as a function of laser fluence.

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## Share and Cite

**MDPI and ACS Style**

Abdelmalek, A.; Kotsedi, L.; Bedrane, Z.; Amara, E.-H.; Girolami, M.; Maaza, M. Optical and Thermal Behavior of Germanium Thin Films under Femtosecond Laser Irradiation. *Nanomaterials* **2022**, *12*, 3786.
https://doi.org/10.3390/nano12213786

**AMA Style**

Abdelmalek A, Kotsedi L, Bedrane Z, Amara E-H, Girolami M, Maaza M. Optical and Thermal Behavior of Germanium Thin Films under Femtosecond Laser Irradiation. *Nanomaterials*. 2022; 12(21):3786.
https://doi.org/10.3390/nano12213786

**Chicago/Turabian Style**

Abdelmalek, Ahmed, Lebogang Kotsedi, Zeyneb Bedrane, El-Hachemi Amara, Marco Girolami, and Malik Maaza. 2022. "Optical and Thermal Behavior of Germanium Thin Films under Femtosecond Laser Irradiation" *Nanomaterials* 12, no. 21: 3786.
https://doi.org/10.3390/nano12213786