Comparison study of the femtosecond laser-induced surface structures on silicon at an elevated temperature

Mochou Yang; BingYi Li; Guoliang Deng

doi:10.1364/OE.475169

1. Introduction

The femtosecond laser is a versatile tool to fabricate surface micro/nano-structures on various materials. Micro-meter-sized quasi-periodic sharp conical spikes [1,2], high regular arrays of nano-rods [3], laser-induced periodic surface structures (LIPSS) [4–10], and micro-meter size grooves [11,12] have been successfully identified in recent years. Controlling the qualities of structures, such as periodicity and crystallinity, is critical for applications. Typically, properties’ tailoring is achieved either by changing the femtosecond laser properties such as polarization, fluence, pulse number, and incident angle, among others, or by changing the environmental condition such as immersing the sample in liquid and gases [13,14]. We recently discovered that the substrate’s temperature significantly impacts surface structures [15–17]. The morphology and crystallinity can be modified by changing the substrate temperature. Surface structures with superior crystallinity can potentially be fabricated with high substrate temperature for improved device performance. These findings show that controlling the temperature of the substrate can be a novel way to alter the characteristics of surface structures and a new scenario for studying the ultrafast laser-mater interaction. We previously investigated the origin of the morphological difference between micro/nano structures created by femtosecond laser irradiation of silicon at various temperatures. We expand the scope of our earlier research to include features of surface structures such as crystallinity and periodicity at elevated temperatures. The mechanisms underlying the temperature dependency of the characteristics of surface structures are examined. Our findings will improve the understanding of the femtosecond laser-material interaction at high substrate temperatures and open the door for high crystallinity surface structure manufacturing.

2. Experimental setup

Before the laser irradiation, an n-type (100) silicon wafer was ultrasonically cleaned using acetone and methanol for 5 minutes each. Then, the wafer was rinsed with distilled water and blow-dried with N₂. An electrical heater and a thermal imager (NEC H2640) were used to control and monitor the substrate temperature. The temperature of the substrate may be raised to 400 °C. In our experiment, we employed a 20 Hz linear polarized femtosecond Ti: sapphire laser with a center wavelength of 800 nm and a pulse duration of roughly 35 fs. With a 20 cm focal length lens, the laser pulse was focused and incident normally on the silicon surface. The beam exhibits a spatially Gaussian distribution with diameters of around 153 µm and 119 µm in these two orthogonal directions, respectively (measured with a CCD beam analyzer and defined at 1/e² of the highest intensity). A half-waveplate and a polarizer were used to control the incident energy of the femtosecond laser light. The fluence used in this work is 2.8 kJ/m² in the center of the beam. The pulse number was counted and controlled by a mechanical shutter (Thorlabs, SH1). The morphology of the irradiated samples was analyzed by scanning electron microscopy (JSM-7500F). Using a Raman spectroscope (LabRAM HR), the crystallinity was studied with a 50× objective and a 2.5-mW 532-nm laser.

3. Experimental results and discussions

The electron micrographs of the surface structures on silicon irradiated with 30, and 500 femtosecond laser pulses at 25 °C and 400 °C are presented in Fig. 1(a)-(d), respectively. A crater formed after the femtosecond laser irradiation. One should notice that only the left and center parts of the crater are illustrated in Fig. 1. The arrow implies the direction of the laser polarization. Figure 1 shows that fs-LIPSS has distinct temperature dependence, consistent with our previously published results [15–17]. Low spatial frequency fs-LIPSS (LSFL) with a direction perpendicular to the laser polarization may be noticed on the crater’s edge after 30 laser shots in both circumstances (25 °C and 400 °C). Almost the whole crater formed at 400 °C is covered in micrometer-sized grooves with a direction parallel to the laser polarization. They can only be seen in the middle part of the crater created at 25 °C, as illustrated in Figs. 1(a) and 1(b). In the structures generated at 400 °C, we discovered that the LSFL ripples and grooves superimposed each other. Figure 2(a) and (b) illustrate the surface morphology of the crater’s center on silicon following irradiation with 100 laser pulses at 25 °C and 400 °C, respectively. In both circumstances, micrometer-sized grooves parallel to the laser polarization are evident. At 25 °C, ablated nanoparticles cover the whole groove surface, but at 400 °C, there is hardly little ablated material. This behavior can also be seen in Fig. 1(a) and (b), where nanoparticles have covered the whole structured region in Fig. 1(a). In contrast, the surface features in Fig. 1(b) are considerably smoother. Our findings suggest that using a high substrate temperature, ultra-smooth micro/nano surface structure manufacturing may be conceivable.

Fig. 1. Scanning electron micrograph of surface structures formed at different temperatures and laser shots: (a) 25 °C, 30 shots; (b) 400 °C, 30 shots; (c) 25 °C, 500 shots; (d) 400 °C, 500 shots. The arrow indicates the direction of the laser polarization. Figure 1(a), (b), (c), and (d) share the same scale bar.

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Fig. 2. Grooves formed in the crater’s center at to 400 °C after being irradiated with 100 laser pulses. (a) 25 °C, (b) 400 °C, (c) The grooves’ spacing as a function of position. The arrow indicates the direction of the laser polarization. Figure 2(a) and (b) share the same scale bar.

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Figure 2(c) depicts the groove spacing as a function of location. Grooves created at higher temperatures have larger periodicities than those formed at lower temperatures. When the substrate temperature is 25 °C, the grooves are spaced between 2.5 and 3.5 µm apart, with a covered area diameter of roughly 35 µm. At 400 °C, the spacing is between 1.5 and 5 µm, with a substantially larger groove coverage area. It has a diameter of roughly 60 µm. Considering that the beam has a Gauss distribution, the phenomenon shown in Fig. 2(c) indicates that the grooves are sensitive to the local fluence, especially at 400 °C. Micrometer-sized grooves have been shown to favor form with a greater dosage, in which higher fluence or more laser pulses are used. The formation of these grooves can also indicate that more energy is deposited in the material. We recently established that energy deposition at various temperatures causes the phenomenon that grooves are easier to develop at higher substrate temperatures, which is consistent with the findings here that the grooves cover a bigger area at 400 °C [15]. Figure 1(c) shows rods or spikes generated when a silicon surface was bombarded with 500 laser pulses at 25 °C. These asymmetric structures, whose short axis is parallel to the polarization of laser light, have previously been reported [1]. Meanwhile, grooves are the most common surface structures generated at 400 °C, as seen in Fig. 1(d). After irradiating with 30 laser pulses, LSFL formed at both temperatures, as shown in the left part of Fig. 1(a) and (b). Figure 3(a) displays the temperature dependency of the LSFL ripples’ average spatial period. When the temperature is raised from 25 °C to 400 °C, the statistics show a consistent increase from around 622 nm to about 748 nm (20.3%).

Fig. 3. (a) Dependence of the spatial periods of LSFL as a function of the substrate temperature; (b) Periodicities of the LSFL feature as a function of the damping rate. The inset shows the 2D gray map of efficacy factor η for crystalline silicon as a function of the normalized LIPSS wave vector when the carrier density is 4.7 × 10²¹/cm³.

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A Drude-Sipe model may explain the variation in periodicities of the LSFL ripples vs temperatures observed in Fig. 3(a). The energy deposition on the silicon surface is proportional to ${\rm{\eta \times |b|}}$, in which $\eta $ is a response function describing the efficacy and has sharp peaks leading to inhomogeneous absorption. b is a slowly changing function for a surface with homogeneously distributing roughness. Previously, we have demonstrated that the significant electron-phonon scattering at high substrate temperature increases the damping rate and then reduces the oscillation of the surface plasma polaritons (SPPs) [17]. The periodicities of the LSFL ripples are also affected by the damping rate. The dielectric function’s real part has to be less than -1 to allow the excitation of SPPs, which requires a carrier density of about 4.7 × 10²¹/cm³ in silicon. We assume the carriers’ effective mass ${{\rm{m}}_{{\rm{opt}}}}^{\rm{\ast }}$ is 0.18, and the Drude damping rate is $\frac{{\rm{1}}}{{{{\rm{\tau }}_{\rm{D}}}}}{\rm{ = 1f}}{{\rm{s}}^{{\rm{ - 1}}}}$ [18]. The values s = 0.4 and f = 0.1 were chosen based on Ref. [19], indicating the assumption of spherically shaped islands and a 10% filling factor. In the inset of Fig. 3(b), the two sickle-like spots in the horizontal direction are connected with LSFL in a direction perpendicular to the polarization. The LSFL ripples’ periodicities are related to the position of the maximum. The periodicities as a function of damping rate are shown in Fig. 3(b) for various carrier densities. The periodicities rise as the damping rate increases (the damping rate increases with temperature) and are consistent with the experimental result in Fig. 3(a).

Raman spectroscopy is beneficial for evaluating many vital phenomena on single-crystal and amorphous silicon [20–23]. Our previous results show that the surface structures fabricated at 400 °C have better crystallinity than those at 25 °C [16]. Here, we extend the scope to a broader laser parameters range and the origin of this phenomenon. Figure 4 shows the Raman spectra between 100 cm^-1 and 600 cm^-1 measured in the crater’s center irradiated with 100 and 500 laser shots at 25 °C and 400 °C, respectively. The spectra are given on a logarithmic intensity scale for clarity. As a comparison, the spectrum of non-irradiated monocrystalline silicon is displayed. The prominent peak at 520 cm^-1 can be identified in all the spectra shown in Fig. 4. This peak is the first-order Raman spectrum that originates from the longitudinal optical phonon (LO) and transverse optical phonon (TO) mode degenerated at the Brillouin zone center.

Fig. 4. Raman spectroscopy of surface structures irradiated with different pulse numbers and temperatures. The reference is the Raman spectrum of non-irradiated monocrystalline silicon.

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The q-vector selection rule does not apply in amorphous silicon due to the loss of long-range order. It gives rise to two humps around 140 cm^-1 and 480 cm^-1 related to the transverse acoustic (TA) phonon and TO phonons [20]. The broad feature around 140 cm^-1 can be observed in the Raman spectra of the surface structure fabricated at 25°C, marked as A and B in Fig. 4. The conspicuous peaks at 520 cm^-1 of these two spectra are asymmetric and have been broadened. The tail at the lower wavenumber side, marked as D and E in Fig. 4, can be identified as the amorphous silicon peak (480 cm^-1) merged with the 520 cm^-1 crystalline silicon peak. The Raman spectra of the structures formed at 400 °C are almost comparable to the reference. According to the Raman data, structures generated at 400 °C had higher crystallinity than those formed at room temperature. We should notice that the Raman spectrum’s intensity of the structures fabricated with 500 laser shots at 400 °C has a slight increase at 140 cm^-1, marked as C in Fig. 4. This feature could imply forming a small amount of amorphous or poly-crystalline silicon.

The surface reflective is also an indicator of amorphization because of the refractive index difference between crystalline silicon and amorphous silicon [21]. An amorphous silicon area with higher reflectivity is formed at proper fluence and pulse number. The optical microscope images of silicon surface irradiated at different temperatures and laser shots are shown in Fig. 5. We observed the formation of amorphous silicon after silicon was irradiated by several laser pulses at room temperature, as shown in the ring in Fig. 1(a). After being irradiated by 50 laser pulses, the surface was covered by a structured area with low reflectivity. The lower surface reflectivity is because of the surface structure formation. We should notice this does not mean there was no amorphous silicon formation, which was mentioned in Ref. [22]. Amorphous ring can not be identified when the sample surface was irradiated with the same laser pulses at 400 °C, as shown in Fig. 5 (c) and (d), which indicates the surface structures formed at 400 °C has better crystallinity.

Fig. 5. Optical micrograph of surface at different temperatures and laser shots: (a) 25 °C, 5 shots; (b) 25 °C, 50 shots; (c) 400 °C, 5 shots; (d) 400 °C, 50 shots. The arrow indicates the direction of the laser polarization.

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We should notice that the thickness of the amorphous silicon layer can not be deduced directly from the Raman spectra and the optical micrographs. From the Ref. [21,22], the amorphous silicon is usually on the order of several tens of nanometers at room temperature. The thickness of the amorphous silicon would be thinner at higher temperatures since it shows better crystallinity. The amorphous layer’s exact thickness is beyond this paper’s scope and will be studied further in the future.

The absorption of laser light by silicon needs phonon assistance because silicon is an indirect bandgap semiconductor. At higher temperatures, higher momentum phonons can be generated to enhance the excitation of electrons. The bandgap energy of silicon also decreases as the temperature rises. These effects increase the absorption coefficient and reduce penetration depth at elevated temperatures [24,25]. To estimate the energy deposition quantitatively, the carrier density ${{\rm{N}}_{\rm{e}}}$ is calculated using partial differential Eq. (1). The first term represents the diffusion process of free carriers. ${{\rm{G}}_{\rm{e}}}$ represents the number of free carriers generated in unit time and unit volume. The expression ${{\rm{G}}_{\rm{e}}}$ is expressed as Eq. (2), in which the linear absorption, two-photon absorption, and collision ionization of silicon are contained. ${{\rm{R}}_{\rm{e}}}$ represents the carriers reduced due to Auger recombination, which is defined by Eq. (3). ${{\rm{\tau }}_{{\rm{AR}}}}$ is the minimum Auger recombination time, slowing down to the rate of carrier reduction. ${{\rm{C}}_{{\rm{AR}}}}$ is the Auger recombination rate.

(1)$$\frac{{\partial {N_e}}}{{\partial t}} = \nabla ({{k_B}{T_e}{\mu_e}\nabla {N_e}} )+ {G_e} - {R_e}$$

(2)$${G_e} = \left( {\frac{{{\sigma_1}I}}{{h\upsilon }} + \frac{{{\sigma_2}{I^2}}}{{2h\upsilon }} + {\delta_I}{N_e}} \right) \times \left( {1 - \frac{{{N_e}}}{{{N_0}}}} \right)$$

(3)$${R_e} = \frac{{{N_e}}}{{{\tau _{AR}} + {{({C_{AR}}{N_e}^2)}^{ - 1}}}}$$

The Gaussian beam at normal incidence is used to calculate the laser intensity. Laser intensity on the material surface can be calculated through Eq. (4) [26]:

(4)$$\begin{aligned} I(r,z = 0,t) =& (1 - R) \cdot \frac{{2\sqrt {\ln 2} }}{{\sqrt \pi {\tau _p}}}{F_0}\exp \left( { - \frac{{{r^2}}}{{{r_0}^2}}} \right)\\ {\rm{ }} &\times \exp \left( { - 4\ln 2{{\left( {\frac{{t - {t_0}}}{{{\tau_p}}}} \right)}^2}} \right) \end{aligned},$$

where R is the surface reflectivity, which is related to the lattice temperature ${{\rm{T}}_{\rm{l}}}$. ${{\rm{F}}_{\rm{0}}}$ is the laser fluence. ${{\rm{\tau }}_{\rm{p}}}$ is the pulse duration (full width at half maximum, FWHM). ${{\rm{r}}_{\rm{0}}}$ is the radius of the laser spot (defined as 1/e² of the maximum intensity). ${{\rm{t}}_{\rm{0}}}$ is the time when the laser intensity reaches the maximum, and ${{\rm{t}}_{\rm{0}}} = 3{{\rm{\tau }}_{\rm{p}}}$ is applied in our simulation. For simplicity, only the center position of the spot is considered, ie.${\rm{r = 0}}$. Laser intensity will decrease with the depth z from the surface because of the absorption. The distribution of I at different depths can be calculated using Eq. (5) [11]:

(5)$$\frac{{\partial I}}{{\partial z}} ={-} ({{\sigma_1}I + {\sigma_2}{I^2}} )\times \left( {1 - \frac{{{N_e}}}{{{N_0}}}} \right) - \varTheta {N_e}I. $$

The significance of the parameters involved in Eqs. (1)–(5) and their values used in our simulations are shown in Table 1.

Table 1. Values of the parameters involved in Eqs. (1)–(5)

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After laser excitation, the electron-hole system absorbed the energy leaving a high density of electrons with a much higher temperature than the lattice. Then, these excited electrons recombine and gradually transfer the energy to the lattice. The time of this process is much longer than the pulse duration ${{\rm{\tau }}_{\rm{p}}}$. The interaction between carriers and lattice can be described using a two-temperature model [31,32], in which the “two-temperature” refers to carrier temperature ${{\rm{T}}_{\rm{e}}}$ and lattice temperature ${{\rm{T}}_{\rm{l}}}$, respectively. Their variation with time can be calculated through partial differential Eqs. (6) and (7) [11,31]:

(6)$${C_e}\frac{{\partial {T_e}}}{{\partial t}} = \nabla ({{\kappa_e}\nabla {T_e}} )- \gamma ({{T_e} - {T_l}} )+ Q, $$

(7)$${C_l}\frac{{\partial {T_l}}}{{\partial t}} = \nabla ({{\kappa_l}\nabla {T_l}} )+ \gamma ({{T_e} - {T_l}} ), $$

where ${{\rm{C}}_{\rm{e}}}$ and ${{\rm{\kappa }}_{\rm{e}}}$ are the free carriers’ specific heat capacity and thermal conductivity, respectively. ${{\rm{C}}_{\rm{l}}}$ and ${{\rm{\kappa }}_{\rm{l}}}$ are the specific heat capacity and thermal conductivity of the lattice, respectively. ${\rm{\varGamma }}$ is the coupling coefficient between the free carriers and lattice. Q represents the laser source term, which is used to describe the influence of the laser beam on the carriers’ temperature change and is defined by [11] :

(8)$$\begin{array}{c} Q = \left[ {({h\nu - {E_g}} )\frac{{{\sigma_1}I}}{{h\nu }} + ({2h\nu - {E_g}} )\frac{{{\sigma_2}{I^2}}}{{2h\nu }} - {E_g}{\delta_I}{N_e}} \right]\\ {\rm{ }} \times \left( {1 - \frac{{{N_e}}}{{{N_0}}}} \right) + \varTheta {N_e}I + {E_g}{R_e} - \frac{3}{2}{k_B}{T_e}\frac{{\partial {N_e}}}{{\partial t}} \end{array}, $$

${{\rm{T}}_{\rm{l}}}$ gradually rises with the increase of lattice energy. The melting temperature ${{\rm{T}}_{\rm{m}}}$ of silicon is 1414 °C at ambient pressure [33]. For the conditions that ${{\rm{T}}_{\rm{l}}}$ is above ${{\rm{T}}_{\rm{m}}}$, the specific heat capacity and thermal conductivity of free carriers and lattice will change. Moreover, Eqs. (6) and (7) need to be replaced by the following two equations to describe the heat transfer between electrons and lattice [33]:

(9)$${C_e}^L\frac{{\partial {T_e}}}{{\partial t}} = \nabla ({{\kappa_e}^L\nabla {T_e}} )- \gamma ({{T_e} - {T_l}} ), $$

(10)$${C_l}^L\frac{{\partial {T_l}}}{{\partial t}} = \nabla ({{\kappa_l}^L\nabla {T_l}} )+ \gamma ({{T_e} - {T_l}} ), $$

To distinguish the parameters of solid silicon and liquid silicon, “S” and “L” are used as the superscript of the heat capacity C, thermal conductivity κ, and energy relaxation time τ_γ (“S” represents the parameters of solid-phase while “L” represents the parameters of liquid phase). Compared with Eq. (6), there is no laser source term in Eq. (9) since silicon’s melting process occurs after the laser pulse. In our simulations, the values of the parameters involved in Eqs. (6)-(10) are shown in Table 2.

Table 2. Values of the parameters involved in Eqs. (6)–(10)

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The energy absorption distribution for the first 50 nm on silicon surface at various temperatures is shown in Fig. 6(a). The energy absorbed within the first 20 nm increases from about 2.86 µJ to 2.95 µJ, shown in the black dash line in Fig. 6(b), assuming a 150 µm diameter of the beam size with a uniform fluence distribution at 2.8 kJ/m². The red dot-dash line gives the corresponding penetration depth in Fig. 6(b), which drops monotonously from 408 nm to 346 nm in the temperature range investigated. As a result, more energy is deposited in a thinner layer on the silicon surface, which heats the surface to a higher temperature. Nanoparticles can then be melted more easily than at room temperature, resulting in grooves with a smoother surface, as seen in Fig. 1.

Fig. 6. (a) Energy absorption distribution for the top 50 nm on silicon surface at different initial temperatures; (b) Energy absorption in the top 20 nm on the silicon surface and penetration depth at different temperatures.

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After absorbing femtosecond laser energy, the solid phase of silicon melts into a liquid phase through non-thermal and thermal melting [39]. After equilibrium, the liquid silicon re-solidifies as a solid through the resolidification process. An epitaxial growth process occurs when the resolidification velocity lower than 15 m/s results in re-solidification as crystalline silicon, while amorphous silicon prefers to form if the speed exceeds 15 m/s [40–42]. The variations of solid-liquid interface position in silicon after the laser pulse under different initial temperatures can be obtained by solving Eqs. (6)-(10). Figure 6 (a) shows the solid-liquid interface for the resolidification process, which is defined as the position where ${{\rm{T}}_{\rm{l}}}$ decreases from above ${{\rm{T}}_{\rm{m}}}$ to below ${{\rm{T}}_{\rm{m}}}$, versus time at different substrate temperatures. The melting duration is extended as the substrate temperature rises. The gradient of the position curves gives the velocity of the resolidification. One can observe that the resolidification velocity decreases rapidly over time and become nearly constant at the last stage of the resolidification process. We plot the resolidification velocities at the last stage versus the substrate temperatures in Fig. 7(b), which shows they decrease monotonously with the increase of initial substrate temperature. The liquid silicon re-solidify to an amorphous phase through a rapid resolidification caused by the steep lateral temperature gradients at room temperature. The decrease of the resolidification velocity at elevated temperatures, which originated from the smaller temperature gradient and higher energy absorption, allows the liquid material to recover to a better crystalline solid phase, consistent with the Raman results in Fig. 4.

Fig. 7. (a) Simulation results of solid-liquid interface position in silicon varies with time after the laser excitation (2.6 kJ/m²) under different initial substrate temperatures; (b) Simulation results of resolidification rates of silicon versus the increase of initial substrate temperature under different laser fluences.

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Overall, the crystallinity of the re-solidified melting silicon mainly depends on the resolidification velocity, which is closely connected with the temperature gradient and the energy deposited into the material. These two factors are determined by the combination of laser parameters (local fluence, pulse duration, pulse number, wavelength) and environment conditions (temperature, refractive index of the environment). From the controllability of the crystallinity, laser fluence and pulse number are the two laser parameters that can be easily controlled. The temperature of the silicon wafer can also be heated up to a certain temperature with a heater easily, which indicates the temperature is an excellent parameter to control the crystallinity of the surface structures. Further study is still needed if one wants to control the thickness of the amorphous layer more precisely.

4. Conclusion

In summary, we compared the morphology and crystallinity of the micro/nano surface structures induced by femtosecond laser irradiation of silicon at elevated temperatures. At room temperature, low spatial frequency laser-induced periodic ripple structures (LSFL), micrometer size grooves, and spikes are fabricated consecutively as pulses number increases. In contrast, the grooves, which are parallel to the laser light polarization, dominate the structures formed at 400 °C because of the enhanced absorption. Additionally, the roughness of the surface structures formed at 400 °C has been significantly reduced. The periodicity of the LSFL ripples increases at elevated temperatures. We attribute these effects to an increase in the damping rate at the higher temperature. The Raman spectra results show that the surface structures formed at 400 °C have better crystallinity, indicating that our method can potentially be used to fabricate crystalline micro/nano-structures to help to enhance the device performance. The numerical calculation shows that the slower resolidification velocity, originating from the smaller temperature gradient at a higher temperature, is responsible for better crystallinity.

Funding

National Natural Science Foundation of China (61705148, 61705149).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61705149).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Symbol	Significance	Value	Unit
$m_{o p t}^{*}$	Effective optical mass	$0.18 m_{e}$ [18]	$kg$
$ν_{e}$	Collision frequency	$1 \times 10^{15}$ [18]	$s^{- 1}$
$μ_{e}$	Carrier mobility	$e / (m_{o p t}^{*} ν_{e})$ [11]	$C \cdot s / kg$
$σ_{1}$	linear absorption (800 nm)	$1.12 \times 10^{5} \exp (T_{l} / 430)$ [27]	$m^{- 1}$
$σ_{2}$	Two-photon absorption (800 nm)	$9 \times 10^{- 11}$ [27]	$s \cdot m/J$
$δ_{I}$	Impact ionization coefficient	$3.6 \times 10^{10} \exp (- 1.5 E_{g} / k_{B} / T_{e})$ [28]	$s^{- 1}$
$Θ$	Free-carrier absorption coefficient (800 nm)	$2.9 \times 10^{- 22} T_{l} / T_{0}$ [27]	$m^{2}$
$τ_{A R}$	Minimum Auger recombination time	$6 \times 10^{- 12}$ [29]	$s$
$C_{A R}$	Auger recombination rate	$3.8 \times 10^{- 43}$ [28]	$m^{6} / s$
$R$	Reflectivity (800 nm)	$0.329 + 5 \times 10^{- 5} T_{l}$ [30]	-

Symbol	Significance	Value	Unit
$E_{g}$	Bandgap energy	$e (1.167 - 0.0258 T_{l} / T_{0} - 0.0198 (T_{l} / T_{0})^{2})$ [27]	$J$
${κ_{e}}^{S}$	Thermal conductivity of the free-carriers (Solid)	$- 0.5552 + 7.1 \times 10^{- 3} T_{e}$ [34]	$W / (m \cdot K)$
${C_{l}}^{S}$	Lattice heat capacity (Solid)	$1.978 \times 10^{6} + 354 T_{l} - 3.68 \times 10^{6} {T_{l}}^{- 2}$ [35]	$J / (m^{3} \cdot K)$
${κ_{l}}^{S}$	Lattice thermal conductivity (Solid)	$1.585 \times 10^{5} {T_{l}}^{- 1.23}$ [35]	$W / (m \cdot K)$
${τ_{γ}}^{S}$	Energy relaxation time (Solid)	$\begin{array}{l} τ_{γ 0} [1 + (N_{e} / N_{c r})^{2}] \\ τ_{γ 0} = 0.5 p s, N_{c r} = 2 \times 10^{27} m^{- 3} \end{array}$ [34]	$s$
${C_{e}}^{L}$	Heat capacity of the free-carriers (Liquid)	$100 T_{e}$ [36]	$J / (m^{3} \cdot K)$
${κ_{e}}^{L}$	Thermal conductivity of the free-carriers (Liquid)	$67$ [36]	$W / (m \cdot K)$
${C_{l}}^{L}$	Lattice heat capacity (Liquid)	$1.06 \times 10^{9} (3.005 - 2.629 \times 10^{- 4} T_{l})$ [37]	$J / (m^{3} \cdot K)$
${κ_{l}}^{L}$	Lattice thermal conductivity (Liquid)	$50 + 0.029 (T_{l} - T_{m})$ [38]	$W / (m \cdot K)$
${τ_{γ}}^{L}$	Energy relaxation time (Liquid)	$10^{- 12}$ [38]	$s$
$γ$	Energy coupling rate between free-carriers and lattice	$C_{e} / τ_{γ}$	$J / (m^{3} \cdot K \cdot s)$

Symbol	Significance	Value	Unit
$m_{o p t}^{*}$	Effective optical mass	$0.18 m_{e}$ [18]	$kg$
$ν_{e}$	Collision frequency	$1 \times 10^{15}$ [18]	$s^{- 1}$
$μ_{e}$	Carrier mobility	$e / (m_{o p t}^{*} ν_{e})$ [11]	$C \cdot s / kg$
$σ_{1}$	linear absorption (800 nm)	$1.12 \times 10^{5} \exp (T_{l} / 430)$ [27]	$m^{- 1}$
$σ_{2}$	Two-photon absorption (800 nm)	$9 \times 10^{- 11}$ [27]	$s \cdot m/J$
$δ_{I}$	Impact ionization coefficient	$3.6 \times 10^{10} \exp (- 1.5 E_{g} / k_{B} / T_{e})$ [28]	$s^{- 1}$
$Θ$	Free-carrier absorption coefficient (800 nm)	$2.9 \times 10^{- 22} T_{l} / T_{0}$ [27]	$m^{2}$
$τ_{A R}$	Minimum Auger recombination time	$6 \times 10^{- 12}$ [29]	$s$
$C_{A R}$	Auger recombination rate	$3.8 \times 10^{- 43}$ [28]	$m^{6} / s$
$R$	Reflectivity (800 nm)	$0.329 + 5 \times 10^{- 5} T_{l}$ [30]	-

Symbol	Significance	Value	Unit
$E_{g}$	Bandgap energy	$e (1.167 - 0.0258 T_{l} / T_{0} - 0.0198 (T_{l} / T_{0})^{2})$ [27]	$J$
${κ_{e}}^{S}$	Thermal conductivity of the free-carriers (Solid)	$- 0.5552 + 7.1 \times 10^{- 3} T_{e}$ [34]	$W / (m \cdot K)$
${C_{l}}^{S}$	Lattice heat capacity (Solid)	$1.978 \times 10^{6} + 354 T_{l} - 3.68 \times 10^{6} {T_{l}}^{- 2}$ [35]	$J / (m^{3} \cdot K)$
${κ_{l}}^{S}$	Lattice thermal conductivity (Solid)	$1.585 \times 10^{5} {T_{l}}^{- 1.23}$ [35]	$W / (m \cdot K)$
${τ_{γ}}^{S}$	Energy relaxation time (Solid)	$\begin{array}{l} τ_{γ 0} [1 + (N_{e} / N_{c r})^{2}] \\ τ_{γ 0} = 0.5 p s, N_{c r} = 2 \times 10^{27} m^{- 3} \end{array}$ [34]	$s$
${C_{e}}^{L}$	Heat capacity of the free-carriers (Liquid)	$100 T_{e}$ [36]	$J / (m^{3} \cdot K)$
${κ_{e}}^{L}$	Thermal conductivity of the free-carriers (Liquid)	$67$ [36]	$W / (m \cdot K)$
${C_{l}}^{L}$	Lattice heat capacity (Liquid)	$1.06 \times 10^{9} (3.005 - 2.629 \times 10^{- 4} T_{l})$ [37]	$J / (m^{3} \cdot K)$
${κ_{l}}^{L}$	Lattice thermal conductivity (Liquid)	$50 + 0.029 (T_{l} - T_{m})$ [38]	$W / (m \cdot K)$
${τ_{γ}}^{L}$	Energy relaxation time (Liquid)	$10^{- 12}$ [38]	$s$
$γ$	Energy coupling rate between free-carriers and lattice	$C_{e} / τ_{γ}$	$J / (m^{3} \cdot K \cdot s)$

Comparison study of the femtosecond laser-induced surface structures on silicon at an elevated temperature

Abstract

1. Introduction

2. Experimental setup

3. Experimental results and discussions

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (10)

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