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Abstract
Pursuing ever-smaller feature size in laser-based lithography is a research topic of vital importance to keep this technique competitive with other micro-/nano-fabrication methods. Features smaller than the diffraction-limited spot size can be obtained by “thresholding”, which utilizes the deterministic nature of damage threshold with ultrashort laser pulses and is achieved by precisely tuning pulse energies so that only the central portion of the focal spot produces permanent modification. In this paper, we examine the formulation commonly used to describe thresholding and show that the relationship between feature size (r) and laser fluence (F) is invariant with respect to the nature of laser absorption. Verified by our experiments performed on metal, semiconductor, and dielectric samples, such invariance is used to predict the smallest feature size that can be achieved for different materials in a real-world system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The deterministic nature of ultrafast laser-induced material modification paves the way for high-precision processing of a wide range of materials with feature sizes similar to Gaussian focal spots, which are limited by light diffraction [1–10]. Techniques to reduce feature size below the diffraction limit include nano-grating [11], stimulated emission depletion (STED) [12], partially overlapping focal spots [13], and many other methods [14–16]. The most straightforward way is so-called “thresholding”, by which the pulse energy is controlled so precisely that only a portion of the focal fluence distribution exceeds the threshold for permanent modification [17]. Thresholding, in conjunction with nonlinear (multiphoton) absorption in wide-bandgap materials, is often invoked as the reason for feature size below the diffraction-limited spot size.
In principle, thresholding itself is capable of achieving a very small feature size according to the formula derived from a Gaussian profile for materials with deterministic thresholds [18]. Thus the question is how different absorption mechanisms (linear or nonlinear) affect thresholding and as a result the relationship between feature size and pulse energy. This topic has been explored by Fischer et al. in the context of multiphoton polymerization [19], and M. Gedvilas et al. explored multiphoton absorption enhancement by dual-wavelength double-pulse laser irradiation [20]. Recently, Garcia-Lechuga et al. have shown that such relationship is invariant for dielectrics with different bandgaps [21]. In this paper, we will theoretically and experimentally show that such invariance is consistent with the well-known single-photon model and provides the ability to predict the smallest feature size in experiments.
2. Theory
We start from the model derived by Liu [18], which relates feature size rth with peak laser fluence F0 as
where w0 is the 1/e2 radius of the Gaussian focus, and Fth is the threshold fluence at which features start to form on the surface. Originally developed for determining the size of laser focal spot w0, Eq. (1) has been commonly used to determine the damage threshold Fth for various types of materials such as semiconductors and metals [22–24], which predominantly absorb laser energy through single-photon processes. This is due to the presence of a large amount of initial free electrons [9,25].However, the mechanism of laser ablation in dielectrics is different since the density of initial free electrons is small, and feature generation is related to the accumulation of free electrons through multiphoton ionization (MPI) and avalanche ionization (AI) processes. Here, we consider a feature forms when the density of free electrons is equal to or higher than the critical density, which is in accordance with the previous results [4–7]. It is worth pointing out that the total laser energy coupled to the electron subsystem is also reported as the laser breakdown threshold when high-intensity few-cycle laser pulses are applied [26]. To understand the relationship between feature size and laser intensity in dielectrics, the following derivation is conducted for two cases depending on which process is dominant. When MPI is dominant, a rate equation can be written as
Here, ne is the density of conduction-band (CB) free electrons excited from the valence band (VB) through MPI, $\sigma $ is the MPI coefficient [4] and k is the order of MPI; I(r,t) is the laser intensity with peak intensity I0, FWHM (full-width-at-half-maximum) pulse duration $\tau $, and focal spot size w0 measured at the 1/e2 level. For simplicity, in Eq. (2), recombination of electron-hole pairs is ignored. This is justified because we are interested in the events leading to the maximum ne at which features are formed [4]. We also exclude free-carrier diffusion because the diffusion length is estimated to be small (<100 nm) compared to w0 used in this experiment [27,28]. This exclusion of free-carrier diffusion needs to be examined when features much smaller than the laser wavelength are desired. The total density of CB free electrons ne(r,t=∞) can be solved by integration over time as
Taking the natural logarithm on both sides, we have
Note that mathematically the argument of the ln() function should be dimensionless. Writing Eq. (6) in the current form requires adding factors such as 1 W/cm2 which are omitted for simplicity. Such factors are only for a mathematical reason and ignoring it will not affect the relationship between rth and F0. Since the relationship between peak intensity I0 and peak fluence F0 follows ${I_0} = \frac{{2{F_0}}}{\tau }\cdot \sqrt {\frac{{ln2}}{\pi }} $, Eq. (6) can be rewritten in terms of F0 as
The last term in Eq. (7) is independent of F0, so we can rename it as CMPI, and Eq. (7) can be simplified as
We can see from Eq. (8) that the nonlinearity factor k vanishes from the first term which dictates how feature size rth changes as peak fluence F0 is varied, resulting in a relationship similar to Eq. (1).
Following a similar procedure, the relationship between rth and F0 for the AI-dominated case can be derived from the following rate equation
where $\eta $ is the avalanche coefficient. Different from the MPI-dominated case, ne is increasing exponentially with respect to t and at the same time, has a spatial distribution due to the Gaussian beam profile, giving rise to a double-exponent asAgain, using the Gaussian integral and setting ne(r = rth,t=∞)=nth and r = rth, we get
Since all the parameters in the last term are constants, we can rename it as CAI, and Eq. (12) can be simplified as
Once again, we arrive at a relationship similar to what Eq. (1) predicts for the change of feature size as a function of laser fluence. It is worth mentioning that although the derivations above deal with two limiting cases, the conclusion still applies when both MPI and AI are present. This can be shown by separating the entire carrier-generation process into two stages, in which MPI provides “seed electrons” whose density is further increased by AI. This technique has been used before to simplify the analysis of the temporal component of the rate equation [4]. In this model, the feature is considered to have a direct response to the local laser fluence. This is because the thermal effects associated with femtosecond laser is low [1–3,29] and the diffusion length of free carriers is much smaller than the laser spot size.
3. Experimental
Experimentally, we perform single-shot laser lithography experiments on three samples, aluminum 6061, monocrystalline Si (100-oriented), and fused silica (FS, Corning 7980). They represent three types of materials - metal, semiconductor and dielectrics, respectively. Figure 1(a) shows the experiment setup. A femtosecond laser (Pharos, Light Conversion) delivers 167 fs (FWHM) pulses at a center wavelength of 1030 nm with polarization in the x-y plane. The beam is first expanded to a diameter of 10 mm by a Galilean telescope (Lenses 1 and 2) with a magnification of two. The expanded beam is focused by Lens 3 (focal length 200 mm). A variable neutral-density (VND) filter is used to control the laser energy. A three-axis stage with a step resolution of 1 μm is used to adjust the position of the samples to ensure the beam is accurately focused on the front surfaces of the samples. A CMOS camera is used to observe the damage morphology in situ and ensure that the front surfaces of the samples are at the same position. The focal spot on the samples is shown in Fig. 1(b), and the black dashed line marks the 1/e2 boundary of the focus. The inset shows a Gaussian fitting of the focus along the vertical direction with a diameter 2w0 = 30 μm. Here we choose the fundamental laser wavelength at 1030 nm because it can test the theory in a wide range of optical nonlinearity in bandgap materials.
Fig. 1. (a) Experimental setup. M1, M2: mirrors; VND: variable neutral density filter; CMOS: CMOS camera. (b) Profile of the laser focus on the samples. The black dashed line indicates the 1/e2 boundary of intensity. The inset shows the Gaussian fitting of experimental data along the vertical direction.
4. Discussion
Figure 2(a) shows typical morphologies of features created by single pulses. The shapes are nearly circular, similar to the beam profile shown in Fig. 1(b). Features on fused silica have clear boundaries which make it easy to measure feature diameter 2rth. For silicon and aluminum, the boundaries are defined as the observable transition between laser modified and pristine surfaces. The feature generated on silicon has a dark spot at the center, which has been attributed to ablation [22]. In our measurement, we use the “modification” size as the feature size 2rth in accordance with that work. For the aluminum sample, the laser ablation experiments were conducted right after polishing so an oxide layer, whose thickness is estimated to be a few nanometers [30,31], is not considered in the model. Thick layers could affect the linear and nonlinear absorption of laser energy, as has been reported before [32]. In all the experiments, the feature diameter 2rth is measured along the vertical direction, which is marked by a black dashed line in Fig. 2(a). Figure 2(b) shows a standard way of extrapolating the damage threshold from the measured 2rth and incident laser fluence F0 in a semi-log plot [18]. All three data sets have almost the same slope, which is consistent with the theory when the same focal spot size w0 is used. w0 is fitted to be 14.6${\pm} $0.2 µm, which is close to the experimental value. This can be seen more easily in Fig. 2(c), which normalizes F0 (or I0) with the corresponding threshold value for each sample. As expected, all data points collapse to a single linear line. It is worth pointing out that the laser ablation threshold would be influenced if multiple pulses are used [33].
Fig. 2. (a) Optical microscopy images of single-shot laser written features on three different materials (dielectrics, semiconductor, and metal). (b) Relation between square of feature diameter (2rth)2 and incident peak fluence F0 (intensity I0) plotted in the semi-log scale. (c) relation with F0 (I0) normalized to the corresponding threshold fluence (intensity) of each material.
An interesting consequence of the invariance relationship between feature size and laser fluence is the predictability of minimum feature sizes achievable for different types of material using one single formula. Replotting the result of Fig. 2(b) to Fig. 3(a) using the linear scale, we can see that feature diameter 2rth changes more quickly above the threshold fluence for material with a smaller threshold. To understand how this affects the smallest feature diameter (2rmin) [the lowest data point for each material in Fig. 3(a)], we point out that in our and many others’ experiments, a gradual (and discrete) change in laser fluence is used to “approach” (2rmin), with a step size ΔF. ΔF should be sufficiently large compared to the pulse-to-pulse fluctuation of a system so that repeatable features can be generated. Our laser has an RMS (root mean square) fluctuation <0.5%.
Fig. 3. (a) Square of feature size (2rth)2 as a function of incident laser fluence F0 and laser intensity I0 for three materials used in the experiment. This is the same result as in Fig. 2(b) but plotted in the linear scale. (b) Minimum feature size 2rmin (nondimensionalized to the laser focal size w0) that can be achieved in experiments depends on the damage threshold Fth of the material used. Data points are from our own results and the literature. The solid lines are fittings based on Eq. (14).
In Fig. 3(b), we plot the minimum feature sizes, (2rmin), achieved in our experiment with values found in the literature ((2rmin) is normalized with respect to $\sqrt 2 $w0 reported in those experiments). The curves are obtained from the relationship between feature diameter 2rth and incident fluence F0 for a material with a certain threshold fluence Fth. From Eq. (1), we can derive the relationship between the minimum feature size (2rmin) and the threshold fluence Fth for a given material with $\Delta F$ extracted from the experiments
The minimum controllable laser fluence $\Delta F$ is 0.08 J/cm2, which is due to the control resolution of the VND in our system. The minimum feature size (2rmin) according to different threshold Fth is shown in Fig. 3(b), which is marked as the blue solid line. In addition to our data, data from the work of Liu [18], Korte et al. [34], and Garcia-Lechuga et al. [21], are included and marked as pink/blue, red, and green. $\Delta F$ is read from the plots in those papers and the values are shown in the legend. Solid lines in Fig. 3(b) are fitting results calculated from Eq. (14) with the corresponding $\Delta F$ values. Most of the data points fall onto the predicted lines; a few exceptions, such as the SiO2 and CaF2 data from Ref. [27], are due to inaccuracy in determining $\Delta F$. Nevertheless, the overall agreement demonstrates the prediction power of Eq. (14) for a variety of materials. Particularly, materials with a large bandgap such as FS, CaF2, sapphire, and soda-lime glass have a high damage threshold Fth [35], which makes them a better choice for high-resolution laser lithography, provided that there is flexibility in choosing substrate materials (which is often not the case). In this study, the control resolution in the experimental setup determines ΔF, which in turn determines the smallest feature size 2rmin that can be produced repeatedly. In addition, it is worth pointing out that there is also a fluctuation of 2rmin due to laser fluence fluctuation [36] which is not considered in our model.
5. Conclusions
To conclude, we have derived a relationship between feature size and laser fluence for band-gap materials, an extension to the well-known formula derived by Liu [18]. We have shown that such a relationship remains the same for single-photon absorption in metals and multiphoton/avalanche absorption in band-gap materials. Single-shot laser lithography experiments on fused silica, silicon, and metal samples have been conducted to verify such invariance, which is later utilized to determine the smallest features that can be fabricated repeatably with different materials. This result can be applied in laser-lithography-based mass production because every laser processing system is subject to a certain level of control resolution, which needs to be taken into consideration in the initial design and the choice of processing parameters.
Funding
National Science Foundation (1846671).
Acknowledgments
The authors thank Dr. Xinpeng Du for his helpful discussion.
Disclosures
The authors declare no conflicts of interest.
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