# Scalable Thermal Engineering via Femtosecond Laser-Direct-Written Phononic Nanostructures

Hiroki Hamma, Roman Anufriev, Kazuyoshi Fushinobu, Masahiro Nomura, Byunggi Kim

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Hiroki Hamma, Roman Anufriev, Kazuyoshi Fushinobu, Masahiro Nomura, Byunggi Kim. Scalable Thermal Engineering via Femtosecond Laser-Direct-Written Phononic Nanostructures. Advanced Functional Materials, 2025, 10.1002/adfm.202525269. [https://doi.org/10.1002/adfm.202525269](https://doi.org/10.1002/adfm.202525269)

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# Scalable Thermal Engineering via Femtosecond Laser-Direct-Written Phononic Nanostructures

Hiroki Hamma, Roman Anufriev, Kazuyoshi Fushinobu,* Masahiro Nomura,* and Byunggi Kim*

Modern electronics demands innovative thermal management strategy beyond conventional approaches. Here, A transformative platform bridging laboratory-scale phononic nanostructures and industrial-scale thermal engineering is demonstrated via femtosecond laser-induced periodic surface structures (fs-LIPSS). Laser-direct-written phononic metamaterials is achieved with throughputs exceeding 10⁻⁵ m² s⁻¹—three orders of magnitude faster than electron beam lithography—while maintaining nanoscale resolution. Applied to silicon membranes, the fs-LIPSS reduce thermal conductivity from 10¹ to 76 W m⁻¹ K⁻¹, surpassing thin-film limit predicted by Fuchs-Sondheimer model. Monte Carlo simulations corroborate these findings, showing that long mean free paths are limited by the fs-LIPSS, while ablation-induced roughness mediates isotropic modulation of thermal conductivity via enhanced diffuse scattering. Thus, this maskless and resist-free process enables the fabrication of phononic nanostructures that are CMOS-compatible and scalable to wafer-level dimensions. This work establishes the fs-LIPSS as a versatile approach that makes phonon engineering practical for real-world devices, unlocking applications in quantum technologies, on-chip energy conversion, and high-performance computing.

H. Hamma, K. Fushinobu, B. Kim
Department of Mechanical Engineering
Institute of Science Tokyo
Tokyo 152-8550, Japan
E-mail: fushinobu.k.4c25@m.isct.ac.jp; kim.b.4dbc@m.isct.ac.jp

R. Anufriev, M. Nomura
Institute of Industrial Science
The University of Tokyo
Tokyo 153-8505, Japan
E-mail: nomura@iis.u-tokyo.ac.jp

R. Anufriev
LIMMS, CNRS-IIS IRL 2820
The University of Tokyo
Tokyo 153-8505, Japan

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adfm.202525269

© 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/adfm.202525269

## 1. Introduction

Controlling heat transport is essential for energy-efficient computing,[¹,²] chip-scale energy conversion,[³⁻⁵] and suppression of thermal noise in quantum information technologies.[⁶,⁷] As device dimensions shrink and power densities increase, microscopic control of thermal conduction at the level of phonon, a quantum of lattice vibration, has become a central challenge in thermal management. Over the past two decades, the concept of phonon engineering has revealed that heat can be controlled beyond the limits of Fourier's law by tailoring the spectral, spatial, and coherence properties of phonons.[⁸,⁹] Phononic nanostructures such as periodic nanohole arrays demonstrate that dispersion flattening can suppress thermal transport, while coherent interference effects emerge at sub-100 nm length scales.[¹⁰⁻¹²] Experiments in nanostructured semiconductors have revealed phonon mean free paths (MFP) spanning from tens of nanometers to microns, confirming that nanostructures within this range can manipulate transport through boundary scattering and ballistic effects.[⁸,¹³] These advances establish phononic nanostructures as promising candidates for nanoscale thermal insulation,[¹⁴] rectification,[¹⁵] and thermoelectric conversion.[¹⁶,¹⁷]

Despite these advances, the lack of scalable fabrication methods remains a major barrier for practical implementation. Standard nanostructuring methods, such as electron beam lithography (EBL) and focused ion beam (FIB) milling, offer sub-100 nm resolution for phononic nanostructure fabrication in a maskless manner. Zhou et al. have demonstrated that the self-assembly of block copolymers can facilitate phononic crystals and a significant reduction in thermal conductivity.[¹⁸] However, those conventional methods strongly rely on slow, multi-step, and highly precise fabrication procedures, limiting high-throughput fabrication of phononic nanostructures. Nanoimprint lithography has been proposed as a repeatable approach, yet the fabrication and maintenance of nanoscale molds remain challenging.[¹⁹] As a result, phononic nanostructures have remained as laboratory-scale demonstrations, hindering their translation into practical devices that require wafer-scale integration.
Here, we showcase that femtosecond laser-induced periodic surface structures (fs-LIPSS) can overcome this scalability bottleneck by enabling maskless, one-step fabrication of phononic nanostructures over centimeter-scale regions with throughputs orders of magnitude higher than electron-beam lithography. The fs-LIPSS is a laser-direct-writing scheme that spontaneously produces sub-wavelength periodic ripples on surfaces via interference between incident light and surface electromagnetic waves,[20-22] and has been widely applied for optical functionalization,[23,24] wettability control,[25-27] tribology,[28,29] and bioengineering.[30] Applied to single-crystalline silicon thin films, we demonstrate that the fs-LIPSS can reduce the thermal conductivity significantly below the thin-film limit predicted by the Fuchs-Sondheimer (FS) model.[31-33] To elucidate the underlying mechanism, we further perform Monte Carlo simulations, which reveal the role of boundary scattering in the fs-LIPSS. Our results position the fs-LIPSS as a scalable, CMOS-compatible platform for phononic thermal engineering, bridging the gap between fundamental studies of nanoscale heat conduction and real-world applications.

## 2. Results and Discussion

### 2.1. Phononic Nanostructures Fabricated Via fs-LIPSS

The LIPSS are characterized by their spatial frequency relative to the incident laser. In general, the low-spatial-frequency LIPSS exhibit a period slightly shorter than the laser wavelength (0.8–0.9 $\lambda_{laser}$), while the high spatial frequency LIPSS have periods far below the laser wavelength (≈0.1 $\lambda_{laser}$). In this work, we focused on the low-spatial-frequency LIPSS, as they offer large depth contrast with well-defined edges.

For silicon of which bandgap is 1.1 eV, near-infrared wavelengths with large optical penetration depths are advantageous to realize the high-aspect ratio LIPSS. In our experiment, we scanned the 220-fs laser pulses at 1030 nm (Figure 1a) on a 1-µm-thick device layer of a silicon-on-insulator (SOI) substrate to form the LIPSS. Subsequently, the anisotropic reactive ion etching (RIE) using sulfur hexafluoride (SF₆) and oxygen (O₂) under bias was performed to tune the top silicon thickness (Figure 1b). This RIE process makes the groove dimension as short as the phonon MFP, thereby enhancing phonon boundary scattering (Figure 1c).

Figure 1d,e show the scanning electron microscope (SEM) images of the fs-LIPSS. The grating structures form perpendicularly to the laser polarization, with a period of 830 nm (0.81 $\lambda_{laser}$). The inset of Figure 1d shows the 2D fast Fourier transform (2D-FFT) spectrum, where two clear symmetric peaks in the y-direction confirm the presence of a single well-defined spatial frequency. Cross-sectional SEM image (Figure 1e) reveals that the RIE shaped the fs-LIPSS rectangular grating structures. The average neck thickness was 314 nm, which was small enough to strongly enhance phonon boundary scattering in silicon, whose bulk phonon MFP spans 100–1,000 nm.[34]

To maximize the regularity of the LIPSS, we optimized laser irradiation parameters, including fluence and repetition rate. Figure 2a presents the spatial frequency spectrum along the dashed line in the inset of Figure 1d. As a figure of merit, we used the product of the quality factor ($k_y$, LIPSS/$\Delta k_y$) and the peak

intensity to account for both the dominance of a single frequency and pattern contrast. For the formation of the regular LIPSS, the polarization should be fixed perpendicular to the scanning direction, ensuring polarization-perpendicular mode to initiate the parallel-line ablation along the scanning direction.[35]

Figure 2b shows that the optimum condition was a repetition rate of 0.5 MHz and a fluence of 1.59 J cm⁻². With a 1 e² spot diameter $d_{spot}$ = 13 µm, the pulse overlap number, defined by $\frac{f_d d_{spot}}{v_i}$, was calculated as 6.5, where $v_s$ and $f_r$ are the scanning speed and the repetition rate, respectively. The optimized pulse overlap number efficiently guides the LIPSS growth while preventing excessive ablation. The LIPSS were formed slightly below the uniform ablation threshold. At higher fluences, thermal and fluid-dynamic instabilities dominate over the interference-driven periodic ablation, erasing periodicity. At lower fluences, the periodic heating induces amorphization, leading to only a few-nanometers corrugation in height.[36]

Raman spectroscopy further confirmed crystallinity (Figure 2c). The fs-LIPSS exhibited the sharp silicon peaks at 520 cm⁻¹, without the peak broadening associated with the amorphization. Finally, the RIE effectively removed the surface oxide layer, leaving a monolithic crystalline silicon system.

### 2.2. Fabrication Throughput Benchmark

To assess the effectiveness of the fs-LIPSS for phononic nanostructuring, we compared their fabrication throughput with representative top-down nanofabrication techniques (Figure 3). Throughput is defined as the processed area per unit time (m² s⁻¹), which allows direct comparison across approaches with different mechanisms and resolutions.

FIB offers direct writing capability, but its throughput for 300–400-nm depth is limited to the order of $10^{-11}$–$10^{-9}$ m² s⁻¹.[37,38] EBL provides higher resolution and greater scalability, but still remains relatively slow, with typical single-beam throughputs of $10^{-10}$–$10^{-8}$ m² s⁻¹.[39] Recent developments in multibeam EBL have demonstrated significant improvement, reaching the $10^{-5}$–$10^{-4}$ m² s⁻¹ regime, though such systems are still at the development and pre-commercial stage.[40,41]

Advanced laser-direct-writing techniques based on structured illumination using a digital micromirror device (DMD) or multibeam interference inherently enable parallel patterning. Their throughputs typically range from $10^{-7}$ to $10^{-4}$ m² s⁻¹,[42–44] depending on the projection optics and illuminated area. These methods offer high speed but require specialized optical setups, which limit their flexibility and robustness for rapid prototyping or direct patterning on non-planar substrates.

In contrast, single-beam fs-LIPSS fabrication does not rely on beam structuring optics, but takes advantage of self-constructed interference fields on material surfaces, achieving surprisingly competitive throughput. Reported values span from $10^{-6}$ up to $10^{-4}$ m² s⁻¹, depending on the scan speed and line spacing.[45–48] For instance, our experimental condition (1 m s⁻¹ scan speed, 9-µm line spacing) corresponds to $9 \times 10^{-6}$ m² s⁻¹, which already surpasses single-beam EBL by more than three orders of magnitude. Notably, high-speed laser scanning, where optimized scanning strategies such as polygon or galvo-based approaches are employed, has pushed LIPSS throughput into the $10^{-4}$ m² s⁻¹
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Fabrication and structural characterization of fs-LIPSS on silicon. a) Schematic illustration of LIPSS formation by the fs-laser scanning on SOI. b) Post-LIPSS for thickness control by the anisotropic RIE. c) Conceptual illustration of phonon scattering in LIPSS-modulated membranes. d) SEM image and FFT analysis (inset) of the surface morphology confirming the periodicity. e) Cross-sectional SEM image showing the neck thickness of 314 nm.</figcaption></figure>

regime, approaching or even exceeding structured photolithography techniques.[49]

Further enhancement is possible using multibeam or diffractive optical element (DOE) configurations, which scale throughput linearly with the number of beamlets. Literature demonstrations report effective ranges of $10^{-4}–10^{-3}$ m$^2$ s$^{-1}$.[50–52] Similar to multibeam EBL, these configurations are still in the development stage, but they highlight the strong scalability potential of the fs-LIPSS, possibly with lower cost required for facilities.

Overall, this benchmark shows that the fs-LIPSS uniquely combines resist-free, maskless process with a one-step operation. While not matching the resolution of EBL, the huge advantage in fabrication speed makes the fs-LIPSS an attractive choice for phononic nanostructures. Further investigations to realize finer-resolution LIPSS will enrich the fs-LIPSS as a promising fabrication route for phonon engineering and thermal management devices.

## 2.3. Thermal Conductivity Measurement

The thermal conductivity of the fs-LIPSS was measured using the micro time-domain thermoreflectance (μTDTR) method. Figure 4a,b show the experimental schematic of the μTDTR and the SEM image of the sample, respectively. A 10-μs pump pulse at 488 nm heated up the gold transducer, while a continuous-wave probe laser at 514 nm monitored the transient temperature change. To make the measurement solely sensitive to the thermal conductivity of the fs-LIPSS, we utilized a suspended air bridge structure of silicon membranes. We changed the orientation of the fs-LIPSS, which is defined as the angle between the heat dissipation direction and the grooves.

Figure 4c shows representative TDTR curves and their fitting through the finite element method (FEM) simulation. The fs-LIPSS had longer cooling times than the pristine film, indicating reduced effective thermal conductivity. The decay time increased with increasing orientation angle, as the grating structures
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Characterization of fs-LIPSS. a) Spatial frequency spectrum of the surface morphology showing a distinct LIPSS peak; inset: Lorentzian fit used to evaluate the periodicity. b) Optimization map of the LIPSS regularity as a function of laser fluence and repetition rate, where the product of peak intensity and quality factor highlights the optimal condition. c) Raman spectra of the silicon surface before and after fs-LIPSS fabrication and subsequent RIE, confirming the preservation of the crystalline silicon peak.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Throughput comparison of maskless nanofabrication techniques expressed as processed area per unit time (m² s⁻¹). Color represents the different lithography scheme. Gray, FIB;[37,38] Blue, EBL;[39–41] Green, Structured photolithography;[42–44] Orange, LIPSS.[45–52] Shaded bars indicate methods still at the development stage (EBL multibeam and LIPSS multibeam/DOE). The star marks the throughput achieved in this study.</figcaption></figure>

enlarge the heat dissipation paths when aligned perpendicular to the heat dissipation.

The results in Figure 4d show that the material thermal conductivity of the 780-nm silicon thin film decreased from 101 W m K (gray shaded region) to 76 W m K (symbols) on average over all the orientation angles. Interestingly, the thermal conductivity reduction was independent of the orientation angle, showing isotropy in the in-plane direction. Thus, phonon guiding along the grating is absent at room temperature, indicating that diffuse boundary scattering is dominant.

To clarify the role of the boundary scattering in the fs-LIPSS, we compared the measured thermal conductivity with the thin-film FS model:[33,53]

$$ \kappa = \int_{0}^{\infty} K_{\Lambda_{\text{bulk}}} S_{\text{FS}}(\xi) d\Lambda_{\text{bulk}} \quad (1) $$

where $K_{\Lambda_{\text{bulk}}} = -\sum_{s} \frac{1}{3} C \nu \Lambda_{\text{bulk}} \frac{d\omega}{d\Lambda_{\text{bulk}}}$ is the differential MFP distribution[54] over each polarization branch s, with volumetric heat capacity C, group velocity v and MFP Λ. The dimensionless parameter $\xi = t/\Lambda_{\text{bulk}}$, where t is the film thickness. For fully diffuse boundaries, the suppression factor $S_{\text{FS}}$ is

$$ S_{\text{FS}}(\xi) = 1 - \frac{3}{8\xi} [1 - 4E_3(\xi) + 4E_5(\xi)] \quad (2) $$

where is the exponential integral $E_n(\xi) = \int_1^\infty e^{-\xi u} u^{-n} du$. Given the MFP distribution, the thermal conductivity of the membrane can be calculated from Equation 1.[53] We used a phonon MFP distribution calculated by Esfarjani et al., and bulk single-crystalline silicon thermal conductivity (148 W m K).[55] The thermal conductivity predicted by the FS model for a 780-nm membrane was 98.8 W m K, showing good agreement of the experimental results (black dashed line in Figure 4d).

Because the FS model assumes a uniform film, we define the effective thickness $t_{\text{eff}}$ of the LIPSS as

$$ t_{\text{eff}} = t_{\text{ridge}} - \varphi d_{\text{groove}} \quad (3) $$
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Thermal transport measurement of suspended LIPSS membranes. a) Schematic illustration of µTDTR measurement on the suspended LIPSS structures. b) SEM image of the air-bridge sample for µTDTR with the LIPSS-rotated relative to the heat dissipation direction. c) Experimental and FEM temperature decay curves for non-structured membrane (780 nm) and different LIPSS orientations. d) Extracted thermal conductivity as a function of the LIPSS orientation, compared with the FS model prediction for three thicknesses, 780 nm (the plain membrane), 673 nm (LIPSS effective thickness), and 314 nm (LIPSS neck thickness).</figcaption></figure>

where $\varphi$ and $d_{groove}$ are duty cycle ((groove width)/(LIPSS period)) and groove depth, respectively. Substituting $t_{eff} = 673$ nm into Equation 1 yields 96.6 W m K (blue dashed line in Figure 4d), well above the measured fs-LIPSS conductivity ($\approx 76$ W m K). The LIPSS thermal conductivity was even smaller than FS-model-predicted value for 314 nm (red dashed line in Figure 4d, which is the neck thickness shown in Figure 1e. As the FS model cannot capture additional lateral boundary scattering introduced by the periodic grooves, the difference between the FS model and experimental results corroborates that the boundary scattering in the periodic nanostructures. Phonon trapping and backscattering contribute to the significant reduction of thermal conductivity, consistently observed in other phononic nanostructures such as nanocones,[56] and nano-porous membranes.[57,58]

## 2.4. Monte Carlo Simulation

To further analyze phonon transport, we employed phonon Monte Carlo simulation (see Methods). Figure 5a shows phonon ray-tracing results in the fs-LIPSS-like rectangular grating. We injected phonons from the hot side and collected them at the cold side, enabling the extraction of trajectories and MFP statistics.

Figure 5b,c are phonon scattering angle distributions at the vertical sidewalls of the fs-LIPSS. Most scattering was diffuse fol-

lowing the Lambertian distribution, while the specular scattering was negligible. The ablation-induced rough surface reduced the specular scattering probability to nearly zero. Lack of the specular scattering underlies the absence of phonons guiding and anisotropy in the LIPSS. For $\theta = 0^{\circ}$ (Figure 5b), scattering was symmetric, while for $\theta = 90^{\circ}$ (Figure 5c), the back scattering dominated due to the grating geometry perpendicular to the heat flux direction.

To calculate the thermal conductivity, we integrated phonon dispersion and sampled the phonon MFP on each branch j at the wavevector q

$$ \kappa = \frac{1}{6\pi^2} \sum_{j,q} f \frac{\hbar^2 \omega_j^2(q)}{k_b T^2} \frac{\exp\left[\frac{\hbar\omega_j(q)}{k_b T}\right]}{\left(\exp\left[\frac{\hbar\omega_j(q)}{k_b T}\right] - 1\right)^2} v_j(q) \Lambda_j(q) q^2 dq \quad (4) $$

where $\hbar$, $k_b$, and $\omega$ are the reduced Planck constant, Boltzmann constant, and angular frequency, respectively. As shown in Figure 5d, the thermal conductivities calculated by the Monte Carlo simulations showed an accurate agreement with the experimental values within their uncertainty, confirming validity of the numerical method.

Figure 5e intuitively depicts the mechanism of the thermal conductivity reduction in the fs-LIPSS. The cumulative thermal
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Monte Carlo simulation of phonon transport in LIPSS. a) Simulation of phonon trajectories across LIPSS structures. b,c) Polar plots of phonon scattering distributions for parallel ($\theta = 0°$) and perpendicular ($\theta = 90°$) orientations. d) Thermal conductivity from experiments and Monte Carlo (MC) simulations. e) Cumulative thermal conductivity vs phonon MFP, highlighting the MFP-limitation effect by the LIPSS period.</figcaption></figure>

conductivity clearly showed the limitation of the phonon MFP by the fs-LIPSS period. Thus, the period served as a limiting dimension, so that the long MFP phonon transport which contributes largely to thermal conduction is suppressed. This MFP limitation effect can be observed for both the $\theta = 0°$ and $90°$, confirming that the diffuse scattering equally limits thermal conduction regardless of the LIPSS orientation angle.

For trench structures with small groove widths like the LIPSS, the trench width and the neck thickness work synergistically to manipulate phonon transport. Meanwhile, the MFP cutoff can be directly related to a laser wavelength (0.81 $\lambda_{laser}$ in the present context). Considering that long-MFP phonons contribute significantly to thermal conduction, use of shorter wavelength lasers is required to achieve greater control over phonon transport. For wide-bandgap, high-thermal-conductivity semiconductors and insulators such as silicon carbide, gallium nitride, and diamond, the use of visible to UV femtosecond lasers will enable dramatic modulation of thermal conductivity for two reasons. First, the large optical penetration depths of those mate-

rials mediate deep ablation depth required for high-aspect ratio LIPSS. Second, the LIPSS period can be significantly shorter than phonon MFP. High-aspect-ratio and short-period LIPSS will enhance phonon scattering and confinement, potentially modulate phonon dispersion. In this work, the fs-LIPSS achieved a neck thickness of 314 nm in silicon, where phonons with MFP larger than 400 nm contributes 50% of thermal conduction in its bulk form.[34,59] Huang et al. demonstrated that 300-fs pulsed with a 520-nm wavelength could be used for fabrication of high-uniformity LIPSS on silicon and 4H silicon carbide by manipulating half-periodic mismatched optical enhancement. Further structural control along the thickness direction could therefore provide an additional route to engineer phonon transport.

Another promising approach is use of a metallic hard mask with LIPSS. Metallic materials with short surface-plasmon-polaritons MFP enable formation of high-uniformity LIPSS.[60] The uniform structures can be then transferred to the underlying semiconductor or insulator via anisotropic dry etching. This
two-step process mitigates the dependence of LDW on optical properties of target materials.

At cryogenic temperatures, where quasi-ballistic phonon transport becomes prominent, the LIPSS are expected to exhibit anisotropic thermal conduction and phonon directionality. Since the LIPSS orientation is inherently determined by the laser polarization, this laser-direct-writing scheme offers a unique opportunity to encode phonon pathways by on-demand polarization control. Meanwhile, the surface profile is not perfectly flat (Figure 1d,e), which may further influence phonon directionality. Accordingly, advanced topographic analysis, such as atomic force microscopy with a high-aspect-ratio cantilever or stereoscopic SEM, will be necessary for the design of phononic metasurfaces. With precise control over the LIPSS morphology, it could facilitate the construction of phononic circuits, thereby opening new avenues for integrated phonon-based thermal management.

Beyond these fundamental insights, the LIPSS also provide a practical advantage in terms of manufacturing processes. This maskless and resist-free process is compatible with wafer-scale silicon technology and can be directly integrated into semiconductor fabrication workflows. More broadly, the scalable laser-written nanostructures can serve as a versatile platform, where thermal functionality is combined with additional optical, electronic, or tribological properties to realize multifunctional platform.

### 3. Conclusion

In summary, we have demonstrated that fs-LIPSS offer a simple, maskless route to fabricate phononic nanostructures directly on crystalline silicon with a throughput several orders of magnitude higher than conventional nanolithography techniques. Whereas electron-beam lithography, focused ion beam, and block copolymer approaches remain indispensable for ultra-high-resolution patterning, albeit with a low-throughput, fs-LIPSS enable scalable nanostructuring over millimeter-scale areas at speeds suitable for practical device integration. Crucially, the laser-written grooves function as effective phononic nanostructures, suppressing thermal conductivity through limitation of long MFP, as confirmed by the TDTR measurements and Monte Carlo simulations. Our study establishes the fs-LIPSS as a versatile and CMOS-compatible platform for large-area thermal management and phonon engineering, bridging the gap between fundamental studies of nanoscale heat conduction and scalable device technologies.

### 4. Experimental Section

**LIPSS Fabrication:** First, we cleaned the SOI wafer using organic solvents (acetone, isopropyl alcohol, and water) in a sonic bath. We utilized the Galvano scanner (Han's Scanner, Ultrascan) and f-theta lens to scan the focused femtosecond laser beam at 1030 nm with 220-fs pulse duration (EKSPLA, FemtoLux 3). The fabrication was performed under ambient air conditions (20–22 °C, 30%–45% relative humidity). Before we conduct the RIE (Figure 1b), we cleaned the surface using acetone in a sonic bath, to remove ablation residuals.

**μTDTR Sample Preparation:** To fabricate the air-bridge sample shown in Figure 4b, we first conducted a photolithography and deposited the 250-nm-thick gold with a 20-nm-thick chromium adhesion layer by using the electron beam evaporator, followed by lift-off to form the gold transducer.

Then, the LIPSS was fabricated on the silicon surface as described in the sub-section LIPSS fabrication. As the gold transducer has larger ablation threshold than silicon, no mask was required to protect the gold transducer during the fs-laser scanning. The LIPSS orientation was controlled by rotating a sample stage. Next, we conducted the second photolithography and the RIE to make etching holes. Finally, we suspended the silicon membrane by removing the silica sacrificial layer with vapor hydrofluoric acid etching through the etching holes.

**μTDTR Measurement:** We employed the μTDTR method to measure the thermal conductivity of the suspended silicon membranes. A 488-nm pump pulse heated the gold transducer by ≈10 K, while a weak continuous-wave 514-nm probe laser monitored the resulting transient reflectance change, assuming linear dependence on temperature. Analysis of the transient temperature decay enabled measurement of the thermal conductivity. Given the density and specific heat capacity values from literatures for gold (19,320 kg m⁻³, 129.7 J kg K) [61] and silicon (2,329 kg m⁻³, 714 J kg K), [62] the thermal conductivity of the silicon membranes was determined. We set the thermal conductivity of thin gold as 250 W m K considering the size effect. [63,64] To extract the material thermal conductivity considering the periodic nanostructures, we reconstructed the LIPSS geometry in our FEM simulations. The normalized transient curves were fitted to the experimental data using least-squares minimization. The 1/e² diameters of the pump and probe beams were 2.0 and 1.9 µm, respectively. Measurements were repeated on five identical samples to obtain statistical standard deviations.

**Monte Carlo Simulation:** The MC method has been well established for simulating thermal conduction in nanostructures. [65–70] We utilized a FreePATHS simulator. [68–70] In this framework, phonon wave packets are treated as quasi-particle heat carriers, hereafter referred to simply as phonons. Phonons propagate along in straight trajectories until they encounter scattering events without coherent interference. In our MC simulations, we injected 1000 (Figure 5d,e) or 10 000 (Figure 5a–c) phonons from hot side in random directions, where their frequencies ω were sampled from Planck's distribution under the Debye approximation. The group velocity v was evaluated as dω/dq from the bulk phonon dispersion ω(q), with the polarization branch assigned randomly. At surface, phonons underwent either specular or diffuse boundary scatterings. The probability of specular scattering p is given as

$$p = \exp \left[ - \frac{16\pi^2 \sigma^2 \cos^2 (\phi)}{\lambda^2} \right] \quad (5)$$

where σ is the surface roughness, Φ is the incident angle, and λ is the phonon wavelength. The specular scattering was treated as elastic reflection at surfaces, whereas diffuse scattering randomized phonon propagation directions according to the Lambert cosine law.

The MFP for phonon-phonon scattering events was considered as $-\ln(r)\Lambda_{bulk}$, where r is a random number between zero and one. [68] We assumed $\Lambda_{bulk} = 300$ nm at 300 K, based on reported literature values. [34,71] The assumption of a frequency-independent MFP is not ideal, and future work should incorporate more rigorous frequency-dependent models.

### Acknowledgements

The authors thank J. Laurent for contributions to the experimental setup. This work was supported by Japan Society for the Promotion of Science (25K00035, 25K01168, and 21H04635), Amada Foundation (AF-2022237-C2), Hirose Foundation, and ENEOS Tonengeneral Research/Development Encouragement & Scholarship Foundation. Korea Institute of Industrial Technology, Key-Tech project (EH250008).

### Conflict of Interest

The authors declare no conflict of interest.
# Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

# Keywords

femtosecond laser, LIPSS, phonon, thermoreflectance

Received: September 22, 2025

Revised: November 13, 2025

Published online:

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