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Using optical resonances to control heat generation and propagation in silicon nanostructures

Here we propose a new computational approach to light-to-matter interactions in silicon nanopillars, which simulates heat generation and propagation dynamics occurring in continuous wave laser processing over a wide temporal range (from 1 fs to about 25 hours). We show that visible light can be exploited to selectively crystallize internal region of the pillars, which is not possible by conventional treatments. A detailed study on lattice crystallization and reconstruction dynamics reveals that local heating drives the formation of secondary antennas embedded into the pillars, highlighting the importance of taking into account the spatial and temporal evolution of the optical properties of the material under irradiation. This approach can be easily extended to many types of nanostructured materials and interfaces, offering a unique computational tool for many applications involving optothermal processes.

Three-dimensional integrated circuits (3D-ICs) offer promising routes for pushing the development of electronics beyond the limits of the Moore's law. (1)(2)(3) In this context, photonic devices based on 3D nanostructures can play a key role in fostering the technology breakthrough. (4)(5)(6) The fabrication of integrated photonic devices passes through an accurate control of silicon doping and lattice reconstruction, which are normally achieved by specific treatments involving multiple steps.
For example, dopants are usually introduced into the silicon lattice by ion implantation. (7) This procedure yields local lattice damages and formation of point defects, which can further migrate, giving rise to uncontrolled segregations and extended dislocations. In many cases, silicon-based structures are also preliminary amorphized in order to prevent ion channeling. In both the cases, post-implantation damage recovery and silicon lattice reconstruction are realized by thermal treatment. However, conventional thermal treatment like furnace annealing and solid-phase epitaxial regrowth (SPER) are carried out at high temperature (T>600°C), which poses severe limitations for process design, as real devices include different circuit components that can undergo thermal degradation. (8,9) Laser-based thermal processing is a valuable tool to localize heating at specific positions, as the thermal effects can be limited to the range of few micrometers. In general, the optical annealing of silicon chips is realized by means of UV lasers (λ=266 nm). In the UV range silicon exhibits high ohmic losses, allowing the laser power to be uniformly distributed all over the chip. Because of its non-resonant nature, UV laser processing does not allow heat generation and propagation in silicon nanostructures to be controlled, which would require the extension of spatial resolution at the level of nanoscale. (10) On the other hand, at optical frequencies silicon is characterized by low ohmic losses, which makes heat generation ineffective. (11) However, recent works demonstrate that very strong local heating can be generated when the frequency of the laser beam is coupled with that of the optical resonance of a silicon nanostructure.
Under these conditions, the local temperature can exceed 600°C, which is a typical temperature for inducing lattice reconstruction. (12) Moreover, the overall opto-thermal effects depend not only on optical excitation but also on thermal dissipation, which is deeply influenced by the nanostructure morphology. These effects have been mainly applied for Raman spectroscopy (13,14) and optical thermometry purposes, (15) but their direct exploitation for controlling silicon lattice reconstruction has not been devised so far.
Here we developed a theoretical and computational model which is able to predict the spatial and temporal evolution of lattice crystallization and reconstruction in silicon nanopillars irradiated with a CW visible laser (λ=532) and we compared these results with those obtained by conventional annealing processes performed by either furnaces or UV lasers. We analyzed the role of different key parameters, including size, aspect ratio and type of substrate. This study reveals that, unlike thermal or UV processing, visible light can induce the formation of sub-wavelength domains of crystalline silicon (c-Si) at the interior of the pillar amorphous (a-Si) matrix, which act as secondary optical antennas, providing unprecedented level of spatial and temporal control of lattice reconstruction in regions that cannot be accessed by conventional methods.

Description of the system
Amorphous silicon can undergo crystallization below its melting temperature, with the obvious advantage that the overall shape of the silicon structure is preserved. (16)(17)(18) Here we investigate the possibility to induce heat generation and control the spatial crystallization in amorphous silicon nanostructures by directly exploiting the coupling of light with optical resonances. Amorphous-Si nanostructures are characterized by low thermal conductivity (1.5 W/mK). Our reference model is represented by a silicon pillar (Si-P), vertically grown onto a given substrate and surrounded by air. Material S1). The role of the aspect ratio is also studied over a wide range of values. The diameter of the Si-Ps ranged from 60 to 140 nm and the height varied from 150 to 640 nm. Light excitation is conveyed through the Si-P in form of a Gaussian beam, focused by a 0.9 NA optical objective. In typical simulations, the light fluence was set to 0.11 mW/μm 2 , which is sufficient to induce local crystallization avoiding electronic ablation. The optically-driven crystallization kinetics was simulated by Finite Element Modelling (FEM) in a wide temporal range (from 1 fs to 25.3 hours, see Methods). We note that this interval is typically out of reach for methods based on molecular dynamics, which are limited to a few hundreds of nanoseconds. (20) Optical vs thermal annealing

Role of the substrate
Nanostructured antennas can be grown on different substrates. In order to clarify the role of the substrate in light-structure opto-thermal interactions and crystallization dynamics, we compare the use of different substrate materials. We chose gold, silicon and silica substrates as examples of conductors, semiconductors and insulators, respectively. The simulation of the behavior of freestanding SiPs was obtained by replacing the solid substrate with air. Figure 3 shows the results for a pillar of AR 3.3.
The resonance of the free standing pillar is located in the center of the structure, while the resonance of the pillar supported by SiO 2 , Si and Au, is shifted toward the top. A similar trend is observed by decreasing the wavelength of the exciting radiation (see Supplementary Material S4).
On the other hand, when the AR of the pillar is larger than 5 the resonance is located far from the substrate, and the optical nature of the substrate does not significantly affect the optical resonance.

No significant differences are observed in the crystallization dynamics. (See Supplementary
Material S4)

Role of aspect ratio
Aspect ratio (AR) is a key parameter to determine the optical properties of a Si-P and, as a consequence, to control heat generation. Figure 1h shows the results of simulations of the optothermal response to a visible light irradiation (λ=532 nm) for Si-Ps with different aspect ratios (from 12 to 4.28), achieved by fixing the height of the Si-P to 600 nm and varying the radius from

Light-induced crystallization dynamics
Light-induced crystallization proceeds through a progressive modification of the degree of crystallinity of the material. The crystallization kinetics of Si-Ps was derived from the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model modified by the Nakamura theory to account for nonisothermal conditions (see Methods). Crystallization follows a nucleation-growth process that is characterized by a stochastic nature. In this context, the terms X c (the degree of crystallization in the JMAK model) can be here regarded as the probability to find a portion of fully crystalline silicon in a given region of the Si-P. Panels a and b of Figure 2 show a comparison between the maxima value of X c,max , which represents the presence/absence of crystalline grains, and the corresponding percentage of crystalline phases averaged over the whole Si-P volume (X c,av ), for a range of AR spanning from about 2.6 to 6. A direct comparison between the two maps reveals that X c,ave is always lower than X c,max , which confirms that the light-driven crystallization is not homogeneous across the pillar and small differences in AR can strongly affect the crystallization efficiency and its spatial extension. In general, these data show a close correlation between structure-light coupling and the crystallization efficiency. The stronger is the pillar-light-coupling, the higher the temperature, and the faster the The key role of the optical resonances is further discussed in Supplementary Material S6, which shows that non-resonant UV processing is not able to produce any secondary antenna-effects.
The absorbed power Q e as a function of time for the two systems, A and B is shown in Figure 3. By multiplying the absorbed power density by the Xc field and integrating over the pillar volume, it is possible to decouple the power absorbed by the crystalline phase from that of the amorphous phase.
At t=4.5 s, as a result of the appearance of the first crystalline seeds in the pillar with AR ∼4 (sample A), there is a strong enhancement in the absorbed power of the crystalline inclusion (blue curve). A similar trend can be observed for the pillar with AR ∼6 but at t=80 s. In general, two evolution regimes, can be distinguished: I) a fast regime, where the system is quickly kept out of equilibrium in a metastable state. This persists for few seconds after crystallization of the first seeds (light blue region in Figure 3a and b).
In particular, it lasts for 4 s (from 0.4 s to 4.4 s) in A and for 62 s (from 31 s to 93 s) in B.
II) a slow regime where the system moves from the metastable towards the equilibrium state. This persists until the whole pillar has been crystallized (light orange region in Figure 3a and b).

Conclusions
This study demonstrates that a rational exploitation of the optical resonances in silicon nanostructures allows the precise generation of localized heating, which can be controlled by size, substrate and aspect ratio. In particular, visible lasers can be utilized to drive local crystallization and lattice reconstruction in regions that cannot be reached by other methods, offering a powerful tool for developing new photonic devices based on silicon nanostructures (25)(26)(27). The theoretical model developed in this work is able to predict the spatial evolution of the light-driven opto-thermal effects over an extended temporal range, which spans from fs to hours. The model also revealed that the crystallization dynamics can give rise to secondary antennas embedded into the primary nanostructures, which broadens the horizons of light harvesting and thermal management at the nanoscale. This approach can be extended to many different materials and thermally-activated processes, including laser resonant printing (28), diffusion-based doping of semiconductors, nanofabrication of all-dielectric and hybrid devices for enhanced vibrational spectroscopy (29)(30)(31) and photothermal therapy.

Methods
The whole process of light absorption, heat transfer and phase change was investigated by process. For a given material and a fixed temperature T, the percentage of crystallized substance ! ! is described by the following equation: where ! is called isothermal constant and ! the Avrami number. If temperature is not constant, eq.
1 is no longer correct. Nakamura et al. (22) provided a more general theory, extending the Avrami treatment for non-isothermal processes. For a given temperature !(!, !), the ! ! (!, !) can be obtained by solving the following differential equation: where ! is the exponent of the JAM equation, and ! !(!, !) is called non-isothermal constant, which can be related to the JAM crystallization constant !(!) by the following expression: For the spatial description of crystallization, we assume that if a temperature gradient is present in the material, the Nakamura model can still be used, but now X c is a space-dependent quantity, assuming any possible values between 0 and 1. Given the stochastic nature of nucleation-growth process (23), we attribute to X c a statistical interpretation, as the probability to find a crystalline region in a given point of the system.
The temperature field !(!, !) is iteratively recalculated every time step, solving the time-dependent heat transfer equation: where !(!, ! ! ) is the density, ! ! (!, ! ! ) is the heat capacity at constant pressure, ! !, ! ! is the thermal conductivity and ! ! is the heat source. All the coefficients involved in eq. 4 vary with temperature and with ! ! (!, !). See supporting information for further details.
The heat source ! ! is obtained from the EM absorption, described by the expression: where ! is the complex electric field, and ! !! !, ! ! is the imaginary part of the relative dielectric permeability. The electromagnetic absorption has been considered as the only heat source in the system. The numerical solution of Maxwell's equations provides at each time step the values for electric and magnetic fields !, ! in all the points of the system. While the system undergoes crystallization, the electric and magnetic fields are recalculated and the losses are updated. This enables to follow the dynamical modification of optical proprieties, while crystallization takes place.
The X c dependence of optical and thermal constants are detailed in the Supplementary Material S1.
The modification of optical properties as a function of granularity is negligible in comparison to the variation of optical proprieties from the amorphous-to-crystalline phase (23). For simplicity, in the model we consider this evolution is accompanied by the corresponding modification of the optical properties of the Si-P, which in turn affects light absorption and heat generation. As a result, the overall kinetics of heat propagation and its spatial extension depend on this mutual feedback. Thus, in evaluating the dynamics of crystallization and heat propagation, we must consider that the optical properties of the pillars are changing as a function of the crystallization degree that has been reached at a given time and the temperature distribution is not constant across the pillar (see Supplementary Material S1).