1. Introduction

Heat transfer issues are vital in the research and application of photothermal (PT) and electrothermal devices [1,2]. With the development of micro/nano fabrication technology, the demand for researching and fabricating thermal devices with ultra-large array is increasing [3]. As the array scale increases, thermal performance prediction could be a tough challenge, since the flow field and temperature distribution will be very variable and complex. Finite element method (FEM) provides a powerful numerical approach for solving complex thermophysics and optical issues [4,5]. In the design and evaluation of space-based optical systems, Structural, Thermal, and Optical Performance (STOP) analysis is an important approach to predict the performance of a large number of optical systems [6,7]. Developing an accurate thermodynamic model to predict and evaluate the performance is a very challenging problem, and many researchers utilize FEM to carry out the numerical calculation. For example, Aleksander et al. proposed a thermodynamic model by combining FEM modeling, heat transfer modeling, and model order reduction techniques, which enables accurate prediction of transient properties of temperature for several hours [8]. This modeling method provides a promising reference direction for the thermodynamic modeling of large-scale optical and photothermal systems. However, for some thermal devices with large scale as well as large array, it is very memory-consuming and time-consuming to build an equal scale three-dimensional (3D) FEM model to predict their thermal performance. Appropriate simplicity of the computational model provides a solution that significantly reduces computational costs, but the simplified method must meet the requirements of accuracy, reliability, efficiency, and experimental verification.

Periodic boundary conditions (PBCs) are a common simplification method used in FEM. In optical and acoustic devices, PBCs are often used to calculate the transmission and reflection properties of photonic and phononic crystals [9–14]. In addition, PBCs are also of great use in the field of heat transfer. Plenty of researchers have used PBCs to calculate the heat transfer properties of many macroscale devices and microscale materials [15–19]. Although PBCs effectively solve the flow and heat distribution problem in periodic elements without significant calculation cost, not all the heat transfer issues in solid meet the conditions of use. The use of periodic boundary conditions in heat transfer requires the following three conditions: (1) there are repetitive and periodic structures in the simulation area; (2) the distribution of temperature and flow field in the periodic direction is periodic, and the flow field is fully developed; (3) the simulation area should be large enough and periodic boundary conditions do not affect the accuracy of the simulation results [20]. For the devices with ultra-large array irradiated by a local heating source, it is inaccurate to use periodic boundary conditions because it does not meet the conditions (2) and (3).

In this paper, a PT transducer with an array scale of over 4000 × 4000 is proposed and fabricated. This PT transducer consists of repetitive and periodic microstructures. To obtain the photothermal properties, a small laser point is used as a local heating source to irradiate on the surface of PT transducer. This transducer is fabricated on a 4-inch Si substrate and the heating laser has a diameter of 8.8 mm. In order to promote the photothermal conversion efficiency of PT transducer and optimize its structural design, it is essential to use numerical simulations to predict its thermal performance. However, it is impossible to directly build a FEM model with equal proportions, since this would require a huge amount of memory to create tens of millions of microstructures. Moreover, the heating area also covers over four hundred thousand microstructures, making it impossible to construct an FEM model with such large array. The aforementioned PBCs are suitable under condition of uniform surface illumination rather than with a localized small heating source. Here, heat diffusion gradually spreads across the whole substrate from the heating position, resulting in a temperature field similar to a Gaussian distribution and different to a periodic arrangement.

To simplify the model and improve computational efficiency, a linear extrapolation method based on multiple equiproportional models (LEM-MEM) is proposed in this paper. This prediction strategy uses multiple FEM models with finite arrays (within 50 × 50 arrays) to calculate and predict the thermal properties of the proposed PT transducers. With appropriate structural simplification and linear extrapolation, this modeling method can be used to predict the steady-state temperature characteristics on wafer-level substrate with tens of millions of pixels. This paper is organized as follows: Section 2 is the theory of the proposed PT transducer and calculation method of the proposed LEM-MEM, as well as the FEM models and setting parameters. Section 3 is the experimental results and discussion, including the fabrication process of the proposed PT transducer, introduction of experimental setup, and simulated and measured steady-state temperature results. The validity of the LEM-MEM is verified by comparing the simulated and measured results of the PT transducers with four different pixel patterns. Section 4 is the analysis of optimizing the design of microstructures by LEM-MEM. Section 5 is the summary of this work and prospects for the future research.

2. Theory and method

Figure 1(a) shows schematics of the proposed PT transducer with ultra-large pixel array based on light-driven technology and the heat transfer setting of the FEM model. COMSOL Multiphysics is utilized as the computational software for the simulations. This transducer absorbs visible light energy, and with the increase of its temperature, the corresponding infrared radiation will be radiated from its surface to the environment. Each pixel contains a radiation area and a supporting structure. The heating energy is applied to the surface of middle radiation area. The heat transfer in this issue includes the heat radiation from the surface, heat conduction from silicon (Si) substrate and the air convection. The PT transducer is put in the air environment to test, so all of the surfaces of the whole structure is set as air convection heat transfer. Heat transfer equation is used to describe the heat transfer mechanism of PT transducer [21].

The left term of the formula represents the transient temperature change, where ρ, d, c_p and T are the density, thickness, specific heat capacity and temperature of the materials, respectively. The right terms of the formula describe the absorption, conduction, radiation and convection. Q is the absorbed light power density. k is thermal conductivity and ∇²T represents the second order partial derivatives of temperature in the x and y directions. For the radiation term, σ is the Stefan-Boltzmann constant of 5.67 × 10⁻⁸ W/(m²× K⁴) and ε_up and ε_down are the emissivity from the up and bottom surfaces, respectively. T_amb is the ambient temperature. For the convection term, h_up and h_down are the convection heat transfer coefficient from the up and bottom surfaces, respectively. When calculating the steady-state thermal characteristics, the Eq. (1) will change to the following form.

 

Fig. 1. (a) Schematics of working principle of PT transducer and the heat transfer setting of the model; (b) Principle diagram of LEM-MEM

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Since the order of magnitude of heat convection and heat radiation is much smaller than that of heat conduction, the steady-state thermal equation can be simplified as applied heating energy and heat conduction. When the applied heating energy and heat conduction change proportionally, this equation still holds. When a heating source (laser) with a certain power density is incident on the surface of PT transducer, a larger irradiated area indicates the device absorbs more heat. Si substrate acts as a heat sink, so the larger volume indicates more heat is conducted. Since the steady-state equation is constant, we can control the heating energy and heat conduction term equiproportionally by varying the irradiated area and the dimension of Si, and further obtain the corresponding steady-state temperature.

Figure 1(b) presents a principle diagram of LEM-MEM. Firstly, we calculate the dimension ratio of thickness of Si (t_Si), diameter of Si substrate (D_Si) and diameter of laser point (D_laser), which conforms to t_Si : D_Si : D_laser= 1 : 200 : 17.6. The model is set as this dimension ratio to get the simulated result of steady temperature. Since it is difficult to build a circular array to simulate the light spots, we build the model in a square array of the same length. The pixel array is built as 3 × 3 array, and then expand to 10 × 10, 20 × 20, ……, 50 × 50 array. Then linear fitting curve can be plotted by the simulated results of these arrays. Utilizing this linear fitting curve, interpolation and extrapolation can be calculated. Through positioning the point to the actual laser diameter (8.8 mm), the final deductive point can be obtained. It should be noted that to reduce the calculation amount, the pixels outside the irradiation area are neglected. These pixels mainly perform the heat conduction function. Since the thermal conductivity of Si material in pixels is the largest and most of the heat is dissipated through the Si substrate, this neglect has little impact on the result.

The radiation area has two layers, including aluminum (Al) black on the top and silica (SiO₂) on the bottom. Here we design four pixel patterns for the radiation area, as shown in Fig. 2 (a). One pattern is square shape with different side lengths (8 µm or 7 µm), and another pattern is circle shape with different diameters (8 µm or 7 µm). The spacing between adjacent pixel patterns is 4 µm. Different from precious PT transducers with micro-bridge or micro-cavity structure on the Si, the proposed PT transducer utilizes a Si columnar support to suspend the radiation area. This support structure is a single column with narrow top and wide bottom. Therefore, the contact area between the radiation area and support becomes smaller and then the design area of radiation pattern can be further reduced. For the precious PT transducers [22–25], their period (includes pixel length and spacing) are more than 30 µm, while that of the proposed PT transducer is less than 12 µm. Consequently, the pixel array scale can be extended from 2000 × 2000 to 4000 × 4000.

 

Fig. 2. (a) Schematic of four design pixel patterns (squares and circles with 8 µm/7 µm side lengths/diameters); (b) Simulated results of pixels with the four patterns based on LEM-MEM.

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The simulation results of four pixel patterns based on LEM-MEM is presented in Fig. 2 (b). The material parameters used in FEM simulation is shown in Table 1 and the structural parameters of four pixel patterns is presented in Table 2.

Table 1. Material parameters used in FEM simulation.

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Table 2. Structural parameters of four pixel patterns (d_c = diameter of circle, l = side length of square).

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The physical parameters of thermal conductivity for SiO₂ and Si are from the material library in COMSOL Multiphysics. Thickness of Si and SiO₂ are provided by the Si wafer manufacturer. According to literature, the average emissivity of SiO₂ with around 300 nm thickness on Si substrate is around 0.6 in the region of 1∼10 µm within the temperature of 200 °C [26,27]. Therefore, all the exposed surfaces of SiO₂ are set with the emissivity of 0.6 in the radiation heat transfer module. The Si substrate is a n-type low doped Si (doping concentration < 1 × 10¹⁵cm^-3), and its measured emissivity is less than 0.05 in the region of 1.5∼8 µm within the temperature of 200 °C [27]. This value could be neglected. Thus, no emissivity is set for the surface of the Si wafer.

Researchers have proposed detailed methods for measuring the temperature-dependent h [8,28]. For example, h can be obtained by heating the entire sample placed vertically at different temperatures and calculating its Nusselt number. The formulas used in these methods are well valid when the surface of the sample in contact with air has the same temperature, i.e., the sample has an isothermal surface [29]. When the surface temperature is uniform, the fluid dynamics and flow velocity distribution near this surface are also uniform, and the related thermophysical parameters of air can be easily obtained from published papers and textbooks. However, for the case of local heating surface, the physical parameters such as density, dynamic viscosity and thermal conductivity of the air will vary with the positions on the surface. Therefore, h is dependent on both temperature and position. Furthermore, the proposed PT transducer has microstructures on its front side, while its back side is a smooth surface. Since the surface roughness also has an impact on h, h is different on both sides. Obviously, it is very complicated and cumbersome to verify the actual h by experiments. Instead, following common practice in engineering, we simplify the modelling and calculations by introducing an equivalent parameter h_eq for describing the convection heat transfer coefficient of air. h_eq is an average and approximate value representing the overall h of the sample. To find a proper value of h_eq, we selected a fabricated sample (circle with d_c= 7 µm) to measured its steady-state temperature and performed FEM with different h_eq. The simulated results of different h_eq are shown in Fig. 3 (a), and the measured temperature of sample and extrapolation results according to the simulated points are shown in Fig. 3 (b), respectively. Comparing the results of measurement and simulation, we can find that h_eq= 5 W/m²×K leads to the closest agreement.

 

Fig. 3. (a) Simulated average temperature vs. side length of the heating areas of the 3 × 3, 10 × 10, 20 × 20, 30 × 30, 40 × 40 and 50 × 50 FEM arrays at different h_eq; (b) the measured temperature of the fabricated sample and simulated extrapolation temperature results after linear fitting the simulated points in Fig. 3 (a)

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The four samples are circle shaped or square shaped with a diameter or side length of 7 µm or 8 µm, and a spacing between the pixels of 4 µm. For the same 4-inch substrate, the dimension and design differences of the four samples are not significant, so h_eq can be expected not to change much. So, we take h_eq= 5 W/m²·K and apply this value further in all the simulations. In section 3, the simulated and measured results again agree well.

The procedure of modeling is summarized as follows.

3. Experimental results and discussion

To verify the accuracy of deductive point, we fabricate the four pixel patterns as shown in Fig. 2(a). Figure 4 (a-d) are the SEM images of PT transducers with four pixel patterns, respectively. The processing flow is shown in Fig. 4(e). First, a SiO₂ layer with 300 nm thickness is grown on a 4-inch clean Si wafer through thermal oxidation. After sequential photolithography and dry etching, the pixel pattern is transferred from the photoresist to the SiO₂ layer. Then, Si is etched using the photoresist and SiO₂ layer as a mask until the photoresist is completely removed. Finally, a layer of Al black is deposited on the SiO₂ layer.

 

Fig. 4. SEM images of PT transducers with pixel patterns of (a) circle with d_c= 7 µm, (b) circle with d_c= 8 µm, (b) square with l = 7 µm, (d) square with l = 8 µm and (g) the processing flow of PT transducers.

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Figure 5 (a) is the experimental setup of steady thermal performance. A 532 nm laser is modulated by the signal generator, then collimated and focused on the surface of PT transducer. Al black layer can increase absorption and emit strong infrared signal, but it is quite difficult to obtain a uniform and thickness-controllable Al black layer on different 4-inch samples by thermal evaporation. To exclude the influence of Al black and explore the effect of pixel pattern on steady thermal performance, the PT transducer is tested without Al black. Since the incident laser power density will be partly reflected and scattered by the sample surface, it is necessary to measure the laser power density actually absorbed by the sample surface. The incident laser power density is measured by an optical power meter, then the sample is fixed at the same distance and the reflected laser power density is measured accordingly. As the reflectance of the four samples is almost the same and their transmittance is zero, we subtracted the reflectance from 1 to obtain the absorption of the sample surface. Figure 5(b) shows the incident and absorbed laser power density at different supply voltages of the laser, and the average absorption of the sample is around 50.6%. The laser power is proportional to the voltage and the diameter of laser spot incident on the transducer surface is 8.8 mm. The measured value of absorbed laser power density is utilized as heating power density in the simulation. The generated infrared radiation signal, i.e. temperature distribution, is observed by thermal imager (VarioCAM HO head, temperature measurement accuracy is ±1 K) and the corresponding temperature data is read and saved in computer. Figure 5(c-f) shows the simulated and measured results of four pixel patterns for PT transducer in different laser power density. All of the pixel patterns conform to a same trend that the temperature rises as the laser power density increases. It is obvious that the simulated points are very close to the measured points.

 

Fig. 5. (a) Experimental setup of steady thermal performance; (b) Given laser power density and the absorbed power density of the PT transducer at different applied voltages; (c-f) Simulated and measured results of PT transducers with four pixel patterns under different laser-supply voltages. The insets are the corresponding deductive point with the measured data when the laser voltage = 5 V and laser-spot diameter D_laser= 8800 µm.

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Table 3 shows the summary of radiation area, contact area, simulated temperature, measured temperature and error for the four pixel patterns. Here the radiation area is the area of pixel pattern in SiO₂ layer, and contact area is the area between SiO₂ and Si support. As shown in Table 1, the maximum percentage error of average temperature between simulated and measured results is within 5.22%, which indicates that the proposed LEM-MEM has a good prediction accuracy for the PT transducer. The simulated and measured temperature rise with the increasing radiation area generally. This is understandable since the larger radiation area corresponds to more light absorption. However, for the pattern of circle with diameter of 8 µm, both the measured and simulated temperature drops slightly. The pixel patterns of circle with d_c= 8 µm has larger radiation area and contact area than that of square with with l = 7 µm. This is because the larger pattern provides a larger mask for the Si in the same etching time, so the etched Si support will form a wider top surface. The larger contact surface means a lower contact thermal resistance and therefore more heat flux is dissipated [23]. Thus, although the square with l = 7 µm and the circle with d_c= 8 µm are very close in radiation area, the difference in the contact area results in a lower temperature for the latter.

Table 3. Summary of radiation area, contact area, simulated temperature, measured temperature and error of the four pixel patterns. (T = temperature, d_c = diameter of circle, l = side length of square)

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The transient response of the four pixel patterns is measured by a point source infrared detector (PVI-4TE-5, response band: 2.5∼4.5 µm). However, due to the high reflection at visible light spectral range and the relatively low emissivity of SiO₂ at the measured spectral range, the collected infrared radiation signal was too small and the impact of noise was significant. To increase the absorption of the pixel patterns and reduce the impact of noise, a layer of Al black is deposited on the surfaces of the four samples, respectively. The absorption of the Al black layer in the visible light spectral range is more than 90% and its infrared emissivity reach 0.8 [30], which can greatly improve the absorption and infrared radiation of the four samples. Figure 6 (a, b) are the SEM images of circle with d_c= 7 µm and square with l = 8 µm after depositing Al black, respectively. Figure 6 (c, d) are the photos of the PT transducer before and after depositing Al black layer. The steady-state average temperature of the four samples on the heated area is measured utilizing the experimental setup in Fig. 5 (a). Figure 6 (e) shows the measured temperature results before and after Al black layer. Each sample is tested under the same laser power. Due to the high absorption of Al black, the absorption power density is higher than that of the sample without Al black. Therefore, the absorbed laser power density before and after Al black deposition are different at the same given laser power. Al black layer significantly increases the temperature of the four pixel patterns. When the laser-supply voltage is 5 V, the temperature of samples with Al black layer will increase by more than 120% and the photothermal conversion efficiency is improved by ∼6.2%, compared to the samples without Al black layer.

 

Fig. 6. SEM images of (a) circle pattern with d_c= 7 µm and (b) square pattern with l = 8 µm after depositing Al black layer on the top of pixels; Photos of PT transducer (c) before and (d) after depositing Al black layer: (e) Steady state temperature distribution of four pixel patterns before and after deposition of Al black layer; (f) transient response of four pixel patterns after deposition of Al black layer.

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To obtain the transient responses of the four samples with Al black layers, the signal generator provides a square wave pulse signal with a frequency of 500 Hz and a duty cycle of 70% to the laser, and then the infrared radiation signals from the samples are obtained by the point source infrared detector, respectively. The measured results are shown in Fig. 6 (f), and the radiation intensity has been normalized for comparison. The rising time and falling time of the four samples are within 0.2 ms, and all of the samples meet the response time within 2 ms. Since the Al black layer is a loose and porous film, its density and thermal conductivity are quite different from those of bulk Al material, and its surface structure full of gaps will also affect the convection heat transfer. Since the thermal conduction and convection heat transfer mechanism of this porous structure are very complex, it is difficult to use the LEM-MEM method to simulate the transient response of the four samples with Al black layer.

4. Optimization of the design of microstructures

To investigate the influence of microstructures on the steady thermal performance, a series of simulations in different controllable variable conditions are carried out, as shown in Fig. 7. To save the calculating time and memory, all simulated models are based on the pixel pattern of circle with 7 µm diameter and the array scale is 3 × 3. It should be noted that the simulation model changes only one parameter at a time and other parameters remain fixed.

 

Fig. 7. Simulated steady temperature results of PT transducers with circle pattern in different control variable conditions. Average temperature of nine pixels in (a) varied area of SiO₂; (b) varied contact area of Si and SiO₂; (c) varied thickness of SiO₂; (d) varied spacing of adjacent SiO₂ pixel pattern.

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As shown in Fig. 7 (a), when the area of SiO₂ increasing, the average pixel temperature rises, which is consistent with the conclusion obtained in Table 1. In Fig. 7 (b), when the contact area between Si and SiO₂ increasing, the average pixel temperature decreases. As mentioned above, when the contact area becomes larger, the contact thermal resistance of the interface becomes smaller. As a result, more heat will be conducted through the Si support and the pixel temperature will drop. When the contact area increases from 1 µm² to 7 µm², the temperature drops slightly, while with the contact area reduces to below 1 µm², the temperature dramatically rises. However, the small contact area will lead to poor mechanical stability. To ensure most of the supporting structure are robust, the contact area should be greater than 1 µm².

In addition, the thickness of the SiO₂ also plays a key role in the steady thermal characteristics, as shown in Fig. 7(c). As the thickness of SiO₂ reduces, the average pixel temperature will rise. This is because as the thickness of SiO₂ increases, the heat conduction term will increase accordingly, resulting in more heat flux dissipation, as described in formula (1). However, it is not the case that the thinner the SiO₂ film is, the better. SiO₂ and photoresist are both used as etching mask layers, as presented in Fig. 3(g). With the increasing of the etching time, the photoresist will be used up and SiO₂ will also be consumed. If the SiO₂ film is too thin, the etching process will be challenging. To obtain the optimal thickness, the etching time and the etching rate of the photoresist need to be considered sophistically. Figure 7(d) indicates the spacing of adjacent pixels has little influence on thermal properties, since the steady temperature remains stable as the spacing changes. Large spacing will reduce the array scale in a limited area, but very small spacing will affect the etching efficiency. Combined with the aforementioned considerations, the spacing of around 2 to 3 µm is appropriate.

5. Conclusion

In conclusion, a prediction strategy based on LEM-MEM is proposed to study the thermal performance of ultra-large pixel array. To verify the accuracy of LEM-MEM, PT transducers with more than 4000 × 4000 pixels are designed and fabricated. Four different kinds of patterns are fabricated and tested to explore the influence of microstructures on the steady thermal properties. The experimental results show that LEM-MEM is highly predictable, with a maximum percentage error of 5.22% for the average temperature in four different pixel patterns. To measure the transient response of the PT transducer, a layer of Al black is deposited on the surface of the PT transducer. Measured results prove that Al black layer increases the photothermal conversion efficiency of PT transducer and all four pixel patterns meet the response time within 2 ms. A series of control variable simulations and systematic analysis show that the area of SiO₂, contact area of SiO₂ and Si and thickness of SiO₂ play key roles in thermal performance, but the spacing of adjacent pixels has little influence. These results prove that the microstructure design is vital to the performance of PT transducer. The proposed LEM-MEM provides optimal design guidance for the PT transducers with ultra-large array. The wider universality of LEM-MEM in other microstructures or materials will be further explored in future studies.