Abstract
Bringing bodies close together at submicron distances can drastically enhance radiative heat transfer, leading to heat fluxes greater than the blackbody limit set by Stefan–Boltzmann law. This effect, known as nearfield radiative heat transfer (NFRHT), has wide implications for thermal management in microsystems, as well as technological applications such as direct heat to electricity conversion in thermophotovoltaic cells. Here, we demonstrate NFRHT from microfabricated hotplates made by surface micromachining of \(\hbox {SiO}_2\)/\(\hbox {SiN}\) thin films deposited on a sacrificial amorphous Si layer. The sacrificial layer is dry etched to form wide membranes (\({100}\,\upmu \hbox {m} \times {100}\,\upmu \hbox {m}\)) separated from the substrate by nanometric distances. Nickel traces allow both resistive heating and temperature measurement on the microhotplates. We report on two samples with measured gaps of \({610}\,\hbox {nm}\) and \({280}\,\hbox {nm}\). The membranes can be heated up to \({250}\,^{\circ }\hbox {C}\) under vacuum with no mechanical damage. At \({120}\,^{\circ }\hbox {C}\) we observed a 6.4fold enhancement of radiative heat transfer compared to farfield emission for the smallest gap and a 3.5fold enhancement for the larger gap. Furthermore, the measured transmitted power exhibits an exponential dependence with respect to gap size, a clear signature of NFRHT. Calculations of photon transmission probabilities indicate that the observed increase in heat transfer can be attributed to nearfield coupling by surface phononpolaritons supported by the \(\hbox {SiO}_2\) films. The fabrication process presented here, relying solely on wellestablished surface micromachining technology, is a key step toward integration of NFRHT in industrial applications.
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Introduction
Nearfield radiative heat transfer (NFRHT) occurs between objects separated by a distance smaller than the wavelength of thermal photons. Some materials (as \(\hbox {SiO}_2\) or \(\hbox {SiC}\)^{1}) support Surface Phonon Polaritons (SPhPs), which are electromagnetic surface waves resulting from coupling between light and crystal vibrations. These waves are strong carriers of electromagnetic energy and are confined on the interfaces. By approaching two materials which support similar SPhPs, a coupling of surface waves could drastically enhance the electromagnetic energy transfer, leading to NFRHT^{2}. In this regime, radiative coupling from evanescent waves emitted by both bodies enhances transmitted power by up to a factor 100 at nanometerscale gaps (\({50}\,\hbox {nm}\)) compared to the blackbody limit of conventional farfield heat transfer^{3,4}. Several studies have shown that NFRHT can be harnessed for applications such as thermophotovoltaic energy conversion^{5,6,7,8,9}, enhanced radiative cooling^{10} and thermal rectification^{11,12,13,14,15}. Keeping two objects with a large temperature difference separated by only a few hundred nanometers remains a challenging task. Successful approaches include precise piezoelectric actuators with feedback control loops to keep both objects apart^{16,17,18} or arrays of spacers between the surfaces^{19,20,21}. The first option is difficult to scale to commercial applications as it requires bulky and expensive equipment, whereas the second implies high heat conduction losses through the spacers and mechanical stress from thermal expansion. In both cases, gap inhomogeneities due to surface curvatures is an issue as well. Microfabrication processes used to manufacture microelectromechanical systems (MEMS) can mitigate many of the previously mentioned problems by allowing high precision suspended structures with submicrons gaps, compatible with industrial production. MEMS technology has been successfully used to observe NFRHT across submicron distances between suspended beams^{22,23} or membranes^{7,24,25}. However, to increase the heat transfer while minimizing conduction losses, large areas combined with good thermal insulation are required. One such type of thermal MEMS is the microhotplate, a thin film membrane suspended by long thin legs with a resistive heater on top. The high thermal resistance of the legs provides good insulation from thermal losses to the environment through conduction, allowing the membrane to reach high temperatures for low heating powers. Due to this insulation and the high surface to mass ratio, the thermal response of the microhotplate is very sensitive to gas conduction and radiative heat transfer. This property is exploited in highly accurate Pirani pressure gauges^{26,27} and microbolometers for uncooled infrared imagery^{28}. Moreover, inplane integration allows fabrication of large membrane arrays thanks to surface micromachining techniques. Although this geometry is widespread, few experimental studies of NFRHT from microhotplates have been realized. Feng et al. demonstrated that the presence of a second membrane \({1}\,\upmu \hbox {m}\) above a microhotplate affects the heating rate^{29}. They attribute this effect to NFRHT between the suspended structures. However, they did not characterize radiative heat exchange to the substrate from the bottom of the membrane, which is expected to dominate heat losses. Furthermore, nearfield heat fluxes decrease exponentially with distance and are expected to be weak at \({1}\,\upmu \hbox {m}\). Thus, for thermal rectification or thermophotovoltaic applications, the required separation distance should be below \({500}\,\hbox {nm}\) to insure good device performance. Song et al.^{24,25} demonstrated NFRHT enhancement between polar dielectric thin films on piezoelectric actuators with an emitter to receiver separation of \({20}\,\hbox {nm}\). However, they had to keep a temperature rise below \({10}\,\hbox {K}\) to prevent thermal buckling.
In this article, we present clear evidence of strong NFRHT both across submicron gaps and with more than \({100}\,\hbox {K}\) temperature differences between microhotplates and the bulk substrate. We characterize the thermal properties of the device, which we name the NFhotplate, and directly measure the gap depth over the span of the membrane with a laser optical interferometer. This data draws an accurate picture of gap inhomogeneities according to realistic design parameters for industrial fabrication. We confirm the enhancement of radiative heat transfer due to nearfield effect through an exponential increase in radiative heat flux as the distance between the membrane and the substrate decreases, consistent with electrodynamic simulations using the measured gap distributions. Our study points to microhotplates as a viable option to leverage NFRHT in practical applications.
Results
Fabrication and gap characterization
The samples in this study are square membranes, \({100}\,\upmu \hbox {m}\) × \({100}\,\upmu \hbox {m}\) wide and \({300}\,\hbox {nm}\) thick, with four supporting arms at the corners, as shown schematically on Fig. 1a. These structures were fabricated by plasma enhanced chemical vapor deposition (PECVD) and etching of thin films on \(\hbox {Si}\) wafers^{30}. The detailed process flow is provided in Supplementary Information S1. The crosssection in Fig. 1a shows that a membrane consists of three layers: \({50}\,\hbox {nm}\) of \(\hbox {SiO}_2\), \({200}\,\hbox {nm}\) of \(\hbox {SiN}\) and \({50}\,\hbox {nm}\) of \(\hbox {SiO}_2\). Such a symmetric stack prevents buckling due to unbalanced outofplane mechanical moments. The \({300}\,\hbox {nm}\) total stack thickness is a tradeoff to maximize radiation/conduction ratio and to have a significant radiative heat flux to measure (see Supplementary Information S2). The \(\hbox {SiN}\) membrane tensile core ensures mechanical stability, while the \(\hbox {SiO}_2\) shell purpose is to sustain SPhPs surface waves for radiative heat transfer. To provide coupling of the evanescent fields with the substrate, it is coated with the same three layers. The deposited \(\hbox {SiN}\) is under a tensile stress of approximately \({400}\,\hbox {MPa}\) according to the measured deformation of a \(\hbox {Si}\) wafer coated with a single \(\hbox {SiN}\) film. The deposited \(\hbox {SiO}_2\), on the other hand, is under \({280}\,\hbox {MPa}\) compressive stress. Thus, the trilayer membrane is expected to be under a net \({170}\,\hbox {MPa}\) tensile stress. Such a high tension is beneficial to achieve a flat suspended structure. A PECVD amorphous silicon (a–Si) film acts as the sacrificial layer on which the \(\hbox {SiO}_2\)/\(\hbox {SiN}\)/\(\hbox {SiO}_2\) membrane trilayer is deposited. \(\hbox {SiN}\) core was patterned using \(\hbox {SF}_6\) etching prior to depositing the second \(\hbox {SiO}_2\) layer. By this way, \(\hbox {SiN}\) walls are protected with \(\hbox {SiO}_2\) and no etching of the \(\hbox {SiN}\) core was observed during membrane release. The square membrane of the NFhotplate and its supporting arms are defined by photolithography followed by a \(\hbox {SF}_6\) plasma etch. Prior to release, a Ni resistive heater is fabricated on the hotplate by liftoff. This heater is connected to aluminum pads by four traces on the arms to provide high quality ohmic contacts. Finally, the a–Si layer is etched by an isotropic \(\hbox {SF}_6\) ion coupled plasma to release the membrane. To facilitate the release, the etching pressure is maintained at \({7}\,\hbox {Pa}\) with no platen power to maximize the ratio of chemical to mechanical etch rates^{31}. At \({600}\,\hbox {W}\) of coil power, the a–Si etch rate is about \({3}\,\upmu \hbox {m}\)/min with a–Si/\(\hbox {SiO}_2\) selectivity of more than 100:1. We remark that this process does not consume expensive \(\hbox {XeF}_2\), contrary to other nanogap fabrication techniques involving silicon dry etching. A scanning electron microscope (SEM) image of a typical NFhotplate is shown on Fig. 1b. We studied two devices, M1 and M2, with designed gaps of \({250}\,\hbox {nm}\) and \({750}\,\hbox {nm}\) respectively, as well as another sample, M3, used as a farfield reference. As shown on Fig. 1b, we performed a deep etch of the substrate under the membrane of M3 to create a large cavity preventing nearfield transfer.
The gap between the released membrane and the substrate is subject to variations because of unequal mechanical stress distribution. Since a precise value of this parameter must be known to accurately model NFRHT, we used a VK Keyence 3D laser confocal scanning microscope to map the gap across the fabricated membrane, as shown on Fig. 1c. This method was already use to characterize membrane planarity for inplane far field heat transfer^{32} but was never used for gap characterization for NFRHT measurement. The gap can only be measured on the Ni heater because it is the only material reflecting the laser beam on the membrane. The gap is computed by subtracting the membrane thickness from the heater height measured relative to the substrate. We studied the gap distribution within a region of interest (ROI) which excludes the edge of the membrane. Pixel heights distribution in the ROI were fitted with a normal distribution to extract the mean gap value (\(\mu\)) and variance (\(\sigma\)) of the gap distribution. We determine the following gap distribution for M1 and M2 devices : \(610\pm {20} \,\hbox {nm}\) and \(280\pm {50} \,\hbox {nm}\) respectively, for initial aSi sacrificial layer thicknesses of \({750} \,\hbox {nm}\) and \({250} \,\hbox {nm}\).
Thermal characterization
We measured heat transfer from the NFhotplates with the fourpoint probe method, each electrical probe being a wire on a supporting arm of the membrane. We wirebonded the NFhotplates on custom Al core printed circuit boards (PCBs). This assembly was introduced in a vacuum chamber evacuated at \(10^{2}\) Pa to limit heat losses through the air. A DC electric current (\(i_{in}\)) flows through the nickel and the membrane heats up. By locally measuring the voltage in the membrane center, thanks to probes (inset in Fig. 2a), we deduced the heater electrical characteristic \(R_{Ni}^{mem}=f(i_{in})\) (Fig. 2a). Another approach, based on frequency thermal analysis (3 omega), offers highly accurate measurements for low temperature variations^{33}. However, in our case the AC technique is not applicable because the devices reach high temperatures (\({250}\,^{\circ }\hbox {C}\)). Multiple harmonics of the output voltage prevent a direct link of the \(3\omega\) temperature rise to the \(2\omega\) output voltage. Therefore, we chose to use a steadysate 4point probe measurement of temperature to determine the thermal losses. We estimated the accuracy on the probe voltage to be \({200} \,\upmu \hbox {V}\) leading to a \({4} \,\hbox {K}\) error at low power and less than \({1} \,\hbox {K}\) for high power. To stabilize the nickel thermistor for high temperatures, current sweeps with a high maximum value were applied to heat up the membrane until the electrical characteristic was stable. Ni’s Temperature Coefficient of Resistance (TCR) was measured after annealing and permits to deduce the thermal characteristic for the three tested membranes (see “From electrical characteristic to thermal characteristic” section in “Methods” section). Input thermal power is the Joule heating of the Nickel thermistor \(P_{in}^{mem}=R_{mem}^{Ni} i_{in}^2\). Temperature elevation of the membrane \(\Delta T_{mem}=T_{mem}T_{room}\) is deduced from the Nickel TCR and the zeropower electrical resistance. The thermal characteristic \(P_{in}^{mem}=f(\Delta T_{mem})\)) is plotted on Fig. 2b. The input power is dissipated by conduction through the legs and by radiation with the substrate or with the environment (depicted on FEM model inset in Fig. 2b). For the far field reference, radiation is assumed much smaller than conduction for low temperature elevation. We fitted the equivalent thermal conductivity of the legs \(\lambda _{leg}^{eq}= 5.2 \, W m^{1} K^{1}\) (detailed in “Thermal balance and conduction losses through the legs” section in “Methods” section). By applying a thermal balance on legs and the membrane, the conduction loss for each membrane is found and plotted with dashed lines on Fig. 2b. Conduction losses through the legs decrease when the cross section area is reduced or the length is increased. We fabricated membranes with \({340}\,\upmu \hbox {m}\) legs but deformation was noticeable and membranes move away from the substrate. In addition, conduction losses can be significantly decreased by increasing the electrical resistance, since more Joule heating will occur along the legs. This has for effect of increasing the leg temperature and therefore reducing the tendency for heat to conduct away from the membrane. We observe that for the studied membranes as the conduction loss deviates from a linear trend with temperature. By subtracting membrane input power and modelled conduction accounting for heat generation along the legs, we determine the radiative heat transfer for the three membranes. The membrane over a cavity (M3) only exhibits far field radiative heat transfer, so it acts as a reference. We notice that radiation losses increase as the gap decreases due to near field thermal radiation (arrows in Fig. 2b), proving existence of a strong thermal coupling at submicron separation distances. In addition, the membranes show good mechanical stability up to \({270}\,^{\circ }\hbox {C}\), with a monotonic and a reversible behavior.
Theoretical modelling
We present in this section a theoretical model of heat transfer of the fabricated NFhotplates for comparison with experimental data, the net heat flux radiated by the membrane is the sum of the thermal energy exchanged with the substrate across a submicron distance and the environment, which is considered as an ideal blackbody located far above the membrane. Thermal radiation within a nanoscale gap between flat surfaces is affected by wave interference due to multiple reflections as well as photon tunneling from evanescent coupling. Both effects are taken into account by the fluctuational electrodynamic formalism first introduced by Rytov^{3} and further developed by several authors^{4,34,35}. According to this theory, the radiative heat exchange between two objects at respective temperatures \(T_1\) and \(T_2\) is given by a Landauertype formula^{36}. The transmission probabilities of a complex multilayer structure such as the NFhotplate can be decomposed as a product of scattering matrices describing photon propagation through each layer. From the scattering matrices calculations, detailed in Supplementary Information, we obtain photons transmission probabilities, noted \({\hat{T}}\), with values between 0 and 1. We assume that the environment is a blackbody at a temperature \(T_{room}\), and that the substrate is at the same temperature. Due to the input heating power, the membrane is at a higher temperature \(T_{mem}\). Thus, the difference in Planck thermal distributions \(\Delta n(\omega )\), between the hot membrane and its cold environment is:
with \(k_b\) the Boltzmann constant, \(\hbar\) the reduced Planck constant and \(\omega\) is the photon frequency. It is important to note that the membrane is semitransparent, meaning that the membrane not only absorbs or emits photons but can also transmit them. For instance, one photon could be emitted directly by the membrane and absorbed or reflected by the substrate. Reflected photons are then absorbed, transmitted to the environment or reflected back by the membrane. The probability of photon \((\omega ,k_{})\) transmission from the membrane to the substrate/environment depends of:

The probability \({\hat{T}}_s\) (or \({\hat{T}}_e\)) of photon emission from the membrane; environment assembly (or membrane, substrate assembly) multiplied by the probability of photon absorption on the substrate (or environment).

Transmission of photons from environment to substrate and vice versa through the membrane according to probabilities. These probabilities are associated to \({\hat{T}}_m^+\) (or \({\hat{T}}_m^\)) coefficient.
The overall spectral flux density \(q(\omega )\) radiated away by the membrane is:
where the integration is performed over \(k_{}\), the wave vector component parallel to the surface, and the contributions from both polarization states TE and TM are summed. As the membrane stack is symmetric and as the environment is considered as a blackbody, we have in this work \({\hat{T}}_m^+= {\hat{T}}_m^\)^{37}. As the distance between the substrate and the membrane is less than \({1} \,\upmu \hbox {m}\), evanescent waves occur only for the \({\hat{T}}_s\) calculation.
To calculate each probability in the \((\omega ,k_{})\) space and to ensure a good stability for evanescent waves, we used the scattering matrices formalism to calculate transmission and reflection complex coefficients \((\tau ,\rho )\) for the membrane and for the twobody assemblies. The detailed theory for a general geometry with three body at three different temperature is shown in Supplementary Information. We used the measured refractive index of \(\hbox {SiO}_2\) and \(\hbox {SiN}\) to lead the theoretical calculations (Fig. 3—ellipsometry measurements in “Complex refractive index of SiO_{2} and SiN in mid infrared” section in “Methods” section). Figure 3b represents the overall transmission probability \({\hat{T}}_{tot}={\hat{T}}_{s}+{\hat{T}}_{e}2{\hat{T}}_{m}\) from the membrane to the environment for a \({280} \,\hbox {nm}\) gap. \(\beta =k_{}c/\omega\) is the dimensionless value of the wave vector parallel component. The total transmission probability matrix has 0 as lower limit and 0.7 for the upper limit proving the good code stability. Representation of \({\hat{T}}_{tot}\) probability is separated in two domains:

For \(\beta <1\), the wave vector is a pure real number corresponding to propagative waves. We observe a strong transmission probability for waves near the absorption region of silicon nitride and silicon oxide (around 0.7 near \({10} \,\upmu \hbox {m}\)). For comparison, a perfect blackbody has a transmission probability of 1 for all propagative waves.

For \(\beta >1\), the wave vector is a pure imaginary number corresponding to evanescent waves as frustrated modes or surface phononpolaritons (SPhPs). Frustrated modes are closed to the light line and exist for both polarizations regardless of the material. The existence of these evanescent waves are not take into account for the blackbody theory.
SPhPs exist in the frequency range where the complex refractive index of silicon oxide satisfies the condition \(k > n\), meaning that the real part of the permittivity is negative (highlighted on Fig. 3a). For TM polarization, SPhPs extend far away from the light line. These surface waves have a major impact on the total radiated heat flux since the transmission probability remains nonzero for large \(\beta\) (highly confined waves but high energy carrier). The heat flux spectral density (Fig. 3c) is the integral over \(k_{}\) of the overall transmission probability at a particular frequency and polarization (TM: red; TE: blue; total: black). We attribute the double peak in the TM contribution to a splitting of the SPhPs dispersion relation into longrange (LR) and shortrange (SR) branches due to coupling between surface waves at both interfaces of each \(\hbox {SiO}_2\) film^{38}. This effect increases with decreasing \(\hbox {SiO}_2\) thickness, as shown in Supplementary Information S2. As a reference, we also plot in Fig. 3c the radiated heat flux from a perfect blackbody according to Planck’s law (dash line). Because of SPhPs, \(q(\omega )\) greatly exceeds the blackbody limit from \(\omega =1.2\times 10^{14}\) to \(2.4\times 10^{14}\) rad/s. The overall heat flux lost by the membrane to the substrate and environment is calculated by integrating over frequency.
Discussion
Integration of the theoretical heat flux spectral density over \(\omega\) gives us a direct estimation of the net heat flux from the membrane to the environment or the substrate, uncertainty due to the gap distribution (\(\mu \pm \sigma\)) is represented with the color fills on Fig. 4a. Since the radiation flux increases exponentially as the gap decreases, the gap uncertainty has a strong impact for the smaller gap (\({280} \,\hbox {nm}\)) compared to the other tested gap. The experimental radiation flux is deduced from the difference between the input power and conduction losses and then normalized to the membrane surface (points on Fig. 4a). Direct comparison of experimental data and modelling confirm the observation of the radiative flux increase due to the nearfield radiation. For example, at \(\Delta T_{mem}=100 \text{K}\), the membrane with a gap around \({610} \,\hbox {nm}\) (M1) and \({280} \,\hbox {nm}\) (M2) have respectively a net measured radiative heat flux of \({720} \,\hbox {Wm}^{2}\) (theory : \({770} \,\hbox {Wm}^{2}\)) and \({1260} \,\hbox {Wm}^{2}\) (theory : \({1680} \,\hbox {Wm}^{2}\)). These values represent respectively a 3.5 and 6.4fold enhancement compared to the far field reference (M3). For \(\Delta T_{mem}=20\,\text{K}\), the measured experimental radiation is in good agreement with the theoretical one (Fig. 4b). Above \(\Delta T_{mem}=50\,\text{K}\) the experimental behavior of the two NF hotplates seems to deviate progressively from the theoretical value. We believe that thermal deformation during heating coupled with the high sensitivity of radiative exchange for small gaps is the main reason to the disparity between experimental and theoretical values at higher temperatures. As shown by StGelais et al^{23}, high tensile stress is required to avoid thermal buckling for large temperatures. For a given \(\Delta T_{mem}\), the average stress in the structure is given by:
with \(\sigma _i\) the prestress, \(t_i\) the thickness of layers, \(\alpha _i\) the thermal expansion coefficient (\(\alpha _{SiN}=1.6\,\text{ppm/K}\), \(\alpha _{SiO_2}=0.5\,\text{ppm/K}\) and \(\alpha _{Ni}=13\,\text{ppm/K}\))^{39} and \(E_i\) the Young modulus (\(E_{SiN}=300 GPa\), \(E_{SiO_2}=87 \text{GPa}\)^{39} and \(E_{Ni}=205 \text{GPa}\)^{40}). We estimated that the structure falls in compressive stress, so in possible thermal buckling, around \(\Delta T_{mem}=200\, \text{K}\). According to Fig. 4b, for \(\Delta T_{mem}=150\, \text{K}\), the deviation of experiment from theory correspond to a membrane displacement of \({50}\,\hbox {nm}\) and \({80}\,\hbox {nm}\), respectively for \({280}\,\hbox {nm}\) (M2) and \({610}\,\hbox {nm}\) (M1) initial gaps. Moreover, contribution of the nonideal geometry (holes and nickel serpentine heater) could also explain the slight deviation of experimental results from theory^{41}.
Overall, our study offers a step forward to achieve thermal management with static MEMS using NFRHT. The objective is to build thermal equivalents of standard microelectronic components such as transistors, diodes and memories^{14,42}. Instead of using voltages and electric currents, these thermotronic devices perform calculations and information treatment with temperatures and thermal heat fluxes. These new concepts are relevant for thermal imaging using uncooled bolometer technology; the integration of thermal logic gates directly on the bolometers could avoid the use of CMOS readout circuits for insitu and realtime image processing. In addition, the thermal circuit could perform autonomous calculations by directly using radiation from the environment as a primary energy source instead of the electrical energy used in microelectronics^{37}. To build a thermal transistor with MEMS technology, the first requirement is to have a threebody configuration with a semitransparent membrane as we have shown in this work, prior to integrate a metal insulator transition material (MIT) such as vanadium dioxide.
Conclusion
We have demonstrated in this work evidence of near field thermal radiation on static semitransparent membranes. Use of a scalable microfabrication process permits to avoid the use of micro actuators to bring the membrane close enough to the substrate to observe a thermal near field regime. These new devices could facilitate the integration of near field thermal radiation in real application, as we removed the complexity of micro actuator control. The process also allows large membranes to be formed with submicron gaps for increased heat transfer. In comparison of propagative radiation (far field reference), we observed a 6.4 times enhancement for a gap around \({300} \,\hbox {nm}\) and 3.5 times higher for larger gap (around \({600} \,\hbox {nm}\)) which is in agreement with theoretical calculations based on transmission probability theory for a three body configuration. Membranes reach a maximum temperature of \({270}\,^{\circ }\hbox {C}\) and a good reversibility after thermal cycling. In addition, the membranes are fabricated in the substrate plane, which allows both MIT material integration as well as stack of different membranes coupled in near field to achieve multi body coupling. This study is a step forward to thermal logical gates for autonomous calculation using radiative heat flux, a class of thermotronics.
Methods
From electrical characteristic to thermal characteristic
Nickel TCR was measured directly on the MEMS devices after annealing using a heating stage fourpoint probe station (Fig. 5a). It was also verified on a blanket Nickel thin film annealed at \({350}\,^{\circ }\hbox {C}\) under nitrogen using a Van Der Pauw setup. TCR is not linear above \({100}\,^{\circ }\hbox {C}\) and the resistivity follows the empirical law:
Nominal values of Nickel resistor \(R_{Ni} (293 \text{K})\) are determined by curve fitting of the electrical characteristic (Fig. 2a). Temperature rise of the membrane \(\Delta T_{mem}\) is then found by determining the positive polynomial root of Eq. (3). Membrane input heating power is determined by Joule heating:
Thermal balance and conduction losses through the legs
The input heat flux, \(P_{in}\), is dissipated by conduction, \(P_{cond}\), through legs or by radiation, \(P_{rad}\), to environment or substrate. These two dissipation heat fluxes depend on the temperature rise \(\Delta T_{mem}\) of the membrane, and the thermal balance on the membrane is:
To determine radiation losses, conduction losses should be perfectly known. As two legs are used as current probes and nickel presents an electrical resistance \(R_{leg}^{Ni}\), Joule power is dissipated in these legs. In addition, radiation losses exist along the four legs. These two thermal fluxes could change the temperature profile in the legs, and therefore affect the conduction losses from the membrane. Thermal balance was done on a slice of width dx along the leg to formulate the differential equation for the temperature rise in the leg \(\Delta T_{leg}\) (Fig. 5b). As boundary conditions, we supposed that the anchor (\(x=0\)) is at room temperature and the membrane/leg frontier (\(x=L_{leg}\)) is at the temperature of the membrane. The temperature rise \(\Delta T_{leg}\) follows the differential equation system:
where \(\lambda _{leg}\), is the equivalent thermal conductivity of the legs, \(R_{leg}^{Ni}\) is the nickel electrical resistance in the leg, \(TCR_{Ni}\) is the linearized TCR coefficient (around \(0.47 \text{{K}}^{1})\), \(h_{rad}\) is the linearized radiation coefficient (in \(\,\hbox {Wm}^{2}\)), \(L_{leg}\), \(w_{leg}\), \(t_{leg}\) are respectively the length, the width and the thickness of the leg, and \(\Delta T\) is the temperature rise of the membrane. Figure 5b depicts the analytical solution of Eq. (6) for the heated (current probes—red) and unheated (voltage probes—blue) legs of the far field membrane with a heating current \(i_{in}=175 \mu A\), corresponding to a temperature rise of the membrane \(\Delta T_{mem}=62 K\). We observe that the unheated leg profile is linear, showing that radiation has low influence on the temperature rise profile in the leg. This justifies the assumption that the radiation could be linearized. We obtained the thermal conductivity by fitting experimental data of heated 2point probe leg resistance for the far field reference membrane Fig. 5c. We found \(\lambda _{leg}^{eq}= 5.2 \, W m^{1} K^{1}\). As thermal radiation is low for this membrane, 10 times lower than Joule dissipated power in the legs, we have \(\beta \approx \ TCR_{Ni}\alpha\) in Eq. (7) and the temperature profile only depends on geometric parameters, dissipated power (experimentally estimated) and \(\lambda\). However, the heated legs deviate from a linear profile showing the importance to consider Joule heating in the legs. After analytical resolution of the temperature profile in the legs, the conduction losses, \(P_{cond} (\Delta T_{mem} )\), in the membrane is found:
Deduced \(P_{cond} (\Delta T_{mem} )\) is plotted on Fig. 2b. We observe that for a large temperature rise, conduction is lower for the membrane presenting high radiation losses (M2). For a given temperature rise, this membrane has a higher input current (as there is more radiation losses), so more heating power is dissipated in the legs compared to the far field reference. The result is that the conduction losses along the heated legs are lower for this membrane.
Complex refractive index of \(\hbox {SiO}_2\) and \(\hbox {SiN}\) in mid infrared
Ellipsometry data of silicon oxide \(\hbox {SiO}_2\) and silicon nitride \(\hbox {SiN}\) were acquired with an IRVase ellipsometer with respectively \(40^{\circ }\) and \(65^{\circ }\) incident angle on thin blanket films directly deposited on silicon substrates with a thickness around \({200}\,\hbox {nm}\). The wavelength range was [\({2}\,\upmu \hbox {m}\) ; \({17}\,\upmu \hbox {m}\)]. Samples were premeasured with a Woolam visible ellipsometer for which both films are completely transparent. We determined the exact thickness of the thin film t and the film high frequency permittivity \(\varepsilon ^{inf}\), which facilitate IR ellipsometry data fitting. We fitted the IR ellipsometry data with Brendel oscillators^{43} (Fig. 5d \(\Psi\) and \(\Delta\) angles for \(\hbox {SiO}_2\) (black) and \(\hbox {SiN}\) (grey)):
We fitted the ellipsometry data with one oscillator for \(\hbox {SiN}\) and with two oscillators for \(\hbox {SiO}_2\); oscillator values are resumed on the Table 1.
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Acknowledgements
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada for the financial support of this research. Authors would also like to thank the technical staff of the 3IT clean room and the team of FCSEL lab at Polytechnique Montreal for IR ellipsometry measurements.
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The funding was provided by NSERC.
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O.M. developed the MEMS process, led thermal characterizations and coded the theoretical calculations. A.J.F. brought his expertise in optical calculations, material characterizations and his relevant thoughton fabrication of thermal logic gates. L.G.F. was the scientific supervisor of this study and is holder of the NSERC/Teledyne DALSA Industrial Research Chair in Next Generation MEMS and Microphotonics.
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Marconot, O., JuneauFecteau, A. & Fréchette, L.G. Toward applications of nearfield radiative heat transfer with microhotplates. Sci Rep 11, 14347 (2021). https://doi.org/10.1038/s41598021936957
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DOI: https://doi.org/10.1038/s41598021936957
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