Thermal radiation dominated heat transfer in nanomechanical silicon nitride drum resonators

Nanomechanical silicon nitride (SiN) drum resonators are currently employed in various fields of applications that arise from their unprecedented frequency response to physical quantities. In the present study, we investigate the thermal transport in nanomechanical SiN drum resonators by analytical modelling, computational simulations, and experiments for a better understanding of the underlying heat transfer mechanism causing the thermal frequency response. Our analysis indicates that radiative heat loss is a non-negligible heat transfer mechanism in nanomechanical SiN resonators limiting their thermal responsivity and response time. This finding is important for optimal resonator designs for thermal sensing applications as well as cavity optomechanics.

Nanomechanical silicon nitride (SiN) drum resonators are currently employed in various fields of applications that arise from their unprecedented frequency response to physical quantities. In the present study, we investigate the thermal transport in nanomechanical SiN drum resonators by analytical modelling, computational simulations, and experiments for a better understanding of the underlying heat transfer mechanism causing the thermal frequency response. Our analysis indicates that radiative heat loss is a non-negligible heat transfer mechanism in nanomechanical SiN resonators limiting their thermal responsivity and response time. This finding is important for optimal resonator designs for thermal sensing applications as well as cavity optomechanics.
Since their emergence, nanomechanical resonators have shown distinct advantages in various fields of application due to their high amplitude and frequency response to external physical quantities. 1 In order to achieve optimal performance, material properties are essential, and silicon nitride (SiN) has proven to be well suited for nanomechanical resonators. The large intrinsic stress results in unprecedented high quality factors based on so-called "damping dilution", which has been observed in silicon nitride strings 2,3 as well as drums. [4][5][6] The combination of high quality factors and excellent optical properties has made nanomechanical SiN drums interesting devices for cavity optomechanics. 5,7 Among other things, they have been used for fundamental research, 8 and as transducers between optical and radio wave 9 or microwave 10 signals. Recent developments of optimized trampoline [11][12][13] and phononic crystal designs [14][15][16] are pushing the quality factors of SiN resonators into the realm of room temperature quantum optomechanical experiments.
From cavity optomechanics experiments it is well known that the local heating of the laser causes a frequency detuning of SiN drums. 17,18 Such photothermal detuning has been extensively used for sensing applications, such as infrared absorption spectroscopy, [19][20][21][22] nanoparticle analysis, 23,24 single-molecule detection, 25 and, recently, electromagnetic radiation detection. 26,27 Generally, it has been assumed that heat transfer is dominated by conduction, as it was concluded for nanomechanical torsional paddle resonators. 28 Recently, evidence for significant radiative heat transfer in large SiN drums has been presented. 29 Despite the proliferation of nanomechanical SiN resonators, the underlying heat transfer mechanisms, which cause the thermal frequency a) Electronic mail: silvan.schmid@tuwien.ac.at response and ultimately determine the performance limit, has not been studied in detail.
In this work, we investigate the heat transfer in nanomechanical SiN drum resonators by means of computational simulations and experiments to gain a better understanding of the dominating mechanism. We assume the situation of an experiment under vacuum in which heat convection is negligible, and heat transfer happens solely by radiation and conduction. Our study is conducted by local heating of SiN drums with a laser and analysing the resulting frequency and time response. We show that the frequency response as well as the response time are dominated by radiative heat transfer, which is a function of the lateral size of the drums. This is an important finding to be considered for the optimal design of thermal sensors as well as cavity optomechanics experiments.
The relative frequency shift δf for an even temperature change ∆T = T − T 0 from an initial temperature T 0 of a resonator under tensile stress σ with a resonance frequency f (T ), such as a drum or a string, is given by 1,30 with the thermal expansion coefficient α and Young's modulus E. This results in the relative temperature responsivity (relative frequency shift per change in temperature) of From (2) it is obvious that the observed temperature induced frequency detuning is enhanced for resonators with a low tensile stress σ. Therefore, we performed our study with nanomechanical SiN drum resonators made of low-stress silicon nitride. The drums with a thickness of h = 50 nm are supported by a silicon frame with a thickness of 380 µm. We present results from square drums of different sizes L × L with L = 0.5 mm, 1.0 mm, 2.5 mm and 4.0 mm. We used a silicon wafer with silicon-rich SiN grown by low-pressure chemical vapor deposition. The square drum shapes were defined on the backside of the wafer by a standard photolithography process and etched by reactive ion etching. To finally release the drum structures, the silicon wafer was etched through from the backside with potassium hydroxide. In order to vary the thermal conductivity for comparative measurements, a 50 nm thick aluminum layer was deposited on one side of some SiN drums by means of physical vapor deposition process. The tensile stress of the drums was calculated from the fundamental mode frequency for a mass density of 3000 kg/m 3 and 2700 kg/m 3 for SiN and Al, respectively. 1 The experimental setup, schematically depicted in Fig. 1(a), shows the silicon nitride drum resonator on a piezo actuator and a laser-Doppler vibrometer (LDV) (MSA-500 from Polytec GmbH) to readout the vibrational motion. The signal from the vibrometer is fed into a lock-in amplifier (HF2LI from Zurich Instruments) with an integrated phase-locked loop to control the piezo actuator and to drive the drum at its resonance frequency. A power controllable laser diode (LPS-635-FC from Thorlabs GmbH), with a center wavelength λ = 638 nm, was attached to the LDV unit to photothermally heat the drums. The exact laser power values were recorded using a silicon photodiode (S120C from Thorlabs GmbH). Simulations shown in this work are performed with COMSOL Multiphysics Version 5.5. All measurements were conducted in a high vacuum at a pressure below 1 × 10 −5 mbar. Fig. 1(b) schematically depicts the heat flux in a drum resonator when locally heated in its center. The incident laser with power P in is absorbed by the drum, producing a heating power P abs = A λ P in for a wavelength specific absorbance A λ . According to Fourier's law, the resulting temperature gradient across the drum causes a conductive heat flux q cond = −κdT /dx from the drum center towards the frame, for a specific thermal conductivity κ. The radiative heat transfer from the drum surface is given by the Stefan-Boltzmann law q rad = εσ SB (T 4 − T 4 0 ), for the special case of having a large surrounding at temperature T 0 and the assumption of a gray surface with an emissivity ε with the Stefan-Boltzmann constant σ SB and the surface temperature T at a specific location on the drum. 31 Taking into account thermal radiation, whereby part of the thermal power is emitted, leads to a reduced effective temperature of the drum. It is obvious from equation (1) that a lower average temperature results in a smaller frequency detuning ∆f 1 , compared to the case of negligible thermal radiation ∆f 2 , as schematically depicted in Fig. 1(c). Fig. 2(a) shows simulated frequency responses of a 1 mm square drum with different emissivities for an increasing absorbed power P abs . As predicted by the simplified model (1), the resonator frequency decreases with increasing absorbed laser power P abs , which corresponds to a rise of the drum's effective temperature. It also shows that the slope of the frequency detuning, and hence SiN drums compared to a fit from the analytical responsivity model for a thermal conduction-dominated heat transfer given in equation (3). The fit parameter was found to be δR = ξ/σ with ξ ≈ 1.88 · 10 3 MPa W −1 according to the drum stress σ. (b) Relative responsivity of bare SiN drums compared to the conduction-dominated model (3), with σ = 30 MPa, α =2.2 × 10 −6 K −1 , and κ = 3 W m −1 K −1 , and FEM simulations assuming a surface emissivity ε = 0.05, α =2.2 × 10 −6 K −1 , 32 . The error band represents the uncertainties in κ and σ originated for a range from 2.7 to 3.5 W m −1 K −1 and 28 to 30 MPa, respectively. the responsivity, becomes smaller when more heat is radiated due to a higher emissivity of the drum.
For small changes of temperature ∆T , that is for small P abs , the frequency response (1) can be linearized to a good approximation. From Fig. 2(b) it can be seen that this linear approximation is valid for P abs < 10 µW. Considering the applied laser powers and the assumed A λ ≈ 0.4% for SiN, which is close to the reported absorbance value of 0.5% 25 our maximal absorbed power is P abs ≈ 1 µW. Fig. 2(c) shows that the measured frequency detuning is indeed linear with the absorbed power. Based on this presented method, we measured the relative power responsivity δR = δf /P abs for Al coated and bare SiN drum resonators, shown in Fig. 3(a) and Fig. 3(b), respectively.
For the case of negligible thermal radiation, an analytical power responsivity model for the fundamental mode has been derived for the case of local heating in the center of a circular drum 22 with the Poisson's ratio ν. A comparison to finite element method (FEM) simulations show that the analytical model is a good approximation for square drums. 22 According to (3), the power responsivity is independent of the lateral drum size. This is the case for the Al coated SiN drums, as can be seen in Fig. 3(a). Compared to the thermal conductivity of SiN of κ = 3 W m −1 K −1 , 33 the effective conductivity of the Al coated drums is ∼32 times higher; thus, the heat transfer in the Al coated drums seems to be dominated by conduction. The fit of (3) for the Al coated drums with different effective tensile stress is of good quality. In contrast, as seen in Fig. 3(b), the measured responsivities for bare SiN drum resonators decreases with increasing drum size. Even the smallest drums show a deviation from the pure conductive model (3), suggesting that all measured bare SiN drums are dominated by radiative heat transfer. In order to take radiative heat transfer into account, we used finite element method simulations to model the responsivity of the bare SiN drum resonators. Prior to this step, we measured the absorption spectrum of our SiN drums and calculated their emissivity to be ε = 0.05 (for details see Supplementary Information), which agrees well with values predicted by Zhang et al. 29 The simulated responsivities plotted in Fig. 3(b) follow the measured values with good agreement. Finally, we studied the response time using frequency response measurements for an increasing laser modulation frequency as shown in Fig. 4(a). By increasing the modulation frequency of the heating laser, the recorded resonance frequency detuning from the phase-locked loop will start to decay. The response time τ is evaluated for the frequency amplitude at f −3dB = f max / √ 2. The extracted response times of Al covered SiN drums are plotted in Fig. 4(b). Compared to the measurements of bare SiN drums, shown in Fig. 4(c), the Al coated drums show a significantly faster response time, due to their higher thermal conductivity and hence faster thermalization. Besides the magnitude, the scaling of τ with lateral size is notably different for the two drum types.
Our theoretical model (see Supplementary Information for the derivation) for the response time of square drums made of multiple layers, considering both thermal conduction and thermal radiation, yields with the mass density ρ, the specific heat capacity C p for each layer i. The model (4) is in excellent agreement with the measured response times for both drum types, as seen from Fig. 4(b)&(c). In order to dissect the dominating heat transfer mechanism at play, we additionally plotted the model taking into account only heat conduction. In that case, (4) predicts a quadratic scaling with drum size L, and is a good approximation in the case for the conduction-dominated Al coated drums, as seen in Fig. 4(b). In comparison, only the full model (4), taking into account both conduction and radiation, predicts the measured response times accurately, as seen in Fig. 4(b). This is another clear sign that heat transfer in Al coated SiN drums is dominated by conduction in contrast to the bare SiN drums that are radiation-limited. The higher radiative heat loss for bigger drums leads to a reduced response time that levels off, compared to what would be expected from heat conduction alone.
In conclusion, it has been demonstrated that the thermal frequency response of nanomechanical SiN drum resonators is significantly affected by radiative heat transfer. This results in a size-dependent responsivity of bare SiN drums in contrast to Al coated resonators where thermal conduction dominates and no size-dependency on responsivity was observed. The dominating heat transfer mechanism of the resonator is also reflected in response time measurements. The Al-covered drums are dominated by thermal conduction and show a significantly faster response time due to the high thermal conductivity of the metal layer. An analytical model for this case corresponds well with the measurements and show that the response time scales quadratically with the drum size. For bare SiN drums, where thermal radiation plays a nonnegligible role, measurements also showed good agreement with the response time model, predicting faster response times due to the additional radiative heat loss. Larger drums are more affected by radiation, exhibiting an even higher deviation for response times compared to the case of a thermal conduction-dominated heat transfer. The obtained results show the significance of thermal radiation in finding the optimal performance of SiN drum resonators for specific sensor applications and fundamental research.

ACKNOWLEDGMENTS
The authors wish to thank Johannes Hiesberger, Sophia Ewert, Patrick Meyer, and Michael Buchholz for their support with the sample fabrication as well as Hendrik Khler and Miao-Hsuan Chien for many fruitful discussions. We would also like to thank Dr. Pavel Grinchuk of HMTI, Belarus for his support. This work is supported by the European Research Council under the European Unions Horizon 2020 research and innovation program (Grant Agreement-716087-PLASMECS) and (Grant Agreement-875518-NIRD). We further acknowledge funding from Invisible-Light Labs GmbH.

A. Calculation of emissivity
In order to obtain the emissivity of our SiN drums we employed Kirchoff's law for a non ideal radiator, where the emissivity corresponds to the absorption, ε = α. 36 Therefore, we needed to measure the optical spectra via Fourier transformed IR spectroscopy (Bruker Tensor 27) with a specified transmittance and reflectance unit (Bruker A510/Q-T). From the measured transmittance and reflectance spectra, as shown in Fig. 5(a), we calculated the absorption spectrum by α = 1 − ρ − τ , which is shown in Fig. 5(b), where ρ is the reflectance and τ the transmittance. To obtain the emissivity at T = 300K we first calculated the total emissive power of a blackbody E b as 31 where h the Planck constant, k the Boltzmann constant, c 0 the speed of light and λ the wavelength. Then the spectral emissive power E λ,b (λ, T ) is calculated as following 31 Considering the absorption α as ε λ (λ, T ) we can calculate the effective emissivity by In our case we integrated over the measured spectrum from 1.4 µm to 20 µm and obtained a emissivity ε ≈ 0.05. Fig. 6 shows the spectra for the measured wavelength range.

B. Response time measurements
For response time measurements the frequency of the modulated heating laser is constantly increased while recording the frequency via the phase-locked loop. At low values of the modulation frequency, the drum resonator can follow this modulation. However, if the modulation frequency of the heating increases, the resonator is, due to its material properties and thermal transfer capabilities, not able to follow. The amplitude of the recorded time signal will decrease as a consequence. To compare the results from measurements we derived an analytical model, assuming a setting similar to the schematic in Fig. 7. The drum size is given by its lateral dimension L and we assume a constant surrounding temperature T 0 that also acts as a boundary condition for the temperature of the silicon frame supporting the silicon nitride and additional layers.
Each layer by index i ∈ [1, 2, ..., n] has a certain height given by h i , a material specific thermal conductivity κ i and a heat capacity per volume c i . The temperature T in the center of the drum resonator originates from absorbed photons with an absorbed power P . Depending on the pressure p there might be a heat loss to the ambient gas which is given by some heat flux density J Q . We denote the origin from the heat flux density by its subscript i, e.g. J Q1 for the bottom side and J Qn for the top side of the drum.
The heat continuity equation is given by where q(r, t) is the dissipated heat rate per unit volume, J Qi = −κ i ∇T the heat current density. The temperature in the vertical direction z can be assumed to be constant due to the great lateral dimension compared to the thickness.
Integrating equation (8) over the thickness leads to, Within a certain range of ambient pressures the heat flux J Qi can be expected to vary linearly with the drum temperature when each molecule is expected to remove energy by its impinging molecular flux density 1 4 nv, where n = p/ (k B T 0 ) is the molecular density and v the mean molecular speed. The removed energy is given by f 2 k B (T −T 0 ), where f is the number of degrees of freedom for the molecules (e.g. f = 6). The heat flux is assumed to be equal on the top and bottom surface given by With equation (10) For the sake of simplicity, the temperature T 0 is set to zero. The temperatures T can now be seen as the deviating temperatures. It follows that C ∂T ∂t = K∇ 2 ⊥ T − HT + Q(x, y, t) (12) For symmetric solutions in x and y we can assume product solutions in the form of ϕ n = A n cos (k n x) cos (k n y) exp − t τ n (16) with cos (k n L/2) = 0 ⇒ k n = π(2n + 1)/L. The homogeneous solution of the partial differential equation is then Finally we are interested in the first mode (n = 0), since for higher modes the decay is faster and obtain Dependent on the quality of the vacuum, (18) can be reduced by assuming H 0, describing a thermal conductivity dominated heat transfer. However, to include thermal radiation, the heat flux density from the top and bottom surface needs to be considered as For small deviations from the ambient temperature, the nonlinear temperature dependency can be replaced by a first-order Taylor expansion with reasonable accuracy given by The thermal radiation can then be accounted for by replacing H in equation (18) with H SB = 8 σ SB T 3 0 , i.e.,