Self-adaptive near-field radiative thermal modulation using a thermally sensitive bimaterial structure

In the last decades, near-field radiative heat transfer (NFRHT) has aroused significant attention in both theoretical and experimental work. The long-standing radiative heat transfer limit set by the Stefan–Boltzmann law can be broken through by several orders of magnitude in the near-field regime. When the separation between two bodies is smaller than the thermal wavelength, evanescent waves (or photon tunneling), which are not taken into account in the far-field regime, dominate the small-scale radiative heat transfer. In addition to the simplest plate-plate configuration, nanostructures have gained tremendous attention for the purpose of further enhancement of radiative heat transfer and spectral tunability, including multilayer structures and photonic crystals,^1–3 metamaterials,^4–6 gratings,^7–9 and graphene.^10,11 Moreover, NFRHT in many-body systems has also been intensively explored recently, unraveling new physical and transport behaviors.^12,13

Moreover, the active control of NFRHT is another research focusing nowadays. One of the methods is to leverage the phase-change materials, whose spectral properties can be tuned by a temperature change or applying an external field. Vanadium dioxide (VO₂) with the critical temperature of 341 K is one of the most popular candidates. VO₂ is in anisotropic insulator state while below its critical temperature; otherwise, it is in metallic state. Yang et al. numerically investigated the thermal rectification based on VO₂ and discussed the influence of film thickness.^14,15 Ghanekar et al. demonstrated the effects of VO₂ inclusions on thermal rectification as the form of bulk, thin film, and grating.¹⁶ It has also been achieved the electronic control of both the direction and magnitude of NFRHT between the two bodies by regulating the chemical potential of photons.¹⁷ Moreover, magneto-optical materials open new avenue to control the radiative heat transfer with an external magnetic field.^18,19 Based on the active control of NFRHT, functional devices for thermal management and rectification analogous to electronic counterparts are designed, such as thermal diodes, thermal transistors, switches, and memory elements.^20–23 

The bimaterial structure is extensively utilized in microelectromechanical systems for thermal measurement. This structure can provide thermally bended deformation due to the thermal expansion coefficient mismatch between two bonded materials, the so-called bimorph effect.²⁴ This kind of thermal device has higher thermomechanical sensitivity and smaller spatial resolution than those of traditional thermal devices. Ever since being introduced as a novel sensing paradigm, technology has emerged to find important applications in chemical, physical, and biological sensing areas. The applications of the bimaterial structure include stress sensor,²⁵ thermal sensor,²⁶ biological sensor,²⁷ thermal actuator,²⁸ spatially resolved calorimetry,²⁹ and thermomechanical data storage.³⁰ The bimaterial cantilever beam was first introduced as a calorimeter to measure the heat generated by chemical reactions with power and energy resolutions in the range of 1 nW and 1 pJ, respectively.³¹ In addition, working as a thermal sensor of infrared radiation, the bimaterial microcantilevers turn heat into a mechanical response and can be referred to as thermomechanical detectors. This type of thermal sensor is essentially free of intrinsic electronic noise and can be combined with many different readout techniques with extremely high sensitivity.³² Consequently, the bimaterial structure can be used to detect very small temperature variations, aided by its small dimension and thermal mass combined with most advanced techniques to monitor the high-precision bending. In other words, a small change of temperature can induce a tiny shape deformation of the bimaterial structure, which may have a huge influence on the NFRHT.

To date, NFRHT has been discussed in geometric cases including two spheres,³³ a sphere in front of a plate,³⁴ gratings,^9,35 and twisted configurations.^36,37 However, studies regarding curved plates driven by temperature change are still scarce. In this work, we study the dynamic NFRHT between a bimaterial structure and a bulk material. Due to different thermal expansion coefficients and Young's moduli of the two materials bonded together, the bimaterial structure has the tendency to bend upon a small temperature change, thus altering the separation gap. As is known to all, NFRHT largely depends on the distance between the two spaced objects. Therefore, this temperature-induced deformation cannot be simply ignored. Combining the NFRHT with the far-field thermal radiation, here, we propose a self-adaptive radiative thermal modulator. Originally, the bimaterial strip is planar and in thermal equilibrium. When a sudden temperature drift occurs, the strip will bend outward or inward. Due to the unbalance between NFRHT and the far-field thermal radiation, the strip tends to spontaneously recover back to its original planar state. During the recovering process, a significant enhancement or reduction of NFRHT is observed due to the curved surface.

The structure of this near-field radiative thermal modulator is composed of a bulk boron nitride (BN) and a bimaterial strip, separated by a gap d = 100 nm originally, as shown in Fig. 1. The bimaterial strip consists of gold (Au) and silicon nitride (Si₃N₄), and their thicknesses are h and H, respectively. In the original state, the temperature of BN is fixed at $T1=905.6$ °C, and it keeps unchanged for all following discussions. The temperature T₂ of the bimaterial strip equals to $900$ °C. With specific temperature and geometric parameters selection, the NFRHT Q_nf and the far-field thermal radiation Q_ff from the upper surface to the ambient could be identical. Thus, the bimaterial structure is in equilibrium. When a small temperature drift $ΔT$ occurs (scenario 1), due to different Young's moduli and thermal expansion coefficients of the bimaterial strip, the planar structure bends outward, thus enlarging the separation gap between the bimaterial strip and BN, which has a significant influence on the NFRHT. Moreover, the temperature difference between the bulk BN and the bimaterial strip decreases by $ΔT$⁠. The aforementioned two factors lead to smaller Q_nf, while Q_ff increases in the meantime. Owing to Q_ff > Q_nf, more heat is removed from the bimaterial structure than the received heat; therefore, the temperature tends to recover to the original T₂. In other words, Q_ff > Q_nf holds true until the bimaterial strip recovers to the original planar state. Similarly in scenario 2, the temperature of the bimaterial strip drops by $ΔT$⁠, and the planar structure bends inward. A smaller separation gap and a larger temperature difference contribute to a bigger Q_nf, which exceeds Q_ff through the recovering process. Eventually, the whole structure will maintain as the original equilibrium state. To sum up, when there is a temperature drift for the planar bimaterial strip and a bending phenomenon occurs, it will recover to the original state spontaneously, aided by a self-adaptive near-field radiative thermal modulation using thermally sensitive bimaterial structure.

A schematic is displayed in Fig. 2 to get a deeper understanding of the bending process and its impact on the NFRHT. The left figure is for scenario 1, while the right one is for scenario 2. Since the bimaterial strip is quite thin, it can be regarded as a line in Fig. 2 for the sake of simplicity. The black straight line stands for the original planar structure with a length L. The purple arc shows the bending state of the bimaterial strip upon a temperature drift, while the blue arc represents an intermediate state during the recovering process. r and $r′$ are the curvature radii of these two states, respectively. The arc can be differentiated into multiple pieces in which each piece has a constant central angle $dθ$⁠, and a length dL equals to $r′dθ$⁠. The distance between each piece and the black line is referred to as dl in which dl_max is also called sagitta. The bimaterial structure with the blur effect is exhibited as a reference as well.

In this study, the bimaterial strip is composed of Au and Si₃N₄. Due to the fact that Au possesses a larger thermal expansion coefficient than Si₃N₄, Au will expand more than Si₃N₄ when heated, causing the bonded strip to bend toward the Si₃N₄ side and bend back when cooled. Vice versa, Au shrinks more, and the strip bends toward the Au side when cooled. In order to quantitatively analyze this bending phenomenon, a general expression is introduced herein for the curvature arising from the imposition of a uniform misfit strain,³⁸ 

where r is the curvature radius, $Ed$ and $Es$ are the Young's modulus of the deposit (Au) and the substrate (Si₃N₄), respectively, h and H are the thicknesses of Au and Si₃N₄, respectively. The misfit strain $Δε=ΔαΔT$⁠, caused by the mismatch of the thermal expansion coefficients $Δα$ and the temperature drift $ΔT$⁠. The thermal expansion coefficients of Au and Si₃N₄ are $14.2×10−6$ and $3×10−6$ K^–1, respectively. The Young's moduli $Ed=78$ and $Es=271$ GPa.³⁹ In addition to the aforementioned known parameters, it is shown that the curvature radius is related to the temperature drift and the thickness of each layer.

In Fig. 3(a), the relationship between the curvature radius and the thickness is studied when $ΔT$ is fixed at 1 °C. As the thickness of Si₃N₄ remains constant at 1 μm (blue curve), the curvature radius first drops rapidly with the increasing h. When h is larger than 50 nm, the pace of decline slows, and the radius remains relatively stable below 1 m. This can be explained by the beam bending theory. First, the misfit strain is removed by the application of two equal and opposite forces. When the two materials are bonded together, these forces lead to an unbalanced moment, and balancing this moment generates curvature of the bimaterial strip. If the Au layer is too thin, the moment does not require a large deformation to counteract. Therefore, the bimaterial strip does not bend so much that the radius has a quite large value. As the Au thickness is fixed at 30 nm (red curve), the radius rises monotonically with thicker Si₃N₄. It is concluded that the bimaterial strip is harder to bend with a great disparity between the thickness of each layer. Moreover, as shown in the inset, the curvature radius is proportional to the substrate thickness for a given h/H ratio equal to 0.03, demonstrating the bending phenomenon is more obvious at smaller scale.

The relationship between the curvature radius and the temperature difference is shown in Fig. 3(b). The geometric parameters are fixed as H = 1 μm and h = 30 nm. It is observed that the curvature radius decreases monotonically as the temperature deviates from the original state. Here, the length, L, is set as 1 mm and the corresponding dl_max is in direct proportion to $ΔT$⁠. The thicknesses are finally determined as H = 1 μm and h = 30 nm due to the reasons as follows. First, it is feasible in practical fabrication. Moreover, the bending range is moderate and suitable; for example, in scenario 2, the bending is inward, so that more attention needs to be paid to avoid the contact of BN and the bimaterial strip. In other words, dl_max should be smaller than the original d, 100 nm.

In the calculation, the original length L of the strip is set as 1 mm, and the bending arc is divided into 200 pieces. The original temperature T₂ is $900$ °C. The detailed geometric parameters are displayed in Table I at six specific temperature points, including curvature radius r, differential central angle $dθ$⁠, and sagitta dl_max, which are all in the moderate and suitable range. It is apparent that the length of the arc surface should be larger than the original length L, as shown in Fig. 2. However, for the temperature points in the table, the differential unit lengths dL are all equal to 5.00 μm, calculated by $rdθ$⁠. Therefore, the total length is very close to the original length of 1 mm, and this variation can be neglected.

The expressions for near-field radiative heat fluxes are obtained through the dyadic Green's function formalism.⁴⁰ The heat flux between planer objects can be calculated by

where T₁ and T₂ are the temperatures of two objects. $Θ(ω,T)=(ℏω/2)coth(ℏω/2kBT)$ is the energy of the harmonic oscillator. $∫0∞k∥dk∥2πZ(ω,k∥)$ is known as the spectral transmissivity in radiative transfer between media 1 and 2 with gap L, where $k∥$ is the parallel component of wavevector and $Z(ω,k∥)$ is known as the energy transmission coefficient.⁴⁰ 

Figure 4 demonstrates the variations of the NFRHT Q_nf and the far-field thermal radiation Q_ff against the temperature drift of the bimaterial strip, while the temperature of the bulk BN keeps fixed at 905.6 °C. The far-field thermal radiation is calculated by the Stefan–Boltzmann law $εσ(T24−Tamb4)$ in which ε is the emissivity of Au and T_amb is the ambient temperature. Here, ε and T_amb equal to 0.025 °C and 27 °C, respectively. Though it was reported from previous study that the thermal radiation from a thin film was more confined in the lateral direction than the vertical direction if the film thickness was much smaller than the thermal wavelength,⁴¹ this angle-dependent characteristic is not significant in this work since the thickness (1.03 μm) is not much smaller than the thermal wavelength (around 2.5 μm). Moreover, the area of the top surface is thousand-fold as large as that of the side surface. Therefore, the far-field thermal radiation from the side surface can be neglected. In the inset of Fig. 4(a), the heat flux against a separation gap from 10 nm to 10 μm is plotted at three T₂ points. It is clearly observed that the heat fluxes rise monotonically as the separation gap is reduced, and this is caused by the presence of evanescent waves in the near-field regime. It should be noted that here the unit of the heat flux is Watt per unit area, while in the main figure, it is integrated along the line; therefore, the unit becomes Watt per unit length. At the original temperature 900 °C, the NFRHT Q_nf received by the bimaterial strip and the thermal radiation Q_ff removed from it are all equal to around 2.672 W m⁻¹. Therefore, the bimaterial strip is in an equilibrium state. In Fig. 4(a), suppose that T₂ suddenly drifts to 905 °C, Q_ff will increase due to a larger temperature difference from the ambient, while Q_nf is reduced significantly. Due to Q_ff > Q_nf, more heat is removed, and the temperature drops spontaneously. Q_ff > Q_nf always holds true at any temperature higher than 900 °C, until it finally recovers to 900 °C. For scenario 2 [Fig. 4(b)], if the bimaterial strip switches to a temperature lower than 900 °C, Q_ff < Q_nf will contribute to the recovering process. Compared with Q_ff, the variation of Q_nf is more significant. For scenario 1, Q_nf is dramatically reduced by a factor of 24 with a 5 °C rise, while for scenario 2, Q_nf is 4.4 times as large as its original value via only 1 °C drop. Thus, the results validate an outstanding tuning range of NFRHT.

In this work, we propose the self-adaptive modulation of near-field radiative thermal transport. Without applying an external field, the mechanism of thermal modulation is based on the different Young's moduli and thermal expansion coefficients of bi-materials bonded together. Upon a small temperature drift, the bimaterial structure will bend outward or inward, which greatly influences the separation gap between the bulk substrate and itself. The large tuning range of the NFRHT combined with the far-field thermal radiation facilitates the recovering process to its original planar state. Therefore, the whole system reaches a stable equilibrium condition regardless of an initial temperature rise or drop. This self-adaptive radiative thermal modulation using a thermally sensitive bimaterial structure sheds light on nanoscale thermal harvesting and management.

This project was supported by the National Science Foundation through Grant No. CBET-1941743.