Near-Field Radiative Heat Transfer between Disordered Multilayer Systems

Peng Tian(田鹏)$^{1\dag}$, Wenxuan Ge(葛文宣)$^{1\dag}$, Songsong Li(李松松)$^{1}$, Lei Gao(高雷)$^{1,2}$,\\ Jianhua Jiang(蒋建华)$^{1}$, and Yadong Xu(徐亚东)$^ {1,3\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/6/067802

⇦Chinese Physics Letters, 2023, Vol. 40, No. 6, Article code 067802⇨ Near-Field Radiative Heat Transfer between Disordered Multilayer Systems Peng Tian (田鹏)1†, Wenxuan Ge (葛文宣)1†, Songsong Li (李松松)1, Lei Gao (高雷)1,2, Jianhua Jiang (蒋建华)1, and Yadong Xu (徐亚东)1,3* Affiliations 1Institute of Theoretical and Applied Physics, School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2Department of Photoelectric Science and Energy Engineering, Suzhou City University, Suzhou 215104, China 3Key Lab of Modern Optical Technologies of Ministry of Education, Soochow University, Suzhou 215006, China Received 28 March 2023; accepted manuscript online 17 May 2023; published online 29 May 2023 ^†These authors contributed equally to this work.
^*Corresponding author. Email: ydxu@suda.edu.cn Citation Text: Tian P, Ge W X, Li S S et al. 2023 Chin. Phys. Lett. 40 067802 Abstract Near-field radiative heat transfer (NFRHT) research is an important research project after a major breakthrough in nanotechnology. Based on the multilayer structure, we find that due to the existence of inherent losses, the decoupling of hyperbolic modes (HMs) after changing the filling ratio leads to suppression of heat flow near the surface mode resonance frequency. It complements the physical landscape of enhancement of near-field radiative heat transfer by HMs and more surface states supported by multiple surfaces. More importantly, considering the difficulty of accurate preparation at the nanoscale, we introduce the disorder factor to describe the magnitude of the random variation of the layer thickness of the multilayer structure and then explore the effect on heat transfer when the layer thickness is slightly different from the exact value expected. We find that the near-field radiative heat flux decreases gradually as the disorder increases because of interlayer energy localization. However, the reduction in heat transfer does not exceed an order of magnitude, although the disorder is already very large. At the same time, the regulation effect of the disorder on NFRHT is close to that of the same degree of filling ratio, which highlights the importance of disordered systems. This work qualitatively describes the effect of disorder on heat transfer and provides instructive data for the fabrication of NFRHT devices.

DOI:10.1088/0256-307X/40/6/067802 © 2023 Chinese Physics Society Article Text Thermal radiation with photons as energy carriers is one of the most ubiquitous physical phenomena.[1,2] In recent years, near-field thermal radiation between two noncontact bodies with different temperatures has attracted extensive attention because a small gap or separation between two bodies can drastically enhance the radiative heat transfer. It is significant to overcome the far field limit set by the Stefan–Boltzmann law,[3-12] which has been found to play important roles in many thermal technologies, such as thermophotovoltaics,[13-16] thermal rectifiers,[17-19] and coherent thermal sources. In the near-field regime, when the separation is much smaller than the thermal wavelength, the photon tunneling of evanescent waves with high wave vectors dominates the efficiency of near-field radiative heat transfer (NFRHT).[20,21] In the last decade, various geometry configurations have been proposed to manipulate and enhance the photon tunneling effect, including two homogeneous semi-infinite plates, nanopore structures, and periodic gratings.[22-31] In particular, much effort has recently been devoted to multilayer structures consisting of alternating metal and dielectric layers, often called hyperbolic metamaterials (HMMs), due to their exceptional features of an open hyperbolic dispersion relationship. Researchers have shown that the open hyperbolic dispersion of HMMs can greatly enhance the local photonic density of states and the photon tunneling of evanescent waves with high wave vectors, thus providing a good platform to significantly enhance NFRHT.[32-34] Additionally, metal-dielectric interfaces support strong local surface modes of surface plasmon polaritons (SPPs), whose excitations can efficiently harvest the photon tunneling of evanescent waves with high wave vectors, particularly around the plasma frequency.[35,36] However, in most previous studies, the geometric models of HMMs considered are strictly one-dimensional layered periodic structures, and the typical thickness of each layer is approximately 10 nm, i.e., a deep subwavelength scale compared with the wavelength of thermal radiation, which poses a serious challenge for current nanofabrication techniques[37] because the thickness of the layer film at such a small nanoscale is difficult to accurately control experimentally. In practice, the thickness of the prepared dielectric or metal layers is usually within a certain range centered on the target value, presenting a disordered distribution to a certain extent. This fact motivates us to consider the following question: What will happen for NFRHT in disordered multilayer structures? In this work, we theoretically explore the NFRHT effect in disordered multilayer structures by designing and studying a series of alternating metal and dielectric layers with deep-subwavelength thicknesses. In the framework of effective medium theory (EMT),[37,38] when the characteristic thickness is in the deep-subwavelength regime, the influence of small perturbations of any photonic structure on the electromagnetic wave scattering and energy transfer, no matter whether disordered or not, can be almost negligible. The electromagnetic wave or light should behave as passing through a homogeneous material with an effective refractive index. However, this is not always the case.[39,40] In this Letter, we introduce disorder into the multilayer heat transfer system for the first time to explore its influence on heat transfer. By comparing the filling ratio, we find that disorder has a strong and non-negligible regulating effect on heat transfer. In addition, we also show that the introduction of disorder results in a similar localization effect in the energy transfer coefficient (ETC), which can alter the radiation spectrum to design desired thermal devices. Compared with the standard model, which is completely ideal and difficult to realize, the more practical disordered system will lead to a slight decrease in the net radiant heat flux. However, due to the existence of hyperbolic modes (HMs), compared with the traditional blocky system, the heat transfer is still enhanced by nearly two orders of magnitude, which provides guidance for development and applications of practical devices.

Fig. 1. Diagram of multilayer radiation structure with the high temperature part ($T_{\rm H}$) and low temperature part ($T_{\rm L}$) in the structure being $T_{\rm H} =301$ K and $T_{\rm L} =300$ K, respectively.

Figure 1 schematically illustrates the considered disordered multilayer system for NFRHT, which consists of two layered structures separated by an air gap with thickness $d_{0}$. Each side of the structure consists of $N$ unit cells, and each unit cell consists of a lossless dielectric layer (the blue area) with a permittivity $\varepsilon_{\rm d} =1$ and a metal layer (the gray area) with its dielectric constant described by the Drude model $\varepsilon_{\rm m} (\omega)=\varepsilon_{\infty} -{\omega_{\rm p}^{2}}/{\omega (\omega +i\gamma)}$, where $\omega_{\rm p} =2.5\times 10^{14}$ rad/s, $\varepsilon_{\infty } =1$, and $\gamma =1\times 10^{12}$ rad/s.[35] The thicknesses of the metal and dielectric layers are $d_{\rm m}$ and $d_{\rm d}$, respectively. For the ordered cases discussed in the previous works,[32,35] the typical sizes are $P=20$ nm with $d_{\rm m} =d_{\rm d} =10$ nm and $d_{0} =10$ nm. The filling ratio of the metal layer is $f={d_{\rm m}}/P$, which is usually 0.5 in all previous studies, with $P$ referring to the period of space. The temperatures on both sides of the system were set as $T_{\rm H} =301$ K and $T_{\rm L} =300$ K, respectively. Theoretically, at the temperature the material can handle, the phenomenon described holds true for any temperature difference. For easy discussion, we assume that the space period and temperature settings in the work are the same as the values given above, unless specified otherwise. To reveal the NFRHT effect in multilayer HMMs from ordered systems to disordered ones, we consider a disorder model of a random distribution of layer thickness, and a disorder factor is introduced, defined by $\eta ={\max (|{d_{\rm e} -d_{\rm r}}|)} / {d_{\rm e}}$, where $d_{\rm e}$ is the thickness expected and $d_{\rm r}$ is the simulated random thickness. Thus, the thickness of each metal layer of the multilayer structure takes a random value between $d_{\rm e} -\eta d_{\rm e}$ and $d_{\rm e} +\eta d_{\rm e}$. When $\eta =0$, the considered disordered multilayer structure reduces back to the ordered system studied previously. In theory, the heat transfer in the near-field condition can be described by using fluctuation electrodynamics and fluctuation dissipation theory. Specifically, the net heat flux can be obtained by performing integral calculation through dyadic Green's function, which is expressed as[3,6] \begin{align} Q=\frac{1}{4\pi^{2}}\int_0^\infty {d\omega [\varTheta (\omega,T_{\rm H})-\varTheta (\omega,T_{\rm L})]} \int_0^\infty {\xi (\omega,\beta,d)} \beta d\beta. \tag {1} \end{align} Here, $\varTheta (\omega,T_{i})$ is the average energy of the Planck harmonic oscillator at angular frequency $\omega$, and it can be calculated by the following formula: \begin{align} \Theta (\omega,T_{i})=\frac{\hslash \omega }{[\exp ({\hslash \omega } / {k_{\scriptscriptstyle{\rm B}} T_{i} })-1]},~~(i={\rm H,L}) \tag {2} \end{align} where $\hslash$ and $k_{\scriptscriptstyle{\rm B}}$ are the reduced Planck constant and the Boltzmann constant, respectively. The term $\xi$ is ETC consists of both propagating waves ($\beta < \omega / c$) and evanescent waves ($\beta >\omega / c$) in s-polarization and p-polarization, respectively. We have[33] \begin{align} \xi_{\rm prop} =\,&\frac{(1-|{r_{1}^{\rm s}}|^{2})(1-|{r_{2}^{\rm s}}|^{2})}{|{1-r_{1}^{\rm s} r_{2}^{\rm s} e^{i2\gamma_{0} d}}|^{2}}\notag\\ &+\frac{(1-|{r_{1}^{\rm p}} |^{2})(1-|{r_{2}^{\rm p}}|^{2})}{|{1-r_{1}^{\rm p} r_{2}^{\rm p} e^{i2\gamma_{0} d}}|^{2}}, ~~~~~~~~~~\,~~ \tag{3a}\\ \xi_{\rm evan} =\,&\frac{4{\rm Im}(r_{1}^{\rm s})\cdot {\rm Im}(r_{2}^{\rm s})\cdot e^{-2{\rm Im}(\gamma_{0})d}}{|{1-r_{1}^{\rm s} r_{2}^{\rm s} e^{i2\gamma_{0} d}} |^{2}}\notag\\ &+\frac{4{\rm Im}(r_{1}^{\rm p})\cdot{\rm Im}(r_{2}^{\rm p})\cdot e^{-2{\rm Im}(\gamma_{0})d}}{|{1-r_{1}^{\rm p} r_{2}^{\rm p} e^{i2\gamma_{0} d}}|^{2}}, ~~~~~ \tag{3b} \end{align} where $r_{1}$ and $r_{2}$ are the reflection coefficients of materials from the middle gap to both sides, respectively, and $d$ is the gap width of the intermediate gap with $\gamma_{0} =\sqrt {({\omega/c})^{2}-\beta^{2}}$. For comparison, let us revisit the NFRHT effect in an ordered multilayer system by setting $\eta =0$ in Fig. 1. We calculate the ETC by the transfer matrix method for the case with $f=0.5$ when $N=100$ and $N=1000$ as shown in Figs. 2(a) and 2(b), respectively. When the number of layers increases sufficiently large, the ETC is saturated because the contribution of layers far from the gap to heat transfer is negligible. Here it is numerically found that $N=1000$ is enough for saturation. In both the cases, as demonstrated in Ref. [35], the multiple surface modes of SPPs bounded at metal–air interfaces do lead to an enormous enhancement of near-field thermal radiation through evanescent waves with $\beta >\omega / c$, particularly around the plasmon resonance frequency of approximately $1.7767\times 10^{14}\,{\rm rad/s}$. Meanwhile, the HMs in HMMs greatly contribute to the heat transfer enhancement through propagating waves with $\beta < \omega / c$. As the number of layers increases, the ETC in the high-$k$ modes is suppressed. This is because increasing the number of layers $N$ results in more energy loss of the whole system, and the SPPs are suppressed, resulting in the impossibility of high-$k$ mode waves with weak penetrating ability.

Fig. 2. [(a), (b)] ETC with $f=0.5$ and (a) $N=100$, (b) $N=1000$. (c) The net spectral heat fluxes corresponding to (a) and (b). The blue, red, and yellow lines correspond to $N=100$, $N=500$, and $N=1000$, respectively. [(d), (e)] ETC with $f=0.6$ and (d) $N=100$, (e) $N=1000$. The frequency range where the ETC is missing is marked by the green dashed box. (f) Corresponding to net spectral heat fluxes in the structures of (d) and (e). The blue, red, and yellow lines correspond to $N=100$, $N=500$, and $N=1000$, respectively.

Fig. 3. (a) The components of the effective permittivity tensor with $f=0.5$. The gray area represents the type-I hyperbolic metamaterial (HMM-1) interval, and the yellow area represents the type-II hyperbolic metamaterial (HMM-2) interval. (b) The components of the effective permittivity tensor with $f=0.6$. The elliptic curve dispersion relationship interval corresponds to the green dashed box in Fig. 2(d), where the dielectric function component does not satisfy the “single negative” condition. The physical meaning of the rest of the graphs is the same as that of (a).

Furthermore, Figs. 2(d) and 2(e) display the same results as Figs. 2(a) and 2(b) but for the $f=0.6$ case. Interestingly, some energy transfer modes have disappeared, i.e., the ETC vanishes in a frequency range marked by the green dashed box in the plots. Moreover, with the increase of the number of layers, this vanished phenomenon becomes more obvious, with the SPPs that are suppressed seriously [see Fig. 2(d)]. The in-depth explorations show that this phenomenon is universal and always exists as long as $f\ne 0.5$, for instance, $f=0.3$ and $f=0.7$. It means that by adjusting the filling ratio, we can simply suppress the energy transfer modes of a specific frequency band to adjust the frequency band distribution of the transmission state. This phenomenon is caused by the suppression of the SPPs by HMs. Here, we use EMT to give an intuitive physical explanation. Note that the standard effective medium theory usually only works for the HMMs of multilayer systems with infinite layers, not for finite-layer systems and even worse in the case of a few layers. Nevertheless, we will show that the EMT is an effective tool used to study the NFRHT effect in HMMs with finite layers. In fact, this aforementioned phenomenon reveals the NFRHT enhancement in HMMs; when the hyperbolic metamaterial composed of multilayer structures participates in the NFRHT, the enhancement effect is not only contributed by the increased surface states. The SPPs are suppressed due to more loss when there are many layers. Moreover, due to the filling ratio change up to 0.6, HMs do not meet hyperbolic dispersion near the SPP frequency, which leads to a decrease in heat flux at the resonance frequency of SPPs. To describe this conclusion more clearly, we treat the multilayer structure with the effective medium theory. As a hyperbolic material, its effective dielectric function can be written as[41,42] \begin{align} &\varepsilon_{xx} =\varepsilon_{yy} =f\varepsilon_{\rm m} +(1-f)\varepsilon_{\rm d}, ~~~~~~~~~~~~\,~~ \tag{4a}\\ &\varepsilon_{zz} =\Big({\frac{f}{\varepsilon_{\rm m} }-\frac{1-f}{\varepsilon_{\rm d} }}\Big)^{-1}. ~~~~~~~~~~~~~~~~~~~~~~ \tag{4b} \end{align} It is worth noting that the effective medium theory generally is built for the case with infinite number of layers. However, in this study, it is found that this EMT is in good agreement with the transfer matrix method for a finite number or few layers. To illustrate this point, Fig. 3 plots the components of the effective permittivity tensor of dielectric function for different filling ratios. It is obvious that HMs dominate NFRHT in low-$k$ vector states. When the equivalent dielectric function of the multilayer structure does not satisfy the hyperbolic condition, in our viewpoint, the decoupling of HMs makes a missing region near the surface plasmon resonance frequency appear in the ETC. This physical mechanism directly and quantitatively illustrates the reason for the disappearance of energy transfer modes. SPPs will be suppressed by HMs when there are many layers, which can be used to adjust the distribution of heat transfer frequency bands. In the following, we consider the NFRHT in the disordered multilayer system to show the influence of the disorder factor on it. We can observe the effect of metal layer thickness variation in a certain range on NFRHT by changing $\eta$. Figures 4(a) and 4(b) show the ETC in the disordered HMM with $f=0.5$ when $\eta =0.1$ and $\eta =0.5$, respectively. In our study, the thickness of each period is fixed to 20 nm, and the number of layers $N=100$. It is obvious that the ETC is localized on some dispersion curves when we introduce disorder into the multilayer system, and the greater the disorder factor is, the more obvious the localization is. This is because the thickness of metal layers in each period is random, which leads to different strengths of mutual coupling between metal layers. When the disorder factor on both sides of the structure is inconsistent, or when one side is ordered and the other side is disordered, ETC will also be localized, but the degree of localization varies, which is related to the disorder factor on both sides. As a comparison, the introduction of disorder $\eta$ into structural systems with a filling ratio $f$ not equal to 0.5 is worth considering. We set the filling ratio of the whole structure $f=0.6$ and plotted the ETC as shown in Figs. 4(c) and 4(d). The disappearance of energy transfer modes still occurs at the resonance frequency of SPPs. The interval where the energy transfer modes are suppressed when the structure is orderly with the same filling ratio ($f=0.6$) is marked with a green dotted line box. It is obvious that with the increase of disorder, the interval where the energy transfer modes are suppressed gradually narrows by comparing Figs. 4(c) and 4(d). Note that the transfer modes only occupy a very narrow frequency range in the green dashed line box in Fig. 4(c), but with the increase of disorder, there are obvious discrete bright bands in the green dashed line frame in Fig. 4(d), which reduces the interval where the energy transfer modes are suppressed, and the frequency band distribution of the transfer modes is readjusted. It is not difficult to find the reason for this phenomenon from the conclusion of Fig. 3(b); that is, after the introduction of disorder into the hyperbolic structure, the coupling strength between metal layers becomes uneven. With the increase of disorder, the disorder of the layer thickness makes the local area in the structure not completely consistent with the filling ratio $f=0.6$. Therefore, some allowable hyperbolic energy transmission states would appear in the elliptical dispersion interval in Fig. 3(b), which will fill a part of the interval where the energy transfer modes are suppressed.

Fig. 4. The ETC with different disorder factors, different filling ratios and different numbers of layers: [(a), (b)] for the filling ratio $f=0.5$, the expected metal layer thickness $d_{\rm e} =d_{\rm m} =10$ nm, $\eta =0.1$ and $\eta =0.5$, indicating that the real metal layer thicknesses $d_{\rm r}$ under the two structures are randomly selected within (9 nm, 11 nm), (5 nm, 15 nm), respectively; [(c), (d)] the same as (a) and (b) but $f=0.6$.

This conclusion can also be obtained from Fig. 5(a). We select three representative layers, i.e., $N = 100$, $N = 500$, and $N = 1000$, and plotted the changes of net radiative heat flux with disorder factor in the three cases. As the disorder gradually increases, the net radiative heat flux also decreases gradually. The greater the disorder, the more obvious the localization, and the lower the heat flux, but note that the decrease in net radiative heat flux is within the acceptable range. On the whole, when the disorder factor increases to a considerable extent, that is, $\eta =0.7$, the reduction of heat flux is also within half an order of magnitude. In addition, a large number of energy transfer modes occur in the interval where the energy transfer modes are suppressed when the disorder factor is large enough, leading to similar heat flux at different filling ratios. It provides a new idea for adjusting NFRHT, such as inhibiting heat transfer at specific frequencies. At the same time, changing the disorder factor has the lowest impact on the structure with fewer layers when $f=0.6$. For example, when the disorder factor increases from 0.1 to 0.2, the heat flux of $N = 100$ decreases by 19% and that of $N = 1000$ decreases by 23%. It is also easy to understand that when the number of layers increases, there will be more local structures that do not meet the requirements of $f=0.6$, leading to a decrease in its heat flux. In addition, for the same disorder factor, different disordered distributions correspond to slightly different ETCs, but their integrated total heat flux is almost indistinguishable, which is consistent with our conclusion. Finally, the responses of NFRHT to disordered systems and systems with different filling ratios are noteworthy. Figure 5(b) shows the heat flux under different vacuum gaps with different structures. It is obvious that in a multilayer hyperbolic system, the response of NFRHT to vacuum spacing is nonmonotonic. When the vacuum gap is equal to the layer thickness, that is, 10 nm, the heat flux of the ordered system has a peak value, which is also consistent with the previous research results; that is, the heat transfer of the system is nonmonotonic with the gap spacing.[35] This is because the multilayer system is similar to a bulk in the small $d_{0}$ limit, the influence of the layer far from the middle gap on heat transfer is weakened, then the gap width is equal to the layer thickness, the coupling strength between layers is consistent, and the damping ratio becomes $\gamma/2$, which leads to the maximum NFRHT. In the large $d_{0}$ limit, the behavior of the multilayer system approaches that of the effective medium, where the heat transfer is mostly carried by waves with relatively small wave vector components parallel to the interface. Similarly, when $f=0.6$, the optimal thickness becomes 8 nm, as shown by a purple line.

Fig. 5. (a) Net radiative heat flux at different disorder factors with $f=0.5$ (solid line) and $f=0.6$ (dotted line). (b) Heat flux under different vacuum gaps based on different structure configurations with $N=100$. The inset is the normalized data based on the bulk system.

When the filling ratio is constant, the heat flux is significantly regulated by the disorder factor. The increase of the disorder factor reduces the heat flux. Because the coupling between layers is no longer the same, the peak value of the heat flux is also suppressed at the optimal gap distance. At the same time, comprehensively comparing the influences of disorder and filling rate on NFRHT, we find that the two influences are comparable. The purple line and the blue line correspond to the disorder, or the filling ratio has change by 20% compared with the ordered structure. It is obvious that the inevitable disorder during preparation has a more obvious control effect on the heat flux. In addition, the inserted graphs show the normalized heat flux for various configurations, and the NFRHT of the multilayer structure is always much larger than that of the bulk material because of the existence of HMs. At the same time, although the disorder introduced by the inevitable error during manufacturing has some suppression on heat transfer, making its NFRHT lower than the perfect multilayer system, its heat flux is always far higher than that of the bulk structure, which can enhance one or two orders of magnitude while ensuring a large absolute value of heat flux. In conclusion, as a comparison, we have studied the heat transfer of an ordered multilayer system and revealed the mechanism of SPPs suppressed by HMs. Then, considering the difficulty of preparing fine structures, we introduce disorder to study the effect on heat transfer when the structural parameters are different from those we expected. We find that the net radiant heat flux will decrease to some extent within the acceptable range, but due to the HMs, the heat transfer is still enhanced by nearly two orders of magnitude compared with the bulk system. At the same time, we note that compared with the filling ratio, the disorder factor also plays an important role in adjusting the heat transfer of the multilayer system, and its control effect even exceeds the filling ratio. Our results provide valuable theoretical data for the production of actual structures and are of great significance in the development of thermal and other devices. Controlling the precise size of disorder, regulating heat transfer to maximize heat transfer or rectification efficiency to maximize, and balancing the contradiction between the actual preparation requires further in-depth research in the future. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11974010 and 12274313), and the National Key R&D Program of China (Grant Nos. 2022YFA1404400 and 2022YFA1404300). 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