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Abstract
Understanding energy transfer via near-field thermal radiation is essential for applications such as near-field imaging, thermophotovoltaics and thermal circuit devices. Evanescent waves and photon tunneling are responsible for the near-field energy transfer. In bulk noble metals, however, surface plasmons do not contribute efficiently to the near-field energy transfer because of the mismatch of wavelength. In this paper, a giant near-field radiative heat transfer rate that is orders-of-magnitude greater than the blackbody limit between two ultrathin metallic films is demonstrated at nanoscale separations. Moreover, different physical origins for near-field thermal radiation transfer for thick and thin metallic films are clarified, and the radiative heat transfer enhancement in ultrathin metallic films is proved to come from the excitation of surface plasmons. Meanwhile, because of the inevitable high sheet resistance of ultrathin metal films, the heat transfer coefficient is 4600 times greater than the Planckian limit for the separation of 10 nm in ultrathin metallic films, which is the same order or even greater than that in other 2D materials with low carrier density. Our work shows that ultrathin metallic films are excellent materials for radiative heat transfer, which may find promising applications in thermal nano-devices and thermal engineering.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thermal radiation is an inevitable physical phenomenon; any heated substance emits electromagnetic waves due to the fluctuating currents induced by the thermal motion of charge carriers. For thermally isolated structures, thermal equilibrium with other surrounding structures and the electromagnetic environments is achieved due to the absorption and thermal emission. When the dimensions involved in a thermal radiation problem are larger than the thermal de Broglie wavelengths $\lambda _T\sim \hbar c/k_B T$, the radiative heat flux between two separated blackbodies is governed by the well-known Stefan-Boltzmann law [1], setting an upper limit for the power which can be transmitted by real materials. However, since only propagating modes are taken into account in this law, it is itself a limit for the far-field only. In contrast, if the separation distance $d$ is much smaller than $\lambda _T$, near field radiative heat transfer (NFRHT) can exceed the blackbody limit by several orders of magnitude due to the tunneling of evanescent waves, such as frustrated total internal reflection (FTIR), surface phonon polaritons (SPhPs) and surface plasmon polaritons (SPPs) [2–5]. As a result, further opportunities of implementing thermal radiation are opened into a plethora of applications, such as thermophotovoltaic (TPV) cell [6,7], thermal modulation [8], thermal rectification [9], and thermal transistor [10,11].
Recently, the NFRHT between two-dimensional (2D) materials such as graphene [12–18] and black phosphorus(BP) [19,20] has attracted a lot of interests. Since the frequency of SPPs in doped graphene [21–23] and black phosphorus [24,25] lie in the mid-infrared to terahertz range at room temperature or above, giant thermal radiation enhancement far beyond the blackbody limit can be achieved. Moreover, metal films down to atomic layer thickness can support strong tunable plasmons [26–28] and possess new quantum optical effects [29,30], demonstrating them as excellent materials for plasmonics. However, noble metal materials are always not recognized as good choices for the radiative heat flux enhancement. The reason is that, naively speaking, the carrier density in noble metals is so high that the wavelength of SPPs does not match the thermal wavelength of the emitted photon at room temperature. Specifically, the NFRHT between two identical thick metallic plates is dominated by transverse electric (TE) evanescent mode because the transverse magnetic (TM) wave contribution is below the blackbody limit for $d\;>\;5$ nm [31]. Nevertheless, it is not true for ultrathin noble metal films. Recently, the plasmon resonance in deposited ultrathin metal films is reported experimentally at 1.5–5 $\mu$m wavelength range with apparent wavelength and amplitude tunability through gating [28], outlining the potential applications of ultrathin metal films in the near-infrared to mid-infrared regimes. As a result, the NFRHT enhancement utilizing ultrathin metallic films becomes possible.
In this work, the NFRHT between two identical silver monolayer sheets is investigated. Based on the fluctuation-dissipation theory, the heat transfer coefficient between two separated sheets is calculated. Enhancement due to SPPs supported in silver monolayers can be orders-of-magnitude greater than the blackbody limit for $d\ll \lambda _{\textrm {T}}$. In addition, the influence of the number of layers, carrier density, carrier scattering rate, and the substrate of silver film on the NFRHT are examined. We find that the inevitable high loss in percolated silver monolayer will greatly enhances the heat transfer coefficient to 4600 greater than the blackbody limit, which is on the same order or even larger than that in other 2D sheets with low carrier density. However, this effect can be compensated by high index substrate. Our results pave the way for heat management based on ultrathin metal films.
2. Results and discussion
The system under study is depicted in Fig. 1(a), consisting of two parallel sheets of single silver atomic layer (111) supported on substrates brought into close proximity with a vacuum gap $d$ between silver planes. The thickness of the silver monolayer equals to the separation between (111) atomic planes in bulk silver (that is, $d_{111}$=$a_0/\sqrt {3}$=2.36 Å). The absolute temperatures at the bottom and upper sheets are $T_1$ and $T_2$, respectively. Heat is radiatively transferred from the hotter silver sheet ($T_1$) to the colder one ($T_2$) as a result of thermal fluctuations in both sheets. The net radiative power per unit of area $S$ exchanged between the two parallel plates at temperatures of $T_1$ and $T_2$ is given by the following Landauer-like expression [32,33]:
Fig. 1. (a) The schematic diagram of near-field radiative heat transfer (NFRHT) between two silver monolayer sheets. The gap size is $d$. The heat of the bottom sheet (temperature $T_1$) transfers to the upper sheet (temperature $T_2$) via radiation. (b) Heat transfer coefficient at room temperature (300 K) as a function of the gap size $d$. The different curves correspond to the total contribution (olive solid curve) and the contributions of propagating (dotted curves) and evanescent (dashed curves) waves for transverse electric (TE, colored in blue) and transverse magnetic (TM, colored in red) polarizations. The solid black line shows the result for the case of two blackbodies ($h_{\textrm {BB}}$=6.124 W/m$^2$K).
We first consider the system with two suspended monolayer silver sheets, and set $\gamma =\gamma _{\textrm {bulk}}=0.021$ eV. To illustrate the effect of NFRHT, the heat transfer coefficients $h$ as a function of the gap size $d$ are plotted in Fig. 1(b). We find that the Planckian heat transfer limit can be overcome provided that $d\lesssim 2\mu m$. As the two silver sheets come closer to each other, the near-field enhancement gets more prominent by orders-of-magnitude for sufficiently small gaps. Incidentally, the contributions of propagating waves in our configuration are $h_{\textrm {TM}}=0.01h_{\textrm {BB}}$ and $h_{\textrm {TE}}=0.02h_{\textrm {BB}}$, which are independent of the gap distance in near field range as expected. In contrast to heat transfer between two semi-infinite bulky metallic plates, where the NFRHT is dominated by TE evanescent waves originated from frustrated total internal reflection waves [31], the contribution of TE evanescent waves is two or three orders-of-magnitude smaller than TM evanescent waves. This result shows that the NFRHT between silver monolayer sheets is truly dominated by TM evanescent waves originated from SPPs similar to other 2D materials such as graphene and BP sheets.
Apart from the intensity enhancement of HTC with subwavelength gaps, the near-field contribution can also strongly modify the spectra of HTC. Figure 2(a) shows the spectral heat flux (i.e., heat conductance per unit frequency) with various gap sizes. By setting $\xi (\omega ,k_\|)=\theta (\omega -k_\|c)$, here $\theta$ is the step function, one can find that the peak of spectral HTC between blackbodies satisfies generalized Wien’s displacement law $\omega _{\textrm {max}}/T=5.01\times 10^{11}$rad/sK, which implies the photon frequency of maximum spectral HTC at $T=300$ K is 0.099 eV (black vertical dashed line). As the gap increases, the maxima of the spectral HTC blueshifts firstly from 0.072 eV ($d=10$ nm) to around 0.11 eV ($d=800$ nm) and then redshifts to 0.081 eV ($d=2$ $\mu$m). Moreover, one can also find that, the spectral HTC of the propagating waves is almost independent of gap sizes for $d\ll \lambda _{T}$ with the maximum spectral HTC located at around 0.037 eV.
Fig. 2. (a) Spectral heat flux for different gap sizes as a function of the radiation frequency corresponding to Fig. 1(a). The solid curves correspond to the total contributions and the dotted lines denote the contributions of propagating waves only for separation $d=10$ nm$-2\mu$ m, while the magenta dashed curve denotes the result for two blackbodies. Additionally, the black vertical dashed curve and the blue arrow represent the maximum positions for blackbodies and propagating waves, respectively. (b) $k_\|-\omega$ dependence of the energy transmission coefficient $\xi$ between two silver monolayers for $d=50$ nm. The white dashed curve corresponds to the dispersion relations of plasmons in a single silver monolayer, and the black dashed curves correspond to the acoustic and optical plasmonic dispersion relations of the silver double layers by using Eq. (7).
The physical mechanism of enhanced heat transfer can be understood by analyzing of the energy transmission coefficient. Figure 2(b) gives $\xi (\omega ,k_\|)$ for two suspended silver monolayers. The large wave-vector ($k_\|\gg \omega /c$) indicates that the NFRHT is mainly contributed by the evanescent waves, originating from tightly confined plasmon polariton modes. Moreover, two distinct bands exist with larger tunneling probability. These two bands are associated with the coupled SPPs caused by the coupled silver monolayers. The dispersion curve of the combined system splits into optical and acoustic branches provided that two silver monolayer are sufficiently close [36], which can be derived from the equation $1-r_1r_2e^{2ik_zd}=0$ and governed by [37,38]
Next we turn to focus on the influence of film thickness on the NFRHT. Recently, atomically thin gold films [28] and silver films [39] are experimentally demonstrated down to 1 nm thanks to the advance of physical vapour deposition technique. These outstanding results show that the optical conductivity of metallic films varies greatly with thickness and doping. More importantly, since the physical mechanism of NFRHT between two thin films (evanescent TM modes) is different from that between thick films (evanescent TE modes), it is important to consider the effect of the number of layers $N$ from both theoretical and practical viewpoints. Figure 3(a) shows the normalized HTC at room temperature for two ultrathin metal films with different layer number $N$ at different gap sizes. We first notice that the NFRHT between the atomically thin silver systems far exceeds the blackbody limit. With the increase of $N$, the HTC decreases quickly when $N\leq$ 4 firstly, and then increases slightly for larger $N$, which makes silver monolayer a preferred candidate to enhance the NFRHT at room temperatures. This phenomena can be intuitively understood by examining the equivalent carrier areal density of metal films. Specifically, the $N$-layer conducting monolayer can be regarded as an equivalent monolayer with an equivalent sheet conductivity $N\sigma _0$[40], which indicate the equivalent carrier areal density $Nn_0d_{111}$ grows linearly with $N$, where $n_0=m_e\omega _p^2/4\pi e^2$ is the $s$-band bulk carrier density. Noting that high carrier areal density indicates higher frequencies of plasmonic response, and considering the fast decay of the term $\partial \Theta (\omega ,T)/\partial T$ at high frequency, the high frequency contribution of SPPs to the NFRHT appears to be very small, thus the contribution from SPPs decreases with the increase of $N$. To get insight into the roles of $N$, we show the spectral HTC for the gap distance $d=50$ nm. As shown in Fig. 3(b), the peak of near field spectral heat flux appears at around 0.08 eV and the magnitude $declines$ quickly when $N\leq$3, thereby reducing the HTC at first. However, a new peak appears at around 0.02 eV for $N\geq$3, which rises quickly with the increase of $N$. This contribution dominates the NFRHT in thick metal films, originating from frustrated total internal reflection waves that are evanescent in the vacuum gap but propagate inside the metal films. Because the peaks originated from SPPs are far broader than the peaks from the TE evanescent modes, the total HTC in silver monolayer is still larger than that in thick silver films. Incidentally, in the semi-infinite bulk silver limit ($N\rightarrow \infty$), the contribution from frustrated total internal reflection wave is peaked at 4.5 meV.
Fig. 3. (a) Normalized HTC at room-temperature for the two atomically thin silver films with different number of layers $N$ at different gap sizes $d$. (b) Spectral heat flux for different number of layers $N$ with $d=50$ nm. The magenta curve corresponds to results between blackbodies, and the arrow denotes the spectral maximum for semi-infinite bulk silver separated by the gap size $d=50$ nm.
It is known that ultra-thin metal films will experience strong surface scatter and percolation resistance, indicating that their electrical resistance sharply increases when the thickness goes down to few layers as previous studies have indicated [41,42]. Even with the excellent crystal quality of metal films, the sheet resistance has been a factor of 2 larger than the estimated value based on the bulk resistivity. Seeded atomically thin gold films has a sheet resistance of 1.5 k$\Omega /\Box$ for $t$=1 nm, which is comparable to single-layer graphene, whereas for $t$=3 nm it substantially decreases to 74$\Omega /\Box$ [28]. Similar phenomena also occur in atomically thin silver films [39]. As a result, the electrical scattering time $\tau$ in atomically thin metallic film will appear two orders-of-magnitude smaller than in bulk metal, which limits its applications based on the propagating SPPs. Fortunately, the high-resistivity in few-layer metal films will not decrease the NFRHT but greatly enhance it, which makes ultrathin metal films a very good candidate in heat management. Figure 4(a) illustrates the HTC as a function of scattering rate $\gamma$ for silver monolayer. Clearly, the HTC increases dramatically with the damping rate. Take $d$=50 nm as an example, for $\gamma =0.021$ eV, the silver monolayers have a normalized HTC of 130, whereas for $\gamma =2$ eV it substantially increases to 4600, which is comparable to single-layer graphene sheets [43] and single-layer BP sheets [19], although the surface carrier density in silver monolayer far outweighs that in graphene or BP sheets.
Fig. 4. (a) Normalized HTC at room-temperature for the two monolayer silver sheets with different damping rates $\gamma$ at different gap sizes $d$. (b) Normalized HTC as a function of doping charge density at different gap sizes $d$, here $\gamma =1$ eV is adopted.
We consider next the effect of electrical doping. One of the exciting advantages of silver monolayer compared with the bulk silver is the efficient electro-optic tunability of the former. In metals, the bulk carrier density $n_0$ is so large that the additional doping charge density $\Delta n$ adding up to the undoped $s$-band density $n_0$ will not remarkably modify the total charge density of silver. Nevertheless, the situation will change for silver monolayer, where the electron areal density $n_0d_{111}\approx 1.44\times 10^{15}/\textrm {cm}^2$ is rather close to the $s$-band electron areal density in neutral monolayer silver $4/\sqrt {3}a_0^2\approx 1.38\times 10^{15}/\textrm {cm}^2$. Here, the doping charge $\Delta n$ is introduced by changing the bulk plasma frequency to $\omega _p=\sqrt {(4\pi e^2/m_e)(n_0+n)}$ in Eq. (1). Now, similar to $N$, the addition of a moderate amount of doping electrons ($\approx \pm$10% of $n_0$) results in the decline of HTC as expected. Obviously, the injection of similar amounts of holes will increase the HTC. However, due to the high carrier areal density, the modulation of HTC based on electrical doping is not very remarkable.
Technically, suspended graphene and BP sheets can be realized in the experiments for their mechanical and thermal stability [44,45]. Monolayer silver sheets, however, have to be deposited on a substrate in real experiments. Except for increasing the refractive index contrast, the effect from the dielectric permittivity of the substrate on the plasmons supported in ultrathin films can be approximately described by a homogenous environment ($\epsilon _s$ + 1)/2 [23]. The substrate effect for NFRHT with different electron scattering rates is demonstrated in Fig. 5(a). As the permittivity of the dielectric substrate (non-polar and non-dispersive materials) $\epsilon _s$ increases, the peaks of high resistant silver monolayer undergo a redshift, and the amplitudes of spectral HTC decline remarkably. As a result, the total HTC shown in Fig. 5(b) declines sharply for high scattering rate when the permittivity is increased. It is worth noting that for low loss sheets, the HTC grows up with the increase of permittivity. In conclusion, the high index substrate compensates the effect of electron scattering on the NFRHT.
Fig. 5. (a) Spectral heat flux for silver monolayers placed on top of semi-infinite substrates with different permittivity $\epsilon _s$ and electron scattering rates $\gamma$. (b) The substrate permittivity dependent HTC for different $\gamma$. Here $d=10$ nm is adopted.
3. Conclusion
In summary, we have demonstrated that the radiative heat transfer in a system composed of two ultrathin metallic layers at room temperature is greatly enhanced in the near-field region. Compared to thick metallic films, the plasmon coupling rather than the inertial frustrated total reflection dominates this enhancement of heat transfer. Moreover, the performances of NFRHT due to thickness of layers and electrical doping are examined as well. Because of the different physical origins of NFRHT for thick and thin metallic films, the NFRHT is weaker between ultrathin silver films with greater number of layers at a small gap size, while this trend reverses for thicker metallic films because of the higher magnetic local density of electromagnetic states near the metallic interfaces. Finally, the scattering rate, doping electron density, separation distance and substrate effect are demonstrated as well. The high sheet resistance of ultrathin silver films due to strong surface scatter and percolation resistance increases the normalized HTC from 130 to 4600, which is on the same order or even larger than that between single-layer graphene sheets and single-layer BP sheets. Our results pave the way to apply monolayer metallic sheets for active thermal management between the hot and the cold sides at the nanoscale.
Funding
National Natural Science Foundation of China (11604283, 11604282, 91750204, 11774185, 11504184, 61775106); National Key R&D Program of China (2017YFA0303800, 2017YFA0305100); 111 Project (B07013); Education Department of Henan Province (16A140048); Natural Science Foundation of Tianjin City (18JCQNJC02100); Fundamental Research Funds for the Central Universities; Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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