1. Introduction

The study of both experimental and theoretical behavior of near-field radiative heat transfer (NFRHT) at a distance smaller than the thermal wavelength of radiation was investigated in the past decades [1–7]. The enhancement mechanism of NFRHT is due to the strong coupling between surface phonon polaritons (SPhPs) or surface plasmon polaritons (SPPs) [8,9], or hyperbolic surface plasmon polaritons supported by graphene meta-surface [10]. Owing to its numerous applications for heat flux modulation [10–12], near-field thermophotovoltaics [13], energy conversion and management [14], NFRHT was widely investigated between two plates [8,15], sphere-plate [16], two-sphere [17], gratings [18], etc.

Near-field thermal rectification, also known as thermal diode, is a two-terminal device with the thermal conductance dependent on the direction of heat flux. It has been predicted in the systems including the asymmetric interface with different materials [19], dissimilar materials having temperature-dependent properties [8,9,15] or asymmetric geometric structures [20]. For practical applications, highly efficient thermal rectification and/or highly NFRHT is of much interest. For instance, Otey et al. [15] proposed the thermal diode made by SiC-3C and SiC-6H plates, relying on the coupling and decoupling of their SPhPs at different temperatures and frequencies. Joulain et al. investigated high heat flux between two bodies when the exchanging temperatures of the two bodies are from ambient to high temperatures or in a reverse manner, and large rectification was interpreted due to the weakening of the surface polariton with increasing temperature [8].

On the other hand, composite metamaterials provide us more adjustable parameters to enhance NFRHT and/or the thermal rectification. In general, these composite metamaterials are designed or fabricated using nanoparticles embedded in a host (or dielectric) medium or a planar substrate. Zhang et al. [21] studied the enhancement of NFRHT between composite nanostructures, and they demonstrated that the significant enhancement of NFRHT results from thermally excited SPhPs, and SPPs, and even hyperbolic phonon polaritons. In addition, Zhou et al. [22] proposed a thermal diode using a single material (i.e. 3C-SiC nanoporous and 3C-SiC plates) by adopting the effective medium theory (EMT) and obtained rectification coefficient above 0.6 with a volume fraction of 0.25. Ogundare et al. [23] proposed a thermal rectifier using composite metamaterials and a rectification coefficient above 0.8 was achieved.

With the development of science and technology, the tip-sample distances can be down to a few nanometers [24]. In order to fill the gap between experimental and theoretical results of near-field heat transfer, the nonlocal description of the permittivity was introduced [24]. The spatial dispersion or nonlocality means that the permittivity (or the permeability) of the material is dependent both on the incident frequency $\omega $ and the wave vector k. Later, some scientists included the nonlocal effect of the material in the study of NFRHT between homogeneous structures. Chapuis et al. [25] investigated the effects of spatial dispersion/nonlocality in NFRHT between two parallel metallic surfaces. They showed that the radiative heat flux between two metallic plates saturates when the gap size reaches a distance equivalent to skin depth. Singer et al. [26] studied NFRHT between two n-doped semiconductors using Lindhard-Mermin nonlocal permittivity model. They observed that the evanescent heat transfer coefficient saturates at a shorter distance, and the nonlocal effect is significant when the distance is shorter than the Thomas-Fermi length for high doped concentration. Venkataram et al. [27] investigated the thermal radiation between molecules and macroscopic bodies at the mesoscopic scales. They uncovered phonon polaritons can be strongly influenced by the nonlocal response to electromagnetic fluctuations. For composite metamaterials, the size of particle inclusions is also about a few nanometers, and it is natural to expect the existence of the nonlocal properties of nanoparticle inclusions. However, the NFRHT between composite metamaterials containing nonlocal nanoparticles has not been considered. Due to the interaction between the nonlocality and the inhomogeneity, NFRHT from such kind of system may be of much interest.

In this paper, we study the NFRHT between a semi-infinite SiO₂ (bottom) plate and a semi-infinite plate with composite metamaterials containing semiconductor nanoparticles randomly embedded in the host medium. In this connection, we introduce the nonlocal permittivity $\varepsilon ({\omega ,k} )$ of nanoparticle inclusions, and adopt nonlocal full-wave electromagnetic scattering theory to derive the effective permittivity of the composite metamaterials [28,29]. With the aid of fluctuational electrodynamics theory, we take one step forward to investigate the performance of NFRHT. We shall observe the strong coupling between the localized surface plasmon polaritons (LSPPs) in the composite metamaterials and surface phonon polaritons (SPhPs) (excited between polar dielectric plate and air). In addition, we analyze the contributions of the nonlocality and the volume fraction of semiconductor nanoparticles to the near-field strong coupling effects. Our work provides us an original coupling mechanism to control heat flux and enhance the thermal rectification, which shows potential applications in NFRHT.

2. Theoretical model

Let us consider the near-field thermal radiation between a semi-infinite (top) plate with composite metamaterial and a semi-infinite SiO₂ (bottom) plate of permittivity ${\varepsilon _2}\; $ separated by a vacuum gap d. We aim at the study of near-field thermal radiation for such a system. In the forward temperature biased scenario, the composite metamaterial is maintained at a high temperature ${T_\textrm{H}}$ and the SiO₂ plate is at a low temperature ${T_\textrm{L}}$, and one yields the forward radiative heat flux ${Q_f}$. In the reverse case, the temperature of the two semi-infinite plates is swapped, and the reverse radiative heat flux ${Q_r}$ can be achieved. The composite metamaterial consists of semiconductor (InSb) nanoparticles with radius a and the volume fraction f randomly dispersed in the host medium of permittivity ${\varepsilon _s}$. For such a composite metamaterial, we take into account the effect of nonlocality or spatial dispersion due to the small size of the InSb nanosphere. In this connection, the permittivity of the nonlocal InSb nanoparticles consists of the transverse (${\varepsilon _T}$) and the longitudinal (${\varepsilon _L}$) parts, given by [28–30],

In what follows, we concentrate on the radiative heat flux and the rectification efficiency $\eta $ to characterize the thermal diode performance.

2.1 Near-field thermal rectification efficiency

For the structure shown in Fig. 1, since the radiative heat flux in the forward-biased scenario is ${Q_f}$ with the top plate at a high temperature ${T_\textrm{H}}$, and the reverse scenario is ${Q_r},$ the characteristic of thermal diode performance is measured by its rectification efficiency [8,23,32],

According to the fluctuational electrodynamics theory for a two-body system, the net heat flux of the system ${Q_f}$ or ${Q_r}$ is defined as [32],

In Eq. (6), ${r_{1\alpha }}\; \textrm{and}\; {r_{2\alpha }}$ represent Fresnel reflection coefficients, which have the form,

 

Fig. 1. Schematic of two semi-infinite plates at a vacuum distance d maintain at low and high temperatures ${T_\textrm{L}}$ and ${T_\textrm{H}}$, respectively.

Download Full Size | PDF

2.2 Effective permittivity of composite metamaterials

The composite metamaterial is composed of nonlocal semiconductor spherical nanoparticles with volume fraction f, randomly embedded in the host medium. To study the effective permittivity of the composite, we adopt nonlocal effective medium theory [28,29,33], which deals with the scattering problem of coated spheres with nonlocal semiconductor core of radius a and the host shell of radius b embedded in an effective medium with permittivity ${\varepsilon _{\textrm{eff}}}$. Note that when the nonlocality is taken into account, both longitudinal and transverse waves exist in the nonlocal semiconductor nanoparticles. In detail, the wave vector of the longitudinal electromagnetic wave ${k_L}$ propagating in the nonlocal semiconductor nanoparticles is determined by ${\varepsilon _L}({\omega ,{k_L}} )= 0$, while the wave vectors of the transverse mode in the core, the shell, and the host effective medium obey the dispersion relation $k_{s,\; \; T}^2 = \; {k_0}^2{\varepsilon _{s,\; T}}(\mathrm{\omega } )\mu (\mathrm{\omega } )$, and $k_{\textrm{eff}}^2 = \; {k_0}^2{\varepsilon _{\textrm{eff}}}\; {\mu _{\textrm{eff}}}$ with $\mu (\omega )= 1$ and ${\mu _{\textrm{eff}}} = 1$.

Here, the governing equations for the fields which will be expanded in spherical unit vectors $({e_r}$, ${e_\theta }$, and ${e_\phi })$, are

On the other hand, the electric fields related with the longitudinal and transverse waves excited in the nanosphere are written as,

By applying the boundary conditions for the outer radius (i.e. $r = b$) and inner radius (i.e. $r = a$) of the core-shell nanosphere, one gets the scattering coefficient of the electric field (${b_n}$),

For the local case, Eq. (15) is naturally reduced to the classical Maxwell-Garnett theory with the form,

Here, we would like to mention that Eq. (15) and Eq. (16) are valid for the long-wavelength limit ${k_{\textrm{eff}}} \cdot a \ll 1$ and ${k_0} \cdot a \ll 1$. In general, the particle size a needs to be much smaller than the operating wavelength. On the other hand, $a < d \ll \lambda $ must be satisfied when investigating the near-field thermal radiation between composite metamaterials [21,34,35]. Indeed, Eq. (16) gives the size-independent effective permittivity. Beyond the long-wavelength limit, the effective permittivity with Eq. (16) shall be dependent on the radius of nanoparticles explicitly even though the nonlocality is not taken into account.

3. Results and discussion

We will start with the analysis of the effective permittivity ${\varepsilon _{\textrm{eff}}}$ against the angular frequency $\omega $ by considering nonlocal effects, as shown in Fig. 2. For the nonlocal case, it is observed that as the radius of semiconductor nanoparticles is increased, the real part of the effective permittivity Re(${\varepsilon _{\textrm{eff}}}$) always exhibits a ripple-like line shape, while the resonant peak is red-shifted accompanied with the enhanced magnitudes for the imaginary part $\textrm{Im}({\varepsilon _{\textrm{eff}}})$. It is evident that for further increasing the size, the nonlocal effect can be neglected, and hence the results should tend to be the local case, as expected (see Fig. 2(a) and (b)). For the local case, the effective permittivity is independent of the radius indeed under the long-wavelength approximation limit. As far as the temperature effect is concerned, the decrease in temperature leads to both the decreased magnitude and the red-shift of the resonant peak (see Figs. 2(c) and (d)).

 

Fig. 2. The effective permittivity ${\varepsilon _{\textrm{eff}}}$ against angular frequency $\omega $ for real part (a) and imaginary part (b) at temperature $T = 450\textrm{K}$, $f = 0.1$ for local and nonlocal cases with different a. The effective permittivity ${\varepsilon _{\textrm{eff}}}$ against angular frequency $\omega $ for real part (c) and imaginary part (d) at different temperatures with a = 25 nm. The inset figure in (a) is the permittivity of SiO₂ with several phonon modes [8].

Download Full Size | PDF

Next, we show the net spectral heat flux, denoted by ${q_\omega }$, and contour plots $q({\omega ,{k_p}} )$ in Fig. 3. Figures 3(a) and (b) depict ${q_\omega }$ for the forward and reverse biased scenarios, respectively. To elucidate on the physical mechanism, the parameters $d = 30\textrm{nm}$, $f = 0.1$, $a = 25\textrm{nm}$, and ${T_\textrm{L}} = 300\textrm{K}$ are maintained unless otherwise stated. For the forward scenario as shown in Fig. 3(a), we use different vertical lines to represent the frequency positions of the resonant modes at different temperatures. For local composite plate (blue dotted line) at ${T_\textrm{H}} = 600\textrm{K}$, its LSPP takes place at $\textrm{Re}({{\varepsilon_{T}}} )={-} 2$, marked with a black dashed line. For nonlocal composite metamaterials (lines red-triangle and -square symbols), the LSPPs at ${T_\textrm{H}} = 450\textrm{K}$ and $600\textrm{K}$ are determined by $\textrm{Re}\{{P{{[{{k_s}a \cdot {y_1}({{k_s}a} )} ]}^\mathrm{^{\prime}}}{\varepsilon_{T}}} \}= \textrm{Re}\{{{y_1}({{k_s}a} ){\varepsilon_{h}}} \}$ (i.e. by equating Eq. (15b) to be zero), are marked with black-triangle and -square symbols, respectively. The peak positions of ${q_\omega }$ coincide with these black lines, suggesting that they are caused by LSPPs of the semiconductor nanoparticles. Conversely, in the reverse scenario, the local and nonlocal composite metamaterials are at the receiving temperature ${T_\textrm{L}}$. As shown in Fig. 3(b), since ${\varepsilon _2}$ does not change with ${T_\textrm{H}}$, the reverse heat fluxes ${q_\omega }$ are almost the same for different ${T_\textrm{H}}$ and their peaks are all at $\omega = 0.5 \times {10^{14}}\; \textrm{rad}\,{\textrm{s}^{ - 1}}$, which is just the LSPP frequency position for the nonlocal semiconductor nanoparticles at ${T_\textrm{L}}$. Note that the reverse heat fluxes are much smaller than those of the forward bias, resulting in thermal rectification.

 

Fig. 3. The net spectral heat fluxes for local, and nonlocal at different ${T_\textrm{H}}$ for forward-biased (a) and reverse-biased (b) scenarios. The contour plots $q({{k_\rho },\omega } )$ for forward-biased scenarios (c) ${T_\textrm{H}} = 450\textrm{K}$ and (e) ${T_\textrm{H}} = 600\textrm{K}$ and their reverse-biased counterpart (d) and (f) for nonlocal conditions, respectively.

Download Full Size | PDF

To better understand the physical mechanism of the thermal rectification effect in our system, we plot the $q({{k_\rho },\omega } )$ with angular frequency $\omega $ and normalized wave vector ${k_\rho }$, as shown in Figs. 3(c)-(f). In these figures, the resonant frequency of the LSPPs supported by nonlocal semiconductor nanoparticles is marked with solid black lines. Meanwhile, it can be seen from the inset figure of Fig. 2(a) that two SPhPs modes, which are ascertained by $\textrm{Re}({{\varepsilon_2}} )={-} 1$, are supported in SiO₂ with angular frequencies ${\omega _1} = 0.9 \times {10^{14}}\; \textrm{rad}\,{\textrm{s}^{ - 1}}$ and ${\omega _2} = 2.2 \times {10^{14}}\; \textrm{rad}\,{\textrm{s}^{ - 1}}$ and their frequencies are marked by white solid lines in Figs. 3(c)-(f). In Fig. 3(c) when the composite metamaterials are at ${T_\textrm{H}} = 450\textrm{K}$, it is found that the solid white line and solid black line are very close to each other, and the energy flow is concentrated on near them, indicating that the forward heat flux is caused by the coupling between LSPP and SPhP at ${\omega _1}$. In Fig. 3(e), the LSPP mode supported by semiconductor nanoparticles is located at $\omega = 1.5 \times {10^{14}}\; \textrm{rad}\,{\textrm{s}^{ - 1}}$ at ${T_\textrm{H}} = 600\textrm{K}$ while the SiO₂ plate does not support SPhPs near this frequency. However, there is a weak phonon mode of SiO₂ at this frequency, which leads to the occurrence of the forward heat flux peak in this case. Following the above principle, due to the existence of coupling modes, energy can propagate through the evanescent wave. In Figs. 3(d) and (f), the heat flux almost exists in the range ${k_\rho }c/\omega < 1$ (propagating wave). The nonlocal composite plates at 300 K support LSPPs at $\omega = 0.5 \times {10^{14}}\; \textrm{rad}\,{\textrm{s}^{ - 1}}$ in the reversed scenario, which are quite far from their corresponding coupling positions ${\omega _1}$ and ${\omega _2}$. As a consequence, one yields the decoupling between these modes, resulting in less energy transmission by propagation wave as shown in Figs. 3(b), (d) and (f).

Then, we concentrate on the analysis of the thermal rectification efficiency in Fig. 4. Figure 4(a) shows the variation between rectification efficiency as the function of the high temperature ${T_\textrm{H}}$ at volume fraction $f = 0.01$. We observe that the rectification efficiency for the nonlocal case is higher than the one for the local case when ${T_\textrm{H}}$ range from 300K to 500K. The high rectification efficiency at these temperatures is quite interesting due to the fact that small temperature difference (since ${T_\textrm{L}} = 300\textrm{K}$) is enough to get high rectification effect. This behavior can be interpreted as follows. As seen in the inset figure in Fig. 4(a), LSPPs frequencies excited from the nonlocal case are closer to the resonant coupling frequency ${\omega _1}$ than the local case in the low temperature region, and hence the coupling between the LSPPs and SPhPs is stronger to enhance the rectification efficiency. In Fig. 4(b), we plot the rectification efficiency as a function of separation distance d. The rectification falls off due to the exponential decay of the surface polaritons. Even for this, the rectification efficiency for the nonlocal case is quite larger than the one for the local case for d less than 100nm.

 

Fig. 4. (a) Rectification efficiency for nonlocal and local against high temperature ${T_\textrm{H}}$. (b) Rectification efficiency for nonlocal and local at ${T_\textrm{H}} = 450\textrm{K}$ against the vacuum distance d.

Download Full Size | PDF

For the present system, the rectification efficiency can be further enhanced with increasing the volume fraction of nonlocal semiconductor nanoparticles. This is shown in Fig. 5. Figure 5 depicts the influence of volume fraction f on the rectification efficiency. The increase in the volume fraction f leads to the increase in the rectification efficiency. To one’s interest, the maximal rectification efficiency $\eta = 0.92$ occurs when ${T_\textrm{H}} = 420K$ with the volume fraction $f = 0.1$. A higher rectification efficiency can also be seen at higher temperature but a small temperature difference may be quite important for practical applications. Figure 5(b) shows the variation of $\eta $ with the separation d. For large volume fraction, high efficiency can be well kept as d is reduced. For instance, for $f = 0.1$, 50% efficiency can still be found even $d = 100\textrm{nm}$. This is quite similar as the one reported in Ref. [26]. Although at larger separations, the rectification efficiency here is lower than those between VO₂ phase transition materials [36], the use of composite metamaterials containing InSb nanoparticles allows us more degrees of freedom to tune the plasmonic properties of semiconductor dynamically, the volume fraction, and so on.

 

Fig. 5. (a) The rectification efficiency with different volume fraction f for the nonlocal case against high temperature ${T_\textrm{H}}$. (b) The rectification efficiency with different volume fraction f at ${T_\textrm{H}} = 450\textrm{K}$ against the vacuum distance d.

Download Full Size | PDF

 

Fig. 6. The variation of rectification efficiency against volume fraction f for both local and nonlocal conditions at ${T_\textrm{H}} = 450\; \textrm{K}$.

Download Full Size | PDF

In the end, we plot the variation of the rectification efficiency against the volume fraction in Fig. 6. We show that the rectification efficiencies for both local and nonlocal cases increase monotonically with increasing the volume fraction, and the rectification efficiency for the nonlocal case (denoted with red dashed line) is always larger than the one for the local case. Although the localized surface plasmon resonant frequency is weakly dependent on the volume fractions, large absorption can be found for the composite metamaterials with large volume fractions, which may lead to large rectification efficiency for large volume fractions.

4. Conclusion

In conclusion, we demonstrate the modulation of thermal rectification between nonlocal composite metamaterial containing semiconductor nanoparticles and polar dielectric plates. The enhancement of near-field thermal transfer results from the coupling of localized surface plasmon polariton (LSPP) in composite metamaterials and surface phonon polariton (SPhP) from the surface of SiO₂ plates. The introduction of the nonlocality is quite helpful to yield high thermal rectification efficiency at low high-temperature region. Moreover, the thermal rectification efficiency can be further enhanced with increasing the volume fraction of nonlocal semiconductor nanoparticles. We believe that beyond their potential applications in thermal management, our study may provide alternative insights into the design of thermal devices with composite metamaterials.