## Abstract

Thermal radiative transport yields unique thermal characteristics of microscopic thin films—wavelength selectivity. This chapter focuses on a methodology about adjusting the wavelength selectivity of thin films embedded with nanoparticles in the far‐field and near‐field regimes. For nanostructured layered media doped with nanoparticles, Maxwell‐Garnett‐Mie theory is applied to determine the effective dielectric function for the calculation of radiative thermal transport. The thermal radiative wavelength selectivity can be affected by volume fraction and/or the size of the embedded nanoparticles in thin films. To characterize wavelength selectivity and optical property of nanostructured materials, both real and imaginary parts of effective refractive index need to be analyzed. It has been shown that the nanoparticles made of polar or metallic materials have different influence on thermal radiative wavelength selectivity of microscopic thin films.

### Keywords

- thermal radiation
- wavelength‐selective
- far field
- near field
- nanostructured layered media

## 1. Introduction

Most naturally occurring materials exhibit a broad range of emission spectrum. However, thermal and optical properties of nanomaterials and nanostructures are significantly different than that of bulk materials. They are the basis of development of selective thermal emitters and absorbers that are crucial for the development of solar cells and thermophotovoltaics (TPVs) [1]. Wavelength‐selective emitters also have a wide range of potential applications such as biosensors, chemical sensors [2, 3], thermal cooling, and thermal detectors [4]. It has been shown that one‐dimensional (1‐D) metallo‐dielectric periodic structures display great selective emission properties in the infrared and visible regions [5]. Multilayered structures of thin films (1‐D photonic crystals) of polar materials can also be used to develop selective emitters [6]. The property of multilayered structures to exhibit wavelength selectivity can be explained by the presence of surface phonon polaritons (SPhPs) for polar materials and surface plasmon polaritons (SPPs) for metals [5, 7]. It has also been demonstrated that two‐dimensional (2‐D) or three‐dimensional (3‐D) photonic crystals can be used to develop selective emitters [8, 9]. However, thin‐film‐layered structures are easy to design and fabricate. Calculation of their emission spectra is also relatively simple.

Wavelength selectivity of 1‐D photonic crystals is a far‐field phenomenon. When the distance between two objects is of the order of the dominant thermal wavelength, the radiative heat transfer is enhanced to many orders of magnitude due to the coupling of surface waves and is referred to as near‐field thermal radiation [10]. If the materials support SPhPs or SPPs, the near‐field radiative flux can be shown to be inversely proportional to the square of the distance. The enhancement of heat transfer does not take place at all wavelengths but only at specific wavelengths [10]. This wavelength selectivity in the near field is exhibited by thin films as well as bulk materials. Wavelength selectivity in the near‐field limit is due to the coupling of SPPs or SPhPs across the two surfaces [10].

While many articles dedicated to the design and fabrication of selective emitters can be found in the literature, the use of nanoparticles specifically for the application of selective emitters is relatively sparse [11]. Optical properties of materials doped with nanoparticles have been investigated before [12, 13]. Experimental and analytical study of thermal coatings doped with nanoparticles such as Gonome et al. [14, 15] can also be found in literature. However, emissive properties of nanoparticles embedded thin films have not been studied in detail to the best of our knowledge. This chapter presents the multilayered structures, which contain nanoparticles (NPs) doped into the thin films, which are suitable for any of the potential applications in both the far field and the near field. In this chapter, we investigate a methodology that can be used to develop selective thermal emitters for a desired wavelength band. Ideally, one may want to develop a selective emitter for specific wavelength band as per the requirements. As the emission spectrum displays peaks at the wavelengths, which are characteristics of the refractive index of the material, changing the thickness of the film allows control over only a narrow spectral band. We propose to dope the top layer (thin film) with nanoparticles to change the dielectric properties of the material. The usual Maxwell‐Garnett equation for effective medium approximation is often employed for such an analysis disregarding the sizes of doped materials [16]. Here, we apply the Maxwell‐Garnett‐Mie formulation [17] for effective medium approximation to calculate the dielectric function of a composite that consists of a host material embedded with nanoparticles of various sizes and volume fractions, and extend the same approach to calculate radiative heat transfer for thin films doped with nanoparticles. Thin‐film structure with nanoparticles would be easy to fabricate as submicron thin films embedded with nanoparticles have been fabricated before [18, 19]. We aim to study the effect on the wavelength selectivity of thin films due to combination of surface polaritons of the films and the nanoparticles and their effects in the near‐field radiative heat transfer and spectral heat flux. We consider hypothetical cases of thin film embedded with nanoparticles although the fabrication of these particular NP‐embedded films discussed here is relatively unknown. We choose SiC and polystyrene (PS) as the host materials (for thin films). SiC is chosen as a host because it has high permittivity in the infrared region and supports SPhP. Polystyrene is chosen because it does not support either SPPs or SPhPs. BN, which supports SPhP, and Au, which supports SPPs, are picked for the material of inclusion (NPs).

The structure of this chapter is as follows. In Section 2, we present the theoretical background and analytical expressions used for the calculation of emissivity of thin‐film structures, calculation of heat transfer between closely placed half spaces, and the application of Maxwell‐Garnett‐Mie theory. In Section 3, we discuss the results of our calculations obtained using the formulations outlined in Section 2. Subsequently, the ideas and conclusions of this chapter are narrated in Section 4.

## 2. Theoretical fundamentals

Consider a structure having *N*‐layer media having (*N* - 1) interfaces. By solving the boundary conditions at the interfaces, one can obtain the expression for the generalized reflection coefficient at the interface between region *i* and region *i* + 1 and is given by [20]

where *i* and *i* + 1, and *i* + 1 and *i* + 2, *p*) refers to transverse electric (or magnetic) polarization, z = ‐ *i*th interface. *z*‐component of the wave vector in medium *i* wherein *i* as a function of angular frequency *ω*, *c* is the speed of light in vacuum, and

where

where

Far‐field radiative heat transfer is described by Planck's law of thermal radiation [22]. This description, however, does not account for the presence of evanescent (surface) waves that dominate only near the boundaries. The expression of radiative transfer between closely spaced objects can be derived using dyadic Green's function formalism [23], and is given by

where *ω* and temperature *T*, *h* is the reduced Planck constant, and *ω*) corresponds to the spectral transmissivity in radiative transfer between media 1 and 2 separated by distance *L* and is expressed as [23]

where *khz* effective reflection coefficients of the two half spaces (calculated in the absence of other half space) and is the *z*‐component of wave vector in vacuum. The first term in Eq. (5) corresponds to propagating waves, whereas the second term describes the thermal transport due to evanescent waves, and its contribution is significant only for small values of gap *L*.

Clausius‐Mossotti equation for the effective dielectric function, of the composite medium containing nanoparticle inclusions in a host material, is given by [24, 25]

where *r* and *f* are the radius and volume fraction of nanoparticles, respectively. The size‐dependent extension of Maxwell‐Garnett formula can be obtained by deriving an expression for electric dipole polarizability using Mie theory [17]

where

where *x*) = *x* *x*) and *x*) = *x**x*), where

Effective medium approximation method is applicable when average distance between inclusions is much smaller than the wavelength of interest. If the dielectric inclusions of radius *r* can be imagined to be arranged in simple cubic lattice of lattice constant *a*, the condition for validity of the effective medium approximation is *a* > 2*r*. Where

## 3. Results

The dielectric function is related to real (*n*) and imaginary (*κ*) parts of refractive index as *n + jκ*. For SiC and BN, the dielectric function has the form as [29, 30]

where *γ* is the damping constant. For SiC, the constants *γ* are equal to 6.7, 9.83 × 10^{-2} eV, 0.12 eV, and 5.90 × 10^{-2} eV, respectively. The values of ^{-2} eV, respectively. Data for the bulk gold (Au) are taken from Johnson and Christy [31]. Figure 2(a) considers the case of SiC film doped with NPs of BN. SiC film of 0.4 µm is on the top of Au film of 1 µm deposited on a substrate. The effect of change in NPs volume fraction (*f*) is studied. The volume fraction of BN nanoparticles is changed from 0% to 30% while maintaining the radius of 25 nm. Thin film of pure SiC exhibits emission peaks at *n*) is large while the imaginary part of refractive index (κ) is small [6]. These peaks are attributed to the presence of SPhPs, and the characteristic wavelengths of the dielectric function of SiC. The appearance of new peaks upon 5% inclusion of BN nanoparticles has been observed at *λ* ≈ 8.5 µm and *λ* ≈ 11.5 µm. When the volume fraction of NPs is increased further, each of these peaks splits into two giving rise to a total of six peaks. Locations of these peaks do not correspond to the characteristic wavelengths of BN (*n* is large while imaginary part of refractive index is small, namely

These wavelengths correspond to the additional peaks. While the additional peaks are at the location of the characteristics of the refractive index of the mixture, it is interesting to note that peaks at ≈10.33 µm and ≈12.98 µm have no or little shift even at large volume fraction of 30%, because they are characteristic wavelengths of the host. Inclusion of BN leads to new SPhP leading to new peaks. Figure 2(b) shows the effect of Au nanoparticles in SiC thin film. When particle size is small, especially when the size is comparable to the mean free path of the free electrons, confinement effects become significant [25, 33]. The optical properties of metallic nanoparticles have shown size dependence [34]. We utilize a size‐dependent dielectric function for Au nanoparticles of radius *r* that takes care of electron scattering, which is given by [35]

where ^{6} m/s, respectively. The proportionality constant A that depends on the electron‐scattering process at the surface of nanoparticles is assumed to be unity. Volume fraction is varied from 0% to 30% while NPs radius is maintained constant (25 nm). Multiple oscillatory peaks are seen in the lower wavelength region upon the addition of Au nanoparticles. A shift in the original peak of SiC at ≈13 µm is seen when volume fraction is large (30%). While the presence of a peak at ≈10.33 µm and a peak at ≈13 µm can be related to *n* and *κ* plots shown in Figure 3(a) and (b), multiple peaks between 0.35 and 8 µm cannot be explained using the refractive index characteristics. While the change in refractive index is seen around 500 nm which corresponds to surface plasmon resonance of Au, one may expect a peak around 500 nm. Multiple peaks are observed instead. Figure 2(c) and (d) shows emission spectra of polystyrene thin film doped with BN nanoparticles and Au nanoparticles, respectively. The dielectric function of PS is in the form of [36]

where the parameters

The appearance of emission peaks at the locations of

Figure 5 shows the effect of NPs size on the emission spectra. In Figure 5(a), SiC film of 0.4 µm is doped with BN nanoparticles and volume fraction of BN NPs is maintained constant at 10% and the radius is varied from 1 to 50 nm. The majority of emission spectrum shows no effect of BN particle size. However, the effect of size is noticeable at wavelengths less than 1 µm. This is due to the fact that Mie scattering becomes important at shorter wavelengths giving rise to higher peaks for larger particles. Figure 5(b) presents the calculation of emissivity for 0.4‐µm thick polystyrene (PS) film doped with Au NPs. The volume fraction of NPs is fixed at 10% and the particle size is changed from 1 to 50 nm. Unlike Figure 5(a), Figure 5(b) shows a strong influence of particle size on the emissivity. While the spectrum in the visible region shows a negligible response to particle size, gold NPs greatly influence the near‐infrared region between 1 and 4 µm. As the NP size is increased from 1 nm, the emissivity peaks reduce in magnitude, showing smaller peaks for 10 and 25 nm. Emissivity for larger particles of 50 nm, however, is increased again and is comparable to that of NPs of 1 nm. This is due to the presence of two counteracting phenomena here. First is the change in dielectric function of Au NPs leading to decreased emissivity of larger particles and the second being Mie scattering of electromagnetic (EM) waves in the host causing an increased emissivity of larger particles.

Next, we present the effect of the doped nanoparticles on radiative heat transfer. We analyze radiative heat transfer between two identical multilayered structures at 300 and 301 K as shown in Figure 1(b). Each structure has a top layer of 0.4 µm deposited on 1 µm of Au. The top layer is doped with nanoparticles of 25 nm and different volume fractions. Figure 6 shows radiative heat flux versus distance between the structures and the normalized spectral density (defined as the ratio of *dq/dω* to the maximum value over the range of wavelengths considered) at a distance of 100 nm is shown in the inset of figures. Consider a structure with SiC layer doped with BN nanoparticles. Figure 6(a) shows very little change in overall heat transfer. While thin film of pure SiC shows nearly monochromatic heat transfer, selectivity is seen at additional bands of wavelength. These locations are wavelengths where the effective refractive index of the mixture becomes zero (Figure 3(a) and (b)). While the locations of the new peaks depend on the volume fraction, the peak corresponding to the host material is relatively unchanged. In Figure 6(b), SiC film is doped with Au nanoparticles of radius 25 nm with different volume fractions. The change in total heat transfer characteristics is not significant with the addition of nanoparticles. Selectivity is observed near *λ* =

## 4. Conclusion

We have demonstrated that nanoparticles influence the emission spectra of the multilayered structures. Wavelength selectivity can be altered and controlled by the size and/or volume fraction of the NPs. The presence of NPs in a host material gives rise to an appearance of new emission peaks and a shift in the existing peaks in the emission spectra. When the metallic NPs are used, the effect of size is stronger as the dielectric function of metallic NPs has a strong dependence of particle size due to electron scattering. We have also shown that the volume fraction of the nanoparticles plays an important role in the near‐field radiative heat transfer. If the NPs support SPhP, wavelength selectivity of thin films in the far field is at the locations where the real part of effective refractive index of the mixture becomes zero or the imaginary part of refractive index is small while the real part of the index is large. If the material of inclusion supports SPPs, as in metallic nanoparticles multiple emission peaks are seen which cannot be related to *n*‐ and *κ*‐values of the mixtures. (Our observation is limited to the case where the host material is thin film.)

In the near field, for NPs supporting SPhPs or SPPs the heat transfer is nearly monochromatic around the wavelength at which *n* for the mixture becomes zero. It is observed that only SPhP supporting inclusions can influence the location of

## Acknowledgments

This work is partially funded by the Rhode Island STAC Research Grant under grant number AWD05085 and NIH RI‐INBRE Pilot Research Development Award supported in part by an Institutional Development Award (IDeA) Network for Biomedical Research Excellence from the National Institute of General Medical Sciences of the National Institutes of Health under grant number P20GM103430.