## Abstract

Microscopic thin films have shown wavelength selectivity in the context of radiative heat transfer. We propose a methodology to shift the wavelength selectivity in the desired location. This work deals with the far-field and near-field radiation from thin films embedded with nanoparticles. The calculations of emission spectra are performed using the Fresnel equations in the far-field limit, and using the dyadic Green’s function formalism for transmissivity between the closely spaced objects in the near-field regime. For the media doped with nanoparticles, an effective dielectric function using the Maxwell-Garnett-Mie theory is used to calculate emissivity and radiative heat transfer. It has been shown that the wavelength selectivity in the emission spectra can be influenced by varying the size and/or the volume fraction of nanoparticles. We characterize the wavelength selectivity using real and imaginary parts of the effective refractive index. We show that the influence of nanoparticles on wavelength selectivity is different depending on whether the particles are of polar materials or are metallic.

© 2015 Optical Society of America

## 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 development of solar cells and thermophotovoltaics (TPV) [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 region [5]. Multi-layered structures of thin films (1-D photonic crystals) of polar materials can also be used to develop selective emitters [6]. The property of multi-layered structures to exhibit wavelength selectivity can be explained by 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 wavelengths 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 design and fabrication of selective emitters can be found in the literature, 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 paper presents the multi-layered structures which contain nanoparticles (NPs) doped into the thin films which are suitable for any of the potential applications in both far-field and near-field. In this paper, we investigate a methodology which 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 nano-particles 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 near-field radiative heat transfer and spectral heat flux. We consider hypothetical cases of thin-film embedded with nanoparticles although fabrication of these particular NPs embedded films discussed here is relatively unknown. We choose SiC and Polystyrene 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 the paper is as follows. In section 2 we present the theoretical background and analytical expressions used for 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 the paper 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],

*i*and

*i*+ 1, and ${\tilde{R}}_{i+1,i+2}^{(\mu )}$ is the generalized reflection coefficient at the interface between the layer

*i*+ 1 and

*i*+ 2,

*µ*=

*s*(or

*p*) refers to transverse electric (or magnetic) polarization,

*z*=

*−d*is the location of the

_{i}*i*th interface. ${k}_{i,z}=\sqrt{{\epsilon}_{i}(\omega ){\omega}^{2}/{c}^{2}-{k}_{\rho}^{2}}$ is the normal

*z*-component of the wave vector in medium

*i*wherein

*ε*(

_{i}*ω*) is the relative permittivity of the medium

*i*as a function of angular frequency

*ω*,

*c*is the speed of light in vacuum and

*k*is the magnitude of the in-plane wave vector. With ${\tilde{R}}_{N,N+1}^{(\mu )}=0$, the above equation provides a recursive relation to calculate the reflection coefficients ${\tilde{R}}_{i,i+1}^{(\mu )}$ in all regions. The generalized transmission coefficient for the layered slab is given by

_{ρ}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 which 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

*ω,T*) = (

*ħω/*2)coth(

*ħω/*2

*k*) is the energy of harmonic oscillator at frequency

_{B}T*ω*and temperature

*T*,

*ħ*is the reduced Planck constant, and

*k*is the Boltzmann constant. The function

_{B}*T*

_{1}

_{→}_{2}(

*ω*) corresponds to the spectral transmissivity in radiative transfer between media 1 and 2 separated of distance

*L*and is expressed as [23]

*k*is the

_{hz}*z*-component of wavevector in vacuum. The first term in Eq. (5) corresponds to propagating waves, while 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, *ε _{eff}*, of the composite medium containing nanoparticle inclusions in a host material is given by [24, 25]

*ε*is the dielectric function of the matrix,

_{m}*α*is the electric dipole polarizability,

_{r}*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

*a*

_{1,r}is the first electric Mie coefficient given by

*ψ*

_{1}and

*ξ*

_{1}are Riccati-Bessel functions of the first order given by

*ψ*

_{1}(

*x*) =

*x j*

_{1}(

*x*) and ${\xi}_{1}(x)=x{h}_{1}^{(1)}(x)$ where

*j*

_{1}and ${h}_{1}^{(1)}$ are first order spherical Bessel functions and spherical Hankel functions of the first kind, respectively. Here, ‘′’ indicates the first derivative. ${x}_{m}=\omega r\sqrt{{\epsilon}_{m}}/c$ and ${x}_{np}=\omega r\sqrt{{\epsilon}_{np}}/c$ are the size parameters of the matrix and the nanoparticles, respectively;

*ε*being the dielectric function of nanoparticles.

_{np}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 *λ _{h}* >> a > 2

*r*. Where

*λ*is the wavelength in the host material [26]. We emphasize that since we use the approximation for thin-films doped with nanoparticles, its use may not be correct when the particle size is comparable to the thickness of the films. Also, as discussed in by Liu et al. [27], it can be argued that the use of effective medium theory (EMT) is questionable at nanoscale distances. Although such might be the case for the near-field calculations presented here, its detailed analysis is beyond the scope of this work and is left for future work. Despite of the limitations of Maxwell-Garnett-Mie theory and its application in the near-field regime, the results obtained should provide general trends and give considerable insight into optical properties of artificial materials. Further investigations by direct numerical simulation may be necessary to confirm the validity of EMT [28]. Moreover, these results will be constructive when judging the validity of the EMT in near-field by direct numerical calculations. We would like to keep these points open for speculation.

_{h}## 3. Results

The dielectric function is related to real (*n*) and imaginary (*κ*) parts of refractive index as
$\sqrt{\epsilon}=n+j\kappa $. For SiC and BN, the dielectric function has the form as [29, 30]

*ω*and

_{TO}*ω*are transverse and longitudinal optical phonon frequencies and

_{LO}*γ*is the damping constant. For SiC, the constants

*ε*

_{∞},

*ω*,

_{TO}*ω*and

_{LO}*γ*are equal to 6.7, 9.83 × 10

^{−}^{2}eV, 0.12 eV and 5.90

*×*10

^{−}^{4}eV respectively. The values of

*ε*

_{∞},

*ω*,

_{TO}*ω*and

_{LO}*γ*for BN are 4.46, 0.1309 eV, 0.1616 eV and 6.55

*×*10

^{−}^{4}eV respectively. Data for the bulk gold (Au) is 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

*µ*m is on the top of Au film of 1

*µ*m deposited on a substrate. Effect of change in NPs volume fraction (

*f*) is studied. Volume fraction of BN nanoparticles is changed from 0% to 30% while keeping the radius of 25 nm. Thin film of pure SiC exhibits emission peaks at ${\lambda}_{n}^{SiC}\approx 10.33$

*µ*m and ${\lambda}_{k}^{SiC}\approx 13$

*µ*m.

*λ*is the wavelength at which the real part of the refractive index becomes zero (zero-index material) [6, 32].

_{n}*λ*is the wavelength at which the real part of refractive index (

_{κ}*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. Appearance of new peaks upon 5% inclusion of BN nanoparticles has been observed at

*λ ≈*8.5

*µ*m and

*λ ≈*11.5

*µ*m. 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 ( ${\lambda}_{n}^{BN}\approx 7.6$

*µ*m and ${\lambda}_{\kappa}^{BN}\approx 9.8$

*µ*m). In addition, there exists a small shift in the emission peak at ${\lambda}_{\kappa}^{SiC}$. This suggests an interaction between SiC matrix and BN NPs. Consider the case with 30% inclusion of BN. Figure 3(a) shows that the mixture has two additional locations where the refractive index is zero ( ${\lambda}_{n1}^{mixture}\approx 8.5$

*µ*m and ${\lambda}_{n2}^{mixture}\approx 11.7$

*µ*m). Moreover, at two more points

*n*is large while imaginary part of refractive index is small, namely ${\lambda}_{\kappa 1}^{mixture}\approx 9$

*µ*m and ${\lambda}_{\kappa 2}^{mixture}\approx 11.1$

*µ*m 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

*µ*m and

*≈*12.98

*µ*m 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]

*ε*,

_{b}*ω*,

_{p}*v*and

_{f}*γ*

_{0}are the bulk dielectric function, the plasma frequency, the Fermi velocity of free electrons and the electron damping, respectively. The values of

*ε*are taken from Johnson and Christy [31]. The parameters

_{b}*ω*,

_{p}*v*and

_{f}*γ*

_{0}are taken to be 9.06 eV, 0.077 eV and 1.4

*×*10

^{6}m/s, respectively. The proportionality constant

*A*that depends on the 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 kept constant (25 nm). Multiple oscillatory peaks are seen in the lower wavelength region upon addition of Au nanoparticles. A shift in the original peak of SiC at

*≈*13

*µ*m is seem when volume fraction is large (30%). While the presence of a peak at

*≈*10.33

*µ*m and a peak at

*≈*13

*µ*m can be related to

*n*and

*κ*plots shown in Figs. 3(a) and 3(b), multiple peaks between 0.35

*µ*m to 8

*µ*m cannot be explained using the refractive index characteristics. While 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. Figures 2(c) and 2(d) show 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]

*f*,

_{i}*w*, and

_{i}*g*are, in the units of eV, given by

_{i}*f*= [14.6,96.9,44.4,136.9],

_{i}*w*= [6.35,14.0,11.0,20.1] and

_{i}*g*= [0.65,5.0,3.5,11.5], respectively. In case of BN, appearance of new peaks is quite similar to that in Fig. 2(a) and its relation with the refractive indices shown in Figs. 4(a) and 4(b) is obvious. Appearance of emission peaks at the locations of ${\lambda}_{\kappa}^{mixture}$ and ${\lambda}_{n}^{mixture}$ is evident. However, when polystyrene film is doped with Au nanoparticles we once again see multiple peaks produced in the region 0.35

_{i}*µ*m to 6

*µ*m which are not related to the refractive index characteristics. Since polystyrene does not support either of SPhPs or SPPs, interaction between SPPs of Au and surface polariton of the host are not responsible for the multiple peaks. We hypothesize that the origin of multiple peaks is due to interaction of SPPs of Au and the boundaries of the thin film. In case of SiC film doped with Au nanoparticles the shift in the peak of

*≈*13

*µ*m is due to interaction between SPPs of Au and SPhPs of SiC, in either cases (SiC and PS) the inclusion material does not produce new polaritons as seen in refractive index characteristics.

Figure 5 shows the effect of NPs size on the emission spectra. In Fig. 5(a), SiC film of 0.4 *µ*m is doped with BN nanoparticles and volume fraction of BN NPs is kept constant at 10% and the radius is varied from 1 nm to 50 nm. The majority of emission spectrum shows no effect of BN particle size. However, effect of size is noticeable at wavelengths less than 1 *µ*m. This is due to the fact that Mie scattering becomes important at shorter wavelengths giving rise to high peaks for larger particles. Figure 5(b) presents the calculation of emissivity for 0.4 *µ*m thick polystyrene (PS) film doped with Au NPs. The volume fraction of NPs is fixed at 10% and particle size is changed from 1 nm to 50 nm. Unlike Fig. 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 *µ*m to 4 *µ*m. As the NP size is increased from 1 nm the emissivity peaks reduce in magnitude, showing smaller peaks for 10 nm 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 EM waves in the host causing an increase emissivity of larger particles.

Next, we present the effect of the doped nanoparticles on radiative heat transfer. We analyse radiative heat transfer between two identical multilayered structures at 300 K and 301 K as shown in Fig. 1(b). Each structure has a top layer of 0.4 *µ*m deposited on 1 *µ*m of Au. The top layer is doped with nanoparticles of 25 nm and different volume fractions. Figure 6 shows radiative heat flux vs 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 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 (Figs. 3(a) and 3(b)). While the locations of the new peaks depends on the volume fraction, the peak corresponding to the host material is relatively unchanged. In Fig. 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* = *λ*_{1} *≈* 10.33 *µ*m as in Mulet et al. [10], which is one of characteristic wavelengths of SiC. Moreover, the inclusion of Au nanoparticles has only a small impact on the selectivity in the near-field limit and this can be related to the refractive indices of the mixture (Figs. 3(a) and 3(b)). When the top layer is polystyrene doped with BN nanoparticles (Fig. 6(c)), near-field heat flux is clearly dependent on volume fraction of the inclusion both in the near-field and far-field regime. Since polystyrene does not support SPPs/SPhPs, inclusion of BN clearly makes significant enhancement in heat transfer. The surface becomes selective at the wavelengths at which the real part of the effective refractive index becomes zero. When the PS film is doped with Au nanoparticles instead, the radiative heat transfer in Fig. 6(d), shows an increment with an increase in NPs volume fraction, both in the near-field as well as far-field limit. However, the normalized spectral density does not show any selectivity in the near-field. In summary, the wavelength selectivity of thin films in the near-field can be related to its effective refractive index in all four cases. This is logical as the selectivity is due to the presence of SPPs/SPhPs across the interfaces. It is interesting to note that, unlike in far-field regime, the selectivity is affected only when BN particles are used as inclusions. Addition of Au particles shows little or no impact on the selectivity in the near-field. This supports the idea that metallic nanoparticles do not induce new SPPs/SPhPs in the surfaces while dielectric nanoparticles such as BN produce new SPhP in the material.

## 4. Conclusion

We have demonstrated that nanoparticles influence the emission spectra of the multilayered structures. Wavelength selectivity can be altered and controlled by size and/or volume fraction of the NPs. 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 dielectric function of metallic NPs has a strong dependence of particle size due to electron scattering. We have also shown that 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 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 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 *λ _{n}* of the mixture, hence wavelength selectivity of thin films in near-field has little or no effect due to the presence of metallic nanoparticles. This can be understood as the presence of NPs in the thin film does not induce new kind of SPPs/SPhPs resonance across the interfaces. This work broadens the range of designs and methods for wavelength selective emitters in both the far-field and near-field regime.

## Acknowledgments

This work is funded by the Start-up Grant through the College of Engineering at the University of Rhode Island.

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