PHOTONIC METAMATERIALS: SPECTRAL CONTROL AND MODULATION OF NANOSCALE THERMAL RADIATIVE TRANSPORT

Micro/nano scale radiative transport phenomena present new opportunities to design spectrally selective radiative surfaces and configurations that enable an active control of radiative transfer. Several applications such as thermophotovoltaic energy technology, local thermal management and sensing can benefit from fundamental research in the field. An investigation into photonic metamaterials and near-field thermal radiation was conducted to explore various ways to achieve spectral control and modulation of radiative heat transfer. This was followed by designing novel metamaterials and configurations to consider their feasibility for specific applications of thermophotovoltaics (TPV) and nanoscale thermal management. An experimental investigation was conducted to estimate optical and radiative properties of SU-8 thin films for various film thicknesses. Samples consisted SU-8 films of thickness ranging from 10 μm to 157 μm deposited on gold coated silicon substrates and were prepared using spin coating. Thickness dependent reflective properties were confirmed using Fourier Transform Infrared Spectrometer measurements. Dielectric function of SU-8 in the range 2 μm to 15 μm was calculated using the reflectance spectra of the samples. Optical properties of SU-8 in mid-infrared (mid-IR) region were reported before and after UV treatment. Measurements imply a change in optical properties of SU-8 upon exposure to UV and heat treatment. Using microscopic thin films is one of the simplest ways to achieve a change in emission spectra. A methodology was proposed to shift the wavelength selectivity in the desired location using thin films embedded with nanoparticles. For the media doped with nanoparticles, an effective dielectric function using the MaxwellGarnett-Mie theory is employed to calculate emissivity and radiative heat transfer. Influence of parameters such as particle size and volume fraction was studied. It was also shown that wavelength selective behavior of such nanocomposite thin films can be related to their effective refractive indices. A theoretical study to explore Mie-resonance metamaterials (nanocomposites) for possible use in TPV technology was conducted. Metamaterials were designed to achieve spectral matching of thermal emitter and photovoltaic (PV) cell. The emitter consists of a thin film of SiO2 on the top of tungsten layer deposited on a substrate. Both near-field and far-field configurations were considered. The methodology followed Maxwell-Garnett-Mie theory discussed earlier. The results showed that the proposed Mie-metamaterial thermal emitter can significantly improve the efficiency of TPV system. It was shown that, by changing volume fraction of nanoparticles and thickness of SiO2 it is possible to tune the near-field thermal radiation to obtain enhanced output power and high thermal efficiency. Methods to achieve dynamic control of radiative heat transfer and thermal rectification characteristics were investigated using a phase-change material called vanadium dioxide (VO2). For a far-field configuration, a tri-layer structure consisting a thin film of KBr sandwiched between a thin film of VO2 and a reflecting layer of gold was proposed. The structure is highly reflective when VO2 is in insulating state (below 68 ◦C), while it is highly absorbing when VO2 is in its metallic state. Thermal rectification achieved by such a structure is greater than 11 a temperature bias of 20 K, which is the highest rectification ever predicted for far-field radiative diode configurations. A near-field thermal diode configuration was also considered. Possible configurations using bulk, thin film and gratings of VO2 were studied. It was shown that using 1-D rectangular or triangular grating of VO2, a high degree of contrast can be achieved in the tunneling of surface waves across the two interface of thermal diode. For minimal temperature difference of 20 K, rectification ratio as high as 16 was obtained and it is maximum in existing literature to date for comparable operating temperatures and separation.

Influence of parameters such as particle size and volume fraction was studied. It was also shown that wavelength selective behavior of such nanocomposite thin films can be related to their effective refractive indices.
A theoretical study to explore Mie-resonance metamaterials (nanocomposites) for possible use in TPV technology was conducted. Metamaterials were designed to achieve spectral matching of thermal emitter and photovoltaic (PV) cell. The emitter consists of a thin film of SiO 2 on the top of tungsten layer deposited on a substrate. Both near-field and far-field configurations were considered. The methodology followed Maxwell-Garnett-Mie theory discussed earlier. The results showed that the proposed Mie-metamaterial thermal emitter can significantly improve the efficiency of TPV system. It was shown that, by changing volume fraction of nanoparticles and thickness of SiO 2 it is possible to tune the near-field thermal radiation to obtain enhanced output power and high thermal efficiency.
Methods to achieve dynamic control of radiative heat transfer and thermal rectification characteristics were investigated using a phase-change material called vanadium dioxide (VO 2 ). For a far-field configuration, a tri-layer structure consisting a thin film of KBr sandwiched between a thin film of VO 2 and a reflecting layer of gold was proposed. The structure is highly reflective when VO 2 is in insulating state (below 68 • C), while it is highly absorbing when VO 2 is in its metallic state.
Thermal rectification achieved by such a structure is greater than 11 a temperature bias of 20 K, which is the highest rectification ever predicted for far-field radiative diode configurations. A near-field thermal diode configuration was also considered.
Possible configurations using bulk, thin film and gratings of VO 2 were studied. It was shown that using 1-D rectangular or triangular grating of VO 2 , a high degree of contrast can be achieved in the tunneling of surface waves across the two interface of thermal diode. For minimal temperature difference of 20 K, rectification Thermal radiation is one of the fundamental modes of heat transfer. Our textbook understanding of thermal radiation is based on Planck's law and Stefan-Boltzmann law. Thermal motion of atoms and molecules lead to fluctuating charges and currents in a material. This in turn leads to a fluctuating electromagnetic field that travels in space and it is what we call thermal radiation. In the infrared and visible part of spectrum, majority of radiation comes from molecular or vibrational motion and the motion of conduction electrons. As this covers a wide range of energy levels, thermal radiation is broadband, incoherent and unpolarized. Meaning, its electromagnetic nature can be ignored and it can be treated as heat rays. However, recent advances in the field of radiative heat transfer have defied the common generalizations observed in the macroscopic world.
When the dimensions of an object or surface features of a material become comparable to the wavelength of radiation, electromagnetic nature of thermal radiation gives rise to two interesting phenomena that cannot be ignored. Micro/nanoscale radiative transfer refers to scenarios where at least one of the two conditions are satisfied: (i) features or dimensions of micro/nanostructured materials are comparable to the characteristic thermal wavelength λ th and (ii) Separation between bodies is smaller or comparable to the thermal wavelength. In the first case, features of a material are comparable to the wavelength of electromagnetic radiation and effects like interference, diffraction, partial coherence and surface scattering come into picture. These nanostructures could be a stack of thin films, nanocomposites, diffraction gratings, 2-D surface patterns or photonics crystals.
As a result, radiative properties of such a nanostructured material can be substantially different than those of a flat surface of the same material(s). This can give rise to unusual properties such as spectrally selective absorption/emission, viii coherent thermal radiation, linearly or circularly polarized thermal radiation, negative refraction and so on. Materials with such unusual properties are known as metamaterials. The second case is known as near-field thermal radiation. When separation between two objects at different temperatures is comparable or smaller than the wavelength of thermal radiation, radiative heat transfer can exceeds what is predicted by Planck's law (the blackbody limit). Derivation of Planck's law is based on the assumption that all dimensions involved in a problem are much larger than the thermal wavelength λ th . Therefore, Planck's law is unable to explain radiative heat transfer between closely spaced bodies as surface waves dominate the radiative heat transfer.
These novel phenomena allow us to spectrally control and modulate overall radiative heat transfer. Spectral control refers to 'tuning' the spectral response of a radiating surface, while modulation of radiative heat transfer refers to attaining an active control of radiative heat transfer coefficient. Spectral control and modulation of nanoscale heat transfer are the two central themes of this dissertation.
Metamaterials and micro/nanoscale thermal radiation in general, have great applications in advanced energy conversion systems, thermal management, thermal cloaking and sensing. Therefore, major goals of this dissertation are: 1. Explore various ways to achieve spectral control and modulation of nanoscale radiative transfer (manuscripts 1 and 2).

Investigate feasible applications of various metamaterials in thermophoto-
voltaic technology (manuscripts 3 and 4) and local thermal management (manuscripts 5 and 6).
This dissertation is prepared using the manuscript format. Sequence of the manuscripts is according to their chief themes rather than chronological.
ix In manuscript 1, optical and radiative properties of a polymer SU-8 in its bulk form and thin-film form are investigated. Reflectivity of SU-8 samples was experimentally measured using fourier transform infrared spectrometer. Thicknessdependent reflectivity (or emissivity) was observed as expected. Reflectivity spectra of samples were used to calculate refractive indices of the polymer in infrared region. As SU-8 is sensitive to UV exposure, its properties before and after UV exposure were reported. The manuscript follows the formatting guidelines specified by Optical Materials Express.
Manuscript 2 investigates a possible way to attain spectral tuning of radiative transfer by using nanoparticle embedded thin films. The Calculations of emission spectra were performed using the Fresnel equations in the far-field limit, while the dyadic Green's function formalism was used to calculate for transmissivity between the closely spaced objects in the near-field regime. Media containing nanoparticle inclusions were modeled using Maxwell-Garnett-Mie theory. It was shown that the spectral selectivity in the emission spectra can be influenced by varying the size and/or the volume fraction of nanoparticles. Characteristic features of the spectra were linked to refractive indices of the composite. Influence of metallic and dielectric inclusions are separately investigated.
Manuscript 3 deals with design of a novel, efficient and cost effective thermal emitter for thermophotovoltaic (TPV) applications. A Mie-resonance metamaterial based on nanoparticle-embedded thin film was used to attain desired spectral response. The emitter consists of a thin film of SiO 2 on the top of tungsten layer deposited on a substrate. Effective dielectric properties are calculated using Maxwell-Garnett-Mie theory. It was shown that, this would significantly improve the efficiency of TPV cells. A new parameter to gauge the efficacy of thermal emitters was also introduced and it was used to compare different emitter designs.
x In manuscript 4, the idea of Mie-metamaterial thermal emitter was implemented to a near-field TPV system. Performance characteristics of a near-field thermophotovoltaic system consisting a Mie-metamaterial emitter and GaSb based photovoltaic cell at separations less than the thermal wavelength were theoretically analyzed. Numerical results were obtained using formulae derived from dyadic Green's function formalism and Maxwell-Garnett-Mie theory. It was shown that, by changing the volume fraction of nanoparticles and thickness of SiO 2 it is possible to tune the near-field thermal radiation to obtain enhanced output power and high thermal efficiency. Materials considered can withstand high temperatures and are suitable for thermal emitter. Improvement in spectral selectivity as well as overall heat transfer was accounted for an increased power output and efficiency.
In manuscript 5, a concept of a far-field radiative thermal rectification device was proposed. The device uses a phase change material to achieve a high degree of asymmetry in radiative heat transfer. The proposed device has a tri-layer structure on one side and a blackbody on other side. The tri-layer structure consists of a thin film of KBr sandwiched between a thin film of VO 2 and a reflecting layer of gold. When VO 2 is in its insulating phase, the structure acts as an infrared mirror due to the two transparent layers on highly reflective gold. When VO 2 is in the metallic phase, Fabry-Perot type of resonance occurs and the tri-layer structure acts like a highly absorbing surface achieved by destructive interference of partially reflected waves making it highly absorptive for majority of spectral range of thermal radiation. The proposed structure forms the active part of configuration that acts like a far-field radiative thermal diode. Thermal rectification greater than 11 was obtained for a temperature bias of 20 K, which is the highest rectification ever predicted for far-field radiative diode configurations.
Manuscript 6 investigates a near-field radiative thermal diode that uses a phase xi change material to achieve asymmetry in radiative heat transfer. The temperature dependent dielectric properties of VO 2 were exploited due to its metal-insulator transition near 341 K. Analogous to an electrical diode, heat transfer coefficient is high in one direction while it is considerably small when the polarity of temperature gradient is reversed. It was shown thermal rectification can be greatly enhanced by using 1-D rectangular and triangular surface gratings of VO 2 . Enhanced rectification in the near-field was accounted to reduced tunneling of surface waves across the interfaces for negative polarity. It was predicted that for minimal temperature difference of 20 K, rectification ratio as high as 16

Introduction
Continuous development in lithography and micromachining has led to invention of a photoresist known as SU-8 by IBM R research [1,2]. SU-8 is a photosensitive epoxy polymer that is now commonly used as negative photoresist for optical lithography, especially for high-aspect ratio lithography [3]. SU-8 has been shown compatible with other nanoscale lithography techniques, such as electron beam lithography and x-ray lithography [4]. SU-8 is of prime importance in fabrication of semiconductor devices, micro-electromechanical systems (MEMS) and microfluidics [5]. Due to its bio-compatibility, SU-8 is also used for bio-MEMS applications [6]. Owing to its good thermal stability and young's modulus, SU-8 can be potentially used for nanoimprint lithography [7]. When exposed to ultraviolet (UV) light, SU-8 undergoes cross-linking leading to polymerization, making it insoluble in solvents such as Acetone, Methyl Ethyl Ketone (MEK) or N-Methyl Pyrrolidinone [2]. While optical properties of SU-8 in visible and UV range have been characterized [8], infrared refractive indices of SU-8 were not estimated until recently [9]. Fourier Transform Infrared Spectroscopy (FTIR) measurements using attenuated total reflectance (ATR) method have also been documented before and various IR absorption bands have been assigned to several different functional groups [10,11,12]. Mid-IR dielectric function of SU-8 was first reported by Motaharifar et al. [9] using reflection and transmission spectra of a free-standing SU-8 sample. Here, we calculate dielectric function of SU-8 using reflectance data for various thickness of SU-8 films. We also observe its reflectance spectra before and after UV treatment.
Polymers in general have been investigated for possible selective thermal emission properties [13]. They can also be embedded with nanoparticles furthering spectral control of thermal emission [14,15,16,17]. Infrared spectral response of polydimethylsiloxane (PDMS) [13], poly(vinyl chloride) (PVC) [18], poly(vinyl fluoride) (PVF) [19] and poly(4-methyl-1-pentene) (PMP) [20] have been studied before. It would be crucial to study spectral response of SU-8 thin films in the infrared region, especially given its sensitivity to UV treatment. We report the These protonated ions react with epoxides in cross-linking reactions upon heating (baking) [21]. Therefore, UV exposure of SU-8 pattern is usually followed by a few baking steps during optical photolithography. In this work, we prepared several samples of SU-8 thin films of various thickness coated on a reflecting layer of gold over silicon wafers and recorded the reflectance spectra of the samples after each step that would normally be undertaken during optical lithography. Reflectance spectra of samples with various thicknesses are used to estimate refractive indices of the polymer in the mid-IR region.

Sample Preparation
Silicon (Si) wafers were cleaned using successive washes of acetone, methanol and deionized water and dried using nitrogen gas. The wafers were then heated to 120 • C on a hot plate to release adsorbed gases. were recorded at each stage as shown Fig. 1.2(a). As the thickness of the gold layer is much larger than the penetration depth (λ/4πκ) of mid-IR radiation, it is thick enough to be opaque. The layer of gold is also highly reflecting (>98%) for the range of wavelengths considered here. Therefore, the spectra show optical properties of SU-8. Reflectance spectrum shows several absorption bands. Each of the absorption bands relate to a specific chemical bond. FTIR spectroscopy using attenuated total reflection (ATR) mode is often performed to identify different chemical bonds in the samples [12]. Here, we have conducted diffuse reflectance measurements instead. While diffuse reflectance spectroscopy is more quantitative than ATR method, the spectrum does not necessarily display all the distinct absorption bands. However, some typical absorptions peaks are clearly visible in the thin-film reflectance spectrum that we briefly discuss. The most notable peaks are However, this is the region where absorption bands corresponding to aromatic C-H bends lie [10]. Being an epoxy, SU-8 layers are sticky, especially for thicker films.
Hence it was not possible to conduct reflectance measurements for the thicker films before UV exposure (as they tend to stick to the apparatus). However, after UV exposure and baking stages, SU-8 becomes completely dry and measurements can be taken. While thinner films show absorption at several bands, thicker samples display a broader absorption throughout the infrared region. Appearance of several sharp peaks in the thin film spectra is due to surface phonon modes [22] corresponding various modes of vibrations. Absorption peaks in thin-film spectra can be related to refractive indices of the material [22]. 8

Estimation of Dielectric Function
To estimate the dielectric function of SU-8, we use the method outlined by Verleur et al. [23]. Inspired by Srinivasan's [13] approach to calculate dielectric function of PDMS, we employ a similar process to calculate dielectric function of SU-8. We picked reflectance measurements of all samples to extract the dielectric function.
Dielectric function ε(ω) of a polymer can be assumed to be of Lorentz-Drude oscillator form given by [23], . (1.1) Here, s k , ω k , Γ k and j are the strength, resonant frequency, damping factor of kth Lorentz-Drude oscillator and the imaginary unit, respectively. N such oscillators are assumed. ε ∞ is the contribution from higher frequencies. Since the angle of incidence in the measurements is 12 • , reflectance values were calculated for that angle of incidence. ω k correspond to several of the vibrational bond resonances inside the material. High absorption and strong dispersion exist around these resonance frequencies.
For an unpolarized incident beam of light, reflectance r c for any angle of incidence is related to polarized reflection coefficients by [24] Here, R T E and R T M are effective reflection coefficients at the given angle of incidence for transverse electric (TE) and transverse magnetic (TM) polarization, respectively.
Consider a structure having 3-layer media: air, SU-8 and gold, respectively. Substrate below the gold layer can be ignored as gold is assumed to be 100% reflective.
By solving the boundary conditions at the interfaces, one can obtain the expression for the generalized (or effective) reflection coefficient at the air-SU-8 interface which is given by [24], 12 is the Fresnel reflection coefficient at the interface between the layer 1 and 2, and R (µ) 23 is the Fresnel reflection coefficient at the interface between the layer 2 and 3, µ = s (or p) refers to TE (or TM) polarization, L is the thickness of SU-8 layer. Since we are dealing with an oblique angle of incidence Dielectric function of SU-8 was obtained by tuning several oscillator parameters by matching reflectance spectra of both thin films and bulk samples. This involved an optimization procedure that aims to minimize the error between calculated and measured spectra. The error between the spectra is given by Here, r m and r c are measured and calculated values of reflectance, respectively.
Index i refers to M different frequencies over which the measurements are taken.
One may use reduced Chi-Squared function instead of using absolute error squared as an objective function. One of the issues with reduced Chi-Squared function is that, it produces large values for low reflectance points. This gives higher weightage to thicker samples (because they have very low reflectance on an average). As a result, it produces a good fit for thick samples but not for films. This method is not practical when dealing with samples of various thicknesses. Therefore, absolute error squared was used as objective function.
The minimization was done in two steps. In the first step, we used Matlab R based genetic algorithm to arrive at an initial guess of oscillator parameters. This brings us closer to the global optimum of the objective function, providing a reasonable fit. This is further improved by using constrained optimization function fmincon. It was found that eighteen such resonant frequencies are needed to achieve a good fit with the experimental data. As observed by Verleur [23], at least one oscillator mode is needed to be outside the spectral range of measurement to achieve a good fit. In our case, this is located at 15.4 µm. The oscillator parameters for the UV exposed and heat treated SU-8 are provided in Average error between calculated and reflected spectra is about 5%. Given that the estimated dielectric function allows us to calculate and match reflectance curves for four different film thicknesses, such an error is reasonable. In Fig. 1 [24] W. C. Chew, Waves and fields in inhomogeneous media. IEEE press, 1995.

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.

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 selec-tive 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 2.2 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 2.3 we discuss the results of our calculations obtained using the formulations outlined in section 2.2.
Subsequently, the ideas and conclusions of the paper are narrated in section 2.4.

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,i+1 is the Fresnel reflection coefficient at the interface between the layer i and i + 1, and R (µ) i+1,i+2 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) i,i+1 in all regions. The generalized transmission coefficient for the layered slab is given by tively, the generalized reflection and transmission coefficients can also be calcu- lated using transfer matrix approach [21]. The hemispherical emissivity is given by the expression [6] e(ω) = where R 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 where Θ(ω, T ) = ( ω/2) coth( ω/2k B T ) is the energy of harmonic oscillator at frequency ω and temperature T , is the reduced Planck constant, and k B is the Boltzmann constant. The function 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] T h2 are polarized effective reflection coefficients of the two half spaces (calculated in the absence of other half space), and k hz is the z-component of wavevector in vacuum. The first term in Eq. (5.2) 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 ef f , of the composite medium containing nanoparticle inclusions in a host material is given by [24,25] where ε m is the dielectric function of the matrix, α r is the electric dipole polarizability, 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] α r = 3jc 3 where a 1,r is the first electric Mie coefficient given by where ψ 1 and ξ 1 are Riccati-Bessel functions of the first order given by ψ 1 (x) = xj 1 (x) and ξ 1 (x) = xh indicates the first derivative. x m = ωr √ ε m /c and x np = ωr √ ε np /c are the size parameters of the matrix and the nanoparticles, respectively; ε np being the dielectric function of nanoparticles.
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 Where λ h 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.

Results
The dielectric function is related to real (n) and imaginary (κ) parts of refractive index as √ ε = n + jκ. For SiC and BN, the dielectric function has the form as [29, 30] where  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 the parameters f i , w i , and g i are, in the units of eV, given by f i =  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 µm. This is due to the fact that Mie scattering becomes important at shorter wavelengths giving rise to high peaks for larger particles. While the spectrum in the visible region shows a negligible response to particle size, gold NPs greatly influence the near-infrared region between 1 µm to 4 µm.
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 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

Conclusion
We have demonstrated that nanoparticles influence the emission spectra of the  [20] W. C. Chew, Waves and fields in inhomogeneous media. IEEE press, 1995. [25] U. Kreibig

Introduction
Thermophotovoltaics (TPV) is a promising technology for heat recovery and an attractive alternative for existing electricity generation technologies [1, 2, 3]. A TPV system consists of a thermal emitter which operates at high temperatures (∼1500 K) [4,5] and a photovoltaic cell and can directly convert thermal energy into electricity [6]. In principle, TPV systems can achieve an efficiency of Carnot engine for monochromatic radiation [7] and such TPV systems with efficiencies approaching the limit have been discussed in the literature [8,9,10]. In reality, efficiencies of TPV systems suffer from the mismatch between emission spectrum of emitter and absorption spectrum of PV cell [11]. External quantum efficiency (EQE) of a PV cell is defined as the fraction of incident photons converted into electron-hole pairs [12] and is indicative of the amount of current generated for a given incident photon wavelength. A PV cell has non-zero EQE above the band-gap, i.e. a PV cell can generate electric current only when incident photons have higher energies than the bandgap of semiconductor material [13]. Moreover, absorption of incident photons having wavelength longer than bandgap wavelength is undesirable as it causes thermal leakage and reduces the efficiency [14]. So, in order to achieve high efficiency of a TPV system, goal is to develop a thermal emitter with high emissivity in the region of high external quantum efficiency (EQE) of PV cell and low emissivity in rest of the spectrum. Several studies have focused on using photonics crystals [15,16], plasmonic metamaterials [17], single micro/nano sized spheres [18], doped materials [19,20], surface gratings in order to obtain spectrally selective emission [21]. 1-D, 2-D and 3-D photonic crystals and surface gratings have also been developed [22,23,24]. Numerous works involve use of intermediate filters which reflect the low energy photons back to emitter [25,26,27,28].
Metamaterials is another class of nanomaterials/nanostructures which has been the topic of many articles focused on selective thermal emitters. Liu et al [29] have demonstrated a narrow-band mid infrared thermal emitter based on a metamaterial. Plasmonic metamaterials have also been explored for solar TPV applications as in [30,31]. Woolf et al [32] have presented and also experimentally demonstrated a heterogeneous metasurface for selective emitters and predicted 22% conversion Mie-metamaterial thin films [42]. In present study we exploit the idea by using thin films of SiO 2 embedded with tungsten (W) nanoparticles to design a novel thermal emitter specifically for GaSb and InGaAs based TPVs.
Schematic of a typical TPV system is shown in Fig. 6.1(a). Thermal radiation from the heat source is absorbed by the blackbody absorber. This heat is emitted by the selective thermal emitter only in the desired band of wavelengths in which the external quantum efficiency of the PV cell is high. emission spectrum. Volume fraction of W nanoparticles was adjusted to 30% to achieve the best result. While the mixture of SiO 2 and W gives rise to spectrally selective emission for the PV, the choice of materials also makes sure that it will withstand high temperatures.

Theoretical Fundamentals
The hemispherical emissivity of the thermal emitter can be expressed as [42] e(ω) = where c is the speed of light in vacuum, ω is the angular frequency and k ρ is the magnitude of inplane wave vector. R SiC [47,48]. Dielectric properties of tungsten were also assumed to be unchaged as the operating temeprature is much less than the melting point. The top layer of the thermal emitter considered in our calculations is either 1-D grating structure or nanoparticle embedded thin-film. In both cases, it can be approximated as a homogeneous layer of an effective dielectric function.
If the period of grating (Λ) is much smaller than the wavelength of interest such as the case considered here, dielectric function of the grating layer can be approximated using effective medium theory [49]. We use second order effective medium approximation to calculate the effective dielectric function which is given by [49] ε where ε A and ε B are the dielectric functions of the two materials in surface gratings, Λ is the period and φ is the filling ratio. The expressions for zeroth order effective dielectric functions ε T E,0 and ε T M,0 are given by For calculating the effective dielectric function the Mie-metamaterial, we use Clausius-Mossotti equation. [50,51].
where ε m is the dielectric function of the matrix, α r is the electric dipole polarizability, r and f are the radius and volume fraction of nanoparticles respectively.
To consider the size effects of nanoparticle inclusions, we use the Maxwell Garnett 42 formula which employs the expression for electric dipole polarizability using Mie theory [52], α r = 3jc 3 a 1,r /2ω 3 ε 3/2 m , where a 1,r is the first electric Mie coefficient given by where ψ 1 and ξ 1 are Riccati-Bessel functions of the first order given by ψ 1 (x) = xj 1 (x) and ξ 1 (x) = xh It is worth mentioning that Maxwell-Garnett-Mie theory is applicable when average distance between inclusions is much smaller than the wavelength of interest [53].
This criteria is satisfied in the calculated presented. Also noteworthy is the fact that nanoparticle diameter (40 nm) is much smaller than the thickness of the thin film (0.4 µm) considered. Thus, effective medium theory holds true for the calculations presented in this study.  of the emissivity are thickness of SiO 2 film, volume fraction of W nanoparticles and particle radius. If size distribution of nanoparticles is available, it can be incorporated into the extended Maxwell-Garnett formula as in [55]. For the present study, the particle radius is fixed to 20 nm. Figure 6.3(a) also highlights the effect  [56]. Also refractive index of SiO 2 has a low temeprature coefficient [57]. Therefore these materials are well suited for high temeprature applications.

Results
The thermal emitter presented here is suitable for a typical TPV system consisting GaSb or InGaAs based photovoltaics. The EQE of the two PV cells is shown in Fig. 6.3 the emission spectrum of 1-D structure (Fig. 6.1(b)) and that of a 2-D grating structure discussed in Zhao et al [13] is also shown. It is apparent that the emission spectrum of the design presented here is comparable to that of 1-D structure and the 2-D structure presented in [13] for wavelengths below 1.7 µm. Although a substantial part of blackbody radiation at high temepraturs (1000 to 1500 K) exists between 2 to 6 µm, very little energy is emitted by proposed emitter in this region as emissivty is very low. Moreover, the proposed design has lower emissivity beyond 1.7 µm when compared to the 2-D design. This indicates much lower losses due to thermal leakage. It is worth mentioning that, while choice of materials in the present study is same as that used in [13], fabrication of thermal emitter based on tungsten embedded SiO 2 thin film is relatively simple. This guarantees low cost of large scale fabrication of such emitters. As refractory materials such as platinum and aluminum oxide (Al 2 O 3 ) have been tested before for thermal emitter [32], they can also be explored for this new class of emitters.

Discussion
While the match between emission spectrum of the emitter and the EQE curve of the PV cell can be a good indicator of TPV system's performance, the emitter temperature has an equally important role as it decides the dominant wavelength and can dictate the overall efficiency. Therefore, discussion of conversion efficiency cannot be left out in this work. Ideal emitter has an e(ω) = 1 above bandgap and e(ω) = 0 below the bandgap. The conversion efficiency of the TPV system could be defined as the ratio of the power output of the PV cell to the power incident on the PV cell given by where P rad is the power radiated by the emitter [58] The power output, P out is proportional to the number density of the photons above bandgap and the EQE of PV cell defined as [32] P out = qV OC F F Now consider a hypothetical TPV system with a thermal emitter which has no radiation below the bandgap but covers only a fraction of spectrum above bandgap with e(ω) = 1. Such a system would have lower power output than the one with ideal thermal emitter despite having maximum efficiency with no losses. It is apparent that conversion efficiency of the TPV system alone cannot be regarded as a criterion to gauge the efficacy of a thermal emitter. Therefore we propose a new parameter to calculate the effectiveness of the thermal emitter for a given PV cell and emitter temperature. The parameter termed as effectiveness index is defined as the ratio of the efficiency of the TPV system to the efficiency of the TPV system with ideal thermal emitter at the same temperature and for the given PV cell, β emitter = η Real /η Ideal , where subscripts Real and Ideal refer to TPV systems 47 with thermal emitter of consideration and an idealized thermal emitter at the same temperature. So, β emitter = P out P rad Real × P rad P out Ideal (3.10) While this norm, β emitter doesn't elaborate anything about the efficiency of the TPV system as a whole, it is a good measure to compare the effectiveness of different thermal emitters for a given PV cell and emitter temperature. Note that if β emitter > 1, it implies that the TPV system is efficient but only a part of the spectrum of the EQE is being utilized and the output power is less than what an ideal emitter would produce. Therefore, the goal is to get β emitter close to 1 but not exceed 1. It is assumed that the emitter has a much higher temperature than the PV cell, which has a negligible reflection coefficient. While calculating effectiveness indices for different thermal emitters and a given PV cell, it is assumed that PV cell temeprature and its EQE do not vary.
We calculate the effectiveness of the thermal emitter for GaSb PV cell and at emitter temperature 1500 K for the designs presented in Fig. 6.3(b). Our proposed design has an effectiveness index β emitter = 0.72 while its value for the 1-D design and the one considered in [13] is 0.78 and 0.65 respectively. It is important to note that despite having lower emissivity, the 1-D design has higher effectiveness index because it has lower emissivity beyond 2 µm, thermal wavelength at 1500 K being 1.93 µm. Structure discussed in [13] shows lower effectiveness index as it has higher emissivity below the bandgap. Effectiveness index of the aforementioned designs at 1300 K is β emitter = 0.6, 0.68 and 0.58 respectively. As the thermal   [43] W. C. Chew, Waves and fields in inhomogeneous media. IEEE press, 1995.

Introduction
Thermophotovoltaics (TPV) have been the focus of several works as an alternative to power generation technologies and a technology for waste heat recovery systems [1,2,3]. A typical TPV system consists of a high temperature (∼1500 K) thermal emitter and a photovoltaic (PV) cell that converts the energy of incident photons into electricity. While ideal TPV systems convert radiative energy into electricity at an efficiency of the Carnot engine, practical TPV systems suffer from mismatch between the emission spectra of the emitter and absorption spectra of PV cell [4,5,6,7,8]. Several studies investigate the use of metamaterials [9,10,11], surface gratings and photonics crystals [12,13,14,15,16,17] and complex surface patterns [18] to improve efficiency of TPV system. In the recent few years, it was demonstrated that near-field thermal radiation has a great potential in improving TPV systems [19,20,21,22,23,24]. Several theoretical studies were focused on exploiting near-field coupling of surface waves between emitter and PV cell at nanometer separation to improve the efficiency. It is challenging to develop materi-als that can withstand high temperatures as well as allow an enhanced coupling of surface waves for the energies above the band gap of PV cell. One-dimensional gratings, hyperbolic metamaterials and photonic crystals have shown a great potential in TPV applications. Dielectric mixtures and nanoparticle embedded thin films can also be utilized to tune near-field thermal radiation [25]. It has been demonstrated earlier that Mie-metamaterials or Mie-resonance metamaterials that utilize Mie resonances of inclusions into host materials can be used for spectral tuning of near-field thermal radiation. Authors earlier demonstrated the possible use of tungsten nanoparticle-embedded thin film of SiO 2 to achieve a selective thermal emitter for far-field TPV system [26]. Effect of nanoparticle inclusions into host material on near-field radiative heat transfer has been investigated in several theoretical studies [27,28,29]. Stemming from earlier work, we demonstrate the use of a Mie-metamaterial thermal emitter that consists of tungsten nanoparticles embedded into a thin film of SiO 2 deposited on thick layer of tungsten for near-field TPV system. We also explore possible alternative of W nanoparticles by refractive materials such as titanium nitride (TiN), tantalum (Ta) and molybdenum (Mo).
We investigate the performance of such a TPV system for various configurations.
While many recent works have dealt with gratings, photonic crystals and other metamaterials, this is the first time nanoparticle-embedded thin films have been investigated for near-field TPV system. The configuration of near-field TPV system considered in the present study is shown in Fig. 6  of 0.726 eV) whose properties can be found in Ref. [30]. Calculations reports in this work are for emitter temperature of 1500 K while PV cell is assumed to be at 300 K. The separation between the emitter and PV cell is comparable or less than the thermal wavelength at 1500 K (λ th = 1.93 µm).
While we concern ourselves with mostly the theoretical side of proposed thermal During the sputtering process, tungsten nanoparticles would form.

Results and Discussion
Refractive indices of plain SiO 2 and that of SiO 2 mixed with 30% nanoparticles of tungsten are shown in Fig. 6.2. Effect of nanoparticle inclusions can be observed.
SiO 2 has a near constant value of refractive index (n) and a negligible extinction coefficient (κ) in the spectral range of our interest. Dielectric mixture of SiO 2 and tungsten displays an overall increased effective refractive index and a higher absorption coefficient for wavelengths shorter than 2.5 µm.
It is crucial to reduce the spectral energy below the band-gap of PV cell in order to improve the overall thermal efficiency of the TPV system. Coupling of surface waves dictates whether radiative transfer would be enhanced or suppressed. The goal is to enhance radiative transfer above the bandgap without significantly enhancing radiation below the bandgap of PV cell. In order to assess the impact of nanoparticle inclusions, we investigate spectral heat flux of the proposed TPV system for various configurations. Spectral heat flux across the proposed thermal emitter at a separation of L = 100 nm for various compositions is plotted in Fig.   6.3. Emitter with a 0.3 µm layer of SiO 2 on tungsten has lower heat flux across the spectrum when compared to plain tungsten as seen in Fig. 6.3 (a). Upon inclu- SiO 2 yield more selective spectral response while maximum spectral heat flux remains relatively same. It can be observed that radiation at longer wavelengths is more sensitive to layer thickness. It is imperative that such a configuration is more desirable for minimizing losses due to long wavelength photons. In principle, it is possible to tune the near-field thermal radiation by changing volume fraction of nanoparticles and thickness of the SiO 2 layer to achieve optimal configuration for a given operating temperature and separation.
We now investigate the power output of the GaSb PV cell as a result of near-field radiative heat transfer at a separation of 100 nm. To assess the overall performance of the TPV system, we model the PV cell as discussed in section 4.3. We calculate spectral density of output power along with the total output power.  are independent of the distance. For separations less than 1 µm (near-field), the overall heat flux increases due to presence of evanescent waves. Consequently, the output power also rises monotonically as separation between the emitter and PV cell is reduced. The inset in Fig. 4.5 shows TPV system efficiency against distance for the same setups. While the trend in efficiencies is not as monotonous, for the separations smaller than 100 nm, the emitter with tungsten nanoparticles shows higher thermal efficiency than the one with pure SiO 2 . This is supported by earlier results that show the increased selectivity at shorter wavelengths leading to lower losses. Interestingly, the emitter with pure SiO 2 has higher efficiency in the far-field. Nevertheless, such a configuration has lower output power. Oscillatory behaviour of efficiency has been observed before and can be attributed to vacuum gap behaving like a waveguide [4]. We would like to emphasize that, to further improve the performance of the TPV system the materials chosen are not necessarily optimal. For example, PV cells with lower band gap such as InGaAs and InGaSb or the quaternary alloys like InGaAsSb can be used. This can allow use of lower and Ta nanoparticles is 3.72×10 5 , 3.81×10 5 , 2.68×10 5 and 2.57×10 5 W/m 2 , respectively. From Fig. 4.6 it appears that emitter with TiN and W nanoparticles display higher heat flux for this particular configuration. Emitter with tungsten nanoparticles shows better selectivity than TiN. Various combinations of material type and dimensions can be investigated to tune the emission spectra.
Overall, we have numerically investigated for the first time, near-field thermophoto-voltaic system that uses Mie-metamaterial based thermal emitter and a PV cell at a separation less than the thermal wavelength. We have theoretically demonstrated an enhanced wavelength selective thermal emitter for near-field thermophotovoltaic system using a Mie-resonance metamaterial. Thermal emitter consists of tungsten nanoparticle-embedded thin film of SiO 2 deposited on thick tungsten substrate.
We analyze performance of such a TPV device for various cases. We study the effect of volume fraction, layer thickness of SiO 2 and separation between emitter and PV cell. The embedded tungsten nanoparticles in the thin film can alter the refractive index of the film and allow spectral control of near-field radiative transfer across the emitter and the PV cell. We evaluate energy conversion efficiency of the proposed near-field thermophotovoltaic system. The results show that the structure of Mie-metamaterial thermal emitter can significantly improve the efficiency of thermophotovoltaic system. Improvement in spectral selectivity as well as overall heat transfer can be accounted for increased power output and efficiency. We show that, by changing volume fraction of nanoparticles and thickness of SiO 2 it is possible to tune the near-field thermal radiation to obtain enhanced output power and high thermal efficiency. Materials considered can withstand high temperatures and suitable for thermal emitter.

Materials and Methods
The expression of radiative transfer between closely spaced bodies can be derived using dyadic Green's function approach [31], and is given by where Θ(ω, T ) = ( ω/2) coth( ω/2k B T ) is the energy of harmonic oscillator at frequency ω and temperature T , is the reduced Planck constant, and k B is the Boltzmann constant. The function 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 [31] T 1→2 (ω) = h2 are polarized effective reflection coefficients of the two half spaces (calculated in the absence of other half space), and k hz is the z-component of wavevector in vacuum. The first term in Eq. (5.2) 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. For a structure having N -layer media having (N − 1) interfaces, the expression for the generalized reflection coefficient at the interface between region i and region i + 1 is given by [32], i,i+1 is the Fresnel reflection coefficient at the interface between the layer i and i + 1, and R i,i+1 in all regions. For calculating the effective dielectric function the Mie-metamaterial, we use Clausius-Mossotti equation. [33,34].
where ε m is the dielectric function of the matrix, α r is the electric dipole polarizability, r and f are the radius and volume fraction of nanoparticles respectively.
To consider the size effects of nanoparticle inclusions, we use the Maxwell Garnett formula which employs the expression for electric dipole polarizability using Mie theory [35], α r = 3jc 3 a 1,r /2ω 3 ε 3/2 m , where a 1,r is the first electric Mie coefficient given by where ψ 1 and ξ 1 are Riccati-Bessel functions of the first order given by ψ 1 (x) = xj 1 (x) and ξ 1 (x) = xh indicates the first derivative. x m = ωr √ ε m /c and x np = ωr √ ε np /c are the size parameters of the matrix and the nanoparticles, respectively; ε np being the dielectric function of nanoparticles. It is worth mentioning that Maxwell-Garnett-Mie theory is applicable when average distance between inclusions is much smaller than the wavelength of interest [36]. This criteria is satisfied in the calculated presented. Since nanoparticle diameter (40 nm) is much smaller than the thickness of the thin film (0.3 µm) considered, effective medium theory holds true for the calculations presented in this study. Dielectric functions of the materials (SiO 2 and W) considered in this paper are taken from literature [37,38]. Having very low temperature coefficients, room temperature values of dielectric function are used for SiO 2 [39]. Dielectric properties of tungsten were also assumed to be unchanged as the operating temperature is much less than the melting point.
Near-field thermal radiation and charge transport in PV cell can be theoretically modelled by accounting for charge density distribution due to number of photons absorbed at different cell depth [22]. Due to limitations of the present study, we employ a simplistic model to calculate output power of PV cell. We assume that quantum efficiency of PV cell in the near-field would be the same as that in the far-field. Therefore, short circuit current can be calculated as [22] where E g is the bandgap of GaSb cell, EQE GaSb is external quantum efficiency, dq/dω is spectral heat flux and e is the electronic charge. Dark current can be calculated by, where D n and D p are diffusivities of electrons and holes, respectively, n i is the intrinsic carrier concentration, N n and N p are concentrations of electrons and holes, respectively, while diffusion lengths L n and L p can be calculated in terms of diffusivity and recombination lifetime τ using, Total recombination lifetime is calculated by, where τ R , τ SHR and τ Au are radiative recombination, Shockley-Hall recombination and Augar recombination, respectively [40]. Open circuit voltage is calculated using, Output power of the PV cell is given by, where z = ln(I SC /I 0 ) and efficiency of the TPV system is given by, η = P out /Q rad (4.12) For our calculations, intrinsic carrier concentration was assumed to be 4.3 × 10 12 cm −3 . Carrier concentration of electrons and holes were assumed to be equal to N n = N p = 10 17 cm −3 . The recombination lifetimes are taken to be τ R = 40 ns, τ SHR = 10 ns and τ Au = 20 µs. Carrier diffusivities are D n = 129 cm 2 /s and D p = 39 cm 2 /s for electrons and holes, respectively.

Abstract
We propose a theoretical concept of a far-field radiative thermal rectification device that uses a phase change material to achieve a high degree of asymmetry in radiative heat transfer. The proposed device has a multilayer structure on one side and a blackbody on other side. The multilayer structure consists of transparent thin film of KBr sandwiched between a thin film of VO 2 and a reflecting layer of gold. When VO 2 is in its insulating phase, the structure is highly reflective due to the two transparent layers on highly reflective gold. When VO 2 is in the metallic phase, Fabry-Perot type of resonance occurs and the tri-layer structure acts like a wide-angle antireflection coating achieved by destructive interference of partially reflected waves making it highly absorptive for majority of spectral range of thermal radiation. The proposed structure forms the active part of configuration that acts like a far-field radiative thermal diode. Thermal rectification greater than 11 is obtained for a temperature bias of 20 K, which is the highest rectification ever predicted for far-field radiative diode configurations.
Analogous to electrical diode, thermal diode is a rectification device wherein magnitude of heat flux strongly depends on the sign of applied temperature bias. To quantify rectification, we employ the widely used definition of rectification ra-tio, i.e., R = (Q f − Q r )/Q r where Q f and Q r refer to forward and reverse heat flux, respectively [17]. Alternatively, rectification coefficient can be defined as There are numerous studies pertaining to near-field and far-field thermal radiation based rectification devices that exploit temperature dependent properties of a phase change materials such as vanadium dioxide (VO 2 ) and La 0.7 Ca 0.15 Sr 0.15 MnO 3 (LCSMO) [18,11,19]. A number of studies deal with far-field thermal radiation [20,21] while several others focus on modulation of radiative heat transfer in the near-field regime [22,23,24,25,18,26,19]. Ben-Abdallah and Biehs introduced a VO 2 based simple far-field radiative thermal diode, while Prodhomme, et al., [27] proposed a far-field thermal transistor that uses a VO 2 base between a blackbody collector and a blackbody emitter. Zhu, et al., [28] showed that temperature dependent optical properties of SiC can be used to attain negative differential conductance. Van Zwol, et al., [22] proposed that one can take advantage of the phase transition from crystalline to amorphous state in AIST (an alloy of Ag, In, Sb, and Te) driven by a current pulse to obtain a large contrast in heat flux. In far-field limit, rectification is due to the change in emissive properties of a phase change material. In near-field limit, the difference in the coupling strength of polaritons or tunneling of surface waves between structures leads to thermal rectification. In general, it is observed that a higher rectification can be achieved in the near-field regime than in the far-field. However, it is challenging to develop such devices operating on the principle of near-field radiative transfer.
Fabry-Perot resonance was achieved at 10 µm wavelength. As discussed later, we show that, by tuning the resonance at right wavelength, maximum rectification can be achieved in the proposed design.
VO 2 has often been used in thermal rectification devices, because its phase-change from an insulator to a metal can be switched reversibly within a short time (∼100 fs) [49]. The common devices use either a bulk VO 2 solid or its thin-film form.
In this work, we present a VO 2 based far-field thermal rectification device with a simple multilayer structure. We predict a record rectification factor of greater than 11 (η >0.91).
A typical far-field thermal diode has two planar components separated by a distance much larger than thermal wavelength. The active component is made of a phase-change solid, whereas the passive component stays inert. to reflective surface upon the reversal of a temperature bias which induces the phase change. This is exactly our design attempts to achieve. Therefore, the passive component is chosen to be a blackbody. Any material other than a blackbody would not yield maximum rectification. Structure 1 and 2 are at temperature T 1 = T C + ∆T and T 2 = T C − ∆T , respectively. The mean temperature is chosen to be the phase transition temperature of VO 2 (T C = 341 K). When T 1 > T 2 (referred to as forward bias), VO 2 layer is in its metallic phase; and when T 1 < T 2 (reverse bias), VO 2 layer becomes insulating with its optical axis aligned along the vertical direction, i.e., z-axis.
Phase transition of VO 2 is not abrupt [49,50] and a complete insulator-metal tran- K as presented in Ref. [49] and Ref. [51], the phase transition is reversible. As we report heat flux values at 10 K above and below the critical temperature of VO 2 , hysteresis behavior is beyond the scope of this study.

Results and Discussion
A multilayer structure can be designed to attain high absorbance or reflectance based on its dimensions and material properties. Multilayers with constituent thicknesses much smaller than the incident wavelength of light have been studied before [52]. We show that in a VO 2 based multilayer structure, the dramatic change in the optical property of VO 2 upon phase-change facilitates an extensive variation in the surface reflectivity.  this gives rise to a high heat flux in forward bias. However, when VO 2 is insulating, the structure has very low emissivity in the broad spectrum. The tri-layer structure behaves like a highly reflecting mirror resulting in very low heat flux.
Consequently, high contrast in heat flow is achieved leading to a high rectification ratio of 11.3 (η =0.918). In order to highlight the diode-like characteristics, heat flux across the device has been plotted against themeprature difference in Fig.   6.3. For comparison, simple case of bulk VO 2 is also shown, it has a rectification coeffcient of η = 0.49. Note that, effect of thermal hystersis is not considered here for simplicity. Angle dependent spectral reflectivity of the active component of the thermal diode is plotted in Fig. 6.4 for the forward and reverse bias cases. When VO 2 is metallic, the tri-layer structure acts like a wide-angle antireflection coating for wavelengths between 4 µm to 10 µm. The dark spot in Fig. 6.4 corresponds to Fabry-Perot type of resonance that occurs around λ = 4n KBr (λ)L 2 = 5.3 µm [47].
High absorption/emission in this wavelength region favors radiative heat transfer as thermal wavelength falls within this range. In reverse bias, the structure is highly reflective in a broad range of wavelengths giving rise to a very low absorption. Note that for thermal wavelength of 8.5 µm, Fabry-Perot resonance occurs (for metallic VO 2 ) when thickness of KBr layer is L 2 = λ th /4n KBr (λ th ) = 1.4 µm.
This configuration however, would not necessarily achieve maximum rectification as the structure may not be purely reflecting when VO 2 is its insulating phase.
Contrasting reflective properties of the structure are due to constructive and de-   interface and R 2,3 is the effective reflection coefficient at VO 2 -KBr interface due to multiple reflections within KBr layer. They can be expressed as geometric series whose terms are relative amplitudes of partial waves due to first, second and third reflection and so on. For both metallic as well as insulating VO 2 , the magnitude of R 2,3 , | R 2,3 |, is large. However, when VO 2 is in metallic phase, each partial reflection results in a phase-shift such that partial waves add up destructively leading to a small value of | R 1,2 | and low reflectivity, especially in the wavelength range centered around thermal wavelength. On the other hand, in reverse bias (insulating VO 2 ) phasors add constructively, giving rise to highly reflective surface properties for a broad range of wavelengths. A similar phenomenon can be observed for TM polarization as well. As KBr is transparent and has a negligible extinction coefficient for most of infrared region, much of the absorption takes place within the VO 2 layer. Transparent layer of KBr mainly influences the reflective properties by altering the phase of the light propagating through the media. Potentially, any other material transparent to infrared light such as magnesium fluoride or intrinsic Figure 5.5: Effective reflection coefficient at air-VO 2 interface ( R 1,2 ) and VO 2 -KBr interface ( R 2,3 ) as phasor sum of reflection coefficients due to each reflection for TE polarized incident plane wave of wavelength λ th = 8.5 µm and angle of incidence 10 • silicon can be used in this concept. However, optimal dimensions of such a device might be different.
In summary, we present a VO 2 based far-field radiative thermal diode structure with a high rectification ratio of 11.3. The active component of the proposed device has a tri-layer structure consisting thin films of VO 2 , KBr and gold. As µm, metallic VO 2 makes the structure behave like a wide-angle antireflection coating while insulating VO 2 makes it highly reflecting. As a result, high degree of asymmetry in radiative heat transfer is predicted across the tri-layer structure and a blackbody. Contrasting reflecting properties of the structure can be explained using constructive and destructive interference of partial reflections across the interfaces. We optimized layer thicknesses to maximize rectification. Thermal rectification greater than 11 is predicted for temperature difference of 20 K and it is highest among far-field radiative diodes that have been studied. Possibility of attaining higher rectification could be investigated in future by using alternate transparent materials, thinner films of VO 2 and/or using more number of alternating VO 2 /dielectric layers. Such devices can find numerous applications such as thermal logic devices and thermal management systems.

Methods
To calculate heat flux in forward and reverse bias across our far-field thermal diode, we use the well known expression of radiative transfer obtained through dyadic Green's function formalism [53]. Radiative transfer between two planar objects is given by where Θ(ω, T ) = ( ω/2) coth( ω/2k B T ) is the energy of a harmonic oscillator at frequency ω and temperature T , is the reduced Planck constant, and k B is the Boltzmann constant. The function T 1→2 (ω, L) corresponds to the spectral transmissivity in radiative transfer between media 1 and 2 with a separation of L and is expressed as [53] T 1→2 (ω, h2 are polarized effective reflection coefficients of the two half spaces (calculated in the absence of other half space), µ = T E (or T M ) refers to transverse electric (or magnetic) polarization and k hz is the z-component of wavevector in vacuum. Here, j is the imaginary unit. For a structure having Nlayer 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 regions i and i + 1 [54], i,i+1 is the Fresnel reflection coefficient at the interface between the layers i and i + 1, and R (µ) i+1,i+2 is the generalized reflection coefficient at the interface between the layers i + 1 and i + 2, z = −d i is the location of the ith interface. i,i+1 in all regions. Note that Eq.
5.2 has only one integral corresponding to propagating waves. The terms due to evanescent waves are ignored as separation between the two half spaces is much larger than the thermal wavelength (L λ th ). The hemispherical emissivity of the active component can be expressed as [32] e(ω) = Note that the term for tranmissivity has been omitted as a layer of gold makes the structure opaque.
Insulating VO 2 (below 341 K) is anisotropic. In a plane (x − y plane in Fig. 6.1) perpendicular to optical axis known as the ordinary mode, its dielectric function is ε O and it is ε E along the optical axis (extraordinary mode). Both ε O and ε E can be calculated using the classical oscillator formula ε(ω) Values of high-frequency constant ε ∞ , phonon frequency ω i , scattering rate γ i and oscillator strength S i are taken from Ref. [55]. There are eight phonon modes for ordinary and nine phonon modes for extraordinary dielectric function. In the metallic state, VO 2 is isotropic and Drude model [55] is used to describe the dielectric function i.e., ε(ω) = −ω 2 p ε ∞ ω 2 − jωΓ . Refractive indices of KBr are taken from Ref. [56], while dielectric properties of gold can be found in Ref. [57]. Blackbody is assumed to have a constant dielectric function ε = 1 + 0.001j.

Abstract
We theoretically demonstrate workings of a near-field thermal rectification device that uses a phase change material to achieve asymmetry in radiative heat transfer.
We exploit the temperature dependent dielectric properties of VO 2 due to metalinsulator transition near 341 K. Analogous to an electrical diode, heat transfer coefficient is high in one direction while it is considerably small when the polarity of temperature gradient is reversed. We show that thermal rectification can be greatly enhanced by using 1-D rectangular and triangular VO 2 surface gratings. With the introduction of periodic grating, rectification ratio is dramatically enhanced in the near-field due to reduced tunneling of surface waves across the interfaces for negative polarity. Our calculations predict that for minimal temperature difference of 20 K, rectification ratio as high as 16 can be obtained and it is maximum in existing literature for comparable operating temperatures and separation.

Introduction
The idea of thermal rectification devices such as thermal diode [1] and thermal equivalents of electronic devices such as thermal transistors [2] and memory element [3] has been around for more than a decade. These conduction (phonon) based thermal rectifiers have some limitations due to speed of acoustic phonons, presence of Kapitza resistances and nonlinear phonon-phonon interaction [4,5]. Devices that utilize property of temperature dependent thermal conductivity [6,7,8] and solid state thermal rectifiers [9,10] have also been demonstrated. Components based on thermal radiation (photons) have also been proposed [5,11,12,13]. In this letter, we discuss radiative thermal rectification devices, which are contactless and do not suffer from above mentioned limitations [14]. They also have weaker nonlinearity [12]. Moreover, physics of thermal transport remains unchanged close and far from equilibrium in photonic devices [4]. Apart from thermal management and energy storage, these thermal analogues of electronic devices could find applications in thermal circuits [15], thermal logic gates [16] and information processing [17].
Thermal diode is a device that is analogous to electrical diode and has a high degree of asymmetry in the magnitude of heat flow depending on the applied temperature bias. To gauge rectification, we use the widely used definition of rectification ratio, i.e. R = (Q f − Q r )/Q r where Q f and Q r refer to forward and reverse heat flux respectively [18]. The goal is to increase R as much as possible so that Q f Q r and rectification is reasonably high to be utilized in a practical device. Studies pertaining to rectification devices other than thermal diodes also deal with achieving manipulation of heat flux. Most thermal rectification devices exploit temperature dependent properties of phase change materials by focusing on the use of bulk materials or thin-films to achieve thermal rectification. Non-contact rectification devices utilize change in dielectric properties of materials such as vanadium dioxide (VO 2 ) and La 0.7 Ca 0.15 Sr 0.15 MnO 3 (LCSMO) [19,12,20]. In far-field, rectification is due to change in emissive properties of a phase change structure. In near-field, difference in the degree of tunneling of surface waves between structures leads to thermal rectification. Consequently higher rectification can be achieved in the near-field regime. Zhu et al [21] utilized temperature dependent optical properties of SiC to put forward a method to attain negative differential conductance. Zwol et al [22] proposed rapid modulation of near-field radiative heat flux at a distance of 100 nm that utilizes fast transition in crystalline and amorphous states of AIST (an alloy of silver, indium, antimony and tellurium) that can be achieved using a current pulse. Zwol et al [23] and Menges et al [24] have experimentally demonstrated that phase change of VO 2 results in change of surface polariton states and significantly alters near-field radiative heat transfer. It has been theoretically calculated that bulk VO 2 and SiO 2 can achieve thermal rectification ratio of about 2 at a gap of 10 nm [19]. Yang et al [25] also proposed a VO 2 based near-field radiative thermal switch whose switching factor can reach up to 80% for gap of 10 nm. Using a combination of two phase change materials -VO 2 and LCSMO, Huang et al [20] achieved the rectification ratio of 7.7 at a gap of 10 nm, hot and cold temperatures being 80 K and 25 K, respectively.
Present study focuses on rectification in a near-field thermal diode and employs VO 2 as phase-transition material. VO 2 can be switched reversibly in a very short amount of time (∼ 100 fs) from an insulating state to a metallic state [26]. To increase the rectification, one may try to use different materials. It has been observed that metamaterials such as mixtures of dielectric materials [27] and grating structures [28] can manipulate radiative transfer in near-field. Therefore they are candidates for enhancement of thermal rectification. Previous studies mainly dealt with bulk materials and thin films. Our calculations indicate that rectification ratio (key results shown in Fig. 6.2) can be raised significantly by using 1-D surface gratings of VO 2 .
A typical near-field thermal diode has two planar structures at a distance less than thermal wavelength [19]. One structure (hereafter referred to as active side) has a phase change material and its counterpart has fixed material properties (passive side). Figure 6.1 introduces the two concepts of thermal diode proposed in this letter that consists of two structures at a distance of L = 100 nm. In both concepts, structure 1 (active side) contains top layer of phase change material VO 2 at temperature T 1 = 341 K + ∆T . On the passive side, structure 2 has its temperature T 2 = 341 K − ∆T . Mean temperature is chosen to be the phase transition temperature of VO 2 at 341 K. When T 1 > T 2 (referred to as forward bias), VO 2 layer is in metallic phase; when T 1 < T 2 (reverse bias), VO 2 layer is in insulator phase with its optical axis aligned along the distance between them.
Proposed configurations will be discussed in more details along with results.
Phase transition of VO 2 is not homogeneous and it happens gradually along temeprature [26,29]. Hence, transition from insulator to metallic phase should be considered complete only at 350 K [25]. We calculate rectification values at a minimal temperature difference of 20 K (∆T = ±10 K), as it reflects intrinsic properties of the proposed device around anchoring temperature. As the temperature dependence of radiative heat transfer is essentially non-linear, rectification ratio is a function of temperature difference (bias). It has been observed before that rectification ratio generally increases with bias near the anchoring temperature [11]. Higher rectification ratios have been predicted earlier, but mostly at larger biases and gaps smaller than 100 nm [19,12,20,21,22,25]. The proposed concepts in this letter exhibit highest rectification ratio in the existing literature for comparable operating conditions and separations.

Theoretical Fundamentals
To calculate heat flux for forward and reverse bias across near-field thermal diode, we use the well known expression of near-field radiative transfer obtained through dyadic Green's function formalism [30]. Radiative transfer between closely spaced objects can be calculated by As our proposed designs involve 1-D grating structure of VO 2 in vacuum, we use second order approximation of effective medium theory to obtain the dielectric properties given by the expressions [31,32,33] ε T E,2 = ε T E,0 1+ π 2 3 Λ λ 2 φ 2 (1 − φ) 2 (ε A −ε B ) 2 ε T E,0 (6.2a) where ε A and ε B are dielectric functions of the two materials (VO 2 and vacuum) in surface gratings, λ is the wavelength, Λ is grating period and filling ratio φ = w/Λ where w is width of VO 2 segment . The expressions for zeroth order effective dielectric functions ε T E,0 and ε T M,0 are given by [34,31] ε T E,0 = φε A + (1 − φ)ε B (6.3a) For triangular gratings as shown in Fig. 6.1(b), gratings can be treated as a composition of multiple layers of rectangular gratings each having decreasing filling ratio and period equal to that of parent grating [34]. We confirm that slicing the triangular structure into 100 layers is sufficient to achieve converging values of near-field heat flux.
Effective medium approximation (EMA) holds true when grating period is much less than wavelength of interest [33]. As this study deals with temperatures around 341 K, grating period (50 nm) is much less than the thermal wavelength (∼ 8.5 µm). Thus the condition of EMA is satisfied. When dealing with near-field radiative transfer pertaining to periodic gratings, criterion for the validity of EMA is somewhat different and effective medium theory holds true as long as gap between the planar structures (100 nm in this case) is greater than the grating period [35] (see discussions about dashed lines in Fig. 6.2).
VO 2 in insulator state (below 341 K) is anisotropic. In a plane perpendicular to optical axis (x − y plane in our case) known as ordinary mode, its dielectric function is ε O and it is ε E along the optical axis (z axis, extraordinary mode).
Both ε O and ε E can be calculated using classical oscillator formula ε(ω) = ε ∞ + N i=1 S i ω 2 i ω 2 i − jγ i ω − ω 2 . Experimental values of high-frequency constant ε ∞ , phonon frequency ω i , scattering rate γ i and oscillator strength S i can be found in Ref. [36]. Here, j is the imaginary unit. There exist eight phonon modes for ordinary and nine phonon modes for extraordinary dielectric function. In metallic state, VO 2 is isotropic and Drude model [36] is used to describe the dielectric function that is given by ε(ω) = −ω 2 p ε ∞ ω 2 − jωΓ . Dielectric function for BN [37] is of the form .
Here  Dielectric properties of gold can be found in Ref. [38].

Results
In order to explain the results of our calculations for the proposed designs shown in Fig. 6.1 we plot rectification ratio against gap for four different configurations in Fig. 6.2 for the temperature difference of 20 K (∆T = ±10 K). Before discussing the main design ( Fig. 6.1(a)) we consider the simplistic case which consists of bulk VO 2 on active side and bulk BN on passive side (case I). As insulator VO 2 and metallic VO 2 have different optical properties, one expects to observe some degree of asymmetry in heat flow. Such a configuration exhibits weak rectification which gradually increases at smaller gaps when surface waves become dominant. Next we consider using thin films. In case II, the active side has 1 µm layer of VO 2 over 1 µm thick gold and the passive side has 1 µm layer of BN on the top of 1 µm gold.
1 µm thick layer of gold is sufficient to block radiation from the substrate and to support the top layer as free standing thin layer is not practical [39]. This design shows an increased near-field rectification, far-field rectification being about the same. Rectification is stronger in near-field because of the asymmetry in heat flux due to different levels of tunneling of surface waves across the two interfaces. We now consider case III where thin film of VO 2 is replaced by a 1-D grating structure of height h = 0.5 µm, period Λ = 50 nm and filling ratio φ = 0.3. The passive side remains the same as in case II. This configuration shows a significant enhancement in rectification and rectification value reaches around 14 at the gap of 100 nm. Figure 6.2 also shows dependence of rectification ratio on gap for an alternative design (case IV) depicted in Fig. 6.1(b). This design has a 1-D triangular structure of height h = 0.5 µm and period Λ = 50 nm on active side. While this structure shows a similar trend as case III, a sharp increase in rectification ratio can be seen at smaller gaps. Rectification ratio of 16 is reached at 100 nm gap. Although results for the distances smaller than 100 nm may not be accurate completely as gap becomes comparable to the grating period (displayed using dashed lines in Fig. 6.2). The trend is noteworthy for the triangular structure (case IV) as rectification ratio keeps increasing to higher values for shorter distances. Further investigations for triangular grating are beyond the scope of this study and left for future work. Numerical methods such as Wiener chaos expansion method [40] can be employed for calculation of near-field heat flux for distances shorter than the grating period. Figure 6.2 inset displays heat flux versus temperature difference for the four cases studied. Difference between slopes for forward and reverse bias is obvious. Diode-like characteristics are apparent especially for designs based on Transmission is high for two prominent frequencies that are close to characteristic wavelengths of BN (7.6 µm and 9.8 µm), as shown by dashed lines. Since metallic VO 2 does not support surface phonon polariton in the infrared region [25], nearfield radiative transfer is mainly attributed to the symmetric and antisymmetric surface phonons supported by the BN layer (k ρ c/ω 1 in Fig. 6.3). In addition, there exists a secondary contribution due to Fabry-Perot modes and frustrated modes (k ρ c/ω ≈ 1) in the near-field regime. High energy transmission in forward bias is due to tunneling of surface waves across interfaces. In reverse bias, although both BN and insulator VO 2 support surface phonon modes, they do not overlap and near-field radiative transfer is dominated by non-resonant surface waves. In addition, surface phonons of insulator VO 2 occur in frequency range where BN has low extinction coefficient (κ ≈ 0), and vice versa. As a result, tunneling between BN and insulator VO 2 is much weaker than that of BN and metallic VO 2 , that leads to thermal rectification. We note that when thin film of VO 2 is replaced by a 1-D rectangular grating, transmission coefficient for reverse bias is reduced as seen in Figs. 6.3(b) and 6.3(d). Reduction in transmission coefficient comes from the presence of grating which suppresses the tunneling of surface waves supported by insulator VO 2 and BN. Tunneling between BN and metallic VO 2 however is relatively unchanged (Figs. 6.3(a) and 6.3(c)). Consequently, a higher rectification is achieved. Resulting difference in spectral heat fluxes can also be observed.
We analyze possible parameters that can influence the rectification for the design using 1-D rectangular grating (case III) for a gap of 100 nm in Fig. 6.4. Filling ratio is fixed to 0.3 and the grating height is varied from 0.1 to 0.9 µm (dashed line). We observe that peak rectification ratio is achieved when grating height is around 0.5 µm. Solid line in Fig. 6.4 shows variation in filling ratio for fixed grating height of 0.5 µm. Dependence of rectification on filling ratio is clearly stronger, as it directly affects the optical properties of grating structure therefore influencing the surface waves across the interfaces. For higher filling ratios, rectification values are lower, and understandably, are close to what is predicted for a thin film design (case II). Thermal rectification even higher than 20 can be predicted at smaller filling ratios. Rectification reaches maximum value when filling ratio is around 0.03, and it becomes zero when filling ratio is zero (Fig. 4 inset). However, filling ratio of 0.03 is impractical as grating period is 50 nm. Another parameter of interest would be grating period. Since we focus on gap of 100 nm we have to limit ourselves to periods less than 100 nm. As the period is much smaller than the dominant thermal wavelength, it has virtually no effect on rectification.
Thus we have demonstrated that an enhanced thermal rectification can be achieved using 1-D grating of phase change material VO 2 . Our calculations indicate that a reasonably high value of rectification ratio (∼ 16) can be obtained at a gap of 100 nm. Rectification ratio can be optimized by tuning parameters such as grating height and especially, filling ratio. Materials and structures on passive side also play a significant role in the rectification. Improved rectification is attributed to reduced tunneling of surface waves across the interfaces for reverse bias. Rectification ratio can be further increased to much higher values for distance shorter than 100 nm and it is a viable candidate for future investigations.