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Shanhui Fan - One of the best experts on this subject based on the ideXlab platform.
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theoretical constraints on reciprocal and non reciprocal many body Radiative Heat Transfer
Physical Review B, 2020Co-Authors: Cheng Guo, Shanhui FanAbstract:We study the constraints on reciprocal and non-reciprocal many-body Radiative Heat Transfer imposed by symmetry and the second law of thermodynamics. We show that the symmetry of such a many-body system in general forms a magnetic group, and the constraints of the magnetic group on the Heat Transfer can be derived using a generalized reciprocity theorem. We also show that the second law of thermodynamics provides additional constraints in the form of a nodal conservation law of Heat flow at equilibrium. As an application, we provide a systematic approach to determine the existence of persistent Heat current in arbitrary many-body systems.
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nonreciprocal Radiative Heat Transfer between two planar bodies
Conference on Lasers and Electro-Optics, 2020Co-Authors: Lingling Fan, Bo Zhao, Yu Guo, Georgia T Papadakis, Zhexin Zhao, Siddharth Buddhiraju, Meir Orenstein, Shanhui FanAbstract:We analyze nonreciprocal Radiative Heat Transfer in two-body planar systems, identify unique nonreciprocal effects and introduce a general constraint from the second law of thermodynamics and reciprocity. Our findings are demonstrated numerically with magneto-optical materials.
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nonreciprocal Radiative Heat Transfer between two planar bodies
Physical Review B, 2020Co-Authors: Lingling Fan, Bo Zhao, Yu Guo, Georgia T Papadakis, Zhexin Zhao, Siddharth Buddhiraju, Meir Orenstein, Shanhui FanAbstract:We study the consequence of breaking reciprocity within the context of near-field Radiative Heat Transfer between two planar bodies. Our findings introduce a thermodynamic constraint, which states that the Heat Transferred from one planar body to another at each frequency and in-plane wave vector is unchanged upon interchanging the two bodies, regardless of whether the materials are reciprocal or not. We further identify a unique signature of nonreciprocity, which is the breaking of the symmetry of the Heat flux density between positive and negative in-plane wave vectors. We numerically demonstrate our findings in an example system consisting of magneto-optical materials. Our formalism applies to both near- and far-field regimes, opening opportunities for exploiting nonreciprocity in two-body Radiative Heat Transfer systems.
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mesh a free electromagnetic solver for far field and near field Radiative Heat Transfer for layered periodic structures
Computer Physics Communications, 2018Co-Authors: Kaifeng Chen, Bo Zhao, Shanhui FanAbstract:Abstract We describe MESH (M ultilayer E lectromagnetic S olver for H eat Transfer), a free software that combines rigorous coupled wave analysis (RCWA) and scattering matrix formalism to simulate the Radiative Heat Transfer both in the near-field and far-field regimes for layered three-dimensional structures made of planar layers. Each layer can have in-plane one-dimensional or two-dimensional periodicity. In this paper, we provide detailed discussions of the algorithms of MESH, which enables it to be a flexible tool for different types of Radiative Heat Transfer simulations. We also discuss aspects of the codes related to parallelization and user scripting. Program summary Program Title: MESH Program Files doi: http://dx.doi.org/10.17632/zx9v3bf3hf.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: C, C++. External routines: Lua[1], Python[2] and LAPACK and BLAS linear-algebra software[3], and optionally MPI message-passing interface[4]. Armadillo[5] is already contained in MESH. Nature of problem: Far-field and near-field Radiative Heat Transfer in layered periodic structures. Solution method: Fourier modal method (rigorous coupled wave analysis) and the scattering matrix method. [1] R. Ierusalimschy, L.H. de Figueiredo, W.C. Filho, Lua an extensible extension language, Software: Practice and Experience 26 (1996) 635652. http://www.lua.org . [2] Python Software Foundation. Available at http://www.python.org [3] MKL: https://software.intel.com/en-us/intel-mkl [4] T.M. Forum, MPI: A Message Passing Interface, in: Supercomputing 93, Portland, OR, 878883, 1993 [5] Conrad Sanderson and Ryan Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software , Vol. 1, pp. 26, 2016. http://dx.doi.org/10.21105/joss.00026
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enhancing near field Radiative Heat Transfer with si based metasurfaces
Physical Review Letters, 2017Co-Authors: Juan Carlos Cuevas, Shanhui Fan, Victor Fernandezhurtado, F J GarciavidalAbstract:We demonstrate in this work that the use of metasurfaces provides a viable strategy to largely tune and enhance near-field Radiative Heat Transfer between extended structures. In particular, using a rigorous coupled wave analysis, we predict that Si-based metasurfaces featuring two-dimensional periodic arrays of holes can exhibit a room-temperature near-field Radiative Heat conductance much larger than any unstructured material to date. We show that this enhancement, which takes place in a broad range of separations, relies on the possibility to largely tune the properties of the surface plasmon polaritons that dominate the Radiative Heat Transfer in the near-field regime.
Juan Carlos Cuevas - One of the best experts on this subject based on the ideXlab platform.
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super planckian far field Radiative Heat Transfer
Physical Review B, 2018Co-Authors: Victor Fernandezhurtado, Johannes Feist, Juan Carlos Cuevas, Antonio I Fernandezdominguez, F J GarciavidalAbstract:We present here a theoretical analysis that demonstrates that the far-field Radiative Heat Transfer between objects with dimensions smaller than the thermal wavelength can overcome the Planckian limit by orders of magnitude. To guide the search for super-Planckian far-field Radiative Heat Transfer, we make use of the theory of fluctuational electrodynamics and derive a relation between the far-field Radiative Heat Transfer and the directional absorption efficiency of the objects involved. Guided by this relation, and making use of state-of-the-art numerical simulations, we show that the far-field Radiative Heat Transfer between highly anisotropic objects can largely overcome the black-body limit when some of their dimensions are smaller than the thermal wavelength. In particular, we illustrate this phenomenon in the case of suspended pads made of polar dielectrics like SiN or ${\mathrm{SiO}}_{2}$. These structures are widely used to measure the thermal transport through nanowires and low-dimensional systems and can be employed to test our predictions. Our work illustrates the dramatic failure of the classical theory to predict the far-field Radiative Heat Transfer between micro- and nanodevices.
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anisotropic thermal magnetoresistance for an active control of Radiative Heat Transfer
ACS Photonics, 2018Co-Authors: Ricardo Martin Abraham Ekeroth, Juan Carlos Cuevas, Philippe Benabdallah, Antonio GarciamartinAbstract:The discovery that the near-field Radiative Heat Transfer enables to overcome the limit set by Planck’s law holds the promise to have an impact in different nanotechnologies that make use of thermal radiation, and the challenge now is to find strategies to actively control and manipulate this near-field thermal radiation. Here, we predict a huge anisotropic thermal magnetoresistance (ATMR) in the near-field Radiative Heat Transfer between magneto-optical particles when the direction of an external magnetic field is changed with respect to the Heat current direction. We illustrate this effect with the case of two InSb particles where we find that the ATMR amplitude can reach values of up to 800% for a magnetic field of 5 T, which is many orders of magnitude larger than its spintronic analogue. This thermomagnetic effect could find broad applications in the field of ultrafast thermal management as well as magnetic and thermal remote sensing.
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thermal discrete dipole approximation for the description of thermal emission and Radiative Heat Transfer of magneto optical systems
Physical Review B, 2017Co-Authors: Juan Carlos Cuevas, Ricardo Martin Abraham Ekeroth, Antonio GarciamartinAbstract:We present here a generalization of the thermal discrete dipole approximation (TDDA) that allows us to describe the near-field Radiative Heat Transfer between finite objects of arbitrary shape that exhibit magneto-optical (MO) activity. We also extend the TDDA approach to describe the thermal emission of a finite object with and without MO activity. Our method is also valid for optically anisotropic materials described by an arbitrary permittivity tensor and we provide simple closed formulas for the basic thermal quantities that considerably simplify the implementation of the TDDA method. Moreover, we show that by employing our TDDA approach one can rigorously demonstrate Kirchhoff's radiation law relating the emissivity and absorptivity of an arbitrary MO object. Our work paves the way for the theoretical study of the active control of emission and Radiative Heat Transfer between MO systems of arbitrary size and shape.
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Study of Radiative Heat Transfer in Ångström- and nanometre-sized gaps
Nature communications, 2017Co-Authors: Longji Cui, Edgar Meyhöfer, Wonho Jeong, Víctor Fernández-hurtado, Johannes Feist, Francisco J. Garcia-vidal, Juan Carlos Cuevas, Pramod ReddyAbstract:Radiative Heat Transfer in Angstrom- and nanometre-sized gaps is of great interest because of both its technological importance and open questions regarding the physics of energy Transfer in this regime. Here we report studies of Radiative Heat Transfer in few A to 5 nm gap sizes, performed under ultrahigh vacuum conditions between a Au-coated probe featuring embedded nanoscale thermocouples and a Heated planar Au substrate that were both subjected to various surface-cleaning procedures. By drawing on the apparent tunnelling barrier height as a signature of cleanliness, we found that upon systematically cleaning via a plasma or locally pushing the tip into the substrate by a few nanometres, the observed Radiative conductances decreased from unexpectedly large values to extremely small ones—below the detection limit of our probe—as expected from our computational results. Our results show that it is possible to avoid the confounding effects of surface contamination and systematically study thermal radiation in Angstrom- and nanometre-sized gaps. Here, Cuiet al. report Radiative Heat Transfer in few Angstrom to 5 nm gap sizes, between a gold-coated probe and a Heated planar gold substrate subjected to various surface cleaning procedures. They found that insufficiently cleaned probes and substrates led to unexpectedly large Radiative thermal conductances.
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enhancing near field Radiative Heat Transfer with si based metasurfaces
Physical Review Letters, 2017Co-Authors: Juan Carlos Cuevas, Shanhui Fan, Victor Fernandezhurtado, F J GarciavidalAbstract:We demonstrate in this work that the use of metasurfaces provides a viable strategy to largely tune and enhance near-field Radiative Heat Transfer between extended structures. In particular, using a rigorous coupled wave analysis, we predict that Si-based metasurfaces featuring two-dimensional periodic arrays of holes can exhibit a room-temperature near-field Radiative Heat conductance much larger than any unstructured material to date. We show that this enhancement, which takes place in a broad range of separations, relies on the possibility to largely tune the properties of the surface plasmon polaritons that dominate the Radiative Heat Transfer in the near-field regime.
Alejandro W. Rodriguez - One of the best experts on this subject based on the ideXlab platform.
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Fundamental limits to Radiative Heat Transfer: Theory
Physical Review B, 2020Co-Authors: Sean Molesky, Prashanth S. Venkataram, Weiliang Jin, Alejandro W. RodriguezAbstract:Radiative Heat Transfer between bodies at the nanoscale can surpass blackbody limits on thermal radiation by orders of magnitude due to contributions from evanescent electromagnetic fields, which carry no energy to the far field. Thus far, principles guiding explorations of larger Heat Transfer beyond planar structures have assumed utility in surface nanostructuring, via enhancement of the density of states, and the possibility that such design paradigms can approach Landauer limits, in analogy to conduction. Here we derive fundamental shape-independent limits to Radiative Heat Transfer, applicable in near- through far-field regimes, that incorporate material and geometric constraints such as intrinsic dissipation and finite object sizes, and show that these preclude reaching the Landauer limits in all but a few restrictive scenarios. Additionally, we show that the interplay of material response and electromagnetic scattering among proximate bodies means that bodies which maximize Radiative Heat Transfer actually maximize scattering rather than absorption. Finally, we compare our new bounds to Landauer limits as well as limits that ignore the interplay between material and geometric constraints, and show that these prior limits lead to overly optimistic predictions. Our results have ramifications for the ultimate performance of thermophotovoltaics and nanoscale cooling, as well as incandescent and luminescent devices.
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fundamental limits to Radiative Heat Transfer the limited role of nanostructuring in the near field
Physical Review Letters, 2020Co-Authors: Prashanth S. Venkataram, Sean Molesky, Weiliang Jin, Alejandro W. RodriguezAbstract:In a previous Letter, we derived fundamental limits to Radiative Heat Transfer applicable in near- through far-field regimes, based on the choice of material susceptibilities and bounding surfaces enclosing arbitrarily shaped objects; the limits exploit algebraic properties of Maxwell's equations and fundamental principles such as electromagnetic reciprocity and passivity. In this Letter, we apply these bounds to two different geometric configurations of interest, namely dipolar particles or extended structures of infinite area in the near field of one another. We find that while near-field Radiative Heat Transfer between dipolar particles can saturate purely geometric ``Landauer'' limits, bounds on extended structures cannot, instead growing very slowly with respect to a material response figure of merit (an ``inverse resistivity'' for metals) due to the deleterious effects of multiple scattering between bodies. While nanostructuring can produce infrared resonances, it is generally unable to further enhance the resonant energy Transfer spectrum beyond what is practically achieved by planar media at the surface polariton condition.
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upper limits to near field Radiative Heat Transfer generalizing the blackbody concept
Proceedings of SPIE, 2016Co-Authors: Owen D Miller, Alejandro W. Rodriguez, Steven G JohnsonAbstract:For 75 years it has been known that Radiative Heat Transfer can exceed far-field blackbody rates when two bodies are separated by less than a thermal wavelength. Yet an open question has remained: what is the maximum achievable Radiative Transfer rate? Here we describe basic energy-conservation principles that answer this question, yielding upper bounds that depend on the temperatures, material susceptibilities, and separation distance, but which encompass all geometries. The simple structures studied to date fall far short of the bounds, offering the possibility for significant future enhancement, with ramifications for experimental studies as well as thermophotovoltaic applications.
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shape independent limits to near field Radiative Heat Transfer
Physical Review Letters, 2015Co-Authors: Owen D Miller, Steven G Johnson, Alejandro W. RodriguezAbstract:We derive shape-independent limits to the spectral Radiative Heat Transfer rate between two closely spaced bodies, generalizing the concept of a blackbody to the case of near-field energy Transfer. Through conservation of energy and reciprocity, we show that each body of susceptibility χ can emit and absorb radiation at enhanced rates bounded by |χ|(2)/Im χ, optimally mediated by near-field photon Transfer proportional to 1/d(2) across a separation distance d. Dipole-dipole and dipole-plate structures approach restricted versions of the limit, but common large-area structures do not exhibit the material enhancement factor and thus fall short of the general limit. By contrast, we find that particle arrays interacting in an idealized Born approximation (i.e., neglecting multiple scattering) exhibit both enhancement factors, suggesting the possibility of orders-of-magnitude improvement beyond previous designs and the potential for Radiative Heat Transfer to be comparable to conductive Heat Transfer through air at room temperature, and significantly greater at higher temperatures.
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fluctuating surface current formulation of Radiative Heat Transfer for arbitrary geometries
APS, 2012Co-Authors: Alejandro W. Rodriguez, M Homer T Reid, Steven G JohnsonAbstract:We describe a fluctuating-surface-current formulation of Radiative Heat Transfer, applicable to arbitrary geometries in both the near and far field, that directly exploits efficient and sophisticated techniques from the boundary-element method. We validate as well as extend previous results for spheres and cylinders, and also compute the Heat Transfer in a more complicated geometry consisting of two interlocked rings. Finally, we demonstrate how this method can be adapted to compute the spatial distribution of Heat flux on the surfaces of the bodies.
Bo Zhao - One of the best experts on this subject based on the ideXlab platform.
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nonreciprocal Radiative Heat Transfer between two planar bodies
Conference on Lasers and Electro-Optics, 2020Co-Authors: Lingling Fan, Bo Zhao, Yu Guo, Georgia T Papadakis, Zhexin Zhao, Siddharth Buddhiraju, Meir Orenstein, Shanhui FanAbstract:We analyze nonreciprocal Radiative Heat Transfer in two-body planar systems, identify unique nonreciprocal effects and introduce a general constraint from the second law of thermodynamics and reciprocity. Our findings are demonstrated numerically with magneto-optical materials.
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nonreciprocal Radiative Heat Transfer between two planar bodies
Physical Review B, 2020Co-Authors: Lingling Fan, Bo Zhao, Yu Guo, Georgia T Papadakis, Zhexin Zhao, Siddharth Buddhiraju, Meir Orenstein, Shanhui FanAbstract:We study the consequence of breaking reciprocity within the context of near-field Radiative Heat Transfer between two planar bodies. Our findings introduce a thermodynamic constraint, which states that the Heat Transferred from one planar body to another at each frequency and in-plane wave vector is unchanged upon interchanging the two bodies, regardless of whether the materials are reciprocal or not. We further identify a unique signature of nonreciprocity, which is the breaking of the symmetry of the Heat flux density between positive and negative in-plane wave vectors. We numerically demonstrate our findings in an example system consisting of magneto-optical materials. Our formalism applies to both near- and far-field regimes, opening opportunities for exploiting nonreciprocity in two-body Radiative Heat Transfer systems.
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mesh a free electromagnetic solver for far field and near field Radiative Heat Transfer for layered periodic structures
Computer Physics Communications, 2018Co-Authors: Kaifeng Chen, Bo Zhao, Shanhui FanAbstract:Abstract We describe MESH (M ultilayer E lectromagnetic S olver for H eat Transfer), a free software that combines rigorous coupled wave analysis (RCWA) and scattering matrix formalism to simulate the Radiative Heat Transfer both in the near-field and far-field regimes for layered three-dimensional structures made of planar layers. Each layer can have in-plane one-dimensional or two-dimensional periodicity. In this paper, we provide detailed discussions of the algorithms of MESH, which enables it to be a flexible tool for different types of Radiative Heat Transfer simulations. We also discuss aspects of the codes related to parallelization and user scripting. Program summary Program Title: MESH Program Files doi: http://dx.doi.org/10.17632/zx9v3bf3hf.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: C, C++. External routines: Lua[1], Python[2] and LAPACK and BLAS linear-algebra software[3], and optionally MPI message-passing interface[4]. Armadillo[5] is already contained in MESH. Nature of problem: Far-field and near-field Radiative Heat Transfer in layered periodic structures. Solution method: Fourier modal method (rigorous coupled wave analysis) and the scattering matrix method. [1] R. Ierusalimschy, L.H. de Figueiredo, W.C. Filho, Lua an extensible extension language, Software: Practice and Experience 26 (1996) 635652. http://www.lua.org . [2] Python Software Foundation. Available at http://www.python.org [3] MKL: https://software.intel.com/en-us/intel-mkl [4] T.M. Forum, MPI: A Message Passing Interface, in: Supercomputing 93, Portland, OR, 878883, 1993 [5] Conrad Sanderson and Ryan Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software , Vol. 1, pp. 26, 2016. http://dx.doi.org/10.21105/joss.00026
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near field Radiative Heat Transfer between doped si parallel plates separated by a spacing down to 200 nm
Applied Physics Letters, 2016Co-Authors: J I Watjen, Bo ZhaoAbstract:Heat Transfer between two objects separated by a nanoscale vacuum gap holds great promise especially in energy harvesting applications such as near-field thermophotovoltaic systems. However, experimental validation of nanoscale Radiative Heat Transfer has been largely limited to tip-plate configurations due to challenges of maintaining small gap spacing over a relatively large area. Here, we report measurements of Heat Transfer near room temperature between two 1 cm by 1 cm doped-Si parallel plates, separated by a vacuum gap from about 200 nm to 780 nm. The measured strong near-field Radiative Transfer is in quantitative agreement with the theoretical prediction based on fluctuational electrodynamics. The largest measured Radiative Heat flux is 11 times as high as the blackbody limit for the same hot and cold surface temperatures. Our experiments have produced the highest Radiative Heat Transfer rate observed to date across submicron distances between objects near room temperature.
Z M Zhang - One of the best experts on this subject based on the ideXlab platform.
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comparison of kinetic theory and fluctuational electrodynamics for Radiative Heat Transfer in nanoparticle chains
Journal of Quantitative Spectroscopy & Radiative Transfer, 2020Co-Authors: Baratunde A Cola, Eric J Tervo, Z M ZhangAbstract:Abstract In chains of closely spaced nanoparticles supporting surface polaritons, near-field electromagnetic coupling leads to collective effects and super-Planckian thermal radiation exchange. Researchers have primarily used two analytical approaches to calculate Radiative Heat Transfer in these systems: fluctuational electrodynamics, which directly incorporates fluctuating thermal currents into Maxwell's equations, and a kinetic approach where the dispersion relation provides modes and propagation lengths for the Boltzmann transport equation. Here, we compare results from the two approaches in the isotropic dipole limit in order to identify regimes in which kinetic theory is valid and to explain differing results in the literature on its validity. Using both methods, we calculate the diffusive Radiative thermal conductivity of nanoparticle chains. We show that kinetic theory is valid and matches predictions by fluctuational electrodynamics if both the surface polariton propagation lengths are greater than the particle spacing and surface polaritons are the dominant contributors to Radiative Heat Transfer.
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validity of kinetic theory for Radiative Heat Transfer in nanoparticle chains
Proceeding of Proceedings of the 9th International Symposium on Radiative Transfer RAD-19, 2019Co-Authors: Eric J Tervo, Baratunde A Cola, Z M ZhangAbstract:In chains of closely-spaced nanoparticles supporting surface polaritons, near-field electromagnetic coupling leads to collective effects and super-Planckian thermal radiation exchange. Researchers have primarily used two analytical approaches to calculate Radiative Heat Transfer in these systems: fluctuational electrodynamics, which directly incorporates fluctuating thermal currents into Maxwell's equations, and a kinetic approach where the dispersion relation provides modes and propagation lengths for the Boltzmann transport equation. Here, we compare results from the two approaches in order to identify regimes in which kinetic theory is valid and to explain differing results in the literature on its validity. Using both methods, we calculate the diffusive Radiative thermal conductivity of nanoparticle chains. We show that kinetic theory is valid and matches predictions by fluctuational electrodynamics when the propagation lengths are greater than the particle spacing.
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application conditions of effective medium theory in near field Radiative Heat Transfer between multilayered metamaterials
Journal of Heat Transfer-transactions of The Asme, 2014Co-Authors: Xianglei Liu, T J Bright, Z M ZhangAbstract:This work addresses the validity of the local effective medium theory (EMT) in predicting the near-field Radiative Heat Transfer between multilayered metamaterials, separated by a vacuum gap. Doped silicon and germanium are used to form the metallodielectric superlattice. Different configurations are considered by setting the layers adjacent to the vacuum spacer as metal–metal (MM), metal–dielectric (MD), or dielectric–dielectric (DD) (where M refers to metallic doped silicon and D refers to dielectric germanium). The calculation is based on fluctuational electrodynamics using the Green's function formulation. The cutoff wave vectors for surface plasmon polaritons (SPPs) and hyperbolic modes are evaluated. Combining the Bloch theory with the cutoff wave vector, the application condition of EMT in predicting near-field Radiative Heat Transfer is presented quantitatively and is verified by exact calculations based on the multilayer formulation.
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energy streamlines in near field Radiative Heat Transfer between hyperbolic metamaterials
Optics Express, 2014Co-Authors: T J Bright, Xianglei Liu, Z M ZhangAbstract:Metallodielectric photonic crystals having hyperbolic dispersions are called indefinite materials because of their ability to guide modes with extremely large lateral wavevectors. While this is useful for enhancing near-field Radiative Heat Transfer, it could also give rise to large lateral displacements of the energy pathways. The energy streamlines can be used to depict the flow of electromagnetic energy through a structure when wave propagation does not follow ray optics. We obtain the energy streamlines through two semi-infinite uniaxial anisotropic effective medium structures, separated by a small vacuum gap, using the Green functions and fluctuation-dissipation theorem. The lateral shifts are determined from the streamlines within two penetration depths. For hyperbolic modes, the predicted lateral shift can be several thousand times of the vacuum gap width.
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near field Radiative Heat Transfer with doped silicon nanostructured metamaterials
International Journal of Heat and Mass Transfer, 2014Co-Authors: Xianglei Liu, Richard Z Zhang, Z M ZhangAbstract:Abstract The objective of this work is to evaluate different practically achievable doped-silicon (D-Si) nanostructured metamaterials (including nanowires and nanoholes, multilayers, and one-dimensional gratings) in terms of their potential for enhancing near-field Radiative Heat Transfer at ambient temperature. It is found that both doped silicon nanowires and nanoholes may achieve an enhancement over bulk doped silicon by more than one order of magnitude in the deep submicron gap region. The enhancement is attributed to either the broadband hyperbolic modes or low-loss surface modes or a combination of both. On the other hand, polarization coupling, which can occur in the grating configuration, contributes little to the Radiative Transfer at the nanometer scale. This work will facilitate the application of nanostructures in more efficient non-contact thermal management, thermal imaging, and near-field thermophotovoltaics.
