Spectral thermal gaps due to plasmon-phonon mode interaction in bilayer systems
J. E. Pérez-Rodríguez Giuseppe Pirruccio Raúl Esquivel-Sirvent




We present a theoretical study of the modification of the near-field radiative heat transfer due to phonon-plasmon coupling in bilayer systems made of a doped semiconductor and a polar dielectric. By tuning the surface-plasmon mode of the former material in resonance with the phonon modes of the latter one, the near-field spectral radiative heat transfer can be suppressed in a frequency band gap.  We distinguish between the interlayer mode coupling within each bilayer and the intragap one between the two bilayers. We elucidate the role of each of them in the formation of the band gap and we determine the distance range at which one mechanism dominates over the other.  Furthermore, we show that the surface plasmon polariton of the top-most layer allows optimizing the total heat transfer as a function of the separation of the two bilayers.


 
Seeded Graph Matching via Large Neighborhood Statistics
    
Elchanan Mossel 

MIT

elmos@mit.edu
Jiaming Xu

Duke University

jiaming.xu868@duke.edu

    26.7.2018
==============================================================================================





§ INTRODUCTION

Near-field radiative heat transfer (NFRHT) between two bodies is characterized by an increase of the radiated power above the prediction of the Stephan-Boltzmann law. At nanometric distances a dependence of the NFRHT on the dielectric properties of the emitting materials and on the separation between the bodies has been predicted and experimentally observed in plate-plate <cit.>, sphere-plate <cit.> and tip-plate <cit.> configuration.

NFRHT is relevant at the micron and submicron scale <cit.>, thus proving crucial in various applications in micro and nanoengineering. The field of thermotronics has been initiated by the introduction of thermal rectifiers <cit.>, thermal diodes <cit.> and thermal transistors <cit.>. Moreover,  proper engineering of the geometry and materials permits the control of local hot spots and heat sinks <cit.> in nanostructures. On the other hand, from the fundamental point of view, understanding the physics behind the NFRHT has led to the study of the thermal properties of novel materials such as graphene <cit.>, graphene multilayers <cit.>, porous materials <cit.>, metamaterials <cit.>. Besides, external magnetic fields have been proposed as a way to actively control the spectral heat flux <cit.>. The basic idea of these works is to modify the dielectric function of the materials  to change the dispersion relation of light in the cavity or gap between the bodies, thus influencing the NFRHT.

Polariton mode interaction plays a role in NFRHT. For two half-spaces of SiC separated by a gap, the spectral heat transfer is maximum close to the transverse bulk phonon frequency. If one of the half-spaces is replaced by a thin film, the phonon modes split, thus increasing the heat transfer <cit.>. By replacing the slabs with layered structures, a super-Planckian behavior arises either from the surface-phonon or hyperbolic modes of the superlattice depending on the dielectric functions of the layers <cit.>.  If the topmost layer does not support surface modes, the strong enhancement in the NFRHT is not present <cit.>.  Metallic films usually exhibit poorer thermal performances in the infrared as compared to dielectric ones, due to their high plasma frequency compared to the thermal one. A way to circumvent this problem is to make use of porosity to redshift the surface plasmon modes <cit.>. In this way, NFRHT can be further modified in systems where plasmon and phonon modes coexist in a narrow frequency range. This situation is realized, for example, in bilayers composed of a NaBr substrate coated by a layer of porous Bi. The coupling between the broad gap surface plasmon and the narrow surface phonon mode gives rise to thermal Fano resonances, at which the spectral heat transfer could be suppressed <cit.>,  paving the way for further exploration of the control of the NFRHT by engineering the polariton mode coupling at the interfaces of the system.

Thin layers of porous materials may be challenging to fabricate. In contrast, the growth of semiconductor layers such as GaAs, InP, InSb, InAs, AlAs, GaP, Si, or Ge is a well-established technology. By continuously varying their free electron density,  doped semiconductors can be made to behave as dielectrics or as metals. This is achieved in several ways, i.e., by thermal tuning of the intrinsic carrier concentration, ion implantation and optical excitation <cit.>. Due to their naturally low plasma frequency, doped semiconductors are ideal materials to control the NFRHT in the near to far infrared region <cit.>. For some of these materials plasmon and phonon modes coexist in a narrow frequency range creating a rich system in which the NFRHT can be modified by the formation of new coupled plasmon-phonon modes.

In this work, we show the formation and control of a thermal band gap in the spectral heat flux between two semiconductor-dielectric bilayers separated by a gap. Correspondingly, almost complete suppression of the heat transfer is observed in a large, frequency-tailorable bandwidth. We distinguish between the strong interlayer coupling taking place at each semiconductor-dielectric interface and the intragap coupling between the two bilayers across the vacuum that separates them. We elucidate that the former plasmon-phonon coupling is responsible for the opening of the thermal band gap while the latter provides a common channel for the energy flow. Interestingly, a non-trivial mode competition between these two arises when the gap distance is varied. A mismatch between the resonant frequencies of the two bilayers results in the inhibition of the heat flow, even though each of them sustains a resonant surface mode. 





§ SYSTEM AND MATERIALS


Tuning of the NFRHT relies on a judicious choice of materials that interact in such a way as to enhance or suppress the heat transfer depending on their dielectric functions an geometry. We consider bilayer systems made by stacking a doped-semiconductor (GaAs) and a polar dielectric (NaBr) so that the plasma frequency of the former material lies close to the longitudinal and transverse phonons of the latter one. The thicknesses of each layer are d_1=50 nm and d_2=500 nm, respectively. We study the NFRHT between two of these bilayers kept at temperature T_1 and T_2, respectively, and separated by a gap L. The system is shown in Fig. 1 from top to bottom, where the left column depicts the individual layers and the total system.


The dielectric functions for the doped GaAs is

    ϵ_1(ω)=ϵ_1∞-ω_p^2/ω^2+iγω+ϵ_1∞ω_1LO^2-ω_1TO^2/ω_1TO^2-ω^2+iγ_1 ω,

where, ϵ_1∞=12.48 and the first Drude term describes the metallic behavior of the semiconductor with carrier effective mass m_eff=0.064m_e, plasma frequency ω_p=123.1 THz, and damping γ=18.84 THz. The second term contains the transverse and longitudinal phonon frequencies ω_1T0=8 THz and ω_1L0=8.54 THz, respectively, and the damping γ_1=0.072 THz.
The dielectric function of NaBr is

    ϵ_2(ω)=ϵ_2∞ω_2LO^2-ω_2TO^2/ω_2TO^2-ω^2+iγ_2 ω,

where ω_2LO=39 THz, ω_2TO=25 THz and γ_2=0.4 THz correspond to the longitudinal phonon frequency, transverse phonon frequency and damping, respectively. The high frequency value of the dielectric function is ϵ_2∞=2.6.





§ NEAR FIELD HEAT TRANSFER

The near-field heat transfer is calculated within the framework of Rytov's theory of fluctuation electrodynamics <cit.>. Throughout this work, the wavevector component normal to the interfaces in vacuum is κ=√(ω^2/c^2-β^2), and for the i^th material is κ_i=√(ϵ_i ω^2/c^2-β^2), where ω is the frequency and β is the parallel component of the wave vector to
the interfaces.
  The total heat flux Q(L,T_1,T_2) is
  
    Q(L,T_1,T_2)=∫_0^∞ dω S_ω

where   S_ω is the spectral heat function given by,


    S_ω(L,T_1,T_2)=[ Θ(ω,T_2)-Θ(ω,T_1) ]∑_j=p,s∫β dβ/(2π)^2 (τ_j^prop+τ_j^evan),

where Θ(ω,T)=ħω /(exp(ħω/K_B T)-1) is the Planckian function.
The sum is over both possible polarizations p and s. τ_ω^prop and τ_ω^evan are the densities of the propagating and evanescent modes, respectively, written in terms of the reflectivities r_p,s  as
 
    τ_p,s^prop=(1-|r_p,s|^2)^2/|1-r_p,s^2 e^2iκ L)|^2,

 
    τ_p,s^evan=4 Im(r_p,s)^2e^-2 |κ|  L/|1-r_p,s^2 e^-2|κ|L|^2.

We define the total transmission  as τ=τ_p,s^evan+τ_p,s^prop.
It is convenient to introduce the spectral heat transfer coefficient <cit.>, h_ω(T), defined as

    h_ω(T)=lim_Δ T → 0S_ω(T+Δ T)-S_ω(T)/Δ T,

where T is a mean temperature that we fix at  T=300 K.

The reflectivities are calculated using the surface impedance approach that captures the interaction of the modes excited in the GaAs and NaBr layers via the transfer matrices of the layers<cit.>.

 

§ RESULTS

To understand the thermal behavior of the cavity, we first analyze the individual layers and then the bilayered composite system. Since for the NFRHT the main contribution comes from the p-polarized waves, we calculate τ_p for the evanescent region (β>ω/c). The thickness of the layers is within the range of values allowed by current experimental techniques <cit.>. Throughout this work the gap separation remains constant and equal to L=50 nm. Figure 1 (a) shows two GaAs slabs of thickness d_1=50 nm and carrier density N=3×10^23 m^-3, that yields a plasma frequency of ω_p=123.1 THz together to  the corresponding τ_p calculated as a function of frequency, ω and normalized wavevector, β/k_0, where k_0=ω_0/c with ω_0=100 THz. Due to the metallic character of doped GaAs this system sustains a plasmonic waveguide mode or gap surface plasmon polariton (GSPP) <cit.>. The red dashed lines indicate the frequency at which the dispersion of the GSPP flattens, i.e. ω_GSPP=ω_p/√(ϵ_1∞+1)∼ 33 THz and the two bulk optical phonon frequencies of GaAs. Similarly, in Fig. 1 (b) two NaBr layers of thickness d_2=500 nm and the corresponding τ_p are shown. The blue dashed lines correspond to the transverse and longitudinal optical phonon modes for the bulk NaBr.  Note that the GSPP lies in the Reststrahlen region of NaBr, i.e., ω_2TO<ω_GSPP<ω_2LO, which provides the base for a complex mode interaction in the composite system.

The consequence of the plasmon-phonon hybridization in the NFRHT is shown in Fig. 1(c), where a clear reduction of the mode density leads to the opening of a thermal band gap around ω∼ 200 THz at which ω_GSPP≃ω_SPhP. We stress that this kind of band gap is not the consequence of spatial translational invariance and Bloch theorem. The opening of the thermal band gap in our non periodic structure is due to destructive interference between the GSPP and the SPhP. 

 This is a purely electromagnetic process related to the interference between the broadband, dispersive GSPP and the narrowband surface phonon polariton, which leads to the opening of a window of suppressed spectral heat flux. Here, we apply this concept to the thermal properties of a nanostructure, demonstrating that the interference between broad and narrow optical modes leads to the opening of an electromagnetically-induced window of suppressed spectral heat flux.


 


The thermal band gap is further analyzed in Fig. 2 where the spectral heat coefficient h_ω(T) is shown. The value of h_ω(T) is shown for two GaAs layers of thickness d_1=50 nm (dashed gray curve), two NaBr layers of thickness d_2=500 nm (dotted grey curve) and the bilayer system (continuous black curve). The mean temperature is kept at T=300 K. The mode hybridization suppresses the intense response of the GaAs and around ω=25 THz where h_ω shows a minimum value. This behavior occurs since ω_GSPP, lies between the longitudinal and transverse phonons. The phase and amplitude condition required to open the gap is realized for frequency values for which the contribution to the NFRHT from the GSPP is much larger than the surface phonon one, leading to the redshift of the thermal band gap with respect to both ω_GSPP and ω_SPhP the optical phonon frequencies.


  

The origin of the band gap in the spectral heat transfer can be explained qualitatively by analyzing the dispersion relation of the  surface modes  at the GaAs-air and NaBr-air interfaces and the coupled mode sustained by the  GaAs-NaBr surface.  We plot the frequency <cit.> ω vs k_x=k_sp=ω/cϵ(ω)/√(ϵ(ω)+1), where ϵ is the dielectric function of either material and for the bilayer the dielectric function for p-polarized light is calculated from <cit.> ϵ^-1=(d_1/(d_1+d_2))ϵ_1^-1+(d_2/(d_1+d_2))ϵ_2^-1. The typical values of N we consider are:  N=5×10^22 m^-3. In Fig. 3 (a) the dispersion relation is shown for NaBr, GaAs, the bilayer, and  the light cone. At high frequencies the dispersion relation of the bilayer follows  the dispersion of the polar material, while  at short frequencies it follows the one of  the GaAs. A zoom  is shown in Fig. 3(b), where  in an intermediate frequency region  the bilayer behaves as a hybrid material. Here  the modes  in the propagating region are responsible for the gap in the spectral thermal emission. 





In Fig. 4 we analyze how the thermal gap forms as the charge carrier density in GaAs changes thus detuning the plasma frequency from the phonon
frequencies of the NaBr, by calculating the  total transmission τ. For the s-polarization the dielectric function of the bilayer is given by <cit.> ϵ=(ϵ_1 d_1+ϵ_2 d_2)/(d_1+d_2) (a),  N=5×10^23 m^-3 (b), and N=5×10^24 m^-3 (c), all experimentally accessible doping levels.  For the lowest value of N ω_GSPP lies below ω_TO2 of NaBr and, therefore, the thermal gap opens. On the other hand, for the highest value of N ω_GSPP lies well above ω_TO2 and no mode coupling occurs. This is confirmed by the spectral heat transfer coefficient shown in Fig. 4(d), where no thermal gap is observed for large carrier densities and the radiative thermal properties of the system are given by the two almost independent contributions from GaAs and NaBr (see Fig. 1). Thus, the doping level provides an efficient and practical way to tailor the thermal band gap and the NFRHT in nanostructures made of doped semiconductors, a characteristic that might be relevant for semiconductor-based devices.



To elucidate the interplay between the intragap and interlayer mode couplings in determining the spectral thermal response of the bilayer system we use the quantity usually measured experimentally, i.e., the total heat transfer, Q (Eq. (3)). We define η_GaAs=(Q_GaAs-Q_GaAs/NaBr)/Q_GaAs, where Q_GaAs is the total heat transfer between two GaAs slabs of thickness d=50 nm, used as a reference. Figure 5 shows the value of η_GaAs as a function of the gap separation, L (solid curve). For all L η_GaAs is positive, so the heat transfer of the bilayer structure is always lower than the reference. We see a peculiar dependence of the total heat transfer with the gap separation L. For L<100 nm η_GaAs decreases rapidly towards zero, meaning that the total heat transfer between two bilayers of GaAs/NaBr approaches the one of the GaAs. Thus, the interaction between the two sides of the cavity is dominated by the metallic layer, the GSPP represents the main channel for the total heat transfer and the NaBr plays no crucial role. Plasmons and surface phonons do not couple efficiently in this case and we conclude that for this distance the intragap coupling dominates over the interlayer one. On the other hand, for L>100 nm, η_GaAs reaches a maximum of approximately 30% at a gap separation of L∼500 nm. We also define η_NaBr=(Q_GaAs-Q_NaBr/GaAs)/Q_GaAs that compares the total heat transfer of the bilayer when the order of the layers is reversed, i.e., the top-most layer is NaBr and the backing is GaAs. Again, the reference is Q_GaAs. In this case, for L<100 nm η_NaBr∼1, meaning that contrarily to the previous case, the contribution from the top-most layer is not important. In this case the interlayer plasmon-phonon coupling dominates over the intragap one. Interestingly, we do not observe a maximum for this layer order with the heat transfer of the bilayer system monotonically approaching the one of GaAs. Both curves go to zero for large L because the systems approach the far-field limit.

The GSPP-hybrid mode competition is explained considering the penetration depth of the surface plasmon polariton at the GaAs-air interface as compared with the thickness d_1. The penetration depth is
δ=1/Im(κ_i).  When the gap separation is L=50 nm we have δ∼30 nm, thus δ<d_1. Conversely, when L=500 nm, δ∼100 nm and δ>d_1. The values of the penetration depth depend in general on both frequency and parallel wave vector β and are calculated at points where τ_p is maximum. This analysis allows us to conclude that when the top-most layer is metallic the mode responsible for the heat transfer is very much confined and the system ignores the presence of other backing layers. While, when the top-most layer is dielectric the bilayer is sensitive to the presence of all the layers. 

 




§ CONCLUSIONS

The NFRHT can be modified by opening thermal band gaps for specific frequency regions in systems formed by the stacking doped semiconductors and polar dielectric materials. The presence of the gap causes a considerable decrease in the total heat flux for a certain frequency window and all wavevectors. By changing the carrier concentration of the doped semiconductor it is possible to continuously shift the plasmon frequency and thus tune the plasmon-surface phonon coupling. This makes doped semiconductors the ideal material choice to tailor the NFRHT in the infrared efficiently. These gaps are the result of the hybridization of surface plasmon polaritons and surface phonon polaritons and are thus very sensitive to the spatial order of the materials in the system.  Two coupling mechanisms are present in the considered structure: the interlayer coupling at the GaAs/NaBr interface and the intragap one between the two bilayers. The former one is responsible for the opening of the thermal band gap, while the latter one provides the channel for heat transfer across the gap. The latter one dominates over the former one for short gap distances and is characterized by higher confinement of the gap plasmon mode to the point that it becomes insensitive to the surface phonon modes of the polar dielectric. 

 

§ ACKNOWLEDGEMENTS: 
This work was supported by UNAM-DGAPA (PAPIIT IA102117, IN110916), CONACYT-postdoctoral fund 291053, PIIF-UNAM and CIC-UNAM. We thank Philippe Ben-Abdallah for his comments and thorough reading of the manuscript.  


G. P. and J. E. P. R. contributed equally to this work.


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f
subitem(mcitesubitemcount)

  
  

[Ottens et al.(2011)Ottens, Quetschke, Wise, Alemi, Lundock,
  Mueller, Reitze, Tanner, and Whiting]Ottens
Ottens, R. S.; Quetschke, V.; Wise, S.; Alemi, A. A.; Lundock, R.; Mueller, G.;
  Reitze, D. H.; Tanner, D. B.; Whiting, B. F. Near-Field Radiative Heat
  Transfer between Macroscopic Planar Surfaces. Phys. Rev. Lett.
  2011, 107, 014301
[Kralik et al.(2011)Kralik, Hanzelka, Musilova, Srnka, and
  Zobac]Kralik1
Kralik, T.; Hanzelka, P.; Musilova, V.; Srnka, A.; Zobac, M. Cryogenic
  apparatus for study of near-field heat transfer. Review of Scientific
  Instruments 2011, 82, 055106
[Kralik et al.(2012)Kralik, Hanzelka, Zobac, Musilova, Fort,
  and Horak]Kralik2
Kralik, T.; Hanzelka, P.; Zobac, M.; Musilova, V.; Fort, T.; Horak, M. Strong
  Near-Field Enhancement of Radiative Heat Transfer between Metallic Surfaces.
  Phys. Rev. Lett. 2012, 109, 224302
[Guha et al.(2012)Guha, Otey, Poitras, Fan, and Lipson]Guha
Guha, B.; Otey, C.; Poitras, C. B.; Fan, S.; Lipson, M. Near-Field Radiative
  Cooling of Nanostructures. Nano Letters 2012, 12,
  4546–4550, PMID: 22891815
[Shen et al.(2012)Shen, Mavrokefalos, Sambegoro, and
  Chen]Sheng12
Shen, S.; Mavrokefalos, A.; Sambegoro, P.; Chen, G. Nanoscale thermal radiation
  between two gold surfaces. Applied Physics Letters 2012,
  100, 233114
[Shen et al.(2009)Shen, Narayanaswamy, and Chen]Shen09
Shen, S.; Narayanaswamy, A.; Chen, G. Surface Phonon Polaritons Mediated Energy
  Transfer between Nanoscale Gaps. Nano Letters 2009, 9,
  2909–2913, PMID: 19719110
[Kittel et al.(2008)Kittel, Wischnath, Welker, Huth, Ruting,
  and Biehs]Kittel08
Kittel, A.; Wischnath, U. F.; Welker, J.; Huth, O.; Ruting, F.; Biehs, S.-A.
  Near-field thermal imaging of nanostructured surfaces. Applied Physics
  Letters 2008, 93, 193109
[Worbes et al.(2013)Worbes, Hellmann, and Kittel]Kittel13
Worbes, L.; Hellmann, D.; Kittel, A. Enhanced Near-Field Heat Flow of a
  Monolayer Dielectric Island. Phys. Rev. Lett. 2013,
  110, 134302
[Hargreaves(1969)]Hargreaves
Hargreaves, C. Anomalous radiative transfer between closely-spaced bodies.
  Physics Letters A 1969, 30, 491 – 492
[Song et al.(2015)Song, Fiorino, Meyhofer, and Reddy]review1
Song, B.; Fiorino, A.; Meyhofer, E.; Reddy, P. Near-field radiative thermal
  transport: From theory to experiment. AIP Advances 2015,
  5, 053503
[Liu et al.(2015)Liu, Wang, and Zhang]review2
Liu, X.; Wang, L.; Zhang, Z. M. Near-Field Thermal Radiation: Recent Progress
  and Outlook. Nanoscale and Microscale Thermophysical Engineering
  2015, 19, 98–126
[Wang et al.(2017)Wang, Hu, Takahashi, Zhang, Takamatsu, and
  Chen]rectifier-grafeno
Wang, H.; Hu, S.; Takahashi, K.; Zhang, X.; Takamatsu, H.; Chen, J.
  Experimental study of thermal rectification in suspended monolayer graphene.
  Nature Communications 2017, 8, 15843 EP –
[Ben-Abdallah and Biehs(2013)Ben-Abdallah, and
  Biehs]rectifier-abdallah
Ben-Abdallah, P.; Biehs, S.-A. Phase-change radiative thermal diode.
  Applied Physics Letters 2013, 103, 191907
[Ghanekar et al.(2018)Ghanekar, Tian, Ricci, Zhang, Gregory,
  and Zheng]Ghanekar:18
Ghanekar, A.; Tian, Y.; Ricci, M.; Zhang, S.; Gregory, O.; Zheng, Y. Near-field
  thermal rectification devices using phase change periodic nanostructure.
  Opt. Express 2018, 26, A209–A218
[Otey et al.(2010)Otey, Lau, and Fan]Otey
Otey, C. R.; Lau, W. T.; Fan, S. Thermal Rectification through Vacuum.
  Phys. Rev. Lett. 2010, 104, 154301
[Ben-Abdallah and Biehs(2014)Ben-Abdallah, and
  Biehs]PhysRevLett.112.044301
Ben-Abdallah, P.; Biehs, S.-A. Near-Field Thermal Transistor. Phys. Rev.
  Lett. 2014, 112, 044301
[Liu and Zhang(2014)Liu, and Zhang]Liugraphene
Liu, X. L.; Zhang, Z. M. Graphene-assisted near-field radiative heat transfer
  between corrugated polar materials. Applied Physics Letters
  2014, 104, 251911
[Zhao et al.(2017)Zhao, Guizal, Zhang, Fan, and
  Antezza]PhysRevB.95.245437
Zhao, B.; Guizal, B.; Zhang, Z. M.; Fan, S.; Antezza, M. Near-field heat
  transfer between graphene/hBN multilayers. Phys. Rev. B 2017,
  95, 245437
[Santiago et al.(2017)Santiago, Peréz-Rodríguez, and
  Esquivel-Sirvent]Esq17
Santiago, E. Y.; Peréz-Rodríguez, J. E.; Esquivel-Sirvent, R. Dispersive
  Properties of Mesoporous Gold: van der Waals and Near-Field Radiative Heat
  Interactions. The Journal of Physical Chemistry C 2017,
  121, 12392–12397
[Esquivel-Sirvent(2016)]Thinfilmraul
Esquivel-Sirvent, R. Ultra thin metallic coatings to control near field
  radiative heat transfer. AIP Advances 2016, 6,
  095214
[Biehs et al.(2011)Biehs, Ben-Abdallah, Rosa, Joulain, and
  Greffet]Biehsporous
Biehs, S.; Ben-Abdallah, P.; Rosa, F. S. S.; Joulain, K.; Greffet, J. Nanoscale
  heat flux between nanoporous materials. Opt. Express 2011,
  19, A1088–A1103
[Basu et al.(2015)Basu, Yang, and Wang]basumeta
Basu, S.; Yang, Y.; Wang, L. Near-field radiative heat transfer between
  metamaterials coated with silicon carbide thin films. Applied Physics
  Letters 2015, 106, 033106
[Moncada-Villa et al.(2015)Moncada-Villa, Fernández-Hurtado,
  García-Vidal, García-Martín, and Cuevas]PhysRevB.92.125418
Moncada-Villa, E.; Fernández-Hurtado, V.; García-Vidal, F. J.;
  García-Martín, A.; Cuevas, J. C. Magnetic field control of
  near-field radiative heat transfer and the realization of highly tunable
  hyperbolic thermal emitters. Phys. Rev. B 2015, 92,
  125418
[Abraham Ekeroth et al.(2018)Abraham Ekeroth, Ben-Abdallah,
  Cuevas, and García-Martín]spheresh
Abraham Ekeroth, R. M.; Ben-Abdallah, P.; Cuevas, J. C.;
  García-Martín, A. Anisotropic Thermal Magnetoresistance for an Active
  Control of Radiative Heat Transfer. ACS Photonics 2018,
  5, 705–710
[Francoeur et al.(2008)Francoeur, Meng, and Vaillon]Francoeur
Francoeur, M.; Meng, M. P.; Vaillon, R. Near-field radiative heat transfer
  enhancement via surface phonon polaritons coupling in thin films.
  Applied Physics Letters 2008, 93, 043109
[Guo et al.(2012)Guo, Cortes, Molesky, and Jacob]Guo12
Guo, Y.; Cortes, C. L.; Molesky, S.; Jacob, Z. Broadband super-Planckian
  thermal emission from hyperbolic metamaterials. Applied Physics
  Letters 2012, 101, 131106
[Biehs et al.(2013)Biehs, Tschikin, Messina, and
  Ben-Abdallah]Biehs13
Biehs, S.-A.; Tschikin, M.; Messina, R.; Ben-Abdallah, P. Super-Planckian
  near-field thermal emission with phonon-polaritonic hyperbolic metamaterials.
  Applied Physics Letters 2013, 102, 131106
[Biehs et al.(2007)Biehs, Reddig, and Holthaus]Biehs2007
Biehs, S.-A.; Reddig, D.; Holthaus, M. Thermal radiation and near-field energy
  density of thin metallic films. The European Physical Journal B
  2007, 55, 237–251
[Pérez-Rodríguez et al.(2017)Pérez-Rodríguez,
  Pirruccio, and Esquivel-Sirvent]Esquivel17
Pérez-Rodríguez, J. E.; Pirruccio, G.; Esquivel-Sirvent, R. Fano
  interference for tailoring near-field radiative heat transfer. Phys.
  Rev. Materials 2017, 1, 062201
[Allen et al.(1977)Allen, Tsui, and Logan]Allen
Allen, S. J.; Tsui, D. C.; Logan, R. A. Observation of the Two-Dimensional
  Plasmon in Silicon Inversion Layers. Phys. Rev. Lett. 1977,
  38, 980–983
[Georgiou et al.(2014)Georgiou, Tyagi, Mulder, Bauhuis,
  Schermer, and Rivas]Georgiou
Georgiou, G.; Tyagi, H. K.; Mulder, P.; Bauhuis, G. J.; Schermer, J. J.;
  Rivas, J. G. Photo-generated THz antennas. Scientific Reports
  2014, 4, 3584 EP –
[Watjen et al.(2016)Watjen, Zhao, and Zhang]dopedsiheat
Watjen, J. I.; Zhao, B.; Zhang, Z. M. Near-field radiative heat transfer
  between doped-Si parallel plates separated by a spacing down to 200 nm.
  App. Phys. Lett. 2016, 109, 203112
[Vinogradov and Dorofeev(2009)Vinogradov, and Dorofeev]Vinogradov
Vinogradov, E. A.; Dorofeev, I. A. Thermally stimulated electromagnetic fields
  of solids. Physics-Uspekhi 2009, 52, 425
[Song et al.(2015)Song, Fiorino, Meyhofer, and Reddy]Song15
Song, B.; Fiorino, A.; Meyhofer, E.; Reddy, P. Near-field radiative thermal
  transport: From theory to experiment. AIP Advances 2015,
  5, –
[Olazagasti et al.(1991)Olazagasti, Cocoletzi, and
  Mochán]Cocoletz
Olazagasti, E. L.; Cocoletzi, G.; Mochán, W. L. Optical properties of
  bimetallic superlattices. Solid State Communications 1991,
  78, 9 – 12
[Moon et al.(2016)Moon, Kim, Kim, Heo, and Lee]Moon
Moon, S.; Kim, K.; Kim, Y.; Heo, J.; Lee, J. Highly efficient single-junction
  GaAs thin-film solar cell on flexible substrate. Scientific Reports
  2016, 6, 30107 EP –
[Smith et al.(2015)Smith, Stenger, Kristensen, Mortensen, and
  Bozhevolnyi.]nanoscale
Smith, C. L. C.; Stenger, N.; Kristensen, A.; Mortensen, N. A.;
  Bozhevolnyi., S. I. Gap and channeled plasmons in tapered grooves: a review.
  Nanoscale 2015, 7, 9355–9386
[Liebsch(1993)]PhysRevLett.71.145
Liebsch, A. Surface plasmon dispersion of Ag. Phys. Rev. Lett.
  1993, 71, 145–148
[Zayats et al.(2005)Zayats, Smolyaninov, and
  Maradudin]ZAYATS2005131
Zayats, A. V.; Smolyaninov, I. I.; Maradudin, A. A. Nano-optics of surface
  plasmon polaritons. Physics Reports 2005, 408, 131 –
  314



