key: cord-0560082-2r4kueg2
authors: Shabbir, Muhammad Waqas; Leuenberger, Michael N.
title: Plasmonically enhanced mid-IR light source based on tunable spectrally and directionally selective thermal emission from nanopatterned graphene
date: 2020-05-19
journal: nan
DOI: nan
sha: bc70846bc89471a3a161bab2976bcf900e365e74
doc_id: 560082
cord_uid: 2r4kueg2

We present a proof of concept for a spectrally selective thermal mid-IR source based on nanopatterned graphene (NPG) with a typical mobility of CVD-grown graphene (up to $3000$ cm$^2$V$^{-1}$s$^{-1}$), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of an in-plane electric field. The localized surface plasmons (LSPs) on the NPG sheet allow for the control and tuning of the thermal emission spectrum in the wavelength regime from 3 $mu$m to 12 $mu$m. The LSPs along with an optical cavity increase the emittance of graphene from about 2.3% for pristine graphene to 80% for NPG, thereby outperforming state-of-the-art pristine graphene light sources operating in the near-infrared (NIR) by a factor of 100. A maximum emission power per area of 11x10^3 W/m$^2$ at $T=2000$ K for a bias voltage of $V=23$ V is achieved by Joule heating. By generalizing Planck's theory and considering the nonlocal fluctuation-dissipation theorem with nonlocal response of surface plasmons in graphene in RPA, we show that the coherence length of the graphene plasmons and the thermally emitted photons can be as large as 13 $mu$m and 150 $mu$m, respectively, providing the opportunity to create phased arrays. The spatial phase variation of the coherence allows for beamsteering of the thermal emission in the range between $12^circ$ and $80^circ$ by tuning the Fermi energy. Our analysis of the nonlocal hydrodynamic response leads to the conjecture that the diffusion length and viscosity in graphene are frequency-dependent. Using finite-difference time domain (FDTD) calculations, coupled mode theory, and RPA, we develop the model of a mid-IR light source based on NPG, which will pave the way to graphene-based optical mid-IR communication, mid-IR color displays, mid-IR spectroscopy, and virus detection.

An object that is kept in equilibrium at a given temperature T > 0 K emits electromagnetic (EM) radiation because the charge carriers on the atomic and molecular scale oscillate due to their heat energy. 1 Planck's law describes quantitatively the energy density u(ω) of the EM radiation per unit frequency ω for black-body radiation, which is u BB (ω)dω = ω 2 π 2 c 3 Θ(ω)dω, where c is the speed of light in vacuum, is the Planck constant, and k B is the Boltzmann constant. Θ(ω, T ) = ω/[exp( ω/k B T ) − 1] is the thermal energy of a photon mode. Consequently, the energy emitted per unit surface area and per unit frequency, also called spectral radiance, of a black body into three-dimensional (3D) space is given by

The total energy density u can then be obtained by integrating over all frequencies and angles over the halfsphere, leading to the Stefan-Boltzmann law for the energy density of black-body radiation,

with a BB = 7.566 × 10 −16 Jm −3 K −4 . The total power emitted per unit surface area P/A of a black-body is

where b BB = 5.67 × 10 −8 Wm −2 K −4 is the Stefan-Boltzmann constant. The factor cos θ is due to the fact that black bodies are Lambertian radiators. In recent years, several methods have been implemented for achieving a spectrally selective emittance, in particular narrowband emittance, which increases the coherence of the emitted photons. One possibility is to use a material that exhibits optical resonances due to the band structure or due to confinement of the charge carriers. 1 Another method is to use structural optical resonances to enhance and/or suppress the emittance. Recently, photonic crystal structures have been used to implement passive pass band filters that reflect the thermal emission arXiv:2005.09560v1 [cond-mat.mes-hall] 19 May 2020 at wavelengths that match the photonic bandgap. 2, 3 Alternatively, a truncated photonic crystal can be used to enhance the emittance at resonant frequencies. 4, 5 Recent experiments have shown that it is possible to generate infrared (IR) emission by means of Joule heating created by means of a bias voltage applied to graphene on a SiO 2 /Si substrate. 6, 7 In order to avoid the breakdown of the graphene sheet at around T = 700 K, the graphene sheet can be encapsulated between hexagonal boron nitride (h-BN) layers, which remove efficiently the heat from graphene. The top layer protects it from oxidation. 8, 9 In this way, the graphene sheet can be heated up to T = 1600 K, 9 or even above T = 2000 K. 8 , 10 Kim et al. and Luo et al. demonstrated broadband visible emission peaked around a wavelength of λ = 725 nm. 8, 9 By using a photonic crystal substrate made of Si, Shiue et al. demonstrated narrowband near-IR emission peaked at around λ = 1600 nm with an emittance of around = 0.07. 10 To the best of our knowledge, there are neither theoretical nor experimental studies on spectrally selective thermal emission from graphene in the mid-IR range.

Here, we present the proof of concept of a method to tune the spectrally selective thermal emission from nanopatterned graphene (NPG) by means of a gate voltage that varies the resonance wavelength of localized surface plasmons (LSPs) around the circular holes that are arranged in a hexagonal or square lattice pattern in a single graphene sheet in the wavelength regime between 3 µm and 12 µm. By generalizing Planck's radiation theory to grey-body emission, we show that the thermal emission spectrum can be tuned in or out of the two main atmospheric transparency windows of 3 to 5 µm and 8 to 12 µm in the mid-IR regime, and also in or out of the opaque mid-IR regime between 5 and 8 µm. In addition, the gate voltage can be used to tune the direction of the thermal emission due to the coherence between the localized surface plasmons (LSPs) around the holes due to the nonlocal response function in graphene, which we show by means of a nonlocal fluctuation-dissipation theorem. The main element of the nanostructure is a circular hole of diameter a in a graphene sheet. Therefore let us focus first on the optoelectronic properties of a single hole.

The frequency-dependent dipole moment of the hole is

where the polarizabilities α 1,2 are given along the main axes x and y of the elliptic hole, and r = r 0 is the position of the dipole moment, i.e. the hole. Graphene's dielectric function is isotropic in the xy-plane, i.e. ε || = ε xx = ε yy . V 0 is the volume of the graphene sheet. In the Supplementary Information we derive the general polarizabilities of an uncharged single-sheet hyperboloid with dielectric function ε(ω) inside a medium with dielectric constant ε m [see Eq. (163)]. The polarizabilities of an elliptical wormhole in x-and y-direction read

. (6) respectively, for which the in-plane polarizabilities lies in the plane of the graphene sheet that is parallel to the xy-plane. ε(ω) is the dielectric function of graphene. We assumed that the thickness d of the graphene sheet is much smaller than the size of the elliptic hole. The geometrical factors in this limit are

In the case of a circular hole of diameter a the polarizability simplifies to

The localized surface plasmon resonance (LSP) frequency of the hole can be determined from the equation

the condition for which the denominator of α || vanishes. Using the linear dispersion relation, the intraband optical conductivity is 11, 12 

which in the case of ε F k B T is reduced to where τ is determined by impurity scattering and electron-phonon interaction

Using the mobility µ of the NPG sheet, it can be presented in the form

where v F = 10 6 m/s is the Fermi velocity in graphene. ω p = e 2 F /2ε m is the bulk graphene plasma frequency.

It is well-known by now that hydrodynamic effects play an important role in graphene because the Coulomb interaction collision rate is dominant, i.e. τ −1 ee τ −1 imp and τ −1 ee τ −1 e−ph , which corresponds to the hydrodynamic regime. τ −1 imp and τ −1 e−ph are the electron-impurity and electron-phonon collision rates. Since for large absorbance and emittance, we choose a large Fermi energy, we are in the Fermi liquid regime of the graphene sheet. Taking the hydrodynamic correction into account, we also consider the hydrodynamically adjusted intraband optical conductivity, 13, 14 

where η 2 = β 2 + D 2 ω(γ + iω), β 2 ≈ 3 4 v 2 F is the intraband pressure velocity, D ≈ 0.4 µm is the diffusion length in graphene, and γ = τ −1 is the relaxation rate. Interestingly, the optical conductivity becomes k-dependent and nonlocal. Also, below we will conjecture that the diffusion length D must be frequency-dependent.

Note that since ε = 1 + χ, where χ is the susceptibility, it is possible to replace ε = χ . Alternatively, using the formula of the polarizability α = ε 0 χ we can write ε = α /ε 0 . The dielectric function for graphene is given by 11, 12 

where g = 2.5 is the dielectric constant of graphite and d is the thickness of graphene. Inserting this formula into Eq. (10) gives

Solving for the frequency and using the real part we obtain the LSP frequency,

which is linear in the Fermi energy F . Let us now consider the 2D array of circular holes in a graphene sheet. Since the dipole moments p j = δp(R j , ω) interact with each other by inducing dipole moments, we need to consider the dressed dipole moment at each site R j as source of the electric field, which

where G jj is the dipole-dipole interaction tensor. Using Bloch's theorem p j = p 0 exp(ik || ·R || ), the effective dipole moment becomes

for each site j, and thus

The lattice some over the dipole-dipole interaction tensor

where P is the lattice period,

Ω 0 is the unit-cell area, and the real part is valid for periods much smaller than the wavelength. The factor g = 5.52 (g = 4.52) for hexagonal (square) lattice. The electric field created by the effective dipole moment is determined byp

from which we obtain the effective polarizability of a hole in the coupled dipole approximation (CDA),

This formula is the same as in Refs. 15 

where the upper (lower) sign and S = 2πω/cΩ 0 cos θ (S = 2πω cos θ/cΩ 0 ) apply to s (p) polarization. Thus, the emittance and absorbance of the bare NPG sheet are given by

The coupling to the interface of the substrate with reflection and transmission amplitudes r 0 and t 0 , respectively, which is located basically at the same position as the NPG sheet, yields the combined reflection and transmission amplitudes 15

where r = r and t = 1 − r are the reflection and transmission amplitudes in backwards direction, respectively. If we include also the whole substrate including cavity and Au mirror, we need to sum over all possible optical paths in the Fabry-Perot cavity, yielding

with

where r Au is the complex reflection amplitude of the Au mirror in the IR regime. δ = 2kL cos θ is the phase accumulated by one back-and-forth scattering inside the Fabry-Perot cavity of length L. k ≈ n SU8 k 0 is the wavenumber inside the cavity for an external EM wave with wavenumber k 0 = 2π/λ. Since the sum is taken over a geometric series, we obtain

Since the transmission coefficient through the Au mirror can be neglected, we obtain the emittance and absorbance A including cavity, i.e. Using these results, let us consider the excitation of the graphene sheet near the hole by means of thermal fluctuations, which give rise to a fluctuating EM field of a localized surface plasmon (LSP). This can be best understood by means of the fluctuation-dissipation theorem, which provides a relation between the rate of energy dissipation in a non-equilibrium system and the quantum and thermal fluctuations occuring spontaneously at different times in an equilibrium system. 17 The standard (local) fluctuation-dissipation theorem for fluctuating currents δĴ ν (r, ω) in three dimensions reads

where the relative permittivity ε(r, ω) = ε (r, ω) + iε (r, ω) = f (r)ε(ω) and µ, ν = x, y, z are the coordinates. Note that since ε = 1 + χ, where χ is the susceptibility, it is possible to replace ε = χ . Alternatively, using the formula of the polarizability α = ε 0 χ we can write ε = α /ε 0 . f (r) = 1 on the graphene sheet and 0 otherwise. Since the fluctuating currents are contained inside the two-dimensional graphene sheet, we write the local fluctuation-dissipation theorem in its twodimensional form, i.e.

where the fluctuating current densities have units of A/m 2 and the coordinates are in-plane of the graphene sheet.

Using the method of dyadic Green's functions, it is possible to express the fluctuating electric field generated by the fluctuating current density by

where Ω is the surface of the graphene sheet. The LSP excitation around a hole can be well approximated by a dipole field such that

where R j = (x j , y j ) are the positions of the holes in the graphene sheet. Consequently, we have

The dyadic Green function is defined as

with the scalar Green function given by

and

Then, the fluctuation-dissipation theorem can be recast into the forms

and thus we obtain

noting that the dielectric tensor ε (r, ω) is diagonal. Since the energy density of the emitted electric field at the point r is

we can write the spectral radiance as

assuming that the dipole current of the LSP is in the plane of the graphene sheet, i.e. the xy-plane, and the

yy , and the same for all holes. N is the number of holes. In order to obtain the spectral radiance in the far field, we need to integrate over the spherical angle. Using the results from the Supplementary Information, we obtain

where we used the definition of the absorbance of a 2D material, i.e.

with 2D complex conductivity σ 2D (ω). According to Kirchhoff's law, emittance (ω), absorbance A(ω), reflectance R(ω), and transmittance T (ω) are related by 18

from which we obtain the grey-body thermal emission formula

whose prefactor bears strong similarity to Planck's black body formula in Eq. (1). Using FDTD to calculate the emittance 2D || (ω), we evaluted the grey-body thermal emission according to Eq. (46) for the thermal emitter structure based on NPG shown in Figs. 1 and 2. Using COMSOL, we calculated the temperature distribution inside the NPG sheet, as shown in Fig. 8 , when a bias voltage V SD is applied, which gives rise to Joule heating. Our results are shown in Figs. 6, 7, and 9 for the temperatures 1300 K, 1700 K, and 2000 K of NPG. After integrating over the wavelength under the curves, we obtain the following thermal emission power per area: Let us consider the dependence of the thermal emission of NPG on the angle θ. Integrating over r 2 ϕ we obtain

which is a clear deviation from a Lambert radiator. The pattern of the thermal radiation can be determined bŷ

which is shown in Fig. 10 . Interestingly, since we assumed that thermal emission is completely incoherent [see Eq. (42)] the thermal emission from NPG is only weakly dependent on the emission angle θ, which can be clearly seen in Fig. 10 . However, the assumption that thermal emission of radiation is incoherent is not always true. Since Kirchhoff's law is valid, thermal sources can be coherent. 19 After theoretical calculations predicted that long-range coherence may exist for thermal emission in the case of resonant surface waves, either plasmonic or phononic in nature, 20, 21 experiments showed that a periodic microstructure in the polar material SiC exhibits coherence over many wavelengths and radiates in well-defined and controlled directions. 22 Here we show that the coherence length of a graphene sheet patterned with circular holes can be as large as 150 µm due to the plasmonic wave in the graphene sheet, thereby paving the way for the creation of phased arrays made of nanoantennas represented by the holes in NPG.

The coherence of thermal emission can be best understood by means of a nonlocal response function. 23 First, we choose the nonlocal hydrodynamic response function in Eq. (13) . Using the 2D version of the fluctuationdissipation theorem in Eq. (33), we obtain the nonlocal fluctuation-dissipation theorem in the hydrodynamic approximation,

where ∆r || = r || − r || and η 2 = β 2 + D 2 ω(γ + iω). This result suggests that the coherence length is given approximately by D, which according to Ref. 13 would be D ≈ 0.4 µm. However, the resulting broadening of the LSP resonance peaks would be very large and therefore in complete contradiction to the experimental measurements of the LSP resonance peaks in Refs. 11, 24, and 25. Thus, we conclude that the hydrodynamic diffusion length must be frequency-dependent with D(ν = 0) = 0.4 µm. Using the Fermi velocity of v F = 10 6 m/s and a frequency of ν = 30 THz, the average oscillation distance is about L = v F ν −1 = 0.033 µm, which is much smaller than D(ν = 0) in graphene. Thus we can approximate D(ν = 30 THz) = 0. We conjecture that there is a crossover for D into the hydrdynamic regime when the frequency is reduced below around ν 0 = 1 to 3 THz, below which the hydrodynamic effect leads to a strong broadening of the LSP peaks for NPG. Consequently, the viscosity of graphene should also be frequency-dependent and a crossover for the viscosity should happen at about the same frequency ν 0 . We plan to elaborate this conjecture in future work. Future experiments could corroborate our conjecture by measuring the absorbance or emittance as a function of wavelength for varying scale of patterning of the graphene sheet. Next, let us consider the coherence of thermal emission by means of the nonlocal optical conductivity in the RPA approximation. Using the general formula

with

in the low-temperature and low-frequency approximation, one obtains Eq. (12) . Now, let us use the full polarization in RPA approximation including only the Coulomb interaction,

from which we obtain

which introduces the nonlocal response via the Coulomb interaction in the denominator. After taking the Fourier transform, we obtain the nonlocal fluctuation-dissipation theorem in RPA approximation,

where the coherence length in RPA approximation is

and the coherence wavenumber is given by

For simplicity, we switch now to a square lattice of holes.

In the case of the LSP resonance for a square lattice of holes at λ = 10 µm, corresponding to ν = 30 THz, F = −1.0 eV, ω = 2πν, and γ = ev 2 F /(µE F ) = 0.3 THz for µ = 3000 cm 2 V −1 s −1 , which results in a coherence length of C RP A = 3 µm. This result is in reasonable agreement with the full width at half maximum (FWHM) values of the widths of the LSP resonance peaks in Refs. 11, 24, and 25. This coherence length would allow to preserve coherence for a linear array of period P = 300 nm and C RP A /P = 10 holes. In order to show the coherence length that can be achieved with graphene, we can consider a suspended graphene sheet with a mobility of µ = 15000 cm 2 V −1 s −1 . Then the coherence length increases to a value of C RP A = 13 µm, which would allow for coherence over a linear array with C RP A /P = 43 holes.

In the case of the LSP resonance for a square lattice of holes at λ = 5 µm, corresponding to ν = 60 THz, F = −1.0 eV, ω = 2πν, and γ = ev 2 F /(µE F ) = 0.3 THz for µ = 3000 cm 2 V −1 s −1 , which results in a coherence length of C RP A = 1.5 µm. Considering again a suspended graphene sheet, the coherence length can be increased to C RP A = 6.7 µm. Since the period in this case is P = 45 nm, the coherence for µ = 3000 cm 2 V −1 s −1 and µ = 15000 cm 2 V −1 s −1 can be preserved for a linear array of C RP A /P = 33 and 148 holes, respectively.

The coherence length and time of thermally emitted photons is larger because the photons travel mostly in vacuum. Taking advantage of the Wiener-Kinchine theorem, 19 we can extract the coherence length C FDTD Figure 12 . Directivity of the thermal emission from NPG where the holes act as nanoantennas in a phased array. This emission pattern for F = −1.0 eV can be used for surfaceemitting mid-IR sources. In the case of a 150x150, 75x75, 56x56, 37x37 square lattice of holes (size of lattice matches coherence length) with period P = 45 nm and hole diameter of 30 nm, introducing a relative phase of 2.43 • , 4.86 • , 7.28 • , 9.71 • between the nanoantennas allows for beamsteering in the range between θ = 12 • and θ = 80 • by tuning the Fermi energy in the range between F = −1.0 eV and F = −0.25 eV. and coherence time τ FDTD of thermally emitted photons by means of the full-width half-maximum (FWHM) of the spectral radiances shown in Figs. 6, 7, and 9. Our results are shown in Fig. 11 . The coherence length of the thermally emitted photons can reach up to C FDTD = 150 µm at a resonance wavelength of λ = 4 µm. This means that the coherence length of the thermally emitted photons is about 37 times larger than the wavelength.

Thus, the latter large coherence length allows for the coherent control of a 150x150 square array of holes with period P = 45 nm, individually acting as nanoantennas, that can be used to create a phased array of nanoantennas. One of the intriguing properties of a phased array is that it allows to control the directivity of the emission of photons, which is currently being implemented for large 5G antennas in the 3 to 30 GHz range. The beamsteering capability of our NPG sheet is shown in Fig. 12 . In contrast, our proposed phased array based on NPG can operate in the 10 to 100 THz range.

The temporal control of the individual phases of the holes requires an extraordinary fast switching time of around 1 ps, which is not feasible with current electronics. However, the nonlocal response function reveals a spatial phase shift determined by the coherence wavenumber K RP A , which is independent of the mobility of graphene. In the case of the LSP resonance at λ = 4 µm, we obtain λ RP A = 2π/K RP A = 6 µm, resulting in a minimum phase shift of 2πP/λ RP A = 0.042 = 2.4 • between neighboring holes, which can be increased to a phase shift of 9.7 • by decreasing the Fermi energy to E F = −0.25 eV. Thus, the phase shift between neighboring holes can be tuned arbitrarily between 2.4 • and 9.7 • by varying the Fermi energy between F = −1.0 eV and F = −0.25 eV. Fig. 12 shows the capability of beamsteering for our proposed structure by means of directional thermal emission, which is tunable by means of the gate voltage applied to the NPG sheet.

Due to the full control of directivity with angle of emission between θ = 12 • and θ = 80 • by tuning the Fermi en-ergy in the range between F = −1.0 eV and F = −0.25 eV, thereby achieving beamsteering by means of the gate voltage, our proposed mid-IR light source based on NPG can be used not only in a vertical setup for surface emission, but also in a horizontal setup for edge emission, which is essential for nanophotonics applications.

In conclusion, we have demonstrated in our theoretical study that NPG can be used to develop a plasmonically enhanced mid-IR light source with spectrally tunable selective thermal emission. Most importantly, the LSPs along with an optical cavity increase substantially the emittance of graphene from about 2% for pristine graphene to 80% for NPG, thereby outperforming stateof-the-art graphene light sources working in the visible and NIR by at least a factor of 100. Combining our proposed mid-IR light source based on patterned graphene with our demonstrated mid-IR detector based on NPG 25 , we are going to develop a mid-IR spectroscopy and detection platform based on patterned graphene that will be able to detect a variety of molecules that have mid-IR vibrational resonances, such as CO, CO 2 , NO, NO 2 , CH 4 , TNT, H 2 O 2 , acetone, TATP, Sarin, VX, etc. In particular, a recent study showed that it is possible to detect the hepatitis B and C viruses label-free at a wavelength of around 6 µm. 26 Therefore, we will make great effort to demonstrate that our platform will be able to detect with high sensitivity and selectivity the COVID-19 virus and other viruses that pose a threat to humanity.

We acknowledge support from NSF CISE-1514089. We thank Gernot Pomrenke, Alireza Safaei, and Sayan Chandra for useful discussions.

Kirchhoff's law of thermal radiation states that emittance is equal to absorbance A, i.e.

(ω, θ, φ, T ) = A(ω, θ, φ, T ).

(57)

In the case of a black body (ω, θ, φ, T ) = A(ω, θ, φ, T ) = 1. Pristine graphene has a very small absorbance of only A = 0.023 and is a nearly transparent body. Shiue et al. used a photonic crystal structure to filter the thermal emission from pristine graphene with an emittance of around A = 0.07. 10 Their spectral radiance is shown in Fig. 13 and exhibits peaks at around λ = 1.55 µm at a temperature of T = 2000 K. After integrating the spectral radiance under the curve, one obtains a emission power per area of about P/A = 100 W/m 2 , which is about 100 times weaker than our proposed thermal radiation source based on NPG at T = 2000 K. Our proposed thermal mid-IR source features an emission power per area of about P/A = 10 4 W/m 2 at T = 2000 K. In addition, our proposed thermal mid-IR source features frequency-tunability and beamsteering by means of a gate voltage applied to the NPG sheet. Using FDTD to calculate the emittance 2D || (ω), we evaluted the grey-body thermal emission according to Eq. (46) for the thermal emitter structure based on NPG shown in Figs. 1 and 2. Our results for the temperature T = 300 K of NPG are shown in Figs. 14, 15, and 16. In these figures we compare our results for NPG with the results for pristine graphene and black body radiation. 

For determining the EM properties of an infinitesimally thin conducting elliptical disk of radius R or an infinitesimally thin conducting plane with a elliptical hole, including coated structures, it is most convenient to perform the analytical calculations in the ellipsoidal coordinate system (ξ, η, ζ), [27] [28] [29] [30] which is related to the Cartesian coordinate system through the implicit equation

for a > b > c. The cubic roots ξ, η, and ζ are all real in the ranges

which are the ellipsoidal coordinates of a point (x, y, z). The surfaces of contant ξ, η, and ζ are ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, respectively, all confocal with the ellipsoid defined by

Each point (x, y, z) in space is determined by the intersection of three surfaces, one from each of the three families, and the three surfaces are orthogonal to each other. The transformation between the two coordinate systems is given by the solutions of Eq. (58), i.e.

defining 8 equivalent octants. The length elements in ellipsoidal coordinates read 

whose elements J ij define the Jacobian matrix. The derivatives are explicitly:

.

(80) Figure 16 . The NPG sheet allows for spectrally selective thermal emission at around λ = 10 µm for a period of P = 450 nm and a hole diameter of a = 300 nm.

The coordinate η is constant on the surfaces of oblate spheroids defined by

The surface associated with the limit η → 0 is an infinitesimally thin circular disk of radius R. In contrast, the surface in the limit η 1 is a sphere of radius r = R cosh η ≈ R sinh η. Thus, the Laplace equation in ellipsoidal coordinates reads

(82)

The surface of the conducting ellipsoid is defined by ξ = 0. Thus, the electric field potential Φ(ξ) is a function of ξ only, thereby defining the equipotential surfaces by confocal ellipsoids. Laplace's equation is then simplified to

The solution outside the ellipsoid is

From the asymptotic approximation ξ ≈ r 2 for large distances r → ∞, i.e. ξ → ∞, we identify R ξ ≈ ξ 3/2 and

using the boundary condition lim ξ→∞ Φ(ξ) = 0. Since the Coulomb field should be Φ(ξ → ∞) ≈ e/r at large distances from the ellipsoid, 2A = e and

is obtained, corresponding to the far-field of a monopole charge.

The solution inside the ellipsoid is

Using the asymptotic approximation R ξ→−c 2 ∝ ξ + c 2 we obtain

This solution satisfies the boundary condition lim ξ→−c 2 Φ(ξ) = 0. The constant B can be found from the boundary condition Φ(ξ = 0) = V , where V is the potential on the surface of the charged ellipsoid.

D. Dipole Moment of Conducting Ellipsoid induced by an External Electric Field in z-direction Following Ref. 30 , let us consider the case when the external electric field is parallel to one of the major axes of the ellipsoid. For the external potential let us choose

Let Φ p be the potential caused by the ellipsoid, with the boundary condition Φ p (ξ → ∞) = 0. Requiring continuous boundary condition on the surface of the ellipsoid, we have

We make the ansatz

which after insertion into the Laplace equation yields

Thus, one obtains for the field caused by the ellipsoid

where

the function we used in the case of the charged ellipsoid (see above). Thus, the field inside the ellipsoid is given by

Using the boundary condition shown in Eq. (91), one obtains the first equation

The boundary condition of the normal component of D at ξ = 0, equivalent to

yields the second equation

Consequently, the potentials are

where

Far away from the ellipsoid for ξ ≈ r 2 → ∞, one can use the approximation

yielding the potential caused by the ellipsoid, i.e.

from which we identify the dipole moment

This result determines the polarizability of the charged ellipsoid, i.e.

If the external electric field is applied along the other major axes of the ellipsoid, x or y, the polarizabilities are

respectively, where

For oblate spheroids (a = b), L 1 = L 2 ,

where e o is the eccentricity of the oblate spheroid. The limiting cases of an infinitesimally thin disk and a sphere are obtained for e o = 1 and e o = 0, respectively. The geometrical factors L i are related to the depolarization factorsL i by

In analogy to Ref. 30 , let us consider the case when the external electric field is parallel to one of the major axes of the ellipsoid, in this case along the x-axis. For the external potential let us choose

. (117) Let Φ p be the potential caused by the ellipsoid, with the boundary condition Φ p (ξ → ∞) = 0. Requiring continuous boundary condition on the surface of the ellipsoid, we have

Thus, one obtains for the field caused by the ellipsoid

with

where

the function we used in the case of the charged ellipsoid (see above). Thus, the field inside the ellipsoid is given by

Using the boundary condition shown in Eq. (118), one obtains the first equation

The boundary condition of the normal component of D at ξ = 0, equivalent to

yields the second equation

Consequently, the potentials are

where

Far away from the ellipsoid for ξ ≈ r 2 → ∞, one can use the approximation

yielding the potential caused by the ellipsoid, i.e.

from which we identify the dipole moment

This result determines the polarizability of the charged ellipsoid, i.e. Contrary to the case of an uncharged ellipsoid, where the solutions when applying the external electric field in x, y, or z direction are similar, the solutions in the case of an uncharged hyperboloid depend strongly on the axis in which the external field E 0 points. While the solutions for E 0 = E 0x and E 0 = E 0ŷ are similar, the solution for E 0 = E 0ẑ is completely different. The reason for this fundamental difference is that the ellipsoid resembles a sphere from far away. However, a single-sheet hyperboloid has elliptical cylindrical symmetry.

Here, let us first calculate the electrostatic potential Φ(ξ, η, ζ) of a conducting single-sheet hyperboloid with an elliptical hole, which can be represented by a limiting hyperboloid from a family of hyperboloids described by the implicit equation

for a > b > c. The cubic roots ξ, η, and ζ are all real in the ranges

which are the ellipsoidal coordinates of a point (x, y, z). The lmiting hyperboloid is a single planar sheet with an elliptical hole, i.e. it belongs to the family of solutions η in the limit η → −c 2 . Therefore, let us choose this limiting case as our origin in ellipsoidal coordinates with c = 0. Then Eq. (133) becomes

for a > b > c = 0. The cubic roots ξ, η, and ζ are all real in the ranges

The surface of the conducting hyperboloid is defined by −b 2 ≤ η = η 1 < 0.

Let us consider the case E 0 = E 0x , which in the limit when the hyperboloid becomes a flat plane is the most relevant one. Therefore

in the lower-half plane, where the negative sign corresponds to positive x values and the positive sign to negative x values. Since the equipotential surfaces are determined by η, let Ψ p be the potential caused by the hyperboloid, with the boundary condition Ψ in (η = 0) = 0.

Requiring continuous boundary condition on the surface of the hyperboloid, we have

where in the second equation the normal component of D at η = η 1 must be continuous. Then we make the ansatz for the electrostatic potential inside the hyperboloid,

where C in is a constant. This ansatz satisfies the boundary condition Ψ in (ξ = 0, η → 0, ζ) = 0. For the outside polarization field we choose

where C p is a constant, and we defined

Note that lim ξ→0+ arctan a √ ξ = π/2, whereas lim ξ→0− arctan a √ ξ = −π/2. Therefore, in order to avoid discontinuity at ξ = 0, we must have arctan

where R η = (η + a 2 )(η + b 2 )(−η). The boundary conditions at z → ±∞ are satisfied:

At large distances r = x 2 + y 2 + z 2 from the wormhole we have ξ ≈ r 2 . Then the far-field potential in the upper half-space, which is given by the pure polarization field, is

The polarization far-field has the form of a dipole field at large distances r from the wormhole. In order to determine the polarizability of the wormhole, let us find the solution at ξ = 0, corresponding to the plane that passes through the center of the wormhole. For ξ = 0, the unit vectorsx andη are parallel. In this near-field limit, the polarization potential has the form

whereC p = C p (π/2 − 1).

Using the boundary conditions shown in Eq. (139), we obtain the first equatioñ

and the second equation

Using the derivatives ∂x ∂η ξ=0,η1 = a 2 (ζ + a 2 ) (η 1 + a 2 )(a 2 − b 2 )(a 2 − c 2 ) , (149)

we can rewrite the second equation as

which is equivalent to

Thus, the potentials are

Then the far-field potential in the upper half-space, which is given by the pure polarization field, is 

where we assumed that a ≈ b. The polarization far-field has the form of a dipole field at large distances r from the wormhole. If the external electric field is applied in y-direction, we obtain the potentials

with F 2 (ξ) = ∞ ξ bdξ 2ξ 1/2 (ξ + b 2 ) − ∞ ξ bdξ 2(ξ + b 2 ) 3/2 , (158)

γ = 18.3 × 10 −3 is a dimensionless constant describing the electron-phonon coupling coefficient, and ω oph ≈ 0.2 eV is the graphene optical phonon energy. It is evident that the plasmon lifetime is reduced due to electronphonon interaction and edge scattering, but the DC conductivity, which is used to calculate the dielectric function of graphene, is invariant if the edge-to-edge distance of the pattern is larger than the carrier mean free path L M F P = v F τ DC . The Drude model and the optical phonon frequency isare not valid for a patterned graphene sheet only if the edge-to-edge distance is much lowersmaller than the carrier mean free paths of electrons and phonons. For the current pattern and carrier mobilities, both the mean free paths L e and L ph are smaller than or of the same order as the edge-to-edge distance, which means that we can safely assume that the Drude model and optical phonon frequencies of our patterned graphene sheet are the same as for pristine graphene. The coupling of plasmon and substrate/graphene phonon can be characterized through the loss function Z, which is the imaginary part of inverse effective dielectric function calculated via the generalized RPA theory

The loss function represents the amount of energy dissipated by exciting the plasmon coupled to the substrate and optical phonons in graphene. The surface plasmons in graphene are damped through radiative and nonradiative processes. Nonradiative damping transfers the plasmon energy to hot electron-hole excitation by means of intraband transition. Fig. 18 (a) shows the loss function for graphene with carrier mobility µ = 3000 cm 2 /V·s and a Fermi energy of F = 1.0 eV. The thickness of the optical cavity is chosen to be λ/4n, where n is the refractive index of the cavity material. 11 The plasmon assisted electron-hole pair generation in this structure lies outside the Landau intraband damping region, indicated by the shaded area in Fig. 18 (a) . A band gap in the plasmonphonon dispersion relation is formed via Frohlich interaction between graphene plasmons and optical phonons. This coupling leads to the splitting of the energy into two distinct branches: surface plasmon phonon polaritons (SPPPs) and graphene plasmons (GPs). The horizontal branch line marked as ω LSP is the LSP mode and is independent of the plasmon wavevector due to the localization of the LSP around a hole. The resonance frequencies of the polar phonons are denoted by ω BN for h-BN and by ω SN for Si 3 N 4 .

H. Integral of dyadic Green function elements over spherical angle

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We defined the geometrical factorswhich are related to the depolarization factors bỹThis result determines the polarizability of the uncharged hyperboloid observable in the far-field, i.e..Similarly, we obtain the polarizability in y-direction, i.e.. (164) Comparing to the polarizabilities of ellipsoids, 30 the polarizabilities of hyperboloids are proportional to ab √ −η 1 , which corresponds to the volume of the ellipsoid abc.In the case of circular wormholes, we have a = b, and therefore α 1 = α 2 = α || , with L 1 = L 2 = L || .

In our proposed mid-IR light source the effective combination of Si 3 N 4 and h-BN behaves as an environment with polar phonons. Polar materials have ions of different valence, whose oscillating dipole moment gives rise to the interaction between electrons and optical phonons, the Frohlich interaction. By placing graphene on a polar substrate the long range Frohlich interaction mediates the interaction between optical phonons and surface plasmons in graphene. 31 The interaction between polar substrate/graphene phonons and electrons in graphene modifies substantially the graphene plasmon dispersion relation, which is shown in Fig. 18 . The dielectric function of graphene in the random phase approximation (RPA) takes the form 11,12The second term represents the effective Coulomb interaction of electrons in graphene, and v c (q) = e 2 2qε 0 is the direct Coulomb interaction. The third term is the effective dielectric function for different phonon modes (l) coming from electron-electron interaction mediated by where |M sph | 2 is the scattering and G 0 l is the free phonon Green function. The last term of Eq. (165) corresponds to the optical phonon mediated electron-electron interaction v oph (q, ω) = |M oph | 2 G 0 (ω).Here |M o ph| 2 defines the scattering matrix element and G o (ω) is the free phonon Green function. In Eq. (165), χ 0 j,j (q, ω) is the current-current correlation function. This description is very general and can be applied to any metallic system. The momentum relaxation time τ can be derived by considering the impurity, electron-phonon interaction, and the scattering related to nanostructure edges τ −1 = τ −1 DC + τ −1 edge + τ −1 e−p , which determines the plasmon lifetime and the absorption spectrum bandwidth. It can be evaluated via the measured DC mobility µ of the graphene sample throughwhere v F = 10 6 m/s is the Fermi velocity and τ DC = µ √ πρ ev F is the charge carrier density. τ edge ≈ 1 × 10 6 (m/s) w − w 0 −1 is due to the scattering from the nanostructure edges, where w is the edge-to-edge distance of the holes and w 0 = 7 nm is the parameter that includes edge effects, and τ e−ph = 2Im( e−ph ) is related to the scattering because of coupling of electrons and phonons. The imaginary part of the electron-phonon self-energy is given byFor the calculation of the spectral radiance we need to integrate the elements of the dyadic Green function over the spherical angle. We can split the total dyadic Green function into a free space term ← → G 0 (r, r ; ω) and a term ← → G SP P (r, r ; ω) that creates surface plasmon polaritons inside graphene. Since the absorbance of the pristine graphene sheet is only 2.3%, we can safely neglect ← → G SP P (r, r ; ω). Our goal is to calculate the gray-body emission of the EM radiation from the LSP around the holes in graphene into free space. Therefore, we need to evaluatewhere can use the approximationIn Cartesian coordinates, we can write down the dyadic Green function as 17Since we are interested only in the far field, we consider only the far-field component of the dyadic Green function, which is The corresponding integrals arê r 2 sin θdθdϕ |G xx (r; ω)| 2 = 2 15π , r 2 sin θdθdϕ |G yx (r; ω)| 2 = 4 15π , r 2 sin θdθdϕ |G zx (r; ω)| 2 = 4 15π , r 2 sin θdθdϕ |G xy (r; ω)| 2 = 4 15π , r 2 sin θdθdϕ |G yy (r; ω)| 2 = 2 15π , r 2 sin θdθdϕ |G zy (r; ω)| 2 = 4 15π.