1. Introduction

Efficient generation and manipulation of single photons radiated by a solid-state quantum emitter are of central importance in a range of quantum information processing applications [1–3]. The ability to control the spontaneous emission by tailoring the photonic density of photon modes is a key to control the efficiency of single-photon sources [4–6]. To enhance the spontaneous emission rate and extraction efficiency into a specific cavity mode, and thus achieve fast, efficient or even directed single-photon emission, various nanophotonic structures have been proposed [7–10]. However, translating such techniques beyond proof-of-principle concepts to technologically relevant implementations remains a challenge. Although various attempts [11–14] have been made to address this challenge, most approaches sacrifice industrial scalability for the sake of core metrics, such as spontaneous emission rate as well as extraction efficiency.

Hyperbolic metamaterials (HMMs) [15–28] are a class of uniaxial metamaterials and their optical properties can be traditionally described in electromagnetic theory by writing the dispersion relation in terms of an anisotropic permittivity tensor with opposite signs of axial (indexed by z) and transverse (indexed by t, or x and y) components. Such metamaterials have elicited a growing attention in the optics and photonics, mostly inspired by the broadband manipulation of high density of electromagnetic states and strong spatial localization. However, two major challenges hinder the practical usage of the HMMs in the realm of single-photon sources. One is the non-radiative plasmonic modes in HMMs [22]. As a result, high-k field waves remain trapped inside HMMs, unless a suitable mechanism is provided that mediates the differences of the momentum supported in HMM and dielectric environment. Another one is the material losses and nonlocal effects related to the finite thickness of the discrete constituent elements during actual realization of metamaterial in reality.

Some approaches such as plasmonic waveguides [29], bullseye gratings [30], photonic crystals [31,32], and engineered multilayer metamaterials [33–36] had been proposed to ease the extraction limitation, but to achieve efficient spontaneous emission and enhanced light extraction simultaneously, we focus on the materials with hyperbolic dispersion. Here, we propose a hyperstructure built with a periodic multilayer system of graphene sheets and hexagonal boron nitride (hBN) films, which is able to realize both efficient spontaneous emission and preferential power extraction. Because of the coupling of surface plasmon polaritons in a monolayer graphene and hyperbolic phonon polaritons in a single hBN film, the hyperstructure could inherit both the tunable behavior of graphene and the natural hyperbolic properties of hBN [37]. Such phenomena can also be found in [38]. In addition to [37], there are several works regarding the plasmon-phonon interactions in the multilayer structures [39–42]. Although the graphene-hBN heterostructure for the enhancement of spontaneous emission has been demonstrated [40], to the best of our knowledge, the extraction from graphene-hBN based structure remains unknown to the academic so far. We focus primarily on the optical topological transitions from an ellipsoid to a hyperboloid in this hyperstructure, which can be achieved by tuning the chemical potential of graphene. In the transition wavelength range, one component of the permittivity tensor of the hyperstructure switches from positive to negative and this could be helpful to extract energy from the non-radiative modes. We also discuss how the effective medium approximation may moderately overestimate the spontaneous emission and extraction.

2. Materials

We start by concentrating the optical properties of graphene-hBN hyperstructure made out of a stack of multilayers as shown in Fig. 1. Each period of the multilayer contains an hBN film with a thickness denoted as ${\text{δ}}_{\text{hBN}}$, a monolayer graphene with a thickness of ${\text{δ}}_{\text{gra}}$, and the thickness of each period is $d={\text{δ}}_{\text{gra}}+{\text{δ}}_{\text{hBN}}$. As a natural material, hBN has two mid-infrared Reststrahlen bands in the infrared region. We assume its optical axis is in the z-direction as shown in Fig. 1(a), accordingly its relative permittivity, i.e.,${\overline{\overline{ϵ}}}_{\text{r,hBN}}=\text{diag}({ϵ}_{\text{t,hBN}},{ϵ}_{\text{t,hBN}},{ϵ}_{\text{z,hBN}})$, where ${ϵ}_{\text{t,hBN}}={ϵ}_{\text{x,hBN}}={ϵ}_{\text{y,hBN}}\ne {ϵ}_{\text{z,hBN}}$, can be demonstrated by receiving the trial information with the thought of practical consideration of realistic loss [37] (For a more detailed analysis of the permittivity of hBN, please see the Appendix, Section A1). One thing to notice is that the nonlocal impact of hBN is dismissed in all estimations in this work since the reciprocal lattice vector of hBN is at least two orders of magnitude larger than the wavevector of polaritons.

 

Fig. 1 (a) Schematic configuration of a graphene-hBN hyperstructure; (b) Real and (c) imaginary parts of the effective permittivity of graphene-hBN hyperstructure at two different graphene chemical potentials (${\text{μ}}_{\text{c}}=0\text{eV}$ and ${\text{μ}}_{\text{c}}=0.3\text{eV}$) when the temperature is set to 300 K. Colored regions indicate different optical topological transitions from the closed (ellipsoid, ${ϵ}_{\text{t}}>0,{ϵ}_{\text{z}}>0$) isofrequency surface to an open (hyperboloid, ${ϵ}_{\text{t}}<0,{ϵ}_{\text{z}}>0)$isofrequency surface.

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Graphene can be considered as an ultrathin layer with conductivity ${\text{σ}}_{\text{s}}$ including the contributions from both the intraband $({\text{σ}}_{\text{I}})$ and interband $({\text{σ}}_{\text{D}})$ transitions [43,44], i.e., ${\text{σ}}_{\text{s}}={\text{σ}}_{\text{D}}+{\text{σ}}_{\text{I}}$, respectively. At temperature T (T = 300K), these surface conductivities can be characterized by the random phase approximation (local RPA) model (For a more detailed analysis of the conductivity of graphene, please see Appendix, Section A2

). As we know, graphene can also be treated as a thin layer of thickness ${\text{δ}}_{\text{gra}}$ with a permittivity tensor ${\overline{\overline{ϵ}}}_{\text{r,gra}}=\text{diag}({ϵ}_{\text{t,gra}},{ϵ}_{\text{t,gra}},{ϵ}_{\text{z,gra}})$ [45–47], where ${ϵ}_{\text{t},\text{gra}}$ is related to the surface conductivity as ${ϵ}_{\text{t},\text{gra}}=1+\text{i}\frac{{\text{σ}}_{\text{s}}}{\text{ω}{ϵ}_0{\text{δ}}_{\text{gra}}}$, and ${ϵ}_{\text{z},\text{gra}}=3$. The thickness of monolayer graphene is assumed to be ${\text{δ}}_{\text{gra}}=0.35\text{nm}$.

According to the optical properties of hBN and graphene, the viable reactions of the whole hyperstructure can be approximated by utilizing the following effective medium theory (EMT) when the individual elements (layers) are much smaller than the wavelengths in the infrared region.

Figures 1(b) and 1(c) illustrate the real and imaginary parts of effective permittivity of graphene-hBN hyperstructure at two different chemical potentials (${\text{μ}}_{\text{c}}=0\text{eV}$ and ${\text{μ}}_{\text{c}}=0.3\text{eV}$). When the chemical potential of graphene is ${\text{μ}}_{\text{c}}=0\text{eV}$, both the axial and transverse dielectric functions of hyperstructure are similar to those of hBN. However, as the graphene chemical potential increases to ${\text{μ}}_{\text{c}}=0.3\text{eV}$ the axial dielectric function of hyperstructure keeps almost the same due to small filling fraction while the transverse dielectric function is changed significantly due to the metallic behavior of graphene as shown in Fig. 1(b). Since the chemical potential of graphene can be tuned via the electrostatic gating, the dispersion of the hyperstructure can be engineered although the structural parameters are fixed. Because we are interested in the optical topological transitions, we separate the whole spectrum into several regions with optical topological transitions from the closed (ellipsoid, ${ϵ}_{\text{t}}>0,{ϵ}_{\text{z}}>0$) isofrequency surface to an open (hyperboloid, ${ϵ}_{\text{t}}<0,{ϵ}_{\text{z}}>0$) isofrequency surface, which are marked with different shades of colors. It is essential to understand how the tunability affects the spontaneous emission and extraction during optical topological transitions. One of the goals of Section 3 is to present and discuss the results of these behaviors. Section 3 also provides the relations of the spontaneous emission and extraction between the EMT- and realistic layered design.

3. Results and discussion

In this Section, we focus on the tunability of spontaneous emission and enhancement of extraction according to the optical topological transitions, when the quantum emitters are located near the graphene-hBN hyperstructure [48]. In general, the density of electromagnetic states is one of the key physical quantities administering an assortment of marvels amid the quantum mechanical process and subsequently its control is the route to new nanophotonic devices. As we know, the density of electromagnetic states describes the number of permitted states within the volume enclosed by the corresponding isofrequency surface [49] at a certain frequency. Any adjustment in the optical properties of the environment where the quantum emitter is located affects the allowed number of states and further changes the spontaneous emission and extraction according to the Fermi’s golden rule. As discussed in Section 2, optical topological transitions from the closed (ellipsoid) isofrequency surface to an open (hyperboloid) isofrequency surface happen in graphene-hBN hyperstructure when the chemical potential of graphene is tuned persistently, which results in a nonintegrable singularity accompanied by a change in the density of states from a finite to an infinite value (in lossless effective medium limit). Light-matter interaction is enhanced because of the presence of these additional electromagnetic states, resulting in a strong effect on related quantum-optical phenomena.

To estimate the spontaneous emission of quantum emitters near the hyperstructure, the following assumption is applied. First, we assume that the nanoscale light-matter interaction with hyperstructure works in the weak-coupling regime [26,28], i.e., that the non-radiative damping of single photon sources is larger than the interaction strength between the quantum emitter and the photonic system. Thus, the outcomes are relied upon to be in a good concurrence with the treatment depends on the principles of classical electrodynamics [50]. Second, the quantum emitter considered in this work is simulated as an oscillating classical point dipole elevated at height h above the hyperstructure upmost layer. With this framework, we can present the following picture of light emission near such hyperstructure. The electric field directly radiated by the quantum emitter can be conveniently expanded in a superposition of plane waves of complex amplitudes for all values of the wavevector (including evanescent terms with $\text{k}>{\text{k}}_0$). The propagating waves (with $\text{k}<{\text{k}}_0$) toward the hyperstructure are scattered into vacuum while the evanescent waves (with $\text{k}>{\text{k}}_0$) are coupled into high-k field waves in the hyperstructure during the transitions. One thing to note is that only a few parts could be able to couple with the surface plasmon polaritons during the transitions, thus we may neglect them. If the distance between the quantum emitter and the hyperstructure is much smaller than the wavelength, $h\ll {\lambda }_0$, the spontaneous emission is dominated by the contribution from the high-k field waves. According to [48], we could deduce the corresponding Purcell factors (${\text{F}}_{\text{P}}$) for a single dipole emitter with in-plane (x and y) or out-of-plane (z) orientation (For a more detailed analysis of the Purcell factor, please see the Appendix, Section A3).

Figure 2 shows the plots of the Purcell factors for different chemical potentials of graphene (from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$). The hyperstructure is described with effective medium approximation. Here, we assume that the total thickness of the layered hyperstructure is about 600 nm, the emitter is located at 10 nm above the hyperstructure, the dielectric environment is a vacuum and the value of Purcell factor for the statistically averaged dipole orientation is considered. Figures 2(a)-2(c) correspond to region I, region II and region III respectively, as shown in Fig. 1. As discussed formerly, optical topological transitions from an ellipsoid to a hyperboloid occur in these regions when the chemical potential of graphene is increased from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$. This results in an enhancement of the spontaneous emission dominated by the contribution from the high-k field waves in a broadband wavelength range. Because only transverse component ${ϵ}_{\text{t}}$ varies when the chemical potential is tuned, we can divide the optical topological transitions into two different circumstances. One is that when the chemical potential of graphene is increased, ${ϵ}_{\text{t}}$ approaches to zero. From the plot in Fig. 2, we can observe that the spontaneous emission reduces in this situation. Another one is that ${ϵ}_{\text{t}}$ becomes negative by further increasing the chemical potential. Although in different topological transitions, losses reduce the sharp transitions to different smooth crossovers, the change in the isofrequency surface topology still leads to an enhanced spontaneous emission, which is shown in the plot. In short, tuning the chemical potential of graphene in a hyperstructure becomes an efficient way to manipulate the spontaneous emission.

 

Fig. 2 Theoretical estimations of the Purcell factor for an emitter located at 10 nm above the graphene-hBN hyperstructure surface. The Purcell factors at different chemical potentials of graphene (${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are shown here. Three different topological transitions (a), (b) and (c) are considered which correspond to region I, region II and region III respectively, as shown in Fig. 1.

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In the following, we turn to study the extraction from the non-radiative modes when the optical topological transitions occur. Generally, when high-k field waves reach the side interface of the graphene-hBN hyperstructure, the conservation of the k-vector component parallel to the interface requires ${k^{\prime }}_{\text{z}}=k_z$, if the dispersion relation for the environment is described by ${k^{\prime }}_t^2+{k^{\prime }}_z^2={k^{\prime }}^2$. According to the dispersion relation of the environment and $k_z\gg k^{\prime }$, we could find that transverse component ${k^{\prime }}_t$ becomes purely imaginary due to ${k^{\prime }}_t=\sqrt{{k^{\prime }}^2-k_z^2}=i\sqrt{k_z^2-{k^{\prime }}^2}$, which leads to an evanescent wave in the environment along transverse directions and prevents the successful energy extraction from hyperstructure into the far-field. As mentioned above, in the proposed hyperstructure, optical topological transitions from an ellipsoid to a hyperboloid occur by increasing the chemical potential of graphene. Amid this procedure, just the transverse dielectric function of the hyperstructure is modified due to the metallic behavior of graphene. The transverse components gradually modify from positive to negative values by tuning the chemical potential continuously as anticipated by the EMT. We find that the optimized value of wavelength and chemical potential of graphene can achieve ${ϵ}_{\text{t}}\to 0^{-}$ at different wavelength ranges, where the hyperstructure’s hyperbolic isofrequency surface gets “flatten” along the transverse direction and “squeezed” along the axial direction. In this case, a band of wavevectors exists in the hyperstructure which possesses a conserved component of $k_z\le k^{\prime }$, and hence hold their propagating nature over the side interface.

On the other hand, when high-k field waves reach the top interface, the conservation of the k-vector component parallel to this interface requires ${k^{\prime }}_t=k_t$. We could find that component $k_z^{\text{'}}$ becomes purely imaginary due to ${k^{\prime }}_z=i\sqrt{k_t^2-{k^{\prime }}^2}$ if $k_t\gg k^{\prime }$, which leads to an evanescent wave in the environment along z direction. As mentioned above, the transverse components of the hyperstructure gradually reduce by increasing the chemical potential continuously as anticipated by the EMT. We find that extremely large value of chemical potential of graphene can achieve ${ϵ}_{\text{t}}\ll 0$ with low losses at different wavelength ranges, thus the hyperstructure’s hyperbolic isofrequency surface gets “flatten” along the z direction. In this regard, a band of wavevectors existing in the hyperstructure could hold their propagating nature over the top interface. This process corresponds to the second circumstance mentioned previously. By combining these two circumstances during optical topological transitions, we could reveal that tuning the chemical potential of graphene in a hyperstructure is also an efficient way to manipulate the extraction from the hyperstructure.

In order to estimate the extraction from both side and top interfaces, we assume that an emitter is located at 10 nm away from the interfaces of graphene-hBN hyperstructure. Figure 3 shows the side and top extractions (normalized collected emission power) from the interfaces of the hyperstructure (For a more detailed analysis of the normalized extraction, please see the Appendix, Section A3). Different chemical potentials (from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are demonstrated here. From the plot, we could observe that the side and top extractions will be enhanced by increasing the chemical potential of graphene except for the side extraction in region III. We could observe that the side extraction in region III becomes complicated. This is because when ${ϵ}_{\text{t}}\to 0^{-}$, the loss of the hyperstructure limit the “flatten” dispersion related effects. By consolidating Figs. 2 and 3, we could conclude that both spontaneous emission and extraction can be controlled by tuning the chemical potential of graphene, and at optimized value of wavelength both of them can be enhanced. Here we show one example of this mechanism to validate our prediction. The system is designed to operate at $\lambda =6\text{μm}$. We conduct field mapping simulations using a commercial finite-element method (FEM) based solver (COMSOL Multiphysics, Wave Optics Module). The hyperstructure is infinite along the y-direction and has a rectangular cross-section of 2 μm$\times$600 nm. We model the quantum emitter using a point dipole. The point dipole emits at $\lambda =6\text{μm}$, and is located at 10 nm above the top surface. Figure 4 illustrates the spatial power distributions of a dipole oriented along x (x-dipole) and z (z-dipole) respectively at two different chemical potential of graphene. From Fig. 4, we could clearly observe the tunability of spontaneous emission and enhancement of extraction according to the hyperstructure by increasing the chemical potential of graphene from 0 to 0.3 eV.

 

Fig. 3 Theoretical estimations of (a-c) side and (d-f) top extractions for an emitter located at 10 nm away from the side and top interfaces of the graphene-hBN hyperstructure surface, respectively. The extractions at different chemical potentials of graphene (${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are shown here. Three different topological transitions (a,d), (b,e) and (c,f) are considered which correspond to region I, region II, and region IV, respectively, as shown in Fig. 1. The extraction power through the interface of hyperstructure is calculated as an integral through a circular area which mimics the emission collection using a commercially available objective lens with numerical aperture (NA) 1.49 (cross-section angle 79.6°).

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Fig. 4 Spatial power distributions (magnitude of the Poynting vector) of a dipole located at 10 nm above the hyperstructure in the air at $\lambda =6\text{μm}$. a, b, c and d represent the power distributions of the dipole along x-direction (x-dipole) and z-direction (z-dipole), respectively, at two different graphene chemical potentials (${\text{μ}}_{\text{c}}$). When ${\text{μ}}_{\text{c}}$ changes from 0 to 0.3 eV, the topological transition from an ellipsoid (${ϵ}_{\text{t}}>0,{ϵ}_{\text{z}}>0$) to a hyperboloid (${ϵ}_{\text{t}}<0,{ϵ}_{\text{z}}>0$) happens respectively.

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The above discussion is based on EMT that holds well in the limit $\text{kd}\ll 1$, which can be obtained according to the Bloch theorem through the Taylor expansion up to the second order. In general, when the electromagnetic momentum approaches the inverse size of the metamaterial unit cell, the effective medium approximation will be broken, therefore the metamaterial can no longer be treated as an effective electromagnetic medium, as the propagating wave cannot ‘resolve’ its internal structure. However, in the proposed hyperstructure, the hBN is a natural hyperbolic material and graphene has atomic size, thus the propagating wave inside the layered hyperstructure could still resolve each layer of the hyperstructure.

In order to estimate the spontaneous emission and extraction of the realistic layered hyperstructure, we compute the generalized reflection and transmission coefficients by using a recursive evaluation of stratified structure (For a more detailed analysis of this approach, please see the Appendix Section A4). According to this method, we obtain the Purcell factor and top extraction for the realistic layered hyperstructure, as shown in Fig. 5. From Figs. 5(a)-5(f), we also reveal that the variation/evolution phenomena of Purcell factor and the extraction for the realistic layered hyperstructure are almost similar to EMT case during the optical topological transitions. This analysis shows that the general method we propose here retains its validity beyond EMT.

 

Fig. 5 Theoretical estimations of the (a-c) Purcell factor and (d-f) top extractions for an emitter located at 10 nm away from the top interfaces of the graphene-hBN hyperstructure surface, respectively. Different chemical potentials of graphene (from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are shown here. Three different topological transitions (a,d), (b,e) and (c,f) are considered which correspond to region I, region II, and region III as shown in Fig. 1 respectively.

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In addition, to test the extraction more practically, a single, isolated quantum emitter has to be chosen. Many different systems are possible candidates, such as atoms, quantum dots, dye molecules and fluorescent defect centers in crystals. The precise distance of the quantum emitter to the interface of the hyperstructure is unknown, however, with certainty it is located within the structure of the quantum emitter. Because of the high refractive index of most host materials which results in strong confinement via total internal reflection, the decay rates mentioned previously will be strongly reduced. On the other hand, as the distance of the quantum emitter to the interface of the hyperstructure is increased, the Purcell factor decreases, which can be found in Fig. 6. The Purcell factor dependence on the distance can be explained by the fact that at short distances the evanescent fields created by the quantum emitter could be better coupled to the $\text{high-k}$modes in the hyperstructure.

 

Fig. 6 Theoretical estimations of the Purcell factor versus different distances between the dipole and the hyperstructure h. The chemical potential of graphene here is ${\text{μ}}_{\text{c}}=0\text{eV}$.

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4. Summary

Although HMMs have the ability to enhance the light-matter interaction due to their open isofrequency surface, these materials have remained elusive for use in practical devices because of their material losses, large size of the unit cell in the realization as well as the non-radiative modes inside the HMMs. Here, we propose the graphene-hBN hyperstructure which inherits the natural hyperbolic properties of the hBN and tunable behavior of graphene. From the theoretical estimation, we show that efficient spontaneous emission and enhanced light extraction could be achieved simultaneously due to the optical topological transition of the isofrequency surface by changing the chemical potential of graphene (For a more detailed analysis of the Purcell factors and extractions for graphene-hBN and graphene-hBN-graphene hetrostructures, please see the Appendix, Section A5).We expect that the graphene-hBN hyperstructure for single-photon sources proposed in this paper may open up new avenues for future research in the field of quantum information processing applications.

5 Appendix

A1. Permittivity of hBN

hBN is a natural hyperbolic material and can accommodate highly dispersive surface phonon-polariton modes. The dielectric constants for hBN can be modeled by permittivity tensor, i.e., ${\overline{\overline{ϵ}}}_{\text{r,hBN}}=\text{diag}({ϵ}_{\text{t,hBN}},{ϵ}_{\text{t,hBN}},{ϵ}_{\text{z,hBN}})$, where ${ϵ}_{\text{t,hBN}}={ϵ}_{\text{x,hBN}}={ϵ}_{\text{y,hBN}}\ne {ϵ}_{\text{z,hBN}}$ [37]. The permittivity components of hBN can be expressed as

(3)$${ϵ}_{\text{hBN}}={ϵ}_{\text{l}}\left(\infty \right)+{\text{s}}_{\text{v},\text{l}}\frac{{\text{ω}}_{\text{v},\text{l}}^2}{{\text{ω}}_{\text{v},\text{l}}^2-{\text{iγ}}_{\text{v},\text{l}}\text{ω}-{\text{ω}}^2},\text{l}=\text{x,}\text{y,}\text{z}$$

For the in-plane (indexed by x and y) direction, i.e., $\text{l}=\text{x,}\text{y}$, we could have a set of parameters: ${ϵ}_{\text{l}}\left(\infty \right)=4.87$, the dimensionless coupling factor ${\text{s}}_{\text{v},\text{l}}=1.83$, the normal frequency of vibration $\hslash {\text{ω}}_{\text{v},\text{l}}=170.1\text{meV}$, and the amplitude decay rate $\hslash {\text{γ}}_{\text{v},\text{l}}=0.87\text{meV}$. $\hslash$ the reduced Planck constant. For the out-of-plane (indexed by z) direction, i.e., $\text{l}=z$, the parameters are: ${ϵ}_{\text{l}}\left(\infty \right)=2.95$,${\text{s}}_{\text{v},\text{l}}=0.61$,$\hslash {\text{ω}}_{\text{v},\text{l}}=92.5\text{meV}$, and $\hslash {\text{γ}}_{\text{v},\text{l}}=0.25\text{meV}$. One thing to notice is that the nonlocal impact of hBN is dismissed in all estimations in this work, since the reciprocal lattice vector of hBN is at least two orders of magnitude larger than the wavevector of polaritons. The permittivity of hBN can be found in Fig. 7(a).

 

Fig. 7 (a) Real and imaginary parts of the permittivity of hBN. (b) Real and imaginary parts of conductivity of graphene as a function of chemical potential. Colored regions indicate the two mid-infrared Reststrahlen bands.

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A2. Conductivity of graphene

Graphene can be displayed as an unendingly thin sheet with conductivity ${\text{σ}}_{\text{s}}$, including the contributions from both the intraband and interband transitions [42], i.e., ${\text{σ}}_{\text{s}}={\text{σ}}_{\text{D}}+{\text{σ}}_{\text{I}}$, respectively. At temperature T (T=300K), these surface conductivities can be characterized by the random phase approximation (local RPA) model, which are expressed as [43]

(4)$${\text{σ}}_{\text{D}}=\frac{\text{i}}{\text{ω}+\text{i}/\text{τ}}\frac{2{\text{e}}^2{\text{k}}_{\text{B}}\text{T}}{\text{π}{\hslash }^2}\text{ln}\left[2\cos \text{h}\left(\frac{{\text{μ}}_{\text{c}}}{2{\text{k}}_{\text{B}}\text{T}}\right)\right]$$

(5)$${\text{σ}}_{\text{I}}=\frac{{\text{e}}^2}{4\hslash }\left[\text{H}\left(\text{ω}/2,\text{T}\right)+\frac{4\text{iω}}{\text{π}}{{\displaystyle \int }}_0^{\infty }\text{dζ}\frac{\text{H}\left(\text{ζ},\text{T}\right)-\text{H}\left(\text{ω}/2,\text{T}\right)}{{\text{ω}}^2-4{\text{ζ}}^2}\right],$$

where $\text{H}\left(\text{ω},\text{T}\right)=\text{sinh}\left(\hslash \text{ω}/{\text{k}}_{\text{B}}\text{T}\right)/\left[\text{cosh}\left({\text{μ}}_{\text{c}}/{\text{k}}_{\text{B}}\text{T}\right)+\text{cosh}\left(\hslash \text{ω}/{\text{k}}_{\text{B}}\text{T}\right)\right]$, e is the elementary charge, $\hslash$ is the reduced Planck constant, ${\text{μ}}_{\text{c}}$ is the chemical potential, ${\text{k}}_{\text{B}}$ is the Boltzmann constant, and $\text{τ}=\frac{{\text{μμ}}_{\text{c}}}{{\text{evF}}^2}$ is the relaxation time characterizing the loss in graphene, where the Fermi velocity ${\text{v}}_{\text{F}}=1\times {10}^6\text{m/s}$ and the electron mobility of $\text{μ}=10000{\text{cm}}^{\text{2}}{\text{.V}}^{\text{-1}}{\text{.s}}^{\text{-1}}$ are used [44–46]. The conductivity of graphene as a function of chemical potential is shown in Fig. 7(b).

A3. Calculations of Purcell effect and normalized collected emission power

In our case we consider the issue as dipole radiation closed to the hyperstructure. According to [26] and [47], we could deduce the corresponding Purcell factors (${\text{F}}_{\text{P}}$) for a single dipole emitter with in-plane (x and y) or out-of-plane (z) orientation at a distance h above the hyperstructure by using the following formulas

(6)$$F_P^{\parallel }=1+\frac{3}{4}\frac{1}{{\epsilon }_{\text{sup}}^{1/2}}{{\displaystyle \int }}_0^{\infty }\text{Re}\left\{\frac{s}{s_{\perp ,\text{sup}}\left(s\right)}\left[{\tilde{r}}^s\left(s\right)-\frac{s_{\perp ,\text{sup}}^2\left(s\right)}{{\epsilon }_{\text{sup}}}{\tilde{r}}^p\left(s\right)\right]e^{2i{k}_0{s}_{\perp ,\text{sup}}\left(\text{s}\right)h}\right\}ds$$

(7)$$F_P^{\perp }=1+\frac{3}{2}\frac{1}{{\epsilon }_{\text{sup}}^{3/2}}{{\displaystyle \int }}_0^{\infty }\text{Re}\left\{\frac{s^3}{s_{\perp ,\text{sup}}\left(s\right)}{\tilde{r}}^p\left(s\right)e^{2i{k}_0{s}_{\perp ,\text{sup}}\left(\text{s}\right)h}\right\}ds$$

The value of Purcell factor for the statistically averaged (ave) dipole orientation is given by

(8)$$F_P^{ave}=\frac{2}{3}{F}_P^{\parallel }+\frac{1}{3}{F}_P^{\perp },$$

The normalized extraction powers from the hyperstructure f_rad for the same dipole orientations are shown below

(9)$$\begin{array}{l}{f}_{rad}^{\left|\right|}=\frac{3}{8}\underset{0}{\overset{{\theta }_{\text{max}}}{{\displaystyle \int }}}{\text{cos}}^2\theta {\left|e^{-i{\epsilon }_{sup}^{1/2}{k}_{0}h\text{cos}\theta }-{\tilde{r}}^p\left(\theta \right)e^{i{\epsilon }_{sup}^{1/2}{k}_{0}h\text{cos}\theta }\right|}^2\\ +{\left|e^{-i{\epsilon }_{sup}^{1/2}{k}_{0}hcos\theta }+{\tilde{r}}^p\left(\theta \right)e^{i{\epsilon }_{sup}^{1/2}{k}_{0}h\text{cos}\theta }\right|}^2\sin \theta d\theta \end{array}$$

(10)$$f_{rad}^{\perp }=\frac{3}{4}\underset{0}{\overset{{\theta }_{\text{max}}}{{\displaystyle \int }}}{\sin }^3\theta {\left|e^{-i{\epsilon }_{sup}^{1/2}{k}_{0}h\text{cos}\theta }+{\tilde{r}}^{\text{p}}\left(\theta \right)e^{i{\epsilon }_{sup}^{1/2}{k}_{0}h\text{cos}\theta }\right|}^{2}d\theta ,$$

(11)$$f_{rad}^{ave}=\frac{2}{3}{f}_{rad}^{\left|\right|}+\frac{1}{3}{f}_{rad}^{\perp }.$$

In the above equations, $s=\frac{k_{\left|\right|}}{k_0}$,${\text{s}}_{\perp ,\text{sup}}\left(\text{s}\right)=\frac{{\text{k}}_{\perp ,\text{sup}}}{{\text{k}}_0}={\text{ε}}_{\text{sup}}-{\text{s}}^{2^{1/2}}$, ${\text{k}}_0=\frac{\text{ω}}{\text{c}}$, and $\theta$ is the polar angle measured from the out-of-plane (indexed by z) direction, the subscript ${\epsilon }_{sup}=1$ represents the permittivity of superstrate. The integrals are numerically evaluated by using an adaptive Gauss–Kronrod quadrature technique [48].

A4. Calculations of generalized reflection and transmission coefficients

In order to estimate the Purcell factor and extraction of real stratified hyperstructure, a recursive evaluation of stratified structure is performed. According to the reflection and transmission problem studied in stratified systems [51], the generalized reflection coefficient has the following form

(12)$${\tilde{R}}_{l,l+1}^t=R_{l,l+1}^t\frac{T_{l,l+1}^t{\tilde{R}}_{l+1,l+2}^t{T}_{l+1,l}^t{e}^{2i{k}_{l+1,z}^t{d}_{l+1}}}{1-R_{l+1,l}^t{\tilde{R}}_{l+1,l+2}^t{e}^{2i{k}_{l+1,z}^t{d}_{l+1}}},$$

where $t=p,s$ stands for s-polarization or p-polarization, $l=1,2,\mathrm{...}$ identifying the layer number, $d_l$ is the thickness of layer $l$, the subscript $l,l+1$ means the coefficient of region $l$ and $l+1$, i.e. $R_{l,l+1}^p=\frac{{ϵ}_{l+1,\left|\right|}{k}_{l,z}^p-{ϵ}_{l,\left|\right|}{k}_{l+1,z}^p}{{ϵ}_{l+1,\left|\right|}{k}_{l,z}^p+{ϵ}_{l,\left|\right|}{k}_{l+1,z}^p}$, $T_{l,l+1}^p=\frac{2{ϵ}_{l+1,\left|\right|}{k}_{l,z}^p}{{ϵ}_{l+1,\left|\right|}{k}_{l,z}^p+{ϵ}_{l,\left|\right|}{k}_{l+1,z}^p}$ or $R_{l,l+1}^s=\frac{k_{l,z}^s-k_{l+1,z}^s}{k_{l,z}^s+k_{l+1,z}^s}$, $T_{l,l+1}^s=\frac{2k_{l,z}^s}{k_{l,z}^s+k_{l+1,z}^s}$ are the local reflection and transmission coefficients, and the wavevector $k_{l,z}^p=\sqrt{\frac{{ϵ}_{l,\left|\right|}}{{\epsilon }_{l,z}}\left({\epsilon }_{l,z}{k}_0^2-k_{\left|\right|}^2\right)},k_{l,z}^s=\sqrt{{\epsilon }_{l,\left|\right|}{k}_0^2-k_{\left|\right|}^2}$. And the coefficients $r^s={\tilde{R}}_{12}^s$ and $r^p={\tilde{R}}_{12}^p$ are the generalized reflection coefficient used in calculating Purcell factor and Extraction.

A5. Graphene-hBN and graphene-hBN-graphene heterostructures

In order to motivate people regarding the advantage(s) of their multilayer periodic design, the Purcell factors as well as the extractions for two heterostructures: graphene-hBN and graphene-hBN-graphene are calculated as shown in Fig. 8 and Fig. 9. Obviously, the extractions remain and will not be enhanced in the optical transition range by tuning the chemical potential of graphene.

 

Fig. 8 Theoretical estimations of the (a-c) Purcell factor and (d-f) top extractions for an emitter located at 10 nm away from the top interfaces of the graphene-hBN heterostructure, respectively. Different chemical potentials of graphene (from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are shown here.

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Fig. 9 Theoretical estimations of the (a-c) Purcell factor and (d-f) top extractions for an emitter located at 10 nm away from the top interfaces of the graphene-hBN-graphene heterostructure, respectively. Different chemical potentials of graphene (from ${\text{μ}}_{\text{c}}=0\text{eV}$ to ${\text{μ}}_{\text{c}}=0.3\text{eV}$) are shown here.

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Funding

National Natural Science Foundation of China (NSFC) (61574127, 61601408, 61625502, 61674128, 11704332, 61731019, 61775193); Natural Science Foundation of Zhejiang Province (LY12F02032, LY13F020036, LY17F010008, LY19F010015); China Postdoctoral Science Foundation (2018M632462).

Acknowledgment

We thank Dr. Lin Xiao, Kalim Ullah and Muhyiddeen Yahya Musa for their valuable comments and suggestions.

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