1. Introduction

Precisely engineered low-dimensional semiconductors embedded in nanophotonic structures have been proposed as highly efficient light emitting devices for the application of sensing, display, and quantum light generation [1]. Recently emerging two-dimensional (2d) material-based quantum dots (QDs) such as hexagonal boron nitride (hBN), graphene, and transition metal dichalcogendies (TMD) QDs [2,3] and Perovskites nanocrystals [4] have been specially focused on account of their easy emission wavelength tunability depending on the size and composition materials and high chemical and photostability compared to those of conventional semiconductor QDs [5].

However, it has been theoretically explained [6–10] and experimentally revealed [11,12] that the commonly used point-like electric dipole approximation is not a valid expression for the light-matter interaction, when highly asymmetric shape and mesoscopic scale of quantum light emitters (QLEs) are exposed to inhomogeneous environments [13,14]. The breakdown of dipole regime was attributed to nonlocal susceptibilities of asymmetrical distribution of quantum-confined excitons [6–8,11,12] and multipolar electric and magnetic moments on atomic scale [9,10]. Completely opposite behaviors shown in the radiative decay rates of small and large QDs in the vicinity of metal surface could only be understood within the framework of nonlocal susceptibility model selectively momentum-matched with that of surface plasmon polaritons [15].

In order to control propagation characteristics of light and to enhance radiative quantum efficiency of quantum light emitters, hyperbolic metamaterials (HMMs) [16,17] and metasurfaces [18,19] with diverse constituents and tunable platforms [20] have been intensively investigated. As a uniaxially anisotropic material, hyperbolic materials (HMs) have the opposite signs of transverse and parallel dielectric constants with respect to the optical axis of materials. As a result, HMs have a hyperbolic dispersion curve at a fixed frequency, distinguished from elliptical ones of ordinary materials. By virtue of hyperbolic dispersion, in principle, all polaritonic modes in bulk HMs are propagating. Thus, HMs are considered to be highly applicable for topological transition material [21], invisible cloaking [22], coherent energy transfer between donor and acceptor [23–25], and radiative heat transfer [26].

Additionally, increasing photonic density of states can lead to large Purcell enhancement of point-like light emitters in/near HMs [27–29]. The maximum Purcell factor of finite-size dipole emitters in homogeneous HM [30], the scaling law of Purcell factor in HMM cavities [31], simultaneous enhancement of spontaneous radiation of dipoles and outcoupling into free space radiation by using nano-patterned HMMs [32–34], and enhanced single photon extraction [35] have been discussed.

Hyperbolic dispersion has also been reported on diverse natural materials composites in a broad frequency range from terahertz (THz) to ultra violet (UV) [36], including Tetradymites [37], organic material [38], 2d materials [39] and their van der Waals heterostructures [40,41]. Natural HMs have an advantage over artificial HMMs, besides uniform quality and relatively simple fabrication processes, in the sense that the available highest mode in the dispersion relation is determined by Fourier transform of the smallest feature size of each material, which is an atomic scale in nature HMs but still limited by several tens of nanometers in HMMs. Further discussion about pro and con between natural and artificial HMs can be found in [36].

In this report, we theoretically study two optical properties, the radiative decay rate (RDR) and frequency shift (FS), of single excitons confined within a highly asymmetric disk-shaped QD representing pyramidal shape of epitaxial self-growth QDs or 2d QDs (see Fig. 1(a)) in a close proximity to the surface of half-infinite tetradymites($\textrm {Bi}_2\textrm {Se}_3$), a natural HM in a visible-to-near infrared (NIR) wavelength range. By using a nonlocal susceptibility model for quantum-confined excitons embedded in anisotropic multilayered structures, we can calculate both properties from the self-energy correction (SEC) as a function of exciton’s lateral confinement and distance from the surface of HM for different quantum eigenstates of exciton. We demonstrate that the SEC of weakly confined (quasi free) 2d excitons are barely influenced by specific choice of the eigenstate and anisotropic properties of the HM even at a close distance, while that of strongly confined excitons shows, on the contrary, strong dependence on those conditions. Our findings are very fundamental to architect most nanophotonic systems containing quantum light emitters.

 

Fig. 1. (a) System configuration considered in this study. A disk-shaped quantum dot with radius $R$ is embedded in an isotropic dielectric $\epsilon _1$ with distance $z_s$ from the surface of a half-infinite uniaxially anisotropic material $\underline {\boldsymbol{\epsilon }}_2$. The optical axis of $\underline {\boldsymbol{\epsilon }}_2$ is assumed to be parallel to the z-axis. The real (b) and imaginary parts (c) of $\textrm {Bi}_2\textrm {Se}_3$ as a natural HM for $\underline {\boldsymbol{\epsilon }}_2$ are plotted as a function of wavelength. $\textrm {Bi}_2\textrm {Se}_3$ shows HM type I for wavelength $\lesssim 0.7\,\mu m$ but HM type II for wavelength $\gtrsim \, 0.85\,\mu m$ but $\lesssim 1.17\,\mu m$. Three wavelength positions are marked by dotted, dot-dashed, and dashed lines in (b), respectively.

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2. Mathematical model

Figure 1(a) shows the configuration of system considered in this study. A shallow QD with a radius $R$ is embedded in an isotropic medium of dielectric constant $\epsilon _1$ and located at a distance $z_s$ from a half-infinite substrate. Due to close similarity to our previous work [15], we can inherit the same mathematical formalism and, without loss of generality, use the general forms of the nonlocal susceptibility $\chi$ and the SEC $\Sigma$ in homogeneous and inhomogeneous environments (Eqs. (7)–(9) in [15]) for this study. In what follows, we demonstrate that $\underline {\boldsymbol{\epsilon }}_2$, which is only the difference from the previous work and now stands for the uniaxially anisotropic dielectric constant of the substrate material as a $3\times 3$ diagonal matrix, can give rise to qualitatively different optical selection rules and optical behaviors of the QD exciton.

However, it would benefit readership to represent the explicit forms of $\chi$ and $\Sigma$ with a brief background of the models. In a shallow QD, a Coulomb-coupled electron-hole pair, i.e. exciton, is assumed to be confined in a very narrow infinite potential well along the z-direction but within a harmonic potential in the xy-plane. Then, quantum kinetic equations of motion for single exciton are accordingly divided into the lateral (xy-plane with spatial vector $\boldsymbol{r}_{\parallel}=(x,\,y)$) and normal components (z-axis). In a weak confinement regime where the radius of QD is sufficiently larger than exciton Bohr radius, the lateral motion of exciton can be further expanded by a product of the relative and center-of-mass (COM) motions [42,43]. For a fixed quantum state $\zeta (z)$ in the normal direction, it can be shown that the light-matter coupling occurs only at the COM motion [44]. Therefore, the polarization of the QD can be expressed as

The GFTs for homogeneous HM [47] and anisotropic multilayered structures [48–50] in the Cartesian coordinate systems have been introduced. In this work we adopt the GFT derived in the cylindrical coordinate system [51,52] because well defined value of the GFT at origin is especially profitable to calculate the SEC. Even though the QD is assumed to be surrounded by an isotropic material in this work, we provide the SEC of exciton in a homogeneous uniaxially anisotropic material, because the reflected field from its surface has mathematically a straightforward connection.

In homogeneous uniaxially anisotropic medium with a dielectric constant $\underline {\boldsymbol{\epsilon }}=\underline {\boldsymbol{\epsilon }}_2$ (see inset of Fig. 1(a)), two different dispersion relations can be obtained by solving Maxwell’s wave equation:

In accordance with the dispersion relations, the GFT in HM is constituted by two parts depending on $h_1=(k_1^{2}-\lambda ^{2})^{1/2}$ and $h_2=(k_2^{2}-\lambda ^{2})^{1/2}$:

Next, we derive a concrete form of Eq. (5) and discuss the resulting optical selection rules. For $\hat {\boldsymbol{u}}_d$, not only linear polarization directions ($\boldsymbol{u}_d=\hat {\boldsymbol{x}}$, $\hat {\boldsymbol{y}}$, or $\hat {\boldsymbol{z}}$), but two circular ones in the xy-plane ($u_x=1/\sqrt {2}$ and $u_y=\pm i/\sqrt {2}$ with $u_z=0$ for $\sigma ^{\pm }$) are included. For $\psi _\alpha (\boldsymbol{r}_{\parallel })$, three energetically lowest eigenstates in a 2d harmonic potential ($V(\boldsymbol{r}_{\parallel })=M\omega _c^{2}\rho ^{2}/2$) are considered.

By expanding Eq. (12) into a product of radial and azimuthal parts $\psi _{ml}(\boldsymbol{r}_{\parallel })=\varphi _{ml}(\rho )e^{il\phi }$ and unifying the coordinate bases, the spatial integration can be separately performed with respect to $\rho$ and $\phi$. In consequence, Eq. (5) is simplified to

3. Numerical results and analysis

For the numerical calculation we adopt the same parameters used in [15] for the QD and the dielectric constant of $\textrm {Bi}_2\textrm {Se}_3$ for $\underline {\boldsymbol{\epsilon }}_2$ of the HM. In Figs. 1(b) and (c), we reproduce the real and imaginary parts of experimentally measured $\epsilon _t$ and $\epsilon _z$ [37], respectively. $\textrm {Bi}_2\textrm {Se}_3$ shows HM type I ($\epsilon _t>0$ and $\epsilon _t<0$) in a wavelength region shorter than $0.7\,\mu m$, but it changes to type II ($\epsilon _t<0$ and $\epsilon _t>0$) in a longer wavelength regime from $0.85\,\mu m$ to $1.15\,\mu m$. In contrast, a constant value of $\epsilon _1=2.5$ is used for the surrounding material of the QD, a typical dielectric constant of materials used to transfer quantum light emitters on substrates or of ionic gel for electrical biasing.

In previous theoretical reports, optical properties of quantum light emitters with finite size have mostly been studied for the Gaussian eigenfunction with zero orbital angular momentum (OAM) for the envelope function of exciton. Meanwhile the optical selection rule determined solely by the spatial integration of the envelope function leads to zero for all the eigenstates with non-zero OAM [42], they are on the contrary allowed in our optical selection rule given by the Hankel transformation of the quantum states (see Eqs. (14) and (16)). Furthermore, it has been experimentally reported that exciton with OAM could be directly excited by using optical four-wave mixing setup [54] and vortex beam [55].

In Fig. 2, the wavelength-dependent frequency shifts (FSs) (Figs. 2(a) and (b)) and radiative decay rates (RDRs) (Figs. 2(c) and (d)) given by the homogeneous (hom) and inhomogeneous (inh) self-energy corrections (SECs) of the two QDs ($R=10\,nm$ and $500\,nm$) are presented for two polarization directions where $\psi _{ml}(\boldsymbol{r}_{\parallel })=\psi _{00}(\boldsymbol{r}_{\parallel })$ for the x-polarization (x-pol, blue) and $\psi _{ml}(\boldsymbol{r}_{\parallel })=\psi _{01}(\boldsymbol{r}_{\parallel })$ for the $\sigma ^{+}$-polarization ($\sigma ^{+}$-pol, red) are selected for both QDs located at the same distance ($z_s=10\,nm$) from the interface.

 

Fig. 2. (a) The frequency shift (FS) and (c) radiative decay rate (RDR) of a small QD with $R=10\,nm$, and (b, d) those of a large QD with $R=500\,nm$ are displayed as a function of wavelength for the two different polarization states (x-pol and $\sigma ^{+}$-pol) of the QD excitons embedded in homogeneous (hom) $\epsilon _1$ and inhomogeneous (inh) $\epsilon _1$ and $\underline {\boldsymbol{\epsilon }}_2$ with $z_s=10\,nm$, respectively.

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The distinctive feature found in Fig. 2 is two orders of magnitude difference in the RDRs of the smaller QD ($R=10\,nm$) between two polarization directions (cp. the red and blue lines in Fig. 2(c)), while the RDRs of the larger QD ($R=500\,nm$) in the homogeneous and inhomogeneous surroundings show almost no difference between the two polarization directions (Fig. 2(d)). According to Eq. (15), only a factor of $1/4$ could be expected in the difference of RDR for the fixed size of QD. In order to understand this behavior, we plot the absolute square of the Hankel-transformed radial functions of both QDs depending on the normalized in-plane wave number $x$ in Figs. 3(a) and (b) where each function is normalized by its maximum for an easy comparison.

 

Fig. 3. The absolute square of the Hankel-transformed exciton envelope functions $|\tilde {\varphi }_{ml}(x)|^{2}$ are normalized by their maximum values and plotted as a function of the normalized in-plane wave number x for (a) $R=10\,nm$ and (b) $R=500\,nm$, respectively. The vertical black dotted lines are located at $x=\sqrt {\epsilon _1}=1.58$ in each figure.

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The RDR of QDs in homogeneous environment, corresponding to the imaginary part of the SEC, is determined by the integration of all propagating modes in the medium ($0\le \,x\,\le \sqrt {\epsilon _1}=n_1=1.58$) as seen in Eq. (13). The vertical black dotted lines at $x=n_1$ in Figs. 3(a) and (b) divide $|\tilde {\varphi }_{ml}(x)|^{2}$ into the propagating ($x<n_1$) and evanescent mode ($x>n_1$) contributions. Because all three radial functions $|\tilde {\varphi }_{ml}(x)|^{2}$ of the larger QD are distributed within $x<n_1$ (Fig. 3(b)), their integrations are determined by the same order of magnitude. On the contrary, the propagating mode contribution of the smaller QD depends critically on the radial function. While $|\tilde {\varphi }_{00}(x)|^{2}$ and $|\tilde {\varphi }_{10}(x)|^{2}$ still have a substantial overlap with the propagating modes, $|\tilde {\varphi }_{01}(x)|^{2}$ is mostly distributed in evanescent modes (Fig. 3(a)). The two-order of magnitude difference in the RDR, e.g. at a wavelength of $0.6\,\mu m$, between the x-pol with $|\tilde {\varphi }_{00}(x)|^{2}$ and $\sigma ^{+}$-pol with $|\tilde {\varphi }_{01}(x)|^{2}$ was numerically verified by using three independent numerical integration programs.

Next, we study the influence of anisotropic property of $\underline {\boldsymbol{\epsilon }}_2$ on the SEC of QD for two different radii $(R=10\,nm$ and $500\,nm)$. In Figs. 4(a) and (c), the FSs and RDRs of the two QDs (solid lines for $R=10\,nm$ and dash-dotted lines for $R=500\,nm$) are compared for the anisotropic (ani, blue) and isotropic (iso, red) $\underline {\boldsymbol{\epsilon }}_2$ where we set intentionally $\epsilon _z=\epsilon _t$ for the latter. A distance of $z_s=10\,nm$ and the x-pol with $|\tilde {\varphi }_{00}(x)|^{2}$ are commonly applied to all cases. A stunning result is that the anisotropic property of $\underline {\boldsymbol{\epsilon }}_2$ can only appear in the SEC of the smaller QD (cp. the red and blue solid lines in Figs. 4(a) and (c)). The FS and RDR of the larger QD are not influenced by the $\epsilon _z$ component (cp. the red dashed and blue dashed-dotted lines in Figs. 4(a) and (c)).

 

Fig. 4. The FSs (a) and RDRs (c) of two different QDs $(R=10\,nm$ and $500\,nm)$ at $z_s=10\,nm$ above the isotropic (iso, $\epsilon _z=\epsilon _t$) and the anisotropic (ani, $\epsilon _t\ne \epsilon _z$) $\underline {\boldsymbol{\epsilon }}_2$. The FSs (b) and RDRs (d) of the small QD $(R=10\,nm)$ at two different distances ($z_s=10\,nm$ and $500\,nm$) from the isotropic and the anisotropic $\underline {\boldsymbol{\epsilon }}_2$. The green solid line in Fig. 4(d) is multiplied by 0.02 for an easy comparison. The x-polarization direction is set for all cases.

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By carefully examining Eqs. (18) and (19), it is discovered that the anisotropic feature is only contained in the first integrand of Eq. (19) when the polarization direction (see Eq. (14) for $\hat {\boldsymbol{u}}_d=\hat {\boldsymbol{x}}$) and the distance from the interface ($e^{2ih_sz_s}\approx 1$) are considered. By defining a new function in the integrand

In Figs. 5(a) and (b), the real and imaginary parts of $B_x$ and the reflection coefficient $B_{N_2}$ as a function of $x$ are compared for two radii of QD on the isotropic (iso) and anisotropic (ani) $\underline {\boldsymbol{\epsilon }}_2$, respectively. The real value of $B_x$ of the larger QD ($R=500\,nm$) is mainly decided by $|\tilde {\varphi }_{00}(x)|^{2}$, but that of the smaller QD ($R=10\,nm$) by the square root terms because of $|\tilde {\varphi }_{00}(x)|^{2}\approx 1$ in the interval $0\le x \le n_1$, where $B_{N_2}$ has almost identical values regardless of the anisotropy of $\underline {\boldsymbol{\epsilon }}_2$ (blue and red lines in Fig. 5(a)). The imaginary part of $B_{N_2}$ at higher modes ($x\ge \,n_1$) shows substantial difference depending on the anisotropic properties. However, $B_x$ of the larger QD overlaps only a very narrow region of $B_{N_2}$ (shaded light blue region in Fig. 5(b)) where the difference is negligibly small. As a result, the anisotropy of $\underline {\boldsymbol{\epsilon }}_2$ can give rise to substantial variation only on the optical properties of of the smaller QD. On the contrary, its optical influence is strongly limited to the larger QD and the QD located far away from the interface since the spatial phase converges rapidly to zero ($e^{2ih_sz_s}\approx 0$), as demonstrated in Figs. 4(c) and (d). In Fig. 4(d), a strongly enhanced RDR of the QD in close proximity to $\underline {\boldsymbol{\epsilon }}_2$ ($z_s=10\,nm$) was numerically observed at a wavelength of $924\,nm$ where $\mathrm {Re}[\epsilon _t]\approx -2.5=-\epsilon _1$ is satisfied. Enhanced fluorescence/spontaneous emission of atomic light emitters inside or near epsilon-near-zero (ENZ) material has already been reported [56].

 

Fig. 5. The real (a) and imaginary (b) parts of the function $B_x$ (see the definition in the main text) for two different sizes of QDs $(R=10\,nm$ and $500\,nm)$ and the reflection coefficient $B_{N_2}$ for the isotropic (iso) and anisotropic (ani) $\underline {\boldsymbol{\epsilon }}_2$.

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4. Summary

In summary, we studied theoretically the frequency shift and radiative decay rate of a shallow quantum dot (QD) exciton located close to the surface of half-infinite Tetradymites($\textrm {Bi}_2\textrm {Se}_3$), a natural hyperbolic material in a visible-to-near infrared wavelength range. By including exciton’s eigenstates with non-zero orbital angular momentum and the Green function tensors for uniaxially anisotropic optical properties of materials in our nonlocal susceptibility and self-energy correction, we were able to investigate both optical properties of exciton, depending on the confinement potential (size of QD), quantum eigenstate, polarization direction, and distance from the surface of hyperbolic material. According to numerically obtained radiative decay rate corresponding to the imaginary parts of the self-energy correction in given environments, the strongly confined exciton in small QDs shows strong dependence on the quantum eigenstate, polarization direction, and anisotropy of the substrate at close distance. On the contrary, the weakly confined exciton has almost no influence from those condition, because negligibly small variations of the self-energy correction for the radiative decay rate occur in narrow propagating modes of radiation. Our results can be crucial for designing nanophotonic devices where quantum light sources are integrated with hyperbolic materials.