1. Introduction

Since graphene was discovered as the first two-dimensional (2D) material in 2004, the variety of 2D materials has been increasing rapidly in the past decade. Great research efforts have been devoted to few-layer graphene [1,2], transition metal dichalcogenides (TMDCs) such as MoS₂ and WS₂ [3,4], hexagonal boron nitride (hBN) [5,6], etc. However, the existing 2D materials are hindered from high-performance optoelectronic and nanophotonic applications due to this mobility/bandgap trade-off [7]. For example, graphene holds extremely high mobilities but has a vanishing bandgap. On the contrary, TMDCs have large direct bandgaps, although their mobilities are usually orders of magnitude lower than those of graphene. Recently, black phosphorus (BP) was reintroduced into the community from the perspective of 2D materials, showing a direct bandgap at the Г point [8], excellent carrier mobility [9], strong light-matter interactions in the mid-infrared (mid-IR) range [10], and unique anisotropic physical properties [11]. These features make it possible to bridge gaps in the 2D material family, complementing important applications in mid-IR optoelectronics and photonics [12,13].

One of the most exciting merits of 2D materials is that they support highly confined surface plasmon polaritons (SPPs), which enable extremely strong light-matter interactions with quantum emitters. For example, the Purcell factors of dipole emitters in the vicinity of graphene can reach 10⁵-10⁷ due to tightly confined, low-loss graphene SPPs [14,15]. Dramatic modification of relaxation rate is also observed in excited erbium ions placed in subwavelength proximity of graphene [16]. Nanostructured graphene can support SPPs with hyperbolic topologies at THz and far-IR frequencies, which gives rise to stronger light-matter interactions with the Purcell factors being up to 10⁷-10¹⁰ [17,18]. Furthermore, a novel theory points out that 2D plasmonics, with its unprecedented level of confinement, presents a unique opportunity to access radiative transitions in atomic-scale emitters that were considered forbidden [19]. Multiplasmon processes, spin-flip radiation, and very high-order multipolar transitions are shown to be competitive with the fastest atomic dipole transitions. Inspired by the intriguing properties of graphene, BP plasmonics promises strong light-matter interactions with quantum emitters as well. A recent study reported that spontaneous emission rate (SER) of emitter is greatly enhanced in BP thin films, comparable to that of graphene, while providing even higher upper limits [10]. However, the high SER comes at the cost of large SPP propagation losses, i. e., most of the emitted photons would be dissipated before being collected or utilized. For example, despite the fact that SER is higher in the hyperbolic regime, the propagation lengths of BP SPPs are hundreds of times smaller than the wavelength in free space, making it inappropriate for applications such as mid-IR photon sources. Comparing to the anisotropic elliptic regime, the SER is higher in the hyperbolic regime only when the emitter is several nanometers away from the BP (see Fig. 8 of Ref [10].). It drops dramatically with farther distance such as 10 nm. Since the non-radiative relaxation, e. g., resonance energy transfer from emitter to BP, boosts at nanoscale distances and is detrimental to the high quantum efficiency of photon sources, we restrict ourselves to separations larger than 10 nm in this work. Obviously, the anisotropic elliptic regime is more suitable than the hyperbolic regime for highly efficient photon sources in terms of SER, SPP propagation losses, and quantum efficiency.

In this paper, we systematically investigate strongly enhanced light-matter interactions in the quantum emitter-monolayer BP coupling system working at 10-30 THz. Since the bandgap of monolayer BP is ~2 eV [20,21], interband transitions in BP which are the primary origin of SPP losses are totally forbidden for the photon frequencies of concern [22]. We also explore the relationships between the SER and the BP doping level, the emitter-BP distance, the permittivity of substrate, and the orientation of transition dipole matrix element. In addition, it is verified that the emitter embedded in double layer BP exhibits even better radiation field confinement and thus higher SER. Therefore, with a judicious design of the emitter-BP coupling system, both high SER and low SPP losses can be achieved at the same time. Different from previous works in the literature studying the basic physics of BP plasmonics, we focus on the specific light-matter interactions in BP and how they can help to build single-photon sources with both record-high Purcell factor and near-unity β factor. Our findings pave the way to developing BP plasmonic platforms for new highly efficient, ultra-compact mid-IR photon sources which could find applications in single-plasmon devices and quantum plasmonics.

2. Device design and analysis

The quantum systems of interest are shown in Fig. 1. A quantum emitter such as a quantum dot or a rare-earth ion is suspended in air above a freestanding monolayer BP or between two layers of freestanding monolayer BP. The three-dimensional schematic diagrams of the emitter coupled to monolayer and double layer BP are illustrated in Figs. 1(a) and 1(c), respectively, while Figs. 1(b) and 1(d) show their cross sections in XOZ plane. The emitter is placed at a distance of d from each monolayer BP. Owing to its deep-subwavelength size, it can be modeled as an electric dipole which corresponds to the transition dipole matrix element of quantum emitter. The monolayer BP can be isolated from bulk BP by means of mechanical exfoliation and plasma thinning [23]. It has an orthorhombic lattice with two clearly distinguishable axes, as depicted in the inset of Fig. 1(a). Here we assume that the armchair and zigzag crystallographic directions of monolayer BP are oriented along x and y axis, respectively. The thickness of monolayer BP is about half of the lattice constant in the out-of-plane direction, i. e., 5.35 Å [24].

 

Fig. 1 Three-dimensional view of a quantum emitter coupled to (a) monolayer and (c) double layer BP. Cross sections in XOZ plane of the systems with (b) monolayer and (d) double layer BP. The distance between emitter and each monolayer BP is denoted by d. Lattice structure and orientations of two crystal axes of monolayer BP are illustrated in the inset.

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For the mid-IR frequencies considered here, the conductivity of monolayer BP is dominated by intraband transitions and can be approximated by the Drude model [21, 22],

 

Fig. 2 Optical conductivity of monolayer BP and in-plane wave vector of quasi-TM SPP supported by monolayer BP as a function of frequency and electron density. Real part of conductivity (a) σ_xx and (b) σ_yy and imaginary part of conductivity (c) σ_xx and (d) σ_yy. Real part of wave vector (e) q_x and (f) q_y and imaginary part of wave vector (g) q_x and (h) q_y. The parameters used in the calculations are m_x = 0.18m₀, m_y = 1.23m₀, η = 3 meV.

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The magnitude of light-matter interactions can be quantified by the SER of nearby emitter, which is calculated by [27]

3. Results and discussions

First, we consider the scenario in Figs. 1(a) and 1(b), with a z-oriented transition dipole moment of quantum emitter, i. e., p = pe_z. Figures 3(a) and 3(b) depict the E_z component in the monolayer BP at 10 and 15 THz, respectively. Here we assume that the emitter is located at a distance d = 10 nm above the monolayer BP, and the doping level of BP is n = 5 × 10¹³ cm⁻². As a result of anisotropic in-plane conductivity, SPP propagation shows a significant canalization along x axis. The plasmon at higher frequency has shorter guided wavelength and propagation length, in agreement with the frequency dependence of in-plane wave vector q shown in Figs. 2(e)-2(f). The magnitude of electric field in XOZ plane is plotted in Fig. 3(c). Obviously, instead of going freely into space, the power generated by emitter is coupled to the monolayer BP and propagates along it as SPPs. Following the methods described in Ref [29], we calculate the percentage of power entering free space, coupling to SPPs, and dissipated in BP, respectively. We find that 6.82% of the total power is radiated into free space while 17.61% is dissipated after propagating along the BP for 1 μm. Only 75.57% is finally emitted in the form of SPPs. That is the reason why controlling SPP losses is of great importance in realizing high-yield photon sources based on BP. The SERs of quantum emitter close to monolayer BP and graphene are compared in Fig. 3(d). To make a fair comparison, the density and scattering rate of electrons in graphene are also n = 5 × 10¹³ and η = 3 meV, respectively, which correspond to the chemical potential${\mu }_c=\hslash v_f\sqrt{\pi n}=0.825$eV and the electron mobility$\mu =\hslash e{v}_f^2/\eta {\mu }_c=2660$cm²/(V·s), where$v_f\approx {10}^6$m/s is the Fermi velocity in graphene [2]. The SER of emitter with monolayer BP is about 3 times larger than that with graphene, which results from the higher photonic density of high-k states near BP. According to Eq. (2), monolayer BP supports evanescent waves with q ranging from q_x to q_y, while graphene only supports those with fixed$q=2i{\epsilon }_0\omega /{\sigma }_g$, where σ_g is the conductivity of graphene. Therefore, the spontaneous emission enhancement is stronger with monolayer BP due to additional coupling of the emitted evanescent waves to a wide variety of high-k SPP states. The mechanism can also explain the frequency dependence of SER. The coupling efficiency is proportional to the spatial overlap of the mode functions of the evanescent waves emitted by dipole and the SPPs supported by BP, hence$\propto e^{-Qd}$, where Q ≈q is the out-of-plane decay coefficient in the z direction [16]. As pointed out in Figs. 2(e) and 2(f), the real part of q increases with frequency, which leads to the decrease in coupling efficiency, and consequently the drop in SER. However, for longer distances and higher frequencies graphene outperforms monolayer BP in the SER enhancement because graphene support less confined evanescent waves [18]. In the inset of Fig. 3(d), we show the SERs of emitter in the vicinity of monolayer BP and graphene at 25-30 THz with d = 100 nm. The SER with graphene is higher in this region and decays slower with frequency than that with BP, which is the feature of less confined SPPs sustained by graphene.

 

Fig. 3 E_z component in monolayer BP at (a) 10 and (b) 15 THz. (c) Magnitude of electric field in XOZ plane at 10 THz. (d) SERs of quantum emitter near monolayer BP (red solid line) and graphene (blue solid line) versus frequency. The emitter-BP distance is d = 10 nm (100 nm in the inset), the electron density is n = 5 × 10¹³ cm⁻², and the transition dipole moment is oriented in the z direction. The graphene has the same electron density and scattering rate as BP.

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Doping provides a degree of freedom to tune the optical properties of monolayer BP, adjust the dispersion relationship of BP SPPs, and in turn modify the SER of nearby emitter. Figure 4(a) illustrates the dependence of SER and SPP propagation length on the electron density in BP, with distance d = 10 nm and frequency f = 10 THz. The propagation length is calculated by$L_j=1/2\mathrm{Im}(q_j)$ [30]. Obviously, the SER is larger at a low doping level, which is attributed to the richness of high-k states. The high-k states include BP SPPs with q ranging from q_x to q_y. This range of q widens when decreasing the electron density (cf. Figures 2(e) and 2(f)), hence giving rise to more decay channels and higher SER of emitter in the vicinity of BP. However, BP SPPs suffer significant losses at a low doping level (cf. Figures 2(g) and 2(h)), which means the emitted photons have a high probability to be lost in the near field rather than be extracted or manipulated in the far field. To achieve a balanced trade-off between SER and propagation length (see Fig. 4(a)), a moderate doping level of BP is desirable for high-yield photon sources. Hereinafter, we assume that the electron density in monolayer BP is maintained at n = 5 × 10¹³ via appropriate doping.

 

Fig. 4 (a) Electron density dependence of the SER of nearby emitter (red solid line), the propagation length along x, L_x (green dashed line), and the propagation length along y, L_y (blue dashed line) at f = 10 THz. (b) SER of emitter versus its distance to the monolayer BP. The frequency is chosen to be f = 10 (red solid line) and 50 THz (blue solid line). (c) Influence of substrates with ε_s = 2.25 (red solid line), 4 (blue solid line), and 6.25 (green solid line) on the SER of nearby emitter at f = 10 THz. Other parameters are the same as in Fig. 3.

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Then we investigate the SER of a z-oriented dipole versus its distance to the monolayer BP at a frequency of f = 10 THz. The distance dependence is shown in Fig. 4(b) where the SER decreases monotonically with growing gap. This is a clear signature of the aforementioned mechanism that the photons emitted by dipole are evanescently coupled to the monolayer BP as the coupling efficiency drops exponentially with increasing emitter-BP distance. To compare with Fig. 8 of Ref [10], we also plot the distance dependence of SER at f = 50 THz in Fig. 4(b). The SERs at d = 10 and 20 nm are 5.3 and 4.6, respectively, in reasonable agreement with the previously reported values [10]. Although the SER is supposed to be larger at even shorter distances, non-radiative relaxation such as resonance energy transfer from emitter to BP may prevail over plasmon-emitter coupling, considering that it scales as d⁻⁴ [16]. Thus, we do not study the system with the distance d smaller than 10 nm in this work.

In practice, monolayer BP always sits on a substrate, so it is necessary to consider the effect of substrate on the SER. In Fig. 4(c), we show the frequency dependence of SER with the permittivity of substrate ε_s = 2.25, 4, and 6.25. The distance is assumed to be d = 10 nm and the frequency is fixed at f = 10 THz. The SER increases with ε_s because higher permittivity leads to more confined SPPs and therefore stronger light-matter interactions. We also observe that the growth of SER at higher frequencies is more obvious than at lower frequencies, which is attributed to the fact that the in-plane wave vectors of SPPs at higher frequencies are more sensitive to the permittivity change of surrounding environment.

We proceed to study how the SER changes when the transition dipole moment is oriented in the x or y direction, i. e., p = pe_x or p = pe_y. Assuming d = 10 nm and f = 10 THz, Figs. 5(a) and 5(b) present the E_z component in the monolayer BP with an x- and y-polarized dipole, respectively. Apparently, for x (y) polarization the emitter is inclined to decay into BP SPPs propagating along x (y) axis. The SERs for x and y polarizations versus frequency are plotted in Fig. 5(c). As expected, they are smaller than that of z-polarized dipole which can radiate BP SPPs in all directions and, therefore, has a higher partial local density of states [27]. We also find that in spite of the canalization observed in the x direction, it is actually the SPPs propagating along y that make greater contributions to the spontaneous emission enhancement, given that the SER of y-polarized dipole is larger than that of x-polarized dipole. As a result, the importance of suppressing propagation losses is more pronounced because BP SPPs propagating along y are more vulnerable to dissipation (cf. Figure 4(a)).

 

Fig. 5 E_z component in monolayer BP with (a) x- and (b) y-polarized dipole at 10 THz. (c) SERs of x- (blue solid line) and y-polarized (red solid line) dipole near monolayer BP versus frequency. Other parameters are the same as in Fig. 3.

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Next, we consider the scenario in Figs. 1(c) and 1(d), i. e., the emitter is coupled to two layers of monolayer BP. Figures 6(a) and 6(b) show the E_z component in individual BP layers and the magnitude of electric field in XOZ plane at 10 THz, respectively. The distance to each layer is d = 10 nm, the electron density in BP is n = 5 × 10¹³ cm⁻², and the transition dipole moment is oriented in the z direction. Similar to the phenomenon observed in monolayer BP, the SPPs of double layer BP exhibit canalization along x axis, arising from the anisotropy of optical conductivity inherited from monolayer BP. In comparison with Fig. 3(c), the radiation field is now localized between two BP layers and thus more confined, resulting in stronger light-matter interactions between emitter and BP SPPs. The amount of power that leaks into free space is 1.32% of the total power radiated by emitter. The power lost in BP after propagation for 1 μm is 13.62% while the remaining 85.06% exists as surface plasmons. As most of the modal energy is concentrated in the air between two BP layers, the SPP propagation losses in double layer configuration are smaller than in single layer, making double layer BP more suitable for applications such as photon sources. The inset of Fig. 6(c) illustrates the E_z field profile of the symmetric (E_s) and anti-symmetric (E_a) SPP modes supported by the double layer structure [31]. Since the transition dipole moment is z-oriented, only the symmetric mode can be excited in the double layer configuration. It is demonstrated by the mode profile monitored between two BP layers, as shown in Fig. 6(c). So the energy of quantum emitter is estimated to be radiated as symmetric mode SPPs at a rate of about 6.75 × 10⁶γ₀. It is then proved in Fig. 6(d) where the SERs of emitter between double layer BP and graphene are plotted as a function of frequency. The parameters of graphene are the same as in Fig. 3(d). As a matter of fact, the SER of emitter between two BP layers is more than 4 times larger than that near single BP layer, whereas the ratio between two SERs derived from double layer BP and graphene is weakly dependent on frequency and hovers around 3. We also depict the SERs of quantum emitter between double layer BP and graphene at 25-30 THz with d = 175 nm in the inset of Fig. 6(d). Since the radiation field is more confined in the double layer structure, the SER with graphene exceeds that with BP at a longer distance and a higher frequency, comparing to the situation in the single layer configuration (see the inset of Fig. 3(d)). The dependence of SER on the BP doping level, the distance between emitter and BP, the permittivity of substrate, and the orientation of transition dipole moment of emitter is quite similar in the cases of monolayer and double layer BP, so we do not repeat those discussions in the context of two BP layers in this paper.

 

Fig. 6 (a) E_z component in each BP layer and (b) magnitude of electric field in XOZ plane at 10 THz. (c) E_z field profile of the SPP mode excited in the double layer BP. Symmetric (E_s) and anti-symmetric (E_a) SPP modes supported by double layers are illustrated in the inset. (d) SERs of quantum emitter between two layers of BP (red solid line) and graphene (blue solid line) versus frequency. The emitter-BP distance is d = 10 nm (175 nm in the inset). Other parameters are the same as in Fig. 3.

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4. Conclusion

In summary, we have studied the enhanced spontaneous emission of quantum emitter in monolayer and double layer BP. Numerical results prove that the SER is multiplied by a factor of about 10⁶, even higher than that of graphene, which is credited to the large density of high-k states supported by BP. We have also unveiled the relationships between the SER and the photon frequency, the BP doping level, the distance between emitter and BP, the permittivity of substrate, the transition dipole orientation, etc. By judiciously designing the system parameters, we have achieved an extremely efficient SPP source with relatively low propagation losses in BP. The proposed device can directly serve as a single-plasmon source or be coupled to a dielectric waveguide via a grating [32], showing promising potential for novel single-plasmon devices and future quantum photonic applications.