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Abstract
Although graphene has been the primary material of interest recently for spontaneous emission engineering through the Purcell effect, it features isotropic and thickness-independent optical properties. In contrast, the optical properties of black Phosphorus (BP) are in-plane anisotropic; which supports plasmonic modes and are thickness-dependent, offering an additional degree of freedom for control. Here we investigate how the anisotropy and thickness of BP affect spontaneous emission from a Hydrogenic emitter. We find that the spontaneous emission enhancement rate i.e. Purcell factor (PF) depends on emitter orientation, and PF at a particular frequency and distance can be controlled by BP thickness. At lower frequencies, PF increases with increasing thickness due to infrared (IR) plasmons, which then enhances visible and UV far-field spectra, even at energies greater than 10 eV. By leveraging the thickness and distance-dependent PF, deep UV emission can be switched between 103 nm or 122 nm wavelength from a Hydrogenic emitter. Additionally, we find that doping can significantly tune the PF near BP and this alteration depends on the thickness of the BP. Our work shows that BP is a promising platform for studying strong plasmon-induced light-matter interactions tunable by varying doping levels, emitter orientation, and thickness.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The pioneering work of Purcell demonstrated that any quantum emitter’s spontaneous emission decay rate can be greatly influenced by the density of optical modes of its environment [1]. Before that, spontaneous emission was thought to be an emitter’s intrinsic property and independent of the environment. Purcell’s discovery has paved the way to engineer spontaneous emission. Since then tunable spontaneous emission has been applied in various fields such as bio-sensing, fluorescence imaging, molecule detection, and LEDs [2–9]. Consequently, various types of atoms, molecules, fluorescent materials [10,11], nanocavities [10], metal-photonic crystal structure [12–14], antennas [15,16], tips [17,18], and metamaterials [19–21] have been proposed to engineer the spontaneous emission rate of an emitter. Among them, two-dimensional (2D) materials are one of the most promising platforms with the exciting merits of supporting highly confined surface plasmon polaritons (SPPs), which enable robust light-matter interactions with quantum emitters.
Particularly, atomically thin 2D materials with striking electronic, optical, mechanical, and thermal properties, have received great research interest recently [22–25]. For instance, the incorporation of two-dimensional materials in the total internal reflection fluorescence microscopy (TIRFM) technique can decrease the detection volume of a target sample, and hence improve the resolution of the obtained image [26,27]. Graphene-based structures have been extensively used to control spontaneous emissions in recent years [28–31]. Recent studies also have shown to enhance the UV emission of a quantum emitter utilizing graphene plasmonics [32]. Another potential 2D material candidate in this field is Black Phosphorus (BP) which can be obtained in monolayer and multi-layer form with the help of mechanical exfoliation and plasma thinning [33–35]. Monolayer BP exhibits unique characteristics including direct bandgap [36], excellent carrier mobility [37,38], and strong light-matter interactions in the mid-infrared frequency range [34]. Compared to graphene, BP has intrinsically high in-plane anisotropic physical properties and so it supports highly localized SPPs with anisotropic dispersions at mid-infrared (mid-IR) and terahertz frequencies [39,40]. On the contrary, at high frequencies, the strong coupling of graphene with optical phonon prevents it from supporting low-loss plasmons [41]. Furthermore, due to the layer-dependent band gap, BP exhibits different electrical and optical properties depending on its thickness. Thus, apart from the strain and electrical field tunable properties, layer thickness of BP also provides additional degree of freedom to engineer light-matter interaction near BP. Hence, researchers have investigated theoretically and experimentally various structures and methods to enhance the light-BP interaction in the terahertz and IR frequency range [42–49].
In the present study, we extensively investigate the interaction between a Hydrogenic emitter and multi-layer BP. Previous studies on BP-enhanced spontaneous emission dealt mainly with monolayer and bilayer Black Phosphorus without taking into account the layer-sensitive bandgap and optical properties of BP [43,50]. Our study incorporates the complete anisotropic effect of layer thickness on the BP-emitter interaction. According to our study, apart from the trivial variables such as distance and wavelength, the enhancement of spontaneous emission of the emitter near BP can be tuned by varying thickness of BP and emitter orientation. The thickness of BP has a significant impact on the Purcell factor (PF) whereas the effect of emitter orientation is less severe. Although the actual value of PF varies with emitter orientation, the order of magnitude remains the same. PF attains highest and lowest values when the emitter is aligned in the vertical and zigzag direction respectively. Next, we find that PF at a particular frequency and distance can be varied by changing layer thickness. Increasing thickness increases the PF at lower frequencies and resonant peaks are observed at different frequencies due to different band-gaps of multi-layer BP. Furthermore, we demonstrate that by utilizing this layer dependent PF control, the distance-dependent output spectrum of the Hydrogenic emitter can be modulated. Our analysis finds that beyond a certain thickness, the distance-dependent output spectrum of the emitter exhibits a shift in wavelength from UV (122 nm) to Deep UV (103 nm) within a certain distance window near BP. In the absence of BP, the output spectrum of Hydrogenic emitter is mostly dominated by 122 nm emission. But in the presence of BP with proper thickness, more than 80% of the total power comes from the 103 nm Deep UV emission. Finally, we show that doping also plays an important role in tailoring the spontaneous emission rate enhancement of an emitter near BP. However, this tunability of light-matter interaction depends largely on the thickness of BP. In case of intrinsic monolayer BP, the output spectrum of the Hydrogenic emitter is from 122 nm UV emission. However, with appropriate Fermi-level shifting via doping, the output spectrum can be switched to 103 nm Deep UV emission at distances below 10 nm. These findings present BP as a suitable platform to tune plasmonics mediated light matter interaction providing several additional degrees of freedom compared to conventional 2D materials.
2. Methodology
In this work, the spectrum of Hydrogenic emitter is considered near multi-layer black Phosphorus with tunable optical properties. The Hydrogenic emitter is taken for the simplicity of calculation. Figure 1(A) shows the schematic of the system with a Hydrogenic emitter near a multilayer BP and the variables used in our numerical calculation. Here, $z_0$ represents the emitter to BP distance, $\varphi$ is the azimuth angle, $\theta$ is the elevation angle, t is the thickness and $n_{\text {L}}$ represents the number of layers. BP possesses an unusual puckered atomic structure with two distinct in-plane directions: ArmChair (AC) and ZigZag (ZZ) as illustrated in Fig. 1(B). In this study, x and y-axes are considered as AC and ZZ directions respectively. The thickness-dependent band-gap $E_{\text {g}}$ of BP is shown in Fig. 1(C). $E_{\text {g}}$ decreases monotonically from approximately 2eV for monolayer with thickness to a narrow band gap value of about 0.3 eV for bulk-black Phosphorus [51–54]. This property is owing to the quantum confinement of the charge carriers in the out-of-plane direction and is stronger in case of BP compared to other 2D semiconducting materials [54]. From Fig. 1(C), it is evident that $E_{\text {g}}$ drastically decreases from monolayer to 10 layers and after that, the change in band-gap with respect to layer number is very small.
Fig. 1. A, Schematic of a quantum emitter (Hydrogenic) placed near BP with coordinates: elevation angle ($\theta$), azimuth angle ($\varphi$), distance of BP sheet from emitter ($z_0$) and sheet thickness (t) B, 2D planar view of BP with Arm chair (AC) and ZigZag (ZZ) direction C, Change of Band Gap ($E_\text {g}$) with the layer number ($n_\text {L}$) i.e. thickness (t) D, Hydrogenic emitter cascade diagram for electron decay pathways from 4d excited state and corresponding transition wavelengths and angular frequencies.
Hydrogen atom has been studied extensively and its spectrum is well-understood and experimentally verified due to having the simplest possible atomic structure. Because of this simplicity, Hydrogenic emitter also serves as an excellent model to explore spontaneous emission in nanophotonics and optics and has been used extensively in literature [32,55,56]. The electronic states of Hydrogen atom are described by a set of quantum numbers labeled by $| {\textit {n, l,m,s}}\rangle$ where n, l, m, and s are the principal, orbital, magnetic and, spin quantum numbers respectively. As the transition energy is mostly dominated by n, the spin transition is not considered in our calculations. The considered Hydrogenic atom is pumped from its ground state $| {g}\rangle = | {1s}\rangle$ to the excited state $| {e}\rangle = | {4d}\rangle$ with a rate of $G=10^4 s^{-1}$ [32]. The allowed transitions in accordance with the selection rule and their corresponding wavelengths are given in Fig. 1(D). Also the transition of 2s to ground state is not considered as this effect is not dominant at first order [32,57] and hence it is shown in black line in the Fig. 1(D). After pumping, the excited atom being unstable tends to return to its ground or lowest energy state. In this process, the electron loses some or all of the excess energy by emitting photons. The process dynamics are governed by the rate equation:
Here $n_i$ represents the occupation number of $i^{th}$ electronic state. The first term in the right hand side represents downward transition from higher $j^{th}$ state to lower $i^{th}$ state, the second term implies downward transition from higher $i^{th}$ state to lower $k^{th}$ state and the third term represents the pumping or generation rate. Here $\Gamma _{ij}$ is the rate of transition between states i and j.
Equation 1 can be compactly written using matrix notation as follows:
where N contains the occupation numbers of the electronic states involved in the overall transitions and A is the matrix containing the first two terms for each state in Eq. (1) and G is the column matrix with entries generation or pumping rate. Matrix G contains non-zero elements for those states which are involved in the pumping process and upward pumping rate is considered to be positive.At steady-state, $\frac {d\mathbf {N}}{dt} = 0$ and so Eq. (2) implies:
where $\textit {N}_\text {s}$ is the steady-state population vector of the emitter levels. The total output power for a particular frequency can be obtained by summing over the transition rates of the same emission frequency to incorporate possible degeneracies. The output power P($\omega$) is where $\omega _{ij} = \omega _j - \omega _i$ is the transition frequency for $| {j}\rangle$ and $| {i}\rangle$ states. The broadening effects of the output spectrum is not considered in this work for simplicity.Near black Phosphorus, the spontaneous emission rate between states $| {i}\rangle$ and $| {j}\rangle$ will be modified due to the Purcell effect [58] and is given as $\Gamma _{ij}=\Gamma _{ij}^{0} F_p (\omega )$ where $\Gamma _{ij}^{0}$ is the rate of transition in the vacuum and $F_p (\omega )$ is the Purcell factor. The considered transition rates in vacuum is taken from [59]. The Purcell factor for an arbitrary polarized mode is given by [60]:
As BP is an anisotropic material, $\sigma _{xx} \neq \sigma _{yy}$. The optical conductivity tensor of BP is calculated using Kubo formula [63]:
3. Results and discussions
3.1 Thickness-dependent optical conductivity
Real and imaginary parts of optical conductivity ($\sigma$) along AC and ZZ direction normalized with respect to $\sigma _0=\frac {e^2}{\hbar }$ (optical conductivity of graphene) [64,65] for different numbers of layers are shown in Fig. 2(A)–(D). The thickness-dependent band-gap results in the shift of the absorption edge and so with an increase in layer thickness i.e. layer number, the transition frequency of the real part of $\sigma _{xx}$ shifts towards lower frequency (lower energy) and the maximum value of Re($\sigma _{xx}$) also increases. A similar trend is observed in the case of real part of $\sigma _{yy}$.
Fig. 2. Thickness-dependent normalized Optical conductivity ($\sigma$) for frequencies $\omega \in [10^{13}, 10^{17}]$ rad/s. Normalized A, imaginary part of $\sigma _{xx}$ B, imaginary part of $\sigma _{yy}$. C, imaginary part of $\sigma _{xx}$. D, imaginary part of $\sigma _{yy}$.
The anisotropy of BP is evident from the optical conductivity along AC (x-axis) and ZZ (y-axis) direction shown in Fig. 2. $\sigma _{yy}$ is always smaller than $\sigma _{xx}$ for all layer numbers. Real part of $\sigma _{xx}$ shows oscillatory behavior with frequency due to the different transition frequencies of the electronic subbands [63] whereas the real part of $\sigma _{yy}$ doesn’t show such behavior. This and the lower value of $\sigma _{yy}$ can be attributed to the absence of inter-subband transitions along the zigzag direction due to the presence of mirror symmetry in the x-z plane [66,67]. Also, the imaginary part of both $\sigma _{xx}$ and $\sigma _{yy}$ are very small at low frequency due to lower dielectric losses. At higher frequencies, dielectric loss increases and so they increase. They also show a dip around the band-gap frequency but the value is smaller than the real parts.
Three distinct regions can be identified from the optical conductivity vs frequency curves of BP. At lower frequencies, both Im($\sigma _{xx}$) and Im($\sigma _{yy}$) are negative, at high frequencies both of these are positive. In between these two regions, there exists a frequency band where Im($\sigma _{xx}$) is negative but Im($\sigma _{yy}$) is positive. The EM responses of BP in these three regions are termed as anisotropic elliptic quasi transverse magnetic, anisotropic elliptic quasi transverse electric and intrinsic hyperbolic topology respectively. Due to the presence of the last frequency band in optical response of BP, it can support SPPs with hyperbolic topology at mid-IR [34]. Also, the frequency band where the hyperbolic topology is found, shifts towards lower frequencies with an increase in the layer thickness and remains unchanged after a certain thickness.
3.2 Effect of distance, orientation and thickness of BP on Purcell factor
The calculated Purcell factor (PF) based on the formalism of Dyadic Green Function shown in Eq. (5) and (6) for different parameters are presented in Fig. 3. In Fig. 3(A)–(C), Purcell factors in the frequency (rad/s) range $\omega \in [10^{13}, 10^{17}]$ for distance $z_0\in [0.1,100]$ nm are presented for 3 different directions of the emitter: x-axis (arm chair), y-axis (zigzag) and z-axis (vertical) respectively near monolayer BP. From these figures, it is evident that the PF decreases with both increasing distance and increasing frequencies showing a resonant peak near band-gap frequency, $\omega _g = \frac {E_g}{\hbar }$. And this trend is the same for all three emitter orientations. Although the order of the PF values are the same for all three directions, the actual values are different which is due to the anisotropy of BP. This can be clearly understood from Fig. 3(D) which shows that PF is the highest for vertically and lowest for zigzag oriented emitter. Also after the band-gap frequency $\omega _g$, the PF starts to decrease even at a smaller distance. This is due to the increased absorption of photons and generation of electron-hole leading to the decay of plasmonic modes. Furthermore, after a certain distance ($\approx 100$ nm) even at lower frequencies than band-gap, the modes cannot exist due to the far distance and the PF tends to become unity for all emitter orientations.
Fig. 3. Purcell factor near monolayer black Phosphorus for frequency $\omega \in [10^{13}, 10^{17}]$ rad/s for different distances for A, Arm chair oriented emitter, B, Zigzag oriented emitter, C, Vertically oriented emitter, D, for $z_0 = 10$ nm for Arm chair, Zig zag and Vertically oriented emitter. Purcell factor for the same frequency range near monolayer BP for emitter at a distance $z_0 = 10$ nm for E, different Elevation angles ($\theta$) F, for $\theta = 0^{\circ}, 30^{\circ}, 60^{\circ} \text { and } 90^{\circ}$ G, different Azimuth angles ($\varphi$) and H, for $\varphi = 0^{\circ}, 30^{\circ}, 60^{\circ} \text { and } 90^{\circ}$. Purcell factor for the same frequency range for emitter at a distance $z_0 =$ 10 nm for BP with layer number $n_L = 1 \text { to } 20000$ for I, Arm chair J, Zigzag and K, Vertically oriented emitter L for $z_0 = 10$ nm for Arm chair, Zig zag and Vertically oriented emitter for $n_L= 38$.
With the change of the orientation of the emitter in x-y and x-z planes, the variations of PFs are observed with the emitter at a distance $z_0$ = 10 nm from monolayer BP (shown in Fig. 3(E) and (G)). The variation of PF with respect to azimuth angle and elevation angle is of the same order but has different magnitudes. From Fig. 3(E), there is a vertical shift of PF values at the lower frequencies with a change in azimuth angle $\varphi$ whereas in Fig. 3(G), the PF values are almost the same with a change in elevation angle $\theta$. This point can also be verified from Fig. 3(F), (H) which show that the variation of PF with a change in azimuth angle is greater than that in elevation angle.
In Fig. 3(I), (L) the layer-dependent PFs for different frequencies for the three directions at a distance $z_0$ = 10 nm are shown. We compute PF for layer numbers from 1 to up to Bulk level (20,000 Layers). For all directions, the PF tends to increase in the low frequency regime and also there are peaks around the band-gap frequency which tends to shift towards low frequency with increasing layer numbers. This reflects the reduction in band-gap with increasing thickness of BP. Also at low frequencies, the PF value increases with an increasing in layer numbers upto $n_L = 10$. After that, the PF becomes almost the same for all layer numbers. Figure 3(D) and (L) show that PF for $n_L = 38$ is more than one order higher than that of $n_L = 1$ for all emitter orientations in lower frequency range. However, PF around the resonant peaks is less for $n_L = 38$ than that of $n_L = 1$. So the number of layers of BP can be a tuning parameter to tailor the spontaneous emission of an emitter. And layer dependent PF also varies with emitter orientation as can be seen from Fig. 3(D) and (L).
3.3 Effect of BP thickness on the output spectrum of hydrogenic emitter
Figure 4(A)–(F) show the calculated normalized spectral output of the emitter at different distance ($z_0$) for undoped monolayer, 7L, 8L, 9L, 38L, and bulk BP. The corresponding photon emission rates (PERs) are given in Fig. 5. Figure 4(A) shows that near monolayer BP, the output power is dominated by UV emission at 122 nm irrespective of the distance. The photon emission rates for monolayer for different transition wavelengths shown in Fig. 5(A) indicate that PERs for 122 nm (2p$\rightarrow$1s) and 486 nm (4d$\rightarrow$2p) dominate in all distances but due to the higher frequency, UV emission at 122 nm is dominant in the output spectrum. With an increase in layer number ($n_L = 7$ and $n_L = 8$), the photon emission rate due to 2p$\rightarrow$1s and 4d$\rightarrow$2p transitions decrease slightly whereas PER due to 103 nm (3p$\rightarrow$1s) transition increases from distance $z_0 \approx 10$ nm to $z_0 \approx 30$ nm. As a result, in the output power spectrum within this distance window, 103 nm contributes around 40% of the total power. After $z_0 \approx 30$ nm, the PER from 2p$\rightarrow$1s and 4d$\rightarrow$2p transition start to dominate and so the output spectrum is dominated by 122 nm emission. However, from $n_L = 9$ and beyond, the spectrum changes significantly from the previous cases. At $z_0 < 10$ nm, the PER of 2p$\rightarrow$1s transition dominates and so 122 nm emission dominates the output spectrum like previous cases. However, the PER from 4d$\rightarrow$2p transition is very low initially. With increasing $z \approx$ from 10 nm to $\approx$ 30 nm, PER from both 3p$\rightarrow$1s and 4d$\rightarrow$2p transitions start to augment at the cost of the depletion of photon emission from 2p$\rightarrow$1s transition. Consequently, deep UV emission i.e. 103 nm emission dominates the output spectrum. For $n_L \geq 9$, more than 80% of the total output is directed into the 103 nm channel from distance $z_0 \approx 10$ nm to $z_0 \approx 30$ nm. The light-matter interaction near multi-layer BP thus provides a way of indirect coupling of IR and UV transition rates to enhance UV emission from an emitter. Monolayer to 8L do not exhibit this sort of behavior. This may be owing to the higher band gap for these BPs. On the other hand, from 9L to bulk the band gap doesn’t change that much and so all of them show the similar type of behavior. And this output spectrum is independent of the orientation of the emitter. However, after $z_0$ = $50$ nm, the output spectrum and PERs remain the same for BPs irrespective $n_L$. So the far-field emission distant from BP is always dominated by 122 nm emission. The BP-emitter interaction thus provides a way to obtain a distance and layer thickness-dependent tunable emission spectrum from the emitter.
Fig. 4. Spectral output power of hydrogenic emitter as a function of distance near A, $n_L = 1$ (monolayer) B, $n_L = 7$, C, $n_L= 8$, D, $n_L = 9$, E, $n_L = 38$, and F, $n_L = 20000$ (bulk) black Phosphorus. Below nine layers ($n_L<9$), the output spectra are dominated by UV emission at $122$ nm in both near and far-field. However, from $n_L = 9$ to bulk, there is a window from around $10$ nm to $30$ nm where Deep UV emission at $103$ nm dominates contributing more than $80\%$ of the total power.
Fig. 5. Photon Emission Rate corresponding to the spectrum in Fig. 4 as a function of distance at different transition wavelengths ($\lambda$) with pumping from 1s to 4d state of Hydrogen, $G=10^4$ $s^{-1}$ is presented for Layer A, $n_L = 1$ (monolayer) black phosphorus, B, $n_L = 7$ , C, $n_L = 8$ , D, $n_L = 9$ , E $n_L= 38$ and F $n_L = 20000$ (bulk) black phosphorus.
3.4 Effect of doping
3.4.1 Effect of doping on optical conductivity of BP and Purcell factor
Doping (both chemical and electrical) provides a way to tune the electronic and optical properties of BP. Here, we consider only the uniformly n-doped case. Figure 6(A), (D) show the normalized optical conductivity $\sigma _{xx}$ and $\sigma _{yy}$ for a doped BP film of monolayer and 38 layers ($t \approx 20$ nm) for chemical potential $\mu$ (defined earlier) = 0.1, 0.2 and 0.3 eV. It can be seen from 6(A), (D) that with an increase in chemical potential $\mu$, the peaks of the optical conductivities shift toward higher frequency which can be attributed to Pauli blocking [63]. This change is more prominent for layer 38 than that of monolayer for both $\sigma _{xx}$ and $\sigma _{yy}$. Due to this blue shift of optical conductivity, the resonant peaks of PF also shift. Figure 6(E)–(F) show the variation of Purcell factor with respect to $\mu$ for different frequencies at a fixed distance $z_0 = 10$ nm. Here we consider chemical potential level less than the corresponding band-gap of the BP and so due to higher band-gap of monolayer $\mu$ has been varied from 0 to 1.3 eV whereas for $n_L = 38$, the variation is up to 0.3 eV only. Figure 6(E)–(F) indicate that with increasing doping, the magnitudes of PF decrease for all frequency values.
Fig. 6. : Effect of doping i.e. tuning the Fermi level ($E_\text {F}$) on BP sheet conductivity and spontaneous emission is presented. Normalized A, real part of $\sigma _{xx}$ B, real part of $\sigma _{yy}$. C, imaginary part of $\sigma _{xx}$ D imaginary part of $\sigma _{yy}$ for Layer (i) $n_L = 1$ and (ii) $n_L = 38$; Purcell factor (PF) in frequency range $\omega \in [10^{13}, 10^{17}]$ rad/s E with chemical potential variation $\mu \in [0.0, 1.3]$ eV for monolayer F with chemical potential variation , $\mu \in [0.0, 0.3]$ eV for Layer Number $n_L=38$.
3.4.2 Effect of doping on the output spectrum of Hydrogenic emitter for multi-layer BP
Finally, we observe the effect of doping on the output power spectrum of Hydrogenic emitter near multi-layer BP. We’ve mentioned earlier that increasing doping level causes a decrease in PF for all frequencies and this contributes to the modification of the output spectrum of Hydrogenic emitter near multi-layer BP. To elucidate this point, we calculate the distance-dependant output spectrum and PERs of Hydrogenic emitter near monolayer, 9L, 38L, and bulk BPs with suitable doping levels. From Fig. 7(A) and 8(A) it is observed that below $z_0 \approx 10$ nm, photon emission rate and hence output spectrum is dominated by deep UV emission (103 nm) which is contrastive to undoped monolayer as the latter case didn’t show any change of output spectrum with distance. Here more than 80% of the output power is contained in 103 nm line. On the other hand, doping of $n_L = 9$ and $n_L = 38$ layered BP causes the suppression of the 103 nm UV emission in the intermediate distance window from $z_0 \approx 10$ nm to $z_0 \approx 30$ nm which can be seen from Fig. 7(B) and 7(C) respectively. PER of these two layered BPs are shown in Fig. 8(B) and (C) respectively. These figures show that the intermediate peaks of PER for 3p$\rightarrow$1s vanish and the photons are mostly emitted for the transitions 2p$\rightarrow$1s and 4d$\rightarrow$2p respectively. Consequently, most of the output power is routed into 122 nm UV channel suppressing the 103 nm emission of undoped $n_L = 9$ and $n_L = 38$ layered BP. In case of bulk, with doping ($\mu = 0.3$eV) the PER reaches a peak within $z_0 \approx 10$ nm and $z_0 \approx 30$ nm distance window at the cost of both 122 nm and 486 nm transitions but the value is lower than the PER of undoped bulk. And hence the dominant 103 nm emission within this distance window is also suppressed. Our analysis shows that doping can significantly modify the spectrum of the Hydrogenic emitter by suppressing or stimulating certain wavelengths in the output power.
Fig. 7. Effect of doping on Spectral Output Power of Hydrogenic emitter as a function of distance near A, doped $n_L = 1$ (monolayer) with $\mu = 1.0$ eV black Phosphorus, B, doped $n_L = 9$ with $\mu = 0.3$ eV , C, doped $n_L = 38$, with $\mu = 0.3$ eV and D, doped $n_L = 20000$ (bulk) black Phosphorus with with $\mu = 0.3$ eV. For monolayer, the output spectra are dominated by Deep UV emission at $103$ nm at near field whereas $122$ nm dominates the far-field. But, from $n_L = 9$ to bulk, there is no such distinction. Everywhere $122$ nm transition dominates for all layers.
Fig. 8. Corresponding photon emission rate of Fig. 7 as a function of distance at different transition wavelengths($\lambda$) doped A, $n_L = 1$ (monolayer) with $\mu = 1.0$ eV black Phosphorus, B, doped $n_L = 9$ with $\mu = 0.3$ eV , C doped $n_L$ = 38 , with $\mu = 0.3$ eV and D doped $n_L= 20000$ (bulk) black Phosphorus with with $\mu = 0.3$ eV.
4. Conclusion
In this work, we show that the far-field emission spectrum of a Hydrogenic emitter can be drastically tailored in the presence of multi-layer black Phosphorus. This anisotropic modification of BP-emitter interaction is dependent on the number of layers of BP and also the doping level. This thickness-dependent interaction is the aftermath of the layer-sensitive band-gap and optical properties of BP. We find that for monolayer to 8 layer BP, the spectrum is the same as that of free space conditions with $122$ nm being dominant whereas BP with $n_L \geq 9$, the deep UV emission at $103$ nm is greatly enhanced within a distance window of $z_0\approx 10$ nm to $z_0\approx 30$ nm. This indicates that layer-dependent BP-emitter interaction can result in the enhancement of spectral lines that are recessive in free space. This environment-mediated modification also depends on the doping level of BP. We notice that with adequate doping level the $103$ nm spectral lines can be enhanced for monolayer BP but suppressed for $n_L = 9, 38$, and bulk. Thus doping can be used to control the output spectrum of the emitter near BP. The Purcell-enhancement type analysis presented in this work can pave the way for exploring similar types of light-matter interactions to detect seemingly weak transitions. Our work can easily be extended to other emitters with much higher energy transitions and other 2D materials with layer dependent properties.
Acknowledgments
The authors acknowledge the support and facilities obtained from the Department of Electrical and Electronic Engineering (EEE), Bangladesh University of Engineering and Technology (BUET).
Disclosures
The authors declare no conflicts of interest related to this article.
Data availability
Results in this paper can be reproduced using the Code 1 written on MATLAB programming language and available on GitHub (Ref. [68]).
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