1. Introduction

Realizing the control of the interaction of light-matter on the nanoscale has been the main goal of nanophotonics research. In addition, the study of the strong interaction between light and matter has also greatly promoted the development of photonics [1–4]. The reason for the formation of Rabi splitting is that the coherent energy exchange rate between light and matter is much larger than its energy dissipation rate. At this time, the system is in a strongly coupled state, and it is no longer the initial separation state, but a hybrid state of plasmon-exciton [5]. Strong coupling is of great significance in fundamental research and many practical applications, such as photon anti-bunching [6], single emitters [7], superconducting resonator [8], and optical detection [9]. Initially, the research and realization of the Rabi splitting were carried out in traditional inorganic semiconductor systems. However, the traditional inorganic semiconductor represented by GaAs has a defect that cannot be ignored, that is, the exciton binding energy is small, which leads to the fact that the Rabi splitting must be observed at low temperature. There will be great restrictions in practical applications [10]. One of the solutions is to use wide-bandgap semiconductors, such as ZnO, GaN and organics [11,12]. Their large exciton energies can ensure that the Rabi splitting can be observed at room temperature. However, the wide bandgap semiconductors are limited to short-wavelength applications and require complex processes, while organic systems are limited by their strong localized effects due to the disordered potential distribution. Therefore, these defects make it difficult for a system based on these two materials to achieve large Rabi splitting at room temperature.

Two-dimensional (2D) materials, such as transition-metal dichalcogenides (TMDCs), graphene and black phosphorus (BP), have attracted widespread attention in recent years due to their atomic thickness and novel physical and chemical properties [13–16]. Because of the high exciton binding energy, 2D materials provide an ideal platform for large Rabi splitting at room temperature [17]. Graphene is a typical 2D material and has been used in research on Rabi splitting. For instance, the Rabi splitting was demonstrated in a coupled semiconductor microcavity, which is composed of distributed Bragg reflectors and graphene nanoribbons. The observed Rabi splitting energy is about 10 to 12 meV, which indicates the possibility of realizing the Rabi splitting for the graphene [18]. However, graphene has a zero-bandgap, so that the turn-off of the field-effect transistor based on graphene cannot be effectively controlled [19]. Therefore, TMDC has attracted widespread attention due to its large bandgap of 1.8 eV for monolayer, which can be used for the research of large Rabi splitting at room temperature. At room temperature, the hybrid structure composed of silver nanodisk array, WS₂ and an optical cavity can achieve large Rabi splitting, almost 300 meV [20]. BP, as a widely studied 2D material, has been used in many applications because of its superior physical properties, such as plasmonics, modulators, hetero-junctions, and transistors [21–25]. However, BP has rarely been explored for the research of Rabi splitting. Therefore, it makes sense to show whether BP is a suitable material for achieving large Rabi splitting. Compared to TMDCs, BP has a stronger interlayer interaction and adjustable band gap for all number of layers [26–29]. Particularly, for monolayer BP, its exciton binding energy (bandgap) can almost reach 2 eV, which makes BP one of the alternative materials in the research of Rabi splitting at room temperature.

It is well known that changes in the light-matter interaction can be achieved by adjusting the optical environment. Two structures are generally used to realize the strong light-matter interaction. The first structure is to embed molecular excitons in the optical microcavity. For example, hybrid polarized exciton states are confirmed in the hybrid structure composed of optical microcavity and MoS₂ [17]. Although Rabi splitting was observed at room temperature in this structure, its Rabi splitting energy was only 46 meV. And the structure of the optical cavity is complicated, and a large number of layers bring a problem of large volume. For instance, a coupled semiconductor microcavity, composed of 32 layers distributed Bragg reflectors and graphene nanoribbons was demonstrated and the total thickness of the structure even reaches a few microns [18]. Another structure is to combine molecular excitons with metallic plasmonic structures that can excite localized surface plasmon polariton (LSPP) modes. In the interaction of light and matter, the coupling of the system and the volume of the system are inversely proportional [30]. The metal nanostructures can support the LSPP mode and can limit the incident light to the nanoscale, which makes it possible to achieve a strong local field in an ultra-small mode volume [31]. In recent years, several studies have reported hybrid systems based on the interaction between metal nanostructures and molecular materials [32–41]. For example, the interaction between the local plasmons in the gold nanorods and the excitons in the J-aggregate was studied, and the strong coupling and large Rabi splitting energy up to 200 meV was demonstrated [42].

In this paper, we report the observation of Rabi splitting by composing BP film and two different metallic plasmonic structures at room temperature. By tuning the dimensions of a plasmonic grating and disk, we can create a spectral overlap between LSPP and BP exciton. In addition, we also demonstrate the strong coupling in the nanograting and nanodisk structure through polarization angular tuning. By establishing a coupled oscillator model, the dispersion curves of the hybrid exciton states are calculated, and the Rabi splitting energy of the system is calculated accordingly. The large Rabi splitting can be realized by composing metallic nanostructures and BP film, even at room temperature, paves the way toward the strong coupling of plasmonic nanostructures with BP based on ultrathin materials in the future for various device applications, including light detection, light-harvesting, single-photon device and optically active device.

2. Results and discussion

To demonstrate the Rabi splitting in different plasmonic nanostructures, we compare the reflection spectra of the heterostructure based on two different metal nanostructures. Figure 1(a) shows the three-dimensional schematic of BP-metal nano-grating heterostructure. Firstly, BP is grown on the prepared substrate, which is the SiO₂ layer with a thickness of 200 nm. There is also the MgF₂ layer with a thickness of 20 nm on the substrate, which is used as a buffer layer. The formation of nanostructures on the BP film can be prepared by electron beam lithography. We can adjust its plasmon resonance by tuning the width (w) of the nanograting arrays. However, previous studies have shown that BP will be rapidly oxidatively degraded under environmental conditions [43]. Later studies have shown that the BP can be preserved well by cracking the sample in dry nitrogen and then transferring it to vacuum [44]. What’s more, the distance (l) between two adjacent gratings is fixed to be 150 nm. This separation distance is to make the interaction between the local plasmons of adjacent gratings negligible, so as to avoid it from affecting the coupling between LSPP and BP excitons. Furthermore, when the grating width is less than the separation distance, the coupling between the adjacent gratings is negligible for the plasmon-exciton coupling. Therefore, a series of nanogratings with a width of 80 ∼ 105 nm was formed on the BP film. The inset in Fig. 1(a) shows a schematic of the BP crystal. BP is anisotropic, with atoms arranged in two directions in the lattice, armchair direction (x-direction) and zigzag direction (y-direction). In this manuscript, the lattice direction of BP is fixed in the direction of the armchair. This is because the effective mass in the armchair direction is less than that in the zigzag direction, and a small effective mass will lead to a higher resonance frequency [30]. Figure 1(b) shows the three-dimensional schematic of BP-metal nanodisk heterostructure. The distance (d) between two adjacent nanodisks is also fixed to be 150 nm, and the radius (r) of the nanodisk ranges from 30 to 45 nm. The refractive index of silver is selected from the results from Johnson [42,45]. Compared to Ag-based plasmon systems, although Au-based plasmon systems exhibit better photochemical stability, the Rabi splitting energy is smaller [46]. In the simulation, the finite-difference time-domain (FDTD) method was used to analyze the reflection spectra of the BP-nanostructure heterostructure [47].

 

Fig. 1. (a) The three-dimensional schematic of BP-nanograting heterostructure. The width of each grating is w, and the distance between two adjacent gratings is l. The inset is a schematic of BP crystal structure. (b)The three-dimensional schematic of BP-nanodisk heterostructure. The radius of each disk is r, and the distance between two adjacent disks is d.

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It is known that BP has an adjustable bandgap, ranging from 0.3 eV for bulk to 2 eV for monolayer, which is dependent on the thickness and near 0.3 eV in thick BP. In this article, the thickness of BP is chosen as 1 nm. In this article, the thickness of BP is chosen as 1 nm. As demonstrated in the gray curve in Fig. 2(a), it can be seen that the binding energy of BP is about 1.72 eV, which is almost the same as the J-aggregate (1.79 eV) [48,49]. As illustrated in Fig. 2(a) and Fig. 2(b), reflection spectra of different grating widths and disk radii are calculated as a function of plasmon energy. The resonance energy shifts from 1.4 to 1.9 eV while its width and radius vary. The reflection spectra of the bare grating and the disk structure completely overlap with the reflection spectrum of the BP exciton, which lays the foundation for the realization of Rabi splitting. Therefore, we plotted the reflection spectra of the complete BP-metal nano-grating heterostructure in Fig. 2(c). It can be seen that from the green line, the resonance energy has two peaks on either side of the exciton energy, these two peaks can be defined as two different resonance modes. The resonance on the right side of the exciton peak is at high energy and can be called the upper branch (UPB), while the resonance on the left side of the exciton peak is at low energy, which is called the lower branch (LPB) [50]. However, we cannot conclude that the system is strongly coupled. Because many reasons can cause double reflection peaks, for example, the corrugations in the metal nanostructures can also lead to the double peaks [51]. In addition, even without these reasons, the double reflection peaks in the reflection spectra still cannot be judged as a sign of Rabi splitting. For the intermediate coupling, there are also two peaks in the reflection spectra, even if they are not in the strongly coupled region, which makes it unreliable to distinguish strong coupling only by the presence of two reflection peaks [51]. Deep research needs to be carried out about these two resonance peaks.

 

Fig. 2. (a) Reflection spectra of the bare grating structure as a function of the grating width. The width of grating sizes range from 80 to 105 nm, and the distance (l) between two adjacent gratings is fixed to be 150 nm. (b) Reflection spectra of the bare disk structure as a function of the disk radius. The radius of risk sizes range from 30 to 45 nm, and the distance (d) between two adjacent disks is fixed to be 150 nm. (c) Reflection spectra of bare BP (gray line), bare Ag nanogratings (red line), and BP-nanograting heterostructure (green line). Two small red dots represent the two split reflection peaks: LPB and UPB. The thickness of BP is 1 nm, the width (w) of the grating is set as 90 nm, and the distance (l) between two adjacent gratings is fixed to be 150 nm.

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The reflection spectra of the BP-grating hybrid structure by varying the widths of the grating are demonstrated in Fig. 3(a). The single reflection peak splits into two separate peaks, separated by a peak near the exciton energy (1.72 eV) revealing a strong coherent coupling between the LSPP of the grating and the exciton in BP. The two peaks redshift as the width of grating increases, in which the LPB resonance has much stronger shifts. In order to judge whether the system is strongly coupled, it is necessary to introduce strong coupling conditions in advance. Assuming that E_pl and E_X are the energies of the uncoupled nanostructures resonance and BP excitons, and hΩ_R is the splitting energy between the UPB and LPB peaks. γ_pl and γ_X correspond to the full linewidth at half maximum of the plasmons and BP exciton reflection spectra, respectively. A Rabi splitting requires this system to match the condition hΩ_R > (γ_pl+γ_X)/2 [20]. That is to say, for strong coupling, the Rabi splitting must exceed half the sum of the half line-width of the plasmon mode and the exciton mode. From the reflection resonant spectra of bare nanogratings and BP exciton, we can obtain γ_pl≈200 meV and γ_X≈180 meV. By fitting the reflection spectra in Fig. 3(a), we can extract the splitting energy hΩ_R = 250 meV. By comparing the linewidths of exciton and plasmon and splitting energies, it is found that their mathematical relationship satisfies the strong coupling condition. The same as the reflection spectra in Fig. 3(a), the reflection spectra in Fig. 3(b) correspond to the BP-disk hybrid structure and also show a single reflection peak splits into two separate peaks with the splitting energy of 202 meV (hΩ_R≈202 meV). The half linewidth of the disk plasmon γ_pl is 100 meV. Therefore, the BP-nanodisk heterostructure is also in a strong coupling state. In addition, we can also find that plasmons with two different metal structures are strongly coupled with BP excitons, resulting in large Rabi splitting energy. This shows that BP has the potential to be used in a variety of metal structures to achieve large Rabi splitting. What’s more, in order to further investigate the relationship between coupling behavior and polarization angle, we analyzed the spectra of these two nanostructures when the polarization angle changed by 15 degrees. As demonstrated in Fig. 3(c), the spectra of BP-grating heterostructure are collected from 0° to 90°, corresponding to the longitudinal and transverse polarizations, respectively. The result for the 105 nm grating width shows a progressive vanish of the two exciton peaks as the polarization is changed from longitudinal (0°) to transverse (90°). As the polarization angle approaches 0 degrees, the UPB peak blueshifts and the LPB peak intensity increases, resulting in the largest difference between UPB and LPB, corresponding to the strongest coupling. The BP-nanodisk heterostructure shows a similar variation trend by varying the polarization angle from 0° to 90° in Fig. 3(d). It can be seen that as the polarization angle increases, the LPB gradually disappears, and the UPB drifts to higher energy. When the polarization angle is 0 degrees, LPB peak intensity increases to the maximum, corresponding to the strongest coupling. These similarities provide strong evidence that the exciton coupling is dependent on the polarization angle.

 

Fig. 3. (a) Reflection spectra of the BP-nanograting heterostructure as a function of the grating width. The width of grating sizes range from 80 to 105 nm, and the distance (l) between two adjacent gratings is fixed to be 150 nm. E_X0 is exciton energy. (b) Reflection spectra of the BP-nanodisk heterostructure as a function of the disk radius. The radius of risk sizes range from 30 to 45 nm, and the distance (d) between two adjacent disks is fixed to be 150 nm. (c) Polarized reflection spectra of exciton grating with detected polarization angles of 0°, 15°, 30°, 45°, 60°, 75°, and 90°. Polarization dependence of exciton properties on BP-nanograting heterostructure with an individual grating of 105 nm and a 150 nm gap. (d) Polarized reflection spectra of exciton disk with detected polarization angles of 0°, 15°, 30°, 45°, 60°, 75°, and 90°. Polarization dependence of exciton properties on BP-nanodisk heterostructure with an individual disk of 40 nm and a 150 nm gap.

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To further illustrate the dependence of the coupling behavior on the polarization angle, Fig. 4 plots the electric field distribution of the BP-nanodisk heterostructure and the bare nanodisk structure at the wavelength of 720 nm in the X-Y plane. From Fig. 4(b) and Fig. 4(d), it can be seen that the electric field intensity of the BP-nanodisk heterostructure is almost 1.5 times larger than that of the bare nanodisk structure. This shows that the coupling process of the LSPP generated by the nanodisk and BP excitons forms a new hybrid polarized exciton, which can also indicate that the heterostructure is strongly coupled. In addition, by comparing the electric field distributions in Fig. 4(c) and Fig. 4(d), we can see that the calculated electric field distributions are found to be well correlated with the polarization dependence in Fig. 3(d). In the longitudinal mode, the electric field intensity reaches the maximum at the polarization angle of 90° and almost disappears at about 0°, while the transverse mode is opposite. In addition, we found that the near field enhancement of the disk gap may also have some influences on the Rabi splitting intensity. It is noticed that the electric field enhancement in the disk gap in Fig. 4(c) is obviously larger than that in Fig. 4(a). This indicates that strong coupling can be realized in the BP-nanodisk heterostructure. These results indicate that the strong coupling is indeed dependent on the strength of the near-field enhancement in the disk gap. Under strong coupling conditions, the intensity of the Rabi splitting depends not only on the binding energy and oscillation intensity of the excitons but also on the local electric field intensity excited by the metal nanostructures.

 

Fig. 4. The electric field distribution of the bare disk structure in the X-Y plane at a wavelength of 720 nm when the polarization angle is (a) 0° and (b) 90°. The electric field distribution of the BP-nanodisk heterostructure in the X-Y plane at a wavelength of 720 nm when the polarization angle is (c) 0° and (d) 90°. The radius (r) of the disk is set as 40 nm, and the distance (d) between two adjacent disks is fixed to be 150 nm.

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We use the Jaynes-Cummings model to describe plasmon-exciton coupling [52]. A simple coupled oscillator model (COM) was used to perform in-depth analysis and description of the observed strong coupling of the BP-metal nanostructure heterostructure [53]. To study the optical properties of heterostructures, it is necessary to analyze their optical transitions, as shown in Fig. 5(a). The exciton excitation of BP is the transition of electron ground state $\mathrm{\beta}$ to exciton excited state $\mathrm{\alpha}$ by photon energy. In heterostructures, BP excitons are coupled with LSPP excited by metal nanostructures, and energy exchange is performed between excitons and plasmons. If the rate of this energy exchange is much greater than the rate of their losses, the system is in a strongly coupled region, also known as Rabi splitting. The interaction of the LSPP excited by the metal structure and BP excitons generates a new hybrid exciton state, which is reflected as a double reflection peak in the reflection spectra and shows anti-crossing behavior. The energies of the exciton states of UPB and LPB are calculated using COM, and the double coupled system was described by the Hamiltonian matrix [54]:

Where E_pl and E_X are the resonance energies of plasmon and exciton modes, γ_pl and γ_X correspond to the full linewidth at half maximum of the plasmons and BP exciton, And g_X represents coupling strength. α and β are Hopfield coefficients of the matrix, which represents the contribution of the plasmon mode and the exciton mode to each polarization branch, where |α|² + |β|² = 1. And E is the eigenvalues of the hybrid plasmon-exciton state. The eigenvalues E are obtained from the equation:

When the linewidths of the plasmon and exciton are less than their energies, the eigenvalues E can be described as:

Where $\hbar {\mathrm{\omega }_0}$ and $\hbar {\mathrm{\omega }_\textrm{p}}$ are the uncoupled exciton and LSPR energies, respectively. $\hbar {\mathrm{\Omega }_\textrm{R}} = 2{g_x}\; $ is the coupling energy, which can be estimated from the spectral distance between these two peaks. As illustrated in Fig. 5(b) and Fig. 5(c), the dispersion curves of the two different hybrid BP-nanostructures plasmon-exciton states are plotted, respectively. The energy of the UPB and LPB exhibits a clear anti-crossing behavior. The Rabi splitting is found to be approximately 250 meV and 202 meV, respectively, which is lower than the highest values previously reported [20]. Although the exciton energies of monolayer BP and monolayer MoS₂ are almost the same, the radiation energy of BP is higher, that is the rate of energy dissipation is larger, which will lead to a greater loss of energy in the process of light-matter interaction [28]. Therefore, the rate of coherent energy exchange between light and matter in BP will be relatively lower than that in MoS₂ and the Rabi splitting energy based on BP system is lower than that based on MoS₂ system. Figure 5(d) shows the change of the reflection cross-section of BP-nanodisk nanostructures with the wavelength and the radius of the nanodisk. The white dotted line represents the energy of uncoupled BP excitons. The blue dashed line in the reflection spectra represents the two polarization branches of UPB and LPB, it can be clearly seen that they exhibit anti-crossing behavior. These results confirm that the resonant coupling of excitons and LSPP is indeed a strong coupling so that the Rabi splitting can be observed in the heterostructure. In addition, in order to understand the contributions of plasmon and exciton to the two polarization branches UPB and LPB, the proportion of plasmon and exciton components in the BP nanodisk heterostructure in UPB and LPB as a function of the energy of the plasmon can be obtained through COM in Fig. 5(e) and Fig. 5(f). When the plasmon energy is larger than the exciton energy (1.72 eV), the exciton mode accounts for a larger proportion of the LPB, and the LPB behaves more like an exciton. Conversely, when the plasmon energy is less than the exciton energy, the plasmon mode accounts for a larger proportion of LPB, and LPB behaves more like plasmon. It is worth mentioning that when the plasmon energy and the exciton energy are equal, both the exciton mode and the plasmon mode participate in the coupling of the exciton and the plasmon. Therefore, when the energy of the plasmon is equal to the energy of the exciton, the coupling between the exciton and the plasmon is the strongest, and it also corresponds to the maximum Rabi splitting energy.

 

Fig. 5. (a) Schematic of the optical transitions in metal/BP hybrid nanostructure. (b) Dispersion curves of the hybrid BP-nanograting exciton states. Energies of reflection peaks as a function of the nanograting width extracted from the reflection spectra. The pink and blue dashed lines represent the uncoupled exciton and plasmon energies, respectively. The double-headed orange arrow stands for the Rabi splitting energy. (c) Dispersion curves of the hybrid BP-nanodisk exciton states. Energies of reflectivity peaks as a function of the nanodisk radius extracted from the reflection spectra. The blue and pink dashed lines represent the uncoupled exciton and plasmon energies, respectively. (d) The reflection peaks (blue dotted curve) as a function of the radius (r) of the disk. (e), and (f) Hopfield coefficients for the polariton branches (UPB in (e) and LPB in (f)) of BP-nanodisk heterostructure, calculated using the COM, from Eq. (1). These provide the weighting of each constituent. Blue stars correspond to the coefficients of the exciton mode and the red spheres to those of the plasmons mode.

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3. Conclusion

In summary, we report the strong coupling between BP excitons and plasmons excited by two different metal nanostructures at room temperature. Rabi splitting was observed in these two hybrid systems with splitting energies of 250 meV and 202 meV, respectively. The strength of the exciton-plasmon coupling is closely dependent on the dimensions of the nanostructure and the polarization angle of the incident light. By tuning the dimensions and polarization, the system will be in a strongly coupled region. In addition, COM is used to analyze the physical mechanism of exciton-plasmon coupling in detail. In general, the heterostructure of BP-metal nanostructures makes it possible to observe large Rabi splitting at room temperature, and the large Rabi splitting indicates a new way to the research of nanophotonics based on the BP-nanostructures heterostructure and its application includes light detection, light-harvesting, single-photon device and optically active device.