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Abstract
We investigate the exciton polaritons and their corresponding optical modes in a hexagonal GaN microrod at room temperature. The dispersion curves are measured by the angle-resolved micro-photoluminescence spectrometer, and two types of exciton polaritons are identified with the help of the finite-difference time-domain simulation. By changing the pump position, the photon part of the exciton polaritons is found to switch between the quasi-whispering gallery modes and the two-round quasi-whispering gallery modes. The exciton polaritons formed by the latter are observed and distinguished for the first time, with a giant Rabi splitting as large as 2Ω = 230.3 meV.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Exciton polaritons are elementary excitations of semiconductor microcavity systems in the strong-coupling regime, where semiconductor excitons and cavity photons strongly interact [1]. These half-matter half-light bosonic quasi-particles can show exciting properties different from what ordinary excitons or photons may show. For instance, we can take advantage of the spontaneous emission of exciton polaritons to study a Bose gas system, as the emission carries energy information, momentum information, density information, and even angular momentum information of the whole system that can be extracted by simple optical measurements [2]. Superfluidity and quantum vortices have been studied in GaAs-based microcavities using this technique [3–7]. However, in these systems, the exciton binding energies are very low, and no stable excitons can exist at room temperature; besides, the matter-light hybrid behaviors are not very distinctive due to the small Rabi splittings. Thus in GaAs-based microcavities, the application of exciton-polariton fluids is limited by the requirement of relatively low temperatures or strong magnetic fields that are applied in order to reduce the exciton Bohr radius [8,9]. On the other hand, Wide-bandgap semiconductors such as GaN [10–16] and ZnO [17–25], along with some organic [26–29] and 2D materials [30–35], have large exciton binding energies capable of forming room-temperature exciton polaritons. While the epitaxy of p-type ZnO is relatively hard and organic materials generally do not favor electronic injection, GaN is a particularly promising material thanks to the technological maturity in the GaN-based light-emitting device industry. Future room-temperature polaritonic devices such as polariton light-emitting diodes [36], polariton lasers [37–39] and polariton gyroscopes [40] are very likely GaN-based. We demonstrate here, a large Rabi splitting in a GaN microrod, and investigate the increase in Rabi splitting when the pumping position changes.
Rabi splitting is one of the key characteristics of an exciton-polariton system. When a single cavity photon mode ${E_{\textrm{cav}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ and a single exciton band ${E_{\textrm{exc}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ are coupled, the dispersion curves of the photons and the excitons give their ways to two new branches of curves, namely the upper polariton branch (UPB) ${E_{\textrm{UPB}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ and the lower polariton branch (LPB) ${E_{\textrm{LPB}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ [41]. In the strong-coupling regime, no matter how the original dispersion curve ${E_{\textrm{cav}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ is tuned across ${E_{\textrm{exc}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$, the new dispersions curves UPB and LPB can never cross each other, there is always a gap between them. The smallest gap ${E_{\textrm{UPB}}} - {E_{\textrm{LPB}}}$ at zero detuning (${E_{\textrm{cav}}}\textrm{ = }{E_{\textrm{exc}}}$) is referred to as the Rabi splitting. This splitting or ‘avoided crossing’ feature is the mark of strong-coupling and what makes exciton polaritons interesting. A larger Rabi splitting indicates a higher-energy UPB and a lower-energy LPB at zero detuning, and thus, these two exciton-polariton branches are further deviated from the original ${E_{\textrm{cav}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ and ${E_{\textrm{exc}}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} } \right)$ curves that sit in-between. This means the phenomenon observed in systems with larger Rabi splitting can differ more from ordinary exciton or photon systems.
A large Rabi splitting requires strong interaction between excitons and photons as well as a high-quality optical cavity [42]. Whispering gallery modes (WGMs) usually have good quality factors owing to the total internal reflections. They also have small mode volumes that are helpful with the formation of exciton polaritons. WGMs exciton polaritons had been realized in GaN microrods, and Rabi splittings as high as 115 meV [43] and 180 meV [44] were reported. In this work, we have observed exciton polaritons formed with quasi-whispering gallery modes (qWGMs) and two-round quasi-whispering gallery modes (2rqWGMs) for the first time. The 2rqWGM exciton polaritons show much larger Rabi splitting than ordinary qWGM exciton polaritons.
2. Experimental observation
2.1 Sample preparation
The GaN microrods are grown by metal-organic vapor phase epitaxy (MOVPE) on a c-plane sapphire substrate. 30 sccm of Trimethylgallium (TMGa) and 240 sccm of ammonia (NH3) are used as the precursors, and pure hydrogen (H2) is used as the carrier gas. The substrate is kept in an ammonia atmosphere at 1100 °C for 20 minutes prior to the growth of the microrods. The rods are formed via self-assembly at a temperature of 1040 °C and a pressure of 200 Torr, without the need for any pre-applied masks. More information about this self-assembly growth method can be found in our previous work [45]. Using this method, we are able to synthesize GaN rods with a perfect hexagonal shape and smooth surface. The vertically grown microrods are then mechanically separated from the sapphire substrate and laid onto a grid-marked fused silica wafer for measurements.
2.2 Measurement setup
In order to observe the dispersion curves of the exciton polaritons, an angle-resolved micro-photoluminescence spectrometer is used. As is shown in Fig. 1, the system consists of a sample stage, an objective lens, an eyepiece lens, a Fourier lens, a spectrometer, two beam splitters, an excitation laser, and a monitoring camera together with some other optical components that can be inserted in the light path.
The gridded silica wafer which holds the GaN microrods is loaded onto the sample stage. A white light source is inserted in the light path to illuminate the sample, and then a real-space microscopic image can be seen on the monitoring camera. Move the stage in y, z, and x direction so that a selected GaN microrod is located in the center of the light path and in focus. Turn the stage ($\varphi$) while adjusting y and z until the microrod is positioned upright or parallel to the spectrometer slit. In this configuration, the angle-resolving ability along the slit corresponds to the wave vector component ${k_\textrm{z}}$ of the exciton-polariton spontaneous emission. The Fourier lens between eyepiece and objective transforms the image at the spectrometer entrance slit from real-space into the k-space, which is then magnified and cast onto the CCD by the imaging system inside the spectrometer. The emission angle $\theta$ is resolved vertically by the slit, while the wavelength is spread horizontally by the gratings. The maximum collection angle is 30 degrees as the numerical aperture of the objective lens is 0.5 (the objective lens has the smallest numerical aperture among all lenses in the light path). Variable attenuators are inserted to protect detectors. In order to make good use of the CCD’s dynamic range, a long-wave pass edge filter with 360 nm cut-on wavelength is applied in front of the spectrometer to block the pumping laser (the excitation laser). The CCD is cooled down before data collection for a better signal-to-noise ratio (SNR). An analyzer is also put inside the light path to select certain polarization.
We can also take real-space pictures with the spectrometer CCD. This is done by removing the Fourier lens to undo the Fourier transformation. The light path is depicted in Fig. 2(a). Open the spectrometer slit widely and disable the dispersion function inside the spectrometer so that it only works as an imaging system, replace the excitation laser with a white light illumination, and real-space microscopic pictures of the sample can be taken on the CCD. Due to the limitation of the CCD’s dynamic range, the excitation laser spot needs to be taken pictures separately with the attenuator set to a large decay.
Fig. 2. Real-space imaging. (a) The experiment setup for real-space imaging with spectrometer CCD. (b) The real-space image of the sample when pumped at its left. (c) The real-space image of the sample when pumped at its center.
The resolution of the real space image in this configuration is about 0.5 µm. The laser used in this experiment is a continuous 325-nm He-Cd laser with a spot size of 1.3 µm (full width at 1/e peak intensity). A grating UV-Vis spectrometer is used, which has a uniform line dispersive power.
2.3 Dispersion curves
A dispersion curve describes the energy versus momentum in the system. The recorded angle-resolved photoluminescence spectra directly relate to the dispersion curves of the exciton polaritons, with the angle represents the wave vector ${k_\textrm{z}}$, and the wavelength represents the energy E. This is because we have collected the spontaneous emission of the exciton polaritons, and each photon has the same energy and momentum as the corresponding exciton-polariton. For the one-dimensional system studied here, we are only interested in the z-direction or the direction along $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over c}$-axis of GaN.
As we move the excitation beam spot in the y-direction and across the GaN microrod, two types of dispersion curves can be observed. Pumping at the center of the rod generates fewer curves or modes than pumping at either side of the rod. Figure 2(b) is the real-space image of the sample when the excitation laser spot lands on the left part of the rod; Fig. 2(c) is the image when the excitation laser locates at the center of the rod. In the image, the laser spot has been false-colored to show its position and size. The corresponding dispersion curves of these two pumping conditions are shown in Fig. 3(a) and (b), respectively. The pictures have been enhanced so that the signals from different dispersion curves have intensity values that are on the same scale and all curves can be seen clearly to the human eye.
Fig. 3. Angle-resolved photoluminescence spectra. Dashed lines are dispersion curves calculated with fitted data. (a) The dispersion curves when pumping at the left part of the sample. (b) The dispersion curves when pumping at the center of the sample. (c) The dispersion curves when pumping at the center of the sample but with half of the pumping power.
Note that a polarization analyzer is applied to extract the light with electric component $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \varepsilon } \bot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over c}$, which is the emission from exciton polaritons of the TE [46] optical modes inside the microrod. We are not able to select and analyze the TM-mode ($\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \varepsilon } \parallel \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over c}$) exciton polaritons in our sample due to their low intensity. A-exciton and B-exciton have high probability of emitting into TE modes, and contribute most to the photoluminescence with A-exciton has the lowest energy. Although the TM optical modes generally have higher quality factors than the TE modes, the emission from A-excitons to TM photons is forbidden by selection rules [47]. C-excitons only emit weakly into TE or TM modes. As a result, the exciton polaritons in our GaN microrod sample are mostly formed by A- or B-excitons interacting with TE-mode photons. The oscillator strengths for A- and B-excitons are 40000 $\textrm{me}{\textrm{V}^2}$ and 39000 $\textrm{me}{\textrm{V}^2}$, respectively [48].
Figure 3(a) and (b) clearly show that two different sets of TE modes exist, which then form two sets of exciton polaritons. The dashed lines are the theoretical dispersion curves calculated with the fitting data. When the pump laser is shone on the center of the rod, the optical modes are quasi-whispering gallery modes (qWGMs). When the pump spot is moved slightly to the left, a new set of modes appears; we call them the two-round quasi-whispering gallery modes (2rqWGMs). We will discuss these modes in detail in the next session. In both cases, the pumping power is 421.3 µW. Figure 3(c) is included as a verification, which shows the dispersion curves of pumping at the same center spot but using only half the excitation power, or 208.8 µW. Despite the relatively low SNR, one can notice the modes are still qWGMs and do not change into 2rqWGMs. This rules out the possibility that the change of modes is caused by a loss of pumping energy when the laser spot is moved off-center. As will be shown by simulation, the change in modes can indeed be caused by the change of pumping position alone.
3. Two-round quasi-Whispering Gallery mode exciton polaritons
3.1 Two-round quasi-Whispering Gallery modes
The concept of a whispering gallery (acoustic) mode was first interpreted by Lord Rayleigh in 1910 [49]. An electromagnetic version of this phenomenon in a dielectric rod was systematically analyzed by James R. Wait [50]. For a hexagonal dielectric prism like our GaN microrod, we can use the rather rudimentary ray optics theory, given that the size of the microcavity is far larger than the light wavelength inside the cavity. Figure 4 shows three categories of whispering-gallery-type modes that can exist in the hexagonal microcavity. When the light in the prism has an angle of incidence of 60 degrees at the microrod-air interface, the reflection is a total internal reflection. This can happen six times before the light finishes a round-trip in the cavity, namely the WGM, which has an optical length of $3\sqrt 3 {n_{\textrm{GaN}}}R$, as is shown in Fig. 4(a). R is the radius of the circumcircle of the hexagon, and ${n_{\textrm{GaN}}}$ is the refractive index of microrod. If the refractive index is big enough, total internal reflection can happen even at a 30-degree incident angle. This is exactly the case of GaN, which has a total-reflection angle of 21.6∼23.1 degrees at the microrod-air interface in the wavelength range we have studied in this experiment. As a result, the cavity mode that consists of three 30-degree total reflections can exist in the GaN hexagonal prism, called the qWGM. It has an optical length of ${\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 2}} \right.}\!\lower0.7ex\hbox{$2$}}{n_{\textrm{GaN}}}R$, as is shown in Fig. 4(b). Notice that six 30-degree total reflections can also form a closed loop of the light path, with an optical length of $9{n_{\textrm{GaN}}}R$, twice as long as the qWGM loop. We call this mode the 2rqWGM, whose light path is depicted in Fig. 4(d). The qWGM can actually be regarded as a special case of the 2rqWGM, where six reflection positions coincide in pairs. Due to this degeneration, the mode spacing of the qWGM is twice as large as that of the 2rqWGM.
Fig. 4. The round-trip optical path of the whispering-gallery-type modes inside a hexagonal GaN microrod. (a) Whispering gallery mode. (b) quasi-Whispering gallery mode. (c) upside-down quasi-Whispering gallery mode (suppressed). (d) two-round quasi-Whispering gallery mode. (e) upside-down two-round quasi-Whispering gallery mode (suppressed).
Using ray optics, we can study the TE modes in our microrod. The incident beam is p-polarized ($\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \varepsilon } \bot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over c}$), and has a reflection phase shift of
where $\alpha$ is the angle of incidence, $n = {{{n_{\textrm{GaN}}}} / {{n_{\textrm{air}}}}}$ is the refractive index ratio of inner material to outer material, namely the GaN refractive index ${n_{\textrm{GaN}}}$ divided by the air refractive index ${n_{\textrm{air}}}$.
The closed-loop optical length ${\delta _{\textrm{round}}}$, the mode’s energy E and order N, and the phase shift ${\delta _\textrm{p}}$, satisfy the following equation:
Apparently, there exists a 2rqWGM in-between every two neighboring qWGMs.
The qWGM can have a light path of a triangular shape or an upside-down triangular shape if all six sides of the hexagonal microrod are exposed to air. In our experiment, however, one side is in contact with SiO2, and no total reflection can happen at 30 degrees at this interface; the upside-down versions are suppressed owing to this energy loss, as shown in Fig. 4(c) and (e). The existence of the SiO2 breaks the symmetry, and for both qWGM and 2rqWGM, only the upright versions of the light path can be seen. In the following simulation, we also verify that the upside-down versions do not appear in either excitation conditions, due to the failing of total reflection and the leakage of light.
3.2 Numerical simulation
We have used the finite-difference time-domain method (FDTD, Lumerical Solutions, Inc.) to simulate the optical modes inside the microrod under different excitation conditions. Since a one-to-one correspondence can be established between the optical-mode dispersion curves and the exciton-polariton dispersion curves, the analysis of the mode spacing and the pumping conditions for the optical qWGMs and 2rqWGMs is also applicable to the corresponding exciton polaritons. In our simulation, the cross-section of the GaN microrod is a regular hexagon with one side touching the SiO2 on the grid-marked wafer. The simulation is performed in this cross-section. For an excitation beam with 325 nm wavelength (3815 meV), the penetration depth in GaN is only about 74 nm, meaning almost all the energy is absorbed in a small region near the surface of the microrod, and photoluminescence due to this pumping is also created at that spot; thus we can simply use a magnetic dipole as the light source in simulation to investigate the TE modes that we want to study.
We have observed in our experiment that when the pumping spot lands at different positions, the measured exciton polaritons originate from different optical modes. This phenomenon is confirmed in the simulation results of the two pumping positions by solving the Maxwell’s equations with the FDTD method. The two rows in Fig. 5 show the optical modes in the microrod when the source is located at the top center and at the top left, respectively. When the source is at the center, only the qWGM can be seen in the microcavity. When the source is shifted to the left, an additional mode shows up between two qWGMs, namely the 2rqWGM.
Fig. 5. The simulated light intensity distribution in the microrod when pumped at the top center or at the top left of the sample.
The qWGM, 2rqWGM, WGM, or even the Fabry–Pérot mode, are all optical modes that are allowed to exist in the GaN microcavity; mathematically, they are all solutions to the Helmholtz equation of electromagnetic waves under certain boundary conditions. Nevertheless, in different pumping conditions, the energy from the source is distributed differently among the optical modes. If the spatial position of the source is at the top center, where the 2rqWGM has zero light intensity, the pumping energy cannot be coupled into the 2rqWGM; thus only the qWGM receives energy and shows up, as is the case depicted in the first row of Fig. 5. On the other hand, because the strongest light intensity in 2rqWGM is reached off-center at the top surface, only when the source is also placed off-center correspondingly, can its energy get into the 2rqWGM easily, as shown in the second row of Fig. 5.
3.3 Larger Rabi splitting
We have discussed the qWGM and 2rqWGM optical modes that exist in the GaN microcavity. Note that there are also excitons in the microrod, and the optical modes can couple with the excitons. When the photon-exciton interaction is strong enough and the photon loss is low enough, the exciton-polariton states become the eigenstates of the system in replacement for the photons and excitons. Rabi splitting appears and two branches of exciton-polariton dispersion curves are formed for each type of optical mode. Thanks to the spontaneous emission of the exciton polaritons, we can measure the LPB dispersion curves via the angle-resolved photoluminescence, which have been shown in Fig. 3. In other words, the dispersion data we collected is not from the optical modes qWGMs and 2rqWGMs, but from the exciton polaritons generated by their coupling with the excitons. The details of fitting the dispersion curves will be discussed later in this section.
Figure 6(a) and (d) are the photoluminescence spectra near zero angle for the qWGM exciton polaritons and the 2rqWGM exciton polaritons. Figure 6(c) and (f) show the fitted dispersion curves of the LPBs and the photon modes, as well as the exciton energy level. The solid lines indicate the LPB dispersion curves, while the dotted lines are the uncoupled photon-mode dispersion curves that have been colored correspondingly. The black dashed line indicates the exciton energy level. The anti-crossing is marked by putting orange crosses at ${E_{\textrm{cav}}}\textrm{ = }{E_{\textrm{exc}}}$, and the (half) Rabi splitting is shown in the figure by the grey arrows. In Fig. 6 (b) and (e), the zero-angle Hopfield coefficient ${|C |^2}$ is plotted for each corresponding LPB; a small ${|C |^2}$ closer to zero indicates a more exciton-like behavior, while a large ${|C |^2}$ closer to one means the polaritons behave more like photons. As the zero-angle LPB energy increases towards the exciton energy, the evolution from strongly photonic to more excitonic nature of the coupled oscillator modes is clearly visible.
Fig. 6. The photoluminescence spectra near zero angle for (a) qWGM exciton polaritons, and (d) 2rqWGM exciton polaritons. The zero-angle Hopfield coefficient ${|C |^2}$ of (b) qWGM exciton polaritons, and (e) 2rqWGM exciton polaritons. The calculated polariton dispersions curves and photon dispersion curves of (c) the qWGMs, and (f) the 2rqWGMs.
For each exciton polariton lost in the system, the corresponding energy and momentum are released out of the system in the form of a photon. This spontaneous emission originates from the exciton loss ${\gamma _{\textrm{exc}}}$ and the photon loss ${\gamma _{\textrm{cav}}}$ in the system. When the loss rate is taken into consideration, the dispersion curves of the exciton polaritons become [42]:
where ${g_0}$ is the exciton-photon interaction strength, ${E_{\textrm{exc}}}$ is the energy of the excitons involved in the strong coupling, and ${E_{\textrm{cav}}}$ is the energy of the involved cavity photons. The dispersion of the $N\textrm{th}$ order cavity mode is [44]:
The refractive index within each optical mode is almost constant, so for simplicity, we can treat the refractive index as a function of N :${n_{\textrm{GaN}}}(\lambda )\approx {n_{\textrm{GaN}}}({{E_{\textrm{cav,0}}}} )\approx {n_{\textrm{GaN}}}(N )$
Moreover, the exciton loss ${\gamma _{\textrm{exc}}}$ due to the nonradiative decay is far less than the photon loss ${\gamma _{\textrm{cav}}}$ due to the out-coupling through imperfect mirror [42], so the dispersion curve of the LPB can be written as:
We take ${E_{\textrm{exc}}}$ as 3412 meV at 300 K based on the data from [52,53]. To fit our exciton-polariton dispersion curves, let ${E_{\textrm{cav,0}}}(N )= {E_{\textrm{cav,0}}}({{N_0}} )+ ({N - {N_0}} )\Delta E$, the mode spacing $\Delta E$ satisfies Eq. (3) and (4). The fitting is performed with different $({N,\theta } )$ data points sampled from different curves in the angle-resolved photoluminescence spectra. All visible branches of the qWGM and 2rqWGM exciton polaritons in Fig. 3 have been sampled; the energy E and $\sin \theta$ are read out by the data cursor from the angle-resolved photoluminescence spectra. We have relied on human eyes to decide where the curves can be seen most clearly; when selecting sample points for each curve, the difference between each point’s $|{\sin \theta } |$ were chosen to be as large as possible. The fitted values are given in Table 1.
Table 1. Dispersion characteristics of exciton polaritons with different cavity modesa
It is thus noticeable that the exciton polaritons with the optical modes of 2rqWGM have a larger ${g_0}$ than the exciton polaritons with the optical modes of qWGM. Which indicates that the 2rqWGMs couple more strongly with the excitons. As for the mode spacings of the optical modes, we can indeed verify that $\Delta {E_{\textrm{qWGM}}} = 2\Delta {E_{2\textrm{rqWGM}}}$. The cavity photon loss rates ${\gamma _{\textrm{cav}}}$ have relatively large intervals of 95% confidence, indicating that during the fitting, tuning their exact values contributes less to minimizing the error function, which deserves a verification. Whether these values reflect the actual experiment condition or not should be checked by other means described below. We have assumed that within the wavelength or energy scope of this work, the loss rate ${\gamma _{\textrm{cav}}}$ of a certain optical mode is a constant; however, what we directly measure is the loss rate ${\gamma _{\textrm{LPB}}}$ of exciton polaritons, which is not a constant. It changes with the Hopfield coefficient [1]:
We can derive the cavity photon loss with ${\gamma _{\textrm{cav}}} \approx {{{\gamma _{\textrm{LPB}}}} / {{{|C |}^2}}}$, where ${\gamma _{\textrm{LPB}}} = 2\pi {\kern 1pt} \Delta \nu$ is measured by the exciton-polariton spontaneous emission linewidth $\Delta \nu$ (full width at half maximum, FWHM). Substitute the spectra data near 3319 meV with good SNR into the above formulas, we can verify that ${\gamma _{\textrm{cav}}}$ are 109.8 meV for qWGMs and 43.80 meV for 2rqWGMs, close to the fitting coefficients we obtained in Table 1. The derived optical cavity quality factor 190 for qWGMs and 476 for 2rqWGMs are not particularly impressive, given the high-quality growth of the GaN hexagonal microrod; This is due to the larger loss of the TE modes which have the electric field $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \varepsilon } \bot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over c}$. Nevertheless, the sample has a reasonably good optical cavity. The geometric size of the GaN microrod can be calculated with:
Take $\Delta {E_\textrm{m}} = 20.25{\kern 1pt} {\kern 1pt} \textrm{meV}$ (mean value of the fitted $\Delta {E_{2\textrm{rqWGM}}}$ and ${{\Delta {E_{\textrm{qWGM}}}} / 2}$ in Table 1). Take ${n_{\textrm{GaN}}} = 2.605$ (near 3188 meV, corresponding to the mode spacing value used). We can calculate the diameter of the circumcircle $D = 2R = 5.22{\kern 1pt} {\kern 1pt} \mathrm{\mu}\textrm{m}$, which agrees with the diameter 5.3 µm measured from the optical microscopic image. The Rabi splitting is defined as the energy difference between the LPB and the UPB at zero detuning, which is:
Thus the Rabi splittings of the qWGM and 2rqWGM exciton polaritons are $2{\Omega _{\textrm{qWGM}}}\textrm{ = }136.9{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{meV}$ and $2{\Omega _{\textrm{2rqWGM}}}\textrm{ = }230.3{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{meV}$, respectively.
4. Conclusion and discussion
We have used MOVPE to grow GaN microrods via a self-assembly method. Strong coupling is observed in these regular hexagonal microrods. The curvature change of the dispersion curves measured by the angle-resolved micro-photoluminescence spectrometer unambiguously shows the presence of the exciton polaritons. Moreover, the optical modes of the exciton polaritons can be manipulated by changing the pump position. Numerical simulation by FDTD shows that when pumping at the center, the optical modes that couple with the excitons are the qWGMs; when pump off-center, the optical modes become what we call the 2rqWGMs, which have twice the close-loop optical length and half the mode spacing as the qWGMs. When coupled with the excitons, they also have a much larger Rabi splitting due to their lower photon decay rate and the non-degenerate light path which overlaps more with the GaN material. For the first time, these 2rqWGM exciton polaritons are observed and identified. The large Rabi splitting of the polariton modes renders them distinguishable from the cavity photons. This pronounced half-matter half-light characteristic at room temperature in the strong-coupling regime shows great potential in prospective polaritonic devices. Last but not least, research on detuning is relatively easy in microrod samples as the diameter can change slightly along the rod, and the tuning can be readily achieved by changing the pumping position. The tuning of the exciton modes can be done by applying pressure [54,55] or magnetic fields. The magnetic field can also be introduced to probe the polaritonic character of emission modes at various detunings by taking advantage of the Zeeman effect [56]. Future polariton studies on detunings in GaN can benefit from the large Rabi splitting we observed, as the less strict strong-coupling regime provides a wider energy range for researchers to tune the modes.
Funding
National Natural Science Foundation of China (61874004); National Key Research and Development Program of China (2017YFB0405000, 2017YFB0405001); Beijing Municipal Science and Technology Commission (Z201100004520004, Z201100006820081, Z201100006820137).
Acknowledgments
The authors would like to thank Prof. Zhanghai Chen and his research group members at Fudan University for providing help with the angle-resolved micro-photoluminescence measurements.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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