1. Introduction

The realization of optical nanocavities possessing a high-quality factor ($Q$-factor) is of great interest in various fields, such as biosensors [1], low threshold nanolasers [2,3] and quantum information technologies [4–7]. Especially, two-dimensional photonic crystal (2D-PhC) cavities, which exhibit a high-$Q$-factor and a small modal volume (V), are promising structures for realizing such novel devices with a small footprint [8]. So far, a large part of research in 2D-PhC cavities has been conducted in the infrared regime using matured semiconductors (e.g., Si) and high-$Q$-factors exceeding 1×10⁶ have been reported [9,10]. Recently, III-nitride-based 2D-PhC cavities have attracted attention for various applications like monolithic laser displays [11] and nonlinear wavelength conversion devices [12,13]. Recently, $Q$-factors exceeding 10000 have been reported in the infrared region [12–14], however, $Q$-factors in the ultraviolet-visible regime are still limited to about 5000 [11,15–23]. Contrarily to cavities designed for the infra-red range, $Q$-factors of III-nitride-based cavities designed for the visible range are strongly limited by various optical losses. Previous studies have suggested that the Q-factor of III-nitride-based cavities are predominantly limited by light absorption of bulk [24], gain medium [quantum well (QW)] [25] and surface [26] for microdisk cavities, structural disorder originating from fabrication imperfections [27], and scattering losses due to surface roughness [27,28] for 1D-nanobeam cavities. Based on these studies various techniques have been utilized to reduce the losses, for example, surface passivation [26] and suppression of light absorption by coupling resonant peaks to longer wavelength region of QW emission [27]. Due to these contributions, a high $Q$-factor of 7950 for a 1D-nanobeam cavity [26], and 10200 for a microdisk cavity [29] have been demonstrated in the visible range. However, the dominant loss of III-nitride-based 2D-PhC cavity has yet to be clarified. In order to improve $Q$-factors of 2D-PhC cavities, it is thus essential to determine the dominant loss process and consider measures to suppress it.

It is well known that air-hole-shifts and changes in size of air-holes near cavities greatly suppress optical losses and contribute to improved $Q$-factors of 2D-PhC defect-type cavities [30]. A high $Q$-factor of 16900 has been reported for a GaN-based L3 cavity in the infrared regime by shifting and shrinking air-holes [14]. However, in-plane disorder (variation of shape and size of air-holes, or periodicity of the PhC), which lead to light scattering [31] and localization [32] in PhC waveguides, drastically decrease $Q$-factors of 2D-PhC cavities particularly for the visible range [33,34]. Very recently, we have investigated the $Q$-factor degradation originating from in-plane disorder in the visible range using finite-difference time-domain (FDTD) simulations. We have reported that hexagonal-defect (HN) cavities, especially H3 cavities, show a more robust stability of $Q$-factor against in-plane disorder as compared to line-defect (LN) cavies [34].

Another approach to achieve high $Q$-factors is to employ 1D-heterostructures formed by modulating air-hole radius or periodicity [35]. However, as it is known that in-plane disorder strongly degrades the $Q$-factor of 1D-heterostructure nanocavities [36], it is not expected to be a successful approach for III-nitrides. Recently, Ge et al. have focused on a 2D-heterostructure which supports a band-edge mode and observed a high $Q$-factor of ∼15000 in the red spectral range using a silicon nitride film deposited on a quartz substrate [37]. Such 2D-heterostructures have a high possibility to show a better robustness to in-plane disorder than 1D-heterostructures, in a similar way as 2D HN cavities show a better robustness to in-plane disorder than line-defect 1D-cavities [34].

In this paper, we focus on two types of structure (H3 cavity and 2D-heterostructure) in order to achieve high $Q$-factors. We present the fabrication and optical characterization and discuss the dominant optical losses for III-nitride-based 2D-PhC cavities in the visible range and demonstrate measures for suppressing such losses to achieve high $Q$-factors.

2. Sample fabrication and characterization method

We used Eu,O-codoped GaN (GaN:Eu,O) [38], which emits red light (λ ∼ 621 nm), as an active layer. The samples in this work were grown on a (0001) sapphire substrate using the organometallic-vapor-phase epitaxy method. The sample growth was initiated with a low-temperature GaN buffer layer, followed by an undoped-GaN (2.7 µm), an Al_0.82In_0.18N sacrificial layer (600 nm), an Eu-doped GaN active layer (220 nm) and a GaN cap layer (10 nm). More details on the growth conditions can be found in Ref. [11]. We fabricated both H3 cavities and 2D-heterostructures on an identical sample.

We designed 2D-PhC cavities with hexagonal air-holes considering the wurtzite structure of GaN [11]. To fabricate 2D-PhC cavities, a 200-nm-thick SiO₂ layer was first evaporated as hard mask and electron-beam resist (ZEP520A, Zeon Corp.) was spincoated subsequently. Triangular 2D-PhC patterns were defined on the resist by electron beam lithography at 125 kV. After development, the 2D-PhC patterns were transferred to SiO₂ mask by CHF₃/Ar reactive ion etching (RIE), and GaN:Eu,O and AlInN layers were etched by inductively coupled plasma RIE with Cl₂ plasma. To form hexagonal air-holes with flat $\{1\overline{1}00\}$ sidewalls, the samples were chemically etched for 1 min at 85 °C in a tetramethylammonium hydroxide (TMAH) aqueous solution with a concentration of 13 wt.%. Subsequently, the Al_0.82In_0.18N sacrificial layer was selectively removed in hot nitric acid (2 mol/L) for 7 h at 120°C.

Microscopic photoluminescence (PL) measurements were performed at room temperature to evaluate the optical properties of the 2D-PhC cavities. A continuous-wave He-Cd laser (λ = 325 nm) was used as excitation source. The laser was focused on the sample with a spot diameter of 1.6 µm through an objective lens (50×, NA = 0.42). The luminescence was collected by the same objective lens, dispersed by a 50-cm-spectrometer and detected by a Peltier-cooled CCD camera. The spectral resolution is about 0.02 nm.

A microscope with a spatial resolution of ∼4 µm was used for spatial mapping of the cavities. A He-Cd laser guided through an objective lens (40×, NA = 0.47) was used to excite samples with a spot diameter of ∼3.2 µm. The luminescence was collected through the objective and was detected by a Peltier-cooled CCD camera on an 80-cm-spectrometer.

To numerically calculate the time evolution of electromagnetic field, we used commercially available Lumerical 3D-FDTD solutions, which directly solves the Maxwell’s equation in time. All simulations were conducted using the perfectly matched layer boundary conditions applied to four sides normal to the membrane, and symmetric boundary applied along the membrane. The 2D-PhC slab was composed of GaN with a thickness of 220 nm. The refractive index of GaN was set to 2.35, which is a typical value in the red region (615–630 nm) [39]. The FDTD grid size was set to less than 1/24 of the wavelength, and the time step was set to 0.022 fs in order to satisfy the stability condition. A point dipole source with transverse-electric polarization was set at the center of cavity. Defect-type cavities were surrounded by triangular-lattice PhCs with 48 rows and columns unless otherwise noted, which are large enough to obtain accurate simulation results.

3. Results and discussions

As discussed above, the experimental $Q$-factor (${Q_{exp}}$) differs from the theoretical ideal $Q$-factor (${Q_{id}}$) that is purely determined by the light confinement characteristics of a non-disordered 2D-PhC cavity structure following the empirical formula:

3.1 Structural characterization

Prior to the sample fabrication, we designed H3 cavity structures with the resonant wavelength tuned to match Eu³⁺ luminescence. We simultaneously modified both radius (r) and the periodicity of the 2D-PhC (a) because the resonant wavelength blue-shifts when the radius r is increased. Figure 1 (a) displays the designed r/a ratio and r is in a linear relationship.

 

Fig. 1. (a) Designed radii (r) of air-holes as a function of the designed r/a, where a is the periodicity of 2D-PhC. (b) SEM image of a fabricated H3 cavity with r/a of 0.31. (c) Fabricated values of r/a and (d) experimentally introduced σ as function of the designed r/a.

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Figure 1 (b) shows a scanning-electron microscope (SEM) image of a fabricated H3 cavity with a of 209 nm, r of 65 nm and r/a of 0.31. The fabricated r/a well matched with the designed r/a [Fig. 1 (c)]. Although hexagonal air-holes with flat sidewalls are obtained due to the TMAH-based etching process, in-plane structural disorder coming from variations in the hole radius is observed, which is known to degrade $Q$-factors [31–34]. We characterized the size-dependence of this degree of disorder from SEM images. As depicted in Fig. 1 (d), the degree of disorder σ (the relative standard deviation of r, assuming a Gaussian distribution [34]) becomes smaller for larger values of r/a because the absolute disorder value Δr is nearly constant (1.4–1.7 nm).

3.2 Optical properties of H3 cavities

We performed PL measurements for H3 cavities with a variety of r/a to investigate the impact of the degree of disorder on the $Q$-factor. Due to the appropriately fabricated r and a (r/a was varied from 0.23 to 0.34), the resonant wavelength of each H3 cavity was in the range from 618 to 629 nm and coupled with Eu³⁺ luminescence (⁵D₀→⁷F₂) [ Fig. 2 (a)]. Two peaks appeared for each cavity corresponding to the fundamental mode of an H3 cavity with different optical polarization orientation. Although these are originally degenerate, they split due to in-plane-disorder [40].

 

Fig. 2. (a) PL spectra of H3 cavities with r/a values of 0.27, 0.29 and 0.31. The black line is a 1.6-times enlarged PL spectrum of an unpatterned GaN:Eu,O film. (b) PL intensity ratio of an H3 cavity with r/a of 0.27. The solid lines indicate the fitted Lorentzian curves of two peaks. (c) Full polarization-resolved normalized PL intensities of the resonant peaks shown in (b). Calculated electric field distribution for the fundamental modes of an H3 cavity polarized in the (d) axis1 [0°] and (e) axis2 [150°] of a hexagonal H3 cavity structure. The scale-bar corresponds to 500 nm.

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Figure 2 (b) depicts the PL enhancement spectrum of an H3 cavity with r/a of 0.27, and the resonant mode at the longer wavelength showed the highest $Q$-factor in this work (7900), which is ∼1.5-times larger than the highest value of III-nitride-based 2D-PhC cavities in the visible ranges ($Q$∼5400 at ∼624 nm [11]). This result supports our suggestion that an H3 cavity is a promising structure for the realization of high $Q$-factors with III-nitride materials [34].

The resonant mode at the shorter wavelength (Peak1) showed a lower $Q$-factor (5200). To clarify the origin of the difference in $Q$-factors between two modes, we investigated the polarization axis of each mode by a PL measurement using a polarizer. Figure 2 (c) presents the polarization-resolved PL intensity, and each peak shows different polarization axis. Using FDTD simulations, we have confirmed that the H3 cavity mode splits into two modes polarized in the axis1 [0°, 60°, 120°, Fig. 2 (d)] and axis2 [30°, 90°, 150°, Fig. 2 (e)] direction of a hexagonal H3 cavity structure. Besides, the mode polarized in the axis1 typically shows a higher $Q$-factor in simulations, which may be due to a smaller interaction between the electric field and the adjacent air-holes resulting in a smaller optical loss. These results suggest that the experimentally observed difference in $Q$-factors is due to the difference of the polarization orientation.

3.3 Dominant optical losses of H3 cavities

Figure 3 shows $Q$-factors of the mode polarized in the axis1, and all cavities show a high ${Q_{exp}}$ (>5800), with an average ${Q_{exp}}$ of ∼7000. Furthermore, we investigated the dominant optical loss limiting $Q$-factors of H3 cavities using the following relationship:$Q_{exp}^{ - 1} = Q_{id}^{ - 1} + Q_{in}^{ - 1} + Q_{other}^{ - 1}\;({Q_{other}^{ - 1} = Q_{out}^{ - 1} + Q_{ba}^{ - 1} + Q_{sa}^{ - 1} + Q_{sc}^{ - 1}} )$. We calculated ${Q_{in}}$ by introducing in-plane-disorder estimated from SEM images [Fig. 1 (b)] into the simulations [34], and calculated ${Q_{other}}$ with above relationship.

 

Fig. 3. ${Q_{id}}$, ${Q_{in}}$, ${Q_{exp}}$, and ${Q_{other}}$ as a function of r/a of H3 cavities. ${Q_{exp}}$ depicts experimentally observed $Q$-factors. ${Q_{in}}$ and ${Q_{id}}$ were calculated using FDTD simulations while Q_other was derived from Eq. (1).

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In Fig. 3, all determined $Q$-factors are given as a function of r/a. While ${Q_{id}}$ decreases for larger r/a, ${Q_{exp}}$ is almost constant over the whole range of r/a. Interestingly, ${Q_{other}}$, which represents out-of-plane disorder, scattering and absorption, slightly increases with r/a. If the scattering was the dominant contribution, it should give a nearly constant contribution. Furthermore, the GaN:Eu,O material applied in this study shows a small bulk absorption coefficient (α ∼ 3.9 cm^-1 at 622 nm) [41]. This corresponds to an estimated ${Q_{ba}}$ of ∼ 6×10⁴, following the relationship: $Q_{b a}=2 \mathrm{\pi} n_{e f f} / \alpha \lambda$ [42], where ${n_{eff}}$ indicates the effective refractive index and is estimated as 2.31. Furthermore, we have confirmed that the ratio of |E|² confined in the PhC membranes is nearly constant (0.87–0.90) for values of r/a ranging from 0.23 to 0.33, therefore the r/a dependence of ${Q_{sa}}$ can be safely neglected. Thus, this trend suggests that ${Q_{other}}$ is mostly determined by ${Q_{out}}$ [43]. This is likely related to the fact that a smaller radius of air-holes results in a larger taper angle of sidewalls [44]. We believe that the large fluctuation in the ${Q_{other}}$ is due to the non-uniformity of the taper angle of sidewalls. Especially, the taper angle of the nearest air-holes from the cavity should have the strong impact on $Q$-factors due to a relatively large overlap of the electric field and the sidewalls.

As a result of the high robustness of H3 cavities to in-plane disorder, the experimental $Q$-factor approaches the theoretical $Q$-factor and is the predominant limiting factor. It is thus imminent to design cavities with larger ${Q_{id}}$, while maintaining the high ${Q_{in}}$ of an H3 cavity.

3.4 Design of 2D-heterostructure

In order to achieve a high ${Q_{id}}$ while simultaneously having a high robustness to in-plane disorder, we hereafter focus on 2D-heterostructures with reference to the research demonstrated by Ge et al. [37].

To form a high-$Q$ 2D-heterostructure, it is preferable to employ the band-edge mode at the K point of the first band (K1) because it supports a full in-plane band gap, and can be expected to have good light confinement [ Fig. 4 (a)]. The inset in Fig. 4 (a) shows an out-of-plane-polarized magnetic field distribution at the K1 point, and the white lines indicate the outline of air-holes.

 

Fig. 4. (a) Calculated band diagram of a PhC in different layers (${r_c}$= 0.38a, ${r_t}$= 0.36a, ${r_o}$= 0.34a). The right panel shows an enlarged diagram around the K point, and the shaded area indicates a region where the K1 mode in the core region is prohibited to couple with the same mode in the outer region. The inset shows a z-polarized magnetic field distribution at the K1 point, white lines indicate the outline of air-holes. Schematic of a (b) CHS and (c) DHS, consisting of clad, transition, and outer layers. Calculated electric field distribution of K1 mode in (d) CHS and (e) DHS. The dot lines represent the boundaries between each range. The scale-bar corresponds to 3 µm.

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The 2D-heterostructure we propose consists of ellipse shaped core, transition, and outer region, with air-hole radii of ${r_c}$, ${r_t}$, and ${r_o}$, respectively. By designing the radii as ${r_c}$ > ${r_t}$ >${r_o}$, light in the shaded area in Fig. 4 (a) is well confined in the core region. As shown in Fig. 4 (b), we defined the size of the regions by the number of air-holes in the major axis of the ellipse (${N_c}$, ${N_t}$, and ${N_o}$), where the regions are described as; core region:${i^2} + {j^2} \le N_c^2$, transition region:${i^2} + {j^2} \le {({{N_c} + {N_t}} )^2}$, and outer region:${i^2} + {j^2} \le {({{N_c} + {N_t} + {N_o}} )^2}$. Integer $i$ and $j$ define the layer number for x (major axis of the ellipse) and y (minor axis) direction, ${x_{i + 1}} - {x_i} = a$ and ${y_{i + 1}} - {y_i} = \sqrt 3 a/2$. We name this structure conventional heterostructure (CHS). In simulations, the CHS shows a great light confinement in the core region [Fig. 4 (d)] and a high ${Q_{id}}$ of 7.1×10⁵ (${r_c}$= 0.27a, ${r_t}$= 0.23a, ${r_o}$= 0.20a, ${N_c}$= 14, ${N_t}$= 11, ${N_o}$= 6), which is at least one order of magnitude larger than the reported ${Q_{id}}$ for a similar heterostructure utilizing the Γ point of the first band [37], despite the number of air-holes constructing the PhC resonator is only 1/3–1/4 of that used in Ref. [34]. This high $Q$-factor is due to the strong light confinement at the K1 point [45].

With the robustness to in-plane disorder in mind, we also designed another type of structure shown in Fig. 4 (c), having a hexagonal defect consisting of 19 missing air-holes (identical to an H3 cavity) at the center. We expected this structure shows a better robustness because it has less air-holes in the core region that could lead to in-plane disorder. We name this structure defected heterostructure (DHS). DHS supports both the K1 band-edge mode and defect modes of an H3 cavity, such as the fundamental modes shown in Fig. 2 (d) and (e). For a K1 mode, we found the electric field is well confined in the core region around the defect [Fig. 4 (e)] showing a ${Q_{id}}$ of 1.1×10⁵ (${r_c}$= 0.27a, ${r_t}$= 0.23a, ${r_o}$= 0.20a, ${N_c}$= 14, ${N_t}$= 11, ${N_o}$= 6).

In the next step, we investigated the robustness to in-plane disorder for the K1 modes of CHS and DHS, and the H3 fundamental mode shown at the central of DHS, by introducing in-plane disorder into simulations and changing the σ value.

In Fig. 5, K1 modes in both CHS and DHS show higher $Q$-factors than an H3 fundamental mode even for large value of σ, suggesting the high robustness to in-plane disorder. Moreover, the K1 mode of DHS shows a slightly higher robustness than that of CHS. With a disorder of σ = 0.13, the K1 mode of DHS shows $Q\;[{{{({Q_{id}^{ - 1} + Q_{in}^{ - 1}} )}^{ - 1}}} ]$ ∼ 21000, which is 1.2-times larger than that of CHS ($Q$ ∼ 18000), and calculated values of ${Q_{in}}$ as 25000 and 18000 for DHS and CHS, respectively. As abovementioned, a smaller number of air-holes is expected to give a higher robustness in DHS. These results suggest that the K1 mode of DHS is a promising mode in order to achieve high $Q$-factors using III-nitride materials.

 

Fig. 5. (a) Calculated $Q$-factors $[{{{({Q_{id}^{ - 1} + Q_{in}^{ - 1}} )}^{ - 1}}} ]$ for the K1 mode in CHS and DHS, and the fundamental mode of the H3 cavity in DHS, as a function of σ value.

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3.5 Optical properties of 2D-heterostructures

Based on the simulation results, we fabricated a DHS (${r_c}$= 0.27a, ${r_t}$= 0.23a, ${r_o}$= 0.20a, ${N_c}$= 14, ${N_t}$= 11, ${N_o}$= 6). The SEM image shown in Fig. 6 (a) reveals that air-holes have the same radii as designed, and the estimated disorder degree introduced to the core region is σ ∼ 0.13. Next, we conducted PL measurements for this sample at room temperature, PL spectra for DHS and GaN:Eu,O film are depicted in Fig. 6 (b). The spectrum from a DHS shows many resonant peaks which originate from band-edge modes coupled to Eu³⁺ luminescence (⁵D₀→⁷F₃ transition). Because higher order band-edge modes appear at k-values somewhat away from the K point, modes appearing in the longer wavelength region correspond to higher order modes. The fundamental mode of the H3 cavity was not observed because it did not couple to Eu³⁺ luminescence.

 

Fig. 6. (a) SEM image of a fabricated DHS (${r_c}$= 0.27a, ${r_t}$= 0.23a, ${r_o}$= 0.20a, ${N_c}$= 14, ${N_t}$= 11, ${N_o}$= 6). (b) PL spectra for a DHS and an unpatterned GaN:Eu,O film, the peaks correspond to the band-edge modes. (c) Enlarged PL spectra for a DHS and an unpatterned GaN:Eu,O film. The inset shows the PL intensity ratio, the solid lines indicate the fitted Lorentzian curves of ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_2^{1{\rm{st}}}$. Micro-PL spatial mapping of the (d) ${\rm{P}}_1^{1{\rm{st}}}$ peak and (e) ${\rm{P}}_2^{1{\rm{st}}}$ peak. The scale-bar corresponds to 3 µm.

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At the shorter wavelength range (660–665 nm), four large peaks labelled as ${\rm{P}}_1^{1{\rm{st}}}$, ${\rm{P}}_2^{1{\rm{st}}}$, ${\rm{\;P}}_1^{2{\rm{nd}}}$, and ${\rm{P}}_2^{2{\rm{nd}}}$ are clearly observed [Fig. 6 (c)]. Because there are no resonant peaks for wavelengths shorter than ${\rm{P}}_1^{1{\rm{st}}}$, it can be identified as the fundamental K1 mode. In order to determine the origin of ${\rm{P}}_1^{1{\rm{st}}}$, ${\rm{P}}_2^{1{\rm{st}}}$, ${\rm{\;P}}_1^{2{\rm{nd}}}$, and ${\rm{P}}_2^{2{\rm{nd}}}$, we focus on the spectral spacing between each mode. Compared to the calculated spectral spacing between the fundamental and second order K1 mode (2.66 nm), the spacing between ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_2^{1{\rm{st}}}$ (0.43 nm) is too small. On the contrary, the spacing between ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_1^{2{\rm{nd}}}$ (2.17 nm), ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_2^{2{\rm{nd}}}$ (2.64 nm) well match with the simulation result. Furthermore, the calculated spacing between the second and third order mode is 3.66 nm, which is much larger than that between ${\rm{P}}_1^{2{\rm{nd}}}$ and ${\rm{P}}_2^{2{\rm{nd}}}$ (0.47 nm). Thus, we conclude that ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_2^{1{\rm{st}}}$ correspond to the fundamental K1 mode while ${\rm{P}}_1^{2{\rm{nd}}}$ and ${\rm{P}}_2^{2{\rm{nd}}}$ are the second order modes. From micro-PL spatial mapping results for ${\rm{P}}_1^{1{\rm{st}}}$ and ${\rm{P}}_2^{1{\rm{st}}}$ [Fig. 6 (d), (e)], it can be seen that each mode has its maximum PL intensity at a different location. For this reason, we assume that these mode-splittings are caused by a field nonuniformity caused by in-plane disorder.

The determined $Q$-factors of each mode (${\rm{P}}_1^{1{\rm{st}}}$-${\rm{P}}_2^{2{\rm{nd}}}$) were 9400, 10500, 9300, and 10300, which surpass the highest $Q$-factor of H3 cavities observed in this study. The inset in Fig. 6 (c) ensures the accuracy of the $Q$-factor estimation because each peak is resolved by several pixels. From these results we confirmed that the K1 resonant mode of a DHS is a promising mode for achieving high $Q$-factors using III-nitride-based 2D-PhC cavities.

The value of ${Q_{exp}}$ of ${\rm{P}}_2^{1{\rm{st}}}$ (10500) matches well with the predicted value of 9400 from the simulations using the value of ${Q_{id}}$ = 110000, ${Q_{in}}$= 21000 (σ = 0.13, Fig. 5), and ${Q_{other}}$ of ∼20000 (r/a = 0.27, Fig. 3), which suggests that ${Q_{exp}}$ of the K1 mode in DHS is dominated by ${Q_{other}}$ and ${Q_{in}}$. Further optimization of the cavity design (e.g., size of the central defect, number of layers) could lead to a further improvement of ${Q_{in}}$. However, ${Q_{exp}}$ will be strongly limited by the relatively small value of ${Q_{other}}$. Thus, in order to achieve still larger $Q$-factors, further improvement of taper angle and roughness of sidewalls is required, for which we think the advancement of the wet-etching technique based on TMAH solutions plays a key role. We also note that by utilizing higher order band-edge modes it is possible to increase feature sizes and suppress disorder-related losses.

For applications where light-matter interactions are required, an H3 cavity is a more promising structure when compared to the DHS, because it possesses a large $Q/V$ ratio. A further advancement of the structure design, for example in the shift and size modulation of air-holes, could improve the $Q$-factor more and open up a larger application potential.

However, the great design flexibility of the 2D-heterostructures and their already demonstrated high $Q$-factors for a wide range of structure designs, makes them good candidates for applications for which higher $Q$-factors are essential.

4. Conclusions

We have systematically analyzed the influence of various sources of optical losses in 2D photonic crystal nanocavities based on GaN in the red wavelength regime by a combination of FDTD simulations and experimental results. We found that for an H3-type cavity, which shows a good robustness against in-plane disorder, there is an influence of the hole radius and lattice constant of the photonic crystal due to limitations in the fabrication accuracy. However, the most important limitation to realize even higher values of the $Q$-factor is the theoretical value of the design itself. Our best performing cavity reached a value $Q$ = 7900, where the theoretical maximum is 12000. Based on this knowledge, we designed 2D-heterostructure type of cavity, which exhibits a band-edge mode with higher theoretical $Q$-factor, and shows a higher robustness to in-plane disorder compared to the H3 cavity. Based on this design we realized a much higher experimental $Q$-factor of 10500, demonstrating the potential of this type of cavity. These results show the potential for application of III-nitride-based 2D-PhC cavities in the visible regime.