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GaN microwires grown by metalorganic vapour phase epitaxy and with radii typically on the order of 1-5 micrometers exhibit a number of resonances in their photoluminescence spectra. These resonances include whispering gallery modes and transverse Fabry-Perot modes. A detailed spectroscopic study by polarization-resolved microphotoluminescence, in combination with electron microscopy images, has enabled to differentiate both kinds of modes and determined their main spectral properties. Finally, the dispersion of the ordinary and extraordinary refractive indices of strain-free GaN in the visible-UV range has been obtained thanks to the numerical simulation of the observed modes.
©2012 Optical Society of America
1. Introduction
Optical microcavities are dielectric structures that confine photons within a certain region of space and whose size, at least in one direction, is on the order of the photon wavelength inside the material (i.e. the wavelength in vacuum divided by the refractive index). Thus, for a material family such as AlGaInN, emitting from the near-IR to the UV, optical microcavities span a size range going from several micrometers to some tens of nanometers.
The most popular GaN-based microcavities are planar Fabry-Perot ones fabricated by stacking the nitride emitter, either bulk material or quantum heterostructures such as quantum wells or quantum dots, in between two distributed Bragg reflectors. The main interest in this kind of cavities has been driven by the possibility of fabricating vertical cavity surface emitting lasers operating at room temperature first optically pumped [1] and, ultimately, electrically injected [2].Furthermore, in the last years the interest in cavities of larger quality factors (i.e. in which the photon lifetime is longer) has been prompted by the proposal for a new kind of laser based on microcavity polaritons [3] for which the laser threshold might be lower. Indeed, room-temperature polariton lasers based on GaN [4] or AlGaN/GaN [5] quantum wells were recently reported. The fabrication of such planar cavities, irrespective of the final device, relies on complex and time-consuming epitaxial growth techniques that are not yet as mature as those used in other semiconductors such GaAs or InP. An alternative and, in principle, easier approach to fabricate GaN cavities consists in exploiting the phenomenon of total internal reflection. In this case there is no need for distributed Bragg reflectors but just for a large enough dielectric discontinuity between two media, e.g. the medium that contains the emitter and air [6]. In the late nineties, GaN-based microdisks were first fabricated [7] and rapidly displayed room-temperature pulsed-mode lasing [8]. In the meantime the improvement of dry and chemical etching techniques adapted to GaN has enabled an effective decoupling of the microdisk from the substrate, by creating an air-gap between them, and the achievement of smoother lateral facets [9, 10]. These two developments have led to whispering-gallery modes (WGMs) with quality factors as large as 7000 for current state-of-the-art nitride microdisks [11], and have paved the way for the first demonstration of room-temperature CW lasing in a GaN/InGaN microdisk [9]. However, even if processing methods improve continuously, the fabrication of such microobjects is still extremely challenging.
Very much similar to GaN in terms of physical properties is ZnO, which is often presented as an alternative to it in many applications. An interesting feature of ZnO is its tendency to form nanostructures, in particular nano- and microwires [12]. Among all these objects, microwires are extremely interesting in the context of optical microcavities as they can sustain WGMs [13], similarly to microdisks, and can work both in the weak- [14] and in the strong-coupling regimes [15].
Inspired by the numerous results in ZnO, we propose to use GaN microwires as an alternative to other nitride-based optical microcavities. In this work we will focus on their linear optical properties within the weak-coupling regime.
2. Experimental details
The GaN microwires were grown by metalorganic vapour phase epitaxy on sapphire (0001) substrates as described in more detail in [16]. It is important to note that in order to get a fast growth rate along the c-axis and form microwires it is necessary to introduce silane during the initial growth steps. This strong Si doping results in typical electron concentrations in the order of ~1020 cm−3, as estimated from the correlation between the full width at half maximum (FWHM) of the near band-edge emission and the carrier concentration [17].Due to the growth along the <0001> direction (Fig. 1(a) ), these microwires present a hexagonal cross-section although deviations from a perfect hexagon are often observed, as discussed later. For their optical characterization, the microwires were dispersed onto an oxidized Si substrate and lie with their longitudinal axis parallel to the sample surface, as shown in Fig. 1(b).
Fig. 1 SEM images of typical GaN microwires viewed at 30° (a) and 90° (b) with respect to the wire longitudinal axis. Schematic representation of WGMs (c) and transverse Fabry-Perot modes (d).
Room-temperature microphotoluminescence (μ-PL) experiments were carried out by exciting the microwires with a CW Ar-doubled laser (excitation wavelength: 244nm) and collecting via a confocal microscope setup enabling micrometer-range spatial resolution. The numerical aperture (NA) of the microscope objective is 0.4, which implies that light rays inclined up to about 23.5° with respect to the microwire normal will be detected. A linear polarizer placed in the collection path allowed polarization-resolved measurements with TE-polarized (TM-) PL corresponding to an electric field perpendicular (parallel) to the microwire axis. Once the optical characterization had been performed, each individual microwire was observed in a scanning electron microscope (SEM) to determine precisely its size and its exact cross-section shape.
3. Microwire resonators
3.1 Whispering-gallery modes
The unpolarized PL spectrum of a microwire with a 1.21μm radius is shown in Fig. 2(a) . It consists of a broad band centered at about 2.25 eV, which is due to the emission from point defects (commonly referred to in the literature as the yellow-band), and a near band-edge emission due to electrons recombining with localized holes [17]and whose broadening is mainly determined by the strong Si doping. It is noteworthy that these two broad bands, especially the visible band (which is the focus of this article), are modulated by a number of peaks evenly spaced in energy and whose density depends on the microwire radius, with larger radii leading to smaller energy spacing between two consecutive maxima (compare wires of 1.21μm and 3.25μm radii in Fig. 2(a)). Moreover, the polarization properties of the microwires emission (Fig. 2(b)) show that it consists of two orthogonal contributions corresponding to polarizations parallel and perpendicular to the microwire axis and with an energy shift between them. These two observations point towards the presence of WGMs, similar to those observed in microdisks.
Fig. 2 Room-temperature (RT) unpolarized μ-PL spectra of microwires with different radii (a), and RT polarization-resolved μ-PL spectra of the microwire with radius 3.25μm (b). The inset in (b) shows an enlarged view of the highlighted area. Energy of the μ-PL maxima (black squares for TE and red circles for TM) for microwires with different radii as a function of the interference order (c). The lines correspond to the WGM dispersion given by Eq. (1).
In the limit of small wavelengths compared to the resonator size, i.e. λ<<R, with R the radius of the circle circumscribing the hexagonal cross-section, the energy (E) of the WGMs resonances can be accounted for by [13, 14]:
where Ri is the microwire in circle radius, i.e. , h and c are Planck’s and the speed of light constants, respectively, n the energy-dependent refractive index, M the interference order and β a polarization-dependent term. For TM polarization, β = n−1, whereas for TE polarization β = n; furthermore, given the uniaxial character of the Wurtzite structure the ordinary (extraordinary) refractive index n = nord (next) should be used for TE (TM) polarization. Equation (1) establishes, therefore, a correspondence between the energy position of the WGMs and the associated refractive index values since for each microwire the geometrical parameters (R) are known thanks to scanning electron microscopy images.Whereas Eq. (1) accurately describes the WGMs resonances in the visible range, as can be seen in Fig. 2(c), two different phenomena limit its applicability elsewhere. On the one hand when the WGMs approach the near-band edge region, a strong interaction between these localized photonic modes and the free-excitons might take place leading eventually to the formation of microcavity polaritons, whose description is out of the scope of the present paper [15, 18].A second limit exists due to the approximation of short wavelengths (compared to the microwire radius) assumed in Eq. (1), which limits its applicability to microwires with radii larger than about 1μm. Nevertheless, it should be noted that these limits are rather diffuse and that, under certain conditions, Eq. (1) can still account for the modes dispersion out of these two limits.
3.2 Transverse Fabry-Perot modes
If we consider the hexagonal cross-section of a microwire and a plane wave bouncing back and forth between the facet lying on the substrate and the opposite one, as depicted in Fig. 1(d), we can also expect the presence of Fabry-Perot modes. Indeed, if we take into account that the microwires have a radius much larger than the photon wavelength, we can see the structure as an infinite GaN slab sandwiched between a Si substrate and air [19]. In this case, the energy position of the PL maxima is simply given by:
where M is again the interference order, d is the Fabry-Perot cavity thickness, i.e. , n is the energy dependent refractive index and E is the peak energy. Again the ordinary and extraordinary refractive indices should be introduced depending on the studied polarization.Figure 3(a) shows the room temperature polarization-resolved μPL spectra of a microwire with a radius R = 3.39μm. While the spectra resemble much those of Fig. 2(a) and 2(b), with two broad bands modulated by a series of narrower peaks and a slight shift between the two orthogonal polarizations, a marked difference appears in terms of mode density, even when comparing microwires of similar radii (3.37μm instead of 3.39μm). In fact, if we try to fit the overall position of the intensity maxima in Fig. 3(a) with Eq. (1) we need to introduce microwire radii largely different from the measured ones. On the other hand, if we use the measured radius and we compute the energy of the modes as a function of the interference order for WGMs, as shown in Fig. 3(b) by the dashed lines, we observe clear discrepancies in the slope of the experimental data and of the simulated curves. Indeed, Eq. (1) and (2) have been written in a form so as to emphasize the different slopes of both kinds of modes, which is in the order of ~1.5 (if we neglect the second term in Eq. (1)).
Fig. 3 RT polarization-resolved μ-PL spectra of a microwire with R = 3.39μm (a). The inset shows an enlarged view of the Yellow-band region. Energy of the μ-PL maxima (black crossed squares for TE and red crossed circles for TM) as a function of the interference order (b). The lines correspond to the Fabry-Perot dispersion given by Eq. (2), whereas the dashed lines correspond to the WGMs dispersion given by Eq. (1) for the microwire of similar radius (R = 3.37μm) shown in Fig. 2(c). The Fabry-Perot resonances and fits have been shifted in M ( + 24 for TE modes and + 25 for TM modes, respectively) in order to compare them to the WGM dispersion.
Instead, Eq. (2) reproduces the modes position with high accuracy as well as the slope of the curve energy as a function of the interference order, pointing towards the presence of transverse Fabry-Perot modes rather than WGMs. It should be noted that, strictly speaking, longitudinal Fabry-Perot modes, i.e. established between the end facets across the length of the microwires, would be also possible. However, their presence in the studied microwires can be discarded due to the poor reflectivity of the microwire bottom facet (we recall that the microwires were mechanically separated from the original substrate). Furthermore, even if the reflectivity of these facets had been large enough, the size of the microwires along this direction (>10μm) would have resulted in a mode density larger, by at least a factor three, than the maximum studied in this work, which corresponds to the microwire shown in Fig. 2(b).
3.3 WGMs Vs Fabry-Perot modes: FWHM
From the theoretical point of view, for a given wavelength (energy) the FWHM of the resonances should decrease with increasing radii and this for both, TE- and TM-polarized modes, and for both kinds of modes, WGMs and Fabry-Perot. Furthermore, for Fabry-Perot modes the FWHM decreases as R [20] while for WGMs the FWHM decreases as R2 [21]. This stronger dependence on microwire radius for WGMs, compared to Fabry-Perot modes, stems from the combination of wave effects and the presence of corners (compare theoretical treatments on [13] and [22]).
All these trends can be found in Fig. 4(a) and 4(b), where the FWHM of modes around 2.35 eV have been plotted for all the studied microwires in which WGMs and Fabry-Perot modes could be unambiguously identified. From our current data, based on more than 30 studied microwires, it is difficult to state under what conditions we observe WGMs only, Fabry-Perot modes only or both concomitantly. Indeed, several microwires, with radii typically around 1.2-1.5μm, exhibit a density of modes that is too large, even for WGMs, and whose positions are neither fitted by Eq. (1) nor by Eq. (2) alone. Consistent with a recent study on ZnO [23],in these intermediate-size microwires both kinds of modes could be visible simultaneously and even quasi-WGMs [24]. Smaller microwires would tend to show mainly WGMs and larger microwires would favor the observation of Fabry-Perot modes (see Fig. 4(a) and (b)). Furthermore, for a given microwire radius, the observation of Fabry-Perot modes rather than WGMs seems to be favored by the deformation of the hexagon as can be seen in Fig. 4(d), where the hexagon is elongated parallel to the substrate surface.
Fig. 4 FWHM of modes around 2.35eV,obtained by deconvoluting the PL spectra after background subtraction and averaging over the three modes closer to 2.35eV, as a function of microwire radius for TE- (full and crossed squares correspond to WGMs and Fabry-Perot modes, respectively) (a) and TM-polarized modes (full and crossed circles correspond to WGMs and Fabry-Perot modes, respectively) (b). The curves in (a) and (b), which are just a guide for the eyes, show 1/R and 1/R2 dependencies and were constructed with the same numerical prefactors. SEM images, tilted at 65° with respect to the microwire longitudinal axis, of microwires exhibiting WGMs (c) or Fabry-Perot modes (d) exclusively.
When considering the figure of merit of an optical microcavity, the most commonly employed quantity is the cavity quality factor (Q = λ/Δλ), which is proportional to the photon lifetime inside the resonator. If we computed the Q of the current microwires based on the FWHMs shown in Fig. 4, we would obtain values in the 30-100 range, which are about one order of magnitude smaller than those determined in similar microwires in reference [18]. The reason for this discrepancy is the lack of angular selection in the microPL setup used in the current study, with a microscope objective integrating rays emitted in a [-23.5°, + 23.5°] angular range around the microwire normal. Since the two kinds of modes we have studied show a parabolic dispersion along the microwire axis, i.e. their energy depends on the value of the wavevector parallel to the microwire axis [15, 18, 23], the lack of angular selection prevents us from measuring their homogeneous linewidth. However, since all the modes under investigation in the current study are purely photonic the lack of angular selection just offsets, in a first order approximation, our measured FWHMs by a given value (typically around 20meV). Thus, the previous qualitative discussion on the FWHM dependence on diameter is still valid, though we cannot give absolute quality factors.
4. GaN refractive indices
Since most GaN-based devices are designed as vertical heterostructures grown along the c-axis, for most applications the knowledge of the ordinary refractive index is sufficient. However, the recent interest on nonpolar and semipolar GaN stresses the need for a precise knowledge of the GaN birefringence. In the past, numerous methods and GaN samples have been used to measure the refractive indices of GaN but just some of them (or some particular technique/sample combination) gave access simultaneously to the ordinary and extraordinary refractive indices: prism coupling techniques [25,26], reflectivity on nonpolar GaN [27], or spectroscopic ellipsometry [28].
Our experimental configuration, with c-oriented wires lying horizontally on a substrate, gives easy access to the two refractive indices (nord and next)for both the WGMs and the Fabry-Perot modes. Indeed, for a given microwire radius Eqs. (1) and (2) establish a correspondence between the position of the energy maxima observed in PL and the couple refractive index/interference order. Thus, to obtain the refractive indices we need to know first the interference order of each resonance. For this purpose we have implemented the following fitting procedure:
1st) With the wire radius measured by SEM and an ansatz refractive index, which we have chosen to be constant, we determine the interference orders for the PL maxima thanks to Eqs. (1) and (2). It is important to note that choosing as initial condition an energy-independent refractive index is a very rough estimation, which tends to increase the number of iterations required for convergence. On the other hand, this choice prevents introducing a priori any functional dependence.
2nd) With these interference orders, which we know are M and M + 1 for consecutive maxima, we obtain point by point the refractive indices that give a best fit between Eqs. (1) and (2)and the data.
3rd) Once we have the refractive indices for all the measured wavelengths, we fit a first or second-order Sellmeier function (Eq. (3)) to the set of refractive indices:
The use of second-order (i.e. with four free parameters) Sellmeier functions leads to coefficients of determination that are larger than those obtained using first-order (i.e. with two free parameters) Sellmeier functions by less than 1% (indeed, the maximum improvement for any of the fit has been 0.55%). This small improvement has prompted us to use first-order Sellmeier functions instead.
4th) We compute the root mean square deviation of the refractive indices determined in the 2nd step with respect to the values given by the Sellmeier equation. If the root mean square deviation is smaller than a threshold value the fit is considered to be satisfactory; if not, the radius of the microwire is varied so as to minimize the root mean square deviation and the procedure is initialized at step 1.For any of the considered microwires, the maximum deviation between the final radius and the one determined by SEM is within the error bar of the corresponding SEM measurement, typically in the order of some tens of nanometers for micrometer range wires.
The refractive indices as well as the first-order Sellmeier curves obtained by fitting the complete set of data are shown in Fig. 5 (a) , while the corresponding fitting parameter values are listed in Table 1 .Overall, our refractive indices reproduce well those obtained by more complex techniques such as spectroscopic ellipsometry or prism-coupling waveguiding. The main difference with the literature (see Fig. 5(b)) is found on the absolute value of the ordinary refractive index dispersion (dnord/dλ) determined from WGMs, which is larger in our case, while the extraordinary refractive index follows almost perfectly the dispersion measured by Ghosh et al. [27] in nonpolar GaN. This larger (dnord/dλ) value for WGMs results in a GaN birefringence that is consistent with previously reported values at 400nm (~0.05), but which exceeds them at 600nm (0.074 compared to 0.043). It should be noted, however, that given the typical uncertainty associated to our refractive index determination (typically 0.05 at 600nm), the determination of an absolute GaN birefringence value is clearly out of the possibilities of the method proposed in the current article.
Fig. 5 Ordinary (black squares) and extraordinary (red circles) refractive indices extracted from WGMs resonances (a) and their fit with a first-order Sellmeier function (full lines). Inset in (a) shows the same information as in (a) but obtained from Fabry-Perot resonances. Ordinary refractive index obtained from WGMs (black full line) compared to literature values (references 26, 27, 29, 30 and 31). (b). The bars correspond to the systematic error associated to a wrong determination (by one unity) of the interference order. Inset in (b) shows the extraordinary refractive index determined from WGMs (red full line) compared to literature values.
Table 1. Parameters of the first-order Sellmeier functions and validity limits. The numbers in brackets correspond to the errors associated to the fitting procedure (not to the error of the refractive index itself).
It is noteworthy that even if our initial refractive index guess is a constant value and even if an error on the interference order can systematically shift upwards or downwards the curve, the final refractive indices that we determine averaging over all microwires are comparable to those previously measured by other authors. This further confirms the validity of our approach.
5. Conclusions
Hexagonal GaN wires, with radii in the micrometer range, act as optical microcavities for photons with wavelengths in the range (380nm-650nm). The microwires can confine photons either between two opposite facets, establishing Fabry-Perot modes, or by total internal reflection on the six lateral facets, creating whispering gallery modes. Analytical expressions for the dispersion of each kind of mode have been used to discriminate among them, as well as to extract the wavelength-dependent ordinary and extraordinary refractive indices of unstrained GaN.
Acknowledgments
We are grateful to J. Y. Duboz, F. Semond and S. Sergent for fruitful discussions. We acknowledge financial support from the EU under FP7 contract SMASH CP-IP 228999-2.
References and links
1. T. Someya, R. Werner, A. Forchel, M. Catalano, R. Cingolani, and Y. Arakawa, “Room temperature lasing at blue wavelengths in Gallium Nitride microcavities,” Science 285(5435), 1905–1906 (1999). [CrossRef] [PubMed]
2. Y. Higuchi, K. Omae, H. Matsumura, and T. Mukai, “Room-temperature CW lasing of a GaN-based vertical surface-emitting laser by current injection,” Appl. Phys. Express 1, 121102 (2008). [CrossRef]
3. A. Imamoglu, R. J. Ram, S. Pau, and Y. Yamamoto, “Nonequilibrium condensates and lasers without inversion: Exciton-polariton lasers,” Phys. Rev. A 53(6), 4250–4253 (1996). [CrossRef] [PubMed]
4. S. Christopoulos, G. B. von Högersthal, A. J. D. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J. F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett. 98(12), 126405 (2007). [CrossRef] [PubMed]
5. G. Christmann, R. Butté, E. Feltin, J. F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in a GaN/AlGaN multiple quantum well microcavity,” Appl. Phys. Lett. 93(5), 051102 (2008). [CrossRef]
6. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992). [CrossRef]
7. R. A. Mair, K. C. Zeng, J. Y. Lin, H. X. Jiang, B. Zhang, L. Dai, A. Botchkarev, W. Kim, H. Morkoç, and M. A. Khan, “Optical modes within III-nitrides multiple quantum well microdisk cavities,” Appl. Phys. Lett. 72(13), 1530–1532 (1998). [CrossRef]
8. S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering-gallery mods of GaN microdisck cavities,” Appl. Phys. Lett. 75(2), 166–168 (1999). [CrossRef]
9. A. C. Tamboli, E. D. Haberer, R. Sharma, K. H. Lee, S. Nakamura, and E. L. Hu, “Room-temperature continuous-wave lasing in GaN/InGaN microdisks,” Nat. Photonics 1(1), 61–64 (2007). [CrossRef]
10. D. Simeonov, E. Feltin, A. Altoukhov, A. Castiglia, J. F. Carlin, R. Butté, and N. Grandjean, “High quality nitride based microdisks obtained via selective wet etching of AlInN sacrificial layers,” Appl. Phys. Lett. 92(17), 171102 (2008). [CrossRef]
11. M. Mexis, S. Sergent, T. Guillet, C. Brimont, T. Bretagnon, B. Gil, F. Semond, M. Leroux, D. Néel, S. David, X. Chécoury, and P. Boucaud, “High quality factor nitride-based optical cavities: microdisks with embedded GaN/Al(Ga)N quantum dots,” Opt. Lett. 36(12), 2203–2205 (2011). [CrossRef] [PubMed]
12. M. Lorenz, A. Rahm, B. Q. Cao, J. Zuniga-Perez, E. M. Kaidashev, N. Zhakarov, G. Wagner, T. Nobis, C. Czekalla, G. Zimmermann, and M. Grundmann, “Self-organized growth of ZnO-based nano and microstructures,” Phys. Status Solidi B 247(6), 1265–1281 (2010). [CrossRef]
13. J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67(2), 023807 (2003). [CrossRef]
14. T. Nobis, E. M. Kaidashev, A. Rahm, M. Lorenz, M. Grundmann, and M. Grundmann, “Whispering gallery modes in nanosized dielectric resonators with hexagonal cross section,” Phys. Rev. Lett. 93(10), 103903 (2004). [CrossRef] [PubMed]
15. A. Trichet, L. Sun, G. Pavlovic, N. A. Gippius, G. Malpuech, W. Xie, Z. Chen, M. Richard, and L. Dang, “One-dimensional ZnO exciton polaritons with negligible thermal broadening at room temperature,” Phys. Rev. B 83(4), 041302 (2011). [CrossRef]
16. B. Alloing, E. Beraudo, Y. Cordier, F. Semond, S. Sergent, O. Tottereau, P. Vennéguès, S. Vézian, and J. Zuniga-Perez, “Fabrication and growth of GaN-based micro and nanostructures,” Int. J. Nanotechnol. 9(3/4/5/6/7), 412–427 (2012). [CrossRef]
17. M. Leroux, B. Beaumont, N. Grandjean, P. Lorenzini, S. Haffouz, P. Vennéguès, J. Massies, and P. Gibart, “Luminescence and reflectivity studies of undoped, n- and p-doped GaN on (0001) sapphire,” Mater. Sci. Eng. B 50(1-3), 97–104 (1997). [CrossRef]
18. A. Trichet, F. Medard, J. Zuniga-Perez, B. Alloing, and M. Richard,“From strong to weak coupling regime in a single GaN microwire up to room temperature,” arXiv 1106.5595 (2011).
19. C. Hums, T. Finger, T. Hempel, J. Christen, A. Dadgar, A. Hoffmann, and A. Krost, “Fabry-Perot effects in InGaN/GaN heterostructures on Si-substrate,” J. Appl. Phys. 101(3), 033113 (2007). [CrossRef]
20. E. F. Schubert, Light-Emitting Diodes(Cambridge University Press, 2003), Chap. 9.
21. C. Czekalla, T. Nobis, A. Rahm, B. Q. Cao, J. Zuniga-Perez, C. Sturm, R. Schmidt-Grund, M. Lorenz, and M. Grundmann, “Whispering-gallery modes in zinc oxide micro- and nanowires,” Phys. Status Solidi B 247(6), 1282–1293 (2010). [CrossRef]
22. A. K. Bhowmik, “Polygonal optical cavities,” Appl. Opt. 39(18), 3071–3075 (2000). [CrossRef] [PubMed]
23. C. P. Dietrich, M. Lange, C. Sturm, R. Schmidt-Grund, and M. Grundmann, “One- and two-dimensional cavity modes in ZnO microwires,” New J. Phys. 13(10), 103021 (2011). [CrossRef]
24. L. Sun, H. Dong, W. Xie, Z. An, X. Shen, and Z. Chen, “Quasi-whispering gallery modes of exciton-polaritons in a ZnO microrod,” Opt. Express 18(15), 15371–15376 (2010). [CrossRef] [PubMed]
25. M. J. Bergmann, U. Ozgur, H. C. Casey Jr, H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1-xN epitaxial layers,” Appl. Phys. Lett. 75(1), 67–69 (1999). [CrossRef]
26. S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103(12), 123112 (2008). [CrossRef]
27. S. Ghosh, P. Waltereit, O. Brandt, H. T. Grahn, and K. H. Ploog, “Polarization-dependent spectroscopic study of M-plane GaN on γ-LiAlO2,” Appl. Phys. Lett. 80(3), 413–415 (2002). [CrossRef]
28. S. Shokhovets, R. Goldhahn, G. Gobsch, S. Piekh, R. Lantier, A. Rizzi, V. Lebedev, and W. Richter, “Determination of the anisotropic dielectric function for wurtzite AlN and GaN by spectroscopic ellipsometry,” J. Appl. Phys. 94(1), 307–312 (2003). [CrossRef]
29. A. Billeb, W. Grieshaber, D. Stocker, E. F. Schubert, and R. F. Karlicek Jr., “Microcavity effects in GaN epitaxial films and in Ag/GaN/sapphire structures,” Appl. Phys. Lett. 70(21), 2790–2792 (1997). [CrossRef]
30. Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1-xN refractive indices,” Appl. Phys. Lett. 79(25), 4103–4105 (2001). [CrossRef]
31. N. Antoine-Vincent, F. Natali, M. Mihailovic, A. Vasson, J. Leymarie, P. Disseix, D. Byrne, F. Semond, and J. Massies, “Determination of the refractive indices of AlN, GaN and AlxGa1-xN grown on (111)Si substrates,” J. Appl. Phys. 93(9), 5222–5226 (2003). [CrossRef]
