Investigation and comparison of optical gain spectra of (Al,In)GaN laser diodes emitting in the 375 nm to 470 nm spectral range

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1 Investigation and comparison of optical gain spectra of (Al,In)aN laser diodes emitting in the 375 nm to 470 nm spectral range Ulrich T. Schwarz and Harald Braun Department of Physics, Regensburg University Regensburg, ermany Kazunobu Kojima, Mitsuru Funato, Yoichi Kawakami Department of Electronic Science and Engineering Kyoto University, Kyoto , Japan Shinichi Nagahama and Takashi Mukai Nitride Semiconductor Research Laboratory, Nichia Corporation 491 Oka, Kaminaka, Anan, Tokushima , Japan ABSTRACT We measure gain spectra for commercial (Al,In)aN laser diodes with peak gain wavelengths of 470 nm, 440 nm, 405 nm, and 375 nm, covering the spectral range accessible with electrical pumping. For this systematic study we employ the Hakki Paoli method, i.e. the laser diodes are electrically driven and gain is measured below threshold current densities. The measured gain spectra are reasonable for a 2D carrier system and understandable when we take into account homogeneous and inhomogeneous broadening. While inhomogeneous broadening is almost negligible for the near UV laser diode, it becomes the dominant broadening mechanism for the longer wavelength laser diodes. We compare the gain spectra with two models describing the inhomogeneous broadening. The first model assumes a constant carrier density, while the second model assumes a constant quasi Fermi level. Both are in agreement with the experimental gain spectra, but predict very different carrier densities. We see our measurements as providing a set of standard gain spectra for similar laser diodes covering a wide spectral range which can be used to develop and calibrate theoretical manybody gain simulations. Keywords: InaN laser diodes, optical gain, inhomogeneous broadening, localized states, Hakki Paoli method 1. INTRODUCTION Nitride based separate confinement heterostructure laser diodes with InaN quantum wells are now able to span the wavelength range from near UV (365 nm) 1 to blue green (482 nm). 2 This extension of the wavelength range from the initial 405 nm laser diodes was possible by a steady improvement of the epitaxial growth technology. In particular for the longer wavelength laser diodes it was necessary to stabilize the growth of the InaN quantum wells (QW) with high indium content. Indium fluctuations in the QW have an impact on the inhomogeneous broadening of the gain spectra and thus on the basic laser parameters like threshold current and differential efficiency. In this article we compare two different models for the inhomogeneous broadening of the optical gain spectra for InaN laser diodes (LD). In the first model the homogeneous broadened gain is calculated and inhomogeneous broadening is introduced by a convolution with a aussian distribution. The physical interpretation is that the carrier density is constant everywhere in the quantum well while the Fermi energy fluctuates synchronous with Further author information: Ulrich T. Schwarz: ulrich.schwarz@physik.uni-regensburg.de, Telephone: +49 (0) Novel In-Plane Semiconductor Lasers VI, edited by Carmen Mermelstein, David P. Bour, Proc. of SPIE Vol. 6485, , (2007) X/07/$18 doi: / Proc. of SPIE Vol

2 Figure 1. Optical gain spectra of a 375 nm laser diode are plotted as thin black lines and are identical for (a) and (b). Simulated gain spectra are plotted as small circles for the constant carrier density model (a) and constant Fermi level model (b). Carrier densities are 5.565, 5.705, 5.845, 5.999, 6.195, 6.426, cm 2 for the constant carrier model and 5.53, 5.67, 5.824, 5.999, 6.16, 6.405, cm 2 for the constant Fermi level model, respectively. The three to four digits in the values for the carrier densities do not reflect the absolute accuracy but are necessary because the fitting of the gain curves are sensitive to the small differences of the carrier densities. the energy gap. This model was often used, e.g. in the discussion of inhomogeneous broadening and temperature dependent gain spectra of 405 nm InaN QW laser diodes. 3, 4 The second model assumes that the Fermi energy is a global constant. As consequence the carrier density fluctuates as function of the local bandgap energy. This implies that the shape of the gain spectra varies spatially and the total gain spectrum has to be calculated by an integration over gain spectra for different local carrier densities. This model was introduced for InaN QW laser diodes by Yamaguchi et al. who pointed out that the localized states can result in an increased differential and absolute gain. 5 This may seem counterintuitive, as usually inhomogeneous broadening leads to a decrease of optical gain. The explanation for the increasing differential gain lies in the low density of the tail states where inversion and gain can be generated at very low carrier densities. These states are localized by bandgap fluctuations of the InaN QW. In this article we extend the second model by including homogeneous broadening. We will then fit both models to the full set of gain spectra for laser diodes with peak gain wavelength ranging from near UV (375 nm) to aquamarine (470 nm). In both models we use one value of the homogeneous broadening, while we compare the impact of the different models on the predicted carrier densities and bandgap energy. The value for the homogeneous broadening was taken from manybody simulations for a different set of laser diodes with InaN QWs OPTICAL AIN SPECTRA: EXPERIMENTS For the optical gain measurements we chose a set of commercially available InaN QW laser diodes spanning the spectral range from near UV (375 nm) to aquamarine (470 nm), which is almost the full spectral range accessible for cw operation by electrically driven InaN QW laser diodes. 2 The high power laser diodes are grown by metalorganic chemical vapor deposition (MOCVD) on bulk an substrates with a low threading dislocation density of 10 5 to 10 6 cm 2. The near UV laser diode, which emits at around 375 nm, has one 10 nm thick quantum well with an estimated indium composition of 2 % to 3 %. The violet (407 nm), blue (440 nm) and aquamarine (470 nm) laser diodes are designed with double quantum wells of In 0.08 a 0.92 N/In 0.02 a 0.98 N, In 0.15 a 0.85 N/In 0.02 a 0.98 N, and In 0.20 a 0.92 N/In 0.02 a 0.98 N composition, and 7 nm, 3 nm, and 3 nm width, respectively. The geometry of the waveguide ridge was optimized to decrease the operating current density at higher output power. Waveguide facets were formed by cleaving along the (1100) face. Cavity length was 650 µm for the near UV and blue laser Proc. of SPIE Vol

3 Figure 2. Optical gain spectra of a 407 nm laser diode. The assignment of lines and symbols is like in Fig. 1. Carrier densities are 5.6, 5.74, 5.915, 5.992, 6.125, 6.244, 6.342, cm 2 for the constant carrier model (a) and 3.92, 4.13, 4.389, 4.48, 4.585, 4.76, 4.914, cm 2 for the constant Fermi level model (b), respectively. diodes and 675 µm for the violet and aquamarine ones. Mirror reflectivities were R 1 /R 2 =0.18/0.93, 0.18/0.98, 0.82/0.98, and 0.82/0.98 from near UV to aquamarine. A detailed description of the laser diodes, including the 1, 2, 6 waveguide and cladding layer dimensions, can be found elsewhere. The optical gain was measured using the Hakki Paoli method. 7, 8 With this method the laser diode is electrically driven below threshold. The individual longitudinal modes of the spontaneous emission are measured with high spectral resolution by a monochromator equipped with a liquid nitrogen cooled CCD detector. The modulation depth of the closely spaced longitudinal modes is a measure for the number of round trips of a photon in the laser cavity. Once the mirror losses are known, the gain or loss can be calculated from the peak to valley ratio of the intensity of the longitudinal modes. 9 The method is very precise when compared to the more common stripe length method. It is limited to carrier densities below threshold. The laser diodes are operated under realistic conditions, i.e. cw electrically driven. The current is provided by a stabilized laser diode driving circuit, and the heat sink temperature is precisely stabilized to T = 298 K. We compared gain spectra measured by the Hakki Paoli method for one individual laser diode at two completely different experimental setups and found quantitative agreement. 10 The measured optical gain spectra for four laser diodes with lasing wavelengths of 375 nm, 407 nm, 440 nm, and 470 nm are plotted in Fig. 1 to 4, respectively. The driving currents for the experimental spectra (thin black curves) are 5 to 35 ma in 5 ma steps for the 375 nm laser diode, 5 to 40 ma in 5 ma steps for the 407 nm and 440 nm laser diodes, and [2, 4, 6, 10, 20, 30, 40] ma for the 470 nm laser diode. The circles in Fig. 1 to 4 are simulated gain curves as discussed below. The experimental gain spectra in the left and right panes are identical. It is obvious that the width of the gain spectra increases with increasing peak gain wavelength. We will see below that homogeneous broadening dominates for the 375 nm laser diode, while inhomogeneous broadening dominates for the 470 nm laser diode. The 470 nm laser diode exhibits a lower energy gain peak at 490 nm which saturates at low carrier densities. This gain peak and saturation was discussed in a separate article. 10 With the exception of the near UV laser diode the internal losses α i are in the range of 20 cm 1 to 26 cm 1. For the 407 nm laser diode the internal losses depend on carrier density, as seen in Fig. 2. For the 375 nm laser diode the internal losses are slightly higher and increase with increasing wavelength. Therefore no stable plateaux is formed at the long wavelength side of the gain spectrum. We estimate the internal losses α i =35cm 1 from the gain value in the energy range of 3.28 to 3.3 ev in the limit of low driving currents. 3. CONSTANT CARRIER DENSITY OPTICAL AIN The first model is based on the standard description of optical gain in quantum wells. We term this model the constant carrier density optical gain model. Detailed derivation of the equations can be found in classic Proc. of SPIE Vol

4 Figure 3. Optical gain spectra of a 440 nm laser diode. Carrier densities are 6.37, 7.07, 7.49, 7.84, 8.057, 8.225, cm 2 for the constant carrier model (a) and 1.54, 2.24, 2.625, 2.94, 3.22, 3.465, cm 2 for the constant Fermi level model (b), respectively. semiconductor optoelectronics textbooks Figure 5 illuminates the situation. All energies with the exception of the bandgap energy are approximately to scale for a carrier density of n = cm 2. The Fermi distribution for electrons and holes is given as: f c (ω) = (1+exp[β ( hω F c )]) 1 (1) f v (ω) = (1+exp[β ( hω F v )]) 1 (2) where β = e/ (k B T) and F c and F v are the quasi Fermi levels for the conduction and valence band, respectively. For simplicity we use the parabolic approximation with one valence band, only. The electron effective mass is m c =0.2m e while the hole effective mass was taken as m v =2m e to account for the combined density of states of light and heavy hole bands. The two dimensional density of states is then given by ρ c = em c π h 2 (3) Figure 4. Optical gain spectra of a 470 nm laser diode. Carrier densities are 8.05, 8.75, 9.03, 9.45, 9.94, 10.29, cm 2 for the constant carrier model (a) and 1.4, 1.855, 2.1, 2.45, 2.87, 3.185, cm 2 for the constant Fermi level model (b), respectively. Proc. of SPIE Vol

5 ρ v = em v π h 2. (4) For a two dimensional quantum well in the parabolic band approximation the quasi Fermi levels F c and F v for electrons and holes, respectively, can be given analytically as a function of the carrier density n: F c (n) = log (exp [nβ/ρ c ] 1) /β (5) F v (n) = log (exp [nβ/ρ v ] 1) /β (6) Because of the large hole effective mass, inversion and thus positive gain can result for a carrier density where the quasi Fermi level F v is above the band edge. As only vertical transitions are allowed (no impulse is transferred to the lattice) the corresponding electron and hole states for a given photon energy hω are depicted in Fig. 5, with the reduced mass m r = m c m v. (7) m c + m v This has to be considered for the electron and hole energies in the Fermi distributions. Then the homogeneous broadened optical gain is expressed as: hom (ω) = C πγ hom E g ( ω ) ω (f c + f v 1) Sech dω (8) γ hom Figure 5. Schematic representation of the electron and hole distribution in conduction and valence band, leading to optical gain at photon energy hω (see text). Proc. of SPIE Vol

6 For the homogeneous broadening we use the Sech function. This expression for the gain does not cause spurious absorption below the band gap, as does broadening by a Lorentzian curve. The broadening parameter γ hom is given by carrier scattering and can be rigorously calculated in a manybody approach. Yet often γ hom is being used as fitting parameter. In the definition of Eq. 8 we use the constant pre factor C = πe 2 n ref c vac ε 0 m 2 e ω m r π h 2 L d M 2 b. (9) Besides the physical constants h, ε 0 and c vac, L d is the width of the QW, e and m e are charge and mass of the free electron, respectively, and n ref is the refractive index. The difficult parameter is the transition matrix element M b which is given by the overlap of the electron and hole wave functions in the QW. Due to piezoelectric fields, resulting in the quantum confined Stark effect (QCSE), this matrix element depends on the carrier density. For large enough carrier densities or by doping the internal fields are screened and the assumption of a constant transition matrix element can be justified. In our discussion below the focus is on inhomogeneous broadening, and we use the assumption of a constant M b. Yet for a rigorous treatment the details of the active layers of the laser diode have to be considered. In this first model inhomogeneous broadening is calculated with the assumption that the shape of the gain curve is identical all over the laser diode. The fluctuation of the energy gap of the InaN QW results then in a shift of the gain spectrum. Assuming a aussian distribution of width σ of the gap energy E,the inhomogeneous broadened gain spectrum is given as convolution of the homogeneous broadened spectrum of Eq. 8 and a aussian distribution: (1) inh (ω) = 1 π 2σ [ hom (ω ω )exp ( ω ω 2σ ) 2 ] dω (10) The last equation presents the material gain of the quantum well. To calculate the modal gain of the laser diode which is being measured in the experiments, we have to include the confinement factor Γ conf and internal losses α i : g mod =Γ conf inh α i (11) 4. CONSTANT FERMI LEVEL OPTICAL AIN In the second model we assume that the lateral diffusion of the carriers is fast enough that an equilibrium with global constant quasi Fermi levels is being reached. We term this the constant Fermi level model. The situation is depicted in Fig. 6. As in the first model, a aussian distribution of the local gap energy E g with width σ around a mean gap energy E is assumed. The fluctuations are distributed to both bands as the ratio of the respective band offsets, i.e. σ c =0.7σ and σ v =0.3σ. The constant Fermi level causes a local varying carrier density by a redistribution of carriers for regions of larger E g to those of smaller E g. The averaged density of states for each band is given by integration over the aussian distribution of E g which results in the so called error function Erf: ρ c (ω) = em ( c 2π h 2 ρ v (ω) = em v 2π h 2 1+Erf ( 1+Erf [ ω 2σc ]) [ ω 2σv ]) The overall carrier densities in conduction and valence bands are given by integration of the averaged density of states times the Fermi distribution with global constant Fermi level: n c = n v = (12) (13) ρ c (ω) f c (ω, F c )dω (14) ρ v (ω) f v (ω, F v )dω (15) Proc. of SPIE Vol

7 Figure 6. Schematic model for the inhomogeneous broadening in the constant Fermi level model (see text). For charge neutrality we have n c = n v = n as in the case of Eqs. 1 and 2. Now, Eqs. 14 and 15 can be solved numerically for the quasi Fermi levels at a given carrier density. In this model the homogeneous and inhomogeneous broadened gain is calculated by a double integral. The averaged band gap energy is given by E, to be distinguished from the local band gap E g, depending on the position within the QW. For a local band gap energy E g the homogeneous broadened gain is calculated by integration from E g to infinity, in analogy to Eg. 8 (see also Fig. 6). The outer integral is over the different local band gap energies E g. This integral ranges form zero to infinity, because due to homogeneous broadening an arbitrary band gap contributes to gain or absorption for a given photon energy hω. Of course, for an efficient numerical implementation the range of the integral has to be chosen carefully. The expression for the gain in the constant Fermi level model is then: (2) inh (ω) = C 2 π 3/2 γ hom σ 0 E g [ (f c + f v 1) exp ( Eg E 2σ ) 2 ] Sech ( ω ω γ hom ) dω de g. (16) 5. DISCUSSION Both models are fitted to the experimental data. The circles in Figs. 1 to 4 mark the simulated gain curves. The left panes show the first model of constant carrier density while the right ones show that of constant Fermi level. With both models the gain spectra can be fitted with similar accuracy. Of course both models predict different carrier densities, the obtained values of which are listed in the figure captions. To be able to compare the carrier densities and the different fitting results we used the following strategy. The basic parameters effective masses m c and m v, and effective refractive index n ref were kept constant. The value for the homogeneous broadening γ hom = 25 mev was taken from manybody simulations fitted to gain spectra of an InaN QW laser diode measured at the same setup. 3 This is certainly a critical issue, as carrier scattering depends on band structure and thus on indium content. But the shape of the gain spectrum does not critically depend on the relative contribution of inhomogeneous and homogeneous gain. Therefore it is impossible to extract their relative strength for the presented gain measurements. One possibility would be to measure gain Proc. of SPIE Vol

8 λ peak [nm] α i [cm 1 ] Γ conf L d [nm] M 2 b /(m e E ) E (1) [ev] σ(1) [mev] E(2) [ev] σ(2) [mev] Table 1. Parameters used in the simulation of the gain spectra for Fig. 1 to 4. The first five columns are identical for both models, while band gap energy and inhomogeneous broadening have different values (see text). spectra as function of temperature, as was demonstrated in an earlier publication. 4 The other possibility are manybody simulations which are outside the scope of this article. Internal losses are extracted from the experimental gain spectra as listed in table 1. For the 407 nm laser diode the losses were linearly interpolated between 20 cm 1 and 23.5cm 1. Other parameters which are identical for both models are the confinement factor Γ conf and QW thickness L d, which are also listed in table 1. The slope at the high energy side of the gain spectra depends critically on the carrier density. This slope was fitted by a variation of the matrix element M b as listed in table 1. For both models one value of M b for each laser diode was used. For the low carrier density spectra an additional red shift was included which increased with increasing indium content. For the 375 nm to 470 nm laser diodes the maximum value of the red shift was 3 mev, 10 mev, 18 mev, and 50 mev, respectively. The red shift was identical for both models. A red shift of this order of magnitude can be explained in terms of the QCSE. The free parameters are the band gap energy E and the inhomogeneous broadening σ. While the inhomogeneous broadening of σ (1) =3.5meV of the UV laser diode was almost negligible, it is the dominating contribution for the 470 nm laser diode (σ (1) = 100 mev). The value σ(1) = 25 mev for the 407 nm laser diode is similar to the value σ (1) = 31 mev found for a different 405 nm laser diode in an earlier work.3, 4 It is noteworthy that the constant Fermi level model results in larger values for the inhomogeneous broadening (see 1). The physical origin is that for low and moderate carrier densities only the lower energy states in the band tail are occupied. As gain originates only from those occupied states, the resulting inhomogeneous broadening of the gain spectra is less than the full width of the broadening of the energy gap σ (2). For the 375 nm laser diode the band gap energy and carrier densities to fit the experimental data are nearly identical for both models. This is no surprise, as the inhomogeneous broadening is only σ (1) =3.5meV or σ (2) = 5 mev. This value depends critically on the chosen value of the homogeneous broadening γ hom = 25 mev. The measurement basically provides an upper boundary for the inhomogeneous broadening for the near UV laser diode of the order of 10 mev. The values of the carrier density are close enough to values published recently for a manybody simulation of our gain data for an InaN QW laser diode. 14 For the longer wavelength laser diodes the difference between both models increases with increasing inhomogeneous broadening. The constant Fermi level model produces a significantly higher band gap energy. For the 470 nm laser diode the shift is by 135 mev from E (1) =2.675 ev to E(2) =2.81 ev. This indicates that the gain originates from the tail band states with a low density of states. The higher band gap energy for the same gain spectra is consistent with the narrowing of the gain spectra as seen by the larger σ (2) values necessary to fit the gain spectra. The predicted carrier density is increasing moderately with increasing peak wavelength for the constant carrier density model. This can be understood because the increasing inhomogeneous broadening causes a decrease of the peak gain in this model. In the constant Fermi level model the carrier density actually decreases with increasing peak wavelength. For the 470 nm laser diode there is factor of three difference in the predicted carrier density for both models. Thus the long wavelength laser diodes clearly profit from the low density of localized states in the constant Fermi level model in the regime of low carrier densities. To understand the impact of carrier localization on optical gain in InaN QW laser diodes it is necessary to be able to distinguish between the two different models of constant carrier density and constant Fermi level. The thus gained knowledge may provide clues to overcome the limitation of long wavelength operation of InaN Proc. of SPIE Vol

9 QW laser diodes. To be able to distinguish both models we need more information on carrier lifetime in these devices. Also manybody simulations of the gain curves with different peak wavelength are mandatory, as they provide the exact value for the homogeneous broadening in each structure. This combination of experiments and simulations is essential to develop a theory which would allow to predict a strategy for the extension of the wavelength range for InaN QW laser diodes. 6. SUMMARY In this article we present experimental gain spectra for commercial InaN QW laser diodes which lase in the range from 375 nm to 470 nm. The optical gain spectra were measured by the Hakki Paoli method which is capable to measure below threshold gain spectra of electrically driven laser diodes with high precision. The gain spectra were compared to two models for the inhomogeneous broadening, one assuming a constant carrier density, the other assuming a constant Fermi level throughout the quantum well. Both models are able to reproduce the measured gain spectra for all laser diodes with good quality and for a similar set of parameters with exception of the gap energy, inhomogeneous broadening, and carrier density. The constant Fermi level model predicts a significantly higher band gap and a up to three times lower carrier density for the 470 nm laser diode. For that laser diode with a inhomogeneous broadening of σ = 140 mev lasing originates from localized tail band states with a low density of states. To understand the limitation to long wavelength operation of InaN QW laser diodes it is necessary to be able to distinguish between both models for the inhomogeneous broadening. We also see our measurements as providing a set of standard gain spectra for a set of similar laser diodes covering a wide spectral range which can be used to develop and calibrate theoretical manybody gain simulations. ACKNOWLEDMENTS The authors thank A. A. Yamaguchi for stimulating discussions. Ulrich T. Schwarz acknowledges support by an invited fellowship from the Japan Society for the Promotion of Science (JSPS). Kazunobu Kojima is supported by a JSPS research fellowship for young scientists. This work is partially supported by COE. REFERENCES 1. S. Masui, Y. Matsuyama, T. Yanamoto, T. Kozaki, S. Nagahama, and T. Mukai, 365 nm ultraviolet laser diodes composed of quaternary AlInaN alloy, Jpn. J. Appl. Phys. 42, LL1318 (2003). 2. S. Nagahama, Y. Sugimoto, T. Kozaki, Y. Fujimura, S. Nagahama, and T. Mukai, Recent progress of AlInaN laser diodes, Proc. SPIE 5738, 57 (2005). 3. B. Witzigmann, V. Laino, M. Luisier, U. T. Schwarz,. Feicht, W. Wegscheider, K. Engl, M. Furitsch, A. Leber, A. Lell, and V. Härle, Microscopic analysis of optical gain in InaN/aN quantum wells, Appl. Phys. Lett. 88, (2006). 4. B. Witzigmann, V. Laino, M. Luisier, U. T. Schwarz, H. Fischer,. Feicht, W. Wegscheider, C. Rumbolz, A. Lell, and V. Härle, Analysis of temperature dependent optical gain in an/inan quantum well structures, IEEE Phot. Tech. Lett. 18, 1600 (2006). 5. A. A. Yamaguchi, M. Kuramoto, M. Nido, and M. Mizuta, An alloy semiconductor system with a tailorable band tail and its application to high performance laser operation: I. A band states model for an alloy fluctuated InaN material system designed for quantum well laser operation, Semicond. Sci. Technol. 16, 763 (2001). 6. T. Kozaki, H. Matsumura, Y. Sugimoto, S. Nagahama, and T. Mukai, High power and wide wavelength range an baser laser diodes, Proc. SPIE 6133, (2006). 7. B. W. Hakki and T. L. Paoli, cw degradation at 300 K of aas double heterostructure junction laser. II. Electronic gain, J. Appl. Phys. 44, 4113 (1973). 8. B. W. Hakki and T. L. Paoli, ain spectra in aas double heterostructure injection lasers, J. Appl. Phys. 46, 1299 (1974). Proc. of SPIE Vol

10 9. U. T. Schwarz, E. Sturm, W. Wegscheider, V. Kümmler, A. Lell, V. Härle, ain spectra and current-induced change of refractive index in (In/Al)aN diode lasers, phys. stat. sol. (a) 200, 143 (2003). 10. K. Kojima, M. Funato, Y. Kawakami, S. Nagahama, T. Kumai, H. Braun, U. T. Schwarz, ain suppression phenoma observed in InaN QW laser diodes emitting at 470 nm, Appl. Phys. Lett. 89, (2006). 11. A. Yariv, Quantum electronics, Wiley, New York (1988), pp S. L. Chuang, Physics of optoelectronic devices, Wiley, New York (1995), pp W. W. Chow and S. W. Koch, Semiconductor Laser Fundamentals, Springer, Berlin (1999), pp J. Hader, J. V. Moloney, and S. W. Koch, Influence of internal fields on gain and spontaneous emission in InaN quantum wells, Appl. Phys. Lett. 89, (2006). Proc. of SPIE Vol