GaN

JOURNAL OF APPLIED PHYSICS

VOLUME 85, NUMBER 7

1 APRIL 1999

Piezoelectric Franz–Keldysh effect in strained GaInN/GaN heterostructures C. Wetzel,a) T. Takeuchi, H. Amano, and I. Akasaki High-Tech Research Center and Department of Electrical and Electronic Engineering, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, 468-8502 Nagoya, Japan

~Received 21 January 1998; accepted for publication 22 December 1998! Pseudomorphic Ga12x Inx N/GaN single heterostructures in the composition range 0,x,0.2 have been investigated by photoreflectance and photoluminescence spectroscopy. Strong Franz–Keldysh oscillations near the band gap of the ternary film are observed and attributed to a large constant piezoelectric field of up to 0.63 MV/cm. This allows an accurate determination of the electric field. A significant redshift between the optical band gap from photoreflectance and the luminescence maximum is observed. Luminescence is proposed to originate in the indirect transitions between the electric field tilted band edges in GaInN. The presence of this field is expected to dominate the bandstructure and the recombination and transport processes in strained nitride structures. We find no evidence for large inhomogeneities or phase separation in this material. © 1999 American Institute of Physics. @S0021-8979~99!01207-4# I. INTRODUCTION

prisingly thick layers of completely strained GaInN of uniform composition can be grown without lattice relaxation. This is tentatively assigned to the fact that dislocations are predominantly oriented along the growth direction and cannot contribute to relax the in-plane stress. PR and ER were performed using a Xe-arc lamp source in near-toperpendicular reflection. Periodic photomodulation ( f 51.4 kHz! was performed using a 325 nm HeCd Laser at a flux of '0.1 W/cm2 . Electromodulation was applied through an electrolyte and Cu electrode. A sinusoidal modulation voltage of 10 V amplitude but no bias was applied to the electrode. The ac signal component was normalized to the dc part forming DR/R. The phase was fixed to the phase of the PL or modulation voltage. PL under an excitation flux of 1 2100 W/cm2 was performed using the same laser. Spatially resolved PL mapping was performed in a confocal ultraviolet ~UV! microscope at typical power densities of 106 W/cm2 . All data were taken at room temperature.

Application of wide gap GaN and group-III nitrides requires the control of the electronic band and defect structure by means of growth, heteroepitaxy and doping.1,2 At present, however, considerable uncertainty exists about the band structure in high quality GaInN/GaN layers leading to superb light emitting devices. Discrepancies of band-gap energies from emission and absorption experiments amount up to 100 and 200 meV.3,4 This effect is accompanied by a large effective bowing of the band-gap energy.4 It has been proposed that clustering of In should induce large localization effects for radiative transitions3 and phase separation5 would lead to spatial inhomogeneities of the composition and to a distribution of band-gap energies in the alloy. Under thermodynamical equilibrium theory predicts a phase separation6 and indeed, difficulties to incorporate high In fractions support such models. In this work, however, we will show that GaInN alloy material can be obtained with very little fluctuation and that a separation of absorption and emission band gap still exists. This apparent discrepancy can be well explained when pseudomorphic strain and piezoelectric properties are taken into account. In order to elucidate the electronic band structure we employ photoreflectance ~PR!, electroreflectance ~ER! and photoluminescence ~PL! on high quality pseudomorphic GaInN/GaN structures.

III. STRAIN ANALYSIS AND HOMOGENEITY

Heteroepitaxy of wurtzite nitride layers along the unique c axis on GaN leads to large biaxial compressive strain within the basal plane of the ternary layer. Under pseudomorphic conditions, hypothetically freestanding Ga12x Inx N with lattice constants c(x)5c GaN(12x)1c InNx55.184 Å (12x)15.705 Åx, a(x)5a GaN(12x)1a InNx53.188 Å (12x)13.540 Åx is forced onto 2 m m GaN/sapphire with lattice constants c 1 55.189 Å, a 1 53.182 Å. This induces strain in the ternary layer: e xx 5a 1 /a x 21,0. As a consequence within Hooke’s law, an elongation along the c axis occurs, e zz 5c 1 /c x 21522c 13 /c 33e xx .0 ~elastic constants c 13 , c 33 ; values of GaN!.9 For x50.2, we therefore induce strain e zz 50.015, e xx 520.023. Despite these large values and a thickness of 40 nm, GaInN films are found to remain a single alloy component and unrelaxed as observed from high resolution x-ray diffraction of both lattice constants a and c.9 This deformation effect of the unit cell strongly affects the interpretation of the composition from x-ray data of one lattice constant alone, i.e., the c-lattice constant. It appears that

II. EXPERIMENT

Single Ga12x Inx N/GaN heterostructures with an InN fraction x in the range of 0,x,0.2 were grown in metalorganic vapor phase epitaxy ~MOVPE! on ~0001! sapphire substrates using low temperature deposited AlN buffer layers.7–9 Pseudomorphically strained ternary layers at a thickness of 40 nm were grown onto 2 m m GaN under fluxes of TMGa 2 m mol/min, TMIn 0.3–12 m mol/min, NH3 2 slm and temperatures of 680–780 °C. As reported in Ref. 9 sura!

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© 1999 American Institute of Physics

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FIG. 1. Statistical analysis of the PL in Ga12x Inx N/GaN (x50.187) over an area of 50 m m2 . ~a! Distribution of the peak energy. ~b! Peak intensity added peak energy distribution. Disregarding the intensity in some locations PL has its maxima well below a narrow distribution. But taking into account the intensity PL is clearly favored within a very narrow distribution on the high energy side. This indicates a very good homogeneity on the length scale of 1–50 m m.

in many previous works the actual InN fraction has been overestimated by roughly a factor of 2 ~for a discussion see Ref. 4!. Here we take this effect into account for the proper determination of x. These results are in important contrast to earlier work on 500 nm layers of GaInN/GaN5 where several components were identified in symmetric x-ray scattering of the c planes. Those were associated with the coexistence of phases with different alloy composition. Unfortunately, no data was given in regards to a-plane constants and strain conditions. From our present view it is conceivable that InN fractions might have been overestimated and films were partially relaxed. The spatial homogeneity in our material was also tested in high resolution PL. Figure 1 presents a typical statistical analysis of 2500 spectra10 taken on a Ga12x Inx N/GaN (x 50.187) layer over an area of ~50 m m) 2 . Spatial resolution and sampling interval were fixed at 1 m m2 . Test spectra taken at different excitation power densities from 103 to 106 W/cm2 did not show any variation. The emission energy in the peak of each spectrum is shown in Fig. 1~a! and reveals a very narrow distribution of 25 meV ~full width at half maximum, FWHM!. In some locations, however, the peak intensity occurs at energies some 50 meV below. According to the concept of preferred luminescence in localized states3 these lower energy tail states would be expected to dominate the PL intensity distribution. Fig. 1~b! represents the same distribution weighed by the respective peak intensity. In this case, the high energy side is strongly favored over the low energy tail and the distribution exhibits a distribution of 28 meV ~FWHM!. This result indicates that spatial inhomogeneities on the length scale of 1–50 m m do not induce a variation of the peak energy of the most intense PL transitions. In turn, the narrow PL distribution itself defines a clear effective PL band-gap energy. This property can be ex-

FIG. 2. Photoreflectance and photoluminescence of pseudomorphic GaInN/ GaN single heterostructures. Franz–Keldysh oscillations due to the piezoelectric field are clearly observed marking the DOS band gap at i50. For x50.09 the electro-optical function F( h ) is sketched. The luminescence coincides with the PR maximum in i521 within the field induced tunnel transitions. Note the appearance of interference fringes in the PL of some samples due to a smaller probing area.

plained by either the absence of any significant fluctuation or the absence of any efficiency enhancing localization process in our material. Altogether this indicates that homogeneous material, such as presented in this study, can be obtained. IV. PHOTOREFLECTANCE

In total, some 40 samples with different composition have been investigated. PR of five samples with various x is presented in Fig. 2 ~right hand scale, offset for clarity!. The typical signal is in the range of DR/R51023 . At 3.42 eV signal of the GaN layer is seen indicating that the entire thickness of the ternary layer is sampled. Similar narrow features of the excitons in GaN have been obtained in photoreflectance11 and electroreflectance12 and have been used to describe the valence band splitting as a function of strain.13 We ascribe the dominant feature in the range from 3.1 to 2.7 eV to the density of states ~DOS! band gap in the GaInN layer.4,14 A detailed argumentation will be given below. The very large spectral width of the feature increases


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FIG. 3. Comparison of photoreflectance and electroreflectance. A direct correspondence is observed in all features indicating a joint mechanism for the oscillations proposed to lie in the modulation of the internal piezoelectric field.

and the band gap decreases for higher x or c values. Both properties show a clear correlation. We previously interpreted this effect to derive the optical band-gap energy.4 Furthermore, PR oscillations are seen on the high energy side with a decreasing amplitude and period for higher energies within each of the spectra ~note the rescaled portions of the spectra!. These subsidiary oscillations could be observed in about half of the samples studied. In the other samples PR signal below the GaN band gap was in the range of DR/R 51024 . In order to distinguish individual contributions to the PR signal we independently varied the following parameters by about a factor of 10: flux of UV light, flux of white light, modulation frequency, and sampling area. Within these limits there was no observable variation in the obtained spectra. To clarify the modulation process PR and ER in a Ga12x Inx N (x50.2) layer are compared in Fig. 3. The good correspondence in all details indicates a common modulation process. V. ORIGIN OF PR SIGNAL

Oscillatory behavior of the PR has been observed in nitride layers in several cases.11,15,16 In order to identify the nature of the oscillations here below the GaN band gap, the following mechanisms shall be considered and discussed: thickness interference fringes, piezoelectrical thickness variations, thermomodulation, electric field effects acting on a series of discrete electronic levels or on a continuum. Thickness interference fringes whether in the dc or the ac signal are controlled by the GaN epilayer. Examples are seen in the PL of several samples ~Fig. 2! and in PR in the literature, i.e., Refs. 11,16. We find them under high density excitation when only a small sample area is probed.17 For PR, however, we average a larger area of '0.5 cm2 and small thickness variations eliminate any trace of fringes. Assuming a variation of the refractive index in the 2 m m GaN epilayer their separation should be identical in all the samples for a given energy. This is not the case in our data either. Comparing the respective periods such fringes can

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certainly not be the source of the dominant PR features and to a very high probability they cannot be the origin of the subsidiary oscillations either. Piezoelectric thickness variations induced by photoscreening correspond to a variation of the thickness in the model of interference fringes and the arguments are identical to the point above. Such effects can therefore not be the origin of oscillations in our data. Thermomodulation in GaN was shown indeed to produce a wide oscillation below the bandgap but with a clearly different amplitude behavior.15 In addition for all the points above no dependence on x in the thin ternary layer, such as clearly seen in our data, would be expected. Derivative-type spectroscopy of a set of discrete subband levels would result in PR features strongly controlled by the broadening of the individual levels. Spectra should then be very sensitive to the modulation depth. The occurrence of periodic oscillations symmetrical to the zero level would be limited to a very narrow parameter window. This, however, cannot be observed. It is furthermore very unlikely to observe a linewidth in the range of 100 meV caused by a single discrete level. We do not observe features that would indicate that the individual levels were separated by forbidden gaps. Instead all oscillations appear to originate in a single process acting on a continuum of states. We consequently assign the dominant minimum in the PR oscillation to the GaInN DOS band gap and the subsidiary oscillations to be a side effect of the same mechanism. VI. INTERPRETATION OF PHOTOREFLECTANCE

The analysis and interpretation of PR spectra has been performed on a wide variety of compound and alloy systems in the literature14,18 and its theory mainly distinguishes cases of localized ~i.e., excitons, states of spatial confinement! and states of free carriers ~i.e., band electrons!. The primary process of photo modulation is a local variation of the screening conditions leading to a variation of an electric field and quasi Fermi levels. The good correspondence of PR and ER spectra in Fig. 3 indeed indicate that we can perform the interpretation of PR spectra along the established mechanisms for ER. For charge carriers free to move along an electric field F an induced variation DF leads to an energy shift DE band ;DF by means of the Franz–Keldysh effect. Carriers in bound states and states of spatial confinement are not free to follow the field and levels shift merely in a much smaller second order process DE loc;(DF) 2 where DE loc!DE band . The typically narrow energy breadth of localized excitons still allows for a clear detection in modulation spectroscopy. In the vicinity of the GaInN gap narrow excitonic features similar to the GaN signal at 3.42 eV might be expected.18 All the features interpreted in our data, however, extend over some 100 meV and relevant contributions of excitons to the PR have to be excluded. Broad transition bands following the field in second order would be strongly undermodulated producing very small signal only. This furthermore suggests that the strong and broad PR signal at the GaInN gap originates in a first order field effect typically ascribed to free band states.


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Furthermore, transitions between free band states having a dimensionality of three are expected to show electric field induced tail states at lower energy due to spatially indirect tunneling along the electric field ~Franz–Keldysh effect! and an oscillatory behavior on the high energy side ~Franz– Keldysh oscillations, FKOs!. It is generally accepted that the gross features of ER spectra are related to the Franz– Keldysh effect.19 So altogether we propose that the entire modulation of PR in our spectra is induced by ~first order electric field! Franz–Keldysh effect and FKOs. A closed description of the expected signal has been given by Aspnes20 in the form of Airy functions with extrema above, at, and below the band gap. This can be expressed by means of the electro-optical functions F( h ) and G( h ). Above the band gap the PR modulation can be approximated by an exponentially decaying cosine:20

F

1 DR 22G AE2E g } 2 exp R E ~ E2E g ! ~ \Q ! 3/2

FS

3cos

4 E2E g 3 \Q

D G

G

3/2

1x .

~1!

Extrema in E5E i (i>0) are given by integer multiples of p of the phase argument in the cosine function. Index i 50 ( x 5 p ) marks the dominant minimum which in the limit of vanishing damping G corresponds to the DOS band gap E g ~see ticks and labels in Fig. 2!. \Q is the electro-optical energy.21 In a purely descriptive model, line shapes can be approximated by third energy derivatives. In the interest of a full interpretation, however, we shall not perform this approximation here. A very clear example of FKOs has been seen in GaAs where up to 15 extrema are observed covering an energy range of 300 meV.22 Within the model above a field of 0.06 MV/cm can be derived. In that case, however, excitonic features are thought to distort the signal in the range of indices 0 and 1. The scaled function F( h ) is shown together with the PR spectrum for x50.09 ~Fig. 2!. A clear correspondence is seen in all features ~see also Ref. 22!. Spectra for higher x are characterized by a stronger damping of the oscillations on the high energy side but the major features near the gap are well preserved. From the model follows that the observation of higher oscillations is dependent on the homogeneity of the field and the band gap energy in the sample under study. This is expressed by means of the damping energy parameter G. The magnitude of the electric field can be derived from an interpretation of the oscillation behavior. Extrema positions versus index number i are interpreted in Fig. 4 according to Eq. ~1!. Up to six and in some cases eight extrema can be resolved. For each sample points can be approximated by straight lines the slope of which corresponds to (\Q) 3/2, and to the electric field F5(\Q) 3/2A2 m /e\. Herein m 50.2m e is the joint effective DOS mass. Due to the lack of better data we assume it to be constant at the GaN value. This is a valid approximation since m enters only as a square root. The model in Eq. ~1! makes use of the band-gap energy E g . This point should therefore be the origin of the linear interpolation in Fig. 4. In our data, however, E g is only assigned to the

FIG. 4. Interpretation of photoreflectance oscillation extrema according to Aspnes ~see Ref. 20!. A field strength of up to 0.63 MV/cm is observed and attributed to the strain induced piezoelectric field.

dominant minimum and therefore, just as the other points, subject to error bars. Again comparing to the case of GaAs where excitonic contributions distort the description of the first extrema a good linear fit of all extrema is found here in Fig. 4 including index 1 and, in extrapolation to E g , also index 0. This again supports the irrelevance of typical excitons in our case. From the slope of the interpolation in Fig. 4 we identify very large electric fields in the range of F50.1920.63 MV/ cm. Under these huge fields excitons must be ionized and consequently can not contribute in the PR spectrum. F shows a tendency to increase with strain in the layer in parallel to x. The samples with x50.12 and x50.16, however, show similar field strength indicating variations in the screening or polarization conditions of the samples. The complete study of all the samples furthermore supports this point ~see Ref. 10!. F is not affected by a variation of the photoexcitation power by a factor of 10 and together with the very high value of F we conclude that this field is not induced by the PR mechanism itself. Instead we deduce that F is a constant field across the ternary layer and we attribute it to the piezoelectric field induced by the biaxial stress conditions. It should be noted that electric fields caused by composition gradients cannot be distinguished here from piezoelectric effects. As of the presently available data, piezoelectric and polarization coefficients of GaN and InN are very similar23 and their compositional variations can certainly be neglected with respect to the piezoelectric part. Very large piezoelectric constants of the nitrides have been observed in experiment24 and theory.23 Our results based on a larger number of samples indicate that previous values overestimate the piezoelectric effects in this system.25 From an interpretation of the quantum confined Stark effect in GaInN quantum wells Takeuchi et al.26,27 recently found a larger value of 1.2 MeV/cm for


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x50.16. The origin of the discrepancy between single heterostructures and quantum wells is yet unclear and subject of ongoing work. A possible source of discrepancy may lie in the proper description of the quantum well pn structures. The interpretation of damping in the oscillations is affected by the lag of detailed knowledge of the DOS high above the bandedges. Assuming a constant mass, values for G/\Q can be estimated in the range of 0.01 (x50.09), 0.1 (x50.12,0.13,0.16), and 1 (x50.19). This indicates that electric field effects clearly dominate the details of the band structure for the lower composition range while around x 50.2 disorder of any kind gains in importance in the interpretation of PR spectra in these single heterostructures. VII. ELECTRIC FIELD-INDUCED TAILSTATES

A direct consequence of the internal fields is the effective band-gap shrinking. The field-induced tilting of the bands in real space creates a triangular tunnel barrier for spatially indirect transitions below the gap ~Franz–Keldysh effect, photon assisted tunneling!.28 This can be expressed by an exponentially decaying effective DOS with a slope parameter determined by \Q and therefore by the electric field. As described in theory20 by means of the electro-optical functions another PR maximum i521 occurs within this tail. Within the models developed in the literature additional possible admixture of the real part of the dielectric function may further complicate a more accurate description of the line shape in this energy range. In all our spectra extrema i521 and i511 are found symmetrically around i50 and we propose that i521 be the tail state’s equivalent of i511.20 The apparent localization in i521, E g 2E 21 , therefore corresponds to E 11 2E 0 5(3 p /4) 2/3\Q. Assuming a constant electric field across this energy the effective DOS has decayed to e 2 p 54% and for the given high field values the base of the tunnel barrier closely corresponds to an effective Bohr radius a r 53 nm. This limited model furthermore suggests that the maximum in i521 is the extension of the oscillations into the evanescent tail states. Room temperature PL is also shown in Fig. 2 ~left hand scale!. PL is found to exhibit a significant redshift with respect to the above defined band gap E g from PR. Similar observations in thin GaInN layers have been reported by Chichibu et al.3 Several models have been proposed to explain the origin including the formation of In-rich regions and clusters.3,29 From the spatially resolved PL we find no evidence for significant variation of the effective PL band gap. The observed distribution of 28 meV (x50.187) cannot account for a redshift as large as 130–150 meV. In our data the discrepancy closely ~i.e., 620 meV! corresponds to one half-period of the PR oscillation and the maximum of PL lies very close to the maximum labeled i521 ~Fig. 2!. It therefore corresponds to a transition involving the electric field-induced tail states ~Franz–Keldysh effect!. In a simplified model the redshift between the PL gap and PR gap can be well approximated by the band gap tilting across the dimension of the excited particle, i.e., the electron-hole pair: DE5Fea r 535

FIG. 5. Schematic of the bandstructure in the presence of heteroepitaxial strain inducing a piezoelectric field. The DOS is modulated by the electric field ~Franz–Keldysh effect! inducing tail states below the gap. A charge separation of photo carriers is expected in the field.

2170 meV, where a r is the effective reduced Bohr radius. Within the range of our samples (0,x,0.2) our model can well explain the observed redshift in the material studied. We do not find evidence for discrete levels possibly associated with In-rich clusters. Other broadening mechanisms such as alloying disorder and composition fluctuations can not be excluded but do not dominate the above effects within the composition range considered in our material. We find that electric fields of this strength and the corresponding quantization energies dominate over exciton binding energies ~in GaN! and higher order multiparticle effects. Consequently, the piezoelectric field gains a high priority in the description of the band and defect structure. A schematic of the band structure neglecting band bending is given in Fig. 5. At the heteroepitaxial interface, the step in strain induces large fixed piezoelectric charges and a band tilting. Photocarriers will be separated in the field and drift in opposite directions. The charge separation is limited by the built-up of a diffusion field between the accumulated carriers that will compensate the piezoelectric driving force. The separation of carriers should lead to long recombination lifetimes and possibly is the origin of long nonexponential decay behavior observed in time resolved experiments.30,31 This charge separation may also lead to persistent conductivity observed in a wide range of nitride films and structures. VIII. CONCLUSION

In summary, the presence of large piezoelectric fields of up to 0.63 MV/cm (615 %! in pseudomorphic GaInN/GaN single heterostructures has been directly observed by clearly resolved Franz–Keldysh oscillations in PR spectroscopy. Comparing damping and electro-optic energy we identify conditions of the high field regime although the observed fields are found to be smaller than previously expected. We confirm to find a redshift of PL versus the PR band-gap energy and propose that this originates in spatially indirect transitions between the electric field tilted band edges. Electric field effects are therefore proposed to control the electronic band edges, recombination and transport mechanisms. We find no evidence that this PL would be associated with


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excitons nor do we find that the PL near the band gap is dominated by large inhomogeneities in this material. ACKNOWLEDGMENTS

This work was partly supported by the Ministry of Education, Science, Sports and Culture of Japan ~Contract Nos. 09450133 and 09875083, and High-Tech Research Center Project! and JSPS Research for the Future Program in the Area of Atomic Scale Surface and Interface Dynamics under the project of Dynamic Process and Control of Buffer Layer at the Interface in a Highly-Mismatched System. 1

For a recent review, see I. Akasaki and H. Amano, Jpn. J. Appl. Phys., Part 1 36, 5393 ~1997!. 2 H. Amano, M. Kitoh, K. Hiramatsu, N. Sawaki, and I. Akasaki, Jpn. J. Appl. Phys., Part 2 28, 2112 ~1989!. 3 S. Chichibu, T. Azuhata, T. Sota, and S. Nakamura, Appl. Phys. Lett. 70, 2822 ~1997!. 4 C. Wetzel, T. Takeuchi, S. Yamaguchi, H. Katoh, H. Amano, and I. Akasaki, Appl. Phys. Lett. 73, 1994 ~1998!. 5 N. A. El-Masry, E. L. Piner, S. X. Liu, and S. M. Bedair, Appl. Phys. Lett. 72, 40 ~1998!. 6 I. H. Ho and G. B. Stringfellow, Appl. Phys. Lett. 69, 2701 ~1996!. 7 H. Amano, T. Takeuchi, S. Sota, H. Sakai, and I. Akasaki, Mater. Res. Soc. Symp. Proc. 449, 1143 ~1997!. 8 H. Amano, N. Sawaki, I. Akasaki, and T. Toyoda, Appl. Phys. Lett. 48, 353 ~1986!. 9 T. Takeuchi, H. Takeuchi, S. Sota, H. Sakai, H. Amano, and I. Akasaki, Jpn. J. Appl. Phys., Part 2 36, L177 ~1997!. 10 C. Wetzel, S. Nitta, T. Takeuchi, S. Yamaguchi, H. Amano, and I. Akasaki, MRS Bull. 3, 31 ~1998!. 11 W. Shan, T. J. Schmidt, X. H. Yang, S. J. Hwang, J. J. Song, and B. Goldenberg, Appl. Phys. Lett. 66, 985 ~1995!. 12 C. F. Li, Y. S. Huang, L. Malikova, and F. H. Pollak, Phys. Rev. B 55, 9251 ~1997!. 13 A. Shikanai, T. Azuhata, T. Sota, S. Chichibu, A. Kuramata, K. Horino, and S. Nakamura, J. Appl. Phys. 81, 417 ~1997!.

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C. Alibert, G. Bordure, A. Laugier, and J. Chevallier, Phys. Rev. 6, 1301 ~1972!. 15 Y. Li, Y. Lu, H. Shen, M. Wraback, M. G. Brown, M. Schurman, L. Koszi, and R. A. Stall, Appl. Phys. Lett. 70, 2458 ~1997!. 16 C. Wetzel, H. Amano, and I. Akasaki, Proc. IEEE 24th International Symposium Compound Semiconductor, San Diego, CA, 7–11 September 1997, p. 239. 17 In the strongly focused condition of the micro PL beam divergence again eliminates interference fringes. 18 F. H. Pollak and H. Shen, Mater. Sci. Eng., R. 10, 275 ~1993!, and references therein. 19 M. Cardona, K. L. Shaklee, and F. H. Pollak, Phys. Rev. 154, 696 ~1967!. 20 D. E. Aspnes, Phys. Rev. B 10, 4228 ~1974!; Phys. Rev. 153, 972 ~1967!. 21 M. Cardona, in Solid State Physics, Suppl. 11, edited by F. Seitz, D. Turnbull, and H. Ehrenreich ~Academic, New York, 1969!. 22 X. Yin, H.-M. Chen, F. H. Pollak, Y. Cao, P. A. Montano, P. D. Kirchner, G. D. Pettit, and J. M. Woodall, J. Vac. Sci. Technol. A 10, 131 ~1992!. 23 F. Bernardini, V. Firoentini, and D. Vanderbilt, Phys. Rev. Lett. 79, 3958 ~1997!. 24 A. D. Bykhovski, V. V. Kaminski, S. Shur, Q. C. Chen, and M. A. Khan, Appl. Phys. Lett. 68, 818 ~1996!. 25 C. Wetzel, T. Takeuchi, H. Amano, and I. Akasaki, Mater. Res. Soc. Symp. Proc. 512, 181 ~1998!. 26 T. Takeuchi, S. Sota, M. Katsuragawa, M. Komori, H. Takeuchi, H. Amano, and I. Akasaki, Jpn. J. Appl. Phys., Part 2 36, L382 ~1997!. 27 T. Takeuchi, C. Wetzel, S. Yamaguchi, H. Sakai, H. Amano, I. Akasaki, Y. Kaneko, S. Nakagawa, Y. Yamaoka, and N. Yamada, Appl. Phys. Lett. 73, 1691 ~1998!. 28 K. H. Bo¨er, Survey of Semiconductor Physics ~Van Nostrand Reinhold, New York, 1990!, p. 968. 29 Y. Narukawa, Y. Kawakami, M. Funato, S. Fujita, S. Fujita, and S. Nakamura, Appl. Phys. Lett. 70, 981 ~1997!. 30 C. Wetzel, W. Walukiewicz, and J. W. Ager III, Mater. Res. Soc. Symp. Proc. 449, 567 ~1997!. 31 B. Monemar, J. P. Bergman, N. Saksulv, G. Pozina, H. Amano, and I. Akasaki, Proceedings of the 2nd International Symposium on Blue Laser and Light Emitting Diodes, Chiba, Japan, 29 September–2 October, 1998, p. 210.
