Introduction

There is considerable current interest in the (InGa)N alloy system because (InGa)N layers form the active region of most of the present green-to-violet light emitting devices based on group III-nitrides [Reference Nakamura and Fasol1]. However, in spite of the technological importance of (InGa)N, so far published data on the composition dependence of the band gap energy of the (InGa)N alloy show significant scatter, with values for the bowing parameter ranging from 1.02 eV [Reference Shan, Little, Song, Feng, Schurmann and Stall2] to 3.5 eV [Reference McClusky, Van de Walle, Master, Romano and Johnson3]. For an accurate determination of the composition dependence of the (InGa)N band gap, the gap energy has to be derived from photoreflection (PR) [Reference Shan, Little, Song, Feng, Schurmann and Stall2] or spectroscopic ellipsometry (SE) [Reference Wagner, Ramakrishnan, Behr, Obloh, Kunzer and Bachem4] measurements rather than from photoluminescence (PL) data. This is because the (InGa)N near band-edge PL spectrum is Stokes-shifted relative to the band edge, leading to an underestimate of the gap energy [Reference Chichibu, Azuhata, Sota and Nakamura5]. The (InGa)N alloy composition should be assessed not only by X-ray diffraction but also by chemical analysis, as strain effects complicate the composition analysis by X-ray diffraction, requiring precise knowledge of the elastic parameters of the (InGa)N. (InGa)N layers grown on top of GaN have been found to be strained to the in-plane lattice parameter of the underlying GaN for layer thicknesses and compositions up to, at least, 100 nm and x=0.15, respectively [Reference Amano, Takeuchi, Sota, Sakai and Akasaki6,Reference Scholz, Off, Kniest, Görgens and Ambacher7].

In the present study pseudomorphically strained (InGa)N layers on GaN, grown by metal-organic chemical vapor deposition (MOCVD), were analyzed by PR spectroscopy and SE to determine the composition dependence of the (InGa)N band gap energy. Secondary ion mass spectroscopy (SIMS) and high-resolution X-ray diffraction (HRXRD) were used to assess the (InGa)N layers` composition and strain, respectively.

Experiment

The hexagonal (InGa)N-on-GaN samples used in this study were grown by low-pressure MOCVD on c-plane 2′′ sapphire substrates. Details on growth can be found in Ref. 8. Data will be presented on a series of samples with varying In content (x≤0.15) and (InGa)N layer thicknesses ranging from 15 to 60 nm, grown on top of a 2 to 3 μm thick GaN layer. The (InGa)N composition was determined by SIMS using appropriate standards calibrated by energy dispersive X-ray analysis (EDX).

Selected samples were further analyzed by HRXRD using both the symmetric 00.6 and the asymmetric 10.5 reflections, in order to determine the c and a lattice parameters parallel and perpendicular to the growth axis. Fig. 1 shows a plot of c versus a for (InGa)N layers with a thickness of 30 nm as well as for the underlying GaN. All the a lattice parameters obtained for the (InGa)N are close to that of the GaN layer, indicating that the (InGa)N layers are under biaxial in-plane compression due to pseudomorphic growth. The GaN layer itself is also under slight compression due to the thermal mismatch with respect to the sapphire substrate. Using linear interpolations between the c and a lattice parameters of unstrained GaN [Reference Balkas, Baskeri and Davis9] and InN [Reference Kubota, Kobayashi and Fujimoto10], the composition dependent lattice parameters of unstrained (InGa)N are obtained, shown in Fig. 1 by the short-dashed line. Based on the elastic parameters given by Wright [Reference Wright11], the effect of in-plane biaxial compressive or tensile strain on c and a can be calculated, as indicated by the dash-dotted lines for In contents of x=0, 0.05, 0.10, and 0.15. To extract information on the In content of the present strained (InGa)N layers, the measured data have to be projected parallel to the dash-dotted lines onto the c-versus-a curve for unstrained (InGa)N (short-dashed line), as indicated by arrows. The (InGa)N alloy composition deduced in this way agrees for x<0.1 within 10% with the SIMS data. For the highest In contents the HRXRD analysis yields a somewhat lower In content than SIMS, e.g. x=0.124 (HRXRD) versus x=0.147 (SIMS). Considering the large uncertainties in the (InGa)N elastic parameters required for the HRXRD based analysis of the alloy composition for strained (InGa)N, the SIMS data are used throughout the remaining paper.

Fig. 1: c versus a lattice parameters as determined by HRXRD for 30 nm thick In_xGa_1−xN layers on GaN (filled squares) with different In content x. Error bars are indicated. For comparison c and a lattice parameters of the underlying GaN are also shown (open circles). The full vertical line marks the a lattice parameter of unstrained GaN. The short-dashed line indicates the calculated cversus-a relation for unstrained (InGa)N. For further details see text.

Results and Discussion

Room-temperature PR spectra are plotted in Fig. 2 for three different In_xGa_1−xN-on-GaN samples with an (InGa)N layer thickness of 30 nm and In contents of x=0.02, 0.08, and 0.15. Besides the very intense and spectrally sharp fundamental gap resonance of GaN at 3.43 eV, the corresponding resonance of the (InGa)N is well resolved at lower energies as a derivative-like structure. The (InGa)N resonance shows a low-energy shift and a significant broadening with increasingIncontent. Its line shape is somewhat distorted by Fabry-Perot interference oscillations. The band gap energy of the (InGa)N was deduced from a fit to the PR resonance [Reference Aspnes12], as shown by the dashed curves. The resulting band gap energies E_G are plotted in Fig. 3 versus the In content determined by SIMS (full circles). The experimental data can be fitted by the linear relationship E_G(x)=3.43-3.28·x (eV) (x≤0.15), as indicated by the full line in Fig. 3. There is fair agreement with the composition dependence of the (InGa)N band gap energy reported by McClusky et al. [Reference McClusky, Van de Walle, Master, Romano and Johnson3] (E_G(x)=3.42-3.93·x (eV)), which was obtained from optical transmission experiments on 0.25 μm thick In_xGa_1−xN layers (x≤0.12) strained to the in-plane lattice parameter of the underlying GaN. For comparison the PL peak energy at room-temperature is also plotted in Fig. 3 (filled squares). There is an increasing Stokes-shift between the band gap energy derived from the PR measurements and the PL peak energy upon increasing the In content. Alloy fluctuations [Reference Chichibu, Azuhata, Sota and Nakamura5] and the confined Stark effect due to piezoelectric fields [Reference Takeuchi, Sota, Katsurgawa, Komori, Takeuchi, Amano and Akasaki14] have been discussed as possible mechanisms responsible for this Stokes-shift.

Fig. 2: Room-temperature PR spectra of 30 nm thick In_xGa_1−xN layers on GaN with various In contents x given in the figure. Dashed curves indicate fits to the (InGa)N fundamental gap resonance.

Fig. 3: Composition dependence of the room-temperature band gap energy of strained In_xGa_1−xN layers on GaN as obtained by PR spectroscopy (filled circles), and after correction for the strain induced band gap shift (open circles, see text). (InGa)N alloy composition was determined by SIMS. The full and dashed curves indicate linear and quadratic fits to EG for strained In_xGa_1−xN and for numerically relaxed In_xGa_1−xN, respectively. For comparison, roomtemperature PL peak positions are also shown (filled squares).

As pointed out by McClusky et al. [Reference McClusky, Van de Walle, Master, Romano and Johnson3], the strain induced shift ΔE of the (InGa)N band gap has to be subtracted from the present E_G(x) data obtained on pseudomorphic (InGa)N films on GaN, to obtain the composition dependence of the band gap energy of unstrained (InGa)N. Using the relationship for the strain induced band gap shift ΔE=1.02·x (eV) given in Ref. 3, the expected band gap energies of unstrained (InGa)N have been calculated, which are shown in Fig. 3 by open circles. Taking a value of 1.89 eV for the room-temperature band gap energy of InN [Reference Tansley and Foley15], the present data on the composition dependence of the band gap energy of unstrained (InGa)N can be fitted by E_G(x)=3.43−4.74·x+3.2·x² (eV). The bowing parameter of 3.2 eV compares favorably well with that of 3.5 eV calculated in Ref. 3 for x=0.125. There is also consistency with a very recent study by Wetzel et al. [Reference Wetzel, Takeuchi, Yamaguchi, Katoh, Amano and Akasaki16], which reports on the basis of a combined HRXRD and PR spectroscopic analysis a bowing parameter of 3.8 eV for unstrained In_xGa_1−xN (x≤0.2).

It has been shown recently that SE measurements allow a determination of the (InGa)N band gap energy too [Reference Wagner, Ramakrishnan, Behr, Obloh, Kunzer and Bachem4]. Fig. 4 shows the real <ε₁> and imaginary part <ε₂> of the pseudo-dielectric function of a 60 nm thick In_0.1Ga_0.9N-on-GaN layer. For comparison the pseudo-dielectric function spectrum of a high-quality bulk-like GaN reference layer is also shown, which is very similar to that reported in Ref. 17 for hexagonal GaN. A pronounced peak in <ε₁> and minimum in <ε₂> at 3.40 eV marks the superposition of the A and B excitonic transitions at the fundamental band gap of GaN [Reference Fischer, Shan, Song, Chang, Horning and Goldenberg18,Reference Yamaguchi, Mochizuki, Sunakawa and Usui19]. At the high-energy side of this peak a shoulder appears in <ε₁> which is tentatively assigned to the C excitonic transition. The oscillations observed in the pseudo- dielectric function spectrum for photon energies below the fundamental energy gap are caused by Fabry-Perot interferences due to multiple internal reflections in the epitaxial layer. In the <ε₁> spectrum of the (InGa)N-on-GaN sample, the fundamental gap interband transition of the GaN is attenuated. A new peak appears in both <ε₁> and <ε₂> at about 3.07 eV, which is in the transparency region of the GaN. The new peak is assigned to the fundamental gap interband transition of the In_0.1Ga_0.9N.

Fig. 4: Real <ε1> and imaginary part <ε2> of the pseudo-dielectric function spectrum of a 60 nm thick In0.1Ga0.9N layer on GaN and, for comparison, of a GaN reference sample. Data were derived from room-temperature SE measurements performed at an incident angle of 75° and a polarizer-azimuth of 30°.

This assignment is confirmed by a direct comparison of the energy position of the (InGa)N related peak in the <ε₁> spectrum with the band gap energy derived from the PR measurements. In Fig. 5 the (InGa)N peak position in <ε₁> is plotted versus the (InGa)N band gap energy for (InGa)N-on-GaN samples with In concentrations up to x=0.15 and (InGa)N layer thicknesses of 15, 30, and 60 nm. There is an almost one-to-one correspondence between these two quantities with the peak in <ε₁> being on the average only slightly higher in energy than the band gap.

Fig. 5: Energy position of the (InGa)N related peak in the <ε1> spectrum versus (InGa)N band gap energy derived from PR measurements. Data are shown for different (InGa)N layer thicknesses d given in the figure. The dashed line indicates a one-toone correspondence.

Conclusion

The composition dependence of the band gap energy of strained In_xGa_1−xN on GaN (x≤0.15) has been studied by PR spectroscopy and SE, using SIMS in combination with HRXRD for an accurate determination of the (InGa)N layers` composition and strain. The composition dependence of the band gap energy of (InGa)N strained to the in-plane lattice parameter of the underlying GaN was found to be given by E_G(x)=3.43-3.28·x (eV). After correction for the strain induced band gap shift, a bowing parameter of 3.2 eV was obtained for the composition dependence of the band gap of unstrained (InGa)N.