1. Introduction
In spite of recent successes in device development, Reference Goldenberg, Zook and Ulmer[1], Reference Nakamura, Mukai and Senoh[2], Reference Nakamura, Senoh, Nagahama, Iwasa, Yamada, Matsushita, Kiyoku and Sugimoto[3], very little is known about the optical constants of the AlxGa1−xN system, especially for Al contents larger than 50 %. Following a quite extensive analysis of AlxGa1−xN grown by molecular beam epitaxy by Yoshida et al. Reference Yoshida, Misawa and Gonda[4] more than a decade ago, significant improvements concerning deposition techniques and the structural quality of epitaxial films have been made, so that a critical revision of the optical constants in the AlGaN alloy system is certainly justified. A precise knowledge of optical constants is particulary important in view of the use of AlGaN films in optical filters, light-emitting and laser diodes.
The fabrication of highly reflective and smooth mirrors for a horizontal cavity laser based on hexagonal Group-III nitrides on sapphire substrate is complicated because of the difficulty of cleaving for parallel mirrors. Therefore a vertical cavity surface emitting laser (VCSEL) operating from the UV to the blue spectral region could be recognized as a useful device for optical imaging systems such as full-color display panels, high density optical recording and photolithography Reference Honda, Katsube, Sakaguchi, Koyama and Iga[5].
Highly reflective mirrors are required for blue-emitting low-threshold laser operation and can be realized by dielectric multilayer mirrors or AlGaN-based Bragg reflectors. Nakamura et al. realized stripe- and ridge-geometry InGaN multi-quantum well laser diodes, which showed stimulated emission at a wavelength of 411 nm under pulsed current injection at room temperature Reference Nakamura, Senoh, Nagahama, Iwasa, Yamada, Matsushita, Sugimoto and Kiyoku[6]. High-reflection facet coatings (reflectivity 70%) composed of 4 pairs of quarter-wave TiO2/SiO2 dielectric multilayers were used to reduce the threshold current. The absorption edge of TiO2 is about 3.1 eV (400 nm) and therefore can not be used for the fabrication of mirrors or filters working in the near UV. AlGaN Bragg reflectors for GaN-based VCSELs can be fabricated using Alx1Ga1−x1N/Alx2Ga1−x2N alternating layers, which can be deposited monolithically during the epitaxial growth of the laser diode.
To calculate the design and the reflectivity of Bragg reflectors for different emission wavelengths of the laser diodes, the refractive index for AlxGa1−xN is needed. Therefore we have determined the refractive index from transmission and ellipsometry measurements. To evaluate the bandgap and to take the self-absorption of the Bragg reflectors into account, the absorption coefficients of epitaxial AlGaN films covering the whole range of composition were also determined.
2. Experiments
Epitaxial films of AlxGa1−xN were grown on c-plane sapphire by plasma-induced molecular beam epitaxy (PIMBE). The Al molar fraction x was varied in approximately equidistant steps between 0 ≤ x ≤ 1. Conventional Al and Ga effusion cells and a r.f.-plasma source (Oxford Applied Research CARS) for nitrogen radicals were used in the deposition process. The nominally undoped epitaxial films were grown without a buffer layer at temperatures between 810 °C and 1000 °C. The FWHM of the (002) rocking curve observed by X-ray diffraction increased from 0.18° to 0.43° by increasing the Al content from x = 0 to x = 0.8. For alloys with x > 0.8, the substrate temperature was changed from 810 to 1000 °C, which resulted in improved structural properties of the films and a FWHM rocking curve of 0.2° Reference Angerer, Ambacher, Dimitrov, Metzger, Rieger and Stutzmann[7]. The thickness (≈ 1 μm) and the growth rate (between 0.5 and 0.6 μm/h) of the films were determined by scanning electron microscopy with an accuracy of ±10 nm Reference Brunner, Angerer, Bustarret, Freudenberg, Höpler, Dimitrov, Ambacher and Stutzmann[8]. The AlxGa1−xN films had rms surface roughnesses between 5 and 15 nm, as obtained by atomic force microscopy. The Al molar fraction was examined by elastic recoil detection analysis (ERDA) Reference Angerer, Brunner, Freudenberg, Ambacher, Stutzmann, Höpler, Metzger, Born, Dollinger, Bergmaier, Karsch and Körner[9]. The ERDA measurements were generaly in good agreement with estimates based on the lattice constants a0 and c0 (taking into account a biaxial stress, σ ≤ 0.4 GPa), obtained by high resolution X-ray diffraction (HRXRD) using Vegard's law Reference Angerer, Ambacher, Dimitrov, Metzger, Rieger and Stutzmann[7], Reference Vegard[10].
In order to determine the absorption coefficient versus photon energy at room temperature, photothermal deflection spectroscopy (PDS) Reference Ambacher, Rieger, Ansmann, Angerer, Moustakas and Stutzmann[11], Reference Ambacher, Brunner, Dimitrov, Stutzmann, Scholz and Sohmer[12] and transmission measurements, using a Perkin Elmer Lambda 900 double beam spectrometer with a resolution of 1 nm, have been carried out in the spectral range from 195 nm to 2000 nm. For the calculation of the absorption coefficient α from the transmission data, we used the procedure described by Freeman and Paul Reference Freeman and Paul[13]. The reflectivity between the epitaxial AlGaN film and the sapphire substrate was determined using the results of Malitson, who developed a Sellmeir equation to permit calculation of the index of refraction of sapphire between 200 and 6000 nm. The index of refraction, n, of the Group-III nitrides was obtained from examination of interference fringe minima and maxima in the transmission spectrum Reference Brunner, Angerer, Bustarret, Freudenberg, Höpler, Dimitrov, Ambacher and Stutzmann[8].
The dielectric function of the epitaxial AlGaN films were measured in the spectral range 3-25 eV using an ellipsometer operating with synchrotron radiation at the Berlin electron storage ring BESSY I Reference Wethkamp, Wilmers, Esser, Richter, Ambacher, Angerer, Jungk and Cardona[14]. A more detailed description of the general experimental design, the beamline polarization and the polarization properties of the triple-reflection-analyser is given in Ref. Reference Barth, Johnson and Cardona[15].
3. Results and Discussion
3.1 Dependence of the Absorption Edge on the Al molar Fraction
Figure 1 shows the transmission measurements of different epitaxial AlxGa1−xN films versus photon energy at room temperature. To determine the bandgap of the samples, the absorption coefficients α of the AlGaN films were calculated from the transmission and PDS measurements, Figure 2. The effective bandgap of the AlxGa1−xN films was defined as the photon energy E4.8 at which the absorption coefficient equals a value of 104.8 cm−1. The physical justification for this definition is given by photoluminescence and reflection measurements of GaN, showing that E4.8 lies between the photon energy of the free excitons FXA and FXB in the temperature range between 5 K and 300 K Reference Brunner, Angerer, Bustarret, Freudenberg, Höpler, Dimitrov, Ambacher and Stutzmann[8]. Moreover, the effective bandgap E4.8(GaN) follows the temperature dependence of the free excitons and increases from 3.420 eV to 3.480 eV with decreasing temperature from 300 K to 5 K. The energy offset E4.8(GaN)-FXA and E4.8(GaN)-FXB is approximately 2 and −5 meV. With increasing Al content x, the effective bandgap E4.8 shifts toward higher energies, following the phenomenological quadratic dependence on the Al molar fraction Reference J.C. Phillips[16]:
Figure 1. Room temperature transmission measurements of 1 μm thick epitaxial wurtzite AlGaN films grown by plasma-induced molecular beam epitaxy.
Figure 2. Absorption coefficient α of epitaxial AlxGa1−xN films versus photon energy measured by transmission and photothermal deflection spectroscopy at room temperature. The insert shows the dependence of the absorption edge determined from E4.8 (energy at which the absorption coefficient is equal to 104.8 cm−1) versus the Al content of the films.
For E4.8(GaN) and E4.8(AlN) we observed 3.42 and 6.13 eV at room temperature. From the energy position of the absorption edge versus Al concentration a bowing parameter b = 1.3 ± 0.2 eV can be determined (see insert in Figure 2). This value agrees within the experimental error with the bowing parameter of 1.0 ± 0.3 eV obtained by Koide Reference Koide, Itoh, Khan, Hiramatu, Sawaki and Akasaki[17]. The biaxial stress σ in the samples, due to mismatch of the lattice constants and the thermal expansion coefficients of the substrate and epitaxial films, was calculated from the results of HRXRD-measurements and is estimated to be below 0.4 GPa. Therefore, an upper limit of 10 meV for the stress-induced shift of the bandgap can be calculated using dEg/dσ ≈ 25 meV/GPa Reference Rieger, Metzger, Angerer, Dimitrov, Ambacher and Stutzmann[18]. As a consequence, an upper limit of the bandgap shift towards higher energies caused by biaxial compressive stress should be approximately 10 meV at room temperature.
3.2 Index of refraction
In Figure 3 the index of refraction (calculated from transmission measurements) versus photon energy is shown for different AlxGa1−xN films (T = 300 K) in the range between 1.5 and 5.5 eV. The expected increase of the index of refraction with increasing photon energy towards the bandgap, and the overall decrease with Al content, is observed for the whole series of samples. The dependence of refractive index on energy obtained for the GaN and the Al0.1Ga0.9N samples is in excellent agreement with the results of Amano et al. Reference Amano, Watanabe, Koide and Akasaki[19], Reference Akasaki and Amano[20] and Vidal et al. Reference Vidal, Ramirez-Flores, Navarro-Contreras, Lastras-Martinez, Powell and Greene[21]. For 0.7 < x < 1 there is a tendency for the experimentally determined refractive indexes to be too low, most probably due to a decrease of the effective film density compared to an ideal crystal. Thus, a decrease of the index of refraction for example from 2.05 to 2.00 would require a density deficit of approximately 2 %.
Figure 3. Index of refraction n of different AlxGa1−xN films versus photon energy at room temperature (solid lines are fits to equation (2a)).
For the case of the direct absorption edge of AlxGa1−xN films the index of refraction can be calculated for energies lower than the bandgap by Reference Yu and Cardona[22]:
where:
εr is the dielectric function, mC and mV are the effective masses of the conduction and valence band, me and e are the electron mass and charge, C is photon energy independent for a fixed Al-content, hν is the photon energy, 8π2|PCV(x)|2/mehν is the oscillator strength of the optical transition and Eg(x) is the direct bandgap of AlxGa1−xN. We used equation (2.a) to fit the measured index of refraction n(hν,x) for different AlxGa1−xN films, by using the measured effective bandgap E4.8(x) = Eg(x) and A(x) and C(x) as fitting parameters. We found a linear increase of C(x) with increasing Al-content of the films and a square root dependence of A(x) (motivated by (2a): A(x) ∝ Eg(x)1/2 ∝ x1/2) which can be described by the following equations:
The combination of the equations (2a), (3a) and (3b) provides an analytical relation for the index of refraction as a function of photon energy (hν < Eg) and Al content (0 ≤ x ≤ 1).
The reflectivity of the AlGaN films above the bandgap (hν ≥ Eg) can be calculated from ellipsometry mesurements, which allow the determination of the effective dielectric function <ε(hν)> = <ε1(hν)> + i <ε2(hν)>. The quantity <ε2(hν)> is a measure of the absorption and is related to the joint density of states. It therefore is directly connected with the electronic bandstructure of the material Reference Wethkamp, Wilmers, Esser, Richter, Ambacher, Angerer, Jungk and Cardona[14]. The refractive index n(hν) = n(hν) + ik(hν) and the reflectivity were calculated using the equations: ε1 = n2 − k2, ε2 = 2nk. In Figure 4 and Figure 5, the dielectric functions of GaN and AlGaN are shown in the energy region between 2.5 and 17 or 25 eV.
Figure 4. Dielectric functions and reflectivity of GaN. Labeled are the interband transitions Eg, E1, E2, and E3, as well as the broad feature above 13 eV, which is not assigned to one distinct point or region of the Brillouin zone.
Figure 5. Imaginary parts of dielectric functions of hexagonal AlGaN films. Datasets are shifted for clarity.
Below the fundamental gaps, Fabry-Perot interferences were observed and the absorption maxima labeled with E1, E2 and E3 could be associated with interband transitions at specific points or regions of the Brillouin zone (discussed in more detail in Ref. Reference Wethkamp, Wilmers, Esser, Richter, Ambacher, Angerer, Jungk and Cardona[14], Reference Logothetidis, Petalas, Cardona and Moustakas[23] and Reference Olson, Lynch and Zehe[24]). The calculated absorption coefficients and reflectivities of the GaN and AlGaN films for photon energies above the bandgap can be applied to determine the influence of thin buffer layers on the optical properties of AlGaN-based Bragg reflectors and filters.
3.3 AlGaN-based Bragg Reflectors
The measured refractive index difference of AlN and GaN is comparable to that of AlAs and GaAs Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25]. Therefore, AlGaN-based Bragg reflectors (DBR) are considered to be suitable for mirrors for the GaN-based VCSEL operating in the UV spectral regions.
When calculating the design of Bragg reflectors for different wavelengths, two limitations for practical Alx1Ga1−x1N/Alx2Ga1−x2N stacked layers have to be taken into account. First the self-absorption of the reflector for a given wavelength α(x1)mdx1 (α(x1): absorption coefficient of Alx1Ga1−x1N, m: number of periods of the DBR, dx1: thickness of one Alx1Ga1−x1N layer) has to be below 103 to achieve reflectivies above 90%. This limits the minimum Al content of the Alx1Ga1−x1N layers with the lower bandgap. We determined x1, assuming that the dependence of the absorption coefficient on photon energy for the AlGaN films (determined by PDS and transmission spectroscopy) are also valid for stacked layers with thicknesses between 30 and 60 nm. Stress or quantum confinement effects will cause a shift of the bandgap to higher energies and will require raising the Al content x1. Stress-induced defects or high defect densities at the interfaces of the DBR will cause an increase of subbandgap absorption and require an increase of x1, which was not taken into consideration in our calculation. Second, the lattice mismatch (a0(x1)−a0(x2))/a0(x1) between the Alx1Ga1−x1N and Alx2Ga1−x2N layers was chosen to be equal or below 1%, which is a prerequisite for epitaxial growth of heterostructures with 20 or more periods of Alx1Ga1−x1N/Alx2Ga1−x2N layers. This assumption limits the maximum Al content x2 for a given x1 or a given wavelength λ at which the reflectivity of the DBR should have a maximum.
The theoretical reflectivity of the DBRs for different wavelength was calculated using Equations 2 and 3 for the wavelength dependence of the refractive index for different AlGaN compositions applying a matrix method for the multilayer system Reference Rochus[26]. The structure of the calculated DBRs consisted of periodically repeated Alx1Ga1−x1N/Alx2Ga1−x2N layers grown on a c-plane sapphire substrate, using a 20 nm thick GaN buffer layer. In Figure 6, the Al contents x1 and x2 are shown versus wavelength, which were calculated (under consideration of self-absorption and lattice mismatch) to give the highest reflectivity. Below 370 nm, the Al contents x1 and x2 increase with decreasing wavelength. At 290 nm, x2 reaches the value for AlN and a further increases of photon energy causes a decrease in the difference between the refractive indices n(x1) and n(x2) and in the reflectivity. Above 390 nm, Bragg reflectors with the highest reflectivity can be fabricated by growing Al0.5Gao.5N/Al0.91Ga0.09N heterostructures. With increasing wavelength the maximum reflectivity Rmax can be reached only by increasing the thicknesses of the AlGaN layers.
Figure 6. Al content x1 and x2 of optimized Alx1Ga1-x1N/Alx2Ga1-x2N Bragg reflectors versus wavelength. The results of Redwing et al. Reference Redwing, Loeber, Anderson, Tischler and Flynn[27] (violet symbols) and Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] (red symbols) are given for comparison.
The Al contents of DBRs already realized by Redwing et al. Reference Redwing, Loeber, Anderson, Tischler and Flynn[27] and Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] are given for comparision. The thicknesses d1 and d2 of the corresponding Alx1Ga1−x1N and Alx2Ga1−x2N quarter-wavelength layers are given in Figure 7. Above 390 nm the optimal thicknesses d1 and d2 increase linearly with increasing wavelength for fixed Al contents. Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] used 20 periods of Si-doped Al0.2Ga0.8N/Al0.5Ga0.5N multilayers grown on an AlN buffer layer. By changing the thickness of the layers, Bragg reflectors for 456, 409 and 369 nm were realized. We calculated the quarter wavelength thicknesses of the AlGaN layers for this case to be between 36.2 and 47.6 nm for d1 and between 39.6 and 50.7 nm for d2, increasing linearly with the wavelength. Peak reflectivities of higher than 60 % were estimated by Kung Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] from dips in transmission measurements at room temperature. By the matrix method, we calculated maximum reflectivities from 45% to 75% for wavelengths from 456 to 369 nm (Figure 8).
Figure 7. Quarter-wave thicknesses d1 and d2 of the optimized Alx1Ga1-x1N/Alx2Ga1-x2N Bragg reflectors versus wavelength. The thicknesses calculated for the reflectors fabricated by Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] are given for comparison.
Figure 8. Calculated maximum reflectivity of the AlGaN-based Bragg reflectors with 20 and 30 periods of Alx1Ga1-x1N/Alx2Ga1-x2N. The reflectivities calculated for the reflectors fabricated by Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] are given for comparison.
By realizing Al0.5Gao.5N/Al0.91Ga0.09N DBR structures, the reflectivities should be improved to between 83% and 94% for 20 periods and between 97% and 99.6 % for 30 periods in the same range of wavelength. Maximum reflectivities Rmax above 98% or 99.8% can be realized between 310 and 350 nm with 20 or 30 periods of AlGaN layers. For photon energies above 4.2 eV, the reflectivity of the AlGaN-based Bragg reflectors decreases drastically, indicating an upper photon energy limit for laser operation that can be realized by AlGaN VCSELs.
To test the validity of our model and the measured indices of refraction of the AlGaN layers, the transmission curves of Bragg reflectors measured by Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25] were simulated. From the published data of x1 = 0.2 and x2 = 0.5, we calculated the refractive index and the quarter-wave thicknesses of the AlGaN layers for the wavelengths of 456, 409 and 369 nm. The matrix method provides the reflectivity Rmax in dependence of the photon energy. The measured transmission spectra of the Bragg reflectors and the calculated transmission (T(hν) = 1%Rmax(hν)) are shown in Figure 9. The measured transmission spectra are in good agreement with the calculated results, indicating the validity of the applied model and the accuracy of the measured refractive indices shown in Figure 3.
Figure 9. Simulation of the transmission spectra of Bragg reflectors obtained by Kung et al. Reference Kung, Saxler, Walker, Zhang, Lavado, Kim and Razeghi[25]
4. Conclusion
In summary the influence of the Al content on the optical properties of epitaxial AlxGa1−xN layers deposited by plasma-induced MBE on sapphire substrates has been studied by combining transmission, ellipsometry and photothermal deflection spectroscopy measurements. The absorption edge was varied between 3.42 and 6.13 eV by changing the Al content of the epitaxial AlxGa1−xN films. The corresponding bandgap value obeys a quadratic relation with a bowing parameter of 1.3 0.2 eV. From the interference fringes obtained in transmission and from ellipsometry measurements, the index of refraction and the reflectivity was determined in dependence of Al content and photon energy. Using a formalism based on the Kramers-Kronig-dispersion relation, a full set of equations is given to describe the refractive index of AlGaN for photon energies below the bandgap. The determined optical properties provide an experimental basis for the design of optical filters and Bragg reflectors. By applying a matrix method for multilayer systems the design of optimized Bragg reflectors and their reflectivity were calculated, providing for the fabrication of DBRs working in the range of wavelength between 450 and 300 nm.
Acknowledgments
The authors would like to thank M. Kelly for helpful discussions. The work was supported by the Deutsche Forschungsgemeinschaft (Stu 139/3−1) and the Bayerische Forschungsstiftung (FOROPTO II).
