Open Access
Add to BibSonomy
Abstract
Precise measurements of the index of refraction for oriented single crystal samples of gallium nitride and aluminum nitride are reported. Dispersion and birefringence were determined from 450 to 1950 nm using internal reflection from a calibrated rutile prism. Values for dn/dT were measured between 20 and 105 °C. Measurements of refractive dispersion allow for predictions of the conditions needed for nonlinear frequency conversion. The stronger birefringence and weaker dispersion displayed by AlN are shown to allow for a limited range of bulk birefringent phase matching. Both crystals are promising candidates for quasi-phase matched nonlinear frequency conversion.
1. Introduction
Recently Gallium Nitride (GaN) and Aluminum Nitride (AlN) have become commercially available in the form of single crystals. These wide band semiconductors have broad transparency and high thermal conductivity making them interesting candidates for high power optical devices. Their strong polarizability and lack of inversion symmetry also make these crystals interesting candidates for nonlinear frequency conversion. This paper examines the intrinsic properties of these two crystals.
While the optical properties of GaN and AlN have been well studied in thin films and waveguides, there are few reported measurements on bulk single crystals [1,2]. In particular, precise measurements of optical refractive indices are needed in order to predict nonlinear optical performance. Here we describe new measurements of optical dispersion in GaN and AlN and explore the implications for nonlinear optical three-wave mixing.
In order to improve the precision of the measured refractive indices, a new measurement approach is described. In this study, a spectrally dispersed super-continuum laser was used to probe the nitride samples in a prism coupler refractometer. This approach allows for many measurements of the samples dispersion reducing the error in the Sellmeier fit. The use of a single laser source should also reduce systematic errors that can occur with interchangeable laser sources.
2. Background
Single crystals of GaN and AlN have many similar properties. Both crystals have a wurtzite structure (space group of P63mc) and similar lattice constants. They are both uniaxial birefringent with their optical axis along the c-axis. They both possess strong internal fields along the c-axis and exhibit polar behavior along that axis. They both exhibit exceptional hardness and chemical stability. Importantly, both crystals provide excellent heat conduction, as illustrated in Table 1.
Table 1. Highlighted properties of Aluminum Nitride and Gallium Nitride
Although similar in many respects, these crystals do have some important differences. While both crystals exhibit broad transparency when prepared as semi-insulators, the much larger bandgap of AlN provides the potential for deep UV transmission [3]. Conversely, the lower phonon energies in GaN provide longer mid-wave transmission [4,5].
While both crystals possess high melting points, the nitrogen pressure required to stabilize molten GaN is too high for practical growth from the melt. Large single crystals of GaN are currently produced via solution growth techniques. The lower pressures required to stabilize an AlN melt allow for single crystal growth via vapor sublimation techniques. The difference in growth techniques leads to significant differences in crystal purity which strongly affects both UV and IR transparency [6–8].
Finally, despite the similarity in crystal structure, GaN and AlN display quite different nonlinear optical behavior. While their crystal symmetry determines the form of the nonlinear coupling, the magnitude of the interaction depends on the second-order nonlinear susceptibility, χ(2), as well as the optical dispersion. The non-zero components of χ(2) shown in Table 1 will be combined with the new dispersion measurements in a later section to predict the strength of nonlinear conversion performance [9].
3. Transmission and refraction experiments
For these experiments, GaN and AlN crystals were oriented and cut from commercially grown material. The GaN sample was prepared via ammonothermal solution growth by Ammono-SP Inc. The AlN crystal was grown via vapor phase transport by Hexatech Inc. Both crystals were grown to be semi-insulating. The largest face of the 1 x 10 x 10 mm samples were cut and polished in a plane containing the c-axis and a-axis. Both samples were amber colored and free of visible defects.
Transmission spectra of the samples were measured using a Perkin Elmer 950 spectrophotometer and a Cary 760 FTIR spectrometer. The spectra shown in Fig. 1, reveal the wide transparency of the crystals and the expected red-shift of the GaN. Note, however, that the AlN UV transmission does not extend below 300 nm, as might be expected from the bandgap. The cause of the UV absorption in some AlN crystals is not yet clear [6]. Lower reflection losses in the AlN sample illustrate its significantly lower index of refraction.
Fig. 1 Room temperature transmission of 1 mm thick bulk nitride crystals. The sharp lines at 2.7 µm and 5 µm are due to background atmospheric absorption.
Refractive indices of the nitride samples were measured using a Metricon 2010M refractometer outfitted with 60 degree Rutile (TiO2) coupling prism and a sample temperature control system. The laser illumination source for these measurements was a spectrally filtered NKT 20W super-continuum fiber laser. The spectral bandwidth of this laser, 0.45 to 2.1 µm,is well matched to the Germanium detector in the Metricon refractometer. A pair of dispersive prisms and broadband turning mirrors were used to adjust the alignment and select the wavelength of the refractometer probe beam. The wavelength of the probe beam was monitored with Yokogawa AQ6873 and AQ6875 optical spectral analyzers. Probe linewidths increased with wavelength from 2 nm FWHM in the blue to 20 nm FWHM at 2 µm. A Glan-Taylor polarizing cube was use to select either TM or TE polarization of the probe beams in the coupling prism. Samples were oriented for TE polarization parallel to the c-axis and TM polarization parallel to the a-axis. Repeated angular scans were averaged at each wavelength and polarization to yield the measured refractive index.
The Metricon refractometer measures the critical angle of the optically contacted interface of the sample material with the Rutile coupling prism. The uniaxial dispersion of the coupling prism was calibrated using a commercial wafer of single crystal lithium niobate. The refractive index of lithium niobate is well characterized over the spectral range of interest and the magnitude of its index is comparable to that of GaN and AlN. Our LiNbO3 crystal was undoped and possessed the standard congruent stoichiometry. The 25 °C prism calibration was conducted using the Sellmeier equation reported by Zelmon [10]. Unlike the previously reported calibration of rutile by Rams et.al., [11] we used Zelmon rather than that of Schlarb and Betzler [12]. This is because Zelmon’s dispersion equations cover a much broader spectral range and match more closely to previously reported measurements of dispersion and phase matching [13].
Sample indices were measured at roughly thirty wavelengths uniformly spaced over the spectral range of the super-continuum laser. The samples were held at 25 °C during these measurements. The measured values shown in Fig. 2 reveal the magnitude of the dispersion and birefringence in the nitride crystals. Both materials are positive uniaxial with the higher extraordinary polarization index parallel with the c-axis. As expected, AlN exhibits weaker dispersion over this spectral range than GaN. Clearly, however, AlN displays significantly higher birefringence than GaN.
Fig. 2 Measured indices of the nitride samples at 25 °C. Extraordinary rays are solid dots and ordinary rays are open dots. The lines are the fits to the two pole Sellmeier equations discussed in the text.
The fits to the measured dispersion shown in Fig. 2 were obtained using a two-pole Sellmeier equation of the form:
Only a single UV and IR pole were needed to obtain minimal standard deviations for the fits. The sensitivity to the IR pole was weak as reflected in the significant figures for the best-fit parameters summarized in Table 2. This fit applies to the spectral range of 450 nm to 1950 nm.Table 2. Best-fit parameters for Sellmeier Eq. (1) at 25 C.
Comparisons of our measurements with previous reports for GaN reveal reasonable agreement over the relevant spectral ranges [2,14–16]. Best overall agreement was found with the dispersion equations reported by Pezzagna for thin-film GaN on sapphire. Over the common spectral range, 0.45 µm to 1.6 µm, the maximum difference in the refractive index between these two reports are found to be Δne = ± 0.003 and Δno = ± 0.007. A comparison with our previously reported measurement of GaN shows maximum departures of Δne = ± 0.004 and Δno = ± 0.003 over the 0.63 µm to 2.0 µm range. However, for shorter wavelengths, our earlier report possessed weaker dispersion by as much as 0.018. Since the same crystal was used for both experiments, we conclude that this departure reflects some systematic difference between the prism-coupling and prism deflection measurement techniques. The revised dispersion relations derived above should be considered more accurate than our previous results over the common spectral band.
Figure 3 illustrates that the birefringence in AlN is as much as 30% larger than that of GaN. Comparison to the smooth Sellmeier expressions reveals the scale of the measurement errors. These errors are of the same magnitude as the standard deviations reported for the scanning average of the Metricon refractometer.
Fig. 3 Measured birefringence at 25 °C for AlN (solid dots) and GaN (open dots). The difference of the Sellmeier equations are shown as solid lines.
Variations in the refractive index with temperature were also measured at several wavelengths with the technique described above. As before, the Rutile coupling prism was calibrated at each temperature with the congruent LiNbO3 wafer, this time using the temperature dependence reported by Schlarb and Betzler [12]. Measurements were taken above ambient temperature at 20 °C intervals up to 105 °C. As shown in Fig. 4, the measured dn/dT values are all positive, with GaN being more sensitive to temperature than AlN. Note that the polarization dependence of dn/dT decreases dramatically at longer wavelengths. Based on these results birefringence in the infrared is expected to be weakly dependent on temperature.
Fig. 4 Measured temperature rate of change in refractive index for the nitride samples between 25 and 105 °C. Solid lines connect AlN data and dashed lines connect GaN data. Solid dots mark e-ray measurements, open dots mark o-ray measurements. Error bars reflect the uncertainty in the linear fit for dn/dT.
4. Nonlinear frequency conversion and phase matching
The refractive indices measurement in the previous section allow us to now predict the nonlinear performance of GaN and AlN. Knowledge of the crystal symmetry allows for calculation of the polarization dependent nonlinear coupling. Polarizations with good nonlinear coupling can then be evaluated for phase matching.
Since both crystals are hexagonal (6mm), they share the same form of the second-order susceptibility tensor [17]. The standard contracted form of the susceptibility tensor is:
Here it has been assumed that absorption is negligible for all three waves so that Kleinman symmetry has been applied. With this assumption, there are only two nonzero values for the components. The values reported for χ (2)31 and χ (2)33 in GaN and AlN are listed in Table 1 [9].The form of χ (2) matrix above determines the effective nonlinear coefficients, deff, for all the possible polarization combinations of the three-wave mixing process where (λ1 ≤ λ2 ≤ λ3) [18]. While there are in general eight polarization combinations, the crystal symmetry here allows for only two non-zero values for the nonlinear couplings. For collinear propagation at an angle θ relative to the optic axis, the nonlinear coupling coefficient, deff, in the nitrides can be summarized as follows:
This expression allows for identification of the optimal polarization states for three-wave nonlinear coupling. As expected, deff vanishes entirely for any rays propagating parallel to the c-axis. For all other possibilities, non-zero second-order nonlinear coupling requires the process to have an even number of ordinary rays.The second step in evaluating the nonlinear performance is to examine the potential for phase matching the three-wave process. Birefringent phase matching allows the waves to conserve both energy and momentum according to:
This is often possible by exploiting the birefringence to offset normal dispersion. Phase matching is therefore most readily obtained for positive uniaxial crystals via the (oee) process. A simple analysis of the refractive dispersion curves shows that phase matching over a broad spectral range is possible in both AlN and GaN for this process. Unfortunately, this process does not generate nonlinear coupling in crystals with hexagonal symmetry. Nevertheless, Eq. (1) predicts that AlN can also birefringently phase match the (oeo) process. This process will experience a nonlinear coupling of deff = ½χ (2)31 sin(θ).A simple laser difference frequency generation experiment was conducted to check the predicted phase matching as shown in Fig. 5(a). The experiment utilized a q-switched, doubled Nd:YAG laser to drive a tunable KTA optical parametric oscillator. The signal and idler beams were injected into the 1 mm thick AlN crystal normal to the c-axis. The OPO was tuned varying λ1and λ2 while a monochromator coupled to an InGaAs detector monitored the difference frequency signal at λ3. Birefringently phase matched difference frequency generation was observed with an o-ray at 0.848 µm, an e-ray at 1.427 µm, and an o-ray at 2.092 µm as shown in Fig. 5(b).
Fig. 5 (a) Difference frequency experiment used to test AlN phase matching. (b) Monochromator resolution limited spectrum of the generated difference frequency. (c) The solid line is the predicted angle-tuned phase matching for (oeo) process in AlN at 25C. The dot shows the observed λ1 which generated phase matching for θ = 90°.
The observed and predicted angle phase matching are shown in Fig. 5(c). The predicted tuning curve assumes that λ3 is fixed at 2.092 µm for direct comparison with the experimental point. Clearly the normal incidence prediction differs from the observation by roughly 80 nm. While this is a substantial wavelength error, it corresponds to a possible birefringence error of only −0.0019, which is consistent with variation seen in Fig. 3. This experiment illustrates the need for precise knowledge of the refractive indices in the design of nonlinear devices.
The (ooe) three-wave process also exhibits a nonlinear coupling of deff = ½χ (2)31 sin(θ). Analysis shows that the (ooe) process cannot be birefringently phase matched in AlN or GaN for wavelengths shorter than 2 µm. However, Eq. (1) suggests that AlN may birefringently phase match over a limited spectral range when 2.1 µm ≤ λ3 ≤ 2.5 µm. Broader spectral measurements will be required to test this projection.
While bulk phase matching is clearly possible in AlN, the nonlinear coupling is expected to be modest, with a maximum value of deff ≈1 pm/V. In order to obtain higher nonlinear coupling and broader spectral ranges, an alternative phase matching approach will be needed. One promising technique is quasi-phase matching (QPM). In the nitrides, this can be accomplished by creating a spatially periodic structuring in the crystal. By matching the periodic structure to the coherence length of the desired nonlinear process, QPM allows efficient conversion of power between the beams. This technique has been demonstrated in GaN through spatially periodic inversion of the crystal along the c-axis [19-20]. Fig. 6 shows the predicted coherence length of both the (eee) and (ooe) process in GaN and AlN. The reported SHG resonance in room temperature QPM GaN (shown as the solid square dot in Fig. 6) lies close to the predicted curve [21].
Fig. 6 Calculated coherence length for second harmonic generation of the fundamental wavelength for crystals at a temperature of 25 °C. Solid dots highlight the AlN processes and open squares highlight the GaN processes. Solid lines display the (eee) processes and dashed lines display the (ooe) processes. The reported resonance of (ooe) SHG in periodically oriented GaN is shown with the solid square.
With QPM it is possible to access the larger nonlinear coupling of the (eee) process. When all three waves are extraordinary and incident normal to the c-axis, the nonlinear coupling obtains its maximum allowed value; deff = 1/π χ (2)33. The 2/π reduction is due to reduced spatial gain duringthe QPM process [22]. Therefore, an optimal nonlinear coupling of deff ≈3 pm/V should be expected from both GaN and AlN for the (eee) QPM process. One advantage of the QPM technique over birefringent phase matching is that this optimum nonlinearity is available over the full transparency range. Another advantage is that fixed orientation crystals can be used for all processes. The significantly longer coherence lengths predicted for AlN may also simplify fabrication of the QPM structures.
5. Summary
We have reported new measurements of the refractive indices of single crystal GaN and AlN. A prism-coupled refractometer was combined with a super-continuum laser to obtain accurate dispersion relation in the spectral range from 0.45 µm to 1.95 µm. The results for both uniaxial crystals were well fit with a two-pole Sellmeier equation with standard deviations 9E-5 for GaN and 3E-5 for AlN. Refractive indices reported here are in reasonable agreement with previous reports. Polarized measurements show that birefringence is significantly higher in AlN than GaN. Also dn/dT in AlN is significantly smaller than in GaN. Both crystals exhibit weak temperature dependence of their birefringence, <1E-5 °C−1, thus indicating that frequency conversion will be relatively insensitive to temperature variations.
Nonlinear frequency conversion was examined as well. The hexagonal crystal symmetry limits the nonlinear coupling options. Frequency conversion via angle-tuned birefringent phase matching is shown to be possible in AlN, but not in GaN. Combined with its other optical and thermal properties, this suggests that single crystal AlN is a promising material for high average power frequency conversion. Nevertheless, optimum nonlinear coupling for both crystals was found to require a quasi-phase matching technique, such as periodic orientation inversion. The predicted coherence length for resonant QPM second harmonic generation in GaN are found to closely match the reported experiment.
Funding
Office of Naval Research; Joint Technology Office for High Energy Lasers.
Acknowledgments
Nitride samples were provided by Hexatech and Ammono-SP.
References and links
1. W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100(22), 223501 (2012). [CrossRef]
2. S. R. Bowman, C. G. Brown, M. Brindza, G. Beadie, J. K. Hite, J. A. Freitas, C. R. Eddy Jr, J. R. Meyer, and I. Vurgaftman, “Broadband Measurements of the Refractive Indices of Bulk Gallium Nitride,” Opt. Mater. Express 4(7), 1287–1296 (2014). [CrossRef]
3. L. Liu and J. H. Edgar, “Substrates for gallium nitride epitaxy,” Mater. Sci. Eng. Rep. 37(3), 61–127 (2002). [CrossRef]
4. V. Yu. Davydov, Yu. E. Kitaev, I. N. Goncharuk, A. N. Smirnov, J. Graul, O. Semchinova, D. Uffmann, M. B. Smirnov, A. P. Mirgorodsky, and R. A. Evarestov, “Phonon dispersion and Raman scattering in hexagonal GaN and AlN,” Phys. Rev. B 58(19), 12899–12907 (1998). [CrossRef]
5. M. Welna, R. Kudrawiec, M. Motyka, R. Kucharski, M. Zając, M. Rudziński, J. Misiewicz, R. Doradziński, and R. Dwiliński, “Transparency of GaN substrates in the mid-infrared spectral range,” Cryst. Res. Technol. 47(3), 347–350 (2012). [CrossRef]
6. M. Strassburg, J. Senawirtne, N. Deitz, U. Haboeck, A. Hoffmann, V. Noveski, R. Dalmau, R. Schlesser, and Z. Sitar, “The growth and optical properties of large, high-quality AlN single crystals,” J. Appl. Phys. 96(10), 5870–5876 (2004). [CrossRef]
7. R. Dwilinski, R. Doradzinski, J. Garczynski, L. Sierzputowski, R. Kucharski, M. Zaja, M. Rudzinski, R. Kudrawiec, J. Serafinczuk, and W. Strupinski, “Recent achievements in AMMONO-bulk method,” J. Cryst. Growth 312(18), 2499–2502 (2010). [CrossRef]
8. R. Schlesser, R. Dalmau, D. Zhuang, R. Collazo, and Z. Sitar, “Crucible materials for growth of aluminum nitride crystals,” J. Cryst. Growth 281(1), 75–80 (2005). [CrossRef]
9. N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97(5), 053512 (2005). [CrossRef]
10. D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide–doped lithium niobate,” Opt. Soc. Am. B. 14(12), 3319–3322 (1997). [CrossRef]
11. J. Rams, A. Tejeda, and J. M. Cabrera, “Refractive indices of rutile as a function of temperature and wavelength,” J. Appl. Phys. 82(3), 994–997 (1997). [CrossRef]
12. U. Schlarb and K. Betzler, “Refractive indices of lithium niobate as a function of temperature, wavelength, and composition: A generalized fit,” Phys. Rev. B Condens. Matter 48(21), 15613–15620 (1993). [CrossRef] [PubMed]
13. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]
14. M. J. Bergmann, Ü. Özgür, H. C. Casey Jr, H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary indices fir AlxGa1-xN epitaxial layers,” Appl. Phys. Lett. 75(1), 67–69 (1999). [CrossRef]
15. N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1-xN films grown on sapphire substrates,” J. Appl. Phys. 94(5), 2980–2991 (2003). [CrossRef]
16. 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]
17. F. Zernike and J. Midwinter, Applied Nonlinear Optics, (John Wiley & Sons New York, 1973) Chap. 2.
18. A. V. Smith, Crystal Nonlinear Optics with SNLO Examples, (AS-Photonics, 2015) Chap. 15.
19. A. Chowdhury, H. M. Ng, M. Bhardwaj, and N. G. Weimann, “Second-harmonic generation in periodically poled GaN,” Appl. Phys. Lett. 83(6), 1077–1079 (2003). [CrossRef]
20. J. Hite, M. Twigg, M. Mastro, J. A. Freitas Jr, J. R. Meyer, I. Vurgaftman, S. O’Connor, N. J. Condon, F. Kub, S. R. Bowman, and C. Eddy Jr., “Development of periodically oriented gallium nitride for non-linear optics,” Opt. Mater. Express 2(9), 1203–1208 (2012). [CrossRef]
21. C. G. Brown, S. R. Bowman, J. K. Hite, J. A. Freitas, F. J. Kub, C. R. Eddy Jr, I. Vurgaftman, J. R. Meyer, J. H. Leach, and K. Udwary, “Frequency Conversion Efficiency in Free-Standing Periodically Oriented Gallium Nitride,” Proc. SPIE 9731, 97310E (2016). [CrossRef]
22. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]
