PHYSICAL REVIEW B VOLUME 59, NUMBER 4 15 JANUARY 1999-II
Ultrafast second-harmonic generation spectroscopy of GaN thin films on sapphire
W. E. Angerer, N. Yang, and A. G. Yodh Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104
M. A. Khan and C. J. Sun APA Optics, Blaine, Minnesota 55449 ͑Received 23 June 1998͒
We have performed ultrafast second-harmonic generation spectroscopy of GaN/Al2O3 . A formalism was developed to calculate the nonlinear response of thin nonlinear films excited by an ultrashort laser source (2) (2) (Ti:Al2O3), and then used to extract zxx(ϭ2o) and xzx(ϭ2o) from our SHG measurements over a two-photon energy range of 2.6–3.4 eV. The spectra are compared to theory ͓J. L. P. Hughes, Y. Wang, and (2) J. E. Sipe, Phys. Rev. B 55, 13 630 ͑1997͔͒. A weak sub-band-gap enhancement of zxx(ϭ2o) was (2) observed at a two-photon energy of 2.80 eV; it was not present in xzx(ϭ2o). The enhancement, which may result from a three-photon process involving a midgap defect state, was independent of the carrier concentration, intentional doping, and the presence of the ‘‘yellow luminescence band’’ defects. In addition, we determined sample miscuts by rotational SHG; the miscuts did not generate observable strain induced nonlinearities. The linear optical properties of GaN from 1.38 to 3.35 eV were also determined. ͓S0163-1829͑99͒03204-X͔
I. INTRODUCTION The remainder of this paper is organized as follows. First, we present our experimental techniques and measurements The optical, electrical, structural, and defect properties of of the index of refraction of GaN. Second, we discuss our GaN ͑Refs. 2–6͒ are important to understand, in part, as a photoluminescence measurements of GaN and show how our result of GaN’s attractiveness as a material for electro-optic samples may be distinguished by the presence or absence of devices.7 In this paper we have explored the nonlinear opti- the yellow luminescence band. Third, we discuss our experi- cal properties of GaN and its interfaces. Several second- mental technique for measuring and normalizing our SHG harmonic generation ͑SHG͒ measurements of GaN ͑Refs. spectra from GaN. In this section, we also present a calcula- 8–12͒ have been reported. These works measured the non- tion of the nonlinear response of a thin slab and a thin film linear susceptibility elements of GaN at a single fundamental stack to an ultrafast laser pulse. We then apply this analysis to our quartz and GaN measurements. Finally we present and wavelength of 1064 nm ͑Refs. 8–10,12͒ and for two-photon discuss our SHG spectra and symmetry measurements of energies above the band gap.11However, none of these mea- GaN. surements have probed the second-order susceptibility as a function of wavelength below the band gap. In this spectral region the contributions of defect states, such as those asso- II. EXPERIMENTAL AND THEORETICAL BACKGROUND ciated with yellow band luminescence, may be detectable. In this section we describe sample preparation, transmis- We have measured the second-order nonlinear response sion experiments to determine the linear optical properties of of GaN over a two-photon spectral range from 2.6 to 3.4 eV GaN ͑index of refraction͒, and photoluminescence experi- in two polarization configurations, and compared these re- 1 (2) (2) ments to probe the defect structure of GaN. We then describe sults with theory. In order to extract xzx and zxx from the our experimental apparatus for second-harmonic generation measurements, we developed an analysis that incorporates spectroscopy, and show how the nonlinear optical signals the interference effects that arise in SHG spectroscopy of generated using an ultrafast fundamental light source depend thin films with ultrafast light pulses. We find the variation of on wave-vector mismatch and group velocity mismatch. This these nonlinear susceptibilities to be independent of the con- latter effect must be included to accurately determine the centration of defect states associated with yellow band lumi- absolute value of the second-order susceptibility elements (2) nescence. GaN shows a modest enhancement of zxx at a when using ultrafast pulses for nonlinear spectroscopy. two-photon energy of 2.80 eV. This enhancement is absent in (2) and may result from point defect states with energies xzx A. Samples that lie only in the band gap. In addition, we separated the contribution of bulk and interface nonlinearities; no resonant The procedures used to grow our samples are described in interface states were observed in the spectral range explored. the literature.13 Here we briefly review the procedure. A Finally, as part of these studies, we measured the linear op- basal plane, i.e., ͓0001͔, sapphire substrate was degreased tical properties of the GaN/Al2O3 samples from 370 to 900 and etched in hot H3PO4 :H2SO4 . After placing the sapphire nm and we determined a Sellmeier equation for the index of substrate onto a silicon carbide coated graphite susceptor in refraction of GaN. the reaction chamber, a 0.05-m buffer layer of AlN was
0163-1829/99/59͑4͒/2932͑15͒/$15.00PRB 59 2932 ©1999 The American Physical Society PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2933
TABLE I. Properties of GaN/Al2O3 samples. ‘‘Yellow band’’ refers to the observation of a yellow photoluminescence band. The uncertainty in the thickness measurements is Ϯ0.004 nm. The film thickness has units of m and the mobility has units of cm2/Vs.
No. Dopant Thickness Mobility Carrier Density Yellow band
G1984.1 none 1.020 20 4.47ϫ1017 no G927.3 none 4.885 346 8.52ϫ1017 yes G921.3 Si 3.765 192 2.13ϫ1018 yes
deposited by low-pressure metalorganic chemical vapor GaN/Al2O3 sample and detected using a Si photodiode. At deposition ͑LPMOCVD͒. GaN films with various thick- each wavelength we measured the transmission through the nesses were deposited using triethylgallium ͑TEG͒ and am- GaN/Al2O3 sample as a function of the angle of incidence monia as the Ga and N sources, respectively. Some samples from 0° to 75°. This was compared to the intensity detected were intentionally n doped with Si using SiH4 as a source with the sample removed from the optical line. gas. After growth, the samples were electrically character- We determined a Sellmeier dispersion equation for the ized by Van der Pauw and Hall measurements. The results of index of refraction of GaN from our linear transmission ex- these measurements are displayed in Table I. The sample periments. The transmission of light through our GaN/Al2O3 thicknesses ranged from 1.020 to 4.885 m. samples was analyzed using matrix methods18,19 using the 20 In principle, the transmission of linear and second- known linear optical properties of Al2O3 . The details of harmonic light through our samples is affected by the AlN this analysis are presented in ͑Ref. 19͒. Using this theory, we buffer layer. In this paper we determine the dispersion of fit our transmission data for the index of refraction. Figure 1 both the second-order susceptibility and the index of refrac- displays our data and exhibits the fits to the optical transmis- tion of GaN. All of our calculations model the sion of a GaN/Al2O3 sample. From the results we derived a GaN/AlN/Al2O3 layered structure as a GaN/Al2O3 structure. two term Sellmeier dispersion relation for the index of re- We determined the fractional error in the calculated transmit- fraction of GaN, i.e., ted linear and second-harmonic light that results from using the simplified two-film model in lieu of the three-film model. 2 2 A1 A2 Our model parameters include the index of refraction of n2͑͒ϭ1ϩ ϩ . ͑1͒ 14 15 2 2 2 2 AlN, the second-order susceptibility of AlN, and our Ϫ1 Ϫ2 measured values for the index of refraction and the second- order susceptibility of GaN. The index of refraction calcu- The parameters for this fit are listed in Table II. lated from the GaN/Al2O3 model has an effective random Note, our transmission theory also includes the effect of a fractional error of less than 1.4% as compared with the broadband light source. The wavelength intensity distribu- GaN/AlN/Al2O3 model. Similarly, we calculated the frac- tion of our light source, I(), has a functional form well tional error in the second-order susceptibility elements using approximated by the two-slab model. This model error is less than 6%, which is about equal to our experimental uncertainty. Therefore, we 2 8͑Ϫo͒ implemented the two slab model for calculating the second- I͑͒ϭI exp Ϫ , ͑2͒ o 2 order susceptibilities for GaN in the experimental samples. ͩ ⌬ ͪ
B. Linear optical properties We performed a series of optical transmission measure- ments from 370 to 900 nm to determine the index of refrac- tion of our GaN samples. Accurate knowledge of this param- eter and of the sample thickness is essential for deriving the nonlinear susceptibility from the SHG spectra. Our disper- sion relation for GaN is consistent with other measurements of the index of refraction16,17 and shows a slightly larger value near the band gap. ͓We measure nϭ2.613 as compared with 2.595 ͑Ref. 16͒ and 2.589 ͑Ref. 17͒ at a wavelength of 382 nm.͔ The experimental setup for our optical transmission mea- surement employs a 450 W Xe lamp with an f/1.0 UV en- hanced lens as the light source. A monochromator selected the desired probe wavelength. The light was then p polarized FIG. 1. Optical transmission data for GaN/Al2O3 with a film and amplitude modulated by an optical chopper. A fraction thickness of 1.020 m. Transmission spectra are for ϭ777 nm of the light was separated and detected with a Si photodiode (छ) and ϭ549 nm (ᮀ). The solid ͑dashed͒ line is the best fit to in order to normalize temporal fluctuations of the light the ϭ777 ͑549͒ nm data using a matrix methods with a finite source. The remaining light was transmitted through the bandwidth light source. 2934 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
TABLE II. Empirical constants for GaN Sellmeier equation. The i parameters are expressed in m.
Sellmeier constant Value
A1 4.081 1 0.1698 A2 0.1361 2 0.3453
where o is the central wavelength of the monochromator and ⌬ϭ4 nm is the wavelength bandwidth. In practice, the interference of light with a broadband spectral intensity dis- tribution of Eq. ͑2͒ exhibits dramatically different behavior in thin and thick films. Thin films are characterized by a thickness dӶ22/n⌬, where n for GaN ranges from 2.32 to 2.68 over our measurement. The transmission of light through a thin film is very sensitive to the film’s index of FIG. 2. Photoluminescence spectra from the unintentionally refraction and its thickness as a result of the interference of doped 4.885-m sample. The salient features include a donor- multiply reflected waves in the film. Conversely, in thick 2 acceptor-pair ͑DAP͒ recombination peak centered at 3.28 eV and a films, where dӷ2 /n⌬, both constructive and destructive broad yellow luminescence band centered at 2.2 eV. The inset dis- interference of the multiply reflected light occur; the light plays the photoluminescence spectra of our 1.020 m sample. The transmitted through the thick film is relatively insensitive to DAP peak is present in this sample, but the yellow luminescence the film thickness and is only mildly sensitive to the index of band is absent. refraction via Fresnel coefficients. For our system, GaN is a thin film (dn⌬/2Ͻ0.2), while sapphire is a thick film (dn⌬/2ϳ10). This is the ideal combination for measur- D. SHG experiments ing the index of refraction of GaN because the transmitted intensity is highly sensitive to the index of refraction of GaN We performed SHG spectroscopy and SHG rotational but is relatively insensitive to the optical properties of sap- measurements of our GaN samples using the experimental phire. setup displayed in Fig. 3. The fundamental light generating source is a Ti:Al2O3 laser. This laser produces ultrashort near C. Photoluminescence measurements IR laser pulses ͑700–1000 nm͒ at a rate of 76 MHz. These ultrashort pulses have a time duration of 80–150 fs and a To characterize the defect structure of GaN, we measured peak power of ϳ50 kW. The fundamental light from the the photoluminescence spectrum of our GaN samples at 82 Ti:Al O laser was polarized and amplitude modulated with K. A cw krypton-ion laser optically excited electrons from 2 3 the GaN valence band to the conduction band. Photolumines- cent light from the GaN sample was collected with a lens and focused on the entrance slit of a single pass monochromator. The light was detected with a PMT. The monochromator slits were set to 0.25 mm to give a spectral resolution of 0.5 nm. The sample was cooled by liquid nitrogen in a dewar. As a result of the limited sensitivity of our detection system for light with wavelengths shorter than 370 nm, the band-edge features were not resolved. Figure 2 displays photoluminescence data from our unin- tentionally doped 1.020-m and 4.885-m GaN samples. Both samples show a strong feature centered at 3.28 eV. This feature has been observed in several other low-temperature photoluminescence experiments,21–26 and has been assigned to donor-acceptor-pair ͑DAP͒ transitions. In addition, the 4.885-m sample has a broad feature centered at 2.2 eV, which is commonly called the yellow band. This band has been observed in the photoluminescence of both bulk and epitaxial layers of GaN grown by a variety of methods. Re- cent theoretical27 and experimental work28,29 suggests that this feature may be related to transitions between a shallow donor and a compensation center or between the conduction band and a compensation center. Neugebauer and Van de 27 Walle have suggested the gallium vacancy VGa as the most FIG. 3. Experimental setup for second-harmonic generation likely candidate for this defect state. spectroscopy of GaN. PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2935 an optical chopper. A 90%/10% beam splitter directed the fundamental light to sample and reference arms containing identical optics. The reference line contained a plate of -cut quartz, which was used to normalize any spectral and tem- poral fluctuations of the fundamental source. With the quartz positioned normal to the Ti:Al2O3 beam, the fundamental (2) light coupled to xxx(ϭ2o), and the second-harmonic fields propagated through the crystal free of the effects of optical activity or birefringence.30 A long pass filter removed any second-harmonic signal generated in the optical line prior to the sample. Achromatic lenses focussed the Ti:Al2O3 laser light to a spot size of ϳ100 m on the GaN/Al2O3 sample and on the quartz plate. The resulting second- harmonic light was polarized and separated from the funda- mental light by short pass filters and a monochromator. The SHG light was detected by a photomultiplier tube, measured by a lock-in amplifier, and analyzed by a computer. A typical signal from a GaN/Al2O3 sample with a p-polarized 200-mW fundamental beam was 2ϫ105 photons/s. Because SHG is so sensitive to the characteristics of the laser and detection systems, all of our SHG measurements are normalized to the nonlinear response of quartz. There are three essential features required of a properly normalized spectral measurement: minimization of the effects of tempo- ral fluctuations of the source, compensation for the spectral variations of the source intensity and pulse width, and elimi- nation or compensation of the inherent spectral response of the detection system. The simultaneous SHG measurements of quartz in the reference arm and GaN/Al2O3 in the sample arm address the first two issues. However, although the op- tics in both the reference and sample arms are nominally identical, identical spectral response is not ensured. We eliminated the spectral response systematics of the detectors FIG. 4. ͑a͒ Comparison of SHG data from a quartz wedge ex- by referencing the measured nonlinearity of GaN to an cited by a Ti:Al2O3 laser (छ) and calculated SHG using a mono- equivalent measurement of the nonlinearity of quartz, which chromatic theory ͑solid line͒. The quartz wedge is 3 mm thick and was placed at the same position in the sample arm. has faces that are aligned 1° from parallel. The Ti:Al2O3 pulse travels through a quartz thickness that varies as the quartz is trans- lated perpendicularly to the beam. The calculated response displays E. Analysis of SHG from an ultrafast fundamental source oscillations ͑Maker fringes͒ that result from variations of the inter- ference between the bound and free waves as the quartz thickness Before analyzing the nonlinear spectra of GaN, it was changes. The fundamental wavelength is 800 nm. ͑b͒ Calculated necessary to consider several features of nonlinear optics that transmitted SHG intensity from a quartz slab excited by an ultrafast arise when using an ultrafast laser source, e.g., laser sources Ti:Al2O3 laser. The Maker fringes are damped as the crystal be- with pulse durations of ϳ100 fs. Consider for example the comes thicker due to the incomplete overlap of the bound and free 31 phenomena of Maker fringes, in which the measured trans- wave pulses ͑group velocity mismatch͒. mitted intensity of the SHG light emerging from a quartz plate oscillates strongly as a function of the orientation of the mitted through our reference quartz slab and show that quartz plate with respect to the fundamental beam direction. The observed alternating maxima and minima result from Maker fringes are absent in this material. In addition, we interference between the bound and free second-harmonic must apply this theory to our spectroscopic measurements of fields. We do not observe Maker fringes from a similar GaN. GaN is highly dispersive, and group velocity disper- quartz plate excited by our Ti:Al O laser ͓see Fig. 4͑a͔͒. sion significantly affects SHG spectroscopy of our thicker 2 3 (2) Interference is suppressed because the bound and free wave samples. Our ultrafast analysis was necessary to deduce zxx 32,33 (2) packets propagate through the crystal with different and xzx from our raw spectroscopic data. Note that while group velocities. Group velocity mismatch between the nonlinear ultrafast pulse propagation has been addressed by bound and free waves leads to incomplete overlap between several groups, particularly with regard to SHG conversion the bound and free wave packets, and hence modifies the efficiency,34–38 our treatment here is important because it interference of these waves. Similar effects occur in our GaN explicitly considers how ultrafast pulses affect the measured samples. nonlinearity in a spectroscopic measurement. We developed a formalism to analyze second-harmonic For our analysis we measured our laser pulse source spec- generation from an ultrafast fundamental light source. We tral intensity, I(), and found that it had an approximate apply these techniques to the second-harmonic light trans- Gaussian angular frequency distribution, i.e., 2936 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
2 Ϫ8͑Ϫ ͒ n͑2o͒2o ͑Ϫ2o͒ o k ϩ 10 I͑͒ϭIoexp , ͑3͒ f͑͒Ϸ ͑ ͒ ͩ ⌬2 ͪ c v f with where o is the center angular frequency and ⌬ is the k͑͒ץ k͑͒ץ angular frequency bandwidth, which can range from 0.021 to 0.053 fsϪ1 for our system. The electric field of a pulse v ϭ and v ϭ . ͑11͒ ͯ ץ ͯo fץ b propagating in the y direction and polarized along the x di- 2o rection then has the form The zeroth-order terms with respect to (Ϫ2o) in Eqs. ͑9͒ 2 and ͑10͒ lead to wave-vector mismatch and the familiar Ϫ4͑Ϫo͒ E͑r,͒ϭxˆ eik͑͒y exp . ͑4͒ Maker fringes, while the first-order terms generate a group E ͩ ⌬2 ͪ velocity mismatch and damp the Maker fringes. The nonlinear wave equation for the transmitted SHG in- The time evolution of the pulse is obtained by Fourier trans- tensity through quartz leads to the following expression for forming E(r,). To facilitate these calculations, we Taylor the temporal variation of the SHG intensity: expand the wave vector k about its central frequency o , i.e., c 2 2 2o I͑t͒ϭ C1ϩC2ϩ2C1C2 cos ͑n f Ϫnb͒d 2 2 8ͫ ͩ c ͪͬ k͑͒ ץ n͑o͒o ͑Ϫo͒ ͑Ϫo͒ k͑͒Ϸ ϩ ϩ ͑12͒ 2 ͯ ץ c vg 2 o with ͑5͒ ⌬22 Ϫ⌬2 d 2 with C ϭ 2͑2͒ ͑ϭ2 ͒␣ exp tϪ 1 E xxx o ͫ 8 ͩ v ͪ ͬ ͱ2͑⑀bϪ⑀ f ͒ f k͑͒ ͑13͒ץ 1 ϭ . ͑6͒ ͯ ץ v g o and vg is the group velocity of the packet. The first term gives ⌬22 Ϫ⌬2 d 2 the phase velocity of the wave, the second term depends on C ϭ 2͑2͒ ͑ϭ2 ͒ exp tϪ . 2 E xxx o ͫ 8 ͩ v ͪ ͬ the velocity at which the pulse propagates, and the third term ͱ2͑⑀bϪ⑀ f ͒ b accounts for pulse broadening. ͑14͒ Suppose our laser pulse is incident on a quartz slab of In these above equations, ⑀ f and ⑀b are the free and bound thickness d. We determine the nonlinear fields within the wave dielectric susceptibilities, i.e., quartz by applying boundary conditions32 at each angular frequency ͑see Appendix A for more details͒. The free and ⑀bϵ⑀͑o͒ and ⑀ f ϵ⑀͑2o͒, ͑15͒ bound wave second-harmonic fields propagating through the quartz are expressed as nb and n f are the bound and free wave indices of refraction, is the fundamental field strength, and ␣ and  are Fresnel- E 2 like coefficients that have values Ϫ2͑Ϫ2o͒ ˆ ikj͑͒y Ej͑r,͒ϭxE„k͑͒…e exp , ͑7͒ ͩ ⌬2 ͪ Ϫ2n n ϩ1 n ϩn f͑ b ͒ b f ␣ϭ and ϭ . ͑16͒ 2 n ϩ1 where j is either b or f, denoting bound or free wave, respec- ͑n f ϩ1͒ f tively, and E k() depends primarily on boundary condi- „ … In the spectroscopic measurement, the detected signal is the tions at the various interfaces. time integral of Eq. ͑12͒: The SHG intensity measured in transmission through the quartz film is simplified by expanding the wave vector coef- ϱ c 2 2 2o ficients, E„k()…, to zeroth-order in (Ϫ2o), and by ex- Iϭ C1ϩC2ϩ2C1C2 cos ͑n f Ϫnb͒d dt. ik ()y 8 ͵Ϫϱͫ ͩ c ͪͬ panding the wave-vector phases, e j , to first-order in ͑17͒ (Ϫ2o). In other words, we approximate any wavevector that appears as a multiplicative coefficient of the field ampli- Consider the consequences of Eq. ͑17͒ for ultrafast SHG. C1 tude by its value at the central angular frequency, 2o .In and C2 represent the amplitudes of the free and bound wave the case of SHG, we thus have fields, respectively. The interference of these fields occurs through 2C1C2cos͓(2o /c)(nfϪnb)d͔. For a monochromatic E„k͑͒…ϷE„k͑2o͒…. ͑8͒ light source, ⌬ϭ0, the wave packets become plane waves, Bound and free wave vectors in the field phase factors, how- and the free and bound waves always interfere. With a ever, are written as broadband fundamental source, ⌬Þ0 and the bound and free wave packets are temporally centered at tϭd/vb and t n͑o͒2o ͑Ϫ2o͒ ϭd/v f , respectively. Increasing the length of the sample or k ͑͒Ϸ ϩ ͑9͒ b c v increasing the dispersion of the material decreases the over- b lap of these waves, and hence suppresses the Maker fringes. and Figure 4͑b͒ displays our calculation of the SHG transmitted PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2937
TABLE III. Parameters used for ultrafast Maker fringe calcula- tion. Note the use of ‘‘ultrafast units.’’ is the time duration of the Ti:Al2O3 pulse and is the wavelength of the fundamental pulse. The indices of refraction and the group velocities are calculated for the ordinary index of quartz.
Symbol Value
c 0.3 m/fs 0.800 m Ϫ1 o 2.35 fs ⌬ 0.053 fsϪ1 80 fs
nb 1.5384 n f 1.5577 vb 0.19 316 m/fs v f 0.18 695 m/fs
FIG. 5. Geometry of nonlinear waves in GaN/Al2O3 .Eb is the intensity from a quartz slab as a function of quartz thickness. bound wave. E f 1 is the free wave that is generated at the air/GaN The parameters for this calculation are displayed in Table III. interface and propagates to the GaN/Al2O3 interface. E fr1 is the free Because our quartz sample is 3 mm thick, we are never in a wave that is generated at the GaN/sapphire interface and propagates regime where Maker fringes are observed during excitation to the air/GaN interface. E f 2 is the free wave that is generated at by our Ti:Al O laser. and propagates away from the GaN/sapphire interface. ⌬ and ␦ are 2 3 relative phases. Points A and B represent the path traversed by E While ultrafast pulses dramatically affect SHG from thick f 2 while points A and C represent the path traversed by E , which quartz plates, their influence on SHG from our GaN films is fr1 undergoes a single reflection in the GaN thin film before transmis- of intermediate importance. We define the error, , in the sion through the sapphire substrate. calculated ultrafast SHG intensity compared to the mono- chromatic theory as terface. Respectively, these fields are illustrated in Fig. 5 as the fields that propagate from point A to point B and from N mono ultra 2 1 ͑Ii ϪIi ͒ point A to point C. The full derivation of the transmitted ϭ , 18 ͚ mono ͑ ͒ second-harmonic field is provided in Appendix B. ͱNiϭ1 ͑I ͒2 i The key results are expressed in terms of the incident mono ultra fields, the nonlinear susceptibilities, Fresnel factors, and the where Ii (Ii ) is the calculated SHG intensity from GaN using the monochromatic ͑ultrafast͒ theory and i labels bound and free wave vectors given below: a wavelength at which Imono and Iultra are evaluated. For our i i 2nfi 4.885-m GaN sample, ϭ0.35 over a fundamental wave- k ϭ , ͑19͒ fi c length range of 740–755 nm. The error between the mono- chromatic and ultrafast theories is especially large in this 2nb1 region due to the thickness of the sample and the high dis- kb1ϭ , ͑20͒ persion of GaN at the second-harmonic wavelength of 370– c 378 nm. This error is much greater than the statistical and systematic errors in our measurements. Therefore, an ul- k fizϭϪcos fikfi , ͑21͒ trafast analysis of SHG from GaN is required to accurately extract the nonlinear susceptibilities from our spectroscopic kb1zϭϪcos b1kb1 , ͑22͒ data. k ϭsin k , 23 In order to extract the wavelength dependent nonlinear fix fi fi ͑ ͒ susceptibilities, we compared our measurements to theoreti- and k ϭsin k , ͑24͒ cal formulas derived using ultrafast analysis and boundary b1x b1 b1 value techniques. The formulas depend on the linear and where i is equal to 0, 1, or 2 and denotes the air, GaN, and nonlinear optical properties of GaN and sapphire, as well as sapphire regions, respectively. The angle of incidence in re- the angle of incidence of the fundamental field, 0 , and the gion i is i and is determined from o by Snell’s law. Also fundamental and second-harmonic field polarizations. We note that n fiϵni(2o) and nb1ϵn1(o). computed the solution in terms of monochromatic fields and The second-harmonic field for the s-in/p-out geometry is then modified these solutions to include ultrafast effects. Fig- ͑2͒ ure 5 displays the geometry of our GaN/Al2O3 sample and Esp͑2o͒ϭYyyzyyEyEy , ͑25͒ labels the incident and the nonlinear fields. Our model relates where the measured second-harmonic field to the SHG field ini- tially created and transmitted through the GaN film, and to Y ϭt2 ͑ ͒ ͑26͒ the SHG field that undergoes one multiple reflection in the yy s01 o Y12 GaN film prior to transmission through the GaN/Al2O3 in- and 2938 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
2 Ϫ1 2 2 Ϫ1 2 4tp20͑2o͒ kf1kf2z kf1 kf0kf1z kf0 ϭeˆ ϩk eikf2⌬ k ϩ eϪikb1zdϩ2k eϪikf1zd Ϫk k Ϫ Ymn ͑⑀ Ϫ⑀ ͒ ͩͩk k f2ͪ Cmͫͩ b1z k ͪ f1 ͩk k f1ͪ ͩ b1z k ͪͬ b f f1z f2 ͭ f1z f0z f1 f0z 2 Ϫ1 2 Ϫ1 kf0kf1z kf2kf1z ϩ Ϫk eϪikb1zdϪ2k k eϪikf1zd Ϫk Ϫr ͑2 ͒t ͑2 ͒eikf1␦ ϩk Cnͫ b1x f1 b1x ͩk k f1ͪ ͬ p10 o p12 o ͩk k f1ͪ f0z f1 ͮ f2z f1 2 2 2 Ϫ1 2 kf2 kf2kf1z kf0kf1z kf0 ϫ k Ϫ eϪikb1zdϩeϪikf1zd k Ϫ Ϫk k Ϫ Cmͫͩ b1z k ͪ ͩ f1 k k ͪͩk k f1ͪ ͩ b1z k ͪͬ ͭ f2z f2z f1 f0z f1 f0z 2 2 Ϫ1 kf2kf1z kf0kf1z ϩ Ϫk eϪikb1zdϪk eϪikf1zd k Ϫ Ϫk . ͑27͒ Cnͫ b1x b1x ͩ f1 k k ͪͩk k f1ͪ ͬ ͪ f2z f1 f0z f1 ͮ
The coupling of the fields in the medium determines mn The transmitted SHG wave is composed of a sum of with Y waves that propagate through the film stack with various phase velocities. For example, we can rewrite the transmitted kb1xkb1z second-harmonic wave, Esp(2o), to explicitly display its ϭϪ 1 2 ͑28͒ phase dependence as C kf1 b i͑kf2⌬Ϫkb1zd͒ f i͑kf2⌬Ϫkf1zd͒ and Esp͑2o͒ϭDABe ϩDABe
f i͑kf2␦Ϫkb1zd͒ b i͑kf2␦Ϫkb1zd͒ 2 ϩDACe ϩDACe . kb1z 2ϭ 1Ϫ . ͑29͒ ͑35͒ C ͩ k2 ͪ f 1 Ultrafast effects are incorporated into the theory by expand- The phases in Eq. ͑27͒ of the wave that propagates from ing the wave-vector phases to first order in (Ϫ2o)asin point A to point B and the wave that propagates from point A Eqs. ͑9͒ and ͑10͒. Comparison of Eq. ͑35͒ with Eqs. ͑25͒– to point C in Fig. 5 are (2) ͑29͒ reveals that the D coefficients consist of zyy ,Ey , Fresnel coefficients, dielectric constants, and wave vectors in ⌬ϭ2d tan f 1 sin f 1 ͑30͒ b air, GaN, and sapphire. DAB corresponds to the wave that and propagates from point A to point B in Fig. 5 and has a phase that depends on the bound wave vectors in GaN. The other D 2d ␦ϭ , ͑31͒ coefficients may be derived similarly. The time required for cos b f b f 1 the DAB and DAB waves to propagate to point B and the DAC f respectively. and DAC waves to propagate to point C is effectively con- Equations ͑26͒ and ͑27͒ include Fresnel transmission and tained in the phase of each wave ͑i.e., a group velocity is reflection coefficients of the fundamental and second- associated with each wave vector͒. We calculated this time by replacing each wave vector with the inverse of its group harmonic fields. For example, ts01(o) is the Fresnel trans- mission coefficient for s-polarized light from layer 0 to layer velocity and Fourier transforming the resulting expression. After carrying out this procedure, we see that ultrafast effects 1. The Fresnel reflection coefficient is denoted similarly with b r substituted for t. Equation ͑25͒ explicitly displays the three are included in the DAB wave by multiplying each term by a factors on which the nonlinear response of the media de- Gaussian centered at the time required to propagate the wave (2) from point A to point B, i.e. pends. These factors are: the nonlinearity of the media, zyy ; the applied fundamental field, Ey ; and the propagation of the b i͑kf2⌬Ϫkb1zd͒ b i͑kf2⌬Ϫkb1zd͒ DABe DABe nonlinear fields, Y yy . Note that Y yy is a function only of the → 2 2 linear optical properties of the media and the GaN film thick- Ϫ⌬ ⌬ d cos b1 ϫexp tϪ Ϫ . ͑36͒ ness. ͫ 8 ͩ v v ͪ ͬ The second-harmonic response of GaN in the p-in/p-out f 2 b1 geometry may be written similarly as The ultrafast effects are incorporated similarly into the other D factors. ͑2͒ ͑2͒ ͑2͒ Epp͑2o͒ϭYxxzxxExExϩ2YzxxzxEzExϩYzzzzzEzEz ͑32͒ III. RESULTS with A. Rotational symmetry 2 Because second-harmonic generation is mediated by a YxxϭYzzϭtp01͑o͒ 12 ͑33͒ Y third rank tensor, it is inherently more sensitive to material and symmetry than linear processes.39 By measuring second- harmonic generation as a function of the angle between the Y ϭt2 ͑ ͒ . ͑34͒ zx p01 o Y21 incident beam polarization and the crystal axes, we probed PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2939
(2) TABLE IV. Azimuthal dependence of P (ϭ2o) from GaN ͓0001͔. Ei is defined as the ith compo- nent of the electric field in the GaN film.
(2) Polarization P (ϭ2o)
(2) (2) (2) p in/p out 2xzxEx(o)Ez(o)ϩzxxEx(o)Ex(o)ϩzzzEz(o)Ez(o) p in/s out 0 (2) s in/p out zyyEy(o)Ey(o) s in/s out 0 the symmetry of our GaN thin films. Using this technique, miscut into our theory to account for this modulation by we were able to measure film miscuts with an accuracy of modifying the form of the rotation matrix to ϳ0.05°. GaN has C6v ͑6mm͒symmetry and the following second- cos sin 0 cos ␣ 0 Ϫsin ␣ order nonlinear susceptibility elements in the dipole approxi- Rϭ Ϫsin cos 0 01 0 , mation: ͩ 001ͪͩsin ␣ 0 cos ␣ ͪ ͑2͒ zzz ͑41͒
2 2 where ␣ is the miscut angle of the film. Note, we have as- ͑ ͒ϭ͑ ͒ xzx yzy sumed that the GaN film is miscut in the xz plane. Because x and y are equivalent in C symmetry, our measurement is ͑2͒ϭ͑2͒ 6v xxz yyz sensitive only to the magnitude of the miscut but not the direction of the miscut. The fits to the p-in/p-out and ͑2͒ϭ͑2͒ . ͑37͒ zxx zyy s-in/p-out data yield a sample miscut of 0.78° and 0.82°, For second-harmonic generation, the last two indices are in- respectively. SHG rotational symmetry measurements in the terchangeable, e.g., (2)ϭ(2) . Note that our analysis con- p-in/s-out and s-in/s-out polarizations were complicated xzx xxz 40 siders only dipole second-order processes. Because GaN further by the sapphire birefringence and were therefore lacks inversion symmetry, higher-order multipole processes, not performed. such as electric quadrupole processes, are weaker than the dipole allowed process by at least a factor of (ka)2Ϸ10Ϫ5, B. Spectroscopy where a is the atomic spacing. In addition, our analysis does In this section we discuss our SHG spectra from not consider any harmonics above second-harmonic genera- GaN/Al2O3 in the s-in/p-out and p-in/p-out polarizations. tion. Third-harmonic and higher-harmonic light, if generated, (2) (2) is removed from our measured intensity by our monochro- From these spectra we extract zxx and xzx , respectively. In mator. The second-order susceptibility may be transformed from the crystal frame, where the elements are defined, to the lab frame, where the nonlinear signal is measured, according to
͑2͒,lab ͑2͒,crystal ijk ϭR͑͒i␣R͑͒jR͑͒k␥␣␥ ͑38͒ with
cos sin 0 Rϭ Ϫsin cos 0 . ͑39͒ ͩ 001ͪ Note, the nonlinear polarization is related to second-order susceptibility as
͑2͒ ͑2͒,lab Pi ͑ϭ2o͒ϭijk Ej͑o͒Ek͑o͒, ͑40͒ where E j(o) and Ek(o) are the fields inside GaN. Table IV displays the results of the calculations of the rotational (2) FIG. 6. Transmitted SHG intensity from GaN/Al2O3 as a func- symmetry of P generated from GaN ͓0001͔. For both the tion of crystal orientation at an angle of incidence of 15°. is the p-in/p-out and s-in/p-out measurements, the nonlinear po- angle between the miscut direction and the plane of incidence. larization, and hence the SHG intensity, is rotationally in- छ(ᮀ) is the data from the p-in/p-out(s-in/p-out) polarized variant. SHG is dipole forbidden in the s-in/s-out and second-harmonic field. The solid ͑dashed͒ line is a fit to the p-in/s-out polarizations. In contrast to the theoretical predic- p-in/p-out(s-in/p-out) data with the miscut angle as a free param- tions, our p-in/p-out and s-in/p-out rotational data ͑see Fig. eter. The fitted miscut angle ␣ is 0.78° ͑0.82°) for the 6͒ show a onefold anisotropy. We incorporated a sample p-in/p-out(s-in/p-out) polarizations. 2940 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
FIG. 8. Measured nonlinear susceptibility elements for GaN. (2) (2) छ(ᮀ) denotes zxx(xzx). Data represent the average values for all of the GaN/Al2O3 samples. The solid line ͑dashed͒ is the calculated (2) (2) value of zxx(xzx) from Ref. 1.
age value of 2.45ϫ10Ϫ8 esu measured at a single fundamen- tal wavelength of 1064 nm.10 This is lower than the other (2) Ϫ8 reported values for ͉zxx͉ of 9.25ϫ10 esu ͑Ref. 8͒ and 3.92ϫ10Ϫ8 esu,12 but is not inconsistent with the predicted (2) Ϫ8 41 value for ͉xzx͉ϭ1.64ϫ10 esu. The data also exhibits a weak but distinct peak centered at a two photon energy of 2.80 eV. This feature is inconsistent with the expected spec- tral dependence based on Miller’s rule.42 For measurements in the p-in/p-out polarizations, SHG (2) (2) (2) couples to zzz and xzx in addition to zxx . Therefore, we measured the SHG response of GaN at several angles of incidence to fit the two unknown susceptibility elements ͑see Fig. 7͒. We observed that our spectra were very weakly de- (2) (2) pendent on zzz . The insensitivity of our spectra to zzz FIG. 7. ͑a͒ SHG spectroscopy from the 1.020-m GaN sample in the s-in/p-out polarization. The inset shows SHG spectroscopy for the same sample in the p-in/p-out polarization. Both spectra are referenced to the nonlinear response of quartz in the same polariza- tion configuration as the corresponding GaN spectra. ͑b͒ SHG spec- troscopy from the 4.885-m GaN sample in the s-in/p-out polar- ization. The inset shows SHG spectroscopy for the same sample in the p-in/p-out polarization. Although both samples have approxi- mately the same nonlinear susceptibility, interference dramatically affects the transmitted second-harmonic generation intensity. addition, from our rotational symmetry measurements as a function of the angle of incidence, we placed an upper bound (2) (2) on any symmetry forbidden nonlinearities, e.g., zzx or xzx , induced by the sample miscut. In order to accurately calculate the effects of group veloc- ity mismatch in our GaN films, we measured ⌬ at each wavelength by determining the wavelength distribution of the laser pulse with a monochromator. Figure 7 displays our nonlinear optical spectra from our undoped 1.020 m and 4.885 m GaN samples. Using the techniques developed in the previous sections and our measured values for the linear optical properties of FIG. 9. Band symmetries of GaN at the point. The subscript (2) ⌫ GaN, we extracted zxx from the data. This parameter is notation labels the symmetry of the band. The superscript notation (2) displayed in Fig. 8. The magnitude of zxx is approximately is ours to distinguish between bands of the same symmetry. EF 2.4ϫ10Ϫ8 esu, which is consistent with Miragliotta’s aver- represents the Fermi energy. PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2941 results from the fact that our nonlinear signal is proportional good, although the calculated data exhibits a larger disper- ˆ (2)ˆˆ 2 (2) 2 sion in the second-order susceptibility elements than the ex- to ͉e2zzzee͉ р0.004͉zzz͉ . Therefore, we assume the (2) (2) perimental data. In addition, the calculated (2) lacks a reso- theoretically predicted relationship between xzx and zzz , zxx (2) (2) nance at 2.80 eV. We speculate this feature may result from i.e., zzzϭϪ2xzx . Figure 8 compares the measured value of (2) to (2) averaged over all of our samples. Although virtual transitions involving an intrinsic midgap defect state. zxx xzx (2) these elements are approximately equal, as predicted by the Since, the calculation of the dispersion of zxx includes only 43 (2) bulk band structures, the calculation cannot predict spectral bound charge model, xzx lacks a feature at a two-photon energy of 2.80 eV. features that couple to defect states. (2) We compared our data with a theoretical calculation of By analyzing the nature of a resonance in ijk , the mag- (2) (2) the dispersion of zxx and xzx Ref. 1 in Fig. 8. Hughes, nitude of this feature, and the symmetry of states that may Wang, and Sipe1 calculated the second-order susceptibility participate in this resonance, we are able to make some elements with the full-potential linearized augmented plane- speculative suggestions about the nature of this feature. First, wave method within the local-density approximation. The we consider possible origins of the feature at a two photon agreement between the magnitudes of the calculated and ex- energy of 2.80 eV. The second-order nonlinear susceptibility perimental second-order susceptibility elements is quite for second-harmonic generation is of the form44
3 e ͗g͉ri͉n͗͘n͉rj͉nЈ͗͘nЈ͉rk͉g͘ ͑2͒͑ϭ2 ͒ϭϪN ϩ ͑0͒ , ͑42͒ ijk o 2 ͚ ••• g ប g,n,n ͫ͑Ϫ ϩi⌫ ͒͑ Ϫ ϩi⌫ ͒ ͬ Ј ng ng o nЈg nЈg
(2) where N is the number of electrons per unit volume; ͉g͘,͉n͘, assumes that the spectral dependence of zxx(ϭ2o) re- and ͉nЈ͘ are states of the system, បng is the energy differ- sults from the denominator of the Lorentzian term. The other (0) ence between states n and g; ⌫ng is a dephasing rate; g is terms in Eq. ͑43͒ are expected to be slowly varying functions the equilibrium density of the initial state g; Ϫerk is the of o . Fitting this function to the data determines Ϸ dipole operator along direction k; and the dots represent (2) Ϫ9 Ϫ/2 and zxx,resϭ3.8ϫ10 esu at a two-photon energy unique permutations of the states. of 2.80 eV. If we assume the resonant component of (2) Because GaN is a direct band-gap semiconductor with a zxx results from defect states with a density of Nϳ1020 cmϪ3, band-gap energy of 3.4 eV, the two-photon resonance at 2.80 then the hyperpolarizability associated with these states is eV cannot result from virtual transitions between valence- Ϫ29 and conduction-band states. In addition, the requirement that ␣zxxϳ10 esu. This speculative analysis yields numbers the process begin with a transition between an occupied and that are large but not impossible. an unoccupied state precludes any process that occurs only in We also determined possible three photon processes that (2) (2) the valence band or only in the conduction band. Thus, it is contribute to zxx(ϭ2o), but not to xzx(ϭ2o), by unlikely that the 2.80-eV feature results purely from the bulk analyzing the symmetry of the proposed midgap defect state. band structure of GaN. On the other hand, it is well known Defects in GaN are frequently characterized by s-like or that GaN is characterized by a variety of defect states.45 The p-like symmetry.46 Our model assumes a defect state lies in spectral feature could result from a three-photon process that the GaN band gap with s-like or p-like symmetry and that a involves both a defect state and the GaN bands. For example, three-photon process originates at either the valence band a bulk defect state ͑or defect band͒ with energy 2.80 eV maximum or at the defect state. Group theory47 was used to above the valence band ͑a midgap state͒ could play a role in calculate the nonzero matrix elements in Eq. ͑42͒. In this this resonance. calculation, we used the band symmetries determined (2) 48 A simple analysis of the zxx spectrum can constrain by Bloom et al. Figure 9 displays the symmetries of the somewhat the properties of such a defect state. We decom- GaN bands at the ⌫ point and their approximate energies. 49 pose the susceptibility into resonant and nonresonant terms Assuming C6v symmetry, s-like states have A1(⌫1) sym- as metry and p-like states have E2(⌫6) symmetry. Table V ͑2͒ ͑2͒ ͑2͒ shows all nonzero resonant three-photon processes for zxx͑ϭ2o͒ϭzxx,res͑ϭ2o͒ϩzxx,non (2) (2) zxx(ϭ2o) and xzx(ϭ2o). Of the cases examined, i only the three-photon process that originates from a p-like Ae ͑2͒ ϭ ϩzxx,non , ͑43͒ defect state is inconsistent with our observation of a 2.80-eV ͑2oϪngϩi⌫ng͒ (2) (2) resonance in zxx(ϭ2o) but not in xzx(ϭ2o). Fig- where the symbols have the same meaning as in Eq. ͑42͒ and ure 10 displays three-photon processes that may contribute to ei is a phase difference between the resonant and nonreso- this resonance. nant components. The resonant term is attributed to a two- We may draw a few more qualitative conclusions from photon resonance involving transitions to a midgap defect our spectroscopic data. First, the spectroscopic feature at a state and the nonresonant term is attributed to all other non- two-photon energy of 2.80 eV is probably not associated resonant bulk three-photon transitions. This decomposition with yellow band defects. This conclusion results from the 2942 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
(2) 50 TABLE V. Possible contributions to zxx(ϭ2o) and from vicinal Si(111)/SiO and Si(100)/Si N interfaces (2) 2 3 4 xzx(ϭ2o) for a defect state in the band gap of GaN. All pro- suggests that strain-induced defect states may be generated at cesses are at the ⌫ point of the Brillouin zone. ⌫defect refers to a 6 or near the GaN/Al2O3 interface. Strain is expected to couple defect state with ⌫6 (p-like͒ symmetry. The labels g, n, and nЈ most strongly to the components of the susceptibility refers to the states in Eq. ͑42͒. elements that are perpendicular to the interface.51 Thus, (2),film Susceptibility Defect symmetry Origin gnnЈ this effect may be most easily observed in xzz and (2),film .52 The film superscript denotes that these ele- (2) a b defect zzx zxx s like valence ⌫1 ⌫5 ⌫1 ments are defined with respect to the film and not to the (2) defect b b zxx s like defect ⌫1 ⌫5 ⌫1 crystal axes, i.e., z ϭz Þz where z , z , and (2) a b defect film lab crystal film lab zxx p like valence ⌫6 ⌫3 ⌫6 z are the film, lab, and crystal axes. Note that these a b defect crystal ⌫6 ⌫5 ⌫6 elements are defined to include only the effect of the strain- (2) zxx p like defect induced nonlinearity and do not result from a coordinate (2) xzx s like valence transformation. We performed rotational symmetry measure- (2) xzx s like defect ments in the p-in/p-out polarizations to probe GaN for these (2) p like valence (2),film xzx elements. We simplified our analysis by assuming zzx (2) p like defect ⌫defect ⌫b ⌫b (2),film xzx 6 6 3 ϭxzz . We deduced the miscut angle of our films from the rotational symmetry data at an angle of incidence of 15° (2),film assuming zzx ϭ0. We repeated our rotational measure- presence of the 2.80-eV feature in all samples with and with- ment over a two photon spectral range of 2.6 to 3.4 eV at an out yellow band luminescence. Second, the defect is likely to (2),film angle of incidence of 65° to fit for with the miscut be a point defect and not a defect complex. Defect com- zzx angle fixed from the 15° rotational scan. We were unable to plexes could arrange themselves in various bonding geom- detect any strain or miscut induced nonlinearity within the etries within the lattice, and hence, the hyperpolarizabilities (2),film of the individual complexes would tend to cancel in an en- limit of our measurement, and determined zzx (2) (2),film semble average. Third, several symmetries and energies may Ͻ0.005zxx . Figure 8 also displays zzx . exist within the band gap of the defect state that may con- Interface states that do not result from miscut induced (2) (2) tribute to zxx but not xzx , but it is impossible to determine strain are not observable by our rotational symmetry tech- the energy level with only SHG measurements. A possible nique. These states could result from bonding between the 53 origin of the 2.80-eV resonance may be the NGa defect state. substrate and the thin film. Due to symmetry at the inter- (2) Jenkins and Dow have predicted the N on Ga site defect state face, these states most strongly affect zzz . Unfortunately, will have s-like symmetry and will lie approximately 3.0 eV sample geometry precludes a direct measurement of this 46 above the valence band. This defect state may contribute to second-order susceptibility element. the 2.80-eV resonance by the three photon process illustrated in Fig. 10͑a͒. Finally, we consider the influence of the sample miscut on IV. CONCLUSION possible strain-induced nonlinearities in GaN. Observation of strain and disorder effects on second-harmonic generation We have used second-harmonic generation to probe the bulk second-order nonlinear susceptibility elements of GaN over a two-photon energy range of 2.6–3.4 eV. We charac- terized the defect structure of our samples by photolumines- cence and measured the index of refraction by a novel trans- mission method. Using various combinations of the fundamental and second-harmonic beams, we were able to (2) (2) separately measure both zxx and xzx . Spectroscopy reveals (2) a resonant enhancement in zxx at a two-photon energy of (2) 2.80 eV, which is absent in xzx . The subband gap enhance- ment is inconsistent with Miller’s rule and may result from an intrinsic midgap defect state. In addition, we demon- strated sensitivity to the crystal miscut of GaN by rotational SHG, and we determined that the miscut strain does not in- duce forbidden bulk nonlinearities within our experimental (2) (2) limits (zzxϽ0.005zxx). Finally, we developed new techniques for analyzing ul- FIG. 10. Possible resonant three photon processes for the 2.80 trafast second-harmonic generation. We employed zeroth- (2) order and first-order expansions of the wave vectors with eV resonance in zxx . ͑A͓͒͑B͔͒ represents a process that originates at the valence band ͑defect state͒. Only the bands that define the respect to angular frequency for the coefficients and phases, band gap and bands explicitly involved in the three-photon pro- respectively. Group velocity mismatch is significant for SHG cesses are displayed. ͑A͓͒͑B͔͒ corresponds to the first ͑second͒ spectroscopy with ultrafast pulses in GaN samples as thin as entry in Table V. 5 m. PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2943
ACKNOWLEDGMENTS kץ o n͑o͒o 1 k1ϭ and ϭ . ͑A5͒ 1ͯץ We would like to thank E. J. Mele and M. S. Yeganeh for c vb 1 stimulating discussions. We thank J. L. P. Hughes and J. E. o Sipe for providing us with their calculated nonlinear suscep- The subscript b denotes a bound wave. The second-order tibilities data. We would also like to thank C. S. Koeppen term in the expansion of k(1) is responsible for the broad- and A. F. Garito for technical help and for use of their fa- ening of the pulse. Assuming a 130-fs fundamental pulse, we cilities. A.G.Y. gratefully acknowledges the support of the determined that the second-harmonic and fundamental pulses NSF through Grant No. DMR-97-01657. broaden less than 0.4% for a 5-m film of GaN and less than 2ϫ10Ϫ6 for a 3-mm slab of quartz over a range of frequen- 55 APPENDIX A: SECOND-HARMONIC GENERATION cies probed in this experiment. Thus, the first-order ap- FROM A QUARTZ SLAB EXCITED proximation is accurate. Using this first-order phase approxi- BY AN ULTRAFAST LASER mation, the nonlinear polarization is
In this appendix, we solve for the second-harmonic gen- 2 Ϫ2͑Ϫ2o͒ eration from a quartz slab excited by an ultrafast laser P͑2͒͑r,͒ϭP eikb͑͒y exp ͑A6͒ x o 2 source. All of the approximations used in this appendix are ͩ ⌬ ͪ also valid for SHG from GaN thin films. The fundamental with field propagates from the vacuum to the nonlinear quartz crystal at ϭ0, where it generates a second-harmonic field. ͱ This field propagates through the crystal to the second 2 ͑2͒ Poϭ ⌬ xxx͑ϭ2o͒. ͑A7͒ crystal/vacuum interface at yϭd. A second-harmonic field is 2ͱ2 E transmitted through this interface and has a magnitude that depends on the boundary conditions. We solve for the trans- As expected, the nonlinear polarization has a Gaussian angu- mitted SHG intensity using a first-order expansion with re- lar frequency distribution that is centered at 2o . Note the use of the symbol k () in the phase P(2) . This symbol spect to (Ϫ2o) in the phase factors and a zeroth-order b x denotes that k () is the bound wave vector, which has the expansion in (Ϫ2o) for the multiplicative field coeffi- b cients. This model does not account for the effects of mul- functional form tiple reflections of the second-harmonic or fundamental Ϫ2 fields. Even without incorporating these effects however, the o ͑ o͒ kb͑͒ϭ2k ϩ . ͑A8͒ model accurately captures both wave-vector mismatch and vb group velocity mismatch. This is twice the wave vector of the fundamental field ͓Eq. For bookkeeping purposes, we assume that our fundamen- (2) tal fields are labeled 1 and 2. This notation is used to explic- ͑A4͔͒. We have assumed in this calculation that xxx is a itly exhibit the mixing of different frequency components of slowly varying function of frequency, assuming its center the same pulse in the SHG process. The form of the funda- frequency value over the bandwidth of the pulse. Using Mill- (2) mental fields in angular frequency space is er’s rule, we calculated that xxx varies less than 0.4% over the bandwidth of the Ti:Al2O3 laser pulse. Since the spectra 2 of GaN varies slowly, this assumption is valid for GaN. 1 Ϫ4͑1Ϫo͒ E ͑r, ͒ϭxˆ E eik͑1͒y exp ͑A1͒ The second-harmonic fields may be calculated from the 1 1 2 2 ͩ ⌬ ͪ nonlinear wave equation and the nonlinear polarization. The nonlinear wave equation may be written in angular fre- and quency space as
2 Ϫ4͑ Ϫ ͒ 2 2 2 ik͑ ͒y 2 o ⑀͑͒ 4 E ͑r, ͒ϭxˆ E e 2 exp . ͑A2͒ ͑2͒ .2 2 ٌϫٌ͑ϫ͒Ϫ E͑r,͒ϭ P ͑r,͒ 2 2 ͩ ⌬ ͪ ͫ c2 ͬ c2 A9 The general expression that relates the second-order suscep- ͑ ͒ 54 tibility to the fundamental fields is We assume that the bound wave solution ͑particular solu- tion͒ to the nonlinear wave equation is ϱ ϱ P͑2͒͑͒ϭ42 ͑2͒͑ϭ ϩ ͒E ͑ ͒ i ͵ ͵ ijk 1 2 j 1 ˆ ˆ ikb͑͒y Ϫϱ Ϫϱ Eb͑r,͒ϭxEbx͑r,͒ϭxb͑͒e . ͑A10͒
ϫEk͑2͒␦„Ϫ͑1ϩ2͒…d1d2. ͑A3͒ Thus, the wave equation simplifies to
We can simplify this expression by expanding the phase of 2 2 2 ⑀͑͒ 4 ͑2͒ the fundamental fields to first order in (1Ϫo), i.e., kb͑͒Ϫ Ebx͑r,͒ϭ Px ͑r,͒. ͫ c2 ͬ c2 ͑A11͒ ͑1Ϫo͒ k͑ ͒ϭkoϩ , ͑A4͒ 1 1 v We simplify the equation further by assuming that the mul- b1 tiplicative coefficients of the field can be approximated by where their zeroth order values, i.e., 2944 ANGERER, YANG, YODH, KHAN, AND SUN PRB 59
4 2 This is the same approximation used in the simplification of 2 ⑀bo kb͑͒Ϸ ͑A12͒ the bound wave field. The transmitted wave envelope may be c2 expressed as and i͓k f 1͑͒Ϫk f 0͔͑͒d f 2͑͒ϭ␣b1͑͒e
2 2 ϩ ͑͒ei͓kb1͑͒Ϫk f 0͔͑͒d. ͑A19͒ ⑀͑͒ 4⑀ f o b1 Ϸ ͑A13͒ c2 c2 with with ⑀bϵ⑀(o) and ⑀ f ϵ⑀(2o). Thus, Eq. ͑A11͒ reduces to Ϫ2n f 1͑nb1ϩ1͒ nb1ϩn f 1 ␣ϭ and ϭ . ͑A20͒ 2 n ϩ1 ͑2͒ ͑n f 1ϩ1͒ f 1 4Px ͑͒ Ebx͑͒ϭ . ͑A14͒ By Fourier transforming the transmitted SHG field, ⑀bϪ⑀f Ef 2(r,), and using the first-order phase approximation This relationship between the nonlinear polarization and the again, the expressions in the text may be generated. bound wave field is the same as for the monochromatic case. 2 2 2 Although kb and ͓⑀() ͔/c vary over the pulse width, APPENDIX B: SECOND-HARMONIC GENERATION their difference is nearly constant. We calculated that this FROM GaN/Al2O3 approximation introduces an error of less than 0.17% in the In this appendix, we derive the SHG field transmitted magnitude of the field. through GaN/Al O by boundary value techniques. We use We now consider the homogeneous solutions to the wave 2 3 the notation employed by Yeganeh et al.33 to label our non- equation. These solutions are required to satisfy boundary linear fields. These fields are displayed in Fig. 5. The layers conditions at both quartz/vacuum interfaces. There are four are denoted 0, 1, and 2 for air, GaN, and sapphire, respec- free wave fields: E (r,), which is generated at the first f 1 tively. Our model includes all fields that affect the magnitude quartz/vacuum interface and traverses the quartz crystal, of the transmitted intensity by at least 10%. For example, we E (r,), which is reflected off of the first quartz/vacuum f 0 assume no reflection of the linear light field, and hence, the interface into the vacuum, E (r,), which is reflected off f 1r bound wave field, from the GaN/Al O interface. This as- of the second quartz/vacuum interface, and E (r,), which 2 3 f 2 sumption is valid because r ( ) 2Ͻ0.015, where is the transmitted second-harmonic field. We assume that the ͉ 12p o ͉ r ( ) is the Fresnel reflection coefficient for the fields have the form 12p o p-polarized light at angular frequency o with 1 ͑GaN͒ as the incident medium and 2 ͑sapphire͒ as the substrate me- ˆ Ϯikfj͑͒y Efj͑r,͒ϭxfj͑͒e , ͑A15͒ dium. Although we ignore the reflection of linear light, we where j is 1, r1, 0, or 2 and the sign of the phase is positive include the free wave field reflected from the GaN/Al2O3 if the free wave propagates in the direction of the bound interface. This field, E fr1, is related to the free wave field, wave field or negative if it propagates in the direction oppo- E f 2 , that is generated at the GaN/Al2O3 interface and propa- site of the bound wave field. By applying continuity of the gates through the sapphire substrate as ͉E fr1͉Ϸ0.1͉E f 2͉. tangential components of the electric and magnetic fields at The p-polarized fundamental light field generates a non- both yϭ0 and yϭd, an expression for the transmitted free linear polarization of the form wave amplitude is determined as ͑2͒ 2 ͑2͒ Px ϭ2t p01͑o͒xzxExEz ͑B1͒
b1͑͒ and ͑͒ϭ Ϫ2k ͑͒ f 2 k ͑͒ϩk ͑͒ͫ f 1 f 1 f 0 ͑2͒ 2 ͑2͒ ͑2͒ Pz ϭt p01͑o͒͑zxxExExϩzzzEzEz͒. ͑B2͒ k ͑͒ϩk ͑͒ b1 f 0 i͑k ͑͒Ϫk ͑͒d ϫ e f 1 f 0 Note that E is the incident fundamental field and t p01(o)E ͩ k f 1͑͒ϩk f 0͑͒ ͪ is the fundamental field transmitted into the GaN film. The nonlinear polarization generated by an s-polarized funda- i͓kb1͑͒Ϫk f 0͔͑͒d mental field is ϩ͓kb1͑͒ϩk f 1͔͑͒e ͬ. ͑2͒ 2 ͑2͒ ͑A16͒ Pz ϭts01͑o͒zyyEyEy . ͑B3͒ Using the zeroth-order approximation for the phases, the re- The bound wave fields are determined by the nonlinear po- lationship between the angular frequency and the wave vec- larization, the nonlinear wave equation, and tor is ͑2͒ Ϫ4ٌ•P E ϭ . ͑B4͒ ٌ • b1 ⑀ n f 12o nb12o b k ͑͒ϭ and k ͑͒ϭ ͑A17͒ f 1 c b1 c The nonlinear bound wave fields are with 2 4 kb1x kb1xkb1z ϭ ͑2͒ Ϫ Ϫ ͑2͒ Eb1x Px 1 2 2 Pz ͑B5͒ ⑀bϪ⑀ fͫ ͩ k ͪ k ͬ n f 1ϵn͑2o͒ and nb1ϵn͑o͒. ͑A18͒ f 1 f 1 PRB 59 ULTRAFAST SECOND-HARMONIC GENERATION... 2945
and ˆ ikf2⌬ Et͑r,2o͒ϭetp20͑2o͒͑E f 2e
2 ikf1␦ 4 kb1z kb1xkb1z ϩrp10͑2o͒t p12͑o͒E f 1re ͒ ͑B10͒ ϭ ͑2͒ Ϫ Ϫ ͑2͒ Eb1z Pz 1 2 2 Px . ͑B6͒ ⑀bϪ⑀ fͫ ͩ k ͪ k ͬ f 1 f 1 with
In order to satisfy boundary conditions, a free wave is 2d generated at the air/GaN interface and propagates with the ␦ϭ ͑B11͒ bound wave to the GaN/sapphire interface. This wave is cos f 1
2 Ϫ1 2 and k f 0k f 1z k f 0 E ϭ Ϫk k Ϫ E Ϫk E . f 1 ͩ k k f 1 ͪ ͫͩ b1z k ͪ b1x b1x b1zͬ f 0z f 1 f 0z ⌬ϭ2d tan f 1 sin f 1 . ͑B12͒ ͑B7͒ At the GaN/sapphire interface, two additional free waves are Note that ␦ is the round-trip distance that E f 1r travels in the GaN film and ⌬ is the distance that E f 2 travels in the sap- generated: a field transmitted through the GaN/Al2O3 inter- face, E , and a free wave reflected from the GaN/Al O phire substrate during the round trip of E f 1r . Equation ͑B10͒ f 2 2 3 is rewritten in the text as Eqs. 25 and 32 to explicitly interface, E . These waves have the forms ͑ ͒ ͑ ͒ fr1 demonstrate the transmitted wave’s dependence on the non- 2 Ϫ1 2 linear susceptibilities of GaN, the applied linear fields, and k f 1k f 2z k f 1 E ϭ ϩk k ϩ E eϪikb1zd propagation effects. f 2 k k f 2 b1z k b1x ͩ f 1z f 2 ͪ ͫͩ f 1z ͪ Equation ͑B10͒ does not include the effect of the ‘‘walk off’’ between the bound and free waves. Walk off refers to Ϫikb1zd Ϫikf1zd Ϫkb1xEb1ze ϩ2kf1Ef1e ͑B8͒ the spatial separation of the bound and free waves that results ͬ from different directions of propagation of these waves in the and nonlinear crystal. Walk off decreases the overlap of the bound and free waves, and hence modifies the interference of 2 Ϫ1 2 kf2kf1z kf2 these waves. For an 800-nm beam with an angle of incidence Ϫikb1zd E f 1rϭϪ ϩkf1 kb1zϪ Eb1xe ͩk k ͪ ͫͩ k ͪ 0 of 46.0°, the corresponding free and bound waves propa- f2z f1 f2z gate in GaN with angles of ϭ16.4° and ϭ18.0°, re- 2 f 1 b1 kf2kf1z spectively. After propagating through 5 m of GaN, the Ϫk E eϪikb1zdϩ k Ϫ E eϪikf1zd . b1x b1z ͩ f1 k k ͪ f1 ͬ spatial separation of the center of the free and the bound f2z f1 waves is 0.15 m. Therefore, walk off is insignificant in ͑B9͒ our experiment. In addition, no walk off occurs in our SHG Therefore, the field transmitted through GaN/Al2O3 is measurements of quartz because bϭ0° and f ϭ0°.
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