Far-field Imaging of Optical Second-Harmonic Generation in Single

NANO LETTERS

Far-field Imaging of Optical Second-Harmonic Generation in Single GaN Nanowires

2007 Vol. 7, No. 3 831-836

J. P. Long,* B. S. Simpkins, D. J. Rowenhorst, and P. E. Pehrsson U.S. NaVal Research Laboratory, Washington, D.C. 20375 Received October 17, 2006; Revised Manuscript Received January 3, 2007

ABSTRACT Means for assessing the nonlinear optical properties of nanoscale materials are of key importance for the advancement of active nanophotonics. By correlating second-harmonic generation (SHG) with electron backscattered diffraction from single GaN nanowires (NWs), we demonstrate that far-field microscopic imaging of SHG offers an approach for distinguishing crystallographic orientations of NWs lying on a substrate. The quasi-static approximation, which should prove useful in describing many nanophotonic behaviors, is shown to satisfactorily account for the SHG data.

The nonlinear optical (NLO) response of nanostructured materials is of current interest because of the need for active elements in nanophotonic applications and because the nonlinear response itself can serve as a useful diagnostic that probes, for example, intense nanoscale plasmonic fields or the nanostructure itself. A nanostructure of particular photonic interest is the semiconducting nanowire (NW), which can serve as a “self-wired” nano-light-emitting diode, laser, or photoconductive detector.1 Several important NW materials, such as CdS, ZnO, and GaN often grow in noncentrosymmetric forms and so can be expected to exhibit a bulk NLO response, and in fact, studies of second-harmonic generation (SHG) both from NW ensembles2 and from single3 NWs of ZnO have been reported. Because the NLO response of materials can depend sensitively on the excitation and emission polarizations relative to the crystallographic orientation, single-particle studies are especially valuable. For example, compared with ensemble measurements, singleparticle investigations avoid averaging not only over material variances among individual nanostructures but also over variances in orientation that can obscure underlying structure in the NLO response of individual particles. By combining electron backscattered diffraction (EBSD) and SHG measurements on the same set of GaN NWs, we demonstrate that the crystallographic orientation of individual NWs can be distinguished Via SHG measurements recorded in the far field with an optical microscope. A far-field approach offers flexibility over more complex, albeit powerful,3 near-field scanning methods. Like polarized Raman spectroscopy,4 SHG offers an alternative to electron diffraction methods and hence could serve as a fully optical in situ

diagnostic of crystallographic orientation, for example, in surveys of NW collections, in NWs under test, or even in NWs sealed in devices. In addition to use in NLO studies, the ability to optically distinguish crystallographic orientation may have utility in other NW applications that also rely on crystallographic orientation, for example, those that depend on the piezoelectric tensor. Piezoelectric properties have been exploited to produce piezoelectric-field-induced carriers in GaN NW-based field-effect transistors5 and also have potential for use in nanoscale components such as motors, resonators, and power generators.6 We find that the main polarization features of the SH generation and emission are adequately explained for our wurtzite NWs by neglecting surface contributions and treating the bulk crystal within the quasi-static approximation,7 wherein one assumes that the NW transverse dimension is much less than the first and second harmonic wavelengths and thus responds as if excited by a spatially uniform electric field. Our data also emphasizes the importance of including the transverse depolarization effect, which reduces the component perpendicular to the NW of not only the pump electric field8 but also9,10 the second-order polarization, P(2ω). The GaN NWs were grown with the vapor-liquid-solid process.11 Oxidized Si wafers, coated with a 2 nm catalyst film of permalloy (80% Ni, 20% Fe), were placed in a quartz tube furnace with solid Ga. After evacuation and N2 purging, wire synthesis occurred at 940 °C under 20 sccm NH3 at atmospheric pressure. The NWs were dispersed in ethanol by sonication and deposited by drop casting on a 100 nm Au film coating an oxidized Si wafer for SHG and EBSD measurements. The crystallographic orientation of a NW was determined by indexing EBSD patterns (Figure 1d) from at

10.1021/nl0624420 This article not subject to U.S. Copyright. Published 2007 by the American Chemical Society Published on Web 02/28/2007

Figure 1. Comparison of a typical (a) SEM image and (b) SHG image of the same NW. Scale bar, applicable to both, is 2 µm. Panel c is a SEM image showing the characteristic triangular cross section of the NWs studied here. Scale bar is 100 nm. Panel d shows a representative electron backscattered diffraction pattern from a single NW showing Kikuchi bands used to determine crystallographic orientation.

least two points along a NW and correlating the results with the scanning electron microscopy (SEM) image of the NW. All NWs were found to be single crystal with a growth axis nominally along [2 1 0], so that the [0 0 1] c-axis was approximately perpendicular to the NW axis. However, variations in the orientation of the c-axis with respect to the experimental optic axis, as well as small variations with respect to the NW axes, resulted in significant effects on the SHG measurements. As illustrated in Figure 1c, all NWs had a triangular cross section with a side length of about 75 nm. Based on size, cross section, and growth axis, these NWs are quite similar to GaN NWs reported elsewhere.11 The experimental geometry for the SHG measurements is shown in Figure 2. For purposes of analysis, a NW is assumed to lie along the x-axis as sketched in Figure 2a. Pulsed, linearly polarized pump light from a femtosecond Ti-sapphire laser (860 nm, 80 MHz repetition rate, 8-30 mW just before the objective)12 was focused to a submicrometer spot on the sample through a dry objective (100×, numerical aperture 0.9) specialized for polarization work. The incident electric field vector Eo could be variably rotated with a half-wave plate to angle R measured with respect to the NW axis. The SH emission, Figure 2b, was collimated by the same objective and isolated from the fundamental pump light and multiphoton-excited photoluminescence by a combination of dichroic mirror and bandpass filter (420 ( 20 nm). A polarizing filter then selected SH radiation polarized either parallel or transverse to the NW axis, after which a tube lens focused the light directly onto an electronmultiplying charge-coupled device (CCD). A SHG image of the NW was recorded by using a laser-scanning unit to raster the focused pump light over a frame (12 µm × 20 µm) containing the NW, while the CCD integrated the emitted light for typically 2 s (so-called non-descanned imaging). That the image was due only to SH radiation was confirmed with a spectrometer attached to a microscope side port. The data points in Figure 3 show the dependences of the SHG signals Sa(R) and St(R) on pump light polarization angle 832

Figure 2. Geometries used for quasi-static analysis of the farfield SHG signals Sa(R) and St(R). Panel a illustrates excitation: incident optical electric field Eo in the diffraction-limited focus of the objective lens lies in the xy plane at a variable angle R to the NW axis; the angles θc and φc define the orientation of the crystallographic c-axis. Panel b illustrates emission: a representative SH ray is shown originating at the origin and intersecting a reference sphere at r, where it refracts to travel parallel to the z-axis. The ray propagates the vector sum of electric fields radiated from the discrete dipole P(2ω) and its image P(2ω)′ . The coordinate system d d aids in computing the SH power that is collected by the objective and passed by a polarizing filter, giving rise to signals Sa(R) and St(R) when the filter is aligned to transmit, respectively, radiation polarized along or transverse to the NW axis.

R. Here Sa and St refer, respectively, to emission polarized along and transverse to the NW axis [see Figure 2b]. The plotted signals represent the CCD charge averaged over a rectangular region-of-interest (typically 1.2 µm × 5 µm) placed with analysis software over a NW image, minus the background averaged over an identically sized region adjacent to the NW. As the analysis below shows, the main features of the markedly different polarization behaviors in Figure 3 are explained by bulk SHG from NWs that lie on the substrate with different crystallographic orientations. More specifically, the polarization behaviors fall into two broad classes, with the Sa(R≈0°) signal exhibiting either a peak, when the (001) facet adheres to the substrate [Figure 3a], or a minimum, when one of the other two side facets adhere [Figure 3b,c]. Nano Lett., Vol. 7, No. 3, 2007

3, which plot results of a simultaneous fit to Sa(R) and St(R) over four separate NWs (only three are shown). The model successfully reproduces the overall shapes of the curves and can detect small deviations in the c-axis orientation from high-symmetry directions, though the degree of approximation presently precludes a fully quantitative analysis for our sample configuration. Here we sketch an outline of our model and provide additional detail in the Supporting Information. In the quasistatic picture, one assumes that the NW cross section is small enough that the NW response to the driving optical electric field is in phase throughout the illuminated volume V. Then the SH radiation may be simply regarded as dipole radiation14 driven by a discrete dipole, P(2ω) d V, oscillating at the SH is the secondfrequency 2ω (hereafter V ) 1). Here P(2ω) d order polarization induced by the driving optical field at frequency ω, corrected as described below for depolarization effects associated with the NW shape. Because our NWs lie on a Au substrate, we also include the SH radiation from an image dipole,15 P(2ω)′ , as sketched in Figure 2b. To deterd mine P(2ω) , we first must compute the uncorrected bulk SH d polarization, P(2ω), which is given by the product of the second-order susceptibility tensor, χ(2ω), and the components of the electric field, Ei, internal to the NW. For the 6mm point-group symmetry of wurtzite, P(2ω) is given by16,17 P(2ω) ) cχcccEcEc + cχcbbEbEb + 2ebχbcbEcEb

Figure 3. SHG signals Versus the polarization angle R of the incident pump light as defined in Figure 2. Sa(R) (O) and St(R) (9) refer, respectively, to SH emission polarized along and transverse to the NW long axis. Relative amplitudes within each panel are as measured, but arbitrary scaling has been applied between panels. Curves show results of a global fit based on a quasi-static model for excitation and for SH emission from a wurtzite NW; the fitting parameters describing NW crystallographic orientation were seeded with values obtained from EBSD measurements for each NW. Panel a shows polarization patterns for a NW lying with its c-axis nominally perpendicular to the substrate. The large intensity difference between Sa(R) and St(R) can be attributed mainly to a slight tilt of the c-axis off the optic axis. The inset to panel a shows SHG images recorded for the two selected emission polarizations when the pump polarization is parallel to the NW (R ) 0). The properly focused image for St(0) exhibits the bifurcation signature of an emitting dipole (P(2ω) d ) oriented parallel to the optical axis, consistent with the c-axis being nearly perpendicular to the substrate. Scale bar 2 µm. Panel b shows results for a NW lying on a side facet, such that the c-axis is tilted ∼114° from the optic axis. The symmetric pattern confirms that the c-axis is perpendicular to the NW to within a degree (φc ≈ 0). Panel c shows results for a NW with its c-axis ∼54° from the optic axis (θc ≈ 54°) but tilted ∼3° from perpendicularity to the NW.

A rigorous analytical treatment of the SHG emission would require solving the wave equations at the first and second harmonics13 for a truncated cylinder in the presence of a reflecting substrate. Fortunately, we find that many features of the data are explicable by applying the much simpler quasi-static approximation, as shown by the curves of Figure Nano Lett., Vol. 7, No. 3, 2007

(1)

where Ei has been decomposed into its components parallel (Ecc) and perpendicular (Ebeb) to c, the unit vector along the c-axis.18 The unit vector eb points along Eb and is parallel to the crystallographic basal plane. Note that in eq 1, the three elements of χ(2ω) are, in order of appearance, equal to the conventionally notated19 χzzz, χzxx, and χxzx; our notation avoids confusion with the lab-frame (x, y, z) axes of Figure 2. The quasistatic approximation allows one to directly relate Ei, and hence Ec and Eb in eq 1, to the external optical field Eo. Because the NW cross section is taken to be much smaller than the pump wavelength, at any instant in time the NW experiences a spatially uniform incident optical electric field Eo eiωt. For an infinitely long dielectric cylinder (qualified below) immersed in an uniform field, a well-known result from electrostatics7 finds that Ei inside the cylinder is also spatially uniform and given by Ei ) Eoxxˆ + Rt1Eoyyˆ , where xˆ and yˆ are unit vectors and Rt1 is a field reduction factor that applies in the direction transverse to the NW axis. The factor Rt1 arises from the so-called depolarization field, that is, an internal field originating from the charge impressed on the cylindrical walls by the abrupt termination there of the dielectric polarization induced by the applied field.20 It is given by7,9,10 Rt1 )

2 1 + (ω)

(2)

where (ω) is the dielectric constant at the pump frequency ω. Similarly, as emphasized by Dadap et al.,13 one must also 833

account for depolarization at the second harmonic frequency. Electromagnetic theory finds that the component of P(2ω) that is perpendicular to the NW is reduced by a factor Rt2 with the same form as eq 2 but evaluated for (2ω).9,10 Hence the SH dipole radiation is driven by an effective dipole P(2ω) ) P(2ω) + Rt2P(2ω) , where P(2ω) is the axial component d a t a (2ω) is the transverse of P (i.e., along the NW axis), and P(2ω) t component. As described in the Supporting Information, the signals Sa(R) and St(R) can be computed by numerically integrating the appropriate polarization component of the dipole radiation and its image over the collection aperture of the from P(2ω) d microscope objective or, equivalently,21 over the reference sphere pictured in Figure 2b. The curves in Figure 3 were obtained by simultaneously fitting eight data sets [Sa(R) and St(R) for four NWs] with a fitting function that incorporated a numerical integration of the light collected by the objective. In the fits, we fixed Rt1 ) 0.28 and Rt2 ) 0.31 as appropriate for the frequencydependent dielectric constant.22,23 (Below we comment on the favorable comparison of field-reduction factors for a circular and triangular cross section.) The elements χijk in eq 1 could be represented in the fits only by the ratios fcbb ) χcbb/χccc and fbcb ) χbcb/χccc, which were treated as fitting parameters common to all NWs. Considering the approximations involved, the values returned, fcbb ≈ -0.3 and fbcb ≈ -0.4, are consistent with the theoretical19 and experimental16,17 value of -0.5 for both fcbb and fbcb, though we do not construe the present data and analysis as a means of quantifying fijk. The SHG behavior can depend sensitively on the angles (θc, φc) that define the orientation of the c-axis (Figure 2), and thus one might expect SHG to distinguish the crystallographic orientation of NWs under certain conditions. To test this possibility, we seeded the fits with (θc, φc) values obtained from EBSD measurements and permitted them to vary. (Exceptions are discussed below.) We found that the EBSD and SHG measurements taken from the same NW produced mutually supporting values for (θc, φc). For example, consider the NW of Figure 3a, which rested on a (001) facet so that the c- and optical axes are nearly coaligned along the z-axis of Figure 2. This nominal alignment is seen immediately from the bifurcated image in the inset to Figure 3a, which was recorded with SH light polarized transverse to a NW that was pumped with light polarized along the NW [i.e., St(R)0°)]. The bifurcated image is the linear (NW) analog to the donut shape that results when an objective of finite aperture forms a diffraction-limited image of a radiating dipole aligned with the optical axis.24 Polarized detection permits observation of the bifurcation for a scanned image such as that in Figure 3a. However, if the c-axis were perfectly aligned with the optical axis, then the intensities Sa(R) and St(R) would be equal by symmetry because only χcbb (eq 1) would be operative and P(2ω) and its image d would be aligned with the optical axis. That Sa(R) and St(R) differ substantially in intensity can be attributed to a slight tilt of c, θc ≈ 4° as determined by EBSD and θc ≈ 7° from the fits. For such small θc, however, the azimuthal angle φc 834

is less well-defined: φc(EBSD) ) 50°; φc(SHG) ) 0°. The discrepancy is within the expected error for an EBSD measurement when θc is near 0°; similarly, when θc is only 7°, computations show that SHG is also rather insensitive to φc (until it approaches 30°). Like the NW of Figure 3a, the NWs of Figure 3b,c also have the [2 1 0] direction along the NW axis, but the c-axis is no longer nominally perpendicular to the substrate. Instead, c is tilted off the z-axis by angles roughly consistent with expectations (θc ) 58.5°) if our NW triangular cross section corresponds to that of Kuykendall et al.11 so that a (112) or (1h1h2) plane rests on the substrate. Because the SHG fits were relatively insensitive to θc for this case, we fixed θc to the EBSD values of 114.6° and 54.5° for Figure 3, panels b and c, respectively. For such large θc, the relative peak heights of the lobes at R ) 45° in Figure 3, panels b and c, are quite sensitive to φc, and we find a gratifying correspondence between the trends in EBSD and SHG values for the two NWs: for the NW of Figure 3b, φc(EBSD) ) 89.5° and φc(SHG) ) 90°; for the NW of Figure 3c, φc(EBSD) ) 92.9° and φc(SHG) ) 92°. We note that inclusion of the image dipole, while affecting relative amplitudes to some extent, does not strongly affect the general behavior of Sa(R) and St(R), even though the components of P(2ω) and P(2ω)′ parallel to the substrate d d oppose each other [Figure 2b]. The cancellation is imperfect because Au is not highly metallic at 430 nm and because was placed at z ) d ≈ 40 nm, rather than z ) 0. By P(2ω) d contrast, inclusion of depolarization is crucial, as illustrated in Figure 4, which compares curves for Sa(R) and St(R) that were computed with and without depolarization. The c-axis was assigned the orientation angles θc ) 7° and φc ) 0°, that is, c nearly parallel to the optical axis, corresponding to the NW of Figure 3a. In Figure 4, curves a and a′ were computed with no depolarization (Rt1 ) Rt2 ) 1). In curves b and b′, depolarization of the applied field is turned on (Rt1 ) 0.31), and both Sa(R) and St(R) plummet when the incident polarization is perpendicular to the NW axis (R ) 90°) because the incident field inside the NW is screened by polarization charge on the surface. In curves c and c′, depolarization at the SH is also turned on (Rt2 ) 0.28); the consequent reduction in the component of P(2ω) transverse d to the NW causes a large reduction in St(R) as seen in curve c′. The reduction in Sa(R) is less because it has a contribution from an unscreened component of P(2ω) parallel to the NW, d a consequence of the 7° tilt of c. Note that similar depolarization should strongly influence both the excitation and emission of photoluminescence in NWs and nanorods.8-10 The success of the quasi-static model in reproducing the general features of the far-field signals Sa(R) and St(R) justifies the approximations employed. Nonetheless, approximations in the model may account for several departures from expectations, in addition to the deviations between fit and data seen in Figure 3. We have already noted, for example, that the fitted values for |fcbb| and |fbcb| are smaller than literature values.16 In addition, the strong variation of Sa(R) with R required use of a numerical aperture ∼0.5 in the fitting function instead of the manufacturer’s value of Nano Lett., Vol. 7, No. 3, 2007

Figure 4. Importance of including field-reduction factors Rt1 and Rt2 at, respectively, the fundamental and SH wavelengths, as illustrated by computations with and without depolarization for c-axis angles θc ) 7° and φc ) 0°, and NA ) 0.5, fcbb ) fbcb ) -0.5. Curves a and a′ show results for Rt1 ) 1 and Rt2 ) 1 (no depolarization). Curves b and b′ show results for Rt1 ) 0.31 and Rt2 ) 1, showing the strong reduction in signal when the incident polarization is perpendicular to the NW, that is, when R ) 90°. Curves c and c′ show results for Rt1 ) 0.31 and Rt2 ) 0.28, showing the strong reduction in SH emission polarized transverse to the NW (curve c′). The reduction is less severe for SH emission polarized along the NW because the 7° tilt produces a component of P(2ω) along the NW that is not affected by depolarization.

0.9. Finally, we measured stronger relative signals than predicted for NWs that had the c-axis perpendicular to the substrate, even after inclusion of the image dipole. The requirement that the NW diameter be much less than the internal wavelengths is reasonably satisfied for our pump wavelength λ1 ) 860 nm, for which the wavelength inside the ∼75 nm diameter NWs is λ1/n1 ) 375 nm, given an index of refraction25 n1 of22,23 2.32. The assumption is less well satisfied, but evidently serviceable, at the SH wavelength λ2 ) 430 nm, for which the internal wavelength is λ2/n2 ) 173 nm when22,23 n2 ) 2.49. The phase coherence between pump and SH implicit in these assumptions also requires the NW diameter to be much less than the coherence length,26 λ2/ [(2(n1 - n2)] ≈ 1.5 µm, which is very well satisfied. Perhaps the most significant approximation in our model is an incomplete treatment of the metallic substrate. Suggestive evidence for this is given by the rather large displacement of P(2ω) above the substrate (d ) 40 nm) that d we found necessary to avoid washing out fitted features in Figure 3. (In fact, neglecting the image dipole altogether does not degrade the fits.) While this value is roughly half the NW facet width and thus not unreasonable, a value closer to the center of mass (23 nm) leads to poor results. In addition, a self-consistent treatment of the substrate, lacking in our model, can be expected to alter the polarization response (e.g., the depolarization fields) within the NW at both the fundamental and SH frequencies. We have also considered the approximation of treating the NW’s triangular cross section as a circle, which enabled Nano Lett., Vol. 7, No. 3, 2007

use of a uniform internal field and the specific form for the shape-dependent field-reduction factor of eq 2. Laplace’s equation was solved with a commercial finite-element code for Ei(y,z) inside an infinitely long triangular NW immersed in a constant electric-field applied perpendicular to the NW. To simulate the conducting substrate, the field component normal to the substrate was set to zero as a boundary condition. The field Ei(y,z) is sensibly uniform and largely parallel to the applied field; the average magnitude of this parallel component is only ∼13% smaller than that predicted by eq 2. The internal electric field also acquires small components unaccounted for in our model that are perpendicular to the applied field (i.e., along z), especially near edges. The magnitude of this component, when averaged over the cross section, amounts to only 6% of the parallel component. Since the radiated SH intensity depends on the fourth power of the field, any contribution from these edgeeffect fields to the SHG signal is expected to be small. Our description of the SHG process in NWs, based on the quasi-static approximation, differs from that of ref 3, which employed geometrical optics to treat near-field scanning optical microscopy (NSOM)-detected SHG from ZnO NWs of similar size to ours (but with hexagonal cross section). Because the geometrical-optics picture is most suitable for structures large compared with a wavelength, it represents an approximation of the opposite extreme to ours. While geometrical optics is often useful beyond its range of applicability,7 its success in treating the polarization dependence of the pump in NSOM-derived SHG images3 raises interesting issues, since, for example, depolarization effects are not included.27 In the language of the quasi-static picture, it is an important question whether localized, nonradiating fields14 generated by P(2ω) contribute materially to NSOM measurements. If so, nonuniformity of P(2ω) due to noncylindrical shape may be detectable. Because nonradiating fields can dominate near nanophotonic structures,15 NSOM should prove invaluable for elucidating such nanoscale photonic behavior. Our far-field approach, on the other hand, provides a flexible format for separating out the radiative NLO response of nanostructures in a wide array of conditions. Given the potentially important role that NLO materials may play in future nanophotonics, additional theoretical effort to treat more rigorously the response in both the near and far field seems justified. In summary, we have demonstrated the utility of SHG imaging by far-field microscopy to distinguish the crystallographic orientation of individual GaN NWs. Under favorable conditions, the method is sensitive to deviations in the c-axis orientation of only a few degrees. To understand the SHG polarization properties, we have introduced the quasistatic approximation to the analysis of SHG from NWs. Hence the NW volume illuminated by the diffraction-limited pump spot of frequency ω is presumed to experience a spatially uniform electric field at any instant and to emit as a discrete linear dipole oscillating at 2ω. In addition, we have outlined the computations that are required to account for the finite collection aperture of a microscope objective in far-field investigations. 835

Acknowledgment. We gratefully acknowledge instructive consultations with Drs. Guy Beadie and Igor Vurgaftman. This was work sponsored by the Office of Naval Research. Supporting Information Available: A more detailed description of the computation of the SHG signals, as well as computed signals for other NW growth habits. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Sirbuly, D. J.; Law, M.; Yan, H.; Yang, P. J. Phys. Chem. B 2005, 109, 15190. (2) Chan, S. W.; Barille, R.; Nunzi, J. M.; Tam, K. H.; Leung, Y. H.; Chan, W. K.; Djurisic, A. B. Appl. Phys. B 2006, 84, 351. (3) Johnson, J. C.; Yan, H.; Schaller, D.; Petersen, P. B.; Yang, P.; Saykally, J. Nano Lett. 2002, 2, 279. (4) Livneh, T.; Zhang, J.; Cheng, G.; Moskovits, M. Phys. ReV. B 2006, 74, 035320. (5) Li, Y.; Xiang, J.; Qian, F.; Gradecek, S.; Wu, Y.; Yan, H.; Blom, D. A.; Leiber, C. M. Nano Lett. 2006, 6, 1468. (6) Wang, Z. L.; Song, J. Science 2006, 312, 242. (7) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley-VCH: Weinheim, Germany, 2004. (8) Wang, J.; Gudiksen, M. S.; Duan, X.; Cui, Y.; Lieber, C. M. Science 2001, 293, 1455. (9) Maslov, A. V.; Bakunov, M. I.; Ning, C. Z. J. Appl. Phys. 2006, 99, 24314. (10) Ruda, H. E.; Shik, A. Phys. ReV. B 2005, 72, 115308. (11) Kuykendall, T.; Pauzauskie, P.; Lee, S.; Zhang, Y.; Goldberger, J.; Yang, P. Nano Lett. 2003, 3, 1063. (12) We estimate that the ∼100 fs laser pulses broaden to between 200 and 300 fs after passing through a pulse extractor and the objective. (13) Dadap, J. I.; Shan, J.; Heinz, T. F. J. Opt. Soc. Am. B 2004, 21, 1328. (14) Jackson, J. D. Classical Electrodynamics; John Wiley & Sons, Inc.: New York, 1962.

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(15) Wu, C.; Ye, M.; Ye, H. J. Opt. A 2004, 6, 1082. (16) Fujita, T.; Hasegawa, T.; Haraguchi, M.; Okamoto, M.; Fukui, M.; Nakamura, S. Jpn. J. Appl. Phys. 2000, 39, 2610. (17) Angerer, W. E.; Yodh, A. G.; Khan, M. A.; Sun, C. J. Phys. ReV. B 1999, 59, 2932. (18) Because SHG is insensitive to the polarity of the wurtzite c-axis, when referring to the c-axis or other vectors, we mean a direction parallel to the stated vector. (19) Hughes, J. L. P.; Wang, Y.; Sipe, J. E. Phys. ReV. B 1997, 55, 13630. (20) Our approximations also neglect a weak depolarization that exists even for the component of P(2ω) parallel to the NW. This depolarization arises because of volume charge associated with the nonzero divergence of P(2ω) at the boundary of the illuminated volume. A more accurate treatment might assume that the emitting region acts as an emitting ellipsoid, but the aspect ratio of the emitting volume of ∼9 for excitation and ∼6 for SH emission (based on field, not intensity, profiles) significantly reduces the depolarization along the NW. (21) Gu, M. AdVanced Optical Imaging Theory; Springer: Berlin, 1999. (22) Sanford, N. A.; Robins, L. H.; Davydov, A. V.; Shapiro, A.; Tsvetkov, D. V.; Dmitriev, A. V.; Keller, S.; Mishra, U. K.; DenBaars, S. P. J. Appl. Phys. 2003, 94, 2980. (23) Yu, G.; Wang, G.; Ishikawa, H.; Umeno, M.; Soga, T.; Egawa, T.; Watanabe, J.; Jimbo, T. Appl. Phys. Lett. 1997, 70, 3209. (24) Bo¨hmer, M.; Enderlein, J. J. Opt. Soc. Am. B 2003, 20, 554. (25) We neglect the birefringence of wurtzite GaN and use the ordinary index of refraction n to determine the dielectric constant  ) n2 at optical frequencies. Because the difference between the ordinary and extraordinary index of refraction is small, negligible error in the depolarization factors is incurred. (26) Shen, Y. R. The Principles of Nonlinear Optics; John Wiley & Sons: New York, 1984. (27) We also note that χxzx was neglected in the analysis of the near-field SHG from ZnO NWs in ref 3. For those NWs, which grew along the c-axis, χxzx would have contributed a weak component of P(2ω) perpendicular to the NW that varied with incident polarization.

NL0624420

Nano Lett., Vol. 7, No. 3, 2007