Small-angle X-ray scattering from GaN nanowires on Si(111): facet truncation rods, facet roughness, and Porod's law
Vladimir M. Kaganer, Oleg V. Konovalov, Sergio Fernández-Garrido
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug research papers Acta Crystallographica Section A
Foundations ofCrystallography
ISSN 0108-7673c (cid:13)
Small-angle X-ray scattering from GaN nanowires onSi(111): facet truncation rods, facet roughness, andPorod’s law
Vladimir M. Kaganer, a Oleg V. Konovalov b and Sergio Fern´andez-Garrido a , c a Paul-Drude-Institut f¨ur Festk¨orperelektronik, Leibniz-Institut im Forschungsverbund Berlin e. V., Hausvogteiplatz5–7, 10117 Berlin, Germany, b European Synchrotron Radiation Facility, 71 avenue des Martyrs, 38043Grenoble, France, and c Grupo de electr´onica y semiconductores, Dpto. F´ısica Aplicada, Universidad Aut´onomade Madrid, C/ Francisco Tom´as y Valiente 7, 28049 Madrid, Spain
Small-angle X-ray scattering from GaN nanowires grown on Si(111) is studiedexperimentally and modeled by means of Monte Carlo simulations. It is shownthat the scattering intensity at large wave vectors does not follow Porod’s law I ( q ) ∝ q − . The intensity depends on the orientation of the side facets withrespect to the incident X-ray beam. It is maximum when the scattering vector isdirected along a facet normal, as a reminiscence of the surface truncation rod scat-tering. At large wave vectors q , the scattering intensity is found to be decreased bysurface roughness. A root mean square roughness of 0.9 nm, which is the height ofjust 3–4 atomic steps per micron long facet, already gives rise to a strong intensityreduction. Keywords: small-angle scattering ; GISAXS ; nanowires ; Porod’s law ; facet truncationrods.
1. Introduction
GaN nanowires (NWs) spontaneously form in plasma-assistedmolecular beam epitaxy (PA-MBE) on various substrates atelevated temperatures under excess of N (Fern´andez-Garrido et al. , 2009; Fern´andez-Garrido et al. , 2012). In contrast to thevapor–liquid–solid (VLS) growth approach followed to synthe-size the majority of semiconductor NWs, PA-MBE growth ofGaN NWs takes place without a metal particle on the top (Risti´c et al. , 2008). Advantages of the spontaneous formation are theabsence of contamination from foreign metal particles and thepossibility to fabricate axial heterostructures with sharp inter-faces by alternating the supply of different elements.GaN NWs on Si(111), which is the most common substrate,grow in dense ensembles ( & cm − ) and initially possessradii of tens of nm as well as broad radius and length distri-butions (Consonni, 2013). As they grow in length, they bundletogether, which results in an enlargement of the radii at their topparts (Kaganer et al. , 2016 a ). Regarding their epitaxial orienta-tion, GaN NWs on Si(111) possess a 3–5 ◦ wide distributionof their orientations with respect to both the substrate normal(tilt) and the in-plane crystallographic orientation of the sub-strate (twist) (Jenichen et al. , 2011).For dense NW ensembles on Si(111), the radius distributioncan be obtained from the analysis of top-view scanning elec-tron micrographs (Brandt et al. , 2014). However, this methodprovides the radius distribution of only the top part of the NWs,which notably differs from the radius distribution at their bot-tom part because of NW bundling. In addition, the use of scan-ning electron micrographs for the statistical analysis of the NWradii becomes much more laborious for NW ensembles withlow densities as those formed on TiN (van Treeck et al. , 2018),since when the magnification of the scanning electron micro-graphs is chosen to quantify the NW diameters, only a few NWs fall into the field of view.Small-angle X-ray scattering is potentially better suited thanscanning electron microscopy for the determination of theradius distribution of GaN NWs ensembles grown on Si(111)because it probes the entire NW volume. From the stand-point of small-angle scattering, GaN NWs are long hexagonalprisms with a substantial distribution of their cross-sectionalsizes and orientations. Since these NWs are, on average, alignedalong the substrate surface normal, the incident X-ray beamis to be directed at a grazing incidence to the substrate sur-face. Grazing incidence small angle X-ray scattering (GISAXS)has been employed to study Si (David et al. , 2008; Buttard et al. , 2013), GaAs (Mariager et al. , 2007), and InAs (Eymery et al. , 2007; Eymery et al. , 2009; Mariager et al. , 2009) NWsgrown by the VLS growth mechanism with Au nanoparticlesat their tops. Unlike spontaneously formed GaN NWs, NWensembles prepared by VLS are characterized by very narrowdistributions of the NW sizes and orientations. The scatteringintensity from such NW ensembles possesses the same featuresas the scattering intensity from a single NW: it exhibits oscil-lations due to the interference caused by reflections at oppositefacets and a pronounced intensity dependence on the facet ori-entation.In the case of GaN NWs, despite the potential advantagesof GISAXS to assess the distribution of NW radii, we arenot aware of any GISAXS study. The closest report is thework of Hor´ak et al. (2008), who performed an in-plane X-ray diffraction study of GaN NWs using a laboratory diffrac-tometer. Their analysis implies the absence of strain in theNWs. If so, the NW diameters can be obtained from ω /2 θ scans in the same way as it can be done in GISAXS. However,this analysis cannot be applied to dense arrays of GaN NWs,which are inhomogeneously strained as a result of NW bundling Acta Cryst. (2020). A , 000000 Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires esearch papers (Jenichen et al. , 2011; Kaganer et al. , 2012; Fern´andez-Garrido et al. , 2014; Kaganer et al. , 2016 b ). We do not discuss here otherX-ray diffraction studies of NWs devoted to the determinationof strain and composition since they are out of the scope of thepresent work.The aim of the present paper is to develop the approachesrequired for the analysis of GaN NW arrays by GISAXS usingdense NW ensembles grown on Si(111) as a model exam-ple. Since GaN NWs are faceted crystals (their side facetsare (cid:8) (cid:9) planes), we expected that the GISAXS inten-sity at large wave vectors follows Porod’s law. Porod’s law(Porod, 1951; Debye et al. , 1957) states that, at large wavevectors q , the small-angle scattering intensity I ( q ) from parti-cles with sharp boundaries (i. e., possessing an abrupt change ofthe electron density at the surface) follows a universal asymp-totic law I ( q ) ∝ q − . Sinha et al. (1988) pointed out a com-mon origin of Porod’s law in small-angle scattering and Fres-nel’s law for reflection from flat surfaces. Namely, the scatteringintensity from a planar surface in the xy plane is proportionalto q − z δ ( q x ) δ ( q y ) . An average over random orientations of theplane gives rise to the q − law just because the delta function δ ( q ) has a dimensionality of q − . Sinha et al. (1988) performedan explicit calculation of the orientational average. Deviationsfrom Porod’s law are caused by fractality or the roughness ofthe surfaces in porous media (Bale & Schmidt, 1984; Wong &Bray, 1988; Sinha, 1989).In this paper, we show that the GISAXS intensity from GaNNWs at large wave vectors depends on the azimuthal orienta-tion of the NW ensemble with respect to the incident X-raybeam. The intensity is maximum when the scattering vector isdirected along the facet normal, and minimum when the scatter-ing vector is parallel to the facet. In other words, the azimuthaldependence of the GISAXS intensity reveals the facet trunca-tion rods. They are well established in X-ray diffraction fromnanoparticles (Renaud et al. , 2009) and stem from crystal trun-cation rods from planar surfaces (Robinson, 1986; Robinson &Tweet, 1992). We also show that the large- q intensity revealsthe roughness of the side facets of the GaN NWs. We determinea root mean squared (rms) roughness of about 0.9 nm, corre-sponding to the height of a few atomic steps on a micron longNW sidewall facet.
2. Experiment
For the present study, we have selected three samples with dif-ferent NW lengths from the series A studied by Kaganer et al. (2016 a ). The GaN NWs were synthesized in a molecular beamepitaxy system equipped with a solid-source effusion cell for Gaand a radio-frequency N plasma source for generating active N.The samples were grown on Si ( ) substrates, which were pre-liminarily etched in diluted HF (5%), outgassed above 900 ◦ Cfor 30 min to remove any residual Si x O y from the surface, andexposed to the N plasma for 10 min. The substrate growth tem-perature was approximately 800 ◦ C, as measured with an opti-cal pyrometer. The Ga and N fluxes, calibrated by determiningthe thickness of GaN films grown under N- and Ga-rich con-ditions (Heying et al. , 2000), were 0 .
29 and 0 .
75 monolayers per second, respectively. The growth time is the only parameterthat was varied among the samples to obtain ensembles of NWswith different lengths.
500 nm
Figure 1
Bird’s eye view (left column) and top-view (right column) scanning electronmicrographs of samples 1 (a,b), 2 (c,d), and 3 (e,f). The average NW lengthsare 230, 650, and 985 nm, respectively. The scale bar in (a) is applicable to allmicrographs.
Figure 1 presents scanning electron micrographs of samples1–3. Sample 1 corresponds to the end of the NW nucleationprocess. The NW density is 3 . × cm − , while the averagelength and diameter of the NWs are 230 and 22 nm, respec-tively. The NWs are mostly uncoalesced hexagonal prisms.Samples 2 and 3 display the further growth of the NWs, withaverage NW lengths of 650 nm for sample 2 and 985 nm forsample 3. The average NW diameters, as determined by theanalysis of the top-view micrographs (right column of Fig. 1),increase with increasing NW lengths, while the NW densitydecreases. Kaganer et al. (2016 a ) showed that the increase inthe diameter is a result of NW bundling, rather than their radialgrowth. A decisive proof of the absence of radial growth comesfrom the measurement of the fraction of the total area thatis covered by NWs. The area fraction covered by the NWs, Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires
Acta Cryst. (2020). A , 000000 esearch papers derived from the top-view micrographs shown in the right col-umn of Fig. 1, does not change from one sample to the otherand remains always at 20%.The distribution of the NW orientations was determined witha laboratory X-ray diffractometer. We have measured the fullwidth at half-maximum (FWHM) of the GaN 0002 reflection todetermine the tilt range with respect to the substrate surface nor-mal and the GaN 1¯100 reflection to determine the twist rangewith respect to the in-plane orientation of the substrate. TheFWHM of the tilt distribution is found to decrease with the NWlength from 5.1 ◦ for sample 1 to 4.0 ◦ and 3.9 ◦ for samples 2and 3, respectively, as a consequence of bundling. The FWHMof the twist distribution is found to be 2.8 ◦ , 2.7 ◦ , and 3.1 ◦ forsamples 1, 2, and 3, respectively.The GISAXS measurements were performed at the beamlineID10 of the European Synchrotron Radiation Facility (ESRF)using an X-ray energy of 22 keV (wavelength λ = . ) . The incident beam was directed at grazing incidence to thesubstrate. The chosen grazing incidence angle was 0.2 ◦ , i. e.,about 2.5 times larger than the critical angle of the substrate,to avoid possible complications of the scattering pattern typicalfor grazing incidence X-ray scattering (Renaud et al. , 2009).A two-dimensional detector Pilatus 300K (Dectris) placed at adistance of 2.38 m from the sample provided a resolution of8 . × − nm − . Figure 2
GISAXS intensity from sample 1 as measured by a two-dimensional detector.The scattering around the transmitted beam, the scattering around the beamreflected from the substrate surface, and the Yoneda streak are labeled as T, R,and Y, respectively. The vertical blue bar in the middle of the scattering patternis the beamstop. The three vertical dotted lines mark the positions of the scanspresented in Fig. 3. The color-coded scale bar represents the intensity in counts.
Figure 2 shows the GISAXS intensity measured from sam-ple 1. The scattering pattern comprises three horizontal streaks.The small-angle scattering around the transmitted beam islabeled as “T”, while the scattering around the beam reflectedfrom the substrate surface is labeled as “R”. Both streaksreveal the same scattering intensity dependence on the lateralwavevector q x . The scattering around the transmitted beam pos-sesses larger intensity. For that reason, the T streaks is cho-sen here for the further analysis. Besides the T and R streaks,the intensity distribution in Fig. 2 contains the Yoneda streak,marked with “Y”, which is located at the critical angle for totalexternal reflection. The chosen incidence angle allows us to sep-arate well the three different streaks, which facilitates the anal-ysis of the GISAXS intensity in the framework of kinematicalscattering.
3. Analysis of the measured intensities
We use the specific features of the NWs as oriented long prismsto improve the accuracy of the determination of the GISAXSintensity I ( q x ) from the measured maps. Since a single NW is aneedle-like object, its scattering intensity in the reciprocal spaceconcentrates in the plane perpendicular to the long axis of theNW. A random tilt of a NW results in the respective tilt of theintensity plane. Hence, one can expect that the spread of 4–5 ◦ in the directions of the long axes of the NWs results in a sectorof intensity in Fig. 2 with the width ∆ q z increasing proportionalto q x .Figure 3 presents intensity profiles along the dotted lines indi-cated in Fig. 2, i.e., scans at constant values of q x . These profilesare fitted by a Gaussian plus a background that may linearlydepend on q z . The FWHMs of these profiles ∆ q z are plottedin Fig. 4(a). As expected, ∆ q z linearly increases with q x . Theslopes ∆ q z / q x give the angular ranges of the NW orientations tobe 5.9 ◦ , 5.1 ◦ , and 4.6 ◦ for samples 1, 2, and 3, respectively.These values are close to (albeit somewhat larger) the widths ofthe orientational distributions measured by Bragg diffraction, asdescribed in Sec. 2. x = 0.1 nm -1 (a) i n t en s i t y ( c oun t s ) q x = 0.9 nm -1 (b) q z (nm -1 ) q x = 1.7 nm -1 (c) Figure 3
Measured intensity profiles along the lines of constant q x values marked bydotted lines in Fig. 2 (circles) and the respective Gaussian fits (lines). Acta Cryst. (2020). A , 000000 Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires esearch papers The fits in Fig. 3 help to improve the determination of thescattering intensity I ( q x ) both at small and large momenta q x .At small q x , the intensity profiles are narrow and the peak inten-sity has to be determined from just a few data points. At large q x , the intensity is low and the background is comparable to thesignal. After performing the fits of the cross-sectional profiles(i.e., along the q z direction) shown in Fig. 3 and establishingthe linear dependence of the FWHM ∆ q z on q x , we make onemore step to improve the accuracy. Linear fits are made for the ∆ q z on q x dependencies plotted in Fig. 4(a). Then, the fits of the q z profiles shown in Fig. 3 are repeated, now with the FWHMsfixed at the values obtained from the linear fits. In this way, thenumber of free parameters in the Gaussian fits is decreased, andthe intensity I ( q x ) is determined more accurately. This intensityis used in the further analysis. q z ( n m - ) sample 1 sample 2 sample 3 (a) = 0(cid:176) i n t en s i t y ( c oun t s ) = 0(cid:176) 5(cid:176) 10(cid:176) 15(cid:176) 20(cid:176) 25(cid:176) 30(cid:176) I (cid:215) q x ( a r b . un i t s ) q x (nm -1 ) = 0(cid:176)5(cid:176)10(cid:176)15(cid:176)20(cid:176), 25(cid:176), 30(cid:176) (c) Figure 4 (a) FWHMs of the intensity profiles ∆ q z as a function of the wave vector q x . (b)GISAXS intensity profiles of sample 1 as a function of the wave vector q x independence of the azimuthal orientation ψ . (c) The same intensity profiles as in(b) but plotted as I ( q x ) q x versus q x . Figure 4(b) presents the GISAXS intensity I ( q x ) measuredon sample 1 in dependence on its azimuthal orientation ψ . The sample orientation ψ = (cid:10) (cid:11) direction, so that the scattering vector(the x -axis direction) is along (cid:10) (cid:11) , which is the normal to theNW facets. Figure 4(b) comprises the measurements obtainedon the rotation of sample 1 about the normal to the substratesurface (i.e., about the direction of the long axes of the NWs)from ψ = ◦ to 30 ◦ with a step of 5 ◦ . Since the sample hasa rectangular shape and the illuminated area varies on rotation,the curves are scaled to obtain the same intensity in the small- q x range. The scaling factors differ less than by a factor of 2.The azimuthal dependence of the intensity at large q x is evidentfrom the plot.In the case of the reflected beam (the streak “R” in Fig. 2),an identical analysis of the intensity (not shown here) results incurves close to those shown in Fig. 4(b). Thus, we observe thesame azimuthal dependence of the intensity but with a smallertotal intensity and a higher level of noise. Because of this rea-son, for the further analysis presented in the paper, we exclu-sively consider the intensity distributions around the transmittedbeam.Since we expect Porod’s law I ( q x ) ∼ q − x to be satisfied atlarge q x , we plotted in Fig. 4(c) the same data as I ( q x ) q x versus q x , which would tend to a constant value for large q x . Surpris-ingly, a strong deviation from Porod’s law is observed. Further-more, the data do not only deviate from Porod’s law, but alsoexhibit a strong azimuthal dependence. In other to explain thisunexpected behavior, in the next section, we develop a MonteCarlo method to calculate the scattering intensity.
4. Calculation of the scattering intensity
We calculate first the scattering amplitude (form factor) of aNW A ( q ) in a coordinate system linked to the NW, i.e., with z -axis in the direction of the long axis of the NW. Hence, thecross-section of the NW is in the xy plane. Next, we will con-sider in Sec. 4.3 a transformation of the wave vectors from thelaboratory frame to the NW coordinate system, and performan average of the intensities | A ( q ) | over different NW orien-tations.The scattering amplitude of a NW is given by its form factor A ( q ) = ˆ V exp ( i q · r ) d r , (1)where the integral is calculated over the NW volume V . Sincethe NW is a prism, the scattering amplitude can be representedas a product of the components along the NW axis and in theplane perpendicular to it, A ( q ) = A k ( q k ) A ⊥ ( q ⊥ ) . The longitu-dinal component is simply A k ( q k ) = sinc ( q k L /2 ) , (2)where sinc ( x ) = ( sin x ) / x and L is the NW length. Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires
Acta Cryst. (2020). A , 000000 esearch papers l j n j r j r j+ h j
20 nm (a) (b)
Figure 5 (a) A hexagon with vertices r j and the unit vectors along and normal to the side l j and n j . The distance from the hexagon center to its side is h j . (b) Examplesof randomly generated hexagons used to simulate the scattering from sample 1. The calculation of the transverse component A ⊥ ( q ⊥ ) canbe reduced to a sum over the vertices, as it was initiallyshown for faceted crystals by von Laue (1936) and used nowa-days to calculate form factors of nanoparticles (Vartanyants et al. , 2008; Renaud et al. , 2009; Pospelov et al. , 2020). Specif-ically, von Laue (1936) proposed to reduce, using Gauss’ the-orem, the volume integral (1) to the integrals over the facets;application of Gauss’ theorem to these area integrals reducesthem to integrals over the edges, which, in turn, can be taken byparts and expressed through the coordinates of the vertices.For a planar polygon, the form factor reads A ⊥ ( q ⊥ ) = q ⊥ X j q ⊥ · n j q ⊥ · l j (cid:0) e i q ⊥ · r j + − e i q ⊥ · r j (cid:1) , (3)where the sum runs over the vertices and, as illustrated inFig. 5(a), r j are coordinates of the vertices, l j and n j are unitvectors along the polygon side between the vertices r j and r j + and normal to it, respectively. Lee & Mittra (1983) proposedanother expression for the form factor, A ⊥ ( q ⊥ ) = X j e i q ⊥ · r j ( l j × l j − ) · N ( q ⊥ · l j ) ( q ⊥ · l j − ) , (4)where N is the unit vector normal to the polygon plane, andWuttke (2017) explicitly showed the identity of the expressions(3) and (4). Equation (3) makes it possible to easily resolvethe numerical uncertainty 0/0 that arises at q ⊥ · l j =
0. Since l j = ( r j + − r j ) / | r j + − r j | , we have in the limit q ⊥ · l j → q ⊥ · l j (cid:0) e i q ⊥ · r j + − e i q ⊥ · r j (cid:1) → ie i q ⊥ · r j | r j + − r j | . (5)Figure 6(a) shows the intensity distribution calculated byEq. (3) for a regular hexagon with a side length of 12 nm. Theintensity is higher in the directions of the side normals and oscil-lates due to interference from opposite sides of the hexagon.Figure 6(b) shows a Monte Carlo calculation of the averageintensity from hexagons of different sizes. A lognormal distri-bution of the lengths of the hexagon sides is taken with the samemean value of 12 nm and a standard deviation of 4 nm. (a) (b) (c)(d)(e)(f) qxqy I × q x ( a r b . un i t s ) q x (nm -1 ) Figure 6
Scattering intensity from (a) a regular hexagon with a side length of 12 nm, (b)a distribution of regular hexagons with the average side length of 12 nm anda standard deviation of the side lengths of 4 nm, (c) a distribution of distortedhexagons as shown in Fig. 5(b), (d) the same distribution as in (c) but with aside facet roughness σ = . Iq is plotted. Thecurves are labeled by the same symbols as the respective maps. The effect ofroughness is illustrated in (f) by two curves: the full red curve corresponds tothe geometric distribution of the atomic steps on the side facets and the dashedred curve to the Poisson distribution, both possessing the same rms roughnessof σ = . The radial intensity distribution in the direction along theintensity maximum is presented in Fig. 6(f) by the black line.The intensity distribution is presented as the product Iq x , whichwould be constant at large q x for an ensemble of randomly ori-ented hexagons (as well as for other two-dimensional objectswith rigid boundaries) after averaging over all possible orienta-tions. As stated above, the intensity maxima in Fig. 6(b) corre-spond to the directions normal to the sides of the hexagon. Theypossess, at large q x , a I ∝ q − x dependence due to a steplike vari-ation of the density at a planar surface. Hence, in Fig. 6(f), weobserve a linear increase of the intensity for large values of q x (black line).The local maximum at q m = .
175 nm − in Fig. 6(f) isrelated to the mean length of the side facet of the hexagons a =
12 nm as a ≈ . q m , which allows us to determinethe hexagon size directly from the plots of Iq x versus q x (inthe three-dimensional case of hexagonal prisms, the same for-mula is applicable for the maximum in Iq x versus q x plot, seeSec. 4.3). For comparison, the form factor of a circle of radius R gives I ( q ) q ∝ J ( qR ) , where J ( x ) is the Bessel function.The first maximum of J ( x ) at x ≈ .
84 gives the circle radius R ≈ . q m . We can relate a hexagon and a circle even closer,by defining an effective radius R a of a circle possessing the Acta Cryst. (2020). A , 000000 Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires esearch papers same area as the hexagon with a side length a . Then, we have R a = ( √ π ) a ≈ . a and R a ≈ . q m , with the propor-tionality coefficient very close to the case of a circle.With the form factor defined by the positions of the verticesaccording to either Eq. (3) or Eq. (4), we are not restrictedto regular hexagons but can take into account the real cross-sectional shapes of the NWs. Since the side facets of the NWsare GaN { } planes making an angle of 60 ◦ to each other,we build the hexagons as shown in Fig. 5(a): random heights h j are taken in the directions normal to the facets. Then, wecheck that the generated hexagon is convex, and discard it oth-erwise. Figure 5(b) presents examples of randomly generatedhexagons with the same orientation of their sides. The distribu-tion of the hexagon shapes is chosen to simulate sample 1 andfurther described in Sec. 5. The intensity map obtained from thisdistribution of hexagons is shown in Fig. 6(c). The respectiveradial intensity distribution is presented in Fig. 6(f) by the blueline. One can see that the black and the blue lines in Fig. 6(f)are remarkably different. In particular, the hexagon shape dis-tribution notably reduces the dip in the intensity. Therefore, thedistortion of the hexagons can be deduced from the intensityplots. The side facets of GaN NWs are atomically flat (Stoica et al. , 2008; Risti´c et al. , 2008) but may have atomic steps.The radial growth of these NWs presumably proceeds by stepflow, with the motion of steps from the NW top, where theyare nucleated, down along the side facets (Fern´andez-Garrido et al. , 2013). Random steps across the side facets can be treatedas facet roughness in the same way as it is done in the calcu-lation of crystal truncation rods (Robinson, 1986; Robinson &Tweet, 1992).A step of height d shifts the j th side of the polygon inFig. 5 by a vector d n j in the direction of the facet normal.Hence, the j th term in the sum (3) acquires an additional fac-tor exp ( id q ⊥ · n j ) . Random steps give rise to a factor R j = R ( q ⊥ · n j ) , where the function R ( q ) is defined as R ( q ) = ∞ X m = p m exp ( imqd ) , (6)here p m are the probabilities of the shift of the side facet by m steps. Hence, the function R ( q ) is the characteristic function ofthe probabilities p m .Consider the geometric probability distribution p m = ( − β ) β m with the parameter β <
1. It describes a flat surfacewith a fraction β one step higher, its fraction β is, in turn, onestep higher, and so on (Robinson, 1986). The root mean squared(rms) roughness is σ = d √ β / ( − β ) and the correspondingcharacteristic function is R ( q ) = − β − β exp ( iqd ) . (7)The Poisson probability distribution p m = exp ( − µ ) µ m / m !gives rise to the rms roughness σ = d √ µ and the characteris- tic function is R ( q ) = exp (cid:2) − µ (cid:0) − e iqd (cid:1)(cid:3) . (8)We stress here that the j th term in the sum (3) is multipliedwith a complex factor R j = R ( q ⊥ · n j ) that depends on theorientation of the respective facet. This is different from a com-mon treatment of the surface roughness, which involves a sin-gle factor | R | . Particularly, the Poisson probability distributiongives for qd ≪ | R | = exp (cid:0) − σ q (cid:1) . Buttard et al. (2013) used such a factor to describe the effect of the rough-ness on the scattered intensity from Si NWs, by analogy to theroughness of planar surfaces, and arrived at an rms roughness σ of 1 nm for their samples. We use Eq. (3) in further calculationswith the complex factors R j in each term of the sum.Figure 6(d) shows the scattering intensity distributionobtained with the roughness factors given by Eq. (7). The rmsroughness is taken to be σ = . d isthat of the atomic steps on the GaN(1¯100) facet, d = a √ a = .
319 nm is the GaN lattice spacing. Strictly speak-ing, the roughness factors given by Eq. (7) or Eq. (8) are derivedfor a prism which has in each cross-section a hexagon withstraight sides. It describes a variation of the cross-section ofthe prism along its length and does not make sense for two-dimensional objects. Hence, the intensity distribution shown inFig. 6(d) corresponds to the prisms with perfectly aligned longaxes.The solid red line in Fig. 6(f) shows the radial intensity dis-tribution obtained from the map shown in Fig. 6(d) in the direc-tion of maximum intensity, calculated using the roughness fac-tors for the geometric probability distribution given by Eq. (7).The dashed red line in Fig. 6(f) shows the intensity from thesame distribution of hexagons but calculated using the rough-ness factors derived from the Poisson probability distribution,Eq. (8). The rms roughness is taken the same in both cases, σ = . q σ >
1. Hence, the rms roughness σ can be obtained from the intensity plots. Moreover, the intensitycurves are fairly sensitive to the choice of the probability distri-bution. The crystal truncation rods from planar surfaces pos-sess a similar sensitivity to the choice of the roughness model(Walko, 2000). Our modeling of the scattering from GaN NWspresented in Sec. 5 shows that the geometric probability distri-bution provides a better agreement with the experimental data. The scattering intensity is measured as a function of the wavevector q in the laboratory frame (see Fig. 2). We need to find thecomponents ( q k , q ⊥ ) of this vector in the frame given by thelong axis of the NW and the normal to one of its side facets. Letus consider first the simple case of the two-dimensional rotationof the hexagons (or perfectly aligned prisms in the plane normalto their long axes). The unit vector normal to a hexagon side (orthe prism facet) can be written as n = ( cos ψ , sin ψ , ) , (9) Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires
Acta Cryst. (2020). A , 000000 esearch papers where ψ is a polar angle (defined modulo 60 ◦ ) with respect toa reference orientation. The unit vector along the hexagon sideis, respectively, l = ( − sin ψ , cos ψ , ) . The components q n , q l of the two-dimensional vector q ⊥ are determined simply as q n = q ⊥ · n and q l = q ⊥ · l .Figure 6(e) presents a Monte Carlo calculation of the inten-sity for the distribution of distorted hexagons described aboveand sketched in Fig. 5(b), after an average over the orientations ψ uniformly distributed from 0 ◦ to 360 ◦ . The correspondingradial intensity distribution, shown in Fig. 6(f) by a gray line,follows the two-dimensional Porod’s law I ( q ) ∝ q − at large q . At small q , it coincides with the intensity distribution for theoriented hexagons.For the three-dimensional distribution of the NW orienta-tions, inherent to the spontaneous formation of GaN NWs onSi(111) (see Fig. 1), we define a unit vector in the direction ofthe long NW axis as e k = ( sin θ cos φ , sin θ sin φ , cos θ ) , (10)where φ and θ are the azimuthal angle of tilt and its polar angle,respectively. The unit vector normal to the facet is defined tobe orthogonal to e k and possessing the same projection on thehorizontal plane as in Eq. (9): n = ( ξ cos ψ , ξ sin ψ , s p − ξ ) , (11)where ξ = (cid:2) + tan θ cos ( φ − ψ ) (cid:3) − and s is the sign of − cos ( φ − ψ ) . Since the tilt angle θ does not exceed a fewdegrees, the difference between Eqs. (9) and (11) is unessen-tial. The vector l is defined as a vector product l = e k × n .We calculate the scattering intensity by the Monte Carlomethod. It enables a simultaneous integration over the distribu-tions of the NW lengths, their cross-sectional sizes and shapes,and the orientations of the NW long axis as well as those oftheir side facets. The calculations take fairly little time. It takesless than a minute on a single CPU core of a standard PC to cal-culate an intensity curve with the accuracy sufficient to makeestimates. The smooth curves presented in the paper took lessthan an hour of CPU time each.We take the mean NW lengths obtained from the scanningelectron micrographs and given in Sec. 2. A large scattering inthe NW lengths is evident from Fig. 1. The length distributionis assumed to be lognormal with a standard deviation of 20%from the respective average lengths. For the facet orientationangle ψ , we take a normal distribution with the FWHM deter-mined by the in-plane X-ray diffraction scans (see Sec. 2). Theaverage value of ψ is given by the orientation of the incidentX-ray beam with respect to the NW facets [see Figs. 4(b) and4(c)].The integration over the orientations of the long axis of theNWs is an integration over a solid angle, i.e., the integral ofthe intensity from a single NW with P ( θ ) sin θ d θ d φ , where P ( θ ) is the probability density distribution of the tilt angle θ . The azimuthal angle φ is uniformly distributed from 0 to2 π . We take a normal distribution of the tilt angles, P ( θ ) = √ π ( ∆ θ ) − exp [ − θ /2 ( ∆ θ ) ] . The standard deviation ∆ θ can be obtained from the slopes of the straight lines in Fig. 4(a)or from the FWHMs of the symmetric Bragg reflections GaN0002, as discussed in Sec. 3. The widths ∆ q z in Fig. 4(a) andthe FWHMs of the Bragg reflections take into account the tiltsin all directions, so that ∆ θ is obtained from the half widthat half maximum (HWHM) and varies from 2.5 ◦ for sample1 to 2 ◦ for sample 3. Since the tilt angle θ is small, we cantake sin θ ≈ θ and proceed to an integral over a new vari-able y = θ . Then, the integral is taken with ˜ P ( y ) d y d φ , where˜ P ( y ) ∝ exp [ − y /2 ( ∆ θ ) ] . Hence, we generate y as an exponen-tially distributed random number with the unit dispersion andcalculate θ = √ y ∆ θ. If the NW orientations are completely random, i.e., the angles φ and ψ vary from 0 to 2 π and the angle θ from 0 to π uni-formly and independently, the small-angle scattering intensityat q ≫ π / a , where a is the width of the side facet, followsPorod’s law I ( q ) ∝ q − . However, since the NWs are longprisms, the scattering intensity from a single NW of length L with its long axis in z -direction concentrates in the reciprocalspace in a disk of the width ∆ q z = π / L . We have seen inSec. 3 that the scattering from the oriented NWs is limited by ∆ q z / q x < ∆ θ , where ∆ θ is the angular range of orientations. Aslong as 2 π / ( Lq x ) < ∆ θ , the oriented NWs give the same inten-sity in the x -direction as fully randomly oriented ones. There-fore, Porod’s law is satisfied for q x > π / ( L ∆ θ ) . I · q x ( a r b . un i t s ) q x (nm -1 )L=100 nmL=300 nmL=1000 nmL=200 nm Figure 7
Monte Carlo calculation of the small-angle scattering intensity from an ensem-ble of NWs with a 5 ◦ wide range of orientations of the long axes and randomorientation of the side facets. The width of the side facets is 12 nm and NWlengths vary from 100 to 1000 nm. Figure 7 presents the Monte Carlo calculation of the small-angle scattering intensity from NWs of different lengths and thesame width of the tilt angle distribution ∆ θ = . ◦ correspond-ing to that of sample 1. Particularly, for the NWs with a length L =
200 nm, the condition derived above reads q x > . − ,which is in a good agreement with the region of constant I ( q x ) q x in Fig. 7. Hence, the limited range of the tilt angles in the NWensemble does not prevent reaching Porod’s law even for therelatively short NWs of sample 1. We remind that the curves inFig. 7 are calculated by averaging over the tilt azimuth φ andthe facet orientation ψ varying from 0 to 2 π . Further Monte Acta Cryst. (2020). A , 000000 Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires esearch papers Carlo calculations, taking into account the orientational order-ing of the side facets of the epitaxially grown GaN NWs, arepresented in the next section.
5. Results
Figure 8 presents the results of the systematic GISAXS mea-surements on samples 1–3. The measurements and their anal-ysis are described in the sections 2 and 3, respectively. The samples have been measured with the azimuthal orientation ψ varying from 0 ◦ to 90 ◦ with steps of 5 ◦ . Each measurementprovided a map similar to the one in Fig. 2, and the intensity I ( q x , q z ) around the transmitted beam has been analyzed by fit-ting every scan of a constant q z by a Gaussian, as shown inFig. 3. The peak values of the q z scans obtained in this fit pro-vided the intensity I ( q x ) . It is presented in Fig. 8 as the product I ( q x ) q x versus q x , to reveal deviations from Porod’s law. I · q x ( a r b . un i t s ) x (nm -1 ) 0(cid:176)30(cid:176)(l)q x (nm -1 ) 60(cid:176)30(cid:176)(j)q x (nm -1 ) 0(cid:176)30(cid:176)(i) q x (nm -1 )60(cid:176)90(cid:176)(k) Figure 8
The measured GISAXS intensity (three left columns) and Monte Carlo simulations (right column) for samples 1 (a–d), 2 (e–h), and 3 (i–l). The measurements areperformed for different mean orientation angles ψ of the side facets of the NWs with respect to the X-ray beam, namely, from 0 ◦ to 90 ◦ with steps of 5 ◦ . Forclarity, these measurements are presented in three different panels (from 0 ◦ to 30 ◦ , from 30 ◦ to 60 ◦ , and from 60 ◦ to 90 ◦ ). The intensities are plotted as I ( q x ) q x versus q x to highlight deviations from Porod’s law. The curves calculated at ψ = ◦ for each sample are repeated as blue curves in the left column, for a directcomparison of the calculated and the measured curves. Since GaN NWs grow epitaxially on Si(111), the ensemblepossesses a 6-fold orientational symmetry. A systematic varia-tion of the intensity curves depending on the azimuthal sampleorientation ψ is evident from Fig. 8: within the statistical errorof the measurements, the orientations ψ and ψ + ◦ , as well as ± ψ or 30 ◦ ± ψ , are equivalent. Hence, the Monte Carlo model-ing presented in the right column of Fig. 8 has been performedfor the angle ψ from 0 ◦ to 30 ◦ with the same step of 5 ◦ .In the Monte Carlo calculations, we take the values of theNW lengths L , the range of the tilt angles θ , and the range ofthe side facet orientations ψ at the values experimentally deter-mined in Secs. 2 and 3. The parameters of the NW ensemble to be determined from the modeling are the mean width ofthe side facets a , its variation as well as the variation of thecross-sectional shapes of the NWs, and the roughness of theside facets. We have seen in Sec. 3 that these parameters affectthe calculated curves in qualitatively different ways. The meanfacet size a determines the position of the local maximum ofthe curves at q x ≈ .
17 nm − , which corresponds to a sidefacet width of about 12 nm. The depth of the dip between thismaximum and the rise of the curves at larger q x is controlled bythe width of the facet size distribution and the shape distributionof the cross-sections. The decrease of Iq x at large q x is causedby the roughness of the side facets. Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires
Acta Cryst. (2020). A , 000000 esearch papers The distorted cross-sections of the NWs are modeled in theMonte Carlo study as described in Sec. 4.1. The heights h j shown in Fig. 5(a) are generated on random around a meanvalue. Figure 5(b) exemplifies the shapes of the NWs used in thesimulation of sample 1. The right column in Fig. 8 presents theMonte Carlo calculation of the small-angle scattering intensityfor samples 1–3. For a direct comparison of the calculated andthe measured intensities, the curves calculated for each sampleat ψ = ◦ are repeated as blue lines in the left column of thefigure.For each generated NW, we calculate the cross-sectionalarea A and the perimeter P . Then, we determine out of theseparameters the radius R from A = π R and the circularity C = π A / P . The circularity thus defined is C = C = π √ ≈ .
907 for a regular hexagon, and C ≪ et al. , 2014). The lines in Fig. 9 show the distributions of theradius and the circularity obtained in the simulations.The distributions of the cross-sectional radii and circularitiesof the NW ensembles have also been interdependently obtainedby analyzing top-view scanning electron micrographs similarto those shown in Fig. 1(d–f). The analysis has been performedusing the open-source software ImageJ (Schneider et al. , 2012),as described in detail by Kaganer et al. (2016 a ) in their Support-ing Information. The distribution of the radius obtained fromthe modeling of the GISAXS intensity for sample 1 in Fig. 9(a)is fairly close to the distribution derived from the micrographs.The circularity distribution obtained from the micrographs is,however, extended towards smaller values indicating a higherdensity of NWs with elongated cross-sectional shapes. Such adiscrepancy can be attributed to an artifact caused by the NWtilt. Specifically, the scanning electron micrographs exhibit avery little difference in brightness between the top facet and thetop part of the side facet of the NW, so that ImageJ treats bothregions together, i. e., as extended intensity spots.In contrast to sample 1, the NW radii obtained from theMonte Carlo simulations of the scattering intensity from sam-ples 2 and 3, see Figs. 9(c) and 9(e), are smaller than thosederived from the analysis of the scanning electron micrographs,and the discrepancy increases with increasing NW length. Weremind that the mean NW radius can be directly derived fromthe position q x of the local maximum in the experimental curvespresented in Fig. 8. It remains at about q x ≈ .
17 nm − and onlyslightly shifts to smaller values (and hence to larger radii) as theNW length increases from sample 1 to sample 3.The origin of the discrepancy between the NW radii deter-mined from the scanning electron micrographs and from themodeling of the GISAXS intensity is in the bundling of NWs.The bundling is almost absent for sample 1, and the cross-sections of the NWs obtained from the micrographs charac-terize the NWs along their full length. As the NWs grow inlength, they bundle together, which causes an apparent radialgrowth. Simultaneously, the NW density decreases, so that thefraction of the surface covered by the NWs remains constant (Kaganer et al. , 2016 a ). The GISAXS provides a statistics ofthe NW radii averaged over their lengths, while the top-viewmicrographs reveal their distribution at the top. That results ina progressive difference between the distributions obtained bythe two methods.The widths of the circularity distributions in the right columnof Fig. 9 slightly reduce with the growth of the NWs. The NWimages in the scanning electron micrographs become more cir-cular since, during NW growth, the bundled nanowires attain acommon shape that tends to a regular hexagon. Also, the low-circularity wing of the circularity histogram reduces, becausethe effective radii of the bundled NWs increase, and the dis-tinction between the top facets and the top parts of the sidefacets becomes more pronounced for the ImageJ analysis. Thecircularity distributions obtained from the GISAXS intensitycurves are sharper than the ones obtained from scanning elec-tron micrographs, because the former takes into account bothsingle NWs in their bottom part and bundled NWs in their toppart, while the latter counts only the NW tops. We also remindthat the circularity of a distorted hexagon is always smaller thanthe circularity C ≈ .
907 of a regular hexagon. Larger circular-ities obtained from the analysis of the scanning electron micro-graphs in Figs. 9(d) and 9(f) are due to the finite resolution ofthe micrographs as well as to the algorithm used by ImageJ thattends to round faceted objects.We have seen in Sec. 4.1 and particularly in Fig. 6(f) that,when the scattering vector is oriented normal to the side facetsof the NWs ( ψ = ◦ ) and the facets are atomically flat, the facettruncation rod scattering would result in a linear increase of theintensity on the I ( q x ) q x vs. q x plot at large q x . The decrease ofthe experimental curves indicates a roughness of the side facets.We obtain in the Monte Carlo modeling an rms roughness of σ = .
9, 0.95, and 0.85 nm for samples 1, 2, and 3, respectively.According to the height of the atomic steps on a GaN(1¯100)facet d = a √ = .
276 nm (here a = .
319 nm is theGaN lattice spacing), the rms roughness is less than 3.5 steps.
6. Discussion and summary
GaN NWs nucleate spontaneously on Si(111) and grow witha substantial disorder with respect to their orientations. Theirgrowth is, nevertheless, epitaxial: the NWs inherit the out-of-of plane and in-plane orientations of the substrate. Since theseNWs are typically long (from hundreds of nanometers to a fewmicrons) and thin (tens of nanometers), the range of orienta-tions of their long axes of 3–5 ◦ is sufficient to provide the sameaverage in the small-angle scattering intensity as if they wouldhave all orientations. However, an angular range of orientationsof the side facets of 3 ◦ gives rise to features in the GISAXSintensity distribution that are reminiscent of the crystal trunca-tion rod scattering from flat surfaces of single crystals.We have found that the GISAXS intensity depends on theorientation of the side facets with respect to the incident X-raybeam direction. In our experiment, the incident beam is keptnormal to the average direction of the long axes of the NWs. Theorientation of the incident beam with respect to the side facetsis varied by rotating the sample about the substrate surface nor-mal. The scattering intensity is maximum when the incident Acta Cryst. (2020). A , 000000 Kaganer, Konovalov, and Fern´andez-Garrido · SAXS from GaN nanowires esearch papers beam is along the facets, or in other words, when the scatter-ing vector is in the direction of the facet normal.The X-ray scattering intensity from a planar surface is pro-portional to q − . Porod’s law q − is a result of a full averageover all orientations of the plane (Sinha et al. , 1988), i.e., theintegration over the two angles defining the plane orientation.For GaN NWs on Si(111), the range of orientations of the longaxes is large enough to provide an integration over the tilt angleand give rise to a q − dependence when the scattering vector isalong the facet normal. In the Iq vs. q plots shown in Fig. 8, thisdependence is seen as a linear increase at ψ = ◦ or 60 ◦ . Theintensity decreases as the sample is rotated about the normal tothe substrate surface. The minimum intensity value is reachedat ψ = ◦ , i.e., in the direction between facets. nu m be r o f nano w i r e s (c) sample 2(d)radius (nm)(e) sample 3 circularity(f) Figure 9
Distributions of the NW radii and circularities of samples 1–3 obtained fromthe analysis of the top-view scanning electron micrographs (histograms) andthe Monte Carlo modeling (lines).
The surface roughness gives rise to a decrease in the intensityat q σ &
1, where σ is the rms roughness. The Monte Carlo mod-eling of the experimental curves in Fig. 8 gives σ from 0.85 to0.95 nm, which is just 3.5 times the height of the atomic steps.Nevertheless, this small roughness strongly modifies the inten-sity curve for high values of q .The GISAXS curves vary fairly little from one sample toanother, despite the large difference between the cross-sectionalsizes of the NWs observed in the scanning electron micrographsshown in Fig. 1. This apparent discrepancy is explained bythe NW bundling, which is an essential effect in their growth(Kaganer et al. , 2016 a ). While GISAXS reflects the distribu-tion of the cross-sectional sizes of the NWs over their wholevolume, the top-view micrographs shown in the right columnof Fig. 1 reveals the cross-sectional sizes of the NWs at theirvery top part. As a result, the distributions of the NW radii andcircularities obtained from the scanning electron micrographsand the GISAXS intensity curves only coincide for sample 1,which is free of bundling. As the NWs grow in height and theirbundling increases (samples 2 and 3), the discrepancies betweenthe results obtained by these two different methods increases.Finally, we conclude that GISAXS, together with the MonteCarlo modeling of the intensity curves, is well suited for the determination of the distributions of the cross-sectional sizes ofthe NWs. The methods developed in the present paper are notspecific to GaN NWs on Si(111) and can be applied to otherNW distributions and material systems. Particularly, they willbe applied in a separate work to the assessment of the radius dis-tributions of GaN NW ensembles grown on TiN, which exhibita much lower density that hinders the analysis of the NW cross-sectional shapes by scanning electron microscopy.
7. Acknowledgments
We thank Vladimir Volkov (Institute of Crystallography,Moscow) for useful discussions, and Oliver Brandt (Paul-Drude-Insitut, Berlin) for useful discussions and a criticalreading of the manuscript, Lewis Sharpnack (ESRF, Greno-ble) for his assistance during preliminary measurements atthe ESRF beamline ID02, Shyjumon Ibrahimkutty (Rigaku,Neu-Isenburg) for the measurement of the NW twist angles,Carsten Stemmler (Paul-Drude-Institut, Berlin) for his helpwith the preparation of the samples, and Anne-Kathrin Bluhm(Paul-Drude-Institut, Berlin) for providing the scanning elec-tron micrographs. S.F.-G. acknowledges the partial financialsupport received through the Spanish program Ram´on y Cajal(co-financed by the European Social Fund) under grant RYC-2016-19509 from the former Ministerio de Ciencia, Innovaci´ony Universidades.
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Small-angle X-ray scattering intensity from GaN nanowires on Si(111) depends on the orientation of the side facets with respect to the incidentbeam. This reminiscence of the truncation rod scattering gives rise to a deviation from Porod’s law. A roughness of just 3–4 atomic steps per amicron long side facet notably changes the intensity curves.