Letter pubs.acs.org/NanoLett
Diameter and Polarization-Dependent Raman Scattering Intensities of Semiconductor Nanowires Francisco J. Lopez,*,§ Jerome K. Hyun, Uri Givan,‡ In Soo Kim, Aaron L. Holsteen, and Lincoln J. Lauhon* Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States ABSTRACT: Diameter-dependent Raman scattering in single tapered silicon nanowires is measured and quantitatively reproduced by modeling with finite-difference time-domain simulations. Single crystal tapered silicon nanowires were produced by homoepitaxial radial growth concurrent with vapor−liquid−solid axial growth. Multiple electromagnetic resonances along the nanowire induce broad band light absorption and scattering. Observed Raman scattering intensities for multiple polarization configurations are reproduced by a model that accounts for the internal electromagnetic mode structure of both the exciting and scattered light. Consequences for the application of Stokes to anti-Stokes intensity ratio for the estimation of lattice temperature are discussed. KEYWORDS: Nanowire, Raman, modeling, FDTD, optical properties
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be considered. Convergence of a full numerical calculation and integration of the incoherent radiation fields from induced dipoles may take impractically long times with commonly available computing resources. Therefore, it is of interest to develop approximate models of Raman scattering that match experiments for useful ranges of experimental conditions. Prior studies of Raman scattering from semiconductors of submicrometer crystal dimensions have examined diameterdependent resonant behaviors.8,9,13−17 One of the first studies on an individual nanowire correlated polarization behavior with crystallographic selection rules, indicating the possibility of identifying nanowire growth direction and crystal orientation,14 although the effects of electromagnetic resonances were not considered. Two reports that followed noted a substantial enhancement of the Raman signal compared to an equivalent volume in a bulk crystal, and predicted8 and measured15 diameter-dependent oscillations of the Raman intensity in silicon, although the tensorial aspect of Raman scattering was not fully considered. Electromagnetic resonances were additionally demonstrated by single nanowire photocurrent spectroscopy experiments that explored the effects of the substrate and nanowire cross-sectional shape on the observed resonant behavior, obtaining good correspondence between theory and experiment.18,19 More recently, Wu et al.16 developed a more sophisticated model for Raman scattering by including both crystallographic selection rules and small particle scattering anisotropies. They found that for small diameters, the response
aman spectroscopy is a versatile characterization tool that can provide information about crystallographic structure1−4 and thermal,5 mechanical,6,7 optical8,9 and electronic properties10,11 of semiconductor nanowires. Structural variations between and within individual nanowires often make it desirable to perform single nanowire measurements, but the weak nature of Raman scattering can make it challenging to obtain good quality spectra. Additionally, retrieval of crystallographic information encoded in the anisotropic nature of Raman scattering intensities, though well established in bulk crystals, is complicated at crystal sizes of the order of the excitation wavelength; at these dimensions, electromagnetic resonances strongly modulate polarized light scattering. In order to fully exploit the potential of Raman microscopy to investigate structure, a model is needed that accounts for both crystallographic and small particle polarization-dependent selection rules. Such a model will facilitate experimental design for optimal signal throughput and will further understanding of inelastic light scattering from nanostructures. Assuming that spontaneous Raman scattering dominates the response of nanowires excited with low powers, emission from a given phonon mode can be constructed from the incoherent superposition of radiating dipoles distributed inside the nanowire volume. The orientation of each dipole is given by the transformation of the local internal electric field by the Raman tensors. The starting point of any effort to model Raman scattering is therefore the solution for the internal electric field at the exciting frequency in the nanowire. This can be obtained by analytical solutions for a plane wave incident on a freely suspended infinitely long cylinder12 or by numerical calculations if the effects of a substrate and focused beam are to © 2012 American Chemical Society
Received: December 23, 2011 Revised: April 11, 2012 Published: April 12, 2012 2266
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deposition, and growth substrates were annealed for 20 min at 550 °C under 100 standard cubic centimeters per minute (sccm) of H2 flow. The growth step of 20 min was carried out at 650 °C and total pressure of 20 Torr with gas flows of 2 sccm SiH4 and 100 sccm H2. Nanowires showed faceting characteristic of a ⟨111⟩ growth orientation as well as some surface roughness. Despite efforts to optimize growth conditions, the homoepitaxial shells of most nanowires showed evidence of polycrystalline and/or amorphous material in both their Raman spectra20 and their dark appearance in an optical microscope, as is common for highly absorbing disordered silicon. Some nanowires, however, exhibit strong light dispersion and can be easily identified in the optical microscope due to the striking color patterns (Figure 1b). Raman spectra indicated pure diamond cubic structure with no evidence of stacking faults.1,2 Further optimization of the growth conditions to increase the yield of highly crystalline tapered nanowires could be beneficial for a number of photonic applications. For optical studies, nanowires were deposited on Au-coated silicon substrates to eliminate substrate contributions to the Raman spectra. The optical image in Figure 1c was obtained using a white light illumination source filtered with a band-pass filter (10 nm) centered at 532 nm. Regions of dark contrast are associated with diameters corresponding to strong extinction resonances at this wavelength. Atomic force microscopy (AFM) indicates a taper of ∼11 nm/μm (Figure 1d). A line profile from the optical image in Figure 1c is also shown in Figure 1d. The diameter-dependent optical properties can be explained qualitatively by analytical solutions that give scattering and absorption cross sections for untapered nanowires of different diameters. Analytical solutions for a plane wave incident on an infinite cylinder are available for a freely suspended nanowire.12 Additionally, we performed FDTD simulations of untapered nanowires excited with a focused Gaussian beam (NA = 0.9). The simulations were done with the commercial package Lumerical FDTD Solutions and employed perfect match layer (PML) and periodic boundary conditions. We considered both transverse magnetic (TM) and transverse electric (TE) excitation, in which the electric field polarization is parallel or perpendicular to the nanowire axis, respectively. The simulation mesh had cells 4 × 2 × 2 nm3 in x, y, and z (see Figure 4a inset) for diameters smaller than 146 nm and 5 × 3 × 3 nm3 for larger diameters. Figure 2a shows the diameter dependence of absorption cross sections for TM excitation for both analytical and FDTD simulations with and without a substrate. For the FDTD simulations the reported quantity is |E|2 integrated over a nanowire volume defined by the focused spot size (0.7 μm). This quantity is proportional to the absorption cross-section. The FDTD simulation of the free-standing nanowire (without substrate) reproduces the analytical result. In the presence of the Au substrate, the resonances shift toward longer wavelengths and are enhanced. Also plotted in Figure 2b−f is crosssectional views of |E|2 inside the nanowire for diameters near several resonance maxima from simulations on the Au-coated substrate. Note that the strongest resonances are reminiscent of standing waves produced in thin film interference. Raman scattering involves two wavelengths, the incoming laser excitation and the inelastically scattered light, shifted by a frequency corresponding to the phonon energy. For 532 nm excitation and the optical phonon in silicon, the Stokes scattered light has a wavelength of 547 nm. Both the initial excitation as well as the Raman scattered light are subject to resonant enhancement, and this double enhancement has been
is dominated by dipolar emission parallel to the nanowire axis. Their model was able to reproduce the polarization dependence of the emitted light for nanowire of different diameters with one reported exception. Here we report a model of Raman scattering from nanowires that considers the change in wavelength of the inelastically scattered light and accounts for the modal structure associated with electromagnetic resonances. Distinct incident and scattered wavelengths have not been included in prior treatments, although we find this to be important for nanowires above a characteristic diameter. We employ finite-difference time-domain (FDTD) simulations in full field calculations to include the effects of a convergent beam geometry and a reflecting substrate. Raman scattering measurements performed along the length on a gently tapered nanowire enable the diameter dependence to be clearly resolved, and measurements are in good agreement with calculations except for the case of incident and scattered polarizations perpendicular to the nanowire axis. Consequences for the interpretation of the Stokes to anti-Stokes intensity ratio in terms of the lattice temperature are discussed. Nanowires were grown by the vapor−liquid−solid (VLS) process in a hot-wall chemical vapor deposition reactor using 50 nm Au particles as catalyst, silane (SiH4) as precursor gas, and He and H2 as carrier gases. Growth conditions (see below) were chosen to concurrently deposit a homoepitaxial shell on the VLS-defined nanowire, producing a slightly tapered ‘nanocone’ (Figure 1a) that allowed diameter-dependent absorption resonances to be spatially resolved using a convergent beam of 0.7 μm diameter; the change in nanowire diameter within the spot size (∼8 nm) is smaller than the breadth of the resonances (∼20 nm at half-maximum). Si (111) substrates were cleaned and HF etched prior to Au nanoparticle
Figure 1. (a) SEM image of a tapered silicon nanowire deposited on a Au-coated substrate (scale bar 5 μm). Dark field (b) and 532 nm filtered (c) light images of the same nanowire. (d) AFM height profile and optical extinction profile obtained from image in (c). The diameter is 80 ± 2 nm at the tip and 345 nm at the base. The AFM topograph indicates a root-mean-squared roughness of 3−5 nm, which does not appreciably impact the optical response. 2267
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Figure 2. (a) Comparison of analytical solutions (solid line) and FDTD simulations for a nanowire that is free-standing (red circles) or lying on a Au-coated substrate (blue circles) for TM excitation. The presence of the metallic substrate shifts the resonances to larger diameters by about 10 nm and modifies their magnitudes. (b−f) Cross-sectional view of |E|2 profiles at diameters near the absorption maxima of the FDTD simulation on gold. The color scale is normalized to the maximum intensity at each diameter. The fields outside of the nanowire are not shown.
approximated by integration of |E|4 at the laser wavelength within the excited volume,15,21 based on the assumption that the Raman shift may be considered small compared to the excitation wavelength. We find this assumption to hold below a certain characteristic diameter. Figure 3a compares the diameter dependence of analytical absorption cross sections
for 532 and 547 nm excitation. At diameters smaller than about 80 nm the two curves practically overlap, while the separation between resonance peaks increases with increasing diameter, to the point that they are almost completely ‘out of phase’ near 400 nm. Accordingly, one expects the assumption I ∝ |E|4 to be very good for diameters below 80 nm but poor for diameters above 150 nm as a consequence of an increase in the frequency gap between absorption and emission resonances for higher order modes due to increasing path length differences; the spatial overlap of the in-phase components of the fields is overestimated if only a single frequency is considered. The criterion for neglecting the wavelength shift should therefore consider the shift between resonance peaks compared to their widths. For example, the two curves in Figure 3a are shifted by the ratio of the two wavelengths inside the nanowire, which is 1.042 in this case or 4% of the diameter. Note also that the differing enhancements at two distinct wavelengths broaden the calculated resonance peaks. To understand the impact of wavelength-dependent scattering on the Raman enhancement, it is instructive to examine the internal electric field magnitude distributions for simulations at the two wavelengths, as shown in Figure 3b,c for a 214 nm diameter nanowire, corresponding to a minimum in the Raman intensity in the (xx) configuration (the notation is described further below). The two profiles overlap substantially only at the central intensity maximum, as is also manifest in Figure 3d where the point by point product of field intensities at the two wavelengths is shown. The highly localized nature of this mixed product intensity profile, confined within less than 10% of the nanowire volume, underscores the importance of considering modal structure when characterizing radially inhomogeneous nanowires; different regions will be probed depending on the choice of wavelength. Figure 3e compares the experimental Raman intensities (black solid line) with three calculations: the integrated |E532|4, the integrated |E547|4, and the integrated product |E532|2·|E547|2. The match is best for the
Figure 3. Comparison of the diameter-dependent electromagnetic response of silicon nanowires at 532 and 547 nm. (a) TM absorption cross sections at the two wavelengths show resonance maxima that are shifted by ∼4% of the diameter. (b−d) Cross-sectional view of |E|4 along nanowire axis at the indicated wavelengths for a nanowire with d = 214 nm, corresponding to a minimum in the Raman intensity. The overlap between intensities at the two wavelengths is poor except in the central spot. (e) Volume-integrated |E|4 fields for 532 nm (▲) and 547 nm (■) excitations as well as mixed-product integration of |E532|2·| E547|2 fields (●) are compared to experimental Raman data (black solid line). The mixed-product calculation agrees best with experimental data, in both width and peak maximum location. 2268
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Figure 4. Experimental and calculated Raman intensities of a tapered silicon nanowire are compared for indicated scattering configurations (see text for notation). The inset in (a) indicates the scattering geometry. Agreement with experiment is very good except for the z(yy)z configuration, in ̅ which deviations are significant at diameters larger than 150 nm. We note that the z(xy)z configuration was especially sensitive to small ̅ misalignments of the analyzer polarization filter, as expected. The agreement with experiments for this configuration was improved by assuming a misalignment of 2°.
dependence of absorption, the absolute Raman intensities at the peak resonances increase with decreasing diameter for the (xx) configuration (data have not been normalized by volume), in part due to the better phase match between the excitation and Raman frequencies that occurs for small diameters. Following the scattering geometry from Figure 4a, the field components at the excitation frequency inside the nanowire Ein are taken from FDTD simulations:
mixed product calculation, which is therefore used in the more complete comparison with experiments described below. Confocal Raman measurements were performed in a backscattering geometry (inset Figure 4a) using a 100× objective with 0.9 numerical aperture. Raman scattered light was isolated by an edge filter, collected via an optical fiber whose core defines the confocal aperture, and dispersed onto a charge coupled detector for analysis. The sample was placed on a piezoelectric scanning stage, and a line scan of varying height was defined to ensure the nanocone was in focus throughout the scan. The gentle taper of the nanocone enabled all experimental variables, including the nanowire orientation and the instrumental parameters, to be well controlled. The Raman intensity was measured in four scattering configurations z(xy)z, z(yx)z, and denoted in Porto’s notation:22 z(xx)z, ̅ ̅ ̅ z(yy)z, where the symbols in the parentheses refer, from left to ̅ right, to the incident and scattered light polarization, while the symbols outside refer to the propagation direction of the incident light (left-hand symbol) and the measured scattered light (right-hand symbol). In the following discussion, the configurations will be denoted only by the symbols inside the parentheses, since the incident and scattered light propagation directions are held constant for all experiments. Integration times (powers) of 4 s (17 μW), 10 s (17 μW), 4 s (81 μW), and 4 s (81 μW) per point were used for the (xx), (xy), (yx), and (yy) configurations, respectively. For the strongest resonances, the nanowire was locally heated to ∼340 K, which broadened and down-shifted the Raman peak. For this reason, the peak area rather than the amplitude is plotted as the intensity (Figure , solid lines); the mild heating has a negligible effect on the intensity profiles. In contrast to the diameter
E in = cos θi·E532,TM + sin θi·E532,TE
(1)
These fields are then transformed by the Raman tensors of the three degenerate modes Ra, Rb, and Rc describing phonon polarizations parallel to the ⟨100⟩ crystal directions in diamond cubic silicon. The tensors are first rotated to the laboratory coordinate system, assuming a [111] growth direction and defining the axial rotation angle α to be between [21̅1̅] and the y axis in the lab frame (counter-clockwise rotation about x axis looking to the origin). The growth direction is assumed to be ⟨111⟩, although a ⟨112⟩ direction could also fit experiments if ⟨110⟩ facets are present. The two orientations are not distinguishable in the present work. An advantage of doing measurements on a ⟨111⟩ oriented nanowire is that, for the scattering configurations considered here, the Raman intensities do not depend on the axial rotation angle, which simplifies comparison with experiments. The transformation from the crystal to the lab frame is given by ⎡ 2 ⎤ 2 2 ⎥ 1 ⎢ ⎢ 2cos α −cos α − 3 sin α −cos α + 3 sin α ⎥ T= 6⎢ ⎥ ⎣ 2sin α −sin α + 3 cos α −sin α + 3 cos α ⎦ 2269
(2)
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The resulting Raman scattered fields inside the nanowire, denoted Ea, Eb, and Ec (still functions of position), are then ‘enhanced’ IHa by the direct point-by-point product with the field intensities from FDTD simulations at the excitation frequency:
IH ai = |Eai|2 ·|Hi|2
Stokes/anti-Stokes ratio are greater than 50% for a 170 nm nanowire, not considering effects of absorption or changes in the Raman polarizability with wavelength. In summary, Raman scattering intensities of silicon nanowires were measured for normal incidence and multiple polarization configurations. The data were explained by a model considering the effects of the internal electromagnetic mode structure for both the excitation and the Raman scattered light. The dependence of the Raman signal on the fourth power of the electromagnetic field makes it extremely sensitive to geometric structure in the subwavelength regime, and it provides an excellent opportunity to benchmark models of light−matter interactions. Experiments and the model are in good agreement except for the case of incident and scattered light polarized perpendicular to the nanowire at diameters larger than 150 nm. Further refinement of the model to account for surface scattering, nanowire faceting, and the angular dependence of light emission may resolve these discrepancies, ultimately enabling recovery of information about nanowire diameter and crystal orientation from the polarization dependence of the Raman scattered light. The work informs the design of Raman experiments on nanowires and related nanostructures by establishing the basis for optimized matching between excitation wavelength and diameter to maximize signal throughput.
(3)
where H = cos θs·E547,TM + sin θs·E547,TE·
(4)
i represents Cartesian components, and θs is the angle between the nanowire axis and the polarization of the analyzed scattered light. The simulations at the scattered frequency are identical in geometry to the simulations at the excitation frequency, except for the change in wavelength; they represent an approximation for the enhancement and emission of the Raman scattered light. Adding contributions from x and y components for the three Raman tensors (ignoring the z component) we get IH = IHax + IHbx + IHcx + IHay + IHby + IHcy. Finally we obtain the total intensity by integrating IH over the excitation volume within the nanowire: I∝
∫V IH dV
(5)
■
The calculated Raman intensities are compared with experiment in Figure 4. The model is in consistently good agreement with experiment for the (xx), (xy), and (yx) configurations. The (yy) calculation deviates from experiment for nanowire diameters greater than ∼150 nm. A possible explanation lies with the transmission of the Raman scattered light across the nanowire−air interface, which has a complex diameter dependence for the angular distribution of intensities.7,23 Surface facets and the exact cross-sectional shape of the nanowire (which was simulated as a circle) may need to be considered, even though photocurrent experiments appeared to be relatively insensitive to the facet geometry,19 because facets could modify scattering and emission more strongly than they affect absorption. The (yy) configuration, which presents larger field components perpendicular to the nanowire surface, could also be more sensitive to preferential surface absorption and modified transmission in the presence of disorder, which is not accounted for in the simulations. We note that the general trend of increasing Raman intensity with increasing wire thickness is what one might expect given the similarity of the (yy) configuration to a simple thin film geometrythe antenna effect is less pronounced in this configuration. Anti-Stokes scattering processes involve energy gain of a photon by absorption of an optical phonon. Because the probability of such event is proportional to the available optical phonons, whose population is small at room temperature, antiStokes scattering in Si is ∼10 times weaker than Stokes scattering. The intensity ratio, which becomes closer to one with increasing temperature, can be used to probe phonon populations and thereby monitor temperature.24 The findings presented above demonstrate that phonon populations in nanowires should not be deduced directly from Stokes/antiStokes intensity ratios without taking into account a diameterdependent correction factor. This is especially true for diameters greater than the reduced wavelength λ/n, as the resonance peaks at the Stokes (547 nm) and Anti-Stokes (517.7 nm) Raman frequencies are shifted by ∼9% of the diameter for 532 nm excitation. For example, deviations in the
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]; lauhon@northwestern. edu. Present Addresses §
Nanoscience Open Research Initiative and Materials Science Institute, University of Oregon, Eugene, Oregon 97403 ‡ Max Planck Institute for Microstructure Physics, Halle, Germany Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Department of Energy, Office of Basic Energy Sciences DE-FG02-07ER46401 and the Northwestern University MRSEC under NSF grant DMR0520513 and DMR-1121262. F.J.L. acknowledges partial support from the National Council for Science and Technology (Mexico). L.J.L. and A.H. acknowledge the support of the Camille and Henry Dreyfus Foundation.
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