Letter pubs.acs.org/NanoLett
Optical Far-Field Method with Subwavelength Accuracy for the Determination of Nanostructure Dimensions in Large-Area Samples Nicklas Anttu,*,† Magnus Heurlin,† Magnus T. Borgström,† Mats-Erik Pistol,† H. Q. Xu,†,‡ and Lars Samuelson† †
Division of Solid State Physics and The Nanometer Structure Consortium (nmC@LU), Lund University, Box 118, S-22100 Lund, Sweden ‡ Key Laboratory for the Physics and Chemistry of Nanodevices and Department of Electronics, Peking University, Beijing 100871, China S Supporting Information *
ABSTRACT: The physical, chemical, and biological properties of nanostructures depend strongly on their geometrical dimensions. Here we present a fast, noninvasive, simple-toperform, purely optical method that is capable of characterizing nanostructure dimensions over large areas with an accuracy comparable to that of scanning electron microscopy. This far-field method is based on the analysis of unique fingerprints in experimentally measured reflectance spectra using full three-dimensional optical modeling. We demonstrate the strength of our method on large-area (millimeter-sized) arrays of vertical InP nanowires, for which we simultaneously determine the diameter and length as well as cross-sample morphological variations thereof. Explicitly, the diameter is determined with an accuracy better than 10 nm and the length with an accuracy better than 30 nm. The method is versatile and robust, and we believe that it will provide a powerful and standardized measurement technique for large-area nanostructure arrays suitable for both research and industrial applications. KEYWORDS: Optical metrology, spectroscopy, nanowire array, semiconductor, nanostructure, indium phosphide
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damaging the nanostructure under investigation.20 In contrast, conventional optical imaging of large areas is fast and noninvasive, but the resolution is limited by diffraction21 to λ/2 (where λ ≈ 400−700 nm is the wavelength of visible light). This resolution is insufficient for the characterization of nanostructure dimensions. In principle, more detailed information could be obtained by analyzing wavelength-dependent optical spectra instead of optical images.22−24 However, technologically important large-area nanostructure arrays usually have geometrical variations along both in-plane directions, prohibiting the use of common and fast twodimensional analysis25 or semianalytical methods23,24 for extracting the geometrical dimensions from a measured spectrum. Here we report on a versatile characterization methodology to overcome these obstacles. This method incorporates full three-dimensional optical modeling and experimentally measured wavelength-dependent optical reflectance spectra to determine the lateral and vertical geometrical dimensions of nanostructures in large-area arrays. We demonstrate the
arge-area arrays of nanostructures have gained considerable recent interest for applications, especially in optoelectronics.1−11 Both bottom-up1−6,12,13 and top-down approaches9,10,14,15 can be used to create nanoscale building blocks with designable physical properties and small spread in dimensions. The high areal packing-density often required by device aspects is made feasible by use of predefined ordered pattern templates for processing and synthesis which allows further tuning and optimization of the physical properties to enhance device performance. In devices based on such arrays of highly uniform nanostructures, the sum of the individual signals from each nanostructure results in a strongly enhanced output signal. In general, the physical, chemical, and biological properties of nanostructures depend strongly on their geometrical dimensions.16,17 Therefore, the development, fabrication, processing, and optimization of nanostructure-based devices require efficient tools for assessing these dimensions. Scanning electron microscopy (SEM) is currently the predominant choice for this characterization because of its high resolution and noncontact nature, which circumvents the limited depth of view in nearfield-probe18,19 based methods. However, SEM is not feasible for the characterization of large area samples because of its small field of view and slow scanning rate. Furthermore, SEM is expensive and time-consuming and involves the risk of © XXXX American Chemical Society
Received: March 5, 2013 Revised: April 10, 2013
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NWs with, as compared to samples A−C, significantly larger diameter of approximately 200 nm. All VLS NW growth was done at 395 °C at a total reactor pressure of 100 mbar using molar fractions χTMIn = 8.2 × 10−6, χPH3 = 1.9 × 10−2, and χHCl = 1.0 × 10−5, while the regrowth step was done at 550 °C at a total reactor pressure of 400 mbar using χTMIn = 18.0 × 10−6 and χPH3 = 3.6 × 10−2. A SEM operating between 10 and 15 kV was used for characterizing the samples and to determine geometrical dimensions and size dispersion. The tilt angle was calibrated by adjusting the reference 0° position in top view so that no NW sidewalls could be observed. We estimate this calibration to be accurate within ±0.5°. The NW length L was determined by tilting the sample to 30° for samples A−C (see Figure 1a for a SEM image of sample B). For sample D, due to the larger values of d, a tilt of only 10° could be used. The NW diameter d was extracted from the same images. Samples A−C consisted of NWs with a cross-section close to circular in shape. In contrast, the cross-sectional shape of most NWs on sample D was found to vary between a regular hexagon and structures with up to 12 facets. To obtain a single value for the diameter d from these SEM images, the largest distance d′ between any two points in the cross-section of a NW was measured, and this value was multiplied by 0.9 to give the value of d for sample D. In this way, the cross-sectional area of a hexagonal NW with a largest distance of d′ between any two points in the cross-section equals that of a NW with circular cross-section of diameter d used in the electrodynamic modeling described below. The optical reflectance R of the samples was measured in the wavelength range of 400 < λ < 850 nm with a wavelength step of Δλ = 0.7 nm using a Filmetrics F40 probe attached to a Carl Zeiss M1m optical microscope. A 5× objective, oriented normal to the sample surface, with a numerical aperture of NA = 0.13, was used both for the illumination and collection of the reflected light. The reflected light was collected from a 100 × 100 μm2 large square spot. The reflectance measurement and corresponding SEM measurement at nominally same position were done within 100 μm from each other. To theoretically describe the interaction of light with the NW arrays, we employ electrodynamic modeling and solve the Maxwell equations with a scattering matrix method.28 Each modeled InP NW array consists of NWs with length L and a circular cross-section of diameter d on top of an optically infinitely thick InP substrate. The NWs are placed in a square array with period p = 400 nm in both directions.11 Tabulated values of the refractive index n are used for InP.29 For the air between and on top of the NWs, the value of n = 1 is used. The incident light of a given wavelength is modeled as a plane wave incident from the top (air) side at normal angle to the array/ substrate: A numerically much heavier integration/averaging4 over the small 7.5° half angle of the incidence cone of the experimental NA = 0.13 caused only very minor changes in modeled spectra. The wavelength range in the modeling is 400 < λ < 850 nm with a wavelength step of Δλ = 5 nm, and linear interpolation is used for the optical spectra between the modeled wavelengths. To demonstrate the capabilities of our purely optical method for measuring nanostructure dimensions, we start by showing characterization results based on optical reflectance measurements and consequent modeling of samples A−C. These samples have NWs with diameter d of approximately 125 nm, but the NW length L varies between 500 and 1500 nm (the
potential of this far-field optical analysis method for the specific case of periodic arrays of vertical semiconductor nanowires (NWs), which are promising for applications in broad technology areas such as light-emitting diodes,2 solar cells,3,5,6,10 photodetectors,7 and lasers.13 Using this method, we determine simultaneously both the NW diameter and length, as well as cross-sample morphological variations, with accuracy comparable to that of SEM measurements. Explicitly, the diameter is determined with accuracy better than 10 nm and the length with accuracy better than 30 nm. The optical measurements are fast, simple to perform, and noninvasive and can in principle be done in situ during nanostructure synthesis and device fabrication.26 For this study, four InP NW arrays (samples A−D) were fabricated using the VLS method with Au catalyst particles defined by nanoimprint lithography (NIL) on (111)B InP substrates (Supporting Figure S1). The NIL pattern has a period of 400 nm with aligned columns of Au particles where each column is slightly offset (inset in Figure 1a). A high quality
Figure 1. (a) SEM image of sample B at 30° tilt from top-view showing high local uniformity of the NWs. The inset shows a top-view SEM image of the NIL pattern of Au particles prior to growth. The Au particles are placed with a period of 400 nm within aligned columns. The columns are also placed in a periodic pattern of 400 nm in period, but with an offset between adjacent columns as compared to a perfect square pattern of Au particles. (b) Light incident on and reflected at the air-NW top interface and the NW-substrate bottom interface gives rise to unique fingerprints in the measured spectra. This allows extraction of NW dimensions from comparison with reflectance spectra obtained from three-dimensional optical modeling.
pattern was obtained on 2” wafers using an Obducat AB developed process with an intermediate polymer stamp (IPS) and soft press technology. Smaller samples were subsequently cleaved from the 2” wafer and used in the growth experiments. All NW samples were grown in a horizontal metal−organic vapor phase epitaxy reactor using trimethyl-indium (TMIn) and phosphine (PH3) as precursor gases. HCl gas was used in order to minimize diameter changes otherwise induced by parasitic radial growth.27 Samples A−C were grown by using different NW growth times of 5.5, 11, and 22 min, respectively, resulting in NWs with a diameter d of approximately 125 nm, and a length L varying between 500 and 1500 nm (depending on the growth time). At the end of the NW array fabrication, the Au particles were removed ex situ by a wet chemical process. Sample D was grown using a two-step approach where NWs were first grown by the VLS mechanism (growth time of 22 min). The Au particles were then removed ex situ by a wet chemical process, before a regrowth step of 30 min was performed to define the final NW dimensions. This resulted in B
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Figure 2. Measured reflectance R of three periodic InP NW arrays (samples A−C) with period p = 400 nm, NW diameter d ≈ 125 nm, and different design for the NW length L (see Table 1). Here, we show for each array also the modeled reflectance spectrum of the NW array geometry that reproduces the measured spectrum most accurately in the sense that it minimizes the sum of squared discrepancies s between the measured and modeled spectra. The modeled system consists of NWs with diameter d = 121 nm and length L = 533 nm for sample A; d = 124 nm and L = 855 nm for sample B; and d = 124 nm and L = 1445 nm for sample C (see Table 1).
sample D (starting and ending approximately 0.3 mm from the edges of the NW array; see Supporting Figure S8 for a discussion of strong edge effects). The measured reflectance spectra in Figure 3a show clear interference fringes that are redshifted close to one end point of the sample (x = 0) while displaying only a weak wavelength-dependence at the other end (x = 2.2 mm). These continuous variations in NW morphology revealed by our optical measurements are reproduced very accurately in our calculated spectra (Figure 3b), where similar reflectance amplitudes, fringe patterns, and red-shifts are present. The cross-sample NW dimensions extracted from the best-fit calculated spectra (Figure 3b) are in excellent agreement with our control measurements performed with SEM (Figure 4). From Figure 4a it is clearly seen that the NW diameter d decreases from approximately 235 to 190 nm when moving from the sample midpoint at x = 1.1 mm to the sample end point at x = 2.2 mm. In contrast, the diameter decreases much less as a function of position in the opposite direction (toward x = 0). The length of the NWs (Figure 4b) is approximately L = 1100 nm at the midpoint x = 1.1 mm and increases by approximately 200 nm (100 nm) when moving toward the end point at x = 0 (x = 2.2 mm). A large (≈0.5 mm wide) unpatterned region without predefined Au catalysts, and consequently without NWs, resides at the edge of sample D located beyond x = 0. This is expected to cause these different types of edge effects in the NW growth at x = 0 and x = 2.2 mm. Knowledge of the variations of d and L across sample D enables us to understand the wavelength-dependencies of the reflectance fringes in Figure 3 in terms of an interference description. The peak of a given reflectance fringe occurs due to the constructive interference between light reflected at the airNW top interface and light reflected at the NW-substrate bottom interface, which travels an additional optical path length of xOPL = 2Lneff because of the round-trip through the NW layer. Here, xOPL depends on both the NW length as well as the NW diameter through the effective refractive index, neff, which describes the optical response of the NW layer. This effective refractive index is expected to increase with increasing d, because the volume-averaged refractive index of the NW layer increases. In this picture, a given reflectance peak at λm occurs due to constructive interference when xOPL = mλm, where m is an integer. Thus, the wavelength position λm of a given reflectance peak is expected to red-shift with both increasing L
length was adjusted by varying the growth time). SEM analysis shows that the NWs have a high local uniformity in both diameter and length (Figure 1a). The experimental reflectance spectra for 400 < λ < 850 nm show clear differences in amplitude and oscillation pattern for samples A−C (Figure 2a−c). To extract the NW dimensions from these fingerprints, we first calculated, with full threedimensional optical modeling,28 the reflectance spectra, Rsim(d,L), of an InP NW square array with period p = 400 nm while varying d and L with a step-size of 1 nm in the ranges 0 < d < 400 nm and 0 < L < 2000 nm. The determination of the NW dimensions was then carried out by finding, for each measured reflectance spectrum Rexp, the values of d and L that minimized the sum of squared discrepancies s = ∑i[Rexp(λi) − Rsim(λi,d,L)]2 between the measured and the calculated spectra, where λi are the wavelengths used in the measurement. With this method of fitting d and L we found for all three samples an excellent agreement in the oscillation pattern between the measured and calculated spectra (Figure 2). To verify the accuracy of our method, we measured d and L of the NW arrays in samples A−C with SEM and found excellent agreement between our purely optical method and the results obtained from SEM (Table 1). For all three samples we found that the discrepancy between our optical method and SEM is less than 10 nm in the value of d and less than 30 nm in the value of L. To explore the applicability and strength of our optical metrology method for the characterization of large-area samples, we have carried out measurements and corresponding calculations of the reflectance along a 2.2 mm line across Table 1. Comparison of the NW Diameter d and Length L of Periodic NW Arrays (Samples A−C) Determined with the Purely Optical Method (Opt. Met.) Based on Far-Field Reflectance Measurements (Figure 2) and the d and L Determined from SEM Imagesa
sample A sample B sample C
d (with SEM)
d (Opt. Met.)
L (with SEM)
L (Opt. Met.)
125 ± 3 nm 126 ± 5 nm 131 ± 2 nm
121 nm 124 nm 124 nm
507 ± 14 nm 862 ± 27 nm 1456 ± 23 nm
533 nm 855 nm 1445 nm
a
Here, in the dimensions measured from the SEM images, the uncertainty denotes the standard deviation in the values of the dimensions of the 50 NWs measured for each sample. C
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Figure 3. (a) Reflectance spectra of NW array sample D of period p = 400 nm measured from a 100 × 100 μm2 square spot at different positions along a 2.2 mm straight line. The separation between neighboring measurement points is approximately 50 μm and 0 < x < 2.2 mm denotes the position along the line. (b) Modeled reflectance spectra of the NW geometries that reproduce most accurately the experimental spectra in (a) in the sense that they minimize the sum of squared discrepancies s between the measured and modeled spectra. The values of the NW diameter d and length L determined with this purely optical method are shown in Figure 4.
Figure 4. (a) Optically measured values of the NW diameter d (red squares) obtained by finding the NW geometry (diameter d and length L) that gives the best agreement between modeled and measured reflectance spectra as shown in Figure 3. Here, also the NW diameters (blue circles) and their standard deviations (dark gray area) measured from SEM images are shown. (b) Similar to part a but for the NW length L (black circles) and their standard deviations (light gray region). Due to the limitation of a maximum tilt angle of 10° in the SEM imaging of sample D, the length of the slightly tapered NW base (Supporting Figure S7) could not be directly measured. To correct (blue circles) for this we approximate, from SEM images taken in side view after cleaving the sample (Supporting Figure S7), the length of this tapered base to approximately 10% of the total NW length. We also show the length distribution after this correction (dark gray region). The SEM characterization was performed with a step size of 100 μm and for each point along the line 15 NWs were measured.
and increasing d. This explains why (i) the reflectance fringes in Figure 3 red-shift when approaching the sample measurement end point at x = 0 mm because here the length of the NWs increases while their diameter stays relatively constant (Figure 4) and (ii) the fringes, in contrast, do not shift strongly when moving toward the other measurement end point at x = 2.2 mm, because here the effects on the fringe position due to the increase of L and the decrease of d counteract each other. The successful determination of nanostructure dimensions relies on having clearly distinguishable fingerprints in the measured reflectance spectra. For the specific case of NW arrays characterized here: (i) a given interference fringe shifts with varying d and L (Figure 3), (ii) an increase of d increases the reflectance of the air-NW top interface (Supporting Figure S2), and (iii) the wavelength regions where the fringes are weak
coincide with regions of strong absorption in the NW array, and these regions shift with d4,11 (Supporting Figure S2). These three dependencies allow, as demonstrated, for the simultaneous determination of d and L. However, for a successful determination, the detected reflectance signal must differ strongly enough from the background signal, which here is the bare substrate reflectance. For NW arrays this requires that the array is not too sparse or the NWs too short or too thin. For example, our modeling shows that individual variations of the parameters in sample B (d = 124 nm, L = 855 nm, and p = 400 nm) result in the mean difference (for 400 < λ < 850 nm) between the bare substrate and the NW array reflectance to decrease to below 0.01 either when p is increased beyond 2500 nm, L is decreased to below 25 nm, or d is decreased to below 40 nm. Another potential limitation can arise if a continuous D
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Foundation for Strategic Research (SSF), the Swedish Energy Agency, the Nanometer Structure Consortium at Lund University (nmC@LU), Nordic Innovation program NANORDSUN, Knut and Alice Wallenberg Foundation, and the National Basic Research Program of the Ministry of Science and Technology of China (Nos. 2012CB932703 and 2012CB932700).
variation of the geometrical dimensions is too strong over the measurement spot (Supporting Figures S8−S10). However, we note that the results presented for d and L in Figure 4 differ negligibly (by less than 2%) from results obtained by using a triangular array pattern in the modeling instead of the square array pattern (Supporting Figure S11). This demonstrates that the optical method is robust to slight offsets in the experimental NW positions (as in the present samples, see inset of Figure 1a) as compared to the modeled perfectly ordered array. We believe that this optical metrology method is versatile and applicable for the characterization and analysis of a wide range of nanostructure arrays. To support this we note that we have found strong evidence that the method is applicable also when the Au catalyst is still present at the top of the NWs (Supporting Figure S3), and we have verified that it can be used for measuring the height of metal disks (Supporting Figure S4). Note that the method is not limited to the characterization of periodic arrays: We have used it successfully for determining the length of randomly positioned NWs of a broad diameter distribution (Supporting Figure S5). Furthermore, the range of possible applications for the method is broadened by the fact that differences in dielectric functions between materials should allow it to be used for determining the material composition of semiconductor nanostructures (Supporting Figure S6) in addition to their geometrical dimensions. Therefore, we believe that this purely optical far-field method will provide a powerful and standardized measurement technique for large-area nanostructure arrays suitable for both research and industrial applications.
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ASSOCIATED CONTENT
S Supporting Information *
Fabrication of NW arrays, additional details of absorption and reflection of light in NW arrays, reflectance in the presence of the Au catalyst, optical measurement of Au disk height, optical measurement on nonperiodic array, material dependent reflection from NW arrays, and additional details on SEM measurements and optical measurements. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
N.A. and M.H. contributed equally to this work. L.S., M.H., M.T.B., and N.A. planned the experiments and designed the samples. M.H., M.T.B., and N.A. wrote the manuscript. M.H. performed the nanowire array fabrication and the SEM characterization. N.A. performed the optical measurements, the optical modeling, and the consequent extraction of geometrical dimensions. H.Q.X. initiated and supervised the optical modeling of nanostructures. M.E.P. initiated and supervised the reflectance measurements of nanostructures. M.T.B. and L.S. supervised the project. All authors discussed the results and commented on the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge Gang Luo from Obducat AB for supplying the nanoimprint patterns. This work was supported by the Swedish Research Council (VR), the Swedish E
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(29) Glembocki, O. J.; Piller, H. Handbook of Optical Constants of Solids; Palik, E. D., Ed.; Academic Press: New York, 1985; pp 503− 516.
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