Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling

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Characterization of Si nanorods

by spectroscopic ellipsometry

with efficient theoretical modeling

Shih-Hsin Hsu*, 1

, En-Shao Liu1

, Yia-Chung Chang1

, James N. Hilfiker2

, Young Dong Kim3

, Tae Jung Kim3

, Chun-Jung Lin4

, and Gong-Ru Lin5

1_{Research Center for Applied Sciences, Academia Sinica, 128 Academia Road, Sec. 2, Taipei 11529, Taiwan} 2_{J. A. Woollam Co., Inc., 645 M Street, Suite 102, Lincoln, NE 68508, USA}

3_{Department of Physics, Kyung Hee University, 1 Hoeki-dong Dongdaemoon-gu, Seoul 130-701, Korea}

4_{Department of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan} 5_{Graduate Institute of Photonics and Optoelectronics, National Taiwan University, 1 Roosevelt Road, Sec. 4, Taipei 10617, Taiwan} Received 11 June 2007, revised 12 December 2007, accepted 17 December 2007

Published online 26 March 2008

PACS 78.20.Bh, 78.20.Ci, 78.66.Db, 78.67.Bf

*_{Corresponding author: e-mail}_{[email protected]}_{, Phone: + 886-2-27898000 ext. 56, Fax: + 886-2-27826680}

1 Introduction Among various kinds of nanostruc-tures which have been widely used in solar energy conver-sion, nanorods are very attractive because they naturally provide a directed path for electrical transport [1, 2]. In particular, Si nanorods are considered one of potential can-didates for future solar cells because they can efficiently absorb light along the length of rods and collect the gener-ated electricity [3]. Thus, fully understanding nanorods’ structures and corresponding optical properties become an urgent issue. Conventionally, micro- and/or nanostructures are characterized using different types of microscopies, such as scanning electron microscopy, transmission elec-tron microscopy, atomic force microscopy, etc. However, these microscopies either require additional sample prepa-rations, or exert energetic electron beams or mechanical forces on samples, which inevitably produce irreversible damages. In contrast, spectroscopic ellipsometry (SE) is a non-contact and nondestructive optical technique for the characterization of thin films and bulk materials [4]. Its applications have extended from measuring optical con-stants and thicknesses toward more sophisticated

charac-terizations such as electronic structures, chemical composi-tions, and micro-/nanostructures [5 – 7].

In this paper we report the application of variable-angle spectroscopic ellipsometry (VASE) to characterize Si nanorods. The Bruggeman effective medium approxi-mation (EMA) [8] theory is employed in the analysis to model porous columnar structures. Moreover, we explore the capability of describing the nanorods’ geometry and structures with the rigorous coupled-wave analysis (RCWA) [9, 10] as well as an efficient finite-element Green’s function approach [11].

2 Experimental

2.1 Sample preparation The Si nanorods under in-vestigation were fabricated by a Ni/SiO₂ nano-mask as-sisted dry-etching procedure. A 5 nm thick Ni film was first e-beam evaporated on the SiO₂/Si substrate, where the SiO₂ buffer layer (20 nm in thickness) was deposited by using plasma-enhanced chemical vapor deposition. Subsequently, a rapid thermal annealing process at 850 °C was performed to self-aggregate the Ni nano-dots on the Spectroscopic ellipsometry (SE) is applied to characterize Si

columnar nanostructures. By employing effective medium approximation (EMA) theory, Si nanorods are treated as a graded layer with each sub-layer modeled as a mixture of Si and voids with varying porosity fraction. In addition, the

rigorous coupled-wave analysis and finite-element Green’s function method were used in modeling Si nanorods as a stack of disks with varying diameters and thicknesses, and the calculations are in satisfactory agreement with the measure-ment results.

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100 nm 100 nm

Figure 1 Scanning electron micrograph of Si nanorods.

SiO₂/Si substrate. By using the Ni dots as a nano-mask, the sample was dry-etched in a planar-type ICP-RIE system. The SiO₂ layer was then removed from the surface together with Ni nano-dots, leaving only Si nanorods standing on the Si substrate. The nanorods samples were stored in methanol to avoid gradual oxidation, and were dried with N₂ purge prior to measurements. Figure 1 dis-plays one typical SEM image of the Si nanorods samples. As depicted in the figure, the diameter and height of nano-rods are around 30 – 45 nm and 200 – 315 nm, respectively.

2.2 Variable-angle spectroscopic ellipsometry

The interaction of polarized light with a sample can be rep-resented by the Jones matrix formalism [12, 13]:

out in p pp sp p out in s ps ss s , E R R E E R R E È ˘₌È ˘_◊È ˘ Í ˙ Í_Î ˙_˚ Í ˙ Î ˚ Î ˚ (1)

where R and pp R are complex Fresnel reflection coeffi-ss

cients of the sample for p- and s-polarized light, respec-tively. The off-diagonal terms R and _ps R describe the _sp cross-coupling of the p- and s-polarized light and are zero for an isotropic sample. Thus, ellipsometric measurements are normally described by two parameters Ψ and ∆ in the following form: pp ss tan ( ) e R R ∆ Ψ ◊ ı = _. ₍₂₎

Because of the charge screening effect developed by the columnar structure, as shown in Fig. 1, it can be expected that the Si nanorods may introduce anisotropy. Moreover, if the rods are inclined with respect to the sample surface, there may also be cross-polarization [7].

To characterize such a sample, we began with a Muel-ler-matrix measurement to test if there exist significant cross-polarization and depolarization for the Si nanorods. The SE data were acquired over a broad range from near

Figure 2 (online colour at: www.pss-a.com) Mueller-matrix measurements of Si nanorods at an incidence angle of 60°.

infrared to vacuum ultraviolet (VUV) using a rotating-analyzer ellipsometer (RAE) system with an adjustable re-tarder (VUV-VASE, J. A. Woollam Co.), which can meas-ure up to 11 Mueller-matrix elements (normalized to m11)

except the fourth row [12]. Since the m22 elements

meas-ured at several incident angles were close to unity, the sample could be regarded as non-depolarizing. Besides, the off-diagonal blocks of the Mueller-matrix elements (m₁₃, m14, m23, m24, m31, m32) were basically zero to within the standard deviations of the measurement, as shown in Fig. 2. In terms of generalized ellipsometry, the off-diagonal ellipsometric ratios R R and ps/ pp R R were found to be sp/ ss

almost zero as well. This suggests the Si nanorods can be treated as vertically standing columns on the substrate surface, and it is possible for us only to do normal ellipsometry measurement, which is sufficient for uniaxial anisotropy with the optical axis perpendicular to the sur-face.

3 Results and discussion

3.1 Analysis employing effective medium ap-proximation In order to extract structural parameters of the Si nanorods, a physical model was established employ-ing effective medium theory, which is developed to explain the connection between the microstructure of a heteroge-neous thin film and the macroscopic dielectric re-sponses [8]. Si nanorods can be considered as a thin film composed of comparable amount of crystalline Si and void, thus is suitable for the Bruggeman EMA. The optical con-stants of crystalline Si analyzed in Johs et al. [12] were used in the model.

Moreover, due to the anisotropic nature of columnar structures, charge screening factors must also be taken into consideration. Because different shapes of microstructures will induce distinct boundary conditions to the electrical and magnetic field in the medium, the depolarization factor q can be used to model the screening effect in an EMA mixture. For the Si nanorods, which are almost perpen-dicular to the surface (along z-direction), qz must be close

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Figure 3 (online colour at: www.pss-a.com) Measurement re-sults and model fits based on the graded, uniaxial EMA theory.

splitting between the other two components (qx= ª /qy 1 2)

can describe the in-plane symmetry [5].

It was found that applying only the above two consid-erations would not give us reasonable results. As the SEM image displays (Fig. 1), the diameter of nanorods is by no means uniform along the vertical direction. Therefore, the Si nanorods were treated as a graded layer, in which each sub-layer is modeled as a mixture of crystalline Si and voids with varying porosity fraction. The uniaxial aniso-tropy of the nanorods was described by modeling the direc-tional-dependence of the EMA screening factor. Figures 3 and 4 show the model fitting based on a 3-node, graded EMA layer and the corresponding depth profile of dielec-tric constants for nanorods, respectively. As expected, this model fit works well at longer wavelengths, where the EMA theory still holds for the feature sizes of the Si nano-rods.

3.2 Analysis based on rigorous coupled-wave analysis and Green’s function method Because the EMA theory only works well when the finite-wavelength effects are negligible [8], alternative modeling approaches are required in the SE analysis of nanostructures from short wavelengths comparable to the critical dimension of sam-ples. Here, we explore the capability of modeling the Si

Figure 4 (online colour at: www.pss-a.com) Depth profile of anisotropic dielectric constants ε (real part) and ₁ ε (imaginary ₂ part) for Si nanorods at 2.48 eV.

Figure 5 (online colour at: www.pss-a.com) (a) Triangular unit cell used in RCWA and GF calculations; (b) the cross-sectional view of the rods model with 5 cylinders.

nanorods using techniques conventionally applied only for periodic structures.

The RCWA may be the most popular method for the accurate diffraction analysis of periodic structures [9, 10], but it becomes terribly inefficient when applied to three-dimensional gratings. Hence, an efficient numerical method based on the Green’s function (GF) approach [11] was employed as well in the modeling. Since the nanorods are very closely packed (Fig. 1), the unit cell for the RCWA and GF modelings was chosen to be a regular tri-angle, as shown in Fig. 5(a), which is the most tightly packed periodic structure in two dimensions.

To describe the structure, we started with partitioning the rods into a stack of 5 to 7 lamellar cylinders with dif-ferent diameters and thicknesses. The unit vector length or the pitch p in a triangle unit cell was fixed at 50 nm at this stage because the same diameter-to-pitch ratio but different

Figure 6 (online colour at: www.pss-a.com) SE measurement and model calculations of Si nanorods based on the GF method with a rod model shown in Fig. 5(b).

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sets of values lead to very similar results. The diameter and thickness of each slice were first converted from the EMA result and then adjusted manually to find a better fit.

It is found that the RCWA and finite-element GF method give almost the same results, but calculations based on the GF approach are about one order of magni-tude faster. Figure 5(b) illustrates the best nanorod geome-try we found till now, and the comparisons between SE measurements and the GF approach calculations are de-picted in Fig. 6. Although the simulated spectra do not match exactly with experimental data, the shapes are simi-lar and trends are in good agreement.

3.3 Discussion The SE studies of columnar Si nano-structures from VUV to near-IR spectrum show that this technique could provide useful information about nano-structures, such as the effective dielectric constants or the geometric structure. The information found from the SE analyses not only agrees with what SEM micrograph revealed qualitatively but also quantitatively to a certain degree. The above analyses could be further improved by increasing the number of stacks and further fine-tuning the porosity fraction (when using EMA theory) or their thick-nesses and diameters (when using the other two ap-proaches). However, the more the stacks and/or other refinement employed, it is more likely that severe parame-ter correlation becomes unavoidable and an unique model could not be determined.

It should be noted that either the RCWA or GF ap-proach has been applied only for periodic structures in the past. But the above model calculations (periodic) indicate that they could also be applied as an approximation for this kind of non-periodic nanostructures with good success.

4 Conclusion Fabrication of micro- and/or

nanostruc-tures requires accurate and reliable methods to characterize the processed structures such that their physical properties as well as fabrication quality can be controlled. Here we explore the capability of SE as a noninvasive characteriza-tion tool for columnar nanostructures. By applying EMA in the model fittings, the Si nanorods were found to present porosity variation along the vertical direction. The SE measurement and model-generated results can reach good

agreement when introducing grading by dividing the nano-rod layer into several sub-layers with varying thicknesses and void fractions. To overcome the EMA’s inherent limi-tation due to the finite-wavelength effects, we employed the RCWA and an efficient GF method as alternative ap-proaches to analyze SE measurements of Si nanorods. These modelings give reasonable fittings, and also support the results obtained from the EMA analysis. These findings demonstrate that SE could be a useful technique for the characterization of various kinds of structures in nano scale.

Acknowledgements The authors would like to thank the referees for their helpful suggestions and comments. The work at Kyung Hee University was supported by the National Research Laboratory Fund through Nano Optical Property Laboratory.

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