Spectroscopic Ellipsometry Study on Refractive Index Spectra of

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Spectroscopic Ellipsometry Study on Refractive Index Spectra of Colloidal β-FeOOH Nanorods with Their Self-Assembled Thin Films Hideatsu Maeda*,† and Yoshiko Maeda‡ † ‡

National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi, Tsukuba, Ibaraki, 305-8566 Japan The University of Tsukuba, 1-1-1 Tennodai,Tsukuba, Ibaraki, 305-8574 Japan ABSTRACT: We prepared monodisperse colloidal β-FeOOH (square prismatic crystalline) nanorods. Self-assembled dry thin films of the rods were formed on quartz substrates by evaporating drops of repulsive-rod suspensions. Using spectroscopic ellipsometry, the complex refractive index spectra of the βFeOOH nanorod films were deduced, which are further corrected on the basis of absorption spectra measured for βFeOOH nanorod suspensions, with aid of the Kramers-Kr€onig relation and an optical two-phase mixing model. Simultaneously, in this correction process, the rod volume fraction in the films and the complex mean refractive index spectra of purely the rod were extracted together. The real part of the refractive index spectra of the β-FeOOH rod, n0 c(λ), shows a maximal value of 2.365 (aver.) at 410 nm (λm) (hence a normal dispersion at λ > λm and an anomalous dispersion at λ < λm). At λ = 589.3 nm (λD for “sodium light”), the n0 c(λ) value is consistent with refractive indices predicted by Gladstone-Dale’s and Anderson’s refractivity formulas with the density of the β-FeOOH rod, 3.638 ( 0.104 g/cm3, measured in this work.

1. INTRODUCTION Mineral species are characterized by their crystal structures and chemical compositions, and thus their refractive indices, which reflect these properties, have been used as a useful measure for the identification and classification of minerals. The refractive indices of R-FeOOH (goethite) and γ-FeOOH (lepidocrocite) minerals for “sodium light” (λD = 589.3 nm) were reported,1 but that of the βFeOOH mineral (akaganeite) has not yet been determined. This is because the β-FeOOH crystals are too small to measure the refractive indices with traditional methods. Recently, ellipsometry technologies have been developed that enable the deduction of the refractive indices of thin-film materials (insulators, semiconductors, and metals) of approximately 1 to 103 nm thick by analyzing changes in the polarization state of reflected light from the films with respect to that of incident light.2 Using spectroscopic ellipsometry, wavelength λ-dependent complex refractive indices of thin film materials are derived whose real part reveals the dispersion behavior of refractive indices of the films, whereas the imaginary part, called the “extinction coefficients,” can yield information, for example, on the electronic or molecular structures of the film materials that cause the dispersion of the refractive indices. Spectroscopic ellipsometry can also determine λ-independent parameters introduced to optical models for the films, such as film thickness, thickness variation rate, surface roughness, constituent fraction, porosity, and so on.2,3 Utilizing these excellent capabilities of spectroscopic ellipsometry, we have attempted to deduce the mean refractive index spectra of purely the β-FeOOH nanorods using their thin film-like assemblies. We have been investigating the liquid-crystalline structure formation mechanism for colloidal β-FeOOH nanorods in their dense suspensions and also determined a phase diagram as a r 2011 American Chemical Society

function of rod concentration and ionic strength of rod suspensions (corresponding to an interrod interaction from attractive to repulsive).4-8 Moreover, we have recently found some fundamental motion modes of colloidal rods in their dense suspensions.9 Utilizing this spontaneous assembling nature of β-FeOOH nanorods, thin films of the rods were formed on quartz substrates.6 Using spectroscopic ellipsometry, we have deduced the complex refractive index spectra of the β-FeOOH nanorod films and purely the rods themselves in the near UV-vis range. Moreover, the comparison of the refractive index of the rod at λ = 589.3 nm (“sodium light”) with those calculated from Gladstone-Dale’s10 and Anderson’s11,12 refractivity formulas has been done.

2. EXPERIMENTAL SECTION 2.1. Synthesis of Colloidal β-FeOOH Nanorods. Following a procedure of Zocher et al.,13 β-FeOOH sols were prepared from 30.1 mmol solutions of FeCl3 at 15.5 C. Actually, the β-FeOOH nanorods appeared to grow in the solutions4,7,8 and eventually settled down on the bottom of a vessel (conical flask). Several months later, the rod sediment was put in a cellophane tube to wash repeatedly with pure deionized water for 2 to 3 weeks. The suspension pH finally increased to ∼4. The ion that determines the surface potential of the rod is proton, and thus the interrod interaction can be controlled by changing the pH value of rod suspensions. Figure 1 shows a TEM image of the β-FeOOH rods synthesized, whose dimensions are 117.1 ( 11.9 nm in length and 30.3 ( 3.2 nm in width. Their cross sections perpendicular to the long axis are square.5 Received: October 2, 2010 Revised: December 28, 2010 Published: February 10, 2011 2895

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Figure 1. TEM image of β-FeOOH nanorods synthesized, whose dimensions are 117.1 ( 11.9 nm long and 30.3 ( 3.2 nm wide.

Figure 2. Cross-sectional profiles of β-FeOOH rod films formed, whose thickness are 83-98 nm (lower) and 130-140 nm (upper). The rods are a single crystal,14,15,29 whose surfaces have various stepped structures.6 Moreover, the surfaces of β-FeOOH rod films formed by the evaporation method have a microscopic height variation, which was estimated to be 0-10 nm from the height profiles of AFM surface images of the films.6

2.2. Formation of β-FeOOH Rod Films and Measurements of Film Thickness. Thin films of the β-FeOOH nanorods were

formed by evaporating drops of nanorod suspensions with volumes of 10-50 μL on quartz and glass substrates; the pH value of the suspensions was ∼4.0, where interrod interactions are long-range electrostatic repulsions.7,8 We have found that interrod repulsions act more effectively to form ordered thin films of the β-FeOOH rods for this evaporation method, whereas interrod attractions can induce large sideby-side clusters of the rods,7,8 resulting in the formation of the films with macroscopically rough surfaces. However, the thickness of the β-FeOOH rod films formed is not uniform. In fact, formed disk-like films showed a macroscopic positional variation of light transmission. Also, a thickness variation in a β-FeOOH film was determined by single wavelength ellipsometry. Height profiles of the films were measured with a stylus profilometer (Talistep, Rank Taylor Hobson). Figure 2 shows typical profiles of two different films whose thicknesses are 83-98 (lower) and 130-140 nm (upper), respectively; considering that each rod lies orienting its long axis parallel to the surface of substrates, these film-thicknesses correspond to three- and four-rod stacking, respectively. The thickness of the films was also deduced by ellipsometry (Figure 4b). 2.3. Ellipsometry and Absorption Measurements. We used a spectroscopic ellipsometer with a beam diameter of 5 mm (M2000U, J. A. Woolam) to measure ellipsometric parameters, Ψ(λ) and Δ(λ), of the β-FeOOH rod films on quartz substrates at λ = 245-1000 nm at incidence angles of 58 and 63. Analysis software “WVASE32” is mounted on the ellipsometer3. In addition, we used a single-wavelength ellipsometer (λ = 632.8 nm, He-Ne laser) with a beam diameter of 1 mm (DHA-OLX3, Mizojiri Optical) to deduce the variations of the

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refractive index and thickness in the measured areas for each β-FeOOH rod film. Absorption spectra of β-FeOOH nanorods suspended in pure deionized water were measured at λ = 200-900 nm using a dual beam UV-vis spectrophotometer (Hitachi type 320), which were then converted to extinction coefficient spectra. 2.4. Calculations. Calculation software, “Mathematica 6” (Wolfram Research) was used to correct the refractive indices of the β-FeOOH nanorod films (deduced only by spectroscopic ellipsometry) on the basis of the absorption spectra of the rod suspensions and to extract the refractive indices of purely the β-FeOOH nanorod from the corrected ones. 2.5. Measurement of β-FeOOH Rod Density. To our knowledge, a precise measurement for the β-FeOOH crystal density has not yet been carried out. Therefore, we have determined the density, Fβ-FeOOH, with the β-FeOOH nanorods by the following traditional method: (1) The volume of β-FeOOH nanorod suspensions (Vs) was determined with pycnometers (specific gravity bottles) whose inner volume was calibrated with the weight of pure water measured by a precision electronic balance (Mettler AT 100) and its density of 0.99652 g/cm3 at 27 C. (2) The weight of the suspension (ws) in the respective pycnometers was also determined with the electronic balance. (3) In a drying oven, the pycnometers containing the suspensions were heated to 100-110 C for 3 to 4 days to evaporate completely the solvent (water) in the suspensions; then, the total weight of all rods left in the respective pycnometers (wr) was determined with the electronic balance. The volume of water in the suspension, Vw, is expressed as V w ¼ ðws -wr Þ=Fw where Fw is the density of water, and thus Fβ-FeOOH ¼ wr =ðV s - V w Þ Actually, we measured the density of β-FeOOH nanorods with their five different suspensions, whose measured Vs and wr values ranged from 12.78710 to 14.00634 mL and from 0.31515 to 0.73041 g, respectively. Resultingly, for the β-FeOOH nanorods washed repeatedly for 2 to 3 weeks (as mentioned above) Fβ-FeOOH ¼ 3:638 ( 0:104 g=cm3 A tunnel structure formed in the β-FeOOH crystal is stabilized by Cl ions.14 On the basis of the cell constants of the β-FeOOH crystal, a = 10.587 nm, b = 3.031 nm, and c = 10.515 nm,14,15 if the Cl ion sites in the crystal are all occupied by the ions, then the crystal density is calculated to be 3.852 g/cm3. If the Cl ion sites are all vacant, then the crystal density is calculated to be 3.498 g/cm3. Considering these calculated densities, together with the measured one (Fβ-FeOOH), the amount of Cl ions remaining in the nanorods washed is estimated to be 3.58 wt %, which is in good agreement with previously reported values, that is, 3.5 wt % after 8 days of washing by Jonson and Logan16 and 3.6 wt % by Post et al.14,15

3. FUNDAMENTALS OF ELLIPSOMETRY2,3 A linearly polarized incident light is generally elliptically polarized after reflection on sample surfaces. The electric field of incident light (E0) is decomposed into a ‘p’ component, E0p, and ‘s’ component, E0s, which are polarized parallel and perpendicular (senkrecht) to the plane of incidence, respectively. When the amplitude of the s and p components after reflection, normalized to their amplitude of incident light, are denoted by rs and rp, respectively, the ratio of rp to rs is written using ellipsometric parameters, Ψ and Δ, as follows r p =r s ¼ tanðΨÞ expði 3 ΔÞ ð1Þ Therefore, tan(Ψ) is the amplitude ratio of reflection and Δ is the relative phase change. Spectroscopic ellipsometry measures the 2896

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Figure 3. Optical model, which consists of air (the top phase with the refractive index of 1), the β-FeOOH rod film (the middle layer with a refractive index denoted by nf), and a thick quartz substrate (the bottom layer). E0 is the electric field of incident light, and E0S and E0p are the components of E0 perpendicular and parallel to the incidence plane of light, respectively. The coefficients rs and rp are the complex total reflection coefficients for s- and p-polarized light, respectively (see the text), which consider the interference of multiple-reflected light between the air/film and film/substrate interfaces.

parameters, Ψ and Δ, as a function of λ, from which the refractive indices of thin films on substrates are derived using appropriate optical models (e.g., Figure 3). Figure 3 shows an optical model used for estimating the refractive indices of the β-FeOOH nanorod thin films, where the top layer is air, the middle layer is the film, and the bottom layer is a thick quartz substrate. The coefficients rs and rp in eq 1 are the total reflection coefficients for s- and p-polarized light, respectively, in which the multireflections between the air/film and film/substrate interfaces are considered.

4. RESULTS AND DISCUSSION 4.1. Single-Wavelength Ellipsometry. Using a single wavelength ellipsometer (with a light source of He-Ne laser with λ = 632.8 nm and a light beam diameter of 1 mm), the parameters Ψ and Δ for the β-FeOOH rod films on a quartz substrate were measured to deduce nf0 and d of the films (because kf for the films is almost 0 at λ = 632.8 nm). Figure 4a shows a typical 0.025 step contour diagram for refractive index, nf0 , of a β-FeOOH rod film, measured at 1 mm intervals over an area of 11 8 mm2. The refractive indices of the film determined are 1.625 to 1.675 in 91% of the measured area, whereas in the remaining 4 and 5% areas, they are 1.600 to 1.625 and 1.675 to 1.700, respectively, resulting in a mean refractive index of 1.650 ( 0.022 over the entire measured area. At the same time, a 30 nm step contour diagram for thickness at the same measured area of the same film was determined as in Figure 4b, where the maximal, minimal, and mean thickness are 193.7, 55.0, and 93.2(34.9 nm, respectively. From the comparison of both contour diagrams, it is suggested that there is no explicit correlation between the variations in thickness and refractive index, indicating that they can be treated as independent parameters in the optical model for the β-FeOOH rod films formed. Moreover, d and nf0 in more flat regions of 3 3 mm2 in the measured area of the film were determined to be 64.2 ( 3.9 nm and 1.654 ( 0.0055, respectively, indicating that the mean value of n0 for the non-uniform-thickness film is almost the same as that for the uniform-thickness film. We also applied single-wavelength ellipsometry to thicker β-FeOOH rod films whose mean thicknesses are 600-850 nm, and the resulted mean

Figure 4. Variations of refractive index and thickness in a measured area (11 8 mm2) of a β-FeOOH nanorod film: (a) a 0.025 step contour diagram for refractive index and (b) a 30 nm step contour diagram for thickness. Both diagrams were determined using a single-wavelength ellipsometer (λ = 632.8 nm, He-Ne laser).

refractive indices mostly lie between 1.62 and 1.66 (e.g., 1.629 for 829.0 nm thick and 1.658 for 672.5 nm). 4.2. Spectroscopic Ellipsometry. Using a spectroscopic ellipsometer, the Ψ(λ) and Δ(λ) pairs for areas in the β-FeOOH rod films on a quartz substrate were measured at λ = 245-1000 nm. Using “WVASE32”, the Ψ(λ) and Δ(λ) pair of the optical model for the films (Figure 3) was generated. By minimizing a weighted mean square error (MSE) of the difference between the measured and model-generated ψ(λ) and Δ(λ) pairs, the wavelength-dependent complex refractive indices, nf(λ) = n0 f(λ) þ i 3 kf(λ), and thickness of the films can be determined.2,3 However, the light beam diameter of the spectroscopic ellipsometer used is ∼10 mm at incidence angles around 58, and thus the thickness nonuniformity in the measured areas of the films cannot be neglected (e.g., as seen in the former section). Besides, because the films consist of separate rods with dimensional distributions, their surfaces are naturally not smooth. By AFM observations,6 it was observed that the surface of βFeOOH rod films formed has a microscopic height variation; the variation width was estimated to be 0-10 nm from the height profiles of AFM surface images of the films. Therefore, in addition to nf(λ) and d, for the films formed, two λ-independent 2897


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Figure 5. Two types of ψ(λ) and Δ(λ) pairs for a β-FeOOH film formed on a quartz substrate: one is measured at λ = 245-1000 nm (black solid line) and the other (red O) is generated for a modified optical model (Figure 3) with fitting parameters, d = 294.6 nm, σ = 53%, and η = 5.0 nm for the film, using a “point-by-point fit” method at the same λ range. Both Ψ(λ) and Δ(λ) pairs almost overlap visually.

parameters, that is, a film thickness variation rate (σ) and a thickness of the thin surface layer for expressing the surface roughness (η), were introduced to modify the original optical model in Figure 3. We expressed the surface roughness of the β-FeOOH rod films by adding a thin layer with a thickness of η on the top surface of the films, where 50% air phase and 50% β-FeOOH rod film phase are mixed. Actually, we determined these unknown parameters by minimizing the MSE of the differences between the measured Ψ(λ) and Δ(λ) pair and the model-generated one with a Cauchy equation n0 f(λ) = A þ B/λ2 þ C/λ4 at the λ range of kf(λ) = 0 or λ > 630 nm. The resulted MSE showed a small value for the film thicknesses around 300 nm, for example, 1.822 (for 294.6 nm thick), 2.184 (for 295.2 nm), and 2.644 (for 356.4 nm). For MSE = 1.822, the parameters determined are d = 294.6 nm, σ = 53.0%, and η = 5.00 nm, and the generated Ψ(λ) and Δ(λ) pair excellently fit the measured one at the λ range of kf(λ) = 0. The refractive indices nf(λ) in the λ range of 245-1000 nm (not only the λ range of kf(λ) = 0) were extracted by a “point-bypoint fit (ppf)” method built in “WVASE32” (where n0 f(λ) and kf(λ) were calculated for each successive wavelength from 1000 to 245 nm). Incidentally, for the films of 200-350 nm thick, the mean value of their n0 f(λ) extracted by the ppf method was 1.657 ( 0.010 at λ = 632.8 nm, being in good agreement with the value, 1.654 ( 0.005, obtained by single-wavelength ellipsometry in the former section, indicating that our optical model for the β-FeOOH rod films is adequate and resulting nf(λ) is reliable. Figure 5 shows the Ψ(λ) and Δ(λ) pair measured at the λ range of 245-1000 nm (black solid line) for a β-FeOOH rod film of 294.6 nm thick and the Ψ(λ) and Δ(λ) pair (red O) generated for its modified model (with σ = 53.0% and η = 5.00 nm) using the ppf method. The resulting n0 f(λ) and kf(λ), called hereafter “original” ppf n0 f(λ) and kf(λ), of the film are shown in Figure 6 (solid line) (together with its corrected ppf n0 f(λ) and kf(λ) (O), as briefly described just below, but in detail in the subsequent sections). To correct the original ppf n0 f(λ) and kf(λ), we used the following method: (1) First, measure the absorption coefficient Rc(λ) (suffix “c” means crystal) of the β-FeOOH nanorods suspended in pure

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Figure 6. Two types of refractive indices, nf0 (λ), and extinction coefficients, kf(λ), for a β-FeOOH film: (a) one is derived from the measured and model-generated ψ(λ) and Δ(λ) pairs in Figure 5 using a “point-bypoint fit” method, here called “original ppf n0 f(λ) and kf(λ)” (—), and (b) the other is corrected using an absorption spectrum of a β-FeOOH rod suspension, called “corrected ppf n0 f(λ) and kf(λ)” (O). Both types of nf0 (λ) and kf(λ) pairs almost overlap at λ > ∼630 nm.

deionized water (the top in Figure 7a) and convert it using eq 3 into the extinction coefficient kc(λ) with one unknown parameter (denoted by “b” in eq 6). (2) Second, further convert kc(λ) to n0 c(λ) with two unknown parameters (“a” and “b” in eq 7) using the Kramers-Kr€onig relation (eq 4a). (3) Finally, determine these two unknown parameters in n0 c(λ) using the original ppf n0 f(λ) at the λ range of kf (λ) = 0 and an optical two-phase mixing model (eq 2) described just below. 4.3. Optical Two-Phase Mixing Model. The β-FeOOH nanorod films can be expressed by an optical two-phase composite model, in which one is the β-FeOOH phase, and the other is the air phase (between the rods). Usually, for two-phase composite systems where one phase approximated by spheres with a refractive index is randomly dispersed in another matrix phase with a different refractive index, the formulas for estimating the effective refractive index of the systems were given by Lorentz17 and Lorenz,18 Bruggeman,19 and Maxwell-Garnet,20 using the “effective medium approximation (EMA)”. These EMA formulas are a function of only the refractive index and volume fraction of the constituents. Therefore, the EMA formulas can be used to estimate the index and parameter of the constituents from the optical data of the systems measured by spectroscopic ellipsometry, often together with other spectroscopic techniques (transmission or absorption, reflection, etc.) to reduce the ambiguity of fitting analyses. For examples, with silica-based xerogel films,21,22 Si-nanorod films,23 and self-assembled silica nanosphere films,24 the refractive index of the SiO2 or Si phase and the porosity of the films were deduced by spectroscopic ellipsometry with aid of the EMA formulas. The β-FeOOH rods take smectic structures in their dense suspensions.4-8 The rods have a square prismatic shape, and the aspect ratio of the rods used is ∼3.6. The β-FeOOH rod films consist of the rods aligning parallel to one another (also to the substrate plane) and air gaps between the parallel rods.6 Effective (averaged) refractive index, nav, for the β-FeOOH rod films can be estimated using the EMA formula for a two-phase system 2898


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whereas k(λ) is related to the real part of n(λ) by the KramersKr€onig relation (eq 4a). In this work, β-FeOOH nanorod suspensions of pure deionized water were used to obtain the rod absorption coefficient Rc(λ). In Figure 7a, the topmost curve is an absorption spectrum measured at λ = 200-900 nm for a β-FeOOH rod suspension and is quite similar to those for colloidal suspensions of goethite (R-FeOOH) and lepidocrocite (γ-FeOOH) minerals determined by Sherman and Waite.26 These spectra show common five absorption peaks (apparently four peaks and one shoulder, as seen from Figure 7a), which are assigned as follows.26 Bands near 290-310, 360-380, and 430 nm are assigned to ligand field transitions of Fe3þ. A low intensity band near 485-550 nm is assigned to “double excitation” processes involving the simultaneous excitation of magnetically coupled adjacent Fe3þ cations. Bands at wavelengths below 270 nm are assigned to ligand to metal (O2- f Fe3þ) charge transfer (LMCT) transitions. Absorption spectra of materials are presented by plotting the optical density (OD) versus λ, and the OD is defined as the following formula (the law of Lambert-Beer) OD ¼ logðI=I 0 Þ ¼ A þ S

Figure 7. (a) Absorption spectrum for a colloidal β-FeOOH nanorod suspension measured at λ = 200-900 nm (the topmost, filled gray square) and corrected ones (four solid lines and one red square), which were obtained by subtracting five different Rayleigh scattering components, respectively, from the topmost curve. (b) Reduced extinction coefficient spectrum, kc*(λ) ([), converted from one of the corrected absorption spectra (filled red square in part a), five decomposed Gaussians (O), and the total of the five Gaussians (red solid line).

where cubes disperse in a matrix25 nav 2 ¼ nc 2 =fð1- nc 2 Þv-1=3 þ nc 2 v-2=3 gþ ð1- v2=3 Þ

ð2Þ

where nc is the λ-dependent complex refractive index of purely the β-FeOOH phase whose volume fraction is denoted by a parameter, v. On the basis of the observed structures of the β-FeOOH rod films, we adopted the EMA formula of eq 2 because this formula can be used for rods of aspect ratios >1 within the framework of the EMA. The effective refractive index, nav, is a λ-dependent complex value and assumed to be equivalent to original ppf nf(λ) at the λ range of kf(λ) = 0. 4.4. Absorption of and Scattering from β-FeOOH Nanorods Suspended in Pure Water. The complex refractive indices, n(λ), of materials can be more correctly deduced together using spectroscopic ellipsometry and absorption spectroscopy because the absorption coefficient R(λ) is directly related to the imaginary part of n(λ), that is, the extinction coefficient, k(λ), by the following equation kðλÞ ¼ RðλÞλ=4π

ð3Þ

where I0 and I are the intensity of incident and transmitted light, respectively, A is the absorption component (coefficient), and S is the scattering component (coefficient). If suspensions of particles are dilute and the particle dimensions are smaller than the wavelength of incident light, then the scattering component has characteristic 1/λ4 dependence (the Rayleigh scattering), which can be seen as a sharp rise in the near-UV absorption spectra, for example, as seen in Figure 7a (the topmost curve). Figure 7a shows an absorption spectrum measured for a colloidal β-FeOOH nanorod suspension (the topmost curve) and absorption spectra corrected by subtracting the Rayleigh scattering component, S = q 3 λ-4, for five different trial q values, from the topmost one. Sherman and Waite26 suggested that d-d transitions of Fe3þ in the O2- octahedral unit are allowed, as the spin and Laporte selection rules are relaxed by the magnetic coupling of adjacent Fe3þ, and thus the absorptions due to the d-d transitions (near 290-310, 360-380, and 430 nm) are intensified to be comparable to the absorptions due to LMCT transitions appearing at λ < 270 nm. In fact, Babin et al.27 measured absorption spectra of colloidal goethite suspensions in the near-UV and VIS regions with a spectrophotometer mounting an integrating sphere and verified the idea of Sherman and Waite. Consulting with the absorption spectra of Babin et al., the third corrected spectrum (filled red square) in Figure 7a was adopted as the net absorption spectrum for the β-FeOOH nanorod, where the absorption intensity due to the LMCT transition at ∼250 nm is comparable to those due to the d-d transitions. However, four corrected curves in Figure 7a (except for the bottom curve) are almost the same at λ > 350 nm. The absorption coefficient, Rc(λ), is linked to the absorbance, R*c(λ), by Rc(λ) = b 3 Rc*(λ). The factor b is normally a constant determined by the volume (or weight) fraction of the rods in the suspension and the optical path length of a cell containing the suspension, but it is regarded here as an unknown adjustable parameter determined by a fitting method described in the next section. Also, the extinction coefficient, kc(λ), is linked to the reduced extinction coefficient, kc*(λ), by kc(λ) = b 3 kc*(λ), and the curve of kc*(λ) for a β-FeOOH nanorod suspension is shown in Figure 7b ([). 2899


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€ nig Relations. The real part of the com4.5. Kramers-Kro plex refractive index, n0 (λ), and the imaginary part (extinction coefficient), k(λ), are mutually connected by the following relations, called the “Kramers-Kr€onig relations”.2,3 Z ¥ n0 ðλÞ ¼ 1 þ ð2=πÞP kðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0 ð4aÞ 0

Z k0 ðλÞ ¼ -ð2=πÞP

¥

kðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0

ð4bÞ

0

To calculate the Cauchy principal integral in eq 4a, for k(λ0 ), we substitute a continuous function of λ that fits well to experimentally obtained k(λ) or R(λ)λ/4π. This is because the measuring wavelength intervals of usual spectrophotometers for R(λ) are too large to numerically perform the Cauchy principal integral with high precision. By trial and error, we found that the experimental kc*(λ) spectrum having four peaks and one shoulder ([ in Figure 7b) can be approximated by a composite function of following five different Gaussians (rather than Lorentzians) as follows 5 X kc ðλÞ ¼ ai fexpð-bi ð1=λ-1=ci Þ2 Þ i ¼1 2

-expð-bi ð1=λþ1=ci Þ Þg

ð5Þ

where ai, bi, and ci denote the height, width, and peak wavelength of the ith Gaussian, respectively, which were determined so that the composite kc*(λ) function (eq 5) fits well with the experimental kc*(λ) spectrum. (The second term in the right-hand side of eq 5 is almost 0.) In fact, Figure 7b also shows the five decomposed Gaussians and their total (red solid line) almost coinciding with the experimental kc*(λ) spectrum. In eq 4a, the integration range of the Cauchy principal integral is from 0 to ¥, but the measuring λ range for kc*(λ) is limited to 200-900 nm. To estimate n0 (λ) in the near UV-vis range, we divided the Cauchy principal integral in eq 4a into three λ ranges as follows Z 200 kc ðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0 n0 ðλÞ ¼ 1 þ b 3 ð2=πÞP Z þb 3 ð2=πÞP

0 900

200 ¥

Z þb 3 ð2=πÞP

kc ðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0

kc ðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0

ð6Þ

900

where k(λ) is substituted with b 3 kc*(λ). In eq 4a, if k(λ0 ) is a single Gaussian function, then its (λ-dependent) integration value sharply decreases first from its maximum (where λ = λm) with increasing λ in the normal dispersion region but turns to a gradual decrease approaching a constant when λ increases by ∼150 nm from λm, whereas in the anomalous dispersion region, the (λ-dependent) integrated value tends to almost 0 when λ decreases by ∼150 nm from the peak wavelength of the single Gaussian (slightly lower than λm). This behavior indicates that absorption bands existing at λ > ∼950 nm scarcely affect n0 (λ) in the VIS λ range (380-780 nm). In fact, for the β-FeOOH nanorods, the experimental kc*(λ) spectrum (200-900 nm) is almost 0 at λ > 600 nm, and no tails of absorption bands in the infrared λ range are observed (Figure 7a), indicating that the contribution of the third integration value in eq 6 to n0 c(λ) must be negligible at λ < ca. 800 nm. If absorption bands exist at λ < 200 nm, then the contribution of their first integration value in eq 6 to

Figure 8. Refractive index n0 c(λ) and extinction coefficient kc(λ) of purely the β-FeOOH nanorod, which are extracted using the “corrected ppf nf(λ)” in Figure 6 and a cubic two-phase mixing model (eq 2).

n0 c(λ) must be almost λ-independent at λ > ∼350 nm, together considering a larger slope of nc0 (λ) in the normal dispersion region (Figure 8). Consequently, n0 c(λ) of the β-FeOOH nanorod can be approximated by the following equation at least in the VIS λ range Z 900 0 kc ðλ0 Þ=λ0 ð1-ðλ0 =λÞ2 Þ dλ0 ð7Þ n cðλÞ ¼ aþ b 3 ð2=πÞP 200

The parameters a and b in this equation were determined so that n0 av(λ) of eq 2 (which includes n0 c(λ)) fits well the original ppf nf0 (λ) (solid line in Figure 6) at the λ range of original ppf kf(λ) = 0 by a fitting method as described in the next section. In this work, the parameter b (which is introduced as Rc(λ) = b 3 Rc*(λ) or kc(λ) = b 3 kc* (λ)) is regarded as an adjustable unknown parameter. This is because the absorbance of the β-FeOOH nanorods is expected to depend on their environment. Actually, in our work, the environment of the β-FeOOH nanorods is water for the absorption measurement, whereas it is air for the measurement of the refractive indices. Babin et al. found that absorption strength of colloidal iron hydroxide particles suspended in water greatly depends on suspension-pH and -ionic concentration, but their λ-dependent profile is scarcely altered.27 Therefore, on the basis of their result, we assumed that the parameter b is independent of λ as the first approximation. 4.6. Volume Fraction of β-FeOOH Nanorods in Films. The unknown parameters, a, b (in eq 7), and v (in eq 2) can be determined using the following nonweighted MSE N X ðn0av ðλi Þ-nn0f ðλi ÞÞ2 =N ð8Þ MSE ¼ i ¼1

where n0 av(λ) is obtained by substituting nc in eq 2 with n0 c(λ) þ i 3 kc(λ) and nnf0 (λ) is a Cauchy dispersion equation of λ that fits well the original ppf n0 f(λ) in the λ range of original ppf kf(λ) = 0. This nn0 f(λ) function was used to interpolate the values of original ppf n0 f(λ) at λ values with the measuring intervals of 1.58 nm (for the ellipsometer used) into those at λi values with the intervals of 1 nm. Actually, the parameters, a, b, and v, were determined so that n0 av(λ) fits well with nn0 f(λ) in the λ range of original ppf kf(λ) = 0 in Figure 6, where the MSE of eq 8 becomes minimal; as a result, the v values determined were 0.697 to 0.708. 2900


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For instance, the resulting n0 av(λ) and kav(λ) (i.e., corrected ppf n0 f(λ) and kf(λ)) and n0 c(λ) and kc(λ) for v = 0.703 are shown in Figures 6 (O) and 8, respectively. In Figure 6, the original ppf n0 f(λ) almost agrees with n0 av(λ) at λ > 580 nm, whereas the peak of λ = 525 nm in the original ppf kf(λ) shifts to 490 nm in kav(λ). In addition, at λ < 400 nm, the peak positions in the original ppf kf(λ) approximately coincide with those in kav(λ). However, the profiles of the original ppf n0 f(λ) and kf(λ) are obviously different from those of n0 av(λ) and kav(λ), particularly at λ < 300 nm. This must be due to the effect of light scattering from the film that consists of the β-FeOOH nanorods and air gaps between them. In Figure 8, n0 c(λ) shows a maximal value of 2.365 at λ = 410 nm (λm) and hence the normal dispersion at λ > λm and the anomalous dispersion at λ < λm. This dispersion is caused by electronic transitions of Fe3þ in O2--forming octahedral units in the β-FeOOH crystal. Moreover, n0 c(λ) is 2.138 at λ = 589.3 nm (for “sodium light”) and 2.111 at λ = 632.8 nm (for He-Ne laser). At ∼510 nm, n0 c(λ) shows a small peak with a height of 2.220, which results from the simultaneous excitation of adjacent Fe3þ cations magnetically coupled.26 The difference between n0 c(410 nm) and n0 c(900 nm) is ∼0.317. Moreover, the n0 c(λ) spectra for v = 0.697 and 0.708 were also obtained using their adjusted a and b values; the resulting value of n0 c(589.3 nm) is 2.159 for v = 0.697 and 2.125 for v = 0.708.

5. CONCLUSIONS The β-FeOOH crystals are too small to measure the refractive indices with traditional methods. Therefore, we formed thin films of the β-FeOOH nanorods on quartz substrates utilizing spontaneous self-assembling nature of the nanorods. Applying spectroscopic ellipsometry to the nanorod films, we successfully have derived the mean refractive index and extinction coefficient spectra of purely the nanorod with the help of absorption spectra of the β-FeOOH nanorod suspensions, a two-phase mixing model, and the Kramers-Kr€onig relation. The results of our work also suggest that using spectroscopic ellipsometry together with structural observation techniques (AFM, electron microscopy, etc.) the refractive index spectra of nanoparticles can be deduced with their thin film-like assemblies, even if the film thickness are nonuniform. ’ APPENDIX Refractive Indices of β-FeOOH Crystal by GladstoneDale’s and Anderson’s Formulae. To confirm the validity

of our analysis method described above, it is of significance to compare n0 c(589.3 nm) for the β-FeOOH nanorod deduced by spectroscopic ellipsometry with those calculated from Gladstone-Dale’s (G-D) empirical formula (eq A1)10 and expected by Anderson’s theoretical formula (eq A2).11,12 For the G-D formula, the chemical compositions of minerals are expressed using the terms of basic oxides (common in many minerals), whose weight fractions are consistently determined from the weight fraction of the elements comprising the minerals. The G-D formula is written as follows. X fi 3 ki ¼ K ðA1Þ ðn-1Þ=F ¼ i

where n and F are the refractive index and density of minerals, respectively, and fi and ki are the weight fraction and specific refractivity (called the G-D constant) of the ith component

basic oxide, respectively. The specific refractivities of the basic oxides are determined by a statistical analysis (least-squares method) using the n values measured (with “sodium light”) and F values calculated from the crystal structure for many between minerals and are tabulated;28 in fact, the agreement P the value of (n - 1)/F and the K value calculated from fi 3 ki is excellent for a variety of minerals.28 Using the ideal point-dipole approximation based on the electromagnetic theory, Lorentz and Lorenz linked the refractive index and density of materials to their polarizability.17,18 Furthermore, Anderson et al. improved the Lorentz-Lorenz formula by introducing an “electronic overlap parameter” to be applicable to compounds with intermediate ionic-covalent bonding between atoms as follows11,12 ðn2 - 1Þ=ð4π þ ξðn2 - 1 ÞÞ 3 F ¼ R=M ¼ μ

ðA2Þ

where R and M are the polarizability and molecular weight of a compound, respectively, and ξ = ξL - ξe, where ξL and ξe are the Lorentz local field factor (= 4π/3 for a spherically symmetrical field around a molecule) and electronic overlap field factor, respectively. The overlap-field arises from electrons sharing the same molecular orbital state, and hence if ξe = 0, then eq A2 is reduced to the Lorentz and Lorenz formula. The ξ value ranges from 0 (covalent bonding) to 4π/3 (ionic bonding). Anderson and Schreiber applied this formula to a variety of oxide and silicate minerals (various combinations of SiO2, MgO, Al2O3, Na2O, and K2O) and found that the G-D relation is valid in the range of 2.2 < F < 3.6 and 1.45 < n < 1.74 for the minerals with a mean atomic weight of ∼21, which is common for mantle minerals.11 They analyzed their data with a ξ value of 0.62 or 2.0. In the FeOOH crystal polymorphs, the Fe3þ ions are surrounded by three each of O2- and OH- anions, giving a FeO6 octahedral unit. The FeOOH polymorphs have topologically different types of frameworks of corner- or edge-linked octahedral units.29 The refractive indices of R- and γ-FeOOH crystals measured for “sodium light” are reported;1 for the R-FeOOH crystal, nR = 2.260 to 2.275, nβ = 2.393 to 2.409, and nγ = 2.398 to 2.515 (aver. n = 2.375), and for the γ-FeOOH crystal, nβ = 1.940, nβ = 2.200, and nγ = 2.510 (aver. n = 2.217). In Figure A1b, the pairs of n (the mean value measured) and F (calculated from the crystal structure) for the R- and γ-FeOOH crystals,1 the pair of n and F (both determined in this work) for the β-FeOOH crystal nanorod, and the pair of n (predicted by the G-D formula) and F (calculated from the crystal structure) for the ideal β-FeOOH crystal (which has no Cl ions in its structure) are plotted, together with G-D and Anderson curves. The constants K (in eq A1) for R- and γ-FeOOH crystals are assumed to be equal (because of the same chemical composition) and calculated to be 0.3175 cm3/g using k1 = 0.315 cm3/g (Mandarino28) and f1 = 89.86 wt % for basic component Fe2O3, and k2 = 0.340 cm3/g (Mandarino) and f2 = 10.14 wt % for basic component H2O. Therefore, the mean refractive indices of the R- and γ-FeOOH crystals were predicted to be 2.327 and 2.257, respectively, from the G-D formula using the calculated densities of 4.18 (R-FeOOH) and 3.96 g/cm3 (γ-FeOOH) from their crystal structures. These predicted values are in good agreement with the measured mean values of 2.375 (R-FeOOH) and 2.217 (γ-FeOOH) with an error of 2.1 and 1.8%, respectively. The β-FeOOH crystal includes a small amount of Cl ions, which are necessary to stabilize a tunnel structure formed in the crystal.14,15 As described in the Experimental Section, the 2901


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Figure A1. (a) G-D curve (eq A1) with K = 0.3175 cm3/g and an Anderson curve (eq A2) with R/M = 0.0591 cm3/g and (ξL - ξe) = 1.20. The Anderson curve fits well the G-D curve in the density range of 3.0 to 4.8 g/cm3 at the deviation rates e0.21%. (b) The pairs of refractive index (measured for “sodium light”) and density (calculated from the crystal structure) for the R- and γ-FeOOH crystals, the pair for the β-FeOOH crystal (determined in this work), and the pair for the “ideal” β-FeOOH crystal (β0) (predicted by the G-D formula) together with the G-D and Anderson curves in the density range of 3.0 to 4.5 g/cm3 in part a.

amount of Cl ions in the β-FeOOH crystalline nanorods synthesized is estimated to be 3.58 wt % from the crystal structure and the measured density of the nanorods. Therefore, considering the Cl ions as the third basic component in eq A1, the K value of the β-FeOOH crystal is calculated using k3 = 0.318 cm3/g (Mandarino) to be 0.31755 cm3/g, which is almost the same as that of the R- and γ-FeOOH crystals, 0.31754 cm3/g. Consequently, from the G-D formula, the mean refractive index of the β-FeOOH crystal is predicted to be 2.155, and thus the deviation rate of deduced n0 c(589.3 nm) from 2.155 is 0.2 to 1.4%. In addition, for the “ideal” β-FeOOH crystal, β0 in Figure A1b (defined here as the β-FeOOH crystal with no Cl ions, but it actually does not exist), the refractive index at λ = 589.3 nm is predicted using its density, 3.498 g/cm3 (calculated from the crystal structure), to be 2.111 from the G-D formula. The Anderson curve has three characteristic density (F) regions; it behaves like (1 þ μ 3 F)1/2 in a low F region and exhibits a

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divergence nature like (μ 3 F/(1 - μ 3 ξ 3 F))1/2 in a high F region, whereas in an intermediate F region, it is very close to a linear curve in a range of the ξ and μ pair values, where Anderson’s general refractivity formula can be approximated by the G-D empirical formula. (See Figure A1a). In fact, with 191 samples of isochemical series comprising several densified silicate glasses (including SiO2, TiO2-SiO2, Na2O-SiO2, etc.), Arndt and Hummel showed linear relations between n and F in a density region of approximately 2.2 to 2.8 g/cm3, where the ξ values (varying between 0.5 and 2.0, depending on the series) converged at 1.3, and the μ value is 0.0365 cm3/g.30 Marler also found a linear relation between n and F in a density range of 1.76 to 2.92 g/cm3 with 13 SiO2 polymorphs, where the ξ and μ values are 1.2 and 0.0369 cm3/g, respectively.31 However, for the FeOOH polymorphs, only three refractive indices, that is, for the R, γ, and β crystals, are given for now, and thus we estimate an allowable range for ξ and μ values that the deviation rates of the measured three refractive indices from the Anderson curve are less than ∼2.0%. (This value is based on the deviation rates of these three indices from the G-D curve, mentioned above.) As a result, the estimated range of the ξ value is approximately 1.0 to 2.0, and the μ values (pairing with the respective ξ values) range from 0.0618 (for ξ = 1.0) to 0.0494 (for ξ = 2.0) cm3/g, leading to an allowable refractive index range of 2.154 to 2.126 (at λ = 589.3 nm). In particular, as shown in Figure A1a, for ξ = 1.2 and μ = 0.0591 cm3/g, the Anderson curve best fits the G-D curve with K = 0.3175 cm3/g in the density range of 3.0 to 4.8 g/cm3 at the deviation rates e0.21%. It is of interest that the ξ value for the FeOOH polymorphs is very close to that for the SiO2 polymorphs. According to Pauling’s electronegativity estimates, the Fe-O bond is about half ionic and half covalent like the Si-O bond. In brief, the refractive index of the β-FeOOH crystal predicted at λ = 589.3 nm by the G-D formula is 2.155, and allowable ones expected by the Anderson formula are 2.126 to 2.154. These refractive indices are in good agreement with n0 c(589.3 nm) = 2.125 to 2.159 (aver. 2.138) deduced using spectroscopic ellipsometry in this work.

’ REFERENCES (1) Lide, D. R.; Haynes, W. M. CRC Handbook of Chemistry and Physics, 90th ed.; CRC Press: Boca Raton, FL, 2009-2010 (2) Fujiwara, H. Spectroscopic Ellipsometry: Principles and Applications; John Wiley and Sons: Hoboken, NJ, 2007. (3) Woolam, J. A. Guide to Using WVASE32; J. A. Woolam: Lincoln, NE, 2002. (4) Maeda, Y.; Hachisu, S. Colloids Surf. 1983, 6, 1; 1983, 7, 353. (5) Maeda, H.; Maeda, Y. Langmuir 1996, 12, 1446. (6) Maeda, H.; Maeda, Y. Nano Lett. 2002, 2, 1073. (7) Maeda, H.; Maeda, Y. Phys. Rev. Lett. 2003, 90, 18303. (8) Maeda, H.; Maeda, Y. J. Chem. Phys. 2004, 121, 12655. (9) Maeda, H.; Maeda, Y. Nano Lett. 2007, 7, 3329. (10) Gladstone, J. H.; Dale, T. P. Philos. Trans. R. Soc. London 1864, 153, 317. (11) Anderson, O. L.; Schreiber, E. J. Geophys. Res. 1965, 70, 1463. (12) Anderson, O. L. Fortschr. Mineral. 1975, 52, 611. (13) (a) Zocher, H. Z. Anorg. Allg. Chem. 1925, 147, 91. (b) Zocher, H.; Heller, W. Z. Anorg. Allg. Chem. 1930, 186, 75. (14) Post, J. E.; Buchwald, V. F. Am. Mineral. 1991, 76, 272. (15) Post, J. E.; Heaney, P. J.; Dreele, R. B.; Hanson, J. C. Am. Mineral. 2003, 88, 782. (16) Jonson, J. H.; Logan, N. E. J. Chem. Soc., Dalton Trans. 1997, 13. (17) Lorentz, H. A. Wiedem. Ann. 1880, 9, 641. (18) Lorenz, L. Wiedem. Ann. 1881, 11, 70. (19) Bruggeman, D. A. G. Ann. Phys. (Leipzig, Ger.) 1935, 24, 636. 2902


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