Determination of texture by infrared spectroscopy in titanium oxide–anatase thin films

Determination of texture by infrared spectroscopy in titanium oxide–anatase thin films Carlos Pecharroma ´na) Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain Francisco Gracı ´a, Juan P. Holgado, Manuel Ocan ˜a, and Agustı ´n R. Gonza ´lez-Elipe Instituto de Ciencia de Materiales de Sevilla (CSIC-University Sevilla), Avda Ame ´rico Vespucio s/n, 41092 Sevilla, Spain J. Bassas Serveis Cientificote `cnics de la U.B., Sole ´i Sabarı ´s, 1-3, 08028 Barcelona, Spain J. Santiso and A. Figueras Institut de Cie `ncia de Materials de Barcelona-CSIC, 08193 Bellaterra, Spain 共Received 29 May 2002; accepted 21 January 2003兲 A theoretical model to determine the effective dielectric tensor of heterogeneous materials composed by anisotropic microcrystallites has been introduced to explain the infrared spectral features of textured thin films of uniaxial materials as the function of a textural parameter. This theoretical treatment is able to satisfactorily reproduce the experimental absorbance spectra of TiO2–anatase thin films chosen as a model system. Comparison of texture data obtained from infrared spectroscopy and x-ray diffraction are in good agreement which support the validity of the proposed model. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1560858兴 I. INTRODUCTION Infrared absorption/reflection infrared 共IR兲spectroscopy on thin films is commonly used as an analytic technique to identify the nature of the deposited materials. When done at normal incidence, transverse optical phonons of isotropic compounds can be detected as absorption maxima, while at grazing angles and perpendicular polarization these modes vanish substituted by longitudinal optical phonons. This effect, firstly reported by Berreman,1has been widely used to experimentally determine the longitudinal frequencies of several cubic substances.2,3 However, in the case of anisotropic samples, this effect is only verified under very restrictive conditions, which depend on the orientation of the optical axes of the layer. In fact, in anisotropic layers pure longitudinal or transverse phonons4only propagates in very specific directions, related with the optical axes of the crystal. Most thin film growth techniques produce polycrystalline films. The physical properties of these layers can be considered isotropic 共at a length scale larger than the crystallite size兲even or anisotropic materials if the crystallites are randomly distributed and oriented. The optical properties of highly diluted composites of these materials can be calculated by an effective dielectric constant approximation5and the original Berreman’s formulation1can still be used. However, this approximation fails in anisotropic substances when the preparation procedure introduces a preferential texture, i.e., when the film depicts a preferential orientation of some of their crystal planes. In these cases, thin films behave as anisotropic media. Preferential texturing of thin films is easily identified by x-ray diffraction 共XRD兲in a Bragg–Brentano configuration because the diffraction peaks of the planes preferentially oriented according to the thin film surface present an enhanced intensity. There is some previous publications which tried to link the effective6and average7,8 dielectric constant with textural parameters, but there is not a systematic treatment of this subject. In a previous publication, we used a qualitative approach by Fourier transform IR spectroscopy 共FT-IR兲to deduce preferential orientation phenomena in Fe2O3pressed pellets9and thin films.10 In the present article we introduce a theoretical treatment of FT-IR spectra of thin films that enables us to gain information about the preferential orientation of crystal planes. For a sake of simplicity we have restrict the range of application of this model to uniaxial compounds, i.e., those that crystallize into the tetragonal, rhombohedral, and hexagonal systems, but the extension to biaxial ones is straightforward. To verify the possibilities of the model, characterization experiments by FT-IR, with polarized and unpolarized light, and XRD of TiO2anatase thin films have been carried out. The good concordance of the results by the two techniques, indicating the preferential orientation of a given type of crystal planes, has confirmed the predictions of the model and open the possibility of systematically using FT-IR for proving preferential orientation phenomena in thin films. II. EXPERIMENT Very compact and dense TiO2thin films have been prepared by ion beam induced chemical vapor deposition. A thorough description of this procedure can be found in a previous publication.11 Briefly, it consists of inducing the decomposition of a volatile precursor of the metal 共in this a兲Author to whom correspondence should be addressed; electronic mail: [email protected] JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 8 15 APRIL 2003 46340021-8979/2003/93(8)/4634/12/$20.00 © 2003 American Institute of Physics Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp case Ti兲with a beam of accelerated O2 ⫹ions. Here, Ti isopropoxide has been used as volatile precursor of Ti and the beam energy and current at the substrate position were 400 eV and 100 ␮ Acm ⫺2, respectively. They were grown at room temperature on a flat Si 共100兲substrate. The film thickness was 450 nm. After preparation, the films were amorphous but they crystallized into the anatase structure of TiO2 after annealing in air at T⭓300°C. Samples annealed at 300 and 600°C were studied by XRD and FT-IR 共hereafter these samples will be called TiO2-300 and TiO2-600). X-ray powder diffraction patterns were obtained in a Bragg–Brentano ␪ /2 ␪ diffractometer, Siemens D500; a divergence slit of 1° was used, resulting on an incident beam of 3.8 mm at the sample position. The CuK ␣ radiation was used as excitation source at a working power of 40 kV⫻30 mA.FT-IR spectra with unpolarized light were recorded from 200 to 1000 cm⫺1in the transmission mode in a Nicolet 510 Fourier transform spectrometer. FT-IR spectra with polarized light were recorded 400–1000 cm⫺1in a Nicolet 20 SXC by using a wire KBr polarizer. The angle between the beam and the normal to the thin film surface was changed for different measurements. III. THEORETICAL PROCEDURE A. Effective medium approximation for dielectric composites In contrast to isotropic polycrystalline materials, the optical properties of dense anisotropic solids can be substantially different from those of single crystals, as a consequence of the decisive role of the orientation of the crystallites in these materials. Thus, a sort of average of the optical properties of the single crystallites along different directions of the dielectric tensor should be expected in macroscopic samples. In order to properly calculate the properties of these materials, an effective medium theory has been introduced, based on an effective medium formalism developed by the authors in some previous publications.9,12 This approach, which can be considered as a generalization of several others models previously presented,5,13,14 was formulated for isotropic and anisotropic particles randomly distributed into a heterogeneous material, in such a way that the whole composite presents an isotropic macroscopic behavior. In this context, it was assumed for the isotropic case that the average electric field in the vicinity of the particles was constant, homogeneous and identical to the average electric field 具 E 典 E0,共effective medium theory兲. In addition, it is supposed that particle shape is ellipsoidal, so that the electric field inside this particles is also constant and can be easily calculated. However, when crystallites present textured distributions, the composite may be anisotropic and in this case the electric field inside of a ellipsoidal article is no longer isotropic nor homogeneous even if the external field is assumed to be homogeneous. However, a detailed calculation shows 共Appendix A兲that although the heterogeneous medium can display an anisotropic behavior, each component of the macroscopic effective dielectric tensor along its optical axes can be reasonably approximated by an effective medium approximation. Thus, according to Refs. 12 and 9, and the conclusions obtained in Appendix A, the general equation which determines the effective dielectric constant of a heterogeneous composite formed by two anisotropic phases 共labeled by subindex ‘‘p’’ and ‘‘m’’兲along the direction l, is given by: 具 ⑀ 典 l⫽ 共1⫺f兲兺 k⫽1 3cmkl ⑀ mk 共1⫺Lmk兲 具 ⑀ 典 l⫹Lmk ⑀ mk ⫹f兺 k⫽1 3cpkl ⑀ pk 共1⫺Lpk兲 具 ⑀ 典 l⫹Lpk ⑀ pk 共1⫺f兲兺 k⫽1 3cmkl 共1⫺Lmk兲 具 ⑀ 典 l⫹Lmk ⑀ mk ⫹f兺 k⫽1 3cpkl 共1⫺Lpk兲 具 ⑀ 典 l⫹Lpk ⑀ pk , 共1兲 where subindex krefers to the axes 共x,yand z兲, which for spheroidal shapes, are restricted to xand z;fis the filling factor, Lmk and Lpk the depolarization factors of the matrix and particle spheroids along the kaxis, and cmkl and cpkl the average of the projection of a unit vector parallel to the optic axis onto the applied electric field. B. Application of the effective medium approximation to dense materials In the case of a dense aggregate, i.e., f⫽1, the mphase is absent so that it can be dropped from Eq. 共1兲. Then the effective dielectric constant along the laxis notably simplify into: 兺 k⫽1 3cpkl共 ⑀ pk⫺ 具 ⑀ 典 l兲 共1⫺Lpk兲 具 ⑀ 典 l⫹Lpk ⑀ pk ⫽0. 共2兲 It should be noted that these expressions of the effective dielectric tensor components depend on geometrical factors of the crystalline grains, i.e., the grain shape through Lpk and on the average orientation given by cpkl . In this sense, any texture effect on the IR spectra should be well described by these coefficients. In order to greatly simplify the notation, we have assumed that crystallites are optically uniaxial so that each crystallite has its own optical axis 共denoted by ‘‘o’’ and ‘‘e’’ for ordinary and extraordinary rays兲, which defines an angle ␪ with the zaxis of the layer. If cylindrical symmetry is assumed for the thin layer, it must become optically uniaxial, being 具 ⑀ 典 zthe dielectric constant along the zaxis 4635J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp and 具 ⑀ 典 xthat corresponding to the x–yplane. The relationships between optical axes of the composite, xand z, with those of the crystallite can be deduced from Fig. 1共a兲. Using the definition of cpkl enclosed in Eq. 共1兲as it appears in Appendix A, we get the analytic expression of these geometric coefficients cpex⫽1/4 ␲ 冕 sin2共 ␪ 兲 ␥ pe共 ␪ 兲d⍀,共3兲 cpez⫽1/4 ␲ 冕 cos2共 ␪ 兲 ␥ pe共 ␪ 兲d⍀,共4兲 cpox⫽1/4 ␲ 冕 sin2共 ␪ 兲 ␥ po共 ␪ 兲d⍀,共5兲 cpoz⫽1/4 ␲ 冕 cos2共 ␪ 兲 ␥ po共 ␪ 兲d⍀,共6兲 where d⍀is the solid angle differential, and ␥ pe and ␥ po are the probability density function corresponding to the angular orientation of the extraordinary and ordinary optical axis. It should be noted that the terms sin2( ␪ ) and cos2( ␪ ) in Eqs. 共3兲–共6兲are obtained as a result of the projection of each component of the dielectric tensor to ␥ kl,E0( ␪ ), i.e., the density function corresponding to the probability of finding that the laxis of the ellipsoid makes an angle ␪ with E0. Due to the fact that axial symmetry has been assumed along the perpendicular to the sample surface, then, ␥ pe and ␥ po only depend on the angle ␪ . Moreover, in uniaxial substances the ordinary axes are degenerated into a plane, then 2 兰 ␥ pod⍀⫹ 兰 ␥ ped⍀ 4 ␲ ⫽1. 共7兲 Thus we get the following relationships between the four different coefficients: 再 2cpox⫹cpex⫽1 2cpoz⫹cpez⫽1 2cpox⫹cpoz⫽1 2cpex⫹cpez⫽1. 共8兲 If we consider that cpez can take any value from 0 to 1, then Eqs. 共8兲impose the limiting conditions for the rest of the ccoefficients 冦 cpez⫽0...1 cpex⫽1⫺cpez 2⫽1 2...0 cpoz⫽1⫺cpez 2⫽1 2...0 cpox⫽1⫹cpez 4⫽1 4...1 2. 共9兲 At this point it is possible to simplify Eq. 共2兲to obtain two expressions for determining the effective dielectric constant along the xand zaxes 共1⫹cpez兲 2 共 具 ⑀ 典 x⫺ ⑀ po兲 共1⫺Lpo兲 具 ⑀ 典 x⫹Lpo ⑀ po ⫹共1⫺cpez兲 2 共 具 ⑀ 典 x⫺ ⑀ pe兲 2Lpo 具 ⑀ 典 x⫹共1⫺2Lpo兲 ⑀ pe ⫽0, 共10兲 共1⫺cpez兲共 具 ⑀ 典 z⫺ ⑀ po兲 共1⫺Lpo兲 具 ⑀ 典 z⫹Lpo ⑀ po ⫹cpez 共 具 ⑀ 典 z⫺ ⑀ pe兲 2Lpo 具 ⑀ 典 z⫹共1⫺2Lpo兲 ⑀ pe ⫽0. 共11兲 These expressions look very similar to that obtained in the well known Bruggemann effective medium theory.15,16 That approximation was formulated for heterogeneous media composed by two isotropic phases, being fthe volume concentration of one of them. In our model the composite only has one single anisotropic phase but two different components of the dielectric tensor. In this case the role of the phases is replaced with the dielectric tensor components of crystallites. Moreover, the role of the volume concentration, f, is assumed in a dense anisotropic composite by two expressions related to the orientation coefficient cpez . It should be noted that Eqs. 共10兲and 共11兲are two second degree equations in 具 ⑀ 典 xand 具 ⑀ 典 z. ax 具 ⑀ 典 x 2⫹ 具 ⑀ 典 x共 ⑀ obox⫹ ⑀ ebex兲⫺ ⑀ e ⑀ ocx⫽0, 共12兲 az 具 ⑀ 典 z 2⫹ 具 ⑀ 典 z共 ⑀ oboz⫹ ⑀ ebez兲⫺ ⑀ e ⑀ ocz⫽0. 共13兲 FIG. 1. 共a兲Geometry of an ellipsoidal particle in the layer frame; 共b兲geometry of a plane thick layer. 4636 J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp The coefficients of these equations are given by 再 ax⫽1⫹Lp⫹cpez共⫺1⫹3Lp兲 box⫽⫺Lp⫺3cpezLp bex⫽⫺Lp⫹2cpez⫺3cpezLp cx⫽1⫺Lp⫹cpez共1⫺3Lp兲, 共14兲 再 az⫽2Lp⫹cpez共1⫺3Lp兲 boz⫽⫺2Lp⫹3cpezLp bez⫽1⫺2Lp⫺2cpez⫹3cpezLp cz⫽1⫺2Lp⫺cpez共1⫺3Lp兲. 共15兲 To illustrate the relationship between these coefficients with textural parameters given by ␥ pe ,关Eqs. 共3兲–共6兲兴 let us consider two extreme cases: 共i兲a layer made by crystallites randomly oriented and 共ii兲a layer formed by crystallites totally oriented towards a specific orientation, which describes an angle ␣ with the layer normal. In the first case ␥ pe( ␪ ) is a constant independent of ␪ ,so that cpez⫽cpex⫽1/3 and according to Eqs. 共10兲and 共11兲the heterogeneous medium presents an isotropic behavior ( 具 ⑀ 典 x ⫽ 具 ⑀ 典 z). This results coincides with the isotropic previous models.13,14 In the second case, the function ␥ pe( ␪ ) can be written as a Dirac’s delta ␥ pe共 ␪ 兲⫽g• ␦ 共 ␪ ⫺ ␣ 兲,共16兲 where gis the normalization constant. By using Eq. 共4兲we get: cpez⫽g 冕 0 ␲ /2 cos2共 ␪ 兲sin共 ␪ 兲 ␦ 共 ␪ ⫺ ␣ 兲d ␪ ⫽gcos2共 ␣ 兲sin共 ␣ 兲.共17兲 The coefficient cpex can be obtained operating in the same way with Eq. 共3兲. Then, by using Eq. 共8兲, it is possible to determine gto get the final expression of cpez as a function of ␣ . cpez共 ␣ 兲⫽cos2共 ␣ 兲 1⫹sin2共 ␣ 兲.共18兲 Equations 共10兲and 共11兲in combination with Eq. 共18兲 allow us to relate the optical anisotropy of the polycrystalline material with the angle of preferential orientation. Assuming a quasispherical particular shape (Lp⫽1/3), we can state that different anisotropy schemes 共Table I兲could appear in a polycrystalline anisotropic material as a function of the orientation coefficient cpez . If all the crystallographic axes of the crystallites are perpendicularly oriented to the normal of the surface sample, i.e., ␣ ⫽90° then, by using Eq. 共18兲, it results that cpez⫽0. The substitution of this value in Eqs. 共10兲and 共11兲determines that 具 ⑀ x 典 , the dielectric tensor coefficient corresponding to the plane parallel to the surface sample, is equal to the arithmetic mean value of ordinary and extraordinary components of crystallites ( ⑀ oand ⑀ e), while the 具 ⑀ z 典 component is identical to the ordinary one of the microcrystals ( ⑀ o). Thus, the absorbance IR spectrum detected for the sample under normal incidence will be a combination of transverse modes of the ordinary and extraordinary rays. However by tilting the sample, longitudinal modes of the ordinary ray begin to appear, while longitudinal modes of the extraordinary component remain absent. On the contrary, if all the crystallites axes are perpendicular to the sample surface we get ␣ ⫽0°, and cpez⫽1. This geometric configuration is identical, from an optical point of view, to that of a single 共anisotropic兲crystal with its optical axis parallel to the normal to the crystal surface. Under these circumstances, we must observe the transverse ordinary modes by IR absorption spectroscopy at normal incidence, while only the longitudinal extraordinary modes must be detected at grazing angles. Finally, if crystallites are randomly oriented, according to Eq. 共4兲, it results that cpez⫽1/3, a value which coincides with that of crystallite axes oriented 45° with respect to the normal to the plane. Under these conditions, Eqs. 共10兲and 共11兲determine that the whole sample behaves as an isotropic material, being identical to all the components of the dielectric tensor. It should be noted that this angle is different from the so called ‘‘magic angle’’ 关cos2( ␪ M)⫽1/3 or ␪ M ⫽54°44⬘] which corresponds to an angular average of the considered magnitude7,8 共permittivity, refractive index, reflectivity, etc兲. The discrepancy in our case is due to the fact that the calculation of effective dielectric constant takes into account the projection of the internal field of the particles 共which in our case satisfies the conditions of magic angle兲 over the xy plane and the zaxis, instead of the average permittivity tensor. These results are the limit cases that can appear in a dense anisotropic polycrystalline material. Thus, they can be used as a quick test to detect texture in polycrystalline thin layers measured by IR absorption spectra at normal and grazing angles. TABLE I. Effective dielectric constant, anisotropic character, and weight of each kind of phonon 共transverse, longitudinal, ordinary, and extraordinary兲in the absorption spectrum as a function of the crystallite orientation angle. ␣ cpez ⑀ x ⑀ zAnisotropy W( ␻ TO) ␪ i⫽0° W( ␻ LO) ␪ i⫽90° W( ␻ TE) ␪ i⫽0° W( ␻ LE) ␪ i⫽90° 90° 0 ⑀ o⫹ ⑀ e 2 ⑀ oYes 1/2 1/2 1 0 45° or random 1/3 2 ⑀ o⫹ ⑀ e 3 2 ⑀ o⫹ ⑀ e 3 No 2/3 1/3 2/3 1/3 0° 1 ⑀ o ⑀ eYes 1 0 0 1 4637J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp However, a quantitative evaluation of the optical properties of anisotropic thin layers is somehow more complicated17 and requires the use of the 4⫻4 transfer matrix formulation.18,19 This method has been recently employed in ellipsometry in order to determine the optical constants of single crystal anisotropic layers.20–22 We have used the explicit formulation given by Schubert23 as it appears in Appendix B. Once the full set of equations is implemented we get a calculation procedure which is able to estimate the theoretical spectra of dense heterogeneous anisotropic thin layers, where anisotropy is a consequence of preferential orientation of its crystallites. In order to obtain its theoretical spectrum the following data are necessary: thickness of the layer, incidence angle and polarization degree of incident beam, refractive index of microparticles, usually measured on single crystals, approximate shape of crystallites (Lp), and orientation degree (cpez). It is clear that a dense material composed by anisotropic grains can present a rich variety of different spectra especially if ⑀ oand ⑀ ehave large differences in a wide spectral range. IV. RESULTS A. Textural analysis of TiO2thin films by XRD The XRD patterns corresponding to the two samples studied here 共i.e., TiO2-300 and TiO2-600) are shown in Fig. 2. The two diagrams show a diffraction pattern typical of that of anatase. For a comparative analysis of the two diagrams, the observed integrated intensities (Iobs) of the main reflections are corrected by taking into account both layer thickness and sample size: Ic Iobs ⫽Chkl 2 ␮␶ ,共19兲 where Chkl is a normalization factor calculated as Chkl ⫽sin ␪ /sin ␪ L共if ␪ ⬎ ␪ L); and Chkl⫽1共if ␪ ⬍ ␪ L); the incident angle cutoff, ␪ L, depends on the sample size and the incident beam cross section ( ␪ L⫽21.0° and 24.7° for the TiO2-300 and TiO2-600 samples respectively兲; ␶ , the layer thickness and the absorption coefficient for TiO2–anatase, is ␮ ⫽0.0485 ␮ m⫺1. Observed and corrected intensities are indicated in Table II, as well as the relative corrected intensities (Rc⫽Ic/I101 c) referred to the 101 reflection. The degree of orientation for a given diffracting plane can be calculated considering the deviation from the relative corrected intensities with respect to the same planes in a random distributed anatase. The used coefficient Qhkl is Qhkl⫽Rhkl c ␤ hkl ⫺1, 共20兲 where ␤ means the relative integrated intensity of the hkl reflection with respect the 101 one 共100%兲when the material is randomly oriented. The relative density of planes of a given orientation, Nhkl , can be also estimated from relative corrected intensities, using the normalized expression Nhkl⫽Rhkl c/ ␤ hkl 兺Nmx i⫽1Rc i/ ␤ i,共21兲 where the summation extends over all the considered reflections along the diffraction pattern. For the present work, we have considered Nmx⫽6 different reflections, as it appears in Table II. Consequently, Nhkl⫽1/6 would correspond to a random orientation and Nhkl⬎1/6 indicates that (hkl) planes are preferentially oriented parallel to the sample surface, the higher the value the stronger the orientation. The Qhkl and Nhkl values obtained for both samples studied are summarized in the Table III. As observed, the most significant difference between the two samples regarding the relative orientation of their crystallographic planes corresponds to the 共101兲plane. In fact, this plane has Nhkl FIG. 2. XRD of anatase thin films prepared at 300 and 600 °C. TABLE II. Intensities, normalization factors, and corrected intensities of the diffraction peaks of anatase thin films. TiO2-300 TiO2-600 ( ␶ ⫽0.45 ␮ m兲,2 ␪ L⫽42.0°兲( ␶ ⫽0.32 ␮ m2 ␪ L⫽49.4°) hkl 2 ␪ Chkl Tobs IcRcChkl Iobs IcRc 101 25.308 1.000 53.0 1214 1.000 1.000 73.0 2352 1.000 004 37.791 1.000 11.0 252 0.208 1.000 4.0 129 0.055 200 48.047 1.137 34.0 886 0.730 1.000 21.0 677 0.288 105 53.885 1.265 7.5 217 0.179 1.084 2.5 87 0.037 211 55.073 1.291 1.8 53 0.044 1.106 3.3 118 0.050 204 62.692 1.452 3.0 99 0.082 1.245 2.5 132 0.056 4638 J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp values of 0.16 and 0.32 in the TiO2-300 and TiO2-600 samples, respectively. These values indicate that while this family of planes is randomly oriented in the first sample, it presents a clear preferential orientation parallel to the thin film surface in the second sample. A smaller change 共i.e., from 0.40 to 0.32兲in the Nhkl value is also found for the 共200兲planes when comparing the two samples, although in the two cases the parameter was almost double that of 0.16, the typical value of a randomly oriented sample. This means that in the two samples the 共200兲family of planes is preferentially oriented parallel to the thin film surface and that there are no significant differences between them in the relative orientation degree. For the 共004兲planes the situation is the opposite since these planes lose orientation degree when comparing the TiO2-300 with the TiO2-600 samples 共i.e., Nhkl values of 0.16 and 0.09兲. The other reflections whose intensities have been considered for the calculations are very small and do not present any preferential orientation in the two samples 共i.e., Nhkl values smaller than 0.16 in all cases兲. Considering these data, we will refer to the 共001兲planes for the use of FT-IR data to account for preferential orientation phenomena. B. FT-IR analysis of TiO2thin films Figure 3 shows the IR absorption spectra of TiO2-300 and TiO2-600 samples taken at different incident angles. The series of bands appearing in these spectra are attributed to lattice vibrational modes. Their assignment according to the data reported in the literature for anatase24 appear in Table IV. The sample TiO2-300 are characterized by two bands at 262 and 435 cm⫺1ascribed to the transverse optical phonons. There is also a feature, corresponding to the longitudinal modes in the spectral region from 750 to 830 cm⫺1, which becomes more prominent at higher incidence angles. The spectrum of sample TiO2-600 displays similar bands at normal incidence. However, the largest difference of these spectra compared with the previous ones is the shape of the band corresponding to longitudinal mode measured at highest incidence angles. In the high temperature samples, a well defined maximum appear at 856 cm⫺1for incidence angles larger than 45°. A proper analysis of the IR spectra at different incidence angles with incident polarized light can give information about the preferential orientation of crystal planes. Figure 4 shows the FT-IR absorption spectra of TiO2-300 and TiO2-600 samples recorded with s-共perpendicular兲and p-共parallel兲polarized light. In this case only the spectral region from 400 to 1000 was accessible to the analysis due to the absorbance of the KBr polarizer at lower frequencies. The spectra of the TiO2-300 sample have the same aspect as those in Fig. 3 共taken with unpolarized light兲, either for parallel and perpendicular polarization. However, in the case of the sample TiO2-600, although the spectra for spolarization do not depict any significant difference with the incidence TABLE III. Texture parameters of the TiO2samples. TiO2-300 TiO2-600 hkl ␤ hkl Rhkl cQkhl Nhkl Rhkl cQkhl Nhkl 101 1.000 1.000 0.00 0.16 1.000 0.00 0.32 004 0.206 0.208 0.01 0.16 0.055 ⫺0.73 0.09 200 0.290 0.730 1.52 0.40 0.288 ⫺0.01 0.32 105 0.188 0.179 ⫺0.05 0.15 0.037 ⫺0.80 0.06 211 0.187 0.044 ⫺0.76 0.04 0.050 ⫺0.73 0.09 204 0.148 0.082 ⫺0.45 0.09 0.056 ⫺0.62 0.12 FIG. 3. Experimental absorbance of anatase thin layers prepared at 300 °C 共a兲and 600 °C 共b兲measured with unpolarized light. The band corresponding to the Si substrate has been marked. 4639J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp angle of the radiation, those for ppolarization show a net enhancement of the longitudinal mode at 856 cm⫺1,inaway similar to that found in the spectra with unpolarized light. V. DISCUSSION To interpret the absorbance of thin layers of anatase we have calculated the theoretical absorbance of anatase thin films deposited on silicon by the above described model by using its IR optical constants, measured by Gonza ´lez et al.24 The calculated theoretical absorbance spectra corresponding to two different crystallite orientations and several different incidence angles for p-polarized light are displayed in Fig. 5. The spectra calculated for s-polarized light were omitted because they do not appreciably change with incidence angle.1 In the first case considered cpez⫽1/9, the thin layer is anisotropic and composed of crystallites whose optical caxes is preferentially tilted 90° to the normal surface vector. In this case, two large maxima of absorbance with small shoulders are present at 242 and 440 cm⫺1corresponding to the transverse optical 共TO兲Euphonons24 共258 and 435 cm⫺1兲.At high values of incidence angle, an absorbance band ranging from 750 to 900 cm⫺1is clearly visible. The maximum of this band shifts from 835 to 875 cm⫺1for higher values of incidence angle. This final value corresponds to the highest frequency longitudinal Euphonon 共876 cm⫺1兲. In the case where the crystallites are randomly oriented, cpez⫽1/3, it can be seen that the general aspect of the absorbance curves in the region of transverse modes are similar to those of cpez⫽1/9. However, in the spectral area of longitudinal frequencies corresponding to high incidence angle spectra, the maximum of the longitudinal Euphonon 共876 cm⫺1兲is now flanked by the longitudinal A2umode 共755 cm⫺1兲, which now appears as a shoulder at 800 cm⫺1. In Fig. 5 we have also plotted the absorbance curves corresponding to cpez⫽7/9, which represents the case of crystallites with its caxes preferentially orientated parallel to the normal of the surface sample. In this situation, the spectral region from 200 to 600 cm⫺1displays two strong absorbance bands corresponding to the Eu-TO modes. However, the high frequency band corresponding to the longitudinal optical 共LO兲phonons is different. In this case, the A2umode 共755 cm⫺1兲is similar or even more prominent than the Eu 共876 cm⫺1兲especially in case of the largest incidence angles. The use of Table I in addition to a close observation of the spectral region corresponding to the LO modes could help to determine if an anisotropic polycrystalline sample is textured. In the case that cpez⬍1/3 we see in Table I that along the zaxis the dielectric tensor takes the value of the ordinary dielectric constant. It implies that at higher incidence angles and with parallel polarization, only the LO frequencies of modes of degeneracy 2 could be visible 共i.e., Eu modes兲.Ifcpez⫽1/3, the sample is isotropic and under both parallel and perpendicular polarization the same spectra should be obtained at grazing angles. Finally, when cpez ⬎1/3 the spectral area corresponding to the LO modes measured at high incidence angles and under parallel polarization are dominated by the longitudinal extraordinary modes. This simple scheme agrees very well with the experimental data. For the sample prepared at 300°C and measured at high incidence angles with the polarized light 共Fig. 4兲, bands around 800 cm⫺1appear similar to those of theoretical spectra calculated for cpez⫽1/3. In this sample no indication of anisotropy was found since the spectra recorded with both the parallel and perpendicular components of the polarized spectra were similar 关Figs. 4共a兲and 4共c兲兴. On the contrary, the sample prepared at 600°C, parallel and perpendicular polarized spectra are no longer similar. In this case, the composite exhibits a true anisotropic behavior. The shape of the LO band indicates that the Eu(876 cm⫺1) mode is active at higher incidence angles in agreement with a partial orientation of the sample corresponding to cpez⬍1/3. Spectra taken with unpolarized light 共Fig. 3兲, are experimentally easier to record, but cannot be used to perform quantitative analysis because they are a linear combination of parallel and perpendicular components. However, from a qualitative point of view, it should be noted that unpolarized spectra at higher angles 共60° and 70°兲共Fig. 3兲look similar to those taken with the parallel component of the electric field but at a lower angle 共50°兲关Figs. 4共a兲and 4共b兲兴. Thus, it can be concluded that unpolarized spectra taken at near grazing angles have qualitative but valuable information about textures in the region of longitudinal modes in a way similar to the parallel polarized spectra. According to our interpretation of IR spectra, it seems that caxes of crystallites of the TiO2-600 sample present a certain preferential orientation parallel to its external surface, a fact that is corroborated by x-ray diffraction results. This orientation corresponds to a preferential growth of 共101兲 planes parallel to the surface. The crystallographic caxis of the anatase structure (a⫽3.776 and c⫽9.486 Å) forms an angle of ␾ (101)⫽21.70° with respect to the 共101兲plane. Therefore, in the oriented 共101兲TiO2-600 sample, the caxes of the crystallites describe a cone-shaped distribution around the azimutal axes perpendicular to the sample, being ␣ th the angle between the axis and the generatrix of the cone. According to Eq. 共18兲, and taking ␣ th⫽90°⫺ ␾ (101)⫽68.28°, we get cpez⫽0.07. This value is close to cpez⫽1/9⫽0.11, whose IR theoretical spectra are represented in Fig. 4共a兲. In order to get a closer view of the validity of the method, we have plotted together 共Fig. 6兲the experimental and calculated absorbance spectra 共ppolarization兲of anatasa films for different incidence angles. The experimental plotted spectra correspond to samples heated at 300 and 600°C and the theoretical ones to those calculated assuming that crystallites are oriented according to values of cpez , 1/9 and 1/3 共random distribution兲. A direct comparison of the spectra reveals the advantages and weaknesses of the proposed method are clearly visible. Firstly, it should be noted that calculated spectra follow the same general trend as the experimental TABLE IV. Vibrational IR modes of anatase. Transverse Longitudinal Symmetry frequency ␻ T共cm⫺1兲frequency ␻ L共cm⫺1兲 Eu262 366 Eu435 876 A2u367 755 4640 J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp FIG. 4. Experimental absorbance of anatase thin layers prepared at 300 °C 共a兲and 共c兲and 600 °C 共b兲and 共d兲measured with parallel 共a兲and 共b兲and perpendicular light 共c兲and 共d兲. The band corresponding to the Si substrate has been marked. 4641J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp ones. However, discrepancies between them appear around 610 and 750 cm⫺1. These bands are due to multiphonon processes of the silicon substrate.25 In the same way, the longitudinal frequency of calculated spectra is shifted 25 cm⫺1to higher frequencies in both kinds of samples. The origin of this mismatch remains unclear and we attribute it to absorbance fluctuations versus frequency of the silicon substrate in this spectral area,25 or more likely, to a possible error in the fitted longitudinal frequency measured on the anatase single crystal.24 In this sense, it should be noted that the calculated maximum of the high incidence angle curve exactly coincides with the longitudinal frequency of the anatase single crystal. Thus, in order to increase the accuracy of this method, in the future it will be necessary to properly characterize the substrate and to know with good precision the IR refractive index of the deposited film. VI. CONCLUSIONS In this article we have shown that longitudinal phonon modes measured by absorption infrared spectroscopy at gracFIG. 5. Calculated values of absorbance for Lp⫽1/3 and cpez⫽1/9 共a兲,cpez⫽1/3 共b兲, and cpez⫽7/9 共c兲. 4642 J. Appl. Phys., Vol. 93, No. 8, 15 April 2003 Pecharroma ´n et al. Downloaded 26 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp