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Infrared spectroscopic ellipsometry of micrometer-sized SiO2 line gratings
Applied Surface Science 416 (2017) 397–401
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Infrared spectroscopic ellipsometry of micrometer-sized SiO2 line gratings Cordula Walder a,∗ , Matthias Zellmeier b , Jörg Rappich b , Helge Ketelsen c , Karsten Hinrichs a a
Leibniz-Institut für Analytische Wissenschaften - ISAS - e.V., Department Berlin, Schwarzschildstraße 8, 12489 Berlin, Germany Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Institut für Silizium Photovoltaik, Kekuléstraße 5, 12489 Berlin, Germany c SENTECH Instruments GmbH, Schwarzschildstraße 2, 12489 Berlin, Germany b
a r t i c l e
i n f o
Article history: Received 13 March 2017 Accepted 16 April 2017 Available online 18 April 2017 Keywords: Scatterometry Metrology Lamellar Grating Process control Wood anomaly
a b s t r a c t For the design and process control of periodic nano-structured surfaces spectroscopic ellipsometry is already established in the UV–VIS spectral regime. The objective of this work is to show the feasibility of spectroscopic ellipsometry in the infrared, exemplarily, on micrometer-sized SiO2 line gratings grown on silicon wafers. The grating period ranges from 10 to about 34 m. The IR-ellipsometric spectra of the gratings exhibit complex changes with structure variations. Especially in the spectral range of the oxide stretching modes, the presence of a Rayleigh singularity can lead to pronounced changes of the spectrum with the sample geometry. The IR-ellipsometric spectra of the gratings are well reproducible by calculations with the RCWA method (Rigorous Coupled Wave Analysis). Therefore, infrared spectroscopic ellipsometry allows the quantitative characterization and process control of micrometer-sized structures. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Emerging technologies like integrated circuits, light management in solar cells [1], metamaterials [2], biotemplates [3] and biosensors [4] make metrology as a measurement technique of micro- or even nanostructured surfaces more and more important. In contrast to other common methods for the structural characterization of surfaces, like atomic force microscopy, transmission electron microscopy or scanning electron microscopy, ellipsometry is a fast, inexpensive and non-destructive method which can be applied in-situ [5]. Consequently, semiconductor industry has already discovered ellipsometry for in-line process control [6–8]. Until now there has been an extensive focus on ellipsometric scatterometry in the UV–VIS spectral range for the geometrical characterization and process control of nano-sized structures [6,9–11]. The objective of this work is to contribute to the ellipsometric characterization of periodic structures in the infrared spectral regime in order to become sensitive to microstructured surfaces and the vibrational fingerprints of the materials. Moreover, structures which are sensitive to illumination with visible light can be investigated without altering the sample. We show the general
∗ Corresponding author. E-mail address: [email protected] (C. Walder). http://dx.doi.org/10.1016/j.apsusc.2017.04.118 0169-4332/© 2017 Elsevier B.V. All rights reserved.
applicability of the method on micrometer-sized SiO2 line gratings. Numerical modeling with RCWA is used for the analysis of the ellipsometric spectra. This way, the whole experiment is simulated and interference effects as well as complex non-linear or anisotropic optical properties can be described correctly. 2. Methods The investigated samples consist of closed and striped SiO2 layers on silicon wafer substrates. Firstly, uniform SiO2 layers were deposited on the silicon substrates by chemical vapor deposition (CVD). Then the samples were spincoated with a photoresist and illuminated by UV light through a patterned mask (sample layout see Fig. 1). After removal of the illuminated area of the photoresist the structure of the mask is transferred to the SiO2 layer by an HF-etching process down to the wafer. Finally the remaining photoresist is removed as well. The resulting sample layout is illustrated in Fig. 1. SiO2 stripes of varying widths and periods were produced and each striped subsample is accompanied by a closed SiO2 reference layer. Fig. 2 shows SEM pictures of the cross sections of two striped SiO2 samples with similar geometries and different layer thicknesses. The wet chemical etching process leads to a trapezoidal form of the cross sections. Although both samples were produced with the same mask the difference in their layer thickness and cor-
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compared with the IR ellipsometric measurement results which only contain this order. If light is diffracted into several orders by an optical grating sudden variations in the intensity occur at certain wavelengths. This effect was discovered by Wood [15] and studied by Rayleigh [16]. Rayleigh found that, if a new scattering order emerges at a diffraction angle of ±90◦ with respect to the surface normal, the scattered field becomes singular because the light intensity is abruptly redistributed among the existing diffraction orders. Consequently, these Rayleigh singularities appear at wavelengths which follow from the grating equation with a diffraction angle of m = ±90◦ [17]. Since our samples were prepared on opaque silicon substrates, only the Rayleigh singularities in reflection influence the spectra:
m = P/m −sini ± 1 m = m/P
Fig. 1. Sample layout of SiO2 line gratings produced with photolithography on a Si-wafer.
responding etching time results in slightly different geometries of the stripes. The samples were measured with a custom-built photometric IR ellipsometer using the FTIR device “Tensor 37” from BRUKER. The angle of incidence was chosen to be 50◦ and the plane of incidence was approximately perpendicular to the stripes. The ellipsometric signal was recorded with a DTGS (deuterated triglycine sulfate) detector in the range of 800–3500 cm−1 . Further information about the measurement principle can be found in [12]. The measured spectra of Psi and Delta were modeled with the software “SpectraRay/3” from SENTECH Instruments GmbH, Germany. The software describes any sample as a layer stack with dielectric function models for each material involved. The real dielectric background of the silicon substrate was chosen to be constant (n∞ = 3.55). The closed SiO2 layer was modeled with two Brendel oscillators [13] at 1053 cm−1 and 1171 cm−1 to describe the coupled asymmetric Si O2 stretching modes [14] in addition to a constant real dielectric background (n∞ = 1.46). The modeled dielectric functions of the closed SiO2 layers were adapted to the corresponding measured spectra in a best fit procedure and the geometrical information of the SiO2 line gratings was extracted from SEM images (see Fig. 2). The resulting optical data and line geometry were used as input parameters for RCWA (Rigorous Coupled Wave Analysis) calculations with the software “SpectraRay/3”. By computing 11–30 scattering orders of the SiO2 line gratings the zeroth order can be approximately determined and
1 −sini ± 1
(1)
Here m and m are the wavelength and wavenumber of the Rayleigh singularities in reflection. P is the period of the grating, m the diffraction order of the Rayleigh singularity and i the incident angle of the light. The plane of incidence is perpendicular to the lines of the grating. 3. Results and discussion In order to extract geometrical information from ellipsometry spectra it is useful to know the dielectric functions of the involved materials. For this purpose a closed SiO2 layer of each sample was measured and its dielectric function was extracted from a best fit simulation of the respective layer stack. The optical data extracted from the simulation of the closed SiO2 layer is depicted in Fig. 3. Fig. 4 shows measured infrared Psi spectra of a closed SiO2 layer and SiO2 line gratings with different dimensions. These dimensions are expressed as (trapezoidal baseline (BCD) x trapezoidal top line (TCD) x period) in the legend. The layer thickness of 1.030 m is comparable for all samples. Apparently, different grating dimensions have a strong effect on the spectra. In comparison to the closed layer the interference peaks of the gratings are shifted to higher wavenumbers and reduced in intensity (see Fig. 4a)). This observation is due to the decrease of the spatial density of the SiO2 lines from a closed layer to gratings which also reduces the effective optical density. Consequently the interference conditions change. In Fig. 4b) an excerpt of the spectra is depicted from 800 cm−1 to 1500 cm−1 . The spectrum of the closed layer shows the influence of the coupled Si O2 stretching modes at 1053 cm−1 and 1171 cm−1 [14]. Depending on the layer thickness the effect of a longitudinal surface wave (Berreman mode) becomes visible around 1266 cm−1 [12]. The peak at 1266 cm−1 shifts to lower wavenumbers with decreasing spatial density of the SiO2 lines and
Fig. 2. SEM images of the cross sections of two SiO2 line gratings with similar geometries and different layer thicknesses. The lower pictures are magnifications of the upper pictures.
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Fig. 3. a) Refractive index and b) absorption coefficient of closed SiO2 layers.
Fig. 4. Measured infrared Psi spectra of a closed SiO2 layer (solid line) and SiO2 line gratings with different dimensions (dashed and dotted lines). The SiO2 layer thickness was determined to approximately 1.030 m. The legend refers to the sample geometry as (trapezoidal baseline (BCD) x trapezoidal top line (TCD) x period). a) Mid-infrared range, b) Si O2 stretching mode regime.
the remaining peaks become smaller. These effects can be simulated with the closed SiO2 layer by reducing the oscillator strength of the Si O2 stretching modes. Since the number of Si O2 oscillators decreases with reduced spatial density of the SiO2 lines, the observed trends of the peaks become explainable. The spectrum of the grating with 2.5 m line width exhibits a special feature at 1132 cm−1 which can be identified as a Rayleigh singularity in reflection (see Fig. 4b)). Rayleigh singularities become discernible in the ellipsometric spectra at the emergence or disappearance of rather low diffraction orders because higher orders often do not cause much intensity change of the zeroth order. The effect of an intense Rayleigh singularity on the spectrum is remarkable, especially, if it superimposes interference fringes or vibrational bands as demonstrated in Fig. 4b). Fig. 5a)–d) shows measured and calculated Psi spectra of striped SiO2 samples with different geometries of the stripes. The geometrical parameters of the stripes were extracted from SEM images (see Fig. 2) and used as start parameters for the RCWA calculations of the spectra. There is good agreement between the measured and calculated spectra in Fig. 5a)–d) for different geometries and layer thicknesses. Minor deviations originate from statistical variations of the sample geometry or systematic errors of the measurement device such as non-ideal polarizers or a non-linear response of the detector. Moreover, the incident beam is not perfectly parallel causing a small distribution of the angle of incidence.
The main features of the spectra in Fig. 5a)–d) are the bands of the Si O2 stretching vibration at 1053 cm−1 and 1171 cm−1 as well as the Berreman mode or surface wave around 1240 cm−1 . At high wavenumbers the spectra follow a rather smooth trend with some indications of remaining interference effects as can be seen in Fig. 5d). The calculated and measured Psi spectra in Fig. 5c) exhibit kinks in regular intervals of the wavenumber. These kinks are caused by Rayleigh singularities in reflection. Their spectral positions were calculated according to Eq. (1), marked by vertical black lines and labeled with the diffraction order. The calculated spectra in Fig. 5a), b) and d) also show single Rayleigh singularities at 1970 cm−1 , 2514 cm−1 and 2740 cm−1 , respectively. However, these Rayleigh singularities are not discernible in the corresponding measured spectra. The reason might be statistical variations of the sample geometry or the opening angle of the incident beam, which causes a distribution of the angle of incidence. Since the variation of the angle of incidence shifts the spectral position of the Rayleigh singularities, the opening angle of the incident beam might lead to the elimination of sharp features. No further Rayleigh singularities are visible in Fig. 5a), b) and d) because they cause too little intensity change at higher scattering orders or they blend into the oxide bands. Yet, if the superposition with the oxide bands is favorable, this can also lead to a prominent kink induced by a Rayleigh singularity. This becomes obvious in Fig. 5c) where the kink at 1132 cm−1 is much more pronounced than the remaining features of Rayleigh singularities. Consequently, if a Rayleigh sin-
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Fig. 5. Measured and calculated infrared Psi spectra of SiO2 line gratings with different dimensions and thicknesses. The integers are diffraction orders of Rayleigh singularities in reflection marked by vertical black lines. a): Unique layer thickness and dimensions. b) and d): Similar period (31–34 m). c) and d): Similar layer thickness (1.030–1.034 m) but very different dimensions.
Fig. 6. Calculated Psi spectra of SiO2 line gratings with varying dimensions (trapezoidal baseline BCD x trench width). a) Variation of BCD from 2 to 4 m and constant trench width of 7.5 m. b) Constant BCD of 2.5 m and variation of trench width from 6.5 to 9.5 m. Trapezoidal top line TCD: 0.4 m. Layer thickness: 1.030 m.
gularity of a low scattering order coincides with the oxide bands this provides a convenient measure to monitor the sample geometry. The spectral position of the Rayleigh singularity shifts with the grating period and the intensity of the oxide bands varies with the spatial density of the grating.
Fig. 6 shows calculated Psi spectra of SiO2 line gratings with slightly varying geometries. The grating geometries are marked as (trapezoidal baseline (BCD) x trench width) in the legend. The trapezoidal top line (TCD) and the thickness are kept constant. In Fig. 6a) the trapezoidal baseline is varied, whereas in Fig. 6b) the
C. Walder et al. / Applied Surface Science 416 (2017) 397–401
trench width is changed. The observable Rayleigh singularity of the second negative order in reflection is marked by a vertical line for each grating period. If the grating period is changed, either by increasing the line width or the trench width, the spectral position of the Rayleigh singularity shifts to smaller wavenumbers. This trend follows from Eq. (1). Moreover, the Rayleigh singularity may appear as an upward or a downward kink in the spectrum depending on the superposition with the oxide bands. Fig. 6b) shows the influence of the trench width variation at a constant trapezoidal baseline of 2.5 m. If the Rayleigh singularity occurs at an unfavorable wavenumber, like in the spectrum with 6.5 m trench width, the superposition with the oxide bands can even reduce the visibility of the Rayleigh peak. However, the spectrum in that wavelength range changes significantly with the grating geometry if a Rayleigh singularity superimposes the oxide bands (see Fig. 6a) and b)). 4. Conclusion In this work we presented infrared spectral ellipsometry for the characterization of periodic micrometer-sized SiO2 structures. The infrared ellipsometric spectra of one-dimensional SiO2 line gratings on Si wafers can successfully be modeled with the RCWA method. The spectrum in the range of the vibrational oxide bands is sensitive to the geometrical parameters of the gratings especially if a Rayleigh singularity appears at this spectral position. Therefore, the presented method qualifies for the extraction of the sample geometry as well as for the process control of periodic structures at the micrometer scale. Acknowledgements We kindly thank Ilona Engler and Özgür Savas for their technical assistance with the measurement devices. Financial support by the Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen, the Senatsverwaltung für Wirtschaft, Technologie und Forschung des Landes Berlin, and the Bundesministerium für Bildung und Forschung is gratefully acknowledged. We acknowledge the financial support by the European Union through the EFRE program (ProFIT grant, contract no.: 10160255, 10160265 and 10160256). References [1] O. Isabella, H. Sai, M. Kondo, M. Zeman, Full-wave optoelectrical modeling of optimized flattened light-scattering substrate for high efficiency thin-film silicon solar cells, Prog. Photovoltaics Res. Appl. 22 (2014) 671–689, http://dx. doi.org/10.1002/pip. [2] S. Viaene, V. Ginis, J. Danckaert, P. Tassin, Transforming two-dimensional guided light using nonmagnetic metamaterial waveguides, Phys. Rev. B 93 (085429) (2016), http://dx.doi.org/10.1103/physrevb.93.085429.
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[3] M.D. Guillemette, B. Cui, E. Roy, R. Gauvin, C.J. Giasson, M.B. Esch, P. Carrier, A. Deschambeault, M. Dumoulin, M. Toner, L. Germain, T. Veres, F.A. Auger, Surface topography induces 3D self-orientation of cells and extracellular matrix resulting in improved tissue function, Integr. Biol. 1 (2009) 196–204, http://dx.doi.org/10.1039/b820208g. [4] C. Lv, Z. Jia, Y. Liu, J. Mo, P. Li, X. Lv, Angle-resolved diffraction grating biosensor based on porous silicon, J. Appl. Phys. 119 (2016) 1–6, http://dx.doi. org/10.1063/1.4943072. [5] H.J. Patrick, T.A. Germer, Y. Ding, H.W. Ro, L.J. Richter, C.L. Soles, Scatterometry for in situ measurement of pattern reflow in nanoimprinted polymers, Appl. Phys. Lett. 93 (2008) 1–4, http://dx.doi.org/10.1063/1.3046117. [6] S. Liu, X. Chen, C. Zhang, Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology, Thin Solid Films. 584 (2015) 176–185, http://dx.doi.org/10.1016/j.tsf.2015.02.006. [7] H.T. Huang, F.L. Terry, Spectroscopic ellipsometry and reflectometry from gratings (Scatterometry) for critical dimension measurement and in situ, real-time process monitoring, Thin Solid Films 455–456 (2004) 828–836, http://dx.doi.org/10.1016/j.tsf.2004.04.010. [8] M. Losurdo, M. Bergmair, G. Bruno, D. Cattelan, C. Cobet, A. De Martino, K. Fleischer, Z. Dohcevic-Mitrovic, N. Esser, M. Galliet, R. Gajic, D. Hemzal, K. Hingerl, J. Humlicek, R. Ossikovski, Z.V. Popovic, O. Saxl, Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: State-of-the-art, potential, and perspectives, J. Nanopart. Res. 11 (2009) 1521–1554, http://dx.doi.org/10.1007/s11051-009-9662-6. [9] R. Antos, I. Ohlidal, J. Mistrik, K. Murakami, T. Yamaguchi, J. Pistora, M. Horie, S. Visnovsky, Spectroscopic ellipsometry on lamellar gratings, Appl. Surf. Sci. 244 (2005) 225–229, http://dx.doi.org/10.1016/j.apsusc.2004.09.160. [10] X. Niu, N. Jakatdar, J. Bao, C.J. Spanos, Specular spectroscopic scatterometry, IEEE Trans. Semicond. Manuf. 14 (2001) 97–111, http://dx.doi.org/10.1109/ 66.920722. [11] X. Chen, S. Liu, C. Zhang, H. Jiang, Z. Ma, T. Sun, Z. Xu, Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry, Opt. Express 22 (2014) 15165–15177, http://dx.doi.org/10.1364/OE.22.015165. [12] A. Röseler, Spectroscopic infrared ellipsometry, in: H.G. Tompkins, E.A. Irene (Eds.), Handb. Ellipsom, William Andrew publishing, 2005, pp. 763–798. [13] R. Brendel, D. Bormann, An infrared dielectric function model for amorphous solids, J. Appl. Phys. 71 (1992) 1–6, http://dx.doi.org/10.1063/1.350737. [14] C.T. Kirk, Quantitative analysis of the effect of disorder-induced mode coupling on infrared absorption in silica, Phys. Rev. B 38 (1988) 1255–1273. [15] R.W. Wood XLII, On a remarkable case of uneven distribution of light in a diffraction grating spectrum, Philos. Mag. Ser. 6 (4) (1902) 396–402, http://dx. doi.org/10.1080/14786440209462857. [16] O.M. Rayleigh, On the dynamical theory of gratings, Proc. R. Soc. A79 (1907) 399–416. [17] D. Maystre, Theory of wood’s anomalies, in: S. Enoch, N. Bonod (Eds.), Plasmon. From Basics to Adv. Top., Springer-Verlag, Berlin Heidelberg, 2012, pp. 39–83, http://dx.doi.org/10.1007/978-3-642-28079-5.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Infrared spectroscopic ellipsometry of micrometer-sized SiO2 line gratings Cordula Walder a,∗ , Matthias Zellmeier b , Jörg Rappich b , Helge Ketelsen c , Karsten Hinrichs a a
Leibniz-Institut für Analytische Wissenschaften - ISAS - e.V., Department Berlin, Schwarzschildstraße 8, 12489 Berlin, Germany Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Institut für Silizium Photovoltaik, Kekuléstraße 5, 12489 Berlin, Germany c SENTECH Instruments GmbH, Schwarzschildstraße 2, 12489 Berlin, Germany b
a r t i c l e
i n f o
Article history: Received 13 March 2017 Accepted 16 April 2017 Available online 18 April 2017 Keywords: Scatterometry Metrology Lamellar Grating Process control Wood anomaly
a b s t r a c t For the design and process control of periodic nano-structured surfaces spectroscopic ellipsometry is already established in the UV–VIS spectral regime. The objective of this work is to show the feasibility of spectroscopic ellipsometry in the infrared, exemplarily, on micrometer-sized SiO2 line gratings grown on silicon wafers. The grating period ranges from 10 to about 34 m. The IR-ellipsometric spectra of the gratings exhibit complex changes with structure variations. Especially in the spectral range of the oxide stretching modes, the presence of a Rayleigh singularity can lead to pronounced changes of the spectrum with the sample geometry. The IR-ellipsometric spectra of the gratings are well reproducible by calculations with the RCWA method (Rigorous Coupled Wave Analysis). Therefore, infrared spectroscopic ellipsometry allows the quantitative characterization and process control of micrometer-sized structures. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Emerging technologies like integrated circuits, light management in solar cells [1], metamaterials [2], biotemplates [3] and biosensors [4] make metrology as a measurement technique of micro- or even nanostructured surfaces more and more important. In contrast to other common methods for the structural characterization of surfaces, like atomic force microscopy, transmission electron microscopy or scanning electron microscopy, ellipsometry is a fast, inexpensive and non-destructive method which can be applied in-situ [5]. Consequently, semiconductor industry has already discovered ellipsometry for in-line process control [6–8]. Until now there has been an extensive focus on ellipsometric scatterometry in the UV–VIS spectral range for the geometrical characterization and process control of nano-sized structures [6,9–11]. The objective of this work is to contribute to the ellipsometric characterization of periodic structures in the infrared spectral regime in order to become sensitive to microstructured surfaces and the vibrational fingerprints of the materials. Moreover, structures which are sensitive to illumination with visible light can be investigated without altering the sample. We show the general
∗ Corresponding author. E-mail address: [email protected] (C. Walder). http://dx.doi.org/10.1016/j.apsusc.2017.04.118 0169-4332/© 2017 Elsevier B.V. All rights reserved.
applicability of the method on micrometer-sized SiO2 line gratings. Numerical modeling with RCWA is used for the analysis of the ellipsometric spectra. This way, the whole experiment is simulated and interference effects as well as complex non-linear or anisotropic optical properties can be described correctly. 2. Methods The investigated samples consist of closed and striped SiO2 layers on silicon wafer substrates. Firstly, uniform SiO2 layers were deposited on the silicon substrates by chemical vapor deposition (CVD). Then the samples were spincoated with a photoresist and illuminated by UV light through a patterned mask (sample layout see Fig. 1). After removal of the illuminated area of the photoresist the structure of the mask is transferred to the SiO2 layer by an HF-etching process down to the wafer. Finally the remaining photoresist is removed as well. The resulting sample layout is illustrated in Fig. 1. SiO2 stripes of varying widths and periods were produced and each striped subsample is accompanied by a closed SiO2 reference layer. Fig. 2 shows SEM pictures of the cross sections of two striped SiO2 samples with similar geometries and different layer thicknesses. The wet chemical etching process leads to a trapezoidal form of the cross sections. Although both samples were produced with the same mask the difference in their layer thickness and cor-
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compared with the IR ellipsometric measurement results which only contain this order. If light is diffracted into several orders by an optical grating sudden variations in the intensity occur at certain wavelengths. This effect was discovered by Wood [15] and studied by Rayleigh [16]. Rayleigh found that, if a new scattering order emerges at a diffraction angle of ±90◦ with respect to the surface normal, the scattered field becomes singular because the light intensity is abruptly redistributed among the existing diffraction orders. Consequently, these Rayleigh singularities appear at wavelengths which follow from the grating equation with a diffraction angle of m = ±90◦ [17]. Since our samples were prepared on opaque silicon substrates, only the Rayleigh singularities in reflection influence the spectra:
m = P/m −sini ± 1 m = m/P
Fig. 1. Sample layout of SiO2 line gratings produced with photolithography on a Si-wafer.
responding etching time results in slightly different geometries of the stripes. The samples were measured with a custom-built photometric IR ellipsometer using the FTIR device “Tensor 37” from BRUKER. The angle of incidence was chosen to be 50◦ and the plane of incidence was approximately perpendicular to the stripes. The ellipsometric signal was recorded with a DTGS (deuterated triglycine sulfate) detector in the range of 800–3500 cm−1 . Further information about the measurement principle can be found in [12]. The measured spectra of Psi and Delta were modeled with the software “SpectraRay/3” from SENTECH Instruments GmbH, Germany. The software describes any sample as a layer stack with dielectric function models for each material involved. The real dielectric background of the silicon substrate was chosen to be constant (n∞ = 3.55). The closed SiO2 layer was modeled with two Brendel oscillators [13] at 1053 cm−1 and 1171 cm−1 to describe the coupled asymmetric Si O2 stretching modes [14] in addition to a constant real dielectric background (n∞ = 1.46). The modeled dielectric functions of the closed SiO2 layers were adapted to the corresponding measured spectra in a best fit procedure and the geometrical information of the SiO2 line gratings was extracted from SEM images (see Fig. 2). The resulting optical data and line geometry were used as input parameters for RCWA (Rigorous Coupled Wave Analysis) calculations with the software “SpectraRay/3”. By computing 11–30 scattering orders of the SiO2 line gratings the zeroth order can be approximately determined and
1 −sini ± 1
(1)
Here m and m are the wavelength and wavenumber of the Rayleigh singularities in reflection. P is the period of the grating, m the diffraction order of the Rayleigh singularity and i the incident angle of the light. The plane of incidence is perpendicular to the lines of the grating. 3. Results and discussion In order to extract geometrical information from ellipsometry spectra it is useful to know the dielectric functions of the involved materials. For this purpose a closed SiO2 layer of each sample was measured and its dielectric function was extracted from a best fit simulation of the respective layer stack. The optical data extracted from the simulation of the closed SiO2 layer is depicted in Fig. 3. Fig. 4 shows measured infrared Psi spectra of a closed SiO2 layer and SiO2 line gratings with different dimensions. These dimensions are expressed as (trapezoidal baseline (BCD) x trapezoidal top line (TCD) x period) in the legend. The layer thickness of 1.030 m is comparable for all samples. Apparently, different grating dimensions have a strong effect on the spectra. In comparison to the closed layer the interference peaks of the gratings are shifted to higher wavenumbers and reduced in intensity (see Fig. 4a)). This observation is due to the decrease of the spatial density of the SiO2 lines from a closed layer to gratings which also reduces the effective optical density. Consequently the interference conditions change. In Fig. 4b) an excerpt of the spectra is depicted from 800 cm−1 to 1500 cm−1 . The spectrum of the closed layer shows the influence of the coupled Si O2 stretching modes at 1053 cm−1 and 1171 cm−1 [14]. Depending on the layer thickness the effect of a longitudinal surface wave (Berreman mode) becomes visible around 1266 cm−1 [12]. The peak at 1266 cm−1 shifts to lower wavenumbers with decreasing spatial density of the SiO2 lines and
Fig. 2. SEM images of the cross sections of two SiO2 line gratings with similar geometries and different layer thicknesses. The lower pictures are magnifications of the upper pictures.
C. Walder et al. / Applied Surface Science 416 (2017) 397–401
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Fig. 3. a) Refractive index and b) absorption coefficient of closed SiO2 layers.
Fig. 4. Measured infrared Psi spectra of a closed SiO2 layer (solid line) and SiO2 line gratings with different dimensions (dashed and dotted lines). The SiO2 layer thickness was determined to approximately 1.030 m. The legend refers to the sample geometry as (trapezoidal baseline (BCD) x trapezoidal top line (TCD) x period). a) Mid-infrared range, b) Si O2 stretching mode regime.
the remaining peaks become smaller. These effects can be simulated with the closed SiO2 layer by reducing the oscillator strength of the Si O2 stretching modes. Since the number of Si O2 oscillators decreases with reduced spatial density of the SiO2 lines, the observed trends of the peaks become explainable. The spectrum of the grating with 2.5 m line width exhibits a special feature at 1132 cm−1 which can be identified as a Rayleigh singularity in reflection (see Fig. 4b)). Rayleigh singularities become discernible in the ellipsometric spectra at the emergence or disappearance of rather low diffraction orders because higher orders often do not cause much intensity change of the zeroth order. The effect of an intense Rayleigh singularity on the spectrum is remarkable, especially, if it superimposes interference fringes or vibrational bands as demonstrated in Fig. 4b). Fig. 5a)–d) shows measured and calculated Psi spectra of striped SiO2 samples with different geometries of the stripes. The geometrical parameters of the stripes were extracted from SEM images (see Fig. 2) and used as start parameters for the RCWA calculations of the spectra. There is good agreement between the measured and calculated spectra in Fig. 5a)–d) for different geometries and layer thicknesses. Minor deviations originate from statistical variations of the sample geometry or systematic errors of the measurement device such as non-ideal polarizers or a non-linear response of the detector. Moreover, the incident beam is not perfectly parallel causing a small distribution of the angle of incidence.
The main features of the spectra in Fig. 5a)–d) are the bands of the Si O2 stretching vibration at 1053 cm−1 and 1171 cm−1 as well as the Berreman mode or surface wave around 1240 cm−1 . At high wavenumbers the spectra follow a rather smooth trend with some indications of remaining interference effects as can be seen in Fig. 5d). The calculated and measured Psi spectra in Fig. 5c) exhibit kinks in regular intervals of the wavenumber. These kinks are caused by Rayleigh singularities in reflection. Their spectral positions were calculated according to Eq. (1), marked by vertical black lines and labeled with the diffraction order. The calculated spectra in Fig. 5a), b) and d) also show single Rayleigh singularities at 1970 cm−1 , 2514 cm−1 and 2740 cm−1 , respectively. However, these Rayleigh singularities are not discernible in the corresponding measured spectra. The reason might be statistical variations of the sample geometry or the opening angle of the incident beam, which causes a distribution of the angle of incidence. Since the variation of the angle of incidence shifts the spectral position of the Rayleigh singularities, the opening angle of the incident beam might lead to the elimination of sharp features. No further Rayleigh singularities are visible in Fig. 5a), b) and d) because they cause too little intensity change at higher scattering orders or they blend into the oxide bands. Yet, if the superposition with the oxide bands is favorable, this can also lead to a prominent kink induced by a Rayleigh singularity. This becomes obvious in Fig. 5c) where the kink at 1132 cm−1 is much more pronounced than the remaining features of Rayleigh singularities. Consequently, if a Rayleigh sin-
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Fig. 5. Measured and calculated infrared Psi spectra of SiO2 line gratings with different dimensions and thicknesses. The integers are diffraction orders of Rayleigh singularities in reflection marked by vertical black lines. a): Unique layer thickness and dimensions. b) and d): Similar period (31–34 m). c) and d): Similar layer thickness (1.030–1.034 m) but very different dimensions.
Fig. 6. Calculated Psi spectra of SiO2 line gratings with varying dimensions (trapezoidal baseline BCD x trench width). a) Variation of BCD from 2 to 4 m and constant trench width of 7.5 m. b) Constant BCD of 2.5 m and variation of trench width from 6.5 to 9.5 m. Trapezoidal top line TCD: 0.4 m. Layer thickness: 1.030 m.
gularity of a low scattering order coincides with the oxide bands this provides a convenient measure to monitor the sample geometry. The spectral position of the Rayleigh singularity shifts with the grating period and the intensity of the oxide bands varies with the spatial density of the grating.
Fig. 6 shows calculated Psi spectra of SiO2 line gratings with slightly varying geometries. The grating geometries are marked as (trapezoidal baseline (BCD) x trench width) in the legend. The trapezoidal top line (TCD) and the thickness are kept constant. In Fig. 6a) the trapezoidal baseline is varied, whereas in Fig. 6b) the
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trench width is changed. The observable Rayleigh singularity of the second negative order in reflection is marked by a vertical line for each grating period. If the grating period is changed, either by increasing the line width or the trench width, the spectral position of the Rayleigh singularity shifts to smaller wavenumbers. This trend follows from Eq. (1). Moreover, the Rayleigh singularity may appear as an upward or a downward kink in the spectrum depending on the superposition with the oxide bands. Fig. 6b) shows the influence of the trench width variation at a constant trapezoidal baseline of 2.5 m. If the Rayleigh singularity occurs at an unfavorable wavenumber, like in the spectrum with 6.5 m trench width, the superposition with the oxide bands can even reduce the visibility of the Rayleigh peak. However, the spectrum in that wavelength range changes significantly with the grating geometry if a Rayleigh singularity superimposes the oxide bands (see Fig. 6a) and b)). 4. Conclusion In this work we presented infrared spectral ellipsometry for the characterization of periodic micrometer-sized SiO2 structures. The infrared ellipsometric spectra of one-dimensional SiO2 line gratings on Si wafers can successfully be modeled with the RCWA method. The spectrum in the range of the vibrational oxide bands is sensitive to the geometrical parameters of the gratings especially if a Rayleigh singularity appears at this spectral position. Therefore, the presented method qualifies for the extraction of the sample geometry as well as for the process control of periodic structures at the micrometer scale. Acknowledgements We kindly thank Ilona Engler and Özgür Savas for their technical assistance with the measurement devices. Financial support by the Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen, the Senatsverwaltung für Wirtschaft, Technologie und Forschung des Landes Berlin, and the Bundesministerium für Bildung und Forschung is gratefully acknowledged. We acknowledge the financial support by the European Union through the EFRE program (ProFIT grant, contract no.: 10160255, 10160265 and 10160256). References [1] O. Isabella, H. Sai, M. Kondo, M. Zeman, Full-wave optoelectrical modeling of optimized flattened light-scattering substrate for high efficiency thin-film silicon solar cells, Prog. Photovoltaics Res. Appl. 22 (2014) 671–689, http://dx. doi.org/10.1002/pip. [2] S. Viaene, V. Ginis, J. Danckaert, P. Tassin, Transforming two-dimensional guided light using nonmagnetic metamaterial waveguides, Phys. Rev. B 93 (085429) (2016), http://dx.doi.org/10.1103/physrevb.93.085429.
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