ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 290–291 (2005) 120–123 www.elsevier.com/locate/jmmm
Modeling of magneto-optical properties of periodic nanostructures M. Foldyna, K. Postava, D. Ciprian, J. Pisˇ tora Department of Physics, Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic Available online 8 December 2004
Abstract Optical and magneto-optical (MO) response from nanometer-size grating is calculated using improved Fourier modal method. The MO grating is modeled by an anisotropic MO layer of effective medium. The complex refractive indices and MO parameter of the effective layer are compared with simple effective medium approximation. Dependence of the optical and MO parameters on the filling factor of the grating is presented. MO grating with period in the subwavelength scale could be used as an effective medium with desired optical and MO properties. r 2004 Elsevier B.V. All rights reserved. PACS: 78.20.Ls; 75.75.+a; 78.20.e Keywords: Magneto-optical effects; Kerr effect; Anisotropy-optical
1. Introduction Magneto-optical (MO) spectroscopic ellipsometry and MO vector magnetometry are powerful techniques for monitoring of thin-film magnetic properties. In order to apply these methods also for periodically nanostructured samples and to characterize the sample quantitatively, the advanced modeling of MO response is needed. Moreover, MO sub-wavelength gratings could be used as artificial effective media with desired new anisotropic optical and MO properties, for instance, for sensor applications. Sub-wavelength MO gratings exhibit new phenomena in comparison with continuous MO layer. The dependence of parameters on the filling factor is a
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well known fact, which has been described for the case of isotropic gratings [1]. Combining different types of magnetic materials (with different MO response) leads to a new type of anisotropy, which consists of the part in the optical anisotropy and part in the MO anisotropy. In this paper we will introduce the example of 1D MO grating which consists of the cobalt stripes prepared on silicon substrate. The stripes are surrounded by air. This configuration makes possible to enhance or weaken the MO response of the whole layer by a change of the filling factor, together with changing the optical anisotropy, mostly in the direction of the stripes. In the next section we define the MO response of the grating and expressions for the MO Kerr rotation y and ellipticity are presented. After that we introduce the effective medium model [1], which is used for comparison and methods used for expressing wave propagation through medium. Results describing the dependency of MO parameters on fill factor could be find in the next section.
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.228
ARTICLE IN PRESS M. Foldyna et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 120–123
2. Theory Fig. 1 shows the system considered in this paper, which consists of isotropic substrate, MO grating and isotropic superstrate. The modes in the superstrate can be easily separated to up and down modes with s or p polarization. The reflection coefficients are then defined as the ratios between the amplitudes of the reflected and incident waves and can be written in the form rxy ¼
Ay;up ; Ax;down
x; y 2 fs; pg:
(1)
In terms of the complex MO parameters rsp rps ; ws ¼ ; wp ¼ rss rpp
(2)
the MO rotation y and ellipticity are in the form tan 2yx ¼
2 Rðwx Þ ; 1 jwx j2
sin 2x ¼
2 Iðwx Þ ; 1 þ jwx j2
(3)
where x denotes s or p polarization. For small values of : y and we can assume that wx ¼yx þ ix : In general, the MO response is defined as the difference between MO parameters of configurations with and without applied magnetic field. For the case of layered media we can use directly Eq. (3) in the configuration with applied magnetic field. This is caused by the fact, that nonmagnetic layers are not anisotropic, so that all off-diagonal elements of the permittivity tensor are zeros. However, in the generalized cases of diffraction gratings this assumption could be broken. In such cases it is necessary to define the MO response of the structure as yx ¼ yx;M yx;M¼0 ;
x ¼ x;M x;M¼0 ;
(4)
where M denotes the magnetization of the used configuration and x is either s or p. Because in usual applications we are only interested in the linear MO effects, it is necessary in most cases to measure (model) MO response as the one half of the difference between two configurations with opposite direction of the magnetic field. This approach separate linear effects from the quadratic ones. The propagation of light waves through the grating multilayered medium is described using the Fourier Ap,down As,down
Ap,up As,up
air superstrate
MO grating layer isotropic substrate Fig. 1. Schematic picture of the studied structure with introduction of the up and down propagating modes.
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modal method (FMM), which is suited for problems of the sub-wavelength gratings. The stability and the convergence of the method is improved by applying the scattering matrix algorithm [2,3]. This removes the problem with computer’s finite number precision in the structure with the thicker layers after applying boundary conditions. Idea is to reorder propagation matrices in such way that we compare only waves propagating in the same direction. Moreover, the algorithm applied for the continuous layers reduces to the Berreman’s approach [4]. The rigorous model is compared with the effective medium model [1,5]. Nanostructured grating is represented, according to this model, by uniaxial anisotropic layer described by the effective parameters pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi no;eff ¼ f eH þ ð1 f ÞeL ; (5) ne;eff ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eH eL =½f eL þ ð1 f ÞeH ;
(6)
where no and ne denote the ordinary and extraordinary refractive indices and f is the filling factor (volume fraction of the MO material to the cell volume). The permittivities of both materials in the grating layer are represented by eH ¼ n2H and eL ¼ n2L :
3. Results and discussions This section deals with the numerical modeling of optical and MO response from the 20 nm height cobalt stripes on the silicon substrate at the chosen wavelength of 310 nm. In this spectral range the silicon substrate is absorbing and the MO Kerr effects are well pronounced. Optical and MO properties of silicon and cobalt are characterized using refractive indices nSi ¼ 5:01 þ i3:59 [7], nCo ¼ 1:44 þ i2:31 [6] and the off-diagonal element of the cobalt permittivity tensor e12 ¼ e21 ¼ in2Co Q; where Q ¼ 0:0262 i0:0058 [6] denotes the MO Voigt parameter in the polar configuration. Figs. 2 and 3 show MO angles and reflectivity as a functions of the filling factor f. The continuous cobalt film and silicon wafer without MO cobalt layer correspond to the limiting cases of f ¼ 1 and 0, respectively. The optical and MO response of the nanostructured MO grating can be described using a layer of uniaxial MO effective medium described by the effective permittivity tensor 1 0 ieo Qeff 0 eo B 0 C eeff ¼ @ ieo Qeff ee (7) A; 0 0 eo where eo ¼ n2o ; ee ¼ n2e are squares of the ordinary and extraordinary effective refraction indices corresponding to the approximative values given by Eqs. (5) and (6).
ARTICLE IN PRESS M. Foldyna et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 120–123
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15
2.5
10
εs,0
2
εp,0
5
0
-5
no,fit no,eff ne,fit ne,eff
θp,0
0
0.2
0.4
0.6
0.8
1
Filling factor
Imaginary part of refractive indices
MO parameters [mrad]
θs,0
1.5
1
0.5
Fig. 2. Change in MO Kerr parameters as a function of the filling parameter f of the grating (incidence angle is 651).
0 0.5
1
1
1.5
2
2.5
Real part of refractive indices Fig. 4. no and ne of the effective layer are shown in the complex plane. Fit of the rigorous FMM to the effective layer is compared with Eqs. (5) and (6).
0.8
Rpp,0 0
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Filling factor Fig. 3. Reflectivity (square norm of reflection coefficients) of the studied structure dependent on the filling factor (incidence angle is 651).
Imaginary part of Voigt parameter
Reflectivity
Rss,0
0.6
−0.005
−0.01
−0.015
−0.02
−0.025 −0.005
The fitted parameters of the permittivity tensor (7) are shown in Figs. 4 and 5, where the mapping into the complex plane was used. As the fitting data we used MO parameters calculated for the incidence angles between 401 and 801 using the rigorous FMM. Fig. 4 shows comparison between the effective medium model given by Eqs. (5), (6) and rigorous calculation based on FMM. In Fig. 5 we can see development of the Voigt parameter as the filling factor is increasing. Filling factor is changed from f ¼ 1 (continuous MO film) to f ¼ 0 (continuous air layer). Triangles and squares show steps of 0.1. We have shown that the effective medium theory roughly describes dependencies of the ordinary and the
0
0.005
0.01
0.015
0.02
0.025
0.03
Real part of Voigt parameter
Fig. 5. Effective Voigt parameter Q in complex plane as a function of the filling factor. Squares denote steps 0.1 in the filling factor.
extraordinary refractive indices on the filling factor. Differences come from multiple reflection and interference effects in the periodic grating system, which are neglected in effective media approximation. Modeling of the MO response by FMM presented here can be used for interpretation of the optical and MO spectroscopic measurements, as well as for
ARTICLE IN PRESS M. Foldyna et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 120–123
processing of the data obtained from MO diffractometry or vectorial magnetometry of the MO gratings.
Acknowledgements This work has been partially supported by the Ministry of Education, Youth and Sport of the Czech Republic under the projects KONTAKT ME 507 and ME 508, by the Grant Agency of the Czech Republic (202/03/0776), and by the grant MJM 272 4000 19.
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W. Stork, et al., Opt. Lett. 16 (1991) 1921. L. Li, J. Opt. Soc. Am. A11 (1994) 2829. L. Li, J. Opt. Soc. Am. A13 (1996) 1024. D.W. Berreman, J. Opt. Soc. Am. 62 (1971) 502. D. Ciprian, J. Pisˇ tora, Proc. SPIE 5445 (2004) 214. Sˇ. Visˇ nˇovsky´, et al., J. Magn. Magn. Mater. 127 (1993) 135. [7] E.D. Palik, Handbook of Optical Constants of Solids III, Academic Press, New York, 1998.
