## Abstract

Magneto-optical Kerr effect (MOKE) spectroscopy in the -1st diffraction order with *p*-polarized incidence is applied to study arrays of submicron Permalloy wires at polar magnetization. A theoretical approach combining two methods, the local modes method neglecting the edge effects of wires and the rigorous coupled wave analysis, is derived to evaluate the diffraction losses due to irregularities of the wire edges. A new parameter describing the quality of the edges is defined according to their contribution in the diffracted MOKE. The quality factor, evaluated for two different samples, is successfully compared with irregularities visible on atomic force microscopy pictures.

© 2005 Optical Society of America

## 1. Introduction

Optical-spectroscopic techniques play an important role in characterizing the quality of grating lithography [1–3]. These techniques are becoming of comparable importance to the classical ones such as scanning electron microscopy (SEM) or atomic force microscopy (AFM) because of the destructive character of cross-sectional SEM or limitations of AFM to study features beneath a surface. The principles of monitoring the quality of surfaces and thin films are suitably established using statistical quantities determined by AFM [4,5] and by optical techniques [6,7]. The height deviation and autocorrelation length of surface irregularities measured either by AFM or optically foremost produce a difference due to the convolution of the AFM tip with the surface. To evaluate imperfections of the wire-edge parts of surfaces in laterally textured films using AFM is obviously very difficult even with highestquality tips.

In the case of gratings made of magnetic materials, the optical techniques can be completed with magneto-optical (MO) analyses, particularly by measuring the magneto-optical Kerr effect (MOKE). Recently micromagnetic properties of periodically arranged magnetic wires, dots, and holes in magnetic films have received remarkable attention [8,9]. It has been reported that the diffracted MOKE (D-MOKE) hysteresis loops, i.e., the loops recorded on beams diffracted by gratings, can help to investigate the magnetization distribution in magnetic nanostructures. Particularly, the D-MOKE contains more detailed information on the magnetic domains in the vicinity of wire or dot edges or in films with holes. To achieve faithful analysis of magnetic-domain behavior including the switching process, authors applied theoretical models with different levels of accuracy. In the case of shallow gratings, with a small depth-to-period ratio, approximate analytical models neglecting the internal diffraction edge-effects were demonstrated as adequate to describe the D-MOKE response [9–11]. Those models are based on the far-field Fourier analysis of the lateral amplitude-reflectance distribution and on the assumption of the optical and MO uniformity within depth. However, if the lateral magnetic-domain features to be described are of small sizes, i.e., comparable to the depth, then the D-MOKE response might be affected by the edge-effects, and hence such a model becomes incorrect. Moreover, to monitor more than the lateral distribution of magnetization, could obviously be possible with theoretical approaches of the relevant capability. Since rigorous theoretical models require long-time and large-memory numerical computation, several authors developed approximation approaches more precise than those for shallow gratings, e.g., the perturbation approximation to the Rayleigh method [12,13]. Nevertheless, those models cannot generally be used for deep gratings [14].

So far, the D-MOKE analyses have been limited to a single wavelength with coherent wave sources. A few attempts were made to measure the incidence-angle dependence [12,15] and the dependence on the orientation of the magnetization vector [16]. Experimental arrangements employing the multiple wavelength range, i.e., the “D-MOKE spectroscopy,” suggest an improvement of the sensitivity and accuracy, as well as an increase of the number of features detectable by the MO measurement, which is the aim of the present work.

In this paper we demonstrate that the D-MOKE response can strongly be affected by the reduced quality of a laterally patterned structure, namely by random irregularities of the wireedge parts of the structure’s surface, which may cause both the approximate and rigorous models inaccurate. The quality of such magnetic gratings is here evaluated by using one parameter identifying the amount of the edges’ internal-diffraction contribution to the DMOKE response in a broad spectral range.

## 2. Samples and measurements

A set of samples with similar fabrication parameters was investigated. The gratings were prepared from a nominally 10-nm-thick Permalloy (Ni_{81}Fe_{19}) film deposited on an Si substrate and protected by 2-nm-thick Cr capping. The patterning was made by means of *e*-beam lithography with subsequent ion milling.

The AFM measurements on two chosen samples are displayed in Fig. 1. For the purposes of this paper, the D-MOKE results obtained on the sample with the most evident irregularities in the set are reported. An AFM picture of this sample is shown in Fig. 1(a) compared with a better one (Fig. 1(b)). The AFM measurement, performed at several positions of the grating, yielded the information on the period of about 900 nm, wire top linewidth of 536 nm, and grating depth of 13 nm. According to former analyses on the samples, the structure was determined consisting of 11 nm of Permalloy on the substrate covered by 3 nm of a native SiO_{2} overlayer, and with 2 nm of capping completely oxidized into Cr2O_{3} [17–

The D-MOKE experiments were performed on an MO spectrometer employing the azimuth modulation and compensation technique. The measurement reported here was achieved in the -1st diffraction order for *p*-polarized incidence in the spectral range of 1.8–4.1 eV at polar magnetization with applied out-of-plane magnetic field of 1.4 T, sufficient for saturated magnetization [20]. The propagation angles of higher diffraction orders are wavelength dependent; a special measurement configuration was therefore chosen. The angle between the incident and the diffracted beams was fixed at 20°, and the sample was rotated while the wavelength was swept.

## 3. Theoretical approach

Two different theoretical approaches were chosen for modeling the D-MOKE response of the gratings, the rigorous coupled wave analysis (RCWA) implemented as the transfer-matrix approach for anisotropic media [21] and the local modes method (LMM), which is an approximate analytical method based on the far-field Fourier analysis of the lateral amplitude-reflectance distribution assuming the saturated magnetization [17]. According to the applied magnetic filed, we assume the homogenous saturated magnetization of the Permalloy wires without any domain structure in modeling throughout this paper. Since none of both approaches describes the measured spectrum presented here correctly, a third model is derived involving both the RCWA and LMM calculations and assuming the reduced quality of the wire edges.

Here we provide a short description of the LMM including an evaluation of its error. Let *x* be the coordinate along the wires of the grating and y the coordinate along its periodicity. Any shallow optical element, with all its lateral-texture sizes considerably higher than its depth, can be described by a complex amplitude reflectance *r*_{αβ}
(*y*), which is a function of only the lateral coordinate *y* for we are working with one-dimensional patterning. The indices *α* and *β* denote the polarization-basis indices of reflected and incident waves, respectively. According to the LMM, in which the edge effects are considered negligible, the function *r*_{αβ}
(*y*) uses only two parameters, the amplitude reflectances of wires, *r*
_{w,αβ}, and of the system of air/substrate between wires, *r*
_{b,αβ}. In accordance with the principles of the far-field Fraunhofer diffraction, the amplitude reflectances in the separate diffracted orders are determined by formulae where *n* corresponds to the *n*th diffraction order, while *f* denotes the grating’s filling factor given by the ratio of the wire linewidth to the period. The term Δ${r}_{\beta \beta}^{\left(n\right)}$ represents the influence of the internal diffraction at wire edges in the rigorously calculated optical response or, in other words, the error of the LMM with respect to the RCWA. Formerly we have reported that only ${r}_{\mathit{\text{pp}}}^{\left(n\right)}$
at oblique angles of incidence contains a significant amount of Δ${r}_{\mathit{\text{pp}}}^{\left(n\right)}$
, especially in higher diffraction orders [18].

## 4. Results and discussion

The material constants used in all simulations were taken from literature [22–24]. Examples of absolute values of two simulated amplitudes ${r}_{\mathit{\text{sp}}}^{(-1)}$
and ${r}_{\mathit{\text{pp}}}^{(-1)}$
in the configuration presented here are shown in Fig. 2. The D-MOKE ellipsometric parameters in the -1st diffraction order for *p*-polarized incidence, the Kerr rotation ${\theta}_{p}^{(-1)}$
and ellipticity ${\epsilon}_{p}^{(-1)}$
, are determined for small D-MOKEs as the real and imaginary parts of the ratio of the corresponding amplitude reflectances, i.e.,

Considering the small values of ${r}_{\mathit{\text{pp}}}^{(-1)}$ in Fig. 2, the D-MOKE parameters are obviously highly sensitive to Δ${r}_{\mathit{\text{pp}}}^{\left(1\right)}$ , or, in other words, both the LMM and RCWA produce remarkably different values, as shown in Fig. 3 together with the experimental data. Since even the rigorous method does not match the measurement, the most acceptable explanation follows from Fig. 1. The AFM picture in Fig. 1(a) indicates considerable irregularities in the wire edges, which naturally reduce the value of Δ${r}_{\mathit{\text{pp}}}^{(-1)}$ in Eq. (2) due to diffraction losses. It is straightforward to replace this value by a reduced value

and to determine the wavelength-dependent function *η*(*λ*) by using real values expected between 0 and 1. In the case of our samples, however, no wavelength dependence was detected. Each sample in the set was successfully characterized with a constant value of *η* by applying the least-square method in a one-parameter fit of the D-MOKE rotation and ellipticity. We refer to this parameter as the “quality factor of the grating with respect to the wire edges,” and to the new theoretical approach as the “combined RCWA-LMM model.” Analogous to the height deviation and autocorrelation length of random irregularities of non-patterned surfaces, which are connected with the reduction of the reflectivity due to diffraction losses [6,7], the *η* factor describes a reduction of the diffracted light according to similar principles. The value of *η*=1 corresponds to ideal periodical smooth edges, whereas *η*=0 implies that no edge effects are observed. The realistic value according to the measured D-MOKE parameters was found *η*=0.53, with the fitted spectrum displayed in Fig. 4. The same procedure applied to the sample of obviously higher quality, with the AFM picture displayed in Fig. 1(b), yielded a value of *η*=0.70.

## 5. Conclusions

Diffracted MO spectroscopy was successfully applied to evaluate the quality of periodically patterned magnetic structures. The method appears to be profitable since the wire edges measured by AFM can hardly be analyzed statistically, as it is conventional in the case of flat surfaces. The principle can be applied for any optical or MO configuration where the edge effects are not negligible compared to the surface/thin film response. The results also suggest advantageous possibilities in other applications, e.g., analyzing the non-linear MO effect or photo-elastic and magneto-elastic effects, since any effect usually negligible may be enhanced in a particular experimental arrangement.

## Acknowledgments

This work is a part of the research plan MSM 0021 620834 that is financed by the Ministry of Education of the Czech Republic. Supported by the Japanese Ministry of Education. S.V. thanks Shizuoka University for the hospitality. The authors acknowledge the group of Claude Fermon (CEA Saclay, France) for the patterning of the samples.

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