1 Introduction

Magneto–optic (MO) effects linear in magnetization are widely used for magnetometry and spectroscopic studies of magnetic surfaces, interfaces, ultrathin structures and multilayers [1]. They are sensitive to both magnitude and orientation of magnetization (M) and are therefore suitable for investigations of both spatial and temporal development of magnetization [2]. In order to extract useful information from various MO techniques, like studies of depth dependent structural and magnetization profile, like MO vector magnetometry [3, 4, 5], or like MO studies of the dynamics of M, MO spectroscopic ellipsometry, etc., a detailed analysis of the MO response is necessary.

The matrix formalisms for anisotropic media characterized by a general permittivity tensor provide a suitable means for the calculation of MO effects in multilayers with arbitrary M. One of them has been developed by Yeh [6]. The approach has been applied to anisotropic absorbing magnetic multilayers and provided the solutions to reflection problems in the film-substrate system with an interface between two magnetic media [7]. It can be extended to periodic structures [8] and to laterally structured multilayers [9]. Then the Yeh’s formalism can be shown to be a limiting case for the Fraunhofer diffraction in multilayered anisotropic gratings. Special cases of reflection MO effects with arbitrary M up to the second order were considered by Postava et al. [10] and Kopřiva et al. [11]. Zak et al. derived off–diagonal elements of Jones matrices for multilayers with M in the plane of incidence in the ultrathin film limit [12]. Due to the restriction to circularly polarized eigenmodes the MO Kerr transverse effect could not be included. The separation of the polar and longitudinal components has been experimentally demonstrated by Ding et al. [13]. These authors based their interpretation on the formulae for polar and longitudinal Kerr effect at a single interface. The explanation of the spectroscopic MO experiments using modelling the multilayer MO response were the subject of several works, e.g. [14, 15, 16].

The experiment and matrix calculations of the MO response may be related by means of the Jones reflection matrix of a multilayer structure [17]. In general, the Jones matrices do not directly provide the information on the structural and magnetization profile of the multilayer. This can be obtained by modelling the Jones matrix starting with a rough idea on the multilayer structure. This approach is similar to that employed in the ellipsometry [18]. In order to identify the effect of various parameters, e.g. layer thickness, details of deposition process, orientation of M, photon energy, polarization state and angle of incidence of the sensing radiation, it is useful to start with simple situations leading to analytical representations. The main advantage is great simplicity. The analytical formulae are much easier to use than the matrix formalism and allow a physical insight. With bulk optical data and estimated thicknesses for individual layers, they provide a first approximation to the MO response. This can serve as a convenient starting point for the fitting procedure leading to a more accurate information on the multilayer.

The present work provides an extension of the Abèles characteristic matrix [19] to media with arbitrary (3D) M. The analysis is restricted to the MO effects linear in M. In order to establish a correct phase relationship using a consistent sign convention among the polar, longitudinal and transverse components, the calculation are performed from the beginning with assumption of general orientation of M without any restrictions on the eigenmode polarizations. The characteristic matrix represents a basic element for the best fit analysis of MO response in multilayers. In simple systems, e.g. ultrathin film with a non–magnetic cover layer and substrate, two magnetic films separated by a non–magnetic spacer, etc., this enables a derivation of analytical formulae for the MO reflection response. As an example, an original completely analytical representation of the Jones matrices for a fundamental case of a magnetic layer on a non–magnetic substrate is provided. The work is motivated by the recent MO investigations of simple ultrathin magnetic structures considered for spin electronics and magnetic random access memories.

The paper is organized as follows. The next section summarizes the 4×4 matrix representation for multilayers with arbitrary M. In the third section the generalized Abèles characteristic matrix is obtained. It allows a more compact multilayer representation and provides a bridge to the Jones reflection matrix for a system consisting of a magnetic film on a non–magnetic substrate (section 4). In section 5 the numerical simulations of MO response are presented for an iron garnet film on a non–magnetic garnet substrate. The conclusions are listed in the last section.

2 4×4 matrix representation

2.1 Permittivity tensor and wave equation

In order to account for the MO effects linear in M, the relative permittivity tensor in the n-th layer medium is assumed in the form [20, 21]

The orientation of M is specified by spherical coordinate angles θ_M and ϕ_M (see

Figure 1). According to the Onsager principle, ${\epsilon }_1^{\left(n\right)}$, which enters the off–diagonal elements of the tensor, is odd in M. At ${\epsilon }_1^{\left(n\right)}$=0 the tensor reduces to the scalar permittivity ${\epsilon }_0^{\left(n\right)}$. For a plane wave solution, E ⁽ⁿ⁾=$E_0^{\left(n\right)}$ exp [i(ωt-γ ⁽ⁿ⁾·r)], the wave equation in a medium characterized by ${\overleftrightarrow{\epsilon }}^{(n)}$ takes the form

Here, $E_0^{\left(n\right)}$ is the complex electric field amplitude, which specifies the wave polarization, γ ⁽ⁿ⁾ is the complex wave vector, t, ω, c, and r denote the time, angular frequency, phase velocity of the wave in vacuum, and the position vector, respectively. The relative magnetic permeability is assumed to take its vacuum value. Without loss of generality the wave vector may be written as ${\gamma }^{\left(n\right)}=\frac{\omega }{c}(\hat{\mathbf{y}}{N}_y+\hat{\mathbf{z}}{N}_z^{\left(n\right)})$, where ŷ and ẑ are the Cartesian unit vectors and $\frac{\omega }{c}{N}_y$ and $\frac{\omega }{c}{N}_z$ are the corresponding components of the wave vector. The plane of light incidence is set normal to the Cartesian unit vector x̂ with ẑ normal to the interfaces. Because of the Snell law, the y-component of the wave vector, which is parallel to the interface planes, remains the same in all layers of the stack. The normal component of the wave vector becomes a solution of the quartic eigenvalue equation obtained from the condition for non–trivial solutions of Eq. (2)

For ${\epsilon }_1^{\left(n\right)}$≪${\epsilon }_0^{\left(n\right)}$, the solutions $N_{\mathit{\text{zj}}}^{\left(n\right)}$ (j=1, …, 4) to second order in ${\epsilon }_1^{\left(n\right)}$/${\epsilon }_0^{\left(n\right)}$ of the eigenvalue equation become

for the two modes propagating in the positive z direction, and

for the two modes propagating in the negative z direction. We have denoted ${\epsilon }_0^{\left(n\right)}$-$N_y^2$ =N ^{(n) 2} _z0. The solutions are valid to the second order in ${\epsilon }_1^{\left(n\right)}$ provided either cos${\theta }_M^{\left(n\right)}$ or sin ${\varphi }_M^{\left(n\right)}$ is not equal to zero. Therefore, the second order perturbation to the complex index of refraction in the transverse configuration of magnetization is not included. In this case, the eigenvalues of $N_z^{\left(n\right)}$ can be obtained directly from the eigenvalue equation (3). In the present work it is sufficient to consider only the terms linear in ${\epsilon }_1^{\left(n\right)}$. The eigenvalue equation (3) is invariant with respect to the reversal of M represented either by ${\epsilon }_1^{\left(n\right)}$ (M) → ${\epsilon }_1^{\left(n\right)}$ (-M)=-${\epsilon }_1^{\left(n\right)}$(M) or by ${\theta }_M^{\left(n\right)}$ → π-${\theta }_M^{\left(n\right)}$ and ${\varphi }_M^{\left(n\right)}$ → ${\varphi }_M^{\left(n\right)}$ +π. In the special cases of the polar (θ_M =0 or π), longitudinal (θ_M =π/2, ϕ_M =π/2 or 3π/2) and transverse (θ_M =π/2, ϕ_M =0 or π) configurations the eigenvalue equation reduces to a bi–quadratic one. It becomes also a bi–quadratic one in two more general cases, where M is restricted to the interface plane (θ_M =π/2) or to the plane normal to the plane of incidence (ϕ_M =0 or π). For M restricted to the plane of incidence (ϕ_M =π/2 or 3π/2), the eigenvalue equation for $N_z^{\left(n\right)}$ remains quartic. The analytical solution to the general eigenvalue equation has been obtained by Schubert et al. [5].

 

Fig. 1. The magnetization M displayed as a cartesian vector sum of polar, M_P, longitudinal, M_L, and transverse M_T. In the spherical coordinates M is specified by its magnitude |M| and the angles θ_M and ϕ_M .

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2.2 Dynamic and propagation matrices

The relation between the four eigenmode amplitudes in the half spaces sandwiching the multilayer, indexed 0 and N+1, is given by [6]

where ${\mathit{E}}_0^{\left(0\right)}$ and ${\mathit{E}}_0^{(𝓝+1)}$ are 4×1 column vectors of eigenmode amplitudes and M is the 4×4 matrix representing the stack of 𝓝 uniform layers sandwiched between the half spaces 0 and 𝓝+1. The M-matrix is given by the matrix product

Here D ⁽ⁿ⁾ and P ⁽ⁿ⁾, for n=1, …,N are the dynamic and propagation matrices, respectively. Up to a common factor the elements of the dynamic matrix for a medium with arbitrary M are given by

The propagation matrix P ⁽ⁿ⁾ becomes

where d⁽ⁿ⁾ is the thickness of the n-the layer. In the isotropic layers (M=0)

The Jones reflection matrix relates the complex electric field mode amplitudes of the incident (i) and reflected (r) waves linearly polarized perpendicular (s) and parallel (p) with respect to the plane of wave incidence

In terms of the M-matrix elements the elements of the Jones matrix are then expressed as [6]

We have adopted the sign convention for the electric fields of the incident and reflected waves according to which both incident and reflected amplitudes are oriented in the same direction at the limit of normal incidence. The complex amplitudes of linearly polarized eigenmodes in the sandwiching half spaces are identified as follows (superscript denotes the transposed vector)

The matrices given in this section provide complete information for the numerical evaluation to first order in ${\epsilon }_1^{\left(n\right)}$/${\epsilon }_0^{\left(n\right)}$ of the electromagnetic interactions in a multilayer consisting of layers with arbitrary M.

3 Characteristic matrix for a layer with arbitrary magnetization

The characteristic matrix is defined by the product [8, 19]

where D ⁽ⁿ⁾ and P ⁽ⁿ⁾ are given by Eqs. (7) and (8). With restriction to the terms linear ${\epsilon }_1^{\left(n\right)}$/${\epsilon }_0^{\left(n\right)}$ it assumes the form

where

where

and characterize the contributions of the polar, longitudinal or transverse magnetization to the MO effects, respectively. When the incident medium is vacuum or air, N_y <1. As a result, the polar M component can be usually more easily detected than the longitudinal or transverse one. For the three special configurations of M the characteristic matrix S ⁽ⁿ⁾ reduces to the previous results [23, 24] and at ${\epsilon }_1^{\left(n\right)}$=0 transforms to the isotropic case [19]. When the n-th layer is ultrathin,

4 Jones reflection matrix for magnetic film – substrate system

We consider the system consisting of a magnetic film with arbitrary magnetization on a non–magnetic substrate. The electromagnetic wave response is represented by

where D ⁽⁰⁾ and D ⁽²⁾ are given by Eq. (9) and S ⁽¹⁾ follows from Eq. (14).

Using Eqs. (11) we obtain for small MO rotations and ellipticities in reflection

In Eq. (18b) for the off–diagonal elements, the upper sign pertains to r _ps and lower one to r _sp. Note that r _pp consists of two components. The dominating one corresponds to the isotropic case while the smaller one, linear in q ⁽¹⁾, accounts for the MO transverse effect. The single interface isotropic reflection and transmission coefficients are defined by the relations [17]

where i, j=0, 1, 2, and $N_{z0}^{\left(0\right)}$=N ⁽⁰⁾ cosθ ⁽⁰⁾, θ ⁽⁰⁾ is the angle of incidence, $N_{z0}^{\left(1\right)}$=(${\epsilon }_0^{\left(1\right)}$-$N_y^2$ )^1/2 and $N_{z0}^{\left(2\right)}$=(${\epsilon }_0^{\left(2\right)}$-$N_y^2$ )^1/2. The single interface off–diagonal elements of the Jones reflection matrix in the polar and longitudinal configurations denoted as $r_{\text{ps}}^{(01,\text{pol})}$ and $r_{\text{ps}}^{(01,\text{lon})}$, respectively, follow from

From the Jones reflection matrix elements in Eq. (18), expressed in terms of the single interface Fresnel coefficients, the anomalies in the region of the principal angle of incidence are easily understood. The diagonal and off–diagonal elements are of zero and first order in ${\epsilon }_1^{\left(1\right)}$, respectively. Note that the first term in the curly brackets of r _{ps, sp} represents the propagation across the film while the second one accounts for the interface effects. From the formulae the symmetry of the linear MO Kerr effects can be appreciated. The off–diagonal elements r _ps and r _sp change the sign when the magnetization is switched to opposite one. The same behavior is manifested by the MO component of r _pp. When the thickness of the absorbing magnetic layer is much higher than the penetration depth of the radiation, exp(-2iβ ⁽¹⁾≪1, Eqs. (18) reduce to the case of a single interface between a non–magnetic half–space and a half–space at arbitrary magnetization [22]. In another limiting case, when the magnetic layer is ultrathin we obtain

Apart from a small M independent term proportional to the ultrathin film thickness d⁽¹⁾ we can replace r _ss in Eq. (18a) and r _pp(${\epsilon }_1^{\left(1\right)}$=0) in Eq. (21b) by $r_{\text{ss}}^{\left(02\right)}$ and $r_{\text{pp}}^{\left(02\right)}$ according to Eqs. (19a) and (19b), respectively. This does not affect, to first order in ${\epsilon }_1^{\left(1\right)}$, the MO rotations and ellipticities in reflection, defined as r _ps/r _ss and r _sp/r _pp, as well as the relative changes of r _pp when the transverse M is switched to -M. The ultrathin film optical parameters and thickness have negligible effect on r _ss and r _pp(${\epsilon }_1^{\left(1\right)}$=0). This is consistent with the fact that the magneto–optics is more sensitive to the presence of magnetic layers than the classical ellipsometry. This advantage of magneto–optics is even more strongly manifested for the ultrathin magnetic layers buried deeper in the multilayer structure. For the polar, longitudinal or transverse M, as well as for M oriented arbitrarily in the plane of incidence the equations (21) reduce to the previous results [23, 24].

5 Examples

In this section the MO response in a magnetic film on a non–magnetic substrate is evaluated numerically using the analytical formulae. The results are consistent with those obtained with the 4×4 formalism. The magnetization vector evolves on a cone shaped surfaces about three mutually perpendicular axes chosen (a) normal to the plane of interface, (b) parallel both to the interface plane and to the plane of incidence, and (c) parallel to the interface plane and normal to the plane of incidence. The orientation of the axes coincides with that of the polar, longitudinal, and transverse magnetizations, respectively. In our coordinate system, these axes correspond to the Cartesian z, y, and x axes, respectively (Figure 2). We put in contrast two situation: one corresponding to purely interface effects and the second one where the propagation gives dominant contribution thanks to the magnetic film thickness bigger than the radiation wavelength. The results of modelling are computed in terms of the ratio r _ps/r _ss directly observable in experiment. We consider the case of an epitaxial iron garnet film deposited on a Gd₃Ga₅O₁₂ (GGG) substrate sensed by a near infrared wavelength of 813 nm (photon energy of 1.525 eV). The optical and magneto–optical data for Bi_1.07Y_1.86Pb_0.07Fe_4.962Pt_0.038O₁₂ (BiLuIG) film and the Gd₃Ga₅O₁₂ substrate, selected respectively from the literature [25] and [26], are collected in Table 1. They reasonably represent a Bi_0.96Lu_2.04Fe₅O₁₂ system investigated experimentally by means of time resolved MO reflection [2]. We shall restrict ourselves to the case of reflection of an s-polarized wave at an angle of incidence of 50 deg (unless otherwise stated). The thickness of the substrate is assumed infinite and its MO activity is ignored. The real part of the ratio r _ps/r _ss, ℜ(r _ps/r _ss), representing the weighted sum of polar and longitudinal Kerr rotations, is displayed.

Table 1. The permittivity tensor elements of the materials used in modelling.

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At first, a single interface vacuum/BiLuIG has been considered. The elements of the reflection matrix follow from equations (18) for an infinite thickness of the (absorbing) film. Figure 3(a) shows the effect of the rotation of M about the normal to the interface (i.e., about the polar or z-axis). The angle between the normal and M, θ_M , varies between zero (pure MO polar Kerr effect) and 90 deg. For a fixed value of θ_M the polar component of r _ps/r _ss remains constant. The rotation of M about the normal is measured by an azimuthal angle ϕ_M ranging from zero to 360 deg. Then, for θ_M =90 deg, and ϕ_M =90 deg or 270 deg M pertains to a pure longitudinal configuration. For θ_M ≈20 deg, in a limited range of ϕ_M around ϕ_M =270 deg ℜ(r _ps/r _ss) exceeds that for the pure MO polar Kerr effect (θ_M =0 deg).

 

Fig. 2. The geometry used in the modelling. The magnetization vector evolves on cone shaped surfaces about polar (a), longitudinal (b), and transverse (c) axes.

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Figure 3(b) displays the effect of the rotation of M by an angle ϕ_y about the y-axis from the starting position specified by a fixed angle ϕ_M =90 deg and an angle θ_M assuming the values between 0 and 90 deg, i.e., starting position of M is in the plane of incidence specified by a unit vector (0, sin θ_M , cos θ_M ). Each curve corresponds to a constant value of M_y ∝ sin θ_M , while M_z ∝ cos θ_M cos ϕ_y and M_x ∝ cos θ_M sin ϕ_y vary with ϕ_y . In agreement with the previous Fig. 3(a), there is a range of ϕ_y , here centered near ϕ_y =180 deg, where |ℜ(r _ps/r _ss)|exceeds the value for the pure MO polar Kerr effect, which occurs at θ_M =0 and ϕ_y =0 or 180 deg.

Figure 3(c) displays the effect of the rotation of M by an angle ϕ_x about the x-axis from the starting position specified by an angle ϕ_M =0 deg and an angle θ_M assuming the values between 0 and 90 deg, i.e., the starting position of M is in the plane normal to the plane of incidence specified by a unit vector (sin θ_M , 0, cos θ_M ). Each curve corresponds to a fixed value of M_x ∝ sin θ_M , while M_z ∝ cos θ_M cos ϕ_x and M_y ∝ cos θ_M sin ϕ_x vary with ϕ_x . Here r _ps/r _ss=0 for θ_M =90 deg in the whole range of ϕ_x (transverse configuration). There exist two fixed values of ϕ_x where all curves cross the zero indicating the cancellation of the polar and longitudinal components of ℜ(r _ps/r _ss). The independence of the zero crossing point on θ_M is a consequence of the θ_M independent M_y /M_z ratio.

 

Fig. 3. Magneto–optical Kerr effect for an interface between vacuum and Bi_0.96Lu_2.04Fe₅O₁₂ magnetic garnet expressed in terms of the real part of the ratio r _ps/r _ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕ_M (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕ_y (b), and normal to the plane of incidence specified by an angle ϕ_x (c). The initial position of M is given by an angle θ_M between M and the interface normal at a fixed azimuth ϕ_M =0 deg (a), ϕ_M =90 deg (b), and ϕ_M =0 deg (c). The incident radiation is s-polarized.

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For the same situations, figures 4(a), 4(b), and 4(c) display the reflection characteristics ℜ(r _ps/r _ss) in a BiLuIG film 1.5 µm thick on a GGG substrate. At the wavelength of 813 nm, where the absorption of the magnetic film is very weak, the main contribution to the MO effects originates from the wave propagation across the film while the interface contribution is almost negligible. This explains why the MO effects are two orders in magnitude higher than in the case of the single interface MO Kerr effect as shown in Fig. 3. At an angle of incidence of 50 deg, the refraction angle in the film is approximately 20 deg and the longitudinal contribution to r _ps/r _ss becomes much less important than the polar one compared to the previous single interface case.

Figure 5 shows the effect on ℜ(r _ps/r _ss) of the angle of incidence, θ ₀, ranging from -90 deg to +90 deg. The magnetization vector M is restricted to the plane of incidence corresponding to ϕ_M =90 deg. For a single interface (Fig. 5(a)), the dependence is even (odd) in θ₀ for θ_M =0 deg (θ_M =90 deg). The dependence becomes more involved for the BiLuIG film, the optical thickness of which is higher than the radiation wavelength. Figure 5(b)) shows several zero crossings. Because of the reduced effect of the longitudinal components the curves are of a nearly even symmetry except for θ_M approaching 90 deg.

Note that Schubert et al. carried out the numerical simulation for a system Au(2 nm)/Co(20 nm)/Au(60 nm)/Si in terms of related parameters at the normal and oblique (65 deg) incidence. Their dependencies show lower symmetry obviously because of a partial effect of the MO effects quadratic in M [5]. This corresponds to the inclusion of the terms proportional to ε ^{(n) 2} ₁ in Eq. (4). A complete account of the MO effects quadratic in M, which would require additional information on the permittivity tensor in magnetic layer, presents much interest.

6 Conclusions

The magneto–optic effects in magnetic multilayers for an arbitrary orientation of magnetization were presented in a simplified way. This was enabled by the restriction to the terms linear in the off–diagonal permittivity tensor element. The corresponding characteristic matrix was obtained, which leads, using Yeh’s formalism, to a completely analytical formulae for the Jones matrices in a magnetic film/non–magnetic substrate system. In this way the behavior of the basic building block of any multilayer, a layer with arbitrary magnetization surrounded by two different adjacent media can be understood. The formulae are useful for the best fit analysis of the experimental magneto–optic spectroscopic ellipsometry data. They are expressed in terms of single interface Fresnel coefficients, which can be phenomenologically adjusted in order to account for the interface roughness. The formulae are relevant for the study of the three dimensional temporal evolution of magnetization. In ultrathin film approximation, they may serve for a rapid evaluation of trends in magneto–optic response when the layer parameters change. An example of magnetic garnet film on non–magnetic garnet substrate with the magnetization varying in three mutually perpendicular planes was evaluated numerically. The results are consistent with those obtained with the Yeh’s 4×4 matrix formalism.

 

Fig. 4. Magneto–optical reflection characteristics at a film/substrate system consisting of a Bi_0.96Lu_2.04Fe₅O₁₂ magnetic garnet film 1.5 µm thick deposited on a Gd₃Ga₅O₁₂ substrate expressed in terms of the real part of the ratio r _ps/r _ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕ_M (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕ_y (b), and normal to the plane of incidence specified by an angle ϕ_x (c). The initial position of M is given by an angle θ_M between M and the interface normal at a fixed azimuth ϕ_M =0 deg (a), ϕ_M =90 deg (b), and ϕ_M =0 deg (c). The incident radiation is s-polarized. Note that the MO effect values are of two orders in magnitude higher than in the case of a single vacuum/BiLuIG interface.

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Fig. 5. The effect on ℜ(r _ps/r _ss) of the angle of incidence, θ ₀, ranging from -90 deg to +90 deg at an interface between vacuum and Bi_0.96Lu_2.04Fe₅O₁₂ magnetic garnet (a) and in a film/substrate system consisting of a Bi_0.96Lu_2.04Fe₅O₁₂ magnetic garnet film 1.5 µm thick deposited on a Gd₃Ga₅O₁₂ substrate (b). The magnetization vector M is restricted to the plane of incidence. Its orientation is specified by an angle θ_M between M and interface normal. The incident radiation is s-polarized.

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