Theoretical Background

The magneto-optic Kerr effect (MOKE) describes the rotation of the polarization plane of an electromagnetic wave when reflected off a ferromagnetic sample. The rotation angle is proportional to the magnetization of the sample. This effect was discovered and empiri-cally described for the first time by John Kerr in 1877.

Microscopically, the MOKE in ferromagnets can be explained by the concurrence of the exchange interaction and the spin-orbit interaction. For a detailed description of the quan-tummechanical approach to explain magneto-optic effects the reader is referred to Refs. [98-100].

A qualitative explanation can also be given by Lorentz’ model of electrons elastically coupled to the atomic cores (see also Ref. [101]). Here the electric field of an incident light wave 𝐸𝐸�⃗_𝑖𝑖 stimulates a harmonic oscillation of the electron. Let us consider the specif-ic case of the polar Kerr geometry as shown in Figure 2.1(a). Here the magnetization of the sample is oriented perpendicular to the surface and the impinging light is parallelly (p) polarized inside the plane-of-incidence. This linearly polarized light induces an oscil-lation of the electrons in the same direction of polarization. Without magnetization the light would be reflected with the same polarization indicated by the light blue arrow and denoted by 𝐸𝐸�⃗_𝑟𝑟. However, an additional magnetization comes along with the Lorentz force that induces a small oscillating component perpendicular to the primary motion given by the electric field of the incident light and perpendicular to the direction of magnetization.

Therefore, the oscillating motion is proportional to 𝑣𝑣⃗_𝐿𝐿∝ 𝐸𝐸�⃗_𝑖𝑖×𝑀𝑀��⃗ (right-hand rule, see green arrow). According to Huygen’s principle (see, e.g., Ref. [102]) this secondary mo-tion is starting point of the Kerr amplitude 𝐸𝐸�⃗_𝐾𝐾 (little red arrow). The superposition of the primary and secondary motion yields magnetization-dependent polarization rotations.

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The same line of argument can be applied to the longitudinal MOKE geometry shown in Figure 2.1(b) for p polarized light. Here the magnetization is oriented parallel to the sam-ple surface and parallel to the plane of incidence. The direction of polarization rotation follows from the right-hand rule again.

Figure 2.1(c) shows the case of the transverse MOKE geometry. Here the right-hand rule yields a Lorentz force pointing along the direction of light propagation for p polarized light. Therefore, no Kerr rotation is observed in the transverse geometry. For p polarized light a change in intensity of the reflected wave is observed.

Figure 2.1: The different experimental magneto-optic geometries illustrating the effect of polarization and magnetization direction on the Kerr rotation. Figure inspired by Ref. [101].

2.2.1.1 Phenomenological Approach

The following considerations presented in this subsection are based on Refs. [103-105].

The dielectric tensor ε of an optically isotropic non-magnetic material has diagonal ele-ments with the value of the dielectric permeability ε of the material whereas the non-diagonal elements are zero. In contrast, the dielectric tensor of a magnetic material is not symmetric and in the case of a ferromagnetic material the Onsager relation 𝜀𝜀_{𝑖𝑖𝑖𝑖}�−𝑀𝑀��⃗�= 𝜀𝜀_{𝑖𝑖𝑖𝑖}�𝑀𝑀��⃗� applies. In this case the dielectric tensor can be written as:

𝜀𝜀 =𝜀𝜀 �

1 𝑖𝑖𝑄𝑄𝑚𝑚_𝑧𝑧 −𝑖𝑖𝑄𝑄𝑚𝑚_𝑦𝑦

−𝑖𝑖𝑄𝑄𝑚𝑚_𝑧𝑧 1 𝑖𝑖𝑄𝑄𝑚𝑚_𝑥𝑥

𝑖𝑖𝑄𝑄𝑚𝑚𝑦𝑦 −𝑖𝑖𝑄𝑄𝑚𝑚𝑥𝑥 1 � ^(2.6)

Here 𝑚𝑚_{𝑥𝑥,𝑦𝑦,𝑧𝑧} denote the components of the unit vector 𝑚𝑚��⃗ in the direction of magnetization and 𝑄𝑄 is the complex Voigt constant. Here we consider only first-order magneto-optic coefficients. The relation between the electric displacement field 𝐷𝐷��⃗ and the electric field 𝐸𝐸�⃗ is then given as

𝐷𝐷��⃗= 𝜀𝜀𝐸𝐸�⃗= 𝜀𝜀𝐸𝐸�⃗+𝑖𝑖𝜀𝜀𝑄𝑄𝐸𝐸�⃗×𝑚𝑚��⃗ (2.7)

2.2 Magneto-Optic Kerr Effect

23 For a simplistic illustration, in the following derivation we consider solely the polar ge-ometry with the magnetization oriented in z direction. Furthermore, we assume that the electromagnetic wave impinges perpendicularly to the surface so that the wave vector 𝑘𝑘�⃗ is parallel to the 𝑧𝑧 direction. (A more general approach facilitating the calculation of the Kerr rotation for any arbitrary geometry is presented in the next subsection 2.2.1.2.) Us-ing expression (2.7) for the described polar geometry together with the Maxwell equa-tions yields the eigenvalue problem

where 𝑁𝑁 denotes the complex refractive index and is defined by 𝑘𝑘�⃗ =𝜔𝜔

𝑐𝑐 𝑁𝑁𝑒𝑒⃗^{𝑘𝑘 (2.9)}

Here 𝜔𝜔, 𝑐𝑐, and 𝑒𝑒⃗𝑘𝑘 are the angular frequency, the speed of light, and the unit vector in the direction of the wave vector 𝑘𝑘�⃗, respectively.

The eigenvalues of expression (2.8) are the roots of the characteristic polynomial:

(𝑁𝑁² − 𝜀𝜀)²+ (𝑖𝑖𝜀𝜀𝑄𝑄)² = 0 ⟹ 𝑁𝑁_±² =𝜀𝜀(1 ±𝑄𝑄). ^(2.10) The corresponding eigenvectors can be expressed as

𝐸𝐸�⃗_±= 𝐸𝐸₀𝑒𝑒⃗_∓𝑒𝑒^{𝑖𝑖�𝜔𝜔𝑠𝑠𝑁𝑁±𝑧𝑧−𝜔𝜔𝑡𝑡�} with 𝑒𝑒⃗_±= 1

√2�1

±𝑖𝑖0�, ^(2.11) where 𝑒𝑒⃗_+,− and 𝑁𝑁_+,− represent the unit vectors and complex refractive indices of left and right circularly polarized light waves. The superposition of a left and right circularly po-larized light wave with the same amplitude results in a linearly popo-larized light wave. The left and right circularly polarized waves pass through a magnetized material with slightly different velocities given by the different refractive indices. After transmitting a distance 𝑙𝑙 through the magnetized material (Faraday effect) the two waves have a phase difference of ∆𝜑𝜑 and therefore the linearly polarization exhibits a Faraday rotation of

Φ𝐹𝐹 =1

2∆𝜑𝜑 =𝜔𝜔𝑙𝑙

2𝑐𝑐(𝑁𝑁₊− 𝑁𝑁−)≈ 𝜔𝜔𝑙𝑙

2𝑐𝑐 √𝜀𝜀𝑄𝑄 ^(2.12)

The expression on the right side of the “approximate equal” sign is obtained by using expression (2.10). The complex Faraday rotation Φ_𝐹𝐹 has a real part and a imaginary part stemming from the real part and the imaginary part of the complex refractive indices 𝑁𝑁+,−. The real part of Φ_𝐹𝐹 describes the rotation of the polarization plane whereas the imaginary part represents the ellipticity of the transmitted wave.

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In the case of reflection (Kerr effect) the relation between complex refractive indices and the complex Kerr rotation angle is obtained by using the Fresnel equations. These equa-tions yield the reflection coeffcients 𝑟𝑟 describing the ratio of the amplitudes of the re-flected and the incident waves (see, e.g., [102]). If we assume that the light propagates perpendicularly to the surface, the reflection coefficients of the two circularly polarized waves are given by

𝑟𝑟±= 𝑁𝑁_±−1

𝑁𝑁±+ 1 ^(2.13)

If we assume that the incident wave is linearly polarized parallel to the x direction, 𝐸𝐸�⃗_𝑖𝑖 = (𝐸𝐸_𝑥𝑥, 0,0), the reflected wave will have a component in x direction and an additional com-ponent in y direction due to the magnetization: 𝐸𝐸�⃗_𝑟𝑟 = (𝑟𝑟_𝑥𝑥𝐸𝐸_𝑥𝑥,𝑟𝑟_𝑦𝑦𝐸𝐸_𝑥𝑥, 0). The quantities 𝑟𝑟_𝑥𝑥 and 𝑟𝑟_𝑦𝑦 are the reflection coeffcients in x and y direction. Using the Jones calculus [106, 107]

they can be expressed by the reflectivities in the circularly polarized basis: 𝑟𝑟_𝑥𝑥 = (𝑟𝑟₊+ 𝑟𝑟₋)/√2 and 𝑟𝑟_𝑦𝑦= (−𝑖𝑖𝑟𝑟₊+𝑖𝑖𝑟𝑟₋)/√2. The complex Kerr rotation is defined as

The approximate equality is obtained by using expression (2.10). Second order terms in 𝑄𝑄 are neglected because |𝑄𝑄|≪1. A quantum-mechanical treatment shows that the non-diagonal elements of the conductivity tensor σ, which is directly proportional to the die-lectrical material tensor ε, are proportional to the spin polarization and therefore also to the magnetization [108]. Looking at expression (2.15), this means that the Kerr rotation is also proportional to the magnetization.

2.2.1.2 General Approach to Magneto-Optics

In this subsection a formalism is presented that enables the calculation of the Kerr rota-tion for magnetic single layer or multilayer systems with arbitrary orientarota-tion of the mag-netization in each layer. This formalism is based on Fresnel’s equations and was devel-oped in its entirety by Zak et al. [109-111]. In the following, a brief summary of this for-malism will be given.

Let us assume a boundary between two media in the x-y plane. The tangential compo-nents of the electric 𝐸𝐸_𝑥𝑥, 𝐸𝐸_𝑦𝑦 and the magnetic 𝐻𝐻_𝑥𝑥, 𝐻𝐻_𝑦𝑦 fields of an electromagnetic wave travelling from medium 1 into medium 2 are conserved. By defining the magneto-optic coeffcients these fields can also be expressed using a set of the electric fields of the inci-dent (i) and reflected (r) waves for s and p polarization: 𝐸𝐸_𝑠𝑠^(𝑖𝑖), 𝐸𝐸_𝑝𝑝^(𝑖𝑖), 𝐸𝐸_𝑠𝑠^(𝑟𝑟), and 𝐸𝐸_𝑝𝑝^(𝑟𝑟). Here s

2.2 Magneto-Optic Kerr Effect

25 and p denote the perpendicular and the parallel components of the electric field with re-spect to the plane of incidence. One can define a matrix 𝐴𝐴 connecting the two different sets of fields

𝐴𝐴 is also denoted as the medium boundary matrix. The elements of this 4 × 4 matrix are composed of the complex refractive index 𝑁𝑁 and the complex Voigt constant 𝑄𝑄 describ-ing the electromagnetic properties of the medium and also contain geometrical angles describing the orientation of the magnetization of the sample. The exact form of the ma-trix is given in Refs. [109-111]. The boundary between two media can be matched by the condition

𝐴𝐴1𝑃𝑃�⃗1 =𝐴𝐴2𝑃𝑃�⃗2. ^(2.17) In the case of more than two media – which is generally the case for a thin film system (vacuum, magnetic film, substrate) – the wave propagates through the medium between two boundaries. The medium propagation matrix 𝐷𝐷 describes the wave propagation in-side the 𝑛𝑛th medium at the depth z from the boundary and is defined by the following condition

𝑃𝑃�⃗𝑛𝑛(𝑧𝑧= 0) =𝐷𝐷𝑛𝑛𝑃𝑃�⃗𝑛𝑛(𝑧𝑧) (2.18) Applying the propagation matrix to the field matrix 𝑃𝑃�⃗_𝑛𝑛(𝑧𝑧) at the position 𝑧𝑧 inside the 𝑛𝑛th medium yields the field 𝑃𝑃�⃗_𝑛𝑛(𝑧𝑧= 0) at position zero (at the boundary between the two me-dia) of the 𝑛𝑛th medium. Considering the general case of a multilayer system one can write

𝐴𝐴𝑖𝑖𝑃𝑃�⃗𝑖𝑖 = � �𝐴𝐴𝑛𝑛𝐷𝐷𝑛𝑛𝐴𝐴𝑛𝑛−1�

𝑚𝑚 𝑛𝑛=1

𝐴𝐴𝑓𝑓𝑃𝑃�⃗𝑓𝑓, ^(2.19)

where 𝑖𝑖 and 𝑓𝑓 denote the initial and final medium, respectively. Equation (2.19) can also be written as

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The Fresnel transmission and reflection coefficients are obtained from the 2 × 2 matrices in the following way:

𝐺𝐺⁻¹ =�𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑠𝑠𝑝𝑝

𝑡𝑡𝑝𝑝𝑠𝑠 𝑡𝑡𝑝𝑝𝑝𝑝� ⋀ 𝐼𝐼𝐺𝐺⁻¹= �𝑟𝑟_{𝑠𝑠𝑠𝑠} 𝑟𝑟_{𝑠𝑠𝑝𝑝}

𝑟𝑟_{𝑝𝑝𝑠𝑠} 𝑟𝑟_{𝑝𝑝𝑝𝑝}�. (2.22) The complex Kerr rotations for s and p polarized light are then given by

Φ𝐾𝐾,𝑠𝑠 =𝜑𝜑𝐾𝐾,𝑠𝑠+𝑖𝑖𝜂𝜂𝐾𝐾,𝑠𝑠 = 𝑟𝑟𝑝𝑝𝑠𝑠

𝑟𝑟𝑠𝑠𝑠𝑠 ∧ Φ𝐾𝐾,𝑝𝑝 = 𝜑𝜑𝐾𝐾,𝑝𝑝+𝑖𝑖𝜂𝜂𝐾𝐾,𝑝𝑝 = 𝑟𝑟𝑠𝑠𝑝𝑝

𝑟𝑟𝑝𝑝𝑝𝑝. (2.23) The formalism presented here is applied in chapter 7 to analyze MOKE data. In all calcu-lations conducted the complex refractive indices 𝑁𝑁 for a laser wavelength of 𝜆𝜆= 632.8 nm and the complex Voigt constant 𝑄𝑄 for Fe from Refs. [109, 112] are used:

GaAs: 𝑁𝑁= 3.856 +𝑖𝑖 ⋅0.196

Fe: 𝑁𝑁= 2.87 +𝑖𝑖 ⋅3.36 𝑄𝑄= 0.0376 +𝑖𝑖 ⋅0.0066