Surface Magneto-Optic Kerr Effect: 

An introduction


Reflection of a beam of linearly polarised light from a magnetised surface causes the polarisation to become elliptical, with the principal axis rotated with respect to the incoming light. Usually the amount of rotation (in radiants) and of ellipticity (ratio between the minor and major axis of the ellipse) induced in the reflected beam is of the order of 1/1000, i.e. relatively small.  This phenomenon is known as the magneto-optic Kerr effect (MOKE). In the case of ultrathin films with total thickness below about 10 nm, the effect is proportional to the film thickness and the technique assumes the name Surface-MOKE (SMOKE). SMOKE is mainly used to measure the hysteresis loops of thin magnetic films, by plotting the signal (rotation or ellipticity) as a function of the applied magnetic field. Unfortunately, such a signal is proportional to the magnetisation, but an absolute quantitative estimation of it requires the use of other magnetometry techniques, such as vibrating sample magnetometry. MOKE can also be used as a scanning microscopy technique, so that magnetic domain imaging with micrometric resolution becomes possible. Even submicrometric resolution can be achieved combining magnetooptic measurements with a scanning near-field optical microscopy (SNOM) apparatus. 

Although a microscopic explanation of magneto-optic effects would have to consider the coupling between the electrical field of the polarized light and the electron spin within the magnetic medium which occurs through the spin-orbit interaction, a simple interpretation is usually achieved using a macroscopic point of view, where magneto-optic effects arise from the antisymmetric, off-diagonal elements in the dielectric tensor. In particular, the magneto-optical effect can be understood on the basis of different response of the electrons to left and right polarized electromagnetic waves. In fact, a linearly polarised beam of light can be thought as a superposition of such two kinds of waves. Electrons will be driven into left circular motion by the left-polarized wave while the right-polarized wave will drive them into right circular motion, with equal radii of the orbits. Now, since the dipole moment is proportional to this radius,  an external magnetic field applied to in the propagation direction will cause an additional Lorentz force to act on each electron. As a consequence, the radius for the left-circular motion will be reduced and the radius for the right circular motion will be increased, so that the emerging light is elliptically (instead of linearly) polarized.

Experimentally, there are three principal configuration geometries to operate MOKE, as shown in the Fig.

In the longitudinal configuration the magnetic field is applied parallel to both the plane of the film and the plane of incidence of the light. The measured signal (either rotation or ellipticity) is proportional to the component of the magnetisation contained in the plane of incidence of  light. This configuration is very useful when one wants to measure the hysteresis cycle of a film characterised by an in-plane magnetisation. Polar geometry, instead, is exploited to gain information one component of magnetisation that is perpendicular to the film plane. The magnetic field is applied perpendicular to the plane of the film.  The experimental setup normally involves passing laser light through a polariser and then reflecting the light off the sample. The light then passes through another cross-polarising analyser. Slight changes in the plane of polarisation will thus cause variations in the detected light intensity after the second filter.  Modulation of the laser source or of the light polarisation, in combination with lock-in amplification of the detected signal helps to increase the signal-to-noise ratio. Use of an additional quarter-wavelength plate permits to measure the Kerr ellipticity. As a more effective alternative to the crossed-polarisers method, a differential method using a Wollanston prism and two identical detectors can also be exploited, as shown in Fig.
Again, either the rotation or the ellipticity of the reflected light can be measured, depending on the use of either half-wavelength or quarter-wavelength retardation plate. Finally, in transverse MOKE the magnetic field is applied perpendicular to the plane of incidence of light. This geometry provides a signal proportional to the component of magnetisation that is parallel to the film plane but perpendicular to the plane of incidence of the light. In this case there is no polarisation modification of the reflected light, but a change in the intensity of linearly polarised incident light is detected.
   A recent development of MOKE is the so called quantitative vector-magnetometry (MOKE-VM), which permits to quantitatively evaluate the different magnetisation components, including the modulus of the magnetisation vector. This method, which relies upon measurements of the MOKE signal in at least three different configurations of the light polarisation and of the applied field, yields an accurate and complete characterisation that enables a three-dimensional tracking down of the magnetisation vector while the external field is swept along certain directions. In this way one can go beyond a qualitative description of the switching process giving a deeper insight into the mechanism of magnetisation reversal during the hysteresis loop cycle for materials displaying anisotropy (both crystalline and shape originated), revealing to what extent magnetisation reversal occurs via coherent magnetisation rotation and allowing for the identification of the easy and hard axes. This is relevant, especially in the case of magnetic domains formation, where the modulus of the magnetisation vector exhibits a marked decrease.