3. Magneto Optical Kerr Effect (MOKE)
Nano-meter scale magnetic particles constitute a rich and rapidly growing area in condensed matter physics, due to the interest both in scientific and technological application.
These so-called nano-magnets or nano-elements are characterized by quite different properties with respect to their parent bulk material, because their size becomes comparable with the intrinsic length scale that determines the magnetic behaviour of a material.
Recent improvements in nano-fabrication allow the creation of arrays of such nano-elements, all equal to each other, making so possible to use standard averaging techniques to investigate properties of large arrays of identical nano-structures, overcoming the problems associated to the study of a single nano-element, like low signal or low size.

4. One of the most important properties of a magnetic material is its magnetic anisotropy, that describes the presence of preferred magnetization directions within the material, so determining the way in which such magnetic material behaves.
In nano-magnets the anisotropy depends not only on the characteristic of the parent bulk material, such as its crystalline structure, but also on the shape, size and thickness of the nano-elements.
Therefore the anisotropy of a nano-structure can be controlled using its geometry, or shape. The angular dependence of the total energy of the nanostructure in the uniform magnetization state is traditionally called shape anisotropy.

5. As the size of the nano-elements is progressively reduced, large deviations from the uniform magnetization state occur, because the magnetization tends to align along the sides of the nano-structure, even if its shape anisotropy is zero.
This effect, that origins the so-called configurational anisotropy, become more important as the size of the nano-structure is reduced, because the weight of the border regions (mainly affected by the configurational term) is large in comparison with the interior regions (mainly affected by shape and intrinsic anisotropic terms).

6. Magneto optical effects derive from the material optical anisotropy and from the magnetization in the surface domains, which can be influenced by external agents as a magnetic field.
The optical anisotropy induces a change in the state of polarization of the light reflected from a magnetic material.
This effect, observed for the first time by John Kerr, is generally known as the Magneto Optical Kerr Effect and it is analogous to the Faraday effect where the polarization of the light transmitted by a transparent material subjected to a magnetic field is rotated, as observed by Michael Faraday.

7. MOKE is an important technique in the study of surface magnetism since
it is highly sensitive to the surface magnetization
and relatively simple to implement.
At the microscopic level, the reflectivity of a medium is determined by the forced motion of the electrons induced by the electrical field of the incident light. If the medium is magnetized the electrons are subjected to an additional Lorentz force that changes their optical response.
The magneto-optical Kerr effect is the small difference in the optical response induced by the magnetization of the medium.
On a macroscopic point of view MOKE can be described by the dielectric tensor theory.

8. Let’s describe the principles of MOKE only in a qualitative way, to provide a phenomenological overview of the Kerr effect.
Let’s define the most relevant points associated to the effect, such us the light plane of incidence, its polarization state and the different geometries in which the effect can be investigated.
 The plane of incidence is the plane containing the incident and the reflected light beams.
Fig. 1. Definition of s- and p-polarized light

9. It is usually measured in two linear polarization configurations, namely perpendicular (s) or parallel (p) to the reflection plane, where the directions define the electric field directions with respect to the plane of incidence of the light, as shown in Fig. 1
When the p or s-polarized light is reflected from a metallic surface the reflected light will still be linearly polarized (p or s). This is because the reflecting surface is a plane of symmetry for the system.
This symmetry is broken when the linearly polarized light is reflected from a magnetized surface.
Magneto-optical Kerr effect is the (small) change in the polarization state of the incident light after reflection by a magnetized medium.

10. In fact the general form of the dielectric tensor, which represents the effect of a magnetic material, is given by:
i = o Q mi
where Q is the Voigt magneto optical constant, a complex number with a modulus of the order of 10-2 and a small (< 5° )negative phase.
Mi represents the magnetization along the i-th direction and Msat is the saturation magnetization.

11. Since the non-diagonal terms are proportional to the magnetization, the polarization state of the reflected light will depend on the magnetization state of the material.
A mathematical way to describe the change in the polarization of the incident light is through Jones matrix formalism. Here the reflection from magnetic material is represented by the Fresnel reflection 2×2 matrix:
where the term rij represents the ratio of the incident j polarized electric field to the reflected i polarized electric field

12. Basically, what MOKE measures directly, is the magneto optic response of the medium, through a change in the polarization of the incident light.
This magneto optic response consists in two parts: a change in the polarization of the in-phase component of the reflected light which give rise to the Kerr rotation, and a change in the polarization of the out-of-phase component of the reflected light which gives rise to the Kerr ellipticity.
There are principally three different geometric configurations in which MOKE measurements can be performed. They are defined by the relative orientations of the magnetization, the sample surface and the plane of incidence of the light, as shown in Fig. 2.

13. Fig. 2 Longitudinal, Transverse and Polar configurations for MOKE measurements
In the longitudinal configuration, the magnetization M lies both in the sample surface plane and in the incidence plane.
In the transverse configuration the magnetization is still in the surface plane of the sample, but is perpendicular to the incident plane;
Finally in the polar configuration the magnetization is perpendicular to the sample surface and parallel to the plane of incidence.

14. In both longitudinal and polar geometries if the incident light is linearly polarized the reflected light is elliptically polarized .
If the magnetization is perpendicular to the incidence plane (transverse) and the incident light is linearly polarized in the incidence plain, a change in intensity and a dephasing occurs, depending on the magnetization intensity,
Whereas polar and longitudinal MOKE occurs in any kind of material, transverse MOKE occurs only in adsorbing media.

15. The longitudinal and the transverse configuration are usually used to study material with in-plane magnetization, for example in thin films with in-plane shape anisotropy, while the polar one is used to study thin films which exhibit perpendicular components of the magnetization.
For the study of continuous or patterned films the longitudinal configuration is used, in which the applied field H is parallel to both the plane of incidence and the sample surface. In this configuration with a suited choice of the light polarization it is possible to obtain all the magnetization components.

16. In the SESAMO – STN Lab it has been built a versatile MOKE apparatus for the recording of hysteresis loops both in-situ, for the films growth in the MBE chamber, and ex-situ, for FIB patterned samples.
Fig.3 Experimental set-up for the in-situ Kerr effect measurements. The grey surface represents the plane of interface between the UHV Chamber and the optical table.

18. - For the in-situ measurements the laser beam is not focused, and the sample can rotate perpendicularly to the incidence plane
- For the ex-situ measurements the laser beam can be focused to down to 10 μm in diameter and the sample-holder has 6 degree of freedom; this allows a reduction of the FIB patterned area extension, usually limited to 30÷50 μm2, and therefore in the required FIB operation time.
The interfaces between the UHV chamber and the optical table viewports are placed after the polarizer in the incident optical line and before the PEM in the reflected line.

19. Light is provided by a 1 mW, frequency-stabilized He-Ne laser (λ=633 nm) at an incidence angle of 22.5° with respect to the sample surface normal.
Before the reflection, the polarization of the light beam is defined by a rotatable Glan- Thompson polarizer in the p, s or sp polarization state. The reflected light beam passes through a Photo-Elastic Modulator (PEM) to modulate the reflected light at a frequency  = 50KHz , then the beam is passed through an analysing rotable Glan-Thompson polarizer.
At the end of the optical line, the polarized modulated reflected beam is detected by a pre-amplified photo-diode. The signal components modulated at  and 2 are detected by the photodiode and amplified and measured by a phase sensitive amplifier (lock-in), locked at the same frequency.

20. All the optical components are mounted onto an optical table suspended on anti-vibration air bearing.
Measurements are fully automatized through a computer aided system which drives the electromagnet (by means of a Digital-to-Analog Converter: DAC), records the magnetic field intensity measured by an Hall probe, detects the signals coming from the lock-in amplifier (by an Analog-to-Digital Converter: ADC) and finally displays the hysteresis loop directly on the computer screen.
The Photo-Elastic Modulator (PEM) represents also a retarder in which the
phase delay  is sinusoidal with a frequency of 50; its axis is rotated by  with respect to the horizontal.
The analyzer is a polarizer rotated by  with respect to the horizontal, whereas  represents the polarization:  = 0: p,  = 90: s.

21. Independently from the optical elements orientation, the signal measured from the detector is given by the sum of three factors:
the first is a continue component IDC;
the second is an oscillating component, with the modulation frequency  ;
a third component with a double frequency with respect to the modulation frequency.
These intensity components contain information about the magnetization of the magnetic reflective medium; their explicit form depends on the optical element orientation given by the angles ,  and .

22. Magnetic loops of 3 (left) and 5 (right) ML of Fe/MgO(001) measured along different crystallographic directions in longitudinal configuration with the incidence light s-polarized.