Application of magneto-optical Kerr effect to first-order reversal curve measurements

While hysteresis loops are ubiquitously used for magnetic sample characterization, they only yield access to averaged properties. The measurement of first-order reversal curves (FORC) is a well established method to gain access to the local magnetization behavior of a sample. It is wide spread in the geomagnetism and paleomagnetism sciences,^1–3 but is also used in thin film physics,⁴ material sciences, and research into nanostructured systems.⁵ Since the introduction of the scalar hysteresis model by Preisach⁶ and further developments by Mayergoyz,⁷ theoretical description and mathematical modeling of FORC has significantly improved for particles,^8,9 multi-component samples,¹⁰ or nanostructures⁵ in recent years.

Due to its coverage of all magnetic states of the investigated systems and its capacity to separate them in a relatively clear and intuitive way by resolving the system's different coercive fields and also its internal interactions, FORC is especially interesting for multi-component systems like combinations of soft and hard magnetic layers, for systems with internal magnetic interactions (e.g., exchange bias systems), but also nanostructured materials.

As the acquisition of FORC diagrams requires the measurement of a large number of minor hysteresis loops, it typically relies on magnetic measurement techniques that yield an absolute magnetization value like VSM or SQUID where the individual loops can be easily merged. A major drawback of these methods is their rather slow measurement process of seconds per data point,¹¹ resulting in FORC acquisition times well above 24 h if a high field resolution is required. Up to now, faster methods that only yield relative magnetization values like magneto-optical Kerr effect (MOKE) have so far only rarely been applied to the FORC technique, e.g., by Bonanni et al.¹² Here, we present a MOKE approach for FORC acquisition, accompanied by several examples proving the relevance of this combination for physics and material sciences.

The term first-order reversal curve describes a minor hysteresis loop measured at an arbitrary magnetic sample and is characterized by a so called reversal field H_r and an applied field H. As a first step, the sample is magnetically saturated in a positive field

The so called FORC-density

a two-dimensional distribution function, is calculated from the collected datasets for different reversal fields in the range from

To obtain the FORC density as a function of coercive fields H_c and internal interaction fields H_u, usually a transformation of the basic H- and H_r-axes is performed in a way that

and

representing a clockwise 45°-rotation and expansion of the coordinates.^2,13 The FORC diagram is confined to the positive half-plane of the coercivity axis because H > H_r and therefore H_c > 0. Different examples of FORC density distributions will be shown later in Sec. IV.

Despite the large utility of the FORC method, different issues, especially in the calculation of the FORC density, must be considered. At first, the magnetization of a sample can show a reversible or irreversible behavior, dependent on the samples magnetic field history. Due to the fact that the FORC distribution involves a mixed second derivative, the reversible component is totally excluded and the normalization of the distribution fails, meaning

This issue is solved by using the extended FORC method,¹⁴ assuming the magnetization value M(H, H_r) = M(H = H_r, H_r) for all H < H_r, i.e., extending the FORC diagram to values of H_c ⩽ 0, and thereby fully capturing the reversible contributions to M.

Besides the contour plot giving information about irreversible processes, reversible processes can be characterized by a so called reversibility indicator

the quotient of the susceptibilities of the reversal curve and the enveloping hysteresis loop.¹⁵ η = 1 describes a fully reversible, η = 0 a completely irreversible magnetic state of the investigated system.

The interpretation and simulation of FORC diagrams of complex systems can yield insight into a diverse set of magnetic interactions and properties. While we focus here on the acquisition of high resolution FORC diagrams, further details regarding their evaluation can be found elsewhere.^16,17

MOKE measurements are a wide spread and easy to use technique for surface sensitive magnetization measurement. The measurement of the magnetization relies on the rotation of the polarization plane of light as it is reflected at the sample surface.¹⁸ This can be phenomenologically described by

where D is the induced displacement vector, ε is the dielectric constant, E is the electrical vector of the incoming plane light wave, Q is a material parameter, and m is the magnetization vector. The cross product in Eq. (6)m × E links the magnetization to a change in the displacement vector and thus the electrical vector of the reflected light wave.¹⁸ Further details can be found elsewhere.^18–20 

Typical geometries for MOKE measurements are shown in Fig. 2. The longitudinal and polar Kerr effect are utilized to measure in-plane and out-of-plane magnetization components, respectively. Most MOKE systems use a polarizing beam splitter, like a Glan-Thompson prism or a Wollaston prism, and two photo diodes as detectors. Thus, they are only sensitive to relative changes of the polarization of the reflected beam.¹⁸ 

Conventional approaches to FORC acquisition are geared towards techniques like VSM and SQUID, while MOKE systems are typically used for single hysteresis measurement only. Thus, some adaption of MOKE to FORC is necessary.

In the following, we will first describe the necessary adjustments of the experimental technique and then go into some data evaluation details.

As mentioned before, the magneto-optical Kerr rotation is detected by a polarizing beam splitter and a set of two photo diodes, thus they only yield a relative magnetization signal. There is also a superimposed background signal from Faraday rotation in optical elements, nonlinear optical effects and so on, that is typically approximated by a straight line fitted to the positive and negative saturation part of the hysteresis. While some MOKE systems feature very fast data acquisition, the measured absolute value of the polarization drifts significantly over the course of several minutes.

To accommodate the fact that MOKE magnetometers are typically optimized for single hysteresis measurements, we propose to treat every minor loop as an individual measurement while providing anchor points to combine many minor loops to a FORC map. This requires that the background slope, the zero offset of the measured polarization rotation, and the saturation amplitude for each individual minor loop is determined.

For this purpose, a special shape of the magnetic field during minor loop acquisition is necessary. During the initial part of the acquisition cycle, shown in Fig. 3, the background slope, the zero offset, and the saturation and amplitude are determined and subsequently the reversal field (shown in pink, marked as 1) is set and the minor loop is measured (shown in blue, marked as 2).

To determine the background, the sample magnetization is saturated (shown in green, marked as 3) and the change of the polarization rotation signal with changing field is measured (shown in red, marked as 4). This is done both for positive and negative saturation for every cycle to have two anchor points for combining measurements for a large number of reversal fields. Furthermore, this means that the electromagnet needs to achieve at least 110% of the magnetic field necessary for sample saturation as to allow for sufficient background determination. Fitting a straight line to the measured slope yields the Faraday background and both saturation values allow to correct the zero offset. Using two anchor points, i.e., positive and negative saturation, results in a more stable automated background and offset correction which is required for the acquisition of a large number of minor loops, although it comes at the price of an increased data overhead.

For the acquisition of a typical minor loop, we average over several cycles with a frequency of about 0.2 Hz, i.e., taking about 10 s for each minor loop. This acquisition speed should be achievable with most modern MOKE magnetometers. Thus, they allow for a large number of minor loops to be combined into one FORC map and a field resolution that is not achievable with other methods in a timely manner. On the other hand, this leads to a large amount of data that needs to be evaluated after data acquisition.

One way to reduce the amount of generated data and the acquisition time is to reduce the field resolution in parts of the FORC map where no significant signals are expected. This can be achieved by choosing non-equidistant reversal fields when acquiring the minor loops and using large reversal field steps in regions of low interest. This approach and the subsequent data treatment has been demonstrated before by Harrison² for FORC measurements with VSM magnetometers. The use of two anchor points for background and drift compensation actually allows to stitch post performed additional minor loops into MOKE FORC maps, in case data evaluation results show that a higher reversal field resolution is desired or necessary.

In general, the sophisticated FORCinel² and VarioFORC¹³ approaches using locally weighted regression smoothing and non-integer, nonlinear smoothing factors for data evaluation are very useful for FORC maps acquired by SQUID, VSM, or AGM. For these methods, there is only a limited number of data points due to measurement time constraints, while for FORC maps acquired by MOKE it can be desirable to be able to reduce the number of data points to speed up calculation for an initial inspection of the FORC density.

While it has been shown that interpolation is not the ideal approach for FORC maps that are data point density limited, this is not the case for MOKE FORC and in many instances downsampling is convenient for rough data inspection. For example, using the profile shown in Fig. 3, we routinely acquire FORC maps with more than 250.000 data points. Therefore, initial interpolation to a square grid allows to achieve said downsampling and prepares the data for an easy determination of the FORC density at the same time. In case the full resolution of the measured data is required, the square grid can still be chosen in such a way that there is no downsampling.

As the FORC density is a double derivative it is rather susceptible to noise and usually some kind of data smoothing is necessary to suppress large spikes due to signal noise. Thus, after interpolating the data points to a square grid the smoothing factor defines the size of the sector that is used to subsequently fit the FORC density of a point of interest. The number of data points N used for each point of interest and a smoothing factor SF is given by² 

Together the field steps ΔH of the interpolation grid and the smoothing factor determine the resolution R

of the calculated FORC density, thus using a minimal smoothing factor is desirable.²¹ 

As we follow a rather conventional FORC processing the actual FORC density is determined by fitting the FORC map sector defined by the smoothing factor for each point of interest to a polynomial function. The used polynomial function is given by² 

where the parameter a₆ yields the desired FORC density² 

To demonstrate the utility and feasibility of FORC measurements for fundamental physics, we performed several reference experiments. First, using a permalloy sample we show the general concept and speed of MOKE FORC. Second, we show the separation of coercivity and bias field in an exchange bias sample,^22,23 consisting of an antiferromagnetic (AFM) and a ferromagnetic layer (FM), grown in a magnetic field high enough to saturate the FM and therefore showing a shift of the ferromagnetic hysteresis loop show a shift of the FORC distribution along the H_u-axis.^16,17 Finally, we demonstrate the flexibility of the MOKE based technique and the necessity for high resolution in material science with the measurement of a gadolinium/iron sample with perpendicular magnetic anisotropy.

All experiments were conducted using a Durham Magneto Optics NanoMOKE3 that is equipped with an air cooled electromagnet that is capable of generating a magnetic field of up to 500 mT and a data acquisition card with 16 bit resolution and a sampling rate of 100 kS/s. Both longitudinal and polar measurement geometry, as illustrated in Fig. 2, were used whereas the beam was focused on the sample using a 30 mm and 11 mm focus length lens, respectively.

Data acquisition and evaluation was done with a custom software, written in MathWorks MATLAB. The Parallel Computing Toolbox was used for fitting of the FORC density according to Eq. (9) to minimize the computing time of high resolution FORC diagrams. The parallelized application scales well on compute clusters using the MATLAB Distributed Computing Server, although even on desktop computers the computation time is small compared to the acquisition time.

Figure 4 shows FORC diagrams of a thin film permalloy reference sample with uniaxial anisotropy with its easy and hard axis oriented along the field direction. The easy and hard axis FORC map are composited of 101 and 76 reversal curves, respectively. Each reversal curve was averaged over 50 measurement cycles with a frequency of 2.1 Hz resulting in a total measurement time of 40 and 30 min, respectively.

The easy axis FORC shows a single very sharp peak at H_c = 2 Oe and H_u = 0 Oe. This corresponds to a concise irreversible domain wall movement through the region measured by the laser beam. Also there is no bias field (H_u = 0 Oe) as expected, as there is no exchange bias or other origins of bias fields in this simple test sample. On the other hand, the hard axis FORC shows a broad peak that is elongated along the H_u direction at H_c = 0 Oe. This is called the reversible ridge, as mentioned in Sec. II A, that occurs due to the extended FORC formalism used for data evaluation.¹⁴ 

It is noteworthy that the irreversible peak shows a FORC density of

The measurements shown in Fig. 4 were optimized for highest resolution and maximal signal-to-noise ratio as a proof of concept, sacrificing some measurement time, but still keeping it less than 1 h. As the major benefit of using MOKE measurements for the acquisition of FORC maps is the superior speed over traditional methods, a measurement of the same permalloy thin film sample, optimized for minimal acquisition time, was performed and is shown in Fig. 5. The FORC map is built up with non-equidistant minor loops with a minimal reversal field step of ΔH_r = 1 Oe. A single minor loop is acquired in 0.9 s, resulting in a total measurement time of 56 s. The FORC map shows the same peak at a small coercivity of H_c = 3 Oe and no interaction field H_u = 0 Oe, although the peak is slightly blurred out as the resolution after smoothing was marginally smaller. However, this proves the ultimate capability of MOKE measurements to deliver FORC maps in a short time and opening up the method to a large number of samples and allows to investigate complex variations as a function of additional parameters like temperature, strain, electrical currents, or voltages and so on.

Fig. 6 shows a FORC diagram of a 4 nm thick ferromagnetic cobalt layer on a 8 nm thick antiferromagnetic iron manganese layer supported by a 20 nm copper and a 50 nm platinum layer on MgO(100), measured along its exchange bias direction. The iron manganese layer was field cooled after preparation to achieve a unidirectional shift of the hysteresis loop.^22,23 The FORC map is composed of 42 minor loops with non-equidistant reversal field steps and a minimal reversal field step of ΔH_r = 4 Oe. Each loop is averaged over 5 cycles with a repetition frequency of 0.21 Hz, resulting in a total measurement time of 38 min.

The FORC diagram with a resolution of 10 Oe after smoothing exhibits a single, well defined peak at H_c = 35 Oe and H_u = 185 Oe. As expected for an exchange bias system there is a large shift in H_u direction, representing an interaction field of the ferromagnet with its antiferromagnetic substrate; while the coercivity, as indicated by the peak's position in H_c direction, is small as expected for metallic cobalt. This shows that, in contrast to simple hysteresis measurements, both coercivity and exchange-bias can be separated onto two independent axis by FORC measurements.

While the signal to noise ratio is sufficient in the measurement, the absolute FORC peak height of

To demonstrate the requirement of a high field resolution, two FORC maps of a gadolinium/iron multilayer with a resolution of 24 and 55 Oe after smoothing are shown in Fig. 7. The maps are based on 125 reversal curves averaged of 3 cycles each at a frequency of 0.55 Hz, resulting in a measurement time of 15 min. Without smoothing (SF = 1) this measurement would yield a resolution of 8 Oe. For the lower resolution FORC diagram, the resolution was artificially reduced by interpolation to a lower resolution field grid as described in Sec. III A 2.

The lower resolution FORC diagram shows one broadened peak centered at H_u = 0 Oe with slight shoulders extending off the H_u = 0-axis. The higher resolution map also features a peak at H_u = 0 Oe and a maximum at about the same coercivity value H_c = 10 Oe. However, the previously broadened feature off the H_u = 0-axis can be fully resolved as individual sharp peak at H_c = 50 Oe and H_u = 50 Oe. This feature indicates that the magnetization reversal occurs as a two-step process. Whereas the magnetization first partially rotates (H_c = 10 Oe) to an axis slightly angled towards the surface normal and in a second step rotates to a fully perpendicular orientation (H_c = 50 Oe). The shift of the H_u = 0-axis can also be attributed to the canting of the magnetic moments as there is an altered interaction between the canted magnetic moments and the external field, shifting the effective field experienced by the local moments.^16,17

As FORC measurements can yield a wealth of information about the magnetic states in a sample, there is a benefit in applying them to thin film samples and questions of fundamental solid state physics. Making these measurements accessible as basic characterization method for a large number of samples can be achieved by reducing the measurement time to a few minutes instead of hours or even days. Both can be reached by using MOKE for FORC measurements.

We have shown that the major problem of non-absolute magnetization values and also the drift can be circumvented by using a special field shape that provides two anchor points for the minor loop measurements. Furthermore, we have shown that the basic features of FORC diagrams like the position of the irreversible peaks in the H_u/H_c-coordinate system and the reversible ridge of the extended FORC formalism are also conserved for MOKE acquired FORC maps.

As the measurement time of an individual minor loop is reduced to some seconds, the total acquisition time is reduced to a few minutes, compared to SQUID or VSM measurements lasting several hours. This routinely allows for measurements of FORC maps with much smaller reversal field steps and thus for resolution of minor features of the FORC diagrams as shown for a Gd/Fe multilayer sample.

The increased speed of MOKE FORC measurements may well make this resourceful method accessible to a broader scientific scope in material science and basic physical research. Furthermore, it allows to follow variations of FORC diagrams depending on additional parameters, resulting in the possibility of FORC movies.

The authors would like to thank Bernd Ludescher for preparing the Gd/Fe sample and Michael Kopp for fruitful discussions about different applications of FORC and the efficient handling of large datasets in MATLAB.

Financial support by the Baden-Württemberg Stiftung within the Kompetenznetz Funktionelle Nanostrukturen is gratefully acknowledged.