research papers
Measuring magnetic hysteresis curves with polarized soft Xray resonant reflectivity
^{a}Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxon OX11 0DE, United Kingdom, and ^{b}Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
^{*}Correspondence email: paul.steadman@diamond.ac.uk
Calculations and measurements of polarizationdependent soft Xray scattering intensity are presented during a magnetic hysteresis cycle. It is confirmed that the dependence of the intensity on the
can be linear, quadratic or a combination of both, depending on the polarization of the incident Xray beam and the direction of the With a linearly polarized beam, the scattered intensity will have a purely quadratic dependence on the when the is parallel to the However, with the perpendicular to the there is also a linear component. This means that, when measuring the hysteresis with linear polarization during a hysteresis cycle, the intensity will be an even function of the applied field when the change in the (and field) is confined within the but becomes more complicated when the is out of the Furthermore, with circular polarization, the dependence of the scattered intensity on the moment is a combination of linear and quadratic. With the moment parallel to the the linear component changes with the helicity of the incident beam. Surprisingly, in stark contrast to absorption studies, even when the is perpendicular to the there is still a dependence on the moment with a linear component. This linear component is completely independent of the helicity of the beam, meaning that the hysteresis loops will not be inverted with helicity.Keywords: reflectivity; scattering; magnetism; soft Xrays; polarization; hysteresis; helicity; thin films.
1. Introduction
The measurement of the reversal process in modern magnetic materials is essential for understanding their magnetic properties, particularly for materials used for spintronics (Kuroda et al., 2005; Hirohata et al., 2020; Khan et al., 2021). There are many ways of measuring magnetic hysteresis. For example, bulk measurements can be achieved using the extremely sensitive technique of vibrating sample magnetometry (VSM), where the total moment is measured using pickup coils (Hurt et al., 2013). By combining this technique with the superconducting interference effect in the measurement circuit, it has become even more sensitive, now being able to measure tiny signals down to the 10^{−7} electromagnetic unit (e.m.u.) level. Surface sensitive measurements can be achieved using the magnetooptical Kerr effect (MOKE) technique pioneered by Bader et al. (Qiu & Bader, 1999, 2000; Osgood et al., 1998). This technique has been developed enormously over the last few years to include microscopy measurements yielding important spatially resolved measurements on thinfilm materials (Corb, 1988). This has proven to be an excellent technique for the visualization of magnetic domains in thin films (Hussain et al., 2017) and other interesting nanosized structures (Stupakiewicz et al., 2014). The surface sensitivity of MOKE measurements and VSM measurements, which measure the magnetization of the whole sample, have been combined to separate out surface and bulk effects [see, for example, Hendrych et al. (2014)].
For measuring microscopic effects more directly in field, neutron (Ankner & Felcher, 1999) and Xray scattering (Hill & McMorrow, 1996; Hannon et al., 1988) are both established probes. The more bulksensitive thermal neutron reflectometry technique contrasts starkly with the surfacesensitive Xray scattering. For neutrons, the for magnetic scattering, based on the internal B, differs with the complexity of the Xray interaction, particularly in the case of which is related to the imbalance of spinup and spindown electrons at the The offers a distinct advantage in element specificity as it is the only technique that is able to do this directly. Soft Xray scattering has the further advantage of having a high sensitivity to magnetic structures with often very small micrometrescalesized beams.
Measurements of elementsensitive hysteresis loops with Xray absorption have already yielded interesting results, e.g. on rareearth transitionmetal exchange springs (Stenning et al., 2012, 2015) and in other thinfilm systems (Chakarian et al., 1995; Hellwig et al., 2011). When using circular polarization, the absorption is proportional to the of the element projected onto the incidentbeam direction; as a result, many studies measure the absorption directly to obtain the hysteresis curve, without the need to calculate the e.g. see Hellwig et al. (2011). The values of atomic moments can be probed almost directly using the well known optical sum rules (Altarelli, 1993; van der Laan & Figueroa, 2014; Thole et al., 1992).
Soft Xray scattering and particularly reflectivity have also been used several times to monitor hysteresis behaviour. By keeping the detector fixed in scattering angle to monitor the intensity of a diffraction peak (Chmiel et al., 2019) or reflectivity (Marrows et al., 2005), much can be learned microscopically about the reversal process. In some exchange bias systems it is possible to measure the behaviour of the antiferromagnetic layers near the interface due to uncompensated moments that can be manipulated through the ferromagnetic layer using an applied magnetic field (Engel et al., 2004; Fan et al., 2022).
Notwithstanding the advantages of soft Xray scattering in measuring the magnetic reversal process, some complexities need to be addressed so that the form of the scattering during the hysteresis cycle can be properly understood (Burn et al., 2022). The main purpose of this work is to demonstrate the nonlinear dependence of the scattered intensity. The shape of the reflectivity or scattering that results during a hysteresis cycle is therefore in general different to that of the magnetization during a hysteresis loop, and strongly dependent on geometry and the polarization state of the incident beam. This work also emphasizes the differences between absorption and scattering when using circular polarization. One of the highlights of this work is the demonstration that circularly polarized Xrays can be used to measure the magnetic reversal when the moment is changing perpendicular to the In this geometry, the form of the scattering during the hysteresis is independent of the helicity of the Xray polarization.
The article is organized as follows. Section 2 begins with subsection 2.1 that explains the samples, their characterization and the main experimental setup of soft Xray scattering. The next subsection 2.2 describes the theory of resonant magnetic soft Xray reflectivity. The following subsection 2.3 describes the case of linear polarization, which in turn has subsections 2.3.1 and 2.3.2 describing the cases of moments in the and out of the respectively, i.e. the effects of the hysteresis cycle on the reflectivity when the moments are forced to change parallel to the and perpendicular to the Following this, in Section 2.4, the case of circular polarization is developed for moments changing in the (Section 2.4.1) and perpendicular to the (Section 2.4.2). Section 2.5 then contains some calculations from a thin film using an optical theory to support some of our findings. This is then followed by the Conclusions.
2. Results and discussion
2.1. Experimental preliminaries
All measurements were carried out on a thin film of 10 nm of Py, an alloy of 80% Ni and 20% Fe, grown on Si(001) with a cap of 3 nm of Pt using DC magnetron sputtering. The hysteresis loop was measured with a Quantum Design vibrating magnetic sample magnetometer (MPMS3). This is shown in Fig. 1(a). The coercive field is less than 10 Oe, making it easy to saturate in our small magnet. In Fig. 1(b), Xray reflectivity from our thin film is shown, which was measured at the Cu Kα energy (8.05 keV). The fit to these data is achieved using the parameters in Table 1. The thicknesses match close to those expected from the growth rates. The rootmeansquare roughnesses at the interfaces are all lower than 1 nm.

Soft Xray reflectivity was measured on the I10 beamline at Diamond Light Source using the soft Xray diffractometer RASOR (Beale et al., 2010). The beamline consists of two helical undulators and employs a planegrating monochromator design [we direct the interested reader to the work of Follath et al. (1998)]. The beamline focuses polarized light of 10^{12} photons s^{−1} at 780 eV with a bandpass of 0.1%. The light here is focused in a spot size of 200 µm × 200 µm (σ). The beamline is capable of linear or circular polarization (rates of circular polarization >99.8%). Reflectivities from the beamline are shown in Fig. 1. Each reflectivity was measured from our thin film at 707 eV with circularly polarized photons at opposite helicities. The difference between these two reflectivities arises because of the magnetization of the film (which was parallel to the film and in the scattering plane). The energy of 707 eV corresponds to the L_{3} resonance of Fe. All subsequent measurements in this article were carried out at this energy, where the photon scattering process is most sensitive to the of the Fe atoms. The measurements could also be done on the Ni L_{3} resonance, where the Ni atomic moments could be probed, but as the Ni and Fe atoms share the same magnetic environment the results and conclusions would be the same. The fringes in the reflectivity are the result of interference between the interfaces of the Py and Pt thin films. The magnetic field was applied using a small electromagnet attached to the sample holder. During application of the magnetic field, the sample and magnet were kept at room temperature by using a diffractometer flow cryostat with liquid nitrogen. A specially built sample holder enabled us to manually rotate the magnet so that fields could be applied parallel and perpendicular to the scattering plane.
2.2. Theory preliminaries
In the dipole approximation, the resonantscattering form factor can be presented by the following expression (Hill & McMorrow, 1996; Hannon et al., 1988) as
Here, e_{i} and e_{f} are directional vectors representing the incident and scattered polarization, respectively. Furthermore, M is the and the coefficients F^{(0)}, F^{(1)} and F^{(2)} depend on the matrix elements involved in the resonant process. The first term is the charge scattering and the second term is the magnetic scattering to first order. The third term, which is second order in magnetization, is assumed to be negligible and will not be considered in the rest of this work.
We can now represent equation (1) in the following 2 × 2 matrix representation, where each element represents a particular incident and outgoing polarization change (ignoring the secondorder term in the magnetic moment),
Using the righthanded coordinate system in Fig. 2, assuming specular reflectivity (so that θ_{i} and θ_{f} are equal to θ), the first two terms in equation (1) become
where
Here, the diagonal terms correspond to no rotation of the polarization between incident and scattered Xrays, whereas the offdiagonal terms correspond to rotations of the polarization.
We can now acknowledge that the charge scattering and magnetic scattering factors are complex quantities by making both F^{(0)} and F^{(1)} complex. This enables us to write equation (3) in the following form (see the supporting information for details),
In the above we have made both the charge scattering and magnetic form factors complex to allow for the phase changes as the energy is adjusted in the vicinity of the resonance. Using equation (5), we can work out the intensity (I = f*f) for the different polarizations (circular and linear) with the magnetization in different directions (in the and out of the scattering plane).
The calculation of magnetic reflectivity requires a knowledge of the values of the charge [ and ] and magnetic [ and ] form factors. The real values are calculated using Kramers–Kronig (KK) transformation, assuming that the absorption spectra from the total electron yield provide the imaginary part of the form factor (see the supporting information for more details). All the calculations in this work have used these form factors, unless mentioned otherwise. Since the total electron yield is only sensitive to the first few nanometres of the thin film, it will be dominated by the Pt capping layer. The KK analysis will therefore be approximate and not yield accurate values of the scattering factors. Nevertheless, there is significant sensitivity of at least part of the NiFe film, as we clearly see the Fe resonance. Since the calculation of the exact reflectivity is not required, the resulting approximation to the scattering factors is appropriate enough here to demonstrate qualitatively the dependence of reflectivity during the hysteresis process.
2.3. Linear polarization
2.3.1. Case 1: moments in the scattering plane
From setting the magnetic field to change along the i in Fig. 2), the resulting reflectivities are shown in Fig. 3. These were measured by positioning the detector at each point along the reflectivity and then cycling the magnetic field. At the top are shown reflectivities for the linear horizontal light (σ) for both branches of the hysteresis curve. At the bottom are shown reflectivities for the linear vertical light (π) for both branches. The fringes that are seen in the reflectivity in Fig. 1 are shown as vertical streaks here, in all of the plots. At the coercive field of roughly 10 Oe, there is a pronounced minimum along most of the reflectivity, although for some select parts, such as close to 15°, there is a maximum.
parallel to the surface (parallel toThe hysteresis curves at 9° are shown in Fig. 4. These can be obtained by tracing vertical lines at 9° in Fig. 3. During the hysteresis cycle, the reflectivity is constant as a function of applied fields, apart from around the coercivity where there is a drop in intensity.
If the moments are kept in the i.e. terms containing the factors π_{MR22} and π_{MI22} can be set to zero. This can be further simplified. In firstorder electric dipole transitions, with the moments in the σ polarized light will give rise to π polarized magnetic scattering leading to the following equation (see the supporting information),
only the offdiagonal terms contribute to the magnetic scattering,In the same way, for π incident polarization (where magnetically scattered light is all in the σ channel) we obtain
Here, in equations (6) and (7) we see that the intensity depends quadratically on π_{MR21}, π_{MI21}, σ_{MR12} and σ_{MI12}. Since both of these terms are linear in the intensity depends quadratically on as mentioned before. There are no interference terms between charge and magnetic scattering as it would not occur unless the also rotated the polarization of the incident light. This has been seen in many materials, such as the normally forbidden [001] diffraction peak from Cu_{2}OSeO_{3}, e.g. Burn et al. (2021). At resonance, this peak appears and the polarization of the scattered light is rotated. In this case, the above expressions would need to be modified and would include linear interference terms where the outgoing magnetic and charge scattering mix and add together (see the supporting information). The strength of these linear terms would depend on the imaginary and real components of the charge scattering. Combined with the quadratic terms, the reflectivity measured during a hysteresis would have a different form as a function of the magnetization.
To calculate the reflectivity using equation (7), we will use a simple hysteresis loop where the moment magnitude changes completely linearly in one dimension parallel to the sample surface with a coercivity of 0.5 arbitrary units. This is shown in the top plots of Fig. 5. Although unrealistic, its simplicity is sufficient to demonstrate the complexities of measuring the magnetic reversal process. In all models, the magnetization will change in one dimension, either parallel or perpendicular to the (parallel to the surface), unless stated otherwise. There will be no component of the in the direction perpendicular to the surface (i.e. in the direction in Fig. 2) in any of the models.
The calculations of the scattering are shown in Fig. 5 during the hysteresis cycle. To calculate the scattering , , and were set to values corresponding to those at 707 eV (8.50, 11.35, 2.56 and 3.83, respectively; see the supporting information) and the sample angle θ was set to 9°. The calculations shown in Fig. 5 are for both σ polarization using equation (6) and π polarization using equation (7).
The behaviour of the scattering with linear light is constant apart from two parabolic dips near the coercive fields. The reflectivity from our thin film will have a more complex behaviour in the vicinity of the coercive field than that predicted from our simple theory, as the moment will not change linearly with the applied field during the reversal process.
2.3.2. Case 2: moments perpendicular to the scattering plane
Measurements of the hysteresis cycle with linear light as the magnetic applied field is applied perpendicular to the j in Fig. 2) are shown in Fig. 6. Apart from some noise, there is no clear signal when measuring with σ polarized incident light. In contrast to this, when measuring with π polarized incident light, there is a clear hysteresis behaviour indicative of a strong linear dependence on the Interestingly, the hysteresis curves switch signs depending on the incident angle.
(parallel toIf we take a slice through Fig. 6 at 9° we obtain the hysteresis loops in Fig. 7. The reflectivity measured with σ polarized light is very noisy with no clear dependence on applied field, while that with π incident polarization shows a clear signal dependent on the of the film, as shown at many of the angles in Fig. 6.
If the . The equation describing the intensity is now
is perpendicular to the then only the one diagonal component is present in the magnetic part of the equation. There is therefore no rotation of the polarization when the Xrays are scattered by the magnetic ion, unlike the previous case in Section 2.3.1thus demonstrating that the dependence of the scattered intensity has both a linear and quadratic component. The relative sizes of the linear and quadratic dependences will depend on the sizes of the imaginary and real components of the charge scattering, which in turn will depend on the energy of the incident beam. By using the form factors calculated from the KK analysis (see the supporting information) we can look at the energy dependence, which will change the relative sizes of the quadratic and linear dependences to the Results of these calculations using these form factors (and the behaviour of the dictated by the hysteresis loop in Fig. 5) are shown in Fig. 8. At 706.5 eV, the loop generated resembles the hysteresis. However, at 707 and 707.5 eV, a quadratic dependence on is clearly visible. In addition, the loop switches sense at 707.5 eV.
2.4. Circular polarization
2.4.1. Case 3: moments in the scattering plane
Results for the magnetic field being applied parallel to the . The reflectivity exhibits a strong linear behaviour with with circular polarization. This is particularly strong at the lower angles in the plots of Fig. 9. As in the linear vertical case with the applied field perpendicular to the (see Fig. 6), the sense of the loop, i.e. which sign of the applied field has the larger reflectivity, is strongly dependent on the incident angle of the beam. In addition, the loops can be inverted when the helicity of the beam is changed, i.e. the light areas become dark and vice versa. Slices through these data at 21° are shown in Fig. 10, demonstrating the clear linear dependence on the Also demonstrated is the dependence on the helicity of the circularly polarized beam.
during the hysteresis cycle are shown in Fig. 9To further examine some individual loops, slices were taken from the negative circular data, as shown in Fig. 11. Out of all the loops measured along the reflectivity profile, six types of hysteresis loops were found when measured with just one incident helicity of circular polarization. It is clear from the loops that there is a strong linear component of the reflectivity on the In addition to this strong component, there are other components that either cause minima or maxima in the vicinity of the coercive field. As mentioned before, the sign of the loop depends on the angle θ of the incident beam.
In the case of circularly polarized Xrays, the amplitudes can be modelled as two orthogonal polarizations phase shifted by π/2 radians. In terms of σ and π polarization, this would take the form σ_{i} + iπ_{i} and σ_{i} − iπ_{i} for both helicities. When this phase difference is taken into account and the moments are kept entirely within the equation (5) becomes (see the supporting information)
The first four terms (σ_{CR11}, π_{CR22}, σ_{CI11} and π_{CI22}) are charge terms and do not change with magnetic field. The next four terms (σ_{MR12}, π_{MR21}, σ_{MI12} and π_{MI21}) are quadratic in and are independent of the helicity of the beam. The last four terms (σ_{CR11}σ_{MI12}, π_{CR22}π_{MI21}, σ_{CI11}σ_{MR12} and π_{CI22}π_{MR21}) are linear in They are the result of interference between charge and magnetic scattering caused by a combination of circular polarization and the π/2 radian phase difference between the charge and magnetic form factors. The sign of these linear terms is dependent on the helicity.
Using the form factors at the resonance Fe L_{3} (707 eV), and also around the resonance (706.5 and 707.5 eV), we obtain the results given in Fig. 12 for one helicity. There is a strong linear component in all three loops as expected and nonlinear effects are not visible.
If there are nonlinear effects, they can be removed by measuring the scattering from two hysteresis loops with opposite helicities and then subtracting one from the other (see the supporting information).
The results shown in Fig. 11 are examples of the six qualitatively different hysteresis loops we found in the data from Fig. 9. They have all been measured with the same helicity of circular polarization. The top row contains a hysteresis loop with no significant quadratic dependence on moment (6°, on the left), one with nonlinear effects appearing at the bottom (8°, in the centre) and one with the nonlinear effects at the top (17°, on the right). The nonlinear effects appear as minima or maxima centred around the coercive fields on the loop. In the bottom row, the other three hysteresis loops appear qualitatively as a reflection through the y (moment) axis at zero field of the three loops on the top row. Whilst our simple theory can describe the loops at 6° (mostly linear dependence on – see, for example, calculations in Fig. 12) and 8° (linear with significant quadratic dependence on magnetic moment), the other four loops need further explanation.
Thus far, the theory only changes the size of the π_{MR22} and π_{MI22} components that have been neglected have to be taken into account. These terms will be maximum when the moment is actually perpendicular to the This means that as the moment rotates from parallel to the to perpendicular then back to parallel these terms will increase from zero to a maximum (90°) to the and back to zero as the moment rests back in the scattering plane.
along one direction: either in the or perpendicular to the If the theory is extended so that a moment can rotate in the surface, which would perhaps be more realistic, then, as the moment points out of the theTo model a system where the moment rotates at a constant rate in the surface plane, we introduce a parameter that is the angle to the see the supporting information). This switching takes place during the same interval of applied magnetic field as the linear model as shown in Fig. 5, i.e. 0.25 to 0.75 arbitrary units and −0.25 to −0.75 arbitrary units. The theory represented in equation (9) also needs to be extended to include the effects of these outofplane moments. This is done by including the quadratic terms in π_{MR22} and π_{MI22}, which are simply added to equation (9) (see the supporting information). Since our model is very simple and is only rotating the moments, we have assumed that the moments perpendicular to the would sum to zero during the magnetic reversal process. This would mean that only the quadratic components of the moments would be finite and that all linear components are assumed to be zero.
This parameter then increases from 0 to 180°, then back to 0°. This can be modelled by having the moment in the depending on the cosine of this angle, whilst the moment out of the depends on the sine of this angle (With this model we should be able to simulate, at least qualitatively, the maximum around the coercive fields by taking into account both components of the magnetic moments parallel and perpendicular to the i.e. equation (9) with the additional quadratic components of π_{MR22} and π_{MI22}, have been done and are shown in Fig. 13. The top plots calculated at 2° and 80° were carried out with the moment in the varying as demonstrated by the hysteresis loop in Fig. 5, i.e. the loop is linear around the coercive field. The scattering factors have been set to those corresponding to the Fe L_{3} resonance in our KK analysis. To make the quadratic effects visible, the calculations needed to be executed at 80°, but this could have also been done by artificially increasing the magnetic structure factors at a much lower angle. Here, the calculations have been carried out using equation (9) without the extra terms π_{MR22} and π_{MI22}, since these are zero. These calculations simulate qualitatively the loops at 6° and 8°, respectively, in Fig. 11. On the right at the top of Fig. 13 is shown a calculation where the is rotating constantly from parallel to the to perpendicular and back to parallel, from 0.25 to 0.75 arbitrary units of field. This exact process is then repeated when the moment is forced in the opposite direction from −0.25 to −0.75 arbitrary units of field. Due to the moment now being out of plane, π_{MR22} and π_{MI22} are now finite and need to be taken into account. The results of the calculations are dependent on incident angle θ. At 60°, strong maxima are seen around the coercive field if the size of the charge is reduced by a factor of ten. If we do not decrease the charge scattering relative to the magnetic scattering, these effects are too small to be visible. This indicates that these effects only occur at points along the reflectivity where the charge form factors are small compared with the magnetic form factors. This simple atomic model is only qualitative in order to demonstrate the main features of the scattering. The angular dependence of the features from the thin film in the experiment shown in Fig. 11 is very different to that from our simple calculation involving a smooth rotation of all of the moments.
Calculations using the general case for circular polarization,So, qualitatively, three of the experimental loops in Fig. 11 have been described. It is now possible to describe the loops that look like the reflectivity hysteresis curves from conventional magnetometry, i.e. have a strong linear component in (such as at a θ of 6°). It is also possible to qualitatively explain the loops with minima around the coercive fields (such as at a θ of 8°). By taking into account moments perpendicular to the we are also able to account for maxima around the coercive fields (e.g. at 16°). However, there are still three loops that have switched sense, which have not been described. All these loops were measured with the same helicity (it is understood that switching the helicity can also switch the sense of the loops).
To switch the sense of the loops, a phase needs to be introduced. In scattering, the exact phase is lost, but since there are terms linearly dependent on (see the supporting information). The origin of this phase, designated as ϕ, will be commented on later. Its effect is introduced as a cosine, which only affects the terms that are linear in This is justified in our qualitative model since the sign change resulting from this phase would be lost in the quadratic terms.
these terms could be reversed by the effect of a phase. This can be done by introducing the phase in equation (9)For the reflectivity curves we have already calculated in the top of Fig. 13, the phase ϕ is effectively set to 0°. If we set the phase ϕ to 180°, we can reverse the signs of the linear terms in equation (9). This is shown for the bottomleft and bottommiddle plots, which are the equivalent of the topleft and topmiddle plots at incoming angles of 2° and 80° but with the phase ϕ set to 180°. By using the phase factor in equation (9), but including the additional quadratic components of π_{MR22} and π_{MI22}, the sense of the linear components of the hysteresis loop in the top right of Fig. 13 can also be changed, as can be seen in the calculated curve underneath it.
With our simple atomic model, we have thus been able to qualitatively explain all the main features of the six families of loops in Fig. 11. This has been done by utilizing the relative dependence of linear and quadratic terms as a function of angle, invoking the dependence of the outofplane component for more complex magnetic reversal, and introducing a simple phase factor that will only influence the linear magnetic terms.
The nonlinear effects in the reflectivity hysteresis loops become very strong when the charge scattering is weaker. The hysteresis loops measured at 16°, 17° and 23° all occur at or near minima in the reflectivity (see Fig. 9), whereas those measured at 6° and 12° occur where the charge scattering is relatively stronger. The loop measured at 8°, which also shows a lot of nonlinearity, has very strong charge scattering, but there is a slight minimum occurring at the reflectivity. If the magnetic structure of the film is different to that of the chemical structure, as in the case of the heterostructure measured here (Pt is maybe slightly magnetized at the interface but our experiment would not be sensitive to this), then at angles where the magnetic scattering is relatively weak, the linear effects, which depend on both the magnetic and charge scattering being significant, will also be weak. This could lead to nonlinear components of the scattering being more prevalent.
2.4.2. Case 4: moments perpendicular to the scattering plane
Reflectivity measured during the hysteresis cycle with the applied field perpendicular to the . If one looks at the top two plots, which were taken for the same handedness of circular light, there is a clear linear dependence on the Since the applied field is now driving the magnetization perpendicular to the this seems to be counterintuitive. If the same experiment was done by measuring the absorption, rather than scattering, you would not have any sensitivity to the moments in this direction. This behaviour is similar to the case where the applied field is in the in Section 2.4.1. In stark contrast, the reflectivity loops measured with the opposite helicity of circular polarization, shown in the bottom of Fig. 14, demonstrate that this scattering is independent of helicity.
is shown in Fig. 14Slices from the data in Fig. 14 at 20° are shown in Fig. 15. Here, the two square hysteresis loops clearly demonstrate that there is a strong linear dependence on the moment perpendicular to the but that this effect is independent of the helicity of the circularly polarized beam.
For moments perpendicular to the , for both helicities, simplifies to (see the supporting information)
the general equation (5)The equations for both helicities are identical, which completely corroborates the data in Figs. 14 and 15, where the data from the two helicities are also identical. Furthermore, equation (10) is identical to that of equation (8), apart from the charge terms σ_{CR11} and σ_{CI11} (which do not change with magnetic field).
Fig. 16 represents calculations using equation (10) at different energies. The form factors are shown on the left. Whilst the loop at 706.5 eV looks mostly linear, quadratic effects are evident in 707 and 707.5 eV. The loop at 707.5 eV is also in the opposite sense to the other two loops. It is no accident that Fig. 16 resembles that of Fig. 8. Since in both cases it is only the π polarization that interacts with the this is not surprising.
In the same way as for circular polarization, but with the magnetic field applied parallel to the
there will be nonlinear effects in the hysteresis curves. These will have the same origin, which has already been discussed in detail.2.5. Optical modelling of the thin film
The calculations so far have been carried out using a simplified model, as a demonstration that the general shapes of the loops can be calculated and described qualitatively. By using software (Macke et al., 2014) based on an optical theory, which can be used to simulate scattering from thin films and multilayers (Zak et al., 1990a,b, 1991), we have modelled the reflectivity from our thin film. Although the theory is based on classical optical theory rather than a quantum mechanical approach (Hannon et al., 1988), it reproduces the basic trends. It can be shown that the optical theory is symmetrically equivalent to the atomic theory represented by equation (1) to the first order. In Figs. 17 and 18, we have plotted the results from the model for circular polarization with the field applied parallel and perpendicular to the respectively. Whilst the calculation is missing some of the finer details, it does show that the loop has a strong linear component, which switches sign (loop is reflected through the vertical moment axis) with helicity in the parallel case but does not switch sign when the field is perpendicular.
Also using the same optical theory, we tested the idea that the switching of the loops, which depends on the angle θ, is due to interference with the substrate and the Pt capping layer. This was achieved by removing them from the calculation and making the Py semiinfinite. This suppresses the effect of all of the interfaces from the calculation. The result is shown in Fig. 19. Here, there is no change in the sense of hysteresis up until θ = 45°, where, as was stated earlier, there is a change in the sign given by the interference of charge and magnetic terms in equation (1) for π to π scattering, i.e. the magnetic and charge terms in equation (3). This causes the charge scattering to change sign at θ = 45°.
3. Conclusions
In this article, we have taken some simple examples and calculated the reflected intensity during hysteresis cycles under various polarizations and magnetization directions. The results highlight possible problems with interpretation owing to the nonlinearity of the scattering dependence on the magnetic moment.
With linear polarization where the moment is within the
there will be no interference between the charge and magnetic scattering. This is the case for both linearvertical and linearhorizontal polarization, in and out of the However, if the charge scattering causes the polarization to rotate (anisotropic anomalous scattering), there will be interference between the magnetic and charge scattering. In the absence of interference, the dependence on the moment will be purely quadratic. When the moment is changing perpendicular to the in addition to this quadratic contribution there is an interference term, which depends linearly on the and both the imaginary and real parts of the charge form factor.With circular polarization and the magnetization in the π/2 phase difference between the real part of the charge scattering and the magnetic scattering, there is a strong interference term with a linear dependence on magnetization. This flips with the helicity. When the magnetization is changing perpendicular to the there is also a strong linear dependence. This arises due to the interference between the σ components of the charge and magnetic scattering. Rather counterintuitively, this dependence does not change with the helicity of the incident beam. As well as the linear component, there is also a quadratic dependence. With the magnetization within the this quadratic dependence can be filtered out by subtracting two hysteresis loops taken with opposite helicities (dichroism), which will recover the original form of the hysteresis loop, i.e. the dichroism has a linear dependence on the Finally, when the magnetic reversal is confined within the we classify six different shapes of hysteresis curves, which can be explained by relative contributions of the linear and nonlinear dependence on dependence on moments out of the and interference at the interfaces.
due to the4. Related literature
The following references are only cited in the supporting information for this article: Brück (2009) and Henke et al. (1993).
Supporting information
Supporting information. DOI: https://doi.org/10.1107/S160057752400119X/ye5035sup1.pdf
Acknowledgements
We would like to acknowledge beam time at Diamond Light Source Ltd on beamline I10. We would like to thank Mr Mark Sussmuth for his valuable technical assistance.
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